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title listlengths 0 18	author listlengths 0 4.41k	authoraffiliation listlengths 0 6.45k	venue listlengths 0 9	abstract stringlengths 1 37.6k	doi stringlengths 10 114 ⌀	pdfurls listlengths 1 3 ⌀	corpusid int64 158 259M	arxivid stringlengths 9 16	pdfsha stringlengths 40 40	text stringlengths 66 715k	github_urls listlengths 0 36
[ "OpenAGI: When LLM Meets Domain Experts", "OpenAGI: When LLM Meets Domain Experts" ]	[ "Yingqiang Ge \nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n\n", "Wenyue Hua \nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n\n", "Jianchao Ji \nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n\n", "Juntao Tan \nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n\n", "Shuyuan Xu \nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n\n", "Yongfeng Zhang \nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n\n" ]	[ "Rutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n", "Rutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n", "Rutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n", "Rutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n", "Rutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n", "Rutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\nRutgers University\n" ]	[]	Human intelligence has the remarkable ability to assemble basic skills into complex ones so as to solve complex tasks. This ability is equally important for Artificial Intelligence (AI), and thus, we assert that in addition to the development of large, comprehensive intelligent models, it is equally crucial to equip such models with the capability to harness various domain-specific expert models for complex tasksolving in the pursuit of Artificial General Intelligence (AGI). Recent developments in Large Language Models (LLMs) have demonstrated remarkable learning and reasoning abilities, making them promising as a controller to select, synthesize, and execute external models to solve complex tasks. In this project, we develop OpenAGI, an open-source AGI research platform, specifically designed to offer complex, multi-step tasks and accompanied by task-specific datasets, evaluation metrics, and a diverse range of extensible models. OpenAGI formulates complex tasks as natural language queries, serving as input to the LLM. The LLM subsequently selects, synthesizes, and executes models provided by OpenAGI to address the task. Furthermore, we propose a Reinforcement Learning from Task Feedback (RLTF) mechanism, which uses the task-solving result as feedback to improve the LLM's task-solving ability. Thus, the LLM is responsible for synthesizing various external models for solving complex tasks, while RLTF provides feedback to improve its task-solving ability, enabling a feedback loop for self-improving AI. We believe that the paradigm of LLMs operating various expert models for complex task-solving is a promising approach towards AGI. To facilitate the community's long-term improvement and evaluation of AGI's ability, we open-source the code, benchmark, and evaluation methods of the OpenAGI project 2 .	10.48550/arxiv.2304.04370	[ "https://export.arxiv.org/pdf/2304.04370v2.pdf" ]	258,049,306	2304.04370	9fd980237e7fdfa4c103a2dc08657e73adf847c4	OpenAGI: When LLM Meets Domain Experts Yingqiang Ge Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Wenyue Hua Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Jianchao Ji Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Juntao Tan Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Shuyuan Xu Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Yongfeng Zhang Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University Rutgers University OpenAGI: When LLM Meets Domain Experts * "May the Force be with LLM and Domain Experts." -Generated by ChatGPT Human intelligence has the remarkable ability to assemble basic skills into complex ones so as to solve complex tasks. This ability is equally important for Artificial Intelligence (AI), and thus, we assert that in addition to the development of large, comprehensive intelligent models, it is equally crucial to equip such models with the capability to harness various domain-specific expert models for complex tasksolving in the pursuit of Artificial General Intelligence (AGI). Recent developments in Large Language Models (LLMs) have demonstrated remarkable learning and reasoning abilities, making them promising as a controller to select, synthesize, and execute external models to solve complex tasks. In this project, we develop OpenAGI, an open-source AGI research platform, specifically designed to offer complex, multi-step tasks and accompanied by task-specific datasets, evaluation metrics, and a diverse range of extensible models. OpenAGI formulates complex tasks as natural language queries, serving as input to the LLM. The LLM subsequently selects, synthesizes, and executes models provided by OpenAGI to address the task. Furthermore, we propose a Reinforcement Learning from Task Feedback (RLTF) mechanism, which uses the task-solving result as feedback to improve the LLM's task-solving ability. Thus, the LLM is responsible for synthesizing various external models for solving complex tasks, while RLTF provides feedback to improve its task-solving ability, enabling a feedback loop for self-improving AI. We believe that the paradigm of LLMs operating various expert models for complex task-solving is a promising approach towards AGI. To facilitate the community's long-term improvement and evaluation of AGI's ability, we open-source the code, benchmark, and evaluation methods of the OpenAGI project 2 . Introduction The acquisition and reuse of skills is a fundamental aspect of human intelligence that enables the formation of complex skills for addressing novel or intricate problems [4]. We posit that machine intelligence should incorporate this capacity to synthesize various skills by composing them into complex skills for complex task-solving. In computer science parlance, each skill is referred to as a domain expert "model" -a reusable network with a defined function. The domain expert models can be synthesized into a larger "plan" for performing more complex tasks. The model synthesis process is adaptable to the input or task, such that for a given task, the models are synthesized into the most suitable plan to address the task at hand. As a result, different inputs or tasks may necessitate distinct synthesized models as a plan for task-solving. Recent advancements in Large Language Models (LLMs) have showcased exceptional learning and reasoning capabilities, rendering them well-suited for selecting, synthesizing, and executing external expert models to address complex tasks. These LLMs, such as GPT-3 [2], LLaMA [35] and Flan-T5 [6], have exhibited a profound understanding of natural language and the ability to generate coherent and contextually relevant responses. This has opened up new possibilities for their application in complex tasks involving multi-modality data, such as image and text processing, as well as the integration of domain-specific knowledge. In this process, LLMs play a crucial role as they can understand and generate natural language, which helps AI to better comprehend and handle various problems. By integrating knowledge and skills from different domains, Open-domain Model Synthesis (OMS) holds the potential to drive the development of artificial general intelligence (AGI), enabling AI to solve a diverse array of problems and tasks. While current research in this field has made some preliminary attempts, there are several notable challenges that need to be addressed: 1) Extensibility: Several existing works employ a fixed number of models, such as WebGPT [18] and ToolFormer [32], resulting in difficulties when attempting to expand their capabilities; 2) Nonlinear Task Planning: The majority of current research is limited to solving tasks with linear task planning solutions [37,11], meaning that each sub-task must be completed before the next sub-task can start. However, linear planning of models may not suffice for solving complicated tasks, besides, many tasks involve multiple multi-modal inputs. 3) Quantitative Evaluation: Many existing works only provide qualitative results, such as HuggingGPT [33]. This makes it difficult to assess the planning capabilities of LLMs to determine whether the strategies employed are optimal. In order to mitigate the above limitations, we develop a platform that encompasses a diverse array of domain-specific expert models and intricate multi-step tasks with single or multiple multi-modal inputs, supported by corresponding datasets. Notably, we employ numerous expert models from the widely recognized Hugging Face's transformers and diffusers libraries 3 and Github repositories 4 , thereby facilitating the expansion of our model set. Additionally, the datasets (also from Hugging Face datasets library) have been meticulously selected to align with or resemble the training datasets of the respective models. Ultimately, we implement a variety of data augmentation techniques to enhance these datasets, enabling the construction of sophisticated multi-step tasks designed to assess the planning and task-solving capabilities of a given LLM. To promote the community's long-term advancement and assessment of AGI's abilities, we open-source all code and datasets, and hence, name this platform OpenAGI. The entire pipeline of OpenAGI is depicted in Fig. 1. Specifically, 1) a natural language task description is chosen along with the task-related dataset; 2) the task description is fed as input into LLM to generate a solution, which may require mapping the solution to functional model names, or using constrained generation to generate model names directly; 3) the models are selected and synthesized, and subsequently executed to process the data samples; 4) the task-solving ability of the LLM can be evaluated by comparison between the output and the ground-truth labels. Although the OpenAGI platform offers numerous advantages and enhanced accessibility, it also gives rise to a variety of novel research challenges, such as: • Out-of-Distribution (OOD) Generalization. Domain-specific expert models may exhibit limited generalization ability due to their strong dependence on the distribution of the training data. As demonstrated in Fig. 2, when processing images from disparate sources exhibiting a distributional shift, the original model sequence to address the task in Fig. 1 becomes ineffective. In the majority of instances, only a few colors are accurately restored, while most remain incorrect. Furthermore, noise and blurring persist, remaining highly perceptible to human observers. • Optimal Task Planning. There is a compositional number of ways to combine different models to generate solutions, which can make it difficult to identify the best approach. Additionally, it is possible for multiple valid solutions to exist for a given task, but the quality of each solution can vary greatly. For instance, as depicted in Figure 3, executing the same four models in different sequences can lead to noticeably different outcomes. The results from the second approach (i.e., the second row in the figure) exhibit significantly more noise and color inconsistencies compared to the first approach. Therefore, it is crucial for the LLM to identify and implement the optimal task plan from among the various possibilities. • Nonlinear Task Structures. During model execution, a model may need more than one inputs and each input need to be produced by a prerequisite model, resulting in a nonlinear (tree) structure for the solution. In this context, employing a nonlinear task planning may enable more effective integration of the diverse inputs and more efficient parallel processing of the models to achieve the desired outcome. However, incorporating such nonlinear task planning ability into LLMs presents unique challenges beyond the LLM's existing task-solving capabilities. In consideration of the first two challenges, we introduce a mechanism referred to as Reinforcement Learning from Task Feedback (RLTF). This approach capitalizes on the performance feedback procured from tasks following the execution of the solution devised by the LLM. Consequently, the RLTF mechanism effectively refines the LLM's planning strategy, resulting in an enhanced and more adaptive system. Indeed, relying solely on input text for learning proves insufficient for Figure 3: Examples of different model sequences for solving the same task (task description is the same as Fig. 1). Both are valid model sequences but they result in very different task-solving quality. LLMs when confronted with real-world tasks. Task feedback, on the other hand, supplies additional information that steers the learning trajectory of LLMs towards improved and efficient solutions. For the third challenge, we propose Nonlinear Task Planning, which utilizes beam search as an efficient semi-autoregressive decoding method [29] such that for each decoding step in beam search, different hypotheses are treated as parallel actionable solutions for different inputs instead of competing hypotheses. If a task requires parallel processing for multiple inputs, such as both text and image, then in generation time, an actionable solution taking text as input and another solution taking image as input will be generated and executed in parallel. In summary, the key contributions of the work include: • We introduce OpenAGI, an AGI research platform, specifically designed to offer complex, multistep tasks accompanied by their respective datasets, evaluation methods, and a diverse range of extensible models which can be synthesized to effectively solve these tasks. The purpose of this platform is to aid in the quantification of the overarching planning and task-solving abilities of LLMs. OpenAGI embraces AGI by focusing on LLM-driven, (open-domain) model synthesis, predominantly utilizing models and datasets on Hugging Face and Github. • We propose the LLM+RLTF approach for OpenAGI, which leverages a Large Language Model as a controller to select, synthesize and execute various external expert models for complex task-solving. The feedback obtained from these tasks is then employed to refine the LLM's planning strategy, thereby enhancing the LLM's overall performance and task-solving ability. • We evaluate a variety of well-established LLMs 5 with differing scales (ranging from 770 million to 175 billion parameters) utilizing distinct learning schemas and the proposed OpenAGI pipeline. Our preliminary findings suggest that even smaller-scale LLMs, when paired with an appropriate learning schema such as RLTF, are able to possess the potential to outperform competitors that equip a significantly greater magnitude of model parameters. 2 Related Work Large Language Models With the advancement of highly parallelizable transformer architectures, pre-trained language models (PLMs) have demonstrated remarkable capabilities in comprehending, generating, and manipulating natural language [24,17]. These models were pre-trained on a large corpora of unlabeled text data and commonly subsequently fine-tuned for specific downstream tasks. Shortly, the scaled-up PLMs, known as LLMs [26,2,21,5,42,35], encompassed a substantially greater number of parameters and leverage vast amounts of training data. Consequently, LLMs exhibited enhanced capacity for learning intricate language patterns and structures, along with a notable reasoning ability. This results in superior performance across diverse natural language processing tasks [2,35]. The achievements of LLMs are possibly due to the scaling laws for neural language models [12], which suggests the performances of the models depend primarily on the amount of model parameters, training data, and computing budget. T5 [26] is one of the first noticeable LLMs with up to 11 billion model parameters. It is an encoder-decoder model pre-trained on multiple tasks and reformulates them as text-to-text problems. Later, the release of GPT-3 [2] drew significant interest due to its surprising large model size containing 175 billion parameters, which made it the largest language model of its time. It was trained on a diverse range of text using unsupervised learning and revealed impressive capabilities on zero-shot and few-shot learning, which suggested LLMs are able to perform well on tasks even without explicit fine-tuning. T5 and GPT-3, along with their re-fined versions [6,21] inspired the research community to keep extending the capacity of LLMs. Recently, more and more LLMs were released with continuously improved model size [25,5,42,35,20] and, as them should be, achieved superior performance on reasoning and language understanding. Augmented Language Models Although LLMs exhibit a robust capacity for comprehending complex human language, they may occasionally produce seemingly plausible yet inaccurate predictions and face challenges when addressing problems that require specialized domain expertise [16]. Consequently, the emerging field of Augmented Language Models (ALMs) focuses on addressing the limitations of conventional LLMs [6,5,2] by equipping them with enhanced reasoning capabilities and the ability to employ external resources [16]. The process of reasoning involves breaking down intricate assignments into smaller, more manageable subtasks that can be independently or collaboratively tackled by LLMs with the assistance of tools. What's more, LLMs can also invoke external tools or models to accomplish the relevant tasks. For exmaple, ToolFormer [33] introduces external API tags within text sequences, facilitating LLMs' access to external tools. Visual ChatGPT [39] is a new model that combines ChatGPT with Visual Foundation Models (VFMs) like Transformers, ControlNet, and Stable Diffusion, which acts as a bridge between users, allowing them to communicate via chat and generate visuals. HuggingGPT [32] integrates the Hugging Face hub with task-specific models around ChatGPT to tackle generalized AI tasks. Augmented language models may use these enhancements separately or joint them in a specific order to finish the specific task, which ultimately results in superior generalization capabilities. Different from prior works in this field, we propose OpenAGI, an open-source AGI research platform designed to address the challenges commonly encountered in existing works, such as extensibility, nonlinear task planning, and quantitative evaluation. Furthermore, we introduce innovative methods into the learning schema of LLMs, including Reinforcement Learning from Task Feedback (RLTF) and nonlinear task planning, which address challenges on out-of-distribution (OOD) generalization, optimal task planning, and nonlinear task structures. We hope the OpenAGI platform can facilitate the open and long-term improvement and evaluation of AGI abilities in the community. The OpenAGI Platform Problem Definition Given a set of natural-language-based task descriptions T and a set of datasets D, where each element D t represents the corresponding dataset for a specific task t, alongside a collection of functional models on D, represented as M, and their corresponding name set N , the objective for a given LLM, denoted as L, is to take a particular task description t as input and produce a multi-step solution s. This solution can be mapped to an arrangement of functional models (linearly or non-linearly) based on the model name set, ultimately working on the task-related dataset to accomplish the task. Consequently, one can employ any LLM derived from any learning schema to assess the planning capability of the LLM within this context, provided that T , D, M, and N are supplied. In this work, our primary objective is to assist AGI researchers in constructing an open-source pipeline, which will contribute to the community's long-term advancement and foster collaborative progress in the field. We introduce the details of our construction in the following sections. Specifically, instead of building complicated, multi-step tasks from scratch, we first explore the models 3.2 and datasets 3.3 that can be easily achieved, then create such tasks based on them. Model Set We now present the domain tasks and the corresponding models that can be employed in our platform. This set is designed to be flexible, allowing users to easily incorporate their own domain tasks and models. Our domain tasks are as follows: • Language-related Models (corresponding models are shown in Table 1): Sentiment Analysis classifies the sentiment polarity of a given sentence; Text Summarization creates a text summary that represents the most important or relevant information within the original text content; Machine Translation converts a sentence from a source language to a target language; Fill Mask involves replacing masked words within a given text; Question Answering (QA) provides a textual answer of a question based on the given context. • Vision-related Models (corresponding models are shown in Table 2): Image Classification aims to comprehend an entire image as a whole and assign it to a specific label; Object Detection identifies and localizes specific objects within an image by detecting their instances of a particular class; Colorization refers to the technique of adding plausible color information to monochromatic photographs or videos; Image Super-resolution generates a high-resolution (HR) image from a low-resolution (LR) image; Image Denoising aims to remove unwanted noise from an image while preserving its important features; Image Deblurring aims to recover a clear image from a blurred input image. • Vision-Language Models (corresponding models are shown in Table 3): Visual Question Answering (VQA) involves answering questions based on an image; Image Captioning generates textual descriptions of the visual content depicted in an image; Text-to-Image Generation aims to generate images from a given input sentence or sequence of words. [9] Object Detection Image Text DETR 12 [3] Colorization Image Image Colorizer 13 [41] Image Super-Resolution Image Image Swin2SR 14 [7] Image Denoising Image Image Restormer 15 [40] Image Deblurring Image Image Restormer [40] Text-to-Image Generation Text Image StableDiffusion 18 [28] 3.3 Tasks and Datasets Raw Datasets After selecting the appropriate models, choosing the raw datasets becomes a more straightforward process, provided that we ensure proper alignment between the datasets and the models' training sets. Raw datasets are provided as follows: • ImageNet-1K [30] is a large-scale image dataset, derived from the broader ImageNet database, containing approximately 1 million images. These images are categorized into 1,000 distinct classes, with each class representing a specific object or concept. The dataset has been instrumental in the development and evaluation of state-of-the-art deep learning algorithms for image classification, object recognition, and transfer learning. • • Stanford Sentiment Treebank (SST2) [22] is a corpus with labeled parse trees that allows for the analysis of the compositional effects of sentiment in language. The corpus consists of 11,855 single sentences extracted from movie reviews. It was parsed with the Stanford parser and includes a total of 215,154 unique phrases from those parse trees, each annotated by 3 human judges. • TextVQA [34] serves as a benchmark for evaluating visual reasoning based on text present in images. In order to answer questions pertaining to the images, TextVQA necessitates models to read and reason about the text contained within them. The incorporation of text as a new modality in images demands that models be able to reason over this modality to address TextVQA queries. Thus, TextVQA poses a unique challenge for models to integrate both visual and textual cues to arrive at a comprehensive answer. • Stanford Question Answering Dataset (SQuAD) [27] is a collection of question-answer pairs sourced from Wikipedia articles. A distinguishing characteristic of SQuAD is that the correct answers to the questions can be any sequence of tokens in the corresponding text. This flexibility is a result of the dataset's construction through crowdsourcing, which results in a diverse set of questions and answers compared to other question-answering datasets. Data Augmentation Methods Upon determining the raw datasets, our next objective is to augment them from various perspectives to construct complex, multi-step tasks. For instance, we can introduce noise and reduce the resolution of an image from ImageNet-1K to create new datasets that may require "Image Denoising" and "Image Super-Resolution" for initial recovery before doing classification. The data augmentation methods employed are as follows: • Gaussian Blur is a prevalent image processing technique that involves convolving an image with a Gaussian filter kernel. This filter is applied to smooth the image and reduce noise, yielding a blurred output image. • Gaussian Noise refers to the addition of Gaussian-distributed noise. • Grayscale entails converting the colorful image to a grayscale image. • Low Resolution pertains to images with a reduced pixel density (pixels per inch, or ppi). • Translation denotes the process of converting a text from one language, such as English, to another, such as German. In this work, we only use English-to-German translator for simplicity. • Word Mask randomly replaces a single word in a given sentence with the "[MASK]" token. Multi-step Tasks Drawing from the models presented in Tab. 1, 2, and 3, we categorize them according to input and output modalities as follows: 1) image in, image out; 2) image in, text out; 3) text in, image out; 4) text in, text out; 5) image-text pair in, text out; 6) text-text pair in, text out. We employ data augmentation techniques discussed above to augment the raw datasets. Specifically, for tasks with image inputs, we can choose one or more techniques from the image augmentation method set {Gaussian Blur, Gaussian Noise, Grayscale, Low Resolution} to generate a compositionally augmented image input, which necessitates a multi-step image restoration process for recovery. Similarly, for tasks with text inputs or outputs, we choose one or more from {Translation, Word Mask} to generate a compositionally augmented text input or output. Furthermore, Visual Question Answering (VQA) and Question Answering (QA) are tasks with multiple multi-model inputs, resulting in natural tasks that cannot be solved with linear task planning solutions. Lastly, we integrate both aspects to construct complex, multi-step tasks. In total, we generate a total number of 185 complex multi-step tasks, with 117 tasks featuring a linear task structure and the remaining 68 tasks exhibiting a non-linear task structure. A selection of task samples, along with their corresponding input and output data samples, can be found in Table 4. For illustration, consider the third row of Table 4, which represents a machine translation domain task (i.e., translating from English to German). In this case, we apply the "Word Mask" augmentation technique on the text inputs to create a multi-step task, which can be described as "Given clozed English text, how can the text be translated into German step by step?". For instance, given an original data sample, "A big burly grizzly bear is shown with grass in the background", the word "with" has been chosen to be masked to generate the augmented data sample, "A big burly grizzly bear is shown [MASK] grass in the background". Evaluation Metrics Given that OpenAGI comprises a diverse range of domain tasks with multi-modal data, we classify them according to domain tasks as well as input and output types. We then assess their performance using the following three metrics: • CLIP Score 19 is a reference-free metric used to assess the correlation between a generated image caption and the actual content of the image. • BERT Score 20 uses contextual embeddings from the pre-trained BERT model to compare words in candidate and reference sentences through cosine similarity. Additionally, BERT Score calculates precision, recall, and F1 measure, making it a valuable tool for evaluating various language generation tasks. In this work, we use the value of F1 score. • ViT Score 21 is a metric designed to assess the visual similarity between two images. By calculating the cosine similarity of their respective embeddings, which are generated using a Vision Transformer, the ViT Score offers a quantitative measure of their likeness. In particular, we employ the CLIP Score only for Text-to-Image Generation-based tasks, the BERT Score is utilized to assess tasks with text outputs, and the ViT score is applied to measure image similarity for the remaining tasks with image outputs. We also normalize the BERT and CLIP scores. Reinforcement Learning from Task Feedback (RLTF) While learning solely from input text is a powerful method for training LLMs, it is not sufficient for handling real-world tasks that require a deeper understanding of context and environment. One potential method to improve the capabilities of LLMs is to incorporate reinforcement learning (RL) techniques. By merging the strengths of RL, LLMs can gain additional insights from trial-and-error experiences. This leads to more robust and adaptive models, especially in situations where labeled data is scarce or when tasks involve physical interactions. In this work, we propose Reinforcement Learning from Task Feedback (RLTF), which utilizes task feedback to supply more information that guides the learning direction of LLMs, resulting in improved and more efficient strategies. In the setup of RLTF, the environment is the proposed OpenAGI platform and the agent is the LLM L parameterized with Φ. The solution s generated by the LLM can be seen as a set of instructions that solve the input task t and can be executed on the corresponding augmented dataset D t . We can use the performance (provided in Sec. 3.4) on that dataset as the reward signal R and use reinforcement learning to fine-tune the LLM. More concretely, to find the optimal solution, we require the LLM to maximize its expected reward on the training set T train , represented by J(Φ): J(Φ) = E strain∼L(Ttrain\|Φ) [R](1) Since the reward signal R is non-differentiable, we need to use a policy gradient method to iteratively update Φ. In this work, we use the REINFORCE in [38] as follows, ∇ Φ J (Φ) = E P (strain\|Φ) [∇ Φ log P (s train \|Φ) · R](2) An empirical approximation of the above quantity is: ∇ Φ J (Φ) ≈ 1 \|T train \| t∈Ttrain ∇ Φ log P (s train \|Φ) · R(3) The above update is an unbiased estimate for our gradient, but has a very high variance. In order to reduce the variance of this estimate, following [43,23], we employ a baseline function b, which is the moving average of the previous reward signals: ∇ Φ J (Φ) ≈ 1 \|T train \| t∈Ttrain ∇ Φ log P (s train \|Φ) · (R − b)(4) Nonlinear Task Planning To generate the solution for a natural language task description, we require the LLM to generate an actionable solution consisting of sequences of model names. For tasks that require only one input, the model only needs to generate one actionable sequence of models. For tasks that require multiple inputs, such as Visual Question Answering, the LLM needs multiple steps in order to accomplish the task, where each step is either a sequence of models or a parallel of several sequences of models. Towards this end, the LLM must satisfy three conditions: 1) generate only model names without irrelevant tokens, 2) generate valid sequences of models, and 3) generate paralleled sequences of models for different inputs when necessary. Condition 1: For the LLM to generate only model names, instead of tuning the model to teach-force it what names are available, we adopt constrained beam search [8], which only allows generating tokens from the M at every decoding step. More specifically, we define our constraints as a prefix trie such that each model name is a path from the root to some leaf node. For each node t in the tree, its children indicate all the allowed continuations from the prefix defined traversing the trie from the root to t. Thus in each decoding step, the next token can only be selected from either all possible continuations allowed based on generated tokens or the first tokens of all possible next model names. For example, if "Text" is already generated, based on the set of model names, the next tokens can only be either "Summarization" due to the "Text Summarization" model or "Generation" due to the "Text Generation" model, as shown in Fig. 4. Condition 2: For the LLM to generate valid sequences of models, consecutive models should have input and output modalities matched. If the output modality of a model is text, then the next model can only be models that take text as input. This is also achieved by constrained beam search such that when finishing generating one model, the constraint function will determine the output modality of this model and find out all possible next models in M, excluding the models that are already generated. It will dynamically construct a new trie for all these model names based on the output modality. For example, if the first generated model name is "Text Summarization", then the next possible models can be "Sentiment Analysis", "Text Generation", etc., as shown in Fig. 5. If a task requires only one input, Conditions 1 and 2 can guarantee a valid sequence. However, if the task requires multiple inputs to generate the final result, each input may require a valid sequence before utilizing a multi-input model such as Question Answering and Visual Question Answering. In this scenario, a sequential solution is unsatisfying because different inputs should be processed in parallel. To handle this problem, we have the following Condition 3. Condition 3: Autoregressive decoding in language models is generally unsuitable for generating parallel valid sequences. In this work, we use beam search to conduct semi-autoregressive generation. Beam search is originally proposed such that multiple hypotheses are generated to compete with each other in order to obtain the highest-scored output. We instead utilize beam search as an efficient semi-autoregressive decoding method [29] such that for each decoding step in beam search, different hypotheses are treated as parallel valid solutions for different inputs instead of competing hypotheses. If a task requires multiple inputs, such as both text and image, then in generation time, a model taking text as input and a model taking image as input are almost equally likely to be generated. Since based on constrained generation, each beam is a valid model sequence eventually, thus, multiple valid sequences with different input types will be generated in parallel. When parallel processing is conducted, multi-input models and subsequent models are required. We concatenate the generated sequences with the natural language task description to generate a new prompt to prompt subsequent models. This process can be done recursively until the end-of-sentence token is generated without any more models, as illustrated in Fig. 6. Experiments Backbone LLMs We employ both ChatGPT (GPT-3.5-turbo) and two other open-source large language models for experimentation. • GPT-3.5-turbo. The GPT (Generative Pre-trained Transformer) series [2], developed by OpenAI, consists of advanced language models. GPT-3.5, a fine-tuned version of GPT-3, boasts over 175 billion model parameters. • LLaMA-7b. LLaMA [35] is a lightweight, open-source language model developed by researchers at Meta. It is designed to be efficient and performant, and can be run on a single GPU. In this work, we use the 7-billion size model of LLaMA. • Flan-T5-Large. Flan-T5 [6] is a series of language models developed by Google. Flan-T5 models are fine-tuned using a technique called instruction finetuning, which allows them to learn from a wider range of data and improve their performance on a variety of tasks. Flan-T5-Large has 770 million parameters. Learning Schema of LLMs We also employ the following LLM learning Schema for experimentation. • Zero-shot Learning (Zero) is to simply feed the task description to the model and ask for results. • Few-shot Learning (Few) presents a set of high-quality demonstrations, each consisting of both input and desired output, on the target task. As the model first sees good examples, it can better understand human intention and criteria for what kinds of answers are wanted. • Fine-tuning involves using manually labeled data samples as additional training signals to refine and adapt pre-trained LLMs to specific tasks or domains. In this setting, we also use constrained generation method introduced in Sec. 5. • RLTF is our proposed method in Sec. 4. In the context of zero-shot and few-shot learning paradigms, LLMs are allowed to produce free-form output solutions. To transform these outputs into viable task planning solutions, we employ text similarity models to map them to our model name set N , which is an established method in existing works [10]. For fine-tuning and RLTF approaches, we utilize constrained generation to directly generate the task planning solution. All the mapped or constrained generated task planning solutions are then fed to OpenAGI, to get executed and evaluated. Datasets We divide the tasks in OpenAGI into training and testing sets. In particular, we randomly select 10% of tasks, along with their corresponding datasets, based on input and output modalities for training purposes. For Few-shot Learning and Fine-tuning, we supply manually curated, feasible solutions as ground-truth labels. In the case of RLTF, we employ the Fine-tuning checkpoint as a reasonable initialization for LLM to reduce the likelihood of producing infeasible solutions. Moreover, considering the fact that the imbalanced number of tasks with different input and output modalities could lead to skewed measurement results, we choose an additional 10% of tasks, adhering to the same selection criteria as mentioned above, to serve as the test set. To counteract the influence of randomness, the test set is randomly sampled multiple times, and the average performance is calculated. Experimental Analysis The experimental results are presented in Tab. 5, the overall performance is calculated as the weighted average of CLIP, BERT and ViT scores. GPT-3.5-turbo exhibits superior performance in both zeroshot and few-shot learning settings compared to LLaMA-7b and Flan-T5-Large. This is evident from the higher scores it achieves in BERT, ViT score, and the overall performance. LLaMA-7b, while not performing as well as GPT-3.5-turbo, demonstrates better overall performance in few-shot learning compared to its zero-shot learning performance. However, its performance is still much lower than that of GPT-3.5-turbo in the same settings. Flan-T5-Large shows significant improvement when using fine-tuning or Reinforcement Learning from Task Feedback (RLTF) compared to zero-shot and few-shot learning strategies. To facilitate a comprehensive analysis of the results, we present the zero-shot and few-shot solutions in Tab. 6 and 7, respectively. Initially, it is evident that in the zero-shot setting, most LLMs struggle to generate valid task planning, let alone optimal solutions. In particular, GPT-3.5 tends to generate repetitive content, which subsequently maps to identical model names. Meanwhile, LLaMA-7b and Flan-T5-Large, constrained by their zero-shot capabilities, fail to produce a reasonable plan. In the few-shot setting, we incorporate several manually labeled task plans as instructions to guide the generation, resulting in a remarkable improvement in the quality of the task plans. As observed in Tab. 7, all three LLMs can produce solutions that are semantically similar to the provided examples. In fact, many solutions can be utilized directly, even without the need for mapping. Conclusion and Future Work In this work, we introduce OpenAGI, an open-source AGI research platform designed to facilitate the development and evaluation of large language models (LLMs) in solving complex, multi-step tasks through manipulating various domain expert models. OpenAGI provides a wide range of extensible models and datasets, predominantly utilizing resources from Hugging Face and GitHub. We also propose the LLM+RLTF approach, which combines LLMs with reinforcement learning to optimize task-solving performance. The evaluation of various LLMs using the OpenAGI pipeline and different learning schema demonstrates that smaller-scale LLMs can potentially outperform larger models when combined with the appropriate learning approach, such as RLTF. In future research, we aim to incorporate multiple models within each single-step task, thereby providing an expanded selection of options for LLMs to address out-of-distribution (OOD) problems. Additionally, we intend to integrate datasets from alternative modalities, such as video and audio, into our OpenAGI platform. These datasets will facilitate the development of more sophisticated tasks to further investigate the planning capabilities of LLMs. We will also endeavor to enhance the evaluation mechanism to enable a more accurate and comprehensive assessment of performance. Another promising direction is to involve humans in the loop in the resolution of complex tasks. In such scenarios, LLM may prompt human experts for answers as one step of the task-solving plan when a suitable model is unavailable, thus enabling better human-machine collaboration. Lastly, we aim to explore automated task generation techniques that empower OpenAGI to generate complex tasks independently, facilitating self-prompting and improvement in its task-solving capabilities. Figure 1 : 1The task-solving pipeline in OpenAGI. Figure 2 : 2Examples of the Out-of-Distribution (OOD) Generalization issue. Figure 4 : 4Model name based constrained generation. Figure 5 : 5Model type based constrained generation. Figure 6 : 6Parallel recursive sequence generation Table 1 : 1Language-related modelsDomain Task Input Modality Output Modality Model Sentiment Analysis Text Text FinBert 6 [1] Text Summarization Text Text BART 7 [13] Machine Translation Text Text T5 8 [26] Fill Mask Text Text DistilRoberta 9 [15] Question Answering Text, Text Text DistilBERT 10 [31] Table 2 : 2Vision-related modelsDomain Task Input Modality Output Modality Model Image Classification Image Text ViT 11 Table 3 : 3Vision-language modelsDomain Task Input Modality Output Modality Model Visual Question Answering Image, Text Text GIT 16 [36] Image Captioning Image Text Vision Encoder Decoder 17 Common Objects in Context (COCO)[14] is a large-scale, richly-annotated image dataset designed to advance the fields of object detection, segmentation, and captioning. Released in 2014, it contains over 200,000 labeled images with 1.5 million object instances from 80 different object categories. The dataset features complex, real-world scenes with multiple objects per image, various object scales, and diverse contexts.• CNN/Daily Mail [19] is a valuable resource for text summarization, which consists of human- generated abstractive summaries, created by transforming news articles from CNN and Daily Mail websites into questions, with one entity concealed, and generating summaries from the corresponding passages. The authors have made available the scripts used to crawl, extract, and generate question-answer pairs from these websites. The corpus contains 286,817 training pairs, 13,368 validation pairs, and 11,487 test pairs, as defined by the scripts. On average, the source documents in the training set span 766 words across 29.74 sentences, while the summaries are composed of 53 words and 3.72 sentences. Table 4 : 4Examples of multi-step tasks and their augmented data samples. Table 5 : 5OpenAGI task-solving performances under different settingsMetrics GPT-3.5-turbo LLaMA-7b Flan-T5-Large Zero Few Zero Few Zero Few Fine-tuning RLTF CLIP Score 0 0 0 0 0 0 0.3059 0.3059 BERT Score 0.1914 0.3820 0 0.1781 0 0.2488 0.1166 0.2554 ViT Score 0.2437 0.7497 0 0 0 0 0.6285 0.6551 Overall 0.2284 0.4335 0 0.1272 0 0.1777 0.1957 0.3446 https://huggingface.co/ 4 https://github.com/ Our OpenAGI project started in 2022 based on the T5 language model series, which was before the release of ChatGPT. After ChatGPT was released, we still decided to keep the open-source LLMs (Flan-T5-Large and LLaMA) rather than only using ChatGPT (GPT-3.5) because we aim to contribute an open-source AGI research platform and benchmark to the community. AppendixLLMOutput Mapped Task Planning Prompt Problem: Given low-resolution blurry grayscale image, how to return the regular image step by step? What is its solution? Use "Step" to mark.GPT-3.5Step 1: Collect more data or images to create a higher resolution version of the blurry image.Step 2: Use a deblurring algorithm to remove the blurriness and increase the sharpness of the image.Step 3: Apply a denoising algorithm to reduce any noise in the image that may have been introduced during the deblurring process.Step 4: Use an upsampling method to increase the resolution of the image to the desired level.Step 5: Apply a contrast enhancement technique to make the image more visually appealing.StepFlan-T5-LargeThe blurry image is a blurry image with a blurry background. To return the regular image, you can use the following steps: First, you need to find the blurry image. Then, you need to find the regular image. Then, you need to find the blurry image. Then, you need to find the regular image. Then, you need to find the blurry image. Then, you need to find the regular image. Then, ... Non-sense. 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Solution: Step 1:Image Super Resolution. how to return the regular image step by step? SolutionBarret Zoph and Quoc V. Le. 2017. Neural Architecture Search with Reinforcement Learning. In 5th ICLR. https://openreview.net/forum?id=r1Ue8Hcxg LLM Mapped Task Planning Prompt Problem: Given low-resolutioned noisy blurry gray image, how to return the regular image step by step? Solution: Step 1:Image Super Resolution, Step 2: Image Denoising, Step 3: Image Deblurring, Step 4: Colorization, ... ... ... Problem: Given clozed English text, how to translate the text in German step by step? Solution: Step 1:Fill Mask, Step 2: Machine Translation, Problem: Given low-resolutioned blurry grayscale image, how to return the regular image step by step? Solution: Step 1: Image Super Resolution. 2Step 3: Image DenoisingStep 1: Image Super Resolution, Step 2: Image Deblurring, Step 3: Image Denoising. Problem: Given low-resolutioned blurry grayscale image, how to return the regular image step by step? Soltuion: Step 1:Image Super Resolution, Step 2: Image Deblurring, Step 3: Colorization, Step 4: Image Classification, Step 5: Machine Translation, Image Super Resolution, Image Deblurring, Colorization, Image Classification, Machine Translation Flan-T5-Large Step 1: Image Super Resolution. . . . . , Image Super Resolution, Image Deblurring, Image Denoising LLaMA-7b Prblem: Given low-resolutioned noisy blurry gray image, how to return the regular image step by step? Solution: Step 1:Image Super Resolution. Machine Translation2Image Super Resolution. Image Deblurring, Colorization, Image Captioning, Machine Translation Table 7: Example of Few-shot PromptImage Super Resolution, Image Deblurring, Image Denoising LLaMA-7b Prblem: Given low-resolutioned noisy blurry gray image, how to return the regular image step by step? Solution: Step 1:Image Super Resolution, Step 2: Image Denoising, Step 3: Image Deblurring, Step 4: Colorization, Prblem: Given noisy blurry gray image, how to return the caption in Chinese step by step? Solution: Step 1:Image Denoising, Step 2: Image Deblurring, ... ... ... Problem: Given low-resolutioned blurry grayscale image, how to return the regular image step by step? Soltuion: Step 1:Image Super Resolution, Step 2: Image Deblurring, Step 3: Colorization, Step 4: Image Classification, Step 5: Machine Translation, Image Super Resolution, Image Deblurring, Colorization, Image Classification, Machine Translation Flan-T5-Large Step 1: Image Super Resolution, Step 2: Image Deblurring, Step 3: Colorization, Step 4: Image Captioning, Step 5: Machine Translation, Image Super Resolution, Image Deblurring, Colorization, Image Captioning, Machine Translation Table 7: Example of Few-shot Prompt.	[]
[ "A GLOBAL PHOTOIONIZATION RESPONSE TO PROMPT EMISSION AND OUTLIERS: DIFFERENT ORIGIN OF LONG GAMMA-RAY BURSTS?", "A GLOBAL PHOTOIONIZATION RESPONSE TO PROMPT EMISSION AND OUTLIERS: DIFFERENT ORIGIN OF LONG GAMMA-RAY BURSTS?" ]	[ "J Wang \nKey Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina\n\nSchool of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "L P Xin \nKey Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina\n", "Y L Qiu \nKey Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina\n", "D W Xu \nKey Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina\n\nSchool of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "J Y Wei \nKey Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina\n\nSchool of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "J Wang ", "Wj@bao Ac Cn \nSchool of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina\n", "Wang " ]	[ "Key Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina", "School of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina", "Key Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina", "Key Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina", "Key Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina", "School of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina", "Key Laboratory of Space Astronomy and Technology\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina", "School of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina", "School of Astronomy and Space Science\nUniversity of Chinese Academy of Sciences\nBeijingChina" ]	[]	By using the line ratio C IVλ1549/C IIλ1335 as a tracer of ionization ratio of the interstellar medium (ISM) illuminated by a long gamma-ray burst (LGRB), we identify a global photoionization response of the ionization ratio to the photon luminosity of the prompt emission assessed by either L iso /E peak or L iso /E 2 peak . The ionization ratio increases with both L iso /E peak and L iso /E 2 peak for a majority of the LGRBs in our sample, although there are a few outliers. The identified dependence of C IV/C II on L iso /E 2 peak suggests that the scatter of the widely accepted Amati relation is related with the ionization ratio in ISM. The outliers tend to have relatively high C IV/C II values as well as relatively high C IVλ1549/Si IVλ1403 ratios, which suggests an existence of Wolf-Rayet stars in the environment of these LGRBs. We finally argue that the outliers and the LGRBs following the identified C IV/C II−L iso /E peak (L iso /E 2 peak ) correlation might come from different progenitors with different local environments.	10.3847/1538-4357/aaad00	[ "https://arxiv.org/pdf/1802.01575v1.pdf" ]	59,406,540	1802.01575	3e9d05714d615a30c3b283c91ad3c7b05f62a66c	A GLOBAL PHOTOIONIZATION RESPONSE TO PROMPT EMISSION AND OUTLIERS: DIFFERENT ORIGIN OF LONG GAMMA-RAY BURSTS? 5 Feb 2018 July 28, 2018 J Wang Key Laboratory of Space Astronomy and Technology National Astronomical Observatories Chinese Academy of Sciences 100012BeijingChina School of Astronomy and Space Science University of Chinese Academy of Sciences BeijingChina L P Xin Key Laboratory of Space Astronomy and Technology National Astronomical Observatories Chinese Academy of Sciences 100012BeijingChina Y L Qiu Key Laboratory of Space Astronomy and Technology National Astronomical Observatories Chinese Academy of Sciences 100012BeijingChina D W Xu Key Laboratory of Space Astronomy and Technology National Astronomical Observatories Chinese Academy of Sciences 100012BeijingChina School of Astronomy and Space Science University of Chinese Academy of Sciences BeijingChina J Y Wei Key Laboratory of Space Astronomy and Technology National Astronomical Observatories Chinese Academy of Sciences 100012BeijingChina School of Astronomy and Space Science University of Chinese Academy of Sciences BeijingChina J Wang Wj@bao Ac Cn School of Astronomy and Space Science University of Chinese Academy of Sciences BeijingChina Wang A GLOBAL PHOTOIONIZATION RESPONSE TO PROMPT EMISSION AND OUTLIERS: DIFFERENT ORIGIN OF LONG GAMMA-RAY BURSTS? 5 Feb 2018 July 28, 2018(Received July 1, 2016; Revised September 27, 2016; Accepted July 28, 2018) Submitted to ApJDraft version Typeset using L A T E X twocolumn style in AASTeX61 Corresponding author:gamma-ray burst: general -methods: statistical -galaxies: ISM By using the line ratio C IVλ1549/C IIλ1335 as a tracer of ionization ratio of the interstellar medium (ISM) illuminated by a long gamma-ray burst (LGRB), we identify a global photoionization response of the ionization ratio to the photon luminosity of the prompt emission assessed by either L iso /E peak or L iso /E 2 peak . The ionization ratio increases with both L iso /E peak and L iso /E 2 peak for a majority of the LGRBs in our sample, although there are a few outliers. The identified dependence of C IV/C II on L iso /E 2 peak suggests that the scatter of the widely accepted Amati relation is related with the ionization ratio in ISM. The outliers tend to have relatively high C IV/C II values as well as relatively high C IVλ1549/Si IVλ1403 ratios, which suggests an existence of Wolf-Rayet stars in the environment of these LGRBs. We finally argue that the outliers and the LGRBs following the identified C IV/C II−L iso /E peak (L iso /E 2 peak ) correlation might come from different progenitors with different local environments. INTRODUCTION Long gamma-ray bursts (LGRBs) are the most powerful explosions occurring from local universe (the nearest one is GRB 980425 at z = 0.008, Galama et al. 1998) to very high redshift (e.g., Salvaterra et al. 2009;Tanvir et al. 2009). Up to date, the most distant one reported in literature is GRB 090429B with a photometric redshift of z = 9.4 (Cucchiara et al. 2011). The detection of the associated supernova in a few LGRBs (see Cano et al. 2016 for a recent review) strongly supports that LGRBs originate from the core-collapse of young massive stars (≥ 25M ⊙ ) (e.g., Hjorth & Bloom 2012;Woosley & Bloom 2006 and references therein). The GRB's afterglow at a wide wavelength range from radio to X-ray is produced through the synchrotron radiation when the jet ignited in the core-collapse impacts and shocks the surrounding medium (e.g., Meszaros & Rees 1997;Sari et al. 1998). Within the first hours after the onset of a burst, the powerful afterglows of LGRBs illuminate not only the interstellar medium (ISM) of their host galaxies, but also the intergalactic medium, which produces an afterglow optical spectrum associated with multiple strong absorption lines of metals with different ionization stages at different redshifts (e.g., Fybno et al. 2009;de Ugarte Postigo et al. 2012). The spectra provide us an opportunity to study the properties of both medium at GRB's local environment and intervening absorption clouds located between the host galaxy and the observer (e.g., Savaglio et al. 2003;Butler et al. 2003;Tejos et al. 2007;Vergani et al. 2009;Vreeswijk et al. 2007;Kawai et al. 2006;Totani et al. 2006;D'Elia et al. 2009D'Elia et al. , 2010Wang 2013). A response of the host galaxy environment to both prompt and afterglow emission has been proposed and observed for a long time. The theoretical study in Perna et al. (2003) suggested that the silicates can be destroyed by the strong X-ray/UV radiation (see also in Waxman & Draine 2000). The photoionization effect due to the prompt and afterglow emission of GRBs has been identified in the X-ray spectra of a few bursts (e.g., Amati et al. 2000, Antonelli et al., 2000Piro et al. 2000). Due to the afterglow evolution, an evolution of photoionization of the medium around the progenitors has been put forward in Perna & Loeb (1998), which is subsequently supported by the observed time variability of absorption lines in a few bursts (e.g., GRB 010222, Mirabal et al. 2002;GRB 020813, Dessauges-Zavadsky et al. 2006;GRB 060418, Vreeswijk et al. 2007;GRB 060206, Hao et al. 2007;GRB 080310, Vreeswijk et al. 2013;GRB 100901, Hartoog et al. 2013). In addition, with a sample of 69 low-resolution afterglow spectra, de Ugarte Postigo et al. (2012) revealed a global weak dependence of the ionization ratio quantified by the line strength parameter on the restframe isotropic prompt energy E iso in 1-1000keV band. In this paper, we identify a global photoionization response of the ISM to GRB's prompt emission by instead focusing on the role of high energy ionizing photon flux. The result further motivates us to suspect that there are two kinds of origin of LGRBs. The paper is organized as follows. The sample selection is presented Section 2. Section 3 shows the results and implications. SAMPLE Our aim is to study the global photoionization response of the host galaxy ISM to LGRB's prompt emission. We adopt the line ratio of C IVλλ1548,1550 and C II/C II * λλ1334,1335 as a tracer of ionization ratio of the ISM of individual LGRB, both because the two lines are usually quite strong in the afterglow spectra and because the ionization potential of C IV is as high as 47.89eV. We compile a sample of Swift (Gehrels et al. 2004) LGRBs with reported measurements of both two absorption lines and prompt emission. Our sample is mainly complied from de Ugarte Postigo et al. (2012) which published a sample of low-resolution afterglow spectra of 69 LGRBs. For the bursts with measurements (including both detection and upper limit) of both C IV and C II, the common objects with a measurement of prompt emission is extracted from Ghirlanda et al. (2017) and Nava et al. (2012). The sample is finally composed of 20 LGRBs and is tabulated in Table 1, along with the references. Columns (1) and (2) list the identification and the measured redshift of each LGRB, respectively. The measured equivalent widths (EWs) of C IIλ1335, C IVλ1549 and Si IVλ1403 absorption lines are given in Columns (3), (4) and (5), respectively. The line ratio in logarithmic of C IV/C II is listed column (6). Columns (7) and (8) are the rest-frame isotropic prompt luminosity and peak energy based on the standard Band spectrum. All the errors reported in the table correspond to the 1σ significance level after taking into account the proper error propagation. RESULTS AND DISCUSSION C IV/C II: A Global Photoionization Response of ISM to LGRBs' Prompt Emission A global ionization response of ISM to the prompt emission of LGRBs is shown in Figure 1 in which the line ratio of C IV/C II is used as a tracer of ionization ratio of the ISM within the line-of-sight of an observer. In stead of using E iso as an indicator of the strength of the prompt emission in the study of de Ugarte Postigo Note-References in the last column: (1) et al. (2012), the left panel in Figure 1 plots C IV/C II as a function of L iso /E peak . Based on the definition of the Band spectrum, L iso /E peak is the isotropic photon numbers emitted per second at the characteristic photon energy defined as E peak , which is equivalent to the ionization parameter that is widely used in the photoionization models (e.g., Osterbrock & Ferland 2006) if the densities of the ISM of different LGRBs are comparable. GRB zGRB EW(CIVλ1549) EW(CII/CII * λ1335) EW(SiIVλ1549) log(CIV/CII) log Liso log E peak References AÅÅ erg s −1 keV (1) (2) (3) (4) (5) (6) (7) (8)(9) For the bursts with a detection of EW of C II, one can see that there is a dependence of C IV/C II ratio on L iso /E peak , except for two outliers with relatively large C IV/C II ratios. Generally speaking, higher the C IV/C II ratio, larger the L iso /E peak will be, which indicates that the ionization ratio of the ISM around the bursts increases with the ionizing photon flux assessed from the prompt emission of LGRBs. A statistical test yields a Kendall's τ = 0.238 and a Z-value of 1.237 at a significance level with a probability of null correlation of P = 0.216. The significance of the dependence is considerably enhanced to be τ = 0.539, Z = 2.562 and P = 0.0104 when the two outliers are excluded from the statistics. The significance is further enhanced obviously in the right panel of Figure 2, which plots line ratio C IV/C II as a function of L iso /E 2 peak . The physical meaning of L iso /E 2 peak can be understood as the specific photon numbers emitted per second with a photon energy of E peak . The same statistical test results in a significantly improved statistics with a τ = 0.352, Z = 1.836 and P = 0.0664 and a τ = 0.603, Z = 2.873 and P = 0.0041 when the outliers are excluded. Both C IV/C II−L iso /E peak and C IV/C II−L iso /E 2 peak correlations suggest a photoionization effect in which the circumburst medium in the line-of-sight is photoionized by the GRB's prompt and afterglow emission. In fact, the photoionization effect is revealed in some previous case studies focusing on individual GRBs. A transient absorption edge at ∼ 3.8keV, which is produced by the the circumburst medium highly ionized by the GRB's prompt emission, is discovered in the X-ray spectrum of the prompt emission of GRB 990705 (Amati et al. 2000). Emission features (e.g., Fe Kα and Lyα lines) resulted from photoionization by the GRB's prompt and afterglow emission is identified in the X-ray afterglow spectra of a few GRBs (e.g., Antonelli et al., 2000;Piro et al. 1999, Yoshida et al. 1999. The dependence of C IV/C II on L iso /E 2 peak shown in the right panel in Figure 1 is quite interesting. In fact, previous statistical studies firmly established a tight correlation between L iso and E peak , which results in a relationship of L iso ∝ E 2 peak in both homogeneous and wind ISM (e.g., Amati et al. 2002;Yonetoku et al. 2004;Ghirlanda et al. 2010Ghirlanda et al. , 2017Nava et al. 2012). Sub-sequent studies suggested that the relationship is physically driven by the initial Lorentze factor (e.g., Nava et al. 2012;Ghirlanda et al. 2017 and references therein). The dependence revealed by us therefore suggests that the scatter of the L iso −E peak relationship is related with the C IV/C II ratio. A linear fitting FITEXY with uncertainties in both x and y coordinates yields a relationship of log L iso E 2 peak = (47.15 ± 0.07) + (2.38 ± 0.28) log CIV CII (1) The D'Agostini fitting method (D'Agostini 2005, see also in e.g., Guidorzi et al. 2006, Amati et al. 2008 is alternatively used to model the linear relationship as (2) with an extra scatter of ǫ = 0.19. Both best-fitted relationships are overplotted in the right panel of Figure 1. Figure 2 shows the E peak versus L iso (E iso ) correlation for the sample used in this study. The same D'Agostini fitting method returns best fits: log E peak = (−14.91 ± 1.64) + (0.33 ± 0.03) log L iso (3) associated with an extra scatter of ǫ = 0.32, and log E peak = (−26.13 ± 2.86) + (0.54 ± 0.05) log E iso (4) associated with an extra scatter of ǫ = 0.27. A comparison of the obtained scatters enables one to definitely see that the dispersion of the L iso /E 2 Peak versus photoionization ratio correlation is smaller than both of the E peak − L iso and E peak − E iso correlations. y = β 0 + β 1 x + ǫ, Outliers: Different Origin of LGRBs? By including both outliers and bursts with an upper limit of EW of C II, the distribution on the C IV/C II versus L iso /E peak (L iso /E 2 peak ) diagram further suggests that the bursts listed in our sample could be divided into two groups: a majority of the bursts that follow the ionization ratio versus ionizing photon flux dependence and a few outliers with relatively either large C IV/C II ratios or small L iso /E peak (L iso /E 2 peak ) (or both). The three outliers in Figure 1 are: GRB 050908, GRB 070411 and GRB 080810. In the first case, Martone et al. (2017) argued that the outliers in the E peak − E iso correlation is possibly Right panel: the same as the left one but for Liso/E 2 peak . The black solid line shows the best fit for the C IV/C II versus Liso/E 2 peak sequence through the FITEXY method (i.e., Eq.(1)). The 3σ deviation from the best fit in both intercept and slope is shown by the black short-dashed lines. The green long-dashed and dotted lines presents the best fit and 1σ deviations, respectively, which is obtained through the D'Agostini fitting method (i.e., Eq.(2)). Table 1 and Section 3.2) are marked by the red dots. Right panel: the same as the left one but for Eiso. due to an overestimation of E peak resulted from an underestimation of X-ray prompt emission. In the current sample, Figure 2, however, shows that all the three outliers generally follow the fitted E peak − E iso correlation, which suggests that the observational bias could be not a favorite explanation for the outliers. Alternatively, the large C IV/C II ratios suggest a abnormally high ionization ratio in the ISM for the outliers. A possible explanation of the high ionization ratio is an additional photoionization contributed by the underlying intensive starformation, especially the massive Wolf-Rayet (WR) stars in the environment of the LGRBs. Taking into account the WR outflow model, Berger et al. (2006) argued that the C IVλ1549/Si IVλ1403 line ratio is a good tracer of an existence of WRs because the outflow from the massive stars would increase the carbon metallicity in the ISM. In fact, a tendency of higher C IV/Si IV ratio in LGRBs than in QSOs is tentatively revealed in de Ugarte Postigo et al. (2012) through a large sample. In order to check if the WR star scenario is working for the outliers, Figure 2 shows the cumulative distribution of C IV/Si IV ratio for the current sample. The vertical lines mark the C IV/Si IV values for the three outliers with the largest deviation from the C IV/C II versus L iso /E peak (L iso /E 2 peak ) sequence. Clearly, both outliers with a firm measurement of EW of C II have quite high C IV/Si IV ratios, which agrees with the prediction of the WR scenario quite well. The outliers with a property of both large C IV/C II and C IV/Si IV ratios motivates us to suspect that the bursts following and deviating the C IV/C II−L iso /E peak (L iso /E 2 peak ) sequence are produced within different environments, or on other worlds, produced by different progenitors. In fact, on the theoretical ground, both single-star model with a central engine of either a blackhole or a magentar (e.g., Woosley & Heger 2006;Dai & Lu 1998;Zhang & Dai 2010;Wang et al. 2017) and close interacting binary models (e.g., Fryer et al. 2007;van den Heuvel et al. 2007) have been proposed as the origin of LGRBs . On the observational ground, features emitted from WR stars have been detected in the spectra of the host galaxies of 8 nearby LGRBs (Han et al. 2010). With the measurements of metallicity of host galaxies of LGRBs up to z ∼ 2, a metallicity threshold of Z th = 0.7Z ⊙ is suggested for the origin of LGRBs (e.g., Japelj et al. 2015;Vergani et al, 2017 and refer-ences therein). This threshold is, however, higher than the requirement of 0.2Z ⊙ of the single-star model, which suggests an alternative progenitor of close interacting binary for some LGRBs, because the binary model is less sensitive to the metallicity of the progenitor. CONCLUSION A correlation between line ratio C IV/C II and L iso /E peak (L iso /E 2 peak ) is identified for a majority of LGRBs listed in this study, which suggests a global response of the ionization ratio to the ionizing photon luminosity assessed from the prompt emission. The outliers of the correlation, which have both high C IV/C II and C IV/Si IV ratios, motivate us to suspect that their progenitors differs from the bursts following the identified correlation. de Ugarte Postigo et al. 2012; (2) Cucchiara et al. 2011; (3) Xin et al. 2017; (4) Ghirlanda et al. 2017; (5) Nava et al. 2012. The three outliers in Figure 1 are marked with asterics in Column (1). Figure 1 . 1Left panel: line ratio C IV/C II plotted against Liso/E peak . The bursts with a determination of EW of C II are shown by the red solid circles, and the bursts with an upper limit of EW of C II by the blue open triangles and arrows. The errorbars correspond to the 1σ significance level. Figure 2 . 2Left panel: E peak plotted against Liso. The best fitted linear relationship obtained through the D'Agostini fitting method is presented by the solid line, and the 1σ scatter by the two dashed lines. The three outliers (see Figure 3 . 3Cumulative distribution of C IV/Si IV line ratio. The values of the three outliers inFigure 1are marked by the vertical lines and labels. Table 1 . 1Sample of SwiftLGRBs with Measurements of Both Absorption Lines and Prompt Emission. . L Amati, F Frontera, M Tavani, A&A. 39081Amati, L., Frontera, F., Tavani, M. et al., 2002, A&A, 390, 81 . 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[ "CLUSTERLLM: Large Language Models as a Guide for Text Clustering", "CLUSTERLLM: Large Language Models as a Guide for Text Clustering" ]	[ "Yuwei Zhang \nUniversity of California\nSan Diego\n", "Zihan Wang \nUniversity of California\nSan Diego\n", "Jingbo Shang [email protected] \nUniversity of California\nSan Diego\n" ]	[ "University of California\nSan Diego", "University of California\nSan Diego", "University of California\nSan Diego" ]	[]	We introduce CLUSTERLLM, a novel text clustering framework that leverages feedback from an instruction-tuned large language model, such as ChatGPT. Compared with traditional unsupervised methods that builds upon "small" embedders, CLUSTERLLM exhibits two intriguing advantages: (1) it enjoys the emergent capability of LLM even if its embeddings are inaccessible; and (2) it understands the user's preference on clustering through textual instruction and/or a few annotated data. First, we prompt ChatGPT for insights on clustering perspective by constructing hard triplet questions <does A better correspond to B than C>, where A, B and C are similar data points that belong to different clusters according to small embedder. We empirically show that this strategy is both effective for fine-tuning small embedder and cost-efficient to query ChatGPT. Second, we prompt ChatGPT for helps on clustering granularity by carefully designed pairwise questions <do A and B belong to the same cat-egory>, and tune the granularity from cluster hierarchies that is the most consistent with the ChatGPT answers. Extensive experiments on 14 datasets show that CLUSTERLLM consistently improves clustering quality, at an average cost of ∼$0.6 1 per dataset. * Corresponding author. 1 The cost is calculated with gpt-3.5-turbo.	10.48550/arxiv.2305.14871	[ "https://export.arxiv.org/pdf/2305.14871v1.pdf" ]	258,866,119	2305.14871	a7beaf4ad0c59ad6c91a03af6eceaafd2d44cef9	CLUSTERLLM: Large Language Models as a Guide for Text Clustering Yuwei Zhang University of California San Diego Zihan Wang University of California San Diego Jingbo Shang [email protected] University of California San Diego CLUSTERLLM: Large Language Models as a Guide for Text Clustering We introduce CLUSTERLLM, a novel text clustering framework that leverages feedback from an instruction-tuned large language model, such as ChatGPT. Compared with traditional unsupervised methods that builds upon "small" embedders, CLUSTERLLM exhibits two intriguing advantages: (1) it enjoys the emergent capability of LLM even if its embeddings are inaccessible; and (2) it understands the user's preference on clustering through textual instruction and/or a few annotated data. First, we prompt ChatGPT for insights on clustering perspective by constructing hard triplet questions <does A better correspond to B than C>, where A, B and C are similar data points that belong to different clusters according to small embedder. We empirically show that this strategy is both effective for fine-tuning small embedder and cost-efficient to query ChatGPT. Second, we prompt ChatGPT for helps on clustering granularity by carefully designed pairwise questions <do A and B belong to the same cat-egory>, and tune the granularity from cluster hierarchies that is the most consistent with the ChatGPT answers. Extensive experiments on 14 datasets show that CLUSTERLLM consistently improves clustering quality, at an average cost of ∼$0.6 1 per dataset. * Corresponding author. 1 The cost is calculated with gpt-3.5-turbo. Introduction Text clustering has been studied for years and it has recently gained new significance in identifying public perception from social media (Park et al., 2022), analysing cause of accidents or detecting emerging research topics (Martínez et al., 2022). A common practice in text clustering is to apply clustering algorithms (MacQueen, 1967;Zhang et al., 2021a) on top of pre-trained embedders (Muennighoff et al., 2022;Su et al., 2022) which could achieve higher performance with better pre-training quality. However, ChatGPT (API-based) A should be closer to C than B Traditional Not Applicable ClusterLLM Figure 1: Traditional text clustering often employs clustering algorithms on top of the pre-trained embedders which could produce sub-optimal clustering. LLMs like ChatGPT are not applicable for text clustering directly because of the inaccessible embeddings. CLUSTER-LLM resolves the dilemma by leveraging LLM as a guide on text clustering. recent instruction-tuned LLMs such as ChatGPT, that demonstrated extraordinary language capabilities for various natural language applications by following textual instructions, can only be utilized through the APIs without accessible embedding vectors for clustering. Hence, LLMs can not be directly applied on text clustering tasks. In this paper, we wish to provide insights on the following question: • Can we leverage API-based LLMs to guide text clustering efficiently? To approach such a challenge, we first draw inspiration from an observation that humans represent an instance through comparing with others (Nosofsky, 2011). For instance, people often classify a new piece of music into a specific genre by relating to familiar ones. In fact, pairwise relationships have been utilized in spectral clustering (Donath and Hoffman, 1972;Cheeger, 1970) before. Nonetheless, naively traversing all the pairs within dataset is obviously intractable for querying ChatGPT. In this paper, we propose CLUSTERLLM, a generic framework that utilizes LLM to guide a small embedder for finding text clusters with a low cost (see Figure 1 for an overview). To improve clustering quality, we propose to prompt LLMs with a triplet task that predicts which one of the two candidates is closer to the anchor instance. The reason we choose triplet task over others is that, it only needs to predict the ranking between two choices, and thus is easier for LLMs when the prior knowledge of granularity is unknown. For example, when asking "Is the same kind of shape as ?", it is hard to answer because the scope of category is unknown. However, when asking "Which shape looks more like ? or ?", it is much easier to pick up . We query LLMs with a collection of such triplet tasks along with a task-specific instruction. These instructions are used to indicate a desired perspective of clustering, such as topic, intent, emotion or even relation extraction, that is usually indicated by the user. The predicted triplets thereafter are used to fine-tune small embedders with the objective function following (Su et al., 2022). We expect that after fine-tuning, the embedding space is refined by the triplets and will produce cluster assignments according to user's perspective. In order to reduce the amount of API queries, we further propose a method to sample the most informative triplets from the dataset according to the current clustering structure. Our intuition is that hard triplets can better benefit from the high-level language ability of LLMs. Specifically, we first calculate entropy for each instance based on cluster assignment probabilities, and then identify those with highest entropy. Two candidate choices are then sampled from its nearest clusters to guarantee that they are close enough to the anchor. Furthermore, when a few annotated data pairs are available, we propose a way to determine the number of clusters by deducing whether a pair of data belong to the same category with in-context demonstrations. The candidate pairs are sampled from intermediate steps of hierarchical clustering that are crucial in determining whether to split/merge clusters. Finally, the decision is made by finding the best corresponded granularity in the hierarchy with the predicted pairs from LLMs. We extensively evaluate CLUSTERLLM on 14 datasets that includes intent discovery, topic mining, information extraction, domain discovery and emotion detection. Furthermore, these datasets span a wide range of granularities that have 10 to 150 number of clusters. We show that CLUSTER-LLM is effective on improving clustering quality, where the clustering performance is improved on 11 of 14 datasets over a deep clustering baseline. Moreover, the ablation study shows that CLUSTER-LLM also outperforms a self-supervise baseline and our sampling strategy is effective compared to a random sampling baseline. Finally, CLUSTER-LLM also outperforms the best baseline that rely on clustering errors on determining cluster granularity. In summary, our contributions are three-fold: (i) We study leveraging API-based LLMs on text clustering tasks. We propose a framework CLUSTER-LLM that utilizes accurately predicted sentence relations from LLMs to guide small embedders for clustering. Furthermore, CLUSTERLLM allows users to provide textual instructions and/or fewshot annotations to specify preferences on clustering. (ii) In order to reduce cost on API-queries, we propose a novel entropy-based sampling strategy to find those most informative triplets from the studied dataset. Extensive experiments show that our proposed method can improve clustering performance at ∼$0.2 for perspective and ∼$0.4 for granularity with GPT-3.5. (iii) Additionally, we propose to determine cluster granularity with LLMs prompted by pairwise data sampled from hierarchical clustering. Methodology Problem Statement. Text clustering takes an unlabeled corpus D = {x i } N i=1 as input, and outputs a clustering assignment Y = {y i } N i=1 that maps the input text to cluster indices. To specify user's needs, CLUSTERLLM integrates additional textual instruction to understand perspective and few-shot annotations to determine cluster granularity. Overview of Our Approach. CLUSTERLLM is based on a pre-trained small embedder Su et al., 2022) (denoted as f ) which usually represents sentences individually. In contrast, inspired by human cognitive ability, CLUSTERLLM considers a pair or a triplet of sentences through prompting LLMs. Specifically, CLUSTERLLM is a two-stage framework. In Section 2.1 we first introduce triplet task to improve clustering quality with respect to user-specified perspectives, along with a sampling strategy that reduces the number of API queries. In Section 2.2, we introduce pairwise task that determines the cluster granularity based on pairs sampled with hierarchical clustering and Figure 2: Our proposed framework CLUSTERLLM has two stages. In the first stage, we start with clustering on top of a small embedder. Entropy-based sampling is applied to identify the most informative triplets to query LLMs. The predicted triplets thereafter are used for fine-tuning small embedders that produce better clustering. In the second stage, we sample pairwise data from hierarchical clustering and then query LLMs. A consistency measure is then utilized to determine the best granularity from cluster hierarchy. predicted by LLMs. See Figure 2 for an overall framework. Triplet Task for Perspective In this section, we explore how to harness triplet task to refine the cluster structures for a userspecified perspective. A triplet task takes as input a tuple of three sentences t = (a, c 1 , c 2 ). The LLM is then prompted to select one of the choices from (c 1 , c 2 ) that better corresponds with the anchor a with a prompt P T . Moreover, in order to specify the user's perspective, P T also requires a task instruction I T as input argument (refer to Table 12 for complete prompts we used in the experiments). Thus the LLM prediction process is, c j = P T (I T , t),(1) where c j ∈ {c 1 , c 2 } indicates one of the choices that is selected by the LLM as positive and we denote the other (or negative) one as c \j . Entropy-based Triplet Sampling In this section, we resolve the question of how to mine the most informative triplets to both save the costs from querying LLMs and optimally improve the clustering. The intuitions of our algorithm is that (1) the candidate choices should be close to the anchor (in other words, it should be either a positive or a hard negative) and (2) the sampled triplets should be challenging enough so that potentially LLMs can provide more information to the embedder f . To achieve this, we resort to the current clustering results from the extracted embeddings Z = {z i = f (x i )} N i=1 . Specifically, since the granularity is unknown at current stage, a clustering algorithm with fixed hyperparameters (such as number of clusters in Kmeans (MacQueen, 1967) or maximum distance to merge clusters in hierarchical clustering) is performed on top of Z. These hyperparameters are consistent across datasets and are only specific to the embedder model f . A cluster center µ k will thereafter be calculated for cluster k by averaging embeddings assigned to it. Following (Xie et al., 2016;Van der Maaten and Hinton, 2008), the soft assignments for each instance are calculated with Student's t-distribution, p ik = (1 + \|\|z i − µ k \|\| 2 /α) − α+1 2 k ′ (1 + \|\|z i − µ k ′ \|\| 2 /α) − α+1 2(2) where α = 1 is the degree of freedom. After that the instance-wise entropy is calculated, H i = − k p ik log(p ik )(3) To identify the most challenging triplets, we sort the entropies in descending order and only keep a fraction γ of the top instances as anchors since they are the most ambiguous ones. And then for each anchor, we first select 2% of the total number of clusters as nearest clusters (with a minimum number of two which include its own cluster). And then two clusters are randomly chosen from these nearest clusters, where an instance will be sampled from each of them to form the two candidate choices. We keep sampling new triplets until a maximum number is reached. Remarks. (1) Notice that, since the maximum number here is fixed by the user and it is not depended on the dataset size, our sampling is costefficient. For example, in our experiments, using 1, 024 queries can improve performance on both dataset scales of ∼ 3, 000 and ∼ 50, 000. (2) From the view of ground truth, the sampled triplets might contain "both are correct" or "none of the above". However, we argue that even these triplets might still provide soft aligning information. For instance, when both are incorrect, one still be closer than the other. (3) Our sampling method may also be utilized in active learning to acquire human annotations when no prior knowledge is available on the categories. Finetuning Embedder Now that we have an accurate estimation of the triplet relationships, it is still not clear how to utilize them in clustering. Previous research on deep constrained clustering Manduchi et al., 2021) are often sensitive to noisy labels (Basu et al., 2008) which is unfortunately the case in our scenario. In this paper, we instead focus on finetuning the base embedder f towards producing an embedding space that better explains the user's perspective. In order to avoid the embedder being biased towards hard examples after fine-tuning, we need to exploit both hard and in-batch negatives. Specifically, following (Su et al., 2022;Ni et al., 2022b), for a triplet t = (a, c j , c \j ), we optimize the following training objective, l j = exp (s(a, c j )/τ ) c l ∈B exp (s(a, c l )/τ )(4) where B combines positive c j , hard negative c \j and other in-batch negatives. τ is a temperature parameter. Following the original implementation, we also compute the loss with a and c j swapped. Finally fine-tuned embedders can be applied to find even more informative triplets with the algorithm in Section 2.1.1 which will further boost the clustering performance in an iterative manner. Pairwise Task for Granularity In this section, we build on top of the refined embedding space in Section 2.1 and determine cluster granularity. Determining granularity can be converted into a problem of finding the best hyper-parameters for clustering algorithms (such as number of clusters or maximum distance). It is non-trivial, since different clustering granularities can be applied to the same dataset (such as coarse-grained domains or fine-grained topics). To tackle such a challenge, we query LLM with pairwise task that predicts whether a pair of instances p = (x i , x j ) belong to the same cluster or not with a prompt P P , w = P P (I P , {p d } D d=1 , p)(5) where w ∈ {same, different} is the binary decision, I P is the task instruction and {p d } D d=1 are few-shot demonstration pairs used for in-context learning. We assume these demonstration pairs are annotated by users who have a desired cluster granularity in mind, and the cluster granularity can be effectively inferred from them. Determine Granularity with Pairwise Hierarchical Sampling To find candidate pairs to be predicted by LLMs, we first use hierarchical clustering to generate clustering hierarchy with fine-tuned embeddings. A clustering hierarchy starts from instance level clusters to a single cluster (see Figure 2 left), and at each intermediate step, the closest pair of clusters are merged into a new cluster. Therefore, each step correspond to a number of cluster. Notice that, here we assume a maximum and a minimum of number of clusters (denoted as k max and k min ) similar to Pelleg et al. (2000) which are depended on the user's expectation on the granularity. We then randomly sample λ pairs of data from the two clusters to be merged at each step to form candi- date pairs {p i } Np i=1 . Here N p = λ(k max − k min ). After querying, each level of granularity can be examined against LLM predictions to choose the one with the highest consistency measure M, k * = argmax k M(W p , W k )(6) where W p = {w p i } Np i=1 denotes the predictions obtained from Eq. 5. And W k represents whether a candidate pair are in the same cluster at granularity (or step in Figure 2) k. Empirically, we found that using F-beta score, a weighted harmonic mean of precision and recall, as measurement M performs better in our framework. Scaling up to Larger Datasets. A major drawback of applying hierarchical clustering is its O(N 3 ) time complexity which makes the algorithm hard to be deployed on larger datasets. However, in our scenario, since we are only interested in a specific range of the hierarchy, the hierarchical clustering can start from an intermediate step. Specifically, a (mini-batch) K-means clustering is run on the largescale dataset with the number of clusters set to the maximum number the users desire. And then the hierarchical clustering takes the current clustering assignments as inputs and return the hierarchy for the interested levels of granularity. Remarks. Similar to Section 2.1.1, pairwise hierarchical sampling can also be used to acquire human annotations. Nonetheless, the reliability of the algorithm is still depended on the quality of clusters. In an extreme case where the clusters are completely random, it is unable to find granularity even though all the pairwise predictions are correct. Experimental Results In this section, we first evaluate CLUSTERLLM on various clustering tasks and then provide results on clustering quality with the ground truth number of clusters in Section 3.4. Ablation studies are conducted in Section 3.6 and Section 3.7 to further analyze the effectiveness of CLUSTERLLM. Finally, we show the results of determining cluster granularity in Section 3.8. Datasets We provide a high-level summary of evaluated datasets in the section. In this paper, we evaluate CLUSTERLLM extensively on a broad range of clustering tasks and datasets, which include various perspectives and granularities. Furthermore, to better analyze the effect of scale, each dataset has a small-scale and large-scale version (we keep number of clusters the same for both versions and restrict the maximum number to be 50, 000). A summary of dataset statistics is shown in Table 1 Experimental Details Query LLMs. The prompts for querying LLMs only contain a task-specific instruction (see Appendix). For all experiments, we use a temperature of 0.5 with gpt-3.5-turbo or gpt-4. We suppress the LLMs to generate long sequences by adding a postfix in the prompts, such as "Please respond with 'Choice 1' or 'Choice 2' without explanation". We use the Python API tool provided by OpenAI. Triplet Sampling. For all the experiments including small-or large-scale, we set a maximum number of triplets 1, 024, making CLUSTERLLM efficient. We keep a fraction of γ = 20% largestentropy instances as anchors and shuffle the order for large-scale experiments. To find clusters, we choose agglomerative clustering with fixed max distance (to merge clusters) 67 for small-scale experiments on Instructor, and 77 on E5 (the embeddings are preprocessed by standard scaler). For largescale datasets, we choose mini-batch K-means with fixed number of clusters 100 because of its light weight computation. Clustering algorithms are implemented by scikit-learn (Pedregosa et al., 2011). Fine-tune Embedders. In this work, we focus on two state-of-the-art pre-trained embedders: Instructor (Su et al., 2022) and E5 . We refer to Appendix C for more details. Evaluation. We run (mini-batch) K-means for 5 different seeds on embeddings with ground truth K to analyze clustering quality. We report clustering accuracy after Hungarian alignment for better interpretability. Baselines Generalized Category Discovery (GCD). GCD (Vaze et al., 2022) (or open-world semisupervised learning (Cao et al., 2022)) studies the problem of leveraging partial known classes as a prior to discover new categories from an unlabeled dataset that contains both known or novel classes. One of our contributions is to show that directly applying K-means on recent pre-trained text embedders is even better than these algorithms that have access to few-shot labels. We first adapt one computer vision algorithm: Contrast (Vaze et al., 2022) to natural language. And then we show the results with CLNN (Zhang et al., 2022), DAC and DPN (An et al., 2022) from intent discovery. All methods here employ bert-base-uncased (Devlin et al., 2019) from Huggingface (Wolf et al., 2020) and are given random 16-way-8-shot labels except for domain discovery. For more training and evaluation details, refer to Appendix A. Pre-trained Text Embedders (E5 and Instructor). We directly apply (mini-batch) K-means on top of extracted embeddings from Instructor and E5 in a zero-shot manner. For Instructor, we use the same or similar prompts as provided by the original paper. SCCL-I. We also combine Instructor with SCCL (Zhang et al., 2021a), which is a deep clustering algorithm for short text. Refer to Appendix C for more details. Main Results on Clustering Quality In Table 2 and Table 6, we compare CLUSTER-LLM with baselines on clustering quality with known granularity. We only conduct main experiments on small-scale datasets to reduce the computational costs. We show three variants of our method: CLUSTERLLM-E and CLUSTER-LLM-I adopt E5 and Instructor as their embedders respectively. CLUSTERLLM-I-iter applies the entire framework in iterative manner for twice. All of these variants use GPT-3.5 for prediction. We have the following observations: (1) Both E5 and Instructor significantly outperform GCD methods with few-shot annotations. This implies pretraining quality is especially important to discovering new categories. (2) Both CLUSTERLLM-E and CLUSTERLLM-I improves upon or perform similar with their original embedders consistently. For example, CLUSTERLLM-I increases the performance by 6.28% on Bank77 and 6.71% on FewRel. However, we do observe that on Massive(D) and CLINC(D), there are no improvements. (3) With the same embedder, CLUSTERLLM-I outperforms SCCL-I by 2.36% on average. However, on GoEmo, Massive(D) and CLINC(D), SCCL-I is more effective than CLUSTERLLM-I. (4) CLUS-TERLLM-I-iter further improves upon CLUSTER-LLM-I for most datasets. For instance, on FewNerd, the performance is improved by 3.28% over CLUSTERLLM-I. Analysis on Prediction Accuracy We attribute the improvements on clustering performance to the improved accuracy on predicting triplet tasks. In Table 3, we show the accuracy on predicted triplets that have ground truth (exactly one positive and one negative choices based on ground truth). For Instructor, we select those choices that have closer euclidean distances with the anchor in the embedding space. We also show both random triplet sampling (uniformly sample three random instances from the dataset, where the two choices are different from anchor) and entropy-based sampling on top and bottom parts respectively. First of all, with entropy-based sampling, GPT-3.5 consistently improves upon Instructor. However, with random sampling, the accu- Instructor (Su et al., 2022) 64 Table 3: Analysis on the triplet accuracy ( † is used to produce the results of CLUSTERLLM-I in Table 2). Red and green mean decreased or increased performances respectively. "#GT Triplets" means triplets that have ground truth (see Section 3.4 for details). We show results on more datasets in Table 10. racy gap is much smaller. Second, random triplet sampling retrieves significantly fewer ground truth triplets, especially on fine-grained datasets such as Bank77. Finally, with GPT-4, the accuracy is further improved on most datasets except for GoEmo. Ablation Study on Clustering Quality In this section, we show ablation studies on CLUS-TERLLM based on Instructor in Table 4 and Table 7. Specifically, we first use the predictions from Instructor embedding space (see Section 3.5) to fine-tune embedder (denoted as self-supervise). We then use the randomly sampled triplets (see Section 3.5) to fine-tune embedder (referred as CLUSTERLLM-random). We also show the results fine-tuned with predictions from GPT-4. Finally, to show a performance upper-bound, we fine-tune embedder with ground truth triplets for those that are available and the others are kept with GPT-3.5 predictions (denoted as CLUSTERLLM-GT&GPT3.5). We can observe that self-supervise also increases the performance upon Instructor but is lower than CLUSTERLLM overall. CLUSTER-LLM-random significantly decreases from original clustering performance which demonstrates the cruciality of our sampling strategy. We also observe that in spite of the high accuracy of GPT-4 for triplet prediction, the clustering performance after fine-tuning might not be higher. Finally, when provided with human labels, CLUSTERLLM can achieve the highest performance that demonstrates the possibility for further improving clustering quality with more accurate predictions. In addition, we also show the results of large-scale datasets in Table 8 and Table 9 where we can observe similar performance trends. Ablation Study on Sampling Strategy In this section, we show the ablation study on the fraction of entropy we sample from. In Figure 3, we can observe that overall, clustering accuracy decreases when increasing mean of interval (or equally decreasing entropies). This implies when sampling large entropy instances, the clustering performance generally increases. We attribute this to two reasons: (1) LLMs are much better than Table 4: Ablation study on clustering quality with Instructor as backbone and known granularity for evaluation. See results on more datasets in Table 7. See more results with large-scale datasets in Table 8 and Table 9 where the x-axis shows the mean of interval and the interval length is set to 20%. For example, "mean of interval= 50%" means that we only keep the data from 40% to 60% where γ% means the instances with the largest γ% entropy. Markers with ♦ denote the setting we used in the main experiments. small embedders on harder instances. (2) highentropy instances are generally more informative. However, we do see the performance first increases when moving from 10% to 20%. We hypothesize the highest entropy instances in the dataset might also be the most confusing data or even outliers. Determining Cluster Granularity In this section, we show the results for determining cluster granularity. We evaluate on a subset of 8 datasets with large-& small-scales which include various cluster granularities. We compare with different methods that rely on clustering errors and refer to more details in Appendix B. For all the methods, we use the same embeddings fine-tuned from CLUSTERLLM-GPT3.5 with Instructor for one iteration. And for methods except for X-means, we use the same cluster hierarchy to calculate scores. The cluster hierarchy is either acquired from hier-archical clustering for small-scale or our proposed two-step clustering algorithm in Section 2.2 for large-scale. For our method, we show results with GPT-3.5, GPT-4 predictions or the ground truth (GT) under λ = {1, 3} (except for GPT-4 because of the high costs). In Table 5 and Table 11, we show the results on small-and large-scale datasets respectively. We evaluate these methods by showing number of clusters determined and the relative errors with the ground truth number of clusters. We also show the rank of methods on the last column. Because of the high running latency, we do not show results for Silhouette and X-means on large-scale datasets. First, we can observe that our methods have more reasonable determined granularity. As a result, our methods can distinguish between intent and domain while baselines can not. For instance, on MTOP(I)/(D) in Table 5, the best baseline predicts number of clusters 82/85 while our method (GPT-3.5, λ = 3) predicts 92/18. Second, increasing λ generally helps on determining granularity. And finally, GPT-4 significantly improves the quality of determined granularity upon GPT-3.5. Related Works Clustering with Images or Texts As a fundamental task in machine learning, clustering has been applied on diverse data types, including texts (Xu et al., 2015;Hadifar et al., 2019;Zhang et al., 2021a), images (Yaling Tao, 2021Yang et al., 2016;Caron et al., 2018;Niu et al., 2020;Xie et al., 2016) and graphs (Huang et al., 2014;Chiang et al., 2019). Among them, several research have been proposed to utilize relational supervision (Yang et al., 2016;Niu et al., 2020;Van Gansbeke et al., 2020;Chang et al., 2017). However, all of them rely on pseudo-labeled pairs from self-supervisory which can be noisy. Another line of works that closely related to 77 64 59 18 102 11 150 10 -Silhouette (Rousseeuw, 1987) 118 ( X-means (Pelleg et al., 2000) 69 ( (1660) 7 ClusterSize (errors)". The "Rank" column is computed with 1-level ranking (Colombo et al., 2022). "GT" denotes ground truth. ours is constrained clustering. It usually incorporates must-link or cannot-link constraints (Basu et al., 2004;Wagstaff et al., 2001;Basu et al., 2008;Manduchi et al., 2021) that indicate clustering assignments. Nonetheless, theses constraints are sampled from labels as a prior knowledge which significantly limits the application of these algorithms in real-world scenarios. In this work, we study how to utilize contemporary LLMs for inferring sentence relationships in order to guide clustering. A concurrent work (Wang et al., 2023) also utilizes LLMs for clustering by assigning instances to different explanations. Generalized Category Discovery Generalized Category Discovery (GCD) (Vaze et al., 2022;Lin et al., 2020;Zhang et al., , 2022Mou et al., 2022; assume partial known classes with annotations which can also be used to infer user's requirement on clustering. However, GCD relies on sufficient annotated & unlabeled data for training. In contrast, CLUSTER-LLM seeks for minimal supervision and studies a setting with controlled computation-& data-cost. Pre-trained Embedding Model Generic pre-trained text embedding models (Reimers and Gurevych, 2019;Gao et al., 2021;Ni et al., 2022a,b) have been applied in many tasks such as text similarity, classification and information retrieval. Recently, two embedding models Su et al., 2022) have been released that show superior performance on a popular benchmark (Muennighoff et al., 2022) including clustering. Specifically, E5 is pre-trained on web-scraped data pairs with contrastive objective and Instructor (Su et al., 2022) is pre-trained on 330 tasks with instructions. CLUSTERLLM improves upon these models with the feedbacks from LLMs. LLMs as Annotators Recent instruction-tuned LLMs, such as ChatGPT, have been shown to have the ability to reproduce or improve human-generated labels (Gilardi et al., 2023;He et al., 2023;Zhu et al., 2023). Furthermore, several works have been dedicated to finetune models with feedbacks from LLMs (Cheng et al., 2023;Bai et al., 2022). In this work, we focus on clustering tasks and study how to utilize predictions from LLMs on sentence relations to improve clustering quality and determine cluster granularity. Conclusion In this paper, we study how to leverage API-based LLMs to guide small embedders for text clustering in order to benefit from high-level language capability of LLMs and user's instructions on clustering. We propose to prompt LLMs with two kinds of sentence relationship tasks: triplet task and pairwise task. Triplet task chooses the sentence that is most similar with query and is combined with the perspective instruction from users. The predicted triplets are used for improving clustering quality. Pairwise task judges whether a pair of sentences belong to the same category hinted by few-shot demonstrations, and then the predictions are used to determine cluster granularity with hierarchical clustering. Extensive experiments show that our proposed framework CLUSTERLLM can improve clustering quality and propose reasonable cluster granularity at a negligible cost. Limitations We list several limitations of our work that we hope to be improved in the future: Reliance on pre-trained embedder. To find the most informative data, we have to rely on a pretrained embedder that can indicate the largest clustering assignment entropy. We hope that selfsupervise triplets and LLM-predicted triplets can be combined to solve such an issue. Computational cost for fine-tuning. Our initial idea is to utilize constrained clustering which is a light-weight algorithm that only needs to be trained without small embedders. However, the inevitable unstable training will be heavily affected by the errors in LLM predictions. We make a comprise by introducing embedder into fine-tuning to temporarily solve the issue, but we hope to reduce the computational cost in a future work. Sub-optimal performance on domain discovery. We notice that on domain discovery datasets such as Massive(D) and CLINC(D), the performance is usually sub-optimal compared with original Instructor embedding. Furthermore, when using ground truth (for those triplets that have it), the performance is still not noticeably improved. We hypothesize that even though it is not a problem for most datasets, the triplets that do not have ground truth can affect performance on these specific datasets. Ethics Statement Our work employs LLMs which are accessed through OpenAI APIs. For some applications, uploading privacy-sensitive data is risky and might require efforts to remove sensitive information. References B More Details about Determining Cluster Granularity Previous methods often employ clustering errors as a metric and they ignore user's need on the granularity. Silhouette coefficient (Rousseeuw, 1987) indicates the clustering quality without ground truths, which exploits the inter-cluster distance with nearest clusters and the intra-cluster distance. We find the granularity by choosing the one with the best silhouette coefficient. Elbow method (Thorndike, 1953) is a heuristic method that plots the clustering error with respect to different levels of granularity in the hierarchy. And then the best granularity is determined with the largest elbow length. X-means (Pelleg et al., 2000) is a variation of K-means that starts with the lowest number of clusters, and then repeatedly attempt to split the clusters by running 2-means on them and evaluate with Bayesian Information Criterion (BIC) . BIC calculates BIC for each of the granularity. Cluster-Size ) uses a confidence threshold to filter small clusters. For all the methods, we use the same embeddings fine-tuned from CLUS-TERLLM-GPT3.5 with Instructor for one iteration. And for methods except for X-means, we use the same cluster hierarchy to calculate scores. For our methods, the weight in F-beta score is set to 0.92. To acquire annotations, we randomly sample 4 pairs (including 2 positive and 2 negative) from the same set of pairs we query LLMs on small-scale datasets. C More Fine-tuning Details For fine-tuning, we adopt the same hyperparameters as in (Su et al., 2022), but modify the learning rate to 2e − 6, the maximum gradient steps to 3, 840 for Instructor (∼ 15 epochs) and 1, 280 for E5, and batch size to 4. Training is conducted with a single NVIDIA Quadro RTX 8000 GPU. For SCCL-I, we change the maximum token length to 128 due to the limited compute resource. We use the same learning rate 2e − 6 as before and batch size 16 and evaluate representations with K-means after 200 iterations. D Description of Datasets Bank77 is a popular dataset in this problem that focuses on creating fine-grained intent categories for a single-domain, "banking". CLINC(I) is originally created for detecting utterances that falls outside of supported intents. The dataset also contains multiple domains, such as "travel", "utility" and "work". In this experiment, we discard all the outof-scope utterances. Moreover, we create a domain discovery dataset CLINC(D) that uses domains as labels. Massive(I)/(D) and MTOP(I)/(D) are both from MTEB (Muennighoff et al., 2022). Here "I" denotes intent and "D" for domain. These datasets are originally used for classification but are adapted here for clustering. We also remove those intents with only a few instances and keep English-only data. Instructor (Su et al., 2022) 33 Table 3. Figure 3 : 3Relative clustering accuracy (divided by maximum for better aligning across datasets) of CLUSTER-LLM-GPT3.5 with different range of entropy we select, . Refer to Appendix D for more descriptions. Intent Discovery & Domain Discovery. Intent discovery aims at discovering unknown intents in the unlabeled customer utterances. We evaluateTask Name #clusters #data(small) #data(large) Intent Bank77 77 3,080 10,003 CLINC(I) 150 4,500 15,000 MTOP(I) 102 4,386 15,638 Massive(I) 59 2,974 11,510 IE FewRel 64 4,480 40,320 FewNerd 58 3,789 50,000 FewEvent 34 4,742 18,969 Topic StackEx 121 4,156 50,000 ArxivS2S 93 3,674 50,000 Reddit 50 3,217 50,000 Emotion GoEmo 27 5,940 23,485 Domain CLINC(D) 10 4,500 15,000 MTOP(D) 11 4,386 15,667 Massive(D) 18 2,974 11,514 Table 1 : 1Dataset statistics. For the last three datasets, we also use the domains as labels to convert the problem into domain discovery. Information Extraction (IE). IE addresses the problem of mining structural information from unstructured or semi-structured texts(Chen et al., 2022). When the labels become fine-grained(Choi et al., 2018), it is especially important to discover new labels and expand the supported types of information. In this work, we only focus on three aspects of IE, including relation, entity and event typing. We adapt FewRel (Gao et al., 2019),FewNerd (Ding et al., 2021) and FewEvent (Deng et al., 2020) for each of the task. To indicate the specific mentions (entities or event triggers) of interest, we append them behind the sentences with a natural language format, such as "The relation between olympia and league of ireland". Topic Mining. We adapt StackEx, Reddit(Geigle et al., 2021) and ArxivS2S from MTEB(Muennighoff et al., 2022). Emotion. We adapt GoEmo (Demszky et al., 2020), a fine-grained emotion detection dataset by removing multi-label or neutral instances.on popular datasets including Bank77 (Casanueva et al., 2020), CLINC(I) (Larson et al., 2019), Mas- sive(I) (FitzGerald et al., 2022) and MTOP(I) (Li et al., 2021). Table 2 : 2Comparison of clustering accuracy with known granularity for evaluation. The top part of the table shows GCD results with 16-way 8-shot labels based on BERT and the bottom part shows the unsupervised clustering results based on two most recently released pre-trained embedders. † denotes methods adapted from computer vision. The average over all 14 datasets are shown in the last column. We show results on more datasets inTable 6.Type Model Bank77 CLINC(I) FewRel FewNerd FewEvent StackEx ArxivS2S GoEmo Random #GT Triplets 23 6 41 156 105 14 22 117 Instructor 100 100 80.49 71.15 98.10 85.71 95.45 68.38 GPT3.5 100 100 85.37 82.05 94.29 71.43 81.82 68.38 ∆ (+0) (+0) (+4.88) (+10.90) (-3.81) (-14.28) (-13.63) (+0) Entropy-based #GT Triplets 510 462 266 347 259 271 145 206 Instructor 64.12 76.19 62.41 59.65 70.66 68.27 59.31 64.08 GPT3.5 † 76.67 79.44 76.69 68.88 83.78 71.22 73.79 64.56 ∆ (+12.55) (+3.25) (+14.28) (+9.23) (+13.12) (+2.95) (+14.48) (+0.48) GPT4 79.41 80.74 87.22 82.13 85.71 79.70 77.93 61.65 ∆ (+15.29) (+4.55) (+24.81) (+22.48) (+15.05) (+11.43) (+18.62) (-2.43) .10% 20% 30% 40% 50% 60% 70% 80% 90% Mean of Interval 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Relative Clustering Accuracy Bank77 FewRel StackEx GoEmo Table 5 : 5Inferred granularity on small-scale datasets. We use the same fine-tuned Instructor with GPT3.5 and set maximum & minimum number of clusters to 200 and 2. The numbers are shown in the format of "[#clusters] Swetha Ranganath, Laurie Crist, Misha Britan, Wouter Leeuwis, Gokhan Tur, and Prem Natarajan. 2022. Massive: A 1m-example multilingual natural language understanding dataset with 51 typologically-diverse languages. Tianyu Gao, Xu Han, Hao Zhu, Zhiyuan Liu, Peng Li, Maosong Sun, and Jie Zhou. 2019. FewRel 2.0: Towards more challenging few-shot relation classification. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pages 6251-6256, Hong Kong, China. Association for Computational Linguistics. Tianyu Gao, Xingcheng Yao, and Danqi Chen. 2021. SimCSE: Simple contrastive learning of sentence embeddings. 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Jack FitzGerald, Christopher Hench, Charith Peris, Scott Mackie, Kay Rottmann, Ana Sanchez, Aaron Nash, Liam Urbach, Vishesh Kakarala, Richa Singh, For all datasets, we use the train & test sets as large-& small-scale datasets respectively. For FewRel and FewEvent, we randomly split datasets into large-& small-scale versions. For FewNerd, we use the original train & test splits. For StackEx and Reddit, we combine all the splits into a single dataset and remove topics that only have few instances. Finally the datasets are randomly splitted into large-& small-scale versions.lab. Method Intent Discovery Topic Mining Domain Discovery Avg All MTOP(I) Massive(I) Reddit MTOP(D) Massive(D) CLINC(D) ✓ Contrast (Vaze et al., 2022) † 29.25(0.90) 33.07(2.26) 19.57(0.82) - - - - - DAC (Zhang et al., 2021b) 31.43(1.42) 34.45(3.57) 16.80(2.72) - - - - - DPN (An et al., 2022) 33.64(2.93) 33.86(4.50) 14.69(1.36) - - - - - CLNN (Zhang et al., 2022) 29.77(1.07) 46.22(1.63) 29.01(4.00) - - - - - ✗ E5 (Wang et al., 2022) 33.54(0.92) 52.52(0.62) 39.03(0.62) 91.23(1.23) 63.70(0.86) 59.64(2.73) 56.61 47.70 CLUSTERLLM-E 34.66(1.31) 54.80(0.72) 47.72(1.58) 89.18(5.25) 60.73(2.81) 58.35(2.92) 57.61 50.92 Table 6 : 6Comparison of clustering accuracy with known granularity for evaluation. The top part of the table shows GCD results with 16-way 8-shot labelled data based on BERT and the bottom part shows the unsupervised clustering results based on two most recently released pre-trained embedders. † denotes methods adapted from computer vision. The average over all 14 datasets are shown in the last column.Method Intent Discovery Topic Mining Domain Discovery Avg All MTOP(I) Massive(I) Reddit MTOP(D) Massive(D) CLINC(D) Instructor 33.35(1.32) 54.08(1.53) 54.98(1.51) 90.56(3.34) 61.81(2.56) 52.50(2.44) 57.88 49.90 +self-supervise 34.06(0.64) 55.07(1.25) 55.41(0.93) 92.12(2.66) 62.28(1.55) 58.58(2.56) 59.58 51.39 +CLUSTERLLM-random 28.05(1.69) 51.66(2.41) 54.60(2.23) 87.0(2.27) 56.40(2.35) 60.27(4.20) 56.34 48.27 +CLUSTERLLM-GPT3.5 35.84(2.07) 59.89(2.05) 56.79(1.90) 93.53(0.10) 61.06(1.91) 52.39(1.84) 59.91 53.09 +CLUSTERLLM-GPT4 34.48(0.38) 59.10(1.12) 55.38 (0.37) 92.04(2.67) 60.16(2.97) 57.45 (2.48) 59.77 53.22 +CLUSTERLLM-GT&GPT3.5 36.86(0.42) 59.27(1.43) 58.33(1.26) 92.26(3.62) 61.65(3.50) 52.87(2.63) 60.21 53.96 Table 7 : 7More results of ablation study on small-scale datasets. Also seeTable 4.Method Intent Discovery Information Extraction Topic Mining Emotion Avg All Bank77 CLINC FewRel FewNerd FewEvent StackEx ArxivS2S GoEmo Instructor 60.30(2.39) 79.52(1.96) 41.38(0.75) 29.62(1.02) 41.42(2.09) 46.76(0.80) 24.55(0.42) 24.02(1.11) 43.45 48.98 +self-supervise 61.48(2.84) 81.87(0.97) 41.09(0.99) 30.57(0.18) 45.54(1.70) 46.24(0.46) 24.49(0.75) 24.34(1.25) 44.45 50.30 +CLUSTERLLM-GPT3.5 65.47(2.28) 82.29(1.09) 47.22(0.89) 33.86(1.19) 47.55(1.51) 47.42(1.35) 25.60(0.51) 25.23(1.21) 46.83 51.21 Table 8 : 8Ablation study on clustering quality for large-scale datasets. See more datasets inTable 9.Method Intent Discovery Topic Mining Domain Discovery Avg All MTOP(I) Massive(I) Reddit MTOP(D) Massive(D) CLINC(D) Instructor 35.53(1.05) 54.72(2.00) 55.04 (2.69) 85.01(2.18) 56.11(5.07) 41.42(2.09) 54.64 48.98 +self-supervise 35.27(1.53) 58.30(1.42) 56.45(1.59) 89.54(4.56) 58.14(3.88) 50.93(4.11) 58.11 50.30 +CLUSTERLLM-GPT3.5 36.80(0.83) 57.70(2.92) 55.47(2.44) 84.08(3.34) 58.14(3.97) 50.12(4.13) 57.05 51.21 Table 9 : 9Ablation study on clustering quality for large-scale datasets. See more datasets inTable 8.Type Model MTOP(I) Massive(I) Reddit MTOP(D) Massive(D) CLINC(D) Random #GT Triplets 102 61 40 184 148 189 Instructor 98.04 88.52 80 96.74 80.41 76.72 GPT3.5 85.29 85.25 70 85.87 82.43 68.25 ∆ (-12.75) (-3.27) (-10) (-10.87) (+2.02) (-18.47) Entropy-based #GT Triplets 140 98 92 144 108 208 Instructor 65.74 63.56 61.98 70.31 63.82 75.06 GPT3.5 67.41 68.76 63.28 69.79 72.09 75.78 ∆ (+1.67) (+5.20) (+1.30) (-0.52) (+8.27) (+0.72) Table 10 : 10Analysis on the triplet accuracy. See other results in TWEAC: transformer with extendable QA agent classifiers. Gregor Geigle, abs/2104.07081Nils Reimers, Andreas Rücklé, and Iryna Gurevych. 2021. CoRRGregor Geigle, Nils Reimers, Andreas Rücklé, and Iryna Gurevych. 2021. TWEAC: transformer with extend- able QA agent classifiers. CoRR, abs/2104.07081. Chatgpt outperforms crowd-workers for textannotation tasks. 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Can chatgpt reproduce human-generated labels? a study of social computing tasks. arXiv preprint arXiv:2304.10145. Method Bank77 FewRel Massive(I) Massive(D) MTOP(I) MTOP(D) CLINC(I) CLINC(D) Rank. Method Bank77 FewRel Massive(I) Massive(D) MTOP(I) MTOP(D) CLINC(I) CLINC(D) Rank . Bic (goutte, 185.9) 148 (150.8) 141 (683.3) 169 (65.69) 170 (1445) 179 (19.33) 178 (1680) 8BIC (Goutte et al., 2001) 183 (137.7) 183 (185.9) 148 (150.8) 141 (683.3) 169 (65.69) 170 (1445) 179 (19.33) 178 (1680) 8 . ( Clustersize, Zhang, ClusterSize (Zhang et al., 2021b) Inferred granularity on large-scale datasets. The setting is the same as in Table 5. Table. 11Dataset Prompt Bank77 Select the banking customer utterance that better corresponds with the Query in terms of intentTable 11: Inferred granularity on large-scale datasets. The setting is the same as in Table 5. Dataset Prompt Bank77 Select the banking customer utterance that better corresponds with the Query in terms of intent. CLINC(I) Select the customer utterance that better corresponds with the Query in terms of intent. FewRel Select the example that better corresponds with the Query in terms of relation type. FewNerd Select the example that better corresponds with the Query in terms of entity typeCLINC(I) Select the customer utterance that better corresponds with the Query in terms of intent. FewRel Select the example that better corresponds with the Query in terms of relation type. FewNerd Select the example that better corresponds with the Query in terms of entity type. Massive(I) Select the user utterance that better corresponds with the Query in terms of intent. MTOP(I) Select the user utterance that better corresponds with the Query in terms of. intent Reddit Select the Reddit question that better corresponds with the Query in terms of topicMassive(I) Select the user utterance that better corresponds with the Query in terms of intent. MTOP(I) Select the user utterance that better corresponds with the Query in terms of intent Reddit Select the Reddit question that better corresponds with the Query in terms of topic. Massive(D) Select the user utterance that better corresponds with the Query in terms of scenario. Massive(D) Select the user utterance that better corresponds with the Query in terms of scenario. Table 12: Prefix of prompts for triplet task. Notice that the prompt for domain and category (such as CLINC(I) and CLINC(D)) can be used interchangeably. Dataset Prompt Triplet Task Select the banking customer utterance that better corresponds with the Query in terms of intent. Query: Should i reinstall the payment app? Choice 1: I've received my card so now I need to know how to sync it to the app. Choice 2: Can I still use the app if I switched phones?. CLINC(D) Select the customer utterance that better corresponds with the Query in terms of domain. Please respond with 'Choice 1' or 'Choice 2' without explanationMTOP(D) Select the user utterance that better corresponds with the Query in terms of domain. CLINC(D) Select the customer utterance that better corresponds with the Query in terms of domain. Table 12: Prefix of prompts for triplet task. Notice that the prompt for domain and category (such as CLINC(I) and CLINC(D)) can be used interchangeably. Dataset Prompt Triplet Task Select the banking customer utterance that better corresponds with the Query in terms of intent. Query: Should i reinstall the payment app? Choice 1: I've received my card so now I need to know how to sync it to the app. Choice 2: Can I still use the app if I switched phones? Please respond with 'Choice 1' or 'Choice 2' without explanation. 1: I would like to see the source of my money. Sentence 2: My source of funds need verified. Yes. Because both intents are verify source of funds. Pairwise Task [Example1] SentencePairwise Task [Example1] Sentence 1: I would like to see the source of my money. Sentence 2: My source of funds need verified. Yes. Because both intents are verify source of funds. Sentence 1: Is there a fee for topping up Sentence 2: What are the top up charges for US cards? Yes. Because both intents are top up by card charge. Sentence 1: Is there a fee for topping up Sentence 2: What are the top up charges for US cards? Yes. Because both intents are top up by card charge. Sentence 1: Can I reactivate my lost card that I found this morning in my jacket pocket? Sentence 2: how to activate card? No. Because Sentence 1 has intent card linking and Sentence 2 has intent activate my card. Sentence 1: Can I reactivate my lost card that I found this morning in my jacket pocket? Sentence 2: how to activate card? No. Because Sentence 1 has intent card linking and Sentence 2 has intent activate my card. Sentence 1: What will I be charged for a physical card? Sentence 2: My card is about to expire and I need to know how much it costs and how long. Sentence 1: What will I be charged for a physical card? Sentence 2: My card is about to expire and I need to know how much it costs and how long ... Because Sentence 1 has intent order physical card and Sentence 2 has intent card. No. Because Sentence 1 has intent order physical card and Sentence 2 has intent card ... Determine whether the intents of two banking customer utterances below belong to the same intent category using above examples. Sentence 1: $1 extra has been charged on my statement, why is that? Sentence 2: Will it automatically top-up if there isn't much money left? Please respond with 'Yes' or 'No' without explanation. Table 13: One example from Bank77 on both triplet task and pairwise taskDetermine whether the intents of two banking customer utterances below belong to the same intent category using above examples. Sentence 1: $1 extra has been charged on my statement, why is that? Sentence 2: Will it automatically top-up if there isn't much money left? Please respond with 'Yes' or 'No' without explanation. Table 13: One example from Bank77 on both triplet task and pairwise task.	[ "https://github.com/PolyAI-LDN/task-specific-" ]
[ "Improved Probabilistic Image-Text Representations", "Improved Probabilistic Image-Text Representations" ]	[ "Sanghyuk Chun \nNAVER AI Lab\n\n" ]	[ "NAVER AI Lab\n" ]	[]	Image-Text Matching (ITM) task, a fundamental vision-language (VL) task, suffers from the inherent ambiguity arising from multiplicity and imperfect annotations. Deterministic functions are not sufficiently powerful to capture ambiguity, prompting the exploration of probabilistic embeddings to tackle the challenge. However, the existing probabilistic ITM approach encounters two key shortcomings; the burden of heavy computations due to the Monte Carlo approximation, and the loss saturation issue in the face of abundant false negatives. To overcome the issues, this paper presents an improved Probabilistic Cross-Modal Embeddings (named PCME++) by introducing a new probabilistic distance with a closed-form solution. In addition, two optimization techniques are proposed to enhance PCME++ further; first, the incorporation of pseudo-positives to prevent the loss saturation problem under massive false negatives; second, mixed sample data augmentation for probabilistic matching. Experimental results on MS-COCO Caption and two extended benchmarks, CxC and ECCV Caption, demonstrate the effectiveness of PCME++ compared to state-of-the-art ITM methods. The robustness of PCME++ is also evaluated under noisy image-text correspondences. In addition, the potential applicability of PCME++ in automatic prompt tuning for zero-shot classification is shown. The code is available at https://naver-ai.github.io/pcmepp/.	10.48550/arxiv.2305.18171	[ "https://export.arxiv.org/pdf/2305.18171v1.pdf" ]	258,959,068	2305.18171	f3fa633693ca0461c1fd17d158948e3889b1f5b8	Improved Probabilistic Image-Text Representations Sanghyuk Chun NAVER AI Lab Improved Probabilistic Image-Text Representations Image-Text Matching (ITM) task, a fundamental vision-language (VL) task, suffers from the inherent ambiguity arising from multiplicity and imperfect annotations. Deterministic functions are not sufficiently powerful to capture ambiguity, prompting the exploration of probabilistic embeddings to tackle the challenge. However, the existing probabilistic ITM approach encounters two key shortcomings; the burden of heavy computations due to the Monte Carlo approximation, and the loss saturation issue in the face of abundant false negatives. To overcome the issues, this paper presents an improved Probabilistic Cross-Modal Embeddings (named PCME++) by introducing a new probabilistic distance with a closed-form solution. In addition, two optimization techniques are proposed to enhance PCME++ further; first, the incorporation of pseudo-positives to prevent the loss saturation problem under massive false negatives; second, mixed sample data augmentation for probabilistic matching. Experimental results on MS-COCO Caption and two extended benchmarks, CxC and ECCV Caption, demonstrate the effectiveness of PCME++ compared to state-of-the-art ITM methods. The robustness of PCME++ is also evaluated under noisy image-text correspondences. In addition, the potential applicability of PCME++ in automatic prompt tuning for zero-shot classification is shown. The code is available at https://naver-ai.github.io/pcmepp/. Introduction Given images and captions, Image-Text Matching (ITM) is the task of retrieving the most relevant images/captions for the given query caption/image [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The applications of ITM include crossmodal retrieval [4] from paired image-caption datasets, such as MS-COCO Caption [20], and zero-shot classification [19], by treating class labels as a text (e.g., "a photo of { · }"). Owing to its significant role in image understanding and language comprehension, ITM has emerged as a fundamental Vision Language (VL) downstream task. However, this problem inherently suffers from the ambiguity caused by many-to-many correspondences and sparse annotations of the ITM datasets. The nature of image-text matching is many-to-many; an image can be described in numerous text explanations, and there are a plentiful number of visual scenes to visualize a text description. However, simultaneously, our datasets are sparsely annotated. The existing ITM datasets are built by collecting paired image-caption, and treating the collected image-caption pairs are the only positives without considering other potential positives in "negative" pairs [20][21][22][23][24]. For example, Chun et al. [25] showed that the MS-COCO Caption dataset has massive missing positives; 88.2% of caption-toimage positives and 72.1% of image-to-caption positives are labeled as "negative". Figure 1 shows an example. While humans judge all images and texts are plausibly matched, the dataset only treats a pair (x i v , x j t ) as positive when i = j. In this paper, we argue that the inherent multiplicity and the sparse annotations lead to the ambiguity of ITM datasets and make ITM problem challenging ( §2.1). This paper aims to design a proper joint embedding space that represents the inherent ambiguity by probabilistic embeddings [14,[26][27][28][29][30][31][32][33][34][35], i.e., encoding an input to a random variable rather than a deterministic vector. Probabilistic embeddings have been introduced for many applications with inherent ambiguity, such as word embeddings [27], face understanding [28,29], 2D-to-3D pose If the match between Z v 1 and Z t ' become "uncertain", the embedding uncertainty (i.e., variance becomes larger) If two distributions have a certain match, an embedding has a low uncertainty value. [30], speaker diarization [31], video understanding [32], and composed image retrieval [33]. Especially, Chun et al. [14] investigated the primitive probabilistic approach for ITM, Probabilistic Cross-Modal Embedding (PCME), based on the approach by Oh et al. [26]. Although PCME shows reasonable retrieval performances and interesting observations through uncertainty measures, PCME suffers from expensive computations due to Monte Carlo approximation and fast loss saturation. Firstly, PCME needs expensive sampling operations for both training and inference. For example, if we randomly draw 7 samples for each input, computing the distance between two samples costs O(7 × 7). Furthermore, due to the sampling operation, PCME retrieval operation cannot be extended to large-scale efficient retrieval systems, such as FAISS [36]. This issue is solved by introducing a new probability distance with a closed-form solution and a new objective function based on the distance ( §2.2). In addition, as the proposed closed-form distance consists of Euclidean distance and the relationship between variance embeddings, we can easily adapt approximated KNN to ours ( §3.5). Experimental results show that the closed-form distance not only makes the operation efficient but also convergences to a better solution by computing an exact solution instead of an approximation. Moreover, this paper demonstrates that PCME suffers from fast loss saturation under abundant false negatives (FNs). The PCME training strategy suppresses the gradient step size if the model prediction and ground truth differ significantly (e.g., if a model predicts an image-caption pair is positive with high confidence, but the ground truth is negative). However, as Chun et al. [25] showed, our datasets have abundant FNs; the gradient of FN pairs converges to zero, and FN samples will not contribute to the model update eventually. The issue is mitigated by introducing two techniques: pseudo-positives ( §2.3) and mixed sample data augmentation for probabilistic matching ( §2.4). This paper conceptually and empirically shows that the proposed techniques can alleviate the zero gradient issue of FNs. PCME++ is evaluated on MS-COCO Caption [20] and its extended benchmarks CxC [37] and ECCV Caption [25] with state-of-the-art ITM methods ( §3.2). In the experiments, PCME++ consistently outperforms the comparison methods on the COCO benchmark. PCME++ is also evaluated on the noisy correspondence benchmark [16], indicating that our method is not only effective for the original task but also holds the potential to address the noisy correspondence problem. Furthermore, this paper shows that the textual uncertainty of PCME++ can be applied to a prompt-tuning for a zero-shot classification with a pre-trained model on large-scale VL datasets, demonstrating the versatility and scalability of our method for a wide range of applications ( §3.5). Finally, the qualitative advantages of the learned uncertainty of PCME++ by capturing dataset uncertainty are shown in §3.4. Contributions. This paper introduces PCME++, an improved probabilistic image-text representation, by introducing: a new closed-form probability distance, named CSD, and a new matching objective function based on CSD for substituting expensive sampling-based approximation of PCME [14]; a pseudo-positive strategy and a mixed sample data augmentation strategy for addressing the loss saturation issue of abundant false negatives. PCME++ shows not only good retrieval performances but also the extensibility to various applications, e.g., mitigating noisy correspondences, prompt tuning for zero-shot classification, and understanding the inherent ambiguity of a dataset. 2 Improved Probabilistic Cross-Modal Embeddings (PCME++) Problem definition: Ambiguity of ITM datasets Let x v and x t be the input image and caption, respectively. For each image text pair, a binary matching indicator m vt ∈ {0, 1} denotes whether x t describes x v well. This paper argues that the inherent multiplicity and the sparse annotations make m vt ambiguous. For example, as shown in Figure 1, x 1 t ("A person on a snowboard flying in the air down the snow") and x 2 t ("A person on a snowboard jumping up in the air.") are semantically almost the same, hence we may assume that x 1 t and x 2 t are mapped to almost the same embedding point z ′ t , i.e., f (x 1 t ) ≈ f (x 2 t ) = z ′ t if we have a proper mapping f (·) between the input space and the embedding space. In this case, if x 1 t and x 1 v are a positive match, x 2 t and x 1 v should be a positive match in the embedding space. However, because our dataset contains only sparse matching relationships [25,37], x 2 t and x 1 v are a negative match. In other words, in the embedding space, the matching between z 1 v and z ′ t (≈ f (x 1 t ) ≈ f (x 2 t )) becomes ambiguous (i.e., it can be either positive or negative). As shown in Figure 1, a deterministic embedding space cannot capture the inherent uncertainty originated by the multiplicity and the sparse annotations. The existing deterministic approaches, therefore, rely on Hardest Negative Mining (HNM) strategy [4], selecting the closest pair as the only negative for computing a triplet loss. The HNM strategy enforces sparse positive pairs to be closer than other false negative (FN) pairs, resulting in a twisted embedding space that cannot capture the inherent uncertainty of VL datasets. We empirically show that the HNM strategy eventually converges to a suboptimal embedding space when the ambiguity intensifies, i.e., under strong noisy correspondences ( §3.2). In contrast, probabilistic embeddings can naturally mitigate the issue by capturing the ambiguity of m vt with a probability distribution. Probabilistic contrastive learning We first define a visual embedding and a text embedding of the given image x v and x t as normally distributed random variables, Z v ∼ N (µ v , Σ v ) and Z t ∼ N (µ t , Σ t ), respectively. For simplicity, we assume diagonal covariance matrices and simplify the notations as N (µ v , σ 2 v ) and N (µ t , σ 2 t ), where µ and σ are D-dimensional vectors. As shown in Figure 1, our purpose is to learn probabilistic embeddings Z v and Z t satisfying the following properties: (a) there exists a proper probabilistic distance between Z v and Z t . (b) if the match m vt is certain, then Z v and Z t have small variances. (c) if the match between x v and x t (m vt ) is ambiguous, then Z v and Z t have large variances. The probabilistic distance d(·) between two probabilistic embeddings Z v and Z t , named closed-form sampled distance (CSD), is defined as follows: d(Z v , Z t ) = E Zv,Zt ∥Z v − Z t ∥ 2 2 = ∥µ v − µ t ∥ 2 2 + ∥σ 2 v + σ 2 t ∥ 1 ,(1) where ∥ · ∥ p is a p-norm operation. To be self-contained, the full derivation of Equation (1) is provided in Appendix A.1. Equation (1) satisfies most of the properties of a metric function (i.e., positivity, symmetry, and triangular inequality) except zero self-distance; d(Z, Z) is 2∥σ 2 ∥ 1 , not zero. I.e., Equation (1) satisfies the condition (a). There are two ways to make Z v and Z t closer/further; making µ v and µ t closer/further, or making σ v and σ t smaller/larger. Hence, if we assume fixed µ v and µ t , we have to decrease σ v and σ t to minimize d(Z v , Z t ); if Z v and Z t are a certain positive match (i.e., m vt = 1), then σ v and σ t will be collapsed to zero (i.e., satisfying the condition (b)), and d(Z v , Z t ) will become Euclidean distance. On the other hand, if the match between Z v and Z t is ambiguous (i.e., m vt can be either positive or negative), then σ v and σ t will not be collapsed to zero, for increasing d(Z v , Z t ) for the negative match case; d(Z v , Z t ) also satisfies the condition (c). CSD has a similar form to Wasserstein 2-distance (WD), CSD (ours) WD inf Zv,Zt E Zv,Zt ∥Z v − Z t ∥ 2 2 = ∥µ v − µ t ∥ 2 2 + ∥σ v − σ t ∥ 2 2 , where WD includes the infimum operation. However, WD is not a proper probabilistic distance in the matching problem, especially WD cannot satisfy the condition (b). Assume the scenario when µ values are fixed again. In this case, σ v and σ t have no motivation to be decreased, but they are just enforced to have the same values. Hence, the learned σ by WD cannot represent the sample certainty. Figure 2 shows a 2-D toy scenario where CSD satisfies the proper uncertainty conditions while WD cannot. In the figure, red, yellow, and green dots are certain samples, and , the average σ 2 value for uncertain/certain samples by CSD are 1.82, while we have 1.04 for WD. More details of the toy experiment are described in Appendix A.2. CSD is also related to the matching probability [26] used by PCME [14], (E Zv,Zt sigmoid(−a∥Z v − Z t ∥ 2 + b)) where the matching probability cannot be computed in a closed-form due to sigmoid but should be computed by an expensive Monte-Carlo approximation. Now, based on Equation (1), the probabilistic matching objective function is defined as the follows: L match = m vt log sigmoid(−a · d(Z v , Z t ) + b) + (1 − m vt ) log sigmoid(a · d(Z v , Z t ) − b),(2) where m vt ∈ {0, 1} is the matching indicator between v and t. a and b are learnable scalar values, following Oh et al. [26] and Chun et al. [14]. In practice, Equation (2) can be easily implemented by binary cross entropy (BCE) loss. We compute L match for all pairs in the mini-batch as contrastive learning objectives, such as InfoNCE [19]. The overview of the comparisons between our objective function, a standard triplet loss, and batch-wise contrastive loss are shown in Figure 3. To prevent the collapse of σ (i.e., σ → 0), PCME++ employs Variational Information Bottleneck (VIB) loss [38], L VIB , following Oh et al. [26] and Chun et al. [14]. As derived by Oh et al. [26], L VIB can be computed by the KL divergence between the learned distribution and N (0, I). Pseudo-positives (PP) for handling numerous false negatives Let −a · d(Z v , Z t ) + b = l vt , then we can derive ∂Lmatch ∂l = 1 − sigmoid(l vt ) and 1 − sigmoid(−l vt ) when m vt = 0 and 1, respectively. Therefore, if there exists a negative pair (m = 0) with a large l vt (i.e., a small probabilistic distance d(Z v , Z t )), the gradient will be converged to zero. Unfortunately, as observed by Chun et al. [25], image-text paired datasets have numerous false negatives (e.g., captions in the COCO Caption dataset have ×8.47 positive images than the "ground-truth" positive images.), i.e., if we have a plausible matching model, then the false negative pairs will not contribute to the objective function Equation (2). We can also observe the same phenomenon for false positives. Note that PCME [14] also suffers from the same issue, as discussed is in Appendix A.3. To tackle the issue, PCME++ employs a simple pseudo-labeling strategy: for a positive match (v, t), t ′ is a pseudo-positive (PP) match with t if d(Z v , Z t ′ ) ≤ d(Z v , Z t ). Using the pseudo-positives, we compute the pseudo-positive matching loss L pseudo-match using (2). The objective function becomes: L match + αL pseudo-match + βL VIB ,(3) where α and β are control parameters of PP matching loss and VIB loss. In the experiments, α = 0.1 and β = 0.0001 are chosen (Appendix C.2). Pseudo-code for Equation (3) is shown in Appendix A.4. Mixed Sample Data Augmentation (MSDA) for probabilistic matching MSDA, such as Mixup [39] or CutMix [40], shows not only great improvements in empirical performances but also shows good theoretical properties, such as generalization [41,42] μ t log σ t 2 Z v ~ N(μ v , σ v 2 ) Z t ~ N(μ t , σ t 2 ) Visual encoder Textual encoder Figure 4: Architecture overview. We use the same visual and textual backbones as CLIP [19]. Each modality encoder encodes D-dimensional ℓ2-normalized mean vector µ and the variance vector log σ 2 , followed by Generalized Pooling Operator (GPO) [15], to represent a normally distributed random variable Z ∼ N (µ, σ 2 ). [43]. MSDA consists of two parts; input mixing (i.e., a generative process to generate a new mixed sample) and label mixing (i.e., modifying the supervision of the mixed sample). The intensity of the augmentation is controlled by λ, usually sampled from a pre-defined Beta distribution. For example, a mixed sample by Mixup is x mix = λx 1 + (1 − λ)x 2 . Usually, it is not straightforward to apply MSDA to metric learning or contrastive learning because their losses are computed in a batch-dependent way (See Figure 3 (a) and (b)). On the other hand, as our objective function is computed in a pair-wise manner (See Figure 3 (c)), it is easier to apply MSDA to our objective function. There are two issues with designing MSDA for probabilistic matching. First, MSDA for the textual modality is not straightforward. Hence, PCME++ only mixes visual inputs using Mixup [39] and CutMix [40] Second, we cannot directly mix labels because our scenario has no class label. Instead, we let m vt smooth in Equation (2), i.e., m vt ∈ [0, 1]. This approach controls the gradient step size by mixing intensity λ: Assuming m vt = λ, then we have ∂Lmatch ∂l = (1−λ)−(1−2λ) sigmoid(l vt ). Here, an extreme case sigmoid(l vt ) ≈ 1 (i.e., model predicts a positive match with high confidence) has a gradient value λ. Therefore, MSDA also makes highly confident false negative samples (m vt = 0, but the model predicts sigmoid(l vt ) ≈ 1) contribute to the parameter updates as pseudo-positives. The overview of the optimization procedure with pseudo-positives and MSDA is illustrated in Figure 3 (d). In the experimental results, 25% of mini-batch images are mixed by sampling the mixing intensity λ from Beta(2, 2). For every mini-batch, Mixup or CutMix is randomly chosen for the mixing strategy. The empirical study shows that this strategy is slightly better than the widely-used batch-wise mixing strategy, i.e., randomly mixing the whole mini-batch or using the original mini-batch (Appendix C.2). Architecture PCME++ trains visual and textual encoders separately, such as visual semantic embeddings [4,15] or CLIP [19]. Each encoder has two heads, µ and log σ 2 heads whose output vectors are D-dimensional. An input is mapped to a normal distribution parameterized by the output of µ and log σ 2 heads. PCME++ employs a Vision Transformer (ViT) [44] as the visual backbone and a 12-layer 512-wide Transformer [45] as the textual backbone, following Radford et al. [19]. PCME++ duplicates the last transformer layer for µ and log σ 2 heads, e.g., a textual backbone has a shared feature extractor with a 11-layer Transformer and µ and log σ 2 are 1-layer Transformer blocks. The log σ 2 head is randomly initialized, while the µ head is initialized as the same as the backbone initialization (e.g., from a pre-trained model). We empirically observe that using more layers for log σ 2 marginally improves the performances, but we set the number of layers for log σ 2 head to 1 for computational efficiency. Finally, we employ Generalized Pooling Operator (GPO) [15] for the feature aggregation with the same parameter setting of Chen et al. [15]. We observe that GPO brings both training stability and performance improvements. The model architecture overview is illustrated in Figure 4. Experiments Experimental protocol Datasets and evaluation metrics. PCME++ is evaluated on MS-COCO Caption [20], a widely used ITM benchmark, containing 123,287 images from MS-COCO [46] and five human-annotated captions per image. 113,287/5,000/5,000 images are used for training/validation/testing [47]. Although Recall@k (R@k) is a common evaluation metric in COCO Caption, as Musgrave et al. [48] showed, R@k is often insufficient to measure retrieval performances. Furthermore, recent studies [37,25] observed that many COCO Caption negatives are actually positives; e.g., Chun et al. [25] showed that 88.2% and 72.1% positive images and captions are annotated as negative in COCO. In other words, COCO R@k, relying on the noisy COCO annotation m vt , is not fully reliable. To mitigate the problem of R@k evaluation, two extended benchmarks, ECCV Caption (EC) [25] and CxC [37], are employed for the test split. Both datasets are validated by human annotators; EC contains more plentiful positives than CxC but its queries are the subset of the original COCO Caption; CxC has fewer positives than EC, but its annotations cover the whole COCO test split, and the annotations are less noisy. Note that the original COCO Caption, EC, and CxC have the same images and captions (x v , x t ) but with different match annotations m vt . The overview of each benchmark can be found in Appendix B.1. In the experiments, following Chun et al. [25], R@k for all benchmarks and mAP@R and R-Precision for EC are reported. The conventional 5-fold 1K COCO R@1 and "rsum", the summation of R@1, R@5, R@10 for image-to-text and text-to-image retrieval are also reported. For the main paper, the averaged scores on each modality is reported, while the full results for each modality and R@5, R@10 results are in Appendix C.6. Comparison methods. VSE∞ [15] is based on a conventional triplet loss and hardest negative mining. InfoNCE is the CLIP [19] pre-training objective. PCME [14] is a primitive probabilistic ITM model with sampling-based matching probability. As we initialize all models by CLIP pre-trained models, CLIP zero-shot (ZS) is also reported as a baseline. All models have the same visual and textual backbones, except probabilistic models; they have an additional log σ 2 head (See Figure 4). All models are trained three times for each setting and the average evaluation metric are reported. Training details and model selection. PCME++ is initialized with the official pre-trained CLIP models [19], while newly introduced modules, such as log σ 2 head and GPO are randomly initialized. All models are trained for 25 epochs using AdamP optimizer [49] by setting the initial learning rate as 0.0005 and weight decay as 0.0001. The learning rate is decayed by a factor of 0.1 for the last 10 epochs. Following Chen et al. [15], different learning rate multipliers are applied for the visual backbone (×0.01) and the textual backbone (×0.1). The visual backbone is frozen for 2 epochs, and a linear learning rate warmup is applied for the first epoch after the freezing. Also, layer-wise learning rate decay (LLRD) for each transformer block is applied by 0.7. The batch size is set to 128. Lastly, for the generalizability of GPO, SizeAugment is employed as Chen et al. [15]. The hyperparameters of PCME++ are set as follows; the affine transform is initialized by a = b = 5 in Equation (2); α for pseudo-positives as 0.1; VIB β as 0.0001. PCME++ mixes 25% of images in the mini-batch by Mixup or CutMix with a mixing ratio drawn from Beta(2, 2). Finally, we adopt stochastic weight average (SWA) [50] on PCME++ for the last 10 epochs to obtain a more generalizable and robust solution [51], except the L/14 backbone due to the GPU memory issue. For comparison methods, The triplet loss margin is set to 0.2 (for VSE∞ [15]) and the initial softmax temperature for InfoNCE [19] is set to 1.0. PCME [14] uses the same initialization of PCME++ for affine transform and VIB, while 8 samples are drawn per input for computing matching probability. For the evaluation, the best model based on the validation rsum is selected. When SWA is applied, models are not selected based on validation scores but the last averaged model is used. More detailed training settings and resource information for the experiments are described in Appendix B.2. COCO ITM results Main results. Table 1 shows the main comparison results of PCME++ and other ITM methods. We first observe that PCME++ consistently outperforms other methods in all evaluation metrics on different backbones. Second, we observe that the scale-up of PCME++ leads to consistent performance increases without hyperparameter tuning, while deterministic counterparts (e.g., VSE∞ and InfoNCE) suffer from performance drops when scaling up from ViT-B/16 to ViT-L/14. The full image-to-text and text-to-image retrieval results, R@5 and R@10 are separately reported in Appendix C.6. Appendix C.1 shows more comparions with other methods using different backbones. Noisy correspondence. Table 2 shows the additional comparisons under noisy correspondence, i.e., by assuming that the training annotations are noisy. Following Huang et al. [16], the image-text Table 1: COCO cross-modal retrieval performances. Comparisons of ITM methods with various backbones in ECCV Caption, CxC and COCO Caption. "Prob?" denotes whether a method is a probabilistic method or not. Each number is the average between the image-to-text retrieval and text-to-image retrieval results, and is the average of three different runs. The full numbers and standard errors are in Appendix C.6. † denotes the re-evaluated results by the official checkpoints, otherwise, numbers are produced by our trained models. ECCV Caption [25] CxC [ relationships are randomly shuffled with probability of 20% and 50%. A specifically designed method for solving the noisy correspondence problem, NCR [16], is also compared with the comparison methods. Following Huang et al. [16], the model selection criterion is also based on the clean validation rsum as the clean dataset scenario. There are three findings in the table. First, the hardest negative mining-based triplet loss (VSE∞) shows vulnerability on strong noisy correspondence, e.g., 50%. Second, although the probabilistic methods, such as PCME and PCME++, are not designed for tackling noisy correspondence, they successfully handle the noisy correspondence scenario, especially showing outperforming precision-based metrics than NCR. Lastly, we observe that under a strong noisy annotation scenario with a 50% noise ratio, PCME shows better scores than PCME++ in some metrics. I presume that it is because the effect of the proposed techniques, such as pseudo-positives and MSDA, can be weakened under an extremely noisy scenario. It will be an interesting topic to combine noisy correspondence and probabilistic embedding, and I leave this for future work. Ablation study Optimization. Table 3 shows that all the proposed techniques effectively improve probabilistic ITM. More detailed hyperparameter studies for each optimization are in Appendix C.2. Table 4 shows the impact of the probability distance on training objective (Equation (2)) by replacing d(Z v , Z t ). For a fair comparison, all newly proposed optimization techniques except VIB for experiments are omitted. As we already observed in Figure 2, we confirm that Wasserstein distance is not a proper uncertainty estimate as a training objective. Also, the table shows that PCME++ outperforms PCME in all metrics. I presume it is because the matching probability is an approximated value by Monte Carlo approximation, therefore, the distance value will have an approximation gap. Architecture. Table 5 shows the architecture ablation study: (1) GPO improves overall performances; (2) if we use a more complex log σ 2 head, ECCV Caption metrics are slightly improved by capturing ambiguity caused by FNs well. However, the performance improvements are marginal, and it shows inferior R@k scores than a shallower log σ 2 head. Therefore, PCME++ uses the number of layers for the log σ 2 head as 1. Uncertainty analysis From Equation (1), we can define the data uncertainty as ∥σ 2 ∥ 1 , i.e., the summation of the variance. Based on the data uncertainty, Figure 5 shows how the uncertainty captures the ambiguity of datasets. The average COCO 1K R@1s for each modality in each of the 10 uncertainty bins are reported in the figure. We observe that by the uncertainty increased, COCO R@1 (the same distribution as the training dataset) is decreased. The results support that the learned uncertainty by PCME++ can capture the inherent ambiguity of the matching annotations. Figure 6 shows examples of uncertain images and captions, and their retrieved items (more examples are in Appendix C.5). The figure shows that data that can be matched with more samples have higher uncertainty values. As shown in the figure, the retrieved items for uncertain inputs are highly plausible even though the retrieved items are not in the COCO ground truth. In Section 3.5 and Section 4, more benefits of the uncertainty-aware learning and the learned uncertainty are discussed. More applications Large-scale retrieval system. Lack of scalability is a common drawback of probabilistic retrieval systems, i.e., it is difficult to apply probabilistic embeddings on a large-scale retrieval system with a billion-scale index. As the proposed probability distance, CSD (Equation (1)), is the summation of Euclidean distance of µ and the intensity of σ 2 of each input, we can easily and efficiently combine PCME++ and approximated KNN (ANN). First, a Euclidean distance-based index system for µ is built as usual, while σ 2 are saved into key-value storage. Then, K items are retrieved by performing ANN on the µ index. Lastly, the retrieved items are re-ranked by computing the summation of the µ distance and σ 2 value of the retrieved items. In Appendix C.3, the comparisons of diverse retrieval strategies are shown, including ANN based on FAISS [36] and the modified ANN for PCME++. CSD is not only stronger than other probability distances but also more practical and scalable. Uncertainty-based prompt-tuning. Zero-shot (ZS) classification is the task of predicting an unseen class during training. Usually, ZS classification is done by converting class information as a text sentence (e.g., "a photo of a cat") and mapping into a shared embedding space with the input. For image ZS classification tasks, large-scale ITM pre-training, such as CLIP [19], has become a standard approach. Despite their usability and generalizability, ZS needs hand-crafted prompt engineering for converting class information into proper text sentences. For example, Radford et al. [19] showed that taking the average of 80 different context prompts improves ImageNet [52] top-1 ZS accuracy by 3.5% over a single prompt ("a photo of { · }"). However, designing the best-performing prompts for every novel task is time-consuming. This paper investigates the potential of PCME++ for automatic prompt engineering using the learned text uncertainty: The uncertainties of prompts for each class are computed, (e.g., "a photo of a cat", "a photo of many cat", . . . ), and the most uncertain text prompts are discarded. Table 6 shows a study on the proposed simple automatic prompt tuning. For the experiment, ViT-S/16 models using InfoNCE loss and PCME++ are trained on the RedCaps dataset [24] for 100K iterations with 1K batch size. Here, "Top-K certain prompts" denotes that every class uses the same top-K for the filtering, and "Best top-K for each class" denotes the best top-K for each class are chosen, e.g., "coral fungus" needs all 80 prompts, while "ringlet butterfly" only needs Top-1 certain prompt while other uncertain 79 prompts are discarded. With this simple strategy, the ZS performance is increased with a significant gap (8.58 → 14.75). The full description of our ZS experiments are provided in Appendix C.4. Limitations and Discussions Normal distribution with diagonal covariance would be insufficient? One can argue that the uncertainty modeling power of PCME++ can be improved by relaxing the diagonal covariance condition. However, Oh et al. [26] showed that if the dimensionality of the embedding space and the number of "hidden features" are the same (e.g., if an image is the combination of two digits, then the number of potential latent features for each input is two), then the diagonal covariance condition can sufficiently capture the inherent uncertainty of the dataset. In practice, we use a very high dimensional embedding space (e.g., 1024) that can sufficiently capture complex relationships between features. Also, in practice, if we relax the condition, the dimensionality of the log σ 2 head output should be about 1M (= 1024 × 1024), which will require expensive computational budgets and large memory. Additional sampling is still required if we use other density functions. The proposed probabilistic distance is defined in distribution-free: E Zv,Zt ∥Z v − Z t ∥ 2 2 . However, the closed-form solution (CSD) is specifically designed for normally distributed embeddings. If one needs probabilistic embeddings with different distributions, such as von Mises-Fisher distribution [35] or Laplacian distribution [34], CSD is no longer applicable. Instead, we can adapt any distribution to PCME++ by using a Monte Carlo approximation, i.e., by computing 1 n×m z n v z i v =z 1 v z m v z j t =z 0 t ∥z i v − z j t ∥ 2 2 , where z i v ∼ Z v and z j t ∼ Z t . This change will share the expensive computation issue of previous approaches [26,14], but the additionally introduced techniques in PCME++ for mitigating the loss saturation issue (i.e., pseudo-positives and MSDA) will still be effective. Applying other probabilistic densities to PCME++ and discovering the effect of different distribution choices will be interesting future work. How does uncertainty help learning image-text representations? As shown in the main experiments, the probabilistic approach is helpful for improving the retrieval performances, but the gaps are not significant (e.g., Table 1 shows that in ViT-B/32, the gap between VSE∞ and PCME++ with SWA is not significant). However, as shown in larger backbone experiments (ViT-B/16 and ViT-L/14) and noisy correspondence experiments (Table 2), PCME++ shows more generalizable performances compared to the existing state-of-the-art ITM methods with the same backbone. Furthermore, as shown in Section 3.4 and Section 3.5, the learned uncertainty by PCME++ shows high interpretability of the datasets as well as the controllability by the users when the rejection of the retrieved items is required. Thus, I believe that the uncertainty-aware learning paradigm and the learned uncertainty will be helpful for image-text matching problems and downstream tasks, such as zero-shot classification. Conclusion This paper addresses the inherent ambiguity of ITM tasks by PCME++. A novel closed-form probability distance and a new matching objective function for efficiency and effectiveness is presented. PCME++ is further enhanced by incorporating a pseudo-positive strategy and a mixed sample data augmentation strategy, successfully addressing the loss saturation issue associated with abundant false negatives. Experimental results demonstrate the extensibility of PCME++ to various applications, such as image-caption cross-modal retrieval, mitigating noisy correspondences, automatic prompt tuning for zero-shot classification, and understanding the inherent ambiguity of a dataset. Acknowledgement Societal Impact This work aims to learn better image-text representations based on a probabilistic approach. As shown in the experiments, PCME++ has the potential to improve the interpretability and the controllability of learned representations by providing an additional degree of freedom to the users. Accordingly, PCME++ shares the potential impact of developing general image-text representations with better interpretability and controllability. For example, as shown by Radford et al. [19], visual-textual representations trained on a large-scale web dataset often suffers from biases in the web; PCME++ can both mitigate or enhance the biases using its interpretability and controllability. A Method Details A.1 Derivation of the closed-form probability distance In this subsection, the full derivation of Equation (1) is shown. We first show two simple well-known lemmas and conclude the full proof using them. Lemma 1. Let X and Y be independent normally distributed random variables where X ∼ N (µ X , Σ X ) and Y ∼ N (µ Y , Σ Y ). Then, the subtraction between X and Y is another normal distribution, i.e., (X − Y ) ∼ N (µ X − µ Y , Σ X + Σ Y ). Proof. Let ϕ X (u) = exp(it ⊤ µ X − 1 2 t ⊤ Σ X t) be a characteristic function of normally distributed random variable X. Using the fact that −Y ∼ N (−µ Y , Σ Y ), we can compute the summation of ϕ X (u) and ϕ −Y (u) as follows: ϕX−Y (u) = exp(it ⊤ µX − 1 2 t ⊤ ΣX t) exp(−it ⊤ µY − 1 2 t ⊤ ΣY t) = exp(it ⊤ (µX − µY ) − t ⊤ (ΣX + ΣY )t). (A.1) Hence, X − Y is another normal distribution, N (µ X − µ Y , Σ X + Σ Y ). Lemma 2. Let X ∼ N (µ, Σ). Then E∥X∥ 2 2 = ∥µ∥ 2 2 + tr(Σ). Proof. We first re-parameterize a random variable X as X = µ + SZ, where S is the square root matrix of Σ, i.e., SS ⊤ = Σ, and Z is a standard normal distribution. Note that S always exists because Σ is a positive semi-definite by definition. Using E[Z] = 0, the property of Frobenius norm ∥A∥ 2 F = tr(A) and the property of trace tr(AB) = tr(BA), we have: E∥X∥ 2 2 = E Z [∥µ∥ 2 2 + 2µSZ + ∥Z ⊤ S ⊤ SZ∥ 2 2 ] = ∥µ∥ 2 2 + E Z ∥Z ⊤ S ⊤ SZ∥ 2 2 = ∥µ∥ 2 2 + E Z tr(Z ⊤ S ⊤ SZ) = ∥µ∥ 2 2 + tr(S ⊤ S E Z [ZZ ⊤ ]) = ∥µ∥ 2 2 + tr(Σ). (A.2) Proposition 1. Let X and Y be independent normally distributed random variables where X ∼ N (µ X , Σ X ) and Y ∼ N (µ Y , Σ Y ). Then we have E∥X − Y ∥ = ∥µ X − µ Y ∥ 2 2 + tr(Σ X + Σ Y ). Proof. By combining Lemma 1 and Lemma 2, the proof is completed. A.2 Toy experiments In Section 2.2, a 2-D toy dataset is introduced for comparing various objective functions under inherent uncertainty. The toy dataset has three classes with "confusing samples" between classes, i.e., a confusing sample randomly can be either class A or class B. The number of confusing samples are 30% of the total data points. To synthesize the samples, a centroid is randomly chosen for each class. Using the centroid, each sample is randomly drawn from µ + 0.1 × N (0, I). To show the effects of different probabilistic distances, the samples are directly updated by the objective function Equation (2) with different metrics, i.e., a sample (µ, σ) is directly updated by Equation (2). The dataset is directly optimized using Adam optimizer [54] with learning 0.02 during 500 epochs. The mini-bath size is set to 128. We employ the same loss function with PCME++ while the probabilistic distance is chosen from either our distance or Wasserstein distance. The animated learning progress of each method can be found in https://naver-ai.github.io/pcmepp/. A.3 Comparisons with PCME and PCME++ objective functions We first recall the definition of matching probability: E Zv,Zt sigmoid(−a∥Z v − Z t ∥ 2 + b) ≈ 1 J 2 zv,zt sigmoid(−a∥z v − z t ∥ 2 + b), (A.3) where J is the number of samples z v and z t . PCME directly optimized the negative log-likelihood: (2) share a similar formulation, but the position of the expectation is different. As the expectation is located at the outside of sigmoid, Equation (A.3) cannot be computed in a closed-form solution, but our distance can. Note that the analysis in Section 2.3 also holds for Equation (A.4), hence the PCME loss suffer from the loss saturation under abundant false negatives. images and 25,000 captions. COCO 5K uses the full 5,000 images and 25,000 captions where each image has five positive captions and each caption only has one positive image. For evaluation, COCO 5K measures image-to-text retrieval performances by setting 5,000 images as queries and 25,000 captions as galleries, while text-to-image retrieval performances are measured in the opposite way. COCO 1K uses the same positive relationships as COCO 5K, but COCO 1K uses the subset of COCO 5K, i.e., there are 1,000 images and their corresponding 5,000 captions for COCO 1K split. COCO 1K measures the performances by taking an average of five different splits. m vt log zv,zt sigmoid (−a∥z v − z t ∥ 2 + b) + (1 − m vt ) log zv,zt sigmoid (a∥z v + z t ∥ 2 + b) (A.4) Equation (A.4) and Equation A.4 PCME++ Pseudo-code CxC [37] and ECCV Caption [25] use the same images and captions of COCO 1K/5K, but with more positive annotations. CxC uses the entire images and valid 24,972 captions among 25,000 captions (by omitting "I cannot see any image" captions). CxC has more positive annotations than COCO, but there are still many missing positives in CxC because their approach is mostly focused on text similarity, not image-text similarity. On the other hand, ECCV Caption is designed for handling false negatives of image-text pairs. ECCV Caption uses the subset of images and captions for the queries, but their retrieval database is the full dataset, i.e., when performing image-to-text retrieval, the number of query images is 1,261 and the number of gallery captions are 25,000; for text-to-image retrieval, the number of query texts is 1,332 and the number of gallery images is 5,000. As discussed by Musgrave et al. [48] and Chun et al. [25], Recall@K is not an informative metric for measuring retrieval performances in terms of precision. Due to this reason, this paper reports mAP@R and R-Precision of ECCV Caption as the main comparison metrics. [55], VinVL [56] and BLIP [57] need heavy computations to perform retrieval because they have to compute pair-wise similarity for all pairs. For example, they need O(5,000×25,000) computation budgets for measuring retrieval performances. On the other hand, methods with separated encoders just need O(5,000+25,000) computation budgets, 4,166 times smaller computation budgets compared to expensive retrieval methods. Therefore, the table only highlights the best retrieval performances among efficient retrieval methods for a fair comparison. PCME++ achieves the best recall scores for all evaluation benchmarks while showing second and third best ECCV mAP@R and R-Precision. I presume that it is because of the capability of the backbone architecture. For example, VSE∞ with CLIP B/16 backbone shows much better recall scores than VSE∞ with WSL backbone, but VSE∞ (WSL) shows better mAP@R and R-Precision than the CLIP backbone. We expect that PCME++ can outperform the previous retrieval methods in precision metrics if we train PCME++ using different backbones, such as large-scale weakly supervised learning (WSL) backbone [58]. Table C.4 shows the comparisons of different retrieval strategies using PCME++ B/32 model. "Mean only" denotes the retrieval strategy only using µ vectors, without σ. "2-Wasserstein" and "CSD" denote that each probabilistic distance is used for the retrieval. In the table, we observe that mean-only retrieval shows sufficiently good performances but using CSD improves the overall performances. B.2 Hyperparameter and resource details C.2 More ablation studies C.3 Comparisons of different retrieval strategies This paper additionally shows the approximated KNN (ANN) results using FAISS [36]. First, a FAISS search index using µ vectors is built. Then, ANN is performed on the FAISS index to get the ranked list. Finally, the ranked list is re-ranked by CSD. Here, CSD can be efficiently computed by storing gallery σ into a fast key-value storage, such as Redis. As shown in the table, ANN can be efficiently and effectively applied to PCME++ with a reasonable computation-performance trade-off. C.4 Details of automatic prompt-tuning by PCME++ For the experiments, a randomly initialized ViT-S/16 backbone is trained by InfoNCE loss and PCME++ loss on RedCaps [24] using hyperparameters in Table B.1. The pre-trained models are evaluated on the ImageNet [52] zero-shot (ZS) classification task. Specifically, 80 prompts provided by CLIP [19] (shown in the next paragraph) are used for the ZS classification. In Table 6, "A photo of a · " denotes that only "A photo of a · " prompt is used for the zero-shot classification, while "All 80 prompts" denotes that all 80 prompts are used for computing text embeddings and the average text embedding is used for the zero-shot classification. Top-5 Retrieved images GT Image \|\| σ \|\| 1 = 9.73e-5 Assorted fruit on display at a fruit market. Top-5 Retrieved images GT Image Query text: \|\| σ \|\| 1 = 9.74e-5 This paper explores the potential of PCME++ for automatic prompt-tuning with a simple uncertaintybased filtering. First, the prompts for every class are sorted by their uncertainty, i.e., ∥σ∥ 1 . Then, uncertain prompts are filtered out, and the remained prompts are used for ZS classification. Here, two strategies are tested. First, the same top-K uncertain prompts for all classes are filtered. As shown in Figure C.1a, this strategy slightly improves the overall performances, but it only shows a marginal improvement against the "all" baseline (+0.04%). To further improve the uncertainty-based filtering, the strategy with different top-K for different prompts is also explored. As shown in Table 6, this strategy shows very effective performance improvement against the baseline (+5.42%). Figure C.1b shows the detailed population of the best top-K filtering per class. Here, the classes whose accuracy is 0% are omitted. Interestingly, we observe that 10% of classes (105) show the best ZS performances when all 80 prompts are used. On the other hand, about half of the classes (499) show the best performance when more than 35 prompts are filtered out. This primitive study on uncertainty-based prompt tuning has two limitations. First, the baseline pretrained model is too weak (13.41% top-1 IN ZS performance with InfoNCE) compared to well-known baselines, such as CLIP [19]. Second, this study has no validation split, i.e., the best top-K prompt for each class is directly searched from the ImageNet validation split. Searching for the best top-K for each class without direct tuning on test split using strong probabilistic pre-trained image-text representations will be an interesting future research direction. C.5 More examples of uncertain samples More examples of uncertain images and captions are shown in Figure C.2. C.6 Full experimental results The image-to-text and text-to-image R@5 and R@10 results are shown in Table C.5 and Table C up in the air. →Figure 1 : 1A probabilistic embedding space captures uncertainty of noisy ITM supervision.If a match is uncertain, then the variance of embeddings becomes larger (Inherent ambiguity of ITM. We assume that the deterministic textual embeddings are mapped to the same point z ′ estimation Figure 2 : 2Learned 2-D embedding spaces by CSD and WD. The full animations can be found in https: //naver-ai.github.io/pcmepp/. Figure 3 : 3Comparisons of different objective functions. For given i-th visual embeddings z i v and j-th textual embedding z j t , we illustrate how each sample contributes to different loss functions. (a) Only two image-caption pairs contributed to the loss in each row/column for triplet loss. (b) Batch-wise contrastive loss, such as InfoNCE, is defined for each row/column. (c) Pair-wise contrastive loss, such as PCME++, is defined for each image-caption pair. Hence, our loss is computed multiple times for each row/column. (d) As our loss is computed pair-wise, it is straightforward to apply pseudo-positives or mixed sample data augmentation (MSDA). others are uncertain samples. The size of each dot denotes the intensity of the learned σ values. Figure 5 : 5∥σ 2 ∥1 vs. R@1. Figure 6 : 6Example of images and captions with high uncertainty. More examples are shown in Appendix C.5. Appendix More additional materials are included. More details of our method are described in §A, including the full derivation of the closed-form probabilistic distance ( §A.1), the toy experiments ( §A.2), comparisons between PCME and PCME++ ( §A.3), and pseudo-code of PCME++ ( §A.4). In experimental protocol details section ( §B), the benchmark dataset details ( §B.1), hyperparameter and resource details ( §B.2) are shown. Finally, additional experimental results ( §C), including comparisons with state-of-the-art ( §C.1), the full ablation studies ( §C.2), the comparisons of different retrieval strategies ( §C.3), automatic prompt-tuning experiments ( §C.4), more qualitative examples ( §C.5), and the full experimental results and error bars ( §C.6) are presented. 500 samples are drawn for each class and 150 samples of them are chosen as "confusing samples", i.e., there are 1500 samples with 1050 certain samples and 450 confusing samples. Then, log σ of each sample is randomly drawn from U(−1.5, 1.5) where U is a uniform distribution. In summary, the dataset has 350 confident samples for class 1, 2 and 3; 150 confusing samples for class (1, 2), (2, 3) and (3, 1). Figure A. 1 : 1PyTorch pseudo-code of PCME++. Here, v_sig and t_sig are computed by taking an exponential to the output of log σ 2 heads. BCE denotes a binary cross-entropy function. Figure A. 1 Figure B. 1 : 11shows the PyTorch style pseudo-code of PCME++. Note that µ and σ are extracted from the augmented inputs, such as MSDA (Section 2.4) and SizeAugment[15].B Experimental Protocol DetailsB.1 More details of benchmark datasetsFigure B.1 illustrates the differences between evaluation datasets. Note that all evaluation benchmarks use the same training dataset described in §3.1. The COCO Caption evaluation split consists of 5Difference between COCO 5K, 1K, CxC[37] and ECCV Caption[25]. All matches not illustrated in the image are negative. ECCV Caption has separated query sets for each modality, while other datasets use the same images and captions for both query and gallery. 80 based prompts. a photo of a { }., a bad photo of a { }., a photo of many { }., a sculpture of a { }., a photo of the hard to see { }., a low resolution photo of the { }., a rendering of a { }., graffiti of a { }., a bad photo of the { }., a cropped photo of the { }., a tattoo of a { }., the embroidered { }., a photo of a hard to see { }., a bright photo of a { }., a photo of a clean { }., a photo of a dirty { }., a dark photo of the { }., a drawing of a { }., a photo of my { }., the plastic { }., a photo of the cool { }., a close-up photo of a { }., a black and white photo of the { }., a painting of the { }., a painting of a { }., a pixelated photo of the { }., a sculpture of the { }., a bright photo of the { }., a cropped photo of a { }., a plastic { }., a photo of the dirty { }., a jpeg corrupted photo of a { }., a blurry photo of the { }., a photo of the { }., a good photo of the { }., a rendering of the { }., a { } in a video game., a photo of one The best Top-K for each class. Figure C. 1 : 1Automatic prompt tuning results. (a) shows the ImageNet (IN) zero-shot (ZS) results when prompts are filtered by the same top-K for every class. The ZS performance shows the best at the number of filtered prompts is 13, but the performance improvement is marginal. (b) shows the population of best top-K filtering for all classes. Here, 105 classes among 1,000 classes show the best performance when there is no filtering, while 77 classes show the best ZS score when filtering out except the most 1 certain prompt. Figure C. 2 : 2More examples of images and captions with high uncertainty { }., a doodle of a { }., a close-up photo of the { }., the origami { }., the { } in a video game., a sketch of a { }., a doodle of the { }., a origami { }., a low resolution photo of a { }., the toy { }., a rendition of the { }., a photo of the clean { }., a photo of a large { }., a rendition of a { }., a photo of a nice { }., a photo of a weird { }., a blurry photo of a { }., a cartoon { }., art of a { }., a sketch of the { }., a embroidered { }., a pixelated photo of a { }., itap of the { }., a jpeg corrupted photo of the { }., a good photo of a { }., a plushie { }., a photo of the nice { }., a photo of the small { }., a photo of the weird { }., the cartoon { }., art of the { }., a drawing of the { }., a photo of the large { }., a black and white photo of a { }., the plushie { }., a dark photo of a { }., itap of a { }., graffiti of the { }., a toy { }., itap of my { }., a photo of a cool { }., a photo of a small { }., a tattoo of the { }. or calibrationPatchified image Transformer blocks Image input Text input Linear projection + Positional embedding "A grey cat is wearing a red hat" GPO GPO L2-norm Transformer blocks GPO GPO L2-norm μ v log σ v 2 Tokenize & embedding + Positional embedding Table 2 : 2COCO noisy correspondence. Noisy correspondence results using the ViT-B/32 backbone, except NCR[16] are shown. NCR scores are re-evaluated by the official weights. Noise ratio 0% is the same asTable 1.ECCV Caption CxC COCO Noise ratio Method mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM 20% VSE∞ 37.1 46.6 80.1 53.3 72.0 51.4 520.2 InfoNCE 35.5 46.0 75.5 47.2 67.8 45.2 513.3 PCME 37.3 47.2 79.2 49.9 69.9 48.1 519.3 PCME++ (ours) 37.9 47.7 79.7 51.3 70.8 49.5 522.4 NCR † [16] 35.9 46.0 78.0 50.6 70.1 48.8 518.6 50% VSE∞ 17.6 28.2 43.6 20.0 38.5 18.4 390.5 InfoNCE 33.2 43.8 72.1 43.3 64.0 41.3 498.2 PCME 34.7 45.2 73.3 45.0 65.8 43.0 505.7 PCME++ (ours) 34.4 44.6 75.0 46.0 65.7 44.0 503.9 NCR † 34.0 44.3 75.1 47.3 66.8 45.5 508.5 Table 3 : 3Effect of optimization methods. Ablation study on VIB[38], Pseudo-Positives (PP), Mixed SampleData Augmentation (MSDA), and SWA [50] with a ViT-B/32 backbone are shown. ECCV Caption CxC COCO VIB PP MSDA SWA mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM ✘ ✘ ✘ ✘ 38.9 48.6 82.2 56.7 75.2 54.9 535.9 ✔ ✘ ✘ ✘ 39.2 49.0 82.2 56.1 74.9 54.3 535.1 ✘ ✔ ✘ ✘ 39.0 48.6 82.7 56.8 75.2 55.0 536.0 ✔ ✔ ✘ ✘ 39.6 49.2 82.6 56.3 74.8 54.5 534.8 ✔ ✔ ✔ ✘ 40.0 49.6 83.3 57.0 75.5 55.3 537.1 ✔ ✔ ✔ ✔ 40.2 49.8 83.6 57.2 75.6 55.5 537.3 Table 4 : 4Effect of probability distance on training objective. Results on ViT-B/32 backbone with VIB loss.ECCV Caption CxC COCO Probability distance mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM Wasserstein 2-distance 26.7 35.5 69.0 46.3 64.5 44.9 484.6 Match probability (PCME [14]) 39.1 48.9 81.4 54.7 73.8 53.0 532.0 Proposed (Equation (1)) 39.2 49.0 82.2 56.1 74.9 54.3 535.1 Table 5 : 5Impact of architecture design choice. Details are the same as the previous tables.ECCV Caption CxC COCO # layers for log σ 2 GPO mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM 1 ✘ 37.4 47.4 79.2 51.0 70.4 49.2 521.8 2 ✔ 40.2 49.7 83.2 56.6 75.3 54.8 536.5 1 ✔ 40.0 49.6 83.3 57.0 75.5 55.3 537.1 A white polar bear is walking on boulders. A white polar bear is walking on some rocks. A polar bear standing on a large rock with its mouth wide open. a close up of a polar bear on a rock formation. A fat polar bear walking along the rocks. a close up of a polar bear on a rock formation. a polar bear yawning while standing on a rock. A polar bear opening its mouth while standing on a rock. A polar bear standing on a large rock with its mouth wide open. A large polar bear stands on rock with an open mouth.\|\| σ 2 \|\| 1 = 1.19e-4 Top-5 Retrieved captions Query image GT captions \|\| σ 2 \|\| 1 = 9.73e-5 Query text: Assorted fruit on display at a fruit market. Top-5 Retrieved images GT Image \|\| σ 2 \|\| 1 = 9.56e-5 Query text: Colored kites sail in the sky above a sandy beach Top-5 Retrieved images GT Image Table 6 : 6ImageNet (IN) Zero-shot (ZS).Model Prompts Top-1 Acc InfoNCE "A photo of { · }" 13.05 All 80 prompts 13.41 PCME++ "A photo of { · }" 8.58 All 80 prompts 9.33 Top-K certain prompts 9.37 Best top-K for each class 14.75 I would like to thank my NAVER AI Lab colleagues for valuable discussions, including Sangdoo Yun, Wonjae Kim, Jiyoung Lee, Dongyoon Han, Byeongho Heo, Taekyung Kim, Song Park and Jung-Woo Ha. NAVER Smart Machine Learning (NSML) platform[53] is used for the experiments. Table B . B1: Hyperparameter details Method CLIP ViT B/32, B/16, L/14 COCO CLIP S/16 RedCaps Epochs 25 25 Batch size 128 1,536 Optimizer AdamP AdamP Initial learning rate 0.0005 0.0005 LR scheduling Step linear warmup and cosine Layer-wise LR decay 0.7 - Visual backbone LR decay 0.01 - Textual backbone LR decay 0.1 - β1, β2, ε 0.9, 0.999, 10 −8 0.9, 0.98, 10 −6 Weight decay 0.0001 0.2 VIB β 0.0001 10 −6 PP α 0.1 0 MSDA CutMix/Mixup λ, mix ratio 2/2/25% -/-/0% Size Augment ✔ ✘ Embedding dimension 1024 128 Initial a and b 5/5 5/5 Resources ViT B/32 1 V100 (38 hours) 8 V100 (84 hours) ViT B/16 1 V100 (75 hours) ViT L/14 8 V100 (62 hours) Table C . 1 : C1Comparisons with state-of-the-art models. All numbers are reproduced by the official weights.We highlight the best scores except expensive retrieval methods, such as BLIP.Efficient ECCV Caption CxC COCO Method retrieval? mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM CVSE [12] ✔ 37.4 47.5 76.7 45.8 67.0 43.8 511.1 VSRN [8] ✔ 42.3 51.8 81.5 48.9 69.5 46.7 515.9 NCR [16] ✔ 36.4 46.3 79.9 51.8 71.0 50.0 522.6 VSE∞ (BUTD region) [15] ✔ 40.5 50.0 82.5 52.4 72.2 50.4 527.5 VSE∞ (WSL) ✔ 42.4 51.4 86.4 60.8 78.3 59.0 545.1 VSE∞ (B/16, our implementation) ✔ 41.7 50.6 86.3 62.3 79.1 60.7 547.2 ViLT [55] ✘ 34.6 44.3 77.8 53.7 72.8 52.2 528.6 VinVL [56] ✘ 40.8 49.6 87.8 67.8 82.4 66.4 555.5 BLIP [57] ✘ 40.5 48.4 91.0 74.3 86.1 73.1 564.4 CLIP Zero-shot (L/14) [19] ✔ 28.0 37.8 72.2 48.1 64.8 46.4 491.6 PCME++ (B/16) ✔ 42.0 51.1 86.6 63.1 79.7 61.6 548.9 PCME++ (L/14) ✔ 42.1 50.8 88.8 65.9 81.8 64.3 554.7 Table C.2: Pseudo-positive α ablation study. ECCV Caption CxC COCO α mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM 0.1 40.2 49.8 83.1 56.5 75.1 54.8 536.0 0.5 40.0 49.5 83.1 56.7 75.4 55.0 536.8 2 40.1 49.7 83.0 56.5 75.1 54.8 535.8 5 40.3 49.9 83.1 55.7 74.7 53.9 534.9 10 40.2 49.9 82.5 54.5 73.7 52.6 531.9 Table B . B1 shows the detailed hyperparameter settings and the detailed GPU resource information.C Additional Experimental Results C.1 Comparisons with state-of-the-arts Table C . C1 shows the comparisons of PCME++ and state-of-the-arts with different backbones. Note that ViLT Table C . C2 shows the ablation study for pseudo positive α. The table shows that our method is not very sensitive to the choice of α. We choose α = 0.1, which shows the second best ECCV Caption Table C . 3 : C3MSDA ablation study. mAP@R and COCO recall measures. Table C.3 shows the ablation study for the mixed sample data augmentation design choice. The design choice for PCME++ shows the best performance.ECCV Caption CxC COCO Mixup λ CutMix λ Mix ratio in-batch? mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM 2 0 25% ✔ 37.3 47.3 79.6 51.9 71.7 50.0 525.5 0 2 25% ✔ 37.5 47.6 79.2 50.5 70.9 48.8 523.4 1 1 50% ✘ 39.7 49.5 81.8 55.2 74.5 53.5 534.0 1 1 25% ✘ 39.9 49.6 82.3 55.5 74.6 53.8 534.4 2 2 25% ✘ 40.0 49.6 82.8 55.8 74.5 54.0 534.4 1 1 50% ✔ 39.9 49.6 82.7 55.4 74.4 53.7 534.1 2 2 25% ✔ 40.1 49.7 82.9 56.5 75.0 54.7 535.9 Table C.4: Effect of inference methods. We compare the mean-only inference and probability distance-based inferences using our ViT-B/32 SWA model. Each number is the average of different three runs. ECCV Caption CxC COCO Method Prob? mAP@R R-P R@1 R@1 1K R@1 5K R@1 RSUM Mean only ✘ 40.2 49.8 83.5 56.9 75.2 55.2 536.3 2-Wasserstein ✔ 40.2 49.9 83.0 56.6 75.2 54.8 535.9 CSD (ours) ✔ 40.2 49.8 83.6 57.2 75.6 55.5 537.3 FAISS [36] (Meany only) ✘ 40.1 49.7 83.5 56.4 74.7 54.6 531.2 FAISS + σ re-ranking ✔ 40.1 49.7 83.2 56.6 74.8 54.8 531.7 \|\| σ \|\| 1 = 0.000111 Several oranges are laying under a few bananas. A bunch of bananas on top of oranges. Closeup of various oranges and bananas in pile. A pile of oranges sitting under a pile of bananas. A plate full of sliced oranges next to a bunch of bananas. Closeup of various oranges and bananas in pile. Several oranges are laying under a few bananas. A bunch of bananas on top of oranges. There are a lot of bananas and oranges. A pile of oranges sitting under a pile of bananas \|\| σ \|\| 1 = 1.13e-4A snowboarder flies through the air in a mountain landscape. A person launching into the air on a snowboard. A person with a snowboard is jumping in the air. a man jumping in the air above a ski slope on a ski board. A man jumping in the air on a snowboard. A snowboarder is flying through the air doing stunts on his snow board. A man with gloves, goggles and a hat on is in the air on his snowboard. A snowboarder is airborne during a trick atop a mountain.A person that is snowboarding through the air as they grab their board. A snowboarder doing a trick after a jump.\|\| σ \|\| 1 = 9.43e-5 Query text: A beach area with people on the sand and various kites flying overhead in the sky.Retrieved captions GT captions Top-5 Retrieved captions GT captions Query image Query text: A large banana tree that has green colored bunches of bananas on it along with various branchesTop-5 Retrieved images GT Image Query text: \|\| σ \|\| 1 = 9.24e-5 Bunches of carrots, beets, radishes, and multiple lettuces displayed on a table Top-5 Retrieved images GT Image Table C . 5 : C5Image-to-text retrieval R@5 and R@10 results.Table C.6: Text-to-image retrieval R@5 and R@10 results.CxC COCO Backbone Method R@5 R@10 1K R@5 1K R@10 5K R@5 5K R@10 ViT-B/32 (151M) VSE∞ 87.1 93.4 96.7 98.8 85.4 92.2 InfoNCE 87.3 93.4 96.5 98.9 85.9 92.3 PCME 87.5 93.5 96.6 98.7 85.8 92.3 PCME++ 88.4 94.0 97.0 99.0 87.0 93.0 PCME++ + SWA 88.5 94.0 97.0 99.0 87.1 92.9 ViT-B/16 (150M) VSE∞ 91.1 95.6 97.8 99.4 89.9 94.8 InfoNCE 90.9 95.8 97.8 99.3 89.7 94.9 PCME 90.5 95.4 97.7 99.3 89.2 94.5 PCME++ 91.4 95.7 97.9 99.3 90.3 94.9 PCME++ + SWA 91.5 95.9 97.9 99.3 90.4 95.1 ViT-L/14 (428M) VSE∞ 58.8 72.6 82.2 91.4 55.7 69.4 InfoNCE 82.8 91.7 95.3 98.6 80.2 90.0 PCME 91.8 95.9 98.1 99.4 90.7 95.2 PCME++ 93.4 96.8 98.5 99.6 92.2 96.2 CxC COCO Backbone Method R@5 R@10 1K R@5 1K R@10 5K R@5 5K R@10 ViT-B/32 (151M) VSE∞ 77.7 86.5 92.2 96.7 75.5 84.8 InfoNCE 77.3 86.5 92.3 96.9 75.1 84.7 PCME 77.3 86.4 92.1 96.9 75.0 84.6 PCME++ 78.5 87.1 92.8 97.1 76.5 85.4 PCME++ + SWA 78.6 87.3 92.8 97.1 76.5 85.5 ViT-B/16 (150M) VSE∞ 82.0 89.5 94.2 97.5 80.3 88.2 InfoNCE 81.3 89.1 94.0 97.7 79.5 87.7 PCME 80.9 88.9 93.9 97.7 79.1 87.5 PCME++ 82.0 89.7 94.4 97.8 80.3 88.3 PCME++ + SWA 82.1 89.7 94.4 97.8 80.4 88.4 ViT-L/14 (428M) VSE∞ 46.4 61.1 74.2 87.4 42.9 57.1 InfoNCE 73.6 84.2 91.3 96.4 71.0 82.3 PCME 82.7 90.2 94.5 97.8 81.1 88.8 PCME++ 84.0 90.8 95.1 98.1 82.6 89.7 .6. The full experimental results, including separated image-to-text and text-to-image retrieval results for the main table, and standard errors, are included inTable C.7, Table C.8 and Table C.9. The Table C.7: Image-to-text retrieval full results. PCME++ + SWA 32.4 (±0.2) 43.5 (±0.2) 77.8 (±0.7) 63.7 (±0.4) 81.4 (±0.2) 62.3 (±0.4) PCME++ + SWA 34.6 (±0.1) 45.2 (±0.1) 81.8 (±0.8) 70.3 (±0.1) 85.6 (±0.1) 69.0 (±0.1) Table C.8: Text-to-image retrieval full results. ±0.5) 55.1 (±0.5) 88.0 (±0.8) 48.0 (±0.3) 67.7 (±0.2) 46.0 (±0.3) PCME 47.1 (±0.2) 55.5 (±0.2) 88.0 (±0.5) 48.0 (±0.1) 67.6 (±0.1) 46.1 (±0.1) PCME++ 48.0 (±0.1) 56.1 (±0.2) 88.8 (±0.3) 49.9 (±0.1) 68.9 (±0.2) 47.9 (±0.0) PCME++ + SWA 48.1 (±0.2) 56.2 (±0.3) 89.2 (±0.3) 50.0 (±0.1) 69.0 (±0.1) 48.0 (±0.1) ViT-B/16 (150M) VSE∞ 49.1 (±0.3) 56.5 (±0.2) 91.3 (±0.4) 55.3 (±0.3) 73.3 (±0.3) 53.4 (±0.3) InfoNCE 48.5 (±0.2) 56.3 (±0.1) 89.9 (±0.2) 53.6 (±0.3) 72.3 (±0.1) 51.7 (±0.PCME++ + SWA 49.8 (±0.1) 57.2 (±0.2) 91.4 (±0.7) 55.5 (±0.2) 73.5 (±0.1) 53.6 (±0.2)full experimental numbers for all experiments, including ablation studies, can be found in https: //naver-ai.github.io/pcmepp/.ECCV Caption CxC COCO Backbone Method mAP@R R-P R@1 R@1 1K R@1 5K R@1 ViT-B/32 (151M) VSE∞ 31.7 (±1.2) 42.8 (±0.9) 75.6 (±3.2) 61.8 (±4.1) 80.4 (±3.2) 60.2 (±4.2) InfoNCE 31.2 (±0.1) 42.3 (±0.1) 75.4 (±1.1) 61.8 (±0.1) 80.3 (±0.6) 60.1 (±0.2) PCME 31.2 (±0.0) 42.3 (±0.0) 74.9 (±0.3) 61.5 (±0.6) 80.1 (±0.2) 59.9 (±0.6) PCME++ 32.2 (±0.1) 43.4 (±0.1) 77.1 (±1.0) 63.3 (±0.2) 81.3 (±0.3) 61.8 (±0.2) ViT-B/16 (150M) VSE∞ 34.4 (±0.1) 44.8 (±0.2) 81.2 (±0.7) 69.4 (±0.2) 84.9 (±0.4) 68.0 (±0.1) InfoNCE 33.7 (±0.1) 44.4 (±0.1) 79.7 (±0.4) 68.2 (±0.6) 84.3 (±0.7) 66.8 (±0.5) PCME 33.2 (±0.3) 44.0 (±0.4) 79.1 (±0.4) 66.8 (±0.6) 83.6 (±0.3) 65.3 (±0.6) PCME++ 34.5 (±0.1) 45.1 (±0.1) 81.5 (±0.2) 69.9 (±0.3) 85.4 (±0.2) 68.7 (±0.4) ViT-L/14 (428M) VSE∞ 15.7 27.2 39.7 28.9 51.2 27.4 InfoNCE L/14 27.8 39.6 69.0 53.9 75.9 51.9 PCME 34.1 44.5 81.5 70.7 86.5 69.5 PCME++ 35.4 45.3 84.0 73.3 87.9 71.8 ECCV Caption CxC COCO Backbone Method mAP@R R-P R@1 R@1 1K R@1 5K R@1 ViT-B/32 (151M) VSE∞ 47.7 (±0.2) 55.9 (±0.3) 88.6 (±0.9) 49.0 (±2.6) 67.9 (±2.2) 46.9 (±2.6) InfoNCE 46.8 (3) PCME 48.7 (±0.2) 56.5 (±0.2) 89.5 (±0.1) 53.1 (±0.9) 72.0 (±0.6) 51.2 (±0.9) PCME++ 49.7 (±0.2) 57.2 (±0.2) 91.4 (±0.6) 55.2 (±0.2) 73.4 (±0.1) 53.4 (±0.2) ViT-L/14 (428M) VSE∞ 24.7 35.8 52.7 19.7 37.9 18.0 InfoNCE L/14 43.4 52.1 82.1 42.1 63.1 39.9 PCME 48.2 56.0 90.5 56.1 74.1 54.3 PCME++ 48.6 56.3 92.5 58.9 75.8 57.1 Table C . 9 : C9Average retrieval full results. PCME++ + SWA 40.2 (±0.1) 49.8 (±0.2) 83.5 (±0.4) 56.9 (±0.2) 75.2 (±0.1) 55.2 (±0.2) PCME++ + SWA 42.2 (±0.0) 51.2 (±0.1) 86.6 (±0.5) 62.9 (±0.1) 79.6 (±0.1) 61.3 (±0.1)ECCV Caption CxC COCO Backbone Method mAP@R R-P R@1 R@1 1K R@1 5K R@1 ViT-B/32 (151M) VSE∞ 39.7 (±0.5) 49.3 (±0.3) 82.1 (±2.0) 55.4 (±3.3) 74.2 (±2.7) 53.6 (±3.4) InfoNCE 41.1 (±0.1) 50.4 (±0.1) 84.8 (±0.3) 60.9 (±0.4) 78.3 (±0.4) 59.3 (±0.3) PCME 39.1 (±0.1) 48.9 (±0.1) 81.4 (±0.4) 54.7 (±0.2) 73.8 (±0.1) 53.0 (±0.3) PCME++ 40.1 (±0.1) 49.8 (±0.1) 83.0 (±0.6) 56.6 (±0.1) 75.1 (±0.2) 54.8 (±0.1) ViT-B/16 (150M) VSE∞ 41.7 (±0.2) 50.6 (±0.2) 86.3 (±0.5) 62.3 (±0.1) 79.1 (±0.3) 60.7 (±0.1) InfoNCE 39.0 (±0.3) 48.7 (±0.2) 81.7 (±1.0) 54.9 (±0.2) 74.0 (±0.3) 53.0 (±0.1) PCME 41.0 (±0.3) 50.3 (±0.3) 84.3 (±0.2) 59.9 (±0.8) 77.8 (±0.4) 58.2 (±0.8) PCME++ 42.1 (±0.1) 51.1 (±0.1) 86.5 (±0.4) 62.6 (±0.1) 79.4 (±0.1) 61.1 (±0.3) ViT-L/14 (428M) VSE∞ 20.2 31.5 46.2 24.3 44.5 22.7 InfoNCE L/14 35.6 45.8 75.6 48.0 69.5 45.9 PCME 41.2 50.3 86.0 63.4 80.3 61.9 PCME++ 42.0 50.8 88.2 66.1 81.8 64.4 t , i.e., z 1 t ≈ z 2 t ≈ z 3 t ≈ z ′ t ,as well as the probabilistic textual embeddings Z 1 t ≈ . . . ≈ Z ′ t . def compute_loss ( v_mu , v_sig , t_mu , t_sig , matched ) : Devise: A deep visual-semantic embedding model. 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[ "Temporal Aware Mixed Attention-based Convolution and Transformer Network (MACTN) for EEG Emotion Recognition", "Temporal Aware Mixed Attention-based Convolution and Transformer Network (MACTN) for EEG Emotion Recognition" ]	[ "Member, IEEEXiaopeng Si ", "Dong Huang ", "Yulin Sun ", "Senior Member, IEEEDong Ming " ]	[]	[]	Emotion recognition plays a crucial role in human-computer interaction, and electroencephalography (EEG) is advantageous for reflecting human emotional states. In this study, we propose MACTN, a hierarchical hybrid model for jointly modeling local and global temporal information. The model is inspired by neuroscience research on the temporal dynamics of emotions. MACTN extracts local emotional features through a convolutional neural network (CNN) and integrates sparse global emotional features through a transformer. Moreover, we employ channel attention mechanisms to identify the most task-relevant channels. Through extensive experimentation on two publicly available datasets, namely THU-EP and DEAP, our proposed method, MACTN, consistently achieves superior classification accuracy and F1 scores compared to other existing methods in most experimental settings. Furthermore, ablation studies have shown that the integration of both self-attention mechanisms and channel attention mechanisms leads to improved classification performance. Finally, an earlier version of this method, which shares the same ideas, won the Emotional BCI Competition's final championship in the 2022 World Robot Contest.	10.48550/arxiv.2305.18234	[ "https://export.arxiv.org/pdf/2305.18234v1.pdf" ]	258,959,385	2305.18234	97cbe4c12e33d45bf32f2383709064c63c9b71a4	Temporal Aware Mixed Attention-based Convolution and Transformer Network (MACTN) for EEG Emotion Recognition Member, IEEEXiaopeng Si Dong Huang Yulin Sun Senior Member, IEEEDong Ming Temporal Aware Mixed Attention-based Convolution and Transformer Network (MACTN) for EEG Emotion Recognition JOURNAL OF L A T E X CLASS FILES, VOL. 14, NO. 8, NOVEMBER 2022 1Index Terms-Emotion recognitionelectroencephalographyattentiontransformer ! Emotion recognition plays a crucial role in human-computer interaction, and electroencephalography (EEG) is advantageous for reflecting human emotional states. In this study, we propose MACTN, a hierarchical hybrid model for jointly modeling local and global temporal information. The model is inspired by neuroscience research on the temporal dynamics of emotions. MACTN extracts local emotional features through a convolutional neural network (CNN) and integrates sparse global emotional features through a transformer. Moreover, we employ channel attention mechanisms to identify the most task-relevant channels. Through extensive experimentation on two publicly available datasets, namely THU-EP and DEAP, our proposed method, MACTN, consistently achieves superior classification accuracy and F1 scores compared to other existing methods in most experimental settings. Furthermore, ablation studies have shown that the integration of both self-attention mechanisms and channel attention mechanisms leads to improved classification performance. Finally, an earlier version of this method, which shares the same ideas, won the Emotional BCI Competition's final championship in the 2022 World Robot Contest. INTRODUCTION E MOTIONS are crucial for human daily life, as they directly influence people's judgment, memory, behavior and social interaction [1]. Emotion recognition is essential for human-computer interaction, as it enables machines to perceive human affective states and make them more "empathetic" in their interactions with humans. Negative emotions may lead to serious brain disorders such as mental illness [2], and emotion recognition can also facilitate doctors' assessment of psychological health and disorders such as autism [3]. Furthermore, emotion recognition plays a key role in cognitive behavioral therapy [4], emotion regulation therapy/emotion-focused therapy [5], [6]. Facial expressions, language, gestures, physiological signals and other modalities are often used for emotion recognition. Based on the type of data, emotion recognition methods can be divided into two categories: one is based on nonphysiological signals, such as facial expression images, body posture and voice signals; the other is based on physiological signals, such as electroencephalogram (EEG), electromyogram (EMG), electrocardiogram (ECG). Among various types of physiological signals, EEG signal is one of the most commonly used ones. It captures directly from the cerebral cortex, thus it is advantageous for reflecting human mental states. EEG also has high temporal resolution, noninvasiveness and high sensitivity to external stimuli [7]. In recent years, EEG-based emotion recognition has attracted wide attention from researchers. Duan et al. [8] • X. Si and D. Huang contributed equally to this work. used differential entropy (DE) features of EEG and support vector machine (SVM) to recognize emotions. Chen et al. [9] proposed a Personal-Zscore feature processing method to process the extracted statistical features, time-frequency features and so on, which improved the accuracy of SVM for emotion recognition. Recent studies have shown that deep learning has achieved good results in EEG classification tasks, such as motor imagery [10], [11], emotion recognition [12], [13], epilepsy detection [14], [15], sleep staging classification [16], [17] and so on. Yang et al. [12] designed a hierarchical network structure and used key subnetwork nodes to improve the performance of emotion recognition. Li et al. [13] mapped the DE features extracted from EEG to a two-dimensional brain space and used a hierarchical convolutional neural network (HCNN) to extract the emotional representation of EEG in the two-dimensional space. Although many machine learning methods have been proposed for emotion recognition, most of them rely heavily on handcrafted features. Convolutional neural networks (CNNs) and attentionbased Transformers have the ability to learn features directly from samples. CNNs have strong local feature extraction capabilities and achieve good results in EEG classification tasks. Schirrmeister et al. [18] proposed deep convolutional neural networks and shallow convolutional neural networks (DeepConvNet and shallow convnet) to process EEG data and obtained better classification performance. Lawhyn et al. [19] proposed EEGNet, which extracts spatial and temporal features by using one-dimensional convolutional kernels on different scales. Transformers can focus on sparse global features and process sequential data such as EEG in parallel, thus attracting the attention of BCI researchers. Moreover, studies have shown that emotions are dynamic, changing in intensity, duration and other attributes [20], and their dura-arXiv:2305.18234v1 [eess.SP] 18 May 2023 tion is highly variable, ranging from a few seconds to several hours or even longer [21]. The Transformer model based on self-attention mechanism can fully consider the temporal dynamics of emotions by assigning different weights to sampling points/segments on the time dimension. Wei et al. [22] transformed EEG into time-frequency domain by wavelet transform and recognized it by capsule (or windowed) Transformer. Peng et al. [23] mapped EEG to twodimensional space and used Transformer for classification after word encoding of two-dimensional space EEG. Considering that CNNs and Transformers have the ability to extract features at different scales, Sun et al. [24] mapped EEG to two-dimensional space and combined 3D CNN with Transformer for emotion recognition. And EEG contains rich spatial information (EEG channels correspond to different brain regions), channel attention as a part of attention mechanism has received researchers' attention. Zhang et al. [25] used cascaded self-attention to extract emotional information on the time dimension hierarchically, using Squeeze-and-Excitation(SE) channel attention mechanism to select task-related channels. Tao et al. [26] proposed an attention-based convolutional recurrent neural network that combines SE channel attention with self-attention to extract more features to improve emotion recognition performance. Although many methods for emotion recognition combining CNNs with Transformers have been proposed, most of them are focused on subject-dependent random-shuffle emotion recognition rather than the more challenging crosssubject and cross-session emotion recognition scenarios [27]. Generally, EEG-based emotion recognition can be divided into subject-dependent emotion recognition and cross-subject emotion recognition. The former relies on building individual classification models for each subject, while the latter relies on the generalization of the model, eliminating the need to build independent classification models for each subject. Hu et al. [28] suggested that there are individual differences in emotions. Although cross-subject emotion recognition has lower accuracy than subject-dependent emotion recognition, it has gained widespread attention due to its higher practicality. Song et al. [29] proposed a new dynamic graph convolutional neural network (DGCNN) model for emotion recognition, which models multi-channel EEG features using a graph. Shen et al. [30] proposed a model called CLISA, which constructs a loss function based on contrastive learning and trains the model using a combination of time and spatial convolutions. The model achieved an accuracy of 47% on the THU-EP dataset. Since EEG signals from the same subject have non-stationarity across different time periods [31], and current emotion induction experiments mainly use video induction, subject-dependent emotion recognition can be divided into cross-trial emotion recognition and random shuffling emotion recognition (random shuffling can also be used on the entire dataset). Ding et al. [32] proposed a multi-scale temporal and spatial convolutional neural network called TSception to capture the temporal dynamics and spatial asymmetry of EEG for cross-trial emotion recognition. Furthermore, researchers are actively exploring the interpretability of deep learning methods. Lawhern et al. [19] have granted the model a certain level of interpretability by visualizing temporal and spatial convolutions. In the aforementioned work, CLISA and TSception also endeavored to offer explanations for the proposed models. However, there are few studies that provide interpretability for methods based on attention mechanisms. To address the above issues, this paper proposes a model called Mixed Attention based Convolution and Transformer Network (MACTN) that combines CNN and Transformer for cross-trial and cross-subject emotion recognition. EEG signals are directly fed into MACTN, which is an end-toend deep learning method that requires minimal domain knowledge for feature engineering. Inspired by the Vision Transformer [33], MACTN extracts local temporal features through 1D convolutions applied to the time dimension. Grouped convolutions map signals from different EEG channels to more feature channels for feature encoding. The Selective Kernel (SK) channel attention module selects channels that are most relevant to the task with different temporal convolution kernels, and the self-attention mechanism is used to extract global sparse emotion features. Experiments on two publicly available benchmark datasets, Emotion Profile (THU-EP) [28] and Database for Emotion Analysis using Physiological signals (DEAP) [34], were conducted to evaluate the performance of MACTN. MACTN was compared with several advanced deep and non-deep methods in the BCI field, and in most experiments, MACTN achieved higher accuracy and F1 scores than other methods. Specifically, an earlier version of MACTN with the same concept won the championship in the Brain-Controlled Robot Contest at the 2022 World Robot Contest. A ablation study was conducted to analyze the contribution of each module in MACTN. In addition, we provide some visualization methods to explain the model, such as feature map visualization, convolution kernel visualization, channel attention visualization, and self-attention weight visualization combined with stimuli materials. These methods intuitively illustrate the working principle of the proposed model. The major contribution of this work can be summarised as: • We propose MACTN, a hybrid model based on CNN and Transformer. MACTN extracts local temporal features through convolution and integrates them using channel attention with different kernel sizes. Finally, MACTN uses self-attention mechanism to extract global, sparse emotional features. • We conducted extensive experiments on the THU-EP and DEAP datasets. In most of the experiments, MACTN achieved higher accuracy and F1 scores than other methods. • Due to the large amount of data in the THU-EP dataset, which includes data from 80 participants, we conducted extensive ablation studies and interpretability experiments on THU-EP to understand the importance of each module in MACTN and demonstrate its working principles. Finally, we explored the relationship between data quantity and emotion recognition tasks. The organization of the remaining parts of this paper is as follows. Section 2 provides a detailed description of our proposed model. Section 3 describes the dataset and experimental settings. Section 4 presents the results and analysis. We discuss the results in Section 5, and conclude the paper in Section 6. PROPOSED MODEL In this section, we specify the proposed model framework, referred to as MACTN, as shown in Figure 1(a). To fully extract the dynamic features of emotions, MACTN is hierarchically divided into two parts, namely (1) the local temporal feature extractor as well as a channel attention part and (2) the global temporal feature extractor part. In the following, these parts will be discussed in detail. Local Temporal Feature Extractor Local temporal feature extractor (LTFE) contains three subblocks, which are (i)depth convolution block ( denoted as Depth Conv-Block or D-Conv Block), (ii)separable convolution block ( denoted as Separable Conv-Block or S-Conv Block) and (iii)channel attention computation block called SK attention [35], and the structure of SK attention is shown in Fig. 1 (c). (i) Depth conv-block The Depth conv-block is mainly based on depth-wise temporal convolution 1D (or D-Conv), which consists of five layers, two D-Conv layers, a batch norm (BN) layer, a ReLU activation function layer, and a dropout layer. We use depth-wise convolution to apply one and more filters for each input channel (input depth). The first D-Conv layer is calculated as follows: H denotes the feature obtained by convolution operation, H (1) ∈ R K2×T1 , W (1) denotes the weight of the filter, W (1) ∈ R K2×P . The length of the filter is P , the number of filters is K 2 , K 2 = C 1 × M , X represents the input raw EEG segment, X ∈ R M ×T , M represents the number of channels of EEG signal. The calculation of the first D-Conv layer can be reduced to H = DConv 1 (X). The second D-Convolution is similar to the first D-Convolution and is formulated as follows: H (1) k2,t = WH (2) k2,t = W (2) k2 H (1) k2,t:t+P −1 ; t = 1, 2, . . . , T ; k 2 = 1, 2, . . . , K 2(2) Compared with the first D-Conv layer, the number of input feature maps is the same as the number of output feature maps instead of being extended by a factor of C 1 . The calculation of the second D-Conv layer can be simplified as H (2) = DConv 2 (H (1) ). After two D-Conv layers, the feature maps are applied with nonlinear activation layer, BN layer, and dropout layer operations, and these calculations can be expressed using δ(•). Thus the calculation of the depth conv-block can be expressed by the following equation: H DConv = δ (DConv 2 (DConv 1 (X)))(3) (ii) Separable conv-block The separable conv-block is mainly based on separable temporal convolution 1D (or S-Conv), which contains five layers, two S-Conv layers, a BN layer, a ReLU layer, and a dropout layer. The whole separable conv-block will be repeated N times, called S-Conv block S i , S i = 1, 2, ..., N . The S-Conv layer is a combined layer, and each S-Conv layer contains two convolutions, the first convolution is D-Conv and the second convolution is a pointwise convolution. The pointwise convolution is calculated as follows: H (3) k2,t = W (3) k2 H (3) k2,t ; t = 1, 2, . . . , T ; k 2 = 1, 2, . . . , K 2(4) The computation of the first S-Conv layer can be simplified as H (3) = SConv 1 (H (2) ). Similar to the depth convblock, BN, ReLU, and dropout layers are added after every two S-Conv layers, so the computation of the S-Conv block S i can be formulated as: H Si SConv = δ SConv Si 2 SConv Si 1 H Si−1 SConv(5) (iii) Channel attention Our model uses SK attention, and the channel attention sub-block can also be called the SK attention sub-block. Compared with the depth conv-block and separable convblock, the SK attention sub-block uses several convolutions with smaller kernel sizes and uses the attention mechanism to select the feature maps computed by different convolutions and fuse them. Specifically, we implement the SK convolution by three operators -(A) Split, (B) Fuse, and (C) Select, as illustrated in Fig. 1 (c), taking two streams as an example. (A) Split: Given the input x, x ∈ R K2×T , x is the feature map computed after separable conv-block with average pooling layer, K 2 represents the number of features or the number of channels in the feature map, T represents the length of time. Use two transforms,F conv : x −→Û ∈ R K2×T andF conv : x −→Ũ ∈ R K2×T , to obtainÛ andŨ , it is noted that F conv includes convolutional, BN and ReLU layers. (B) Fuse: The basic idea of Fuse is to use gates to control the flow of information from multiple streams to the next layer of neurons, which carry information of different scales. To achieve this goal, it is first necessary to integrate multi-stream information, which is achieved by element-byelement summation of: U = U + U(6) We then embed the global information by simply averaging over the temporal dimensions to generate the channel statistics as follows: S = F gp (U ) = 1 T T i=1 U (i)(7) where S ∈ R K2 , and then compress the information through the fully connected layer, the operation is noted as F f c : z = F fc (S) = W S(8) in the equation (8), z ∈ R d , W is the weight of information compression and W ∈ R d×K2 , to control the value of d in Split X � � U V Select a b Fgp Ffc � Kernel=3 � Kernel=5 Softmax EEG Signal Segment (Size=30×1750) D-Conv(15)×120 D-Conv(15)×120 BN ReLU Dropout S-Conv(15)×120 S-Conv (15) order to study its effect on the model efficiency, we use the scaling ratio r. L is a constant value. d = max(K2/r, L)(9) (C) Select: Cross-channel soft attention is used to adaptively select information at different scales, which is guided by the compact feature descriptor z. To be specific, the channel information on the compact channel feature z is extracted using the fully connected layer and the softmax operator. a = e Waz e Waz + e W b z , b = e W b z e W a z + e W b z(10) a, b represent the attention vectors ofÛ andŨ . W a , W b represent the select weights, W a ∈ R K2×d , W b ∈ R K2×d , respectively. The final feature map v is obtained by vectors of attention weights for different kernels: v = a U + b U(11) Global Temporal Feature Extractor Global temporal feature extractor (GTFE) consists of several transformer encoders, Fig. 1(b) shows the structure of the transformer encoder. Besides, GTFE includes operations such as adding a learnable class token, position encoding, and extracting the class token. The transformer encoder mainly consists of two modules, multi-headed self-attentive (MHSA) and feedforward network (MLP), where layer normalization (LN) is applied before each module, and residual connectivity is applied after each module. MHSA can be described as using three matrices, query (Q), key (K), and value (V), to calculate scaled dot-product attention among them, and we calculate the output as: A = Attention(Q, K, V ) = softmax QK T √ d k V(12) where 1/ √ d k denotes the dimensional deflation factor and d k denotes the dimensionality of query and/or key. It is beneficial to linearly project the query, key, and value h times to the d k , d k , and d v dimensions using different learned linear projections. Multi-head self-attention can be represented as: the projectors are the parameter matrices MHSA(x) = concat (A 1 , A 2 , . . . , A h ) W O , where A i = Attention xW Q i , xW K i , xW V i (13)W Q i ∈ R (dseq+1)×d embed ×d k , W O ∈ R (dseq+1)×hdv×d embed , d k = d v . The calculation of the feedforward network MLP can be expressed as follows: MLP(x) = δ xW M LP 1 + b M LP 1 W M LP 2 + b M LP 2(14) Where the transformation parameter matrix of the feedfor- ward network W M LP 1 ∈ R (dseq+1)×d embed ×d M LP , W M LP 2 ∈ R (dseq+1)×d M LP ×d embed . The computed output of the entire transformer encoder is: T E i = MLP LN SA i + SA i where SA i = MHSA LN T E i−1(15) where the transformer encoder output of layer i is T E i and the self-attentive output is SA i . In the paper, a learnable encoding is added to the sequence element by element as a position encoding before feeding into the transformer encoder, as in BERT [36]. EXPERIMENT Dataset In order to evaluate MACTN, we conducted several experiments on two public datasets, one being the Tsinghua University Emotional Profiles (THU-EP) and the other being the Database for Emotion Analysis using Physiological signals (DEAP) [34]. The THU-EP database includes 80 college students (50 females and 30 males) with an average age of 20.16 years old ranging from 17 to 24 years old. There are 28 video clips used as stimuli, which include 9 different emotions: anger, disgust, fear, sadness, amusement, joy, inspiration, tenderness, and neutral. Except for the 4 video clips used to induce neutral emotions, each of the other emotions was induced using 3 video clips. Each subject viewed the video clips in seven blocks, with each block containing four trials. Participants were asked to solve 20 arithmetic problems between two blocks [37]. NeuSen.W32 wireless EEG system was used to record EEG signals with a sampling frequency of 250Hz. The positions and names of the 32 channels are shown in Fig. 2. DEAP is a human emotional dataset that contains multiple physiological signals, including EEG and galvanic skin response (GSR). The dataset consists of 32 participants (17 males and 15 females) with an average age of 27.19 years ranging from 19 to 37 years old. 40 music videos, each lasting one minute, were used as stimuli. Each subject completed 40 trials, and after each trial, they were asked to rate their emotional state using four dimensions: arousal, valence, dominance, and liking, each rated on a discrete scale ranging from 1 to 9. The EEG was recorded using a 32-channel device at a sampling frequency of 512 Hz. Pre-process The same preprocessing methods were applied to each trial for THU-EP. Firstly, a bipolar re-referencing method was used as shown in Fig. 2. The electrodes from the left and right mastoids were discarded (as the left and right mastoid signals were not recorded in the BCI competition), and the remaining 30 channels were reconfigured into 30 montages by subtracting each channel pairwise. Montage 21 and Montage 28 contained signals that were symmetrical about 0 uV. Secondly, each EEG segment was extracted using a sliding window approach with a window length of 14 seconds and a step size of 4 seconds. Thirdly, a 6th order Butterworth filter was used for filtering, which included a 50 Hz notch filter (with a notch width of 48-52 Hz) and a bandpass filter of 0.5-45 Hz. Fourthly, to simplify the subsequent calculations, the EEG signals were downsampled to 125 Hz. Fifthly, a Z-score normalization method was used to normalize each segment. After preprocessing, each subject had a total of 363 EEG segments (min=362, max=366, mode=363). For DEAP, the EEG data from each trial were downsampled to 128Hz, eye artifacts were removed, and bandpass filtering was performed using a 4-45Hz filter. The filtered signals were then re-referenced using common average referencing. The DEAP dataset provides signals that have been processed according to the above steps. We partitioned each trial into 13 segments using a sliding window approach with a window length of 12 seconds and a step size of 4 seconds. Following previous work, we converted the ratings for arousal and valence dimensions into binary labels using a threshold of 5. We extracted 28 channels based on the TSception setup and rearranged them accordingly [32]. Implement Details In the LTFE of the model, all operations are 1-dimensional (e.g., convolution, BN, pooling, etc.), the size of the convolution kernels are all 15, the number is fixed at 120 (i.e., K 2 = 120 in THU-EP or K 2 = 120 in DEAP) in THU-EP and 112 in DEAP. in Depth ConvBlock, C 1 is set to 4, i.e., in Step No. 2 of Table 1, four separate convolutions are used to process the data for each EEG channel. Depth ConvBlock does not use padding for all convolutions, and after two convolutions, the output feature map is reduced by 28 in the temporal dimension. After Depth ConvBlock, we use an average pooling layer to compress the temporal features (i.e., Stem No. 4) with a pooling size of 4. Unlike Depth ConvBlock, Separable Convolution in ConvBlock uses padding to ensure that the temporal dimension is invariant before and after the computation. Similarly, after Separable ConvBlock, we compress the features using an average pooling layer with a pooling size of 5. After the average pooling, SK attention was used in which we used four convolutions with smaller kernel sizes, 1, 3, 5, and 7, to extract finer-grained temporal features and integrate them, with the compression ratio r set to 4 and L set to 32 in keeping with SKNet [35]. In GTFE, the learnable class token and position embedding are initialized using standard Gaussian distribution. We use six layers of transformer encoder, and in each layer of transformer encoder, we set the number of multi-head self-attention heads to 8 and set the dimensions of Q, K, and V to d k = d v = 256. It is noted that d k is not equal to d embed /h, and d embed is 120 (THU-EP) or 112 (DEAP), while d seq is 86 (THU-EP) or 75 (DEAP). The dimension of the internal layer of the feedforward network is set to 128. In the experiments, the optimizer applied is AdamW optimizer [38], the initial learning rate is set to 0.001, the learning rate decay strategy is ReduceOnPlateau, i.e., the loss decays to 10% of the original after ten epochs without decreasing, the weight decay is set to 0.0001, the batch size is 16, the number of epochs is 100, and the early stopping strategy is applied. At the same time, we use the flooding strategy [39] to prevent the model from overfitting and set its parameter b to 1.3. Our model is implemented in Python 3.8 using PyTorch 1.10. The model is configured to run on a GeForce RTX 3090 GPU. The entire model code can be accessed on GitHub 1 . Performance Evaluation For THU-EP, we employed three different cross-validation methods: 10-fold cross-subject-validation (10F-CSV), leaveone-subject-out cross-validation (LOSO), and leave-onetrial-out cross-validation (LOTO). To facilitate comparison, the data partitioning and performance evaluation of 10F-CSV were kept consistent with Shen et al. [30], where some subject data were used for training and the remaining data were used for testing. Each subject in THU-EP participated in 28 trials, and LOTO involved selecting one trial for testing while using the other 27 trials for training. This process was repeated 28 times to ensure that all 28 trials were tested. To investigate the impact of the number of subjects on the performance of MACTN, we added the LOSO method. Unlike the 10F-CSV method, we divided all subjects into three sets: train, validation, and test. One subject was used for testing, and the data of the remaining subjects were randomly divided into 80% and 20%. For example, with 80 subjects, the number of subjects in the training set, validation set, and test set were 63, 16, and 1 (80 = 63 + 16 + 1). This process was repeated 80 times. For DEAP, we adopted two methods for cross-validation: 10-fold cross-trial-validation (10F-CTV) and leave-onesubject-out cross-validation (LOSO). To facilitate comparison, the data partition and performance evaluation of 10F-CTV were consistent with Ding et al. [32], with 40 experiments, in which 4 were used for testing and the other 36 were split into training and validation sets based on the proportion of 80% and 20% of the segments, repeated 10 times. The data partition of LOSO was consistent with Zhang et al. [40], with one subject used for testing and the remaining subjects used for training, repeated 32 times. The evaluation metrics for model performance are accuracy (ACC) and F1 score (F1). MACTN Feature Explainability In this paper, we use four methods to explain our model based on the THU-EP dataset. These methods are as follows: 1) feature map visualization, 2) convolution kernel visualization, 3) channel attention weight visualization, and 4) self-attention weight visualization, where the fourth method is dedicated to explaining the interpretability of individual segments. 1) Feature map visualization: We use T-SNE to reduce the dimensions of specific feature maps and create a scatter plot to display the topological relationships between each feature map. 2) Convolution kernel visualization: This method focuses on directly visualizing and interpreting the convolution kernel weights from the model. In our proposed model, we use temporal convolutions, and due to the limited connectivity of the convolution layers (using depthwise and separable Montage1 Fp1-F7 Montage16 T8-P8 Montage2 Fp2-F8 Montage17 CP1-Pz Montage3 Fz-Cz Montage18 CP2-Pz Montage4 F3-C3 Montage19 CP5-P3 Montage5 F4-C4 Montage20 CP6-P4 Montage6 F7-T7 Montage21 Pz-Oz Montage7 F8-T8 Montage22 P3-PO3 Montage8 FC1-CP1 Montage23 P4-PO4 Montage9 FC2-CP2 Montage24 P7-O1 Montage10 FC5-CP5 Montage25 P8-O2 Montage11 FC6-CP6 Montage26 PO3-O1 Montage12 Cz-Pz Montage27 PO4-O2 Montage13 C3-P3 Montage28 Oz-Pz Montage14 C4-P4 Montage29 O1-Oz Montage15 T7-P7 Montage30 O2-Oz Fig. 2. Bipolar re-referencing montage. convolutions), temporal convolutions can be interpreted as narrowband frequency filters. Additionally, smaller kernels between adjacent convolution layers can be equivalently represented as larger convolutions using a calculation formula to extract richer frequency information. 3) Channel attention weight visualization: This method focuses on directly visualizing and interpreting the attention weights of the channel dimension to display the model's preference for specific channels. Similarly, due to the limited connectivity of the convolution layers in our proposed model, each original EEG channel corresponds to several (C 1 ) feature channels after convolution calculation. Therefore, averaging every C 1 feature channel can demonstrate the model's preference for specific EEG channels and corresponding brain regions through channel attention visualization. 4) Self-attention weight visualization: This method focuses on directly visualizing and interpreting the attention weights of the temporal dimension. In our proposed model, we use a temporal transformer that can focus on different emotional information on a global temporal scale and assign higher attention weights. Notably, these high-weight assignments are sparse, allowing the model to provide certain interpretability when combined with the original stimuli. RESULTS In this section, we report and compare our results in terms of accuracy and F1 score with state-of-the-art methods. We then conduct an ablation study to reveal the role of each component in MACTN. Emotion Recognition Performance We compared our proposed method with the state-of-theart (SOTA) methods that also applied the THU-EP dataset. Table 2 shows the 10-fold cross-subject-validation results on THU-EP, where our method achieved the best performance with an 11.6% higher accuracy than the CLISA method proposed by Shen et al. [30]. Moreover, we compared our method with other convolutional neural networkbased models, including DeepConvNet [19], ShallowCon-vNet [19], EEGNet [19], and TSception [32], in terms of accuracy and F1 score. Our proposed method outperformed these models by 20% in the non-transfer learning setting, and it is worth noting that some parameter settings of these models are consistent with their proposers. Among the many results, we observed that the performance of these models was lower than that of CLISA, whose model complexity was similar to other models. We found that increasing the model complexity of DeepConvNet could still improve its performance. Table 3 shows the leaveone-trial-out cross-validation results on THU-EP, where our proposed MACTN method achieved 62.5% accuracy and 61.6% F1 score. Compared to DeepConvNet, ShallowCon-vNet, EEGNet, TSception, our method achieved an 8% accuracy improvement. Compared to SVM and KNN, MACTN surpassed their accuracy by more than 20%. In DEAP, we analyzed based on the F1 score. Compared to accuracy, F1 score is a more reliable metric that can quantify the performance of classification methods when there are imbalanced classes in the dataset. Looking at Table 4, deep learning methods are generally superior to non-deep learning methods, especially MACTN which outperforms by about 7% in F1 score. MACTN achieved a arousal F1 score of 65.3% and a valence F1 score of 65.2%. TSception method ranks second among other methods with arousal F1 score of 64.1% and valence F1 score of 64.6%, which is about 7% better in F1 score. Among the compared deep learning methods, MACTN leads TSception by 1.2% in arousal F1 score and 0.6% in valence F1 score. Compared to other methods such as DeepConvNet, ShallowConvNet, EEGNet, MACTN outperforms by about 3% in F1 score. The results of the LOSO experiment ( Table 5) also show that deep learning models outperform non-deep learning models in performance. Compared to non-deep learning models, MACTN improves by 6% in arousal F1 score and 11% in valence F1 score. Compared to other deep learning models, MACTN improves by 1% in arousal F1 score and 4% in valence F1 score. According to extensive comparisons with various methods, this approach has demonstrated excellent performance in emotion recognition tasks and exhibits a certain degree of generality. Ablation Study and Parameter Analysis To investigate how our model achieves such performance, we conducted various ablation experiments on the proposed model. Table 6 examines the contributions of the LTFE and GTFE blocks to the model's performance improvement. The experimental results show that without LTFE, the model only achieved an accuracy of 22%, while without GTFE, the accuracy decreased by 22.4% compared to the model with both blocks. This suggests that LTFE plays a more significant role in feature extraction and representation, while the lower performance of the model with only GTFE may be due to the transformer's limited ability to handle longer sequence inputs. Furthermore, we investigated which sub-block in LTFE has a greater impact on the model's performance. In this ablation study, we retained GTFE and compared the models formed by removing any one of the three sub-blocks present in LTFE. The results in Table 7 indicate that removing the Separable Conv-Block results in the maximum decrease in accuracy, by 16.9%, compared to when all three subblocks are present. Removing the SK attention, on the other hand, only leads to a 2.4% decrease in accuracy. The model without Depth Conv-Block achieves an accuracy of 46.2%. It is noteworthy that the Depth Conv-Block is responsible for amplifying the feature channel dimension, and thus removing it results in feature maps with the same channel dimension as the preprocessed EEG channel dimension. We found that the length of EEG segments can affect the model's accuracy and F1 score. Fig. 3 shows that, with a fixed preprocessing pipeline and sliding window stride of 4 seconds, the model's accuracy consistently increases with the window length. However, the rate of change in accuracy slows down after a certain window length. After conducting a Wilcoxon sign rank test, the model accuracy for input windows of 14 seconds was significantly different from those of 12, 10, 8, 6, and 4 seconds, with p-values less than 0.05. In contrast, the accuracy of the 14-second input was not significantly different from those of 16 and 18 seconds. At the same time, we computed the model's computational complexity (GFLOPs) under different window lengths, and Fig. 3 shows a linear relationship between the window length and computational complexity. . Accuracy was tested using the Wilcoxon signed-rank test, with * representing p < 0.05 and n.s. representing not significant. The THU-EP dataset has a sufficient number of subjects that enables the model to learn individual differences between subjects and narrow them down. We were curious about how the number of subjects affects the performance of MACTN, so we conducted experiments by varying the number of subjects as the independent variable. We selected 10,20,30,40,50,60,70, and all subjects from a pool of 80 subjects and performed leave-one-subject-out crossvalidation experiments. The data partitioning and evaluation methods are described in detail in Section 3. Fig. 4 shows the relationship between the number of subjects and the performance of MACTN. As observed from the figure, the accuracy of MACTN increases with an increase in the number of subjects, but the rate of increase in accuracy slows down after a certain point (n=50). Moreover, when the number of subjects is 60, the accuracy of MACTN remains around 55%. The accuracy of MACTN for 70 subjects is only 0.05% higher than that for 60 subjects, and the accuracy for 80 subjects is 55.42%. Thanks to such a large dataset, this work has the potential to establish a highperformance and robust emotion model, thereby expanding the range of applicable populations for emotional braincomputer interfaces. DISCUSSION Impact of Dataset Size on Classification Performance Explainability Analysis In order to investigate the topological changes of the samples after being processed by LTFE and GTFE, we selected two parts of the feature maps for visualization, namely the ones that have undergone LTFE and GTFE respectively. We utilized T-SNE to reduce the dimensionality of the feature maps, and displayed them in a scatter plot. In Fig. 5, each point represents a sample, that is, an EEG segment or a feature map obtained through feature extraction. Each row in Figure 4 represents a subject. From left to right, the inter-class and intra-class distances between preprocessed EEG segments are relatively large. After LTFE, the distances between samples in both intra-class and inter-class have been reduced, while after GTFE, the inter-class distances between samples have been enlarged while the intra-class distances remain unchanged. Through ablative studies, we found that the Separable Conv-Blocks in LTFE have a significant impact on model accuracy. Therefore, we visualized the convolutional kernels within these sub-blocks. Each subplot in Fig. 6 displays the learned temporal kernels for a 0.232-second window. The results indicate that the convolutional filters located after the processing stream extract lower frequency information compared to those in the front stream. This finding is consistent with other electrophysiological studies demonstrating that the human brain attends to low-frequency components when processing emotional information, such as those works by LowryK et al. [41] and Huang et al. [42]. In LTFE, we utilized the channel attention sub-block of SK attention, which mainly extracts information of different time and frequency components based on small convolutional kernels of different sizes, and integrates them according to the computed channel attention. Fig. 7 shows the topological map of the average channel attention weights in the proposed model, which is normalized between -1 and 1 for better visualization. The proposed model uses kernel sizes of 1, 3, 5, and 7, all of which are smaller than those used in previous convolutional networks. The channel attention weights of different kernel sizes can be calculated according to equation (10). Fig. 7 of 7 primarily attends to information in the temporal and occipital regions. Comparing the topological maps of the most accurate fold and the average topological map, the topological map of the most accurate fold shows a similar focus area to the average topological map. The proposed model treats a feature map of a short period as a word in the natural language processing domain, treats the feature channels as embeddings of each "word," and adds an extra learnable class token with sentiment information for each time segment. The attention of each time segment can be calculated based on the equation (12). Fig. 8 shows an EEG fragment from trial 1. Fig. 8(a) shows the screenshot of the stimulus (the stimulus mainly induces anger emotion). Fig. 8(b) shows the self-attention weight curve of the EEG fragment (the weight is normalized). Fig. 8(c) shows the waveform of this EEG fragment The caption shown in Fig. 8(a) is "According to Chinese statistics, the number of soldiers and civilians massacred in Nanjing was 300,000", and Fig. 8(a) corresponds to the part of 8(b) and 8(c) with higher attention weight. 图8 Self attention权重可视化(经过了归一化).此Segment来源于trial 1, 该trial的主要包含情绪为anger, self attention权重较高的2个时间片段分别对应了视频片段中令人愤怒的部分，如军、民被侵略者屠杀. (a) FIGURE 8: Self attention weight visualization (normalized). The Segment is derived from trial 1, which contains the main emotion of anger, and the two time segments with high self attention weights correspond to the outrageous parts of the video clip, such as the massacre of soldiers and civilians by the invaders. According to Chinese statistics, the number of soldiers and civilians massacred in Nanjing was 300,000. Temporal Feature Extractor and Attention Mechanism The reason why MACTN can achieve high performance is related to the cascaded feature extraction blocks. Consistent with most previous works [30], [43], we use CNN as a local temporal feature extractor. In many fields, especially in computer vision, CNN has demonstrated its powerful capability in local feature extraction [44]. Research has shown that the intensity of emotions varies over time. We believe that CNN can extract emotional representations with short-term high intensity. Unlike previous works, we use transformer-based attention mechanism to integrate global temporal features, which assigns higher attention weights to segments with higher emotional intensity, thus achieving higher recognition performance. By visualizing the attention weights of self-attention, we found that the parts with high emotional intensity of stimulus-induced materials correspond to high attention weights. Attention weight assignment is samplespecific, which can extract highly variable emotional representations in terms of duration. Sun et al. [14] found that the origin of epileptic seizures was near the epileptic focus by visualizing the self-attention weights, and the electrodes in those regions had higher self-attention weights. In addition to self-attention, SK attention can further improve the model's performance. By visualizing the attention weights of SK attention, it uses convolution with multiple smallsize kernels to extract features of different scales and selectively integrates features of different scales (different brain regions) by means of channel attention mechanism. Studies have shown that different brain regions have different selectivity for emotions, such as the amygdala's selectivity for negative emotions [45], and direct electrical stimulation of the orbitofrontal cortex can improve the emotional state of depressed patients [46]. Limitations and Future Work Many previous studies have shown that transfer-learningbased methods can improve the performance of crosssubject emotion recognition [47], [48], [49], and domain generalization is a method of transfer learning that can be used for cross-subject emotion recognition in the future. Panagiotis et al. [50] used frontal asymmetry-supported multidimensional directional information analysis with asymmetric indices to discard signal segments with little "emotionrelated" information and retain only valuable signal segments. Future work is expected to be more effective in stimulus analysis. In future work, it is hoped that temporal emotion localization can be achieved by assigning different temporal segments with corresponding emotional intensity information on the stimulus material. EEG contains rich temporal and spatial information, and we highly focus on temporal information in these works. In contrast, more and more works have started to focus on the spatial information of EEG, such as using graph convolutional neural networks to extract spatially relevant emotional features [51]. In future work, we hope to combine models that focus on spatial information, such as graph neural networks to extract more complete emotional features to improve the model's performance. CONCLUSIONS In this study, we propose a model based on a hybrid of CNN and transformer, inspired by research on the temporal dynamics of emotions in neuroscience. The method uses CNN to extract local features with high emotional intensity over time, integrates sparse global features over time using the self-attention mechanism in transformer, and combines channel attention mechanism to focus on features in the channel dimension. It is worth noting that this method is an end-to-end approach that does not require manual feature extraction. Extensive experiments on two public datasets, THU-EP and DEAP, show that in most experimental settings, MACTN achieves higher classification accuracy and F1 scores than other methods. Moreover, an earlier version of the method with the same concept won the championship of the 2022 World Robot Contest -Emotional BCI. In addition, we use four visualization methods to provide insights into the model, and demonstrate its potential in emotion localization over time by visualizing important attention weights. Xiaopeng Si (Member, IEEE) works as an associate Professor at Tianjin University, since 2018. He obtained Ph.D. in biomedical engineering from Tsinghua University, and finished the visiting scholar in biomedical engineering department at Johns Hopkins University. His research interest includes cognitive neuroscience and brain-inspired intelligence, neural engineering and brain-computer interaction, neural information acquisition and intelligent computing, multimodal human brain neuroimaging and regulation, etc. His team used multimodal neuroimaging and machine learning methods to understand the human cognitive processes. They tried to apply these mechanisms in people with brain disorders, and also tried to enhance normal people's ability by creating mind reading technology. Dr. Si has managed and participated many national and international research projects. He has published papers in PNAS, Cerebral Cortex, IEEE JBHI, JNE, Frontiers in Aging Neuroscience, Frontiers in Neuroscience, etc. His team won the first prize of Emotional BCI Competition during 2022 World Robot Contest, and the second prize of BCI Competition during 2020 World Robot Contest. Dong Huang received the B.S. degree in electronics information science and technology from Northeast Petroleum University in 2020. He is currently pursuing the M.E. degree in biomedical engineering, Tianjin University, Tianjin, China. His research interest includes affective computing, and deep learning. ACKNOWLEDGMENTS Yuling Sun received the B.E. degree in biomedical engineering from Tianjin University in 2020. Currently, he is working toward the M.M. degree in biomedical engineering, Tianjin University, Tianjin, China. His research interest includes seizure detection, digital signal processing, affective computing, and deep learning. • X. Si, D. Huang, Y. Sun, and D. Ming are with the Academy of Medical Engineering and Translational Medicine, Tianjin University, Tianjin 300072, China, and also with Tianjin Key Laboratory of Brain Science and Neural Engineering, Tianjin University, Tianjin 300072, China (Email:{xiaopengsi, huang dong, syuri, richardming}@tju.edu.cn). • X. Si and D. Ming are the corresponding authors. m,t:t+P −1 ; m = 1, 2, . . . , M ; t = 1, 2, . . . , T ; k 1 = 1, 2, . . . , C 1 ; k 2 = 1, 2, . . . , C 1 M (1) Fig. 1 . 1Mixed Attention based Convolution and Transformer Network (MACTN) model structure. (a) MACTN consists of 2 main parts, indicated by the green box and the blue box. The green box indicated the low-level local time feature extractor as well as the channel attention part. The blue box indicates the high-level global time feature extractor part. (b) The structure of the Transformer encoder. (c) The structure of Select Kernel (SK) attention (with two streams as an example). FIGURE 5 : 5Fold plots of MACTN for segments with different window lengths (fixed step size of 4s). Accuracy was tested using the Wilcoxon test, with * representing p < 0.05 and n.s. representing not significant.Window length (s)Fig. 3. Fold plots of MACTN for segments with different window lengths. The error bar indicates the standard deviation of the mean (SEM) Fig. 4 . 4Line chart of average accuracy of MACTN under different dataset sizes. FIGURE 5 : 5(a)-(d) are the average topological maps of all 10-fold MACTN, Fig. 7 (e)-(h) are the topologies of the most accurate fold, and Fig. 7 (i)-(l) are the topologies of the least accurate fold among the 10 folds. Based on Fig. 7 (a)-(d), the branch with a convolutional kernel size of 1 mainly focuses on information in the parietal region, the branch with a kernel size of 3 pays more attention to information in the parietal and temporal regions, the branch with a kernel size of 5 focuses on information in the frontal and temporal regions, and the branch with a kernel size 图5 使用T-SNE对特征降维.Sub A-B分别代表被试36、50. Dimensionality reduction of features using T-SNE. Sub A-B represent subjects 36 and 50, respectively. Fig. 6 . 6Visualization of Separable Conv-Block temporal convolution kernel. (a)Top row represents Separable Conv-Block 1. (a)Bottom row represents Separable Conv-Block 2. Convolution kernel A-D represents several typical feature channels. Fig. 7 . 7Channel attention weight topology map of MACTN in THU-EP. The first row (a)-(d) shows the average topology map of all 10-fold MACTN, where (a)-(d) represent the channel attention weight topology map of feature maps after convolution with different kernel sizes. The second row (e)-(h) shows the topology map of one fold MACTN with the highest accuracy. The third row (i)-(l) shows the topology map of one fold MACTN with the lowest accuracy. Fig. 8 . 8Self-attention weight visualization (normalized). This segment is derived from trial 1, which contains the emotion of anger. The time segments with higher self-attention weights correspond to the angry parts of the video segments. (a) Two screenshots from the stimuli and the descriptions corresponding to them respectively. (b) Plot of the normalized self-attention weights. (c) Waveforms of EEG signals, color bar represents normalized self-attention weights. TABLE 1 1Structure of the proposed modelChans: The number of channels in THU-EP is 30, while in DEAP it is 28.Block Sub-block Step No. Step Name Parameters THU-EP Output Shape DEAP Output Shape Local Temporal Feature Extractor 1 Input - (30, 1750) (28, 1536) Depth Conv-Block 2 Depth-wise Conv1D Size=15, Group=Chans, Num=Chans×4 (120, 1736) (112, 1522) 3 Depth-wise Conv1D Size=15, Group=Chans×4, Num=Chans×4, BN, ReLU, Dropout=0.5 (120, 1722) (112, 1508) 4 AvgPool1D Pool size=4 (120, 430) (112, 377) Separable Conv-Block 5 Separable Conv1D Size=15, Group=Chans×4, Num=Chans×4, Padding=7 (120, 430) (112, 377) 6 Separable Conv1D Size=15, Group=Chans×4, Num=Chans×4, Padding=7, BN, ReLU, Dropout=0.5 (120, 430) (112, 377) 7 Separable Conv1D Size=15, Group=Chans×4, Num=Chans×4, Padding=7 (120, 430) (112, 377) 8 Separable Conv1D Size=15, Group=Chans×4, Num=Chans×4, Padding=7, BN, ReLU, Dropout=0.5 (120, 430) (112, 377) 9 AvgPool1D Pool size=5 (120, 86) (112, 75) Channel Attention 10 SK Attention Kernel size=[1,3,5,7], Reduction=4 (120, 86) (112, 75) Global Temporal Feature Extractor 11 Reshape - (86, 120) (75, 112) 12 Extra Class Token Input shape=(1, Chans×4) (1, 120) (1, 112) 13 Concatenate & Position Encoding - (87, 120) (76, 112) Transformer 14-19 Temporal Transformer Encoder Head=8, Head dim=256, MLP dim=128 (87, 120) (75, 112) 20 Extract Class Token - 120 112 21 FC - 9 2 1. https://github.com/ThreePoundUniverse/MACTN8 CP2 Fp1 Fp2 F7 F3 Fz F4 F8 FC5 FC1 FC2 FC6 Cz C4 T8 C3 T7 CP1 CP5 CP6 Pz P3 P4 P7 P8 PO3 PO4 Oz O1 O2 A1 A2 TABLE 2 The 2accuracies (ACC) and F1 scores(mean±STD) on the THU-EP dataset (cross-subject) Model ACC (%) F1 (%) KNN 25.1 ± 4.5 25.0 ± 4.5 SVM 27.5 ± 3.1 27.3 ± 3.2 DeepConvNet 35.6 ± 4.8 34.4 ± 5.1 ShallowConvNet 31.2 ± 3.1 30.8 ± 3.2 EEGNet 33.4 ± 3.8 32.0 ± 4.1 TSception 35.8 ± 4.1 35.2 ± 4.3 CLISA [30] 47.0 ± 4.4 - MACTN (Proposed) 58.6 ± 5.7 58.7 ± 5.9 TABLE 3 The 3accuracies (ACC) and F1 scores(mean±STD) on the THU-EP dataset (cross-trial) Model ACC (%) F1 (%) KNN 40.2 ± 7.8 40.1 ± 7.9 SVM 40.1 ± 7.6 39.9 ± 7.7 DeepConvNet 49.9 ± 7.1 49.4 ± 7.2 ShallowConvNet 44.0 ± 5.9 43.1 ± 5.4 EEGNet 49.6 ± 6.8 49.1 ± 6.9 TSception 54.3 ± 8.0 53.1 ± 7.7 MACTN (Proposed) 62.5 ± 7.1 61.6 ± 7.0 TABLE 4 4The accuracies (ACC) and F1 scores(mean±STD) on the DEAP dataset (cross-trial) Model Arousal Valence ACC (%) F1 (%) ACC (%) F1 (%) KNN 60.2±12.3 58.3±25.0 54.3±9.2 56.3±16.2 SVM 61.2±12.3 58.4±26.6 56.3±7.0 58.8±11.3 DeepConvNet 62.0±8.6 62.6±17.4 60.7±7.9 63.2±11.6 ShallowConvNet 62.1±10.3 62.2±20.1 60.2±8.3 63.1±10.3 EEGNet 59.1±8.6 61.5±15.2 55.7±8.2 58.7±10.5 TSception 63.0±11.0 64.1±16.6 61.6±8.6 64.6±10.2 MACTN (Proposed) 63.6±10.5 65.3±16.1 61.9±8.2 65.2±9.8 TABLE 5 The accuracies (ACC) and F1 scores(mean±STD) on the DEAP dataset (cross-subject) Model Arousal Valence ACC (%) F1 (%) ACC (%) F1 (%) KNN 59.3±12.6 57.2±25.0 54.1±8.3 50.2±20.4 SVM 60.1±12.5 57.8±26.6 55.6±7.9 51.2±22.1 DeepConvNet 65.0±10.0 62.5±23.6 56.3±8.1 45.9±24.5 ShallowConvNet 66.2±13.2 60.9±25.3 61.6±6.7 58.5±16.6 EEGNet 60.6±9.2 60.9±24.5 57.1±7.1 51.3±19.9 DGCNN [40] 61.1 - 59.3 - TSception 63.1±8.8 60.1±22.0 63.6±7.8 60.5±21.7 MACTN (Proposed) 67.8±8.1 63.4±22.3 66.1±6.1 62.2±18.6 TABLE 6 6For the results of ablation experiments applying local and global feature extractor Module Metrics Local Temporal Feature Extractor Global Temporal Feature Extractor ACC (%) F1 (%) 22.0±2.1 20.3±2.1 46.2±5.0 46.1±5.0 58.6±5.7 58.7±5.9 TABLE 7 For the results of ablation experiments applying the sub-blocks in local temporal feature extractor Module Metrics Depth Conv-Block Separable Conv-Block Channel Attention ACC (%) F1 (%) 46.2±6.5 45.9±6.8 41.7±4.9 41.5±4.9 56.2±5.3 56.1±5.4 58.6±5.7 58.7±5.9 2006, and has been promoted to a Full Professor of biomedical engineering, since 2011. He is currently a Chair Professor with the Department of Biomedical Engineering, TJU, where he is also the Head of the Neural Engineering and Rehabilitation Laboratory. His major research interests include neural engineering, rehabilitation engineering, sports science, biomedical instrumentation, and signal/image processing, especially in functional electrical stimulation, gait analysis, and brain-computer interface. Furthermore, he has been an International Advisory Board Member of the The Foot, and the Editorial Committee Member of Acta Laser Biology Sinica, and International Journal of Biomedical Engineering, China. He has managed over ten national and international research projects, organized and hosted several international conferences, as the Session Chair or Track Chair over the last ten years and was the General Chair of the 2012 IEEE International Conference on Virtual Environments, Human-Computer Interfaces and Measurement Systems (VECIMS 12). 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[ "Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows", "Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows" ]	[ "Sergii Kivva " ]	[]	[]	First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented.	10.48550/arxiv.2305.17774	[ "https://export.arxiv.org/pdf/2305.17774v1.pdf" ]	258,959,494	2305.17774	1f5b6b5fc48a6325d6c3ac88abb542ba112fafef	Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows Sergii Kivva Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows fully discrete entropy inequalityflux corrected transportshallow water equationshydrostatic reconstruction schemelinear programming Mathematics Subject Classification (2010) MSC 65M08 · MSC 65M22 · 76M12 First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented. Introduction In this paper, we consider a design of entropy stable flux correction for a hydrostatic reconstruction scheme for shallow water equations with variable bottom topography. For simplicity, without loss of generality, we focus on the Saint-Venant system of one-dimensional shallow water equations, given by ∂ t h + ∂ x Q = 0, ∂ t Q + ∂ x Q 2 h + g h 2 2 = −gh∂ x z,(1.1) subject to the initial conditions h(x, 0) = h 0 (x), Q(x, 0) = Q 0 (x), (1.2) where h(x, t) is the water depth, Q(x, t) is the water discharge, g is the gravitational constant, and z(x) is the bottom topography. The system (1.1) is considered in a certain spatial domain D, and if D ̸ = R, then on the boundary of D a corresponding boundary conditions should be specified. In vector form, the system (1.1) can be written as ∂u ∂t + ∂ ∂x f (u) = s,(1.3) where u = (h, Q) T is the vector of conserved variables, f = (Q, Q 2 /h + gh 2 /2) T is the flux vector, and s = (0, ghz x ) T is the source vector. It is well known [19] that solutions of (1.1)-(1.2) may develop singularities in finite time even for a smooth initial condition. Hence, we should interpret (1.1) in the sense of distribution and search for weak solutions. However, such weak solutions are not unique. To single out a unique physically relevant weak solution, the latter should satisfy ∂U (u) ∂t + ∂F (u) ∂x ≤ 0 (1.4) in the sense of distribution for every entropy pair (U, F ). Here U is a convex function of u, the so-called entropy function, and F is its entropy flux that satisfies F T u (u) = U u (u)f u (u). (1.5) For shallow water equations (1.1) with bottom topography z(x), the total energy U (u) = 1 2 Q 2 h + gh 2 + ghz,(1.6) serves as an entropy function with entropy flux F (u) = Q 3 2h 2 + ghQ + gQz. (1.7) We discretize (1.3) by the difference scheme 1 ∆t (v i − v i ) + 1 ∆x g i+1/2 − g i−1/2 = s i ,(1.8) where the numerical flux g i+1/2 is calculated as g i+1/2 = g L i+1/2 + α i+1/2 g H i+1/2 − g L i+1/2 . (1.9) Here, v i = v(x i , t) = (y(x i , t), q(x i , t)) T is the discrete solution at the grid point (x i = i∆x, t); v i = v(x i , t + ∆t); ∆x and ∆t are the spatial and temporal computational grid size, respectively. g H i+1/2 and g L i+1/2 are a high-order and low-order numerical fluxes such that g i+1/2 = g(v i−l+1 , ..., v i+r ) is the Lipschitz continuous numerical flux consistent with the differential flux, that is g(u, ..., u) = f (u) for all flux-limiters α i+1/2 ∈ [0, 1]. The expression in square brackets on the right-hand side of (1.5) can be considered as an antidiffusive flux. For flux-correction we compute the flux limiters α i+1/2 as an approximate solution to the corresponding optimization problem. The classical two-step Flux-Corrected Transport (FCT) was firstly developed by Oran and Boris [4] to solve the transient continuity equation. The procedure of two-step flux correction consists of computing the time-advanced low-order solution in the first step and correcting the solution by adding antidiffusive fluxes in the second step to produce accurate and monotone results. The antidiffusive fluxes, which define as the difference between the high and low-order fluxes, are limited in such a way that neither new extrema are created nor existing extrema are increased. Later, Zalesak [31,32] extended FCT to multidimensional explicit difference schemes. In [32], using characteristic variables, Zalesak proposed FCT algorithms for nonlinear systems of conservation laws. Several implicit FEM-FCT schemes for unstructured grids were proposed by Kuzmin and coworkers [17,18]. However, the known FCT algorithms do not guarantee entropy solutions for hyperbolic conservation laws. We discretize the entropy inequality (1.4) as follows U (v i ) − U (v i ) + ∆t ∆x G i+1/2 − G i−1/2 ≤ 0 (1.10) where G i+1/2 = G(v i−l+1 , ..., v i+r ) is the numerical entropy flux consistent with the differential one G(u, ..., u) = F (u). A difference scheme (1.8) is called entropy stable if computed solutions satisfy the discrete cell entropy inequality (1.6). We mention here the pioneering studies of entropy stable schemes by Lax [19]. Entropy stable schemes were developed by several authors [7,10,12,13,20,25,29,30]. To single out a physically relevant solution, we use the so-called proper numerical entropy flux, the concept of which was formulated by Merriam [21] and Sonar [27]. Zhao and Wu [33] proved that three-point monotone semi-discrete schemes in conservative form satisfy the corresponding semi-discrete entropy inequality with the proper numerical entropy flux. Fully discrete entropy stable schemes with the proper numerical entropy flux for scalar conservation laws were obtained in [15,16]. The numerical entropy flux G(v i−l+1 , ..., v i+r ) for F is not unique. The distinguishing feature of the proper numerical entropy flux among others is that it satisfies property (1.5) of the differential entropy flux. In this paper, we apply a first-order hydrostatic reconstruction (HR) scheme as a low-order scheme to design flux correction. A first-order HR scheme was originally developed by Audusse et al. [1], and it does not properly account for the acceleration due to a sloped bottom [8] for shallow downhill flow. Morales de Luna et al. [23] improved the original first-order HR scheme for partially wet interfaces. Using a technique of subcell reconstructions, Chen and Noelle [6] proposed a new reconstruction with a better approximation of the source term for shallow downhill flows. The main properties of the original HR scheme or its modifications are positivity preserving, well-balanced, and satisfying a semi-discrete in-cell entropy inequality. Unfortunately, it is well known that semi-discrete entropy inequalities are insufficient to obtain a suitable convergence to the entropy weak solution or to get relevant energy estimates. Audusse et al. [2] showed that the HR scheme combined with a kinetic solver satisfies a fully discrete entropy inequality with an error term coming from the topography. Thus, we can expect the convergence of this scheme for Lipschitz continuous bathymetry. Berthon et al. [3] suggested to introduce artificial viscosity into the HR scheme to get fully discrete entropy inequalities. Using the approach proposed in [15,16], we construct a flux correction for 1D shallow water equations (1.1) to obtain numerical entropy solutions for which the antidiffusive fluxes are maximal. For this, the flux limiters for the hybrid scheme (1.8)-(1.9) are computed from the optimization problem with constraints that are valid for the low-order scheme. An approximate solution to the optimization problem yields the desired flux correction formulas. Moreover, con-sidering the flux limiters as functions of the numerical solution, we prove the unique solvability of the hybrid scheme (1.8)-(1.9) under general assumptions on them. We show that the approximate limiters satisfy the assumptions under which the hybrid scheme has a unique solution. The developed approach is a novel view on the known FCT method. The paper is organized as follows. In Section 2, we present estimates that are valid for an explicit first-order HR scheme with the Rusanov numerical flux. Section 3 defines the proper numerical entropy flux and studies the conditions under which the explicit HR scheme for homogeneous and inhomogeneous shallow water equations satisfies the fully discrete entropy inequality. The unique solvability of flux correction for the HR scheme, the optimization problem for finding flux limiters, and the algorithm for its solution are described in Section 4. An approximate solution of the optimization problem is derived in Section 5. The results of numerical experiments with different HR schemes are given in Section 6. Concluding remarks are drawn in Section 7. First-Order Hydrostatic Reconstruction Scheme We consider an explicit first-order HR scheme of Chen and Noelle [6] in the form v i − v i + ∆t ∆x g L i+1/2 (v − i+1/2 , v + i+1/2 ) − g L i−1/2 (v − i−1/2 , v + i+1/2 ) = ∆t s i ,(2.1) where g L i+1/2 is the Rusanov numerical flux [26] consistent with the differential flux f and given by g L i+1/2 (v − i+1/2 , v + i+1/2 ) = 1 2 f (v − i+1/2 ) + f (v + i+1/2 ) − c i+1/2 (v + i+1/2 − v − i+1/2 ) . (2.2) The vectors of conservative variables v ± i+1/2 are given by v − i+1/2 = y − i+1/2 y − i+1/2 u i , v + i+1/2 = y + i+1/2 y + i+1/2 u i+1 , u i = √ 2y i q i y 4 i + max (y 4 i , ϵ) , (2.3) where ϵ is a small a-priori chosen positive number. The water depths are calculated as y − i+1/2 = min w i − z i+1/2 , y i , y + i+1/2 = min w i+1 − z i+1/2 , y i+1 (2.4) with water levels w i = z i + y i , and the cell interface bottom z i+1/2 = min (max (z i , z i+1 ) , min (w i , w i+1 )) . (2.5) The source term s i = −s + i−1/2 + s − i+1/2 = (0, −s + i−1/2 ) T + (0, s − i+1/2 ) T is discretized as s − i+1/2 = −g y i + y − i+1/2 2 z i − z i+1/2 ∆x , s + i+1/2 = −g y + i+1/2 + y i+1 2 z i+1 − z i+1/2 ∆x . (2.6) Finally, the local speed c i+1/2 in (2.2) is calculated using the eigenvalues of the Jacobian f u (u) as follows c i+1/2 = max \|u i \| + g y − i+1/2 , \|u i+1 \| + g y + i+1/2 . (2.7) The following theorem gives estimates for the numerical solution of the HR scheme (2.1). ∆t ≤ 2∆x max i (c i+1/2 − u i+1 + c i−1/2 + u i−1 ) , (2.8) the following inequalities hold for the numerical solution of the system of equations (2.1)-(2.2) ∆x ∆t min(w i , w − i−1/2 , w + i+1/2 ) ≤ ∆x ∆tŵ i − u i + u i+1 2 z i+1/2 − c i+1/2 − u i+1 2 w i − c i+1/2 + u i 2 w − i+1/2 + u i + u i−1 2 z i−1/2 − c i−1/2 + u i−1 2 w i − c i−1/2 − u i 2 w + i−1/2 ≤ ∆x ∆t max(w i , w − i−1/2 , w + i+1/2 ), (2.9) ∆x ∆t min(q i , q − i−1/2 , q + i+1/2 ) ≤ ∆x ∆tq i − c i+1/2 − u i+1 2 q i − c i+1/2 + u i 2 q − i+1/2 + g 2 1 2 y − 2 i+1/2 + y + 2 i+1/2 + (y i + y − i+1/2 )(z i+1/2 − z i ) − g 2 1 2 y − 2 i−1/2 + y + 2 i−1/2 + (y i + y + i−1/2 )(z i−1/2 − z i ) − c i−1/2 + u i−1 2 q i − c i−1/2 − u i 2 q + i−1/2 ≤ ∆x ∆t max(q i , q − i−1/2 , q + i+1/2 ), (2.10) where w ± i+1/2 = y ± i+1/2 + z i+1/2 . Proof. Let us prove inequalities (2.9). Inequalities (2.10) are proved similarly. We rewrite the equation (2.1) for the conservative variable y i in the form ∆x ∆tŷ i = ∆x ∆t y i − c i+1/2 + u i 2 y − i+1/2 − c i−1/2 − u i 2 y + i−1/2 + c i+1/2 − u i+1 2 y + i+1/2 + c i−1/2 + u i−1 2 y − i−1/2 . (2.11) Substituting the water level w i in (2.11) instead of the water depth y i , we obtain ∆x ∆tŵ i = ∆x ∆t − c i+1/2 − u i+1 2 − c i−1/2 + u i−1 2 w i + c i+1/2 − u i+1 2 w + i+1/2 + c i−1/2 + u i−1 2 w − i−1/2 + u i + u i+1 2 z i+1/2 − u i + u i−1 2 z i−1/2 + c i+1/2 − u i+1 2 w i − c i+1/2 + u i 2 w − i+1/2 + c i−1/2 + u i−1 2 w i − c i−1/2 − u i 2 w + i−1/2 . (2.12) Note that under the condition (2.8), the first three terms in the right-hand side of (2.12) are a convex combination of w i , w + i+1/2 , and w − i−1/2 , which proves the theorem. Remark 2.1. We note that if c i+1/2 satisfies the following inequalities ∆x ∆t y i − c i+1/2 + u i 2 y − i+1/2 − c i−1/2 − u i 2 y + i−1/2 ≥ 0, c i+1/2 − u i+1 ≥ 0, c i−1/2 + u i−1 ≥ 0, (2.13) then the difference scheme (2.11) preserves the non-negativity of the water depth y. Cell Entropy Inequality for Fully Discrete HR Scheme In this section we study the cell entropy inequality for the fully discrete HR scheme (2.1)-(2.2). We consider a homogeneous three-point low-order scheme in the form v i − v i + ∆t ∆x g L i+1/2 (v i , v i+1 ) − g L i−1/2 (v i−1 , v i ) = 0, (3.1) where the low-order numerical flux g L i+1/2 = g(v, w) is consistent with the smooth differential flux f (u) : R m → R m of the conservative variables u = (u 1 , . . . , u m ) T . We define the numerical entropy flux as follows. Definition 3.1. Numerical entropy flux G(v i−l+1 , ..., v i+r ) of the difference scheme (3.1) is called proper if for any v i−l+1 , ..., v i+r ∈ R m we have ∂ ∂v j p G(v i−l+1 , ..., v i+r ) = k ∂U (v p ) ∂v k p ∂ ∂v j p g k (v i−l+1 , ..., v i+r ), p = i − l + 1, ..., i + r. (3.2) Then the proper numerical entropy flux for the difference scheme (1.8) and (1.9) can be written in the form G i+1/2 = G L i+1/2 + α i+1/2 G H i+1/2 − G L i+1/2 ,(3.3) where G L i+1/2 and G H i+1/2 are the low-order and high-order proper numerical entropy fluxes corresponding to the numerical fluxes g L i+1/2 and g H i+1/2 . Theorem 3.1. Suppose that f : R m → R m is hemicontinuosly Gateaux differentiable, U : R m → R is a strictly convex function with a hemicontinuos second Gateaux derivative. If matrices U ′′ (w)g ′ u (u, v i ) and U ′′ (w)g ′ u (v i , u) are positive and negative definite, respectively, for any u, w ∈ R m , ∆t satisfies the inequality ∆t max s∈(v i ,v i ) λ (U ′′ (s)) ⟨ g L i+1/2 − g L i−1/2 , g L i+1/2 − g L i−1/2 ⟩ ≤ 2∆x ⟨U ′ (v i ), g L i+1/2 − g L i−1/2 ⟩ − G L i+1/2 + G L i−1/2 ,(3. 4) then the fully discrete scheme (3.1) satisfies the discrete cell entropy inequality U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 (v i , v i+1 ) − G L i−1/2 (v i−1 , v i ) ≤ 0. (3.5) where G L i+1/2 is the proper numerical entropy flux corresponding to the numerical flux g L i+1/2 , ⟨·, ·⟩ denotes the Euclidean inner product. Proof. Multiplying (3.1) by U ′ (v i ) and subtracting it from the left-hand side of (1.10), we get U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 − G L i−1/2 = U (v i ) − U (v i )−⟨U ′ (v i ), (v i − v i )⟩ + ∆t ∆x G L i+1/2 − G L i−1/2 − U ′ (v i ), g L i+1/2 − g L i−1/2 = 1 2 ∆t ∆x 2 U ′′ (s) g L i+1/2 − g L i−1/2 , g L i+1/2 − g L i−1/2 + ∆t ∆x G L i+1/2 − F (v i ) − U ′ (v i ), g L i+1/2 − f (v i ) + ∆t ∆x F (v i ) − G L i−1/2 − U ′ (v i ), f (v i ) − g L i−1/2 (3.6) where s = θv i + (1 − θ)v i , 0 < θ < 1. It is easy to see that the first term on the right-hand side of (3.6) is non-negative. Now we show that the second and third terms in square brackets are non-positive. Indeed, we rewrite the second and third terms as follows G L (v i , v i+1 ) − G L (v i , v i ) − j ∂U ∂v j (v i ) g L,j (v i , v i+1 ) − g L,j (v i , v i ) = 1 0 j,k ∂U ∂v j (v i + ξ∆v i+1/2 ) − ∂U ∂v j (v i ) ∂g L,j ∂v k (v i , v i + ξ∆v i+1/2 )∆v k i+1/2 dξ = 1 0 1 0 k,l j ∂ 2 U ∂v j ∂v l (v i + ηξ∆v i+1/2 ) ∂g L,j ∂v k (v i , v i + ξ∆v i+1/2 ) ∆v k i+1/2 ∆v l i+1/2 dη ξdξ (3.7) G L (v i , v i ) − G L (v i−1 , v i ) − j ∂U ∂v j (v i ) g L,j (v i , v i ) − g L,j (v i−1 , v i ) = 1 0 j,k ∂U ∂v j (v i − ξ∆v i−1/2 ) − ∂U ∂v j (v i ) ∂g L,j ∂v k (v i − ξ∆v i−1/2 , v i )∆v k i−1/2 dξ = − 1 0 1 0 k,l j ∂ 2 U ∂v j ∂v l (v i − ηξ∆v i−1/2 ) ∂g L,j ∂v k (v i − ξ∆v i−1/2 , v i ) ∆v k i−1/2 ∆v l i−1/2 dη ξdξ (3.8) where ∆v i+1/2 = v i+1 − v i . Thus, according to our assumption, the integrals in (3.7)-(3.8) do not change the sign over the integration interval, which means that the second and third terms are negative. The second and third terms on the right side of (3.6) are linear in ∆t, and the first term is of second-order. Therefore, we can choose the time step small enough that the second and third terms dominate over the first term. Consequently, the right-hand side of (3.5) is non-positive if ∆t satisfies (3.4). This completes the proof of the theorem. The proper numerical entropy flux for the first-order HR scheme (2.1) can be written as follows G L i+1/2 = 1 2 F (U − ) + F (U + ) − c i+1/2 (U + − U − ) . (3.9) Multiplying (2.1) by U ′ (v i ) and subtracting it from the left-hand side of (1.10), we obtain U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 − G L i−1/2 = U (v i ) − U (v i ) − ⟨U ′ (v i ), (v i − v i )⟩ + ∆t ∆x G L i+1/2 − G L i−1/2 − U ′ (v i ), g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 = U (v i ) − U (v i ) − ⟨U ′ (v i ), (v i − v i )⟩ + ∆t ∆x ∆G L,− i+1/2 − ∆G L,+ i−1/2 ,(3.10) where ∆G L,± i+1/2 are defined as ∆G L,± i+1/2 = G L i+1/2 (U − i+1/2 , U + i+1/2 ) − F (U i ) − U ′ (v i ), g L i+1/2 (v − i+1/2 , v + i+1/2 ) − s ± i+1/2 − f (u i ) . (3.11) or substituting the values of the corresponding functions, ∆G L,± i+1/2 can be represented as ∆G L,− i+1/2 = 1 2 u i+1 2 y + i+1/2 (u i+1 − u i ) 2 + g u i+1 y + i+1/2 − u i y i (z i+1/2 − z i ) + g u i 2 (y − i+1/2 − y i ) 2 + (y + i+1/2 − y i ) u i+1 y + i+1/2 − u i 2 (y + i+1/2 + y i ) − c i+1/2 1 2 y + i+1/2 (u i+1 − u i ) 2 + g 2 (y + i+1/2 − y i ) 2 − g 2 (y − i+1/2 − y i ) 2 + g(y + i+1/2 − y − i+1/2 )(z i+1/2 − z i ) , (3.12) ∆G L,+ i−1/2 = 1 2 u i−1 2 y − i−1/2 (u i−1 − u i ) 2 − g u i−1 y − i−1/2 − u i y i (z i − z i−1/2 ) + g u i 2 (y + i−1/2 − y i ) 2 + (y − i−1/2 − y i ) u i−1 y − i−1/2 − u i 2 (y − i−1/2 + y i ) + c i−1/2 1 2 y − i−1/2 (u i−1 − u i ) 2 − g 2 (y + i−1/2 − y i ) 2 + g 2 (y − i−1/2 − y i ) 2 + g(y + i−1/2 − y − i−1/2 )(z i − z i−1/2 ) . (3.13) We consider two cases: (1) homogeneous and (2) inhomogeneous shallow water equations. Theorem 3.2. Assume that c i+1/2 and ∆t satisfy the inequalities c i+1/2 ≥ 1 2 max −u i+1 − u i + (u i+1 − u i ) 2 + 4gy i , u i+1 + u i + (u i+1 − u i ) 2 + 4gy i+1 , (3.14) ∆t max v∈(v i ,v i ) λ (U ′′ (v)) ⟨ g L i+1/2 − g L i−1/2 , g L i+1/2 − g L i−1/2 ⟩ ≤ 2∆x ⟨U ′ (v i ), g L i+1/2 − g L i−1/2 ⟩ − G L i+1/2 + G L i−1/2 . (3.15) where λ (U ′′ (v)) = 1 2y (u 2 + gy + (u 2 + gy) 2 − 4gy. Then for homogeneous shallow water equations, the fully discrete HR scheme (2.1)-(2.2) satisfies the discrete cell entropy inequality U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 (v i , v i+1 ) − G L i−1/2 (v i−1 , v i ) ≤ 0. (3.16) where G L i+1/2 is the proper numerical entropy flux (3.9). Proof. For homogeneous shallow water equations, we have that v − i−1/2 = v i−1 , v − i+1/2 = v + i−1/2 = v i and v + i+1/2 = v i+1 . Then ∆G L,± i+1/2 in (3.12)-(3.13) take the form ∆G L,− i+1/2 = 1 2 u i+1 2 y i+1 (u i+1 − u i ) 2 + g(y i+1 − y i ) u i+1 y i+1 − u i 2 (y i+1 + y i ) − c i+1/2 y i+1 2 (u i+1 − u i ) 2 + g 2 (y i+1 − y i ) 2 , (3.17) ∆G L,+ i−1/2 = 1 2 u i−1 2 y i−1 (u i−1 − u i ) 2 + g(y i−1 − y i ) u i−1 y i−1 − u i 2 (y i−1 + y i ) + c i−1/2 1 2 y i−1 (u i−1 − u i ) 2 + g 2 (y i−1 − y i ) 2 . (3.18) Note that the multipliers in the square brackets for c i±1/2 in (3.17)-(3.18) are non-negative. It is easy to check that when they are zero, then ∆G L,± i∓1/2 are also zero. Let us show that c i±1/2 can be chosen so that ∆G L,− i+1/2 and ∆G L,+ i−1/2 are non-positive and non-negative, respectively. Indeed, we rewrite (3.17) as ∆G L,− i+1/2 = 1 2 u i+1 2 y i+1 (u i+1 − u i ) 2 + g y i+1 (y i+1 − y i ) (u i+1 − u i ) +g u i 2 (y i+1 − y i ) 2 − c i+1/2 y i+1 2 (u i+1 − u i ) 2 + g 2 (y i+1 − y i ) 2 = 1 2 y i+1 2 (u i+1 − c i+1/2 )(u i+1 − u i ) 2 + g y i+1 (y i+1 − y i ) (u i+1 − u i ) + g 2 (u i − c i+1/2 ) (y i+1 − y i ) 2 . (3.19) ∆G L,− i+1/2 is a quadratic form with respect to (u i+1 −u i ) and (y i+1 −y i ). For it to be non-positive, it is sufficient for the leading coefficient and its discriminant to be non-positive. Then, c i+1/2 should satisfy the following inequalities c i+1/2 − u i+1 > 0, c 2 i+1/2 − (u i+1 + u i ) c i+1/2 + u i+1 u i − gy i+1 ≥ 0. (3.20) It is clear that inequalities (3.20) hold for c i+1/2 ≥ 1 2 u i+1 + u i + (u i+1 − u i ) 2 + 4gy i+1 . Similarly, it can be shown that ∆G L,+ i−1/2 is non-negative for c i−1/2 ≥ 1 2 (−u i−1 − u i + (u i−1 − u i ) 2 + 4gy i−1 . We rewrite the discrete cell entropy inequality (3.10) in the form U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 − G L i−1/2 = U (v i ) − U (v i ) − ⟨U ′ (v i ), (v i − v i )⟩ + ∆t ∆x ∆G L,− i+1/2 − ∆G L,+ i−1/2 = 1 2 ∆t ∆x 2 U ′′ (s) g L i+1/2 − g L i−1/2 , g L i+1/2 − g L i−1/2 + ∆t ∆x ∆G L,− i+1/2 − ∆G L,+ i−1/2 . (3.21) Thus, the non-positivity of (3.21) can be achieved by choosing a sufficiently small ∆t so that the second term dominates over the first non-negative term on the right-hand side of (3.21). This completes the proof of the theorem. c i+1/2 ≥ max   a − i u i+1 + b − i u i + (a − i u i+1 − b − i u i ) 2 + 4g a − i y + i+1/2 (w + i+1/2 − w i ) 2 2a − i , −a + i+1 u i − b + i+1 u i+1 + (a + i+1 u i − b + i+1 u i+1 ) 2 + 4g a + i+1 y − i+1/2 (w − i+1/2 − w i+1 ) 2 2a + i+1   . (3.22) ∆t max v∈(v i ,v i ) λ (U ′′ (v)) ⟨ g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 , g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 ⟩ ≤ 2∆x ⟨U ′ (v i ), g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 ⟩ − G L i+1/2 + G L i−1/2 . (3.23) where λ (U ′′ (v)) = 1 2y (u 2 + gy + (u 2 + gy) 2 − 4gy a ∓ i = ±(y + i±1/2 − y − i±1/2 ) w ± i+1/2 + w − i±1/2 − 2w i , b ∓ i = ±(y − i±1/2 − y i ) 2 ∓ (y + i±1/2 − y i ) 2 + 2(y ± i±1/2 − y i )(w ± i±1/2 − w i ). (3.24) Then for inhomogeneous shallow water equations, the fully discrete HR scheme (2.1)-(2.2) satisfies the discrete cell entropy inequality U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 (v − i+1/2 , v + i+1/2 ) − G L i−1/2 (v − i−1/2 , v + i−1/2 ) ≤ 0. (3.25) where G L i+1/2 is the proper numerical entropy flux (3.9). Proof. We rewrite ∆G L,± i+1/2 in (3.12)-(3.13) as follows ∆G L,− i+1/2 = 1 2 u i+1 2 y + i+1/2 (u i+1 − u i ) 2 + g y + i+1/2 (u i+1 − u i )(w + i+1/2 − w i ) + g u i 2 (y − i+1/2 − y i ) 2 − u i 2 (y + i+1/2 − y i ) 2 + u i (y + i+1/2 − y i )(w + i+1/2 − w i ) − c i+1/2 1 2 y + i+1/2 (u i+1 − u i ) 2 + g(y + i+1/2 − y − i+1/2 ) 1 2 (w + i+1/2 + w − i+1/2 ) − w i , (3.26) ∆G L,+ i−1/2 = 1 2 u i−1 2 y − i−1/2 (u i−1 − u i ) 2 − g y − i−1/2 (u i−1 − u i )(w i − w − i−1/2 ) + g u i 2 (y + i−1/2 − y i ) 2 − u i 2 (y − i−1/2 − y i ) 2 + u i (y − i−1/2 − y i )(w − i−1/2 − w i ) + c i−1/2 1 2 y − i−1/2 (u i−1 − u i ) 2 + g(y + i−1/2 − y − i−1/2 ) w i − 1 2 (w + i−1/2 + w − i−1/2 ) . (3.27) Let us show that the coefficients in the square brackets at c i±1/2 in (3.26)-(3.27) are nonnegative. Consider the following cases: (i) In the fully wet case, min(w i , w i+1 ) > max(z i , z i+1 ). According to (2.4)-(2.5), we have w − i+1/2 = w i and w + i+1/2 = w i+1 . Hence, if y + i+1/2 ≥ y − i+1/2 , then w + i+1/2 ≥ w − i+1/2 , and (y + i+1/2 −y − i+1/2 ) 1 2 (w + i+1/2 + w − i+1/2 ) − w i ≥ 0. Otherwise, if y + i+1/2 < y − i+1/2 , then also w + i+1/2 < w − i+1/2 , and the required inequality holds. (ii) In the partially wet case min(w i , w i+1 ) ≤ max(z i , z i+1 ). Depending on which bottom is higher, right or left, we consider two subcases. Let z i ≥ z i+1 . Then z i+1/2 = w i+1 , y − i+1/2 = y i , y + i+1/2 = 0, and w − i+1/2 , w + i+1/2 ≤ w i . Therefore, (y + i+1/2 − y − i+1/2 ) 1 2 (w + i+1/2 + w − i+1/2 ) − w i ≥ 0. If z i < z i+1 , then z i+1/2 = w i , y − i+1/2 = 0, y + i+1/2 = y i+1 , and w − i+1/2 = w i , w − i+1/2 > w i . Thus, we again obtain the required inequality. The non-negativity of the terms in square brackets at c i−1/2 is proved similarly. It is easy to check that when they are zero, then ∆G L,± i∓1/2 are also zero. Let us show that c i±1/2 can be chosen so that ∆G L,− i+1/2 and ∆G L,+ i−1/2 are non-positive and non-negative, respectively. Indeed, we consider ∆G L,− i+1/2 and ∆G L,+ i−1/2 as quadratic equations with respect to (u i+1 − u i ) and (u i−1 − u i ), respectively. ∆G L,− i+1/2 will be non-positive if its leading coefficient and discriminant are non-positive, i.e. c i+1/2 > u i+1 , −a − i c 2 i+1/2 + (a − i u i+1 + b − i u i )c i+1/2 − u i+1 u i b − i + g y + i+1/2 (w + i+1/2 − w i ) 2 ≤ 0, (3.28) where a − i = (y + i+1/2 − y − i+1/2 ) w + i+1/2 + w − i+1/2 − 2w i , b − i = (y − i+1/2 − y i ) 2 − (y + i+1/2 − y i ) 2 + 2(y + i+1/2 − y i )(w + i+1/2 − w i ). (3.29) It is clear that the inequalities (3.28) hold for c i+1/2 ≥ a − i u i+1 + b − i u i + (a − i u i+1 − b − i u i ) 2 + 4g a − i y + i+1/2 (w + i+1/2 − w i ) 2 2a − i . (3.30) Similarly, it is proved that ∆G L,+ i−1/2 is non-negative for c i−1/2 ≥ −a + i u i−1 − b + i u i + (a + i u i−1 − b + i u i ) 2 + 4g a + i y − i−1/2 (w − i−1/2 − w i ) 2 2a + i , (3.31) where a + i = (y + i−1/2 − y − i−1/2 ) 2w i − w + i−1/2 − w − i−1/2 , b + i = (y + i−1/2 − y i ) 2 − (y − i−1/2 − y i ) 2 + 2(y − i−1/2 − y i )(w − i−1/2 − w i ). (3.32) Thus, returning to the discrete entropy inequality (3.21), we can choose ∆t so that the nonpositive terms in square brackets dominate over the non-negative first term on the right-hand side of (3.21). This concludes the proof of the theorem. Finding Flux Limiters The system of equations (1.8)-(1.9) is nonlinear if we consider α as a function ofv, and it can be written in the formv − ∆t Pv =v L , (4.1) where mapping P : R N ×R N → R N ×R N is defined by P iv = α i−1/2 (v)g AD i−1/2 − α i+1/2 (v) g AD i+1/2 /∆x, g AD i+1/2 = g H i+1/2 − g L i+1/2 , andv L i = v i − ∆t/∆x g L i+1/2 − g L i−1/2 + s i . Let O 0 =Ō(v L , δ) be a closed ball with center atv L and radius δ > 0. Furthermore, we define a mapping Sv : O 0 → R N × R N as Sv = v − ∆tP v. (4.2) Let us show that for sufficiently small ∆t the system of equations (4.1) is uniquely solvable in a neighborhood of the first-order HR solution of (2.1). Theorem 4.1. Assume that ∥P (w) − P (v)∥ ≤ M ∥w − v∥, ∀v, w ∈ O 0 . (4.3) If ∆t satisfies ∆t < δ(∥Pv L ∥ + δM ) −1 , (4.4) then the system of equations (4.1) has a unique solution in O 0 . Proof. Our proof mimics the proof of theorem 5.1.6 [24, p.122]. For fixed d ∈ O Sv L , ε , we define the mapping T : O 0 → R N × R N by T y = ∆tP v + d = v − [Sv − d] . Then, Sv = d has a unique solution in O 0 if and only if T has a unique fixed point. For any v, w ∈ O 0 ∥T v − T w∥ = ∆t∥P v − P w∥ ≤ ∆tM ∥v − w∥ (4.5) and S is contractive on O 0 if ∆tM < 1. Moreover, for any v ∈ O 0 , ∥T v −v L ∥ ≤ ∥T v − Tv L ∥ + ∥Tv L −v L ∥ ≤ ∆tM ∥v −v L ∥ + ∥Sv − d∥ ≤ ∆tM δ + ε. (4.6) For ε = δ(1 − ∆tM ), the expression on the right-hand side of (4.6) equal to δ. Hence, T maps O 0 into O 0 , and for any d ∈ O Sv L , ε the equation Sv = d has a unique solution in O 0 . Finally, we have thatv L ∈ O Sv L , ε if ∥Sv L −v L ∥ = ∆t∥Pv L ∥ < ε. (4.7) Combining all restrictions on ∆t, we obtain that the nonlinear system of equation ( Our goal is to find the maximum values of the flux limiters α ∈ U ad = α\| 0 ≤ α i+1/2 ≤ 1 , for which the numerical solution of the hybrid scheme (1.8)-(1.9) satisfies the constraints (2.9)-(2.10) and the discrete cell entropy inequality (3.25). Then finding the flux limiters can be considered as the following optimization problem ℑ(α) = i α i+1/2 → max α∈U ad (4.8) subject to ∆x ∆t (w i − w i ) + c i+1/2 − u i+1 2 (w i − w + i+1/2 ) + c i−1/2 + u i−1 2 (w i − w − i−1/2 ) ≤ −α i+1/2 g AD,y i+1/2 + α i−1/2 g AD,y i−1/2 ≤ ∆x ∆t (w i − w i ) + c i+1/2 − u i+1 2 (w i − w + i+1/2 ) + c i−1/2 + u i−1 2 (w i − w − i−1/2 ), (4.9) ∆x ∆t q i − q i + c i+1/2 − u i+1 2 (q i − q + i+1/2 ) + c i−1/2 + u i−1 2 (q i − q − i−1/2 ) ≤ −α i+1/2 g AD,q i+1/2 + α i−1/2 g AD,q i−1/2 ≤ ∆x ∆t (q i − q i ) + c i+1/2 − u i+1 2 (q i − q + i+1/2 ) + c i−1/2 + u i−1 2 (q i − q − i−1/2 ), (4.10) ∆x ∆t [U (v i ) − U (v i ) − ⟨U ′ (v i ), (v i − v i )⟩] − U ′ (v i ), (g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 ) + G L i+1/2 − G L i−1/2 ≤ α i+1/2 ⟨U ′ (v i ), g AD i+1/2 ⟩ − G AD i+1/2 − α i−1/2 (⟨U ′ (v i ), g AD i−1/2 ⟩ − G AD i−1/2 ), (4.11) ∆x ∆t (v i − v i ) + g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 + α i+1/2 g AD i+1/2 − α i−1/2 g AD i−1/2 = 0, (4.12) where w i = min w i , w − i−1/2 , w + i+1/2 , w i = max w i , w − i−1/2 , w + i+1/2 , and G AD i+1/2 = G H i+1/2 − G L i+1/2 . Due to constraints (4.11) the optimization problem (4.8)-(4.12) is nonlinear. Consequently, finding a numerical entropy solution of shallow water equations with variable bottom topography (1.1)-(1.2) in one time step can be represented as the following iterative process. Step 1. Initialize positive numbers δ, ϵ 1 , and ϵ 2 . Set p = 0,v 0 = v, α 0 = 0. Step 2. Find α p+1 as a solution to the following linear programming problem ℑ(α) = i α p+1 i+1/2 → max α p+1 ∈U ad (4.13) subject to ∆x ∆t (w i − w i ) + c i+1/2 − u i+1 2 (w i − w + i+1/2 ) + c i−1/2 + u i−1 2 (w i − w − i−1/2 ) ≤ −α p+1 i+1/2 g AD,y i+1/2 + α p+1 i−1/2 g AD,y i−1/2 ≤ ∆x ∆t (w i − w i ) + c i+1/2 − u i+1 2 (w i − w + i+1/2 ) + c i−1/2 + u i−1 2 (w i − w − i−1/2 ), (4.14) ∆x ∆t q i − q i + c i+1/2 − u i+1 2 (q i − q + i+1/2 ) + c i−1/2 + u i−1 2 (q i − q − i−1/2 ) ≤ −α p+1 i+1/2 g AD,q i+1/2 + α p+1 i−1/2 g AD,q i−1/2 ≤ ∆x ∆t (q i − q i ) + c i+1/2 − u i+1 2 (q i − q + i+1/2 ) + c i−1/2 + u i−1 2 (q i − q − i−1/2 ), (4.15) ∆x ∆t [U (v p i ) − U (v i ) − ⟨U ′ (v i ), (v p i − v i )⟩] − U ′ (v i ), (g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 ) +G L i+1/2 −G L i−1/2 ≤ α p+1 i+1/2 ⟨U ′ (v i ), g AD i+1/2 ⟩ − G AD i+1/2 −α p+1 i−1/2 (⟨U ′ (v i ), g AD i−1/2 ⟩−G AD i−1/2 ),(4. 16) Step 3. For α p+1 we findv p+1 from the system of linear equations ∆x ∆t v p+1 i − v i +g L i+1/2 −g L i−1/2 −s − i+1/2 +s + i−1/2 +α p+1 i+1/2 g AD i+1/2 −α p+1 i−1/2 g AD i−1/2 = 0, (4.17) Step 4. Algorithm stop criterion ŷ p+1 i −ŷ p i max δ, ŷ p+1 i < ε 1 , q p+1 i −q p i max δ, q p+1 i < ε 1 , α p+1 i+1/2 − α p i+1/2 < ε 2 . (4.18) If conditions (4.18) hold, then setv =v p+1 . Otherwise, set p = p + 1 and go to Step 2. Approximate Solution to the Optimization Problem Solving a linear programming problem is computationally expensive. So, at Step 2, instead of solving the linear programming problem, it is reasonable to use its computationally less expensive approximate solution. In this section our goal is to look for an approximate solution of the linear programming problem (4.13) -(4.16). First, we find a nontrivial α ∈ U ad satisfying inequalities (4.14), which are rewritten in the form −α i+1/2 g AD,y i+1/2 + α i−1/2 g AD,y i−1/2 ≥ Q −,y i , (5.1) −α i+1/2 g AD,y i+1/2 + α i−1/2 g AD,y i−1/2 ≤ Q +,y i , (5.2) where Q +,y i = ∆x ∆t (w i − w i ) + c i+1/2 − u i+1 2 (w i − w + i+1/2 ) + c i−1/2 + u i−1 2 (w i − w − i−1/2 ), Q −,y i = ∆x ∆t (w i − w i ) + c i+1/2 − u i+1 2 (w i − w + i+1/2 ) + c i−1/2 + u i−1 2 (w i − w − i−1/2 ). Denote by α −,y i and α +,y i the maximum values of the components α for the negative and positive terms on the left-hand side of (5.1)-(5.2), respectively. Then −α i+1/2 g AD,y i+1/2 + α i−1/2 g AD,y i−1/2 ≥ α −,y i P −,y i , (5.3) −α i+1/2 g AD,y i+1/2 + α i−1/2 g AD,y i−1/2 ≤ α +,y i P +,y i , (5.4) where P −,y i = min 0, −g AD,y i+1/2 + min 0, g AD,y i−1/2 , P +,y i = max 0, −g AD,y i+1/2 + max 0, g AD,y i−1/2 . Each flux limiter α i+1/2 appears twice in (5.3) and twice in (5.4) with coefficients that differ only in sign. Substituting (5.3)-(5.4) into (5.1)-(5.2), we obtain that α i+1/2 should not exceed α y i+1/2 = min(α +,y i , α −,y i+1 ) = min(R +,y i , R −,y i+1 ), g AD,y i+1/2 < 0, min(α −,y i , α +,y i+1 ) = min(R −,y i , R +,y i+1 ), g AD,y i+1/2 > 0, (5.5) where R −,y i = min 1, min(0, Q −,y i ) P −,y i and R +,y i = min 1, max(0, Q +,y i ) P +,y i . Similarly, it is proved that inequalities (4.15) hold for α q i+1/2 = min(α +,q i , α −,q i+1 ) = min(R +,q i , R −,q i+1 ), g AD,q i+1/2 < 0, min(α −,q i , α +,q i+1 ) = min(R −,q i , R +,q i+1 ), g AD,q i+1/2 > 0, (5.6) where R −,q i = min 1, min(0, Q −,q i ) P −,q i and R +,q i = min 1, max(0, Q +,q i ) P +,q i , Q +,q i = ∆x ∆t (q i − q i ) + c i+1/2 − u i+1 2 (q i − q + i+1/2 ) + c i−1/2 + u i−1 2 (q i − q − i−1/2 ), Q −,q i = ∆x ∆t q i − q i + c i+1/2 − u i+1 2 (q i − q + i+1/2 ) + c i−1/2 + u i−1 2 (q i − q − i−1/2 ). P −,q i = min 0, −g AD,q i+1/2 + min 0, g AD,q i−1/2 , P +,q i = max 0, −g AD,q i+1/2 + max 0, g AD,q i−1/2 . Finally, we rewrite (4.16) in the form A i ≤ α i+1/2 d ii+1 + α i−1/2 d ii−1 , (5.7) where A i = ∆x ∆t (U (v i ) − U (v i ) − ⟨U ′ (v i ), (v i − v i )⟩) + G L i+1/2 − G L i−1/2 − U ′ (v i ), (g L i+1/2 − g L i−1/2 − s − i+1/2 + s + i−1/2 ) , d ik = ⟨U ′ (v i ), g AD (i+k)/2 ⟩ − G AD (i+k)/2 sgn(k − i) . By reasoning similar to the above, we obtain from (5.7) that the upper bound of α i+1/2 is equal toᾱ U i+1/2 = min 1, −A i B i min (0, sgn d ii+1 ) + max (0, sgn d ii+1 ) , −A i+1 B i+1 min (0, sgn d i+1i ) + max (0, sgn d i+1i ) , (5.8) where B i = min(0, d ii+1 ) + min(0, d ii−1 ) Thus, a nontrivial feasible solution to the linear programming problem (4.13)-(4.16) on U ad is equal to α i+1/2 = min(ᾱ y i+1/2 ,ᾱ q i+1/2 ,ᾱ U i+1/2 ). (5.9) Remark 5.2. The approach presented in this paper can be extended to multidimensional and implicit HR schemes. For details we refer the reader to [15]. Theorem 5.1. Let U (v) : R N × R N → R N × R N Numerical Examples In this section, we demonstrate the benefits of the proposed approach and compare numerical results with analytical and previous numerical studies. We also compare numerical results obtained with flux limiters, which are approximate and exact solutions to the corresponding optimization problems. Applying the centered space flux as a high-order flux, we use the following hybrid HR scheme in our calculationŝ v i − v i + ∆t ∆x g L i+1/2 (v − i+1/2 , v + i+1/2 ) − g L i−1/2 (v − i−1/2 , v + i+1/2 ) + 1 2 ∆t ∆x α i+1/2 c i+1/2 (v + i+1/2 − v − i+1/2 ) − α i−1/2 c i−1/2 (v + i−1/2 − v − i−1/2 ) = ∆t s i , (6.1) where g L i+1/2 is the Rusanov numerical flux (2.2). Then the discrete cell entropy inequality (1.10) can be written in the form U (v i ) − U (v i ) + ∆t ∆x G L i+1/2 (v − i+1/2 , v + i+1/2 ) − G L i−1/2 (v − i−1/2 , v + i−1/2 ) +α i+1/2 c i+1/2 2 (U (v + i+1/2 ) − U (v − i+1/2 )) − α i−1/2 c i−1/2 2 (U (v + i−1/2 ) − U (v − i−1/2 )) ≤ 0 (6.2) with the proper numerical entropy flux (3.9). Below we will mark the numerical solutions of scheme (6.1)-(6.2) with a label indicating how the flux limiters are calculated. The letters L and A denote the applying linear programming or approximate solution to the optimization problem, respectively. The letters H, Q and E mean that the flux limiters were calculated using inequalities (4.14), (4.15) and (6.2), respectively. Numerical solutions with flux limiters satisfying inequalities (2.13) are denoted as P P . In the latter case, flux limiters are defined as follows α i+1/2 = max 0, min 1, 1 − u i+1 c i+1/2 , 1 + u i c i+1/2 . (6.3) In addition, we use the following labels: HR1 is a first-order hydrostatic reconstruction scheme with HLL numerical flux given in [6]; HR2 is a hydrostatic reconstruction scheme of second-order spatial accuracy with explicit Euler time integration proposed in [5]; ZL is a characteristic variable implementation of the Boris-Book flux limiter described in [32] and applied to the HR scheme (2.1)-(2.2). To solve linear programming problems we apply GLPK package v.4.65 (https : //www.gnu. org/sof tware/glpk/). One-Dimensional Dam Break Over a Wet Flat Bed In this section, we consider a dam break on a wet flat bed in a frictionless, horizontal, rectangular channel. The channel is 1000 m long. The dam is located in the middle of the channel. The water depth at the left and right hand sides of the dam is 100 m and 1 m, respectively. The dam instantly collapses across its entire width and the resulting flow consists of a shock wave traveling downstream and a rarefaction wave traveling upstream. In this problem there is a transition from subcritical upstream to supercritical downstream flows. The simulation is performed up to time t=10 s. The analytical solution of this problem was given by Stoker (1957) [28]. The 1D dam-break on a wet flat bed is a classical test to verify the shock-capturing ability of numerical schemes. Numerical results obtained with different schemes at time t=10 s on a uniform grid of N=100 cells are shown in Fig. 1-Fig. 3. As shown in Fig. 2, the shock wave resolutions using HR2, LHE, and LHQE are less dissipative (sharper) and better than with the other schemes shown in Fig. 1. The simulated results with PP are close to those obtained with ZL but require much less calculations. In the numerical results with LHE, AHE, LHQE, and AHQE, we observe the so-called "terracing" phenomenon characteristic of FCT methods. Numerical results for water discharge obtained using the LHE and AHE schemes have oscillations that are absent in the velocities (Fig 3). The analytical and numerical solutions were compared quantitatively by the L1 error. The error is defined as L 1 = 1 N N i=1 \|y i − y a (x i )\| (6.4) where y i is the numerical and y a is the analytical solution at point x i , N means the number of these points. Table 1 shows the L 1 -norms of errors of the numerical solutions obtained with different schemes. A comparison of analytical solutions with computed depths, as well as velocities and discharges at t=10 s using LHE(LHQE) and AHE(AHQE) are given in Fig. 3. The flux limiters for LHE(LHQE) and AHE(AHQE) are calculated using exact and approximate solutions to linear programming problems. We note good agreement between these numerical solutions, and the addition of constraints on water discharges to calculate flux limiters leads to suppression of oscillations in the numerical solutions. Figure 2: One-dimensional dam break over a wet flat bed. Comparisons of exact solutions with simulated water depths (left) and discharges (right) using HR2, LHE, and LHQE at t=10 s. The second row is a zoom in the area behind the shock. The number of cells is N=100. Table 1: L 1 -norms of errors for the numerical solutions of the 1D dam break over a wet flat bed at t=10 s with N=100. HR1 ZL P P HR2 H 1.468×10 0 1.679×10 0 1.365×10 0 4.052×10 −1 Q 3.596×10 1 3.882×10 1 3.060×10 1 9.180×10 0 LHE AHE LHQE AHQE H 6.153×10 −1 5.114×10 −1 7.898×10 −1 6.306×10 −1 Q 1.912×10 1 1.619×10 1 2.003×10 1 1.647×10 1 One-Dimensional Dam Break Over a Dry Bed The dry bed dam-break test is usually applied to verify the ability of a difference scheme to propagate a wet/dry front at the correct speed and to keep water depth positive. The analytical solution of this problem was given by Stoker (1957) [28]. We consider a rectangular channel with 1000 m length and a flat bed. The dam is located in the middle of the channel. The water depth at the left and right hand sides of the dam is 100 m and 0 m, respectively. The dam break is instantaneous and there is no friction. The solution consists of a single rarefaction wave with a wet/dry front at its lower end. The flow domain is discretized into 100 uniform cells. The simulation time is t=7 s. Comparisons of exact solutions with simulated depths as well as discharges at t=7 s using the six schemes are presented in Fig. 4-5. Among the proposed schemes, the HR1, ZL, and 1.320×10 0 1.050×10 0 3.684×10 −1 Q 2.838×10 1 3.218×10 1 2.590×10 1 1.038×10 1 LHE AHE LHQE AHQE H 5.463×10 −1 5.216×10 −1 7.690×10 −1 5.786×10 −1 Q 2.129×10 1 1.978×10 1 2.207×10 1 1.820×10 1 PP schemes present more dissipative results than the HR2, LHE, and LHQE schemes. The simulated results with PP are close to those obtained with ZL but require much less calculations. In the numerical results obtained with LHE, AHE, LHQE, and AHQE, we observe the so-called "terracing" phenomenon, which is characteristic of FCT methods. The LHE and AHE schemes produce oscillations in the water discharges that are absent in the velocities (Fig 6). For all the considered schemes, the largest error is observed at the front of the moving water. Adding constraints on water discharges to the LHQE scheme to calculate flux limiters eliminates oscillations in numerical solutions. The numerical results obtained with LHE(LHQE) Figure 6: One-dimensional dam break over a dry bed. Comparisons of numerical results obtained with FCT schemes whose flux limiters are computed using exact and approximate solutions to a linear programming problem with discrete entropy inequality and different constraints at time t=7 s. and AHE(AHQE) are in a good agreement (Fig. 6). The flux limiters for LHE(LHQE) and AHE(AHQE) are calculated using exact and approximate solutions to linear programming problems. Dam Break Over a Step. In this test [5], a dam break over a downward bottom step is considered. The bottom topography and the initial data are given as follows z(x) = 1 if x ≤ 0, 0 otherwise , h(x, 0) = 0.75 if x ≤ 0, 1.0 otherwise , Q(x, 0) = 0,(6.5) After a dam break, the solution consists of a left rarefaction wave, a stationary shock wave at an intermediate height of the bottom step between two stationary contact waves located at the bottom discontinuity, and a right shock wave [11,22]. Comparisons of the numerical results obtained on a uniform grid of 200 cells with a reference solution at t=0.1 after the dam break are shown in Fig. 7-9. The reference solution was calculated using a central-upwind scheme of second-order spatial accuracy [14] on a uniform grid with 2000 cells. In Fig. 7, the PP scheme generates oscillations in the numerical results in the area of the bottom discontinuity. In the numerical results obtained with the ZL scheme, we see an overshoot of the water depth and discharge for the right shock wave. The secondorder HR2 scheme does not reproduce the left rarefaction wave in its whole entirety, as well as the shock wave (Fig. 8). In Fig. 8-9, the right side of the shock wave for the LHE, LHQE, AHE, and AHQE schemes shows an overshoot of the simulated water depth and discharge. Figure 9: Dam break over a step. Comparisons of numerical results obtained with FCT schemes whose flux limiters are computed using exact and approximate solutions to a linear programming problem with discrete entropy inequality and different constraints. (t=0.1 s, N=200). We note that none of the considered schemes reproduces the exact solution, especially in the bottom discontinuity. Table 3 shows the L1-norm error between the reference solution and the numerical solutions at time t = 0.1 for different difference schemes. Table 3: L 1 -norms of errors for the numerical solutions of the 1D dam break over a step at t = 0.1s with N =200. HR1 ZL P P HR2 H 5.219×10 −3 8.647×10 −3 6.039×10 −3 6.840×10 −3 Q 1.390×10 −2 2.489×10 −2 1.806×10 −2 1.703×10 −2 LHE AHE LHQE AHQE H 4.830×10 −3 4.100×10 −3 4.695×10 −3 4.656×10 −3 Q 1.326×10 −2 1.096×10 −2 1.243×10 −2 1.209×10 −2 We also note that the numerical results obtained with LHE(LHQE) and AHE(AHQE) agree well (Fig. 9). The flux limiters for LHE(LHQE) and AHE(AHQE) are calculated using exact and approximate solutions to linear programming problems. Steady Transcritical Flow With a Shock Over a Bump. We consider a test taken from [9] consisting of a transcritical flow with a shock over a bump. The bed topography of a rectangular channel 25 m long is given as follows z(x) = 0.2 − 0.05(x − 10) 2 if 8 < x < 12, 0 otherwise . (6.6) Initial conditions satisfy the hydrostatic equilibrium h + z = 0.33 and Q = 0. (6.7) Discharge Q = 0.18 m 2 /s and water level h + z = 0.33 m were set as upstream and downstream boundary conditions. In the steady-state solution, the flow to the left of the bump is subcritical, then closer to the end of the bump it becomes supercritical, and after a hydraulic jump it is subcritical again. Numerical results for the steady state, obtained on a uniform grid of 100 cells, are shown in Fig. 10-11. In the numerical results obtained with the HR1 scheme, we observe an overshoot of the free surface before the bump and an undershoot of the free surface for the PP scheme. The free water surfaces calculated with HR2, LHE, and LHQE agree fairly well with the analytical Figure 11: Steady transcritical flow with a shock over a bump. Comparisons of numerical results obtained with FCT schemes whose flux limiters are computed using exact and approximate solutions to a linear programming problem with discrete entropy inequality and different constraints. solution, with slight deviations around the hydraulic jump. Numerical oscillations for water discharges near the hydraulic jump are present for all compared schemes. Small oscillations are also present in the calculated discharges with the LHE and LHQE schemes in the whole modeling area. The L 1 -norm error between the exact and numerical solutions are shown in Table 4. Note that none of the considered schemes is well-balanced for moving water steady states with non-zero discharges. We also note that the numerical results obtained with LHE(LHQE) and AHE(AHQE) agree well (Fig. 11). The flux limiters for LHE(LHQE) and AHE(AHQE) are calculated using exact and approximate solutions to linear programming problems. Drainage on a Non-Flat Bottom We consider drainage of a symmetric rectangular reservoir to a dry bed through its boundaries, leaving water in topographic depressions. Due to the symmetry, the flow is computed on half the domain, with wall boundary conditions on the left boundary, and open boundary conditions on the right boundary. The boundary condition on the right side of the domain allows water that was at rest to flow freely through the right boundary into the originally dry region. The bottom topography consists of one hump Numerical results of water flow at different times, obtained on a uniform grid with N=200 cells, are presented in Fig. 12-14. Fig. 12 shows that all numerical schemes, except HR1, give similar results for the water surface level. The first-order HR1 scheme produces a more diffusive water level profile. The most significant difference in the computed discharges is observed over the right side of the hump. In Fig. 13, the numerical results obtained with the LHE and AHE schemes are in good agreement, in contrast to the results presented in Fig. 14. Note that the AHE and AHQE schemes use flux limiters, which are approximate solutions of the corresponding linear programming problems. The numerical results in Fig. 14 show their strong dependence on the numerical diffusion of the applied difference schemes. 2D Partial Dam Break In this section, a partial dam break problem with a nonsymmetrical breach is considered. The spatial domain is defined as a channel with 200 m in length and 200 m in width, the dam is located in the middle of the domain at a distance of 100 m. The bottom is horizontal and frictionless. Initially, the upstream and downstream water depths are set at 10 and 5 m, respectively. The breach is 75 m long, located 30 m from the left bank and 95 m from the right bank. The computational domain is discretized by a 40 x 40 square grid. Fig. 15-20 show a threedimensional view of the water surface levels and water depth contours 7.2 s after the dam failure. The numerical results obtained with the HR1 scheme are the most diffusive of the others. The water surface levels and water depth countours obtained by the LHE, LHQE, AHE, and AHQE schemes are similar to the numerical results of HR2 but are non-smooth. AHE and AHQE, whose flux limiters are approximate solutions to the corresponding optimization problems, produce smoother solutions than LHE and LHQE, but their solutions are nonsymmetric about the center of the breach. Conclusions We presented the flux correction design for a hybrid scheme to obtain an entropy-stable solution of shallow water equations with variable topography. The hybrid scheme is an explicit HR scheme whose numerical flux is a convex combination of a first-order Rusanov flux and a highorder flux. We studied the conditions under which a first-order HR scheme with the Rusanov flux satisfied the fully discrete entropy inequality. The flux limiters for the hybrid scheme can be an exact or approximate solution to the corresponding optimization problem in which constraints valid for the first-order HR scheme are applied to the hybrid scheme. It is proved that in the vicinity of a numerical solution of the first-order HR scheme, there is a unique flux correction with flux limiters that are the proposed approximate solution to the optimization problem. Numerical examples show that the hybrid HR scheme can produce oscillations in numerical results if only water surface level constraints for the optimization problem are used to compute the flux limiters. We also note that numerical results obtained with hybrid HR schemes whose flux limiters are exact and approximate solutions to the optimization problem can differ significantly. Conflict of interest The author declare that he has no conflict of interest. Data Availability Statement All data generated or analysed during this study are included in this published article. Theorem 3 . 3 . 33Suppose that c i+1/2 and ∆t satisfy the inequalities 4.1) has a unique solution if ∆t satisfies (4.4). Remark 4. 1 . 1Note that the mapping S in (4.2) is contractive in the vicinity of a low-order solution with the HR scheme (2.1)-(2.7) if the mapping P in (4.1) is Lipschitz-continuous. Remark 4. 2 . 2It is clear that the linear programming problem (4.13)-(4.16) is solvable if ∆t satisfies inequalities (2.8) and (3.23). It follows from the non-emptiness of the feasible set and the boundedness on U ad of the objective function ℑ(α). be a Lipschitz-continuous function on a closed ball O 0 =Ō(v L , δ), wherev L is a solution of the system of equations (2.1). Then the flux limiters α, defined by (5.9), are Lipschitz-continuous on O 0 . Proof. It is clear thatᾱ y i+1/2 ,ᾱ q i+1/2 , B i , d ik are constants, and A i are Lipschitz-continuous functions on O 0 . Thus, α i+1/2 ,ᾱ U i+1/2 are Lipschitz-continuous on O 0 , since the minimum of Lipschitz-continuous functions is again a Lipschitz-continuous function. Remark 5 . 1 . 51The hypotheses of Theorems 4.1 and 5.1 are satisfied if U (v) is a strictly convex function, and α i+1/2 are calculated using (5.1)-(5.9). In this case, the system of equations (4.1) has a unique solution. Figure 1 : 1One-dimensional dam break over a wet flat bed. Comparisons of exact solutions with simulated water depths (left) and discharges (right) using HR1, ZL, and PP at t=10 s. The number of cells is N=100. Figure 3 : 3One-dimensional dam break over a wet flat bed. Comparisons of numerical results obtained with FCT schemes whose flux limiters are computed using exact and approximate solutions to a linear programming problem with discrete entropy inequality and different constraints. The number of cells is N=100. Figure 4 : 4One-dimensional dam break over a dry bed. Comparisons of exact solutions with simulated water depths and discharges using HR1, ZL, and PP at time t=7 s. The number of cells is N=100. Figure 5 : 5One-dimensional dam break over a dry bed. Comparisons of exact solutions with simulated water depths and discharges using HR2, LHE, and LHQE at time t=7 s. The second row is a zoom in the area of the front of the moving water. The number of cells is N=100. Figure 7 : 7Dam break over a step. Comparisons of reference solutions with simulated water depths and discharges using the HR1, ZL, and PP schemes at time t=0.1 s with N=200 cells. Figure 8 : 8Dam break over a step. Comparisons of reference solutions with simulated water depths and discharges using HR2, LHE, and LHQE at time t=0.1 s with N=200 cells. On the right is a zoom of the area of the bottom discontinuity and the right shock wave. Figure 10 : 10Steady transcritical flow with a shock over a bump. Comparison of exact solutions with computed water depths and discharges obtained by HR1, PP, HR2, LHE, and LHQE with N=100. Figure 12 : 12Drainage on a non-flat bottom. Water levels and discharges at various times t=0.15,0.25,0.5,1.0 s. begins, the solution converges to a steady-state solution in which water exists only to the left of the hump. Figure 13 : 13Drainage on a non-flat bottom. Water levels and discharges at times t=0.5,1.0 s. Figure 14 : 14Drainage on a non-flat bottom. Water levels and discharges at times t=0.5,1.0 s. Figure 15 : 15Water surface levels and depth contours for the partial dam-break flow at t = 7.2 s computed with the HR1 scheme. Figure 16 : 16Water surface levels and depth contours for the partial dam-break flow at t = 7.2 s computed with the HR2 scheme. Figure 17 : 17Water surface levels and depth contours for the partial dam-break flow at t = 7.2 s computed with the LHE scheme. 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[ "Assess and Summarize: Improve Outage Understanding with Large Language Models", "Assess and Summarize: Improve Outage Understanding with Large Language Models" ]	[ "Pengxiang Jin ", "Shenglin Zhang ", "Minghua Ma ", "Microsoft China ", "Haozhe Li ", "China Yu Kang ", "Liqun Li ", "Yudong Liu ", "Bo Qiao ", "Microsoft China ", "Chaoyun Zhang ", "Pu Zhao ", "Shilin He ", "Federica Sarro ", "Yingnong Dang ", "Saravan Rajmohan ", "Qingwei Lin ", "Dongmei Zhang ", "\nNankai University\nChina\n", "\nPeking University\nMicrosoft China\nUniversity College London\nMicrosoftUK, USA\n", "\nMicrosoft China\n\n" ]	[ "Nankai University\nChina", "Peking University\nMicrosoft China\nUniversity College London\nMicrosoftUK, USA", "Microsoft China\n" ]	[]	Cloud systems have become increasingly popular in recent years due to their flexibility and scalability. Each time cloud computing applications and services hosted on the cloud are affected by a cloud outage, users can experience slow response times, connection issues or total service disruption, resulting in a significant negative business impact. Outages are usually comprised of several concurring events/source causes, and therefore understanding the context of outages is a very challenging yet crucial first step toward mitigating and resolving outages. In current practice, on-call engineers with in-depth domain knowledge, have to manually assess and summarize outages when they happen, which is time-consuming and labor-intensive. In this paper, we first present a large-scale empirical study investigating the way on-call engineers currently deal with cloud outages at Microsoft, and then present and empirically validate a novel approach (dubbed Oasis) to help the engineers in this task. Oasis is able to automatically assess the impact scope of outages as well as to produce human-readable summarization. Specifically, Oasis first assesses the impact scope of an outage by aggregating relevant incidents via multiple techniques. Then, it generates a human-readable summary by leveraging fine-tuned large language models like GPT-3.x. The impact assessment component of Oasis was introduced in Microsoft over three years ago, and it is now widely adopted, while the outage summarization component has been recently introduced, and in this article we present the results of an empirical evaluation we carried out on 18 real-world cloud systems as well as a human-based evaluation with outage owners. The results obtained show that Oasis can effectively and efficiently summarize outages, and lead Microsoft to deploy its first prototype which is currently under experimental adoption by some of the incident teams.CCS CONCEPTS• Computer systems organization → Cloud computing; • Software and its engineering → Maintaining software.	10.48550/arxiv.2305.18084	[ "https://export.arxiv.org/pdf/2305.18084v1.pdf" ]	258,959,521	2305.18084	9943bbb97a48d10b70453e62307c1c797ed64012	Assess and Summarize: Improve Outage Understanding with Large Language Models Pengxiang Jin Shenglin Zhang Minghua Ma Microsoft China Haozhe Li China Yu Kang Liqun Li Yudong Liu Bo Qiao Microsoft China Chaoyun Zhang Pu Zhao Shilin He Federica Sarro Yingnong Dang Saravan Rajmohan Qingwei Lin Dongmei Zhang Nankai University China Peking University Microsoft China University College London MicrosoftUK, USA Microsoft China Assess and Summarize: Improve Outage Understanding with Large Language Models *Work done mainly during internship at Microsoft Research Asia.Outage UnderstandingLarge Language ModelCloud Systems Cloud systems have become increasingly popular in recent years due to their flexibility and scalability. Each time cloud computing applications and services hosted on the cloud are affected by a cloud outage, users can experience slow response times, connection issues or total service disruption, resulting in a significant negative business impact. Outages are usually comprised of several concurring events/source causes, and therefore understanding the context of outages is a very challenging yet crucial first step toward mitigating and resolving outages. In current practice, on-call engineers with in-depth domain knowledge, have to manually assess and summarize outages when they happen, which is time-consuming and labor-intensive. In this paper, we first present a large-scale empirical study investigating the way on-call engineers currently deal with cloud outages at Microsoft, and then present and empirically validate a novel approach (dubbed Oasis) to help the engineers in this task. Oasis is able to automatically assess the impact scope of outages as well as to produce human-readable summarization. Specifically, Oasis first assesses the impact scope of an outage by aggregating relevant incidents via multiple techniques. Then, it generates a human-readable summary by leveraging fine-tuned large language models like GPT-3.x. The impact assessment component of Oasis was introduced in Microsoft over three years ago, and it is now widely adopted, while the outage summarization component has been recently introduced, and in this article we present the results of an empirical evaluation we carried out on 18 real-world cloud systems as well as a human-based evaluation with outage owners. The results obtained show that Oasis can effectively and efficiently summarize outages, and lead Microsoft to deploy its first prototype which is currently under experimental adoption by some of the incident teams.CCS CONCEPTS• Computer systems organization → Cloud computing; • Software and its engineering → Maintaining software. INTRODUCTION With the trend of large IT enterprises such as Microsoft, Amazon, and Google deploying services to the cloud platforms, cloud systems have had a booming development in recent years [4,8,23]. Tremendous efforts have been devoted to improving the reliability of cloud systems, however, unplanned incidents or performance degradation are still inevitable due to the complex and dynamic nature of cloud systems. Often these incidents escalate to a so called outage, which impacts multiple services and customers. Once an outage occurs to a cloud system, it is crucial to understand its impact scope as soon as possible in order to promptly notify customers, mitigate issues [4,24], and ultimately resolve the outage, aiming at reducing as much as possible the loss associate with it. Nevertheless, a cloud system is quite complex and involves many services such as across-region infrastructures, virtual machines, networking, and database systems, thus making this task very challenging. To support engineers in monitoring the reliability of the cloud system, each cloud system service has multiple monitors that create an incident each time something wrong occurs. For example, Figure 1 shows the timeline of an incident caused by a flawed configuration change in the Storage service. The failed storage affected several SQL databases, and the failure was further propagated to web application instances that depend on the impaired databases. Finally, the outage is declared and associated with the multiple incidents occurred in the storage, SQL, and web application services. Doing this job manually is not trivial, and in some cases not even feasible. Being able to efficiently aggregate all and only those incidents which are relevant to a given outage, would empower the engineers to promptly investigate the impact scope, as it greatly reduces the number of incidents that need to be investigated. Previous studies [8,11,17,29] have devoted a lot of efforts to dealing with incidents aggregation or linking the relevant incidents to the outage. However, based on our real-world experience in Microsoft, we observe that on-call engineers (OCEs) still need to manually check the detailed information of relevant incidents and write a summary of outages (a real-world example in Section 3.2), which is helpful to further handle the outage in terms of notification, mitigation, diagnosis, and resolution. To the best of our knowledge, extensive studies on outage understanding are lacking. Therefore, in this paper, we first empirically investigate the negative effects of outages in worldwide popular cloud systems in Microsoft and how engineers currently deal with them. To this end, we exploit data collected from the usage of 18 real-world cloud systems (many of which are worldwide popular systems) over the past three years. We found that most outages have a huge negative impact on customers, and the median summarization time is 1 time unit (about one hour) 1 . Therefore, in practice, engineers have to spend significant efforts to understand outages. Besides, the content of outage summaries often contains detailed when, where, who, what, and why. This information is complex and cannot simply adopt as a template because it must be readable by engineers from various component teams. Thus, it is necessary to automatically summarize outages for understanding quickly. These results motivated us to explore automated ways to improve engineers' understanding of outages. To this end, we propose Oasis, which has two components: impact scope assessment and summary generation. As for impact scope assessment, we adopt three techniques i.e., rule-based, historical lookup, and deep learning based to aggregate relevant incidents to the outage. To embed domain knowledge of cloud systems, engineers implement some linking rules from incidents to outages. To automatically learn the correlations among components of cloud systems, we propose a historical lookup algorithm to form a component graph based on the historical incident linkage and match new incidents in the graph. To capture the rapid evolution of cloud systems, the deep learning based linking approach is used. The impact scope of an outage is composed of the relevant incidents aggregated by these three techniques. We have deployed the impact assessment component of Oasis in Microsoft, which is running for over three years and achieve significant results in impact scope assessment. After we obtain relevant incidents of the outage, we adopt the most popular pre-trained large language models GPT-3.x (both GPT-3.0 and GPT-3.5), to automatically generate outage summaries. This task presents two main challenges: 1) identifying which information on relevant incidents is helpful to outage summarization; 2) identifying how to effectively generate domain-specific outage summaries with complex cloud-related information. For the first challenge, our empirical study provides some guidelines on summarizing outages, which reveals the importance of incident severity and description. To tackle the second challenge, we fine-tune the pre-trained large language model, which can generate human-readable sentences and embed with knowledge from cloud systems. To investigate the effectiveness of Oasis, we conduct extensive experiments using real-world outages from Microsoft. The results show that Oasis is able to effectively and efficiently generate outage summaries and titles for cloud systems, and significantly outperform all the compared approaches [22,27]. More specifically, Oasis achieves scores of 0.665 (BLEU-4), 0.742 (ROUGE-L), and 0.734 (ME-TEOR) with its summarization which outperforms state-of-the-art approaches by at least 32.3%. Furthermore, to investigate the usefulness and readability of our generated summaries, we conduct a preliminary human evaluation involving 54 outage owners. Based on the rankings of summaries produced by models and the original OCEs, we find that Oasis can achieve human-level summaries much more quickly (251.2 times faster than the median of manual summarization). 1 Due to the company policy, we hide the actual time and normalize it as time unit. Based on the above results, the Oasis outage assessment component has already been in usage for over three years at Microsoft, while the more recent summarization component has been now prototyped and used by some of the incident teams at Microsoft in a phase preceding the final rolling in production. To sum up, our work has the following contributions: • We are the first to identify outage understanding, a practical scenario for large-scale cloud services. We have conducted an empirical study of 18 cloud systems to investigate this scenario. • We propose Oasis, the first automated approach to tackle the problem of outage understanding based on impact scope assessment and large language models (LLMs). We are the first to propose LLM-based summary generation of outages. • Our impact scope assessment of Oasis has deployed in Microsoft for over three years and achieved significant impact. We conduct an extensive study and human evaluation to demonstrate the efficacy and potential usage of Oasis. BACKGROUND Cloud systems. Cloud systems have become increasingly popular in recent years, as they offer a range of benefits such as scalability, accessibility, and cost-efficiency. To ensure the reliability of these systems, engineers use various monitoring tools and techniques, e.g., Azure Monitor, to track and analyze the performance and health of different levels and components of the cloud system [10,14]. If the monitors detect anomalies, incidents will be reported. Incidents. Incidents are unplanned interruptions to cloud service. Incident Management is the process of logging those interruptions, and resolving those in a timely manner [5,7,10,16,19,20]. An incident is reported with many fields, for example, the service that the incident is defined on, the source of the incident creation, the time of the incident creation, and a text field describing the problem. The text description can be generated by the monitor based on predefined templates or filled in manually by the engineers. Moreover, engineers assign a severity level to each incident, ranging from 0 to 4, where a severity of 0 means highest priority and large customer impact, and a severity of 4 means lowest priority. Outages. Outages are severe incidents that require collaboration across many services or result in customer impact [9,29]. Different products and teams may define outages differently depending on service level agreements (SLAs), customer expectations, or other criteria. When an outage happens, it tends to affect various aspects of the cloud system, causing many incidents to be reported. OCEs need to go through these incidents to fully understand the outage. IcM system. To facilitate mitigating and resolving outages, our collaborated Microsoft has developed an Incident Management system (IcM) for cloud systems. After a monitor reports an incident, an associated incident is created on the IcM. Then engineers can discuss the incident, check the information, and update the status of the incidents on the IcM page, etc. An incident may escalate and is declared as an outage if it impacts multiple services or customers as shown in Figure 1. During these processes, records of incidents and logs of the actions are persistently stored in the IcM database. OUTAGE UNDERSTANDING: A CASE STUDY To better understand the impact of outages and the need for automatic support in outage scope and summary production, we conduct a case study on real-world outages and their summaries. To this end, we collect outages from 18 systems over three years in the IcM database of Microsoft, which serves millions of daily users worldwide, specifically, outages and relevant incidents that occurred between January 1, 2020 and October 1, 2022. To ensure that the outages have undergone careful examination and their summaries are ready, we keep over 6000 outages whose state is 'MITIGATED' or 'RESOLVED' during collection. We are not able to make all the details public due to the company's policy. In this study, we address the following research questions: • RQ1: What is the impact of outages? • RQ2: What are the information included in outage summaries? • RQ3: What is the cost (in terms of time) of manually summarizing outages? 3.1 RQ1: Impact of Outages Impact on customers. When OCEs deal with outages, it is important to decide the impact on customers, especially the number of customers affected. For each outage, OCEs determine whether it impacts a large number of users and record this determination. We statistically analyze the outages that OCEs considered as impacting a large number of users and found that such outages accounted for as much as 86.4% of all outages. Outages usually have a significant impact on cloud systems, resulting in a degraded user experience for a large number of customers. Therefore, it is crucial to quickly and effectively respond to outages. Another aspect of the customer impact is whether an outage has resulted in persistent impacts. OCEs mark the outages that have persistent or intermittent impacts with a flag variable. The number of outages resulting in persistent impacts is 1.81 times more than the number of outages that have intermittent impacts. The impact of an outage on a cloud system is frequently severe. Relevant incidents. Several incidents in cloud systems are continuously reported and escalate to one outage, as they share a common root cause. The distribution of incidents associated with outages is illustrated in Figure 2(a), with 25% of outages having more than 10 associated incidents. The average number of relevant incidents to an outage is 9.36. Based on this data, the outages bring about many incidents, consuming the efforts and time of the OCEs. RQ2: Outage Summary Information To help understand what information needs to be summarized for an outage, we demonstrate a real-world outage summary written by OCEs and its relevant incidents. We mask several details due to confidentiality. The Email Service experienced connectivity issues to their replica database in the West US Region. This affected customer email delivery for approximately 3 internal company services. Due to this issue, System-Cloud customers were not receiving notifications including purchase, renewal, and monitor alert notifications. The Portal team reported that approximately 1 customers were unable to upgrade their subscriptions on URL-Cloud-Portal. Incident We can see from the above example that each relevant incident describes various aspects of the outage, and the information about an outage fall into many different categories. For example, West US Region is a physical location, and {System-Cloud, Email Service, API-Marketplace} are software components at different layers that are affected, and { 1 , 2 , 3 } are specific numbers describing the number of impacted customers or services, and 5xx HTTP error is a software bug that affects the service functionality. Formally, the information of an outage usually involves 5W (when, where, who, what, why): When. When does the outage start impact, get declared, and engaged? Engineers pay attention to several time points and periods of an outage. For example, the time when the outage starts to make an impact, when the outage is declared, and when the OCEs start to engage are important signals for assessing the availability and reliability of the system. Additionally, when assessing the impact of an outage, it is also useful to know the time window period when a certain function is unavailable. Where. Where does the outage come from? The physical location of an outage can lie in various levels of the cloud infrastructure. The physical location can have an impact on the time required to resolve it and the potential for cascading failures. Additionally, the physical location of an outage can be a key factor in determining the impact on customers, as local or nearby customers may be affected more severely. The physical location of the cloud infrastructure at Microsoft is structured in a hierarchical manner [18] with regions and availability zones at the top level, which is directly accessible to customers. Each region can consist of up to three availability zones, each containing one or more datacenters. These datacenters are further divided into clusters. Despite the fact that other cloud systems may exhibit different location hierarchies, it is as important to know the location of outages. Who. Which services are suffering from the outage? Services of cloud systems can be divided into different layers: (1) application layer: this layer contains the actual code and functionality of the cloud system, where frontend and backend services are located, (2) platform layer: this layer provides the operating system, middleware, and runtime environment for the cloud system, which may include virtualization software, container orchestration software, or serverless computing framework, (3) data layer: this layer handles the storage and management of data used by the cloud system, which may include databases, data lakes, and data warehouses, (4) infrastructure layer: this layer provides the underlying physical and virtual resources that are used to run the cloud system, which includes host servers, storage, and networking. Each layer has its fine-grained components. Assessing which parts of the cloud system are affected by the outage is helpful to handle the outages. What. What happens to the cloud system in the outage? Previous research has shown [14] some common symptoms of outages, including: (1) code bugs, such as buggy or incompatible code that generates error results, (2) dependency failures, such as an unhealthy dependent service that impacts the functioning of downstream services, (3) infrastructure issues, such as high CPU utilization of a server that prevents the service from functioning normally, (4) deployment errors, such as an engineer deploying an incorrect certificate. There are also other less frequent symptoms, such as configuration bugs, database/network issues, authentication failures, etc. Why. Why did the outage happen? Previous research has investigated the four most common root causes of outages [21]: (1) insufficient or erroneous mechanism of fault handling (e.g., error component, unresponsive component, and silent corruption), (2) data format incompatibility between different software components, (3) timing(e.g., concurrent) bugs, and (4) misconfigured or outdated constant values. However, the bugs in production cloud systems are highly diversified. The underlying causes of misbehavior require thorough manual investigation by OCEs. Focus of outage summarization. We can see from the example that the summary of an outage is not simply a list of information. OCEs when writing outage summaries are more favorable to highseverity incidents. Moreover, textual description is an important reference in the outage summarization process. For example, the dashed sentences are taken directly from the textual descriptions of two high-severity incidents, i.e., Incident 1 and 2. Finding: High-severity incidents and their textual descriptions are important in outage summarization. RQ3: Time to Summary After an outage starts to make an impact on customers, OCEs need to quickly respond to the outage. One key step is to summarize the context of the outage. Therefore, we investigate the time to manually write outage summaries. Specifically, we retrieved the impact start time (T1 in Figure 1) of the outage and its summary completed time (T2). The time needed to summarize outages is calculated by T2 − T1. Summary According to our empirical study, the impact summary of an outage may include domain specific terminology of physical or logical locations, service name, code change name, etc. Besides, what and why of outages are even more difficult to summarize simply using templates. OCEs must have a thorough grasp of the relevant incidents in order to effectively summarize an outage, and the text descriptions of these incidents provide crucial information for this comprehension. Nowadays, pretrained large language models (e.g., GPT-x) show their ability in many tasks, such as Q&A and summarization in ChatGPT. Therefore, we aim to employ large language models to help outage summarization. In this work, we aim to generate outage summaries with the following goals: Usefulness. Usefulness measures whether the outage summary contains relevant and valuable information. Readability. Readability measures whether the outage summary read fluently, especially considering the context information of the outage and the affected system. Reducing TTS (time to summary). It is desirable to summarize outage in a short time because it helps improve the overall outage handling process, improve communnication, shorten the lifecycle of outage, and in turn, improve customers' satisfaction. OUR PROPOSAL: OASIS 4.1 Overview In this paper, we aim to automatically generate summaries for outages of cloud systems. However, outage summarization faces two challenges. The first challenge is determining which information on relevant incidents is helpful to outage summarization. Since the cloud system is complex and rapidly growing, it is not trivial to extract domain-specific terminologies of incidents. The second challenge is how to effectively generate human-readable outage summaries with complex cloud-related information. To solve these challenges, we propose Oasis. The overview of Oasis is shown in Figure 3, which consists of the following two components. In the first component, i.e., impact scope assessment, Oasis identifies relevant incidents via three types of linking to comprehensively assess the impact of the outage. In the second component, i.e., summary generation, Oasis first performs domain-specific text processing to denoise and prioritize important information from the obtained relevant incidents, thus addressing the first challenge. Then Oasis employs a fine-tuned large language model, i.e., GPT-3.x, to understand the incidents and generate a compact summary for the outage, thus addressing the second challenge. Impact Scope Assessment Assessing the impact scope of an outage is about comprehensively understanding different aspects of an outage such as the when, where, who, what, why, etc. As shown in Section 3, the impact of these aspects of the outage is collectively described by many relevant incidents. However, there is no simple and direct way to identify the set of relevant incidents, since incidents with the same underlying root cause can have different properties and spread across services. Meanwhile, if an OCE determines two incidents are highly relevant to each other, she can formally link the two incidents together, which is a feature provided by the IcM system. Linking an incident with relevant incidents reduce the effort of OCEs in many ways, for example, reducing the number of incidents that require manual examination, auto-resolving less severe incident if a more severe linked incident is resolved, etc. Inspired by the process, Oasis assesses the impact scope of an outage by linking its relevant incidents. To completely link the relevant incidents of an outage, Oasis incorporates domain knowledge and historical linking patterns. Specifically, Oasis performs three types of incident linking: linking by rule, linking by historical lookup, and linking by prediction model. Linking by rules. Automated incident linking is a capability in IcM that correlates and de-duplicates incidents to reduce alert storms and noise. Engineers can set up specific rules to create links between incidents upon various triggers, which represent the domain knowledge of the engineers. For example, an engineer can set up a rule to have incidents that are triggered by the same KPI anomaly be linked. These are structural incident links that can be directly queried from the IcM database. During the operation of cloud systems and the corresponding IcM system, a large number of historical incidents and rule-based links are persistently recorded. These data are a natural source of labeling for learning, which facilitates the following two types of linking. Linking by historical lookup. We propose a heuristic lookup mechanism to utilize the historical links between incidents. The mechanism consists of an offline phase to memorize the historical linking pattern, and an online phase to apply the patterns to current incidents. In each phase, we use the field of component that is reported along with the incident. Components are fine-grained parts of cloud systems that are defined by engineers. In the offline phase, we build a component linking graph by summing up links between incidents, i.e., if incident A and incident B are linked, then we link their component in the component graph. In the online phase, we check whether there are active incidents (incidents within a short time range) on the linked components. Linking by prediction model. Another way to automatically discover the relationship between incidents is by employing deep learning techniques [8,17]. It has the advantage of being highly automated and can be applied to a large number of incidents. Also, it can be applied to scenarios where new incident detection criteria are created and the engineers have not set up rules and historical links fail to apply because of the lack of historical data. We train a neural network to predict the link between incidents. The neural network takes the titles and descriptions of two incidents as input and outputs the similarities between the titles. If the similarity of two incidents is larger than a threshold, we determine the two incidents are linked. The neural network has been trained on pairs of incidents to learn the relationships between incident linking and incidents' titles and descriptions. In the training set, incident pairs Put them together. Oasis periodically assesses relevant incidents to the outage by querying the information of incidents within a time window to the outage. We take advantage of three linking approaches: the rule-based linking has the highest confidence and interoperability; the historical lookup may find hidden dependencies; the prediction model can adapt to the rapid evolution of cloud systems. Together, these three linking approaches link an outage to a set of relevant incidents, which will be used to generate the outage summary. Summary Generation After gathering relevant incidents of an outage, Oasis generates a summary of the outage based on the incidents' information. To overcome the challenge of noisy information, we fine-tune an LLM, i.e., GPT-x, to summarize the relevant incidents. GPT-x. Generative Pre-trained Transformer x (GPT-x) [3] is a large pretrained language model that can tackle a wide range of natural language processing (NLP) tasks. One typical usage scenario of GPT-x is text completion, where the model is given a block of text as context and generates text as the completion of the context. It has been explored to recommend the root causes of cloud incidents [2]. The GPT-x model is based on Transformers [28], which takes advantage of the attention mechanism to assign weights to different parts of the text. Thus it is suitable to summarize noisy information. There are different sizes (number of parameters) of GPT-x model. In our work, we implement Oasis with two parameter sizes: GPT-3-Curie and GPT-3.5-DaVinci (see Section 5.4 for more details). Domain-specific text processing. In this step, we process the structural incident information into a paragraph of text so that the GPT-3 model can take it as input, i.e., context. Inspired by the findings from Section 3, we propose to process incidents in a way that the high-severity incidents and textual descriptions are emphasized. First, we sort the relevant incidents by their severity so that the incidents with higher severity precede the ones with lower severity. Then we transform the incidents into a piece of text in the following way: for the incident with sorted order , the text is Fine-tuning. The GPT-3 model was trained on a general corpus that allows the model the learn various knowledge like linguistics, common knowledge, factual knowledge, basic logical inference ability, etc. To achieve better summary generation, we use our IcM-specific data to fine-tune the GPT-3 model so that it learns the domain knowledge of the applied cloud system and incidents. Moreover, the training samples presented to the model teach it to emphasize the aspects that are of interest to OCEs, thereby improving its ability to summarize information from noisy sources. The data we use to fine-tune the model is in the same form of summary generation, i.e., for each outage, we provide the relevant incidents as context and the outage summary written by engineers as the desired completion. System Implementation We have deployed Oasis as an aid to the IcM system of Microsoft. We will introduce the integration of Oasis with the IcM workflow and the underlying implementation details. The implementation of Oasis in production consists of four parts, as shown in Figure 4. The Oasis backend periodically queries the local database to get active incidents within the time window, as well as rule-based and historical-lookup links of all current outages (1) . On receiving the API call initiated by the IcM Backend querying a specific outage, the Oasis backend applies the prediction model to determine what other incidents should be linked to the outage (2). After that, it performs domain-specific text processing for the outage's relevant incidents and feeds the processed text to the finetuned GPT-3.x model (3). Finally, the backend returns the summary generated by GPT-3.x to the IcM Backend. The local database of Oasis is ingested from the IcM database in a streaming manner. Compared to batch ingestion, streaming ingestion is more smooth in resource utilization. Moreover, the linking requires real-time data records of incidents. The prediction model has already been trained in ML-dedicated servers and exported as a binary file to minimize the operation effort of Oasis in production. OASIS EVALUATION: EMPIRICAL STUDY DESIGN To assess the effectiveness of OASIS, we investigate the following: • RQ4: Is Oasis effective at summarizing outages? Generating the summary of outages is the main task of Oasis. We are interested in the ability of Oasis to generate a reasonable outage summary with automatic impact scope assessment. • RQ5: Is Oasis effective at proposing outage titles? The title of an outage is a short, highly abstracted piece of text stating the problem that is happening. Proposing outage titles also demonstrates the ability of Oasis to understand and summarize the outage. • RQ6: Does Oasis get better at summarizing outages if the outage title is given? In practical settings, OCEs first write the title of an outage and then write the summary of the outage. We are interested in whether Oasis can better summarize an outage if the title written by OCEs is also given as part of the context. • RQ7: What is the time efficiency of Oasis? Since Oasis needs to work in the production environment, it is important for Oasis to summarize outages efficiently. Study data In the study, we applied Oasis to the same 18 cloud systems and the same time range (3 years) in Section 3.1 to evaluate the effectiveness of Oasis. In particular, we split the data in chronological order using a 7:1:2 ratio for the training (fine-tuning), validation, and test sets, respectively. Each data point, representing an outage, is presented as a context-completion pair. The context consists of the processed text from relevant incidents linked by impact scope assessment. The completion, on the other hand, is provided as the summary of the outage written by OCEs. Compared approaches To better answer the RQ 4 to 7, we compare the performance of Oasis with some baseline approaches. We formulate the task as a text generation problem, therefore we compare with 3 methods that have been proven capable of generating summarization. In answering each research question, we provide the same context (information of relevant incidents) to the baselines and to the GPT-3 model of Oasis. • Joint incident summary (Rule-based): A straightforward rulebased method that concatenates all the information of incidents. This method imitates the behavior that OCEs read through relevant incidents when handling outages. • Information retrieval (IR): NNGen [22] leverages bag-of-words embedding and nearest neighbor to retrieve summaries from similar history outages. • GPT-2: Generative Pre-training Transformer 2 (GPT-2) is a language model that is trained to generate coherent text. We use GPT-2 with 117M parameters. Metrics Following the existing work [1,2,15], we use the BLEU-4, ROUGE-L, and METEOR to evaluate Oasis and its baselines in terms of readability. The BLEU-4 compares the matching of n-grams between generated text and the ground truth. The ROUGE-L is widely used in Machine Translation evaluation, which measures the overlap of the longest sequence between hypothesis and reference. The METEOR calculates the harmonic mean of unigram precision and recall with consideration of stemming and synonym matching. Specifically, we get five candidate generated texts from each model, except for the joint incident summary which can only give one piece of generation. To better evaluate the quality of generation models, we calculate the Top1 metrics using the first generated text, and the Top5 metrics using the best of five generated text. We also measure the running time of each approach. Specifically, we record the overall time needed to train/fine-tune the model, and the average time spent on generating a summary for an outage. To further evaluate usefulness and readability, we conduct a human evaluation in Section 7. When summarizing outages, the style of OCEs can vary from generic to specific. Automatic metrics only compare the models' suggestions with a single reference, while other versions of the summary can be useful and relevant as well, so these metrics may not fully capture the performance of models. To better evaluate the model's performance, we went to the owners (responsible engineers) of the outages and presented the outputs of our models and baselines. We will discuss our methodology and findings from the human evaluation in Section 7. Experiment environment Generation model. We implement Oasis with two GPT-3 variants, i.e., Curie and DaVinci: Curie (GPT-3) is a fast GPT-3 model with 6.7 billion parameters, which was pre-trained on a natural language corpus. DaVinci (GPT-3.5) is a large GPT-3 model with 175 billion parameters, which was pre-trained on both text and code. We fine-tune these generation models using the training and validation set from Section 5.1. Experiment environment. We implement all training with one NVIDIA GeForce A100 GPU, PyTorch 1.11, and CUDA toolkit 11.3.1. Implementation of baselines. Baselines are implemented using Python 3.8 and scikit-learn 1.0.2. The number of GPT-2's training epochs is 20. The temperature is GPT-2 is 0.7, which is recommended by a previous study [2]. Table 1 lists the effectiveness of baselines and Oasis in summarizing outages. Oasis with DaVinci, the largest GPT-3 model, achieves the best metrics with both Top1 and Top5 summary generation. The advantage of DaVinci over Curie comes from the larger parameter size and the extra code corpus used in pretraining since some incidents contain API names or investigating code. However, the performance gain of DaVinci over Curie, the fastest GPT-3 model, is modest in both Top1 and Top5 generations. We observe IR method is especially not suitable for outage summary generation. The major reason that the scores of IR are poor is that the rapid evolution of cloud systems has resulted in significant changes in the architecture of the systems over time, so similar outages are not likely to appear repeatedly, therefore historically useful summaries fail to depict the new outages. Although Rulebased summaries have a higher METEOR score than GPT-2, their BLEU-4 score is far lower than that of GPT-2. This is because the METEOR score takes into account the precision and recall of the unigram rather than subsequences, resulting in a more lenient evaluation of summaries. Rule-based summaries are often too long as the original incidents, which is not helpful for engineers to understand the context of outages. RQ5: Performance of Title Generation The title of an outage is a compact description of the outage. The example of an outage title and summary is shown in Section 3.2. The performance of baselines and Oasis at summarizing outages in the form of titles are listed in Table 2. By comparing Table 2 with Table 1, we achieve a higher generation score (0.826-0.857 BLEU-4) at generating titles for outages than generating the whole summary (0.654-0.664 BLEU-4). Similarly, GPT2, which is also a large Transformer based language model, scores higher in summarizing outages in the form of a title than the whole summary. The ROUGE-L and METEOR score exhibit the same trend for Oasis and GPT-2. The reason for this performance improvement lies in the nature of outage title and summary. The title of outage has a stronger pattern than that of outage summary. Firstly, a large portion of outage titles starts with "Outage for". This pattern is easy for LLM to learn, so the titles generated by LLM tend to have more overlapping words, and consequently, higher scores. Secondly, the words used in titles are usually either in a dictionary (e.g., the "Triage" in the example title), or have been mentioned in incidents (e.g., the "Email Service" in the example title). Another observation is that title generation is the only task where IR outperforms the Rule-based method. Since the Rule-based method performs simple concatenation, the generated title is long and contains unnecessary words, thus resulting in lower scores, while IR method retrieves titles from historical outages which conforms better with the pattern of outage titles in general. Oasis achieves significantly high scores, with considerable improvement over baselines (at least 30.0% of BLEU-4, 30.9% of ROUGE-L, 31.4% of METEOR), indicating that applying Oasis to production outages title generation is very promising. RQ6: Performance of Summary Generation Given Title We evaluate the performance of the outage summary generation when the title of the outage is given. Remember that we provide the information of relevant incidents to these methods as context. In this experiment, we include the title written by OCEs as part of the context. For GPT-2 and Oasis where there is an instruction in the context ([The outage summary is:]), we insert the outage title between incident information and the prompt. The results of this experiment are listed in Table 3. Surprisingly, the performance of Oasis, GPT-2, and Rule-based degrade slightly in this setting, with the exception of IR, whose metrics remain as low as before. The relative order of methods in Table 3 keeps the same as Table 1, for their tasks are very similar. The performance degradation is because the title of the outage is not a grammaticallycomplete sentence. Table 4 lists the fine-tuning (if any) and summary generation times for each model. The fine-tuning time reflects the total time spent on all outages from the training set, while the summary generation time is the average time taken to generate a summary for one outage. Despite the differences in parameter size, all models have a very small generation time. The fine-tuning time for LLMs (GPT-2, Curie, DaVinci) increases as the parameter size increases, although not linearly. Please note that Oasis only needs to be fine-tuned once on historical outages and incidents and can then be used for outage summary generation. In other words, the time to generate an outage summary of Oasis is only 13.3 or 39.6 (×10 −5 ) time units, which is at least 251.2 times faster than the median of manual summarization. In conclusion, the fine-tuning time for Oasis is reasonable, and its short generation time of summary demonstrates its practicality in cloud systems. RQ7: Efficiency Comparison OASIS PRELIMINARY HUMAN EVALUATION 7.1 Methodology Summarizing an outage is a challenging task that requires a deep understanding of both the service and the specific outage, as well as a comprehensive knowledge of the relevant context and domain. To ensure the accuracy of the generated summaries, we ask the owners of these outages to evaluate the generated summaries from RQ4. The process is done through Email and eventually, a total of 54 outage owners respond to our email and provide their evaluations. For each outage, we present the generated summaries from all the methods, with the output of Oasis with Curie and DaVinci treated as independent summaries. This results in five summaries for each outage. We present the outage and the summaries in the following order: (1) first we give the IcM link to the outage so that the outage owner can better recall the details of the outage, (2) next we present the original outage summary written by OCEs and tell the outage owners this is the human-written summarization, (3) then we present the five outage summaries generated by models (Oasis-Curie, Oasis-DaVinci, GPT-2, Rule-based, IR), and we shuffle the order of these summaries to minimize the effect of default ordering. To ensure the objectivity of the evaluation and avoid the subjectivity and bias of scoring for each summary, we ask the outage owners to rank the summaries from 1 to 5, where 1 for the most useful and readable and 5 for the least useful and readable. Useful means that the summary contains useful and relevant information on the outage. Readable means the ease with which the summary can be understood, which may be characterized by clear and simple language, logical organization, grammatical correctness, etc. Besides ranking, we also ask outage owners to share their opinions and comments on model-generated summaries. Figure 5 shows the ranking of outage owners. In general, the results of human evaluation are in accordance with automatic metrics presented in Table 1. The outage owners report positive feedback regarding the readability and usefulness of Oasis. Notably, 32 out of 54 OCEs rank the summaries produced by Oasis-DaVinci as their top preference. To investigate whether the rankings of outage owners are consistent with each other, we conduct Friedman Test [12] at the significant level of 0.05. The null hypothesis is that there is no significant difference between the rankings of outage owners. The calculated p-value on our rank by outage owner is larger than the level of significance, which means that the outage owners basically conform to each other in evaluating the summaries. Results More encouragingly, the majority of outage owners have a favorable attitude toward the practice of generating outage summaries using Oasis: "I absolutely believe in the ability of AI to assist with incident management and outage summaries. " DISCUSSION Case Study As described in Section 4.4, Oasis serves as a supportive tool in the production IcM, i.e., OCEs have the option to use or not use Oasis to generate a reference summary when handling outages. Since it is difficult to determine the extent to which Oasis contributes to the generated summary for outages where it is utilized, we randomly selected an outage from recent outages that were handled without the use of Oasis. In this outage, a misconfiguration of the load balancer led to an overwhelming number of requests being directed to a single service endpoint. As a result, this endpoint was unable to function properly, causing the unified account API to fail. This failure cascaded to the downstream account APIs of the Cloud, Productivity, and Partner systems. As a result, the signup, ordering, and billing services of the Cloud and Productivity systems were affected. Below is the actual outage summary, written by an experienced OCE: Outage Summary by OCE: The API failed with HTTP 5xx errors (over 1 fall failures) because of bad gateway errors to the endpoint 1 . Due to this issue, commercial customers could not sign-up for System-Cloud or System-Productivity via endpoint 2 or endpoint 3 , and perform management related actions on endpoint 4 . Additionally, customers could not complete purchases within these ecosystems. Partner system is also impacted. Outage Summary by Oasis: The API failed with HTTP 5xx errors (over 1 fall failures) because of bad gateway errors to the endpoint 1 . Due to this issue, commercial customers could not sign-up for System-Cloud or System-Productivity via endpoint 2 or endpoint 3 , and perform management related actions on endpoint 4 . Additionally, System-Cloud users were not able to access their billing accounts and invoices on System-Cloud portal. Approximately 2 unique users were impacted. To study the ability of Oasis in summarizing online outages, we triggered Oasis manually, limiting it to only knowing the information at the time of the outage. Oasis managed to find six relevant incidents, and the generated summary is presented above. (We indicate sentences that diverge from the OCE outage summary by underlining them with wavy lines) In the above summaries, endpoints 1-4 are URLs that serve the API calls. We notice that Oasis failed to identify System-Partner as impacted because the impact of System-Partner can only be determined by knowing the prefix of endpoint 4 refers to System-Partner. This knowledge is difficult to learn even after the LLM has been fine-tuned using incident and outage corpus. Despite this, we can see from the above example that Oasis is capable of generating reference summaries for outages. In this study, we adopt fine-tuning of the GPT-x models to generate outage summaries, which perform much better than prompt tuning according to our experiment. Because the outage summary is very domain-specific and fine-tuned models may capture the domain knowledge. Threats to Validity Threats to internal validity mainly lie in our implementation of Oasis and compared approaches. To reduce this threat, we implemented these approaches based on well-established frameworks, which have been described in Section 5.4. Additionally, two authors carefully examined the code and configurations. Threats to external validity mainly lie in the subjects used in our study. Our study and evaluation are conducted on 18 major cloud systems of Microsoft. Since the incidents and outages we used are only from Microsoft, modifications may be necessary when applying to other incident management systems. However, the cloud systems we used in our experiments include a variety of types, such as infrastructure, productivity, communication, game, search engine, etc. Moreover, these cloud systems serve millions of customers, thus having a certain degree of representativeness. In the future, we plan to extend our evaluations to include more cloud systems. Threats to the construct validity mainly lie in the evaluation metrics we adopted. Automated evaluation metrics (BLEU-4, ROUGE-L, and METEOR) may not fully reflect the readability and usefulness of the outage summary. To address this limitation, we will consider using additional metrics in the future to better measure these factors. Moreover, we reached out to the owners of outages to conduct a human evaluation, and the evaluation results are basically aligned with automated metrics. RELATED WORK Incident storm / outage handling. Handling outages (incident storms) in cloud systems has been widely studied in previous work [8,11,17,29]. A series of works perform incident linking to provide engineers with more relevant information. LiDAR [8] calculates both textual similarity and component similarity to determine whether two incidents should be linked. LinkCM [17] argues that linking customer incidents (reported by customers) with system incidents (reported by monitors) can lead to more efficient incident triage. The above works utilize neural networks to learn from historical linking patterns. GRLIA [11] also employs graph embedding, with additional concerns about node closeness with KPI trend similarities. COT [29] first builds a heuristic dependency graph for the cloud systems based on historical incidents and links. It then finds the corresponding incidents by searching connected nodes in the graph. Another series of works focuses on alert reduction or prioritization [6,31]. OAS [6] combines semantic and behavioral features of alerts to decide the groups of alerts and then correlate alerts within a time window. Zhao et al. [31] first calculate the textual and topological similarity of alerts to reduce the number of alerts. They then use DBSCAN to group similar alerts and selected the centroid alert of each cluster as the representative incident to show to engineers. Our impact scope assessment is similar to these approaches, and we also include domain-specific knowledge via rule-based incident linking. Large Language Models (LLM) for Software Engineering. In recent years, the rise of LLM has brought new opportunities to the field of software engineering [2,13,25,26,30]. Mastropaolo et al. [26] studied the ability of fine-tuned T5 in the following tasks: automatic bug fixing, generation of assert statements in test methods, code summarization, and injection of code mutants. LANCE [25] uses fine-tuned T5 to automatically generate logging statements for Java methods. VulRepair [13] also fine-tune T5 on vulnerability repairs datasets to automatically propose vulnerability fixes. The above works fine-tune LLM on task-specific datasets. Zhang et al. [30] propose to use prompting for LLM to improve code version control. They further integrate k-shot learning to resolve code merge conflicts. GPT-3.x models are used to recommend root causes and mitigation steps to facilitate cloud incident management [2]. Different from previous studies, Oasis is the first work to leverage the capabilities of LLM in to summarize outages for cloud systems. CONCLUSION In this paper, we identify the problem of outage understanding in real-world cloud systems. Through our empirical study on 18 industrial cloud systems, we show that understanding outage is time-consuming and involves complex contexts. To improve the process of outage understanding, we present Oasis, the first framework to automatically assess impacts and summarize outages. Oasis incorporates an assessment of outage impact scope and a fine-tuned large language model, i.e., GPT-3.x. Our experiments on 18 cloud systems within Microsoft demonstrate that Oasis outperforms baseline approaches. We also received feedback from outage owners, which further validates the effectiveness of Oasis. Figure 1 : 1The timeline of handling an outage, where multiple incidents should be summarized when declare the outage. Figure 2 : 2CDF of (a) Number of relevant incidents to outages. (b) Time to Summary (TTS). Figure 2 ( 2b) shows the CDF of the time needed to summarize outages. From this figure, there are nearly 23% of outages cannot be summarized within two time units after the outage starts. The median time needed to summarize outages is one time unit. Therefore, outage summarization is time-consuming and labor-intensive. Figure 3 : 3Overview of Oasis. that were actually linked by engineers are labeled positive and pairs that were not linked are labeled negative. [ The title of ℎ incident is . . . . The description of ℎ incident is . . . . The service of ℎ incidents is . . . . . . . ] Finally, we append an instruction to the end of the text to hint the GPT-3 model to generate a summary: [The outage summary is:]. Figure 4 : 4Oasis in production. The upper part is the IcM system. The lower part is the architecture of Oasis. Figure 5 : 5The ranking given by outage owners. Rank #1 means the most preferred summary. Table 1 : 1Effectiveness of models at summarizing outagesModel BLEU-4 ROUGE-L METEOR Top1 Top5 Top1 Top5 Top1 Top5 IR 0.042 0.051 0.144 0.180 0.115 0.146 Rule-based 0.277 NA 0.508 NA 0.629 NA GPT-2 0.455 0.51 0.561 0.592 0.536 0.574 Curie 0.654 0.701 0.73 0.777 0.721 0.767 DaVinci 0.664 0.706 0.742 0.782 0.734 0.776 6 OASIS EVALUATION: EMPIRICAL STUDY RESULTS 6.1 RQ4: Performance of Summary Generation Table 2 : 2Effectiveness of models at proposing outage titlesModel BLEU-4 ROUGE-L METEOR Top1 Top5 Top1 Top5 Top1 Top5 IR 0.170 0.211 0.398 0.427 0.342 0.369 Rule-based 0.069 NA 0.211 NA 0.316 NA GPT-2 0.624 0.673 0.672 0.694 0.639 0.688 Curie 0.826 0.88 0.88 0.9 0.84 0.894 DaVinci 0.857 0.893 0.883 0.913 0.869 0.907 Table 3 : 3Effectiveness of models at generating outage summaries given outage titlesModel BLEU-4 ROUGE-L METEOR Top1 Top5 Top1 Top5 Top1 Top5 IR 0.037 0.055 0.156 0.189 0.109 0.142 Rule-based 0.247 NA 0.505 NA 0.614 NA GPT-2 0.428 0.504 0.548 0.59 0.515 0.569 Curie 0.65 0.697 0.729 0.776 0.719 0.764 DaVinci 0.652 0.699 0.734 0.779 0.724 0.77 Table 4: Average time cost of models Time Rule IR GPT-2 Curie DaVinci Fine-tuning (10 −1 ×time unit) NA NA 3.0 3.4 8.7 Generation (10 −5 × time unit) 2.8 13.9 11.1 13.3 39.6 Multilingual training for software engineering. 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[ "DäRF: Boosting Radiance Fields from Sparse Inputs with Monocular Depth Adaptation", "DäRF: Boosting Radiance Fields from Sparse Inputs with Monocular Depth Adaptation" ]	[ "Jiuhn Song \nKorea University\n\n", "Seonghoon Park \nKorea University\n\n", "Honggyu An \nKorea University\n\n", "Seokju Cho \nKorea University\n\n", "Min-Seop Kwak \nKorea University\n\n", "Sungjin Cho \nKorea University\n\n", "Seungryong Kim \nKorea University\n\n" ]	[ "Korea University\n", "Korea University\n", "Korea University\n", "Korea University\n", "Korea University\n", "Korea University\n", "Korea University\n" ]	[]	Neural radiance field (NeRF) shows powerful performance in novel view synthesis and 3D geometry reconstruction, but it suffers from critical performance degradation when the number of known viewpoints is drastically reduced. Existing works attempt to overcome this problem by employing external priors, but their success is limited to certain types of scenes or datasets. Employing monocular depth estimation (MDE) networks, pretrained on large-scale RGB-D datasets, with powerful generalization capability would be a key to solving this problem: however, using MDE in conjunction with NeRF comes with a new set of challenges due to various ambiguity problems exhibited by monocular depths. In this light, we propose a novel framework, dubbed DäRF, that achieves robust NeRF reconstruction with a handful of real-world images by combining the strengths of NeRF and monocular depth estimation through online complementary training. Our framework imposes the MDE network's powerful geometry prior to NeRF representation at both seen and unseen viewpoints to enhance its robustness and coherence. In addition, we overcome the ambiguity problems of monocular depths through patch-wise scaleshift fitting and geometry distillation, which adapts the MDE network to produce depths aligned accurately with NeRF geometry. Experiments show our framework achieves state-of-the-art results both quantitatively and qualitatively, demonstrating consistent and reliable performance in both indoor and outdoor real-world datasets.	10.48550/arxiv.2305.19201	[ "https://export.arxiv.org/pdf/2305.19201v1.pdf" ]	258,967,497	2305.19201	aa2c0f7e8ac0462693e3faa69269b27ea541f618	DäRF: Boosting Radiance Fields from Sparse Inputs with Monocular Depth Adaptation Jiuhn Song Korea University Seonghoon Park Korea University Honggyu An Korea University Seokju Cho Korea University Min-Seop Kwak Korea University Sungjin Cho Korea University Seungryong Kim Korea University DäRF: Boosting Radiance Fields from Sparse Inputs with Monocular Depth Adaptation Neural radiance field (NeRF) shows powerful performance in novel view synthesis and 3D geometry reconstruction, but it suffers from critical performance degradation when the number of known viewpoints is drastically reduced. Existing works attempt to overcome this problem by employing external priors, but their success is limited to certain types of scenes or datasets. Employing monocular depth estimation (MDE) networks, pretrained on large-scale RGB-D datasets, with powerful generalization capability would be a key to solving this problem: however, using MDE in conjunction with NeRF comes with a new set of challenges due to various ambiguity problems exhibited by monocular depths. In this light, we propose a novel framework, dubbed DäRF, that achieves robust NeRF reconstruction with a handful of real-world images by combining the strengths of NeRF and monocular depth estimation through online complementary training. Our framework imposes the MDE network's powerful geometry prior to NeRF representation at both seen and unseen viewpoints to enhance its robustness and coherence. In addition, we overcome the ambiguity problems of monocular depths through patch-wise scaleshift fitting and geometry distillation, which adapts the MDE network to produce depths aligned accurately with NeRF geometry. Experiments show our framework achieves state-of-the-art results both quantitatively and qualitatively, demonstrating consistent and reliable performance in both indoor and outdoor real-world datasets. Introduction Neural radiance field (NeRF) [31] has gained significant attention for its powerful performance in reconstructing 3D scenes and synthesizing novel views. However, despite its impressive performance, NeRF often comes with a considerable limitation in that its performance highly relies on the presence of densely well-calibrated input images which are difficult to acquire. As the number of input images is reduced, NeRF's novel view synthesis quality drops significantly, displaying failure cases such as erroneous overfitting to the input images [18,33], artifacts clouding empty spaces [33], or degenerate geometry that yields incomprehensible jumble when rendered at unseen viewpoints [19]. These challenges derive from its under-constrained nature, causing it to have extreme difficulty mapping a pixel in input images to a correct 3D location. In addition, NeRF's volume rendering allows the model to map a pixel to multiple 3D locations [14], exacerbating this problem. Previous few-shot NeRF methods attempt to solve these issues by imposing geometric regularization [33,19,23] or exploiting external 3D priors [14,41] such as depth information extracted from [50] that distills depths by pretrained MDE to NeRF at seen view only, our DäRF fully exploits the ability of MDE by jointly optimizing NeRF and MDE at a specific scene, and distilling the monocular depth prior to NeRF at both seen and unseen views. input images by COLMAP [43]. However, these methods have weaknesses in that they use 3D priors extracted from a few input images only, which prevents such guidance from encompassing the entire scene. To effectively tackle all the issues mentioned above, pretrained monocular depth estimation (MDE) networks with strong generalization capability [39,38,5] could be used to inject an additional 3D prior into NeRF that facilitates robust geometric reconstruction. Specifically, geometry prediction by MDE can constrain NeRF into recovering smooth and coherent geometry, while their bias towards predicting smooth geometry helps to filter out fine-grained artifacts that clutter the scene. More importantly, NeRF's capability to render any unseen viewpoints enables fully exploiting the capability of the MDE, as MDE could provide depth prior to the numerous renderings of unseen viewpoints as well as the original input viewpoints. This allows injecting additional 3D prior to effectively covering the entire scene instead of being constrained to a few input images. However, applying MDE to few-shot NeRF is not trivial, as there are ambiguity problems that hinder the monocular depth from serving as a good 3D prior. Primarily, relative depths predicted by MDEs are not multiview-consistent [6,12]. Moreover, MDEs perform poorly in estimating depth differences between multiple objects: this prevents global scale-shift fitting [62,30] from being a viable solution, as alignment to one region of the scene inevitably leads to misalignment in many other regions. There also exists a convexity problem [30], in which the MDE has difficulty determining whether the surface is planar, convex, or concave, are also present. To overcome these challenges, we introduce a novel method to adapt MDE to NeRF's absolute scaling and multiview consistency as NeRF is regularized by MDE's powerful 3D priors, creating a complementary cycle. In this paper, we propose DäRF, short for Monocular Depth Adaptation for boosting Radiance Fields from Sparse Input Views, which achieves robust optimization of few-shot NeRF through MDE's geometric prior, as well as MDE adaptation for alignment with NeRF through complementary training (see Fig. 1). We exploit MDE for robust geometry reconstruction and artifact removal in both unseen and seen viewpoints. In addition, we leverage NeRF to adapt MDE toward multiview-consistent geometry prediction and introduce novel patch-wise scale-shift fitting to more accurately map local depths to NeRF geometry. Combined with a confidence modeling technique for verifying accurate depth information, our method achieves state-of-the-art performance in few-shot NeRF optimization. We evaluate and compare our approach on real-world indoor and outdoor scene datasets, establishing new state-of-the-art results for the benchmarks. Related Work Neural radiance field. Neural radiance field (NeRF) [31] represents photo-realistic 3D scenes with MLP. Owing to its remarkable performance, there has been a variety of follow-up studies [3,59,29]. These studies improve NeRF such as dynamic and deformable scenes [35,49,37,2], real-time rendering [59,40,32], unbounded scene [4,46,54] and generative modeling [44,34,56,8]. However, these works still encounter challenges in synthesizing novel views with a limited number of images in a single scene, limiting their applicability in real-world scenarios. Few-shot NeRF. Numerous few-shot NeRF works attempted to address few-shot 3D reconstruction problem through various techniques, such as pretraining external priors [60,11], meta-learning [47], regularization [18,33,19,57,23] or off-the-shelf modules [18,33]. Recent approaches [33,19,57,23] emphasize the importance of geometric consistency and apply geometric regularization at unknown viewpoints. However, these regularization methods show limitations due to their heavy reliance on geometry information recovered by NeRF. Other works such as DS-NeRF [14], DDP-NeRF [41] and SCADE [50] exploit additional geometric information, such as COLMAP [43] 3D points or monocular depth estimation, for geometry supervision. However, these works have critical limitations of only being able to provide geometry information corresponding to existing input viewpoints. Unlike these works, our work demonstrates methods to provide geometric prior even at unknown viewpoints with MDE for more effective geometry reconstruction. Monocular depth estimation. Monocular depth estimation (MDE) is a task that aims to predict a dense depth map given a single image. Early works on MDE used handcrafted methods such as MRF for depth estimation [42]. After the advent of deep learning, learning-based approaches [15,17,20,24] were introduced to the field. In this direction, the models were trained on ground-truth depth maps acquired by RGB-D cameras or LiDAR sensors to predict absolute depth values [27,26]. Other approaches trained the networks on large-scale diverse datasets [9,25,38,39], which demonstrates better generalization power. These approaches struggle with depth ambiguity caused by ill-posed problem, so the following works LeRes [58] and ZoeDepth [5] opt to recover absolute depths using additional parameters. Incorporating MDE into 3D representation. As both NeRF and monocular depth estimation are closely related, there have been some works that utilize MDE models to enhance NeRF's performance. NeuralLift [55] and MonoSDF [61] leverage depths predicted by pretrained MDE for depth ordering and detailed surface reconstruction, respectively. Other works optimize scene-specific parameters, such as depth predictor utilizing depth recovered by COLMAP [52] or learnable scale-shift values for reconstruction in noisy pose setting [7]. As a concurrent work, SCADE [50] utilizes monocular depths for scene reconstruction by providing explicit geometry priors for sparse view inputs. However, these previous approaches were limited in that MDEs were used to provide prior to only the input viewpoints, which constrains their effectiveness when input views are reduced, e.g., in the few-shot setting. In addition, they only used pretrained MDE models without fine-tuning on a specific scene, which inherits the limitations of pretrained MDE, such as estimating relative and view-inconsistent depths [39,38]. In contrast, our method exploits pretrained MDEs even at unknown viewpoints, along with finetuning for absolute depth prediction, for effective suppression of artifacts and divergent behaviors of few-shot NeRF. Preliminaries NeRF [31] represents a scene as a continuous function F θ (·) represented by a neural network with parameters θ. During optimization, 3D points are sampled along rays represented by r coming from a set of input images S = {I i }, whose ground truth camera poses are given, for evaluation by the neural network. For each sampled point, F θ (·) takes as input its coordinate x ∈ R 3 and viewing direction d ∈ R 2 with a positional encoding γ(·) that facilitates learning high-frequency details [48], and outputs a color c ∈ R 3 and a density σ ∈ R such that {c, σ} = F θ (γ(x), γ(d)). With a ray parameterized as r p (t) = o + td p , starting from camera center o along the direction d p , color and depth value at the pixel p are rendered as follows: I(p) = t f tn T (t)σ(r p (t))c(r p (t))dt,D(p) = t f tn T (t)σ(r p (t))tdt,(1) whereĪ(p) andD(p) are rendered color and depth values at the pixel p along the ray r p (t) from t n to t f , and T (t) denotes an accumulated transmittance along the ray from t n to t as follows: Based on this volume rendering, F θ (·) is optimized by the reconstruction loss L recon that compares rendered colorĪ(p) to corresponding ground-truth I(p), with R as a set of pixels for training rays: T (t) = exp − t tn σ(r p (s))ds .(2)L recon = Ii∈S p∈R ∥I i (p) −Ī i (p)∥ 2 2 .(3) Our work explores the setting of few-shot optimization with NeRF [19,23]. Whereas the number of input viewpoints \|S\| is normally higher than one hundred in the standard NeRF setting [31], the task of few-shot NeRF considers scenarios when \|S\| is drastically reduced to a few viewpoints (e.g., \|S\| < 20). With such a small number of input viewpoints, NeRF shows high divergent behaviors such as geometry breakdown, overfitting to input viewpoints, and generation of artifacts that cloud the empty space between the camera and object, which causes its performance to drop sharply [18,19,33]. To overcome this problem, existing few-shot NeRF frameworks applied regularization techniques at unknown viewpoints to constrain NeRF with additional 3D priors [41,14] and enhance the robustness of geometry, but they showed limited performance. Methodology Motivation and Overview Our framework leverages the complementary benefits of few-shot NeRF and monocular depth estimation networks for the goal of robust 3D reconstruction. The benefits that pretrained MDE can provide to few-shot NeRF are clear and straightforward: because they predict dense geometry, they provide guidance for the NeRF to recover more smooth geometry. In cases where few-shot NeRF's geometry undergoes divergent behaviors, MDE provides strong constraints to prevent the global geometry from breaking down. However, there are difficult challenges that must be overcome if the depths estimated by MDE are to be used as 3D prior to NeRF. These challenges, which can be summarized as depth ambiguity problems [30], stem from the inherent ill-posed nature of the monocular depth estimation. Most importantly, MDE networks only predict relative depth information inferred from an image, meaning it is initially not aligned to NeRF's absolute geometry [5]. Global scaling and shifting may seem to be the answer, but this approach leads us to another depth ambiguity problem, as predicted scales and spacings of each instance are inconsistent with one another, as demonstrated in (a) of Fig. 2. Additionally, MDE's weakness in predicting the convexity of a surface, whether it is flat, convex, or concave -also poses a problem in using this depth for NeRF guidance. In this light, we adapt a pre-trained monodepth network to a single NeRF scene so that its powerful 3D prior can be leveraged to its maximum capability in regularizing the few-shot NeRF. In the following, we first explain how to distill geometric prior from off-the-shelf MDE model [39] from both seen and unseen viewpoints (Sec. 4.2). We also provide a strategy for adapting the MDE model to handle ill-posed problems to a specific scene, while keeping its 3D prior knowledge (Sec. 4.3). Then, we demonstrate a method to handle inaccurate depths (Sec. 4.4). Fig. 1 shows an overview of our method, compared to previous works using MDE prior [50,61]. Distilling Monocular Depth Prior into Neural Radiance Field To prevent the degradation of reconstruction quality in few-shot NeRF, we propose to distill monocular depth prior to the neural radiance field during optimization. By exploiting pre-trained MDE networks [38,39], which have high generalization power, we enforce a dense geometric constraint on both seen and unseen viewpoints by using estimated monocular depth maps as pseudo ground truth depth for training few-shot NeRF. We describe the details of this process below. Monocular depth regularization on seen views. We leverage a pre-trained MDE model, denoted as G ϕ (·) with parameters ϕ, to predict pseudo depth map from given seen view image I i as D * i = G ϕ (I i ). Since D * i is initially a relative depth map, it needs to be scaled and shifted into an absolute depth [62] and aligned with NeRF's rendered depthD in order for it to be used as pseudo-depth D * . However, the scale and shift parameters inferred from the global statistic may undermine local statistic [62]. For example, as shown in Fig. 2 (a), global scale fitting tends to favor dominant objects in the image, leading to ill-fitted depths in less dominant sections of the scene due to inconsistencies in predicted depth differences between the objects. Naïvely employing such inaccurately estimated depths for distillation can adversely impact the overall geometry of the NeRF. To alleviate this issue, we propose a patch-wise adjustment of scale and shift parameters, reducing the impact of erroneous depth differences, as illustrated in Fig. 2 (b). The depth consistency loss is defined as follows: L seen = Ii∈S p∈P ∥(w i sg (D * i (p)) + q i ) −D i (p)∥,(4) where w i and q i denote the scale and shift parameters obtained by least square [39] between D * i and D i , P denotes a set of pixels within a patch, and sg(·) denotes stop-gradient operation [10]. Thus patch-based approach also helps to overcome the computational bottleneck of full image rendering. Monocular depth regularization on unseen views. We further propose to give supervision even at unseen viewpoints. As NeRF has the ability to render any unseen viewpoint of the scene, we render colorĪ l and depthD l from a sampled patch of l-th novel viewpoint. Sequentially, we extract a monocular depth map from the rendered image asD * l = G ϕ (Ī l ). Then, we enforce consistency between our rendered depthD l and the monocular depthD * l of l-th novel viewpoint as follows: L unseen = I l ∈U p∈P ∥(w l sg D * l (p) + q l ) −D l (p)∥,(5) where U denotes a set of unseen view images, w l and q l denotes the scale and shift parameters used to alignD * l towardsD l , and P denotes randomly sampled patch. A valid concern regarding this approach is that monocular depth obtained from noisy NeRF rendering may be affected by fine-grained rendering artifacts that frequently appear in unseen viewpoints of few-shot NeRF, resulting in noisy and erroneous pseudo-depths. However, we demonstrate in Fig. 3 that a strong geometric prior within the MDE model exhibits robustness against such artifacts, effectively filtering out the artifacts and thereby providing reliable supervision for the unseen views. It should be noted that our strategy differs from previous methods [14,41,61,50] that exploit monocular depth estimation [38] and external depth priors such as COLMAP [43]. These methods only impose depth priors upon the input viewpoints, and thus their priors only influence the scene partially due to self-occlusions and sparsity of known views. Our method, on the other hand, enables external depth priors to be applied to any arbitrary viewpoint and thus allows guidance signals to thoroughly reach every location of the scene, leading to more robust and coherent NeRF optimization. Adaptation of MDE via Neural Radiance Field Although the patch-wise distillation of monocular depth provides invariance to depth difference inconsistency in MDE, the ill-posed nature of monocular depth estimation often introduces additional ambiguities, such as the inability to distinguish whether the surface is concavity, convexity, or planar or difficulty in determining the orientation of flat surfaces [30]. We argue that these ambiguities arise due to the MDE lacking awareness of the scene-specific absolute depth priors and multiview consistency. To address this issue, we propose providing the scene priors optimized NeRF to MDE, whose knowledge of canonical space and absolute geometry helps eliminate the ambiguities present within MDE. Therefore, we propose to adapt the MDE to the absolute scene geometry, formally written as: L MDE = Ii∈S p∈P ∥sg D i (p) −D * i (p)∥ + ∥(w i sg D i (p) + q i ) −D * i (p)∥ .(6) In addition to the patch-wise loss in Eq. 4, we add an l-1 loss without scale-shift adjustment to adapt the MDE with absolute depth prior. We also introduce a regularization term to preserve the local smoothness of MDE, given by: L reg = Ii∈S p∈P ∥(w i sg D * ,init i (p) + q i ) − D * i (p)∥,(7) where D * ,init i denotes monocular depth map of I i extracted from MDE with initial pre-trained weight. Confidence Modeling Our framework must take into account the errors present in both few-shot NeRF and estimated monocular depths, which will propagate [45] and intensify during the distillation process if left unchecked. To prevent this, we adopt confidence modeling [23,45] inspired by self-training approaches [45,1], to verify the accuracy and reliability of each ray before the distillation process. The homogeneous coordinates of a pixel p in the seen viewpoint are transformed to p ′ at the target viewpoint using the viewpoint difference R i→l and the camera intrinsic parameter K, as follows: p ′ ∼ KR i→l D i (p)K −1 p.(8) We generate the confidence map M i by measuring the distance between rendered depth of the unseen viewpoint and MDE output of seen viewpoint such that M i (p) = ∥(w i D * i (p) + q i ) −D l (p ′ )∥ < τ ,(9) where τ denotes threshold parameter, [·] is Iverson bracket, and D l (p ′ ) refers to depth value of the corresponding pixel at l-th unseen viewpoint for reprojected target pixel p of i-th seen viewpoint. We fit D * i to absolute scale, where scale and shift parameters, w i and q i , are obtained by least square [39] between D * i andD i . Overall Training With the incorporation of confidence modeling, the loss functions for both the radiance field and MDE can redefined. L seen and L unseen can be redefined as: In addition, the loss for the adaptation of the MDE module can be redefined considering M : L seen = Ii∈S p∈P M i (p) (w i sg (D * i (p)) + q i ) −D i (p) ,(10)L unseen = I l ∈U p∈P M l (p) (w l sg D * l (p) + q l ) −D l (p) .(11)L MDE = Ii∈S p∈P M i (p) ∥sg D i (p) −D * i (p)∥ + ∥(w i sg D i (p) + q i ) −D * i (p)∥ .(12) With these losses, we train both NeRF and MDE simultaneously, enhancing both models by complementing each other. MDE provides a strong geometric prior to NeRF while having the inherent limitation of obliviousness to the scene-specific prior, whereas NeRF provides it with its absolute geometry. Experiments Experimental Settings Implementation details. DäRF is implemented based on K-planes [36] as NeRF. We use DPThybrid [38] as MDE model. We use Adam [21] as an optimizer, with a learning rate of 1 · 10 −2 for NeRF and 1 · 10 −5 for the MDE, along with a cosine warmup learning rate scheduling. See supplementary material for more details. The code and pre-trained weights will be made publicly available. Datasets. We evaluate our method in real-world scenes captured at both indoor and outdoor locations. Following previous works [41,50], we use a subset of sparse-view ScanNet data [13] comprised with three indoor scenes, each consisting of 18 to 20 training images and 8 test images. We also conduct evaluations on more challenging setting with 9 to 10 train images. For outdoor reconstruction, we further test on 5 challenging scenes from the Tanks and Temples dataset [22]. The scenes are real-world outdoor dataset, with a wide variety of scene scales and lighting conditions. Note that these setups are extremely sparse compared to full image setups, where we use approximately 0.5 to 5 percent of the whole training inputs. Baselines. We adopt the following six recently proposed methods as baselines: standard neural radiance field method: K-planes [16], few-shot NeRF method: RegNeRF [33], and depth prior based methods: NerfingMVS [52], DS-NeRF [14], DDP-NeRF [41], and SCADE [50]. For methods whose code has not open-sourced, we leave the result as blank. Evaluation metrics. For quantitative comparison, we follow the NeRF [31] and report the PSNR, SSIM [51], LPIPS [63]. We report standard evaluation metrics for depth estimation [15], absolute (a) K-planes [16] (b) DS-NeRF [14] (c) DDP-NeRF [41] (d) DäRF (e) Ground truth relative error (Abs Rel), squared relative error (SqRel), root mean squared error (RMSE), root mean squared log error (RMSE log). To evaluate view consistency, we utilize a single scaling factor s for each scene, which is the median scaling [64] value averaged across all test views. Indoor scene reconstruction. We conducted experiments in two settings: (1) a standard few-shot setup as described in literature [41,50], and (2) an extreme few-shot setup with approximately 0.5 percent of the full images. As shown in Tab. 1, our approach outperforms the baseline methods in both settings in most of the metrics. Additionally, we provide quantitative results of the adapted MDE model in ScanNet dataset in Tab. 2, and qualitative results in Fig. 4. As shown in Fig. 5 for the setting of standard few-shot, DS-NeRF [14] and DDP-NeRF [41] still show floating artifacts in the novel view and show limitation in capturing details in the chair, smoothing into nearby object. Our method shows better qualitative results compared to other baselines, showing better geometry understanding and detailed view synthesis in the small objects near the chair. In the extreme few-shot setup, we conducted a visual comparison between our method and a baseline [16] in Fig. 6. This is a more complex setting than standard, but our method outperforms the baseline, showing better geometric understanding. More qualitative images are included in the supplementary material. Comparisons Outdoor scene reconstruction. We conduct the qualitative and quantitative comparisons on the Tanks and Temples dataset in Tab. 1 and Fig. 7. Since COLMAP [43] with sparse images is not available, we provide comparisons with baselines without explicit depth prior. The quantitative results show that our approach outperforms the baseline methods on this complex outdoor dataset in all metrics. As shown in Fig. 7, our baseline shows limited performance, despite its feasible results of (a) Baseline [16] (b) Baseline-Depth (c) DäRF (d) DäRF-Depth Figure 7: Qualitative results on Tanks and Temples [22]. Figure 8: Visualization of ablation studies on ScanNet [13]. view synthesis in novel viewpoint, its depth results show that the network totally fails to understand 3D geometry. Our method shows rich 3D understanding, even in this real-world outdoor setting which is more complicated than other scenes. More qualitative images are included in the supplementary material. Ablation on core components. In Tab. 3 and Fig. 8, we evaluate the effect of each proposed component. The quantitative results show effectiveness of each component. For qualitative results, we found out that L unseen suppresses the artifacts in novel viewpoint, compared to when only L seen is given. With adaptation of the MDE network to this scene, red basket in the background shows more accurate results and artifacts near the table are removed. In our model, with confidence modeling, view synthesis results show to be more structurally confident in the overall scene. Ablation Study Conclusion We propose DäRF, a novel method that addresses the limitations of NeRF in few-shot settings by fully leveraging the ability of monocular depth estimation networks. By integrating MDE's geometric priors, DäRF achieves robust optimization of few-shot NeRF, improving geometry reconstruction and artifact removal in both unseen and seen viewpoints. We further introduce patch-wise scale-shift fitting for accurate mapping of local depths to 3D space, and adapt MDE to NeRF's absolute scaling and multiview consistency, by distilling NeRF's absolute geometry to monocular depth estimation. Through complementary training, DäRF establishes a strong synergy between MDE and NeRF, leading to a state-of-the-art performance in few-shot NeRF. Extensive evaluations on real-world scene datasets demonstrate the effectiveness of DäRF. should be noted that the results for DDP-NeRF are with out-of-domain priors. The results of DDP-NeRF with in-domain priors are 20.96, 0.737, and 0.236 for PSNR, SSIM, and LPIPS, respectively. However, we were unable to evaluate DDP-NeRF in the extreme settings of ScanNet and Tanks and Temples, as reliable COLMAP 3D points could not be obtained. We utilized the authors' provided official implementations of RegNeRF [33] and K-planes [16], training one model for each scene using two different scenarios on the ScanNet [13] and Tanks and Temples [22] datasets. However, since there is no official code available for SCADE [50], we are unable to provide performance comparisons for this method. B Datasets and Metrics B.1 Datasets ScanNet [13]. We adhere to the few-shot protocol provided by DDP-NeRF [41] in our experimental setup. We noticed that the split contained major overlaps across the train and test sets, which makes the task easier compared to realistic few-shot settings where images exhibit minimal overlap. For this reason, we construct an extreme few-shot scenario, using only half of the training images while maintaining the same test set. Tanks and Temples [22]. To test the robustness of our method in challenging real-world outdoor environments, we conduct further experiments on Tanks and Temples dataset, an real-world outdoor dataset acquired under drastic lighting effects and reflectances. As no existing protocols exist for a few-shot scenario for this dataset, we introduce a new split for the few-shot setting. We carefully selected 5 object-centric scenes -truck, francis, family, lighthouse, and ignatius-with inwardfacing cameras. From each scene, we sample 10 training images that capture the overall geometry of the whole scene. For testing, we use one-eighth of the dataset as a test set, consisting every 8 th repeating image from the entire image set. We run COLMAP [43] on all images to obtain camera poses for NeRF training. However, for the lighthouse scene, which exhibits highly sensitive lighting and specular effects dependent on view pose, we manually preprocess the parts that contain these effects. B.2 Evaluation metrics To evaluate the quality of novel view synthesis, following previous works [31], we measure PSNR, SSIM, and LPIPS. It is mentioned in K-planes that an implementation of SSIM from mip-NeRF [4] results in lower values than standard scikit-image implementation. For a fair comparison per dataset, we use the latter scikit-image SSIM implementation following the relevant prior work. For the evaluation of the MDE module, we use 4 depth estimation metrics as follows: • AbsRel: 1 \|I\| p∈I ∥D(p) − D GT (p)∥/D GT (p); • SqRel: 1 \|I\| p∈I ∥D(p) − D GT (p)∥ 2 /D GT (p); • RMSE: 1 \|I\| p∈I ∥D(p) − D GT (p)∥ 2 ; • RMSE log: 1 \|I\| p∈I ∥ logD(p) − log D GT ∥ 2 ; where p is a pixel in the image I and D GT is ground truth depth map. In addition, following [64], we use single scaling factor s for each scene which is obtained by s = 1 N Ii∈S (median(D GT i /D i )),(5) rather than fit each frame to ground truth, to evaluate view consistency of MDE models. Here, S denotes set of images from single scene. Image-Level Patch-Level Figure 1: Error map visualization of image-level and patch-wise scale and shift adjustment: relative depth map in various viewpoints is fitted in two ways, image-level fitting (first row) and patch-level fitting (second row). C Additional Analysis C.1 Comparison of patch-and image-level scale-shift adjustment We provide additional analysis and visualization results regarding the patch-wise scale and shift adjustment. In Fig.1, we present error maps showing the discrepancies between the ground truth sensor depth and the predicted depth. Additionally, in Fig.2, we present qualitative results of rendered color and depth using each fitting method. It is important to note that in the image-level fitting scheme, a single set of scale and shift values is computed for an entire depth map. Conversely, in our patch-level fitting method, scale and shift values are calculated individually for each 80 × 80 patch within the depth map. The error map clearly demonstrates the significant reduction in misalignment errors achieved by our patch-level fitting method compared to the image-level fitting approach. For Fig. 2 and Tab. 4 of the main paper, we set the scale and shift as learnable parameters per image for image-level fitting and conduct patch-wise scale-shift invariant loss for patch-level fitting. This comparison is conducted only with L seen given and results with patch-level fitting show better performance compared to image-level fitting. The difference between the two methods is especially distinguished in rendered depth maps of these two settings, in that patch-level fitting lets NeRF learn depth more accurately. Figure 3: Comparions on MDE depth map with and without confidence masking. the initial MDE depth map predicted is filtered through mask from our confidence modeling. C.2 Confidence modeling (a) Input Image (b) Initial MDE (c) Conf. Mask (d) Masked MDE In Fig. 3, we demonstrate the effectiveness of our confidence modeling which effectively eliminates inaccurate information present in depth maps from both NeRF and the MDE network through leveraging multi-view consistency of NeRF. MDE depth from the input image contains errors, which can be filtered out by verifying consistency with depth from NeRF's other viewpoint. Likewise, the error of MDE depth from unseen viewpoint can be filtered through consistency check with MDE depth from the seen viewpoint. We conduct an ablation on the Monocular Depth Estimation (MDE) network to assess its impact on our methodology. Considering the recent advancements [39,38] in MDE models that shows strong generalization power for depth estimation in unseen images, we replace our MDE network with state-of-the-art models such as LeReS, MiDaS, and DPT. The results in Tab. 1 show that our method shows consistent performance across different baselines. C.3 Ablation of MDE baselines D Additional Qualitative Results In this section, we show additional qualitative comparisons in Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, and Fig. 9 for ScanNet [13] dataset in two different settings and in Fig. 10, Fig. 11, Fig. 12, Fig. 13, and Fig. 14 for Tanks and Temples [22] dataset. E Limitations and Future Works While our method shows powerful performance quantitatively, its limitations can be noticed in its qualitative results above, where it struggles to reconstruct the fine-grained details present in ground truth images. Also, our usage of depth supervision from various viewpoints does not get rid of the artifacts completely: some artifacts that cloud the space between objects and the camera, are reduced yet still visible in rendering of unseen viewpoints. These limitations may be attributed to fundamental limitations in the few-shot NeRF setting [18], where fine-grained details are often occluded from one viewpoint to another due to an extreme lack of input images, preventing faithful geometric reconstruction of details. Also, since the seen viewpoints view a comparatively small portion of the entire scene, there inevitably occur artifacts in the unseen viewpoint as some depths cannot be perfectly determined from given input information. F Broader Impacts Our work achieves robust optimization and rendering of NeRF under sparse view scenarios, drastically reducing the number of viewpoints required for NeRF and bringing NeRF closer to real-life applications such as augmented reality, 3D reconstruction, and robotics. Our extension of few-shot NeRF to a real-world setting with the usage of monocular depth estimation networks also would enable NeRF optimization under various real-life lighting conditions and specular surfaces due to its increased robustness and generalization power. Figure 1 : 1Motivation. Unlike existing work Figure 2 : 2Visualization of the effectiveness of patch-wise scale and shift adjustment: (a) monocular depth with image-level adjustment, (b) monocular depth with patch-level adjustment, and (c) rendered depth by NeRF trained with patch-level adjustment. For visualization, depth maps are unprojected into 3D space. The proposed patch-wise adjustment helps to minimize the errors caused by inconsistency in depth differences among objects. Figure 3 : 3Robustness of MDE model for multi-view scale ambiguity and artifacts: (a-b) color and depth of NeRF rendered in the early stage of the training, (c-d) monocular depths estimated from rendered imageĪ and input image I. The results show that MDE model ignores the artifacts of rendered images by NeRF, enabling reliable supervision for seen and unseen viewpoint. Figure 5 : 5Qualitative results of on ScanNet[13] with 18 -20 input views. Figure 6 : 6Qualitative results on ScanNet[13] with 9 -10 input views. Figure 4 : 4Error map visualization. MDE adaptation results in a reduction of errors. Figure 2 : 2Comparison of patch-and image-level scale-shift adjustment. Rendered color and depth from NeRF with (a-b) image-level scale and shift adjustment and (c-d) patch-level scale and shift adjustment. the comparison of image-level and patch-level fitting provided in the Figure 4 : 4Qualitative results on Scan 0710 of ScanNet[13] with 9 -10 input views. Figure 5 : 5a) Baseline[16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e)Ground truth Qualitative results on Scan 0758 of ScanNet[13] with 9 -10 input views. Figure 6 : 6Qualitative results on Scan 0781 of ScanNet[13] with 9 -10 input views. Figure 7 : 7a) Baseline [16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e) Ground truth Qualitative results on Scan 0710 of ScanNet [13] with 18 -20 input views. Figure 8 : 8Qualitative results on Scan 0758 of ScanNet[13] with 18 -20 input views. Figure 9 :Figure 10 :Figure 11 :Figure 12 : 9101112a) Baseline [16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e) Ground truth Qualitative results on Scan 0781 of ScanNet [13] with 18 -20 input views. Qualitative results on truck scene of Tanks and Temples [22] with 10 input views. Qualitative results on francis scene of Tanks and Temples [22] with 10 input views. Qualitative results on lighthouse scene of Tanks and Temples [22] with 10 input views. Table 1 : 1Quantitative comparison on ScanNet[13] and Tanks and Temples[22]. The best results are highlighted in bold, while the second best results are underlined. SSIM ↑ LPIPS ↓ PSNR ↑ SSIM ↑ LPIPS ↓ PSNR ↑ SSIM ↑ LPIPS ↓Methods Depth prior ScanNet [13] Tanks and Temples [22] 9 -10 views 18 -20 views 10 views PSNR ↑ NerfingMVS [52] ✓ N/A N/A N/A 16.29 0.626 0.502 N/A N/A N/A K-planes [16] ✗ 16.01 0.618 0.494 18.70 0.708 0.400 12.57 0.453 0.607 RegNeRF [33] ✗ 16.38 0.624 0.493 18.93 0.676 0.450 14.12 0.469 0.580 DS-NeRF [14] ✓ N/A N/A N/A 20.85 0.713 0.344 N/A N/A N/A DDP-NeRF [41] ✓ N/A N/A N/A 19.29 0.695 0.368 N/A N/A N/A SCADE [50] ✓ - - - 21.54 0.732 0.292 - - - DäRF (Ours) ✓ 18.29 0.690 0.412 21.58 0.765 0.325 15.70 0.514 0.583 Table 2 : 2Evaluation of depth quality: (a) quantitative evaluation of the adapted MDE, compared with other monocular depth estimation models and (b) visualization of depth distributions. The adapted MDE by our method shows a similar distribution to that of the ground truth.Methods AbsRel ↓ SqRel ↓ RMSE ↓ RMSE log ↓ LeRes [58] 0.391 0.472 0.999 0.661 MiDaS [39] 0.152 0.095 0.452 0.183 DPT [38] 0.191 0.135 0.563 0.220 DäRF (9 -10 views) 0.154 0.074 0.361 0.171 DäRF (18 -20 views) 0.151 0.071 0.356 0.168 (a) Quantitative comparison (b) Depth distribution comparison Table 3 : 3Ablation study. Table 4 : 4Local fitting ablation.Components PSNR↑ SSIM↑ LPIPS↓ Baseline [16] 18.65 0.706 0.502 w/ global fitting 19.05 0.698 0.399 w/ local fitting (DäRF) 19.71 0.730 0.380 Table 1 : 1Ablation study on MDE baseline.Components PSNR↑ SSIM↑ LPIPS↓ DäRF with LeReS [58] 21.31 0.757 0.343 DäRF with MiDaS [39] 21.48 0.758 0.337 DäRF with DPT [38] 21.58 0.765 0.325 (a) Baseline[16] (b) Baseline -Depth Appendix A Implementation DetailsA.1 ArchitectureWe implement DäRF with K-planes[16]as the base model. It represents a radiance field using tri-planes with three multi-resolutions for each plane: 128, 256, and 512 in both height and width, and 32 in feature depth. This approach also incorporates small MLP decoders and a two-stage proposal sampler. It should be noted that our framework is not restricted to the K-planes baseline, but can be incorporated into any NeRF backbone models[31,32,28]. In our experiments, we implemented our framework on top of the K-planes hybrid version codebase due to its quality, reasonable optimization speed, and model size. For the monocular depth estimation (MDE) module, we choose the pre-trained DPT[38]as our base MDE model due to its powerful generalization ability in a zero-shot setting. Trained on very large datasets, DPT demonstrates impressive prediction quality and generalizes well to novel scenes. However, any MDE model can be utilized within our framework[58,39,38].A.2 Training detailsWe use the Adam optimizer[21]and a cosine annealing with warm-up scheduler for NeRF optimization. The learning rate is set to 1 · 10 −2 , and we perform 512 warm-up steps. For MDE adaptation, we also employ the Adam optimizer[21]with a learning rate of 1 · 10 −5 . NeRF optimization is performed with a pixel batch size of 4,096, totaling 20K iterations. For L seen , we render a 64 × 64 patch, while for L unseen , we render a 128 × 128 patch with a stride of 3.For the loss functions, we set the coefficients of L seen , L MDE , and L reg as 0.01, 0.01, and 0.1, respectively. During the warm-up stage of 5,000 steps, the coefficient of L unseen is initially set to 0 and then increased to 0.01 after 5,000 warm-up steps. For the first 1,000 steps, we employ the ranking loss[58]with a coefficient of 0.1, in addition to L seen . All experiments were conducted using a single NVIDIA GeForce RTX 3090. The training process takes approximately 3 hours.A.3 Training loss detailsIn the following, we describe a least-square alignment[53]used in loss functions for MDE prior distillation in detail. As described in the main paper, we use a scale-shift invariant loss[39]with patch-wise adjustment for depth consistency as follows:where w i and q i are scale and shift values that align D * i (p) to the absolute locations ofD i (p). In this loss function, to calculate w i and q i , we following least-squares criterion[39]:In other words, we can rewrite the above scheme as a closed problem., 1] T , then we can modify our problem aswhich can be solved as follows:A.4 Baseline implementationsWe directly use quantitative results reported in prior literature[50]for the comparison of Nerfing-MVS[52], DS-NeRF[14]and DDP-NeRF[41]. As the setting[50]requires out-of-domain priors, it Pseudolabeling and confirmation bias in deep semi-supervised learning. 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Baseline [16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e) Ground truth. Baseline [16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e) Ground truth Figure 13: Qualitative results on ignatius scene of Tanks and Temples. 22with 10 input views. (a) Baseline [16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e) Ground truthFigure 13: Qualitative results on ignatius scene of Tanks and Temples [22] with 10 input views. (a) Baseline [16] (b) Baseline -Depth (c) DäRF (d) DäRF-Depth (e) Ground truth Qualitative results on family scene of Tanks and Temples [22] with 10 input views. Figure. 14Figure 14: Qualitative results on family scene of Tanks and Temples [22] with 10 input views.	[]
[ "Subequivariant Graph Reinforcement Learning in 3D Environments", "Subequivariant Graph Reinforcement Learning in 3D Environments" ]	[ "Runfa Chen ", "Jiaqi Han ", "Fuchun Sun ", "Wenbing Huang " ]	[]	[]	Learning a shared policy that guides the locomotion of different agents is of core interest in Reinforcement Learning (RL), which leads to the study of morphology-agnostic RL. However, existing benchmarks are highly restrictive in the choice of starting point and target point, constraining the movement of the agents within 2D space. In this work, we propose a novel setup for morphologyagnostic RL, dubbed Subequivariant Graph RL in 3D environments (3D-SGRL). Specifically, we first introduce a new set of more practical yet challenging benchmarks in 3D space that allows the agent to have full Degree-of-Freedoms to explore in arbitrary directions starting from arbitrary configurations. Moreover, to optimize the policy over the enlarged state-action space, we propose to inject geometric symmetry, i.e., subequivariance, into the modeling of the policy and Q-function such that the policy can generalize to all directions, improving exploration efficiency. This goal is achieved by a novel SubEquivariant Transformer (SET) that permits expressive message exchange. Finally, we evaluate the proposed method on the proposed benchmarks, where our method consistently and significantly outperforms existing approaches on single-task, multi-task, and zero-shot generalization scenarios. Extensive ablations are also conducted to verify our design.	10.48550/arxiv.2305.18951	[ "https://export.arxiv.org/pdf/2305.18951v1.pdf" ]	258,967,523	2305.18951	5b73fdfb7752b02f6a09ede09e636a6e13ee7c6a	Subequivariant Graph Reinforcement Learning in 3D Environments Runfa Chen Jiaqi Han Fuchun Sun Wenbing Huang Subequivariant Graph Reinforcement Learning in 3D Environments Learning a shared policy that guides the locomotion of different agents is of core interest in Reinforcement Learning (RL), which leads to the study of morphology-agnostic RL. However, existing benchmarks are highly restrictive in the choice of starting point and target point, constraining the movement of the agents within 2D space. In this work, we propose a novel setup for morphologyagnostic RL, dubbed Subequivariant Graph RL in 3D environments (3D-SGRL). Specifically, we first introduce a new set of more practical yet challenging benchmarks in 3D space that allows the agent to have full Degree-of-Freedoms to explore in arbitrary directions starting from arbitrary configurations. Moreover, to optimize the policy over the enlarged state-action space, we propose to inject geometric symmetry, i.e., subequivariance, into the modeling of the policy and Q-function such that the policy can generalize to all directions, improving exploration efficiency. This goal is achieved by a novel SubEquivariant Transformer (SET) that permits expressive message exchange. Finally, we evaluate the proposed method on the proposed benchmarks, where our method consistently and significantly outperforms existing approaches on single-task, multi-task, and zero-shot generalization scenarios. Extensive ablations are also conducted to verify our design. Introduction Learning to locomote, navigate, and explore in the 3D world is a fundamental task in the pathway of building intelligent agents. Impressive breakthrough has been made towards realizing such intelligence thanks to the emergence * Equal contribution 1 Dept. of Comp. Sci. & Tech., Institute for AI, BNRist Center, Tsinghua University 2 THU-Bosch JCML Center 3 Gaoling School of Artificial Intelligence, Renmin University of China 4 Beijing Key Laboratory of Big Data Management and Analysis Methods. of deep reinforcement learning (RL) (Mnih et al., 2015;Silver et al., 2016;Mnih et al., 2016;Schulman et al., 2017;Fujimoto et al., 2018), where the policy of the agent is acquired through interactions with the environment. More recently, by getting insight into the morphology of the agent, morphology-agnostic RL (Wang et al., 2018;Pathak et al., 2019;Huang et al., 2020;Kurin et al., 2020;Hong et al., 2021;Dong et al., 2022;Trabucco et al., 2022;Gupta et al., 2022;Furuta et al., 2023) has been proposed with the paradigm of learning a local and shared policy for all agents and the tasks involved, offering enhanced performance and transferability, especially in the multi-task scenario. It is usually fulfilled by leveraging Graph Neural Networks (GNNs) (Battaglia et al., 2018) or even Transformers (Vaswani et al., 2017) to derive the policy through passing and fusing the state information on the morphological graphs of the agents. In spite of the fruitful progress by morphology-agnostic RL, in this work, we identify several critical setups that have been over-simplified in existing benchmarks, giving rise to a limited state/action space such that the obtained policy is unable to explore the entire 3D space. In particular, the agents are assigned a fixed starting point and restricted to moving towards a single direction along the x-axis, leading to 2D motions only. Nevertheless, in a more realistic setup as depicted in Figure 1, the agents would be expected to have full Degree-of-Freedoms (DoFs) to turn and move in arbitrary directions starting from arbitrary configurations. To address the concern, we extend the existing environments to a set of new benchmarks in 3D space, which meanwhile introduces significant challenges to morphology-agnostic RL due to the massive enlargement of the state-action space for policy optimization. Optimizing the policy in our new setup is prohibitively difficult, and existing morphology-agnostic RL frameworks like (Huang et al., 2020;Hong et al., 2021) are observed to be susceptible to getting stuck in the local minima and exhibited poor generalization in our experiments. To this end, we propose to inject geometric symmetry (Cohen & Welling, 2016;Cohen & Welling, 2017;Worrall et al., 2017;van der Pol et al., 2020) into the design of the policy network to compact the space redundancy in a lossless way (van der Pol et al., 2020). In particular, we restrict the policy network to be subequivariant in two senses (Han et al., 2022a): 1. the output action will rotate in the same way as the input state of the agent; 2. the equivariance is partially relaxed to take into account the effect of gravity in the environment. We design SubEquivariant Transformer (SET) with a novel architecture that satisfies the above constraints while also permitting expressive message propagation through self-attention. Upon SET, the action and Q-function could be obtained with desirable symmetries guaranteed. We term our entire task setup and methodology as Subequivariant Graph Reinforcement Learning in 3D Environments (3D-SGRL). Our contributions are summarized as follows: • We introduce a set of more practical yet highly challenging benchmarks for morphology-agnostic RL, where the agents are permitted to turn and move in the 3D environments with arbitrary starting configurations and arbitrary target directions. For this purpose, we redesign the agents in current benchmarks by equipping them with more DoFs in a considerate way. • To effectively optimize the policy on such challenging benchmarks, we propose to enforce the policy network with geometric symmetry. We introduce a novel architecture dubbed SET that captures the rotation/translation equivariance particularly when external force fields like gravity exist in the environment. • We verify the performance of the proposed method on the proposed 3D benchmarks, where it outperforms existing morphology-agnostic RL approaches by a significant margin in various scenarios, including single-task, multi-task, and zero-shot generalization. Extensive ablations also reveal the efficacy of the proposed ideas. Background Morphology-Agnostic RL In the context of morphologyagnostic RL (Huang et al., 2020), we are interested in an environment with N agents (a.k.a tasks), where the n-th agent comprises K n limbs that control its motion. At time t, each limb k ∈ {1, · · · , K n } of agent n receives a state s n,k (t) ∈ R d and outputs a torque a n,k (t) ∈ [−1, 1] to its actuator. As a whole, agent n executes the joint action a n (t) = {a n,k (t)} Kn k=1 to interact with the environment which will return the next state of all limbs s n (t + 1) = {s n,k (t + 1)} Kn k=1 and a reward r n (s n (t), a n (t)) for agent n. The goal of morphology-agnostic RL is to learn a shared policy π θ among different agents to maximize the expected return: J (θ) = E π θ N n=1 ∞ t=0 γ t r n (s n (t), a n (t)) ,(1) where a n (t) = π θ (s n (t)), γ is a discount factor, and θ consists of trainable parameters. The objective in Equation (1) is usually optimized via the actor-critic setup of the deterministic policy gradient algorithm for continuous control (Lillicrap et al., 2016), which estimates the Q-function for agent n: Q π θ (s n , a n ) = E π θ ∞ t=0 [γ t r n (s n (t), a n (t))\| s n (0) = s n , a n (0) = a n ]. (2) To uniformly learn a shared policy across all agents and tasks, previous methods (Wang et al., 2018;Pathak et al., 2019;Huang et al., 2020;Kurin et al., 2020;Hong et al., 2021;Dong et al., 2022), take into account the interaction of connected limbs and joints, and view the morphological structure of the agent as an undirected graph G = (V, E), where each v i ∈ V represents a limb and the edge (v i , v j ) ∈ E stands for the joint connecting limb i and j 1 . A graph neural network φ θ is then employed to instantiate the policy π θ , which predicts the action a given the state of all limbs s and the graph topology E as input, i.e., a = φ θ (s, E) .(3) Equivariance and Subequivariance To further relieve the difficulty of learning a desirable policy within the massive search space formed by the states and actions of the agent in 3D space, we propose to encode the physical geometric symmetry of the policy learner φ θ , so that the learned policy can generalize to operations in 3D, including rotations, translations, and reflections, altogether forming the group of E(3). Such constraint enforced on the model is formally described by the concept of equivariance (Thomas et al., 2018;Fuchs et al., 2020;Villar et al., 2021;Satorras et al., 2021;Huang et al., 2022;Han et al., 2022a;. Definition 2.1 (E(3)-equivariance). Suppose ⃗ Z to be 3D geometric vectors (positions, velocities, etc) that are steerable by E(3) transformations, and h non-steerable features. The function f is E(3)-equivariant, if for any transformation g ∈ E(3), f (g · ⃗ Z, h) = g · f ( ⃗ Z, h), ∀ ⃗ Z ∈ R 3×m , h ∈ R d . Similarly, f is invariant if f (g · ⃗ Z, h) = f ( ⃗ Z, h). Built on this notion, Han et al. (2022a) additionally considers equivariance on the subgroup of O(3), induced by the external force ⃗ g ∈ R 3 like gravity, defined as O ⃗ g (3) := {O ∈ R 3×3 \|O ⊤ O = I, O⃗ g = ⃗ g}. By this means, the symmetry is only restrained to the rotations/reflections along the direction of ⃗ g. Such relaxation of group constraint is crucial in environments with gravity, as it offers extra flexibility to the model so that the effect of gravity could be captured. Han et al. (2022a) also presented a universally expressive construction of the O ⃗ g (3)-equivariant functions: f ⃗ g ( ⃗ Z, h) = [ ⃗ Z, ⃗ g]M ⃗ g , s.t. M ⃗ g = σ([ ⃗ Z, ⃗ g] ⊤ [ ⃗ Z, ⃗ g], h),(4) where σ (·) is an Multi-Layer Perceptron (MLP) and [ ⃗ Z, ⃗ g] ∈ R 3×(m+1) is a stack of ⃗ Z and ⃗ g along the last dimension. In particular, f will reduce to be O(3)-equivariant if ⃗ g is omitted in the computation. In this way, f ⃗ g can then be leveraged in the message passing process of the graph neural network φ θ in Equation (3) to obtain desirable geometric symmetry. Our task and method: 3D-SGRL In this section, we present our novel formulation for morphology-agnostic RL, dubbed Subequivariant Graph Reinforcement Learning in 3D Environments (3D-SGRL). We first elaborate on the extensions made to the environment in Section 3.1, then introduce our entire framework, consisting of an input processing module (Section 3.2), a novel SubEquivariant Transformer (SET) for expressive information passing and fusion (Section 3.3), and output modules of actor and critic to obtain the final policy and Q-function (Section 3.4). From 2D-Planar to 3D-SGRL A core mission of developing RL algorithms is enabling the agent (e.g., a robot) to learn to move in the environment with a designated goal. Ideally, the exploration should happen in the open space where the agent is able to move from the arbitrary starting point, via arbitrary direction, towards an arbitrary destination, offering much flexibility which highly corresponds to how the robot walks/runs in the real world. However, in the widely acknowledged setup in existing morphology-agnostic RL literature (Huang et al., 2020;Kurin et al., 2020;Hong et al., 2021;Dong et al., 2022), the agents are unanimously restricted in the fixed choice of starting position, target direction, and even the Degreeof-Freedom (DoF) of each joint in the action space. We summarize the limitations of the existing setup, which we dub 2D-Planar, and compare it with our introduced 3D-SGRL in Table 1 in three aspects, including state space, action space, and the consideration of geometric symmetry. State Space In the 2D-Planar setup, all positions of the limbs are projected onto the xoz-plane, and the agent is always initialized to face the positive x-axis. The agent is also designated to move in the same direction as it is initialized, lacking many vital movements, e.g., turning, that an agent is ! = " ( # ) Torques ! $ ! ( ! , ! ) ! = [ ! , ! , ! , ⋯ ] , ! = [ ! , ! , ! , ⋯ ] State ! O!(3) - invariant Matrix O!(3) - invariant Self-attention Coefficients O!(3) -equivariant Value Message [ , , ] O!(3) -invariant Value Message ! (#) ! (#) O!(3) - invariant Query O!(3) - invariant Key 3D Graph Abstraction ∈ O % (3) ∈ O % (3) State State Action Action External Force SET × , O!(3) - invariant Matrix " ($) O!(3) - equivariant Vectors [ " ($) , , ] Figure 2. The flowchart of our 3D-SGRL. The states of the agents are processed into hi and ⃗ Zi for each limb i, and are updated by L layers of our proposed SubEquivariant Transformer. The actor and critic are finally obtained, which are guaranteed to preserve the geometric symmetry for guiding the agent in arbitrary directions. There is no weight sharing between actor π θ and critic Qπ θ . challenge, we propose to take advantage of the geometric symmetry in the environments by enforcing it as a constraint in the design of φ θ . In particular, we construct φ θ to be an O ⃗ g (3)-equivariant function, which ensures that the policy learned in each direction can generalize seamlessly to arbitrary direction rotated along the gravity axis. Instead of O(3), we resort to subequivariant O ⃗ g (3) to empower the model such that the effect of gravity reflecting in the policy can be well captured. By contrast, existing morphology-agnostic RL works lack the consideration of geometric symmetry, leading to poor performance in a real and more challenging setup like 3D-SGRL. In addition to gravity, we have a target direction ⃗ d ∈ R 3 that is steerable and acts like an attracted force guiding the agent towards expected destinations. The task guidance is not explicitly specified in the previous 2D-Planar setting but comes as an indispensable clue in our 3D-SGRL tasks. Input Processing To fulfill the constraint in geometric symmetry, we need to subdivide the state s i into the directional geometric vectors ⃗ Z i and the scalar features h i for each node i ∈ {1, · · · , \|V\|} in the morphological graph G of the agent. Quantities in ⃗ Z i will rotate in accordance with the transformation g ∈ O ⃗ g (3) while those in h i remain unaffected. To be specific, for our 3D environments generated by MuJoCo (Todorov et al., 2012), the vectors in ⃗ Z i ∈ R 3×6 include the position ⃗ p i ∈ R 3 , the positional velocity ⃗ v i ∈ R 3 , the rotational velocity ⃗ ω i ∈ R 3 , joint rotation x-axis ⃗ x i ∈ R 3 , joint rotation y-axis ⃗ y i ∈ R 3 , and joint rotation z-axis ⃗ z i ∈ R 3 . The values in h i ∈ R 13 consist of the rotation angles κ i , ζ i , δ i of joint x-axis, y-axis, and z-axis, respectively, and their corresponding ranges as well as the type of limb, which is a 4-dimensional one-hot vector representing "torso", "thigh", "shin", "foot" and "other" respectively. As mentioned before, we have a target direction ⃗ d apart from ⃗ Z i and h i . Specifically, ⃗ d := [ ⃗ p xy −⃗ p xy 1 ∥⃗ p xy −⃗ p xy 1 ∥2 , 0], where ⃗ p xy is the xy coordinate of the assigned target and ⃗ p xy 1 is the xy coordinate of limb 1 (torso), each of which is in R 2 , and the resulting ⃗ d ∈ R 3 . SubEquivariant Transformer (SET) Given the states encoded in ⃗ Z i and h i , i ∈ {1, · · · , \|V\|}, we are still in demand of a highly expressive φ θ to learn the policy while ensuring the subequivariance. To this end, we present a novel architecture SET, to conduct effective message fusion between the limbs and joints, where the attention module is carefully designed to meet the symmetry. In particular, our SET processes the following operations in each computation. h (0) i = [h i , ⃗ p z i ],(5)⃗ Z (0) i = ⃗ Z i ⊖ ⃗ Z 1 := [⃗ p i − ⃗ p 1 , ⃗ v i , ⃗ ω i , ⃗ x i , ⃗ y i , ⃗ z i ],(6) where, the binary operation "⊖" transforms the input positions into translation invariant representations by subtracting ⃗ p 1 , the position of the node with index 1, i.e., the torso limb; ⃗ p z i is the projection of the coordinate ⃗ p i to the z-axis, which is indeed the relative height of node i when taking the ground as reference. The superscript 0 indicates the processed input. In the next step, we derive an O ⃗ g (3)-invariant matrix M i ∈ R m×m as the value matrix in self-attention. Formally, M (l) i = σ M σ ⃗ m [ ⃗ m (l) i , ⃗ g, ⃗ d] ⊤ [ ⃗ m (l) i , ⃗ g, ⃗ d] , h (l) i ,(7) where ⃗ m (l) i = ⃗ Z (l) i W (l) ⃗ m is a mixing of the vectors in ⃗ Z (l) i to capture the interactions between channels, with a learnable weight matrix W (l) ⃗ m ; the concatenation with ⃗ g and ⃗ d, and the inner product operation follow the practice in Equation (4); σ ⃗ m and σ M are two separate MLPs, and the superscript l indexes the layer number. With the value matrix M i , we compute the self-attention coefficients α ij ∈ R \|V\|×\|V\| between all pairs of node i and j, by deriving the O ⃗ g (3)-invariant query and key: q (l) i = W (l) q vec(M (l) i ) + b (l) q , (8) k (l) i = W (l) k vec(M (l) i ) + b (l) k , (9) α (l) ij = exp(q (l)⊤ i k (l) j ) m exp(q (l)⊤ i k (l) m ) ,(10) where vec(·) is a column vectorization function of matrix: R m×m → R mm×1 , W (l) q , W (l) k ∈ R mm×mm are the learn- able weights and b (l) q , b (l) k ∈ R mm×1 are the biases in the l-th layer. Finally, the O ⃗ g (3)-equivariant and invariant values are transformed by the attention coefficients α ij and aggregated to obtain the updated information. In detail, ⃗ Z (l+1) i = ⃗ Z (l) i + j α (l) ij [⃗ u (l) j , ⃗ g, ⃗ d] W (l) ⃗ Z ,(11)h (l+1) i = LN   h (l) i + W (l) h j α (l) ij v (l) j + b (l) h   ,(12) where ⃗ u (l) j = ⃗ Z (l) j W (l) ⃗ u is a mixing of the vectors in ⃗ Z (l) j to capture the interactions between channels, v (l) j = W (l) v vec(M (l) j ) + b (l) v is a invariant value message, with learnable weight matrices W (l) ⃗ u , W Actor and Critic With multiple layers of message fusion on the morphological graph of the agent, we are ready to output the actor policy π θ and critic Q-function Q π θ to obtain the training objective of morphology-agnostic RL. Notably, the action in 3D-SGRL setting has been extended to be the three values of the torques projected onto the three rotation axes of each joint, driven by the actuators attached. This is attained by firstly reading out the subequivariant vector from the output of the L-th layer of our SET, namely, ⃗ T i = [⃗ u (L) i , ⃗ g, ⃗ d]σ M M (L) i W ⃗ T ,(13) where ⃗ u (L) i = ⃗ Z (L) i W (L) ⃗ u is a mixing of channels, [⃗ u (L) i , ⃗ g, ⃗ d] ∈ R 3×m ′ is a stack of ⃗ u (L) i , ⃗ g and ⃗ d along the last dimension, σ M is, again, an MLP: R m×m → R m ′ ×m ′ , and W ⃗ T ∈ R m ′ ×1 is a linear transformation. Thanks to the O ⃗ g (3)-equivariance of SET and the readout in Equation (13), the torque matrix ⃗ T i ∈ R 3×1 is also O ⃗ g (3)-equivariant. The scalars of the torques projected on three rotation axes of the joint are then naturally given by taking the inner products: a i ∈ R 3 = [ ⃗ T i · ⃗ x i , ⃗ T i · ⃗ y i , ⃗ T i · ⃗ z i ],(14) where a i is the O ⃗ g (3)-invariant output action of the actuators assigned to limb i. By putting together all actions a i , i ∈ {1, · · · , \|V\|}, the final output action a in Equation (3) is collected. The O ⃗ g (3)-invariant Q-function Q π θ is similarly obtained by directly making use of the invariant M (L) i , given by, Q π θ = W Qπ θ vec(M (L) i ) + b Qπ θ ,(15) where W Qπ θ ∈ R 1×mm , b Qπ θ ∈ R collects the learnable weights and bias. Note that for learning actor policy π θ and critic Q π θ , we employ two separate SETs, since for computing Q π θ we need to additionally concatenate the action a i into the input of the first layer, i.e., h (0) i = [h i , a i ]. Here, we concatenate a i to h i rather than Z (0) i owing to the O ⃗ g (3)-invariance of a i . Formal proof of the equivariance of SET and the invariance of the output action and critic are presented in Appendix A. Benchmark Construction In this section, we introduce technical details in constructing our challenging benchmarks in 3D-SGRL. Environments and Agents The environments in our 3D-SGRL are modified from the default 2D-planar setups in MuJoCo (Todorov et al., 2012). Specifically, we extend agents in environments including Hopper, Walker, Humanoid and Cheetah (Huang et al., 2020) into 3D counterparts. For the multi-task training, we additionally construct several variants of each of these agents, as displayed in Table 5. We create the following collections of environments with these variants, and categorize the collections into two settings: in-domain and cross-domain. animal. However, in 3D-SGRL, the half-cheetah is highly vulnerable to falling over in its locomotion, adding more difficulties to policy optimization. On account of this limitation, we extend the model to a full-cheetah with one torso, four legs, and one tail made of 14 limbs, enabling it stronger locomotion ability to explore in our 3D-SGRL environments. More design details are shown in Appendix C.1. State Space We take the initial position of the agent's torso as the center, and randomly select its initial orientation and the destination within a radius of R. When the agent reaches the assigned target position, we set another destination for it. To relieve the agent from falling down when turning at a high speed, we set the radius R = 10km by default so that the agent will turn less frequently in an episode. We also set R ∈ [10m, 20m] as "v2-variants", which is more difficult since the agent will change the direction more frequently. Action Space The action space is enlarged by changing the type of the joint of torso from "slide-slide-hinge" to "free" and adding two more actuators that rotate around different axes of the joint. This allows the agent to have full DoFs to turn and move in arbitrary directions starting from arbitrary initial spatial configurations. Termination and Reward The goal in 3D-SGRL environments is learning to turn and move towards the assigned destination as fast as possible without falling over. Episode Termination follows that of the morphology-agnostic RL benchmark, but we modify the cheetah's termination to be the time it falls over or squats still. The reward consists of four parts. 1. Alive bonus: Every timestep the agent is alive, it gets a reward of a fixed value 1 (3D Cheetah's is 0 due to the stability of its morphological structure); 2. Locomotion reward: It is a reward for moving towards the assigned target which is measured as (distance before action -distance after action)/dt, where dt is the time between consecutive actions. This reward will be positive if the agent is close to the target position; 3. Control cost: It is a cost for penalizing the agent if it takes actions that are too large. It is measured as 0.001 * K k=1 (a k ) 2 ; 4. Forward reward (not available for 3D Hopper): It is a reward of moving forward measured as (coordinate after action -coordinate before action)·forward direct ion of torso/dt. This reward will be positive if the agent moves in the forward direction of torso. Evaluations and Ablations This section first introduces the baselines and implementations, then compares the performance of different methods on our 3D benchmarks and reports the ablation studies for the design of our method. Baseline, Metric and Implementation Baselines We compare our method SET against state-ofthe-art methods SMP (Huang et al., 2020) and SWAT (Hong et al., 2021). We also compare SET with standard TD3based non-morphology-agnostic RL: Monolithic in singletasks. Please refer to Appendix C.2 for more details about baselines. Metrics 1. Multi-task with different morphologies: For each multi-task environment discussed in Section 4, a single policy is simultaneously trained on multiple variants. The policy in each plot is trained jointly on the training set (80% of variants from that environment) and evaluated on these seen variants. 2. Zero-Shot Generalization: We take the trained policies from multi-task and test on the unseen zeroshot testing variants. 3. Evaluation on v2-variants: We evaluate SET in a transfer learning setting where the trained policies from multi-task are tested and fine-tuned on the v2-variants environments. 4. Single-task Learning: The policy in each plot is trained on one morphology variant and evaluated on this variant. Implementations We adopt the same input information and TD3 (Fujimoto et al., 2018) as the underlying reinforcement learning algorithm for training the policy over all baselines, ablations, and SET for fairness. We implement SET in the SWAT codebase. There is no weight sharing between actor π θ and critic Q π θ . Each experiment is run with three seeds to report the mean and the standard error. The reward for each environment is calculated as the sum of instant rewards across an episode. The value of the maximum timesteps of an episode is 1, 000. Main Results Multi-task with different morphologies As shown in Figure 3, our SET outperforms all baselines by a large margin in all cases, indicating the remarkable superiority of taking into account the subequivariance upon Transformer. The baselines fail to achieve meaningful returns in most cases, which is possibly due to the large exploration space in our 3D-SGRL environments and they are prone to get trapped in local extreme points. Zero-Shot Generalization During test time, we assess the trained policy on a set of held-out agent morphologies. Table 2 records the results of both in-domain and crossdomain settings. The training and zero-shot testing variants are listed on Table 5. For example, SET is trained on 3D Humanoid++ without 3d humanoid 7 left leg and 3d humanoid 8 right knee, while these two excluded environments are used for testing. Table 2 reports the average performance and the standard error over 3 seeds, where the return of each seed is calculated over 100 rollouts. Once again, we observe that SET yields better performance. Evaluation on v2-variants The v2-variants (R = 10 ∼ 20m) are more challenging. We conduct two-stage training in this scenario. In the first stage, we train the policy under the multi-task setting where R = 10km. The results and related demos are in Appendix F. In the second stage, we transfer the currently-trained policy to the R = 10 ∼ 20m setting on 3D Cheetah++ and 3D Humanoid++. It is seen from Figure 4 that SET is able to further improve the performance upon the first stage, while SWAT hardly receives meaningful performance gain especially on 3D Humanoid++. Single-task Learning Apart from SMP and SWAT, we implement another baseline Monolithic for reference. Figure 5 displays the performance on 3d humanoid 9 full and 3d cheetah 14 full. In line with the observations in (Dong et al., 2022), the GNN-based method SMP is worse than the MLP-based model Monolithic; but different from the results in (Dong et al., 2022), SWAT still surpasses Monolithic on 3d cheetah 14 full. We conjecture SWAT benefits from the application of Transformer that is expressive enough to characterize the variation of our 3d cheetah 14 full environments. Our model SET takes advantage of both the expressive power of the Transformer-akin model and the rational constraint by subequivariance, hence it delivers much better performance than all other methods. Comparison with Invariant Methods Invariant methods have been widely utilized in the 3D RL literature. For instance, in humanoid control, the presence of gravity allows for the normalization of state and action spaces in the heading (yaw) direction (e.g., a recent work (Won et al., 2022)). This heading normalization (HN) technique transforms the global coordinate frame into a local coordinate frame, enabling the input geometric information to be mapped to a rotation-and translation-invariant representation. We compare SET with the following invariant variants: 1. SWAT+HN: a state-of-the-art morphologyagnostic baseline that uses the heading normalization, and 2. Monolithic+HN: a standard TD3-based non-morphology- agnostic baseline that uses the heading normalization. As shown in Figure 5 and Figure 6, SET can only be considered on par with SWAT+HN, since heading normalization can achieve heading-equivariance by construction. Indeed, there is a limitation of heading normalization in that it assumes a consistent definition of the "forward" direction across all agents. Without a consistent "forward" direction, the normalization scheme would need to be redefined for each individual agent, which could limit its transfer ability to different types of agents or environments. On the contrary, equivariant methods, such as the one proposed in our work, can be more generalizable as they do not rely on a specific normalization scheme and can adapt to different transformations in the environment. We design a simple experiment to verify the above statement by translating the "forward" direction of the agent via a certain bias angle during testing. Table 3 demonstrates the significant performance degradation caused by adding bias in the heading normalization. Moreover, we can support this point through zero-shot generalization experiments, where we evaluate the trained policies from multi-task on unseen zero-shot testing variants. Table 4 demonstrates that SET has stronger generalization ability compared to SWAT+HN. For more detailed discussions, please refer to Appendix D. Ablation We ablate the following variants in 4. SET in invar: a non-equivariant model without all geometric vectors, instead taking them as the scalar input, h (0) i = [h (0) i , ⃗ Z i , ⃗ g, ⃗ d] ; 5. SET out invar: an O ⃗ g (3)equivariant model by replacing the action output by the projection strategy in Equation (14) with an O ⃗ g (3)-invariant mapping a i = W π θ vec(M (L) i ) + b π θ . 1. SET\g and SET\z, compared with SET, gain close performance on 3d cheetah 14 full, but are much worse on 3d humanoid 9 full. This is reasonable, as the agent 3d cheetah 14 full has four legs and can locomote stably (see Figure 8). It is thus NOT so essential to consider the effect of gravity and the height to the ground on 3d cheetah 14 full. As for 3d humanoid 9 full with 2 legs, however, it is important to sense the direction of gravity and detect the height to avoid potential falling down, hence the correct modeling of gravity and the height are necessary for locomotion policy learning. 2. The performance of SET\gd is poor in both cases, indicating that maintaining the direction information of the task guidance is indispensable. 3. SET in invar behaves much worse than SET, which verifies the importance to incorporate subequivariance into our model design. 4. SET out invar is worse than SET but already exceeds other variants. The equivariant output ⃗ T i in SET contains rich orientation information, and it is more direct to obtain the output torque by projecting ⃗ T i , than SET out invar which uses the invariant matrix M (L) i to predict the action. Discussion In current machine learning research, equivariance and attention are both powerful ideas. To learn a shared graph-based policy in 3D-SGRL, we design SET, a novel transformer model that preserves geometric symmetry by construction. Experimental results strongly support the necessity of encoding symmetry into the policy network, which demonstrates its wide applicability in various 3D environments. We also compare the Monolithic MLP-based model using heading normalization for single-task training in Figure 5. It can be found that a simple MLP with heading normalization may outperform the benefits brought by equivariance and attention. Therefore, in comparison to traditional methods in single-task settings, we cannot guarantee that all humanoids and legged robots will experience considerable enhancement when using our equivariant methods. In this work, our main contribution is extending the 2D benchmark to 3D for morphology-agnostic RL, which mainly addresses challenges in multi-task learning with agents of inhomogeneous morphology where MLP may not be applicable. Although these are just initial steps, we believe that further exploration of this research direction will lead to valuable contributions to the research community. A. Proofs In this section, we theoretically prove that our proposed SubEquivariant Transformer (SET), and the final output action and critic Q-function value preserve the symmetry as desired. We start by verifying our design in SET. Theorem A.1. Let ( ⃗ Z ′ , h ′ ) = φ( ⃗ Z, ⃗ g, ⃗ d, h) , where φ is one layer of our SET specified from Equation (7) to Equation (12). Let ( ⃗ Z ′ * , h ′ * ) = φ(O ⃗ Z, ⃗ g, O ⃗ d, h), ∀O ∈ O ⃗ g (3). Then, we have ( ⃗ Z ′ * , h ′ * ) = (O ⃗ Z ′ , h ′ ), indicating φ is O ⃗ g (3)- equivariant. Proof. In the first place, we have ⃗ m * i = O ⃗ Z i W ⃗ m = O ⃗ m i . For the message M i , we have, M * i = σ M σ ⃗ m [ ⃗ m * i , ⃗ g, O ⃗ d] ⊤ [ ⃗ m * i , ⃗ g, O ⃗ d] , h i ,(16)= σ M σ ⃗ m [O ⃗ m i , ⃗ g, O ⃗ d] ⊤ [O ⃗ m i , ⃗ g, O ⃗ d] , h i ,(17)= σ M   σ ⃗ m       ⃗ m ⊤ i O ⊤ O ⃗ m i ⃗ m ⊤ i O ⊤ ⃗ g ⃗ m ⊤ i O ⊤ O ⃗ d ⃗ g ⊤ O ⃗ m i ⃗ g ⊤ ⃗ g ⃗ g ⊤ O ⃗ d ⃗ d ⊤ O ⊤ O ⃗ m i ⃗ d ⊤ O ⊤ ⃗ g ⃗ d ⊤ O ⊤ O ⃗ d       , h i    ,(18)= σ M   σ ⃗ m       ⃗ m ⊤ i ⃗ m i ⃗ m ⊤ i ⃗ g ⃗ m ⊤ i ⃗ d ⃗ g ⊤ ⃗ m i ⃗ g ⊤ ⃗ g ⃗ g ⊤ ⃗ d ⃗ d ⊤ ⃗ m i ⃗ d ⊤ ⃗ g ⃗ d ⊤ ⃗ d       , h i    ,(19)= σ M σ ⃗ m [ ⃗ m i , ⃗ g, ⃗ d] ⊤ [ ⃗ m i , ⃗ g, ⃗ d] , h i = M i .(20) From Equation (18) u * j = ⃗ Z * j W ⃗ u = O ⃗ Z j W ⃗ u = O⃗ u j is O ⃗ g (3)-equivariant. Finally, we have, ⃗ Z ′ * i = O ⃗ Z i + j α ij [O⃗ u j , ⃗ g, O ⃗ d] W ⃗ Z ,(21)= O ⃗ Z i + j α ij O[⃗ u j , ⃗ g, ⃗ d] W ⃗ Z ,(22)= O   ⃗ Z i + j α ij [⃗ u j , ⃗ g, ⃗ d] W ⃗ Z   ,(23)= O ⃗ Z ′ ,(24) and similarly, h ′ * i = LN   h i + W h j (α ij v j ) + b h   = h ′ i .(25) By going through all nodes i ∈ {1, · · · , \|V\|} the proof is completed. By iteratively applying Theorem A.1 for l ∈ {1, · · · , L} layers, we readily obtain the O ⃗ g (3)-equivariance of the entire SET. As for the actor and critic, we additionally have the following corollary. Corollary A.2. Let a, Q π θ be the output action and the critic of 3D-SGRL with ⃗ Z, ⃗ g, ⃗ d, h as input. Let a * , Q * π θ be the action and critic with O ⃗ Z, ⃗ g, O ⃗ d, h as input, O ∈ O ⃗ g (3) . Then, (a * , Q * ) = (a, Q), indicating the output action and critic preserve O ⃗ g (3)-invariance. Proof. By Theorem A.1, we have ⃗ Z (L) * i = O ⃗ Z (L) i , and M (L) * i = M (L) i . Therefore, ⃗ u (L) * i = ⃗ Z (L) * i W (L) ⃗ u = O ⃗ Z (L) i W (L) ⃗ u = O⃗ u (L) * i . Hence, ⃗ T * i = [O⃗ u (L) i , ⃗ g, O ⃗ d]σ M M (L) i W ⃗ T ,(26)= O [⃗ u (L) i , ⃗ g, ⃗ d]σ M M (L) i W ⃗ T ,(27)= O ⃗ T i ,(28) where Equation (26) to Equation (27), again, leverages the fact that ⃗ g = O⃗ g, given the definition of O ⃗ g . Finally, a * i = [ ⃗ T i O ⊤ O⃗ x i , ⃗ T i O ⊤ O⃗ y i , ⃗ T i O ⊤ O⃗ z i ],(29)= [ ⃗ T i · ⃗ x i , ⃗ T i · ⃗ y i , ⃗ T i · ⃗ z i ] = a i ,(30) and meanwhile, Q * π θ = W Qπ θ vec(M (L) i ) + b Qπ θ = Q π θ ,(31) B. Related Works Morphology-Agnostic RL In recent years, we have seen the emergence and development of multi-task RL with the inhomogeneous morphology setting, where the state and action spaces are different across tasks (Devin et al., 2017;Chen et al., 2018;D'Eramo et al., 2020). The morphology-agnostic approach, which learns policies for each joint using multiple message passing schemes, decentralizes the control of multi-joint robots. In order to deal with the inhomogeneous setting, NerveNet (Wang et al., 2018), DGN (Pathak et al., 2019) and SMP (Huang et al., 2020) represent the morphology of the agent as a graph and deploy GNNs as the policy network. AMORPHEUS (Kurin et al., 2020), SWAT (Hong et al., 2021) and SOLAR (Dong et al., 2022) utilize the self-attention mechanism instead of GNNs for direct communication. In morphology-agnostic RL, both of their investigations demonstrate that the graph-based policy has significant advantages over a monolithic policy. Our work is based on SWAT and introduces a set of new benchmarks that relax the over-simplified state and action space of existing works to a much more challenging scenario with immersive search space. Geometrically Equivariant Models Prominently, there are certain symmetries in the physical world and there have been a number of studies about group equivariant models (Cohen & Welling, 2016;Cohen & Welling, 2017;Worrall et al., 2017). In recent years, a field of research known as geometrically equivariant graph neural networks (Han et al., 2022b), leverages symmetry as an inductive bias in learning. These models are designed such that their outputs will rotate/translate/reflect in the same way as the inputs, hence retaining the symmetry. Several methods are used to achieve this goal, such as using irreducible representation to solve group convolution (Thomas et al., 2018;Fuchs et al., 2020) or utilizing invariant scalarization (Villar et al., 2021) like taking the inner product (Satorras et al., 2021;Huang et al., 2022;Han et al., 2022a). Along with GMN's (Huang et al., 2022) and SGNN's (Han et al., 2022a) To systematically investigate the proposed method applied to multi-task training, we construct several variants from the agents we mentioned above, as shown in Table 5. The morphologies of ten variants of 3D Cheetah are different from that of the 2D-Planar, as is shown in Figure 9. C.2. Baselines This part illustrates the implementations of these baselines. SWAT All of the GNN-like works show that morphology-agnostic policies are more advantageous than the monolithic policy in tasks aiming at tackling different morphologies. However, Kurin et al. (2020) validate a hypothesis that the benefit extracted from morphological structures by GNNs can be offset by their negative effect on message passing. They further propose a transformer-based method, AMORPHEUS, which relies on mechanisms for self-attention as a way of message transmission. Hong et al. (2021) make use of morphological traits via structural embeddings, enabling direct communication and capitalizing on the structural bias. We use the original implementation of SWAT released by Hong et al. (2021). For a fair comparison, SET uses the same hyperparameters as SWAT (Table 6). Monolithic We choose TD3 as the standard monolithic RL baseline. The actor and critic of TD3 are implemented by fully-connected neural networks. C.3. Implementation details For the scalar features h i ∈ R 13 , in addition to retaining the original rotation angle of joint, we also undergo the following processing: the rotation angle and range of joint are represented as three scalar numbers (angle t , low, high) normalized to [0, 1], where angle t is the joint position at time t, and [low, high] is the allowed joint range. The type of limb is a 4-dimensional one-hot vector representing "torso", "thigh", "shin", "foot" and "other" respectively. Besides, note that the torso limb has no joint actuator in any of these environments, so we ignore its predicted torque values. We implement SET based on SWAT codebase (Hong et al., 2021), which is built on Official PyTorch Tutorial. SWAT also shares the codebase with SMP (Huang et al., 2020) and AMORPHEUS (Kurin et al., 2020). Table 6 provides the hyperparameters needed to replicate our experiments. Our codes are available on https://github.com/alpc91/SGRL. D. More Discussion about Invariant Methods Specifically, by choosing the "forward" direction, we can achieve heading-equivariance with heading normalization. In essence, the lack of a predetermined "forward" direction that is consistent across all agents prevents us from transferring experiences between different agents. For example, if we create a duplicate of one agent and redefine the "forward" direction, heading normalization will no longer be applicable. In particular, let's consider two agents that have very similar morphology, with the only difference being that their torso orientations are opposite and both encourage movement along the torso orientation. If the torso orientation is selected as the "forward" direction, the normalization applied to these two agents will vary significantly. As a result, the policy learned by one agent will not generalize to the other agent, unless the other agent's movement mode is to move in the opposite orientation of the torso. Therefore, generalization performance is affected by the choice of the "forward" direction and the agent's movement mode. Besides, there is extensive experimental evidence (Hsu et al., 2022;Jørgensen & Bhowmik, 2022;Schütt et al., 2021;Joshi et al., 2022) indicating that equivariant methods that preserve equivariance at each layer outperform those invariant methods that solely apply transformations at the input layer to obtain invariant features and then use an invariant network. Our framework, falling into the equivariant family, enables the propagation of directional information through message passing steps, allowing the extraction of rich geometric information such as angular messages. In contrast, the invariant methods may result in the loss of higher-order correlations between nodes, which are crucial for modeling the geometric relationships between them. E. More Ablation on Equivariance In addition, we conduct another experiment by fixing the initial orientation as 0°when training, but allowing arbitrary angles when testing. As shown in Table 7, SET generalizes well to all cases. On the contrary, SWAT only obtains desirable performance when the testing angle is fixed to 0°which is the same as that during the training process, and its performance drops rapidly in other cases, especially at 180°. The experiments here justify the efficacy of involving orthogonality equivariance. F. The Evaluation on v2-variants The v2-variants (R = 10 ∼ 20m) are more challenging. We train the policy in the multi-task setting where R = 10km, then we do the test in v2-variants. The results and related demos are shown in Figure 10, Figure 11, Figure 12 and Figure 13. While SWAT fails to perform well, SET has obvious advantages. With more episode timesteps, SET locomotes closer to the destination (a shorter distance) and gets more episode rewards. (b) The last frame illustrating SET-produced demos on morphologies (a) The last frame illustrating SWAT-produced demos on morphologies Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s). Figure 1 . 1Illustrative comparison between previous 2D planar setting and our 3D subequivariant formulation. Notably, the agents in (b) are equipped with more DoFs to allow 3D movement. Code and videos are available on our project page: https://alpc91.github.io/SGRL/. LN (·) is the Layer Normalization (Ba et al., 2016). The operations are stacked over L layers in total, resulting in the final architecture of SET, with the full flowchart visualized in Figure 2. For in-domain, there are four collections: (1) three variants of 3D Hopper [3D Hopper++], (2) eight variants of 3D Walker [3D Walker++], (3) eight variants of 3D Humanoid [3D Humanoid++], (4) ten variants of 3D Cheetah [3D Cheetah++]. The cross-domain environments are combinations of in-domain environments: (1) Union of 3D Walker++, 3D Humanoid++ and 3D Hopper++ [3D WHH++], (2) Union of 3D Cheetah++, 3D Walker++, 3D Humanoid++ and 3D Hopper++ [3D CWHH++]. We keep 20% of the variants as the zero-shot testing set and use the rest for training. In particular, the standard half-cheetah(Wawrzynski, 2007; Wawrzyński, 2009) has been so far designed as a 2D-Planar model with the morphology of a walking Figure 3 . 3Multi-task performance of our method SET compared to the morphology-agnostic RL baselines: SWAT and SMP. Training curves on 6 collections of environments. The shaded area represents the standard error. Figure 4 . 4Training curves of v2-variants on 3D Humanoid++ and 3D Cheetah++. Figure 5 . 5Training curves of single-task on 3d humanoid 9 full and 3d cheetah 14 full. On the left-hand side, we present the comparison with baselines, while on the right-hand side, we present the comparison with invariant methods. Figure 6 . 6Training curves of multi-task on 3D CWHH++. The comparison with invariant methods. Figure 7 : 1 .iiFigure 7 . 717SET\g: an O(3)-equivariant model, where gravity ⃗ g is removed from the external force and concatenated into the scalar input, h , ⃗ g]; 2. SET\gd: an O(3)-equivariant model, where both ⃗ g and ⃗ d are considered as scalars: h , ⃗ g, ⃗ d]; 3. SET\z: an O ⃗ g (3)-equivariant model without Equation(5), by omitting the height ⃗ p z i ; Training curves of ablations of SET on 3d humanoid 9 full and 3d cheetah 14 full. Figure 8 . 8Average height of all limbs. Huang, W., Mordatch, I., and Pathak, D. One policy to control them all: Shared modular policies for agent-agnostic control. 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Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5028-5037, 2017. to Equation (19) we use the fact O ⊤ O = I and O ⊤ ⃗ g = ⃗ g, by the definition of the group O ⃗ g (3). With the O ⃗ g (3)-invariant message M i ,it is then immediately illustrated that the query q i , key k i , value message v j , and the attention coefficient α ij are all O ⃗ g (3)-invariant, and value message ⃗ since concatenating the O ⃗ g (3)-invariant a into the input h does not affect the O ⃗ g (3)-invariance of the message M (L) i . SMP Huang et al. (2020) employs GNNs as policy networks and uses both bottom-up and top-down message passing schemes through the links between joints for coordinating. We use the implementation of SMP in the SWAT codebase, which is the same as the original implementation of SMP provided byHuang et al. (2020). Figure 9 . 9The morphologies of 10 variants of cheetah. Figure 10 .Figure 11 .Figure 12 . 101112The evaluation on v2-variants on 3D Hopper++.(a) The last frame illustrating SWAT-produced demos on morphologies (b) The last frame illustrating SET-produced demos on morphologies The evaluation on v2-variants on 3D Walker++.(a) The last frame illustrating SWAT-produced demos on morphologies (b) The last frame illustrating SET-produced demos on morphologies The evaluation on v2-variants on 3D Humanoid++.(b) The last frame illustrating SET-produced demos on morphologies (a) The last frame illustrating SWAT-produced demos on morphologies Figure 13 . 13The evaluation on v2-variants on 3D Cheetah++. Table 1 . 1Comparison in the problem setup.2D-Planar Our 3D-SGRL Table 2 . 2Comparison in zero-shot evaluation on the test set. Note that we omit the lacking part in the name of morphologies.Environment SET SWAT SMP in-domain (3D Walker++, 3D Humanoid++, 3D Cheetah++) 3d walker 3 276.2 ± 17.4 207.0 ± 52.7 56.8 ± 15.1 3d walker 6 431.3 ± 146.2 358.0 ± 58.9 143.4 ± 50.7 3d humanoid 7 244.8 ± 7.9 170.3 ± 51.7 190.9 ± 16.2 3d humanoid 8 299.6 ± 23.7 141.4 ± 22.1 185.4 ± 9.2 3d cheetah 11 4643.9 ± 292.6 1785.3 ± 999.3 2.0 ± 2.9 3d cheetah 12 916.0 ± 39.7 744.1 ± 317.1 29.8 ± 10.7 cross-domain (3D CWHH++) 3d walker 3 206.8 ± 37.4 17.9 ± 13.7 18.0 ± 22.9 3d walker 6 243.7 ± 32.3 114.9 ± 40.3 103.9 ± 1.8 3d humanoid 7 161.9 ± 3.4 152.0 ± 6.8 124.2 ± 15.7 3d humanoid 8 180.0 ± 6.5 156.6 ± 1.7 129.3 ± 0.1 3d cheetah 11 1078.1 ± 722.8 4.3 ± 1.6 6.2 ± 0.5 3d cheetah 12 3038.3 ± 2803.3 349.7 ± 304.3 6.6 ± 1.2 Table 3 . 3Single-task performance with added bias in the heading normalization. The table header (the first row of the table)represents the environment and the bias.Methods 3d humanoid 9 full 3d cheetah 14 full 0 • 180 • 0 • 180 • Monolithic+HN 13142.2 ± 2840.2 57.8 ± 12.0 11357.4 ± 1933.0 −3.2 ± 0.7 SWAT+HN 8517.7 ± 1796.4 92.3 ± 17.8 15924.9 ± 543.1 −1.2 ± 0.4 SET 9931.9 ± 632.0 10106.4 ± 2023.4 14987.9 ± 710.7 14957.9 ± 758.0 Table 4. Compared with Heading Normalization in zero-shot eval- uation on the test set. Note that we omit the lacking part in the name of morphologies. Environment SET SWAT+HN cross-domain (3D CWHH++) 3d walker 3 206.8 ± 37.4 26.3 ± 72.4 3d walker 6 243.7 ± 32.3 156.8 ± 11.1 3d humanoid 7 161.9 ± 3.4 130.2 ± 2.1 3d humanoid 8 180.0 ± 6.5 152.9 ± 36.8 3d cheetah 11 1078.1 ± 722.8 786.5 ± 779.3 3d cheetah 12 3038.3 ± 2803.3 2517.3 ± 2113.9 action space where policies can thus be optimized in the simpler abstract MDP.van der Pol et al. (2020) attempts to learn equivariant policy and invariant value networks in 2D toy environments. Our work focuses on the realization of this motivation in more complex 3D physics simulation environments.C. More Experimental DetailsC.1. Environments and AgentsWe choose the following environments from morphology-agnostic RL benchmark(Huang et al., 2020) to evaluate our methods: Hopper++, Walker++, Humanoid++, Cheetah++. To facilitate the study of subequivariant graph reinforcement learning across these agents, we modify the 2D-Planar agents and extend them into 3D agents. Specifically, we modifyapproaches to scalarization, our method is a member of this family. In a Markov decision process (MDP) with symmetries (van der Pol et al., 2020), there are symmetries in the state-the joint of front foot:[− 1 180 π, 1 180 ], [− 30 180 π, 30 180 π], [− 20 180 π, 5 180 π]. Table 5 . 5Full list of environments used in this work. 3d cheetah 10 tail leftbleg 3d cheetah 11 leftbkneen rightffoot 3d cheetah 11 leftfleg 3d cheetah 12 tail leftffoot 3d cheetah 11 tail rightfknee 3d cheetah 12 rightbknee 3d cheetah 12 tail leftbfoot 3d cheetah 13 rightffoot 3d cheetah 13 tail 3d cheetah 14 full 3D Walker-3D Humanoid-3D Hopper++(3D WHH++) Union of 3D Walker++, 3D Humanoid++ and 3D Hopper++ 3D Cheetah-3D Walker-3D Humanoid-3D Hopper++(3D CWHH++)Union of 3D Cheetah++, 3D Walker++, 3D Humanoid++ and 3D Hopper++Environment Training Zero-shot testing 3D Hopper++ 3d hopper 3 shin 3d hopper 4 lower shin 3d hopper 5 full 3D Walker++ 3d walker 2 right leg left knee 3d walker 3 left knee right knee 3d walker 3 left leg right foot 3d walker 6 right foot 3d walker 4 right knee left foot 3d walker 5 foot 3d walker 5 left knee 3d walker 7 full 3D Humanoid++ 3d humanoid 7 left arm 3d humanoid 7 left leg 3d humanoid 7 lower arms 3d humanoid 8 right knee 3d humanoid 7 right arm 3d humanoid 7 right leg 3d humanoid 8 left knee 3d humanoid 9 full 3D Cheetah++ Table 6 . 6Hyperparameters of our SET.Hyperparameter Value Learning rate 0.0001 Gradient clipping 0.1 Normalization LayerNorm Total attention layers 3 Attention heads 2 Attention embedding size 128 Attention hidden size 256 Matrix embedding size 32×32 Matrix hidden size 512 Encoder output size 128 Mini-batch size 100 Maximum Replay buffer size 10M Table 7 . 7Fixed initial orientation (about 0°) training, arbitrary initial orientation (any given angle) test on 3d cheetah 14 full. The table header (the first row of the table) represents the progress of training and the initial orientation.SWAT 1886.1 ± 148.9 1005.5 ± 615.3 120.5 ± 178.5 791.0 ± 493.4 1232.3 ± 72.9 2592.6 ± 155.6 1340.2 ± 668.0 -5.6 ± 8.5 1193.5 ± 345.2 1178.6 ± 674.9 SET 1587.4 ± 411.3 1695.6 ± 278.4 1659.9 ± 110.2 1388.3 ± 173.8 1465.2 ± 161.0 4622.0 ± 292.8 4799.5 ± 172.9 4756.3 ± 103.4 4899.8 ± 139.7 4902.8 ± 62.9Methods 500k training steps 1M training steps 0 • 90 • 180 • 270 • random 0 • 90 • 180 • 270 • random For simplicity, we omit the index n and t henceforth in the above notations of agent n at time t, since all agents share the same model for all time, e.g., sn(t) → s and an(t) → a. supposed to learn. In our 3D-SGRL environment, all agents are initialized randomly in the full 3D space, facing a random direction, with the goal of moving towards a random destination. This setup is more like a comprehensive navigation task, which brings significant challenges by permitting an input/output state space with much higher complexity.Action Space For a more detailed granularity, our 3D-SGRL also expands the action space that offers the agent more flexibility to explore and optimize the policy on this challenging task. Specifically, the number of actuators is increased from only 1 on each joint in 2D-Planar to 3 per joint, which implies the DoF on each joint is also enlarged from 1 to 3 correspondingly.Geometric symmetry Since both the state space and action space have been enormously augmented, the functional complexity of the policy network φ θ in Equation (3) scales geometrically in correspondence. This poses a unique challenge, especially in RL, where the skills of the agent are gradually obtained through abundant explorations in the environments. During the learning process, the optimization of φ θ becomes highly vulnerable to getting stuck in local minima, and searching for a good policy within the large space would be notoriously difficult. To tackle this AcknowledgementsThis work is jointly funded by "New Generation Artificial . We sincerely thank the reviewers for their comments that significantly improved our paper's quality. Our heartfelt thanks go to Yu Luo, Tianying Ji, Chengliang Zhong, and Chao Yang for fruitful discussions. Finally, Runfa Chen expresses gratitude to his fiancée, Xia Zhong, for her unwavering love and support.the joint of torso from the combination of "slide-slide-hinge" type to "free" type. Normally, each joint of the agent in the 2D-Planar environment has only one hinge-type actuator to make it rotate around y-axis. In order to make the agent more flexible to explore and optimize the learning process, we expand its action space including increasing the number of hinge-type actuators from 1 to 3, thus the DoF of each joint is also enlarged to 3. The two newly-added actuators enable the joint to basically rotate around x-axis and z-axis, respectively.3D Hopper: The rotation range of the joint's two newly-added actuators is limited to [− 10 180 π, 10 180 π]. 3D Walker: The legs of 3D Walker is designed with reference to the legs of standard 3D Humanoid(Tassa et al., 2012). The rotation range of each joint is limited to new intervals. The rotation range of the joints in left and right leg are the same, we only show the intervals of a joint of the left leg:(Wawrzynski, 2007;Wawrzyński, 2009) is specially designed as a planar model of a walking animal, which would not fall over in 2D-Planar environments, so there is no interruption in each episode. But in 3D-SGRL environments, the half-cheetah very easy to falls over and this will interrupt its learning process, making it more difficult for effective locomotion. So we modify the model of a half-cheetah into a full-cheetah, and its torso, four legs and tail are made of 14 limbs. 3D Cheetah is about 1.1 meters long, 0.6 meters high and weighs 55kg. We limit the "strengths" of its joints within the range from 30 to 120Nm. So it is designed as a 3D model of a large and agile cat with many joints yet smaller strength, making it more stable and less easy to fall over in 3D-SGRL environments while retaining a strong locomotion ability. As a result, the full-cheetah is more adaptable to 3D-SGRL environments. The rotation range of joints is limited to new intervals. . J L Ba, J R Kiros, G Hinton, arXiv:1607.06450E. Layer normalization. arXiv preprintBa, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. 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Learning inverse folding from millions of predicted structures. C Hsu, R Verkuil, J Liu, Z Lin, B Hie, T Sercu, A Lerer, A Rives, International Conference on Machine Learning. PMLRHsu, C., Verkuil, R., Liu, J., Lin, Z., Hie, B., Sercu, T., Lerer, A., and Rives, A. Learning inverse folding from millions of predicted structures. In International Conference on Machine Learning, pp. 8946-8970. PMLR, 2022.	[ "https://github.com/alpc91/SGRL." ]
[ "Towards Universal End-to-End Affect Recognition from Multilingual Speech by ConvNets", "Towards Universal End-to-End Affect Recognition from Multilingual Speech by ConvNets" ]	[ "Journal Of L A T E X Class ", "Files " ]	[]	[]	We propose an end-to-end affect recognition approach using a Convolutional Neural Network (CNN) that handles multiple languages, with applications to emotion and personality recognition from speech. We lay the foundation of a universal model that is trained on multiple languages at once. As affect is shared across all languages, we are able to leverage shared information between languages and improve the overall performance for each one. We obtained an average improvement of 12.8% on emotion and 10.1% on personality when compared with the same model trained on each language only. It is endto-end because we directly take narrow-band raw waveforms as input. This allows us to accept as input audio recorded from any source and to avoid the overhead and information loss of feature extraction. It outperforms a similar CNN using spectrograms as input by 12.8% for emotion and 6.3% for personality, based on Fscores. Analysis of the network parameters and layers activation shows that the network learns and extracts significant features in the first layer, in particular pitch, energy and contour variations. Subsequent convolutional layers instead capture language-specific representations through the analysis of supra-segmental features. Our model represents an important step for the development of a fully universal affect recognizer, able to recognize additional descriptors, such as stress, and for the future implementation into affective interactive systems.	null	[ "https://arxiv.org/pdf/1901.06486v1.pdf" ]	58,981,601	1901.06486	ec72d05b345a3a1fcd9fc07c1cfece3161d8319b	Towards Universal End-to-End Affect Recognition from Multilingual Speech by ConvNets AUGUST 2015 1 Journal Of L A T E X Class Files Towards Universal End-to-End Affect Recognition from Multilingual Speech by ConvNets 148AUGUST 2015 1Index Terms-universal affect recognitionspeechemotionpersonalityend-to-end We propose an end-to-end affect recognition approach using a Convolutional Neural Network (CNN) that handles multiple languages, with applications to emotion and personality recognition from speech. We lay the foundation of a universal model that is trained on multiple languages at once. As affect is shared across all languages, we are able to leverage shared information between languages and improve the overall performance for each one. We obtained an average improvement of 12.8% on emotion and 10.1% on personality when compared with the same model trained on each language only. It is endto-end because we directly take narrow-band raw waveforms as input. This allows us to accept as input audio recorded from any source and to avoid the overhead and information loss of feature extraction. It outperforms a similar CNN using spectrograms as input by 12.8% for emotion and 6.3% for personality, based on Fscores. Analysis of the network parameters and layers activation shows that the network learns and extracts significant features in the first layer, in particular pitch, energy and contour variations. Subsequent convolutional layers instead capture language-specific representations through the analysis of supra-segmental features. Our model represents an important step for the development of a fully universal affect recognizer, able to recognize additional descriptors, such as stress, and for the future implementation into affective interactive systems. are able to detect and use affect, in addition to standard ASR and NLP techniques, to provide advanced services such as personality analysis, counseling, education, medical, or commercial services. We focus on affect recognition from audio and speech in this work through two universal affect characteristics retrievable from speech, namely emotion and personality. In the field of emotion detection, there is no general agreement on the number of basic emotion descriptors [5]. It ranges from three (Anger, Happiness and Sadness, with the eventual inclusion of the Neutral class) to up to 20 for some commercial services. Each available corpus includes a different set of emotions. These emotions are often projected onto a plane formed by two main axes: Valence and Arousal [6]. This way the classification task is reduced to the prediction of these two scores and the identification of a point on the plane. This greatly simplifies the classification process and training procedure, but it is less natural for humans to understand and interpret the meaning of Valence and Arousal compared to discrete emotions labels. Furthermore, it poses difficulties and uncertainties in the annotation process. Various emotion types are usually obtained through clustering the plane. For these reasons, and to provide more detailed analysis on each emotion, we decided to perform classification on discrete emotion values in our work described in this paper. For personality recognition a standard set of descriptors are five personality traits from the Big Five model [7]. Traits are patterns in thought and behavior. An individual scores between 0 (low) and 1 (high) for each trait. Thus, an individual's personality is represented by a 5-dimensional vector of scores for the following traits: • Extraversion refers to assertiveness and energy level. Low scorers in this trait are more reserved and calm. • Agreeableness refers to cooperative and considerate behavior. Low scorers in this trait are less interested in social harmony with those around them. • Conscientiousness refers to behavioral and cognitive selfcontrol. Low scorers in this traits are typically seen as more irresponsible and disorganized. • Neuroticism refers to a person's range of emotions and his/her control over these emotions. Low scorers are often more chaotic and anxious. • Openness to Experience refers to creativity and adventurousness. Low scorers in this trait are typically more conservative and less curious. We are particularly interested in whether affect is languagedependent and whether we can build a universal affect arXiv:1901.06486v1 [cs.CL] 19 Jan 2019 recognition system. Some researchers found that emotions vary only a little from one language or culture to the other [8]. The Big Five traits of personality have been demonstrated to be quite robust over different geographic locations [9]. Our previous work has shown that the manifestation of at least some affect, such as stress, is more gender-dependent than languagedependent [10]. It is interesting for us to investigate what features of speech, if any, are language independent. Finding commonality in the speech of different languages has shown to be effective for multilingual acoustic modeling, where it shares part of their phone set at the model level, the statelevel, or the acoustic feature level [11], [12], [13]. Multilingual models are also beneficial with the scarcity of training data in one or multiple languages. For our task on affect recognition, there is not a huge amount of human-labeled data available in any language. We postulate that a universal end-to-end model shared between different language samples may improve the performance on each individual language. Another objective of our work is to explore machine learning methods that can extract features automatically from raw waveforms, without explicit human design. We are motivated by the fact that the human auditory system is capable of processing audio in different languages without any morphological changes. In addition, previous work on multilingual speech recognition has shown that there are stronger responses in certain groups of the spectral frequencies to phonetic sounds in certain languages. The class of deep learning algorithms called Convolutional Neural Networks (CNN) has shown to be astute in automatic feature extractions in both the image and speech domains [14], [15]. We aim to investigate end-to-end CNN models for affect recognition. An important objective of using automatic feature extraction combined with classification is to bypass the time delay in extracting features. A system based on narrow-band raw waveforms would allow us to avoid any corpus-dependent and language-dependent feature engineering step, would be applicable to any sort of spoken input signal, such as phone calls, and would require less memory and pre-processing overhead. This work is a significant extension of our earlier attempts to detect speech emotions from narrow-band speech raw waveforms [16], [17] to include personality analysis and a multilingual approach. In this paper we significantly revise the model and experimental setup from our previous works and experiments on more datasets in different languages. We do not only limit the application to the emotion detection problem, but we also show the effectiveness on the more difficult personality detection from speech, again in a multilingual setting. We then provide more insights about what the model is actually trying to learn, and how it generalizes across languages. II. RELATED WORK A. Multilingual approaches for speech and language Multilingual approaches from speech and language first appeared in the 1990s with statistical models. These models have been found to help improve the performance for those languages with limited resources, through taking advantage of the similarities among different languages and by borrowing from resource-rich languages [13]. Since we are interested in deep learning methods that can automatically extract features, we also look at recent multilingual neural models that have been proposed in speech processing, including speech recognition [18], [19], [20], [21], [22], [23], [24], [25]. Neural network architectures and training procedures were specifically designed to handle multilingual input and take advantage of multiple languages combined to improve the recognition performance on each of them. For Automatic Speech Recognition (ASR), [18], [19] used a multilayer DNN with an array of language-specific final layers to share the acoustic features across different languages. [20], [21] instead used a similar DNN with a single final layer finetuned on different languages, while [22] applied a progressive layer by layer training first from a multilingual corpus and then from specific languages. Other techniques were used to adapt the final layers such as low-rank factorization of parameter matrices [23] or bottleneck layers and extra features [24], [25]. B. Emotion recognition from speech Previously speech emotion detection was performed through the extraction of many features from the audio sample which are then fed into a supervised classifier [26]. A standard set of features included speech features such as MFCC, psycholinguistic features [27] and other low-level audio features such as pitch, zero-crossing rate, energy and many others [26]. They were extracted from small audio frames, typically of around 25 ms, and then combined together to represent the utterance to analyze. This combination was performed either through many statistical functionals such as mean, standard deviation, skewness, kurtosis, etc. [28], or directly through the classifier [29]. The classifier choice ranged from basic supervised classifiers such as SVM [26], [30] and decision trees [31], to more complex deep learning structures such as DNN [32], CNN [33], ELM [29] or LSTM in the case of continuous emotion detection [34]. Most of the analysis was performed on the valence-arousal plane [26], often grouping multiple discrete emotions as high/low valence and high/low arousal [35], [30]. All those feature sets were often collected and provided in various shared task [36], [37], [38], and used as standard feature sets for affect recognition thereafter. Others have applied more complex feature engineering and feature selection [39], but these processes are often time consuming, add overhead latency or be database dependent. Departing from the traditional feature engineering approach [40], [41] used deep learning models to perform automatic feature extractions from the audio represented as a spectrogram. In these works the spectrograms are described as the "raw audio signal". However, we note that the spectrogram itself, though more limited in scope, is already a feature extraction step where each audio frame is associated with its FFT coefficients, thus it is not the "raw audio signal" as purported. We are interested in investigating whether CNNs can extract features and classify them correctly directly from the time-domain raw waveform. Extending the analysis to the field of multilingual and cross-domain emotion and personality recognition, most works applied traditional feature-engineering with eventual speakernormalization [35], [42]. [43] tried to solve the problem through the extraction of a shared feature representation using Kernel Canonical Correlation Analysis [44], while [45], [46] obtained the shared feature representation training an autoencoder. [30] managed to increase the classification accuracy over various corpora by using unlabeled data. With the exception of [47] and our preliminary work [16], [17], no other work to our knowledge have ever tried to classify paralinguistic traits using raw waveforms as input directly. Even [47] analyzed only a very limited dataset, and only on the valence-arousal dimensions, providing very limited insights on the proposed network architecture. C. Personality recognition from speech The field of Personality Computing is quite young, but some work has been published on recognizing Big Five personality traits from the non-verbal part of speech [48]. Just as in emotion detection, most of the work has focused on extracting lowlevel prosodic features and statistical functions thereof, using a standard classification algorithm such as SVM or Random Forest to determine whether the subject scores above or below the median for each of the five traits [49], [50], [51]. The Interspeech 2012 Speaker Trait Challenge [52] was the first comprehensive effort to compare different approaches to the problem, by benchmarking on the same test dataset. Popular prosodic features are statistics of pitch, energy, first two formants, length of voided and unvoiced segments, as well as Mel Frequency Cepstral Coefficients (MFCC). The winner of the competition extracted thousands of spectral features before doing a feature selection process [53], a method that was very common [54]. These features were then mapped to the five traits through SVM classifiers. Classification accuracies from this challenge are between 60 and 75 percent, depending on the trait. Although many different approaches and machine learning algorithms were tried, none of them clearly outperformed the others. Also, due to the limited number of subjects, the results from these works are often statistically unreliable and could be heavily corpus-dependent. Related work found that the Extraversion and Conscientiousness traits were easiest to classify, and Openness to Experience the hardest [55], [51], [56]. The ChaLearn 2016 shared task challenge released a large corpus of 10,000 extracts from YouTube video blogs [57]. Each clip was labeled with continuous Big Five labels. Each of the participants of the shared task used audio as well as video, and it is not possible to directly look at recognition performance from just audio. This workshop is still interesting for two reasons. The corpus provided is, to our knowledge, the biggest open-domain personality corpus, and the best performing teams used neural network techniques. However, although teams inserted video directly into a neural network, they still extracted traditional audio features (zero crossing rate, energy, spectral features, MFCCs) that were then fed into the neural network [58], [59], [60]. A deep neural network should however be able to extract such features itself (if relevant for the task). An exception was [61], but they used a neural network specialized for image processing and computer vision, and did not adapt it to audio input. The team with the best performance in the challenge extracted openSMILE [26] acoustic features as used in the INTERSPEECH 2013 challenge baseline set [38], which they linearly mapped to the Big Five traits. They did not publish their work. The challenge was aimed at the computer vision community (many only used facial features), thus although many teams analyzed their approaches to vision, not many looked into detail what their deep learning network was learning regarding audio input. III. METHODOLOGY In this paper, we propose a method for automatically recognizing emotion and personality from speech in different languages, without the need for feature extraction upfront. We propose to achieve this with a multi-layer Convolutional Neural Network (CNN) framework, trained end-to-end from raw waveforms. We train models in monolingual and multilingual settings. We compare our model with a similar CNN model that takes spectrogram representations as input. A. Preprocessing We are interested in recognizing affect from a given input audio sample. The very first processing step is to downscale the input sample to a uniform sampling rate. We choose narrowband speech at 8 kHz for our work. There are two main reasons for this choice. The first is to analyze how the system would work under the worst possible conditions, for example to detect emotion over a phone call. The second one is to reduce the eventual transmission time and memory requirements when the speech has to be sent over internet or has to be stored and processed in an embedded system, it also reduces the number of network parameters. An aspect often overlooked while designing models for affect recognition is the input volume. Features such as relative energy within or across frames are important components of affect, as sudden changes may signal high arousal emotions like anger. However absolute energy over the entire sample is not useful, as it mainly depends on the volume the sample was recorded at. The absolute energy level may cause severe overfitting to the model. This is often evident with emotions like anger or sadness, where sometimes the model only learns to distinguish the classes through the amplitude level ignoring the rest of the features. Different language corpora, especially when consisting of spontaneous speech, may contain samples recorded at varying input volumes. The position of the speaker with respect of the microphone may also differ each time. All these aspects cannot be determined a priori. Volume normalization techniques, such as peak or RMS normalization, can be applied but they are not fully suitable for our task for the following reasons: 1) Peak normalization would suffer from isolated peaks which are often not representative of the whole sample; 2) RMS normalization instead would be sensibly different depending on the amount of silence in the audio sample, which is not always related to affect (potentially due either to a low speaking rate, pauses in the recording or the microphone kept open). Starting from the assumption that affect does not change depending on the recording volume, during the training phase, but not during the evaluation, we randomized uniformly the amplitude (volume) of the input audio sample through an exponential random coefficient α at each training iteration, where α is equal to α = 10 U(−a,a)(1) where U(−a, a) is a uniform random variable, and a a hyperparameter. We applied this pre-processing instead of normalizing the volume to a fixed value in order to increase the robustness of the system. A uniform random variable over a wide range of value (compared to a normal distribution) not only as said before helps reducing the overfitting related to the energy component, but also strongly augments the training set size. B. CNN for feature extraction The aim of our work is, given an input utterance or speech segment in the form of raw waveforms, to determine the overall affect state expressed. The Convolutional Neural Network is an ideal architecture for this task as it is able in sequence to learn and perform feature extraction from short overlapping windows regardless of the overall sample length, analyze the variation over neighboring regions of different sizes, and combine all these contributions into an overall vector for the entire audio sample. CNN are typically employed with great success in image recognition task. In acoustic analyses, audio samples can be regarded as 1-dimensional "images". Each component does not represent a pixel but the value of the acoustic waveform, and different input channels may include different signal transformations. Ideally our model should also learn to internally extract features and process audio consisting of different speaker characteristics, such as gender and age, different languages and different input volumes without any prior normalization. This process is similar to that applied by traditional featurebased methods. In these methods a feature extraction tool, such as openSMILE [26], is used to extract a series of features (typically MFCC, pitch, zero-crossing rate and energy) from the audio sample divided into small frames. Framebased features are then merged together with a series of higher-level descriptors (such as mean, standard deviation, skewness, kurtosis, etc.). However, the features and the highlevel descriptors are not statically defined a priori, but are learnt by the network. We expect that low-level features would be mostly extracted by the first layer, and high-level descriptors by the higher layers [62]. The network would also presumably learn to automatically filter the ones less useful, concentrate more on those more useful and eventually extract some other different features which were not usually applied in affect recognition before. C. CNN model description Our CNN consists of a stack of convolutional layers of different sizes. It is followed by a global average pooling operation on the output of each layer, a weighted average combination of all these vectors, a fully connected layer and final activation layer (softmax for emotion and sigmoid for personality). The specific role of each layer will be described in detail in Section VI. A model diagram is shown in Figure 1. The CNN receives as input a raw sample waveform x of narrow-band speech, sampled at 8 kHz, of arbitrary length. We split the input signal into two feature channels as input for the CNN. The first one is the raw waveform as-is, the second one is the signal with squared amplitude. The second channel is mostly aimed at capturing the energy component of the signal and learn an implicit normalization. The two input signal components are then directly fed into a first convolutional layer: x (1) i = f (W (1) x [i,i+v] + b (1) )(2) where v is the convolution window size and f a non-linear function. In this first layer we use a window size of 200, which at 8 kHz sampling rate corresponds to 25 ms, and a stride of 100, which corresponds to around 12 ms. The output size dim(x (1) i ) (the number of filters trained) from each window is set to 512. This first layer acts as a low-level feature extractor, or a customized filterbank learnt over the corpus during training. It ideally replaces the discrete features extraction step or the FFT computation of the spectrogram window. The window length of 25 ms is a common choice for the feature extraction step, as shown in previous works using feature-based or spectrogram based CNNs [36], [33]. It is then followed by several higher convolutional layers, of the same number of filters. Their convolution window size and stride is set to capture increasingly larger time spans. The subsequent convolutions are aimed at combining the features and capturing information at the suprasegmental level, such as phonemes, syllables and words, as well as looking at the difference between contiguous frames. Since our intent is to capture globally expressed emotion and personality characteristics from the entire audio frame, the contributions of the convolutional layer outputs must be combined together. This is done by a global average pooling operation over the output vectors: x AP j = i (x i,j ) L i (3) where i is the window index, L i the number of output windows from the convolution layer, and j the feature vector index within each convolution window. The average pooling is performed over the output vectors of each layer instead of just the final one. This would combine the contributions of both the segmental and suprasegmental features at different temporal granularities for the final emotion and personality decision. The output averagepooling vectors for each layer are then combined through a weighted-average layer where the weights are parameters of the model: x OUT = f l W OUT l x AP l + b OUT (4) where l is the layer index, and f again the non-linear function. We decided to use average pooling instead of the more common max-pooling. This choice yielded higher results on the development sets. It is also meaningful as the objective of our work is to detect the overall affect of an entire utterance or a speech passage. The global average pooling can be seen as merging together all the intermediate affect results []. It sums and accumulates the contributions among all the speech segments considered, instead of just selecting a few salient instances. We empirically noticed that applying max-pooling, even side-by-side with average pooling, makes the network overfit the training data more easily. After obtaining the audio-frame overall vector x OUT by weighted-average of each convolutional layer output (Eq. 4), we then feed it through a fully connected layer, followed by a final softmax/sigmoid layer. This last layer performs the final classification/regression operation and outputs the probability of the sample to belong to each emotion class analyzed as well as the personality trait scores. In each of the intermediate layers the exponential linear activation function is used as non-linearity [63], as it performed better on the development set compared to other popular choices such as the hyperbolic tangent (tanh) or the rectified linear function (ReLU). D. Multilingual adaptation Our CNN is already designed to handle a multilingual setting taking advantage of data in different languages. The duty of the first layer is to learn and extract low-level features common across all languages, such as filterbank features, pitch and energy. More data can improve this step's performance. The subsequent layers are instead delegated to supra-segmental features, some of which are specific to languages or groups of languages. The application of a large layer size, 512 in our architecture, also allows the network to better learn these language-specific features and language acoustic models. Although the model is already adequate to learn affect from multiple languages, further language-specific adaptation is desirable. After the initial training on the full dataset, we retrain the final layers after the average pooling on a single language data. This adaptation, or fine-tuning step, operates by weighting differently the extracted features of each layer, in order to adapt to each specific language analyzed. It is here where different affect states are communicated that can be dependent on language. E. Spectrogram CNN Until recently the idea of using the raw representation of a signal often refers to a spectrogram presented as an image to a CNN [40], [41]. As a comparison baseline, we propose a similar CNN that takes the spectrogram representation as input. The spectrogram CNN is very similar to the one used for raw waveforms. A spectrogram representation is first extracted from a raw input waveform, again sampled at 8 kHz. This is done through a Tukey window of 25 ms, with an FFT-size of 256, and yields a series of 129 power spectral features for each window. This operation replaces the feature extraction done by the first convolution layer of the raw waveform network. The subsequent layers are the same as in the raw waveform network, including several convolutional layers, global average pooling for each layer, weighted-average, fully connected and activation layers. IV. EMOTION RECOGNITION EXPERIMENTS A. Corpora In our experiments we make use of two set of corpora: a multi-domain English corpus with crowdsourced labels, and a set of smaller corpora of acted emotions in various languages. A summary of the number of utterances of each corpus is shown in Table I. The English corpus is made of data we collected and annotated in multiple phases over time [16], [69]. We collected thousands of utterances and short speeches from different sources including monologues (TED talks, YouTube vloggers) and dialogues (TV shows, debates). In the case of TV shows, individual utterances were segmented from the audio track using the subtitles timestamps as references. The monologues instead were cut into segments of around 10 − 15 s, using silences as references. We then labeled them with several emotion descriptors, using student helpers and through Amazon Mechanical Turk. Each audio clip was annotated by a minimum of one annotator (in the case of the student helpers, previously instructed on the task) to a maximum of five annotators. We took the label selected by the majority of the annotators, discarding the sample in case of disagreement. In this work we only consider the subset of utterances classified as anger, sadness, happiness and anxiety. We also annotated the data with other emotions labels. However, some of them were not present in all languages. Others contained a number of samples too limited for training. To train a universal multilingual model and evaluate the performance of our classifier on different languages, we used made of 3 actors and 3 actresses and a total of 2694 utterances, including one longer passage but excluding the isolated word part of the database. It includes five emotions: anger, happiness, fear, sadness and neutral. In our work we analyze a subset of emotion labels common to most of the corpora: namely anger, sadness, happiness and anxiety. As each database is made of slightly different emotions or denominations, we take fear as anxiety and joy as happiness. B. Experimental setup To build the test sets, for the corpora which included different speakers of different genders. For the German, Italian and Serbian corpora one speaker of each gender was used as the test set. For the other three corpora we could not apply speaker separation. In the Spanish and Estonian corpora contained too few speakers for each gender: one male and female the former, and only one female speaker the latter. In the English dataset instead most of the samples did not include any information about the speaker identity. In any case the overall number of speakers and samples in this language was much greater than the other language corpora, since it includes data from a large number of sources. For these three datasets around 10% of samples of each emotion class were taken as test set. The detailed division among training and test set is reported in Table I. The test set was kept the same during the multiclass and fine-tuning training phases, as well as with various network configurations. In order to tune the network structure and hyperparameters, and determine the early stopping condition, a subset of the training set of 10% was each time randomly taken as the development set. Each audio sample was transformed into wav format at 16 bits and downsampled to 8 kHz with sox 1 . To keep the input range of every sample small and avoid parameter overflowing during training, a constant value of k = 5·10 −4 was multiplied to every input audio sample. The k value was chosen in order to approximately normalize the overall standard deviation to 1. The volume randomization hyperparameter a (see Eq. 1) was set to 1.5. We apply four convolutional layers after the first feature extraction layer, the first layer with a kernel size of 8 and a stride of 2, and each subsequent ones with a kernel size of 4 and a stride of 2. This means that each layer from the first to the last analyzes increasingly larger time spans starting from 25 ms. To train our CNN we applied standard backpropagation with Adam optimizer [70]. The initial learning rate was set to 10 −4 , and halved once after the first 25 epochs and subsequently after another 15 epochs. We stopped training when the error on the development set began to increase. During the global multiclass training a minibatch size of 2 was used, while in the single class and fine-tuning we used a minibatch size of 1. C. Results Results of our experiments on multilingual emotion detection are shown in Table II. They are represented in terms of precision, recall and F-score over each emotion and language. We obtained an average F-score of 67.7% (68.5% after finetuning the last layer) across all the languages using our raw waveform CNN trained on multiple languages. We obtained an average of 55.2% with the same model trained on a single language, 58.2% from the multilingual spectrogram baseline and 60% from the same baseline trained on single languages. Overall, this yields a relative improvement of 12.8% of the multilingual raw waveform CNN over the second best model, the spectrogram CNN trained on individual languages. V. PERSONALITY RECOGNITION EXPERIMENTS A. Corpora For the personality recognition task we use three different languages datasets: a bigger one in English and two smaller ones in Mandarin and French. Each sample from each dataset is annotated with five continuous scores between 0 and 1 (for the Big Five personality traits). Each dataset is recorded at a sampling rate of at least 8 kHz. The datasets are: consists of 640 clips taken from French radio shows. Each clip is labeled by 11 unique judges. Final scores are taken as the average of the scores of these judges. It's important to note that the distributions (means and standard deviations) of trait scores differ per dataset. Especially the spread in scores for the Chinese dataset is very small. To combine all data for training, the labels thus need to be normalized. For the English dataset we use the pre-defined ChaLearn Validation Set (2,000 samples) as test set. For Mandarin, we take 60 samples each from male and female speakers, which results in 120 samples in total. For French, we take out 160 random samples to serve as test set. As the development set we used 10% samples from each corpus. B. Experimental setup For the personality recognition experiments we used four convolutional layers in the CNN. Everything else is identical to the architecture used for the emotion recognition experiments. We pre-processed the input samples and trained the network mostly in the same way, and with the same single and multilingual experiments, as described for emotion in the previous section. However, an important exception is represented by the labels. Due to the difference in label distributions (mean and spread), across the three datasets we rescaled all training labels to have the same mean and standard deviation before training. We assumed the labels distribution for each personality trait as a Gaussian random variable. At evaluation time, the output predictions were inversely converted back to the original distribution for each individual language. We trained the model with a regression cost function by minimizing the Mean Squared Error between model prediction and ground truth: MSE = 1 N N i=1 (p i − g i ) 2 (5) where p i is the vector of the five trait predictions for a given sample i and g i is the vector of the five ground truth trait values for that sample. Another difference is the higher learning rate of 2 · 10 −4 . We evaluate the model both from a regression point of view, evaluating the Mean Absolute Error (MAE) between the prediction and the ground truth for each trait, and from a classification point of view by turning the predictions and corresponding labels into binary classes using the average of each trait as the boundary between the two classes. In this setting we compute classification accuracy, precision, recall and F-score. C. Results Results on each corpus, including the average over each trait for each language, are shown in Table IV-A. The fine-tuned multilingual model performs best on the test sets in terms of F-score. For the multilingual model using raw waveforms, we obtained an average F-score over the three languages of 62.4%. Training this same model on each language individually resulted in an average F-score over the languages of 56.7%. Using a spectrogram instead of raw waveforms gives an average F-score of 58.8%. Thus, our multilingual efforts show a relative improvement of 6.3% over the spectrogram approach and 10.1% over the single language approach. VI. DISCUSSION A. Affect recognition performance Results obtained for both emotion and personality recognition show that in all cases the multilingual training with raw waveforms input outperforms both the spectrogram input and the single language training. In some cases, like the German or Serbian emotion corpora, and the Chinese personality dataset, the improvement is particularly significant. Another evident result, in particular on the emotion experiments, is that using raw waveforms improves the performance of the multilingual training, while on the other hand the spectrogram input is better on the single language case. Fine-tuning of the last layers helps in most cases achieving an improvement, although in a minority of cases it is not that beneficial. It seems less useful when the datasets are larger than average (the two English datasets) or very small (the emotion German corpus). Regarding the emotion recognition task, there is no particular emotion that is easier or more difficult across all the languages. Some emotions in specific languages are sometimes mistaken, for example in the English dataset anxiety is often classified as sadness, or German happiness as anger. These misclassifications are often related to the specific corpus characteristics. Above are shown the filters applied to the raw signal, below those applied to the squared signal. In this case the network extracts a wider feature set than in the emotion detection case. These features include energy, pitch, contour variations and also frequency components between 500 and 1000 Hz. B. Low-level feature selection layer The first layer of our CNN has the role of extracting low-level features from the raw waveforms. It is important to visualize and understand which kind of features are extracted, how much these features correspond with those used in traditional featurebased approaches [36], and whether something new or unusual is extracted. To visualize the first layer we follow a similar procedure as used in [62], [17]. We consider each row of the parameter matrix W c , which represents a filtering function applied to each convolution window and whose output is then summed together before the application of the non-linearity. We transform each filter to the frequency domain, taking the absolute values of the FFT: F (W (1) i ) = \|FFT(W (1) i )\|(6) where i is the filter index. Each FFT coefficient represent the activation of the filter to each frequency. We do this for both the raw waveform channel and the squared signal channel. The activation values have been converted to logarithmic scale with the following function: a(i, f ) = 20 log 10 (F (W (1) i,f ))(7) To better show the filter contributions we sorted them according to the frequency with the highest activation, in ascending order. Figures 3 and 4 show the filter activation pattern respectively for the emotion recognition and personality recognition experiments. In the emotion recognition experiments, three kinds of features are evident from the plots. Approximately one-third of the filters applied to the raw waveform, and more than half of those applied to the squared value have their peak at 0 Hz. This first set of filters is likely capturing the signal energy. A second set of filters in reverse proportion over raw signal and squared signal channels has instead its peak over a narrow range of low frequency values, between 0 and 250 Hz. Those filters act as pitch detectors, and this is compatible with the The last fully connected layer shows stronger emotion clustering instead. Some languages, in particular Spanish, Serbian, German and some English samples, seem to be clustered together according to the emotion, thus interacting with each other to build the final predictions. fact that the average human pitch frequency lies below 250 Hz for both males and females [72]. It is interesting to note what happens for frequencies above 250 Hz. In the original waveform signal input channel, very few filters have their central frequency between 250 Hz and 500 Hz, and the higher frequencies in the spectrum are almost ignored. This may be due to an amount of emotion data available too small to capture effectively further information at higher frequency, or might suggest the hypothesis that high frequency components do not carry useful information for emotion detection. If the latter hypothesis is confirmed, there would be no need to use wide-band audio to improve the performance on emotion detection. However, in the squared signal input channel, a small number of filters extend above 500 Hz. These filters may capture the local amplitude variations of the signal, particularly frequent in angry speech. They may also learn an amplitude normalization function to apply to the signal, to remove the effect of variable amplitude levels at input (often due to non-uniform recording volume across samples). This hypothesis is supported by the observation that most filter activate dominantly on 0 Hz. For personality recognition, a similar observation can be made about energy (activations at 0 Hz) and pitch (activations between 0 and 250 Hz). On the other hand, a third of the filters activate between 500 and 1000 Hz, higher than the cutoff frequency for emotion. These higher activation frequencies also result in about a third of the filters for the squared signal input channel activating strongest at higher frequencies. Since the squared signal is likely used for internal normalization, this may indicate a more complex normalization for higher frequency components in the signal. C. Intermediate convolutional layers As mentioned in section III, the second to last convolutional layers are aimed at combining features at supra-segmental level and, among others, selecting the most salient phonetic units that may carry the emotion or the personality information. To visualize the contribution of these layers over a few examples, we estimate from the average pooling vectors a weighing factor w t to each time window taking the RMS value of the difference between the average pooling values and the element-wise average, in the following way: w t = i (x t i − x i ) 2 N(8) where i is the vector element index, N the vector length (512 in the emotion detection experiments) and x i the element-wise average among all time instants. A high w t means that some of the filters have a different value than the average for that specific time-frame, and are more likely to contribute to the final classification. Figure 2 shows the activation of the intermediate convolutional layers over speech segments randomly taken from the corpus respectively for emotion and personality. For emotion, the uniform low activation pattern over the silence regions shows these do not usually carry any emotion-related information. For personality, filters do activate over silence, indicating these regions are correlated with personality. The intermediate layers activate strongly over voiced regions, especially when there is a prosodic change, such as energy or pitch variations. The activation pattern is often similar among the layers, but it is slightly more sparse toward the upper layers. This signals that upper layers tend to select the most important features extracted by the lower layers. The behavior of each layer after the average pooling operation, and the final fully-connected layer, is also worth noticing. We projected the output of each intermediate layer into a two-dimensional space through a t-SNE transformation [73]. The output for the intermediate layers of the emotion and personality detection networks are respectively shown in Figures 5 and 6, highlighting the language of each sample. The figures illustrate that later convolutional layers are grouping each source language into its own cluster, with more defined cluster boundaries going upwards in the layers hierarchy. It seems that, through suprasegmental feature analysis, the network is automatically learning a specific affect model for each single input language. In the emotion recognition experiments (Fig. 5, first and second rows) this pattern is very clearly shown by the t-SNE for all languages, except German due to the low amount of samples in that language. This pattern is also shown in the personality recognition experiments (Fig. 6. We also note that the Mandarin Chinese cluster is clearly distinct from the English and French clusters, which can be explained by the different cultural factors between Europe and China which may affect personality and its perception by annotators. Another factor contributing to this might be that English and French are much more similar phonetically than they are to Mandarin Chinese. In the emotion recognition case instead, Spanish seems to have a more distinct cluster. This dataset also yields the best average performance, which could be because it is acted and emotions are very clearly expressed. Overall, these figures show that, as we expected from previous multilingual acoustic modeling [13], [11], languages do share common features in the low, signal processing level, while they tend to have distinct characteristics at the higher, perhaps phonetic level. All these components are sent to the final classification layers, allowing the network to use both the common and different characteristics of the languages and use them to improve the final predictions. This is evident in the t-SNE representation of the emotion recognition last layer (Fig. 5, last row). Some groups of languages, in particular Serbian, Spanish, German and some English and Italian samples share emotion clusters. This could indicate that emotions from different languages have similar representations inside the network, thus explaining why adding data from other languages improves the model's performance. The exception to this is Estonian, which has a very different root from the other languages. We do not show these projections for personality recognition, as the regression nature of this task prevents clear clusters to form. VII. CONCLUSION In this paper, we proposed a universal end-to-end affect recognition model using convolutional neural networks. It is able to automatically extract features from narrow-band raw waveforms and detects emotions and personality traits regardless of the input language, whose characteristics are automatically learned and distinguished. We have obtained significant improvements both over a spectrogram baseline (+12.8% for emotion and +6.3% for personality), and training it with a multilingual setting as opposed as each single input language (+12.8% for emotion and +10.1% for personality). That is, we have shown that using raw waveforms yields higher performance than using spectrograms as input, and that training on multiple languages increases evaluation performance on each individual one in comparison to training separate models for each language. We have furthermore shown how the first convolutional layer in the model extracts low level features from the audio sample, while higher layers activate on prosodic changes and learn language-specific representations. We have shown that universal affect recognition has the potential to take advantage of each language to improve the performance of other languages, as the affect descriptors studied share features among languages. Furthermore, end-to-end deep learning architectures are able to recognize different affect classes, emotion and personality, automatically learning and processing the most relevant speech features. Fig. 1 . 1Convolutional Neural Network architecture for emotion and personality recognition from raw waveforms. The output consists of either the four emotion classes analyzed or the Big Five personality trait scores. • English -ChaLearn Looking at People 2016 Apparent Personality Analysis (APA) Dataset [57]: consists of 8,000 clips of around 15 seconds, taken from YouTube blogs with diverse conversational content. The videos are labeled by Amazon Mechanical Turk workers. Audio clips are extracted from the videos. • Mandarin Chinese -Beijing Social Personality Corpus (BSPC) [56]: consists of 258 male and 240 female clips taken from 70 Chinese talk shows. Clip length varies from 9 to 13 seconds. The utterances are labeled by three student workers each by filling in a standard NEO-PI-R personality inventory for the speakers. • French -SSPNet Speaker Personality Challenge [71]: Fig. 2 .Fig. 3 .Fig. 4 . 234Spectrogram representation of short speech samples from the corpus (top), and relative RMS activation over time of the intermediate layers for the emotion (left two) and personality (right two) networks. Figures show how the higher network layers activate on the voiced parts with different patterns, especially when there is a change in prosody. For emotion, silences are mostly ignored, whereas for personality one layer also activates heavily on longer durations of silence. Frequency response of each of the 512 filters (horizontal axis) of the first-layer of the CNN for emotion recognition. Above are shown the filters applied to the raw signal, below those applied to the squared signal. It is evident how energy (left) and pitch (middle) are the main features extracted for emotion recognition by the CNN. Frequency response of each of the 512 filters (horizontal axis) of the first-layer of the CNN for personality recognition. Fig . 5. t-SNE projection of outputs from average pooling after the first and fifth convolutional layers, and the last fully connected layer of the emotion recognition CNN. The left column shows the sample points with the languages highlighted in different colors. The right column shows the same projection with emotion labels highlighted in different colors. Higher convolutional layers tend to cluster samples according to the language compared to the first layer. Fig . 6. t-SNE projections of average pooling after each CNN layer for personality recognition, from first to fourth (left to right, top to bottom). The language of each sample is highlighted in different colors and symbols. Language clusters appear especially after the second layer, and get more distinct after each layer. Chinese samples tend to have more distinct clusters than the other two languages (English and French). TABLE I NUMBER IOF UTTERANCES OF EACH CLASS, AND TOTAL NUMBER, IN THE EMOTION CORPORA CONSIDERED. IN PARENTHESIS THE DIVISION AMONG TRAINING AND TEST SET.Speakers Language Anger Sadness Happiness Anxiety Total utterances English 1202 (1092/110) 1246 (1115/131) 2128 (1933/195) 952 (865/87) 5528 Estonian [64] 306 (275/31) 249 (224/25) 271 (243/28) - 826 German [65] 127 (102/25) 62 (54/8) 71 (65/6) 68 (62/6) 328 Spanish [66] 725 (652/73) 731 (657/74) 732 (658/74) 735 (661/74) 2923 Italian [67] 84 (56/28) 84 (56/28) 84 (56/28) 84 (56/28) 336 Serbian [68] 366 (244/122) 366 (244/122) 366 (244/122) 366 (244/122) 1464 Total 2810 2738 3652 2205 11405 TABLE II RESULTS II(PERCENTAGE) ON MULTILINGUAL TASK FOR THE FOUR EMOTIONS ANALYZED (ANGER, SADNESS, HAPPINESS, ANXIETY). P STANDS FOR PRECISION, R FOR RECALL, AND F1 FOR F-SCORE (OR F1 MEASURE).several corpora listed below. Compared to the English database, they contain a limited number of speakers who were actors that generated each emotion in a studio setting. Source sampling rate was usually 16 kHz or higher.English Estonian German Spanish Italian Serbian Method P R F1 P R F1 P R F1 P R F1 P R F1 P R F1 Anger Single lang. spec 0.0 0.0 0.0 45.1 74.2 56.1 82.6 76.0 79.2 92.8 87.7 90.1 93.3 50.0 65.1 56.7 45.1 50.2 Multilingual spec 30.1 25.5 27.6 56.7 54.8 55.7 77.3 68.0 72.3 76.3 83.6 79.7 48.4 53.6 50.8 67.0 50.0 57.3 Single lang. raw 44.6 40.9 42.7 40.0 96.8 56.6 76.9 80.0 78.4 86.5 87.7 87.1 61.9 46.4 53.1 47.8 73.0 57.8 Multilingual raw 49.6 54.5 51.9 54.5 77.4 64.0 82.6 76.0 79.2 93.2 94.5 93.9 68.4 46.4 55.3 58.2 87.7 69.9 Fine-tuned raw 46.2 54.5 50.0 56.4 71.0 62.3 83.3 80.0 81.6 95.8 94.5 95.2 77.8 50.0 60.9 67.2 70.5 68.8 Sadness Single lang. spec 62.4 44.3 51.8 75.0 36.0 48.6 100.0 100.0 100.0 95.9 95.9 95.9 73.3 39.3 51.2 95.7 90.2 92.8 Multilingual spec 62.0 37.4 46.7 50.0 60.0 54.5 100.0 87.5 93.3 93.4 95.9 94.7 85.2 82.1 83.6 80.1 95.9 87.3 Single lang. raw 60.8 47.3 53.2 0.0 0.0 0.0 80.0 100.0 88.9 93.0 89.2 91.0 66.7 71.4 69.0 91.7 90.2 90.9 Multilingual raw 64.2 65.6 64.9 52.2 48.0 50.0 100.0 100.0 100.0 96.1 100.0 98.0 62.2 82.1 70.8 88.3 99.2 93.4 Fine-tuned raw 64.2 60.3 62.2 57.1 48.0 52.2 100.0 100.0 100.0 96.1 100.0 98.0 76.0 67.9 71.7 91.0 99.2 94.9 Happiness Single lang. spec 42.3 93.3 58.2 57.1 42.9 49.0 25.0 50.0 33.3 86.1 91.9 88.9 71.9 82.1 76.7 44.1 68.0 53.5 Multilingual spec 43.2 66.7 52.4 58.3 50.0 53.8 15.4 33.3 21.1 80.3 71.6 75.7 26.1 21.4 23.5 47.0 64.8 54.5 Single language raw 52.6 77.4 62.7 22.2 7.1 10.8 0.0 0.0 0.0 81.3 87.8 84.4 58.6 60.7 59.6 51.5 42.6 46.6 Multilingual raw 61.5 63.1 62.3 58.8 35.7 44.4 28.6 33.3 30.8 89.3 90.5 89.9 68.2 53.8 60.0 61.2 51.6 56.0 Fine-tuned raw 63.2 62.6 62.9 54.2 46.4 50.0 20.0 16.7 18.2 89.6 93.2 91.4 73.9 60.7 66.7 55.7 59.8 57.7 Anxiety Single lang. spec 0.0 0.0 0.0 - - - 66.7 28.6 40.0 91.8 90.5 91.2 46.0 82.1 59.0 69.3 50.0 58.1 Multilingual spec 14.0 8.0 10.2 - - - 50.0 28.6 36.4 87.7 86.5 87.1 61.3 67.9 64.4 72.3 49.2 58.5 Single lang. raw 36.4 13.8 20.0 - - - 83.3 71.4 76.9 90.0 85.1 87.5 28.1 32.1 30.0 69.1 45.9 55.2 Multilingual raw 23.5 18.4 20.6 - - - 87.5 100.0 93.3 98.6 91.9 95.1 64.7 78.6 71.0 84.4 44.3 58.1 Fine-tuned raw 24.7 21.8 23.2 - - - 77.8 100.0 87.5 98.6 91.9 95.1 54.3 89.3 67.6 80.2 63.1 70.6 Average Single lang. spec 26.2 44.4 27.5 59.1 51.0 51.2 68.6 63.7 63.1 91.6 91.5 91.5 71.1 63.4 63.0 66.5 63.3 63.7 Multilingual spec 37.3 34.4 34.2 43.0 54.9 54.7 60.1 54.4 55.8 84.4 84.4 84.3 55.3 56.3 55.6 66.6 65.0 64.4 Single lang. raw 48.6 44.9 44.7 20.7 34.6 22.4 60.1 62.9 61.1 87.7 87.4 87.5 53.8 52.7 52.9 65.0 62.9 62.6 Multilingual raw 49.7 50.4 49.9 55.2 53.7 52.8 74.7 77.3 75.8 94.3 94.2 94.2 65.9 65.2 64.3 73.0 70.7 69.4 Fine-tuned raw 48.9 49.8 49.6 55.9 55.1 54.8 70.3 74.2 71.8 95.0 94.9 94.9 70.5 67.0 66.7 73.5 73.2 73.0 • Estonian -Estonian Emotional Speech Corpus [64]: the corpus consists of 1234 Estonian utterances. They are generated by a single actress in four emotions: Anger, Joy, Sadness and Neutral. • German -Berlin EmoDB [65]: this database consists of 535 German utterances. A total of 5 short and 5 long utterances were generated by 5 actors and 5 actresses in 7 emotions: Anger, Neutral, Fear, Boredom, Happiness, Sadness and Disgust (not all the actors read all the sentences for each emotion). • Spanish -INTERFACE Emotional Speech Syntesis Database [66]: this database includes around 150 items (phonemes, words, short, long sentences and a longer 30 s passage) in Spanish language. Each item is spoken by a male and a female actors in several emotions: Anger, Sadness, Joy, Surprise, Disgust, Fear and Neutral. For the purpose of our work we discarded the phonemes and the individual words. • Italian -EMOVO [67]: emotion corpus in Italian. It includes 6 actors (3 males and 3 females), each acting 14 sentences into 7 different emotions: Anger, Neutral, Disgust, Joy, Fear, Surprise, Sadness. • Serbian -Serbian Emotional Speech Database [68]: TABLE III RESULTS III(PERCENTAGE) ON MULTILINGUAL TASK FOR BIG FIVE PERSONALITY TRAITS ANALYZED. MEAN ABSOLUTE ERROR (MAE), ACCURACY (A), PRECISION (P), RECALL (R), AND F-SCORE (F1) ARE SHOWN.English Mandarin French Method MAE A P R F1 MAE A P R F1 MAE A P R F1 Extraversion Single lang. spec .1093 63.7 67.4 57.9 62.3 .0449 70.8 68.0 64.2 66.0 .1043 69.4 73.3 72.5 72.9 Single lang. raw .1141 61.7 67.3 50.6 57.8 .0513 66.7 58.7 83.0 68.8 .1117 59.4 59.4 90.1 71.6 Multilingual spec .1108 54.4 53.9 83.5 65.5 .0496 60.8 54.1 75.5 63.0 .1090 63.1 69.5 62.6 65.9 Multilingual raw .1080 66.0 66.7 68.8 67.7 .0539 60.0 53.3 75.5 62.5 .1141 56.2 62.1 59.3 60.7 Fine-tuned raw .1122 64.1 60.9 85.5 71.2 .0479 65.8 58.3 79.2 67.2 .1000 73.1 76.1 76.9 76.5 Agreeableness Single lang. spec .0975 59.6 65.9 53.6 59.1 .0518 55.0 61.4 42.2 50.0 .0679 70.0 74.6 61.7 67.6 Single lang. raw .1004 58.8 61.8 63.7 62.8 .0704 46.7 50.0 6.2 11.1 .0749 60.0 71.8 34.6 46.7 Multilingual spec .0989 55.2 56.1 81.1 66.3 .0586 42.5 45.6 40.6 43.0 .0744 56.2 55.7 66.7 60.7 Multilingual raw .0953 60.9 64.4 63.2 63.8 .0555 46.7 50.0 68.8 57.9 .0761 51.9 51.5 84.0 63.8 Fine-tuned raw .0969 62.0 60.8 84.9 70.9 .0508 55.8 55.9 81.2 66.2 .0661 67.5 72.3 58.0 64.4 Conscientiousness Single lang. spec .1194 61.2 67.6 52.1 58.9 .0509 59.2 62.5 18.9 29.0 .0629 68.1 78.6 60.4 68.3 Single lang. raw .1248 59.4 60.8 66.4 63.5 .0521 55.8 50.0 37.7 43.0 .0775 60.0 58.7 100.0 74.0 Multilingual spec .1199 56.3 59.1 57.6 58.4 .0598 49.2 44.7 64.2 52.7 .0645 61.9 64.2 74.7 69.0 Multilingual raw .1160 62.5 65.8 61.2 63.4 .0564 49.2 44.6 62.3 52.0 .0646 60.6 61.9 80.2 69.9 Fine-tuned raw .1168 62.6 60.7 84.2 70.6 .0522 50.0 45.7 69.8 55.2 .0615 65.0 68.8 70.3 69.6 Neuroticism Single lang. spec .1096 64.3 70.1 56.4 62.5 .0427 61.7 62.3 73.8 67.6 .0722 67.5 68.8 65.4 67.1 Single lang. raw .1143 61.1 65.0 56.8 60.7 .0421 68.3 71.4 69.2 70.3 .0828 55.6 64.7 27.2 38.3 Multilingual spec .1121 54.9 55.6 71.8 62.7 .0484 55.0 63.4 40.0 49.1 .0789 58.8 59.0 60.5 59.8 Multilingual raw .1077 64.8 67.9 62.8 65.2 .0513 51.7 56.1 49.2 52.5 .0807 60.0 58.1 75.3 65.6 Fine-tuned raw .1102 65.6 62.6 86.1 72.5 .0426 68.3 74.5 63.1 68.3 .0731 72.5 70.3 79.0 74.4 Openness to Experience Single lang. spec .1048 63.8 67.7 57.2 62.0 .0278 57.5 63.6 44.4 52.3 .0434 60.0 60.7 48.1 53.6 Single lang. raw .1099 61.6 66.9 50.8 57.7 .0353 50.8 52.0 81.0 63.4 .0502 50.6 49.2 81.8 61.5 Multilingual spec .1055 54.1 55.1 60.3 57.6 .0316 51.7 53.7 57.1 55.4 .0444 52.5 50.6 55.8 53.1 Multilingual raw .1024 66.0 67.5 66.2 66.9 .0283 57.5 57.9 69.8 63.3 .0433 60.0 57.5 64.9 61.0 Fine-tuned raw .1067 64.1 61.1 84.3 70.8 .0281 56.7 56.6 74.6 64.4 .0418 66.2 61.4 80.5 69.7 Average over Traits Single lang. spec .1081 62.5 67.8 55.4 61.0 .0436 60.8 63.6 48.7 53.0 .0701 67.0 71.2 61.6 65.9 Single lang. raw .1127 60.5 64.4 57.7 60.5 .0502 57.7 56.4 55.4 51.3 .0794 57.1 60.8 66.7 58.4 Multilingual spec .1094 54.9 56.0 70.9 62.1 .0496 51.8 52.3 55.5 52.6 .0742 58.5 59.8 64.1 61.7 Multilingual raw .1059 64.0 66.5 64.5 65.4 .0491 53.0 52.4 65.1 57.6 .0758 57.8 58.2 72.8 64.2 Fine-tuned raw .1086 63.7 61.2 85.0 71.2 .0443 59.3 58.2 73.6 64.3 .0685 68.9 69.8 73.0 70.9 http://sox.sourceforge.net Towards empathetic human-robot interactions. 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[ "Published as a conference paper at ICLR 2023 MITIGATING DATASET BIAS BY USING PER-SAMPLE GRADIENT", "Published as a conference paper at ICLR 2023 MITIGATING DATASET BIAS BY USING PER-SAMPLE GRADIENT" ]	[ "Sumyeong Ahn [email protected] \nKAIST AI\nKAIST ISysE\n\n", "Seongyoon Kim \nKAIST AI\nKAIST ISysE\n\n" ]	[ "KAIST AI\nKAIST ISysE\n", "KAIST AI\nKAIST ISysE\n" ]	[]	The performance of deep neural networks is strongly influenced by the training dataset setup. In particular, when attributes with a strong correlation with the target attribute are present, the trained model can provide unintended prejudgments and show significant inference errors (i.e., the dataset bias problem). Various methods have been proposed to mitigate dataset bias, and their emphasis is on weakly correlated samples, called bias-conflicting samples. These methods are based on explicit bias labels provided by humans. However, such methods require human costs. Recently, several studies have sought to reduce human intervention by utilizing the output space values of neural networks, such as feature space, logits, loss, or accuracy. However, these output space values may be insufficient for the model to understand the bias attributes well. In this study, we propose a debiasing algorithm leveraging gradient called Per-sample Gradient-based Debiasing (PGD). PGD is comprised of three steps: (1) training a model on uniform batch sampling, (2) setting the importance of each sample in proportion to the norm of the sample gradient, and (3) training the model using importance-batch sampling, whose probability is obtained in step (2). Compared with existing baselines for various datasets, the proposed method showed state-of-the-art accuracy for the classification task. Furthermore, we describe theoretical understandings of how PGD can mitigate dataset bias. Code is available at Link * Equal contribution arXiv:2205.15704v3 [cs.LG]	10.48550/arxiv.2205.15704	[ "https://export.arxiv.org/pdf/2205.15704v3.pdf" ]	249,209,690	2205.15704	35cd12a6fe30e8fc532c37396f581bc8dc0c2a09	Published as a conference paper at ICLR 2023 MITIGATING DATASET BIAS BY USING PER-SAMPLE GRADIENT Sumyeong Ahn [email protected] KAIST AI KAIST ISysE Seongyoon Kim KAIST AI KAIST ISysE Published as a conference paper at ICLR 2023 MITIGATING DATASET BIAS BY USING PER-SAMPLE GRADIENT The performance of deep neural networks is strongly influenced by the training dataset setup. In particular, when attributes with a strong correlation with the target attribute are present, the trained model can provide unintended prejudgments and show significant inference errors (i.e., the dataset bias problem). Various methods have been proposed to mitigate dataset bias, and their emphasis is on weakly correlated samples, called bias-conflicting samples. These methods are based on explicit bias labels provided by humans. However, such methods require human costs. Recently, several studies have sought to reduce human intervention by utilizing the output space values of neural networks, such as feature space, logits, loss, or accuracy. However, these output space values may be insufficient for the model to understand the bias attributes well. In this study, we propose a debiasing algorithm leveraging gradient called Per-sample Gradient-based Debiasing (PGD). PGD is comprised of three steps: (1) training a model on uniform batch sampling, (2) setting the importance of each sample in proportion to the norm of the sample gradient, and (3) training the model using importance-batch sampling, whose probability is obtained in step (2). Compared with existing baselines for various datasets, the proposed method showed state-of-the-art accuracy for the classification task. Furthermore, we describe theoretical understandings of how PGD can mitigate dataset bias. Code is available at Link * Equal contribution arXiv:2205.15704v3 [cs.LG] INTRODUCTION Dataset bias (Torralba & Efros, 2011;Shrestha et al., 2021) is a bad training dataset problem that occurs when unintended easier-to-learn attributes (i.e., bias attributes), having a high correlation with the target attribute, are present (Shah et al., 2020;Ahmed et al., 2020). This is due to the fact that the model can infer outputs by focusing on the bias features, which could lead to testing failures. For example, most "camel" images include a "desert background," and this unintended correlation can provide a false shortcut for answering "camel" on the basis of the "desert." In (Nam et al., 2020;Lee et al., 2021), samples of data that have a strong correlation (like the aforementioned desert/camel) are called "bias-aligned samples," while samples of data that have a weak correlation (like "camel on the grass" images) are termed "bias-conflicting samples." To reduce the dataset bias, initial studies (Kim et al., 2019;McDuff et al., 2019;Singh et al., 2020;Li & Vasconcelos, 2019) have frequently assumed a case where labels with bias attributes are provided, but these additional labels provided through human effort are expensive. Alternatively, the bias-type, such as "background," is assumed in (Lee et al., 2019;Geirhos et al., 2018;Bahng et al., 2020;Cadene et al., 2019;Clark et al., 2019). However, assuming biased knowledge from humans is still unreasonable, since even humans cannot predict the type of bias that may exist in a large dataset (Schäfer, 2016). Data for deep learning is typically collected by web-crawling without thorough consideration of the dataset bias problem. In this paper, we present a resampling method from the perspective of the per-sample gradient norm to mitigate dataset bias. Furthermore, we theoretically justify that the gradient-norm-based resampling method can be an excellent debiasing approach. Our key contributions can be summarized as follows: • We propose Per-sample Gradient-norm based Debiasing (PGD), a simple and efficient gradientnorm-based debiasing method. PGD is motivated by prior research demonstrating (Mirzasoleiman et al., 2020;Huang et al., 2021;Killamsetty et al., 2021b) that gradient is effective at finding rare samples, and it is also applicable to finding the bias-conflicting samples in the dataset bias problem (See Section 3 and Appendix E). • PGD outperforms other dataset bias methods on various benchmarks, such as colored MNIST (CM-NIST), multi-bias MNIST (MBMNIST), corrupted CIFAR (CCIFAR), biased action recognition (BAR), biased FFHQ (BFFHQ), CelebA, and CivilComments-WILD. In particular, for the colored MNIST case, the proposed method yielded higher unbiased test accuracies compared with the vanilla and the best methods by 35.94% and 2.32%, respectively. (See Section 4) • We provide theoretical evidence of the superiority of PGD. To this end, we first explain that minimizing the trace of inverse Fisher information is a good objective to mitigate dataset bias. In particular, PGD, resampling based on the gradient norm computed by the biased model, is a possible optimizer for mitigating the dataset bias problem. (See Section 5) DATASET BIAS PROBLEM Classification model. We first describe the conventional supervised learning setting. Let us consider the classification problem when a training dataset D n = {(x i , y i )} n i=1 , with input image x i and corresponding label y i , is given. Assuming that there are c ∈ N \ {1} classes, y i is assigned to the one element in set C = {1, ..., c}. Note that we focus on a situation where dataset D n does not have noisy samples, for example, noisy labels or out-of-distribution samples (e.g., SVHN samples when the task is CIFAR-10). When input x i is given, f (y i \|x i , θ) represents the softmax output of the classifier for label y i . This is derived from the model parameter θ ∈ R d . The cross-entropy (CE) loss L CE is frequently used to train the classifier, and it is defined as L CE (x i , y i ; θ) = − log f (y i \|x i , θ) when the label is one-hot encoded. Dataset bias. Let us suppose that a training set, D n , is comprised of images, as shown in Figure 1, and that the objective is to classify the digits. Each image can be described by a set of attributes, (e.g., for the first image in Figure 1, it can be {digit 0, red, thin,...}). The purpose of the training classifier is to find a model parameter θ that correctly predicts the target attributes, (e.g., digit). Notably, the target attributes are also interpreted as classes. However, we focus on a case wherein another attribute that is strongly correlated to the target exists, and we call these attributes bias attributes. For example, in Figure 1, the bias attribute is color. Furthermore, samples whose bias attributes are highly correlated to the target attributes are called bias-aligned (top three rows in Figure 1). Conversely, weakly correlated samples are called bias-conflicting (see the bottom row of Figure 1). Therefore, our main scope is that the training dataset which has samples whose bias and target attributes are misaligned. 1 According to (Nam et al., 2020), when the bias attributes are easier-to-learn than the target attributes, dataset bias is problematic, as the trained model may prioritize the bias attributes over the target attributes. For example, for a model trained on the images in Figure 1, it can output class 4 when the (Orange, 0) image (e.g., left bottom image) is given, due to the wrong priority, color which is an easier-to-learn attribute (Nam et al., 2020). 3 PGD: PER-SAMPLE GRADIENT-NORM-BASED DEBIASING Algorithm 1 PGD: Per-sample Gradient-norm based Debiasing 1: Input: dataset D n , learning rate η, iterations T b , T d , Batch size B, Data augmentation operation A(·), Initial parameter θ 0 , GCE parameter α / STEP 1: Train f b / 2: for t = 1, 2, · · · , T b do 3: Construct a mini-batch B t = {(x i , y i )} B i=1 ∼ U . 4: Update θ t as: θ t−1 − η B ∇ θ (x,y)∈Bt L GCE (A(x), y; θ t−1 , α) 5: end for / STEP 2: Calculate h / 6: Calculate h(x i , y i ) for all (x i , y i ) ∈ D n , (1). / STEP 3: Train f d based on h / 7: for t = 1, 2, · · · , T d do 8: Construct a mini-batch B t = {(x i , y i )} B i=1 ∼ h. 9: Update θ Tb+t as: θ Tb+t−1 − η B ∇ θ (x,y)∈B t L CE (A(x), y; θ Tb+t−1 ) 10: end for In this section, we propose a novel debiasing algorithm, coined as PGD. PGD consists of two models, biased f b and debiased f d , with parameters θ b and θ d , respectively. Both models are trained sequentially. Obtaining the ultimately trained debiased model f d involves three steps: (1) train the biased model, (2) compute the sampling probability of each sample, and (3) train the debiased model. These steps are described in Algorithm 1. Step 1: Training the biased model. In the first step, the biased model is trained on the mini-batches sampled from a uniform distribution U , similar to conventional SGD-based training, with data augmentation A. The role of the biased model is twofold: it detects which samples are bias-conflicting and calculates how much they should be highlighted. Therefore, we should make the biased model uncertain when it faces bias-conflicting samples. In doing so, the biased model, f b , is trained on the generalized cross-entropy (GCE) loss L GCE to amplify the bias of the biased model, motivated by (Nam et al., 2020). For an input image x, the corresponding true class y, L GCE is defined as L GCE (x, y; θ, α) = 1−f (y\|x,θ) α α . Note that α ∈ (0, 1] is a hyperparameter that controls the degree of emphasizing the easy-to-learn samples, namely bias-aligned samples. Note that when α → 0, the GCE loss L GCE is exactly the same as the conventional CE loss L CE . We set α = 0.7 as done by the authors of Zhang & Sabuncu (2018), Nam et al. (2020) and Lee et al. (2021). Step 2: Compute the gradient-based sampling probability. In the second step, the sampling probability of each sample is computed from the trained biased model. Since rare samples have large gradient norms compared to the usual samples at the biased model (Hsu et al., 2020), the sampling probability of each sample is computed to be proportional to its gradient norm so that bias-conflicting samples are over-sampled. Before computing the sampling probability, the per-sample gradient with respect to L CE for all (x i , y i ) ∈ D n is obtained from the biased model. We propose the following sampling probability of each sample h(x i , y i ) which is proportional to their gradient norms, as follows: h(x i , y i ) = ∇ θ L CE (x i , y i ; θ b ) r s (xi,yi)∈Dn ∇ θ L CE (x i , y i ; θ b ) r s ,(1) where · r s denotes r square of the L s norm, and θ b is the result of Step 1. Since, h(x i , y i ) is the sampling probability on D n , h(x i , y i ) is the normalized gradient-norm. Note that computing the gradient for all samples requires huge computing resources and memory. Therefore, we only extract the gradient of the final FC layer parameters. This is a frequently used technique for reducing the computational complexity (Ash et al., 2019;Mirzasoleiman et al., 2020;Killamsetty et al., 2021b;a;2020). In other words, instead of h(x i , y i ), we empirically utilizeĥ( x i , y i ) = ∇ θ fc LCE(xi,yi;θ b ) r s (x i ,y i )∈Dn ∇ θ fc LCE(xi,yi;θ b ) r s , where θ fc is the parameters of the final FC layer. We consider r = 1 and s = 2 (i.e., L 2 ), and deliver ablation studies on various r and s in Section 4. Step 3: Ultimate debiased model training. Finally, the debiased model, f d , is trained using minibatches sampled with the probability h(x i , y i ) obtained in Step 2. Note that, as described in Algorithm 1, our debiased model inherits the model parameters of the biased model θ T b . However, Lee et al. (2021) argued that just oversampling bias-conflicting samples does not successfully debias, and this unsatisfactory result stems from the data diversity (i.e., data augmentation techniques are required). Hence, we used simple randomized augmentation operations A such as random rotation and random color jitter to oversample the bias-conflicting samples. EXPERIMENTS In this section, we demonstrate the effectiveness of PGD for multiple benchmarks compared with previous proposed baselines. Detail analysis not in this section (e.g., training time, unbiased case study, easier to learn target attribute, sampling probability analysis, reweighting with PGD) are described in the Appendix E. BENCHMARKS To precisely examine the debiasing performance of PGD, we used the Colored MNIST, Multi-bias MNIST, and Corrupted CIFAR datasets as synthetic datasets, which assume situations in which the model learns bias attributes first. BFFHQ, BAR, CelebA, and CivilComments-WILDS datasets obtained from the real-world are used to observe the situations in which general algorithms have poor performance due to bias attributes. Note that BFFHQ and BAR are biased by using human prior knowledge, while CelebA and CivilComments-WILDS are naturally biased datasets. A detailed explanation of each benchmark are presented in Appendix A. Colored MNIST (CMNIST). CMNIST is a modified version of MNIST dataset (LeCun et al., 2010), where color is the biased attribute and digit serves as the target. We randomly selected ten colors that will be injected into the digit. Evaluation was conducted for various ratios ρ ∈ {0.5%, 1%, 5%}, where ρ denotes the portion of bias-conflicting samples. Note that CMNIST has only one bias attribute: color. Multi-bias MNIST (MB-MNIST). The authors of (Shrestha et al., 2021) stated that CMNIST is too simple to examine the applicability of debiaising algorithms for complex bias cases. However, the dataset that Shrestha et al. (2021) generated is also not complex, since they did not use an real-world pattern dataset (e.g., MNIST) and used simple artificial patterns (e.g., straight line and triangle). Therefore, we generated a MB-MNIST; we used benchmark to reflect the real-worled better than (Shrestha et al., 2021). MB-MNIST consists of eight attributes: digit (LeCun et al., 2010), alphabet (Cohen et al., 2017, fashion object (Xiao et al., 2017), Japanese character (Clanuwat et al., 2018), digit color, alphabet color, fashion object color, Japanese character color. Among the eight attributes, the target attribute is digit shape and the others are the bias attributes. To construct MB-MNIST, we follow the CMNIST protocol, which generates bias by aligning two different attributes (i.e., digit and color) with probability (1 − ρ). MB-MNIST dataset is made by independently aligning the digit and seven other attributes with probabity (1 − ρ). Note that rarest sample is generated with probability ρ 7 . When ρ is set as the CMNIST case, it is too low to generate sufficient misaligned samples. Therefore, we use ρ ∈ {10%, 20%, 30%} to ensure the trainability. Corrupted CIFAR (CCIFAR). CIFAR10 (Krizhevsky et al., 2009) is comprised of ten different objects, such as an airplane and a car. Corrupted CIFAR are biased with ten different types of texture bias (e.g., frost and brightness). The dataset was constructed by following the design protocol of (Hendrycks & Dietterich, 2019), and the ratios ρ ∈ {0.5%, 1%, 5%} are used. Biased action recognition (BAR). Biased action recognition dataset was derived from (Nam et al., 2020). It comprised six classes for action, (e.g., climbing and diving), and each class is biased with place. For example, diving class pictures are usually taken underwater, while a few images are taken from the diving pool. Biased FFHQ (BFFHQ). BFFHQ dataset was constructed from the facial dataset, FFHQ (Karras et al., 2019). It was first proposed in (Kim et al., 2021) andwas used in (Lee et al., 2021). It is comprised of two gender classes, and each class is biased with age. For example, most female pictures are young while male pictures are old. This benchmark follows ρ = 0.5%. CelebA. CelebA (Liu et al., 2015) is a common real-world face classification dataset. The goal is classifying the hair color ("blond" and "not blond") of celebrities which has a spurious correlation with the gender ("male" or "female") attribute. Hair color of almost all female images is blond. We report the average accuracy and the worst-group accuracy on the test dataset. RESULTS AND EMPIRICAL ANALYSIS Accuracy results. In Table 1, we present the comparisons of the image classification accuracy for the unbiased test sets. The proposed method outperforms the baseline methods for all benchmarks and for all different ratios. For example, our model performance is 35.94% better than that of the vanilla model for the colored MNIST benchmark with a ratio ρ = 0.5%. For the same settings, PGD performs better than Disen by 2.32%. As pointed out in (Shrestha et al., 2021), colored MNIST is too simple to evaluate debiasing performance on the basis of the performance of baselines. In Multi-bias MNIST case, other models fail to obtain higher unbiased test results, even though the ratio is high, e.g., 10%. In this complex setting, PGD shows superior performance over other methods. For example, its performance is higher by 36. 15% and 35.63% compared with the performance of vanilla model and Disen for the ratio of 10%. Similar to the results for the bias-feature-injected benchmarks, as shown in Table 2 and Table 3, PGD shows competitive performance among all the debiasing algorithms on the raw image benchmark (BAR, BFFHQ, and CelebA). For example, for the BFFHQ benchmark, the accuracy of PGD is 1.43% better than that of Disen. As in Table 3, PGD outperforms the other baselines on CivilComments-WILDs, much more realistic NLP task. Therefore, we believe PGD also works well with transformer, and it is applicable to the real-world. Unbiased test accuracy on various norms. Since, gradient norm can have various candidates, such as order of the norm, we report four configurations of gradient norms. As shown in Figure 2, all norms have significant unbiased test performance. Amongst them, the L 2 -norm square case shows lower unbiased performance than the other cases. Therefore, it is recommended that any first power of L {1,2,∞} -norms be used in PGD for overcoming the dataset bias problem. This is quite different from the results in (Huang et al., 2021), which suggested that L 1 -norm is the best choice in the research field of out-of-distribution detection. Ablation study. Table 4 shows the importance of each module in our method: generalized cross entropy, and data augmentation modules. We set the ratio ρ as 0.5% and 10% for CMNIST and MB-MNIST, respectively. We observe that GCE is more important that data augmentation for CMNIST. However, data augmentation shows better performance than GCE for MB-MNIST. In all cases, the case where both are utilized outperforms the other cases. MATHEMATICAL UNDERSTANDING OF PGD This section provides a theoretical analysis of per-sample gradient-norm based debiasing. We first briefly summarize the maximum likelihood estimator (MLE) and Fisher information (FI), which are ingredients of this section. We then interpret the debiasing problem as a min-max problem and deduce that solving it the min-max problem can be phrased as minimizing the trace of the inverse FI. Since handling the trace of the inverse FI is very difficult owing to its inverse computation, we look at a glance by relaxing it into a one-dimensional toy example. In the end, we conclude that the gradient-norm based re-sampling method is an attempt to solve the dataset bias problem. PRELIMINARY Training and test joint distributions. The general joint distribution P(x, y\|θ) is assumed to be factored into the parameterized conditional distribution f (y\|x, θ) and the marginal distribution P(x), which is independent of the model parameter θ, (i.e., P(x, y\|θ) = P(x)f (y\|x, θ)). We refer to the model f (y\|x, θ ) that produces the exact correct answer, as an oracle model, and to its parameter θ as the oracle parameter. The training dataset D n is sampled from {(x i , y i )} n i=1 ∼ p(x)f (y\|x, θ ), where the training and test marginal distributions are denoted by p(x) and q(x), respectively. Here, we assume that both marginal distributions are defined on the marginal distribution space M = {P(x)\| x∈X P(x) dx = 1}, where X means the input data space, i.e., p(x), q(x) ∈ M. The space H of sampling probability h. When the training dataset D n is given, we denote the sampling probability as h(x) which is defined on the probability space H 4 : H = {h(x) \| (xi,yi)∈Dn h(x i ) = 1 , h(x i ) ≥ 0 ∀(x i , y i ) ∈ D n }.(2) Maximum likelihood estimator (MLE). When h(x) is the sampling probability, we define MLÊ θ h(x),Dn as follows:θ h(x),Dn = arg min θ − (xi,yi)∈Dn h(x i ) log f (y i \|x i , θ). Note that MLEθ h(x),Dn is a variable controlled by two factors: (1) a change in the training dataset D n and (2) the adjustment of the sampling probability h(x). If h(x) is a uniform distribution U (x), thenθ U (x),Dn is the outcome of empirical risk minimization (ERM). Fisher information (FI). FI, denoted by I P(x) (θ), is an information measure of samples from a given distribution P(x, y\|θ). It is defined as follows: I P(x) (θ) = E (x,y)∼P(x)f (y\|x,θ) [∇ θ log f (y\|x, θ)∇ θ log f (y\|x, θ)].(3) FI provides a guideline for understanding the test cross-entropy loss of MLEθ U (x),Dn . When the training set is sampled from p(x)f (y\|x, θ ) and the test samples are generated from q(x)f (y\|x, θ ), we can understand the test loss of MLEθ U (x),Dn by using FI as follows. Theorem 1. Suppose Assumption 1 in Appendix F and Assumption 2 in Appendix G hold, then for sufficiently large n = \|D n \|, the following holds with high probability: E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x),Dn ) ≤ 1 2n Tr I p(x) (θ U (x),Dn ) −1 Tr I q(x) (θ ) . (4) Here is the proof sketch. The left-hand side of (4) converges to the Fisher information ratio (FIR) Tr I p(x) (θ ) −1 I q(x) (θ ) -related term. Then, FIR can be decomposed into two trace terms with respect to the training and test marginal distributions p(x) and q(x). Finally, we show that the term Tr[I p(x) (θ ) −1 ] which is defined in the oracle model parameter can be replaced with Tr[I p(x) (θ U (x),Dn ) −1 ] . The proof of Theorem 1 is in Appendix D. Note that Theorem 1 means that the upper bound of the test loss of MLEθ U (x),Dn can be minimized by reducing Tr[I p(x) (θ U (x),Dn ) −1 ]. Empirical Fisher information (EFI). In practice, the exact FI (3) cannot be computed since we do not know the exact data generation distribution P(x)f (y\|x, θ). For practical reasons, the empirical Fisher information (EFI) is commonly used (Jastrzębski et al., 2017;Chaudhari et al., 2019) to reduce the computational cost of gathering gradients for all possible classes when x is given. In the present study, we used a slightly more generalized EFI that involved the sampling probability h(x) ∈ H as follows: Î h(x) (θ) = (xi,yi)∈Dn h(x i )∇ θ log f (y i \|x i , θ)∇ θ log f (y i \|x i , θ). (5) Note that the conventional EFI is the case when h(x) is uniform. EFI provides a guideline for understanding the test cross-entropy loss of MLEθ h(x),Dn . UNDERSTANDING DATASET BIAS PROBLEM VIA MIN-MAX PROBLEM Debiasing formulation from the perspective of min-max problem. We formulate the dataset bias problem as described in Definition 1. (6) is a min-max problem formula, a type of robust optimization. Similar problem formulations for solving the dataset bias problem can be found in (Arjovsky et al., 2019;Bao et al., 2021;Zhang et al., 2022a). However, they assume that the training data is divided into several groups, and the model minimizes the worst inference error of the reweighted group dataset. In contrast, the objective of (6) minimizes the worst-case test loss without explicit data groups where the test distribution can be arbitrary. Definition 1. When the training dataset D n ∼ p(x)f (y\|x, θ ) is given, the debiasing objective is min h(x)∈H max q(x)∈M E (x,y)∼q(x)f (y\|x,θ ) − log f (y\|x,θ h(x),Dn ) .(6) The meaning of Definition 1 is that we have to train the modelθ h(x),Dn so that the loss of the worst case test samples (max q(x) ) is minimized by controlling the sampling probability h(x) (min h(x) ). Note that since we cannot control the given training dataset D n and test marginal distribution q(x), the only controllable term is the sampling probability h(x). Therefore, from Theorem 1 and EFI, we design a practical objective function for the dataset bias problem as follows: (7). min h(x)∈H Tr Î h(x) (θ h(x),Dn ) −1 .(7) MEANING OF PGD IN TERMS OF In this section, we present an analysis of PGD with respect to (7). To do so, we try to understand (7), which is difficult to directly solve. It is because computing the trace of the inverse matrix is computationally expensive. Therefore, we intuitively understand (7) in the one-dimensional toy scenario. One-dimensional example. We assume that D n comprises sets M and m such that elements in each set share the same loss function. For example, the loss functions of the elements in set M and m are 1 2 (θ + a) 2 and 1 2 (θ − a) 2 with a given constant a, respectively. We also assume that each sample of M and m has the set dependent probability mass h M (x) and h m (x), respectively. With these settings, our objective is to determine h (x) = arg min h(x)∈H Tr[Î h(x) (θ h(x),Dn ) −1 ] . Thanks to the model's simplicity, we can easily find h (x) in a closed form with respect to the gradient atθ U (x),Dn for each set, i.e., g M (θ U (x),Dn ) and g m (θ U (x),Dn ). Theorem 2. Under the above setting, the solution of (h M (x), h m (x)) = arg min h(x)∈H Tr[Î h(x) (θ h(x),Dn ) −1 ] is: h M (x) = \|g M (θ U (x),Dn )\|/Z, h m (x) = \|g m (θ U (x),Dn )\|/Z, where Z = \|M \|\|g M (θ U (x),Dn )\| + \|m\|\|g m (θ U (x),.1 0 .2 0 .4 0 .5 0 .9 Tr $ ! " $ ! " , ! %& (a) Colored MNIST Tr [IU(θh,D n ) -1 ] 0 2 × 1 0 6 4 × 1 0 6 6 × 1 0 6 8 × 1 0 6 Bias conflict ratio ρ 0 .1 0 .3 0 .5 0 .7 0 .9 Tr $ ! " $ ! " , ! %& (b) Multi-bias MNIST Figure 3: Target objective Tr[Î h (θ h(x),Dn ) −1 ]. PGD : h(x) =ĥ(x), and vanilla: h(x) = U (x). PGD tries to minimize (7). Theorem 2 implies that (7) can be minimized by sampling in proportion to their gradient norm. Because the basis of PGD is oversampling based on the gradient norm from the biased model, we can deduce that PGD strives to satisfy (7). Furthermore, we empirically show that PGD reduces the trace of the inverse of EFI in the high-dimensional case, as evident in 2020) proposed a new overlap loss defined by a class activation map (CAM). The overlap loss reduces the overlapping parts of the CAM outputs of the two bias labels and target labels. The authors of (Li & Vasconcelos, 2019;Li et al., 2018) employed bias labels to detect bias-conflicting samples and to oversample them to debias. In (Liu et al., 2021), a reconstructing method based on the sample accuracy was proposed. Liu et al. (2021) used bias labels in the validation dataset to tune the hyper-parameters. On the other hand, there has been a focus on fairness within each attribute (Hardt et al., 2016;Woodworth et al., 2017;Pleiss et al., 2017;Agarwal et al., 2018). Their goal is to prevent bias attributes from affecting the final decision of the trained model. Debiasing with bias context. In contrast to studies assuming the explicit bias labels, a few studies ( On the other hand, there have been studies (Li & Xu, 2021;Lang et al., 2021;Krishnakumar et al., 2021) that identify the bias attribute of the training dataset without human supervision. CONCLUSION We propose a gradient-norm-based dataset oversampling method for mitigating the dataset bias problem. The main intuition of this work is that gradients contain abundant information about each sample. Since the bias-conflicting samples are relatively more difficult-to-learn than bias-aligned samples, the bias-conflicting samples have a higher gradient norm compared with the others. Through various experiments and ablation studies, we demonstrate the effectiveness of our gradient-normbased oversampling method, called PGD. Furthermore, we formulate the dataset bias problem as a min-max problem, and show theoretically that it can be relaxed by minimizing the trace of the inverse Fisher information. We provide empirical and theoretical evidence that PGD tries to solve the problem of minimizing the trace of the inverse Fisher information problem. Despite this successful outcome and analysis, we are still working on two future projects: release approximations, such as a toy example, for understanding PGD and cases where the given training dataset is corrupted, such as with noisy labels. We hope that this study will help improve understanding of researchers about the dataset bias problem. -Appendix -Mitigating Dataset Bias by Using Per-sample Gradient Due to the page constraint, this extra material includes additional results and theoretical proofs that are not in the original manuscript. Section A demonstrates how to create datasets. Section B.1 contains implementation details such as hyperparameters, computer resources, and a brief explanation of the baseline. Section C and Section D include case studies and empirical evidence of PGD. Section E demonstrates additional experiment results. In Section F, we first provide a notation summary and some assumptions for theoretical analysis. Section G and Section H include proofs of Theorem 1 and Theorem 2 with lemmas, respectively. A BENCHMARKS AND BASELINES We explain the datasets utilized in Section 4. In short, we build MNIST variants from scratch, while others get them directly from online repositories BAR 5 , CCIFAR and BFFHQ 6 . A.1 CONTROLLED BIASED BENCHMARKS Colored MNIST (CMNIST) The MNIST dataset (LeCun et al., 2010) is composed of 1-dimensional grayscale handwritten images. The size of the image is 28 × 28. We inject color into these gray images to give them two main attributes: color and digit shape. This benchmark comes from related works (Nam et al., 2020;Kim et al., 2021;Lee et al., 2021;Bahng et al., 2020). At the beginning of the generation, ten uniformly sampled RGB colors are chosen, {C i } i∈[10] ∈ R 3×10 . When the constant ρ, a ratio of bias-conflicting samples, is given, each sample (x, y) is colored by the following steps: (1) Choose bias-conflicting or bias-aligned samples: take a random sample and set it to bias-conflicting set when u < ρ where u ∼ U(0, 1), otherwise bias-aligned. In experiments, we use ρ ∈ {0.5%, 1%, 5%}. (2) Coloring: Note that each C i ∈ R 3 (i ∈ [10]) is a bias-aligned three-dimensional color vector for each digit i ∈ {10}. Then for bias-aligned images with the arbitrary digit y, color the digit with c ∼ N (C y , σI 3×3 ). In the case of bias conflicting images with the arbitrary digit y, first uniformly sample C Uy ∈ {C i } i∈[10]\y , and color the digit with c ∼ N (C Uy , σI 3×3 ). In the experiments, we set σ as 0.0001. We use 55, 000 samples for training 5, 000 samples for validation (i.e., 10%), and 10, 000 samples for testing. Take note that test samples are unbiased, which means ρ = 90%. Multi-Bias MNIST Multi-bias MNIST has images with size 56 × 56. This dataset aims to test the case where there are multiple bias attributes. To accomplish this, we inject a total of seven bias attributes: digit color, fashion object, fashion color, Japanese character, Japanese character color, English character, and English character color, with digit shape serving as the target attribute. We inject each bias independently into each sample, as with the CMNIST case (i.e., sampling and injecting bias). We also set ρ = 90% for all bias attributes to generate an unbiased test set. As with CMNIST, we use 55, 000 samples for training and 5, 000 samples for validation, and 10, 000 samples for testing. CelebA CelebA (Liu et al., 2015) is a common real-world face classification dataset, and each image has 40 attributes. The goal is to classify the hair color ("blond" and "not blond") of celebrities, which has a spurious correlation with the gender ("male" or "female") attribute. In fact, only 6% of blond hair color images are male. Therefore, ERM shows poor performance on the bias-conflicting samples. We report the average accuracy and the worst-group accuracy on the test dataset. Corrupted CIFAR CivilComments-WILDS CivilComments-WILDS (Borkan et al., 2019) is a dataset to classify whether an online comment is toxic or non-toxic. Each sentence is a real online comment, curated on the Civil Comments platform, a comment plug-in for independent news sites. The mentions of certain demographic identities (male, female, White, Black, LGBTQ, Muslim, Christian, and other religion) cause the spurious correlation with the label. Table 5: For each demographic identity, the portion of toxic comments in the CivilComments-WILDS. A.3 BASELINES In this section, we briefly describe how the baselines, such as LfF, JTT, Disen, GEORGE, BPA, CNC, and EIIL. Please refer to each paper for a detailed explanation because we briefly explain the algorithms. (1) LfF (Nam et al., 2020) trains the debiased model by weighting the bias-conflicting samples based on the "relative difficulty", computed by the two loss values from the biased model and the debiased model. To amplify the bias-conflicting samples, the authors employ generalized cross-entropy loss with parameter α = 0.7. We implement the LfF algorithm following the code officially offered by the authors. The loss functions that this work proposes are as follows: L LfF = W (z)L CE (C d (z, y) + λL GCE (C b (z, y)), W (z) = L CE (C b (z), y) L CE (C b (z), y) + L CE (C d (z), y) . Note that W (z) is a relative difficulty and that GCE is a generalized cross-entropy. z denotes feature, which is the output of the penultimate layer, and C · is a fully connected layer. (2) JTT (Liu et al., 2021) aims to debias by splitting the dataset into correctly and incorrectly learned samples. To do so, JTT trains the biased model first and splits the given training dataset as follows: D error-set = {(x, y) s.t. y given = arg max c f b (x)[c]},(1) The ultimate debiased model is then trained by oversampling D error-set with λ up times. We set λ up for all experiments as 1/ρ. We reproduce the results by utilizing the official code offered by the authors. The main strength of PGD compared to JTT is that PGD does not need to set a hyperparameter λ up . (3) Disen (Lee et al., 2021) aims to debias by generating abundant features from mixing features between samples. To do so, the author trains the biased and debiased model by aggregating features from both networks. This work also utilize the "relative difficulty" that is proposed in (Lee et al., 2021). We reproduced the results utilizing the official code offered by the authors. The loss function proposed in this work is as follows: L total = L dis + λ swap L swap , where L swap = W (z)L CE (C d (z swap , y) + λ swap b L GCE (C b (z swap ,ỹ)) L dis = W (z)L CE (C d (z, y) + λ dis L GCE (C b (z, y)), W (z) = L CE (C b (z), y) L CE (C b (z), y) + L CE (C d (z), y) . Except for the swapped feature, z swap all terms are identical to those in LfF explanation. (4) GEORGE (Sohoni et al., 2020) aims to debias by measuring and mitigating hidden stratification without requiring access to subclass labels. Assume there are n data points x 1 , ..., x n ∈ χ and associated superclass (target) labels y 1 , ..., y n ∈ {1, · · · , C}. Furthermore, each datapoint x i is associated with a latent (unobserved) subclass label z i . George consists of three steps. The author trains the biased model using ERM. Next, to estimate an approximate subclass (latent) label, apply UMAP dimensionality reduction (McInnes et al., 2018) to the features of a given training dataset at the ERM model. Here, GEORGE cluster the output of the reduced dimension for the data of each superclass into K clusters, where K is chosen automatically. The original paper contains a detailed description of the clustering process. Lastly, to improve performance on these estimated subclasses, GEORGE minimizes the maximum per-cluster average loss (i.e., (x, y) ∼Pz), by using the cluster as groups in the G-DRO objective (Sagawa et al., 2019). The loss function proposed in this work is as follows: minimize L,f θ max 1≤z≤K E (x,y)∼Pz [l(L • f θ (x), y)] where f θ and L are parameterized feature extractor and classifier, respectively. (5) BPA (Seo et al., 2022) aims to debias by using the technique of feature clustering and cluster reweighting. It consists of three steps. First, the author trains the biased model based on ERM. Next, at the biased model, cluster all training samples into K clusters based on the feature, where K is the hyperparameter. Here, h(x, y;θ) ∈ K = {1, · · · , K} denote the cluster mapping function of data (x, y) derived by the biased model with parameterθ. At the last step, BPA computes the proper importance weight, w k for the k-th cluster, where k ∈ K and the final objective of debiasing the framework is given by minimizing the weighted empirical risk as follows: minimize θ E (x,y)∼P w h(x,y;θ) (θ)l(x, y; θ) , Concretely, for any iteration number T , the momentum method based on the history set H T , which is defined as: H T = 1 ≤ t ≤ T \| E (x,y)∼P k [l((x, y); θ t )] N k , where N k is the number of the data belonging to k-th cluster. • Weight update based on the gradient of the loss functionL(f θ ; x, y), the detail is like below: L(f θ ; x, y) = λL sup con (x, {x + m } M m=1 , {x − n } N n=1 ; f enc ) + (1 − λ)L cross (f θ ; x, y). Here, λ ∈ [0, 1] is a hyperparameter andL cross (f θ ; x, y) is an average cross-entropy loss over x, the M positives, and N negatives. Moreover, f enc is the feature extractor part of f θ and the detail formulation ofL sup 224). For the CMNIST, MBMNIST, CCIFAR, BAR, and BFFHQ, we utilize random resize crop, random rotation, and color jitter to avoid overfitting. We use normalizing with a mean of (0.4914, 0.4822, 0.4465), and standard deviation of (0.2023, 0.1994, 0.2010) for CCIFAR, BAR, and BFFHQ cases. con (x, {x + m } M m=1 , {x − n } N n=1 ; f enc ) is like below: − 1 M M r=1 log exp(f enc (x) T f enc (x + r )/τ ) M m=1 exp(f enc (x) T f enc (x + m )/τ ) + N n=1 exp(f enc (x) T f enc (x − n )/τ ) .( Implementation For table 1 and Table 3 reported in Section 4, we follow the implementation settings of CelebA and CivilComments-WILDS, suggested by Seo et al. (2022) and Liu et al. (2021), respectively. A summary of the hyperparameters that we used is reported in Table 6. We conduct our experiments mainly using a single Titan XP GPU for all cases. C CASE STUDIES ON PGD In this section, we analyze PGD in many ways. Most analyses are based on the CMNIST dataset, and the experimental setting is the same as the existing setting unless otherwise noted. For example, all experiments used the same data augmentation, color jitter, resize crop, and random rotation. Table 7: Ablation study on GCE parameter α. C.1 STUDY 1: ABLATION STUDY ON GCE PARAMETER α Colored MNIST α = 0.3 α = 0.5 α = 0.7 α = 0.9 Debiased model ρ = 0. The only hyper-parameter used in PGD is the GCE parameter α. We experimented with this value at 0.7 according to the protocol of LfF (Nam et al., 2020). However, we need to compare the various cases of α. To analyze this, we run PGD with various α and report the performance in Table 7. As in Table 7, the debiased model performs best when the GCE parameter is 0.9. This is because the biased model is fully focused on the bias feature, rather than the target feature, which can be seen from the unbiased test accuracy of the biased model, as in the bottom of In other words, PGD can be modeled to run the following loop: updating the biased model, updating the sampling probability, and updating the debiased model. In this section, we justify why we use the multi-stage approach. We report the performance of multi-stage and single-stage PGD on the colored MNIST dataset. As in Table 8, the single-stage method has two characteristics: (1) it requires more training time than the multi-stage method. (2) It has lower unbiased accuracy compared to the multi-stage method. The longer training time is due to the high computational resources required to compute the per-sample gradient norm. Moreover, because the single-stage method's sampling probability changes the training distribution over epochs, the debiased model suffers from unstable training and loses debiasing performance. (Liu et al., 2021) are the two main techniques to debias by up-weighting bias-conflicting samples. PGD is an algorithm that modifies the sampling probability by using the per-sample gradient norm. To check whether PGD works with reweighting, we examine the results of PGD with reweighting on colored MNIST dataset and report them in Table 9. We compute the weight for each sample as follows: w(x i , y i ) = \|D n \| × ∇ θ LCE(xi,yi;θ b ) 2 (x i ,y i )∈Dn ∇ θ LCE(xi,yi;θ b ) 2 . As in Table 9, PGD with resampling slightly outperforms PGD with reweighting. As argued in (An et al., 2020), this gain from resampling can be explained by the argument that resampling is more stable and better than reweighting. Table 10: loss score vs gradient norm score PGD performance improvement comes not only from the two-stage and resampling modules that we wrote about in Appendix C.2 and Appendix C.3, but also from the gradient score. To verify this, we report the following two results: C.3 STUDY 3: RESAMPLING VS REWEIGHTING C.4 STUDY 4: ANALYSIS OF THE PURE EFFECT OF GRADIENT-NORM-BASED SCORE (1) resample based on per-sample loss rather than per-sample gradient norm in PGD, and (2) change the relative difficulty score of LfF to gradient norm. As shown in Table 10, we can conclude the following two results: (1) loss of the last epoch of the first stage in a two-stage approach is not suitable for resampling, and (2) the results of replacing the relative difficulty metric of LfF with a gradient norm show that the gradient norm has better discriminative properties for bias-conflicting samples than loss. C.5 STUDY 5: COMPUTATION COST Vanilla LfF Disen ours Computation time 14m 59s 21m 35s 23m 18s 33m 31s Step 1 Step 2 Step 3 Computation time 15m 19s 1m 26s 16m 46s Debiasing algorithms require an additional computational cost. To evaluate the computational cost of PGD, we report the training time in Table 11. We conduct this experiment by using the colored MNIST with ρ = 0.5%. As in the top of Table 11, we report the training time of four methods: vanilla, LfF, Disen, and PGD. Here, PGD spends a longer amount of training time. This is because there is no module for computing per-sample gradient in a batch manner. At the bottom of Table 11, we report part-by-part costs to see which parts consume the most time. Note that Steps 1, 2, and 3 represent training the biased model, computing the per-sample gradient-norm, and training the debiased model, respectively. We can conclude with the following two facts. (1) Step 2 (computing the per-sample gradient norm and sampling probability) spends 4.3% of training time. (2) Resampling based on the modified sampling probability h(x) requires an additional cost of 1m 27s by seeing the difference between the computing times of Step 3 and Step 1. We examine whether PGD fails when an unbiased dataset is given. To verify this, we report two types of additional results: (1) unbiased CMNIST (i.e., ρ = 90%) and (2) conventional public dataset (i.e., CIFAR10). We follow the experimental setting of CMNIST for the unbiased CMNIST case. On the other hand, we train ResNet18 (He et al., 2016) for CIFAR10 with the SGD optimizer, 0.9 momentum, 5e − 4 weight decay, 0.1 learning rate, and Cosine Annealing LR decay scheduler. As shown in Table 12, PGD does not suffer significant performance degradation in unbiased CMNIST. Furthermore, it performs better than the vanilla model on the CIFAR10 dataset. This means that the training distribution that PGD changes do not cause significant performance degradation. In other words, PGD works well, regardless of whether the training dataset is balanced or unbiased. D EMPIRICAL EVIDENCE OF PGD As same with the setting of Appendix C, in this section, we also use the existing CMNIST setting, such as data augmentation, hyperparameters. D.1 CORRELATION BETWEEN GRADIENT NORM AND BIAS-ALIGNMENT OF THE CMNIST To check if the per-sample gradient norm efficiently separates the bias-conflicting samples from the bias-aligned samples, we plot the gradient norm distributions of the colored MNIST (CMNIST). For comparison, we normalized the per-sample gradient norm as follows: ∇ θ LCE(xi,yi;θ b ) max (x i ,y i )∈Dn ∇ θ LCE(xi,yi;θ b ) . As in Figure 9, the bias-aligned sample has a lower gradient norm (blue bars) than the bias-conflicting samples (red bars). CMNIST ρ= 5% (c) Colored MNIST ρ = 5% Figure 9: Histogram of per-sample gradient norm. D.2 PGD DOES NOT LEARN ONLY THE SECOND-EASIEST FEATURE We provide the results of the following experimental settings: The target feature is color, and the bias feature is digit shape, i.e., the task is to classify the color, not the digit shape. Let us give an example of this task. When one of the target classes is red, this class is aligned with one of the digits (e.g., "0"). In other words, the bias-aligned samples in this class are (Red, "0"), and the bias-conflicting samples are (e.g., (Red, "1"), (Red, "2"),.., (Red, "9")). Note that, as shown in LfF (Nam et al., 2020), color is empirically known to be easier to learn than digit shape; we think that the above scenario reflects the concern: whether PGD is only targeting the second-easiest feature (digit shape). Therefore, if the concern is correct, PGD may fail in this color target MNIST scenario since the model will learn digit shape. However, as shown in the table below, vanilla, PGD, and LfF perform well in that case. We can also support this result by seeing the distribution of the normalized gradient norms, 0,1], extracted from the biased model θ b (trained in Step 1 of Algorithm 1 of Section 3). [0.0,0.1) [0.1,0.2) [0.2,0.3) [0.3,0.4) [0.4,0.5) [0.5,0.6) [0.6,0.7) [0.7,0.8) [0.8,0.9) The numbers filled in Table 14 are the number of data items belonging to each bin category. We can check that there are no bias-conflicting samples whose gradient norm is significantly larger than the bias-aligned samples. In other words, PGD does not force the debiased model to learn the digit shape (i.e., the second-easiest feature) in this scenario. This scenario brings similar performance to Vanilla. For in-depth analysis, we provide the results of 25 tests on CMNIST in Figure 10. We compare with Disen, which shows the best performance except for PGD. However, very few cases overlap, as shown in Figure 10. We conduct a t-test for a more in-depth analysis of this. (Seo et al., 2022). The best worst accuracy is indicated in bold. ∇ θ L CE (x i , y i ; θ b ) 2 / max (xi,yi)∈Dn ∇ θ L CE (x i , y i ; θ b ) 2 ∈ [ We reported the results of CelebA in Table 3 of section 4, following the settings of (Zhang et al., 2022b). For comparison with more diverse algorithms, we further report the CelebA results according to the settings of (Seo et al., 2022). Note that the (Zhang et al., 2022b) and(Seo et al., 2022) used a different model, each using ResNet50 and ResNet18, respectively. As in For convenience, we describe notations used in Section 5, Appendix F, G, and H. x ∈ R d , y ∈ C = {1, ..., c} y true (x) the true label of image x labeled by the oracle model f (y\|x, θ ) θ model parameter - θ oracle model parameter satisfying f (y true (x)\|x, θ ) = 1 for any x D n training dataset composed of {(x i , y i )} n i=1 ∼ p(x, y\|θ ) h(x) sampling probability of each sample in D n satisfying h(x) ∈ H Distributions P(x) general distribution of input image x - p(x) distribution of training image - q(x) distribution of test image - f (y\|x, θ) conditional distribution with model parameter θ - P(x, y\|θ) general joint distribution with model parameter θ P(x)f (y\|x, θ) p(x, y\|θ ) joint distribution of the training dataset p(x)f (y\|x, θ ) q(x, y\|θ ) joint distribution of the test dataset q(x)f (y\|x, θ ) Estimatorŝ θ h(x),Dn MLE solution on the D n with h arg max θ n i=1 h(x i ) log f (y i \|x i , θ) θ U (x),Dn MLE solution on the D n with uniform distribution U solution of ERM Fisher Information I P(x) (θ) Fisher Information E (x,y)∼P(x)f (y\|x,θ) [∇ θ log f (y\|x, θ)∇ θ log f (y\|x, θ)] I h(x) (θ) Empirical Fisher Information n i=1 h(x i )∇ θ log f (y i \|x i , θ)∇ θ log f (y i \|x i , θ) Set H set of all possible h(x) on D n {h(x)\| (xi,yi)∈Dn h(x i ) = 1 and h(x i ) ≥ 0 ∀(x i , y i ) ∈ D n } M set of all possible marginal P(x) on input space X {P(x)\| x∈X P(x) dx = 1} W set of all possible (x, y true (x)) supp(P(x, y\|θ)) Support set of P(x, y\|θ) {(x, y) ∈ X × {1, · · · , c} \| P(x, y\|θ) = 0}, ∀ P(x, y\|θ) (A0). The general joint distribution P(x, y\|θ) is factorized into the conditional distribution f (y\|x, θ) and the marginal distribution P(x), not depend on model parameter θ, that is: Order notations in probability P(x, y\|θ) = P(x)f (y\|x, θ).(2) Thus, the joint distribution is derived from model parameter θ and the marginal distribution P(x), which is determined from the task that we want to solve. Without loss of generality, we match the joint distribution's name with the marginal distribution. (A1). (Identifiability): The CDF P θ (whose density is given by P(x, y\|θ)) is identifiable for different parameters. Meaning that for every distinct parameter vectors θ 1 and θ 2 in Ω, P θ1 and P θ2 are also distinct. That is, ∀θ 1 = θ 2 ∈ Ω, ∃A ⊆ X × {1, · · · , c} s.t. P θ1 (A) = P θ2 (A), where X, {1, · · · , c} and Ω are input, label, and model parameter space, respectively. (A2). The joint distribution P θ has common support for all θ ∈ Ω. (A3). (Model Faithfulness): For any x ∈ X, we assume an oracle model parameter θ that generates y true (x), a true label of input x with the conditional distribution f (y true (x)\|x, θ ) = 1. (A4). (Training joint): Let p(x) denote the training marginal with no dependence on the parameter. Then, the set of observations in D n {(x 1 , y 1 ). · · · (x n , y n )} are drawn independently from the training/proposal joint distribution of the form p(x, y\|θ ) p(x)f (y\|x, θ ), because we do not think the existence of mismatched label data situation in the training data. (A5). (Test joint): Let q(x) denote the test marginal without dependence on the parameter. The unseen test pairs are distributed according to the test/true joint distribution of the form q(x, y\|θ ) q(x)f (y\|x, θ ), because we do not think the existence of mismatched label data situation in the test task. (A9). {(x, y) ∈ supp(q(x, y\|θ )) \| ∇ 2 θ log q(x, y\|θ ) is singular} is a measure zero set. In contrast to (Sourati et al., 2016), we modify (A3) so that the oracle model always outputs a hard label, i.e., f (y true (x)\|x, θ ) = 1 and add (A9) which is not numbered but noted in the statement of Theorem 3 and Theorem 11 in (Sourati et al., 2016). F.3 PRELIMINARIES We organize the two types of background knowledge, maximum likelihood estimator (MLE) and Fisher information (FI), needed for future analysis. F.3.1 MAXIMUM LIKELIHOOD ESTIMATOR (MLE) In this section, we derive the maximum likelihood estimator in a classification problem with sampling probability h(x). Unless otherwise specified, training set D n = {(x i , y i )} n i=1 is sampled from p(x, y\|θ ). For given probability mass function (PMF) h(x) on D n , we define MLEθ h(x),Dn as follows:θ h(x),Dn arg max θ log P(D n \|θ; h(x)) = arg min θ − n i=1 h(x i ) log p(x i , y i \|θ) (3) = arg min θ − n i=1 h(x i ) log f (y i \|x i , θ) (4) = arg min θ n i=1 h(x i ) L CE (x i , y i ; θ).(5) In (3) and (4) For a set of random variables X n and a corresponding set of constant a n , the notation X n = o p (a n ) means that the set of values X n /a n converges to zero in probability as n approaches an appropriate limit. It is equivalent with X n /a n = o p (1), where X n = o p (1) is defined as: lim n→∞ P(\|X n \| ≥ ) = 0 ∀ ≥ 0. The notation X n = O p (a n ) means that the set of values X n /a n is stochastically bounded. That is ∀ > 0, ∃ finite M > 0, N > 0 s.t. P(\|X n /a n \| > M ) < for any n > N . G THEOREM 1 In this section, we deal with some required sub-lemmas that are used for the proof of Lemma 8, which is the main ingredient of the proof of Theorem 1. G.1 SUB-LEMMAS Lemma 1 ( (Lehmann & Casella, 1998), Theorem 5.1). When P − → denotes convergence in probability, and if (A0) to (A7) of the Assumption 1 in Appendix F hold, then there exists a sequence of MLE solutions {θ U (x),Dn } n∈N thatθ U (x),Dn P − → θ as n − → ∞, where θ is the 'true' unknown parameter of the distribution of the sample. Proof. We refer to (Lehmann & Casella, 1998) for detailed proof. Lemma 2 ((Lehmann & Casella, 1998), Theorem 5.1). Let {θ U (x),Dn } n∈N be the MLE based on the training data set D n . If (A0) to (A8) of the Assumption 1 in Appendix F hold, then the MLÊ θ U (x),Dn has a zero-mean normal asymptotic distribution with a covariance equal to the inverse Fisher information matrix, and with the convergence rate of 1/2: √ n(θ U (x),Dn − θ ) D − → N ( 0, I p(x) (θ ) −1 ), where D − → represents convergence in distribution. Proof. We refer to (Lehmann & Casella, 1998) for detailed proof, based on Lemma 1. F hold, we get √ nI p(x) (θ U (x),Dn ) 1/2 (θ U (x),Dn − θ ) D − → N ( 0, I d ), where D − → represents convergence in distribution. Proof. We refer to (Wasserman, 2004) for detailed proof, based on Lemma 2. Proof. We refer to (Serfling, 1980) for detailed proof. Lemma 5. ((Sourati et al., 2016), Theorem 27) Let {θ n } be a sequence of random vectors in a convex and compact set Ω ⊆ R d and θ ∈ Ω be a constant vector such that θ n − θ 2 = O p (a n ) where a n − → 0 (as n − → ∞). If g : Ω − → R is a C 3 function, then g(θ n ) = g(θ ) + ∇ T θ g(θ )(θ n − θ ) + 1 2 (θ n − θ ) T ∇ 2 θ g(θ )(θ n − θ ) + o p (a n 2 ). Proof. We refer to (Serfling, 1980) for detailed proof. Lemma 6. If (A0) and (A3) of the Assumption 1 in Appendix F hold, then ∇ θ log P(x, y true (x)\|θ ) = 0 for any joint distribution P(x, y\|θ ). Proof. ∇ θ log P(x, y true (x)\|θ ) = ∇ θ log f (y true (x)\|x, θ ) = ∇ θ log 1 = 0. On the first equality, (A0) of Assumption 1 in Appendix F is used. At the second equality, (A3) of Assumption 1 in Appendix F is used. Lemma 7. If (A0) to (A8) of the Assumption 1 in Appendix F hold and the case ∇ 2 θ log p(x, y true (x)\|θ ) is non-singular for given data (x, y true (x)) satisfies, then the asymptotic distribution of the log-likelihood ratio is a mixture of first-order Chi-square distributions, and the convergence rate is one. More specifically: n log p(x, y true (x)\|θ ) p(x, y true (x)\|θ U (x),Dn ) D − → 1 2 d i=1 λ i (x, y true (x)) · χ 2 1 ,(13) where {λ i (x, y true (x))} d i=1 are eigenvalues of I p(x) (θ ) − 1 2 −∇ 2 θ log p(x, y true (x)\|θ ) I p(x) (θ ) − 1 2 . Proof. The proof is based on the Taylor expansion theorem. Remind that we deal with the data (x, y true (x)) satisfying ∇ 2 θ log p(x, y true (x)\|θ ) is non-singular. From the property √ n(θ U (x),Dn − θ ) D − → N ( 0, I p(x) (θ ) −1 ) derived from Lemma 3, one concludes that √ n θ U (x),Dn −θ 2 = O p (1) and therefore θ U (x),Dn − θ 2 = O p ( 1 √ n ) by the Lemma 4. Thus, by the Lemma 5, log p(x, y true (x)\|θ U (x),Dn ) = log p(x, y true (x)\|θ ) + (θ U (x),Dn − θ ) T ∇ θ log p(x, y true (x)\|θ ) + 1 2 (θ U (x),Dn − θ ) T ∇ 2 θ log p(x, y true (x)\|θ )(θ U (x),Dn − θ ) + o p 1 n holds. By the Lemma 3 and the property ∇ θ log p(x, y true (x)\|θ ) = 0 derived by Lemma 6, we can obtain n log p(x, y true (x)\|θ ) − log p(x, y true (x)\|θ U (x),Dn ) = − 1 2 √ n(θ U (x),Dn − θ ) T ∇ 2 θ log p(x, y true (x)\|θ ) √ n(θ U (x),Dn − θ ) + o p (1) D − → 1 2 N 0, I p(x) (θ ) −1 T −∇ 2 θ log p(x, y true (x)\|θ ) N 0, I p(x) (θ ) −1 = 1 2 N 0, I d T −I p(x) (θ ) − 1 2 ∇ 2 θ log p(x, y true (x)\|θ )I p(x) (θ ) − 1 2 N 0, I d . Define Γ(x, y true (x)) as −I p(x) (θ ) − 1 2 ∇ 2 θ log p(x, y true (x)\|θ )I p(x) (θ ) − 1 2 and rewrite the righthand-side element-wise 13 as 1 2 N 0, I d T Γ(x, y true (x))N 0, I d = 1 2 d i=1 λ i (x, y true (x)) · N (0, 1) 2 = 1 2 d i=1 λ i (x, y true (x)) · χ 1 2 , where {λ i (x, y true (x))} d i=1 are eigenvalues of Γ(x, y true (x)). Thus, the desired property is obtained. G.2 MAIN LEMMA In this section, we derive the main Lemma, which represents the test cross-entropy loss and can be understood as Fisher information ratio (FIR) (Sourati et al., 2016). 13 Suppose Γ = U Σ U T and Σ = diag(λ1, · · · λ d ). Then, N 0, I d Proof. We prove Lemma 8 via two steps. First we show that the expected cross-entropy loss term can be rewritten in terms of the log-likelihood ratio. Then, we prove that the expected log-likelihood ratio can be asymptotically understood as FIR. Step 1: Log-likelihood ratio We show that the expected log-likelihood ratio can be formulated as the expected test cross-entropy loss as follows: This property holds because, Since (A0) of Assumption 1 in Appendix F, (15) and (16) hold. At (17), the properties, (i) Supp(q(x, y\|θ )) ⊆ W and (ii) f (y\|x, θ ) = 1 ∀ (x, y) ∈ W, was used which is derived by (A3) of the Assumption 1 in Appendix F. E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) log p(x, y\|θ ) p(x, y\|θ U (x),Dn ) = E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) log f (y\|x, θ ) f (y\|x,θ U (x),Dn ) (15) = E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) log f (y\|x, θ ) f (y\|x,θ U (x),Dn ) 1 Supp(q(x,y\|θ )) (16) = E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) log f (y\|x, θ ) f (y\|x,θ U (x), Step 2: FIR Here, we show that the expected test loss of MLE can be understood as FIR. By (A0) in Assumption 1 in Appendix F, trivially {(x, y) ∈Supp(q(x, y\|θ )) \| ∇ 2 θ log q(x, y\|θ ) is singular} = {(x, y) ∈ Supp(q(x, y\|θ )) \| ∇ 2 θ log p(x, y\|θ ) is singular}. holds. Since (18), and (A9) in Assumption 1 in Appendix F, Supp(q(x, y\|θ )) can be replaced by, S Supp(q(x, y\|θ )) \ {(x, y) ∈ W \| ∇ 2 θ log p(x, y\|θ ) is singular} (19) when calculate expectation. We can get a result of Lemma 8 as follows: = E (x,y)∼q(x)f (y\|x,θ ) lim n→∞ E Dn∼p(x)f (y\|x,θ ) n log p(x, y\|θ ) p(x, y\|θ U (x),Dn ) 1 S G.3.2 REPLACING θ BYθ U (x),Dn Lemma 9. Suppose Assumption 1 in Appendix F and Assumption 2 in Appendix G hold, then with high probability: Tr I p(x) (θ ) −1 = lim n→∞ Tr I p(x) (θ U (x),Dn ) −1 .(29) Proof. It is shown in the proof of Lemma 2 in (Chaudhuri et al., 2015) that under the assumptions mentioned in Assumption 2, the following inequalities hold with probability 1 − δ(n): β(n) − 1 β(n) I(θ , x) I(θ U (x),Dn , x) β(n) + 1 β(n) I(θ , x), where β(n) and 1 − δ(n) are proportional to n, which is the size of the training set D n . Because of the independence for the class labels y of I(θ, x), P(x), I P(x) (θ) = E x∼P(x) [I(θ, x)] holds for any marginal distribution P(x). 14 Taking the expectation to the (30) with respect to the marginal p(x) and q(x), then: β(n) − 1 β(n) I p(x) (θ ) I p(x) (θ U (x),Dn ) β(n) + 1 β(n) I p(x) (θ ).(31) β(n) − 1 β(n) I q(x) (θ ) I q(x) (θ U (x),Dn ) β(n) + 1 β(n) I q(x) (θ ). Since I p(x) (θ ) and I p(x) (θ U (x),Dn ) are assumed to be positive definite, we can write (31) in terms of inverted matrices 15 : β(n) β(n) + 1 I p(x) (θ ) From (33), β(n) − 1 β(n) Tr I p(x) (θ U (x),Dn ) −1 ≤ Tr I p(x) (θ ) −1 ≤ β(n) + 1 β(n) Tr I p(x) (θ U (x),Dn ) −1(34) holds when taking n → ∞ to the (34). Note that β(n) is proportional to n. G.4 STATEMENT AND PROOF OF THEOREM 1 Theorem 1. Suppose Assumption 1 in Appendix F and Assumption 2 in Appendix G hold, then for sufficiently large n = \|D n \|, the following holds with high probability: E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x),Dn ) ≤ 1 2n Tr I p(x) (θ U (x),Dn ) −1 Tr I q(x) (θ ) . holds with high probability. It is worth noting that Theorem 1 states that the upper bound of the MLEθ U (x),Dn , D n test loss can be minimized by lowering Tr I p(x) (θ U (x),Dn ) −1 when training marginal p(x), the only tractable and controllable variable. H THEOREM 2 In this section, we introduce the motivation of gradient norm-based importance sampling. To show why this is important, we introduce the debiasing object problem for a given D n under sampling probability h(x) and show how to solve it in a toy example because the problem is difficult. H.1 PRACTICAL OBJECTIVE FUNCTION FOR THE DATASET BIAS PROBLEM Remark that the right-hand side term of (36) are controlled by the training and test marginals p(x), and q(x). Since we can only control the training dataset D n not p(x) and q(x), we can design a practical objective function for the dataset bias problem by using EFI and Theorem 1 as follows: min h(x)∈H Tr Î h(x) (θ h(x),Dn ) −1 ,(37) whereÎ h(x) (θ) is an empirical Fisher information matrix. Remark that EFI is defined as: I h(x) (θ) = n i=1 h(x i )∇ θ log f (y i \|x i , θ)∇ θ log f (y i \|x i , θ).(38) Here, h(x) describes the sampling probability on D n , which is the only controllable term. We deal with (37) in the toy example because of the difficulty of the problem. H.2 ONE-DIMENSIONAL TOY EXAMPLE SETTING For simplicity, we assume that D n comprises sets M and m, and the samples in each set share the same loss function and the same probability mass. The details are as follows: Thus, it is consistent with our intuition that setting the sampling probability h for set M and m in proportion to \|g M (θ U (x),Dn )\| and \|g m (θ U (x),Dn )\| helps to minimize the trace of the inverse empirical Fisher information. Figure 1 : 1Target and bias attribute: digit shape, color. Figure 2 : 2Average PGD results for various of norms, {L 1 , L 2 , L 2 2 , L ∞ }, for the feature-injected benchmarks. The error bars represent the standard deviation of three independent trials. Dn )\|, and \|M \| and \|m\| denote the cardinality of M and m, respectively.The proof of Theorem 2 is provided in Appendix E. Note that h M (x) and h m (x) are computed using the trained biased model with batches sampled from the uniform distribution U (x). It is the same with the second step of PGD. Figure 3 3bias label. In(Goyal et al., 2017; 2020), a debiased dataset was generated using human labor. Various studies(Alvi et al., 2018; Kim et al., 2019; McDuff et al., 2019; Singh et al., 2020; Teney et al., 2021) have attempted to reduce dataset bias using explicit bias labels.Some of these studies (Alvi et al., 2018; Kim et al., 2019; McDuff et al., 2019; Singh et al., 2020; Li et al., 2018; Li & Vasconcelos, 2019), used bias labels for each sample to reduce the influence of the bias labels when classifying target labels. Furthermore, Tartaglione et al. (2021) proposed the EnD regularizer, which entangles target correlated features and disentangles biased attributes. Several studies (Alvi et al., 2018; Kim et al., 2019; Teney et al., 2021) have designed DNNs as a shared feature extractors and multiple classifiers. In contrast to the shared feature extractor methods, McDuff et al. (2019) and Ramaswamy et al. (2021) fabricated a classifier and conditional generative adversarial networks, yielding test samples to determine whether the classifier was biased. Furthermore, Singh et al. ( Geirhos et al., 2018; Wang et al., 2018; Lee et al., 2019; Bahng et al., 2020; Cadene et al., 2019; Clark et al., 2019) assumed that the bias context is known. In(Geirhos et al., 2018; Wang et al., 2018; Lee et al., 2019), debiasing was performed by directly modifying known context bias. In particular, the authors of (Geirhos et al., 2018) empirically showed that CNNs trained on ImageNet(Deng et al., 2009) were biased towards the image texture, and they generated stylized ImageNet to mitigate the texture bias, while Lee et al.(2019) and Wang et al. (2018) inserted a filter in front of the models so that the influence of the backgrounds and colors of the images could be removed. Meanwhile, some studies (Bahng et al., 2020; Clark et al., 2019; Cadene et al., 2019), mitigated bias by reweighting bias-conflicting samples: Bahng et al. (2020) used specific types of CNNs, such as BagNet (Brendel & Bethge, 2018), to capture the texture bias, and the bias was reduced using the Hilbert-Schmidt independence criterion (HSIC). In the visual question answering (VQA) task, Clark et al. (2019) and Cadene et al. (2019) conducted debiasing using the entropy regularizer or sigmoid output of the biased model trained on the fact that the biased model was biased toward the question. Debiasing without human supervision. Owing to the impractical assumption that bias information is given, recent studies have aimed to mitigate bias without human supervision (Le Bras et al., 2020; Nam et al., 2020; Darlow et al., 2020; Kim et al., 2021; Lee et al., 2021). Le Bras et al. (2020) identified bias-conflicting samples by sorting the average accuracy of multiple train-test iterations and performed debiasing by training on the samples with low average accuracy. In (Ahmed et al., 2020), each class is divided into two clusters based on IRMv1 penalty (Arjovsky et al., 2019) using the trained biased model, and the deibased model is trained so that the output of two clusters become similar. Furthermore, Kim et al. (2021) used Swap Auto-Encoder (Park et al., 2020) to generate biasconflicting samples, and Darlow et al. (2020) proposed the modification of the latent representation to generate bias-conflicting samples by using an auto-encoder. Lee et al. (2021) and Nam et al. (2020) proposed a debiasing algorithm weighted training by using a relative difficulty score, which is measured by the per-sample training loss. Specifically, Lee et al. (2021) used feature-mixing techniques to enrich the dataset feature information. Seo et al. (2022) and Sohoni et al. (2020) proposed unsupervised clustering based debiasing method. Recently, contrastive learning based method (Zhang et al., 2022b) and self-supervised learning method (Kim et al., 2022) are proposed. Figure 4 : 4Colored MNIST: The single bias attribute is color, and the target attribute is shape. The top 3 rows represent biasaligned samples, and the bottom row samples are bias-conflicting examples. Figure 5 : 5Multi-bias MNIST: Multiple colors and objects bias, with digit shape as the target. The top 3 rows represent biasaligned samples, and the bottom row samples are bias-conflicting examples. Figure 6 : 6Corrupted CIFAR: corruption is the bias attribute, while target attribute is object. The top three rows are biasaligned samples, while the bottom row are bias-conflicting examples. Figure 7 : 7This dataset was generated by injecting filters into the CIFAR10 dataset(Krizhevsky et al., 2009). The work(Nam et al., 2020; Lee et al., 2021) inspired this benchmark. In this benchmark, the target attribute and the bias attribute are object and corruption, respectively. {Snow, Frost, Fog, Brightness, Contrast, Spatter, Elastic, JPEG, Pixelate, Saturate} are examples of corruption. We downloaded this benchmark from the repository of the official code of Disen(Lee et al., 2021). This dataset contains 45, 000 in training samples, 5, 000 in validation samples, and 10, 000 in testing images. As with prior datasets, the test dataset is composed of unbiased samples (i.e., ρ = 90%). Biased Action Recognition: The biased attribute is background, while the target attribute is action. The top 2 rows are bias-aligned samples, and the bottom row is bias-conflict samples. Figure 8 : 8BFFHQ: stands for biased FFHQ. Target attribute: gender, biased attribute: age. The samples in the top two rows are bias-aligned, while the samples in the bottom row are bias-conflict. ( 6 ) 6CNC (Zhang et al., 2022b) aims to debias by learning representation such that samples in the same class are close but different groups are far. CNC is composed of two steps: (1) inferring pseudo group label, (2) supervised contrastive learning. Get the ERM-based model f and the pseudo predictionŷ first, then standard argmax over the final layer outputs of the model f . Next, CNC trains the debiased model based on supervised contrastive learning using pseudo predictionŷ. The detailed process of contrastive learning for each iteration is as follows:• From the selected batch, sample the one anchor data (x,y). • Construct the set of positives samples {(x + m , y + m )} which is belong to the batch, satisfying y + m = y andŷ + m =ŷ. • Similarly, construct the set of negative samples {(x − n , y − n )} which is belong to the batch, satisfying y − n = y andŷ − n =ŷ. • With the loss of generality, assume the cardinality of the positive and negative sets are M and N , respectively. Figure 10 : 10Histogram of unbiased test accuracy among 25 trials for each. D.3 HISTOGRAM OF THE RESULTS OF 25 TRIALS IN CMNIST O p Big O, stochastic boundedness o p Small o, convergence in probability -Toy example (Appendix H) M set of majority (i.e., bias-aligned) samples m set of minority (i.e., bias-conflicting) samples -g M (θ) gradient of samples in M at θ g m (θ) gradient of samples in m at θ h M (x)Optimal sampling probability of samples in M h m (x)Optimal sampling probability of samples in m -F.2 MAIN ASSUMPTIONHere, we organize the assumptions that are used in the proof of Theorems. These are basically used when analyzing models through Fisher information. The assumptions are motivated by(Sourati et al., 2016). Assumption 1. ( A6). (Differentiability): The log-conditional log f (y\|x, θ) is of class C 3 (Ω) for all (x, y) ∈ X × {1, 2, · · · , c}, when being viewed as a function of the parameter. 11 (A7). The parameter space Ω is compact, and there exists an open ball around the true parameter of the model θ ∈ Ω. (A8). (Invertibility): The arbitrary Fisher information matrix I P(x) (θ) is positive definite and therefore invertible for all θ ∈ Ω. Lemma 3 ( 3(Wasserman, 2004), Theorem 9.18). Under the (A0) to (A8) of the Assumption 1 in Appendix Lemma 4 . 4((Serfling, 1980), Chapter 1) Let {θ n } be a sequence of random vectors. If there exists a random vectorθ such that θ n D − →θ, then θ n −θ 2 = O p (1), where · 2 denote the L 2 norm. TU ∼ N 0, U U T = N 0, I d . Thus, N 0, I d T ΓN 0, I d = N 0, I d T ΣN 0, I d = d i=1 λiN (0, 1) 2 .G.2.1 MAIN LEMMA STATEMENT AND PROOFLemma 8 (FIR in expected test cross entropy loss with MLE). If the Assumption 1 in Appendix F holds, thenlim n→∞ nE (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x)I p(x) (θ ) −1 I q(x) (θ ) . Dn ) 1 1Supp(q(x,y\|θ )) = E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x),Dn ) 1 Supp(q(x,y\|θ )) (17) = E (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x),Dn ) lim n→∞ nE (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x),Dn ) = lim n→∞ nE (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) log p(x, y\|θ ) p(x, y\|θ U (x),Dn ) 1 Supp(q(x,y\|θ )) (20) = lim n→∞ nE (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) log p(x, y\|θ )p(x, y\|θ U (x),Dn ) 1 S 14 I P(x) (θ) = E (x,y)∼P(x,y\|θ) −∇ 2 θ log f (y\|x, θ) = E x∼P(x) E y∼f (y\|x,θ) −∇ 2 θ log f (y\|x, θ) = E x∼P(x) E y∼f (y\|x,θ) [I(θ, x)] = E x∼P(x) [I(θ, x)].15 For ∀ two positive definite matrices A and B, we have that A B ⇒ A −1 B −1 16 If A B, Tr [A] ≤ Tr [B] holds. (∵) A B ⇒ B − A O and B − A := U ΣU T , where U = [u1\| · · · , \|u d ]. Then, Tr(B − A) = d i=1 ui(B − A)u T i ≤ 0 because of the positive definite property of B − A. Tr(B − A) ≤ 0 ⇒ Tr(B) ≤ Tr(A). ( 36 )Tr 36Proof. Because of the (A8) of Assumption 1 in Appendix F, I p(x) (θ ) −1 and I q(x) (θ ) are positive definite matrix. Thus,Tr I p(x) (θ ) −1 I q(x) (θ ) ≤ Tr I p(x) (θ ) −1 Tr I q(x) (θ ) holds. 17From the result of Lemma 8 and 9 in Appendix G,lim n→∞ nE (x,y)∼q(x)f (y\|x,θ ) E Dn∼p(x)f (y\|x,θ ) − log f (y\|x,θ U (x)I p(x) (θ U (x),Dn ) −1 Tr I q(x) (θ ) \|m\| 2\|M \|·\|m\| , h m (x) = \|M \| 2\|M \|·\|m\| . This result is related with the trained modelθ U (x),Dn , where U M (x) = U m (x) = 1 \|M \|+\|m\| . Atθ U (x),Dn , \|M \|g M (θ U (x),Dn ) + \|m\|g m (θ U (x),Dn ) = 0 satisfies and this is equivalent to \|M \| : \|m\| = \|g m (θ U (x),Dn )\| : \|g M (θ U (x),Dn )\|. Table 1 : 1Average test accuracy and standard deviation (three runs) for experiments with the MNIST variants under various bias conflict ratios. The best accuracy is indicated in bold for each case.Dataset ρ Vanilla LfF JTT Disen PGD (Ours) 0.5% 60.94± 0.97 91.35± 1.83 85.84± 1.32 94.56± 0.57 96.88± 0.28 CMNIST 1% 79.13± 0.73 96.88± 0.20 95.07± 3.42 96.87± 0.64 98.35± 0.12 5% 95.12± 0.24 98.18± 0.05 96.56± 1.23 98.35± 0.20 98.62± 0.14 10% 25.23± 1.16 19.18± 4.45 25.34± 1.45 25.75± 5.38 61.38± 4.41 MBMNIST 20% 62.06± 2.45 65.72± 6.23 68.02± 3.23 61.62± 2.60 89.09± 0.97 30% 87.61± 1.60 89.89± 1.76 85.44± 3.44 88.36± 2.06 90.76± 1.84 0.5% 23.06± 1.25 28.83± 1.30 25.34± 1.00 29.96± 0.71 30.15± 1.22 CCIFAR 1% 25.94± 0.54 33.33± 0.15 33.62± 1.05 36.35± 1.69 42.02± 0.73 5% 39.31± 0.66 50.24± 1.41 45.13± 3.11 51.19± 1.38 52.43± 0.14 Table 2 : 2Average test accuracy and standard deviation (three runs) for experiments with the raw image benchmarks: BAR and BFFHQ. The best accuracy is indicated in bold, and for the overlapped best performance case is indicated in Underline.Dataset Vanilla LfF JTT Disen PGD (Ours) BAR 63.15± 1.06 64.41± 1.30 63.62±1.33 64.70± 2.06 65.39± 0.47 BFFHQ 77.77± 0.45 82.13± 0.38 77.93± 2.16 82.77± 1.40 84.20± 1.15 CivilComments-WILDS. CivilComments-WILDS (Borkan et al., 2019) is a dataset to classify whether an online comment is toxic or non-toxic. The mentions of certain demographic identities (male, female, White, Black, LGBTQ, Muslim, Christian, and other religion) cause the spurious correlation with the label. 4.2 IMPLEMENTATION. Baselines. We select baselines available for the official code from the respective authors among debiasing methods without prior knowledge on the bias. Our baselines comprise six methods on the various tasks: vanilla network, LfF (Nam et al., 2020), JTT (Liu et al., 2021) 2 , Disen (Lee et al., 2021), EIIL (Creager et al., 2021) and CNC (Zhang et al., 2022b). Implementation details. We use three types of networks: two types of simple convolutional networks (SimConv-1 and SimConv-2) and ResNet18 (He et al., 2016). Network imeplementation is described in Appendix B. Colored MNIST is trained on SGD optimizer, batch size 128, learning rate 0.02, weight decay 0.001, momentum 0.9, learning rate decay 0.1 every 40 epochs, 100 epochs training, and GCE parameter α 0.7. For Multi-bias MNIST, it also utilizes SGD optimizer, and 32 batch size, learning rate 0.01, weight decay 0.0001, momentum 0.9, learning rate decay 0.1 with decay step 40. It runs 100 epochs with GCE parameter 0.7. For corrupted CIFAR and BFFHQ, it uses ResNet18 as a backbone network, and exactly the same setting presented by Disen (Lee et al., 2021). 3 For CelebA, we follows experimental setting of (Zhang et al., 2022b) which uses ResNet50 as a backbone network. For CivilComments-WILDS, we utilize exactly the same hyperparameters of (Liu et al., 2021) and utilize pretrained BERT. To reduce the computational cost in extracting the per-sample gradients, we use only a fully connected layer, similar to (Ash et al., 2019; Mirzasoleiman et al., 2020; Killamsetty et al., 2021b;a; 2020). Except for CivilComments-WILDS and CelebA, we utilize data augmentation, such as color jitter, random resize crop and random rotation. See Appendix B for more details. Table 3 : 3Average and worst test accuracy with the raw image benchmark: CelebA and raw NLP task: CivilComments-WILDS. The results of comparison algorithms are the results reported in(Zhang et al., 2022b). The best worst accuracy is indicated in bold.Vanilla LfF EIIL JTT CNC Ours CelebA Avg. 94.9 85.1 85.7 88.1 88.9 88.6 Worst 47.7 77.2 81.7 81.5 88.8 88.8 CivilComments Avg. 92.1 92.5 90.5 91.1 81.7 92.1 Worst 58.6 58.8 67.0 69.3 68.9 70.6 L 1 L 2 L 2 2 L ∞ Unbiased Acc. 90 95 100 Portion of bias-conflicting 0.5% 1% 5% (a) Colored MNIST Unbiased Acc. 60 80 100 Portion of bias-conflicting 10% 20% 30% (b) Multi-bias MNIST Unbiased Acc. 30 40 50 Portion of bias-conflicting 0.5% 1% 5% (c) Corrupted CIFAR Table 4 : 4Ablation studies on GCE and data augmentation ( for applied case).GCE Aug. CMNIST (0.5%) MB-MNIST (10%) 84.93 ± 0.79 40.58 ± 3.39 93.18 ± 1.07 45.70 ± 2.91 91.19 ± 0.97 46.70 ± 1.10 96.88 ± 0.28 61.38 ± 4.41 ACKNOWLEDGEMENTThis work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) [No.2019-0-00075, Artificial Intelligence Graduate School Program(KAIST), 10%] and Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) [No.2022-0-00641, XVoice: Multi-Modal Voice Meta Learning, 90%] REFERENCES Alekh Agarwal, Alina Beygelzimer, Miroslav Dudík, John Langford, and Hanna Wallach. A reductions approach to fair classification. In International Conference on Machine Learning, pp. 60-69. PMLR, 2018. Faruk Ahmed, Yoshua Bengio, Harm van Seijen, and Aaron Courville. Systematic generalisation with group invariant predictions. In International Conference on Learning Representations, 2020. 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URL https://www.semanticscholar.org/paper/ Learning-Non-Discriminatory-Predictors-Woodworth-Gunasekar/ 00cda3a1c7f716d6136f0d3c0c1fe1046e685e82. Biased Action Recognition (BAR) This dataset comes from the paper (Nam et al., 2020) for realworld image testing. TThis benchmark aims to classify six actions {Climbing, Diving, Fishing, Racing, Throwing, Vaulting} even though the places are biased.Target and bias attribute pairs are (Climbing,RockWall), (Diving,Underwater), (Fishing, WaterSurface), (Racing, APavedTrack), (Throwing, PlayingField), and (Vaulting, Sky). Bias-conflicting samples, for example, are (Climbing, IceCliff), (Diving, Indoor), (Fishing, Forest), (Racing, OnIce), (Throwing, Cave), (Vaulting, Beach). There are 1, 941 samples for training and 6, 54 samples for testing. To split the training and validation samples, we used 10% validation samples, i.e., 1, 746 images for training and 195 validation. We download training datasets from the online repository of BAR. Biased FFHQ This BFFHQ benchmark was conducted in (Lee et al., 2021; Kim et al., 2021). Target and bias attributes for bias-aligned samples are (Female, Young), and (Male, Old). Here, "Young" refers to people aged 10 to 29, while "old" refers to people aged 40 to 59. The bias-conflicting samples are (Female, Old) and (Male, Young). The number of training, validation, and test samples are 19, 200, 1, 000, and 1, 000, respectively. Table 5 5indicates the portion of toxic comments for each demographic identity.Identity Male Female White Black LGBTQ Muslim Christian Other religions Portion(%) of toxic 14.9 13.7 28.0 31.4 26.9 22.4 9.1 15.3 Table 2 2reported in Setion 4, we reproduce all experimental results referring to other official repositories: 7 8 9 10 .The differences compared to the baseline codes are network architecture for CMNIST and usage of data augmentation. Here, we use the same architecture for CMNIST and data augmentation for all algorithms for a fair comparison. Except for JTT, all hyperparameters for CCIFAR and BFFHQ follow previously reported parameters in repositories. We grid-search for other cases, MNIST variants, and BAR. We set the only hyperparameter of PGD, α = 0.7, as proposed by the original paper (Zhang & Sabuncu, 2018). A summary of the hyperparameters that we used is reported in Table 6. Colored MNIST Multi-bias MNIST Corrupted CIFAR BAR Biased FFHQ CelebA CivilComments-WILDS Optimizer SGD SGD Adam SGD Adam Adam SGD Batch size 128 32 256 16 64 256 16 Learning rate 0.02 0.01 0.001 0.0005 0.0001 0.0001 0.00001 Weight decay 0.001 0.0001 0.001 1e-5 0.0 0.01 0.01 Momentum 0.9 0.9 - 0.9 - - 0.9 Lr decay 0.1 0.1 0.5 0.1 0.1 Cosine annealing 0.1 Decay step 40 - 40 20 32 - - Epoch 100 100 200 100 160 100 5 GCE α 0.7 0.7 0.7 0.7 0.7 0.7 0.7 Table 6 : 6Hyperparameter detailsFor Table 7 . 7C.2 STUDY 2: MULTI-STAGE VS SINGLE-STAGE Training time Test Acc. Colored MNIST Single-stage Multi-stage Single-stage Multi-stage ρ = 0.5% 2h 53m 40s 33m 39s 92.19± 0.12 96.88± 0.28 ρ = 1% 2h 54m 45s 32m 28s 97.23± 0.34 97.35± 0.12 ρ = 5% 2h 49m 31s 34m 13s 98.44± 0.17 98.62± 0.14 Table 8 : 8Multi-stage vs Single-stagePGD computes the per-sample gradient norm only once, between training the biased model and the debiased model. However, an update of the per-sample gradient can be performed repeatedly at each epoch (i.e., single-stage). Table 9 : 9Reweighting vs resamplingTo support our algorithm design, we provide further ex- perimental analysis, i.e., resampling versus reweighting. Reweighting (Nam et al., 2020; Lee et al., 2021) and re- sampling Table 11 : 11Computation cost C . 6 .STUDY 6: WHAT IF PGD RUNS ON UNBIASED DATASETSVanilla LfF Disen Ours ρ = 90% Colored MNIST 99.04± 0.05 98.75± 0.07 99.31± 0.1 98.43± 0.11 Natrual CIFAR10 94.24± 0.01 - - 94.79± 0.02 Table 12 : 12Results on unbiased CMNIST and natu- ral CIFAR10 cases. Table 13 : 13Digit target MNIST vs Color target MNIST Table 14 : 14Number of samples at each bin: Color target MNIST (ρ = 0.5%) Table 15 : 15Average and worst test accuracy with CelebA setting of (Seo et al., 2022). The results of comparison algorithms for CelebA † are the results reported in Table 15 , 15PGD shows Table 16 : 16Notation TableNotation Description , (A0) and (A4) of Assumption 1 in Appendix F were used, respectively. It is worth noting that MLEθ h(x),Dn is a variable that is influenced by two factors: (1) a change in the training F.3.4 STOCHASTIC ORDER NOTATIONS o P AND O P Note that bias-alignment cannot always be strictly divisible in practice. For ease of explanation, we use the notations bias-conflicting/bias-aligned. In the case of JTT(Liu et al., 2021), although the authors used bias label for validation dataset (especially, bias-conflicting samples), we tune the hyperparameters using a part of the biased training dataset for fair comparison. Considering that JTT does not show significant performance gain in the results, it is consistent with the existing results that the validation dataset is important in JTT, as described in(Idrissi et al., 2022).3 Lee et al. (2021) only reported bias-conflicting case for BFFHQ, but we report the unbiased test result. Note that for simplicity, we abuse the notation h(x, y) used in Section 3 as h(x). This is exactly the same for a given dataset Dn situation. https://github.com/alinlab/BAR 6 https://github.com/kakaoenterprise/Learning-Debiased-Disentangled https://github.com/alinlab/LfF 8 https://github.com/clovaai/rebias 9 https://github.com/kakaoenterprise/Learning-Debiased-Disentangled 10 https://github.com/anniesch/jtt We say that a function f : X − → Y is of C p (X), for an integer p > 0, if its derivatives up to p-th order exist and are continuous at all points of X. Note that for simplicity, we abuse the notation h(x, y) used in Section 3 as h(x). This is exactly the same for a given dataset Dn situation. For ∀ two positive definite matrices A and B, Tr [AB] ≤ Tr [A] Tr [B] satisfies. Extended version of FI. Here, we summarize the extended version of FI, which can be derived by making some assumptions. These variants of FI are utilized in the proof of Theorems.• (Hessian version) Under the differentiability condition (A6) of Assumption 1 in Appendix F, FI can be written in terms of the Hessian matrix of the log-likelihood:• (Model decomposition version) Under the factorization condition (A0) of Assumption 1 in Appendix F,(6)and(7)can be transformed as follows:Specifically, (8) and(9)can be unfolded as follows:From now on, we define I p(x) (θ) and I q(x) (θ) as the FI derived from the training and test marginal, respectively.F.3.3 EMPIRICAL FISHER INFORMATION (EFI)When the training dataset D n is given, we denote the sampling probability as h(x) which is defined on the probability space H:Practically, the training dataset D n is given as deterministic. Therefore, (8) can be refined as empirical Fisher information (EFI). This reformulation is frequently utilized, e.g., in(Jastrzębski et al., 2017;Chaudhari et al., 2019), to reduce the computational complexity of gathering gradients for all possible classes (i.e., expectation with respect to f (y\|x, θ) as in(8)). Refer the c y=1 term of (10). Different from prior EFI, which is defined on the case when h(x) is uniform, U (x), we generalize the definition of EFI in terms of h(x) ∈ H as follows:Note that (a) holds owing to (11).2 (θ + a) 2 and 1 2 (θ − a) 2 loss function arise for all data in M and m, respectively.•θ h,Dn denote the trained model from the arbitrary PMF h(x) ∈ H which has a constraint having degree of freedom 2, (h M (x), h m (x)). • Concretely, each samples of M and m has a probability mass h M (x) and h m (x), respectively.i.e., \|M \| · h M (x) + \|m\| · h m (x) = 1, where \|M \| and \|m\| denote the cardinality of M and m, respectively.• Let g M (θ) and g m (θ) denote the gradient of each sample in M and m at θ ∈ R, respectively.• In these settings, our objective can be written as finding h (x) = arg min h(x)∈H Tr Î h (x)(θ h(x),Dn ) −1 and this is equivalent to find (h M (x), h m (x)).H.3 STATEMENT AND PROOF OF THEOREM 2In this section, we introduce the motivation for the gradient norm-based importance sampling in the toy example setting. 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Resound: Towards action recognition without representation bias. Yingwei Li, Yi Li, Nuno Vasconcelos, Proceedings of the European Conference on Computer Vision (ECCV). the European Conference on Computer Vision (ECCV)Yingwei Li, Yi Li, and Nuno Vasconcelos. Resound: Towards action recognition without representa- tion bias. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 513-528, 2018. Discover the unknown biased attribute of an image classifier. Zhiheng Li, Chenliang Xu, Proceedings of the IEEE/CVF International Conference on Computer Vision. the IEEE/CVF International Conference on Computer VisionZhiheng Li and Chenliang Xu. Discover the unknown biased attribute of an image classifier. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 14970-14979, 2021. Just train twice: Improving group robustness without training group information. 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If h(x) is a uniform distribution U (x), then the result of empirical risk minimization (ERM) isθ U (x). Dn) the adjustment of the sampling probability h(x)dataset D n and (2) the adjustment of the sampling probability h(x). If h(x) is a uniform distribution U (x), then the result of empirical risk minimization (ERM) isθ U (x),Dn . .2 FISHER INFORMATION (FI). F , F.3.2 FISHER INFORMATION (FI) denoted by I P(x) (θ), is a measure of sample information from a given distribution P(x, y\|θ) P(x)f (y\|x, θ). General definition of FI. Fisher information (FI). It is defined as the expected value of the outer product of the score function ∇ θ log P(x, y\|θ) with itself. evaluated at some θ ∈ ΩGeneral definition of FI. Fisher information (FI), denoted by I P(x) (θ), is a measure of sample information from a given distribution P(x, y\|θ) P(x)f (y\|x, θ). It is defined as the expected value of the outer product of the score function ∇ θ log P(x, y\|θ) with itself, evaluated at some θ ∈ Ω. ytrue(x))∼q(x)f (y\|x,θ ) lim n→∞ E Dn∼p(x)f (y\|x,θ ) n log. = , = E (x,ytrue(x))∼q(x)f (y\|x,θ ) lim n→∞ E Dn∼p(x)f (y\|x,θ ) n log \|M \| · h M (x) + \|m\| · h m (x) = 1. probability definition\|M \| · h M (x) + \|m\| · h m (x) = 1. (probability definition) Dn ) + \|m\| · h m (x) · g m (θ h(x),Dn ) = 0. \|M \| · h M (x) · g M (θ h(x). Note that convex linear sum of the sample's gradient w.r.t. h(x) = (h M (x), h m (x)) is zero at the trained modelθ h(x),Dn\|M \| · h M (x) · g M (θ h(x),Dn ) + \|m\| · h m (x) · g m (θ h(x),Dn ) = 0. Note that convex linear sum of the sample's gradient w.r.t. h(x) = (h M (x), h m (x)) is zero at the trained modelθ h(x),Dn . \|g M (θ)\| + \|g m (θ)\| = 2a ⇔ g M (θ) − g m (θ) = 2a ⇔ g M (θ) = 2a + g m (θ). Note that this is derived by the property of predefined loss function at θ ∈ [−a, a\|g M (θ)\| + \|g m (θ)\| = 2a ⇔ g M (θ) − g m (θ) = 2a ⇔ g M (θ) = 2a + g m (θ). Note that this is derived by the property of predefined loss function at θ ∈ [−a, a]. 2nd constraint is equivalent to \|M \| · h M (x) · (2a + g m (θ h(x),Dn )) + \|m\| · h m (x) · g m (θ h(x),Dn ) =. 2nd constraint is equivalent to \|M \| · h M (x) · (2a + g m (θ h(x),Dn )) + \|m\| · h m (x) · g m (θ h(x),Dn ) = ⇔ g m (θ h(x),Dn ) = −2a\|M \| · h M (x). Because of the 3rd constraint, g M (θ h(x),Dn ) = 2a(1 − \|M \| · h M (x)). Then the objective is, maximizing \|M \| · h M (x) · {2a(1 − \|M \| · h M (x))} 2 + (1 − \|M \| · h M (x)){2a\|M \| · h M. ⇔ g m (θ h(x),Dn ) = −2a\|M \| · h M (x). Because of the 3rd constraint, g M (θ h(x),Dn ) = 2a(1 − \|M \| · h M (x)). Then the objective is, maximizing \|M \| · h M (x) · {2a(1 − \|M \| · h M (x))} 2 + (1 − \|M \| · h M (x)){2a\|M \| · h M (x)} 2 = 4a 2 \|M \| · h M (x)(1 − \|M \| · h M. = 4a 2 \|M \| · h M (x)(1 − \|M \| · h M (x)) and it means \|m\| · h m (x) = 1 2 . Thus, h M. and it means \|m\| · h m (x) = 1 2 . Thus, h M (x) =	[ "http://github.com/lmcinnes/umap.", "https://github.com/alinlab/BAR", "https://github.com/kakaoenterprise/Learning-Debiased-Disentangled", "https://github.com/alinlab/LfF", "https://github.com/clovaai/rebias", "https://github.com/kakaoenterprise/Learning-Debiased-Disentangled", "https://github.com/anniesch/jtt" ]
[ "New application of decomposition of U(1) gauge potential: Aharonov-Bohm effect and Anderson-Higgs mechanism", "New application of decomposition of U(1) gauge potential: Aharonov-Bohm effect and Anderson-Higgs mechanism" ]	[ "Jian-Feng Li \nCollege of Mathematics and Physics\nNantong University\n226019NantongChina\n", "Yu Jiang \nCollege of Mathematics\nPhysics and Information Engineering\nZhejiang Normal University\n321004JinhuaChina\n", "Wei-Min Sun \nDepartment of Physics\nNanjing University\n210093NanjingChina\n\nJoint Center for Particle, Nuclear Physics and Cosmology\n210093NanjingChina\n", "Hong-Shi Zong \nDepartment of Physics\nNanjing University\n210093NanjingChina\n\nJoint Center for Particle, Nuclear Physics and Cosmology\n210093NanjingChina\n", "Fan Wang \nDepartment of Physics\nNanjing University\n210093NanjingChina\n\nJoint Center for Particle, Nuclear Physics and Cosmology\n210093NanjingChina\n" ]	[ "College of Mathematics and Physics\nNantong University\n226019NantongChina", "College of Mathematics\nPhysics and Information Engineering\nZhejiang Normal University\n321004JinhuaChina", "Department of Physics\nNanjing University\n210093NanjingChina", "Joint Center for Particle, Nuclear Physics and Cosmology\n210093NanjingChina", "Department of Physics\nNanjing University\n210093NanjingChina", "Joint Center for Particle, Nuclear Physics and Cosmology\n210093NanjingChina", "Department of Physics\nNanjing University\n210093NanjingChina", "Joint Center for Particle, Nuclear Physics and Cosmology\n210093NanjingChina" ]	[]	In this paper we study the Aharonov-Bohm (A-B) effect and Anderson-Higgs mechanism in Ginzburg-Landau model of superconductors from the perspective of the decomposition of U(1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A ⊥ in A-B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A , which reflects the Anderson-Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A , while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring.	10.1142/s0217984912501242	[ "https://export.arxiv.org/pdf/1110.1843v2.pdf" ]	119,280,378	1110.1843	4f3d1398eef8998a0c91d99da908632e3f425654	New application of decomposition of U(1) gauge potential: Aharonov-Bohm effect and Anderson-Higgs mechanism May 2012 Jian-Feng Li College of Mathematics and Physics Nantong University 226019NantongChina Yu Jiang College of Mathematics Physics and Information Engineering Zhejiang Normal University 321004JinhuaChina Wei-Min Sun Department of Physics Nanjing University 210093NanjingChina Joint Center for Particle, Nuclear Physics and Cosmology 210093NanjingChina Hong-Shi Zong Department of Physics Nanjing University 210093NanjingChina Joint Center for Particle, Nuclear Physics and Cosmology 210093NanjingChina Fan Wang Department of Physics Nanjing University 210093NanjingChina Joint Center for Particle, Nuclear Physics and Cosmology 210093NanjingChina New application of decomposition of U(1) gauge potential: Aharonov-Bohm effect and Anderson-Higgs mechanism May 2012Key-words: gauge potential decompositionpure gaugephase fieldsuperconducting ring PACS Numbers: 0365Vf, 7135Lk In this paper we study the Aharonov-Bohm (A-B) effect and Anderson-Higgs mechanism in Ginzburg-Landau model of superconductors from the perspective of the decomposition of U(1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A ⊥ in A-B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A , which reflects the Anderson-Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A , while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring. The decomposition of the gauge potential in gauge field theory plays an important role in the study of many physical problems. In the literature there are different types of gauge potential decomposition in different physical contexts [1][2][3]. For the case of U(1) gauge theory, the one with the most definite physical meaning is the decomposition of the vector gauge potential A into its transverse component A ⊥ and longitudinal component A . This type of decomposition has many applications, e.g., it can be used to construct a decomposition of the QED angular momentum into the spin and orbital parts of the electron and the photon with each part satisfying the angular momentum algebra and the requirement of gauge invariance simultaneouly [4], and the proper gauge invariant momentum and Hamiltonian operator for a charged particle in an external electromagnetic field [5,6]. In this paper we shall discuss two physical problems, the Aharonov-Bohm (A-B) effect [7] and the the Anderson-Higgs (A-H) mechanism [8] in Ginzburg-Landau (G-L) model [9] for the superconductors, from the perspective of this type of decomposition. To start, we recall the decomposition of the vector potential A in terms of its transverse and longitudinal parts A = A ⊥ + A ,(1) where A ⊥ and A are defined by ∇ · A ⊥ = 0 ∇ × A = 0.(2) With the boundary condition that A, A ⊥ and A all vanish at spatial infinity, the above two conditions prescribe a unique decomposition of A into A ⊥ and A . Under a gauge transformation of A: A −→ A ′ = A + ∇χ,(3) where χ is a nonsingular function, A ⊥ and A transform as A ⊥ −→ A ′ ⊥ = A ⊥ ,(4)A −→ A ′ = A + ∇χ.(5) The condition ∇ × A = 0 and Eq. (5) tell us that the longitudinal part A is a pure gauge part and it transforms in the same manner as does the full A. The transverse part A ⊥ is gauge invariant under all gauge transformations and should be regarded as the "physical" part of A. Now, let us turn to the discussion of the A-B effect. The A-B effect, which indicates the importance of vector potential in quantum mechanics, has been widely studied over 50 years. In the literature the A-B effect is usually ascribed to the non-trivial topology of the region outside the infinitely long solenoid. Here we shall study this problem from the perspective of the gauge potential decomposition (1). We describe the system using cylindrical polar coordinates with z-axis along the symmetry axis of the infinitely long solenoid. According to Helmholtz theorem, A ⊥ at all points of the space (both outside the solenoid and inside the solenoid) can be expressed as A ⊥ (x) = ∇ × 1 4π d 3 x ′ ∇ ′ × A(x ′ ) \| x − x ′ \| = ∇ × 1 4π d 3 x ′ B(x ′ ) \| x − x ′ \| = ∇ × B 4π e z ρ<R d 3 x ′ \| x − x ′ \| ,(6) where R is the radius of the cross section of the solenoid. In the cylindrical polar coordinates, the field point x = (r cos ϕ, r sin ϕ, z) and the source point x ′ = (ρ cos ϕ ′ , ρ sin ϕ ′ , h). The integral in Eq. (6) can be written as ρ<R d 3 x ′ \| x − x ′ \| = R 0 ρdρ 2π 0 dϕ ′ +∞ −∞ dh 1 r 2 + ρ 2 − 2rρ cos(ϕ ′ − ϕ) + (h − z) 2 = R 0 ρdρ 2π 0 dϕ ′ +∞ −∞ dh 1 r 2 + ρ 2 − 2rρ cos ϕ ′ + h 2 .(7) For an infinitely long solenoid, the integral over h in Eq. (7) is logarithmically divergent. It can be regulated by introducing a large cutoff L: +∞ −∞ dh 1 r 2 + ρ 2 − 2rρ cos ϕ ′ + h 2 → +L −L dh 1 r 2 + ρ 2 − 2rρ cos ϕ ′ + h 2 = ln 4L 2 − ln(r 2 + ρ 2 − 2rρ cos ϕ ′ ) + O( 1 L 2 ).(8)Then ρ<R d 3 x ′ \| x − x ′ \| (9) = πR 2 ln 4L 2 − R 0 ρdρ 2π 0 dϕ ′ ln(r 2 + ρ 2 − 2rρ cos ϕ ′ ) + O( 1 L 2 ). The integral in the right-hand-side of Eq. (9) can be evaluated to be R 0 ρdρ 2π 0 dϕ ′ ln(r 2 + ρ 2 − 2rρ cos ϕ ′ ) = 2π R 0 ρdρ ln r 2 + ρ 2 + \|r 2 − ρ 2 \| 2 = πR 2 ln R 2 − πR 2 + πr 2 r ≤ R πR 2 ln r 2 r > R(10) The divergent term πR 2 ln 4L 2 in Eq. (9) is a constant and does not contribute to A ⊥ (x). The contribution of the third term in Eq. (9) vanishes in the limit L → ∞. We then obtain A ⊥ (x) = Br 2 e ϕ r ≤ R Φ 2πr e ϕ r > R,(11) where Φ = πR 2 B is the magnetic flux inside the solenoid, and e ϕ is the base vector along the ϕ direction of the cylindrical polar coordinate system. Note that outside the solenoid (r > R) one has A ⊥ (x) = Φ 2π ∇ϕ. As is shown by Eq. (6), once the magnetic field B at all points inside the solenoid has been measured, one can determine A ⊥ at all points of the space. In this sense, A ⊥ in A-B effect is completely determined by the magnetic field B and is thus physical. It is not a pure gauge term and cannot be gauged away by a gauge transformation. Here, we note that the transverse vector potential in electrodynamics was also studied in Ref. [10]. In that reference the authors argued that the transverse vector potential is the physical degrees of freedom of electrodynamics and also discussed the transverse vector potential in the region where the field strength vanishes (such as the region outside the solenoid in the case of A-B experiment). Compared with their approach, we do not discuss A ⊥ inside the solenoid and outside the solenoid separately. As is shown above, we derive an explicit expression of A ⊥ in the A-B experiment at all points (both inside the solenoid and outside the solenoid) by means of Helmholtz theorem. Our approach shows that the A ⊥ inside the solenoid and outside the solenoid is a unified physical quantity and is the physical degree of freedom of the electromagnetic field. We also note that in Ref. [11] the authors proposed a reformulation of electrodynamics in terms of a physical vector potential entirely free of gauge ambiguities. Our formulation differs from theirs in that in our formulation the gauge degrees of freedom still exists, whereas in their formulation there are no gauge degrees of freedom at all. Recently, based on the gauge potential decomposition (1), the authors of [5] proposed that the proper quantum mechanical momentum operator for a charged particle in an external magnetic field should be P = p − q A and the corresponding orbital angular momentum operator is L = r × ( p − q A ). Thus, the proper momentum operator for the electron in A-B experiment is P = p − q A = p − q A + q A ⊥ = m v + qΦ 2πr e ϕ ,(12) where m v = p − q A is the mechanical momentum. Note that m v is observable only in classical physics. In quantum mechanics, when the magnetic field B is nonzero, the three components of m v do not commute with each other and therefore cannot be measured simultaneously. Here we also point out that neither the mechanical momentum nor the canonical momentum provides for the true description of the motion of an electron wave passing around an infinite magnetic solenoid, whereas the proper momentum operator given in Eq. (12) does precisely what is needed. It displays that an electron wave passing on one side of the solenoid picks up additional momentum while the electron wave passing on the other side loses momentum. Thus fringes appear when the waves are recombined on the far side. Therefore, A-B effect is due to interaction with A ⊥ , which is shown clearly by the expression (12) for the proper momentum operator. On the other hand, in the discussion of A-B experiment a multi-valued vector potential can also be defined [12] A ′ = −ϕB z (r) r, where B z (r) = Bθ(R − r) is the z-component of the magnetic field. The vector potential in Eq. (13) also satisfies ∇ × A ′ = B z (r) e z = B.(14) It can be easily seen that A ′ differs from A ⊥ by a pure gauge term: A ′ = A ⊥ − ∇( Λ(r)ϕ 2π ),(15) where Λ(r) = 2π r 0 r ′ B z (r ′ )dr ′ .(16) It is obvious that only the purely transverse part A ⊥ of the gauge potential A ′ is physical. In fact, there are various other expressions of the vector potential that are used to describe the A-B effect. All of them are connected to A ⊥ in Eq. (11) by a suitable gauge transformation. Now we turn to the discussion of A-H mechanism in G-L model for superconductors. We begin with the Lagrangian for the relativistic version of G-L model L = − 1 4 F µν F µν + (D µ φ) * (D µ φ) − V (φ),(17) where the complex scalar field φ is the order parameter and D µ = ∂ µ + iqA µ is the covariant derivative with q = −2e being the charge of the Cooper pair. Here the potential V (φ) = α \|φ\| 2 + β 2 \|φ\| 4 . According to Landau phase transition theory, the coefficient β is always positive. For α > 0 the potential has a minimum at \|φ\| = 0 and the system is in the symmetric phase. For α < 0 the potential has a minimum at \|φ\| 2 = − α β . In this case, the vacuum is degenerate and spontaneous symmetry breaking occurs. In the case of symmetry breaking, φ acquires a nonzero vacuum value: φ = φ 0 = − α β .(18) The scalar field can be written as φ (x) = (φ 0 + η (x)) e iθ(x) ,(19) where η (x) and θ (x) are the amplitude and phase fluctuations of the order parameter, respectively, and the latter represents the massless Goldstone mode. When expressed in terms of η and θ, the Lagrangian (17) reads L = − 1 4 F µν F µν + ∂ µ η∂ µ η − V (φ 0 + η) +(φ 0 + η) 2 (∂ µ θ − 2eA µ )(∂ µ θ − 2eA µ ).(20) One can further absorb the Goldstone field θ into A µ by defining a new field A ′ µ : A ′ µ = A µ − 1 2e ∂ µ θ.(21) Then the Lagrangian (20) reads L = − 1 4 F ′ µν F ′µν + ∂ µ η∂ µ η − V (φ 0 + η) + 4e 2 (φ 0 + η) 2 A ′ µ A ′µ ,(22) where F ′ µν = ∂ µ A ′ ν − ∂ ν A ′ µ . In the Lagrangian (22), the Goldstone field does not appear and the original massless gauge field acquires a mass. The degree of freedom of the Goldstone field has transformed into the longitudinal component of the massive gauge field. This is just the A-H mechanism. Now we shall discuss the A-H mechanism from the perspective of the gauge potential decomposition Eq.(1). It is well known that the G-L order parameter for superconductors is represented by the bosonic field φ( x) = ρ( x)e iθ( x) , where the amplitude ρ( x) is meaningful and observable. It can be identified as the density of Cooper pairs. In the following we shall argue that for a bulk superconductor, the gradient of the phase field θ provides the longitudinal component A of the vector potential A, while for a superconducting ring, θ can be decomposed into a nonsingular longitudinal field θ 1 and a singular transverse field θ 2 , the gradient of θ 1 providing the longitudinal component A , and θ 2 being connected with the transverse component A ⊥ which is induced by a constant magnetic flux Φ trapped in the superconducting ring. From the G-L wave function, we can first obtain the electric current density inside the superconductor j = qρ m (h ∇θ − q c A),(23) whereh ∇θ = m V is the usual canonical momentum for the Cooper pair. We now rewrite the current density as j = qρ m (h ∇θ − q c A − q c A ⊥ ).(24) Since j and A ⊥ are gauge invariant quantities, whereas ∇θ and A are not, so for a bulk superconductor one should have the relationh ∇θ = q c A .(25) Eq. (25) shows clearly that in a bulk superconductor the gradient of the phase field provides the longitudinal component of the vector potential. In this sense, this relation just plays the same role as the A-H mechanism does in the previous discussion, i.e., the Goldstone field θ is eaten up and the longitudinal component A appears. Note that owing to Eq. (25) we do not need to adopt Coulomb gauge to eliminate the pure gauge term A in G-L theory. Eq. (25) immediately leads to the famous London equation j = − q 2 ρ mc A ⊥ .(26) Substituting London equation into the Maxwell equation (for static fields) ∇ × B = 4π c j,(27) and using ∇ × ( ∇ × A ⊥ ) = −∇ 2 A ⊥ , we can obtain ∇ 2 A ⊥ − 1 λ 2 A ⊥ = 0,(28) where λ = ( mc 2 4πq 2 ρ ) 1/2 . Eq. (28) leads to the exponentially decaying behavior of the magnetic field in a bulk superconductor and λ is the penetration depth. Therefore, inside a bulk superconductor A ⊥ = 0, B = 0, and j = 0. This is just the Meissner effect. Now let us apply the G-L theory to the case of a multiply connected superconducting ring. We describe the system using cylindrical polar coordinates with z-axis perpendicular to the plane of the ring. We assume that there is a magnetic flux Φ through the ring. Below T c a persistent current will flow around the ring to maintain the constant flux Φ = nΦ 0 in the ring, where Φ 0 = hc 2e is the flux quantum. Due to the A-B effect A ⊥ = 0 inside the superconducting ring, which differs from the case of a bulk superconductor. For a superconducting ring the phase field can be decomposed into two parts: θ = θ 1 + θ 2 , where the longitudinal phase field θ 1 is nonsingular and the transverse phase field θ 2 is singular. θ 1 and θ 2 are independent of each other. Here the nonsingular phase field θ 1 still represents the Goldstone mode and the singular phase field θ 2 = 0 is multi-valued [13]. Such a decomposition of phase field can also be seen in Kosterlitz-Thouless (K-T) transition [14]. In that case, θ 1 is the analytic spin-wave component and θ 2 represents the singular vortex component. Note that K-T transition only occurs in 2D space. For the case of a superconducting ring, similar to Eq. (25), we should havē h ∇θ 1 = q c A ,(29) Eq. (29) leads to j = qρ m (h ∇θ 2 − q c A ⊥ ).(30) Substituting Eq. (30) into Eq. (27), one can obtain ∇ 2 A ⊥ − 1 λ 2 A ⊥ = Φ 0 2πλ 2 ∇θ 2 .(31) The solution to Eq. (31) can be written as A ⊥ = A 1⊥ + A 2⊥ , where A 1⊥ is a solution to the homogeneous equation corresponding to Eq. (31) and A 2⊥ is a specific solution to Eq. (31). For the 1D case it is easy to obtain A 1⊥ = A 1⊥ (0)e − x λ . This is just the Meissner effect. The specific solution A 2⊥ is connected with the A-B effect. If one takes θ 2 = nϕ, where n is an integer (this is due to the single-valuedness condition of the wavefunction), then it can be verified that a specific solution to Eq. (31) is A 2⊥ = − nΦ0 2π ∇ϕ (inside the superconducting ring). Here we have used the equality ∇ 2 ϕ = 0. It is obvious that A 2⊥ is a solution for A-B effect with −nΦ 0 being the flux trapped in the superconducting ring. It gives null magnetic field inside the superconducting ring. The complete solution in this case is the superposition of the Meissner effect solution and A-B effect solution. Therefore, inside a superconducting ring B = 0 and A 1⊥ = 0, whereas A 2⊥ = 0. Note that here we have given a unified description of Meissner effect and A-B effect for the case of a superconducting ring. Owing to Maxwell equation (27), inside the superconducting ring j = 0. From Eq. (30) it follows immediately that inside a superconducting ringh ∇θ 2 = q c A ⊥ .(32) Taking the line integral of both sides of Eq. (32) along a curve Γ around the superconducting ring, one obtains Γ ∇θ 2 · d l = Γ q hc A ⊥ · d l.(33) If one assumes that θ 1 is single valued, then from this one immediately obtains the magnetic flux quantization condition Φ = nΦ 0 by the wave function single-valuedness condition φ(r, ϕ, z) = φ(r, ϕ + 2π, z). Note that in Ref. [10] the authors also gave a discussion on the wave function of the ground state of a superconducting ring, while in our work we have given a unified description of Meissner effect and flux quantization for the case of a superconducting ring from the perspective of gauge potential decomposition. To summarize, in this paper we study the A-B effect and A-H mechanism in G-L model of superconductors from the perspective of the decomposition of U(1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A ⊥ in A-B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ of the order parameter provides the longitudinal component A of the vector potential, which reflects the A-H mechanism, while the transverse part A ⊥ only exists in the surface region, which reflects the Meissner effect. For the case of a superconducting ring, the phase field can be decomposed into two parts: θ = θ 1 + θ 2 , where the longitudinal phase field θ 1 is nonsingular and the transverse phase field θ 2 is multi-valued and singular. The gradient of the phase field θ 1 provides the longitudinal component A , while θ 2 will produce new physical effects, for example, the flux quantization inside a superconducting ring. . Y S Duan, M L Ge, Sci. Sin. 111072Y.S. Duan and M.L. Ge, Sci. Sin. 11, 1072(1979). . Y M Cho, Phys. Rev. Lett. 46302Y.M. Cho, Phys. Rev. Lett. 46, 302 (1981). . L Faddeev, A J Niemi, Phys. Rev. Lett. 821624L. Faddeev and A.J. Niemi, Phys. Rev. Lett. 82, 1624 (1999). . 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[ "Multimodal Graph Transformer for Multimodal Question Answering", "Multimodal Graph Transformer for Multimodal Question Answering", "Multimodal Graph Transformer for Multimodal Question Answering", "Multimodal Graph Transformer for Multimodal Question Answering" ]	[ "Xuehai He \nUC Santa Cruz\nUC Santa Cruz\n\n", "Eric Xin \nUC Santa Cruz\nUC Santa Cruz\n\n", "Wang \nUC Santa Cruz\nUC Santa Cruz\n\n", "Xuehai He \nUC Santa Cruz\nUC Santa Cruz\n\n", "Eric Xin \nUC Santa Cruz\nUC Santa Cruz\n\n", "Wang \nUC Santa Cruz\nUC Santa Cruz\n\n" ]	[ "UC Santa Cruz\nUC Santa Cruz\n", "UC Santa Cruz\nUC Santa Cruz\n", "UC Santa Cruz\nUC Santa Cruz\n", "UC Santa Cruz\nUC Santa Cruz\n", "UC Santa Cruz\nUC Santa Cruz\n", "UC Santa Cruz\nUC Santa Cruz\n" ]	[ "Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics", "Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics" ]	Despite the success of Transformer models in vision and language tasks, they often learn knowledge from enormous data implicitly and cannot utilize structured input data directly. On the other hand, structured learning approaches such as graph neural networks (GNNs) that integrate prior information can barely compete with Transformer models. In this work, we aim to benefit from both worlds and propose a novel Multimodal Graph Transformer for question answering tasks that requires performing reasoning across multiple modalities. We introduce a graph-involved plug-and-play quasi-attention mechanism to incorporate multimodal graph information, acquired from text and visual data, to the vanilla self-attention as effective prior. In particular, we construct the text graph, dense region graph, and semantic graph to generate adjacency matrices, and then compose them with input vision and language features to perform downstream reasoning. Such a way of regularizing self-attention with graph information significantly improves the inferring ability and helps align features from different modalities. We validate the effectiveness of Multimodal Graph Transformer over its Transformer baselines on GQA, VQAv2, and MultiModalQA datasets.	10.48550/arxiv.2305.00581	[ "https://www.aclanthology.org/2023.eacl-main.15.pdf" ]	258,378,175	2305.00581	5f09c0dc0cdbb662be761bd34a87e68e3ca02a57	Multimodal Graph Transformer for Multimodal Question Answering 200 May 2-6, 2023 Xuehai He UC Santa Cruz UC Santa Cruz Eric Xin UC Santa Cruz UC Santa Cruz Wang UC Santa Cruz UC Santa Cruz Multimodal Graph Transformer for Multimodal Question Answering Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics the 17th Conference of the European Chapter of the Association for Computational Linguistics189200 May 2-6, 2023 Despite the success of Transformer models in vision and language tasks, they often learn knowledge from enormous data implicitly and cannot utilize structured input data directly. On the other hand, structured learning approaches such as graph neural networks (GNNs) that integrate prior information can barely compete with Transformer models. In this work, we aim to benefit from both worlds and propose a novel Multimodal Graph Transformer for question answering tasks that requires performing reasoning across multiple modalities. We introduce a graph-involved plug-and-play quasi-attention mechanism to incorporate multimodal graph information, acquired from text and visual data, to the vanilla self-attention as effective prior. In particular, we construct the text graph, dense region graph, and semantic graph to generate adjacency matrices, and then compose them with input vision and language features to perform downstream reasoning. Such a way of regularizing self-attention with graph information significantly improves the inferring ability and helps align features from different modalities. We validate the effectiveness of Multimodal Graph Transformer over its Transformer baselines on GQA, VQAv2, and MultiModalQA datasets. Introduction A myriad of complex real-world tasks require both prior knowledge and reasoning intelligence (Yi et al., 2018a;Ilievski and Feng, 2017). These days, vision-and-language reasoning tasks such as as vision question answering (VQA) (Antol et al., 2015) and multimodal question answering (Multi-ModalQA) (Talmor et al., 2021) post further needs for integrating structured info from different input modalities and thus perform reasoning. Towards this, two questions yield: What is the best way to integrate prior knowledge and reasoning components from multiple modalities in a single model? How would such an integration lead to accurate It takes visual features, text features, and their corresponding generated graphs as inputs. The generated graph is first converted to an adjacency matrix to induce the mask matrix G. The modified quasi-attention score in the Transformer is computed to infer the answer. In the formular, G is the graph-induced matrix constructed by concatenating adjacency matrices both from the vision and the language end.Ĝ is the trainable bias. The input features from different modalities are fused along with graph info to perform downstream reasoning. models, while being more computationally efficient and allowing for significantly more interpretability? Such questions are important to address when scaling reasoning systems to real-world use cases. These years, there are a spectrum of methods in the literature exploring different ways of integrating structured prior information. Graph neural networks (GNNs) , have been widely used in representation learning on graphs. Some experts tried to investigate the embedding of the structured information by resorting to them. However, GNNs are inefficient and they can barely compete with Transformer models. Besides, most GNNs are designed to learn node representations on fixed and homogeneous graphs. Thereby, it is suboptimal to operate GNNs on vision-and-language tasks such as visual question answering (VQA), where graphs encountered in these problems (e.g. scene graphs) can be more complex; Alternatively, knowledge graphs (KGs), such as Freebase (Bollacker et al., 2008), represent world-level factoid information of entities and their relations in a graph-based format, surfaced these years. They have been successfully used in vision and language applications including VQA (Marino et al., 2019). However, they have not been dedicated to be applied to our scenario, more concretely, we aim at filling the gap of capturing prior knowledge in Transformer models. To mitigate deficiencies of the existing methods, this paper proposes a novel plug-and-play graph-involved Transformer-based method for multimodal question answering tasks. Our method is Multimodal Graph Transformer in the sense that it is built upon the well-established Transformer (Vaswani et al., 2017a) backbone, albeit with several key fundamental differences. First, we introduce a systematic scheme to convert text graphs, dense region graphs, and semantic graphs from vision and language tasks to adjacency matrices to use in our method. Second, instead of directly computing the attention score, we learn the newly proposed quasi-attention score with graphinduced adjacency matrices live at its heart, to signify the importance of learning relative importance as a highly effective inductive bias for computing the quasi-attention score. Third, different from previous Transformer methods, where self-attention are fully learned from data, we switch gears to introduce the graph-structured information in the self-attention computation to guide the training of Transformers as shown in Figure 1. The main contributions are summarized below: • We propose a novel Multimodal Graph Transformer learning framework that combines multimodal graph learning from unstructured data with Transformer models. • We introduce a modular plug-and-play graphinvolved quasi-attention mechanism with a trainable bias term to guide the information flow during training. • The effectiveness of the proposed methods is empirically validated on GQA, VQA-v2, and MultiModalQA tasks. 2 Related Works Multimodal question answering Visual Question Answering (VQA) (Antol et al., 2015) has been a prominent topic in the field of multimodal question answering, garnering significant attention and advancing significantly since the introduction of the first large-scale VQA dataset byAntol et al. (2015). To answer VQA questions, models typically leverage variants of attention to obtain a representation of the image that is relevant to the question (Andreas et al., 2016;Yang et al., 2015;Xu and Saenko, 2016;Fukui et al., 2016;Lu et al., 2016). A plethora of works (Liang et al., 2021;Hudson and Manning, 2018;Yi et al., 2018b;Xiong et al., 2016;Kim et al., 2018;Teney et al., 2017a) have attempted to enhance the reasoning capability of VQA models, with Teney et al. (2017a) proposing to improve VQA using structured representations of the scene contents and questions. They developed a deep neural network that leverages the structure in these representations and builds graphs over scene objects and question words. The recent release of MultiModalQA (Talmor et al., 2021), a dataset that demands joint reasoning over texts, tables, and images, has received widespread attention. However, similar to VQA, existing MultiModalQA methods have not fully utilized structured information from the input concepts. To address this, we propose a combination of multimodal graph learning and Transformer models to improve question answering across inputs from multiple different modalities. Attention mechanisms The attention mechanism (Xu et al., 2015a,b;Devlin et al., 2018), has dramatically advanced the field of representation learning in machine learning. The attention mechanism is introduced in Vaswani et al. (2017b) and widely used in language tasks (i.e., abstract summarization (Xu et al., 2020)), machine translation (Bahdanau et al., 2014), reading comprehension (Dai et al., 2020), question answering (Min et al., 2019), etc. proposes using syntax to guide the text modeling by incorporating explicit syntactic constraints into attention mechanisms. Meanwhile, it has seen increasing application in multimodal tasks Nam et al., 2017;Lu et al., 2016), where it is usually used for learning of interactions between multiple inputs. Following their success, multimodal Transformer models Hu et al., 2020;Sun et al., 2019) have also shown impressive results on several vision-and-language tasks. Yun et al. (2019) proposes Graph Transformer Networks (GTNs) that can generate new graph structures and learn effective node representation on the new graphs in an end-to-end fashion. Different from these works, our work incorporates graph information from different modalities into the Transformer to improve the reasoning ability. Exploiting graphs in multimodal reasoning Considering that graph priors can transfer commonalities and mitigate the gap between visual and language domains, researchers explore how to use graphs (Teney et al., 2017b; properly in both tasks. In recent years, many classes of GNNs have been developed for both tasks which are divided into two approaches: spectral (Bruna et al., 2013) and non-spectral methods . Graphs can also be transferred into latent variables by GCN (Yang et al., 2019a;Yao et al., 2018), which can be directly utilized by models. However, the need for aligning graph priors from different modalities to do reasoning limits the use of graph priors. Our work addresses this problem via the graph-involved quasi-attention mechanism. Pretraining Pretrained models in computer vision (Simonyan and Zisserman, 2014;He et al., 2016) and NLP (Devlin et al., 2018;Yang et al., 2019b;, have achieved state-of-the-art performances in many downstream tasks (Thongtan and Phienthrakul, 2019;White et al., 2017;Karpathy and Fei-Fei, 2015;Ren et al., 2015b). Other pretrained models such as VLBERT (Lu et al., 2019;Sun et al., 2019) and ViLT (Kim et al., 2021) also demonstrate their effectiveness on downstream vision-language tasks. Recent works on vision-language pretraining such as OSCAR perform cross-modal alignment in their visual-language pretraining models. Likewise, our proposed method includes cross-modality alignment, which is critical for reasoning. Our proposed modular plug-and-play graph-involved quasiattention mechanism is also model-agnostic and can be also applied to other pretrained Transformerbased vision and language models. Multimodal Graph Transformer Background on Transformers The Transformer layer (Vaswani et al., 2017b) consists of two modules: a multi-head attention and a feed-forward network (FFN). Specifically, each head is represented by four main matrices: the query matrix W q i ∈ R d m ×d q /h , the key matrix W k i ∈ R d m × d k h , the value matrix W v i ∈ R d m × d v h , and the output matrix W o i ∈ R d v h ×d o , and takes the hidden states H ∈ R l×d m of the previous layer as input, where d denotes the dimension of the model, h represents the number of head, and i denotes the index of layer number. The output of attention is given by: Q i , K i , V i = HW q i , HW k i , HW v i (1) Attention (Q i , K i , V i ) = SoftMax Q i K T i d q\|k h V i (2) H i = Attention (Q i , K i , V i ) W o i (3) where Q i ∈ R l× d q h , K i ∈ R l× d k h , V i ∈ R l× d v h are obtained by the linear transformations of W q i , W k i , W v i respectively. Attention(·) is the scaled dot-product attention operation. Then out- put of each head is transformed to H i ∈ R l×d o by W o i . Framework overview The entire framework of the proposed Multimodal Graph Transformer method is depicted in Figure 2. Without loss of generality, we assume the end task is VQA in the following discussion while noting that our framework can be applied to other visionlanguage tasks, such as multimodal question answering. Given the input images and questions, the framework first constructs three graphs, including the semantic graph, dense region graph, and text graph, which will be described in more detail in the following sections. The graph G = (V, E), where V represents the set of nodes in the graph and E represents the edges connecting them, is fed into Transformers to guide the training process. Multimodal graph construction We build three types of graphs and feed them into Transformers: text graph, semantic graph, and dense region graph. We now introduce them in detail. Text graph The task of Visual Question Answering involves a combination of an image, a question, and its corresponding answer. To process the question, we extract the entities and create a text graph representation. We then build the graph G = (V, E) Input Concepts Semantic graph Answer Question What color is the thing under the food left of the little girl with the yellow shirt? Dense region graph MLP … … Features Transformer Figure 2: The figure illustrates the overall framework of our Multimodal Graph Transformer. The input from different modalities are processed and transformed into corresponding graphs, which are then converted into masks and combined with their features to be fed into Transformers for downstream reasoning. In detail, semantic graphs are created through scene graph generation methods, dense region graphs are extracted as densely connected graphs, and text graphs are generated through parsing. Figure 3: The naive demonstration of converting a semantic graph into an adjacency matrix. Cells in blue means '0's for that element in the graph matrix, while white ones means '-inf's. We employ the matrix as the mask when computing the quasi-attention. Masks as shown in the left of Figure 2. The set of nodes, V, represents the entities and the set of edges, E, represents the relationships between the pairs of entities. This results in: • A set of N entities, each represented by a vector of token embeddings, that constitute the nodes of the graph. • A set of pairwise relations between entities, forming the edges of the text graph. The relationship between entities i and j is represented by a vector e ij which encodes the relative relationships. Semantic graph In tasks such as multimodal question answering, there might be additional inputs in the form of tables or lengthy paragraph sentences. To handle these inputs, a linear representation of the table can be created and a semantic graph can be constructed using a similar approach. They are processed using the scene graph parser (Zhong et al., 2021), which transforms the text sentence into a graph of entities and relations, as depicted in Figure 3. The output of the scene graph parser includes: • A set of N words that constitute the nodes of the semantic graph, where N is the number of parsed words in the texts. • A set of possible pairwise relations between words, such as "left" and "on" as shown in Figure 3, which constitute the edges of our graph. An edge between words connecting j to i is represented by e ij , namely, the connectivity is indicated as: e ij = 0, i, j not connected 1, i, j connected . Dense region graph The visual features are extracted by slicing the input images into patches and flattening them. A dense region graph G = (V, E) is then converted into masks, with V being the set of extracted visual features and E being the set of edges connecting each feature node, following the method described in (Kim et al., 2021). This results in a graph that is nearly fully connected. The resulting three graphs are then transformed into adjacency matrices, where the elements are either -∞ or zero. The conversion process is depicted in Figure 3 using the semantic graph as an example. These adjacency matrices are used inside the scaled dot-product attention to control the flow of information, by masking out (setting to −∞) the values. Graph-involved quasi-attention In order to effectively utilize structured graph knowledge in our self-attention computation, we incorporate the graph as an extra constraint in each Figure 4: A naive demonstration of adding the graphinduced mask while computing the quasi-attention when the inputs are from two modalities. The visual mask is the mask converted from the dense region graph and the text mask is converted from the text graph. The crossmodal mask, which is always set as an all-zero matrix, is imposed to encourage the model to learn the crossattention between the image features and text features, thus facilitating the alignment across them. attention head by converting it into an adjacency matrix. The graph matrix, denoted as G, is constructed by combining various masks. An illustration of this process can be seen in Figure 4. The visual mask is generated from the dense region graph, while the text mask is derived from the text graph. Additionally, the cross-modal mask is set to an all-zero matrix to encourage the model to learn the cross-attention between visual and text features, thereby promoting alignment across the different modalities. Within the context of adding graph information, when vision graph mask and text graph mask are concatenated and aligned with image and text features, we believe that a more flexible masking-out mechanism is beneficial, rather than keeping a single constant mask matrix inside the Softmax operation. Drawing insights from , where they include a relative position bias to each head in computing similarity, we also intuitively parameterize a trainable biasĜ and involve it in the training process. Finally, we compute the quasiattention as follows: Attention = SoftMax( Q i K T i d q\|k h + G + λĜ)V i ,(4) where λ is the tradeoff hyper-parameter that controls the contribution ofĜ, and G is our graphinduced matrix constructed by concatenating a graph matrix both from the vision and the language end. Here for clear clarification, we use G andĜ to distinguish the graph matrices fixed and trainable, respectively. During training, G is frozen as before and does not receive gradient updates, whilê G contains trainable parameters. We now introduce the motivation behind adding two types of graph matrices. We perform the masking process by adding G when computing the quasiattention because it can be interpreted as a form of attentional pooling (learning to align), in which each element of G pools all relevant information across all elements of the relative importance matrix computed by Q i K T i d q\|k h . Hence during finetuning, the model ignores redundant features and only focuses on useful information. The mask can also force the model to learn the cross attention between features from the images and questions and perform aligning across them. And the trainable biasĜ captures information gained during the training process. Such information is valuable for fine-tuning, making the Transformer more robust and helping it gain numerical stability. Training The interdependence of output features from various modalities calls for a unified optimization approach for the Transformers in both the visual question answering and multimodal question answering tasks. To accomplish this, we implement a kind of end-to-end training, which ensures the optimality of the models. The final outcome of our models is a classification logit, which is generated by the VQA models that select the best answer from the available candidate answers. To evaluate the accuracy of the models, we compute the cross-entropy loss (Zhang and Sabuncu, 2018) using the output logits produced by the Transformer. This measure helps us determine the difference between the predicted class probabilities and the actual class labels. (Antol et al., 2015) dataset to better balance visual and textual information through the collection of complementary images. Each question in VQA v2 is associated with a pair of similar images with different answers, resulting in a total of 1.1 million QA pairs and 204,000 images. The data split for VQA v2 includes a training set with 83,000 images and 444,000 questions, a validation set with 41,000 images and 214,000 questions, and a test set with 81,000 images and 448,000 questions. The annotated answers are in natural language, but they are commonly converted to a classification task with 3,129 answer classes. Table 1: Accuracy (%) comparison of different methods on the GQA and VQA v2 test-dev. Ours has the second best performance and is comparable to state-of-the-art methods. After applying our proposed quasi-attention mechanism and exploiting the use of graphs, there is also a 2% improvement of overall accuracy on the LXMERT baseline, suggesting the generalization ability of our method. As described by Anderson et al. (2018), the model selects the answer to each question from a set of 3,129 most frequent answers. Following this convention, we fine-tune the multimodal graph transformer model on the VQAv2 training and validation sets, while reserving 1,000 validation images and related questions for internal validation. GQA The GQA dataset contains 22M questions over 113K images. The questions in GQA are designed to require multi-hop reasoning to test the reasoning skills of VQA models. GQA greatly increases the complexity of the semantic structure of questions, leading to a more diverse function set. The real-world images in GQA also bring in a bigger challenge in visual understanding. We conduct experiments on the public splits (Hudson and Manning, 2019a) of the GQA dataset and also treat the task as the classification task reffering to the VQA v2 setting. MultiModalQA MultiModalQA (MMQA) contains 29, 918 questions. We split the dataset into 23,817 training, 2,441 development (dev.), and 3,660 test set examples referring to the official split. Around 60% of the questions in MMQA are compositional. The answer for each question can be a single answer or a list of answers. Baselines We compare with four state-of-the-art VQA models: LXMERT (Tan and Bansal, 2019), NSM (Hudson and Manning, 2019b), OSCAR , and VinVL . • LXMERT (Tan and Bansal, 2019) designs five pretraining tasks: masked language modeling, feature regression, label classification, cross-modal matching, and image question answering to pretrain a large Transformer model. Towards this, a large-scale Transformer (Vaswani et al., 2017b) model is built that consists of three encoders: an object relationship encoder, a language encoder, and a cross-modal encoder. • NSM (Hudson and Manning, 2019b) predicts a probabilistic graph that represents its underlying semantics and performs sequential reasoning over the graph to traversing its nodes to make the inference. • OSCAR uses object tags detected in images as anchor points to significantly ease the learning of alignments, improving previous methods and using self-attention to learn image-text semantic alignments. • VinVL developed a new object detection model to create better visual features of images than previous classical object detection models. We compare with four baselines introduced in the MultiModalQA paper (Talmor et al., 2021): Question-only (Kaushik and Lipton, 2018), Context-only (Kaushik and Lipton, 2018), Au-toRouting, ImplicitDecomp. • Question-only is a sequence-to-sequence model that directly generates the answer given the question. • Context-only first predicts the question type using the classifier and then feed in the relevant context to predict the answer. • AutoRouting first determines the modality where the answer is expected to occur, and then runs the corresponding single-modality module. • ImplicitDecomp is a 2-hop implicit decomposition baseline and so far the state-of-the-art method on the MultiModalQA dataset. Implementation details The input texts undergo preprocessing using a scene graph parser which extracts entities and their relationships. The text features are obtained through a pre-trained BERT tokenizer, allowing us to extract text spans of individual entities and text spans containing two related entities. As for images, we employ the methods described in Dosovitskiy et al. (2020); Kim et al. (2021) to extract visual features and create graph masks. This involves resizing the shorter edge of the input images while preserving the aspect ratio and limiting the longer edge, followed by patch projection and padding for batch training. The resulting patch embeddings are used as inputs along with constructed dense region graph that is densely connected. The Transformer backbone used in this setting is the pretrained VIT-B-32 (Dosovitskiy et al., 2020) version, consisting of 12 layers with a hidden size of H = 768, layer depth of D = 12, patch size of P = 32, a multi-layer perceptron size of 3072, and 12 attention heads. To test this setting, all inputs and graphs are merged and processed by the Transformer backbone, which learns from features from different modalities. MultiModalQA We further investigate the effectiveness of our proposed method on MultiModalQA (Talmor et al., 2021), a recently introduced and demanding task that requires joint reasoning across various modalities such as texts, images, tables, etc. We employ a Multimodal Graph Transformer to tackle the task, using the same approach for extracting vision and text features as in VQA. Additional modalities, such as tables, are encoded by linearizing them and utilizing pre-trained models like RoBERTalarge . After generating text graphs, semantic graphs, and dense region graphs from input questions, text, tables, and images, we feed them along with the extracted features into the Transformer. Unlike the Transformer used in VQA, which takes inputs from two modalities, the Mul-tiModalQA Transformer accepts input from three modalities and performs the final reasoning. Table 1 presents a comparison of the accuracy of our proposed method on the GQA dataset with previous state-of-the-art methods. Our proposed method ranks second in terms of accuracy and outperforms the third best method by a substantial margin, with an absolute improvement of over 3% in overall accuracy. The performance of our method is comparable to the state-of-the-art method. Results and analysis We also conducted experiments on the VQA v2 dataset, and the results are summarized in Table 1 and Table 3. As shown, there are significant improvements over methods without graphs, suggesting that incorporating graph information into the Transformer is effective. Additionally, after incorporating our proposed graph method into LXMERT, we can observe a boost in overall accuracy on the GQA dataset, demonstrating the generalization ability of the proposed method in incorporating graph information into quasi-attention computation. Table 2 compares the Exact Match (EM) and average F1 score of our proposed method on the MultiModalQA dataset with the baseline. The results show that our proposed method outperforms the baseline without the aid of graph information, demonstrating the generalization of our method to more complicated vision-and-language reasoning tasks. Ablation studies We perform ablation studies to verify the necessity of using two-stream inputs with the help of graphs to deal with input from different modalities, with GQA dataset as our testing bed. For all experiments, we use the overall accuracy as the evaluation metric. The results presented in Table 3 show the superiority of our proposed Multimodal Graph Transformer over the method where a single modality input is fed into a Transformer. Our method, which involves dividing the input streams into two separate parts and processing each part through a Transformer, outperforms the Multimodal Transformer without Graph. This demonstrates the beneficial effect of incorporating graph information into the processing of the input data and performing training. The use of different input features with the help of graphs allows for a better alignment of the information from different modalities, which is reflected in the improved performance of our proposed method. Qualitative results One qualitative example is shown in Figure 5. As can be seen, predictions from Multimodal Graph Transformer are more relevant to contents of the input image as the graph information improves the inferring ability of the Transformer, which further indicates the effectiveness of Multimodal Graph Transformer. Conclusions In this paper, we have presented a novel method to integrate structured graph information to guide the Transformers training. Our method can model interactions between different modalities and achieves Figure 5: A qualitative comparison from VQA v2. fresh is the ground truth. Predictions from the Multimodal Graph Transformer (ours) are more relevant to the contents of the input image and achieve a higher confidence score over the ground truth. competitive performance on multimodal reasoning tasks such as VQA and MultiModalQA. Experimental results show that our method outperforms many other methods on the GQA dataset. More importantly, the proposed quasi-attention mechanism is model-agnostic and it is possible to apply it to other Transformer-based methods. We will test our methods on other vision-and-language reasoning tasks and include the comparison with existing graph representation learning methods in our future work. Limitations and Potential Risks The Limitations of the proposed Multimodal Graph Transformer include the potential preservation of fairness and bias issues inherent in the pretrained Transformer models, despite the involvement of graph information. Additionally, the integration of graphs may introduce new biases that can further exacerbate the problem. One potential source of bias is the vision-and-language dataset itself, which may favor majority cases and overlook minority cases. Unfortunately, the proposed method is not equipped to address these biases and issues, making further research and consideration crucial when building upon or directly using this method for vision and language tasks. (2013). The question-answer pairs are created by human annotators who are encouraged to ask "interesting" and "diverse" questions. VQA v2 (Goyal et al., 2017) is extended from the VQA (Antol et al., 2015) dataset to achieve more balance between visual and textual information by collecting complementary images in a way that each question is associated with a pair of similar images with different answers; In the COCO-QA (Ren et al., 2015a) dataset, the question-answer pairs are automatically generated from image captions based on syntactic parsing and linguistic rules; DAQUAR (Malinowski and Fritz, 2014) is built on top of the NYU-Depth V2 dataset (Silberman et al., 2012) which contains RGBD images of indoor scenes. DAQUAR consists of (1) synthetic question-answer pairs that are automatically generated based on textual templates and (2) humancreated question-answer pairs produced by five annotators; CLEVR (Johnson et al., 2017) is a dataset developed on rendered images of spatially related objects (including cube, sphere, and cylinder) with different sizes, materials, and colors. The locations and attributes of objects are annotated for each image. The questions are automatically generated from the annotations; GQA is a new dataset for realworld visual reasoning and compositional question answering, seeking to address key shortcomings of previous VQA datasets. Considering questions in GQA are most objective, unambiguous, compositional, and can be answered by reasoning only on the visual content. We mainly use the GQA dataset in this work as it best fits our goal of reasoning. We also evaluate our methods on the VQA v2 dataset as it is the most common and general VQA dataset so far. Figure 1 : 1Overview of Multimodal Graph Transformer. Figure 6 : 6Examples from the GQA dataset for visual reasoning and compositional question answering. Figure 7 : 7Examples from the VQA v2 dataset for Visual Question Answering. 200 Table 2 2: EM (%) and F1 (%) of Multimodal Graph Transformer and its Transformer baseline on questions in MultiModalQA that require reasoning over multiple modalities. Incorporating graph information into the Multimodal Graph Transformer can boost about 2% F1 and 4% EM performance. Method EM F1 Question-only 16.9 19.5 Context-only 6.6 8.5 AutoRouting 32.0 38.2 ImplicitDecomp 46.5 51.7 Human 84.8 90.1 Multimodal Transformer w/o Graph 50.1 56.4 Multimodal Graph Transformer (Ours) 52.1 57.7 Table 3 : 3Ablation Studies on the GQA and VQA v2 Validation Sets. The figure demonstrates the effectiveness of incorporating graph information into the Transformer architecture through ablation studies performed on the GQA and VQA v2 validation sets. The results of these studies clearly indicate that including graph information can lead to an improvement in performance.Dataset Method Open questions Binary questions Overall accuracy GQA One-modality Transformer 47.7 78.1 62.7 Multimodal Transformer w/o Graph 49.9 81.0 65.4 Ours 60.1 90.2 72.4 VQA v2 One-modality Transformer w/ one Transformer 60.5 85.4 70.1 Multimodal Transformer w/o Graph 64.8 86.3 72.1 Ours 66.7 87.2 74.6 Yash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. 2017. Making the v in vqa matter: Elevating the role of image understanding in visual question answering. In CVPR. He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep residual learning for image recognition. In CVPR. Ronghang Hu, Amanpreet Singh, Trevor Darrell, and Marcus Rohrbach. 2020. Iterative answer prediction with pointer-augmented multimodal transformers for textvqa. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9992-10002. 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[ "Universal Instance Perception as Object Discovery and Retrieval", "Universal Instance Perception as Object Discovery and Retrieval" ]	[ "Bin Yan \nSchool of Information and Communication Engineering\nDalian University of Technology\nChina\n", "Yi Jiang ", "Jiannan Wu \nThe University of Hong\nKong\n", "Dong Wang \nSchool of Information and Communication Engineering\nDalian University of Technology\nChina\n", "Ping Luo \nThe University of Hong\nKong\n", "Zehuan Yuan ", "Huchuan Lu \nSchool of Information and Communication Engineering\nDalian University of Technology\nChina\n\nPeng Cheng Laboratory\n\n" ]	[ "School of Information and Communication Engineering\nDalian University of Technology\nChina", "The University of Hong\nKong", "School of Information and Communication Engineering\nDalian University of Technology\nChina", "The University of Hong\nKong", "School of Information and Communication Engineering\nDalian University of Technology\nChina", "Peng Cheng Laboratory\n" ]	[]	All instance perception tasks aim at finding certain objects specified by some queries such as category names, language expressions, and target annotations, but this complete field has been split into multiple independent subtasks. In this work, we present a universal instance perception model of the next generation, termed UNINEXT. UNINEXT reformulates diverse instance perception tasks into a unified object discovery and retrieval paradigm and can flexibly perceive different types of objects by simply changing the input prompts. This unified formulation brings the following benefits: (1) enormous data from different tasks and label vocabularies can be exploited for jointly training general instance-level representations, which is especially beneficial for tasks lacking in training data. (2) the unified model is parameter-efficient and can save redundant computation when handling multiple tasks simultaneously. UNINEXT shows superior performance on 20 challenging benchmarks from 10 instance-level tasks including classical image-level tasks (object detection and instance segmentation), vision-and-language tasks (referring expression comprehension and segmentation), and six video-level object tracking tasks. Code is available at https://github.com/MasterBin-IIAU/UNINEXT.	10.48550/arxiv.2303.06674	[ "https://export.arxiv.org/pdf/2303.06674v1.pdf" ]	257,496,231	2303.06674	9337e3e508408c84a27fb4628328a8efe56cb5e4	Universal Instance Perception as Object Discovery and Retrieval Bin Yan School of Information and Communication Engineering Dalian University of Technology China Yi Jiang Jiannan Wu The University of Hong Kong Dong Wang School of Information and Communication Engineering Dalian University of Technology China Ping Luo The University of Hong Kong Zehuan Yuan Huchuan Lu School of Information and Communication Engineering Dalian University of Technology China Peng Cheng Laboratory Universal Instance Perception as Object Discovery and Retrieval All instance perception tasks aim at finding certain objects specified by some queries such as category names, language expressions, and target annotations, but this complete field has been split into multiple independent subtasks. In this work, we present a universal instance perception model of the next generation, termed UNINEXT. UNINEXT reformulates diverse instance perception tasks into a unified object discovery and retrieval paradigm and can flexibly perceive different types of objects by simply changing the input prompts. This unified formulation brings the following benefits: (1) enormous data from different tasks and label vocabularies can be exploited for jointly training general instance-level representations, which is especially beneficial for tasks lacking in training data. (2) the unified model is parameter-efficient and can save redundant computation when handling multiple tasks simultaneously. UNINEXT shows superior performance on 20 challenging benchmarks from 10 instance-level tasks including classical image-level tasks (object detection and instance segmentation), vision-and-language tasks (referring expression comprehension and segmentation), and six video-level object tracking tasks. Code is available at https://github.com/MasterBin-IIAU/UNINEXT. Introduction Object-centric understanding is one of the most essential and challenging problems in computer vision. Over the years, the diversity of this field increases substantially. In this work, we mainly discuss 10 sub-tasks, distributed on the vertices of the cube shown in Figure 1. As the most fundamental tasks, object detection [8,9,32,61,85,87,95] and instance segmentation [6,39,66,94,101] require finding all objects of specific categories by boxes and masks respectively. Extending inputs from static images to dynamic * This work was performed while Bin Yan worked as an intern at ByteDance. Email: yan [email protected]. † Corresponding authors: [email protected], [email protected]. videos, Multiple Object Tracking (MOT) [3,77,129,132], Multi-Object Tracking and Segmentation (MOTS) [49,97,112], and Video Instance Segmentation (VIS) [45,104,107,116] require finding all object trajectories of specific categories in videos. Except for category names, some tasks provide other reference information. For example, Referring Expression Comprehension (REC) [119,125,134], Referring Expression Segmentation (RES) [120,123,125], and Referring Video Object Segmentation (R-VOS) [7,90,108] aim at finding objects matched with the given language expressions like "The fourth person from the left". Besides, Single Object Tracking (SOT) [5,53,110] and Video Object Segmentation (VOS) [18,81,111] take the target annotations (boxes or masks) given in the first frame as the reference, requiring to predict the trajectories of the tracked objects in the subsequent frames. Since all the above tasks aim to perceive instances of certain properties, we refer to them collectively as instance perception. Although bringing convenience to specific applications, such diverse task definitions split the whole field into fragmented pieces. As the result, most current instance perception methods are developed for only a single or a part of sub-tasks and trained on data from specific domains. Such fragmented design philosophy brings the following drawbacks: (1) Independent designs hinder models from learning and sharing generic knowledge between different tasks and domains, causing redundant parameters. (2) The possibility of mutual collaboration between different tasks is overlooked. For example, object detection data enables models to recognize common objects, which can naturally improve the performance of REC and RES. (3) Restricted by fixed-size classifiers, traditional object detectors are hard to jointly train on multiple datasets with different label vocabularies [38,63,92] and to dynamically change object categories to detect during inference [23,63,77,84,116,124]. Since essentially all instance perception tasks aim at finding certain objects according to some queries, it leads to a natural question: could we design a unified model to solve all mainstream instance perception tasks once and for all? To answer this question, we propose UNINEXT, a universal instance perception model of the next generation. We first reorganize 10 instance perception tasks into three types according to the different input prompts: (1) category names as prompts (Object Detection, Instance Segmentation, VIS, MOT, MOTS). (2) language expressions as prompts (REC, RES, R-VOS). (3) reference annotations as prompts (SOT, VOS). Then we propose a unified prompt-guided object discovery and retrieval formulation to solve all the above tasks. Specifically, UNINEXT first discovers N object proposals under the guidance of the prompts, then retrieves the final instances from the proposals according to the instance-prompt matching scores. Based on this new formulation, UNINEXT can flexibly perceive different instances by simply changing the input prompts. To deal with different prompt modalities, we adopt a prompt generation module, which consists of a reference text encoder and a reference visual encoder. Then an early fusion module is used to enhance the raw visual features of the current image and the prompt embeddings. This operation enables deep information exchange and provides highly discriminative representations for the later instance prediction step. Considering the flexible query-toinstance fashion, we choose a Transformer-based object detector [135] as the instance decoder. Specifically, the decoder first generates N instance proposals, then the prompt is used to retrieve matched objects from these proposals. This flexible retrieval mechanism overcomes the disadvantages of traditional fixed-size classifiers and enables joint training on data from different tasks and domains. With the unified model architecture, UNINEXT can learn strong generic representations on massive data from various tasks and solve 10 instance-level perception tasks using a single model with the same model parameters. Extensive experiments demonstrate that UNINEXT achieves superior performance on 20 challenging benchmarks. The contributions of our work can be summarized as follows. • We propose a unified prompt-guided formulation for universal instance perception, reuniting previously fragmented instance-level sub-tasks into a whole. • Benefiting from the flexible object discovery and retrieval paradigm, UNINEXT can train on different tasks and domains, in no need of task-specific heads. • UNINEXT achieves superior performance on 20 challenging benchmarks from 10 instance perception tasks using a single model with the same model parameters. Related Work Instance Perception. The goals and typical methods of 10 instance perception tasks are introduced as follows. Retrieval by Category Names. Object detection and instance segmentation aim at finding all objects of specific classes on the images in the format of boxes or masks. Early object detectors can be mainly divided into two-stage methods [8,12,87] and one-stage methods [35,62,85,95,127] according to whether to use RoI-level operations [37,39]. Recently, Transformer-based detectors [9,55,135] have drawn great attention for their conceptually simple and flexible frameworks. Besides, instance segmentation approaches can also be divided into detector-based [8,12,39,51,94] and detector-free [16,101] fashions according to whether box-level detectors are needed. Object detection and instance segmentation play critical roles and are foundations for all other instance perception tasks. For example, MOT, MOTS, and VIS extend image-level detection and segmentation to videos, requiring finding all object trajectories of specific classes in videos. Mainstream algorithms [49,82,105,106,113,128] of MOT and MOTS follow an online "detection-then-association" paradigm. However, due to the intrinsic difference in benchmarks of MOTS [97,124] (high-resolution long videos) and VIS [116] (low-resolution short videos), most recent VIS methods [45,60,104,107] adopt an offline fashion. This strategy performs well on relatively simple VIS2019 [116], but the performance drops drastically on challenging OVIS [84] benchmark. Recently, IDOL [109] bridges the performance gap between online fashion and its offline counterparts by discriminative instance embeddings, showing the potential of the online paradigm in unifying MOT, MOTS, and VIS. Retrieval by Language Expressions. REC, RES, and R-VOS aim at finding one specific target referred by a language expression using boxes or masks on the given images or videos. Similar to object detection, REC methods can be categorized into three paradigms: two-stage [42,64,67,117], one-stage [59,72,118,119], and Transformerbased [24,47,133] ones. Different from REC, RES approaches [11,26,33,43,46,71,123] focus more on designing diverse attention mechanisms to achieve vision-language alignment. Recently, SeqTR [134] unifies REC and RES as a point prediction problem and obtains promising results. Finally, R-VOS can be seen as a natural extension of RES from images to videos. Current state-of-the-art methods [7,108] are Transformer-based and process the whole video in an offline fashion. However, the offline paradigm hinders the applications in the real world such as long videos and ongoing videos (e.g. autonomous driving). Retrieval by Reference Annotations. SOT and VOS first specify tracked objects on the first frame of a video using boxes or masks, then require algorithms to predict the trajectories of the tracked objects in boxes or masks respectively. The core problems of these two tasks include (1) How to extract informative target features? (2) How to fuse the target information with representations of the current frame? For the first question, most SOT methods [5,14,52,53,114] encode target information by passing a template to a siamese backbone. While VOS approaches [18,81,121] usually pass multiple previous frames together with corresponding mask results to a memory encoder for extracting fine-grained target information. For the second question, correlations are widely adopted by early SOT algorithms [5,53,115]. However, these simple linear operations may cause serious information loss. To alleviate this problem, later works [14,19,114,122] resort to Transformer for more discriminative representations. Besides, feature fusion in VOS is almost dominated by space-time memory networks [17,18,81,121]. Unified Vision Models. Recently, unified vision models [13,16,36,39,56,70,86,91,99,113,136] have drawn great attention and achieved significant progress due to their strong generalizability and flexibility. Unified vision models attempt to solve multiple vision or multi-modal tasks by a single model. Existing works can be categorized into unified learning paradigms and unified model architectures. Unified Learning Paradigms. These works [2,36,70,86,91,99,136] usually present a universal learning paradigm for covering as many tasks and modalities as possible. For example, MuST [36] presents a multi-task self-training approach for 6 vision tasks. INTERN [91] introduces a continuous learning scheme, showing strong generalization ability on 26 popular benchmarks. Unified-IO [70] and OFA [99] proposes a unified sequence-to-sequence framework that can handle a variety of vision, language, and multi-modal tasks. Although these works can perform many tasks, the commonality and inner relationship among different tasks are less explored and exploited. Unified Model Architectures. These works [13,16,39,56,113] usually design a unified formulation or model architecture for a group of closely related tasks. For example, Mask R-CNN [39] proposes a unified network to perform object detection and instance segmentation simultaneously. Mask2Former [16] presents a universal architecture capable of handling panoptic, instance, and seman-tic segmentation. Pix2SeqV2 [13] designs a unified pixelto-sequence interface for four vision tasks, namely object detection, instance segmentation, keypoint detection, and image captioning. GLIP [56] cleverly reformulates object detection as phrase grounding by replacing classical classification with word-region alignment. This new formulation allows joint training on both detection and grounding data, showing strong transferability to various objectlevel recognition tasks. However, GLIP [56] supports neither prompts in other modalities such as images & annotations nor video-level tracking tasks. In terms of object tracking, Unicorn [113] proposes a unified solution for SOT, VOS, MOT, and MOTS, achieving superior performance on 8 benchmarks with the same model weights. However, it is still difficult for Unicorn to handle diverse label vocabularies [23,63,77,84,116,124] during training and inference. In this work, we propose a universal prompt-guided architecture for 10 instance perception tasks, conquering the drawbacks of GLIP [56] and Unicorn [113] simultaneously. Approach Before introducing detailed methods, we first categorize existing instance perception tasks into three classes. • Object detection, instance segmentation, MOT, MOTS, and VIS take category names as prompts to find all instances of specific classes. • REC, RES, and R-VOS exploit an expression as the prompt to localize a certain target. • SOT and VOS use the annotation given in the first frame as the prompt for predicting the trajectories of the tracked target. Essentially, all the above tasks aim to find objects specified by some prompts. This commonality motivates us to reformulate all instance perception tasks into a prompt-guided object discovery and retrieval problem and solve it by a unified model architecture and learning paradigm. As demonstrated in Figure 2, UNINEXT consists of three main components: (1) prompt generation (2) image-prompt feature fusion (3) object discovery and retrieval. Prompt Generation First, a prompt generation module is adopted to transform the original diverse prompt inputs into a unified form. According to different modalities, we introduce the corresponding strategies in the next two paragraphs respectively. To deal with language-related prompts, a language encoder [25] Enc L is adopted. To be specific, for categoryguided tasks, we concatenate class names that appeared in the current dataset [63,84,116,124] as the language expression. Take COCO [63] as an example, the expression can be written as "person. bicycle. ... . toothbrush". Then for both category-guided and expression-guided tasks, the language expression is passed into Enc L , getting a prompt embedding F p ∈ R L×d with a sequence length of L. For the annotation-guided tasks, to extract fine-grained visual features and fully exploit the target annotations, an additional reference visual encoder Enc ref V is introduced. Specifically, first a template with 2 2 times the target box area is cropped centered on the target location on the reference frame. Then the template is resized to a fixed size of 256 × 256. To introduce more precise target information, an extra channel named the target prior is concatenated to the template image, forming a 4-channel input. In more detail, the value of the target prior is 1 on the target region otherwise 0. Then the template image together with the target prior is passed to the reference visual encoder Enc ref V , obtaining a hierarchical feature pyramid {C 3 , C 4 , C 5 , C 6 }. The corresponding spatial sizes are 32 × 32, 16 × 16, 8 × 8, and 4×4. To keep fine target information and get the prompt embedding in the same format as other tasks, a merging module is applied. Namely, all levels of features are first upsampled to 32 × 32 then added, and flattened as the final prompt embedding F p ∈ R 1024×d . The prompt generation process can be formulated as F p =      Enc ref L (expression) expression-guided Enc ref L (concat(categories)) category-guided merge(Enc ref V ([template, prior]) annotation-guided Image-Prompt Feature Fusion In parallel with the prompt generation, the whole current image is passed through another visual encoder Enc V , obtaining hierarchical visual features F v . To enhance the original prompt embedding by the image contexts and to make the original visual features prompt-aware, an early fusion module is adopted. To be specific, first a bi-directional cross-attention module (Bi-XAtt) is used to retrieve information from different inputs, and then the retrieved representations are added to the original features. This process can be formulated as F p2v , F v2p = Bi-XAtt(F v , F p ) F v = F v + F p2v ; F p = F p + F v2p(1) Different from GLIP [56], which adopts 6 vision-language fusion layers and 6 additional BERT layers for feature enhancement, our early fusion module is much more efficient. Object Discovery and Retrieval With discriminative visual and prompt representations, the next crucial step is to transform input features into instances for various perception tasks. UNINEXT adopts the encoder-decoder architecture proposed by Deformable DETR [135] for its flexible query-to-instance fashion. We introduce the detailed architectures as follows. The Transformer encoder takes hierarchical promptaware visual features as the inputs. With the help of efficient Multi-scale Deformable Self-Attention [135], target information from different scales can be fully exchanged, bringing stronger instance features for the subsequent instance decoding. Besides, as performed in two-stage Deformable DETR [135], an auxiliary prediction head is appended at the end of the encoder, generating N initial reference points with the highest scores as the inputs of the decoder. The Transformer decoder takes the enhanced multi-scale features, N reference points from the encoder, as well as N object queries as the inputs. As shown in previous works [76,104,108,126], object queries play a critical role in instance perception tasks. In this work, we attempt two query generation strategies: (1) static queries which do not change with images or prompts. (2) dynamic queries conditioned on the prompts. The first strategy can be easily implemented with nn.Embedding(N,d). The second one can be performed by first pooling the enhanced prompt features F v along the sequence dimension, getting a global representation, then repeating it by N times. The above two methods are compared in Sec 4.3 and we find that static queries usually perform better than dynamic queries. The potential reason could be that static queries contain richer information and possess better training stability than dynamic queries. With the help of the deformable attention, the object queries can efficiently retrieve prompt-aware visual features and learn strong instance embedding F ins ∈ R N ×d . At the end of the decoder, a group of prediction heads is exploited to obtain the final instance predictions. Specifically, an instance head produces both boxes and masks of the targets. Besides, an embedding head [109] is introduced for associating the current detected results with previous trajectories in MOT, MOTS, and VIS. Until now, we have mined N potential instance proposals, which are represented with gray masks in Figure 2. However, not all proposals are what the prompts really refer to. Therefore, we need to further retrieve truly matched objects from these proposals according to the prompt embeddings as demonstrated in the right half of Figure 2. Specifically, given the prompt embeddings F p after early fusion, for categoryguided tasks, we take the embedding of each category name as a weight matrix W ∈ R 1×d . Besides, for expressionguided and annotation-guided tasks, the weight matrix W is obtained by aggregating the prompt embedding F p using global average pooling (GAP) along the sequence dimension. W =        F p [i], i ∈ {0, 1, ..., C − 1} category 1 L L i=0 F p (i, j) expression/annotation Finally, the instance-prompt matching scores S can be computed as the matrix multiplication of the target features and the transposed weight matrix. S = F ins W . Following previous work [56], the matching scores can be supervised by Focal Loss [62]. Different from previous fixed-size classifiers [135], the proposed retrieval head selects objects by the prompt-instance matching mechanism. This flexible design enables UNINEXT to jointly train on enormous datasets with diverse label vocabularies from different tasks, learning universal instance representations. Training and Inference Training. The whole training process consists of three consecutive stages: (1) general perception pretraining (2) image-level joint training (3) video-level joint training. In the first stage, we pretrain UNINEXT on the largescale object detection dataset Objects365 [92] for learning universal knowledge about objects. Since Objects365 does not have mask annotations, we introduce two auxiliary losses proposed by BoxInst [96] for training the mask branch. The loss function can be formulated as L stage1 = L retrieve + L box + L boxinst mask (2) Then based on the pretrained weights of the first stage, we finetune UNINEXT jointly on image datasets, namely COCO [63] and the mixed dataset of RefCOCO [125], Re-fCOCO+ [125], and RefCOCOg [80]. With manually labeled mask annotations, the traditional loss functions like Dice Loss [78] and Focal Loss [62] can be used for the mask learning. After this step, UNINEXT can achieve superior performance on object detection, instance segmentation, REC, and RES. L stage2 = L retrieve + L box + L mask(3) Finally, we further finetune UNINEXT on video-level datasets for various downstream object tracking tasks and benchmarks. In this stage, the model is trained on two frames randomly chosen from the original videos. Besides, to avoid the model forgetting previously learned knowledge on image-level tasks, we also transform image-level datasets to pseudo videos for joint training with other video datasets. In summary, the training data in the third stage includes pseudo videos generated from COCO [63], RefCOCO/g/+ [80,125,125], SOT&VOS datasets (GOT-10K [44], LaSOT [30], Track-ingNet [79], and Youtube-VOS [111]), MOT&VIS datasets (BDD100K [124], VIS19 [116], OVIS [84]), and R-VOS dataset Ref-Youtube-VOS [90]. Meanwhile, a reference visual encoder for SOT&VOS and an extra embedding head for association are introduced and optimized in this period. L stage3 = L retrieve + L box + L mask + L embed (4) Inference. For category-guided tasks, UNINEXT predicts instances of different categories and associates them with previous trajectories. The association proceeds in an online fashion and is purely based on the learned instance embedding following [82,109]. For expression-guided and annotation-guided tasks, we directly pick the object with the highest matching score with the given prompt as the final result. Different from previous works [98,108] restricted by the offline fashion or complex post-processing, our method is simple, online, and post-processing free. Experiments Implementation Details We attempt three different backbones, ResNet-50 [40], ConvNeXt-Large [68], and ViT-Huge [28] as the visual encoder. We adopt BERT [25] as the text encoder and its parameters are trained in the first and second training stages while being frozen in the last training stage. The Transformer encoder-decoder architecture follows [135] with 6 encoder layers and 6 decoder layers. The number of object queries N is set to 900. The optimizer is AdamW [69] with weight decay of 0.05. The model is trained on 32 and 16 A100 GPUs for Objects365 pretraining and other stages respectively. More details can be found in the appendix. Evaluations on 10 Tasks We compare UNINEXT with task-specific counterparts in 20 datasets. In each benchmark, the best two results are indicated in bold and with underline. UNINEXT in all benchmarks uses the same model parameters. Object Detection and Instance Segmentation. We compare UNINEXT with state-of-the-art object detection and instance segmentation methods on COCO val2017 (5k images) and test-dev split (20k images) respectively. As shown in Table 1 [125], RefCOCO+ [125], and RefCOCOg [73] are three representative benchmarks for REC and RES proposed by different institutions. Following previous literature, we adopt [email protected] and overall IoU (oIoU) as the evaluation metrics for REC and RES respectively and results are rounded to two decimal places. As shown in Table 3 and Table 4, our method with ResNet-50 backbone surpasses all previous approaches on all splits. Furthermore, when using ConvNeXt-Large and ViT-Huge backbones, UNINEXT obtains new state-of-theart results, exceeding the previous best method by a large margin. Especially on RES, UNINEXT-H outperforms LAVT [120] by 10.85 on average. SOT. We compare UNINEXT with state-of-the-art SOT methods on four large-scale benchmarks: LaSOT [30], LaSOT-ext [29], TrackingNet [79], and TNL-2K [102]. These benchmarks adopt the area under the success curve (AUC), normalized precision (P N orm ), and precision (P) as the evaluation metrics and include 280, 150, 511, and 700 videos in the test set respectively. As shown in Table 5, UNINEXT achieves the best results in terms of AUC and P among all trackers with ResNet-50 backbone. Especially on TNL-2K, UNINEXT outperforms the second best method TransT [14] by 5.3 AUC and 5.8 P respectively. Besides, UNINEXT with stronger backbones obtains the best AUC on all four benchmarks, exceeding Unicorn [113] with the same backbone by 3.9 on LaSOT. Table 6. DAVIS-2017 [83] adopts region similarity J , contour accuracy F, and the averaged score J &F as the metrics. Similarly, Youtube-VOS 2018 [111] reports J and F for both seen and unseen categories, and the averaged overall score G. UNINEXT achieves the best results among all non-memory-based methods, largely bridging the performance gap between non-memory-based approaches and memory-based ones. Furthermore, compared with traditional memory-based methods [18,81], UNINEXT does not rely on the intermediate mask predictions. This leads to constant memory consumption, enabling UNINEXT to handle long sequences of any length. MOT. We compare UNINEXT with state-of-the-art MOT methods on BDD100K [124], which requires tracking 8 classes of instances in the autonomous driving scenario. Except for classical evaluation metrics Multiple-Object Tracking Accuracy (MOTA), Identity F1 Score (IDF1), and Identity Switches (IDS), BDD100K additionally introduces mMOTA, and mIDF1 to evaluate the average performance across 8 classes. As shown in Table 7, UNINEXT surpasses Unicorn [113] by 3.0 mMOTA and 2.7 mIDF1 respectively. MOTS. Similar to MOT, BDD100K MOTS Chal- lenge [124] evaluates the performance on multi-class tracking by mMOTSA, mMOTSP, mIDF1, and ID Sw. This benchmark contains 37 sequences with mask annotations in the validation set. As shown in Table 8, UNINEXT achieves state-of-the-art performance, surpassing the previous best method Unicorn [113] by 6.1 mMOTSA. VIS. We compare UNINEXT against state-of-the-art VIS methods on Youtube-VIS 2019 [116] and OVIS [84] validation sets. Specifically, Youtube-VIS 2019 and OVIS have 40 and 25 object categories, containing 302 and 140 videos respectively in the validation set. Both benchmarks take AP as the main metric. As shown in Table 9, when using the same ResNet-50 backbone, UNINEXT obtains the best results on both datasets. Especially on more challenging OVIS, UNINEXT exceeds the previous best method IDOL [109] original Youtube-VOS [111] and DAVIS17 [83] datasets. As same as semi-supervised VOS, region similarity J , contour accuracy F, and the averaged score J &F are adopted as the metrics. As demonstrated in Table 10 Ablations and Other Analysis In this section, we conduct component-wise analysis for better understanding our method. All models take ResNet-50 as the backbone. The methods are evaluated on five benchmarks (COCO [63], RefCOCO [125], Youtube-VOS [111], Ref-Youtube-VOS [90], and Youtube-VIS 2019 [116]) from five tasks (object detection, REC, VOS, R-VOS, and VIS). The results are shown in Table 11. Fusion. To study the effect of feature fusion between visual features and prompt embeddings, we implement a variant without any early fusion. In this version, prompt embeddings do not have an influence on proposal generation but are only used in the final object retrieval process. Experiments show that early fusion has the greatest impact on VOS, the performance on VOS drops dras- tically by 21.4 J &F without feature fusion. This is mainly caused by the following reasons (1) Without the guidance of prompt embeddings, the network can hardly find rare referred targets like trees and sinks. (2) Without early fusion, the network cannot fully exploit fine mask annotations in the first frame, causing degradation of the mask quality. Besides, the removal of feature fusion also causes performance drop of 2.3 [email protected] and 2.8 J &Fon REC and RVOS respectively, showing the importance of early fusion in expression-guided tasks. Finally, feature fusion has minimum influence on object detection and VIS. This can be understood because both two tasks aim to find all objects as completely as possible rather than locating one specific target referred by the prompt. Queries. We compare two different query generation strategies: static queries by nn.Embedding(N, d) and dynamic queries conditioned on the prompt embeddings. Experiments show that dynamic queries perform slightly better than static queries on the first four tasks. However, static queries outperform dynamic ones by 2.8 AP on the VIS task, obtaining higher overall performance. A potential reason is that N different object queries can encode richer inner relationship among different targets than simply copying the pooled prompt by N times as queries. This is especially important for VIS because targets need to be associated according to their affinity in appearance and space. Unification. We also compare two different model design philosophies, one unified model or multiple taskspecific models. Except for the unified model, we also retrain five task-specific models only on data from corresponding tasks. Experiments show that the unified model achieves significantly better performance than its taskspecific counterparts on five tasks, demonstrating the superiority of the unified formulation and joint training on all instance perception tasks. Finally, the unified model can save tons of parameters, being much more parameter-efficient. Conclusions We propose UNINEXT, a universal instance perception model of the next generation. For the first time, UNINEXT unifies 10 instance perception tasks with a prompt-guided object discovery and retrieval paradigm. Extensive experiments demonstrate that UNINEXT achieves superior performance on 20 challenging benchmarks with a single model with the same model parameters. We hope that UNINEXT can serve as a solid baseline for the research of instance perception in the future. A. Appendix In this appendix, we present more details about the training process and loss functions in A.1 and A.2, network architecture in A.3, as well as more analysis and visualizations for better understanding in A.4. A.1. Training Process The detailed hyperparameters during training are shown in Tab 12. The whole training process consists of three stages. In each stage, the StepLR learning rate scheduler is adopted. The learning rate drops by a factor of 10 after the given steps. For multi-dataset training, we follow the implementation of Detic [131], which randomly samples data from different tasks and then computes them on different GPUs in one iteration. Besides, the multi-scale training technique is used across all datasets in all stages. Take the pre-training on Objects365 [92] as an example, the original images are resized such that the shortest side is at least 480 and at most 800 pixels while the longest side is at most 1333. We use this as the default setting except on Youtube-VOS [111], Youtube-VIS-2019 [116], and Ref-Youtube-VOS [90]. A lower resolution with the shortest side ranging from 320 to 640 and the longest side not exceeding 768 is applied to these datasets [90,111,116], following previous works [18,108,109]. Specifically, in the first stage, the model is pretrained on Objects365 [92] for about 340K iterations (12 epochs) and the learning rate drops on the 11th epoch. In the second stage, we finetune UNINEXT on COCO [63] and RefCOCO/g/+ [80,125] jointly for 12 epochs. In the third stage, UNINEXT is further finetuned for diverse video-level tasks. To guarantee balanced performance on various benchmarks, we set the data sampling ratios as (SOT&VOS):(MOT&MOTS):VIS:R-VOS = 1:1:1:1. For each task, 45K iterations are allocated, thus bringing 180K iterations in total for the third stage. Besides, to avoid forgetting previously learned knowledge on image-level tasks, we also generate pseudo videos from COCO [63] and Re-fCOCO/g/+ [80,125] and mix them with training data of VIS [84,116] and R-VOS [90] respectively. A.2. Loss Functions We present detailed loss functions described in Sec. 3.4 for better readability. First, L retrieve and L box are used across all three stages. Second, to learn mask representations from coarse boxes [92] and fine mask annotations [63,90,111,116,125], UNINEXT uses L boxinst mask in the first stage and L mask in the next two stages respectively. Finally, to associate instances on different frames [84,116,124], UNINEXT additionally adopts L embed in the last stage. L retrieve . Given the raw instance-prompt matching score s, the normalized matching probability p is computed as p = σ(s), where σ is sigmoid function. Then L retrieve can be written as the form of Focal loss [62]. L retrieve (p t ) = −α t (1 − p t ) γ log(p t ).(5)p t = p if matched 1 − p otherwise.(6) γ and α are 2 and 0.25 respectively. L box . Following DETR-like methods [9,135], L box consists of two terms, GIoU Loss [88] and 1 loss: L box (b,b) = λ giou L giou (b,b) + λ L1 b −b .(7)L giou (b,b) = 1 − IoU (b,b) + A c (b,b) − U (b,b) A c (b,b) ,(8) where A c (b,b) is the area of the smallest box containing b andb. U (b,b) is the area of the union of b andb. L mask . For datasets with mask annotations [63,90,111,116,125], Focal Loss [62] and Dice Loss [78] are adopted. L mask (m,m) = λ f ocal L focal (m,m) + λ dice L dice (m,m). (9) L dice (m,m) = 1 − 2mm + 1 m + m + 1 ,(10) where m andm are binary GT masks and predicted masks after sigmoid activation respectively. L boxinst mask . For Objects365 [92] without mask annotations, UNINEXT uses Projection Loss and Pairwise Affinity Loss like BoxInst [96], which can learn mask prediction only based on box-level annotations. S e = S(c i,j , c l,k ) = exp − \|\|c i,j − c l,k \|\| θ ,(15) where y e = 1 means the two pixels have the same groundtruth label. S e is the color similarity of the edge e. c i,j and c l,k are respectively the LAB color vectors of the two pixels (i, j) and (l, k) linked by the edge. θ is 2 in this work. L embed . UNINEXT uses contrastive loss [109] to train discriminative embeddings for associating instances on different frames. (16) where k + and k − are positive and negative feature embeddings from the reference frame. For each instance in the key frame, v is the feature embedding with the lowest cost. L embed = log[1 + k + k − exp(v · k − − v · k + )], A.3. Network Architecture To transform the enhanced visual features F v and prompt features F p into the final instance predictions, an encoderdecoder Transformer architecture is adopted. Based on the original architecture in two-stage Deformable DETR [135], UNINEXT makes the following improvements: • Introducing a mask head for segmentation. To predict high-quality masks, UNINEXT introduces a mask head [94] based on dynamic convolutions. Specifically, first an MLP is used to transform instance embeddings into a group of parameters ω. Then these parameters are used to perform three-layer 1×1 convolu-tions with feature maps, obtaining masks of instances. • Replacing one-to-one Hungarian matching with one-to-many SimOTA [35]. Traditional Hungarian matching forces one GT to be only assigned to one query, leaving most of the queries negative. UNINEXT uses SimOTA [35], which enables multiple queries to be matched with one GT. This strategy can provide more positive samples and speed up convergence. During inference, UNINEXT uses NMS to remove duplicated predictions. • Adding an IoU branch. UNINEXT adds an IoU branch to reflect the quality of the predicted boxes. During training, IoU does not affect the label assignment. During inference, the final scores are the geometric mean of the instance-prompt matching scores (after sigmoid) and the IoU scores. • Adding some techniques in DINO [126]. To further improve the performance, UNINEXT introduces some techniques [126], including contrastive DN, mixed query selection, and look forward twice. A.4. Analysis and Visualizations Analysis. We compare UNINEXT with other competitive counterparts, which can handle multiple instance-level perception tasks. The opponents include Cascade Mask R-CNN [8] for object detection and instance segmentation, SeqTR [134] for REC and RES, VMT [48] for MOTS and VIS, and Unicorn [113] for SOT, VOS, MOT, and MOTS. As shown in Figure 3, UNINEXT outperforms them and achieve state-of-the-art performance on all 10 tasks. Retrieval by Category Names. As shown in Figure of different categories by taking the corresponding category names as the prompts. For example, when taking "dining table. wine glass. cake. knife" as the prompts, UNINEXT would only perceive dining tables, wine glasses, cakes, and knives. Furthermore, benefiting from the flexible retrieval formulation, UNINEXT also has the potential for zero-shot (open-vocabulary) object detection. However, open-vocabulary object detection is beyond the scope of our paper and we leave it for future works. Retrieval by Language Expressions. We provide some visualizations for retrieval by language expressions in Figure 5. UNINEXT can accurately locate the target referred by the given language expression when there are many similar distractors. This demonstrates that our method can not only perceive objects but also understand their relationships in positions (left, middle, right, etc) and sizes (taller, etc). Retrieval by Target Annotations. Our method supports annotations in formats of both boxes (SOT) and masks (VOS). Although there is only box-level annotation for SOT, we obtain the target prior by filling the region within the given box with 1 and leaving other regions as 0. As shown in Figure 6, UNINEXT can precisely track and segment the targets in complex scenarios, given the annotation in the first frame. Figure 1 . 1Task distribution on the Format-Time-Reference space. Better view on screen with zoom-in. Figure 2 . 2Framework of UNINEXT. The whole pipeline is shown on the left side. The schematic diagram of object retrieval is shown on the right side. The instance head predicts both boxes and masks of the objects. Better view in color on screen. by 3.8 AP. When using stronger ViT-Huge backbone, UNINEXT achieves state-of-the-art AP of 66.9 on Youtube-VIS 2019 and 49.0 on OVIS respectively, surpassing previous methods by a large margin. R-VOS. Ref-Youtube-VOS [90] and Ref-DAVIS17 [50] are two popular R-VOS benchmarks, which are constructed by introducing language expressions for the objects in the , UNINEXT outperforms all previous R-VOS approaches by a large margin, when using the same ResNet-50 backbone. Especially on Ref-DAVIS17, UNINEXT exceeds previous best ReferFormer [108] by 5.4 J &F. Furthermore, when adopting stronger ViT-Huge backbone, UNINEXT achieves new state-of-the-art J &Fof 70.1 on Ref-Youtube-VOS and 72.5 on Ref-DAVIS17. Besides, different from offline Ref-Former, UNINEXT works in a flexible online fashion, making it applicable to ongoing videos in the real world. L boxinst mask (b,m) = L proj (b,m) + L pairwise (b,m). (11) L proj (b,m) =L dice (proj x (b), proj x (m))+ L dice (proj y (b), proj y (m)).(12)L pairwise = − 1 N e∈Ein 1 {Se≥τ } log P (y e = 1).(13) P (y e = 1) =m i,j ·m k,l + (1 −m i,j ) · (1 −m k,l ).(14) Figure 3 . 3Better view in color on screen. Figure 4 . 4Illustration of retrieval by category names. UNINEXT can flexibly perceive objects of different categories by changing the input prompts. Better view in color on screen. Prompt: furthest left plane. Prompt: bottom sofa. Prompt: middle apple. Prompt: bus at center. Prompt: left elephant. Prompt: right full cow. Prompt: taller. Prompt: the cat in the mirror. Figure 5 .Figure 6 . 56Illustration of retrieval by language expressions. Better view in color on screen. Illustration of retrieval by target annotations. UNINEXT can flexibly perceive different objects according to the box or mask annotations given in the first frame. Better view in color on screen. Table 1 . 1State-of-the-art comparison on object detection.Model Backbone AP AP50 AP75 APS APM APL Faster R-CNN [87] ResNet-50 42.0 62.1 45.5 26.6 45.4 53.4 DETR [9] 43.3 63.1 45.9 22.5 47.3 61.1 Sparse R-CNN [93] 45.0 63.4 48.2 26.9 47.2 59.5 Cascade Mask-RCNN [8] 46.3 64.3 50.5 - - - Deformable-DETR [135] 46.9 65.6 51.0 29.6 50.1 61.6 DN-Deformable-DETR [55] 48.6 67.4 52.7 31.0 52.0 63.7 UNINEXT 51.3 68.4 56.2 32.6 55.7 66.5 HTC++ [12] Swin-L 58.0 - - - - - DyHead [21] 60.3 - - - - - Cascade Mask R-CNN [8] ConvNeXt-L 54.8 73.8 59.8 - - - UNINEXT 58.1 74.9 63.7 40.7 62.5 73.6 ViTDet-H [74] ViT-H 58.7 - - - - - UNINEXT 60.6 77.5 66.7 45.1 64.8 75.3 Table 2. State-of-the-art comparison on instance segmentation. Methods marked with * are evaluated on the val2017 split. Model Backbone AP AP50 AP75 APS APM APL CondInst [94] ResNet-50 38.6 60.2 41.4 20.6 41.0 51.1 Cascade Mask R-CNN [8] 38.6 60.0 41.7 21.7 40.8 49.6 SOLOv2 [103] 38.8 59.9 41.7 16.5 41.7 56.2 HTC [12] 39.7 61.4 43.1 22.6 42.2 50.6 QueryInst [31] 40.6 63.0 44.0 23.4 42.5 52.8 UNINEXT 44.9 67.0 48.9 26.3 48.5 59.0 QueryInst [31] Swin-L 49.1 74.2 53.8 31.5 51.8 63.2 Mask2Former [16] * 50.1 - - 29.9 53.9 72.1 Cascade Mask R-CNN [8] ConvNeXt-L 47.6 71.3 51.7 - - - UNINEXT 49.6 73.4 54.3 30.4 53.6 65.7 ViTDet-H [74] * ViT-H 50.9 - - - - - UNINEXT 51.8 76.2 56.7 33.3 55.9 67.5 , UNINEXT surpasses state-of-theart query-based detector DN-Deformable DETR [55] by 2.7 box AP. By replacing ResNet-50 [40] with stronger ConvNeXt-Large [68] and ViT-Huge [28] backbones, UNINEXT achieves a box AP of 58.1 and 60.6, surpassing competitive rivals Cascade Mask-RCNN [8] and ViTDet-H [74] by 3.3 and 1.9 respectively. Besides, the results of instance segmentation are shown in Table 2. With the same ResNet-50 backbone, UNINEXT outperforms stateof-the-art QueryInst by 4.3 AP and 6.2 AP L . When using ConvNeXt-Large as the backbone, UNINEXT achieves a Table 3 . 3State-of-the-art comparison on REC.Method RefCOCO RefCOCO+ RefCOCOg val testA testB val testA testB val-u test-u UNITERL [15] 81.41 87.04 74.17 75.90 81.45 66.70 74.86 75.77 VILLAL [34] 82.39 87.48 74.84 76.17 81.54 66.84 76.18 76.71 MDETR [47] 86.75 89.58 81.41 79.52 84.09 70.62 81.64 80.89 RefTR [57] 85.65 88.73 81.16 77.55 82.26 68.99 79.25 80.01 SeqTR [134] 87.00 90.15 83.59 78.69 84.51 71.87 82.69 83.37 UNINEXT-R50 89.72 91.52 86.93 79.76 85.23 72.78 83.95 84.31 UNINEXT-L 91.43 93.73 88.93 83.09 87.90 76.15 86.91 87.48 UNINEXT-H 92.64 94.33 91.46 85.24 89.63 79.79 88.73 89.37 Table 4 . 4State-of-the-art comparison on RES. With ViT-Huge as the backbone, UNINEXT achieves state-of-the-art mask AP of 51.8. REC and RES. RefCOCOMethod RefCOCO RefCOCO+ RefCOCOg val testA testB val testA testB val-u test-u CMSA [123] 58.32 60.61 55.09 43.76 47.60 37.89 - - BRINet [43] 60.98 62.99 59.21 48.17 52.32 42.11 - - CMPC+ [65] 62.47 65.08 60.82 50.25 54.04 43.47 - - MCN [72] 62.44 64.20 59.71 50.62 54.99 44.69 49.22 49.40 EFN [33] 62.76 65.69 59.67 51.50 55.24 43.01 - - VLT [26] 65.65 68.29 62.73 55.50 59.20 49.36 52.99 56.65 SeqTR [134] 71.70 73.31 69.82 63.04 66.73 58.97 64.69 65.74 LAVT [120] 72.73 75.82 68.79 62.14 68.38 55.10 61.24 62.09 UNINEXT-R50 77.90 79.68 75.77 66.20 71.22 59.01 70.04 70.52 UNINEXT-L 80.32 82.61 77.76 70.04 74.91 62.57 73.41 73.68 UNINEXT-H 82.19 83.44 81.33 72.47 76.42 66.22 74.67 76.37 mask AP of 49.6, surpassing Cascade Mask R-CNN [8] by 2.0. Table 5 . 5State-of-the-art comparison on SOT.VOS. The comparisons between UNINEXT with previous semi-supervised VOS methods are demonstrated inMethod Backbone LaSOT [30] LaSOT ext [29] TrackingNet [79] TNL-2K [102] AUC P N orm P AUC P N orm P AUC P N orm P AUC P PrDiMP [22] ResNet-50 59.8 68.8 60.8 - - - 75.8 81.6 70.4 47.0 45.9 LTMU [20] 57.2 - 57.2 41.4 49.9 47.3 - - - 48.5 47.3 TransT [14] 64.9 73.8 69.0 - - - 81.4 86.7 80.3 50.7 51.7 KeepTrack [75] 67.1 77.2 70.2 48.2 - - - - - - - UNINEXT 69.2 77.1 75.5 51.2 58.1 58.1 83.2 86.9 83.3 56.0 57.5 SimTrack [10] ViT-B 69.3 78.5 - - - - 82.3 - 86.5 54.8 53.8 OSTrack [122] 71.1 81.1 77.6 50.5 61.3 57.6 83.9 88.5 83.2 55.9 - Unicorn [113] ConvNeXt-L 68.5 76.6 74.1 - - - 83.0 86.4 82.2 - - UNINEXT 72.4 80.7 78.9 54.4 61.8 61.4 85.1 88.2 84.7 58.1 60.7 UNINEXT ViT-H 72.2 80.7 79.4 56.2 63.8 63.8 85.4 89.0 86.4 59.3 62.8 Table 6. State-of-the-art comparison on VOS. Method YT-VOS 2018 val [111] DAVIS 2017 val [83] G J s F s J u F u J &F J F Memory STM [81] 79.4 79.7 84.2 72.8 80.9 81.8 79.2 84.3 CFBI [121] 81.4 81.1 85.8 75.3 83.4 81.9 79.1 84.6 STCN [18] 83.0 81.9 86.5 77.9 85.7 85.4 82.2 88.6 XMem [17] 86.1 85.1 89.8 80.3 89.2 87.7 84.0 91.4 Non-Memory SiamMask [100] 52.8 60.2 58.2 45.1 47.7 56.4 54.3 58.5 Unicorn [113] - - - - -69.2 65.2 73.2 Siam R-CNN [98] 73.2 73.5 -66.2 -70.6 66.1 75.0 TVOS [130] 67.8 67.1 69.4 63.0 71.6 72.3 69.9 74.7 FRTM [89] 72.1 72.3 76.2 65.9 74.1 76.7 73.9 79.6 UNINEXT-R50 77.0 76.8 81.0 70.8 79.4 74.5 71.3 77.6 UNINEXT-L 78.1 79.1 83.5 71.0 78.9 77.2 73.2 81.2 UNINEXT-H 78.6 79.9 84.9 70.6 79.2 81.8 77.7 85.8 Table 7 . 7State-of-the-art comparison on MOT.Table 8. State-of-the-art comparison on MOTS.Method Split mMOTA↑ mIDF1↑ MOTA↑ IDF1↑ ID Sw.↓ Yu et al. [124] val 25.9 44.5 56.9 66.8 8315 QDTrack [82] val 36.6 50.8 63.5 71.5 6262 Unicorn [113] val 41.2 54.0 66.6 71.3 10876 UNINEXT-L val 41.8 54.9 64.6 68.7 9134 UNINEXT-H val 44.2 56.7 67.1 69.9 10222 Method Online mMOTSA↑ mMOTSP↑ mIDF1↑ ID Sw.↓ SortIoU 10.3 59.9 21.8 15951 MaskTrackRCNN [116] 12.3 59.9 26.2 9116 STEm-Seg [1] 12.2 58.2 25.4 8732 QDTrack-mots [82] 22.5 59.6 40.8 1340 PCAN [49] 27.4 66.7 45.1 876 VMT [48] 28.7 67.3 45.7 825 Unicorn [113] 29.6 67.7 44.2 1731 UNINEXT-L 32.0 60.2 45.4 1634 UNINEXT-H 35.7 68.1 48.5 1776 Table 9 . 9State-of-the-art comparison on VIS.Method Backbone Online VIS2019 val OVIS val AP AP50 AP75 AP AP50 AP75 VisTR [104] ResNet-50 36.2 59.8 36.9 - - - MaskProp [4] 40.0 - 42.9 - - - IFC [45] 42.8 65.8 46.8 13.1 27.8 11.6 SeqFormer [107] 47.4 69.8 51.8 15.1 31.9 13.8 IDOL [109] 49.5 74.0 52.9 30.2 51.3 30.0 VITA [41] 49.8 72.6 54.5 19.6 41.2 17.4 UNINEXT 53.0 75.2 59.1 34.0 55.5 35.6 SeqFormer [107] Swin-L 59.3 82.1 66.4 - - - VMT [48] 59.7 - 66.7 19.8 39.6 17.2 VITA [41] 63.0 86.9 67.9 - - - IDOL [109] 64.3 87.5 71.0 42.6 65.7 45.2 UNINEXT ConvNeXt-L 64.3 87.2 71.7 41.1 65.8 42.0 UNINEXT ViT-H 66.9 87.5 75.1 49.0 72.5 52.2 Table 10 . 10State-of-the-art comparison on R-VOS.Method Backbone Ref-Youtube-VOS Ref-DAVIS17 J &F J F J &F J F CMSA [123] ResNet-50 36.4 34.8 38.1 40.2 36.9 43.5 URVOS [90] 47.2 45.3 49.2 51.5 47.3 56.0 YOFO [54] 48.6 47.5 49.7 54.4 50.1 58.7 ReferFormer [108] 58.7 57.4 60.1 58.5 55.8 61.3 UNINEXT 61.2 59.3 63.0 63.9 59.6 68.1 PMINet + CFBI [27] Ensemble 54.2 53.0 55.5 - - - CITD [58] 61.4 60.0 62.7 - - - MTTR [7] Video-Swin-T 55.3 54.0 56.6 - - - ReferFormer [108] 64.9 62.8 67.0 61.1 58.1 64.1 UNINEXT ConvNext-L 66.2 64.0 68.4 66.7 62.3 71.1 UNINEXT ViT-H 70.1 67.6 72.7 72.5 68.2 76.8 Table 11 . 11Ablations. The settings in our final model is underlined.Experiment Method OD REC VOS RVOS VIS COCO RefCOCO YTBVOS R-YTBVOS VIS19 (AP) ([email protected]) (J &F) (J &F) (AP) Fusion Early Fusion 51.3 89.7 77.0 61.2 53.0 W/o Fusion 51.1 87.4 55.6 58.4 51.0 (+0.2) (+2.3) (+21.4) (+2.8) (+2.0) Queries Static 51.3 89.7 77.0 61.2 53.0 Dynamic 51.9 89.8 77.4 61.6 50.2 (-0.6) (-0.1) (-0.4) (-0.4) (+2.8) Model Unified 51.3 89.7 77.0 61.2 53.0 Task-specific 50.8 87.6 74.2 57.2 50.1 (+0.5) (+2.1) (+2.8) (+4.0) (+2.9) Table 12 . 12Details in training.Step is the time to reduce the learning rate.Stage Task Dataset Sampling Weight Batch Size Short Long Num GPU Lr Max Iter Step I OD&IS Objects365 [92] 1 2 480 ∼ 800 1333 32 0.0002 340741 312346 II OD&IS COCO [63] 1 2 480 ∼ 800 1333 16 0.0002 91990 76658 REC&RES RefCOCO/g/+ [80, 125] 1 2 480 ∼ 800 1333 III SOT&VOS LaSOT [30] 0.20 2 480 ∼ 800 1333 16 0.0001 180000 150000 GOT10K [44] 0.20 2 480 ∼ 800 1333 TrackingNet [79] 0.20 2 480 ∼ 800 1333 Youtube-VOS [111] 0.20 2 320 ∼ 640 768 COCO [63] 0.20 2 480 ∼ 800 1333 MOT&MOTS BDD-obj-det [124] 0.18 2 480 ∼ 800 1333 BDD-box-track [124] 0.72 2 480 ∼ 800 1333 BDD-inst-seg [124] 0.02 2 480 ∼ 800 1333 BDD-seg-track [124] 0.08 2 480 ∼ 800 1333 VIS Youtube-VIS-19 [116] 0.34 4 320 ∼ 640 768 OVIS [84] 0.17 2 480 ∼ 800 1333 COCO [63] 0.51 2 480 ∼ 800 1333 R-VOS Ref-Youtube-VOS [90] 0.33 2 320 ∼ 640 768 RefCOCO/g/+ [80, 125] 0.67 2 480 ∼ 800 1333 4, UNINEXT can flexibly detect and segment objects Prompt: dining table. 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[ "Strain-Driven Thermal and Optical Instability in Silver/Amorphous- Silicon Hyperbolic Metamaterials", "Strain-Driven Thermal and Optical Instability in Silver/Amorphous- Silicon Hyperbolic Metamaterials", "Strain-Driven Thermal and Optical Instability in Silver/Amorphous- Silicon Hyperbolic Metamaterials", "Strain-Driven Thermal and Optical Instability in Silver/Amorphous- Silicon Hyperbolic Metamaterials" ]	[ "Jose L Ocana-Pujol [email protected] \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n", "Lea Forster \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n\nDepartment of Materials\nLaboratory of Multifunctional Ferroic Materials\nETH Zurich\nZürich, Switzer-land\n", "Ralph Spolenak \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n", "Henning Galinski [email protected] \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n", "Jose L Ocana-Pujol [email protected] \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n", "Lea Forster \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n\nDepartment of Materials\nLaboratory of Multifunctional Ferroic Materials\nETH Zurich\nZürich, Switzer-land\n", "Ralph Spolenak \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n", "Henning Galinski [email protected] \nDepartment of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland\n" ]	[ "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory of Multifunctional Ferroic Materials\nETH Zurich\nZürich, Switzer-land", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory of Multifunctional Ferroic Materials\nETH Zurich\nZürich, Switzer-land", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland", "Department of Materials\nLaboratory for Nanometallurgy\nETH Zurich\nZürichSwitzerland" ]	[]	Hyperbolic metamaterials show exceptional optical properties, such as near-perfect broadband absorption, due to their geometricallyengineered optical anisotropy. Many of their proposed applications in thermophotovoltaics or radiative cooling, require high-temperature stability. In this work we examine Ag/a-Si multilayers as a model system for the thermal stability of hyperbolic metamaterials. Using a combination of nanotomography, finite element simulations and optical spectroscopy, we map the thermal and optical instability of the metamaterials. Although the thermal instability initiates at 300 • C, the hyperbolic dispersion persists up to 500 • C. Direct finite element simulations on tomographical data provide a route to decouple and evaluate interfacial and elastic strain energy contributions to the instability. Depending on stacking order the instability's driving force is either dominated by changes in anisotropic elastic strain energy due thermal expansion mismatch or by minimization of interfacial energy. Our findings open new avenues to understand multilayer instability and pave the way to design hyperbolic metamaterials able to withstand high temperatures.	10.1002/adom.202201749	[ "https://export.arxiv.org/pdf/2207.09571v1.pdf" ]	250,698,939	2207.09571	40f07e9ad897113a0a43e961423e404bc1b3690a	Strain-Driven Thermal and Optical Instability in Silver/Amorphous- Silicon Hyperbolic Metamaterials Jose L Ocana-Pujol [email protected] Department of Materials Laboratory for Nanometallurgy ETH Zurich ZürichSwitzerland Lea Forster Department of Materials Laboratory for Nanometallurgy ETH Zurich ZürichSwitzerland Department of Materials Laboratory of Multifunctional Ferroic Materials ETH Zurich Zürich, Switzer-land Ralph Spolenak Department of Materials Laboratory for Nanometallurgy ETH Zurich ZürichSwitzerland Henning Galinski [email protected] Department of Materials Laboratory for Nanometallurgy ETH Zurich ZürichSwitzerland Strain-Driven Thermal and Optical Instability in Silver/Amorphous- Silicon Hyperbolic Metamaterials metamaterialhyperbolic dispersionmultilayerthermal stabilitylarge-scale photonicsamor- phous siliconhigh temperature Hyperbolic metamaterials show exceptional optical properties, such as near-perfect broadband absorption, due to their geometricallyengineered optical anisotropy. Many of their proposed applications in thermophotovoltaics or radiative cooling, require high-temperature stability. In this work we examine Ag/a-Si multilayers as a model system for the thermal stability of hyperbolic metamaterials. Using a combination of nanotomography, finite element simulations and optical spectroscopy, we map the thermal and optical instability of the metamaterials. Although the thermal instability initiates at 300 • C, the hyperbolic dispersion persists up to 500 • C. Direct finite element simulations on tomographical data provide a route to decouple and evaluate interfacial and elastic strain energy contributions to the instability. Depending on stacking order the instability's driving force is either dominated by changes in anisotropic elastic strain energy due thermal expansion mismatch or by minimization of interfacial energy. Our findings open new avenues to understand multilayer instability and pave the way to design hyperbolic metamaterials able to withstand high temperatures. Introduction Hyperbolic metamaterials (HMMs) are periodic designer materials with controllable uniaxial optical anisotropy, where the two components of the permittivity tensor can be engineered to have opposite signs, i.e. ⊥ · < 0. This extreme anisotropy offers an unusual degree of freedom to direct the energy flow of light and gives rise to exotic phenomena, such as abnormal refraction [1,2], super-collimation [3,4], imaging beyond the diffraction limit [5,6], and the realisation of near-perfect broadband absorbers [7,8]. The ascribed control of the optical anisotropy enables to realize different dispersion regimes which for a given frequency manifest as optical topological transitions (OTTs) in the isofrequency surface [9]. The four optical phases or dispersion regimes given by the isofrequency surface topologies are: an effective dielectric ( ⊥ > 0 , > 0), a Type I HMM ( ⊥ < 0 , > 0), a Type II HMM ( ⊥ > 0 , < 0), and an effective metal ( ⊥ < 0 , < 0). They are commonly described within the effective medium approximation (EMA) [10]. For example, in the case of a transition from a dielectric to a Type II HMM the isofrequency surface transforms from a spheroid to a hyperbole. In the absence of loss, the volume of the hyperbolic shell is infinite which feeds back into an infinite local density of optical states (LDOS) [11]. Multilayers are the most common realization of HMMs [12]. Of special interest is the interaction of temperature with multilayer HMMs, either in wavelength-switchable temperature-actuated hyperlenses [13] and super absorbers [14] or in applications with high operating temperatures such as thermophotovoltaics [15,16], hyperthermia therapy [17,18], or radiative cooling [19,20,21]. However, one of the key challenges for all of these proposed applications is thermal stability. Therefore, it is crucial to understand how temperature changes the geometry and related optical properties of HMMs. Here, we examine magnetron sputtered silver/amorphous-silicon (a-Si) multilayers as a model system to study the thermal stability of HMMs. Silicon deposited by magnetron sputtering, which is suitable for large-scale photonics, is typically amorphous. Amorphous silicon is a different allotrope with different properties than the more widely studied crystalline silicon (c-Si). The Ag/Si system is characterized by the absence of a low temperature eutectic [22] and very low boundary solubility on the two sides of the phase diagram [23,24]. To the best of our knowledge, a complete Ag/a-Si phase diagram does not exist, but silver diffusivity is known to be slower than in crystalline silicon [25]. Amorphous silicon also shows distinct optical properties [26,27], which has been exploited in the field of solar energy [28,29]. Silver is considered the preferred metal for HMMs applications in the visible due to its low losses [10]. Moreover, magnetron sputtered stacked Ag/Si layers were one of the first systems for which hyperbolic behavior was demonstrated [30]. In this work, we provide a detailed analysis of the thermal stability of layered Ag/a-Si HMMs. First, the design and optical properties of the structure is presented. Then, we examine the structural and optical changes of the system upon annealing. The driving forces behind the multilayer instability are examined. Finally, we explore design parameters to test the scope of our findings. Results and Discussion Optical Design and Characterization A schematic of the hyperbolic metamaterials realized in this work is illustrated in Figure 1a. HMMs were synthesized by magnetron sputtering keeping the bilayer unit cell size Λ = 40 nm constant for all data shown in Figure 1 to 3. The samples were sputtered on a c-Si wafer with 50 nm SiO 2 and 50 nm Si x N y as thermal barriers. The film thickness in the multilayer was confirmed by milling and imaging cross-sections using focused ion beam assisted scanning electron microscopy (FIB-SEM). Figure 1b reports the optical phase diagram in the visible wavelength range. The values of ⊥ and for selected geometries are given in the Supporting Information ( Figure S2g,b). To include non-local effects [31], the optical phase diagram was obtained by FEM simulations and parameter retrieval along the directions parallel and perpendicular to the surface [32]. The calculated plasmonic dispersion of HMMs with 30 vol.% and 55 vol.% Ag are shown on Figure 1c,d, respectively. The graphs show radiative modes left of the surface plasmon dispersion (SP) which intersect with the light line and modes at higher wavevector β and wavelength λ which do not intersect with the light line and can be considered non-radiative [33]. Figure 1c,d further indicates that radiative modes exist in both the dielectric and Type I regimes. We observe the transition between these optical topologies experimentally by measuring the reflectance ratio R p /R s as a function of incidence angle Ω, since the propagation of hyperbolic modes is limited to ppolarized light. The colormaps in Figure 1e,f show minima in the reflectance ratio for Type I and dielectric optical topologies in accordance with Figure 1b. Due to the non-radiative nature of the plasmonic modes shown in Figure 1c and d, no reflection anisotropy occurs at the Type II HMM and metallic regimes. Moreover, in the case of a system with 30 vol.% Ag (see Figure 1e), the transition between Type I HMM and the dielectric phases should lead to a discontinuity in the pseudo-Brewster angle [1,34]. In agreement with the optical phase diagram (Figure 1b), our measurements reproduce this discontinuity shown in Figure 1e caused by the change of sign in ⊥ at λ = 550 nm. Figure 2c reports a comparison of the reflectance spectrum measured at normal incidence of the as-deposited HMM with several calculated reflectance spectra. Interestingly, the FEM simulations using a geometry without interfacial roughness (see also Figure 2a) do not reproduce the measured reflectance dip at 900 nm, while the reflectance spectrum obtained on the basis of the FIB cross-section does. To understand the origin of this Lorentz-like dip in reflectance of the as-deposited state, we employed the transfer-matrix (TM) method to model the reflectance of the HMM. Scalar scattering theory [35] was used to account for contributions of interfacial roughness (see Supporting Information). Figure 2c shows the calculated response accounting for roughness reproduces the experimental spectral response. This indicates that the reflectance dip at about 900 nm is caused by coupling to the propagating modes with high-k wavevector. In order to test whether this modes are hyperbolic, we placed a vertical oriented dipole inside the 3D tomography and simulated the energy flow, as shown in Figure 2d. The characteristic cone-like pattern of the propagation is a fingerprint of hyperbolic behavior [36]. Furthermore, the energy flow is confined to the multilayer indicating the non-radiative nature of the mode. This result is in line with a previous theoretical work, which identified disorder, such as roughness, in an HMM as viable route to effectively couple to non-radiative highly absorbing modes [37]. Thermal and Optical Instability When subjected to thermal annealing in ultra-high vacuum, our HMMs exhibit two regimes. Upon annealing at 300 • C, the geometry of the HMM is no longer preserved. In both, the FIB nanotomography ( Figure 2e) and the corresponding cross-sectional image (Figure 2f), a redistribution of the lower three Ag-layers is evident. This reorganization is characterized by silver diffusion away from the HMM/substrate interface and by agglomeration and grain growth within the silver phase. Interestingly, the loss in structural integrity is not paralleled by a significant degradation of the optical properties. Despite the structural disorder in the HMM, the metamaterial is robust and exhibits still the characteristic dip in the reflectance spectrum ( Figure 2g) and cone-like electromagnetic energy flow (Figure 2h). Upon annealing at 500 • C, the geometry and distinct optical response of the HMM is lost. The FIB nanotomography (Figure 2i) and the corresponding cross-sectional image (Figure 2j), show a dissolution of the multilayer and the formation of a complex Ag/a-Si composite in its place. The composite contains an interconnected silver phase mostly concentrated in center of the former HMM. Locally, we observe Ag segregation to the surface. At this degree of disorder, the optical response of the HMM is no longer present and the measured and simulated reflectance spectra exhibit a quasi-flat response (Figure 2k). In addition, the electromagnetic flow in the geometry shown in Figure 2l does no longer feature the characteristic cone-like shape, confirming that the system has lost its hyperbolic dispersion at this temperature. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Generally, the inhomogeneous redistribution of silver in the HMM can be understood in the context of a structural instability triggered by temperature [38,39]. Here, we found the onset of this instability to be independent of the unit cell's composition and to occur approximately at 300 • C for all experimentally analyzed HMM-designs ( Figure S5 in the Supporting Information). Our experiments further show that the HMM/substrate interface has a destabilizing role enhancing the kinetics of the fundamental processes in vicinity of this interface. This is an important result as commonly the reorganization of matter due to a thermal or multilayer instability is assumed to be isotropic [40]. Thermodynamics of the Instability Motivated by the experimentally observed anisotropic instability, we aim to identify the main contributions to its driving force. Intuitively, we can link the tendency of reorganization or pattern formation in an inhomogeneous mixture, here the Ag/a-Si multilayer, to the lowering of the systems total free energy [41]. With reference to our layered HMM, we can formulate its total free energy F within the Cahn-Hilliard approach [42,43], F (c, u) = V f (c) + κ(∇c) 2 + E el (c, u)dV.(1) Here, the thermodynamic potential has three main contributions, f (c) is the free energy density of the homogeneous system, κ(∇c) 2 considers contributions from gradients in composition, e.g. interfaces, while E el (c, u) is the anisotropic elastic strain energy density. c is the local concentration and u is the displacement vector. We assume that E el (c, u) originates at the HMM/substrate interface and results from the mismatch in coefficient of thermal expansion (CTE). Taking advantage of our FIB nanotomographies, we performed a series of 3D structural mechanics FEM simulations, as illustrated in Figure 3, to determine main contributions of κ(∇c) 2 and E el (c, u) to the free energy change as function of temperature. Since silver and silicon have a positive enthalpy of formation [22], we can exclude chemical reactions in the system and assume that f (c) does not contribute to the free-energy change [39]. Figure 3a,b,c show the elastic strain energy at different temperatures. In Figure 3a, the elastic strain energy caused by the thermal expansion mismatch of Ag and a-Si was simulated assuming nominal flat layers both at 300 • C and 500 • C with the latter being shown in the color scheme. The subset on the right shows that the elastic strain energy in the metal layer is approximately five times larger than in the dielectric, while the elastic strain energy in the metal decays exponentially from the HMM/substrate interface. The thermal expansion and elastic strain energy density of the thermally degraded multilayers, shown in Figure 3b and 3c are calculated based on the experimental FIB nanotomographies of the annealed state. Overall, the redistribution of the silver phase significantly lowers the elastic strain energy density. A comparison between the subsets in Figure 3a,b,c shows a lower and more homogeneously distributed elastic strain energy within the degraded HMMs. Figure 3c illustrates the concentration of Ag in the center of the HMM when annealed at 500 • C, which leads to a significant reduction of the elastic strain energy. Partial segregation of Ag to the HMM/ambient interface enhances this effect (see Figure 3e and f). To illustrate the reorganization of the silver phase, Figure 3d reports the averaged silver depth profiles as function of temperature derived from classified FIB cross-sectional images. Interestingly, the silver distribution and profiles of the elastic strain energy show a similar thermal response. While in the first regime at 300 • C redistribution is limited to the HMM/substrate interface, the second regime includes agglomeration in the center of the HMM and partial segregation. This dynamics can be explained by considering the diffusion length of Ag in a-Si ( Figure S8) which exceeds the thickness of the HMM for temperatures larger than 425 • C. To determine the main driving force of the instability, we calculated the difference in elastic strain energy ∆E elast and interfacial energy ∆E surf . The change in elastic strain energy is derived directly from the FEM simulations, while the interfacial energy is given by the product of the interfacial area derived from the 3D FIB-nanotomographies times the Ag/a-Si interfacial energy ( Figure S8 in the Supporting Information). Figure 3g shows the change of this two main contributions to the free energy as function of temperature. The net change in free energy is negative for the analyzed temperatures, i.e. the observed instability in fact reduces the system free energy. Notably, at 300 • C, the change in elastic strain energy is negative, while the change in interfacial energy is positive, suggesting that the lowering in elastic strain energy is the main driving force of the observed instability. At 500 • C both contributions are negative, although the change in elastic strain energy being still pre-dominant. Our analysis underlines the importance of anisotropic contributions to the thermodynamic potential that describes thermal instabilities. Here, the anisotropic elastic strain energy due to thermal expansion mismatch is identified as the predominant driving force. Most previous literature on multilayer instabilities disregards this effect and treats the morphological redistribution in the context of interfacial energy minimization, including work on sputtered Ag/a-Si multilayers [44,45], other systems with high CTE missmatch [46,47,48], predictive models [49,39] and reviews [50]. It should be noted that these conclusions are often based on experiments using transmission electron microscopy (TEM). Due to sample preparation these kind of experiments occur in the absence of a substrate and thus most likely fall short to reproduce the dynamics observed in real world materials. Influence of Design and Stacking Order Based on our previous analysis, this section focuses on the question,whether the driving forces can be controlled by changing the multilayer design parameters. Hence, we changed the HMM design and inves- Figure 4b shows that, despite the design change, the structural evolution of the sample with the AB stacking order is in agreement with the instability observed in Figure 2: the onset of the instability at 300 • C occurs at the bottom interface and the multilayer has disappeared at 500 • C. In contrast, in case of an BA architecture (Figure 4d) where amorphous silicon is the material in contact with the substrate, the instability occurs in the vicinity of the HMM/ambient interface (Figure 4c). We attribute this change to lower elastic strain energy accumulation in the lower interface due to the similarity in the CTE of a-Si and the substrate. In this context, the observed degradation on the top of the multilayer is likely dominated by interfacial energy minimization, which leads to island formation as known from phenomena such as dewetting [51]. The changes in the optical properties caused by degradation are shown by means of reflectance spectra at normal incidence in Figure 4e,f for the AB and BA architectures, respectively. Upon annealing, we observe the persistence of the HMM behavior at 300 • C, in line with the results shown in Figure 2. This indicates the robustness of optical phases of the HMM independent of the explored design parameters. At 500 • C the optical properties change and both systems show a flat reflectance. We attribute this ef-fect to loss of hyperbolic behavior due to multilayer degradation, as observed in Figure 2l. It is to note, that we observe hyperbolic behavior in a system composed of only 3 unit cells, which is below the minimum number of components proposed to achieve hyperbolic behavior [10,52,53,54]. For further insight into the optical response of these samples, the reader is referred to Figure S6 in the Supporting Information. To test the limits of the second regime, we annealed the samples for 12 h in a vacuum at 600 • C and 800 • C. Figure 4b,c show that, after annealing at these temperatures, the contrast of the SEM images has dramatically changed. We attribute it to Ag evaporation, which is confirmed by the EDX spectra on Figure 4g,h. This phenomenon leads to lower reflectivity, as shown in Figure 4e,f. The observed evaporation can be understood as a third regime of the thermal instability. Interestingly, our experiments further show that one can impact the evaporation temperature by the stacking order. While in samples with BA stacking order silver evaporates at 600 • C or 0.7 T /T m (Figure 4h), the AB geometry still contains silver after the same treatment ( Figure 4g) and evaporation is not observed until 0.85 T /T m [22]. The difference can also be observed in the reflectance spectra in Figure 4e,f. Together with the change on the pathway to the instability at 600 • C, these results confirm the dependence of the thermal stability on the stacking order. Conclusion If hyperbolic metamaterials are to become widely used in fields such as thermophotovoltaics or radiative cooling, it is crucial to understand their thermal stability. We examine magnetron sputtered Ag/a-Si layered HMMs as a model system to study the limits of stability at elevated temperatures. Using interfacial roughness as design parameter, we achieved near-perfect absorption above 0.45 homologous temperature in the studied HMMs by coupling to non-radiative modes. We show that the system undergoes a thermal instability upon annealing in vacuum at 300 • C. We investigate the driving forces behind this structural degradation by a combination of FIB nanotomographies and FEM simulations. Our results indicate that the main driving force is the inhomogeneous elastic strain energy caused by a mismatch in thermal expansion between the metallic phase of the metamaterial and the substrate. This contribution has often been disregarded [47,39] or assumed to be homogeneous [38]. Notably, the observed instability is tightly related to the HMM design, as co-sputtered Ag/a-Si [55] preserve their structural integrity in the studied temperature range. We believe that our experiments can open new ways to design metamaterials with enhanced thermal stability. In this context the identification of elastic strain energy as the driving force of thermally-induced degradation in multilayered HMMs can be used to select the appropriate materials for applications requiring near-perfect absorption above that temperature. Besides having the required optical properties, the constituents should be immiscible and have similar CTEs. Other interesting approaches to limit elastic strain driven degradation include geometrical designs limiting the contact area with the surface [18,56,57] and the use of natural hyperbolic materials [8]. Experimental Section Synthesis and annealing: The hyperbolic metamaterials in this work were deposited using magnetron sputtering (PVD Products Inc). Silicon (99.99% MaTecK GmbH ) was sputtered using radio frequency (RF) sputtering, while silver (99.99% MaTecK GmbH ) was deposited using direct current (DC) sputtering. Since DC deposition rates are higher than those for RF [58], a precise determination of the sputter rates for both materials is needed to fabricate multilayers with the appropriate metal-to-semiconductor ratios. The calibration of the sputter rate was achieved using focussed ion beam cross sections (FIB-CS) and atomic force microscopy (AFM) experiments. The samples were annealed for 1 h or 12 h in a vacuum at 1 × 10 −9 mbar in a Createc rapid-thermal annealing (RTA) setup. The heating rate was kept at 5 K/min and no active cooling was used. Structural characterization and simulations: Single and multiple cross-sections of the HMMs were cut, polished, and imaged using a Zeiss NVISION 40 FIB. The energy-dispersive X-ray (EDX) spectra were acquired using a EDAX Pegasus XM 2 System. Detailed information on the FIB nanotomography are given in the Supporting Information ( Figure S7). 3D structural mechanics FEM simulations were carried out using Comsol Multiphysics 5.6. Optical characterization and calculations: The refractive index of Ag and a-Si was measured by Variable Angle Spectroscopic Ellipsometry (VASE) with a J.A. Woollam M-2000 system. The optical properties of a-Si are known to be sensitive to the fabrication conditions [27,59]. A comparison between the experimental n and k values of magnetron sputtered Ag and a-Si with the corresponding literature data is shown in Figure S1. M-2000 was also used to measure the angular and polarization-dependent reflectance spectra. The reflectance spectra at near-normal incidence were measured using a fibre-coupled reflectometer (OceanOptics). Optical simulations were performed using Comsol Multiphysics 5.6. The optical phase diagram in Figure 1b is based on the effective permittivities calculated using FEM simulations and S-parameter retrieval [32]. Transfer Matrix (TM) calculations were performed in a Wolfram Mathematica. We refer to the Supporting Information for a more detailed explanation of the Transfer Matrix Method [60] and how it was modified to account for the effect of roughness [61,35]. The plasmonic dispersion was calculated based on Ref. [62]. Supporting Information Supporting Information is available. Figure 2 2shows the optical and structural response of the 30 vol.% Ag multilayers as a function of temperature. The samples were annealed in vacuum at 300 • C and 500 • C for 1 h. A schematic of the as-deposited structure is shown inFigure 2a, whileFigure 2billustrates a focused ion beam (FIB) polished cross-section (CS) of the as-deposited metamaterial. Figure 1 : 1Design of the hyperbolic metamaterial (a) Schematic of a deposited Ag/a-Si multilayer. (b) Optical phase diagram for Ag/a-Si HMMs with Λ = 40 nm unit cell. The color code depicts the different zones: effective dielectric ( ⊥ > 0, > 0), effective metal ( ⊥ < 0, < 0), Type I HMM ( ⊥ < 0, > 0), and Type II HMM ( ⊥ > 0, < 0). The dashed vertical lines illustrate the expected compositions of the samples with 30 vol.% Ag (for (c) and (e)) and 55 vol.% Ag (for (d) and (f)). (c) and (d) Dispersion relationship of the bulk plasmonic modes of the (c) 30 vol.% Ag and (d) 55 vol.%Ag systems. The black continuous line indicates the light line in vacuum and the red line shows the dispersion relation of surface plasmons at a single Ag/a-Si interface. (e) and (f) Measured ratio between the p and s-polarized reflectance of the (e) 30 vol.% Ag and (f) 55 vol.% Ag systems. Figure 2 : 2Thermal degradation (a): Three-dimensional (3D) model of the as-deposited structure with 30 vol% Ag. Silver is depicted in gray. (e) and (i): 3D reconstructed FIB nanotomography of the samples annealed at (e) 300 • C and (i) 500 • C. The gray structure corresponds to the silver phase. (b), (f), and (j) FIB-SEM cross-sections of the as-deposited (b), annealed at 300 • C (f), and 500 • C state (j). The bright color corresponds to silver. (c), (g), and (k) Experimental and simulated reflectance at normal incidence in the as-deposited state (g), after annealing at 300 • C (h), and 500 • C (k). FEM stands for finite element simulations assuming perfectly flat layers, CS-based FEM stands for FEM simulations when the geometry is imported from FIB-SEM cross-sections, and TM is the transfer matrix calculated reflectance including interfacial roughness. The shown simulated reflectance spectra in (g) and (k) represent an average reflectance obtained by averaging over 8 different cross-sections of the HMM. (d), (h) and (l) FEM simulation of the electromagnetic energy flow created by a point dipole (with its position and orientation depicted by an arrow) placed inside HMMs with different degree of disorder. The geometries are based on FIB-SEM cross-sections. Figure 3 : 3Analysis of the driving forces of the instability (a), (b), and (c) FEM simulations of the elastic strain energy density of the geometries at different temperatures, where the deformation of the geometries maps the thermal expansion. In (a) the nominal HMM design was simulated at both 300 • C and 500 • C. In panels (b) and (c) the elastic strain energy density are modelled by using the experimental FIB-nanotomography data of the annealed state. The graphs on the right show the mean values of the elastic strain energy density probed in different locations. (d) Ag probability profiles derived from experimental FIB-CS. (e) and (f) SEM images highlighting the change in surface topography (300 • C and 500 • C) due to silver segregation at 500 • C. (g) Average specific energy change ∆E of the elastic strain energy density and interfacial energy density as function of annealing temperature. The change in elastic strain energy density is negative within the given temperature range, confirming its status as pre-dominant driving force of the instability. Figure 4 : 4Influence of stacking order (a) and (d): Schematic showing the difference in stacking order between the ABABAB (Si on top) and the BABABA (Ag on top) architectures. 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[ "Predicting Parameters for Modeling Traffic Participants", "Predicting Parameters for Modeling Traffic Participants" ]	[ "Ahmadreza Moradipari ", "Sangjae Bae ", "Mahnoosh Alizadeh ", "Ehsan Moradi Pari ", "David Isele " ]	[]	[]	Accurately modeling the behavior of traffic participants is essential for safely and efficiently navigating an autonomous vehicle through heavy traffic. We propose a method, based on the intelligent driver model, that allows us to accurately model individual driver behaviors from only a small number of frames using easily observable features. On average, this method makes prediction errors that have less than 1 meter difference from an oracle with full-information when analyzed over a 10-second horizon of highway driving. We then validate the efficiency of our method through extensive analysis against a competitive data-driven method such as Reinforcement Learning that may be of independent interest.	10.1109/itsc55140.2022.9922467	[ "https://export.arxiv.org/pdf/2301.10893v1.pdf" ]	253,251,904	2301.10893	98b211309944a4097476b7807e439bd7d6d65f9b	Predicting Parameters for Modeling Traffic Participants Ahmadreza Moradipari Sangjae Bae Mahnoosh Alizadeh Ehsan Moradi Pari David Isele Predicting Parameters for Modeling Traffic Participants Accurately modeling the behavior of traffic participants is essential for safely and efficiently navigating an autonomous vehicle through heavy traffic. We propose a method, based on the intelligent driver model, that allows us to accurately model individual driver behaviors from only a small number of frames using easily observable features. On average, this method makes prediction errors that have less than 1 meter difference from an oracle with full-information when analyzed over a 10-second horizon of highway driving. We then validate the efficiency of our method through extensive analysis against a competitive data-driven method such as Reinforcement Learning that may be of independent interest. I. INTRODUCTION Generating safe and efficient behaviors for an autonomous driving (AD) agent requires an ability to model how other traffic participants will behave. Since the behavior of a traffic participant depends on the behavior of surrounding traffic agents, including our autonomous vehicle (ego vehicle), it must be conditioned on these surrounding agents. Moreover, because AD is a safety critical application, the behaviors must be robust to a large variety of circumstances. Recently, data-driven methods have emerged with increasing volumes of dataset, appealing to both the automotive industry and academia with empirically proven performances under complex environments. In particular, variants of deep learning techniques have shown their effectiveness and computational efficiency for predicting AD environments [1], [2], [3]. Although plausible and powerful, deep learning methods share technical and practical limitations that hinder the datadriven approaches being applied to real-world applications. The limitations are represented by interpretability (behind a choice of actions), generalizability (shifting domains), and the need of excessive data (for both general and corner cases), which are critical for safe-sensitive applications. Despite the growing set of the literature, deep learning methods require profound verification and validation until practically implemented. Therefore, simpler but easily implementable methods still remain as a popular choice of baseline in practice as well as research papers [4], [5], [6]. In particular, the Intelligent Driver Model (IDM) [7] has proven a valuable method for modelling traffic behavior and has been adapted to a wide variety of scenarios [8], [9], [10], [11], [12]. Estimating the behavioral parameters of IDM [13] was proposed as a constrained learning problem for discovering the driver-specific behaviors of a traffic participant. This method provides accurate, individualized behaviors, while This work was funded by Honda Research Institute USA, Inc. 1 Honda Research Institute USA, Inc., 2 University of California, Santa Barbara. Email: {ahmadreza_moradipari,alizadeh}@ucsb.edu, {ahmadreza_moradipari,sbae,emoradipari,disele}@honda-ri.com sufficiently constraining the model to prevent most erratic edge cases which could trigger undesirable reactions (i.e. phantom braking) from our ego agent. However there are some limitations to this approach: 1) acceleration is hard to measure accurately as it usually requires either differentiating velocity or double differentiating position which introduces noise to the measurement, 2) under normal driving conditions, times of relatively constant velocity are common, meaning one might not observe changes in velocity. This makes it difficult to accurately estimate an acceleration profile for a given driver. To address these issues we propose a method to predict the IDM parameters from other, more easily observable, traffic behaviors. Specifically, we investigate how accurately we can predict IDM parameters from lateral lane displacements, relative velocity, and the headway spacing of observed traffic participants. This allows us to quickly ascertain an individualized model for traffic participants over fairly long horizons. The primary contribution of this work is presenting a prediction method for optimizing the use of a benchmark method for practical applicability and validating it through comparative analysis against a combination of heuristic and data-driven methods, with an accompanying analysis of the trade-offs of various aspects of the algorithm. This analysis includes a comparison of our heavily-constrained learning problem with a less-constrained learning-based approach to modeling behavior which might be of interest to the broader machine learning community. Specifically we observe how the two most commonly used metrics, average displacement error and final prediction error, when considered in isolation, can obscure the overall quality of an algorithm. While shortcomings of these metrics have been discussed before 1 , our use-case provides a concrete example that many alternative metrics would also fail to recognize. While it has been shown that sometimes machine learning methods can be outperformed by much simpler methods [14], our results show that even when blackbox machine learning methods perform well according to quantitative analysis, they might have obvious failings not apparent from the metrics. This serves to highlight that thorough analysis should consider a diverse set of metrics and an assessment of broader practical issues (i.e data, reliability, stability, interpretability) when deploying a system. While we believe machine learning has tremendous potential, our experiments serve to benchmark current capabilities and highlight the strengths and weaknesses of competing methods. II. PRELIMINARIES Predicting the motion of other agents is essential when safely planning the trajectory for an autonomous vehicle. Understanding the uncertainty of a prediction has been one important line of research [15], but just knowing the uncertainty of predictions often lead to overly conservative behaviors which can cause unnecessary delays and make the ego agent's actions difficult for others to predict [16]. Deep learning techniques have been making huge strides in increasing the accuracy of prediction [1], [2], [3] and will likely become the dominant method of prediction. However, for the time being, they suffer from distribution drift and often struggle to beat simple baselines like constant velocity [14], especially in cases like highway driving where constant velocity is strongly advantaged. For this reason, we turn to constrained learning problems, which while not as flexible as end-to-end learning, are easier to interpret, more robust to out-of-sample operation, and as a result are better able to fail gracefully and in easy-to-predict ways. In this paper we use IDM to constrain our learning problem. By estimating the parameters of IDM we are able to learn driver-specific models for traffic participants. However the IDM parameters, in practice, can be difficult to estimate accurately, so we present a method to predict IDM parameters from more easily measurable features. Before reviewing IDM. we discuss the notation we used in this work. Let T = {X 1 , . . . , X T } denote a sequence of the physical states for the modeled vehicle over a finite horizon T , and let U = {u 1 , . . . , u T }, denote a finite sequence of control inputs. We denote F as the transition model such that for each time step t, we have X t+1 = F(X t , u t+1 ). In this work, we use the kinematic bicycle model from [17]. The detailed explanation of the transition model is presented in Section III-D. A. Intelligent Driver Model The intelligent driver model (IDM) [7] was proposed to better understand the dynamics of traffic participants, and how changes in velocity contribute to traffic jams. IDM has grown into a popular choice for designing behaviors in simulation [18], [19], [20], controlling autonomous agents [9], [10], [21], and predicting traffic participants [11], [22]. In IDM, the change in velocityv is described aṡ v = a 1 − v v 0 φ − d * (v, ∆v) d 2 ,(1) where d * (v, ∆v) = d 0 + d 1 v v 0 + T v + v ∆v 2 √ ab .(2) Here v is the car's current velocity, ∆v is the difference in velocity with respect to the car in front, d * is the desired minimum gap distance, and d is the actual gap distance to the vehicle in front. We assume the desired velocity v 0 is the road's speed limit, and following Treiber [7] we keep the acceleration exponent φ fixed at 4. This leaves five driver specific parameters: safe time headway T , maximum acceleration a, desired deceleration b, and jam distances d 0 and d 1 . Note that Treiber identifies d 1 as being important for accurately modelling the differences in driving behavior. Together these parameters form the IDM behavior parameters θ IDM = {a, b, T, d 0 , d 1 }. B. Parameter Estimation It was shown that these IDM behavioral parameters can be estimated online using a particle filter [13]; however, most of these parameters relate to acceleration which is typically measured as the first derivative of velocity or the second derivative of position, both of which are noisy. Moreover, under normal driving conditions, it is common that a car will maintain a roughly constant velocity, making it difficult to estimate the dynamic behavior. To handle these issues, we propose a method to predict the IDM parameters for each traffic participant, from other more easily observable traffic behaviors. Here for each vehicle, we focus on the three observable features from the trajectory: (i) lateral lane displacement τ ; (ii) relative velocity ν; (iii) headway spacing from the proceeding vehicle ω. We note these features with the vector ϕ = {τ, ν, ω} ∈ R 3 and we refer to it as driving code. Then, we propose a prediction method, that given the vector ϕ, computes the IDM behavior parameters θ IDM for each vehicle. Next, we formally define the problem we seek to solve, then we present our prediction method. III. APPROACH Our goal is to model the behavior of human drivers. To do this, we focus on the two dimensional continuous action space: acceleration and steering. Specifically, we first compute the acceleration and steering of the modeled vehicle at each time step, and then using the transition model in Section III-D, we can generate trajectories for traffic participants. To model the acceleration of the human driven vehicle we use the IDM model in (1). This model, requires the IDM behavioral parameters, where we use a learning approach to predict the IDM parameters from their driving code. For steering we make the strong assumption that a car follows it's current lane, and to do this, we adopt the Pure Pursuit method from [23]. This assumption obviously makes errors whenever a car changes lanes, and provides room for improvement by less restricted models; however, we will show that with this restricted assumption, we gain a robustness that allows us to outperform more flexible models (see Section IV for details). The velocity and steering are then passed to transition model in (6) to allow iteration of our models and rollout trajectories in simulation. Two metrics we use in this work to measure the performance of our proposed algorithms and the baselines are average displacement error (ADE) and final displacement error (FDE). In particular, given ground truth trajectory, T g expressed as a sequence of physical states T g = {X t } T t=1 , we want to minimize the difference between the ground truth trajectory T g and the trajectory generated by our model T m . Two popular metrics for measuring this difference are: ADE(T g , T m ) = 1 T T t=0 (x m t − x g t ) 2 + (y m t − y g t ) 2 (3) FDE(T g , T m ) = (x m T − x g T ) 2 + (y m T − y g T ) 2 ,(4) where x t and y t are Cartesian positions of the vehicles. These two metrics in isolation can obscure important qualities of the behaviors, so we also include a collision count in the performance evaluation. Next, we formally state the problem we seek to solve in this work. A. Problem Definition We consider a human driver driving a car in a congested traffic scene. The goal is to accurately model the behavior of the driver so we can understand how the driver will respond to changes in the scene. Note that this is different from driving optimally as human drivers are in general not optimal, and an autonomous vehicle needs to interact with the participants on the road complete with whatever idiosyncrasies they might have. In the following we first express our prediction problem for learning the individualized IDM behavioral parameters in order to compute the acceleration at each time step, and then we present the method we adopt for computing the steering in Section III-C. For each vehicle, we note its IDM parameters with the vector θ IDM ∈ R 5 that models the velocity of the driver based on its current IDM state S IDM t . The IDM state vector S IDM t consists of the car's current velocity, v, and relative speed ∆v with respect to the car in front, and gap distance d t i.e., S IDM t = {v t , ∆v t , d t }. Therefore, for each vehicle, given the IDM state S IDM t and its behavioral parameter vector θ IDM , its acceleration can be expressed asv = IDM θIDM (S IDM t ). Next, we express the problem we are seeking to solve. Problem 3.1: Given a set of driving codes for each vehicle ϕ, can we learn a function f from expert data that maps ϕ to accurate IDM behavioral parameters θ IDM ? Formally, we seek to solve the following feasibility problem such that for each vehicle given its ground truth T g as well as feature vector ϕ: min f ADE(T g , T m )(5)s.t. θ IDM = f (ϕ) u t = IDM θIDM (S IDM t ), T m = {F t (X t , u t+1 )} T t=0 In particular, in our setting, given ϕ ∈ R 3 , we seek to find the function f : ϕ → θ IDM such that it minimizes the ADE in (3) of the modeled vehicle from its ground truth trajectory. B. Prediction Model For each vehicle i, we denote its IDM behavioral parameters and driving code with θ i IDM and ϕ i , respectively. Our goal is that for the vehicle i * , given its driving code ϕ i * as well as a set of IDM behavioral parameters and driving code pairs from other traffic participants, Θ = {(θ 1 IDM , ϕ 1 ), . . . , (θ N IDM , ϕ N )}, we predict an accurate IDM behavioral parameters θ i * IDM . We employ the idea that the driving code ϕ j , i.e., lateral lane displacement, relative velocity and headway spacing, is a sufficient representative for the driving behavior of the vehicle j in a highway. Because our set of driving codes ϕ has a low dimension, we use nearest neighbor prediction, as this is known to work well for low dimensional problems [24]. We use the k-nearest neighbors (KNN) algorithm to find the k closest driving codes from the set Θ to the ϕ i * . Then, we average their corresponding IDM behavioral parameters to compute θ i * IDM . Given θ i * IDM we compute acceleration using (1). C. Steering Because IDM only provides the acceleration, we need a way to determine our steering for a complete control algorithm. Using the assumption that the vehicle will not change lanes, we can select steering that will keep the vehicle centered in the lane. This could be accomplished with a standard control algorithm like PID, but in practice we use a vector based version of pure pursuit [23] which preserves heading angle [25] and is very easy to tune. D. Transition Model In order to iterate a scene and generate trajectories in simulation, we need to model the vehicle transitions. For this we use the kinematic bicycle model from [17]. Specifically, we define the physical state of the each vehicle at time t as X t = (x t , y t , ψ t , v t ) and we let u t = (v t , δ t ) represents the control input, wherev t is the acceleration input and δ t is the steering. Then we can write the transition model F t (X t , u t+1 ) as: β k = tan −1 lr l f + lr tan(δ k )(6) x k+1 = x k + v k cos(ψ k + β k )∆t y k+1 = y k + v k sin(ψ k + β k )∆t ψ k+1 = ψ k + v k lr sin(β k )∆t v k+1 = v k + a k ∆t, where ψ is the heading angle, l r and l f are the distances from the center of the mass to the front and rear of the vehicle, respectively. This allows us to simulate traffic agents into the future using plausible transition dynamics. While we could simulate an entire traffic scene this way, for evaluation purposes we place one modeled agent in a pre-recorded scene, which we describe in the next section. IV. EXPERIMENTS A. NGSIM data We use the Next Generation Simulation (NGSIM) 2 dataset for US Highway 101. NGSIM provides 45 minutes of driving at 10 Hz. This dataset covers an area in Los Angeles approximately 640m in length. In our experiment, we focus on the five mainline lanes, and we remove the data for auxiliary lanes for the highway entrance and exit. For training dataset, we consider the data from 7:50 a.m. to 8:05 a.m. which includes 1992 cars with unique ID. For testing dataset, B. Baseline methods In this section, we compare the performance of our proposed algorithm with four other baseline methods on the test data. We report the average and standard error for the performance of the algorithms over a 10 seconds horizon (100 frames) in Table I. Because the traffic vehicles are recorded they cannot respond/react to realistic behaviors that differ from the original vehicle's behavior, and this can result in collisions. Since plausible but different behaviors are not inherently bad for a model, we differentiate these collisions from cases where the model vehicle is at fault. In our results, we report the collisions where the model vehicle is at fault. Next, we describe the baseline methods. 1) Constant Velocity: While simple, a constant velocity method has been established as a good baseline approach for the straight driving in highways. In this method, the modeled car drives with a constant velocity with zero input action (i.e., with no acceleration and steering), and it does not receive any feedback from the environment. 2) IRL + RL: To have a less constrained learning agent, we learn behaviors of a traffic agent using Inverse Reinforcement Learning (IRL). In our implementations, we learn the reward function from human data. We collect trajectories of human (expert) drivers from the NGSim dataset, and use IRL to recover a reward function that explains the trajectories. To handle continuous state and action space, we employ continuous inverse optimal control [26]. We model the reward function as a linear combination of predefined features and learn the reward weights corresponding to each feature. Then, we apply the principle of maximum entropy [27] to optimize the reward weights in order to make the human demonstrations more likely. We display the heat map of the features we used in Figure 2. The features we include in the reward function are as follows: • distance to the middle of the lane • distance to the boundaries of the road • higher speed for moving forward • heading: in order to align the heading of the vehicle along with the road direction • collision avoidance: we define a non-spherical Gaussian distribution on each car using its positions and heading to compute the probability of collision to the other cars. The formulation of the non-spherical Gaussian is included in the Appendix V. Then, using the learned reward function, we train a Proximal Policy Optimization (PPO) algorithm [28] as our reinforcement learning agent. We trained the PPO agent using the training data over 1992 cars each with 10 seconds (100 frames) horizon. During the training process, we penalized the learning agent for colliding and driving off the road boundaries in order to reduce the state space exploration of the learning agent away from bad states. After training, we evaluate the performance of the trained model on the test data over 1484 cars for 10 seconds horizon. We report the results in I. 3) IDM Estimation (oracle): Here, we assume that the oracle has access to the full trajectory in the dataset (i.e., full-information method), and it estimates the IDM parameters using "L-BFGS-B" method in order to minimize the euclidean distance of the modeled car from the expert (ADE). In particular, we assume that for each expert in the training (or testing) dataset, the oracle has access to the entire 10 seconds (i.e., 100 frames) of the expert's trajectory, and it computes IDM parameters θ train IDM (or θ test IDM ) to learn driverspecific models for traffic participants. Then, we evaluate the performance of the optimized IDM parameters from the fullinformation method for the 10 seconds horizon and we report the results in Tables I. 4) IDM Average: To confirm that our prediction method is predicting something meaningful, we compare our approach against the average of the IDM parameters from the training data. We report its result in the Table I. Since our prediction approach outperforms the average model, it indicates that the parameter space has meaningful variation. To verify this, we visualize two of the parameters d 0 and d 1 of (2) in figure 1 (middle). Note that [7] identified d 1 as being important for modeling differences in driver behaviors. 5) IDM Prediction: IDM prediction is our proposed approach. Given the trajectories of the experts, we create a vector of driving code for each vehicle representing its driving style, ϕ. For each expert in the training dataset, the driving code is a vector of three dimensions that consists of the average of (i) velocity, (ii) offset from the main lane (iii), headway time that represents the time to travel from the expert vehicle to the proceeding vehicle, over the whole trajectory. We compute the driving code ϕ for experts in the test dataset, given the first 10 frames (1 second) of their trajectories. Then, for each expert in the test dataset, we use KNN to find the k closest driving codes from the set of Θ train from the training experts. Then, we averaged their corresponding IDM parameters (computed from the full-information algorithm on the training dataset) in order to compute our prediction of the IDM parameters for the experts in the test dataset. We note that this approach requires less information and is less computationally expensive than the full-information approach. On average, predicting the Note the parameter space is not uni-modal. Right: A visualization of the behavior or IRL. The modeled car drifts frighteningly close to other traffic participants. While this particular case did not lead to a collision, it is unnatural driving behavior that poorly models the true traffic participant. IDM behavioral parameters for each car in the test dataset, takes 0.001 seconds that shows its real-time applicability. We report the performance of the IDM prediction in Tables I. Moreover, in Table II, we show the effect of each combination of the driving code features on the performance of the IDM prediction method on the test data. In our prediction approach, we assume that we have access to the first 1 seconds (10 frames) of the trajectories in the test dataset. One might ask how robust is this prediction approach to the available information of the test dataset? To answer this, we evaluate the performance of our prediction approach using different number of available frames from the test data. We report the results in Table III. Moreover, in Figure 1 (left), we show the effect of the number of neighbors in KNN on the performance of our prediction approach. Then, we select K = 8 for the rest of our experiments. C. Discussion In Tables I we observe that the our IDM parameter prediction method is within half of a meter of the full optimization on the test data. Note that this is at high speed after a relatively long horizon of 10 seconds. Currently many deep learning prediction dataset predict less than 5 second horizons [2]. We observe approximately a meter of prediction improvement over the averaged IDM parameters. This suggests the parameter space cannot be modelled well with a fixed set of parameters. Figure 1 (middle) plots the models learned from the full-information process. We observe multiple linear trends, we observed similar trends in the other feature dimensions (not shown for space considerations). The clustering around the edges is caused by the optimization bounds. In Table II, we observe that the IDM Prediction method considering only a relative velocity feature as a driving code, has a performance improvement over the IDM average method. Also, we observe that headway spacing and lateral lane displacement features are sufficient for the IDM prediction to achieve a less than half a meter error in the longitude with respect to the oracle in a 10 seconds horizon. Additionally, we observe that RL, outperforms constant velocity on the test dataset. This is because RL benefits from having a learned reward model -as has been discussed in prior literature [29], explicitly modelling a reward function enables an agent to generalize better than methods like behavior cloning [30]. Because RL is less constrained it is able to accomplish many maneuvers not available to IDM, lane changes being one notable example. However this comes at the cost of RL taking much more risky behaviors and often colliding with traffic participants. We also observe that RL has a much greater variance than other methods. The extent of this is very pronounced as there are times when RL is right on top of the traffic agent for the whole run, and times when it speeds off through traffic like an aggressive motorcyclist. With the straightforward implementability, leveraging the proposed method could benefit the AD community in various aspects. Examples include (i) enhancing a simulation environment with more realistic settings for testing AD, comprehending the diversity of drivers' characteristics, (ii) enhancing baseline prediction modules (e.g., replacing naive constant velocity model) while not requiring an ample dataset or complex vehicle dynamics, and (iii) integrating it into a planning algorithm, accommodating the real-time applicability of IDM. One limitation is associated with the validation environment which neglects the inter-agent interactions. Specifically, the trained models (both IDM and RL) are tested by rolling out the behaviors in the (NGSim) data-set offline, without empowering traffic participants to react to the behavior changes of the trained vehicle. This is done for standardization, i.e. IDM might not collide with other IDM agents, but that doesn't mean IDM would be safe with IRL agents. By using recorded behavior, modelled agents are prevented from appearing safe because of other traffic agent's adjustments. It does, however, mean any benefits of interactivity are ignored, possibly to the agents detriment. Another limitation is inherited from the fact that IDM does not model a lanechanging behavior. Thus, IDM is often used together with a standalone lane changing model (e.g., MOBIL [31]). This modification is applicable to our proposed method. V. CONCLUSION In this work, we study the problem of learning/predicting driver-specific models to accurately model the behavior of traffic participants. We constrained this learning problem based on intelligent drive model (IDM), and we propose a prediction method that is able to efficiently learn the individualized IDM behavioral parameters from human data. We then compare the performance of our proposed method with other less-constrained data-driven methods such as reinforcement learning (RL) whose reward function has been learned from human data using inverse reinforcement learning (IRL). APPENDIX: ELLIPSOIDAL GAUSSIAN RISK MEASURE Similarly, we define the traffic vehicle as N (θ, Γ). The risk is then defined: risk = N (µ, Σ) N (θ, Γ) .(8) We let η be a two dimension vector of {x, y} grid points. We can then rewrite 8 as: Multiplying and combining terms we get: κ = η (Σ −1 + Γ −1 )η − η Σ −1 µ − µ Σ −1 η + µ Σ −1 µ − η Γ −1 θ − θ Γ −1 η + θ Γ −1 θ = η (Σ −1 + Γ −1 )η − 2η (Σ −1 µ + Γ −1 θ) + µ Σ −1 µ + θ Γ −1 θ . We then define: Ω = Σ −1 + Γ −1 −1 (11) v = Ω Σ −1 µ + Γ −1 θ .(12) This results in a new form: κ = η Ω −1 η − 2η Ω −1 )v + µ Σ −1 µ + θ Γ −1 θ = (η − v) Ω −1 (η − v) − v Ω −1 )v + µ Σ −1 µ + θ Γ −1 θ Next, since we know the integral of a Gaussian PDF: 1 2π \|Ω\| e −1 2 (η−v) Ω −1 (η−v) = 1 ,(13) the final analytic form for the risk of two Gaussian distributions, N (µ, Σ) and N (θ, Γ), can be written as: risk = \|Ω\| 2π \|Σ\|\|Γ\| e 1 2 (v Ω −1 v−µ Σ −1 µ−θ Γ −1 θ) .(14) Fig. 1 : 1Left: Effect of K on model accuracy. Middle: A visualization of model parameters learned by the offline optimization. Fig. 2 : 2Features used in IRL from the modeled vehicle point of view. We show features corresponding to (i) holding center of the lanes, (ii) respecting the road boundaries, (iii) avoiding collisions with the other cars. Fig. 3 : 3Ellipsoidal Gaussian Risk Measure We define an ellipsoidal Gaussian for the ego vehicle N (µ, Σ), where µ is the {x, y} position of the vehicle.Σ = cos(τ ) −sin(τ ) sin(τ ) cos(τ ) τ ) sin(τ ) −sin(τ ) cos(τ ) κ = (η − µ) Σ −1 (η − µ) + (η − θ) Γ −1 (η − θ) (10) TABLE I : ITest Results over 1484 carsMethod ADE FDE Collisions Const. vel 7.94 ±0.19 14.36 ±0.45 1467 IRL + RL 6.41 ±0.32 17.50 ±0.74 766 IDM Ave. 5.87 ±0.17 8.94 ±0.40 0 IDM Pred. 4.80 ±0.15 7.40 ±0.37 0 IDM Est. (Oracle) 4.38 ±0.15 7.39 ±0.36 0 we consider the data from 8:05 a.m. to 8:20 a.m. which includes 1521 cars with unique ID. In the test dataset, 8 cars have frame ID irregularities, that cause the cars to disappear, and 29 cars have the wrong car in front at some frames, causing IDM to be blind to the true vehicle in front. These cars are removed from the dataset. Therefore, our test dataset has 1484 cars with unique ID. TABLE II : IIEffect of Driving Code on IDM Pred.Method ADE FDE Collisions ϕ = {ν} 5.57 ±0.17 8.50 ±0.39 0 ϕ = {ω} 4.90 ±0.15 7.60 ±0.37 0 ϕ = {τ } 5.35 ±0.17 8.11 ±0.38 0 ϕ = {ν, ω} 4.95 ±0.15 7.60 ±0.37 0 ϕ = {ν, τ } 5.53 ±0.16 8.34 ±0.39 0 ϕ = {ω, τ } 4.76 ±0.15 7.35 ±0.37 0 ϕ = {ν, ω, τ } 4.80 ±0.15 7.40 ±0.37 0 TABLE III : IIIEffect of Frame NumberFrame Number ADE FDE Collisions 2 5.32 ±0.15 8.31 ±0.37 0 4 5.22 ±0.15 8.23 ±0.37 0 6 5.01 ±0.15 8.05 ±0.37 0 10 4.80 ±0.15 7.40 ±0.37 0 20 4.80 ±0.15 7.40 ±0.37 0 https://towardsdatascience.com/why-ade-and-fde-might-not-be-thebest-metrics-to-score-motion-prediction-model-performance-and-what-1980366d37be https://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm Drogon: A trajectory prediction model based on intention-conditioned behavior reasoning. C Choi, S Malla, A Patil, J H Choi, arXiv:1908.00024arXiv preprintC. Choi, S. Malla, A. 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[ "BAYESIAN ESTIMATION OF THERMONUCLEAR REACTION RATES FOR DEUTERIUM+DEUTERIUM REACTIONŚ", "BAYESIAN ESTIMATION OF THERMONUCLEAR REACTION RATES FOR DEUTERIUM+DEUTERIUM REACTIONŚ" ]	[ "A Gómez Iñesta \nDepartment of Physics\nUniversitat Politècnica de Catalunya\n08930BarcelonaSpain\n", "C Iliadis \nDepartment of Physics & Astronomy\nUniversity of North Carolina at Chapel Hill\nChapel Hill27599-3255NCUSA\n\nTriangle Universities Nuclear Laboratory\n27708-0308DurhamNCUSA\n", "A Coc \nCentre de Sciences Nucléaires et de Sciences de la Matière (CSNSM)\nUniv. Paris-Sud\nCNRS/IN2P3\nUniversité Paris-Saclay\nBâtiment 104F-91405Orsay CampusFrance\n" ]	[ "Department of Physics\nUniversitat Politècnica de Catalunya\n08930BarcelonaSpain", "Department of Physics & Astronomy\nUniversity of North Carolina at Chapel Hill\nChapel Hill27599-3255NCUSA", "Triangle Universities Nuclear Laboratory\n27708-0308DurhamNCUSA", "Centre de Sciences Nucléaires et de Sciences de la Matière (CSNSM)\nUniv. Paris-Sud\nCNRS/IN2P3\nUniversité Paris-Saclay\nBâtiment 104F-91405Orsay CampusFrance" ]	[]	The study of d+d reactions is of major interest since their reaction rates affect the predicted abundances of D, 3 He, and 7 Li. In particular, recent measurements of primordial D/H ratios call for reduced uncertainties in the theoretical abundances predicted by big bang nucleosynthesis (BBN). Different authors have studied reactions involved in BBN by incorporating new experimental data and a careful treatment of systematic and probabilistic uncertainties. To analyze the experimental data, Coc et al.(2015)used results of ab initio models for the theoretical calculation of the energy dependence of S-factors in conjunction with traditional statistical methods based on χ 2 minimization. Bayesian methods have now spread to many scientific fields and provide numerous advantages in data analysis. Astrophysical S-factors and reaction rates using Bayesian statistics were calculated byIliadis et al. (2016). Here we present a similar analysis for two d+d reactions, d(d,n) 3 He and d(d,p) 3 H, that has been translated into a total decrease of the predicted D/H value by 0.16%.	10.3847/1538-4357/aa9025	[ "https://arxiv.org/pdf/1710.01647v1.pdf" ]	54,502,242	1710.01647	24ff691ab6475cfdd82b04c41f49f433b2735281	BAYESIAN ESTIMATION OF THERMONUCLEAR REACTION RATES FOR DEUTERIUM+DEUTERIUM REACTIONŚ October 1, 2018 A Gómez Iñesta Department of Physics Universitat Politècnica de Catalunya 08930BarcelonaSpain C Iliadis Department of Physics & Astronomy University of North Carolina at Chapel Hill Chapel Hill27599-3255NCUSA Triangle Universities Nuclear Laboratory 27708-0308DurhamNCUSA A Coc Centre de Sciences Nucléaires et de Sciences de la Matière (CSNSM) Univ. Paris-Sud CNRS/IN2P3 Université Paris-Saclay Bâtiment 104F-91405Orsay CampusFrance BAYESIAN ESTIMATION OF THERMONUCLEAR REACTION RATES FOR DEUTERIUM+DEUTERIUM REACTIONŚ October 1, 20181 Draft version Preprint typeset using L A T E X style AASTeX6 v. 1.0methods: numerical -nuclear reactionsnucleosynthesisdeuteriumabundances -primor- dial nucleosynthesis The study of d+d reactions is of major interest since their reaction rates affect the predicted abundances of D, 3 He, and 7 Li. In particular, recent measurements of primordial D/H ratios call for reduced uncertainties in the theoretical abundances predicted by big bang nucleosynthesis (BBN). Different authors have studied reactions involved in BBN by incorporating new experimental data and a careful treatment of systematic and probabilistic uncertainties. To analyze the experimental data, Coc et al.(2015)used results of ab initio models for the theoretical calculation of the energy dependence of S-factors in conjunction with traditional statistical methods based on χ 2 minimization. Bayesian methods have now spread to many scientific fields and provide numerous advantages in data analysis. Astrophysical S-factors and reaction rates using Bayesian statistics were calculated byIliadis et al. (2016). Here we present a similar analysis for two d+d reactions, d(d,n) 3 He and d(d,p) 3 H, that has been translated into a total decrease of the predicted D/H value by 0.16%. INTRODUCTION Big bang nucleosynthesis (BBN) is responsible for the formation of primordial 2 H, 3 He, 4 He and 7 Li. Considering that the primordial abundances of these isotopes span more than eight orders of magnitude, there is a fair agreement between BBN predictions and observations (see Cyburt et al. (2016) for a recent review). In recent years the uncertainties have been greatly reduced on both the primordial abundances deduced from observations, and on the parameters entering into the BBN model. For instance, observations of the anisotropies of the cosmic microwave background (CMB), e.g. by the Planck space mission (Ade et al. 2016), led to precise estimations of cosmological parameters. In particular, the baryonic density of the Universe was measured with an uncertainty of less than 1%: Ω b ·h 2 = 0.02225±0.00016 (Ade et al. 2016). With this determination, the BBN model becomes parameter free and should be able to make accurate predictions. [email protected] [email protected] However, it is now widely known (see Fields (2011) for a review) that there is a factor of three difference between the calculated 7 Li/H ratio, by number of atoms (Cyburt et al. 2016;Coc et al. 2015), and the corresponding primordial value deduced from observations (Sbordone et al. 2010). The primitive lithium abundance is deduced from observations of low metallicity stars in the halo of our Galaxy, where the lithium abundance is almost independent of metallicity, displaying a plateau both as a function of metallicity and effective temperature. This puzzling discrepancy, known as the lithium problem, has not yet found a satisfactory solution (Coc 2016) and casts a shadow on the model. The uncertainty on the 4 He primordial abundance, which is deduced from the observation of metal-poor extragalactic H II regions, has been reduced by the inclusion of an additional atomic infrared line in the analysis (Aver et al. 2015). For this isotope, BBN predictions agree well with observations, keeping in mind that these predictions rely on the n↔p weak reaction rates. One should note that these calculated rates incorporate various corrections that need to be assessed. The weak rates are also normalized to the experimental neutron lifetime whose recommended value, τ n = 880.3±1.1 s (Olive et al. 2014), has evolved in the last few years (Young et al. 2014). Because of its low abundance, 3 He, has not been observed outside of our Galaxy (Bania et al. 2002). Since it is both produced and destroyed in stars, its galactic chemical evolution is uncertain. It is, hence, presently of little use to constrain BBN. However, the next generation of 30+ m telescope facilities may allow to extract the 3 He/ 4 He ratio from observations of extra-galactic metal poor HII regions (Cooke 2015). Deuterium's most primitive abundance is determined from the observation of few cosmological clouds at high redshift, on the line of sight of distant quasars. Up to a few years ago, there was a significant scatter in observations that lead to an ≈8% (Olive et al. 2012) uncertaininty on the primordial deuterium abundance. BBN prediction were, then, fully compatible with observations. However, recent measurements of primordial D/H, based on observations of damped Lymanα systems at high redshift, led to an uncertainty of 1.3%, D/H = (2.547±0.033)×10 −5 (Cooke et al. 2014(Cooke et al. , 2016. This has to be compared to the most recent predictions of (2.45±0.05)×10 −5 (Coc et al. 2015) and (2.58±0.04)×10 −5 (Cyburt et al. 2016) that quote a 1.6-2.0% uncertainty, but whose central values differ by 5%. However, this difference almost vanishes if the same rates are used for the d(p,γ) 3 He, d(d,n) 3 He and d(d,p) 3 H nuclear reactions (Tsung-Han Yeh, priv. comm.). These small, but significant, differences between obeservations and predictions require further investigations that are currently underway, in particular, the re-evaluations of reaction rates including the particle physics corrections to the weak rates, the comparison between numerical methods used in the network calculations and the comparison with other independent BBN codes and networks (e.g. Cyburt et al. (2016)). This paper concerns one important contribution to this goal, but others are needed before one is able to provide improved BBN predictions. This is why we will, here, only discuss relative effects of these new rates. An improved D/H predicion is also very important for the lithium problem since most proposed solutions lead to an unacceptable increase of the deuterium abundance (Olive et al. 2012;Kusakabe et al. 2014;Coc et al. 2015). Indeed, for the CMB deduced baryonic density, 7 Li is produced, during primordial nucleosynthesis, indirectly by 3 He(α, γ) 7 Be, where 7 Be will decay much later to 7 Li, while 7 Be is destroyed by 7 Be(n,p) 7 Li(p,α) 4 He. The solutions to the lithium problem generally rely on an increased late time neutron abundance to boost 7 Be destruction through the 7 Be(n,p) 7 Li(p,α) 4 He channel. These extra neutrons, inevitably, also boost the deuterium production through the 1 H(n,γ) 2 H channel. Hence, it is very important that the uncertainties on D/H predictions be reduced, because (i) the observational uncertainties of the primordial D/H ratio are smaller than those predicted by simulations, (ii) differences appear between predictions using different prescriptions for the reaction rates, and (iii) deuterium provides strong constraints to solutions of the lithium problem. The precision of these calculations is currently limited by our knowledge of certain key thermonuclear reaction rates. For example, a 10% error in the d(p,γ) 3 He, d(d,n) 3 He and d(d,p) 3 H rates causes a 3.2%, 5.4% and 4.6% uncertainty, respectively, in the predicted D/H ratio (Coc et al. 2015). The aim of our study is to reduce the uncertainties of BBN nucleosynthesis simulations as a continuation of our previous work that included the d(p,γ) 3 He rate (Iliadis et al. 2016). Both d(d,n) 3 He and d(d,p) 3 H are non-resonant reactions, meaning that the S-factor, S(E), varies smoothly with energy. We apply a Bayesian analysis to the most recent experimental d+d S-factor data, and use the resulting improved S-factors to calculate the reaction rates. The theoretical model used for the S-factor (Arai et al. 2011) is assumed to accurately predict the energy dependence but not necessarily its absolute scale. The experimental data is used to scale this S-factor curve. We carry out a multiparametric estimation. The model parameters are the scale factor for the theoretical S-factor (we will refer to it as "overall scale factor" or "scale factor") and a normalization factor for each data set accounting for systematic errors (we will refer to them as "normalization factors"). Hence, there is a total of 6 parameters for each reaction, since there is an overall scale factor and 5 normalization factors, one per data set. The Bayesian model provides a consistent description of all uncertainties involved (statistical and systematic), and yields the probability density for each parameter. Unlike traditional data analysis methods (e.g., Coc et al. 2015), it does not involve ad hoc assumptions or rely on Gaussian approximations for uncertainties. A more detailed explanation of this statistical analysis is given in Section 2. See Iliadis et al. (2016) for further information on these Bayesian models. STRATEGY: BAYESIAN STATISTICS AND MCMC We adopt the ab initio calculation of Arai et al. (2011) for the energy-dependence of the S-factor. This microscopic calculation uses a four-nucleon configuration space with a realistic nucleon-nucleon interaction. Their study was focused on low energies only, where partial waves up to J=2 contribute to the reaction cross section. Therefore, their calculation underestimates the data above a center-of-mass energy of 1 MeV. Consequently, we took only data points below an energy of 0.6 MeV into account in our Bayesian model. We analyzed S-factor data by means of Bayesian statistics and Markov chain Monte Carlo (MCMC) algorithms. We used the software JAGS ("Just Another Gibbs Sampler") (Plummer 2003), specifically the rjags package, withing the R language (R Core Team 2015). The inputs for the program are the experimental data (Brown et al. 1990;Greife et al. 1995;Krauss et al. 1987;Leonard et al. 2006) 1 , the theoretical nuclear model we want to scale, and the prior distributions of the model parameters (i.e., the scale factor of the theoretical Sfactor curve and the normalization factors of each data set). The way of constructing the Markov chain in this project is by a Metropolis-Hastings algorithm. Each step of the chain consists in a set of values for all six parameters (the overall scale factor and the normalization factors of five data sets). The transition from one step to another can be summarized as: 1. Given a state θ (i) , propose a new one θ by drawing a value from a proposal distribution (see Albert (2007)). 2. Accept the transition with a probability P(θ \|θ (i) )=min(1, P(θ \|S) P(θ (i) \|S) ), where S stands for the experimental S-factor data. Moreover, P(θ\|S) ∝ P(S\|θ) · π(θ), where P(S\|θ) is the likelihood function and π(θ) is the prior distribution of the parameters. They are explained in Section 2.1. If the transition is accepted, θ (i+1) = θ . If not, θ (i+1) = θ (i) 4. Repeat 1-3. When the Markov chain reaches the steady state, the values of the parameters taken at every step yield their posterior distributions. With that information, lately it was possible to estimate the reaction rates. For more information about this method, see the Appendices in Iliadis et al. (2016). As a general reference in this topic, see Hilbe (2017). The likelihood distribution of the S-factor given a set of parameters and prior distributions of those parameters are needed to compute the acceptance probabilities in the Markov chain. The central limit theorem states that the probability density function resulting from the sum of independent random variables tends to a Gaussian distribution. By extension, a product of random variables will follow a lognormal distribution. Measured nuclear reaction cross sections and astrophysical S-factors result from the product (or ratios) of different physical quantities. Thus we can assume that the likelihood function for the S-factor (P(S\|θ) in Section 2) will follow a lognormal distribution (Longland et al. 2010): Likelihood and prior distributions Energy (MeV) S−Factor (MeV b) 0.01 0.1 0.05 0.1 0.2 • Leo06 Gre95 Bro90 Kra87(B) Kra87(M) d(d,p) 3 H • • • • • • • •f (x) = 1 σ √ 2πx e −(lnx−µ) 2 /(2σ 2 ) , x > 0 (1) µ = ln(E[x]) − 1 2 ln 1 + V [x] E[x] 2 σ = ln 1 + V [x] E[x] 2 where µ is the location parameter for the normally distributed logarithm of random variable x, i.e., e µ is the median of the distribution of x, and σ is the spread parameter for the normally distributed logarithm of x; E[x] and V[x] denote the expected mean value and the variance, respectively, of the lognormal distribution. One advantage of this type of distribution is that negative S-factor values, which are unphysical, are not allowed. The lognormal likelihood function is then given by: P(S\|f ) = N i=1 1 S i 2πσ 2 L;i exp (ln S i − µ i ) 2 2σ 2 L;i (2a) µ i = ln (f n f s S th ) − 1 2 ln 1 + σ 2 i (f n f s S th ) 2 (2b) σ 2 L;i = ln 1 + σ 2 i (f n f s S th ) 2 (2c) where S i stands for the experimental S-factor data, f are the sampled parameters (f n is the normalization factor for a particular data set and f s is the overall scale factor), N is the number of measurements of the data set, µ i is the location parameter of data point i, σ L;i is the spread parameter of data point i, S th corresponds to the theoretical S-factor and σ i is the reported standard devi-ation of data point i. Notice that there is no degeneracy regarding the product f n ·f s , since f n is different for each data set while f s is the same parameter throughout. Since the scale factor, f s , is expected to be close to unity, we assume for the overall scaling factor a noninformative prior (π(θ) in Section 2), i.e., a normally distributed probability density with a mean of zero and a standard deviation of 100. Therefore we expressed the prior for the scale factor as: π(f s ) =        1 √ 2π100 2 exp (fs−0.0) 2 2·100 2 , for f s > 0 0 for f s ≤ 0 (3) The distribution was truncated at zero since the scaling factor must be a positive quantity. To test the sensitivity of our results, we repeated the analysis using different priors (e.g., uniform distributions and gamma functions), and the results were very similar in all cases. For the normalization factors of each data set, we assumed highly informative priors. It is discussed in Section 2.2. Additionally, we incorporate a robust regression method to avoid the bias that outliers can introduce in the results. Our algorithm accomplishes this by detecting possible outliers (i.e., measurements with overoptimistic uncertainties) and reducing their influence in the analysis (see Section 2.3). Systematic uncertainties A measurement is usually subject to statistical and systematic uncertainties. Statistical uncertainties are inherent to any physical process and cannot be avoided. They can be reduced by combining results from different measurements, leading to different measured values for the same experimental conditions. Conversely, systematic uncertainties will not change if the experimental conditions remain the same. Hence, all of the data points from the same measurement will likely be affected by a systematic effect in a similar manner. We introduce statistical uncertainties in our model by assuming lognormal priors for the individual normalization factors, f n;k , of all five data sets, k. The experimental data considered in this study (Brown et al. 1990;Greife et al. 1995;Krauss et al. 1987;Leonard et al. 2006) provided systematic uncertainties for each data set as normalization factor uncertainties 1+ , with given in Table II of Coc et al. (2015). We include in our Bayesian model a systematic effect as a highly informative, lognormal prior. The parameters of this distribution are a median of 1.0, i.e., e µ = 1, and a systematic factor uncertainty of e σ k . This prior can be written as: π(f n;k ) = 1 f n;k 2π(ln(e σ k )) 2 exp (ln f n;k − ln(1.0)) 2 2(ln(e σ k )) 2 (4) For more information on this choice of prior, see Iliadis et al. (2016). Robust regression Outliers can bias the data analysis significantly and thus need to be treated carefully. In our JAGS code, we model outliers as data points with over-optimistic reported uncertainties. The algorithm designates each data point as either having believable uncertainty (i.e., not an outlier) or over-optimistic uncertainty (i.e., outlier). This operation is done for each step of the chain. Ultimately, data points having smaller outlier probabilities are more heavily weighted in the final results, thus reducing the statistical weight of the outliers (Andreon & Weaver 2015). For the presentation of these results, we average the outlier probabilities for all data points in a given set and list the values in Tables 1 and 2. BAYESIAN ASTROPHYSICAL S-FACTORS The astrophysical S-factor of a nuclear reaction is defined as: S(E) ≡ σ(E)Ee 2πη(5) where σ(E) is the cross-section of the reaction at the center-of-mass energy E and e 2πη is the Gamow factor, which depends on the charges of the projectile and the target, the relative atomic masses, and the energy E (see Iliadis (2015) for details). The theoretical model used here for the energy dependence of the d+d S-factor is based on a multichannel ab initio calculation (Arai et al. 2011). We assume that the nuclear model accurately predicts the energy dependence of the S-factor, but not necessarily its absolute scale. Our model predicts the best estimate of the overall scale factor and its uncertainty. The Bayesian model for the analysis of the S-factor has several parameters. These include the normalization factors for each of the five individual data sets as well as the overall scale factor of the theoretical S-factor curve. We employ the same procedure as Iliadis et al. (2016), and we use three different Markov chains of 7500 steps each, with a burn-in of 2000 steps. These values ensure the convergence of the chains and that the Monte Carlo fluctuations are negligible compared to the statistical and systematic uncertainties. We performed several tests with different chain lengths (e.g., 75000 steps) and the results were the same. Traditional methods based on χ 2 minimization have been applied to the calculation of the d+d reaction rates by Coc et al. (2015). In their analysis, they assumed that the scale factor is given by the weighted average of the normalization factors that independently fit each data set to the theoretical S(E) curve. They made a number of ad hoc assumptions to include systematic errors in their analysis and assumed Gaussian approximations for the uncertainties (see Appendix A in Coc et al. (2015)). Their results were deemed satisfactory by the authors, since the reduced χ 2 was always close to unity. Bayesian S-factors are shown in Figure 1 for d(d,n) 3 He and Figure 2 for d(d,p) 3 H. Grey lines represent credible S-factor curves for different sets of parameters, yielding the shaded region. All of the credible S-factors are close to the median value (blue line). The red lines correspond to the 16th and 84th percentiles. Results from our Bayesian analysis, and the traditional method (Coc et al. 2015) for comparison, are shown in Tables 1 and 2 for the d(d,n) 3 He and d(d,p) 3 H reactions, respectively. Some of the results are also displayed in Figures 3 and 4, where the red data points correspond to the present Bayesian method and the black data points correspond to the traditional χ 2 minimization. The top panels (labeled as "Scale factor") display the overall scale factor. For both reactions, the scale factors are in agreement. It can also be seen that the scale factors are smaller than unity (see Tables 1 and 2), i.e., the theoretical S-factor curve exceeds the data. The bottom regions (labeled as "Normalization factors") of Figures 3 and 4 show the normalization factors of each data set. It can be seen that the Bayesian normalization factors are consistently larger than the traditional analysis values. This is caused by the different methods to calculate these factors, as explained below. In the Bayesian approach, the theoretical S-factor is multiplied by the overall scale factor. We defined our Bayesian model so that each data set is the result of multiplying the scaled S-factor curve by a normalization factor. As explained before, this normalization factor includes the effect of systematic uncertainties. Hence, at each step of the Markov chain, there is a shift in the magnitude of the theory (scaling) and the data sets (to account for the systematic uncertainties). These shifts are performed by multiplying the S-factor theoretical curve by the overall scale factor and dividing each data set by its corresponding normalization factor. Each measurement is affected by a multiplicative error (S exp = f n · S true , where S exp is the experimental datum, f n is the normalization factor and S true is the actual value), so we must divide the experimental value by the normalization factor if we want to cancel it. In this way, the final probability density function for each parameter is influenced by all other parameters. In the traditional analysis performed by Coc et al. (2015), however, the theoretical S-factor is multiplied by a normalization factor for each data set separately. The overall scale factor is then obtained by computing the weighted average of all normalization factors. The systematic uncertainties are introduced in the weights of the average by adding systematic and statistical errors quadratically for each data set (see Eq. (A8) in Coc et al. (2015)). To explain the discrepancies between traditional and Bayesian normalization factors, consider the data presented in Figure 5. This figure shows the measured d(d,n) 3 He S-factors of each data set. The solid curve shows the ab initio S-factor of Arai et al. (2011) before scaling. At each step of the Markov chain, the Bayesian model suggests a new value smaller than unity for the overall scale factor, to displace the curve downwards. The model also samples a new normalization factor for each data set. As an example, look at the suggested normalization factor for Kra (B) in Table 1 (0.922±0.024). It is less than unity since these experimental points should be shifted upwards to correct the effect of the systematic errors. Moreover, each normalization factor is influenced by the overall scale factor: at each step, the normalization factor fits the data to the scaled theory. In the traditional analysis, each normalization factor is calculated independently to fit the original S-factor. In the case of Kra (B), the traditional normalization factor needs to perform a larger shift, i.e. it will be further away from unity. This means a smaller normalization factor in the traditional case than in the Bayesian one. (Leonard et al. 2006), (Greife et al. 1995), (Brown et al. 1990), and (Krauss et al. 1987 a Uncertainties derived from the 16th, 50th, and 84th percentiles. b Data from Coc et al. (2015). c Reference labels of data sets: Leo06 (Leonard et al. 2006), Gre95 (Greife et al. 1995), Bro90 (Brown et al. 1990), Kra87(B) (Krauss et al. 1987), and Kra87(M) (Krauss et al. 1987). d Number of points of each data set. e Normalization factor for each data set (see explanation in text). f Probability that the reported experimental uncertainty is over-optimistic. Calculated from average outlier probabilities of all data points in a given data set. g Normalization factor for each data set (see explanation in text). Uncertainties given represent 1σ. a Uncertainties derived from the 16th, 50th, and 84th percentiles. b Data from Coc et al. (2015). c Reference labels of data sets: Leo06 (Leonard et al. 2006), Gre95 (Greife et al. 1995), Bro90 (Brown et al. 1990), Kra87(B) (Krauss et al. 1987), and Kra87(M) (Krauss et al. 1987). d Number of points of each data set. e Normalization factor for each data set (see explanation in text). f Probability that the reported experimental uncertainty is over-optimistic. Calculated from average outlier probabilities of all data points in a given data set. g Normalization factor for each data set (see explanation in text). Uncertainties given represent 1σ. REACTION RATES The thermonuclear reaction rate per particle pair, N A σv , can be written as: N A σv = 8 πm 01 1/2 N A (kT ) 3/2 ∞ 0 e −2πη S(E)e −E/kT dE (6) where m 01 is the reduced mass of projectile and target, N A represents Avogadro's constant, and the product of Boltzmann constant, k, and plasma temperature, T , is given by kT = 0.086173324 T 9 (MeV)(7) with the temperature, T 9 , in units of GK (see Iliadis (2015) for details). The reaction rates are calculated by numerical integration of Eq. (6) for each set of parameters sampled by the Markov chain, at 60 different temperatures between 1 MK and 10 GK. The reaction rate probability densities at selected temperatures are shown in Figures 6 and 7 in red. The blue lines correspond to a lognormal approximation (Longland et al. 2010), for convenient implementation of the rates in libraries such as STARLIB (Sallaska et al. 2013). Numerical reaction rate values are listed in Table 3. The recommended rates are computed as the 50th percentile of the probability density, while the rate factor uncertainty, f.u., is obtained from the 16th and 84th percentiles. The lognormal parameters, µ and σ, can be calculated from the recommended (median) rate (x med = e µ ) and the factor uncertainty (f.u. = e σ ; for a coverage probability of 68%). The rate factor uncertainty is 1.1% for both reactions at most temperatures. The present rates for d(d,n) 3 He and d(d,p) 3 H agree with the results of Coc et al. (2015) within 1% at most temperatures. However, our rates are more than 15% larger than those of Coc et al. (2015) at very low temperatures (near 1 MK). This is caused by a low-energy cutoff that is too high for the numerical integration of the rates in the previous analysis. The theoretical model of Arai et al. (2011) only applies to low energies, and thus we can derive Bayesian reaction rates only up to a temperature of 2 GK. The results in Table 3 for higher temperatures, shown in italics, are adopted from Coc et al. (2015). The most important temperatures for BBN are near 1 GK, corresponding to an effective kinetic energy range of <250 keV for the d+d reactions. The last step is to calculate the effect of the new reaction rates on the predicted primordial D/H ratio. The Bayesian mean value for the scale factor is larger by 0.21% for d(d,n) 3 He and larger by 0.12% for d (d,p) 3 H compared to Coc et al. (2015). The discrepancies of both reaction rates (0.21% and 0.12%, respectively), weighted by the sensitivity of the D/H abundance ratio to each reaction rate variation (-0.54 and -0.46, respectively) , result in a 0.113% and 0.055% decrease of the central D/H value. Fortuitously, the uncertainties on the scale factors (see Tables 1 and 2) are almost identical to the former ones (Coc et al. 2015). Hence, when using these two new reaction rates, instead of the Coc et al. (2015) ones, this translates to a 0.16% decrease of the predicted D/H value, while its total uncertainty remains unchanged at 2.0%. Half of this error budget originates from the d(p,γ) reaction rate and it would be premature to update the D/H value before new measurements concerning this reaction, done at LUNA, are published (see Mossa (2017)). Only after these new data are made available and the investigation of other sources of uncertainties (numerical, correction to weak rates,...) are completed, it will be relevant to provide new predictions of D/H. 0.008 6.074E+00 1.011 6.198E+00 1.011 Table 3 continued Table 3 continued a Reaction rates in units of cm 3 mol -1 s -1 , corresponding to the 50th percentile of the rate probability density function. The rate factor uncertainty, f.u., is obtained from the 16th and 84th percentiles (see text). The parameters µ and σ of the lognormal approximation to the reaction rate are given by x med = e µ and f.u. = e σ , respectively, where x med denotes the median rate. Values for T > 2 GK, shown in italics, are adopted from Coc et al. (2015). CONCLUSIONS We presented improved reaction rates for d(d,n) 3 He and d(d,p) 3 H based on the Bayesian method discussed in Iliadis et al. (2016). Unlike previous methods that were based on traditional statistics (i.e., χ 2 minimization), our method does not rely on weighted averages or the quadratic addition of systematic and statistical errors. For both reactions, the rate factor uncertainty is 1.1% and agrees with the traditional results. However, the Bayesian scale factors by which the theory needs to be multiplied to fit the data are larger than those of Coc et al. (2015). We obtained scale factors which are 0.20% larger for d(d,n) 3 He and 0.12% larger for d(d,p) 3 H. This translates to a 0.16% decrease of the predicted D/H value, while its total uncertainty remains unchanged at 2.0%. This shows the robustness of the deuterium predictions, provided that the same experimental data and nuclear model are used. It leaves very little room for those solutions to the lithium problem that cannot avoid an increase in D/H. It also calls for improved theoretical calculations. The theoretical work of Arai et al. (2011), used here, was focused on low ener-gies and does not correctly reproduce the experimental data above ≈600 keV. It is highly desirable that these calculations be extended up to ≈2 MeV, to cover the range of experimental data. Here we presented the first statistically rigorous results for d+d reaction rate probability densities. These can be employed in future Monte Carlo studies of big bang nucleosynthesis. where the lognormal parameters µ ('mu") and σ ("sig") are directly calculated from the expectation value and variance of all rate samples, ln(NA σv i ), at a given temperature. T9 is the temperature in GK. Figure 1 . 1Astrophysical S-factor versus center-of-mass energy for the d(d,n) 3 He reaction. The symbols show the data ofLeonard et al. (2006) (circles),Greife et al. (1995) (diamonds),Brown et al. (1990) (squares),Krauss et al. (1987) (B) (down-pointing triangles) andKrauss et al. (1987) (M) (up-pointing triangles). The error bars (1σ) refer to statistical uncertainties only. Grey lines forming the shaded area correspond to credible S-factors that result from different sets of parameter samples (the inset shows a magnification for a clearer view of these lines). The blue line is the median (50th percentile) of all credible S-factors, and red lines correspond to the 16th and 84th percentiles. The credible lines are calculated from the theoretical S-factor ofArai et al. (2011), multiplied by a scale factor that is a parameter of the Bayesian model. Figure 2 . 2Astrophysical S-factor versus center-of-mass energy for the d(d,p) 3 H reaction. The symbols show the data ofLeonard et al. (2006) (circles),Greife et al. (1995) (diamonds),Brown et al. (1990) (squares),Krauss et al. (1987) (B) (down-pointing triangles) andKrauss et al. (1987) (M) (up-pointing triangles). The error bars (1σ) refer to statistical uncertainties only. Grey lines forming the shaded area correspond to credible S-factors that result from different sets of parameter samples (the inset shows a magnification for a clearer view of these lines). The blue line is the median (50th percentile) of all credible S-factors, and red lines correspond to the 16th and 84th percentiles. The credible lines are calculated from the theoretical S-factor ofArai et al. (2011), multiplied by a scale factor that is a parameter of the Bayesian model. Figure 3 .Figure 4 . 34Results for d(d,n) 3 He. (Top) Overall scale factor for the theoretical S-factor. (Bottom) Normalization factors of each data set:Leonard et al. (2006) (Leo06),Greife et al. (1995) (Gre95),Brown et al. (1990) (Bro90),Krauss et al. (1987) (Kra87 (B) and Kra87 (M)). Present and previous(Coc et al. 2015) results are shown in red and black, respectively. The range indicated in red corresponds to the 68% credible interval of the posterior. The range indicated in black shows the 68% confidence interval of the traditional analysis. Results for d(d,p) 3 H. (Top) Overall scale factor for the theoretical S-factor. (Bottom) Normalization factors of each data set:Leonard et al. (2006) (Leo06),Greife et al. (1995) (Gre95),Brown et al. (1990) (Bro90),Krauss et al. (1987) (Kra87 (B)and Kra87 (M)). Present and previous(Coc et al. 2015) results are shown in red and black, respectively. The range indicated in red corresponds to the 68% credible interval of the posterior. The range indicated in black shows the 68% confidence interval of the traditional analysis. Figure 5 . 5Astrophysical S-factor versus center-of-mass energy for d(d,n) 3 He. Experimental points are from h Reduced χ 2 .i Best estimate for the scale factor of the theoretical S-factor fromArai et al. (2011). h Reduced χ 2 .i Best estimate for the scale factor of the theoretical S-factor fromArai et al. (2011). Figure 7 . 7Reaction rate probability density of d(d,p) 3 H at different temperatures. The rate samples (red histograms) are computed using the S-factor samples obtained from the Bayesian analysis. Blue curves represent lognormal approximations, ). The error bars (1σ) refer to statistical uncertainties only. The solid curve shows the ab initio S-factor ofArai et al. (2011) before scaling.Table 1. Results for the d(d,n) 3 He reaction. Data Present a Previous b Ref. c n d norm e outlier f norm g χ 2 ν h Leo 06 8 0.978 +0.012 −0.011 55.1% 0.933 ± 0.007 2.033 Gre 95 8 1.045 +0.017 −0.017 45.4% 1.016 ± 0.013 1.247 Bro 90 9 1.004 +0.010 −0.010 64.6% 0.964 ± 0.003 2.366 Kra 87 (B) 7 0.922 +0.024 −0.024 35.3% 0.868 ± 0.022 0.292 Kra 87 (M) 20 0.964 +0.021 −0.021 27.6% 0.919 ± 0.018 0.624 Quantity Present a Previous b Scale factor i : 0.961 +0.010 −0.010 0.959 ± 0.010 (χ 2 ν = 1.33) S(0) (keVb): 51.70 +0.54 −0.51 Table 2 . 2Results for the d(d,p) 3 H reaction.Data Present a Previous b Ref. c n d norm e outlier f norm g χ 2 ν h Leo 06 8 0.989 +0.013 −0.013 80.4% 0.942 ± 0.006 5.376 Gre 95 8 1.034 +0.017 −0.017 30.6% 0.997 ± 0.013 0.999 Bro 90 9 1.002 +0.011 −0.010 51.9% 0.958 ± 0.002 1.969 Kra 87 (B) 7 0.921 +0.023 −0.023 21.1% 0.864 ± 0.021 0.100 Kra 87 (M) 20 0.944 +0.020 −0.020 8.6% 0.890 ± 0.017 0.177 Quantity Present a Previous b Scale factor i : 0.956 +0.010 −0.011 0.955 ± 0.010 (χ 2 ν = 1.33) S(0) (keVb): 53.26 +0.55 −0.59 Table 3 . 3Present recommended reaction rates. ad(d,n) 3 He d(d,p) 3 H T (GK) Rate f.u. Rate f.u. 0.001 1.322E-08 1.011 1.364E-08 1.011 0.002 5.489E-05 1.011 5.653E-05 1.011 0.003 3.025E-03 1.011 3.110E-03 1.011 0.004 3.737E-02 1.011 3.835E-02 1.011 0.005 2.214E-01 1.011 2.269E-01 1.011 0.006 8.556E-01 1.011 8.755E-01 1.011 0.007 2.508E+00 1.011 2.563E+00 1.011 Table 3 3(continued) d(d,n) 3 He d(d,p) 3 H T (GK) Rate f.u. Rate f.u. 0.009 1.280E+01 1.011 1.304E+01 1.011 0.010 2.427E+01 1.011 2.471E+01 1.011 0.011 4.242E+01 1.011 4.314E+01 1.011 0.012 6.945E+01 1.011 7.055E+01 1.011 0.013 1.078E+02 1.011 1.094E+02 1.011 0.014 1.602E+02 1.011 1.624E+02 1.011 0.015 2.293E+02 1.011 2.322E+02 1.011 0.016 3.183E+02 1.011 3.220E+02 1.011 0.018 5.674E+02 1.011 5.729E+02 1.011 0.020 9.321E+02 1.011 9.395E+02 1.011 0.025 2.507E+03 1.011 2.516E+03 1.011 0.030 5.307E+03 1.011 5.305E+03 1.011 0.040 1.570E+04 1.011 1.558E+04 1.011 0.050 3.373E+04 1.011 3.325E+04 1.011 0.060 6.020E+04 1.011 5.900E+04 1.011 0.070 9.539E+04 1.011 9.298E+04 1.011 0.080 1.392E+05 1.011 1.350E+05 1.011 0.090 1.914E+05 1.011 1.847E+05 1.011 0.100 2.516E+05 1.011 2.418E+05 1.011 0.110 3.194E+05 1.011 3.056E+05 1.011 0.120 3.943E+05 1.011 3.758E+05 1.011 0.130 4.759E+05 1.011 4.518E+05 1.011 0.140 5.638E+05 1.011 5.334E+05 1.011 0.150 6.575E+05 1.011 6.199E+05 1.011 0.160 7.568E+05 1.011 7.111E+05 1.011 0.180 9.702E+05 1.011 9.061E+05 1.011 0.200 1.201E+06 1.011 1.116E+06 1.011 0.250 1.843E+06 1.011 1.691E+06 1.011 0.300 2.555E+06 1.011 2.321E+06 1.011 0.350 3.318E+06 1.011 2.988E+06 1.011 0.400 4.118E+06 1.011 3.681E+06 1.011 0.450 4.944E+06 1.011 4.391E+06 1.011 0.500 5.788E+06 1.011 5.113E+06 1.011 0.600 7.510E+06 1.011 6.573E+06 1.011 0.700 9.251E+06 1.011 8.036E+06 1.011 0.800 1.099E+07 1.011 9.489E+06 1.011 0.900 1.271E+07 1.011 1.092E+07 1.011 1.000 1.440E+07 1.011 1.233E+07 1.011 1.250 1.850E+07 1.011 1.572E+07 1.011 1.500 2.236E+07 1.011 1.893E+07 1.011 1.750 2.599E+07 1.011 2.194E+07 1.011 2.000 2.938E+07 1.011 2.477E+07 1.011 Table 3 ( 3continued) d(d,n) 3 He d(d,p) 3 H T (GK) Rate f.u. Rate f.u. 2.500 3.546E+07 1.012 2.976E+07 1.013 3.000 4.093E+07 1.014 3.440E+07 1.014 3.500 4.585E+07 1.014 3.863E+07 1.014 4.000 5.031E+07 1.015 4.251E+07 1.015 5.000 5.816E+07 1.016 4.946E+07 1.016 6.000 6.488E+07 1.017 5.552E+07 1.017 7.000 7.072E+07 1.018 6.077E+07 1.018 8.000 7.583E+07 1.018 6.529E+07 1.018 9.000 8.037E+07 1.018 6.912E+07 1.018 10.000 8.437E+07 1.018 7.228E+07 1.019 Reaction rate (cm 3 mol −1 s −1 )Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 ) Reaction rate (cm 3 mol −1 s −1 )Reaction rate (cm 3 mol −1 s −1 )1.28e−08 1.32e−08 1.36e−08 0.0e+00 5.0e+08 1.0e+09 1.5e+09 2.0e+09 2.5e+09 3.0e+09 Probability (arb. units) T9=0.001 mu=−18.1418 sig=0.0108714 33000 33500 34000 34500 35000 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 Probability (arb. units) T9=0.05 mu=10.426 sig=0.0108714 14000000 14400000 14800000 0.0e+00 5.0e−07 1.0e−06 1.5e−06 2.0e−06 2.5e−06 3.0e−06 Probability (arb. units) T9=1 mu=16.483 sig=0.0108714 23.5 24.0 24.5 25.0 0.0 0.5 1.0 1.5 Probability (arb. units) T9=0.01 mu=3.18937 sig=0.0108714 245000 250000 255000 260000 0.00000 0.00005 0.00010 0.00015 Probability (arb. units) T9=0.1 mu=12.4357 sig=0.0108714 7.7e+07 7.9e+07 8.1e+07 0e+00 1e−07 2e−07 3e−07 4e−07 5e−07 Probability (arb. units) 1.32e−08 1.36e−08 1.40e−08 0.0e+00 5.0e+08 1.0e+09 1.5e+09 2.0e+09 2.5e+09 3.0e+09 Probability (arb. units) T9=0.001 mu=−18.1105 sig=0.0108714 32500 33000 33500 34000 34500 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 Probability (arb. units) T9=0.05 mu=10.4119 sig=0.0108714 12000000 12400000 0.0e+00 5.0e−07 1.0e−06 1.5e−06 2.0e−06 2.5e−06 3.0e−06 Probability (arb. units) T9=1 mu=16.3276 sig=0.0108714 24.0 24.5 25.0 25.5 0.0 0.5 1.0 1.5 Probability (arb. units) T9=0.01 mu=3.20731 sig=0.0108714 235000 240000 245000 250000 0.00000 0.00005 0.00010 0.00015 Probability (arb. units) T9=0.1 mu=12.3958 sig=0.0108714 6.8e+07 7.0e+07 7.2e+07 0e+00 1e−07 2e−07 3e−07 4e−07 5e−07 6e−07 Probability (arb. units) T9=10 mu=18.0597 sig=0.0108714 The experiments ofKrauss et al. (1987) took place in Münster and at Bochum and so both data sets are considered independently. ACKNOWLEDGEMENTSWe would like to thank Jordi José, Jack Dermigny, Rafa De Souza, Lori Downen and Sean Hunt for their support and feedback. One of us (AGI) would like to express his gratitude to the Department of Physics and Astronomy for hospitality during his visit to UNC-CH, where this project was started. This work was supported in part by NASA under the Astrophysics Theory Program grant 14-ATP14-0007 and the U. . P A R Ade, Planck Collaboration XIIIAstron. & Astrophys. 59413P. A. R. Ade et al. (Planck Collaboration XIII), 2016, Astron. & Astrophys., 594, A13 Bayesian computation with R. J Albert, SpringerNew York1st EditionAlbert, J. (2007), Bayesian computation with R, (1st Edition, New York, Springer) . S Andreon, B Weaver, Bayesian Methods for the Physical Sciences. 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[ "QUANTITATIVE RECURRENCE OF SOME DYNAMICAL SYSTEMS WITH AN INFINITE MEASURE IN DIMENSION ONE", "QUANTITATIVE RECURRENCE OF SOME DYNAMICAL SYSTEMS WITH AN INFINITE MEASURE IN DIMENSION ONE" ]	[ "Nasab Yassine " ]	[]	[]	We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of Z-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs.	10.3934/dcds.2018017	[ "https://arxiv.org/pdf/1609.05791v1.pdf" ]	119,577,351	1609.05791	1ee50c6fabb67f4960f80d5c6bf1ee5ee25fecc8	QUANTITATIVE RECURRENCE OF SOME DYNAMICAL SYSTEMS WITH AN INFINITE MEASURE IN DIMENSION ONE 19 Sep 2016 Nasab Yassine QUANTITATIVE RECURRENCE OF SOME DYNAMICAL SYSTEMS WITH AN INFINITE MEASURE IN DIMENSION ONE 19 Sep 2016 We are interested in the asymptotic behaviour of the first return time of the orbits of a dynamical system into a small neighbourhood of their starting points. We study this quantity in the context of dynamical systems preserving an infinite measure. More precisely, we consider the case of Z-extensions of subshifts of finite type. We also consider a toy probabilistic model to enlight the strategy of our proofs. Introduction The quantitative recurrence properties of dynamical systems preserving a probability measure have been studied by many authors since the work of Hirata [6]. Some properties are defined by estimating the first return time of a dynamical system into a small neighbourhood of its starting point. Results in this concern have been described in [14], let us mention works in this situation [1,15]. This question has been less investigated in the context of dynamical systems preserving an infinite measure. In [3], Bressaud and Zweimüller have established first results of quantitative recurrence for piecewise affine maps of the interval with infinite measure. The case of Z 2 -extension of mixing subshifts of finite type has been investigated in [11]. Results have been also established for random walks on the line [12], for billiards in the plane [10] and for null-recurrent Markov maps in [13]. A measure-preserving dynamical system is given by (X, B, µ, T ) where (X, B) is a measurable set, µ is a finite or σ-finite positive measure and T : X → X is a measurable transformation preserving the measure µ (i.e. µ(T −1 A) = µ(A), for every A ∈ B). We are interested in the case where µ is σ-finite. We assume that X is endowed with some metric d X and that B contains the open balls B(x, r) of X. Our interest is in the first time the orbit comes back close to its initial position. For every y ∈ X, we define the first return time τ ǫ of the orbit of y in the ball B(y, ǫ) as: τ ǫ (y) := inf{n ≥ 1 : T n (y) ∈ B(y, ǫ)} ∈ N ∪ {+∞}. We consider conservative dynamical systems, that is dynamical systems for which the conclusion of the poincaré theorem is satisfied. This ensures that, for every ǫ > 0, τ ǫ < ∞, µ almost everywhere. The main goal of this article is to study the behavior of τ ǫ as ǫ → 0. A classical example of dynamical systems preserving an infinite measure is given by Z-extensions of a probability-preseving dynamical system. Given a probability-preserving dynamical system (X,B, ν,T ) and a measurable function ϕ :X → Z, we construct the Z-extension (X, B, µ, T ) of (X,B, ν,T ) by setting X :=X × Z, B :=B ⊗ P(Z), µ := ν ⊗ l∈Z δ l and T (x, l) = (T (x), l + ϕ(x)). We endow X with the product metric given by d X ((x, l), (x ′ , l ′ )) := max{dX (x, x ′ ), \| l − l ′ \|}. Hence T n (x, l) = (T n x, l + S n ϕ(x)), where S n ϕ is the ergodic sum S n ϕ := n−1 k=0 ϕ •T k . Therefore, for ǫ small enough, T n (x, l) ∈ B((x, l), ǫ) ⇐⇒T n (x) ∈ BX (x, ǫ) and S n ϕ(x) = 0. Our main results concern the case when (X,B, ν,T ) is a mixing subshift of finite type (see Section 3 for precise definition), which are classical dynamical systems used to model a wide class of dynamical systems such as geodesic flows in negative curvature, etc. Consider (X,B, ν,T ) a mixing subshift of finite type and ν a Gibbs measure associated to a Hölder continuous potential. Moreover we have a ν-centered Hölder continuous function ϕ. Then we get (1.1) lim ǫ→0 log τ ǫ log ǫ = −2d, µ-almost everywhere, where d is the Hausdorff dimension of ν. Moreover the following convergence holds in distribution with respect to any probability measure absolutely continuous with respect to µ: (1.2) µ(B(., ǫ)) τ ǫ (.) −→ ǫ→0 E \|N \| , where E and N are two independent random variables with respective exponential distribution of mean 1 and standard normal distribution (see Theorem 2.1 and Theorem 2.2 for precise statements). Roughly speaking the strategy of our proof is that there is a large scale (corresponding to S n ϕ(x)) and a small scale (corresponding toT n (x)), which behave independently assymptotically. To enlight this strategy, we start out this paper with the study of the toy probabilistic model (Y n , S n ), where (S n ) n is the simple symmetric random walk and (Y n ) n is a sequence of independent random variables, with uniform distribution on (0, 1) d and where S n and Y n are independent. For this simple model, we obtain the same results. More precisely, we prove that (1.1) holds almost surely and that (1.2) holds in distribution. toy probabilistic model Let d ∈ N. In this section, we give a real random walk (M n ) n≥0 with values in R×]0, 1[ d−1 ⊂ R d . 2.1. Description of the model and statement of the results. The random process M n is given by M n = (S n , 0) + Y n . (S n ) n≥0 and (Y n ) n≥0 are independent such that: • Y n is uniformly distributed on (0, 1) d . • S n is the simple symmetric random walk on Z given by S 0 = 0, i.e. S n = n k=1 X k , where (X k ) k is a sequence of independent random variables such that: P(X k = 1) = P(X k = −1) = 1/2. We want to study the asymptotic behavior, as ǫ goes to 0, of τ ǫ for the metric associated to some norm on R d . Let c be the Lebesgue measure of the unit ball in R d . We will prove the following: Theorem 2.1. Almost surely, log τǫ − log ǫ converges to 2d as ǫ goes to 0. For this constant c > 0, we have the following result: Theorem 2.2. The sequence of random variables ((cǫ d ) √ τ ǫ ) ǫ converges in distribution to E N , where E and N are two independent random variables, E having an exponential distribution of mean 1 and N having a standard Gaussian distribution. 2.2. Proof of the pointwise convergence of the recurrence rate to the dimension. M 0 is in )0; 1( d , let ǫ so small that B(M 0 ; ǫ) is contained in )0, 1( d . Note that Leb(B(x, ǫ)) = cǫ d . We define for any p ≥ 0 the p th return time R p of (M n ) n in )0; 1( d , setting R 0 = 0, by induction : R p+1 := inf m > R p : S m = 0 . We have the relation: τ ǫ = R Tǫ with T ǫ := min{l ≥ 1 : Y R l ∈ B(Y 0 , ǫ)} We will study the asymptotic behavior of the random variables R n and T ǫ and use the relation between them to prove Theorem 2.1. 2.2.1. Study the return of the random variable R n . Proposition 2.3 (Feller [4]). There exists C > 0 such that: (2.1) P(R 1 > s) ∼ C √ s , as s → ∞ Remark 2.4. Due to the strong Markov property, the delays U p := R p − R p−1 between successive return times are independent and identically distributed. Lemma 2.5. Almost surely, log √ Rn log n converges to 1 as n goes to ∞. Proof. The proof of the lemma directly holds, once the following inequality is proved: ∀α ∈ (0, 1), ∃n 0 , ∀n ≥ n 0 , n 1−α ≤ R n ≤ n 1+α Let α ∈ (0, 1), by independence (using Remark 2.2.1), we have: (2.2) P( R n ≤ n 1−α ) ≤ P(∀p ≤ n, R p − R p−1 ≤ n 1−α ) ≤ P R 1 ≤ n 1−α n . Due to the asymptotic formula given in Proposition 2.1, for n sufficiently large P( R 1 ≤ n 1−α ) n ≤ 1 − C 2n 1−α n ≤ exp −C n α 2 . This allows us to get the first inequality of (2.2) by using the Borel Cantelli lemma. Again, using proposition 2.1, we have P R 1 2+α 1 > s ≤ C ′ s 1+ α 2 for some C ′ > 0, implying obviously that E R 1 2+2α 1 < ∞. Note that one can see, R n = n i=1 U i ≤ n 2+2α 1 n n i=1 U 1 2+2α i 2+2α . But 1 n n i=1 U 1 2+2α i converges almost surely to E R 1 2+2α 1 < ∞ due to the strong law of large numbers. Hence R n = O(n 2+2α ) almost surely, from which we get the second inequality. 2.2.2. Study the return of the random variable T ǫ . In this subsection the asymptotic behavior of the random variable T ǫ is illustrated in the following lemma. Lemma 2.6. Almost surely, log Tǫ − log ǫ → d as ǫ → 0. Proof. Given Y 0 , let ǫ > 0 be such that B(Y 0 , ǫ) ⊂ (0, 1) d . The random variable T ǫ has a geometric distribution with parameter λ ǫ := cǫ d . For any α > 0, a simple decomposition gives: P log T ǫ − log ǫ − d > α = P T ǫ > ǫ −d−α ) + P(T ǫ < ǫ −d+α . The first term is handled by the Markov inequality: P(T ǫ > ǫ −d−α \| Y 0 ) ≤ ǫ α ǫ d λ ǫ = O(ǫ α ). While the second term using the geometric distribution: P(T ǫ < ǫ −d+α ) = 1 − (1 − cǫ d ) ǫ −d+α ≤ 1 − exp[ǫ −d+α log(1 − cǫ d )] ≤ (−ǫ) −d+α log(1 − cǫ d ) = O(ǫ α ). Let us define ǫ n := n −2 α . Thus (ǫ n ) n≥1 is a decreasing sequence of real numbers, and T ǫ is monotone in ǫ, so that: n≥1 P(\| log T ǫn − log ǫ n − d\| > α) < +∞. According to Borel Cantelli lemma Proof of Theorem 2.1 The theorem follows from the two previous lemmas 2.5 and 2.6, since: log √ τ ǫ − log ǫ = log R Tǫ log T ǫ log T ǫ − log ǫ → 1 × d = d a.s. Hence, we get: log τ ǫ − log ǫ → 2d as ǫ → 0 a.s. 2.3. Proof of the convergence in distribution of the rescaled return time. Proposition 2.7. The sequence of random variables ( Rn n 2 ) n converges in distribution to N −2 where N is a standard Gaussian random variable. The proof of this proposition follows from the two following successive lemmas; the proof of which is straightforward and is omitted. Lemma 2.8. n≥0 P(S 2n = 0)s 2n = 1 √ 1−s 2 and P(S 2n = 0) 1 √ πn . Note that P(S 2n = 0) = k=0 P(S k = 0)P(R 1 = 2n−2k). Hence, n>1 P(S 2n = 0)s 2n = n≥0 P(S 2n = 0)s 2n E(s R1 ). And so E s R1 = 1 − √ 1 − s 2 . Lemma 2.9. The moment generating function of N −2 is E e −tN −2 = e − √ 2t , ∀t ≥ 0, where N is standard Gaussian random variable. Proof of Proposition 2.7. Knowing that R 1 , (R 2 − R 1 ), ..., (R n − R n−1 ) are i.i.d., and the fact that E s R1 = 1 − √ 1 − s 2 , we get: E[e − t n 2 Rn ] = E[e − t n 2 R1 ] n = 1 − 1 − e −2 t n 2 n and from Lemma 2.9, we have: ∀t ≥ 0, lim n→∞ E e − t n 2 Rn = e − √ 2t = E[e −tN −2 ]. Hence, ( Rn n 2 ) n converges in distribution to N −2 . Lemma 2.10. (λ ǫ T ǫ ) ǫ converges in distribution to an exponential random variable E of mean 1. Proof. Given Y 0 , T ǫ has a geometric distribution of parameter λ ǫ = λ(B(Y 0 , ǫ)). Let t > 0, P(λ ǫ T ǫ ≤ t \| Y 0 ) = ⌊ t λǫ ⌋ n=1 λ ǫ (1 − λ ǫ ) n−1 = 1 − exp t λ ǫ log(1 − λ ǫ ) , it follows that, for E a random variable which follows exp (1), lim ǫ→0 P(λ ǫ T ǫ ≤ t \| Y 0 ) = 1 − e −t = P(E ≤ t), a.s. Proof of Theorem 2.2. Let us prove that the family of couples λ ǫ T ǫ , RT ǫ Tǫ ǫ>0 converges in distribution, as ǫ → 0, to (E, N −2 ) , where E and N −2 are assumed to be as above and independent. Let s > 0 and t ∈ R ,then using the independence of (T ǫ ) ǫ and (R n ) n , we get: P λ ǫ T ǫ > s, R Tǫ T 2 ǫ > t − P λ ǫ T ǫ > s)P(N −2 > t ≤ n> s λǫ λ ǫ (1 − λ ǫ ) n−1 P R n n 2 > t − P N −2 > t ≤ sup n> s cǫ d P R n n 2 > t − P(N −2 > t) This latter goes to 0 as ǫ goes to 0, due to Proposition 2.7. Moreover by Lemma 2.10, P(λ ǫ T ǫ > s) → P(E > s) as ǫ → 0, hence: ∀s > 0, ∀t lim ǫ→0 P(λ ǫ T ǫ > s, R T 2 ǫ T ǫ > t) − P(E > s, N −2 > t) = 0. This proves that the couple λ ǫ T ǫ , RT ǫ T 2 ǫ ǫ>0 converges in distribution, as ǫ goes to 0, to (E, N −2 ). Knowing that τ ǫ = R Tǫ , we thus find that: (cǫ d ) 2 τ ǫ = cǫ d λ ǫ 2 λ 2 ǫ T 2 ǫ R Tǫ T 2 ǫ Since (x, y) → x 2 y is continuous, λ 2 ǫ T 2 ǫ RT ǫ T 2 ǫ d −→ E 2 N −2 as ǫ → 0. Observe that cǫ d λǫ 2 a.s. −→ 1, hence by Slutzky's Lemma, we end up with : (cǫ d ) 2 τ ǫ d → E 2 N −2 , as ǫ → 0. Z-extension of a mixing subshift of finite type Let A be a finite set, called the alphabet, and let M be a matrix indexed by A × A with 0-1 entries. We suppose that there exists a positive integer n 0 such that each component of M n0 is non zero. The subshift of finite type with alphabet A and transition matrix M is (Σ, θ), with Σ := {w := (w n ) n∈Z : ∀n ∈ Z , M (w n , w n+1 ) = 1} together with the metric d(w, w ′ ) := e −m , where m is the greatest integer such that w i = w ′ i whenever \|i\| < m, and the shift θ : Σ → Σ, θ((w n ) n∈Z ) = (w n+1 ) n∈Z . Let ν be the Gibbs measure on Σ associated to some Hölder continuous potential h, and denote by σ 2 h the asymptotic variance of h under the measure ν. Recall that σ 2 h vanishes if and only if h is cohomologous to a constant, and in this case ν is the unique measure of maximal entropy. For any function f : Σ → R we denote by S n f := Σ n−1 l=0 f • θ l its ergodic sum. Let us consider a Hölder continuous function ϕ : Σ → Z, such that ϕdν = 0. We consider the Z-extension F of the shift θ by ϕ. Recall that F : Σ × Z → Σ × Z (x, m) → (θx, m + ϕ(x)). Recall that Σ × Z is endowed with distance d 0 ((w, l), (w ′ , l ′ )) := max{d(w, w ′ ), \| l − l ′ \|}. Note that, if ǫ < 1, for every (w, l) ∈ Σ × Z, we have µ (B Σ×Z ((w, l), ǫ)) = ν(B Σ (w, ǫ)). We want to know the time needed for a typical orbit starting at (x, m) ∈ Σ × Z to return ǫ-close to the initial point after iterations of the map F . By the translation invariance we can assume that the orbit starts in the cell m = 0. Recall that τ ǫ (x) = min{n ≥ 1 : F n (x, 0) ∈ B(x, ǫ) × {0}} = min{n ≥ 1 : S n ϕ(x) = 0 and d(θ n x, x) < ǫ}. We know that there exists a positive integer m 0 such that the function ϕ is constant on each m 0 -cylinders. Let us denote by σ 2 ϕ the asymptotic variance of ϕ: σ 2 ϕ = lim n→∞ 1 n E[(S n ϕ) 2 ]. We assume that σ 2 ϕ = 0 (otherwise (S n ϕ) n is bounded). We reinforce this by the following non-arithmeticity hypothesis on ϕ: We suppose that, for any u ∈ [−π; π]\{0} the only solutions (λ, g), with λ ∈ C and g : Σ → C measurable with \|g\| = 1, of the functional equation (3.1) g • θ − g = λe iu.ϕ is the trivial one λ = 1 and g = const. The fact that there is no non constant g satisfying (3.1) for λ = 1 ensures that ϕ is not a coboundary and so that σ 2 ϕ = 0. The fact that there exists (λ, g) satisfying (3.1) with λ = 1 would mean that the range of S n ϕ is essentially contained in a sub-lattice of Z; in this case we could just do a change of basis and apply our result to the new reduced Z-extension. We emphasize that this non-arithmeticity condition is equivalent to the fact that all the circle extensions T u defined by T u (x, t) = (θ(x), t + u.ϕ(x)) are weakly mixing for u ∈ [−π; π]\{0}. In this section we obtain the following results: Theorem 3.1. The sequence of random variables log √ τǫ − log ǫ converges ν-almost everywhere as ǫ → 0 to the Hausdorff dimension d of the measure ν. Theorem 3.2. The sequence of random variables ν((B ǫ (.)) τ ǫ (.) converges in distribution with respect to every probability measure absolutely continuous with respect to ν as ǫ → 0 to E \|N \| , where E and N are independent random variables, E having an exponential distribution of mean 1 and N having a standard Gaussian distribution. Corollary 3.3. If the measure ν is not the measure of maximal entropy, then the sequence of random variables log √ τǫ+d log ǫ √ − log ǫ converges in distribution as ǫ → 0 to a centered Gaussian random variable of variance 2σ 2 h . 3.1. Spectral theory of the transfer operator and Local Limit Theorem. In this subsection, we follow [11] to adapt our results. To begin with, let us define: Σ := {w := (w n ) n∈N : ∀n ∈ N, M (w n , w n+1 ) = 1}, the set of all one-sided infinite sequences of elements of A, endowed with the metricd((w n ) n≥0 , (w ′ n ) n≥0 ) := e − inf{m≥0:wm =w ′ m } , and the one-sided shift mapθ((w n ) n≥0 ) = (w n+1 ) n≥0 . The resulting topology is generated by the collection of cylinders: C a0,...,an = {(w n ) n∈N ∈Σ : w 0 = a 0 , ..., w n = a n }. Let us introduce the canonical projection Π : Σ →Σ, Π((w n ) n∈Z ) = (w n ) n≥0 . Denote byν the image probability measure (onΣ) of ν by Π. There exists a function ψ :Σ → Z such that ψ • Π = ϕ • θ m0 . let us denote by P : L 2 (ν) → L 2 (ν) the Perron-Frobenius operator such that: ∀f, g ∈ L 2 (ν), Σ P f (x)g(x)dν(x) = Σ f (x)g •θ(x)dν(x). Let η ∈]0; 1[. Let us denote by B the set of bounded η-Hölder continuous function g :Σ → C endowed with the usual Hölder norm : \|\|g\|\| B := \|\|g\|\| ∞ + sup x =y \|g(y) − g(x)\| d(x, y) η . We denote by B * the topological dual of B. For all u ∈ R, we consider the operator P u defined on (B, \|\|.\|\| B ) by: P u (f ) := P (e iuψ f ). Note that the hypothesis of non-arithmeticity of ϕ is equivalent to the following one on ψ: for any u ∈ [−π; π]\{0}, the operator P u has no eigenvalue on the unit circle. We will use the method introduced by Nagaev in [8] and [9], adapted by Guivarch and Hardy in [5] and extended by Hennion and Hervé in [7]. It is based on the family of operators (P u ) u and their spectral properties expressed in the two next propositions. Proposition 3.4. (Uniform Contraction). There exist α ∈ (0; 1) and C > 0 such that, for all u ∈ [−π; π]\[−β; β] and all integer n ≥ 0, for all f ∈ B, we have: (3.2) \|\|P n u (f )\|\| B ≤ Cα n \|\|f \|\| B . This property easily follows from the fact that the spectral radius is smaller than 1 for u = 0. In addition, since P is a quasicompact operator on B and since u → P u is a regular perturbation of P 0 = P , we have : , for all f ∈ B and for all n ≥ 0, we have the decomposition: P n u (f ) = λ n u ϕ u (f )v u + N n u (f ), with (1) \|\|N n u (f )\|\| B ≤ Cα n \|\|f \|\| B , (2) \|λ u \| ≤ e −c1\|u\| 2 and c 1 \|u\| 2 ≤ σ 2 φ u.u, (3) with initial values : v 0 = 1, φ 0 =ν, λ ′ u=0 = 0 and λ ′′ u=0 = −σ 2 ϕ . Lemma 3.6. There exist γ ′ > 0 and C η > 0 such that, ∀q ≥ m 0 and all 2q-cylinderÂ ofΣ, we have: (3.3) ∀u ∈ [−π, π], \|\|P q u P q (1Â •θ m0 )\|\| B ≤ C η e −γ ′ (2q−m0) . In particular, we haveν(Â) ≤ C η e −γ ′ (2q−m0) . Proof. P q u P q (1Â •θ m0 )(y) = P q e iuSqψ P q−m0 (1Â) (y) = w:θ 2q−m 0 w=y e S2q−m 0 h(w) 1Â(w)e iuSq ψθ q−m 0 (w) = 1 [θ 2q−m 0Â] (y)e S2q−m 0 h(wy) e iuSq ψθ q−m 0 (wy) . where w y ∈Â is the unique element such thatθ 2q−m0 w y = y (it exists if y ∈θ 2q−m0Â ). From this later formula, we can obtain that \|\|\|P q u P q (1Â •θ m0 )\|\| ∞ ≤ e max h(2q−m0) , where max h < 0. Now a step to compute the norm \|\|.\|\| B is to estimate the η−Hölder coefficient. Let x = y ∈Σ, we know that d(x, y) = e −n , for some n ∈ N * . We will consider two cases: the first case when n > m 0 , we note the equivalence x ∈θ 2q−m0Â ⇔ y ∈θ 2q−m0Â . Thus, either x, y / ∈θ 2q−m0Â and hence \|P q u P q−m0 (1Â)(y) − P q u P q−m0 (1Â)(x)\| = 0. Or x, y ∈θ 2q−m0Â , so thatd(w x , w y ) = 2q − m 0 + n. Let us denote for simplicity F h,ψ = S 2q−m0 h(.) + iuS q ψ • θ q−m0 (.) . Introducing the ergodic sum formula, we get: \|S 2q−m0 h(w y ) − S 2q−m0 h(w x )\| ≤ 2q−m0−1 i=0 \|h\| η e −η(2q−m0+n−i) ≤ c\|h\| ηd η (x, y), where c is a constant such that j≥1 e −αj ≤ c < ∞. And in the same way for S q ψ(θ q−m0 (.)), we can see that: \|S q ψ(θ q−m0 (w y )) − S q ψ(θ q−m0 (w x ))\| ≤ c\|ψ\| ηd η (x, y). Thus, from these computations, we verify that: \|P q u P q−m0 (1Â(y)) − P q u P q−m0 (1Â(x))\| = \| e F h,ψ (wy) − e F h,ψ (wx) \| ≤ e max h(2q−m0) c(\|h\| η + \|ψ\| η )d η (x, y). Now, we treat the second case where n ≤ m 0 . Here, if x ∈θ 2q−m0Â , then y / ∈θ 2q−m0Â , \|P q u P q−m0 (1Â)(y) − P q u P q−m0 (1Â)(x)\| ≤ sup \|P q u P q−m0 (1Â)\| ≤ sup w∈Â \|e S2q−m 0 h(w)+iuSqψ•θ q−m 0 (w) \|e ηn e −ηn ≤ e max h(2q−m0) e ηm0dη (x, y). From all this process, setting γ ′ := min(η, − max h) > 0, we get an estimation for the η-Hölder coefficient, ∀n ≥ 0: \|P q u P q−m0 (1Â)\| η ≤ e −γ ′ (2q−m0) max (e ηm0 , c(\|h\| η + \|ψ\| η )) Hence, for C η := (1 + max (e ηm0 , c(\|h\| η + \|ψ\| η ))), we deduce that \|\|P q u P q−m0 (1Â)\|\| B ≤ C η e −γ ′ (2q−m0) . Next proposition is a two-dimensional version of Proposition 13 in [11]. We give a more precise error term in order to accomodate the one-dimensional case. It may be viewed as a doubly local version of the central limit theorem: first, it is local in the sense that we are looking at the probability that S n ϕ = 0 while the classical central limit theorem is only concerned with the probability that \|S n ϕ\| ≤ ǫ √ n; second, it is local in the sense that we are looking at this probability conditioned to the fact that we are starting from a set A and landing on a set B on the base. Proposition 3.7. There exist real numbers C 1 > 0 and γ > 0 such that, for all integers n, q, k such that n − 2k > 0 and all m 0 < q ≤ k, all two-sided q-cylinders A of Σ and all measurable subset B ofΣ, we have: ν A ∩ {S n ϕ = 0} ∩ θ −n (θ k (Π −1 (B))) − ν(A)ν(B) √ n − kσ ϕ ≤ C 1ν (B)k 2 e −γq n − 2k . Proof. Set Q := A∩{S n ϕ = 0}∩θ −n (θ k (Π −1 (B)). The proof of the proposition will be illustrated in estimating the measure of the set Q. Since ϕ • θ m0 = ψ • Π and using the semi-conjugacyθ • Π = Π • θ, we have the identity: {S n ϕ • θ m0 = 0} = {S n ψ • Π = 0}. In addition, Im(ψ) ∈ Z, thus we have: 1 θ −q−m 0 Q = 1Â •θ m0 .1 B •θ q+n−(k−m0) . 1 2π e iu.Snψ•θ q du • Π, withÂ := Πθ −q A( indeed θ −q A = Π −1Â since A is a q-cylinder). Since the measure ν is θ-invariant, then we can verify that: ν(Q) = 1 2π [−π,π] Eν 1Â •θ m0 .1 B •θ q+n−(k−m0) e iu.Snψ•θ q du. Now we want to estimate the expectation a(u) = Eν(...). Introducing the Perron-Frobenius operator P, and using the fact that it is the dual ofθ, we get: a(u) = Eν P q (1Â •θ m0 ) exp(iu.S n ψ)1 B •θ n−(k−m0) = Eν P n u P q (1Â •θ m0 )1 B •θ n−(k−m0) = Eν P k−m0 u (1 B P n−(k−m0) u P q (1Â •θ m0 )) . We will treat two cases concerning the values of u. Let us denote for simplicity l := n − (k − m 0 − q). First, using the contraction inequality given in Proposition 3.4 applied to P l u (1), the fact that \|\|P q u P q (1Â • θ m0 )\|\| B ≤ e −γ ′ (2q−m0) from Lemma 3.6, and the fact that E P k−m0 u (1 B g) ≤ν(B)\|\|g\|\| B , we will show that a(u) is negligible for large values of u, so when u / ∈ [−β, β] we get for γ = 2γ ′ : \|a(u)\| = Eν P k−m0 u (1 B P n−(k−m0)−q u P q u P q (1Â •θ m0 )) = O ν(B)α l e −γq . We now use the decomposition in 3.5 to obtain an estimation of the main term coming from small values of u. Indeed, whenever u ∈ [−β, β], we have: a(u) = Eν P k−m0 u (1 B P l u P q u P q (1Â •θ m0 )) = λ l u ϕ u (P q u P q (1Â •θ m0 ))Eν P k−m0 u (1 B v u ) + Eν P k−m0 u (1 B N l u (P q u P q (1Â •θ m0 )) = a 1 (u) + a 2 (u). Using inequality (1) in Proposition 3.5, one can see that the second term is of order (3.4) a 2 (u) = O(ν(B)α l e −γq ). The mappings u → v u and u → φ u are C 1 -regular with v 0 = 1 and ϕ 0 =ν, from which we find that: a 1 (u) = λ l uν (P q u P q (1Â •θ m0 ))Eν P k−m0 u (1 B ) + λ l uν (P q u P q (1Â •θ m0 ))Eν P k−m0 u (1 B O(u)) +λ l u O(u)(P q u P q (1Â •θ m0 ))Eν P k−m0 u (1 B ) + λ l u O(u)(P q u P q (1Â •θ m0 ))Eν P k−m0 u (1 B )O(u)) To obtain an approximation of the first term a 1 (u), we introduce the formula of P in P u : Eν P k−m0 u (1 B ) −ν(B) = \|Eν P k−m0 (e iu.S k−m 0 ψ − 1)1 B \| ≤ \|\|e iu.S k−m 0 ψ − 1\|\| ∞ \|\|1 B \|\| L 1 (ν) ≤ \|u\|.(k − m 0 )\|\|ψ\|\| ∞ν (B), so that, from this approximation, we get: a 1 (u) = λ l uν (P q u P q (1Â •θ m0 ))Eν P k−m0 u (1 B ) + O(λ l u \|u\|ν(B)e −γq ) = λ l uν (Â)ν(B) (1 + O(\|u\|q)) (1 + O(\| u \| (k − m 0 )) + O(λ l u \|u\|ν(B)e −γq ) = λ l uν (Â)ν(B) + O(λ l u \|u\|ν(B)k 2 e −γq ). Using Proposition 3.5 and that u → λ u belongs to C 3 ([−β; β] → C), hence applying the intermediate value theorem gives: \|λ l u − e − l 2 σ 2 ϕ u 2 \| ≤ l(e −c1\|u\| 2 ) l−1 \|λ u − e − 1 2 σ 2 ϕ u 2 \| = le −c1l\|u\| 2 e c1\|u\| 2 O(\|u\| 3 ) = C 0 l\|u\| 2 e −c1l\|u\| 2 e c1\|u\| 2 \|u\| = O(e −c2l\|u\| 2 \|u\|), for the constant c 2 = c 1 /2. As a consequence, an estimate for a 1 (u) is: a 1 (u) = e − l 2 σ 2 ϕ u 2ν (Â)ν(B) + O(e −c2l\|u\| 2 \|u\|ν(B)ke −γq ), sinceν(Â) = O(e −γ ′ (2q−m0) ). A final step to reach an estimation of ν(Q) is to integrate the approximated quantity of a 1 (u) obtained above. Using the Gaussian integral, a change of variable v = u √ l gives: [−β,β] e − l 2 σ 2 ϕ u 2 du = 1 √ l 2π σ ϕ + O 1 l In the same way we treat the error term to get: [−β,β] \|u\|e −c2l\|u\| 2 du = 1 l [−β √ l,β √ l] \|v\|e −c2\|v\| 2 dv = O 1 l . From these computations, it follows that: [−β,β] a 1 (u)du = 2π lσ 2 ϕν (A)ν(B) + O ν(B)k 2 e −γq l . From this main estimate and (3.1) and (3.4) we conclude that: ν(Q) = 1 2π [−π,π] a(u)du = 1 √ n − kσ ϕν (A)ν(B) + O ν(B)k 2 e −γq n − 2k 3.2. Proof of the pointwise convergence of the recurrence rate to the dimension. Let us denote by G n (ǫ) the set of points for which n is an ǫ-return : G n (ǫ) := {x ∈ Σ : S n ϕ(x) = 0 and d(θ n (x), x) < ǫ}. Let us consider the first return time in an ǫ-neighborhood of a starting point x ∈ Σ : τ ǫ (x) := inf{m ≥ 1 : S m ϕ(x) = 0 and d(θ m (x), x) < ǫ} = inf{m ≥ 1 : x ∈ G m (ǫ)}. Proof of Theorem 3.1. Let us denote by C k the set of two-sided k-cylinders of Σ. For any δ > 0 denote by C δ k ⊂ C k the set of cylinders C ∈ C k such that ν(C) ∈ (e −(d+δ)k , e −(d−δ)k ). For any x ∈ Σ, let C k (x) ∈ C k be the k-cylinder which contains x. Since d is twice the entropy of the ergodic measure ν, by the Shannon-Breiman theorem, the set K δ N = {x ∈ Σ : ∀k ≥ N, C k (x) ∈ C δ k } has a measure ν(K δ N ) > 1 − δ provided N is sufficiently large. • First, let us prove that, almost surely : lim inf ǫ→0 log √ τ ǫ − log ǫ ≥ d. Let α > 1 d and 0 < δ < d − 1 α . Let us take ǫ n := n − α 2 and k n := ⌈− log ǫ n ⌉. In view of Proposition 3.7, whenever k n ≥ N , we have : ν(K δ N ∩ G n (ǫ n )) = ν {x ∈ K δ N : S n ϕ(x) = 0 and θ n (x) ∈ C kn (x)} = C∈C δ kn ν(C ∩ {S n ϕ = 0} ∩ θ −n (C)) = C∈C δ kn ν(C) 2 σ ϕ √ n + O ν(C)k 2 n e −γkn n − 2k n . Notice that for ǫ n and k n taken as above, one can verify that the term k 2 n e −γkn n−2kn = O(n −1− γα 2 (log n) 2 ). In addition, for C ∈ C k δ n , ν(C) ≤ n − α(d−δ) 2 , from which it follows that ν(K δ N ∩ G n (ǫ n )) = O (log n) 2 n min(1+ γα 2 ,1+ α(d−δ) 2 ) but 1+α(d−δ) 2 > 1, so n ν(K δ N ∩ G n (ǫ n )) < ∞. Hence by the Borel Cantelli lemma, for a.e. x ∈ K δ N , if n is large enough, we have τ ǫn > n, which in turn implies that : lim inf n→∞ log √ τ ǫn − log ǫ n ≥ 1 α a.e., and this proves the the lower bound on the lim inf, since (ǫ n ) n decreases to zero and lim inf n→+∞ ǫn ǫn+1 = 1, and since we have taken an arbitrary α > 1 d . • Next, we will prove the upper bound (d) on the lim sup : lim sup ǫ→0 log √ τ ǫ − log ǫ ≤ d. let α ∈ (0, 1 d ) and δ > 0 such that 1 − αd − αδ > 0. Take ǫ n := n − α 2 and k n := ⌈− log ǫ n ⌉. We define for all l = 1, ..., n, the sets A l (ǫ) := G l (ǫ) ∩ θ −l {τ ǫ > n − l} which are pairwise disjoint. Setting L n := ⌈n a ⌉ with a > α(d + δ − γ), we then realize that: (3.5) 1 ≥ n l=0 ν(A l (ǫ n )) ≥ n l=Ln C∈C δ kn ν(C ∩ A l (ǫ n )). But due to Proposition 3.7, for any C ∈ C δ kn and l ≥ L n , whenever k n ≥ N , we have : ν(C ∩ A l (ǫ n )) = ν(C ∩ {S l ϕ = 0} ∩ θ −l (C ∩ {τ ǫn > n − l})) = ν(C)ν(C ∩ {τ ǫn > n − l}) σ ϕ √ l − k n + O ν(C ∩ {τ ǫn > n − l})k 2 n e −γkn l ≥ cǫ d+δ n 1 √ l ν(C ∩ {τ ǫn > n − l}). Indeed, the error is negligible, because for a > α(d + δ − γ), k 2 n e −γkn √ l = O(ǫ d+δ n ). Now, note that: ν K δ N ∩ {τ ǫn > n} ≤ C∈C δ kn ν (C ∩ {τ ǫn > n}) . Next, we will work to prove that ν K δ N ∩ {τ ǫn > n} is summable. Observe that: n l=Ln ν(C ∩ A l (ǫ n )) ≥ cǫ d+δ n ν(C ∩ {τ ǫn > n}) √ n − L n . But, from (3.5), it follows immediately that 1 ≥ C∈C δ kn n l=Ln ν(C ∩ A l (ǫ n )) ≥ C∈C δ kn cǫ d+δ n ν(C ∩ {τ ǫn > n}) √ n − L n from which one gets ν K δ N ∩ {τ ǫn > n} ≤ C∈C δ kn ν(C ∩ {τ ǫn > n}) = O 1 n 1−α(d+δ) 2 . Now let us take n p := p − 4 1−αd−αδ . We have: p≥1 ν(K δ N ∩ {τ ǫn p > n p }) is finite, revealing that, using Borel Cantelli lemma, almost surely x ∈ K δ N ,τ ǫn p (x) ≤ n p , which implies that : lim sup n→+∞ log √ τ ǫn p − log ǫ np ≤ 1 α . This gives the estimate lim sup since (ǫ np ) p decreases to 0 and since lim p→+∞ ǫn p ǫn p+1 = 1. 3.3. Fluctuations of the rescaled return time. Throughout this subsection, we adapt the general strategy of [12,13]. Recall that C k (x) = {y ∈ Σ : d(x, y) < e −k }. Let R k (y) := min{n ≥ 1 : θ n (y) ∈ C k (y)} denote the first return time of a point y into its k-cylinder C k (y), or equivalently the first repetition time of the first k symbols of y. We recall that C k (x) = {y ∈ Σ : d(x, y) < e −k }. There have been a lot of studies on the quantity R k , among all the results we will use the following. Proposition 3.8. (Hirata [6]) For ν-almost every point x ∈ Σ, the return time into the cylinders C k (x) are asymptotically exponentially distributed in the sense that lim k→+∞ ν C k (x) R k (.) > t ν(C k (x)) = e −t for a.e. x, where the convergence is uniform in t. Lemma 3.9. ∀t > 0, lim sup k→+∞ ν τ e −k > t ν(C k (x)) 2 C k (x) ≤ 1 1 + βt , with β := 1 σ . Proof. Let k ≥ m 0 and n be some integers. We make a partition of a cylinder C k (x) according to the value l ≤ n of the last passage in the time interval 0, ..., n of the orbit of (x, 0) by the map F into C k (x) × {0}. This gives the following equality : (3.6) ν(C k (x)) = n l=0 ν C k (x) ∩ {S l = 0} ∩ θ −l (C k (x) ∩ {τ e −k > n − l}) . Let n k := t ν(C k (x)) 2 . We claim that : lim sup k→∞ ν({τ e −k > n k } \| C k (x)) ≤ 1 1 + βt According to the decomposition (3.6) and to Proposition 3.7, there exists c 1 > 0 such that we have : ν(C k (x)) ≥ ν(C k (x) ∩ {τ e −k > n k }) 1 + βν(C k (x)) n k l=2 k+1 1 √ l − k − c 1 ν(C k (x))k 2 e −γk n k l=2 k+1 1 l − 2 k Our claim follows from the fact that βν(C k (x)) l=n k l=2 k+1 1 √ l−k ≃ βt and the term k 2 e −γk l=n k l=2 k+1 1 l−2 k ≪ 1. C k (x)) √ τ e −k \| C k (x) k≥0 is tight. Hence it will be enough to prove that the advertised limit law is the only possible accumulation point of our destination. We hence abbreviate X k := ν(C k (x)) √ τ e −k Lemma 3.11. Suppose that the sequence of conditional distributions of (X kp \| C kp (x)) p converges to the law of some random variable X. Then the limit satisfies the integral equation: 1 = P(X > t) + βt 1 0 P(X > t √ 1 − u) √ u du ∀t > 0. Proof. To begin with, let us set f (t) := P(X > t) • First, we will prove that ∀t > 0 1 ≥ f (t) + βt 1 0 u −1/2 f (t(1 − u) 1/2 )du. The decomposition in (3.6) and Proposition 3.7, implies that there exists c > 0 such that we have: 1 ≥ ν(τ e −k > n k \| C k (x)) + βν(C k (x)) n k l=1 ν(τ e −k > n k − l \| C k (x)) √ l − k − c n k l=1 k 2 e −γk ν(C k (x)) l − 2k We want to estimate the formula of this inequality when k → ∞. We note that through the proof of Lemma 3.9, it has been proved that lim kp→∞ n kp l=1 kpe −γkp l−kp = 0. Thus, if we set B n k := n k l=1 ν(τ e −k >n k −l\|C k (x)) √ l−k , we are left to estimate mainly the lower bound on the lim inf of ν(C kp (x))B n kp as p → ∞. Now, by monotonicity, we have B n k ≥ N ⌊ n k N ⌋ l=⌊ n k N ⌋ ν(τ e −k > n k − l \| C k (x)) √ l − k = N −1 r=1 ⌊ n k N ⌋ l=0 ν(τ e −k > n k − l − (r⌊n k/N ⌋) \| C k (x)) l + r⌊n k /N ⌋ . Observe that the term: ν(τ e −k > n k − l − (r⌊n k/N ⌋) \| C k (x)) ≥ ν(τ e −k > (1 − r/N )n k \| C k (x)) = P X k > ( 1 − r/N t) \| C k (x) . Thus, now by evaluating the following sum: ⌊ n k N ⌋ l=0 1 l + r⌊n k /N ⌋ ≥ n k N 1 √ r + 1 , We obtain B n k = N −1 r=1 n k N 1 √ r + 1 P X k > t (1 − r/N ) \| C k (x) But, from the hypothesis that P X kp > t 1 − r/N kp→∞ −→ f t 1 − r/N , we get: lim inf p→∞ ν(C kp (x))B n kp ≥ t N N −1 r=1 P X > t 1 − r/N (r + 1)/N ≥ t 1 0 f t √ 1 − u √ u du. Combining these estimates and taking the limit when k p → ∞, we establish the desired inequality: 1 ≥ f (t) + βt 1 0 f t √ 1 − u √ u du. • In the same way we treat the converse inequality, using the other half of Proposition 3.7, then there exists c ′ > 0, such that: 1 ≤ ν(τ e −k > n k \| C k (x)) + βν(C k (x)) n k l=1 ν(τ e −k > n k − l \| C k (x)) √ l − k + c ′ n k l=1 k 2 e −γk l − 2k 1 ν(C k (x)) , then using Proposition 3.8, we have ν(τ e −k ≤ 1 − ν τ e −k > m k ν(C k )(x) ν(C k (x)) k→∞ −→ 1 − e 0 = 0. from which we observe that we can forget the first m k term of the following sum, because m k l=1 ν(C k (x) ∩ {S l = 0} ∩ θ −l (C k (x) ∩ {τ e −k > m k − l})) = ν m k l=1 C k (x) ∩ {S l = 0} ∩ θ −l (C k (x) ∩ {τ e −k > m k − l}) ≤ ν({τ e −k ≤ m k } ∩ C k (x)) = o(ν(C k (x))). Furthermore, one verifies that this sum of terms between m k and ⌊n k /N ⌋ is bounded above by 2t ν(C k (x)) √ N . Hence, we get: 1 ≤ ν(τ e −k > n k \| C k (x)) + βν(C k (x)) n k l=⌊ n k N ⌋ ν(τ e −k > n k − l \| C k (x)) √ l − k + c ′ n k l=1 k 2 e −γk l − 2k + o(ν(C k (x))) + β 2t √ N , where N is so large that the last three terms goes to 0 as k → 0. Moreover, if we set B ′ nn k := N ⌊n k /N ⌋ l=⌊n k /N ⌋ ν(τ e −k >n k −l\|C k (x)) √ l , we verify that n k l=⌊ n k N ⌋ ν (τ e −k > n k − l \| C k (x)) √ l − k ≤ (1 + ǫ k ) B ′ n k + N 1 √ n k − N . We now proceed to show the bound on the lim sup of ν(C k (x))B ′ n kp as p → 0 B ′ n k = N −1 r=1 (r+1)⌊ n k N ⌋−1 l=r⌊ n k N ⌋ ν (τ e −k > n k − l \| C k (x)) √ l ≤ N −1 r=1 ⌊ n k N ⌋−1 l=0 ν (τ e −k > n k − l − ((r + 1)⌊n k /N ⌋) \| C k (x)) l + r⌊n k /N ⌋ . It can be easily seen that ⌊ n k N ⌋−1 l=0 1 l + r⌊n k /N ⌋ ≤ n k N 1 √ r , hence, it follows immediately that B ′ n k ≤ N −1 r=1 n k N 1 √ r ν (τ e −k > (1 − (r + 1)/N )n k \| C k (x)) . Applying lim sup when p → ∞, then lim sup p→∞ ν(C kp (x)) N ⌊n kp /N ⌋ l=⌊n kp /N ⌋ ν τ e −kp > n kp − l \| C kp (x) √ l ≤ t 1 0 f (t √ 1 − u) √ u du. Taking the limit when k p → ∞, and combining all these estimates, we get the second inequality: 1 ≤ f (t) + βt 1 0 f (t √ 1 − u) √ u du. Lemma 3.12. We know that the conditional distributions of the X kp converge to a random variable X iff the conditional distributions of the X 2 kp converge to X 2 . The later then satisfies Hence, for any s > 0, we find Proof of theorem 3.2. According to Lemma 3.9, the family of distributions of X k is tight. By Lemmas 3.11, 3.12 and 3.13, the law of c β E \|N \| is the only possible accumulation point of the family of distributions of ν(C k (x)) √ τ e −k \| C k (x) k≥0 . Let P be a probability measure absolutely continuous with respect to µ, with density h. Set H(x) := l∈Z h(x, l). Note that by Z-periodicity, the distribution of τ ǫ under P is the same of that under the probability measure with density (x, l) → H(x) with respect to ν ⊗ δ 0 . Assume first that the density H is continuous. Denote by A k := {y : (ν(C k (y) τ e −k (y) > t}, then we have: 1 = P(X 2 > t) + t 0 P(X 2 > t − v) √ v dv ∀t > 0.E[e −sW ] = ∞ 0 β v 0 P(W ≥ v − w) √ w se −sv dv = ∞ 0 1 √ w β ∞ w P(W ≥ v − w)se −sv dv dw = β ∞ 0 e −P(A k \|C k (x)) ∼ k→∞ ν(A k > t\|C k (x)) ∼ k→∞ P c β E \|N \| . And so, by the dominated Lebesgue theorem, we get: P(A k ) = P(A k \|C k (x))H(x)dν(x) ∼ P c β E \|N \| ). Now, take in general the density H in L 1 (ν). We use the fact that the set of the continuous functions is dense in L 1 (ν), so that there exists H n continuous such that H n L 1 (ν) −→ H. P(A k ) = 1 A k (x)H(x)dν(x) ≤ 1 A k (x)H n (x)dν(x) + \|\|H n − H\|\| L 1 (ν) . We know that there is n such that ∀ ǫ > 0 \|\|H n − H\|\| L 1 (ν) < ǫ 2 . Moreover H n is continuous, then there is k such that ∀ ǫ > 0 1 A k (x)H n (x)dν(x) − P c β E \|N \| < ǫ 2 . Hence the conclusion follows. Proof of Corollary 3.3. Let us set: Y k := log √ τ e −k (.) − kd √ k . We have the case that ν is a Gibbs measure with a non degenerate Hölder potential h. There is a constant c h > 0 such that log ν(C k (x)) = k j=−k h • σ j (x). This Birkhoff sum follows a central limit theorem (e.g. [2]), which implies that: log ν(C k (.)) + kd √ k dist −→ N (0, 2σ 2 h ). Observe that Y k has the following decomposition: Y k = log ν(C k (.)) τ e −k (.) √ k − log ν(C k (.)) + kd √ k . Hence, it will be enough to prove that the first term of Y k converges in distribution to 0, which is true due to Theorem 3.2. Université de Bretagne Occidentale, LMBA, CNRS UMR 6205, Institut des sciences et Techniques, 29238 Brest Cedex 3, France E-mail address: [email protected] log Tǫ n − log ǫn → d almost surely as n → +∞. Hence the proof follows since lim Proposition 3.5. (Perturbation Result). There exist α > 0, β > 0, C > 0, c 1 > 0, θ ∈]0; 1[ such that: there exists u → λ u belonging to C 3 ([−β; β] → C), there exists u → v u belonging to C 3 ([−β; β] → B), there exists u → ϕ u belonging to C 3 ([−β; β] → B * ) such that, for all u ∈ [−β; β] Corollary 3 . 10 . 310The family of conditional distributions of the random variables ν( Lemma 3. 13 .. 13Let W be a random walk variable with values in [0, In particular, the distribution of W coincides with that of c 2 β E 2 N 2 , where the independent variables E and N are the exponential distribution of mean 1 and the standard Gaussian distribution respectively.Proof. Let s > 0. W ≤ v)se −sv dv. sw √ w dw. 1 − E[e −sW ] , and our claim about the Laplace transform of W follows, because up to a change of variable (v 2 = 2sw), we have Hence, as a consequence of the previous computations, we end up with E[e −sW ] = 1 1+c β √ s . Then W has the same Laplace transform of c 2∞ 0 e −sw √ w dw = ∞ 0 e − 1 2 v 2 v/ √ s 2v 2s dv = √ π √ s . β E 2 N 2 . Acknowledgment:I would like to thank my supervisors F. Pène and B. Saussol for their help and advice during this work and their availability for answering all my questions. Inequalities for the occurrence times of rare events in mixing processes. The state of art. M Abadi, A Galves, Markov Process. Related Fields. 7M. Abadi and A. Galves, Inequalities for the occurrence times of rare events in mixing processes. The state of art, Markov Process. Related Fields 7 (2001) 97-112 Equilibirium states and the ergodic theory of Anosov diffeomorphisms. R Bowen, Lecture Notes in Mathematics. 470Springer-Verlagsecond edR. Bowen, Equilibirium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470. Springer-Verlag, Berlin-New York, 1975; 2008, second ed. 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Hirata, Poisson law for Axion A diffeomorphisms, Ergod. Th & Dynam. Sys 13 (1993) 533-556 Limit theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. H Hennion, & L Hervé, Lecture Notes in Mathematics. 1766SpringerH. Hennion & L.Hervé, Limit theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi- Compactness, Lecture Notes in Mathematics 1766, Springer, Berlin. 2001 Some limit theorems for stationary Markov Chains. S V Nagaev, Theor. Probab. Appl. 2Veroyatn. PrimenS.V.Nagaev, Some limit theorems for stationary Markov Chains, Theor. Probab. Appl 2 (1957) 378-406; translation from Teor. Veroyatn. Primen 2(1958) 389-416 More exact statement of limit theores of homogeneous Markov chains. S V Nagaev, Theor. Probab Appl. 6S.V.Nagaev, More exact statement of limit theores of homogeneous Markov chains, Theor. Probab Appl. 6 (1961) 67-86 Back to balls in billiards. F Pène, B Saussol, Comm. Math. Phys. 293837866F.Pène and B. Saussol, Back to balls in billiards, Comm. Math. Phys., 293 (2010) 837866. Quantitative recurrence in two-dimensioinal extended processes, Annales de l'Institut Henri Poincaré. F Pène, B Saussol, Probabilités et Statistiques. 454F.Pène and B. Saussol, Quantitative recurrence in two-dimensioinal extended processes, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 2009 Vol. 45, No. 4, 1065-1084 F Pène, B Saussol, R Zweimüller, Recurrence rates and hitting-time distributions for random walks on the line. 41F. Pène, B. Saussol, R. Zweimüller, Recurrence rates and hitting-time distributions for random walks on the line, The Annals of Probability 41 (2013), 619-635 Return and Hitting time limits for rare events of null-recurrent Markov maps. F Pène, B Saussol, R Zweimüller, Th. Dynam. Sys. 33pto appear in ErgodF. Pène, B. Saussol, R. Zweimüller, Return and Hitting time limits for rare events of null-recurrent Markov maps, to appear in Ergod. Th. Dynam. Sys., 33 p. 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[ "Superhumps and flickering in V1316 Cygni", "Superhumps and flickering in V1316 Cygni" ]	[ "David Boyd ", "Christopher Lloyd ", "Robert Koff ", "Thomas Krajci ", "Bart Staels ", "Jerrold Foote ", "William Goff ", "Tonny Vanmunster ", "Lewis Cook ", "Joseph Patterson " ]	[]	[]	We present analysis and results of a coordinated CCD photometry campaign to observe the 2006 June superoutburst of the cataclysmic variable V1316 Cyg involving 8 longitudinallydistributed observers. The outburst peaked at magnitude 15.03 on June 10, declined at a rate of 0.14 mag day -1 , lasted 11 days and had an amplitude above quiescence of 2.4 magnitudes. We detected common superhumps for the first time, thereby confirming that V1316 Cyg is a member of the UGSU class of dwarf novae. We observed a transition to late superhumps two-thirds of the way through the outburst with an associated phase shift of 0.50 +/-0.06 cycles. The mean common superhump period before this transition was 0.07685 +/-0.00003 d and the mean late superhump period following the transition was 0.07654 +/-0.00002 d. The common superhump period decreased at a rate dP/dt = -5.1 +/-1.7 *10 -5 cycle -1 . At the onset of late superhumps, there was a transient shift in power from the superhump fundamental frequency to its first harmonic and back again. We detected an orbital period of 0.0740 +/-0.0002 d giving a fractional superhump period excess of 0.038 +/-0.003 and a mass ratio of 0.167 +/-0.010. A scalegram analysis of the flickering behaviour of V1316 Cyg found that the α and Σ parameters characterising flickering changed significantly during the superoutburst. We also found flickering to be at a relatively much lower level at the beginning of the superoutburst and during two "normal" outbursts.	null	[ "https://arxiv.org/pdf/0710.1653v1.pdf" ]	117,873,863	0710.1653	17b652e78e4ffbce12c49d49d9677003ed25da1a	Superhumps and flickering in V1316 Cygni David Boyd Christopher Lloyd Robert Koff Thomas Krajci Bart Staels Jerrold Foote William Goff Tonny Vanmunster Lewis Cook Joseph Patterson Superhumps and flickering in V1316 Cygni 1 We present analysis and results of a coordinated CCD photometry campaign to observe the 2006 June superoutburst of the cataclysmic variable V1316 Cyg involving 8 longitudinallydistributed observers. The outburst peaked at magnitude 15.03 on June 10, declined at a rate of 0.14 mag day -1 , lasted 11 days and had an amplitude above quiescence of 2.4 magnitudes. We detected common superhumps for the first time, thereby confirming that V1316 Cyg is a member of the UGSU class of dwarf novae. We observed a transition to late superhumps two-thirds of the way through the outburst with an associated phase shift of 0.50 +/-0.06 cycles. The mean common superhump period before this transition was 0.07685 +/-0.00003 d and the mean late superhump period following the transition was 0.07654 +/-0.00002 d. The common superhump period decreased at a rate dP/dt = -5.1 +/-1.7 10 -5 cycle -1 . At the onset of late superhumps, there was a transient shift in power from the superhump fundamental frequency to its first harmonic and back again. We detected an orbital period of 0.0740 +/-0.0002 d giving a fractional superhump period excess of 0.038 +/-0.003 and a mass ratio of 0.167 +/-0.010. A scalegram analysis of the flickering behaviour of V1316 Cyg found that the α and Σ parameters characterising flickering changed significantly during the superoutburst. We also found flickering to be at a relatively much lower level at the beginning of the superoutburst and during two "normal" outbursts. History V1316 Cyg was discovered by Romano [1], originally labelled GR141, and classified as LP, presumably meaning a long period variable. Romano [2] later reclassified it as a dwarf nova with designation V1316 Cyg. A spectrum obtained during an outburst by Bruch & Schimpke [3] showed Balmer emission lines on a strong blue continuum, confirming it was a dwarf nova. V1316 Cyg has acquired a UGSU classification but it is unclear how. Downes & Shara [4] cite Bruch, Fischer & Wilmsen [5] as the type reference but the latter appear not to have classified it, and indeed not to have observed it in outburst. So far, this classification seems to have been unsupported by observational evidence. Correct identification of V1316 Cyg has also been a problem. In figure 19 of Bruch, Fischer & Wilmsen [5], V1316 Cyg was incorrectly identified as the nearby magnitude 14.8 star. In Downes & Shara [4] this mis-identification was continued. This was finally resolved only in 2000 September when V1316 Cyg was correctly identified as the faint star 12 arcsec to the east of the magnitude 14.8 star [6]. The situation was further complicated by the detection of low amplitude variability in the magnitude 14.8 star in the years before 2000. Downes et al. [7] give a correct identification chart for V1316 Cyg and list the coordinates as RA 20h 12m 13.62s, Dec +42° 45' 51.5" (J2000). These difficulties mean that the reliable historical record of observations of V1316 Cyg only begins in 2000. The AAVSO International Variable Star Database [8] contains one welldocumented outburst which was first reported by Simonsen on 2003 September 21 and lasted about 17 days. During this outburst, estimates were contributed by 11 observers, 6 observing visually and 5 using a CCD. Only one brief time-series photometry run was obtained which did not confirm the presence of superhumps. The maximum reliable magnitude reported during this outburst was 15.2. A search of the BAA Variable Star Section (BAAVSS) database did not reveal any additional observations. Detection of short, low amplitude outbursts As part of the BAAVSS Recurrent Objects Programme [9], V1316 Cyg has been monitored regularly. During 2005 several unusually short outbursts of V1316 Cyg were detected [10]. These were of lower amplitude than the outburst reported in 2003, reaching about magnitude 16. They typically lasted 1-2 days and recurred sometimes after only 10 days. Time-series photometry during one of these outbursts on 2005 September 5 at magnitude 16.0 did not detect any sign of superhumps. Whether these are the "normal" outbursts of V1316 Cyg is unclear. Further observations are needed to understand them better. Photometric observations during 2006 June superoutburst The quiescent magnitude of V1316 Cyg is around 17.4 but varies continuously by several tenths of a magnitude due to flickering. Following a period at quiescence, V1316 Cyg was observed brighter and rising on 2006 June 7 and on June 9 superhumps were detected in the light curve for the first time. This finally confirmed the postulated UGSU classification of V1316 Cyg. Figure 1 shows the field of V1316 Cyg on June 9. Henceforth we will refer to dates in the truncated form JD = JD -2,453,000. The Centre for Backyard Astrophysics [11] is a global network of small telescopes set up to study the periodic behaviour of cataclysmic variables. Following announcement of the outburst of V1316 Cyg, CBA observers began an intensive observing campaign. 8 observers were involved, 3 in Europe and 5 spread across America, collecting between them 32 timeseries photometry runs comprising 123 hours of photon collection and over 5000 magnitude measurements on 18 out of 24 nights. This comprehensive coverage of the outburst continued until the star had returned to quiescence and is an impressive example of the power of a longitudinally-distributed network of observers who can be quickly mobilised and have ready access to capable equipment. CCD images were calibrated and reduced by each observer. All images were dark-subtracted and flat-fielded then measured using differential aperture photometry. Comparison stars from the AAVSO f-chart for V1316 Cyg (dated 020301) were used with magnitudes from photometry by Henden given in Sumner [12]. Close proximity of the star labelled 148 on the AAVSO f-chart necessitated care to ensure that light from that star did not contaminate either the photometric aperture for V1316 Cyg or measurements of sky background. The observations obtained are listed in Table 1 and the instrumentation used in Table 2. In almost every case, observers worked unfiltered to maximise signal to noise ratio as the principal objective was to record the time-varying behaviour of the light curve as accurately as possible. Given the different spectral responses of the cameras used, some variation between the magnitudes measured by different observers is inevitable. The discrepancy between concurrent measurements was generally less than 0.1 magnitude. Figure 2 shows the light curve of the superoutburst. The outburst started on JD 894, or possibly the previous day, and lasted 11 days reaching a maximum magnitude of 15.03 on JD 897, an outburst amplitude of 2.4 magnitudes. Maximum light was followed by a steady decline at a rate of 0.14 mag day -1 for 8 days before a rapid drop at the end of the outburst on JD 905. Immediately following the main outburst, there was a brief rebrightening from 17.1 to 16.5 before the star returned to its previous quiescent level below magnitude 17. Such rebrightenings are not uncommon in UGSU stars. It was similar to one of the short, low amplitude outbursts noted earlier. The coordinates of V1316 Cyg measured from images during the outburst agree with those given in Downes et al. [7] to better than 0.2 arcsec. Analysis of superhumps 13 of the 32 observing runs obtained during the campaign were longer than 4 hours and light curves of these are shown in Figure 3. All the light curves are drawn at the same time and magnitude scale. The presence of superhumps is a prominent feature of the early runs with their amplitude gradually diminishing from 0.5 magnitude at the peak of the outburst to around 0.1 magnitude as the outburst ended. The structure of the light curves also becomes more complex towards the end of the outburst. Times of maximum of the superhumps which could be resolved in the individual light curves were determined using a quadratic fit. For three of the later runs, in which the superhumps were less clearly defined, we subtracted the linear trend then synchronously summed the data on the period 0.07654 d before fitting a maximum. In all, 40 maxima were measured of which 29 are shown in Figure 3. A preliminary analysis showed that the superhump phase remained relatively stable up to the end of JD 901. The times of maximum from JD 896 to 901 inclusive were therefore used to determine a preliminary superhump maximum ephemeris of HJD 2,453,896.49647 + 0.07685 E. Using this ephemeris, times of maximum throughout the outburst were computed and used to assign superhump cycle numbers and calculate O-C values for each observed maximum. These are listed in Table 3 and plotted in Figure 4. The O-C diagram clearly distinguishes between a "common superhump" regime up to at least cycle 70 (JD 901), at which point a transition took place (during JD 902) to a "late superhump" regime (van der Woerd et al. [13]) which was in place by cycle 90 (JD 903). This transition involved a phase shift of 0.50 +/-0.06 cycles. From linear fits to the times of maximum in the common and late superhump regions, we found the mean common superhump period to be 0.07685 +/-0.00003 d and the mean late superhump period to be 0.07654 +/-0.00002 d. We tested both linear and quadratic fits in the common superhump region. The improvement in chi-squared of the quadratic over the linear fit was significant at the 2% level. The quadratic fit gave a period rate of change dP/dt = -5.1 +/-1.7 10 -5 cycle -1 indicating a slowly decreasing period over this interval. Period analysis To ensure stability when subtracting the mean and trend from each time-series, we excluded time-series shorter than 0.08 days from the period analysis. The remaining 25 observing runs had their means and linear trends subtracted and were then combined. To analyse the frequency content of this combined light curve, we separated it into three segments: (a) before transition (JD 896-901 inclusive), (b) from the onset of transition to the end of the outburst (JD 902-905 inclusive), (c) after the outburst (JD 910-917 inclusive -we had no data for JD 906-909). These segments are shown in Figure 5. Period analysis using a data compensated discrete Fourier transform was carried out on each segment separately with the CLEANest algorithm in the PERANSO software [14]. Power spectra of these analyses are shown in Figure 6 and the frequencies containing most power in each spectrum are listed in Table 4. (a) In the early and middle part of the outburst up to the onset of phase transition (JD 896-901) there is a strong superhump signal at 13.00 +/-0.05 cycles day -1 . Removing this superhump signal leaves small signals at 11.95, 12.86 and 13.45 cycles day -1 . There is a weak signal at the first harmonic of the superhump frequency, which is probably due to the non-sinusoidal shape of the light curve. (b) Between the onset of phase transition and the end of the outburst (JD 902-905), power in the superhump signal appears to be divided between its fundamental frequency at 12.96 +/-0.07 cycles day -1 and first harmonic at 26.07 +/-0.07 cycles day -1 , while a strong signal at 13.54 +/-0.14 cycles day -1 appears. Removing this latter signal leaves the superhump and its first harmonic essentially unchanged and the reverse is also true so they appear to be unconnected. (c) Beyond the end of the outburst (JD 910-917), our coverage is relatively poor so discrimination against alias signals in the power spectrum is weak. The strongest signal is the superhump frequency at 13.04 +/-0.04 cycles day -1 . Removing this frequency and its aliases leaves little remaining power. To examine what is happening around the interesting phase transition region in more detail, we analysed the light curves for JD 902, 903, 904 and 905 separately. The power spectra for these four days are shown in Figure 7 and the prominent frequencies listed in Table 5. Over this interval, power appears to shift from the superhump fundamental frequency into its first harmonic and then back again. In the interval JD 896-901, we find a mean common superhump period of 0.07693 +/-0.00027 d with mean peak-to-peak amplitude 0.23 mag. In the interval JD 903-917, the mean late superhump period is 0.07654 +/-0.00010 d and mean peak-to-peak amplitude 0.10 mag. These periods are consistent with the results obtained from linear fits to the times of superhump maximum. Averaged phase diagrams for common and late superhumps are given in Figure 8. The phase diagram for late superhumps shows a double peak as expected from the presence of a strong first harmonic signal immediately after the transition. A weak superhump signal is present for at least 12 days (160 cycles) after the end of the outburst. Based on available knowledge of UGSU cataclysmic variables (see for example Patterson et al. [15] and Pearson [16]) and assuming V1316 Cyg is not an abnormal member of its class, we estimate the fractional superhump period excess is likely to be ~0.04 and therefore the orbital period to be ~0.074 d (frequency 13.5 cycles day -1 ). We note that the signals at 13.54 cycles day -1 for JD 902-905 and 13.48 cycles day -1 for JD 910-917 listed in Table 4, plus the weak residual signal at 13.45 cycles day -1 for JD 986-901 noted in (a) above, are close to this frequency. Analysing all the data together yields a small but persistent periodic signal at 0.0740 +/-0.0002 d (frequency 13.52 +/-0.03 cycles day -1 ) with peak-to-peak amplitude 0.05 mag which we interpret as the orbital period. Radial velocity measurements or further timeseries photometry at quiescence are required to confirm this orbital period. The fractional common superhump period excess ε = (P sh -P orb ) / P orb , where P sh is the common superhump period and P orb is the orbital period, is then 0.038 +/-0.003. Using the relationship ε = 0.18q + 0.29q 2 (Patterson et al. [15]) gives a mass ratio q = 0.167 +/-0.010. The period analysis was repeated using the ANOVA method (Schwarzenberg-Czerny [17]) which gave results fully consistent with the above analysis. All the main features noted above were present in both solutions. What physical explanations can we invoke for these observations? During the early and middle part of the outburst, from JD 896 to JD 901, we see a strong common superhump signal caused by tidal and thermal stresses induced by the secondary as it passes the maximum radius of an eccentric precessing accretion disc (Hellier [18]). The slow reduction in the period may be explained by the accretion disc emptying and shrinking. During JD 902 we see transition to a late superhump regime where the dominant light source is now the hot spot where the accretion stream impacts the disc (Rolfe et al. [19]). Maximum light is produced at the point when the accretion stream generates most energy as it impacts the disc. This occurs where the eccentric disc has minimum radius, thus explaining the 0.5 cycle superhump phase change. We note that in V1316 Cyg this transition from common to late superhumps occurs at a point approximately two-thirds of the way through the outburst rather than at the end of the outburst as in IY UMa (Patterson et al. [20]) or well after the outburst as in 1RXS J232953.9+062814 (Skillman et al. [21]). The temporary increase in power at the first harmonic of the superhump frequency during JD 903 and 904 may possibly indicate a brief resurrection of the common superhump mechanism which then dies away again by JD 905. Analysis of flickering In common with many other dwarf novae, V1316 Cyg exhibits flickering. This is a stochastic variation in the light output of the system covering many timescales which is thought to be caused by irregularities in the flow of material, either in the accretion stream or at the inner edge of the disc. Flickering is most apparent in quiescence and is primarily responsible for the variation in the observed quiescent magnitude. Fritz & Bruch [22]. This potentially offers assistance in classifying variables of unknown type based on their flickering behaviour. The scalegram in Figure 9 shows results of a wavelet analysis of the JD 896 (June 9) data of V1316 Cyg and a 15.1 magnitude comparison star. The results were not sensitive to which base or mother wavelet was used so wavelet C6 was adopted for the analysis. Random noise has no preferred time scale so the scalegram of a constant star is essentially flat in this diagram. At the shortest time scales the scalegram of a flickering source will be increasingly dominated by noise and the flickering spectrum will flatten out to the level of the noise in the data. If the noise level can be determined then this can be subtracted from the values at longer time scales to provide a more reliable spectrum (see Fritz & Bruch [22]). All the scalegrams generated here show some indication of flattening so it has been assumed that the value of log(s) at the shortest timescale is entirely due to noise and this value has been subtracted from the others to produce the noise-corrected spectrum. Generally the noise correction makes a significant difference to only one or two of the weakest points in the spectrum. At the longest time scales the scalegram is under sampled and the values are unreliable. These points correspond to the time scale of the data length or longer and can be safely ignored. The parameters of the flickering spectrum are determined from the shortest possible time scale up to something less than the orbital period, typically 60 minutes, and all the runs used were substantially longer than this. Scalegrams have been constructed for all the data sets and 12 of these which provided reliable values of α and Σ are shown in Figure 10. The uncertainties are typically 0.2 in both parameters. All these values come from the slow decline in magnitude following the maximum. Generally the later runs are fainter and so have poorer signal to noise, also they tend to have longer exposure times and so the shorter time scales of the scalegrams are less well sampled. Weighted mean values of α and Σ have been calculated for each day as plotted in Figure 11, which shows their chronological development as the outburst declined from maximum. In the α-Σ plane the scalegrams tend to cluster in two groups corresponding to the periods JD 896-899 (June 9-12) and JD 900-904 (June 13-17) although there is some overlap. The earlier points have α ~ 2.7 and Σ ~ -2.7, which indicate an unusually steep gradient but a rather weak flickering level for the start of the superoutburst. These values tend to come from the better sampled scalegrams so should be reliable. The later values of α ~ 2.0 and Σ ~ -2.4 are more typical of UGSU-type dwarf novae in superoutburst (see Figure 15 of Fritz & Bruch [22]). This transition in the α-Σ plane occurs about two days before the large phase change in the superhump cycle but it is not clear what association there may be, if any, between these events. Despite the relatively large uncertainties in the individual α and Σ parameters the correlation between them is clear. The Spearman Rank Correlation is -0.62 with a probability of 3% that this is due to chance. However, the negative correlation is very unusual as only 2 systems (TT Ari and RR Pic) of the 19 studied by Fritz & Bruch [22] behave in this way. As well as the positive observation of flickering there are two interesting negative observations to report. On the night the outburst was discovered and before superhumps had developed, JD 894 (June 7), there was no significant flickering, but the signal to noise of the observations was sufficient to see any of the subsequent flickering reported here. The implication of this is that the usual flickering mechanism was either switched off or swamped immediately before the superhumps became visible. The data after JD 904 unfortunately have a high noise level so it is not possible to follow the behaviour of the flickering into quiescence. However, in data obtained on JD 1003 (September 24), when the system was undergoing one of its short, low amplitude outbursts at magnitude 16.1, there was also no significant flickering. Observations reported by Shears Boyd & Poyner [10] of another of V1316 Cyg's many brief outbursts on JD 619, a year prior to the superoutburst, show a suggestion of some activity but this is substantially less than the flickering seen during the superoutburst (see Figure 12). The mechanism generating the flickering in V1316 Cyg appears to be quite fragile; it can switch on and off on a short time scale and also evolves quickly. For most CV's flickering is most visible at quiescence, but in this system that regime has not yet been thoroughly explored. Conclusions A coordinated CCD photometry campaign involving 8 longitudinally-distributed observers has obtained comprehensive coverage of the 2006 June superoutburst of V1316 Cyg. The outburst peaked at magnitude 15.03 on June 10 and subsequently declined at a rate of 0.14 mag day -1 . It lasted 11 days and had an amplitude above quiescence of 2.4 magnitudes. Our detection of common superhumps confirms for the first time that V1316 Cyg is a member of the UGSU class of dwarf novae. We observed a transition to late superhumps two-thirds way through the outburst with an associated phase shift of 0.50 +/-0.06 cycles. This contrasts with other systems in which this transition has occurred either at the end of the superoutburst or later. We measured the mean common superhump period before this transition as 0.07685 +/-0.00003 d and the mean late superhump period following the transition as 0.07654 +/-0.00002 d. We found the common superhump period slowly decreased at a rate of dP/dt = -5.1 +/-1.7 10 -5 cycle -1 . At the onset of late superhumps, we observed a transient shift in power from the superhump fundamental frequency to its first harmonic and back again, possibly indicating a temporary re-growth of common superhumps. We detected a small orbital signal with period 0.0740 +/-0.0002 d giving a fractional common superhump period excess of 0.038 +/-0.003. Using the relationship between superhump period excess and mass ratio published in [15] gives a corresponding mass ratio of 0.167 +/-0.010. From a scalegram analysis of the flickering behaviour of V1316 Cyg, we found that the α and Σ parameters characterising flickering changed significantly during the outburst, evolving towards values similar to those observed in other UGSU-type dwarf novae in superoutburst. At the beginning of the outburst, before superhumps formed, flickering appeared to be absent. A similar analysis of flickering during two short, low amplitude outbursts of V1316 Cyg, respectively one year before and three months after this superoutburst, found flickering to be at a substantially lower level than during the superoutburst. Table 4: Frequencies containing most power in the spectra in Figure 6. Table 5: Frequencies containing most power in the spectra in Figure 7. A well-established technique for analysing flickering behaviour is the scalegram(Fritz & Bruch [22]) which is a log-log plot of time scale against normalised amplitude. It is obtained by a wavelet analysis of the light curve using a base function which is designed to represent well the characteristic transient sharply peaked behaviour of flickering in the light curve. This complements Fourier analysis which is optimised for periodic sinusoidal variation. Analysis of the light curve of an object creates a track in the scalegram. The two important parameters of a track are α, the slope of the linear segment of the track which is a measure of the variation of flickering power with time scale, and Σ, the height of the track at a reference time scale which indicates the overall strength of flickering. A positive value of α means there is more flickering power at longer time scales. The flickering behaviour of an object can therefore be represented by its position in α-Σ space. It appears empirically that cataclysmic variables of the same type tend to cluster together in α-Σ space, as shown in figures10-15 in Figure 1 : 1V1316 Cyg on 2006 June 9, field 8' x 8', north at top. Figure 2 : 2Light curve of 2006 June superoutburst, mean magnitudes for each run. Figure 3 : 3Light curves longer than 4 hours during the outburst. Horizontal axes are JD-2453000. Vertical axes are unfiltered magnitudes. All plots have the same time and magnitude scale. Figure 4 : 4O-C diagram for superhump maxima. Figure 5 : 5Light curves of all runs longer than 0.08 days after subtraction of mean and linear trends -(upper) JD 896-901, (middle) JD 902-905, (lower) JD 910-917. Figure 6 : 6Power spectra -(upper) JD 896-901, (middle) JD 902-905, (lower) JD 910-917. Figure 8 : 8Phase diagrams -(upper) common superhumps averaged over the interval JD 896-901 showing 2 cycles, standard errors per bin are within the data points, (lower) late superhumps averaged over JD 903-917 with standard errors per bin. Figure 9 : 9Scalegram of V1316 Cyg on JD 896 with the variable at magnitude 15.2 (upper, filled circles) and the equivalent scalegram of a 15.1 magnitude comparison star (lower track). The noise-corrected values are shown as crosses and are significantly different for only the weakest values of log(s). The fit to the scalegram up to 60 minutes is shown by the straight line and this is used to determine the values of α and Σ. A similar process is used on all the other scalegrams. Figure 10 : 10The scalegrams of all the data sets that provided reliable values of α and Σ. Figure 11 : 11Evolution of the superoutburst in the α-Σ plane showing the weighted mean values from each day. The line connects the points in chronological order starting with the point on the top-right showing the data from Figure 9. The large square and diamond show the mean positions of the early (JD 896-899) and later (JD 900-904) data. Figure 12 : 12Scalegrams of the data from two brief outbursts, the first on JD 619 at magnitude 16.0 that occurred a year before the superoutburst (squares) and the second on JD 1003 at magnitude 16.1 that occurred shortly after (circles). The scalegrams of two comparison stars of similar magnitudes are shown as open symbols. Table 1: Log of observations.Date (UT) 2006 Start time (HJD) 2,453,000+ Duration (h) No of images Exposure (s) Filter Mean mag Observer June 3 890.41855 0.20 10 60 C 17.17 DB June 4 891.44024 0.07 3 60 C 17.61 DB June 5 892.41353 0.11 6 60 C 17.60 DB June 7 894.42286 3.08 132 60 C 15.82 DB June 9 896.41491 4.40 303 60 C 15.21 DB June 10 897.41802 2.32 132 40 C 15.03 DB June 11 897.67339 6.50 349 60 C 15.10 TK June 11 898.38260 5.16 345 50 C 15.09 BS June 11 898.41097 0.66 53 30 C 15.10 DB June 12 898.67204 7.24 380 60 C 15.18 TK June 12 898.72601 5.60 213 90 C 15.19 JF June 12 899.39276 4.84 194 80 C 15.39 TV June 12 899.42837 2.48 103 40 C 15.33 DB June 13 899.72445 5.18 200 90 C 15.39 JF June 14 900.71855 5.18 110 120 C 15.57 RK June 15 901.66728 6.01 131 120 C 15.65 RK June 15 902.43517 3.85 205 40 C 15.80 DB June 16 902.71841 5.80 220 90 C 15.73 JF June 16 902.82319 2.33 67 75 C 15.72 LC June 16 903.39350 4.74 268 60 C 15.81 BS June 17 903.69402 5.67 102 120 C 15.99 RK June 17 903.76731 3.76 306 30 C 16.04 WG June 17 903.80746 3.09 128 45 C 15.86 LC June 17 904.43174 1.12 34 50 C 16.05 DB June 17 904.43509 3.84 142 80 C 15.93 TV June 17 904.45889 1.31 75 60 C 16.04 BS June 18 904.66404 6.32 151 120 C 16.19 RK June 19 905.80412 2.89 134 60 C 16.87 WG June 21 908.42784 0.27 17 60 C 17.13 DB June 23 909.84538 1.83 46 120 V 16.46 WG June 24 910.65786 6.62 161 120 C 16.76 RK June 26 912.65752 6.61 163 120 C 17.63 RK June 27 914.49703 0.32 21 60 C 17.65 DB June 28 915.43732 1.35 48 60 C 17.65 DB June 30 917.43604 2.77 151 60 C 17.40 DB Observer Instrumentation DB 0.35-m f/5.3 SCT + SXV-H9 CCD camera LC 0.73-m reflector + HSV-H9 CCD camera JF 0.60-m f/3.4 reflector + ST-8e CCD camera WG 0.40-m Newtonian + ST-8 CCD camera RK 0.25-m f/10 SCT + AP47 CCD camera TK 0.28-m f/10 SCT + ST-7E CCD camera BS 0.28-m f/6.3 SCT + MX716 CCD camera TV 0.35-m f/6.3 SCT + ST-7XME CCD camera Table 2 : 2Instrumentation used.Cycle no Time of maximum (HJD) 2,453,000+ O-C (cycles) 0 896.49364 -0.0369 1 896.57119 -0.0277 13 897.49622 0.0099 16 897.72912 0.0408 17 897.80442 0.0206 18 897.88181 0.0277 25 898.41436 -0.0421 26 898.49084 -0.0468 27 898.56732 -0.0516 29 898.72717 0.0285 30 898.80147 -0.0045 30 898.80178 -0.0005 31 898.88115 0.0323 31 898.88054 0.0244 32 898.95674 0.0160 38 899.41656 -0.0002 39 899.49262 -0.0105 40 899.57052 0.0033 43 899.80299 0.0284 44 899.87851 0.0112 56 900.80212 0.0304 57 900.87940 0.0361 68 901.71608 -0.0760 69 901.79846 -0.0040 70 901.87491 -0.0091 81 902.73635 0.2010 83 902.88946 0.1935 83 902.88070 0.0794 94 903.74889 0.3775 95 903.82769 0.4029 96 903.90185 0.3680 96 903.90337 0.3877 96 903.90149 0.3633 104 904.51693 0.3721 109 904.90046 0.3631 121 905.81943 0.3219 122 905.89548 0.3115 185 910.70928 -0.0453 211 912.70490 -0.0758 273 917.45381 -0.2771 Table 3 : 3Observed superhump maxima.JD 896 -901 JD 902 -905 JD 910 -917 Freq c d -1 Power Freq c d -1 Power Freq c d -1 Power 13.00 651 13.54 96 13.04 45 12.04 266 26.07 90 13.48 40 13.91 202 12.96 86 12.61 38 15.09 162 12.03 69 11.99 34 10.00 150 14.54 62 14.10 32 AcknowledgementsWe acknowledge with thanks variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research. We thank Clive Beech for his assistance in extracting information from the BAAVSS data archives. We are also grateful for the constructive comments of the referee. BS: CBA Flanders (Alan Guth Observatory). New Mexico, PO Box 1351 Cloudcroft, New Mexico 88317, USA; B-9308 Hofstade, Belgium; Utah, 4175 East Red Cliffs Drive, Kanah, UT 84741, USA51TK: CBA. [email protected]] JF: CBA. [email protected]: CBA New Mexico, PO Box 1351 Cloudcroft, New Mexico 88317, USA [[email protected]] BS: CBA Flanders (Alan Guth Observatory), Koningshofbaan 51, B-9308 Hofstade, Belgium [[email protected]] JF: CBA Utah, 4175 East Red Cliffs Drive, Kanah, UT 84741, USA [[email protected]] [email protected]] TV: CBAWG: 13508 Monitor Ln. Sutter Creek, CA 95685, USA; Belgium, Walhostraat 1A, B-3401 Landen, [email protected]: 13508 Monitor Ln., Sutter Creek, CA 95685, USA [[email protected]] TV: CBA Belgium, Walhostraat 1A, B-3401 Landen, Belgium [[email protected]] . 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Astrophys., 332, 586 (1998) Figure 7: Power spectra -(upper) JD 902, (upper middle) JD 903, (lower middle) JD 904, (lower) JD 905.	[]
[ "Trees with few leaves in tournaments", "Trees with few leaves in tournaments" ]	[ "Alistair Benford ", "Richard Montgomery " ]	[]	[]	We prove that there exists C > 0 such that any (n + Ck)-vertex tournament contains a copy of every n-vertex oriented tree with k leaves, improving the previously best known bound of n+O(k 2 ) vertices to give a result tight up to the value of C. Furthermore, we show that, for each k, there exists n0, such that, whenever n n0, any (n + k − 2)-vertex tournament contains a copy of every n-vertex oriented tree with at most k leaves, confirming a conjecture of Dross and Havet.	10.1016/j.jctb.2022.02.005	[ "https://arxiv.org/pdf/2103.06229v3.pdf" ]	232,170,586	2103.06229	e93e32be3d5da87ada2fee497cda13696c0805da	Trees with few leaves in tournaments Alistair Benford Richard Montgomery Trees with few leaves in tournaments We prove that there exists C > 0 such that any (n + Ck)-vertex tournament contains a copy of every n-vertex oriented tree with k leaves, improving the previously best known bound of n+O(k 2 ) vertices to give a result tight up to the value of C. Furthermore, we show that, for each k, there exists n0, such that, whenever n n0, any (n + k − 2)-vertex tournament contains a copy of every n-vertex oriented tree with at most k leaves, confirming a conjecture of Dross and Havet. with = + implies, after reversing the edges again, the result with = −. We also use standard hierarchy notation. That is, for a, b ∈ (0, 1], we write a b to mean that there is a non-decreasing function f : (0, 1] → (0, 1] such that the subsequent statement holds whenever a f (b). Median orders Median orders were first used to embed trees in tournaments by Havet and Thomassé [9]. Given a tournament G, an ordering σ = v 1 , . . . , v n of V (G) is a median order if it maximises the number of pairs i < j with v i v j ∈ E(G). The following lemma gives two simple fundamental properties of median orders (see, e.g., [3,Lemma 9]). Lemma 2.1. Let G be a tournament and v 1 , . . . , v n a median order of G. Then, for any two indices i, j with 1 i < j n, the following properties hold. i) v i , v i+1 , . . . , v j is a median order of the induced subtournament G[{v i , v i+1 , . . . , v j }]. ii) v i dominates at least half of the vertices v i+1 , v i+2 , . . . , v j , and v j is dominated by at least half of the vertices v i , v i+1 , . . . , v j−1 . In particular, each vertex v i , 1 i < n, dominates its successor v i+1 . Median orders contain short directed paths from any vertex to any vertex later in the order, as follows (in combination with Lemma 2.1 i)). Corollary 2.2. Let n 2. If v 1 , . . . , v n is a median order of the n-vertex tournament G, then G contains a directed path from v 1 to v n with length at most 2. Proof. Suppose v 1 v n / ∈ E(G), for otherwise such a path exists, and let V = {v 2 , . . . , v n−1 }. Then, by Lemma 2.1 ii), \|N + (v 1 , V )\| = \|N + (v 1 )\| n− 1 2 > \|V \|/2, and, similarly, \|N − (v n , V )\| > \|V \|/2. Therefore, there is some w ∈ V such that v 1 wv n is a directed path. Median orders have been used particularly effectively to embed arborescences in tournaments. An out-arborescence (respectively, in-arborescence) is an oriented tree T with a root vertex t ∈ V (T ) such that, for every v ∈ V (T ), the path between t and v in T is directed from t to v (respectively, from v to t). Dross and Havet [3] used median orders to prove that any (n + k − 1)-vertex tournament contains a copy of any n-vertex arborescence with k leaves. We will use their result in the following slightly stronger form (see [3,Theorem 12]). Theorem 2.3. Let A be an n-vertex out-arborescence with k 1 out-leaves and root r. Let G be a tournament on n + k − 1 vertices and let σ = v 1 , . . . , v n+k−1 be a median order of G. Then, there is an embedding φ of A in G such that φ(r) = v 1 . We will need some linear bound on the number of vertices required in a tournament, which we then apply for small trees. Any linear bound would suffice, but with the value of C in mind we derive Corollary 2.5 from the following theorem of Dross and Havet [3]. Theorem 2.4. For each n 2, every 3 2 n + 3 2 k − 2 -vertex tournament contains a copy of every n-vertex oriented tree with k leaves. Corollary 2.5. Let n, r, k 1, and suppose G is a tournament with at least 3 2 n + 3 2 k − 2r vertices and T is an oriented forest with n vertices, r components and, in total, k leaves and isolated vertices. Then, G contains a copy of T . Proof. Label the components of T as T 1 , . . . , T r , and say, for each i ∈ [r], that T i has n i vertices and, in total, k i isolated vertices and leaves. Note that n i + 3k i 4 for each i ∈ [r]. Take the largest s r for which there are vertex-disjoint subgraphs S i ⊆ G, i ∈ [s] such that, for each i ∈ [s], S i is a copy of T i . Suppose s < r, for otherwise we have already found a copy of T in G, and note that G − i∈[s] V (S i ) \|G\| − n + n s+1 i∈[r]\{s+1} n i + 3k i − 4 2 + 3n s+1 2 + 3k s+1 2 − 2 3n s+1 2 + 3k s+1 2 − 2. Therefore, by Theorem 2.4, G − ∪ i∈[s] V (S i ) contains a copy of S s+1 , a contradiction. For Theorem 1.2 it is convenient to use the following bound, originally proved by El Sahili [4], which can also be recovered from Theorem 2.4 by observing we must have k n − 1. Theorem 2.6. For each n 2, every (3n − 3)-vertex tournament contains a copy of every n-vertex oriented tree. Non-directed paths In both the proofs of Theorem 1.1 and 1.2, we will take a median order, σ = v 1 , . . . , v m say, of an mvertex tournament, G say, and carefully partition this order into intervals before embedding different parts of the tree into each interval. This embedding must thus work when v i v j ∈ E(G) for each 1 i < j m, that is, when G is a transitive tournament. Our embeddings then, will embed the vertices along a directed path into a consistent order under σ. From this, embedding directed paths will be more restrictive than embedding paths which have some changes of direction. Here, we will recall some results of Thomason which we use to embed paths with changes of directions, allowing us to assume later that each maximal bare subpath in the tree is directed. To discuss the changes of direction in a path and recall these results, we use the terminology of blocks. A block of an oriented path P is a maximal directed subpath. When we introduce an oriented path we assume it has an associated overall direction, and thus a first and last vertex as well as a first block and a last block. When the path is a directed path we will always assume the associated direction is the natural one, i.e., the one in which the first vertex has no in-neighbours. With only a couple of exceptions, when a tournament G has one (or two) more vertices than an oriented path P , we can embed P into G, while furthermore embedding one (or two) endvertices into a matching set of two vertices, if each endvertex is next to a block of length 1. That is, we have the following two results of Thomason. Theorem 2.7 ([15, Theorem 1]). Let P be an oriented path of order n with first block of length 1. Let G be a tournament of order n + 1 and X be a subset of V (G) of order at least 2. Then, there is a copy of P in G with first vertex in X. Theorem 2.8 ([15, Theorem 5]). Let P be a non-directed oriented path of order n with first and last block of length 1. Let G be a tournament of order n + 2 and X and Y be two disjoint subsets of V (G) of order at least 2. If P does not consist of three blocks with length one, then there is a copy of P in G with first vertex in X and last vertex in Y . Short directed paths Having found parts of a tree in a median order of a tournament, we will often wish to join two of them with a directed path. The following lemma shows that this is possible across a median order, even in cases where the interval in between the vertices to be joined contains some forbidden vertices. Lemma 2.9. Suppose G is an n-vertex tournament with a median order σ = v 1 , . . . , v n . Then, for any set A ⊆ V (G) \ {v 1 , v n } with \|A\| (n − 8)/6, there is a directed v 1 , v n -path in G − A with length 3. Proof. If there are some distinct x, y ∈ (N + G (v 1 )∩N − G (v n ))\A, then assume, by relabelling if necessary, that xy ∈ E(G) and observe that v 1 xyv n is a path with length 3 in G − A, as required. Therefore, suppose that \|(N + G (v 1 ) ∩ N − G (v n )) \ A\| 1. By Lemma 2.1 ii), we have \|N + G (v 1 ) \ {v n }\|, \|N − G (v n ) \ {v 1 }\| (n − 2)/2. Let B 1 = N + G (v 1 ) \ (A ∪ N − G (v n ) ∪ {v n }) and B 2 = N − G (v n ) \ (A ∪ {v 1 }). Note that \|B 1 \| n/2 − 2 − \|A\| > 0 and \|B 2 \| n/2 − 1 − \|A\|. Let B 0 = V (G) \ (B 1 ∪ B 2 ∪ {v 1 , v n }), so that \|B 0 \| = n − 2 − \|B 1 \| − \|B 2 \| n − 2 − (n/2 − 2 − \|A\|) − (n/2 − 1 − \|A\|) = 2\|A\| + 1.(1) Colour vertices in B 0 , B 1 and B 2 respectively green, red and blue. If any blue vertex, x say, has a red in-neighbour, y say, then v 1 yxv n is a path with length 3 in G − A, as required. Therefore, suppose that every in-neighbour of each blue vertex is a green vertex or a blue vertex, for otherwise we have the desired path. Let j be the largest integer such that v j is blue. Let A 1 = A ∩ {v 2 , . . . , v j−1 } and A 2 = A ∩ {v j+1 , . . . , v n−1 }, so that \|A 1 \| + \|A 2 \| = \|A\|. For the appropriate r, let I 1 , . . . , I r be the maximal intervals of v 2 , . . . , v j−1 consisting of only red and green vertices. Observe that, for each i ∈ [r], the vertex after I i in σ is blue, and has at least \|I i \|/2 in-neighbours in I i by Lemma 2.1 ii), all of which must be green. Thus, every interval I i , i ∈ [r], contains at least as many green vertices as red vertices. As every red or green vertex before v j in σ is in some interval I i , i ∈ [r], we have that there are at least as many green vertices as there are red vertices in {v 2 , . . . , v j−1 }. As \|N + G (v 1 ) ∩ {v 2 , . . . , v j }\| (j − 1)/2 by Lemma 2.1 ii), at least (j − 1)/2 − \|A 1 \| − 1 of the vertices in {v 2 , . . . , v j−1 } are red. Therefore, there are at least (j − 1)/2 − \|A 1 \| − 1 green vertices in {v 2 , . . . , v j−1 }. By (1) and the definition of A 2 , we have that there at most 2\|A\| + 1 − \|A 2 \| green vertices in {v 2 , . . . , v j−1 }. Thus, 2\|A\| + 1 − \|A 2 \| (j − 1)/2 − \|A 1 \| − 1. Rearranging, and using that \|A 1 \| + \|A 2 \| = \|A\|, we get 3\|A\| 2\|A 2 \| + j/2 − 5/2. Now, by Lemma 2.1 ii), \|N − G (v n )∩({v j+1 , . . . , v n−1 })\| (n−1−j)/2, so, as v j is the last blue vertex in σ, there are at least (n−1−j)/2 vertices in A 2 . Thus, 3\|A\| 2\|A 2 \|+j/2−5/2 (n−j)+j/2−7/2 = n − j/2 − 7/2. As j n − 1, we have 3\|A\| (n − 6)/2, contradicting that \|A\| (n − 8)/6. Trees and random sets Here we collect a number of elementary properties of oriented trees, which we use later, before recalling Chernoff's lemma. Our first proposition considers the number of maximal bare paths in a (nonoriented) tree with k leaves, as follows. Proposition 2.10. An n-vertex tree T with k 2 leaves has at most 2k − 3 maximal bare paths, one of which must have length at least (n − 1)/(2k − 3), and at most 2k − 2 vertices whose degree is not 2. Proof. For the appropriate r, let P 1 , . . . , P r be the maximal bare paths in T , and label vertices such that, for each i ∈ [r], P i is an x i , y i -path. Note that the tree T formed from T by replacing each path P i , i ∈ [r], by a single undirected edge has r edges, r + 1 vertices, k leaves and no degree 2 vertices. Therefore, 2(\|T \| − 1) = 2e(T ) = v∈V (T ) d T (v) k + 2(\|T \| − k) + \|{v : d T (v) 3}\|, and thus \|{v : d T (v) 3}\| k − 2. As {v : d T (v) 3} = {v : d T (v) 3} , T has at most 2k − 2 vertices whose degree is not 2. Furthermore, \|T \| = r + 1 k + (k − 2), so that r 2k − 3. Finally, as i∈[r] (P i ) = e(T ) = n − 1, one of the paths P i , i ∈ [r], has length at least (n − 1)/(2k − 3). In the main embedding for both Theorem 1.1 and Theorem 1.2, we will embed collections of small subtrees with directed paths between them. The next two propositions (appropriately applied to an auxiliary oriented tree with vertices representing subtrees and edges representing paths) will give us an order in which these trees and paths will be embedded along a median order of the tournament. We use Proposition 2.11 for Theorem 1.1, and Proposition 2.12 for Theorem 1.2. Proposition 2.11. Every oriented tree T with n 1 vertices has a vertex partition V (T ) = V 1 ∪. . .∪V s of non-empty sets, for some s ∈ [n], such that, for each edge e ∈ E(T ), for some i ∈ [s − 1], e is an edge directed from V i to V i+1 . Proof. Noting that the statement is trivially true if \|T \| 2, we prove this by induction on \|T \|. Suppose then it is true for all oriented trees with fewer than n 3 vertices. We may assume, by directional duality, that T has an out-leaf. Let T be formed from T by removing such an out-leaf, t say, and let s ∈ [n − 1] be such that there is a vertex partition V (T ) = V 1 ∪ . . . ∪ V s of non-empty sets, such that, for each edge e ∈ E(T ), for some i ∈ [s − 1], e is an edge directed from V i to V i+1 . Let V s+1 = ∅. Let j be such that the in-neighbour of t in T is in V j , and add t to V j+1 . Taking the non-empty sets from V 1 , . . . , V s+1 completes the proof of the inductive step, and hence the proposition. j ∈ [n − 1], there is some i 1 , i 2 ∈ [n] with i 1 j < i 2 and e j = t i1 t i2 . Proof. We proceed by induction on n, noting the proposition is trivial for n = 1. For n > 1, we may assume, by directional duality, that T has an out-leaf. Let t n be this out-leaf, and e n−1 its adjacent edge. By the inductive hypothesis, there are labellings V (T − t n ) = {t 1 , . . . , t n−1 } and E(T − t n ) = {e 1 , . . . , e n−2 }, such that, for every j ∈ [n − 2], e j = t i1 t i2 for some i 1 j < i 2 . Taking V (T ) = {t 1 , . . . , t n } and E(T ) = {e 1 , . . . , e n−1 } completes the proof. Finally, in our embeddings we sometimes take small random sets, on which we use a standard Chernoff bound, as follows (see, for example [1]). Lemma 2.13. If X is a binomial variable with standard parameters n and p, denoted X = Bin(n, p), and ε satisfies 0 < ε 3/2, then P(\|X − EX\| εEX) 2 exp −ε 2 EX/3 . Proof of Theorem 1.1 In Section 3.1, we use the results quoted in Section 2.3 to show that it is enough to prove Theorem 1.1 in the case where all bare paths of T are directed. That is, we reduce the proof to showing the following result. Theorem 3.1. There is some C > 0 such that each (n + Ck)-vertex tournament contains a copy of every n-vertex oriented tree with k leaves in which every bare path is a directed path. To prove Theorem 3.1, we first remove O(k) long directed paths from T to leave a forest with size linear in k. The components of this forest we embed into carefully chosen intervals of a median order with O(k) spare vertices in total, using Corollary 2.5. It remains then to embed the long directed paths, where we only have a constant number of spare vertices per path. This we do with Lemma 3.2 in Section 3.2. A simple modification of Dross and Havet's procedure for embedding arborescences into median orders (which they used to prove Theorem 2.3) allows directed paths from specified first vertices to be embedded efficiently into a median order. To embed such paths with both endvertices specified, we adapt this procedure, using it to embed most of the directed paths, but, as soon as all but three edges of any path are embedded, using Lemma 2.9 to connect the path to its desired last vertex. This allows us to find a set of directed paths while having only constantly many spare vertices per path (see Lemma 3.2), which we use to prove Theorem 3.1 in Section 3.3. Reduction to trees with only directed bare paths To prove Theorem 1.1 from Theorem 3.1, we take a tree T , remove most of the middle section of the maximal bare paths with at least 6 blocks, and duplicate each new leaf created by this removal. (Here, a duplicated vertex is a new vertex with exactly the same in-and out-neighbourbood as the matching original vertex.) Calling the resulting forest T , if we have an embedding of T then the duplication of a leaf gives us two options to embed the original vertex from T . This will allow us to use the results in Section 2.3 to embed the deleted path given enough other vertices in the tournament (with no further restriction on these other vertices). Not every maximal bare path in T will be directed, but each such path will have at most 5 blocks. Adding a dummy leaf at any vertex in two blocks will give a forest T containing T whose maximal bare paths are all directed, allowing us to apply Theorem 3.1 to each component. Importantly, T , and hence T , will still have O(k) leaves. Proof of Theorem 1.1 from Theorem 3.1. Using Theorem 3.1, let C 8 be large enough that, for everyn andk, every (n + (C − 8)k)-vertex tournament contains a copy of everyn-vertex oriented tree with (at most) 9k leaves in which every bare path is a directed path. Let G be an (n + Ck)-vertex tournament, and let T be an n-vertex oriented tree with k leaves. For the appropriate r, let P 1 , . . . , P r be the maximal bare paths in T , and label vertices such that, for each i ∈ [r], P i is an x i , y i -path. By Proposition 2.10, we have r 2k − 3. Let I ⊆ [r] be the set of i ∈ [r] such that P i has at least 6 blocks. For each i ∈ I, let P (1) i and P (2) i be the first two blocks of P i from x i , and let P (4) i from y i on P i . Let Q i = (P i − 4 j=1 P (j) i ) + e (1) i + e (2) i . Note that, for each i ∈ I, the first and last block of Q i have length 1, its endvertices have degree 2 in T , and it has at least 4 blocks (and thus length at least 4). Label vertices so that, for each i ∈ I, Q i is a u i , v i -path. Let T be the forest formed from T by, for each i ∈ I, deleting Q i and creating two new vertices, u i and v i , so that u i is a duplicate of u i and v i is a duplicate of v i . Note that u i , u i , v i and v i are all leaves of T . Let B be the set of vertices with degree 2 in T with either no in-neighbour or no out-neighbour, so that they lie in the intersection of two (consecutive) blocks. Observe that each such vertex must lie on some path P i , i ∈ [r]\I, or on P (1) i ∩P (2) i or P (3) i ∩P (4) i for some i ∈ I. Therefore, \|B\| 4(r −\|I\|)+2\|I\|. Now, form T from T by taking each v ∈ B and adding a new out-neighbour as a leaf, calling the new vertex u v . We note here that all bare paths of T are directed paths. Note that, ifT is a component of T , and q is the number of paths Q i adjacent toT that are deleted when forming T from T , thenT has at most k − q + 2q k + \|I\| leaves. Furthermore, T has in total n + 2\|I\| − i∈I (\|Q i \| − 2) n + 2\|I\| − 3\|I\| = n − \|I\| vertices. Therefore, as r 2k − 3, each component of T has at most k + \|I\| + \|B\| 9k leaves and T in total has at most n − \|I\| + \|B\| n + 8k vertices. Iteratively and vertex-disjointly, embed as many different components from T into G as possible. If a component of T , say a treeT withn vertices andk leaves, is left unembedded then there are at least \|G\| − (\|T \| − \|T \|) (n + Ck) − (n + 8k) +n n + (C − 8)k vertices not used in the embedding, andk 9k. Thus, by the choice of C, we can embedT using the unused vertices in G, a contradiction. Thus, G contains a copy of T , S say. For each v ∈ B, delete the copy of u v from S , and let the resulting copy of T be S . Note that, as C 8 and \|I\| 2k − 3, \|V (G) \ V (S )\| = n + Ck − \|T \| = n + Ck − n + 2\|I\| − i∈I (\|Q i \| − 2) i∈I (\|Q i \| − 2), and take vertex disjoint sets A i , i ∈ I, in V (G) \ V (S ) with \|A i \| = \|Q i \| − 2 for each i ∈ I. For each i ∈ I, letū i ,ū i ,v i ,v i be the copy of u i , u i , v i , v i respectively in S . Using Theorem 2.8, for each i ∈ I, find a copy of Q i , say R i , in G[A i ∪ {ū i ,ū i ,v i ,v i }] starting atū i orū i and ending atv i orv i . Take then S , and, for each i ∈ I, delete from T any vertices in {ū i ,ū i ,v i ,v i } which are not an endvertex of R i and add the path R i . Note that this gives a copy of T . Joining vertex pairs with directed paths disjointly We now connect multiple pairs of vertices with directed paths, where the start vertex for each path lies in a set B 1 , and the end vertex lies in another set B 2 , and the vertices of B 1 come before the vertices of B 2 in a median order. With Lemma 2.9 we can find such paths; the challenge here is to find these paths when they collectively must use almost all of the intermediate vertices in the median order. To do this, we find most of the paths using a procedure of Dross and Havet [3] for embedding arborescences, modifying it with Lemma 2.9 to attach each path to the correct end vertex when most of the path has been found. Lemma 3.2. Let G be an (m 0 + m 1 + m 2 )-vertex tournament, and suppose σ = v 1 , . . . , v m0+m1+m2 is a median order of G. Let B 1 ⊆ V (G) be the first m 1 vertices of G according to σ, let B 2 ⊆ V (G) be the last m 2 vertices of G according to σ, and let B 0 = V (G) \ (B 1 ∪ B 2 ). Let (x 1 , . . . , x r ) ∈ B r 1 and (y 1 , . . . , y r ) ∈ B r 2 . For each i ∈ [r], let i 5. Suppose finally that m 0 m 1 + m 2 + i∈[r] i + 22r − 15.(2) Then, there are internally vertex-disjoint directed paths P 1 , . . . , P r in G such that, for each i ∈ [r], P i is a directed x i , y i -path with length i and internal vertices in B 0 . Proof. Let B 1 be the first (m 1 +2r−2) vertices of B 0 according to σ, and let B 2 be the last (m 2 +2r−2) vertices of B 0 according to σ. Choose X = {x 1 , . . . , x r } ⊆ B 1 such that x i ∈ N + (x i ) for each i ∈ [r] . This is possible as, if for i ∈ [r] we have chosen x 1 , . . . , x i−1 , letting U i = {w ∈ B 1 : x i < σ w σ v m1 }, then Lemma 2.1 ii) gives \|N + (x i , B 1 ) \ {x 1 , . . . , x i−1 }\| = \|N + (x i , U i ∪ B 1 ) \ (U i ∪ {x 1 , . . . , x i−1 })\| \|U i \| + \|B 1 \| 2 − \|U i \| − \|{x 1 , . . . , x i−1 }\| = \|B 1 \| − \|U i \| 2 − (i − 1) (m 1 + 2r − 2) − (m 1 − 1) 2 − (r − 1) > 0. Similarly, choose Y = {y 1 , . . . , y r } ⊆ B 2 such that y i ∈ N − (y i ) for each i ∈ [r]. Let A be a digraph formed by taking the disjoint union of directed paths Q i , i ∈ [r], where Q i has length i − 5 for each i ∈ [r]. For i ∈ [r], let b i be the first vertex and c i be the last vertex of Q i . Note that A has i∈[r] ( i − 4) vertices. Let n 1 = m 0 − m 2 − 20r + 13. We now give a procedure which produces a partial embedding φ of A into G[{v m1+1 , . . . , v m1+n1 }]. Throughout, if a vertex v j of G is the image of a vertex of A, we say that it is hit and denote its pre-image by a j ∈ V (A). The sets W j record vertices of G already used for the last two internal vertices of the paths P 1 , . . . , P r found by stage j. • Initially, set W m1+1 = ∅ and φ(b i ) = x i for each i ∈ [r] (so that x 1 , . . . , x r are hit). • For j = m 1 + 1 to m 1 + n 1 in turn, do the following. (a) If v j is hit and a j = c i for some i ∈ [r], then, if possible, let w i,1 , w i,2 ∈ {v j+1 , . . . , v m1+m0 }\ (W j ∪ Y ) be such that w i,1 and w i, 2 are not yet hit, and v j → w i, 1 → w i,2 → y i in G. Set W j+1 = W j ∪ {w i,1 , w i,2 }. If it is not possible to find such a w i,1 and w i,2 , then simply set W j+1 = W j . (b) If v j is hit and a j / ∈ {c 1 , . . . , c r }, then extend φ if possible by assigning the first not-yet- hit out-neighbour of v j in {v j+1 , . . . , v m1+n1 } \ W j to the out-neighbour of a j in A. Set W j+1 = W j . (c) If v j ∈ W j , then set W j+1 = W j . (d) If v j / ∈ W j and v j is not hit, then say that v j is failed. Set W j+1 = W j . Note that, for each m 1 + 1 j m 1 + n 1 , the vertices in W j are never hit, so that this procedure is well-defined. We first show that the paths with length 3 in (a) are always found, as follows. Claim 3.3. For each m 1 + 1 j m 1 + n 1 , if v j is hit and a j = c i for some i ∈ [r] , then the procedure finds vertices w i,1 and w i,2 as described in (a). Proof of Claim 3.3. Suppose j satisfies m 1 + 1 j m 1 + n 1 , v j is hit and a j = c i for some i ∈ [r], so that, at stage j, we carry out (a). Let s denote the number of times (a) was carried out before stage j. As W j contains only vertices found in these previous instances of (a), we have \|W j \| 2s. At stage j, each path Q i has at most one vertex embedded by φ to {v j , v j+1 , . . . , v m1+n1 }. Moreover, if a path Q i has a vertex embedded by φ to {v j+1 , . . . , v m1+n1 }, then (a) has not been carried out for that c i . Thus, at most r − 1 − s vertices in {v j+1 , . . . , v m1+n1 } have been hit. Let W be the union of W j , Y \ {y i }, and the hit vertices in {v j+1 , . . . , v m1+n1 }. Thus, as s r − 1, \|W \| 2s + (r − 1) + (r − 1 − s) 3(r − 1).(3) Let j be such that v j = y i , and note that, as y i ∈ B 2 , j m 1 + m 0 − m 2 − 2r + 3, so that, as n 1 = m 0 − m 2 − 20r + 13, we have j − j + 1 m 1 + m 0 − m 2 − 2r + 4 − m 1 − n 1 = 18(r − 1) + 9 6\|W \| + 9.(4) Therefore, by Lemma 2.9, vertices w i,1 and w i, 2 exist in {v j , v j+1 , . . . , v j } \ (W j ∪ Y ) which have not yet been hit so that v j → w i,1 → w i,2 → v j = y i in G. If the procedure finds a full embedding of A into G[{v m1+1 , v m1+2 , . . . , v m1+n1 }], then observe that, for each i ∈ [r], the image of Q i and the path φ(c i ) → w i,1 → w i,2 → y i together give a path, P i say, with length i − 2 which is directed from φ(b i ) = x i to y i . Furthermore, the paths P i , i ∈ [r], are vertex-disjoint with vertices in B 0 . Taking P i to be the path x i P i y i for each i ∈ [r] gives the desired result. All that remains to show is that the procedure produces a full embedding φ of A. Let W = W m1+n1+1 and note that \|W \| 2r. Assume for a contradiction that the procedure does not yield an embedding of A into G. Then the set, F say, of failed vertices in {v m1+1 , . . . , v m1+n1 } has \|F \| > n 1 − \|A\| − \|W \|. Let U ⊆ V (A) be the set of embedded vertices at the end of the procedure. Let L be the set of vertices of A which are the last embedded vertex on some path Q i . Note we have \|L\| = r. Say a vertex a ∈ V (A) is active at stage j if φ(a) ∈ {v m1+1 , . . . , v j−1 } and a has an out-neighbour b that is not embedded in {v m1+2 , . . . , v j } (i.e., either b is not embedded or φ(b) ∈ {v j+1 , . . . , v m1+n1 }). Now, if v j ∈ F comes before some vertex in X = {x 1 , . . . , x r } ⊆ B 1 , then it is possible there will be no active vertex at stage j. However, because we have assumed that the procedure does not yield an embedding of A into G, if v j ∈ F and j 2m 1 + 2r − 1, then there must be some active vertex at stage j, for otherwise all the vertices of A would be embedded in {v m1+1 , . . . , v j−1 }. LetF = {v j ∈ F : j 2m 1 + 2r − 1}, so that, for each v j ∈F we can define r j to be the largest index such that a rj is active for j. Note, by the definition of an active vertex, r j < j. Furthermore, as \|F \| > n 1 − \|A\| − \|W \|, B 1 = {v m1+1 , . . . , v 2m1+2r−2 } contains at least r vertices in the embedding (those in X ), and \|A\| = i∈[r] ( i − 4), we have \|F \| > n 1 − \|A\| − \|W \| − (m 1 + 2r − 2 − r) m 0 − m 2 − 20r + 13 − i∈[r] i + r − m 1 + 2 (2) 3r. (5) For each v j ∈F , set I j = {v i : r j < i j}. We now bound from above the number of vertices of F in I j , as follows. Claim 3.4. If v j ∈F , then \|I j ∩ F \| \|I j ∩ φ(L)\| + \|I j ∩ W \|. Proof of Claim 3.4. Let J = (I j ∩ N + (v rj )) \ W . As the out-neighbour of a rj is never embedded in I j , all the vertices in J must be hit by the start of stage r j . Thus, as F ∩ W = ∅, we have I j ∩ F ⊆ I j ∩ N − (v rj ), so that \|I j ∩ F \| \|I j ∩ N − (v rj )\|.(6) Now, let A rj and A j−1 be the sub-digraphs of G[v m1+1 , . . . , v j ] which are the image of the partial embedding φ at the end of stage r j and stage j − 1, respectively, restricted to the vertex set {v m1+1 , . . . , v j }. Observe the following. • Each vertex of J is the last vertex of a path of A rj , as it is hit by the end of stage r j and occurs later in σ than r j . • Any vertex in I j which is the last vertex of some path of A j−1 must be the image of some c i , for otherwise it is active for j, contradicting the definition of r j . Thus, because L is the set of vertices of A which are the last embedded vertex on some path Q i , such a vertex is in I j ∩ φ(L). • As r j j − 1, A rj ⊆ A j−1 , and V (A j−1 ) \ V (A rj ) ⊆ I j , so A j−1 must have at least as many paths terminating in I j as A rj does. Combining these three observations we have \|J\| \|I j ∩ φ(L)\|, and hence \|I j ∩ N + (v rj )\| \|I j ∩ φ(L)\| + \|I j ∩ W \|.(7) Now, by (6) and (7), this proves the claim. Lemma 2.1 ii), \|I j ∩ N − (v rj )\| \|I j ∩ N + (v rj )\|. Together with Let M be the set of indices j such that v j ∈F , and I j is maximal for inclusion among the sets I i , with v i ∈F . We will show that the sets I j , j ∈ M are disjoint. If i, j ∈ M with i < j and I i ∩ I j = ∅, then we have r j < i. Observe that, as a rj is active for j and φ(a rj ) ∈ {v 0 , . . . , v i−1 }, a rj is also active for i, and hence r i r j . Thus, I i ⊆ I j and, as i < j, I i = I j , and hence I i is not maximal for inclusion among the sets I i , with v i ∈F , a contradiction. Since v j ∈ I j for all v j ∈F , we haveF ⊆ ∪ j∈M I j . As the sets I j , j ∈ M , are pairwise disjoint, \|F \| j∈M \|I j ∩ F \|. By Claim 3.4, we therefore obtain \|F \| j∈M \|I j ∩ F \| j∈M (\|I j ∩ φ(L)\| + \|I j ∩ W \|) \|φ(L)\| + \|W \| 3r, contradicting (5). This completes the proof of the lemma. Proof of Theorem 3.1 Given Lemma 3.2 it is now straight-forward to prove Theorem 3.1. Given an n-vertex oriented tree T with k leaves whose maximal bare paths are directed, we label such paths with length at least 5 as P 1 , . . . , P r , for the appropriate r (which, by Proposition 2.10, satisfies r = O(k)). We can then consider T to be formed of small vertex-disjoint subtrees T 1 , . . . , T r+1 connected by the paths P 1 , . . . , P r . We use Proposition 2.11 to group these subtrees into classes, with the classes ordered so that each path P i goes from some class to the next class. Given then a tournament G with n + 50k vertices, we divide a median order into intervals, with one interval for each class of subtrees and one for the set of paths between each pair of consecutive classes (see (9)). Then, we then use Corollary 2.5 to embed the subtrees T i into their interval in the median order before using Lemma 3.2 to embed the paths P i with interior vertices in their interval in the median order. Proof of Theorem 3.1. We will prove this with C = 50, so letn = n + 50k. Let T be an n-vertex oriented tree with k leaves in which every bare path is a directed path, and let G be an-vertex tournament. Let B be the set of vertices of T which do not have degree 2, so that, by Proposition 2.10, \|B\| 2k − 2. Remove all maximal bare paths of length at least 5 from T . Let r be the number of removed paths, noting that, by Proposition 2.10, r 2k −3, and label these paths as P 1 , . . . , P r (where we recall (P i ) denotes the length of P i ). Say the remaining forest F has component trees T 1 , . . . , T r+1 , and, for each i ∈ [r + 1], let k i be the number of leaves of T i if \|T i \| 2, and let k i = 1 if \|T i \| = 1. Note that F is a union of (\|B\| − 1 − r) maximal bare paths of T with length at most 4 between vertices in B, resulting in a forest with r +1 components. Thus, we have that \|F \| \|B\|+3(\|B\|−1−r) 8k −3r −11. Observing that every leaf or isolated vertex of F is in B, we have i∈[r+1] k i \|B\| 2k − 2. We also note that \|F \| = i∈[r+1] \|T i \| and i∈[r] (P i ) = \|T \| − \|F \| + r = n − i∈[r+1] \|T i \| + r.(8) Let S be the oriented tree on vertex set [r + 1] with ij ∈ E(S) whenever there is a directed path from T i to T j in T . By applying Proposition 2.11 to S, let s r + 1 be such that there is a partition I 1 , . . . , I s of [r + 1] into non-empty sets such that, for each distinct i, j ∈ [r + 1], and i ∈ [s], if i ∈ I i and there is a directed path from T i to T j in T , then i < s and j ∈ I i +1 . For each i ∈ [s − 1], let J i be the set of indices j ∈ [r] such that P j is directed from T i to T j for some i ∈ I i and j ∈ I i+1 , and note that ∪ i∈[s−1] J i = [r]. Let σ = v 1 , . . . , vn be a median order of G. In this median order take consecutive intervals V 1 , U 1 , V 2 , U 2 , V 3 , . . . , V s−1 , U s−1 , V s ,(9) appearing in that order, such that, for each j ∈ [s], \|V j \| = 3 2 i∈Ij (\|T i \| + k i ) − 2\|I j \| 3 2 i∈Ij (\|T i \| + k i ) + 1 2 − 2\|I j \|,(10) and, for each j ∈ [s − 1], \|U j \| = \|V j \| + \|V j+1 \| + i∈Jj (P i ) + 22\|J j \| − 15.(11) Note that this is possible, as (11), for each j ∈ [s − 1], the \|J j \| paths P i , i ∈ J j , between ∪ i∈Ij T i and ∪ i∈Ij+1 T i can then be embedded in the intervals V j , U j , V j+1 with the appropriate first and last vertex in V j and V j+1 , respectively, and internal vertices in U j . This completes the embedding of T , and hence the proof of the theorem. s j=1 \|V j \| + s−1 j=1 \|U j \| (11) 3 s j=1 \|V j \| + j∈[r] (P j ) + 22 j∈[s−1] \|J j \| − 15(s − 1) (10) 9 2 i∈[r+1] (\|T i \| + k i ) + 3 2 s − 6 s j=1 \|I j \| + j∈[r] (P j ) + 22r − 15(s − 1)(8 Proof of Theorem 1.2 As an illustrative case, let us first sketch Theorem 1.2 for trees consisting of a directed path between two arborescences, as follows. Suppose we have a directed path P , an in-arborescence S with root the first vertex of P , and an out-arborescence S with root the last vertex of P , and suppose that S ∪ P ∪ S is an oriented tree with n vertices. Say S has k in-leaves and S has k out-leaves, and the tournament G has m := n + k + k − 2 vertices and a median order Essentially, all our embeddings will look like this, where P will be a very long path, but with some additional subtrees and paths found within the interval we use to embed P . For example, suppose now the tree T also has a subtree F which shares one vertex, t say, with S, where t only has out-neighbours in F . If P is a long path (compared to \|F \|, \|S\|, \|S \|) then we can embed T = F ∪ S ∪ P ∪ S into a tournament G with m := \|T \| + k + k − 2 vertices as follows. Carry out the above embedding of S and S into the start and end respectively of a median order v 1 , . . . , v m of G and note that the path Q := v \|S\|+k−1 → v \|S\|+k → . . . → v m−\|S \|−k +2 has \|F \| − 1 + \|P \| vertices. If s is the embedding of t ∈ V (S), then by Lemma 2.1 ii) and as \|Q\| \|P \| − 1 \|F \|, \|S\|, s will have many out-neighbours in this path, enough that we can easily embed F − t among the out-neighbours of s in Q (using, in particular, Corollary 2.5). However, we wish to do this so that there is a directed path between v \|S\|+k−1 and v m−\|S \|−k +2 covering exactly the \|Q\| − (\|F \| − 1) = \|P \| vertices of V (Q) which are not used to embed F − t. To do this, before embedding F , we first find a short directed v \|S\|+k−1 , v m−\|S \|−k +2 -path R with vertices in V (Q) so that every vertex in V (Q) has at least one out-neighbour on R occurring after some in-neighbour on R. The path R will be short enough that we can embed F − t in the out-neighbours of s in V (Q) while avoiding V (R). Once F − t has been embedded, we slot the remaining vertices in V (Q) into R one by one. This will be possible from the property of R as we are working in the tournament G (see Claim 4.5). Note that, in the language of absorption (as codified by Rödl, Ruciński and Szemerédi [13]), R is a path which can absorb any set of vertices from the interval of the median order between its first and last vertex. More generally, we can embed small trees attached with an out-edge from S ∪ P ∪ S , as long as the attachment point is not too late in P , and also not in S , by embedding such small trees within the interval for the path P . Similarly, we can embed small trees attached with an in-edge from S ∪ P ∪ S , as long as the attachment point is not too early in P , and also not in S. We can also use Lemma 2.9 to add short paths between vertices in the interval from P that are not too close together. We therefore decompose any n-vertex tree T with k leaves by finding a digraph D which can be built in this way and which contains T . Roughly speaking, we call the digraph D a good decomposition for T if it contains T and can be built from some S ∪ P ∪ S as described above by adding digraphs in these ways; this is defined precisely in Section 4.1. In Section 4.2, we show that there is a good decomposition for any tree without a subpath that we could otherwise deal with using Section 2.3 as before. Then, in Section 4.3, we show it is possible to embed any good decomposition of any n-vertex tree with k leaves into an (n + k − 2)-vertex tournament. Finally, in Section 4.4, we put this together to prove Theorem 1.2. (r, m)-good decompositions We now define a good decomposition precisely, using the follow definition of a path partition. Definition 4.1. Say a sequence of paths P 1 . . . P is a path partition of a path P if P = ∪ i∈[ ] P i and, for each i ∈ [ − 1], the end vertex of P i is the start vertex of P i+1 , and all the paths are otherwise pairwise vertex disjoint. Roughly speaking, as depicted in Figure 1, an (r, m)-good decomposition for a tree T is a digraph D with T ⊆ D, such that D can be constructed by taking a long directed path P from the root of an inarborescence S 1 to the root of an out-arborescence S r+1 , attaching small forests F i to a limited number of well-separated subpaths S i of P , and, finally, attaching short directed paths Q i between some of these well-separated subpaths and forests. More precisely, we define an (r, m)-good decomposition as follows. x y P 1 P 2 P i−1 P i P r−1 P r . . . . . . F + 2 F − 2 F 2 S 2 F + i F − i F i S i F + r F − r F r S r S r+1 F − r+1 S 1 F + 1 . . . A7.1 Q j A 7 .2 Definition 4.2. Say that a digraph D is an (r, m)-good decomposition for an n-vertex oriented tree T if V (D) = V (T ), and, for some distinct x, y ∈ V (D), there is a directed x, y-path P with path partition P = P 1 S 2 P 2 S 3 . . . P r−1 S r P r ,(12) an in-arborescence S 1 with root x, an out-arborescence S r+1 with root y, and • forests F + i , F − i , i ∈ [r + 1], and • for some 0 2r, vertices s i , t i and directed s i , t i -paths Q i , i ∈ [ ], such that, letting F i = F − i ∪ F + i for each i ∈ [r + 1], the following hold. A1 T ⊆ S 1 ∪ P ∪ S r+1 ∪ (∪ i∈[r+1] F i ) ∪ (∪ i∈[ ] Q i ) = D. A2 The following sets, over i ∈ [r + 1] and j ∈ [ ], form a partition of V (T ) = V (D): V (P ), V (F + i ) \ V (S i ), V (F − i ) \ V (S i ), V (S 1 ) \ {x}, V (S r+1 ) \ {y}, V (Q j ) \ {s j , t j }. A3 For each i ∈ [r], P i has length at least 2000m. A4 For each i ∈ [r + 1] and ∈ {+, −}, V (S i ) ⊆ V (F i ), \|F i \| m, and F i is a forest in which each component has exactly one vertex in S i , which furthermore has only -neighbours in F i . A5 E(F − 1 ) = E(F + r+1 ) = ∅ and \|S 1 \|, \|S r+1 \| 2. A6 The total number of in-leaves of S 1 and out-leaves of S r+1 is at most the number of leaves of T . A7 For each i ∈ [ ], one of the following holds. A7.1 For some 1 j < j r + 1, Q i is a directed path from F j to F j with length 3(j − j) + 1. A7.2 For some 2 j r, Q i is a directed path with length 3 from V (F − j ) \ V (S j ) to the last vertex of S j . Finding a good decomposition As noted before, by the results in Section 2.3, we will be able to assume that our n-vertex tree T with k leaves in Theorem 1.2 mostly consists of directed bare paths. To find a good decomposition, we consider these paths and arrange them in order of decreasing length. Identifying a point where the length of these paths drops significantly (perhaps including all the paths), we show that removing these long paths creates a forest in which each component is much smaller than each of the removed paths. Next, we order these paths and components using Proposition 2.12. Taking (essentially) the removed paths as the paths P i , carefully chosen directed subpaths S i of the components of the forest (see B1-B4) and some dummy edges if necessary will form the path in (12). After the careful selection in B1-B4, we will be able to divide naturally the rest of T into the other sets in the decomposition. Lemma 4.3. Let 1/n µ 1/k. Let T be an n-vertex oriented tree with k 2 leaves and no bare path of length at least µn with first and last block of length 1 and whose endvertices have degree 2 in T . Then, for some r 10k and m µn, T has an (r, m)-good decomposition. Proof. We will construct an (r, m)-good decomposition using the notation in Definition 4.2, and confirm that each of A1-A7 hold. Let p be the number of maximal bare paths of T , and let them be T 1 , . . . , T p . By Proposition 2.10, we have p 2k − 3. Observe that each T i has fewer than µn edges that are not contained in the first two blocks or the last two blocks, for otherwise, taking the last edge of the second block, and the first edge of the penultimate block, and all the edges between them on T i , gives a bare path with length at least µn with first and last block of length 1 whose endvertices have degree 2 in T . Let q be the number of maximal directed bare paths of T with length at least µn, and let them be T 1 , . . . , T q with length 1 , . . . , q respectively, so that 1 2 . . . q . By the above observation, we find q 4p 8k − 12, and \|T − T 1 − . . . − T q \| (2k − 3)(4µn + µn) 10kµn. Furthermore, as µ 1/k, we must have that q 1 and 1 n/2q n/20k. Now, let r ∈ [q −1] be the smallest integer such that r > 10 6 k r+1 , if it exists, and r = q otherwise. Let m = r /2500. Note that, as 1 n/20k and µ 1/k, , so that, for every j ∈ [r], T j is a directed path from R i − (j) to R i + (j) , and i − (j) j < i + (j). For each j ∈ [r], label vertices so that T j is an x j , y j -path directed from x j ∈ V (R i − (j) ) to y j ∈ V (R i + (j) ). Let I ⊆ {2, . . . , r} be the set of i with y i−1 ∈ V (R i ), x i ∈ V (R i ), and such that the path between y i−1 and x i in R i is not directed from y i−1 to x i . For each j ∈ [r], let Q + j be the path consisting of the last 3(i + (j) − j − 1) + 1 1 edges of T j . For each j ∈ [r] \ I, let Q − j be the path consisting of the first 3(j − i − (j)) + 1 1 edges of T j . For each j ∈ I, let Q − j be the path consisting of the first 3 edges of T j . Note that the lengths of the paths Q + j , Q − j are always much smaller than the length of the path T j . For each i ∈ [r], let P i be such that T i = Q − i P i Q + i is a path partition. Label vertices so that P i is an x i , y i -path directed from x i to y i . Note that each path P i is T i with up to 3r +1 edges removed from each end. As the original length of such a path was at least r = 2500m, and we have 1/n µ 1/r, we have by (13) that A3 holds. Let x = x 1 and note that Q − 1 = x 1 x. Let S 1 ⊆ R 1 + x 1 x be the maximal in-arborescence in R 1 + x 1 x with root x. Note we have that \|S 1 \| 2. Let y = y r and note that Q + r = yy r . Let S r+1 be the maximal out-arborescence in R r+1 + yy r with root y. Note we have \|S r+1 \| 2. y i−1 Q + i−1 yi−1 Pi−1 y i−1 Q + i−1 x i Q − i x i Q − i yi−1 Pi−1 Q + i−1 yi−1 Pi−1 Q + i−1 Q − i Q − i B1 B2 B3 B4 Figure 2: Cases B1-B4. If k 0 is the number of in-leaves of S 1 , then as its root x is an out-leaf, S 1 has k 0 +1 leaves. Similarly, if k 1 is the number of out-leaves of S r+1 , then S r+1 has k 1 + 1 leaves. Now, take the path, S say, between S 1 and S r+1 in T and note that the tree S 1 ∪ S ∪ S r+1 has (k 0 + 1) + (k 1 + 1) − 2 = k 0 + k 1 leaves. Noting that T has at least as many leaves as S 1 ∪ S ∪ S r+1 ⊆ T completes the proof that A6 holds. Now, for each i ∈ {1, r + 1} and each ∈ {+, −}, let F i ⊆ S i ∪ R i be the digraph formed from the union of the paths in (S i ∪ R i ) − E(S i ) from V (S i ) which start with a -edge, and let F i = F + i ∪ F − i = (S i ∪ R i ) − E(S i ). Note that, by the maximality of S 1 as an in-arborescence and the maximality of S r+1 as an out-arborescence, we have that E(F − 1 ) = E(F + r+1 ) = ∅, completing the proof that A5 holds. For each i ∈ {1, r + 1}, \|F i \| \|R i \| + 1 m/2 + 1 m, so A4 holds as well for i ∈ {1, r + 1}. We now construct y i−1 , x i -paths S i , for each 2 i r. For each such i, we consider Q + i−1 ∪R i ∪Q − i , and add up to two edges (according to the cases below) before finding a directed path S i through the resulting digraph. We divide into cases B1-B4 according to whether y i−1 ∈ V (R i ) (i.e., if i + (i−1) = i) and whether x i ∈ V (R i ) (i.e., if i − (i) = i) . These cases are depicted in Figure 2. Note that, if y i−1 ∈ V (R i ) then Q + i−1 consists of only the edge y i−1 y i−1 , and if x i ∈ V (R i ) with i / ∈ I, then Q − i consists of only the edge x i x i . Precisely, for each 2 i r, we do the following. B1 If y i−1 and x i are both in V (R i ), then do the following. B1.1 If the y i−1 , x i -path in the tree R i is a directed path from y i−1 to x i , then let S i be the directed path from y i−1 to x i in R i + y i−1 y i−1 + x i x i . B1.2 If the y i−1 , x i -path in the tree R i is not a directed path from y i−1 to x i (i.e., if i ∈ I), then let S i be the maximal directed subpath from y i−1 that it contains. Let S i be the path consisting of the edge y i−1 y i−1 , followed by S i , followed by a new edge from the endvertex of S i to x i . B2 If y i−1 ∈ V (R i ) and x i / ∈ V (R i ), then let S i be the path y i−1 y i−1 x i . B3 If y i−1 / ∈ V (R i ) and x i ∈ V (R i ), then let S i be the path y i−1 x i x i . B4 If y i−1 , x i / ∈ V (R i ), then let z ∈ V (R i ) be arbitrary, and let S i be the path y i−1 zx i . Now, for each 2 i r, we choose F + i , F − i and F i = F + i ∪ F − i . To do so, for each 2 i r and each ∈ {+, −}, let F i ⊆ S i ∪R i be the digraph formed from the union of the paths in (S i ∪R i )−E(S i ) from V (S i ) which start with a -edge, and let F i = F + i ∪ F − i = (S i ∪ R i ) − E(S i ). Note that F + i and F − i could consist of a single vertex. For each 2 i r, \|F i \| = \|R i \| + 2 m/2 + 2 m. We now have that A4 holds for each i ∈ [r + 1], as required. Let be the number of paths Q i , i ∈ [r], ∈ {+, −} with length greater than 1, so that 0 2r. Relabel these paths arbitrarily as Q i , i ∈ [ ]. Note that, as we created no new vertices, we have that V (D) ⊆ V (T ) (with equality once we confirm T ⊆ D below). It is left then to prove that A1, A2, and A7 hold and check the properties at the start of Definition 4.2. Note that, for each 2 i r, S i was a directed y i−1 , x i -path. Therefore, as x = x 1 and y = y r , P := P 1 S 2 P 2 S 2 . . . P r−1 S r P r(14) is a path partition of the directed x, y-path P . Furthermore, we have that S 1 is an in-arborescence with root x and that S r+1 is an out-arborescence with root y. Now, by construction, T ⊆ P ∪ S 1 ∪ S r+1 ∪ (∪ i∈[r+1] F i ) ∪ (∪ i∈[r], ∈{+,−} Q i ) = D. Whenever Q + i has length 1 and i < r, we have that i + (i) = i + 1, so S i+1 is chosen in B1.1, B1.2, or B2, and hence Q + i = y i−1 y i−1 ⊆ S i+1 . Note that Q + r has length 1, and Q + r = yy r is in S r+1 . Whenever Q − i has length 1 and i > 1, we must have that i / ∈ I and i − (i) = i, and therefore S i is chosen in B1.1 or B3, so that Q − i = x i x i ⊆ S i . Note that Q − 1 has length 1, and Q − 1 = x 1 x is in S 1 . Therefore, P ∪(∪ i∈[r], ∈{+,−} Q i ) = P ∪(∪ i∈[ ] Q i )+x 1 x+yy r , and so T ⊆ P ∪S 1 ∪S r+1 ∪(∪ i∈[r+1] F i )∪(∪ i∈[ ] Q i ) = D and A1 holds. Furthermore, note that V (R i ), i ∈ [r + 1], and V (T i ) \ {x i , y i }, i ∈ [r], form a partition of V (T ). For each i ∈ [r], V (Q − i ) \ {x i , x i }, V (P i ) and V (Q + i ) \ {y i , y i } form a partition of V (T i ) \ {x i , y i }. For each 2 i r, by the choice of F + i and F − i , V (F + i ) \ V (S i ), V (F − i ) \ V (S i ) and V (S i ) \ {y i−1 , x i } form a partition of R i , while V (F − 1 ) \ V (S 1 ) = ∅, V (F + 1 ) \ V (S 1 ) and V (S 1 ) \ {x 1 } partition V (R 1 ) \ {x 1 }, and V (F − r+1 ) \ V (S r+1 ), V (F + r+1 ) \ V (S r+1 ) = ∅ and V (S r+1 ) \ {y r } partition V (R r+1 ) \ {y r }. As V (P ) = (∪ i∈[r] V (P i )) ∪ (∪ 2 i r (V (S i ) \ {y i−1 , x i })) , the sets listed in A2 form a partition of V (T ). Therefore, we need only show that, for each path i ∈ [ ], either A7.1 or A7.2 holds. If Q i = Q + j for some j ∈ [r], then Q i is a directed y j , y j -path of length 3(i + (j) − (j + 1)) + 1 > 1, so that i + (j) > j + 1. As y j ∈ V (S j+1 ) ⊆ V (F j+1 ) and y j ∈ V (R i + (j) ) ⊆ V (F i + (j) ), A7.1 holds for Q i . If Q i = Q − j for some j ∈ [r] \ I, then Q i is a directed x j , x j -path of length 3(j − i − (j)) + 1 > 1, so that i − (j) < j. As x j ∈ V (R i − (j) ) ⊆ V (F i − (j) ), and x j ∈ V (S j ) ⊆ V (F j ), A7.1 holds for Q i . Finally, if Q i = Q − j for some j ∈ I, then S j was chosen in B1.2. From the choice of the relevant maximal directed path S j , the first vertex x j of Q i is in V (F − j ) \ V (S j ) and the last vertex x j of Q i is also the last vertex of S j , and therefore A7.2 holds. Embedding a good decomposition We now show that it is possible to embed an (r, m)-good decomposition D of a n-vertex tree T with k leaves into an (n + k − 2)-vertex tournament G, when 1/n 1/r, 1/k, m/n. For our sketch we will use the notation of Definition 4.2. We take a median order of G and find within it consecutive disjoint intervals V 1 , U 1 , V 2 , U 2 , . . . , V r , U r , V r+1 with carefully chosen sizes. We will embed S 1 into G[V 1 ] while embedding its root to the last vertex of V 1 under σ, using Theorem 2.3, and similarly embed S r+1 into V r+1 so that its root is embedded to the first vertex of V r+1 under σ. For each i ∈ {2, . . . , r}, we will have \|V i \| = \|S i \| and embed the directed path S i into G[V i ] using the ordering provided by σ. As described at the start of this section, for each i ∈ [r], we then find a short path R i from the last vertex of V i under σ to the first vertex of V i+1 under σ which can 'absorb' any subset of vertices from U i (see Claim 4.5). We then embed the forests F + i , F − i , i ∈ [r + 1] and directed paths Q i , i ∈ [ ], into ∪ i∈[r] (U i \ V (R i )), before incorporating the right number of vertices into each path R i . More specifically, as depicted in Figure 3, for each i ∈ [r], we will divide U i into six parts, U i,1 , . . . , U i,6 , again with carefully chosen sizes. The sets U i,1 and U i,6 will be small and covered by R i (aiding the desired 'absorption' property of R i ). We will embed V (F + i ) \ V (S i ) into U i,2 \ V (R i ), using A4 and that typical vertices in V i (the image of S i ) have plenty of out-neighbours in U i,2 (see Claim 4.6) and Claim 4.6). We will embed paths Q j satisfying A7.2 using the appropriate set U i,5 (see Claim 4.7). We will then embed paths Q j satisfying A7.1 using different sets U i,3 (see Claim 4.8). As we chose the size of the sets U i , i ∈ [r], carefully, for each i ∈ [r], we will then have the correct number of vertices unused in U i to absorb into R i and complete the embedding of P i , and hence also the embedding of T ⊆ D. V (R i ) is small. Similarly, we will embed V (F − i+1 ) \ V (S i+1 ) into U i,4 \ V (R i ) (see also Lemma 4.4. Let 1/n µ, 1/r, 1/k and m µn. Every tournament with n + k − 2 vertices contains a copy of every n-vertex oriented tree with k leaves which has an (r, m)-good decomposition. Proof. Note that we can additionally assume that µ 1/r, 1/k. Let G be an (n + k − 2)-vertex tournament and suppose that the n-vertex tree T with k leaves has an (r, m)-good decomposition D using the notation in Definition 4.2. Let k 0 be the number of in-leaves of S 1 and let k 1 be the number of out-leaves of S r+1 . By A5, we have k 0 , k 1 1 and by A6 we have k 0 + k 1 k. Let I 1 ⊆ [ ] be the set of i ∈ [ ] satisfying A7.1. Let I 2 = [ ] \ I 1 , so that, by A7, each i ∈ I 2 satisfies A7.2. For each i ∈ I 1 , using A7.1, let q i , r i ∈ [r + 1] with q i < r i be such that Q i is a directed path from F qi to F ri with length 3(r i − q i ) + 1. For each i ∈ [r], let a i be the number of j ∈ I 1 for which q j i < r j . For each i ∈ I 2 , using A7.2, let 2 s i r be such that Q i is a directed path from V (F − si ) \ V (S si ) to the last vertex of S si . For each i ∈ [r], let b i be the number of j ∈ I 2 with s j = i + 1 (and note that we always have b r = 0). Let σ = v 1 , . . . , v n+k−2 be a median order of G. Take in v 1 , . . . , v n+k−2 consecutive disjoint intervals V 1 , U 1 , V 2 , U 2 , V 3 , . . . , V r , U r , V r+1 such that \|V 1 \| = \|S 1 \| + k 0 − 1, \|V r+1 \| = \|S r+1 \| + k 1 − 1, and, for each 2 i r, \|V i \| = \|S i \|, and, for each i ∈ [r] , \|U i \| = \|P i \| − 2 + \|V (F + i ) \ V (S i )\| + \|V (F − i+1 ) \ V (S i+1 )\| + 3a i + 2b i \|P i \| − 2 A3 2000m − 1. (15) Note that this is possible, as, by A4 and A5, \|F − 1 \| = \|S 1 \| and \|F + r+1 \| = \|S r+1 \|, so that, using A4, we have r+1 i=1 \|V i \| + r i=1 \|U i \| = k 0 + k 1 − 2 + r+1 i=1 \|S i \| + r i=1 (\|P i \| − 2 + \|F + i \| + \|F − i+1 \| − \|S i \| − \|S i+1 \| + 3a i + 2b i ) A6 k − 2 + \|S 1 \| + \|S r+1 \| + r i=2 \|S i \| + r i=1 (\|P i \| − 2) + r+1 i=1 (\|F + i \| + \|F − i \| − 2\|S i \|) + i∈[r] (3a i + 2b i ) (12) = k − 2 + \|S 1 \| + \|S r+1 \| + \|P \| − 2 + r+1 i=1 (\|F + i \| + \|F − i \| − 2\|S i \|) + 3 i∈I1 (r i − q i ) + 2\|I 2 \| = k − 2 + \|P \| + (\|S 1 \| + \|S r+1 \| − 2) + r+1 i=1 \|(V (F + i ) ∪ V (F − i )) \ V (S i )\| + i∈[ ] (\|Q i \| − 2) A2 = n + k − 2. Next, for each i ∈ [r], partition U i as intervals U i,1 , . . . , U i,6 in that order such that \|U i,1 \| = m, \|U i,2 \| = 10m, \|U i,4 \| = 110m, \|U i,5 \| = 100m, \|U i,6 \| = m(16) and \|U i,3 \| = \|U i \| − 222m (15) 1700m. Note also, by A4, that, for each i ∈ {2, . . . , r}, \|V i \| = \|S i \| \|F i \| m.(18) For each i ∈ [r], let U i be a subset of U i where each vertex is included uniformly at random with probability µ/20. By Lemma 2.1 ii) v 1 v 2 . . . v n+k−2 forms a directed path in that order, so there is a directed path from the last vertex of V i under σ to the first vertex of V i+1 under σ, whose vertex set covers U i,1 ∪ U i ∪ U i,6 and whose vertex order is a suborder of σ. Let R i be a shortest such path. We now prove that, with positive probability, the 'absorption property' we need for R i holds, as well as a bound on \|R i \|. i ∈ [r], \|V (R i ) \ (U i,1 ∪ U i,6 )\| m, so that \|R i \| 3m, and, for any U ⊆ U i ∪ V (R i ) with V (R i ) ⊆ U , there is a directed path with the same start vertex and end vertex as R i but with vertex set U . Proof of Claim 4.5. Let p = µ/20 and i ∈ [r]. Note that, by Lemma 2.13, as \|U i \| n and 1/n µ, 1/r, we have, with probability at least 1 − 1/3r that \|U i \| 2pn. For each v ∈ U i \ (U i,1 ∪ U i,6 ), let E v be the following event. E v : There are u ∈ N − (v) ∩ U i and u ∈ N + (v) ∩ U i with u < σ v < σ u . Now, by Lemma 2.1 ii), for each v ∈ U i \ (U i,1 ∪ U i,6 ), we have \|{u ∈ N − (v) ∩ U i : u < σ v}\| \|{u ∈ U i : u < σ v}\| 2 \|U i,1 \| 2 (16) = m 2 , and \|{u ∈ N + (v) ∩ U i : u > σ v}\| \|{u ∈ U i : u > σ v}\| 2 \|U i,6 \| 2 (16) = m 2 , so that P(E v does not hold) 2(1 − p) m/2 2 exp(−pm/2) 2 exp(−µ 2 n/40). Therefore, as 1/n µ, 1/r, a union bound implies that, with probability at least 1 − 1/3r, E v holds for each v ∈ U i \ (U i,1 ∪ U i,6 ). Thus, with probability at least 1/3, we have, for each i ∈ [r], that E v holds for each v ∈ U i \ (U i,1 ∪ U i,6 ), and \|U i \| 2pn. Assuming these events occur, we now prove that the property in the claim holds for each i ∈ [r]. By Corollary 2.2 and the minimality of R i , any two vertices in U i,1 ∪ U i ∪ U i,6 on R i , with no vertices between them on R i from U i,1 ∪ U i ∪ U i, 6 have at most 1 vertex between them on R i . As the vertices from U i,1 ∪ U i,6 form two intervals on R i , just after the first vertex and just before the last vertex of R i respectively, \|V (R i ) \ (U i,1 ∪ U i,6 )\| 2 + 2\|U i \| + 1 4pn + 3 m. Now, take any set U ⊆ U i ∪ V (R i ) with V (R i ) ⊆ U . Let R U be a directed path with the same endvertices as R i which contains every vertex of R i in order according to σ and for which V (R U ) ⊆ U , and which, under these conditions, has the maximum possible length. Note that this exists as R i itself satisfies these conditions. Suppose, for contradiction, that there is some 6 ). Let be the length of R U and label vertices so that R U = u 0 u 1 . . . u . As E v holds and U i ⊆ V (R i ) ⊆ V (R U ), we can take j = min{j ∈ {0, 1, . . . , } : u j ∈ N − (v)} and find that u j < σ v. Let j ∈ {0, 1, . . . , } be the smallest j > j such that u j ∈ N + (v). v ∈ U \ V (R U ). Note that v ∈ U i \ (U i,1 ∪ U i, Observe that, u j −1 / ∈ N + (v), so that, as G is a tournament, u j −1 ∈ N − (v) and therefore u 0 u 1 . . . u j −1 vu j . . . u , is a directed path with the same endvertices as R U (and hence R i ) which contains every vertex of R i in order according to σ. As this path has vertex set {v} ∪ V (R U ) ⊆ U and v / ∈ V (R U ), this path contradicts the maximality of R U . Thus, V (R U ) = U , so that R U is a directed path with the same endvertices as R i and with vertex set U , as required. Figure 3: Embedding an (r, m)-good decomposition (as depicted in Figure 1) into a median order, with the claims used to embed each part. U i−1,1 U i−1,2 U i−1,3 U i−1,4 U i−1,5 U i−1,6 V i U i,1 U i,2 U i,3 U i,4 U i,5 U i,6 S i F + i \ S i [Claim 4.6] F − i \ S i [ Assume then, that the property in Claim 4.5 holds. We now show three further claims, before embedding T . This embedding, annotated with which part of the embedding is done with each claim, is depicted in Figure 3. For each i ∈ [r + 1], we will use the following claim to embed the vertices in V (F + i ) \ V (S i ) to U i,2 (if i = r + 1) and embed the vertices in V (F − i ) \ V (S i ) to U i−1,4 (if i = 1) so that they attach appropriately to an embedding of S i into the vertex set V i . Claim 4.6. For each i ∈ [r] and v ∈ V i , we have \|N + (v, U i,2 ) \ V (R i )\| 3m, and, for each i ∈ [r] and v ∈ V i+1 , we have \|N − (v, U i,4 ) \ V (R i )\| 3m. Proof of Claim 4.6. Let i ∈ [r] and v ∈ V i , and take V i,v = {u ∈ V i : u > σ v}. By Lemma 2.1 ii), we have that \|N + (v, U i,2 )\| \|N + (v, V i,v ∪ U i,1 ∪ U i,2 )\| − \|V i,v ∪ U i,1 \| \|V i,v ∪ U i,1 ∪ U i,2 \| 2 − \|V i,v ∪ U i,1 \| = \|U i,2 \| − \|V i,v ∪ U i,1 \| 2 \|U i,2 \| − \|V i ∪ U i,1 \| 2 (16),(18) 10m − m − m 2 = 4m. Therefore, by the property from Claim 4.5, \|N + (v, U i,2 )\V (R i )\| \|N + (v, U i,2 )\|−(\|R i \|−\|U i,1 \|−\|U i,6 \|) 3m. Let then i ∈ [r] and v ∈ V i+1 and let V i+1,v = {u ∈ V i+1 : u < σ v}. By Lemma 2.1 ii), we have similarly that \|N − (v, U i,4 )\| \|N − (v, V i+1,v ∪ U i,4 ∪ U i,5 ∪ U i,6 )\| − \|V i+1,v ∪ U i,5 ∪ U i,6 \| \|U i,4 \| − \|V i+1,v ∪ U i,5 ∪ U i,6 \| 2 (16),(18) 110m − 100m − m − m 2 = 4m. Therefore, by the property from Claim 4.5 again, \|N − (v, U i,4 ) \ V (R i )\| \|N − (v, U i,4 )\| − (\|R i \| − \|U i,1 \| − \|U i,6 \|) 3m. We will use the following claim, for each i ∈ I 2 , to embed the path Q i when its first and last vertex have already been embedded into U si−1,4 and V si respectively. Claim 4.7. For each 2 j r, v ∈ U j−1,4 , w ∈ V j and U ⊆ U j−1,4 ∪ U j−1,5 with \|U \| 2m, there is a directed v, w-path in G with length 3 and internal vertices in (U j−1,4 ∪ U j−1,5 ) \ (U ∪ V (R j−1 )). Proof of Claim 4.7. Let A j,v,w,U = {u ∈ U ∪ V (R j−1 ) ∪ V j : v < σ u < σ w} , and note that, by (18) and the choice of R i according to Claim 4.5, \|A j,v,w,U \| 6m. The number of vertices between v and w in σ is at least \|U j−1,5 \| + \|U j−1,6 \| = 101m > 6\|A j,v,w,U \| + 8. Therefore, by Lemma 2.9, there is a directed v, w-path in G with length 3 and internal vertices in {u / ∈ A j,v,w,U : v < σ u < σ w}. Because U j−1,6 ⊆ V (R j−1 ), we have {u / ∈ A j,v,w,U : v < σ u < σ w} ⊆ (U j−1,4 ∪ U j−1,5 ) \ (U ∪ V (R j−1 )), and so the claim holds. For each i ∈ [6], let U 0,i = U r+1,i = ∅, and note that, by A4 and A6, \|V 1 \|, \|V r+1 \| m + k 2m. 2 , and note that, by (16) and (18), \|V i \| 225m. Note thatV 1 U 1,3V2 U 2,3 . . .V r U r,3Vr+1 are consecutive intervals in σ. For each i ∈ [r + 1], letV i = U i−1,4 ∪ U i−1,5 ∪ U i−1,6 ∪ V i ∪ U i,1 ∪ U i, We will use the following claim, for each i ∈ I 1 , to embed the path Q i when its first and last vertex have already been embedded intoV qi andV ri respectively. Claim 4.8. For each 1 i < j r + 1, v ∈V i , w ∈V j and U ⊆ V (G) with \|U \| m, there is a directed v, w-path in G with length 3(j − i) + 1 and exactly 3 vertices in each set U i ,3 \ (U ∪ V (R i )), i i < j. Proof of Claim 4.8. First we will choose vertices u i , i i < j between u i−1 := v and w in the median order, with u j−1 w ∈ E(G) before carefully applying Lemma 2.9 to each consecutive pair of vertices in v, u i , u i+1 , . . . , u j−1 to get, together with u j−1 w, a v, w-path with length 3(j − i) + 1. To do this, first, for each i , i i j − 2, let u i be the last vertex in U i ,3 \ (U ∪ V (R i )) under σ. Let U j−1,3 be the set of the last 250m vertices of U j−1,3 under σ, and letV j,w = {w ∈V j : w < σ w}, so that \|V j,w \| \|V j \| 225m. Note that, by Lemma 2.1 ii), we have \|N − (w, U j−1,3 ) \ (U ∪ V (R j−1 ))\| \|N − (w,V j,w ∪ U j−1,3 )\| − \|V j,w \| − \|U ∪ V (R j−1 )\| \|U j−1,3 \| − \|V j,w \| 2 − \|U ∪ V (R j−1 )\| 250m − 225m 2 − 4m > 0. Let u j−1 then be the last vertex of N − (w, U j−1,3 ) \ (U ∪ V (R j−1 ) ) under σ, noting that there are fewer than 250m vertices in U j−1,3 after u j−1 under σ. Let u i−1 = v. For each i i < j, we will show there exists a directed u i −1 , u i -path T i with length 3 and internal vertices in U i ,3 \ (U ∪ V (R i )). Noting that T i T i+1 . . . T j−1 w is a directed path with length 3(j − i) + 1 and exactly three vertices in each set U i ,3 \ (U ∪ V (R i )), i i < j, will then complete the proof of the claim. Let then i i < j and let A i = {u ∈ U i −1,3 ∪V i ∪ ((U ∪ V (R i )) ∩ U i ,3 ) : u i −1 < σ u < σ u i }. Note that, for each i i < j, by the choice of u i there are at most \|U ∪ V (R i −1 )\| 4m vertices after u i −1 in U i −1,3 under σ, so \|A i \| 4m + 225m + \|U ∪ V (R i )\| 233m. In addition, recall that there are fewer than 250m vertices in U j−1,3 after u j−1 under σ. Therefore, by (17), for each i i < j, there are at least 1700m − 250m > 6\|A i \| + 8 vertices in U i ,3 before u i under σ. So, by Lemma 2.9, there is a directed u i −1 , u i -path T i with length 3 and internal vertices in {u / ∈ A i : u i −1 < σ u < σ u i } ⊆ U i ,3 \ (U ∪ V (R i )), as required. We are now ready to embed the (r, m)-good decomposition D into G, as follows. Begin with the empty embedding φ : ∅ → V (G). For each 2 i r, recalling that \|V i \| = \|S i \|, extend φ to embed the directed path S i onto the vertices in V i in the order given by σ. Note that the vertices of each interval V i form a directed path in this order by Lemma 2.1 ii). Let x be the last vertex of V 1 under σ, and let y be the first vertex of V r+1 under σ. Recall that P , as defined in (12), is a directed x, y-path, S 1 is an in-arboresence with k 0 in-leaves and root x, and S r+1 is an out-arboresence with k 1 out-leaves and root y. Therefore, as \|V 1 \| = \|S 1 \| + k 0 − 1 and \|V r+1 \| = \|S r+1 \| + k 1 − 1, by Theorem 2.3 (applied twice, once with directional duality) we can extend φ to embed S 1 into V 1 such that φ(x) = x and embed S r+1 into V r+1 such that φ(y) = y . Now, for each i ∈ [r + 1] and v ∈ V (S i ), let F − v be the component of \|N − (φ(v), U i−1,4 ) \ (V (R i−1 ) ∪ (∪ u∈V (Si):φ(u)<σφ(v) φ(F − u )))\| 3m − (\|F − i \| − \|V (F − v ) \ {v}\|) A4 3\|V (F − v ) \ {v}\|, so that a copy of F − v − v in N − (φ(v), U i−1,4 ) \ (V (R i−1 ) ∪ (∪ u∈V (Si):φ(u)<σφ(v) φ(F − u ) )) exists by Theorem 2.6. Similarly, for each v ∈ V (S i ), this is also possible for F + v − v. For each i ∈ [ ], say that Q i is a directed path from x i to y i . For each i ∈ [ ] in turn, extend φ to cover V (Q i ) \ {x i , y i }, by doing the following. • If i ∈ I 1 , recall that q i , r i are such that Q i is a directed path from F qi to F ri with length 3(r i − q i ) + 1, where q i < r i , and note that φ(x i ) ∈ φ(V (F qi )) ⊆V qi and φ(y i ) ∈ φ(V (F ri )) ⊆V ri . Embed Q i as a directed φ(x i ), φ(y i )-path with length 3(r i − q i ) + 1 and exactly three vertices in U i ,3 \ (V (R i ) ∪ (∪ j∈[i−1] φ(V (Q j )))), for each q i i < r i . Note that this is possible, by Claim 4.8, as when we look for such a path we have \| ∪ j∈[i−1] φ(V (Q i ))\| · (3r + 2) m as 2r, 1/n µ 1/r and m µn. • If i ∈ I 2 , recall that 2 s i r is such that Q i is a directed path with length 3 from V (F − si )\V (S si ) to the last vertex of S si , and note that φ(x i ) ∈ φ(V (F − si ) \ V (S si )) ⊆ U si−1,4 and φ(y i ) ∈ φ(V (S si )) ⊆ V si . Embed Q i as a directed path with length 3 from φ(x i ) to φ(y i ) with interior vertices in (U si−1,4 ∪ U si−1,5 ) \ (φ(V (F − si )) ∪ (∪ j∈[i−1] φ(V (Q j ))) ∪ V (R si−1 )). Note that this possible, by Claim 4.7, as when we look for such a path we have, by A4, \|φ(V (F − si ))\| + \| ∪ j∈[i−1] φ(V (Q j ))\| m + · (3r + 2) 2m. Finally, we extend φ to cover the internal vertices of P i , for each i ∈ [r]. For each i ∈ [r], let U i = V (R i ) ∪ U i \ φ V (F + i ) ∪ V (F − i+1 ) ∪ (∪ j∈[ ] V (Q j ) ) . Note that V (R i ) ∪ U i contains exactly the vertices in U i and the endvertices of R i . Therefore, \|U i \| = \|U i \| + 2 − (\|F + i \| − \|S i \|) − (\|F − i+1 \| − \|S i+1 \|) − 3\|{j ∈ I 1 : q j i < r j }\| − 2\|{j ∈ I 2 : s j = i + 1}\| (15) = (\|P i \| + 3a i + 2b i ) − 3a i − 2b i = \|P i \|. By Claim 4.5, for each i ∈ [r], there is a directed path between the endvertices of R i with vertex set U i . Using these paths, for each i ∈ [r], extend the embedding φ to cover P i , for each i ∈ [r]. This completes the embedding φ of D = P ∪ S 1 ∪ S r+1 ∪ (∪ i∈[r+1] F i ) ∪ (∪ i∈[ ] Q i ), and hence, by A1, G contains a copy of T . Proof of Theorem 1.2 Given Lemmas 4.3 and 4.4, it is now straight-forward to prove Theorem 1.2. Proof of Theorem 1.2. Note that, due to the result of Thomason [15] quoted in the introduction, we may assume that k 3. Let n 0 and µ be such that 1/n 0 µ 1/k. Let T be a tree with n n 0 vertices and k leaves, and let G be a tournament with n + k − 2 vertices. If there are no vertices x and y with degree 2 in T and a bare x, y-path P with length at least µn with first and last block of length 1, then, by Lemma 4.3, T has an (r, m)-good decomposition for some m µn and r 10k. In this case, then, by Lemma 4.4, G contains a copy of T . Thus, we can assume that T contains vertices x and y with degree 2 in T and a bare x, y-path P with length at least µn with first and last block of length 1. Suppose first, that k = 3. Note that in this case P must lie in a maximal bare path of T with one endvertex that is a leaf. Say this leaf is z, and assume, by relabelling x and y if necessary, that the path, Q say, from x to z in T contains y (and hence P ). Let T = T − (V (Q) \ {x}). Noting that x is a leaf of T , duplicate x to get the tree T with the new leaf x . Note that T has 4 leaves and \|T \| − \|Q\| + 2 n − µn + 1 vertices. Therefore, by Theorem 1.1, as 1/n µ, 1/k, G contains a copy of T , S say. Let s and s be the copy of x and x in S respectively. Note that \|G − (V (S ) \ {s, s })\| = n + 1 − (n − \|Q\|) = \|Q\| + 1. By Theorem 2.7, there is a copy, Q say, of Q with x embedded to {s, s }. Then S ∪ Q gives a copy of T . Therefore, we have that k 4. In this case, let T = T − (V (P ) \ {x, y}). Noting that x and y are leaves of T , create T by duplicating x and y to get the new vertices x and y respectively, and adding the edge xy. Note that T has k + 2 leaves and \|T \| − \|P \| + 4 n − µn + 3 vertices. Therefore, by Theorem 1.1, as 1/n µ, 1/k, G contains a copy of T , S say. Let s, s , t and t be the copy of x, x , y and y in S respectively. Note that \|G − (V (S ) \ {s, s , t, t })\| = n + k − 2 − (n − \|P \|) = \|P \| + k − 2 \|P \| + 2. By Theorem 2.8, there is a copy, P say, of P with x embedded to {s, s } and y embedded to {t, t }. Observing that S ∪ P contains a copy T completes the proof that G contains a copy of T in this case. Proposition 2 . 12 . 212Every n-vertex oriented tree T has labellings V (T ) = {t 1 , . . . , t n } and E(T ) = {e 1 , . . . , e n−1 }, such that, for every first two blocks of P i from y i . Let e (1) i be the furthest edge of P (2) i from x i on P i , and let e v 1 , . . . , v m . Using Lemma 2.1 i) and Theorem 2.3 (via directional duality), we can embed S into G[{v 1 , . . . , v \|S\|+k−1 }] with the root vertex embedded to v \|S\|+k−1 . Similarly, we can embed S into G[{v m−\|S \|−k +2 , . . . , v m }] with the root vertex of S embedded to v m−\|S \|−k +2 . Finally, by Lemma 2.1 ii), we have v \|S\|+k−1 → v \|S\|+k → . . . → v m−\|S \|−k +2 , so we can use this path to embed the n − \|S\| − \|S \| + 2 = m − \|S\| − \|S \| − k − k + 4 vertices of P and complete an embedding of T into G. Figure 1 : 1An (r, m)-good decomposition. , as r q, r 10k and m µn, as required. As T −T 1 −. . .−T r is the union of T −T 1 −. . .−T q and at most 8k −12 paths of length at most r /10 6 k, we have \|T −T 1 −. . .−T r \| 10kµn+m/4 m/2. Note that T − T 1 − . . . − T r has r + 1 components. Say these are R 1 , . . . , R r+1 , and note that\|R i \| \|T − T 1 − . . . − T r \| m/2 for each i ∈ [r + 1].Using Proposition 2.12, relabel the components {R 1 , . . . , R r+1 } and paths {T 1 , . . . , T r }, and define functions i − , i + : [r] → [r+ 1] Claim 4 . 5 . 45With positive probability, for each F − i containing v and let F + v be the component of F + i containing v. For each vertex v ∈ V (S i ) in increasing order of φ(v) under σ, greedily and disjointly extend φ to embed F − v − v into N − (φ(v), U i−1,4 ) \ V (R i−1 ) and F + v − v into N + (φ(v), U i,2 ) \ V (R i ).Note this is possible for each v ∈ V (S i ) as, by A5, if \|E(F − v )\| > 0, then i 2 and thus, by Claim 4.6, The Probabilistic Method. N Alon, J H Spencer, John Wiley & SonsN. Alon and J. H. Spencer. The Probabilistic Method. John Wiley & Sons, 2004. Trees with three leaves are (n + 1)-unavoidable. S Ceroi, F Havet, Discrete Applied Mathematics. 141S. Ceroi and F. Havet. Trees with three leaves are (n + 1)-unavoidable. Discrete Applied Mathe- matics, 141:19-39, 2004. On the unavoidability of oriented trees. F Dross, F Havet, Electronic Notes in Theoretical Computer Science. 346F. Dross and F. Havet. On the unavoidability of oriented trees. Electronic Notes in Theoretical Computer Science, 346:425-436, 2019. Trees in tournaments. A El Sahili, Journal of Combinatorial Theory, Series B. 92A. El Sahili. Trees in tournaments. Journal of Combinatorial Theory, Series B, 92:183-187, 2004. . R Häggkvist, A Thomason, Trees in tournaments. Combinatorica. 11R. Häggkvist and A. Thomason. Trees in tournaments. Combinatorica, 11:123-130, 1991. C B Hanna, Paths in tournaments a simple proof of Rosenfeld's conjecture. arXiv preprint arXivC. B. Hanna. Paths in tournaments a simple proof of Rosenfeld's conjecture. arXiv preprint arXiv 2011.14394, 2020. . F Havet, Trees in tournaments. Discrete Mathematics. 2431F. Havet. Trees in tournaments. Discrete Mathematics, 243(1):121-134, 2002. On unavoidability of trees with k leaves. F Havet, Graphs and Combinatorics. 19F. Havet. On unavoidability of trees with k leaves. Graphs and Combinatorics, 19:101-110, 2003. Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture. F Havet, S Thomassé, Journal of Graph Theory. 354F. Havet and S. Thomassé. Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture. Journal of Graph Theory, 35(4):244-256, 2000. Oriented Hamiltonian paths in tournaments: a proof of Rosenfeld's conjecture. F Havet, S Thomassé, Journal of Combinatorial Theory, Series B. 782F. Havet and S. Thomassé. Oriented Hamiltonian paths in tournaments: a proof of Rosenfeld's conjecture. Journal of Combinatorial Theory, Series B, 78(2):243-273, 2000. A proof of Sumner's universal tournament conjecture for large tournaments. D Kühn, R Mycroft, D Osthus, Proceedings of the London Mathematical Society. 1024D. Kühn, R. Mycroft, and D. Osthus. A proof of Sumner's universal tournament conjecture for large tournaments. Proceedings of the London Mathematical Society, 102(4):731-766, 2010. Embedding oriented n-trees in tournaments. K Reid, N Wormald, Studia Sci. Math. Hungar. 182-4K. Reid and N. Wormald. Embedding oriented n-trees in tournaments. Studia Sci. Math. Hungar, 18(2-4):377-387, 1983. A Dirac-type theorem for 3-uniform hypergraphs. V Rödl, A Ruciński, E Szemerédi, Probability & Computing. 151-2229CombinatoricsV. Rödl, A. Ruciński, and E. Szemerédi. A Dirac-type theorem for 3-uniform hypergraphs. Com- binatorics, Probability & Computing, 15(1-2):229, 2006. Antidirected Hamiltonian paths in tournaments. M Rosenfeld, Journal of Combinatorial Theory, Series B. 121M. Rosenfeld. Antidirected Hamiltonian paths in tournaments. Journal of Combinatorial Theory, Series B, 12(1):93-99, 1972. 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[ "Conceptual quantification of the dynamicity of longitudinal social networks", "Conceptual quantification of the dynamicity of longitudinal social networks" ]	[ "Shahadat Uddin [email protected] \nComplex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia\n", "Mahendra Piraveenan [email protected] \nComplex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia\n", "Arif Khan \nComplex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia\n", "Babak Amiri [email protected] \nComplex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia\n" ]	[ "Complex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia", "Complex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia", "Complex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia", "Complex Systems Research Centre\nFaculty of Engineering & IT\nThe University of Sydney Sydney\nAustralia" ]	[]	A longitudinal social network evolves over time through the creation and/or deletion of links among a set of actors (e.g. individuals or organisations). Longitudinal social networks are studied by network science and social science researchers to understand network evolution, trend propagation, friendship and belief formation, diffusion of innovations, the spread of deviant behaviour and more. In the current literature, there are different approaches and methods (e.g. Sampson's approach and the Markov model) to study the dynamics of longitudinal social networks. These approaches and methods have mainly been utilised to explore evolutionary changes of longitudinal social networks from one state to another and to explain the underlying reasons for these changes. However, they cannot quantify the level of dynamicity of the over time network changes and the contribution of individual network members (i.e. actors) to these changes. In this study, we first develop a set of measures to quantify different aspects of the dynamicity of a longitudinal social network. We then apply these measures, in order to conduct empirical investigations, to two different longitudinal social networks. Finally, we discuss the implications of the application of these measures and possible future research directions of this study.	10.1109/socialcom.2013.131	[ "https://arxiv.org/pdf/1311.0090v1.pdf" ]	11,772,183	1311.0090	e75efcfda1236fee85354e068bfe6a6c03645e1f	Conceptual quantification of the dynamicity of longitudinal social networks Shahadat Uddin [email protected] Complex Systems Research Centre Faculty of Engineering & IT The University of Sydney Sydney Australia Mahendra Piraveenan [email protected] Complex Systems Research Centre Faculty of Engineering & IT The University of Sydney Sydney Australia Arif Khan Complex Systems Research Centre Faculty of Engineering & IT The University of Sydney Sydney Australia Babak Amiri [email protected] Complex Systems Research Centre Faculty of Engineering & IT The University of Sydney Sydney Australia Conceptual quantification of the dynamicity of longitudinal social networks Dynamicitylongitudinal social networknetwork dynamics A longitudinal social network evolves over time through the creation and/or deletion of links among a set of actors (e.g. individuals or organisations). Longitudinal social networks are studied by network science and social science researchers to understand network evolution, trend propagation, friendship and belief formation, diffusion of innovations, the spread of deviant behaviour and more. In the current literature, there are different approaches and methods (e.g. Sampson's approach and the Markov model) to study the dynamics of longitudinal social networks. These approaches and methods have mainly been utilised to explore evolutionary changes of longitudinal social networks from one state to another and to explain the underlying reasons for these changes. However, they cannot quantify the level of dynamicity of the over time network changes and the contribution of individual network members (i.e. actors) to these changes. In this study, we first develop a set of measures to quantify different aspects of the dynamicity of a longitudinal social network. We then apply these measures, in order to conduct empirical investigations, to two different longitudinal social networks. Finally, we discuss the implications of the application of these measures and possible future research directions of this study. INTRODUCTION The study of longitudinal social networks has been the subject of intense research interest in recent years [1,2] because it provides a way to analyse the underlying mechanism in the process of network formation, development and evolution over time [3]. Researchers have been exploring longitudinal social networks in order to understand a wide range of processes in various contexts, such as knowledge creation in co-authorship networks [4,5], spread of virus in computer networks [6] and spread of happiness and obesity in kinship networks [7,8]. Organisations are also interested in studying longitudinal network in order to get inside the decision cycle of major events [9]. There is a growing interest in studying longitudinal social networks in other research areas, for example, education, psychology, health study, childhood and youth study, life history, organisation science and criminology [10][11][12]. In network science and social science literature, the presence of methods and approaches for the analysis of longitudinal social networks has been noticed for quite some time. One of the most notable and earliest approaches to the study of dynamics of longitudinal social network is the Sampson's [13] approach that he followed in his monastery study. In this study of the dynamics of a longitudinal social network, he took snapshots of the same network from different intervals, and observed and analysed the evolution of the network. The other dominant methods for analysing longitudinal social networks are Markov models and Multiagent simulation models. Continuous time Markov chains for modelling longitudinal social networks were proposed as early as 1977 by Holland and Leinhardt [14], which have been significantly improved later by many researchers [15][16][17][18]. The most important property of a Markov model is that the future state of a process is dependent only on the current state but not on any previous state [19]. For modelling longitudinal social networks, exponential random graph and stochastic actororiented models are the two Markovian methods proposed by Robins et al. [17] and Snijders et al. [20] respectively. In these two approaches of network analysis, links are modeled as random variables that can be in different states (e.g. positive, negative or neutral) at different time. The purpose of this linkmodelling approach is to examine which network effect fits the empirical data and better accounts for the observed structural changes. These two Markovian approaches to longitudinal social network analysis are efficient enough to detect and describe network changes and to explain why these changes occur. However, they may have convergence issues in the presence of sufficiently large abrupt endogenous (i.e. structure based) and exogenous (i.e. attribute based) social changes [21]. In the Multi-agent simulation model, members in a social network are often modeled and implemented as computer agents who have the abilities to behave and make decisions based on certain criteria. The collective behaviours' of all members in a network will determine how the network evolves from one structure to another. Evolutionary models often use multi-agent simulation. Carley et al. [22] use multi-agent technology to simulate the evolution of covert networks such as terrorist groups. Moreover, using a multi-agent system called DYNET they perform a 'what-if' analysis to anticipate how a network adapts to environmental changes such as the removal of a central member. A simulation model can be a powerful tool for predicting a network's future. However it often oversimplifies the behavior and decision-making of humans, and may not be able to model the complex reality of social networks [23]. Like Sampson's approach, the Markov and Multi-agent simulation models are also unable to quantify the level of dynamic behaviour shown by an individual network member or a group of network members in any longitudinal setting [24]. The methods and approaches for exploring longitudinal social networks available in the present literature give emphasis mainly to the holistic view of network for studying network dynamics and are therefore unable to quantify different aspects of the dynamicity of a longitudinal social network. This limitation further hinders the introduction of an effective approach to compare and contrast two (or more) different longitudinal social networks [11]. This study aims to overcome this shortcoming by proposing a set of measures to quantify different aspects of network dynamicity of a given longitudinal social network. The rest of this contribution is organised as follows. In section two, we construct a set of measures for quantifying different aspects of the dynamicity of a given longitudinal social network. These measures are utilised to explore two real longitudinal social networks in section three. Section four discusses the contribution of this study to the present literature. Finally, there is a conclusion and an illustration of possible future research directions in section five. II. CONSTRUCTING MEASURES FOR LONGITUDINAL SOCIAL NETWORK Longitudinal social networks are being observed at different time points to collect network data for research analysis purposes. These observed networks are named as short-interval networks. The accumulation of these shortinterval networks creates another bigger network, which is termed as an aggregated network. Based on the concept of static and dynamic social network topology, this study develops a set of measures to quantify different aspects of the dynamicity of longitudinal social networks. Social network topology defines the way that a given longitudinal social network will be analysed in terms of over time aggregation of links among network members [11,25]. In static topology, methods of social network analysis (SNA) are applied on the aggregated network of an entire data collection period; whereas, smaller segments of network data accumulated in less time compared to the entire data collection period are used in dynamic topology for research analysis purposes [25,26]. For instance, a dynamic topology could be exercised on daily or weekly or even monthly network of a university email communication network that evolves over five years, while static topology considers only one network -the single aggregated network of five years. Figure 1 shows a schematic difference between these two types of SNA topologies. In this figure, two longitudinal social networks, that are observed in three points of time (i.e. Day1, Day2 and Day3), evolve over time. According to this figure, for static network analysis purposes SNA methods are applied to the aggregated network (i.e. the upper shaded network inside the square of the first longitudinal social network) at the end of Day3. In contrast, SNA methods are applied to each day network for research analysis purposes in dynamic topology (i.e. the three lower shaded networks inside squares of the second longitudinal social network). There is no aggregated network considered for network analysis in this topology. That means dynamic topology explores structural positions of actors in different sets of network data that are collected in a shorter time period compared to the overall duration of the longitudinal social network. The static topology explores only one network which is constructed by aggregating all links and actors of different sets of network data utilised in the dynamic topology. The level of dynamicity shown by a longitudinal social network relies on the changes of positional behaviours (e.g. degree centrality and closeness centrality) of actors in all shortinterval networks compared to the aggregated network. In order to explore the dynamicity of a longitudinal social network, it is therefore required to observe and analyse the involvements of individual actors (i) in all short-interval networks; and (ii) in the aggregated network. To capture dynamics of networks that emerge in short-interval networks, the dynamic topology needs to be followed. On the other hand, static topology has to be carried out for the single aggregated network. Thus, to explore longitudinal social networks, both static and dynamic topological analyses of networks need to be carried out. (i.e. degree centrality) in the j th short-interval network (SIN) for the i th actor and m indicates the number of short-interval networks considered in the analysis. Since the scaled value of any network attribute (e.g. degree centrality) for an actor in a social network will have the range between 0 and 1 [27], OV AN i and OV SIN(j) i have the range between 0 and 1. Therefore, the range for DDA i is between 0 and 1. In a given longitudinal social network, an actor may not be found in all short-interval networks. An actor may participate in the j th short-interval network; however, it may not participate in the (j-1) th short-interval network. Or, it could be the case that an actor is present in the current short-interval network but will be absent in the subsequent short-interval network. The possible combination of 'presence' and 'absence' of an actor in two consecutive short-interval networks is illustrated in Table 1. When an actor is absent in the j th shortinterval network, the value of OV SIN(j) i in Equation 1 for that actor will be zero. In terms of 'presence' and 'absence' in two consecutive short-interval networks, an actor who is present in the j th and (j-1) th short-interval networks will show higher level of dynamicity compared to any other actor who is present in the j th short-interval network but absent in the (j-1) th shortinterval network. That means a transition from the 'absence' state in the (j-1) th short-interval network to the 'presence' state in the j th short-interval network will negatively impact the shown dynamicity for an actor. In order to capture the contribution of this type of phase transition to the shown dynamicity, a constant is introduced to the Equation 1: The values of α j,j-1 for all the possible combinations of 'presence' state and 'absence' state of an actor in two consecutive short-interval networks are presented in Table 1. When there is a phase transition from the 'absence' state to the 'presence' state in two consecutive short-interval networks for an actor, α j,j-1 will be 0.5. If an actor is absent in the present short-interval network then α j,j-1 will be 0 and it will be 1 when an actor is present in two consecutive short-interval networks. For the first short-interval network (i.e. α , for j=0) it will be 1. Since the maximum value of α j,j-1 can be 1, the range for DDA i in Equation 2 will be between 0 and 1. A. Organisational Communication Network The email communication dataset from Enron, commonly referred as Enron email corpus, has been analysed using the proposed measures in this example. This corpus was released by Federal Energy Regulatory Commission (FERC) in May, 2002. Shetty and Adibi [28] from University of Southern California created a MySQL database of this corpus. They also cleaned the database by removing a large number of duplicate emails, computer generated folders, junk data and invalid email addresses. In the area of organisational science and social networking research, the Enron corpus is of great value because it allows the academic to conduct research on real-life organisation over a number of years. It is well documented in the literature that a drastic form of critical loss, which was being started to flourish in the beginning of the third quarter of 2001, occurs in Enron during the final quarter of 2001 [29]. In this empirical example, we consider a portion of the email communications of Enron which evolve during the second half of the year 2001 (i.e. from July to December 2001). This portion of the Enron dataset contains 735261 messages from 2253 distinctive users. A short-interval network consists of email communications that evolve during a month. Therefore, there are six short-interval networks and one aggregated network considered for research data analysis in this example. Table 2 presents the top 5 actors of the Enron's email network in terms of dynamicity (i.e. DDA i ). In calculating these dynamicity values, we consider only three actor-level network centralities (i.e. degree, closeness and betweenness). That means this table presents top 5 actors that show higher dynamicity in terms of degree centrality, closeness centrality and betweenness centrality in the Enron's longitudinal network. The dynamicities, in terms of all three basic centrality measures, shown by each of the six short-interval networks (i.e. DDN SIN(i) ) for Enron email dataset are presented in Table 3. In regards to basic centrality measures, the level of dynamicity (i.e. DDN) shown by the Enron's longitudinal social network is presented in the second last row of Table 4. There is an overlapping of actors' positions in the topranked lists of degree-dynamicity and betweenness-dynamicity. Two actors with ID 13 and 20 are found (see Table 2) in the lists of top 5 actors showing higher degree-dynamicity and betweenness-dynamicity. Dynamicities shown by fourth and fifth short-interval networks in respect of all three centrality measures are higher, as noted in Table 3, compared to the other short-interval networks (i.e. first, second, third and sixth shortinterval network). It is well documented in the literature that the organisational crisis of Enron was at its peak during this period (i.e. October and November 2001) which resulted in the bankruptcy declaration during the first week of December 2001 [30]. Therefore, the measures proposed in this study are able to explore the underlying external influences (e.g. organisational crisis of Enron) to the different phases (i.e. short-interval networks) of the longitudinal social network. B. Students' Communication Network In this example, we utilise a students' email communication network dataset. This communication network was evolved among 34 students during a university semester consisting of 3 months. These 34 students were doing a masters-degree course, entitled Statistical Methods in Project Management, which was delivered in face-to-face mode. For all course-related communication, students were motivated and advised to communicate with other students as well as with the tutor and the lecturer of the course only through a designated email communication system. Those emails that have a single recipient are considered for research analysis as this type of emails reflect more intensive and directed communications [31]. After conducting all required refinements, 621 emails were found in the research dataset. Three short-interval networks and an aggregated network are considered for longitudinal data analysis. i DDA ) in terms of three basic centrality measures in the students' email communication network. The dynamicities shown by each of three short-interval networks (i.e. DDN SIN(i) ) of the students' email communications are presented in Table 6. The dynamicity (i.e. DDN) shown by the longitudinal students' email network is presented in the last row of Table 4. Three actors (i.e. actor with ID 2, 32 and 4) are found in the top-ranked lists of degree-dynamicity and betweennessdynamicity. Degree-dynamicity shows an increasing pattern among three short-interval networks. With the increase of study load throughout a semester, students make more email communication with their peers [32], which will eventually lead to an increased degree-dynamicity shown by the different short-interval networks. IV. DISCUSSION Based on the concept of static and dynamic network topology, in this study we develop a set of measures to explore dynamicity of a longitudinal social network. We then utilise these measures to explore two longitudinal social networks (i.e. organisational communication network and students' communication network). For these two longitudinal social networks, we show: (i) top 5 actors that reveal higher dynamicity as captured by the Equation 2; (ii) the dynamicity shown by each of the short-interval networks as quantified by the Equation 4; and (iii) the dynamicity shown by the longitudinal social network, which has been calculated by the Equation 6. The proposed measures of this study are able to explore dynamicity of a given longitudinal social network in respect of any actor-level network attribute. Although we only consider basic centrality measures (i.e. degree centrality, closeness centrality and betweenness centrality) for the empirical study of two longitudinal social networks using the proposed measures, other actor-level social network attributes (e.g. information centrality, in-degree and out-degree) can be considered. This will further enable researchers to explore the dynamicity of a given longitudinal social network from a wide range of perspectives. For instance, researchers can utilise the in-degree attribute in the proposed measures of this study to explore dynamicity of the activity of actors in a given longitudinal social network since the in-degree represents activity of actors in a given social network [27]. Similarly, outdegree can be used in the proposed measures of this study to explore dynamicity of the popularity of actors in a given longitudinal social network. Existing methods (e.g. Markov model) of current literature for analysing longitudinal social networks are unable to quantify the contribution of an individual actor to the overall evolutionary dynamicity of a given longitudinal social network [21]. The measure proposed in the Equation 3 is able to overcome this shortcoming. Using this measure, researchers now can explore involvements of actors (e.g. which actor is playing major in the network development process) throughout the evolution of a given longitudinal social network. Moreover, this study proposes another measure in the Equation 6 to calculate the level of dynamicity shown by a longitudinal social network. This measure eventually enables researchers to compare the network-level dynamicity of two or more longitudinal social networks regardless of their network sizes, the number of interactions among their member actors and the number of the short-interval networks constitutes the aggregate network. The measure proposed in the Equation 4 can determine the dynamicity shown by a short-interval network. It can further reveal the network statistics of the corresponding short-interval network. For example, if the measure proposed in the Equation 4 shows a low value for the second short-interval network of a given longitudinal social network then it can concluded that (i) a lower number of actors participate in that short-interval network compared to other short-interval networks and the aggregated network; and/or (ii) most of actors participated in that short-interval do not participate in the first short-interval network ( α j,j-1 will be 0.5 in that case; thus, lowering the numerical value of the Equation 4). V. CONCLUSION AND FUTURE RESEARCH In the present literature, there are many studies that propose methods and approaches to explore longitudinal social networks. Those studies mostly give emphasis to explore the underlying process for network development and evolution. This study takes the initiative to quantify different aspects of the dynamicity of a longitudinal social network. The proposed measures of this study will create opportunities for researchers to explore a given longitudinal social network from different perspectives (e.g. which actor contributes more to the evolution of a longitudinal social network and the dynamicity shown by a short-interval network). The structural positions of an individual actor in shortinterval networks of a longitudinal social network represent the pattern of the changes of its network behaviour. This can further reveal how actors change their network roles (e.g. network positions and level of interactions with other actors) in short-interval networks over time. The structural positions of individual actors of a longitudinal social network can be calculated by using basic social network analysis measures (e.g. degree centrality, closeness centrality and network constraint). This study defines the degree of dynamicity (or level of dynamicity or simply dynamicity) shown by an individual actor as the variability of the structural positions of that actor in all short-interval networks compared to its structural position in the aggregated network. The following equation represents the degree of dynamicity (or dynamicity) shown by an individual actor in a longitudinal social network: ) 1 ( .......... .......... .........degree of dynamicity shown by the i th actor, OV AN i indicates observed value (say degree centrality) in the aggregated network for the i th actor, OV SIN(j) i indicates observed value for the same SNA measure Fig 1 . 1Illustration of static and dynamic topology of social network analysis. LSN stands for Longitudinal Social Network Equation 2 can quantify the level of dynamicity shown by individual actors in a given longitudinal social network regardless of its size and the number of short-interval networks constitutes the aggregated network. It can be conceptualised that the dynamicity shown by a longitudinal social network is the reflection of the dynamicities represented by its all member actors. The dynamicity shown by each actor is normalised using the highest observed dynamicity for an actor in the longitudinal social network. Therefore, the contribution of individual actors to the dynamicity of the longitudinal social network can be quantified by the following equation: ) 3 .( .......... .......... .........DDN i represents the contribution of the i th actor to the degree of dynamicity shown by the longitudinal social network, DDA * is the highest observed degree of dynamicity shown by individual actors in the network, DDA i is the degree of dynamicity for the actor i and n is the number of actors in the network. Since the range for DDA * and DDA i is between 0 and 1, the range for DDN i is between 0 and 1 n .To calculate the degree of dynamicity shown by a shortinterval network, it is required to compare the network position of all actors of that short-interval network with their positions in the aggregated network. This is represented by the following equation:) 4 .....( .........DDN SIN(i) represents the dynamicity shown by the i th short-interval, α i,i-1 j represents α i,i-1 (same as in the Equation 2) for actor j, OV AN j indicates the observed value (say degree centrality) of the j th actor in the aggregated network and OV SIN(i) j indicates the observed value of the j th actor in the i th short-interval network and i w is the total number of actors in the i th short-interval network. Precisely, DDN SIN(i) indicates the average dynamicity shown by an actor of the i th short-interval. The range for DDN SIN(i) will be between 0 and 1 since the range for to calculate the degree of dynamicity shown by the longitudinal social network, the right hand side of Equation (3) needs to be summed up for all actors. Therefore, ........( .......... .......... .................. .......... .........DDN represents degree of dynamicity shown by the longitudinal social network, * DDA is the highest observed degree of dynamicity shown by an individual actor in the ) 2 ( .......... .......... .........the degree of dynamicity for actor i and n is the number of actors in the longitudinal network. III. APPLICATION OF THE PROPOSED MEASURES The measures constructed in the previous section can quantify dynamicity of a longitudinal social network at different levels. In this section, we explore two longitudinal social networks using the measures represented in Equation 2, Equation 4 and Equation 6. The first longitudinal social network is considered from the context of organisational communication network and the second one consists of students' email communications that evolve throughout a university semester. TABLE 2 . 2TOP-5 ACTORS SHOWING HIGHER DYNAMICITY ( ) BASED ON BASIC CENTRALITY MEASURES FOR ORGANISATIONAL COMMUNICATION NETWORK Degree Centrality Closeness Centrality Betweenness Centrality Actor ID Dynamicity Actor ID Dynamicity Actor ID Dynamicity 71 0.00385 157 0.3600 13 0.07110 85 0.00337 129 0.3368 110 0.06392 128 0.00306 450 0.3321 43 0.05517 13 0.00248 132 0.3197 20 0.05242 20 0.00226 199 0.3087 35 0.04141 Table 5 5presents top 5 actors that show higher dynamicity (i.e. TABLE 5 .TABLE 6 . 56TOP-5 ACTORS SHOWING HIGHER DYNAMICITY ( ) BASED ON BASIC CENTRALITY MEASURES FOR STUDENTS' COMMUNICATION NETWORK DYNAMICITY OF SHORT-INTERVAL NETWORK ( ) OF THE STUDENTS' COMMUNICATION NETWORKDegree Centrality Closeness Centrality Betweenness Centrality Actor ID Dynamicity Actor ID Dynamicity Actor ID Dynamicity 2 0.10875 10 0.65227 32 0.23069 32 0.09239 23 0.56904 2 0.18307 23 0.06397 3 0.55960 7 0.12922 6 0.05290 34 0.41451 9 0.09728 4 0.04965 1 0.40255 4 0.06982 SIN ID Dynamicity based on basic centrality measures Degree centrality Closeness centrality Betweenness centrality 1 0.011903 0.08460 0.01083 2 0.012069 0.13616 0.02373 3 0.015999 0.07307 0.01868 This study creates future research opportunities in a number of ways. First, we can explore the dynamicity shown by individual actors in each short-interval network using a variation of the Equation 2 (i.e. without the summation operator). It will further enable to study the changing network behaviour of actors in a longitudinal setting. Second, to capture the dynamicity due to a phase change (i.e. from the 'presence' state to the 'absence' state) of an actor, this study introduced a constant (i.e. α j,j-1 ). The possible values that this constant can take (i.e. 0, 0.5 and 1) for any longitudinal social network are presented inTable 1. This value assignment may not capture the appropriate quantity of a phase-change dynamicity shown by casual or part-time network members. In an organisation, for example, a part-time employee works on Monday, Wednesday and Friday. If short-interval networks consist of a day then she will be found to be responsible for many phase changes in the email communication network of that organisation although she participates in the communication network during her regular office hour. Further research investigations in regards to actors' network membership and connectivity are required in order to assign correct values to this constant for different types of phase changes. Finally, the pattern of over time interaction among actors can be examined by using the measures that capture the dynamicity shown by an actor and the contribution of individual actors to the dynamicity shown by the longitudinal social network (i.e. measures represented by Equation 2 and Equation 3 respectively). For instance, these measures can be utilised to explore whether there is a tendency of link establishments in subsequent shortinterval networks among actors that are highly connected with other actors at the present short-interval network. 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[ "Annular Non-Crossing Matchings", "Annular Non-Crossing Matchings" ]	[ "Paul Drube [email protected] \nDepartment of Mathematics and Statistics\nValparaiso University\n\n", "Puttipong Pongtanapaisan [email protected] \nDepartment of Mathematics and Statistics\nValparaiso University\n\n" ]	[ "Department of Mathematics and Statistics\nValparaiso University\n", "Department of Mathematics and Statistics\nValparaiso University\n" ]	[]	It is well known that the number of distinct non-crossing matchings of n half-circles in the half-plane with endpoints on the x-axis equals the n th Catalan number C n . This paper generalizes that notion of linear non-crossing matchings, as well as the circular non-crossings matchings of Goldbach and Tijdeman, to non-crossings matchings of n line segments embedded within an annulus. We prove that the number of such matchings \|Ann(n, m)\| with n exterior endpoints and m interior endpoints correspond to an entirely new, one-parameter generalization of the Catalan numbers with C n = \|Ann(1, m)\|. We also develop bijections between specific classes of annular non-crossing matchings and other combinatorial objects such as binary combinatorial necklaces and planar graphs. Finally, we use Burnside's Lemma to obtain an explicit formula for \|Ann(n, m)\| for all n, m ≥ 0.	null	[ "https://arxiv.org/pdf/1508.01712v1.pdf" ]	17,715,780	1508.01712	780dad9e7ad3eff498ba776b31cf55a6c589cd4f	Annular Non-Crossing Matchings 7 Aug 2015 Paul Drube [email protected] Department of Mathematics and Statistics Valparaiso University Puttipong Pongtanapaisan [email protected] Department of Mathematics and Statistics Valparaiso University Annular Non-Crossing Matchings 7 Aug 2015 It is well known that the number of distinct non-crossing matchings of n half-circles in the half-plane with endpoints on the x-axis equals the n th Catalan number C n . This paper generalizes that notion of linear non-crossing matchings, as well as the circular non-crossings matchings of Goldbach and Tijdeman, to non-crossings matchings of n line segments embedded within an annulus. We prove that the number of such matchings \|Ann(n, m)\| with n exterior endpoints and m interior endpoints correspond to an entirely new, one-parameter generalization of the Catalan numbers with C n = \|Ann(1, m)\|. We also develop bijections between specific classes of annular non-crossing matchings and other combinatorial objects such as binary combinatorial necklaces and planar graphs. Finally, we use Burnside's Lemma to obtain an explicit formula for \|Ann(n, m)\| for all n, m ≥ 0. Introduction The Catalan numbers are arguably the most studied sequence of positive integers in mathematics. Among their seemingly countless combinatorial interpretations is an identification of the n th Catalan number C n = 1 n+1 2n n with non-crossing matchings of 2n points along the x-axis via n half-circles in the upper half-plane. In an effort to avoid confusion, we will sometimes refer to such arrangements as linear noncrossing matchings of order n. The n th Catalan number is also known to equal the number of ordered rooted trees with n non-root vertices. One bijection between these two interpretations is shown in Figure 1. That map involves placing a vertex in each region of the complement of a matching, with the "external" region receiving the root vertex, and then adding an edge if two regions are separated by a half-circle. For an extended treatment of the many different interpretations of Catalan numbers and the bijections between them, see [6]. ⇔ Figure 1: The bijection between linear non-crossing matchings and ordered rooted trees. Non-crossing matchings admit many interesting generalizations if one restricts curves to a subset of R 2 that is not the upper half-plane. One such modification is what we refer to as circular non-crossing matchings. In a circular non-crossing matching of order n, 2n distinct points on the unit circle are connected by a set of n non-intersecting smooth curves within the unit circle. Circular non-crossing matchings are considered equivalent if they differ by isotopies within the unit circle (including isotopies that "slide" endpoints), as long as those isotopies do not involve in curves or endpoints intersecting. In particular, circular matchings related by rotation about the center of the unit circle are equivalent. However, matchings that can only be related via reflections are considered distinct. We henceforth refer to the number of circular non-crossing matchings of order n, modulo these relations, by C n . One in-depth study of circular non-crossing matchings was undertaken by Goldbach and Tijdeman in [3]. In that work, an involved application of Burnside's Lemma showed that: C n = 1 2n   d\|n φ(n/d) 2d d   − 1 2 C n + 1 2 C (n−1)/2(1) where φ(k) is Euler's totient function and C k is taken to be zero when k is not an integer, so that the final term only appears when n is odd. Although not obvious from Equation 1, there is also a bijection between circular non-crossing matchings of order n and unrooted planar trees with n + 1 nodes (n edges). For an illustration of this bijection, see Figure 2. Notice that this correspondence identifies C n with the (n − 1) st entry of A002995 [1]. See [4] for further discussion of the graph-theoretic interpretations of C n . ⇔ Figure 2: The bijection between circular non-crossing matchings and unrooted planar trees. In this paper we introduce a new, two-parameter generalization of linear non-crossing matchings in which our smooth curves are embedded within an annulus. So let n, m be non-negative integers such that n + m is even. We define an annular non-crossing matching of type (n, m) to be a collection of n+m 2 non-intersecting smooth curves within the annulus whose endpoints lie at n + m distinct points along the annulus, with n of those endpoints on the exterior boundary of the annulus and m of those endpoints on the interior boundary of the annulus. Two annular matchings are considered equivalent if they differ by isotopies within the annulus (including isotopies that "slide" endpoints), as long as those isotopies don't result in curves or endpoints intersecting. Annular matchings that can only be related by reflections are considered distinct; also disallowed are isotopies where an edge must pass through the hole in the middle of the annulus. We denote the set of annular non-crossing matchings of type (n, m), modulo these relations, by Ann(n, m). If we wish to reference the larger collection of all annular non-crossing matchings with N total endpoints, no matter how those endpoints are partitioned between inner and outer boundary components, we write Ann(N ). Thus Ann(N ) = {M ∈ Ann(n, m) \| n + m = N } and Ann(N ) = 0 if and only if N is even. In many settings, it will be advantageous to sub-divide annular matchings according to the the number of curves that do not isotope to half-circles on one boundary component. We define a cross-cut to be a curve in an annular non-crossing matching with one endpoint on the inside of the annulus and one endpoint on the outside of the annulus. We denote the set of annular non-crossing matchings of type (n, m) with precisely k cross-cuts by Ann k (n, m), so that Ann(n, m) = k Ann k (n, m). See Figure 3 for several quick examples. Observe that every annular matching may be isotoped so that all of its cross-cuts appear as straight chords that meet both boundary circles at a right angle. Also notice that \|Ann k (n, m)\| = 0 unless n − k and m − k are both even. When working with an element of Ann k (n, m), we will sometimes refer to the (n − k)/2 curves with both endpoints on the outside of the annulus as "external half-circles", and to the (m − k)/2 curves with both endpoints on the inside of the annulus as "internal half-circles". Outline of Results The primary goal of this paper is to enumerate \|Ann(n, m)\| for arbitrary non-negative integers n and m. This will be accomplished by enumerating \|Ann k (n, m)\| for arbitrary n, m, k and then summing over k. Section 2 will begin the process with a series of basic results about annular non-crossing matchings, laying the theoretical framework for our subsequent enumerations. Along the way we will demonstrate a bijection between elements of Ann k (n, m) and sets of planar graphs that possess a single k-cycle and no cycles of any other size (Proposition 2.2, Theorem 2.5). Section 3 presents our enumerative results. Subsection 3.1 begins with a consideration of the sets Ann k (2n + k, k), a sub-case that we refer to as "maximal cross-cut annular matchings". Burnside's Lemma will be used to prove the following, which later appears as Theorem 3.1: Theorem. Let n and k be non-negative integers, not both zero. Then: \|Ann k (2n + k, k)\| = 1 2n + k d\|(2n+k,n) φ(d) (2n + k)/d n/d Where φ(d) is Euler's totient function and d runs over all common divisors of 2n + k and n. Subsection 3.1 will also exhibit a direct bijection between these maximal cross-cut annular matchings and binary combinatorial necklaces. If N 2 (n 1 , n 2 ) denotes the number of binary combinatorial necklaces with n 1 black beads and n 2 white beads, then Theorem 3.3 will prove: Theorem. Let n and k be non-negative integers. Then \|Ann k (2n + k, k)\| = N 2 (n + k, n). Our results for \|Ann k (2n + k, k)\| are then used in Subsection 3.2 to enumerate \|Ann k (2n + k, 2m + k)\| for the remaining choices of n, m, and k. In particular, Theorem 3.4 will show that: Theorem. Let n, m , and k be non-negative integers with m > 0. Then: 1. \|Ann k (2n + k, 2m + k)\| = \|Ann 0 (2n, 0)\| · \|Ann 0 (2m, 0)\| if k = 0, and 2. \|Ann k (2n + k, 2m + k)\| = k (2n + k)(2m + k) d\|(2n+k,n,m) φ(d) (2n + k)/d n/d (2m + k)/d m/d if k > 0 Where φ(d ) is Euler's totient function and summations run over all common divisors of the given integers. We close the paper with an appendix of tables giving outputs for various Ann(n, m), Ann k (n, m), and Ann(N ), all calculated in Maple using Theorems 3.1 and 3.4. Basic Results About Annular Non-Crossing Matchings In this section we present a series of foundational results about annular non-crossing matchings, some of which will be utilized to prove the more general enumerative results of Section 3. We also take the opportunity to draw bijections between annular matchings and various classes of planar graphs, and relate the number of "zero cross-cut" matchings in Ann(2n, 0) to C n and C n . Proposition 2.1. Let n, m be non-negative integers. Then \|Ann k (n, m)\| = \|Ann k (m, n)\| for all k ≥ 0. In particular, \|Ann(n, m)\| = \|Ann(m, n)\| for all n, m ≥ 0. Proof. Begin by isotoping elements of Ann k (n, m) and Ann k (m, n) so that all cross-cuts appear as straight chords orthogonal to the boundary circles. Reflection across the "core" of the annulus then maps every cross-cut to itself, and defines a bijection between Ann k (n, m) and Ann k (m, n) for any k ≥ 0. The primary use of Proposition 2.1 is that it will allow us to restrict our attention to Ann(n, m) and Ann k (n, m) such that n ≥ m. Yet even within the realm where n ≥ m, there will be specific "easy" choices for n and m that will prove to have the most useful combinatorial interpretations. The first of these special cases is Ann(2n, 0) = Ann 0 (2n, 0), corresponding to the situation where all curves in the matchings are external half-circles. ⇔ Here the distinguished vertex is placed in the sole region that borders the interior boundary of the annulus. In placing the edges for our graph, we disregard whether that edge would have passed through the hole in the center of the annulus (we treat the hole as part of the internal region). As the half-circles in our matchings may be cyclically rotated around the center of the annulus in "blocks", this gives us the desired notion of equivalence for our planar graphs. The class of planar graphs from Proposition 2.2 is equivalent to rooted planar graphs with n nonroot vertices, as long as cyclic reordering of subtrees around the root vertex gives equivalent trees. This interpretation identifies \|Ann(2n, 0)\| with the n th entry of A003239 [1]. See [4] and [5] for further bijections involving \|Ann(2n, 0)\|. Via the planar graph bijections of Section 1 and Proposition 2.2, it is immediate that C n ≤ \|Ann(2n, 0)\| ≤ C n for all n ≥ 0. One can easily verify that C n = \|Ann(2n, 0)\| = C n for both n = 0 and n = 1. For n = 2 we have \|Ann(4, 0)\| = C 2 = 2 yet C 2 = 1. As shown in the following proposition, n = 2 is the largest value of n for which any of the three quantities are equal: Proposition 2.3. For all n ≥ 3 we have C n \|Ann(2n, 0)\| C n . Proof. We define a map φ from the set of all linear non-crossings of order n to Ann(2n, 0) by identifying the endpoints of the x-axis and placing the resulting circle as the outer boundary of the annulus. We then define a map ψ from Ann(2n, 0) to the set of all circular non-crossing matchings of order n by "deleting" the hole in the middle of the annulus. φ( A ) = A ψ( A ) = A Both φ and ψ are clearly well-defined and surjective for all n ≥ 0. To see that neither map is injective for n ≥ 3, let A be some non-empty collection of half-circles and notice that: φ( A ) = φ( A ) ψ( A ) = ψ( A ) With the zero cross-cuts case well-understood, we expand our attention to Ann k (n, m) with k ≥ 1. In what follows we re-index variables to consider Ann k (2n + k, 2m + k), as this alternative notation explicitly references the presence of n external half-circles and m internal half-circles. The cases that will prove most useful are what we refer to as maximal cross-cut annular matchings. In maximal cross-cut annular matchings, the only endpoints on the interior boundary belong to cross-cuts. In our new notation this implies that m = 0, so that we are dealing with sets of the form Ann k (2n + k, k). Notice that the previously considered sets Ann(2n, 0) = Ann 0 (2n + 0, 0) qualify as maximal cross-cut annular matchings. When k = 1, maximal cross-cut annular matchings are in bijection with the Catalan numbers. As shown in Figure 4, a bijection of such annular matchings with linear non-crossing matchings is realized by identifying the outer boundary of the annulus with the real line, in such a way that the outer endpoint of the sole cross-cut corresponds to ±∞. A ⇔ A Figure 4: The bijection between Ann 1 (2n + 1, 1) and linear non-crossing matchings of order n. Maximal cross-cut annular matchings will be directly enumerated in Subsection 3.1 for all k ≥ 0, and those enumerations will be necessary building blocks for the general enumerations of Subsection 3.2. Although the full importance of the maximal cross-cut case will not become obvious until those subsections, we pause to prove one useful property shared by maximal cross-cut matchings and annular matchings with zero cross-cuts: Proposition 2.4. Let n, m, k be non-negative integers. If m = 0 or k = 0 then: \|Ann k (2n + k, 2m + k)\| = \|Ann k (2n + k, k)\| \|Ann k (2m + k, k)\| Proof. We define a map φ : Ann k (2n + k, 2m + k) → Ann k (2n + k, k) ⊕ Ann k (k, 2m + k) that deletes internal half-circles in the first coordinate and deletes external half-circles in the second coordinate, as in the example below: → ( , ) This map φ is clearly a bijection whenever m = 0. To see that φ is a bijection when k = 0, notice that the lack of cross-cuts in the k = 0 case means that the internal half-circles and external half-circles may be isotoped independently around their respective boundary components and hence "do not interact". It can be shown that the equality of Proposition 2.4 holds precisely when n = 0 or m = 0 or k < 2. If k ≥ 2, n ≥ 1, and m ≥ 1 all hold, the left side of the expression always proves to be strictly larger than the right side. However, only the m = 0 and k = 0 cases are needed in Section 3, motivating our omission of the more general result. For our final result of this section, we adapt the planar graph bijection of Proposition 2.2 to the general case of Ann k (2n + k, 2m + k). Although the resulting language of Theorem 2.5 is arguably rather contrived, it combines with Proposition 2.2 to give a succinct geometric characterization of any subset of annular matchings that is not Ann 0 (2n, 2m) with n, m > 0. Theorem 2.5. Let n, m, k be non-negative integers and let k ≥ 1. Then \|Ann k (2n + k, 2m + k)\| equals the number of connected planar graphs such that: 1. The only cycle in the graph is a single k-cycle. 2. There are n edges within the cycle. 3. There are m edges outside the cycle. Proof. The methodology required is extremely similar to what has already been presented in Proposition 2.2. Notice that a collection of k cross-cuts produces a single k-cycle, as below: ⇔ Any internal half-circles are then in bijection with edges inside the k-cycle, while external half-circles are in bijection with edges outside the k-cycle. Enumeration of Annular Matchings We are now ready for the general enumerative results that form the core of this paper. Subsection 3.1 begins with an enumeration of maximal cross-cut annular matchings, and shows that those matchings are in bijection with certain types of binary combinatorial necklaces. Subsection 3.2 then collects all of our results to give an explicit formula for general \|Ann k (2n + k, 2m + k)\|, thus allowing for the direct calculation of \|Ann(a, b)\| and \|Ann(N )\|. Enumeration of Ann k (2n + k, k) and Combinatorial Necklaces Burnside's Lemma has already been mentioned as the method used in [3] to calculate the number of circular non-crossing matchings. Given that our annular non-crossing matchings possess a similar notion of rotational equivalence, it comes as little surprise that the lemma may also be applied to the enumeration of distinct matchings in the annulus. Recall that Burnside's Lemma (also known as the Cauchy-Frobenius Lemma) applies to any situation where a finite group G acts upon a set A. It asserts that the number of orbits \|A/G\| with respect to the action equals the average size of the sets A g = {a ∈ A \| ga = a} when ranging over all g ∈ G: that \|A/G\| = 1 \|G\| g∈G \|A g \|. So fix n, k ≥ 0, and let A equal the set of all non-crossing matchings in the annulus (prior to any notion of equivalence) with 2n + k endpoints located at 2π 2n+k radian intervals about the exterior boundary and precisely k straight cross-cuts that meet both boundary components orthogonally. We may define a (left) action of G = Z 2n+k on A whereby g · a is counter-clockwise rotation of a by 2πg 2n+k radians. Then G/A = Ann k (2n + k, k), with distinct orbits in G/A corresponding to matchings that are equivalent via rotation. This sets up the following application of Burnside's Lemma: Theorem 3.1. Let n and k be non-negative integers, not both zero. Then: \|Ann k (2n + k, k)\| = 1 2n + k d\|(2n+k,n) φ(d) (2n + k)/d n/d Where φ(d) is Euler's totient function and the sum runs over all common divisors d of 2n + k and n. Proof. Using the aforementioned group action, by Burnside's Lemma we merely need to show that g∈Z 2n+k \|A g \| = d\|(2n+k,n) φ(d) (2n+k)/d n/d . So take g ∈ Z 2n+k , and assume that g has order d in Z 2n+k Notice that Lagrange's Theorem guarantees d \| (2n + k), although it may or may not be true that d \| n. The elements of A g are those matchings that can be radially divided into d identical sub-matchings. Each of these sub-arrangements features (2n + k)/d endpoints on the outer boundary of the annulus, k/d of which are the outer endpoints of cross-cuts and n/d of which are left-endpoints of exterior half-circles. Now if d ∤ n, the left-endpoints cannot be sub-divided in this way and we may conclude that \|A g \| = 0. However, d \| n and d \| (2n + k, n) together guarantee that k \| n, making \|A g \| = 0 a possibility. If d \| n, notice that every possible sub-arrangement may be uniquely identified by specifying which of the (2n+k)/d endpoints correspond to left (clockwise) endpoints of exterior half-circles. This bijection, which closely resembles the upcoming construction in the proof of Theorem 3.3, involves recursively connecting each specified endpoint to the nearest available non-specified endpoint on its right and then associating the k unused endpoints with cross-cuts. See Figure 5 for an example. It follows that \|A g \| = (2n+k)/d n/d whenever \|g\| = d and d \| n. If q \| N , basic number theory ensures that there are precisely φ(q) elements i ∈ Z N with greatest common divisor (i, N ) = N/q. As the order of any element in Z N is N/(i, N ), there exist precisely φ(q) elements i ∈ Z N with order \|i\| = q. Letting N = 2n+k, this ensures that there are precisely φ(d) elements g ∈ Z 2n+k such that \|A g \| = (2n+k)/d n/d , thus deriving the summation of the theorem. Figure 5: For every choice of n left-endpoints (white circles) there is a unique annular sub-matching with n half-circles and k cross-cuts. Here the relevant piece of the outer boundary is drawn as the real line. Table 1 of Appendix A exhibits values of \|Ann k (2n + k, k)\| for 0 ≤ n, k ≤ 10, all calculated in Maple via the equation of Theorem 3.1. An examination of that table places \|Ann k (2n + k, k)\| into direct correspondence with the T (n + k, n) entry of OEIS sequence A241926 [1]. Using Proposition 2.1 when necessary to ensure that 2n + k > n, \|Ann k (2n + k, k)\| may also be identified with the "circular binomial coefficient" T (2n + k, n) of OEIS sequence A047996 [1]. Both of those OEIS sequence reveal a bijection between the Ann k (2n + k, k) and binary combinatorial necklaces, and in fact a summation equivalent to the one of Theorem 3.1 has already been shown to equal the number of binary combinatorial necklaces of certain types [2]. Yet before investigating how these results relate to annular non-crossing matchings, we observe that the formula of Theorem 3.1 may be significantly simplified when k is prime: ⇒ ⇒ Corollary 3.2. Let p be a prime integer and let n be any non-negative integer. Then: \|Ann p (2n + p, p)\| = 1 2n + p 2n + p n + p − 1 p C n/p Where C n/p is the Catalan number, and is taken to be zero when n/p is not an integer. Proof. It is a straightforward exercise to calculate that the greatest common divisor of 2n + p and n is (2n + p, n) = 1 when (p, n) = 1, as well as that (2n + p, n) = p when (p, n) = p. Notice that the more straightforward formula for \|Ann k (2n + k, n)\| in Corollary 3.2 simplifies to the sequences A007595 [1] when p = 2 and A003441 [1] when p = 3. Driven by sequences A241926 and A047996, we now work to develop an explicit bijection between the \|Ann k (2n + k, k)\| and binary combinatorial necklaces, providing a new combinatorial identity that supplements the one in [2]. A k-ary combinatorial necklace is a circular arrangement of "beads" of up to k distinguishable varieties (typically referred to as "colors"), such that rotations of the beads around the circle are considered equivalent. Necklaces that are related only via (orientation-reversing) reflection are not considered equivalent. The number of distinct k-ary combinatorial necklace with precisely n total beads is denoted N k (n). Combinatorial necklaces are well-studied in the literature, and different classes of combinatorial necklaces are the focus of many integer sequences [1]. In this paper we deal only with binary (2-ary) combinatorial necklaces, whose colors we refer to as "black" and "white". Our results require increased specificity in that we need to designate the number of beads of each color, so we denote the number of distinct binary necklaces with precisely n 1 black beads and precisely n 2 white beads by N 2 (n 1 , n 2 ). Pause to note that some places in the literature refer to such combinatorial necklaces as "binary necklaces of weight n 1 ". Theorem 3.3. Let n and k be non-negative integers. Then \|Ann k (2n + k, k)\| = N 2 (n + k, n). Proof. Denote the set of all combinatorial necklaces with n 1 black beads and n 2 white beads by S. We define functions φ 1 : Ann k (2n+k, k) → S and φ 2 : S → Ann k (2n+k, k) that are both injective. For φ 1 we follow the procedure exemplified below, placing white beads at the left-endpoints of exterior half-circles and black beads at right-endpoints of exterior half-circles as well as at the exterior endpoints of cross-cuts: ⇒ ⇒ For φ 2 we begin at any point along the combinatorial necklace and proceed counter-clockwise. Every time we encounter a white bead, we add a half-circle connecting that bead to the first black bead (in the counter-clockwise direction) that is not already the right-endpoint of a half-circle. Repeat this procedure, traversing the necklace multiple times if necessary, until every white bead is the left-endpoint of a halfcircle. Then add the inner boundary of the annulus and, for every black bead that is not already the right-endpoint of a half-circle, add a cross-cut whose exterior endpoint is that black bead. ⇒ ⇒ ⇒ ⇒ Both φ 1 and φ 2 are clearly well-defined and injective, as the excess of (n + k) − n = k black beads are in bijection with the k necessary cross-cuts and the rotational notion of equivalence is identical for combinatorial necklaces and annular non-crossing matchings. The result then follows. 3.2 Enumeration of Ann k (2n + k, 2m + k), General Case We are finally ready for the enumeration of general Ann k (2n + k, 2m + k) that do not correspond to the special m = 0 case of Subsection 3.1. There are actually two sub-cases here depending upon whether or not k is nonzero: Theorem 3.4. Let n, m , and k be non-negative integers with m > 0. Then: 1. \|Ann k (2n + k, 2m + k)\| = \|Ann 0 (2n, 0)\| · \|Ann 0 (2m, 0)\| if k = 0, and 2. \|Ann k (2n + k, 2m + k)\| = k (2n + k)(2m + k) d\|(2n+k,n,m) φ(d) (2n + k)/d n/d (2m + k)/d m/d if k > 0 Where φ(d) is Euler's totient function and summations run over all common divisors of the given integers. Proof. Case #1 follows directly from Proposition 2.4. For Case #2 we look to apply Burnside's Lemma by defining an action of Z (2n+k)(2m+k) on a set A of relevant matchings. So fix n, m ≥ 0, k > 0, and consider annular non-crossing matchings with precisely n exterior halfcircles, m interior half-circles, and k cross-cuts. To form our set A, we require that the 2n + k exterior endpoints are located at 2π 2n+k radian intervals about the exterior boundary, and that the 2m + k interior endpoints are located at 2π 2m+k radian intervals about the interior boundary. Unlike in the proof of Theorem 3.1, we do not require that the k cross-cuts appear as straight lines that meet the boundaries at right angles, as that condition could require the re-spacing of endpoints on one of the boundary components. Here we consider cross-cuts up to isotopy that fix their endpoints. Absolutely no rotational isotopy of endpoints or rotation of either boundary component is allowed. Notice that A is composed of exactly k 2n+k n 2m+k m matchings. Here the binomial coefficients are derived from specifying which of the endpoints on the inner and outer boundary component correspond to the left-endpoints of half-circles (as in the proofs of Theorems 3.1 and 3.3), while the additional k term results from the ambiguity in matching up the remaining k endpoints on each side to form k cross-cuts. Notice that specifying both endpoints of a single cross-cuts determines how all remaining cross-cuts are matched amongst the remaining (k − 1) endpoints on each side. We then define a left action of Z (2n+k)(2m+k) on A where g · a is a counter-clockwise rotation of the entire matching by 2πg (2n+k)(2m+k) radians. If \|g\| = d, the elements of A g are matchings that may be radially divided in d identical sub-matchings along both the inner and outer boundaries, with analogous identifications of cross-cuts in each sub-matching. Observe that we do not require that both endpoints of each cross-cut lie in the same sub-matching, merely that each sub-matching exhibits an identical pattern with regards to any cross-cuts involved. For \|A g \| = 0 it is immediate that we must have both d \| (2n + k) and d \| (2m + k). To ensure that left-endpoints of half-circles are mapped to left-endpoints and that cross-cuts are mapped to cross-cuts, it is also required that d \| n, d \| m, and d \| k. It can easily be shown that d \| (2n + k, n, m) is necessary and sufficient to satisfy all of these conditions. 1 Thus \|A g \| = 0 if and only if d \| (2n + k, n, m). So take g ∈ Z (2n+k)(2m+k) with \|g\| = d such that d \| (2n+k, n, m). Via similar reasoning as in Theorem 3.1, there are 2n+k n choices for the outer boundary of each sub-matching and 2m+k m independent choices for the inner boundary of each sub-matching. After the endpoints belonging to cross-cuts have been identified, there are also k independent choices for how the cross-cuts match up across the annulus. These k choices correspond to similarly-symmetrical matchings whose blocks are identical apart from the fact that their cross-cuts uniformly "twist around" the annulus by different amounts. We may conclude that \|A g \| = k (2n+k)/d n/d (2m+k)/d m/d if d \| (2n + k, n, m). Similarly to our argument from the proof of Theorem 3.1, if q\|(2n + k)(2m + k) one may show that there are precisely φ(q) elements i ∈ Z (2n+k)(2m+k) with order \|i\| = q. It follows that there are precisely φ(d) elements g ∈ Z (2m+k)(2n+k) with \|A g \| = k (2n+k)/d n/d (2m+k)/d m/d , yielding the equation of Case #2 via an application of Burnside's Lemma. Theorems 3.1 and 3.4 combine to provide a closed formula for every \|Ann k (2n + k, 2m + k)\| in which n, m, k are not all zero. As we clearly have \|Ann 0 (0, 0)\| = 1, this accounts for all possibilities and allows us to directly enumerate Ann(n, m) = k Ann k (n, m) for all n, m ≥ 0. Table 2 of Appendix A presents values of \|Ann(n, m)\| for all 0 ≤ n, m ≤ 12, calculated in Maple using the equations of Theorems 3.1 and 3.4. In Proposition 2.2, we have already established that the n = 0 row (or m = 0 column) of Table 2 corresponds to A003239 [1]. Also noted in Section 2 what the fact that the n = 1 row (or m = 1 column) of Table 2 corresponds to the Catalan numbers. However, no other rows, diagonals, or triangles of numbers from Table 2 appear to correspond to any known sequences on OEIS. This yields an entire family of new integer sequences with an explicit geometric interpretation. Of particular interest are the rows for n > 2, representing new generalizations of the Catalan numbers that appear as later terms in a sequence of sequences beginning with A003239 and the Catalan numbers. Table 3 of Appendix A shows values of Ann(2n) for small values of n. These values are most easily derived by summing anti-diagonals from Table 2. The sequence of Table 3 also fails to appear as a known integer sequence on OEIS. Figure 3 : 3An element of Ann 2(6,4), and an element of Ann 0 (6, 0). Proposition 2 . 2 . 22Let n be any non-negative integer. Then \|Ann(2n, 0)\| equals the number of unrooted planar trees with a distinguished vertex and n additional vertices.Proof. The required bijection is analogous to the constructions ofFigures 1 and 2for linear and circular matchings. We add one vertex for each region in the matching and connect two vertices with an edge if they are separated by a half-circle: Theorem 3.1 then gives: • \|Ann p (2n + p, n)\| = 1 2n+p 2n+p n when p ∤ n • \|Ann p (2n + p, n)\| = 1 2n+p 2n+p n + 1 2n+p (p − 1) (2n+p)/p n/p when p\|n A simplification of the final term in the second case then yields the desired formula. Other equivalent conditions on d include d \| (2m + k, n, m) and d \| (k, n, m). AcknowledgementsBoth authors would like to thank the Department of Mathematics & Statistics at Valparaiso University, whose MATH 492 Research in Mathematics course provided the framework under which this research took place. The first author would also like to thank Dr. Lara Pudwell, who offered helpful advice in the latter stages of the project.A Tables of ValuesAll tables in this appendix were generated in Maple 18 using the equations of Theorems 3.1 and 3.4. Coding is available upon request from Paul Drube ([email protected]) The On-Line Encyclopedia of Integer Sequences. The On-Line Encyclopedia of Integer Sequences. Combinatorics of necklaces and "Hermite reciprocity. A Elashvili, M Jibladze, D Pataraia, J. Algebraic Combin. 102A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173-188. R W Goldbach, R Tijdeman, Pairings of 2n points on a circle. 38R.W. Goldbach and R. Tijdeman, Pairings of 2n points on a circle, Utilitas Mathematics 38 (1990), 277-284. Frank Harary, Edgar M Palmer, Graphical Enumeration. NYAcademic PressFrank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973. . Richard P Stanley, Enumerative Combinatorics. 2Cambridge University PressRichard P. Stanley, Enumerative Combinatorics vol. 2, Cambridge University Press, 1999. . Catalan Numbers, Cambridge University Press1st edition, Catalan Numbers, 1st edition, Cambridge University Press, 2015.	[]
[ "Complex Analysis of Askaryan Radiation: A Fully Analytic Model in the Time-Domain", "Complex Analysis of Askaryan Radiation: A Fully Analytic Model in the Time-Domain" ]	[ "Jordan C Hanson \nDepartment of Physics and Astronomy\nWhittier College\n\n", "Raymond Hartig \nDepartment of Physics and Astronomy\nWhittier College\n\n" ]	[ "Department of Physics and Astronomy\nWhittier College\n", "Department of Physics and Astronomy\nWhittier College\n" ]	[]	The detection of ultra-high energy (UHE, ≥10 PeV) neutrinos via detectors designed to utilize the Askaryan effect has been a long-time goal of the astroparticle physics community. The Askaryan effect describes radio-frequency (RF) radiation from high-energy cascades. When a UHE neutrino initiates a cascade, cascade properties are imprinted on the radiation. Thus, observed radiation properties must be used to reconstruct the UHE neutrino event. Analytic Askaryan models have three advantages when used for UHE neutrino reconstruction. First, cascade properties may be derived from the match between analytic function and observed data. Second, analytic models minimize computational intensity in simulation packages. Third, analytic models can be embedded in firmware to enhance the real-time sensitivity of detectors. We present a fully analytic Askaryan model in the time-domain for UHE neutrino-induced cascades in dense media that builds upon prior models in the genre. We then show that our model matches semi-analytic parameterizations used in Monte Carlo simulations for the design of IceCube-Gen2. We find correlation coefficients greater than 0.95 and fractional power differences < 5% between the the fully analytic and semi-analytic approaches.arXiv:2106.00804v4 [astro-ph.HE] 1 Jun 2022	null	[ "https://arxiv.org/pdf/2106.00804v4.pdf" ]	235,294,040	2106.00804	1ff1e38dd13bba75063e912d8650561de461a750	Complex Analysis of Askaryan Radiation: A Fully Analytic Model in the Time-Domain Jordan C Hanson Department of Physics and Astronomy Whittier College Raymond Hartig Department of Physics and Astronomy Whittier College Complex Analysis of Askaryan Radiation: A Fully Analytic Model in the Time-Domain (Dated: June 3, 2022) The detection of ultra-high energy (UHE, ≥10 PeV) neutrinos via detectors designed to utilize the Askaryan effect has been a long-time goal of the astroparticle physics community. The Askaryan effect describes radio-frequency (RF) radiation from high-energy cascades. When a UHE neutrino initiates a cascade, cascade properties are imprinted on the radiation. Thus, observed radiation properties must be used to reconstruct the UHE neutrino event. Analytic Askaryan models have three advantages when used for UHE neutrino reconstruction. First, cascade properties may be derived from the match between analytic function and observed data. Second, analytic models minimize computational intensity in simulation packages. Third, analytic models can be embedded in firmware to enhance the real-time sensitivity of detectors. We present a fully analytic Askaryan model in the time-domain for UHE neutrino-induced cascades in dense media that builds upon prior models in the genre. We then show that our model matches semi-analytic parameterizations used in Monte Carlo simulations for the design of IceCube-Gen2. We find correlation coefficients greater than 0.95 and fractional power differences < 5% between the the fully analytic and semi-analytic approaches.arXiv:2106.00804v4 [astro-ph.HE] 1 Jun 2022 I. INTRODUCTION The extrasolar flux of neutrinos with energies between [0.01-1] PeV has been measured by the IceCube collaboration [1]. Previous analyses have shown that the discovery of UHE neutrinos (UHE-ν) will require an expansion in detector volume because the flux is expected to decrease with energy [2][3][4][5][6]. The UHE-ν flux could potentially explain the origin of UHE cosmic rays (UHECR), and provides the opportunity to study electroweak interactions at record-breaking energies [7,8]. Utilizing the Askaryan effect expands the effective volume of UHE-ν detector designs, because this effect offers a way to detect UHE-ν with radio pulses that travel more than 1 km in sufficiently RF-transparent media such as Antarctic and Greenlandic ice [9][10][11]. The Askaryan effect occurs within a dense medium with an index of refraction n. A relativistic particle with v > c/n initiates a high-energy cascade with negative total charge. The charge radiates energy in the RF bandwidth, and the radiation may be detected if the medium does not significantly attenuate the signal [12,13]. The IceCube EHE analysis has constrained the UHE-ν flux to be E 2 ν φ ν ≤ 2 × 10 −8 GeV cm −2 s −1 sr −1 between [5 × 10 15 − 2 × 10 19 ] eV [4]. Arrays of O(100) in situ detectors encompassing effective areas of ≈ 10 4 m 2 sr per station, spaced by O(1) RF attenuation length could discover a UHE-ν flux beyond the EHE limits. The most suitable ice formations exist in Antarctica and Greenland, and a group of prototype Askaryan-class detectors has been deployed. These detectors seek to probe unexplored UHE-ν flux parameter-space from astrophysical and cosmogenic sources [5,6,14,15]. * Electronic address: [email protected] Askaryan radiation was first measured in the laboratory in silica sand, and later ice [16][17][18]. Cascade properties affect the amplitude and phase of the radiation. At RF wavelengths, cascade particles radiate coherently, and the radiation amplitude scales with the total track length of the excess negative charge. The RF pulse shape is influenced by the longitudinal length of the cascade, and the pulse is strongest when the viewing angle is close to the Cherenkov angle, θ C . The excess charge profile describes the excesse negative charge versus longitudinal position on the cascade axis. Radiation wavelengths shorter than the lateral width of the cascade, perpendicular to the cascade axis, are attenuated. At energies far above 10 PeV in ice, however, excess charge profiles generated by electromagnetic cascades experience the LPM effect and can have multiple peaks [19,20]. This theoretical foundation has been constructed from a variety of experimental and simulation results. The field of Askaryan-class detectors requires this foundation for at least two reasons. First, the theoretical form of the Askaryan RF pulse is used to optimize RF detector designs. Askaryan models are incorporated into simulations [21][22][23] in order to calculate expected signals and aid in detector design. For example, reconstruction tools for the radio component of IceCube-Gen2 combine machine learning and insights from Askaryan radiation physics [24][25][26]. Second, Askaryan models are used as templates to search large data sets for signal candidates [5,27]. The signal-to-noise ratios (SNRs) at RF channels are expected to be small (SNR ≈ 3), because the amplitude of the radiated field decreases with the vertex distance (1/r), and the signal is attenuated by the ice [9,28,29]. Low SNR signals reqire correspondingly low RF trigger thresholds, but signals must be sampled for a bandwidth of [0. GHz. Thus, RF channels are triggered at high rates by thermal noise. UHE-ν signals will be hidden within millions of thermal triggers. Template-waveform matching between models and data is a powerful technique for isolating RF signals from high-energy particles [27,30]. Askaryan models fall into three categories: full Monte Carlo (MC), semi-analytic, and fully analytic. The original work by E. Zas, F. Halzen, and T. Stanev (ZHS) [13] was a full MC model. The properties of cascades with total energy ≤ 1 PeV were examined. A parameterization for the Askaryan field below 1 GHz was offered, attenuating modes above 1 GHz via a frequency-dependent form factor tied to the lateral cascade width. The semianalytic approach was introduced by J. Alvarez-Muñiz et al (ARVZ) [31]. This approach accounts for fluctuations in the charge excess profile, and provides an analytic vector potential observed at the Cherenkov angle. The vector potential at the Cherenkov angle is labeled the form factor, and observed fields are derived from the derivative of the vector potential once convolved with a charge excess profile from MC. Recent work also accounts for differences in fit parameters from electromagnetic and hadronic cascades, and other interaction channels, while matching full MC simulations [32]. Finally, fully analytic models of Askaryan radiation from first principles have been introduced. J. Ralston and R. Buniy (RB) gave a fully analytic model valid for observations of cascades in the near and far-field, with the transition encapsulated by a parameter η [33]. The result was complex frequency-domain model. Recently, a model and software implementation was given by J. C. Hanson and A. Connolly (JCH+AC) that built upon RB by providing an analytic form factor derived from GEANT4 simulations, and accounted for LPM elongation [34]. This work connected the location of poles in the complex frequency plane to η and the form factor. The poles combine to form a low-pass filter for the Askaryan radiation. The JCH+AC results match the ZHS results while demonstrating the physical origins of model parameters. The RB and JCH+AC results are given in the Fourier domain, but most UHE-ν searches (like template matching) have taken place in the time-domain. The goals of this work are to produce a fully analytic timedomain model accounting for complex poles, valid for all viewing angles θ and η < 1, and to demonstrate that it matches semi-analytic models. In Section II, the cascade geometry, units, and vocabulary are defined. In Section III, we describe how the JCH+AC form factor fits into the current model [34]. In Section IV, the analytic Askaryan field, observed at θ = θ C (on-cone), is presented. In Section V, the analytic Askaryan field observed for θ = θ C (offcone) is presented. In Section VI, fully analytic fields are matched to semi-analytic fields generated with Nu-RadioMC [23] at 10 PeV (electromagnetic cascades) and 100 PeV (hadronic cascades). Though the LPM effect is activated in NuRadioMC, it has a negligible influence on the waveform comparison at these energies. In Section VII, the results are summarized and potential applications of the model are described. II. UNITS, DEFINITIONS, AND CONVENTIONS The coordinate system of the Askaryan radiation from a vector current density J is shown in Fig. 1 (a)-(b). Primed cylindrical coordinates refer J, and the unprimed spherical coordinates refer to the observer. The zenith or viewing angle is measured with respect to the longitudinal axis (z ). The observer displacement is r = \| x − x \|, in ther direction. The origin is located where the cascade has the highest instantaneous charge density (ICD). The ICD is treated with cylindrical symmetry, so it has no φ -dependence. This assumption is based on the large number of cascade particles and momentum conservation. The lateral extent of the ICD is along the lateral axis (ρ ). The viewing angle is θ in spherical coordinates, and the Cherenkov angle occurs when θ satisfies cos(θ C ) = 1/n ice with n ice = 1.78 ± 0.003 [35]. In Fig. 1 (c), an example excess charge profile n(z ) is shown with characteristic longitudinal length a. The individual ICDs represent the excess charge density for small windows of time, and n(z ) refers to the total excess charge as a function of z . Approximating the central portion of n(z ) as a Gaussian distribution N (µ, σ) corresponds to setting a = 2σ. Askaryan radiation occurs because n(z ) represents excess negative charge [13,34,36]. Cascades may be characterized as electromagnetic, initiated by charged outgoing leptons from UHE-ν interactions, or hadronic, initiated by the interaction between the UHE-ν and the nucleus. Electromagnetic cascades follow the Greisen distribution and hadronic cascades follow the Gaisser-Hillas distribution. An example of such an implementation via the ARVZ semi-analytic parameterization is AraSim [11]. z', z ρ' r θ θ J(z',ρ') r sinθ r z' n(z') ICD ICD Observer (a) (c) x' y' φ' (b) The units of the electromagnetic field in the Fourier domain are V/m/Hz, often converted in the literature to V/m/MHz. To make the distance-dependence explicit, both sides of field equations are multiplied by r, as in r E = ..., making the units V/Hz. Throughout this work, an overall field normalization constant E 0 is used. E 0 may be linearly scaled with energy, as in other Askaryan models. We show that the on-cone field amplitude is proportional to E 0 times a characteristic frequency-squared, so the units of E 0 are V/Hz 2 . For off-cone results, we show that the field amplitude is proportional to E 0 times a characteristic frequency divided by a characteristic pulse width, and the units of E 0 remain V/Hz 2 . In Section III B, we review briefly the energydependence of the longitudinal length a in both the electromagnetic and hadronic cases. For the Greisen distribution with critical energy E crit , it can be shown that if n max = n(z max ), where z max = ln (E C /E crit ), then n max a ∼ E C /E crit . Thus, the area under the curve n(z ) scales with the total cascade energy E C . RB demonstrated that the Askaryan radiation amplitude is proportional to n max a and therefore E C . The cascade develops over a length ≈ a, but the radiation is coherent over a length ∆z coh for which the displacement is constant to first order relative to a wavelength. The η parameter is the square of the ratio of a to ∆z coh : η = a ∆z coh 2 = k r (a sin θ) 2(1) In far-field, η < 1. In the first JCH+AC model, a limiting frequency ω C (Equation 2) was shown to filter the Askaryan radiation [34]: η = ω ω C(2) The effect of ω C is described in Section IV. The Askaryan radiation is primarily polarized in theθdirection, with a small amount alongr [31,34]. The wavevector is k = (2π)/(nλ), where n is the index of refraction. A 3D wavevector was defined by RB, equivalent to q = nk(1, ρ/R). The vector current density is treated by RB as a charge density times the velocity of the ICD: J(t, x ) = ρ(z − vt, ρ ) v. Further, the charge density is factored into n(z ) times ICD: ρ(z − vt, ρ ) = n(z )f (z − vt, ρ ). The form factor F is the three-dimensional spatial Fourier transform of the ICD [33]. The result for F was derived analytically by JCH+AC [34], and that derivation is briefly described in Section III A. JCH+AC define a parameter σ, and F is a function of σ: F (σ). The variable σ is related to the ratio of lateral ICD width to radiated wavelength. In the derivation of F , it is convenient to set σ equal to the ratio of angular frequency to the low-pass cutoff frequency ω CF of F : σ = ω ω CF(3) Armed with F , the longitudinal length a and the corresponding energy-dependence on E 0 , the RB field equations E, and the displacement r, the Askaryan electromagnetic field may be assembled according to the following form [33]: r E(ω, θ) = E 0 ω 2π ψ E(ω, θ) F (ω, θ)(4) The factor E 0 is proportional to cascade energy. The factor ω is the angular frequency. The variable ψ is ψ = −i exp(ikr) sin θ. The function E(ω, θ) contains the vector and complex pole structure of the field (see [33] and [34]). The model represented by Equation 4 is an all-θ, all-ω model. That is, Equation 4 is valid at all frequencies and all viewing angles, provided one accepts the approximation of the central portion of n(z ) as Gaussian. The first goal of this work is to build an all-θ, all-t model in the time-domain, derived from Equation 4, and the second goal is to compare it to semi-analytic parameterizations. III. THE FORM FACTOR AND LONGITUDINAL LENGTH PARAMETER To arrive at the main electromagnetic field in the timedomain, the individual pieces of Equation 4 must first be assembled. The first piece will be the form factor F that accounts for the 3D ICD, followed by some remarks about the energy-dependence of the longitudinal length parameter a. A. The Form Factor The form factor is the 3D Fourier transform of the ICD f (z , ρ ), with q = nk(1, ρ/R) [33]: F ( q) = d 3 x f (z , ρ )e −i q· x(5) The goal is to evaluate F in the Fourier domain for an ICD definition informed by cascade simulations. Simulations of the cascade induced by UHE-ν indicate a thin wave of charge in z spread uniformly in φ , that decreases exponentially in ρ . Using these observations JCH+AC complete the derivation in [34]. The final result was a simple analytic formula: F = 1 (1 + (ω/ω CF ) 2 ) 3/2(6) The form factor acts as a low-pass filter with the cutofffrequency ω CF : F ≈ ω 2 0 (ω + iω 0 )(ω − iω 0 )(7) The definition ω 0 = 2/3 ω CF has been used in the final step. Equation 7 matches the original ZHS parameterization (see Equation 20 of [13]). A Note about the Molière Radius In Section VI B, the decay constant l of the lateral component of the ICD is inferred from best-fit values of ω 0 . The connection between the l-parameter and ω 0 was described by JCH+AC [34]. Put simply, the ICD decays by a factor of 1/e a lateral distance l from the cascade axis. Note, however, that the l-parameter is not the Molière radius. The Molière radius is the lateral radius which forms a cylinder containing 90% of the energy deposition of the cascade. For ice with a density of 0.917 g cm −3 , one can estimate R M ≈ 9.2 cm using standard formulas. Although it is tempting to compare l to R M , these parameters have different definitions. Knowing that l is related to ω 0 , l may be estimated as λ/2 in ice at the cutoff-frequency. At 3 GHz in ice, λ/2 ≈ 2.8 cm, and at 1 GHz in ice, λ/2 ≈ 8.4 cm. Although the results are at the same order of magnitude as R M , there are three effects limiting the high-frequency spectrum of the radiation: ω 0 , ω C , and the viewing angle. Thus, l < R M is possible for a radiation spectrum limited to 1 GHz. B. The Longitudinal Length Parameter The next piece required in the assembly of the main electromagnetic field is the energy-dependence of the overall amplitude, and the energy dependence of the longitudinal length parameter, a, which is a part of E in Equation 4 [33]. What follows are two separate discussions, one for electromagnetic cascades, and one for hadronic cascades. Though we share these calculations for convenience, note that a variety of theoretical and experimental results on this topic are available [16] [37] [38]. Electromagnetic Case The number of charged particles versus distance in radiation lengths n(z ) in an electromagnetic cascade taking place in a dense medium with initial cascade energy E C , critical energy E crit , normalization parameter n 0 , and age s is [34] n(z ) = n 0 ln(E C /E crit ) exp z 1 − 3 2 ln(s)(8) To find the energy-dependent width of the Greisen distribution, four steps are necessary: (1) normalization of n(z ) as a fraction of the maximum excess charge, (2) conversion of n(z ) to n(s), (3) determination of the width of n(s) by approximating the central portion as a Gaussian distribution, and (4) conversion of the width from s units to radiation lengths z , and then converting those results to a distance. Define the ratio R = n(z max ± a/2)/n max , so the FWHM occurs when R = 0.5. The final result in radiation lengths is a = ln(E C /E crit ) −6 ln(R)(9) Since R < 1, ln(R) < 0 and a is real-valued, and a in Equation 9 is in radiation lengths. In solid ice the density is ρ ice = 0.917 g cm −3 , and the electromagnetic radiation length is z 0 = 36.08 g cm −2 [34]. Converting to distance gives a = z 0 ρ ice ln(E C /E crit ) −6 ln(R)(10) Note that a ∝ ln(E C ), as shown by RB and others. The product n max a is proportional to the energy E C /E crit . For this reason RB took n max a as the field normalization rather than E C [33]. As an example, let R = 0.4, and E crit ≈ 10 8 eV, gives a ≈ 4 meters for E C = 10 16 eV. We show in Section VI that our fitted a-values are close to 4 meters when matched to semianalytic parameterizations. Hadronic Case The Gaisser-Hillas distribution describes hadronic cosmic-ray air showers, but has also been applied to hadronic cascades in dense media in codes like AraSim [11,22]. The original function reads n(z ) = n max z − z 0 z max − z 0 (zmax−z0)/λ e zmax−z λ(11) The variables are defined as follows: n max is the instantaneous maximum number of particles in the cascade, z is the longitudinal distance in radiation lengths, z 0 is the initial starting point, λ is the interaction length, and z max is the location of n max . Using the same steps as the electromagnetic case, we find a = λz max −8 ln(R)(12) The a parameter again goes as √ z max ∝ ln(E C ) which produces similar lengths as the electromagnetic case when scaled by the appropriate interaction length and ice density. IV. ON-CONE FIELD EQUATIONS Theθ-component of the electromagnetic field at θ = θ C will now be built in the time-domain from Equation 4. Setting θ = θ C in the general RB field equations (Appendix A), with Equation 6 for F , σ = ω/ω CF and η = ω/ω CF , and letting E 0 be proportional to cascade energy E C produces Equation 45 from JCH+AC [34]: r E(ω, θ C ) = (−iω)E 0 sin(θ C )e iωr/c (1 − iω/ω C ) 1/2 (1 + (ω/ω CF ) 2 ) 3/2 (13) More detail is provided in Appendix A. Let the re- tarded time be t r = t − r/c, and let ω 0 = 2 3 ω CF and E 0 = E 0 sin θ C . Finally, let = ω 0 /ω C . The inverse Fourier transform of Equation 13 is rE(t, θ C ) =Ê 0 iω C ω 2 0 π d dt r ∞ −∞ e −iωtr (2iω C + ω) (ω + iω 0 )(ω − iω 0 ) dω(14) In Equation 14, the derivative with respect to the retarded time d/dt r is introduced to remove a factor of (−iω) from the numerator. Accounting for the complex poles and the sign of t r , complex integration and expansion to first-order in yields rE(t, θ C ) = 1 3Ê 0 ω 2 CF 1 − 1 2 e ω0tr t r < 0 2e −2ωCtr − 1 + 1 2 e −ω0tr t r > 0(15) Equation 15 represents the time-domain solution for the on-coneθ-component of the Askaryan electric field. The expansion to first-order in is only performed so the final result resembles semi-analytic results for E = −∂ A/∂t r [31,32]. Table I summarizes the definitions of the parameters in Equation 15. Fit results for the parameters of Table I are shown in Section VI. Notice that the amplitude is asymmetric, and the the parameter influences the asymmetry. The parameter was studied in JCH+AC in detail. For example, Fig. 10 of [34] shows that ≈ [0.1 − 1] for inverse lateral width l −1 = √ 2πρ 0 ≈ 20 m −1 and a ≈ 4 m. The best-fit results for and a are shown in Section. VI. JCH+AC showed that the expression for is the product of the ratio of the lateral to longitudinal length, and the ratio of the longitudinal length to the observer displacement, making it a physical parameter connecting the event geometry to the cacscade shape [34]. Figure 3 displays normalized examples of Equation 15 for different values of ω 0 , ω C , and . A. Verification of the Uncertainty Principle As a check on the procedures used to perform the inverse Fourier transform that produces Equation 15, we verify below that the uncertainty principle holds, for ∆θ → 0. JCH+AC provide the Gaussian width of the radiation in the Fourier domain: σ ν , where ν represents the frequency in Hz. Generally speaking, Fourier transform pairs must obey σ ν σ t ≥ 1/(2π). The following procedure is used to compute the width σ t of the on-cone field. First, the t r < 0 and t r > 0 cases are each treated as probability distributions and normalized. Next, the average positive and negative retarded times,t r,+ and t r,− , are computed. Finally, subtracting the two averages yields σ t : σ t =t r,+ −t r,− = + 2 ω 0 = 1 ω C + 2 ω 0(16) The result has the correct units and the limiting cases are sensible. Suppose → 0 (ω C ω 0 ), then σ t → 2/ω 0 , which is expected from observing Equation 15 if the ω C exponential disappears. If = 1 (ω C = ω 0 ), then σ t = 3/ω 0 . That is, the pulse is wider if there is more than one relevant cutoff frequency. The expression for σ ν is given by Equation 36 of JCH+AC [34]: σ ν = c 2πa∆ cos θ 1 + η 2 1/2(17) Expanding to first order in ∆ cos(θ) = cos(θ)−cos(θ C ), σ ν ≈ c 2πa sin(θ C )∆θ 1 + η 2 1/2(18) From Table I: [34]). Multiplying σ t and σ ν with the far-field limit (η < 1) gives the inequality ω −1 C = na 2 sin 2 (θ C )/(rc), and ω −1 0 = nl sin(θ C )/c, with l = 3/2/( √ 2πρ 0 ). (Recall that ρ 0 is a parameter discussed inσ ν σ t ≥ n 2π a r sin(θ C ) ∆θ + 2 l a 1 ∆θ(19) Therefore, in order to satisfy σ ν σ t > 1/(2π), n a r sin(θ C ) + 2n l a > ∆θ(20) Although a/r 1 and l/a 1, as long as these expressions do not approach zero as fast as ∆θ → 0 in Equation 20, the uncertainty principle holds. Yet these are exactly the conditions of the problem: a displacement r in the far-field (but not infinitely far away) and a longitudinal length a much larger (but not infinitely larger) than the lateral ICD width l. Thus, σ ν σ t > 1/(2π) holds. V. OFF-CONE FIELD EQUATIONS Turning to the case for which θ = θ C , theθ-component of the electromagnetic field will now be built in the timedomain. The RB field equations for theθ andr components are summarized in both RB and JCH+AC [33,34], and Appendix A. Recall the general form of the electromagnetic field, given in Equation 4: r E(ω, θ) = E 0 ω 2π ψ E(ω, θ) F (ω, θ)(21) The first task is to simplify E(ω, θ) before taking the inverse Fourier transform. The simplification inolves expanding E(ω, θ) in a Taylor series such that u = 1 − iη ≈ 1, restricting η < 1 (far-field). Once E(ω, θ) is simplified, the inverse Fourier transform of Equation 21 may be evaluated to produce the result. Table II contains useful variable definitions, Table III contains useful function definitions, and Table IV contains special cases of the functions in Table III. Variable Definition The original form of E(η, θ) is shown in Appendix A. Changing variables to u and x (Tab. II) and using the function definitions and values in Tabs. III-IV u 1 − iη x cos(θ) xC cos(θC ) q (xxC − x 2 C )/(1 − x 2 ) y 1 2 (ka) 2 (cos θ − cos θC ) 2 p 1 2 a c 2 (cos θ − cos θC ) 2f (u, x) u + 3 (1−u) 2 u x 2 −xx C 1−x 2 −1/2 g(u, x) exp − 1 2 (ka) 2 (x − xC ) 2 u −1 h(u, x) 1−u u q E(u, x) ·θ f (u, x)g(u, x)(1 − h(u, x)), E(u, x) · θ = E(u, x) becomes E(u, x) = f (u, x)g(u, x)(1 − h(u, x))(22) Expanding E(u, x) near u = 1 gives E(u, x) = E(x, 1) + (u − 1)Ė(x, 1) + O(u − 1) 2 (23) The details of the expansion are shown in Appendix B. The result is E(x, u) = e −y 1 − 1 2 jη (2y + 2q − 1)(24)Function (u = 1) Result f (x, 1) 1 f \|u=1 − 1 2 g(x, 1) exp(−y) g\|u=1 y exp(−y) h(x, 1) 0 h\|u=1 −qrE(t, θ) = F −1 E 0 ω 2π F ψE(25) Intriguingly, the result is proportional to the linebroadening function, H (DLMF 7.19, [39]) common to spectroscopy applications. There are three terms in Equation 24. Two terms ultimately vanish, being integrals over odd integrands (see Appendix B). The integral that remains contains H, with ω 1 = t r /(2p): I 0 = 2πi ω C ω 0 e − t 2 r 4p H( √ pω 0 , iω 1 √ p)(26) The line-broadening function is similar to a convolution between a Gaussian function and a Lorentzian function, and cannot be expressed analytically, though there are examples of polynomial expansions [40]. Note that, for situations relevant to the current problem, ω > ω 1 . Requiring that ω > ω 1 amounts to a restriction between ∆θ and \|t r \|: \|t r \| < \|2pω\|(27) It is shown in the next section that √ 2p is the pulse width σ t , so \|2pω\| has units of time. Using the results of Sec. V A below, the restriction on the retarded time may be written \|t r \|/σ t < ωσ t = 2π(σ t /T ). That is, the accuracy of the waveform should be trusted within a number of pulse widths that is less than 2π times the ratio of the pulse width to the period of the lowest frequency. This is not a strong requirement, since the field quickly approaches zero after several pulse widths. Hereafter, this step will be called the symmetric approximation, because the result for r E(t r , θ) in Equation 28 has equal positive and negative amplitude. Evaluating the line-broadening function numerically would account for amplitude asymmetry. The restriction on ∆θ is formalized in Sec. V B. Solving I 0 using the symmetric approximation clears the way for the final result (see Appendix B): zero crossings are at t r = 0 ns as a result of the symmetric approximation. Figure 5 contains contours of the same results as in Fig. 4. As in the on-cone result, the overall field amplitude scales with energy (E 0 ∼ n max a). However, the amplitude scales also with ω 0 /p. The argument of the complementary error function, √ pω 0 , is unitless. This factor is strictly positive, so the range of the complementary error function is (0, 1). The factor √ pω 0 cannot be zero without setting θ = θ C , or setting ω CF = 0. Both cases are not allowed. Equation 28 represents the offcone (θ = θ C ) solution, so p = 0. Setting ω CF = 0 is not physical, for this implies infinite lateral width (l) and cascade particles have finite transverse momentum. Another possibility is that p = 0 if a = 0, but this implies E 0 = 0. Therefore, 0 < erfc( √ pω 0 ) < 1. rE(t, θ) = − E 0 ω 0 sin(θ) 8πp t r e − t 2 A. Verification of the Uncertainty Principle As in Section IV A, the uncertainty principle should be checked. Equation 28 is an anti-symmetric Gaussian function with pulse width σ t = √ 2p. Let ∆ cos θ = (cos θ − cos θ C ). Using Table II, the expression √ 2p evaluates to σ t = 2p = a c (∆ cos θ)(29) Recall that σ ν is given by σ ν = c 2πa∆ cos θ 1 + η 2 1/2(30) The uncertainty product is σ t σ ν = 1 2π 1 + η 2 1/2(31) In the far-field, η < 1, so σ t σ ν ≥ 1/(2π) holds: B. Usage of the On-Cone versus Off-Cone Fields The form of Equation 28, and the restriction between ∆θ and \|t r \| from the symmetric approximation suggests the limit ∆θ → 0 must be examined carefully. Since p ∝ (cos θ − cos θ C ) 2 , probing the model near θ = θ C is equivalent to taking the limit that p → 0. Intriguingly, the p −1 -dependence in the field does not lead to a divergence. As the field grows in amplitude from p −1 as p → 0, the field width, √ 2p, approaches zero. Equations 16 and 29 contain the pulse widths of the on-cone and off-cone fields, respectively. Power in the off-cone case is limited by the pulse width √ 2p, and the observed power increases as ∆θ and √ 2p both decrease. Thus, a reasonable constraint on when ∆θ min is large enough to use Equation 28 is given by setting the offcone pulse width equal to the on-cone pulse width: 1 ω C + 2 ω 0 = 2p(32) Expanding the expression for p near θ = θ C , and evaluating the square root leads to 1 ω C + 2 ω 0 = a c sin θ C ∆θ min(33) Using = ω 0 /ω C , and letting k 0 = ω 0 /c, the formula may be rearranged: + 2 = ak 0 sin θ 0 ∆θ min(34) Squaring both sides, and then dividing both sides by r yields ( + 2) 2 r = k 0 k 0 (a sin θ C ) 2 r ∆θ 2 min(35) The quantity in parentheses on the right-hand side is η, with ω = ω 0 . Setting ω = ω 0 means η = . Solving for ∆θ min gives ∆θ min = + 2 √ k 0 r(36) Assuming ∆θ min ∝ 1 √ kr(37) VI. COMPARISON TO SEMI-ANALYTIC PARAMETERIZATIONS The fully analytic model will now be compared to the ARVZ semi-analytic parameterization used in Nu-RadioMC to predict signals in IceCube-Gen2 Radio [23]. Specifically, the comparison is between Equations 15 and 28 and the NuRadioMC implementation of the semianalytic parameterization given in [32]. To provide concrete comparisons, a small set of waveforms was generated with NuRadioMC, for both electromagnetic and hadronic cascades, on and off-cone. The electromagnetic cascades have E C = 10 16 eV, while the hadronic cascades have E C = 10 17 eV. These choices minimize the impact of the LPM effect, though the LPM effect was activated in the NuRadioMC code. The comparison involves three stages. First, waveforms and a-values are generated for each cascade type, energy, and angle: θ = θ C + 3.0 • , and θ = θ C . Second, Equations 15 and 28 are tuned to match the waveforms. In each fit, the Pearson correlation coefficient (ρ) is maximized, and the sum-squared of amplitude differences ((∆E) 2 ) is minimized. Finally, best-fit parameters are tabulated. Two remarks are important regarding the fit criteria. First, the Pearson correlation coefficient is not sensitive to changes in amplitude because it is normalized: ρ = cov(f data , f model ) σ data σ model(39) Parameters that affect ρ are those that scale t r . Second, parameters that control (∆E) 2 are those that scale the waveform amplitude. If E i represent the samples of the models, then (∆E) 2 = N i=1 (E i,data − E i,model ) 2(40) A. Waveform Comparison: θ = θC Electromagnetic case. Six different electromagnetic cascades and the corresponding Askaryan fields were generated using the ARZ2019 model from NuRadioMC [23] [32] for comparison to Equation 15. The cascades have E C = 10 PeV, and r = 1000 meters. The LPM effect is activated in NuRadioMC for all comparisons in this work. The units of E(t r , θ C ) are mV/m versus nanoseconds, so the units of r E are Volts. The sampling rate of the digitized semi-analytic parameterizations was 100 GHz, with N = 2048 samples. Let f C = ω C /(2π) and f 0 = ω 0 /(2π). The frequencies f C and f 0 were varied from [0.6 -6.0] GHz. The parameter E 0 was varied from [0.05 -5.0] V GHz −2 . In a simple 2-level for-loop, the Pearson correlation coefficient ρ was maximized by varying f 0 and f C . Next, the sum of the squared amplitude differences (∆E) 2 was minimized by varying E 0 , while holding f 0 and f C fixed. Several other schemes were studied, including a 3-level for-loop, but the two-stage process produced the best results. The results are shown in Fig. 6. Maximizing ρ corresponds to minimizing (∆E) 2 . In Fig. 7, (∆E) 2 is graphed versus ρ for one event. Best-fit ρ-values are ≈ 0.97 for this set, corresponding to bestfit (∆E) 2 values of ≈ 7%. Contours of ρ > 0.95 for f 0 versus f C are shown in Fig. 6 (left column). The crosses represent the best-fit location. The dashed gray line at y = x corresponds to f 0 /f C = = 1. Though Equation 15 contains an expansion to first order in , making it resemble the derivative of the vector potential from the ARVZ semi-analytic parameterization [32], the expansion is optional. There is a restriction that = 2 (see Equation A14 of Appendix A). Thus, the best-fit -values avoid the solid black lines ( = 2) in Fig. 6, but are large enough to account for pulse asymmetry. The best-fit waveforms are shown in Fig. 6 (right column). The gray curves correspond to the semi-analytic parameterization, and the black curves represent Equation 15. Table V contains the best-fit results for the Equation 15 parameters, along with best-fit ρ-values and (∆E) 2values. The horizontal and vertical distances from the crosses to the ρ > 0.95 contour are used as error estimates for f 0 and f C in Tab. V. The a-errors typically encompass the a-values from NuRadioMC. The full region in [f 0 , f C ] space for which UHE-ν signals are expected for IceCube-Gen2 radio will be the topic of future studies, along with the apparent difference in -value depending on the electromagnetic or hadronic classification of the cascade (see Figure 8). Hadronic case. Using the same procedure as the electromagnetic case, NuRadioMC was used to generate six hadronic cascades at 100 PeV for comparison to Equation 15. The energy was increased to show that the model describes a range of energies, so the waveform amplitudes are larger by a factor of 10 relative to the 10 PeV case. The LPM effect is activated in NuRadioMC for all comparisons in this work. The main results are shown in Figure 8, and the correlation contours represent ρ = 0.985. The results shown in Figure 8 demonstrate that modeling hadronic cascades at θ = θ C is similar to the electromagnetic case, with one interesting difference. The contours enclose best-fit -values below the dashed line, whereas the fits to the electromagnetic cases were above the dashed line. This could indicate a potential discriminator for cascade classification. Another difference between the electromagnetic and hadronic cases is that the gray contours in Fig. 8 correspond to ρ = 0.985, as op- [23]. Black: Equation 15. posed to ρ = 0.95 in the electromagnetic case. Table VI contains the best-fit parameters corresponding to Figure 8. The typical power difference (∆E) 2 has decreased with respect to the electromagnetic case. The ρ-values all exceed 0.985, and the (∆E) 2 results are typically below 2 percent. Intriguingly, < 1 means higher f C values, which in turn yields systematically low avalues relative to those generated in NuRadioMC, despite the increased energy. Reconstructed a-values are still within a factor of 2 of the MC-true values. Despite the systematic offset, the best-fit a and the NuRadioMC a-values are tightly correlated (see Fig. 11 below). # f 0 (GHz) f C (GHz) E 0 (V GHz −2 ) awave (m), aMC (m) ρ (∆E) 2 (%) B. Waveform Comparison: θ = θC Electromagnetic case. The general comparison procedure of Section VI A was repeated with the same semianalytic parameterization from NuRadioMC, but with twelve new events each viewed at θ = θ C + 3.0 • (six electromagnetic cascades, six hadronic). One difference is that ω 0 only changes the waveform amplitude, along with E 0 . The pulse width σ t = √ 2p connects the longitudinal length a and the viewing angle with respect to the Cherenkov angle. The fit procedure was performed in two stages. First, θ-values and a-values were scanned from [θ C + 1. From left to right, the form-factor cutoff-frequency, coherence cuofffrequency, energy-scaling normalization, longitudinal length parameter, the best-fit correlation coefficient, and the relative power difference between NuRadioMC semi-analytic parameterization and the fully analytic model. The parameter means and errors in the mean are quoted in the bottom two rows. # f 0 (GHz) f C (GHz) E 0 (V GHz −2 ) awave (m), aMC (m) ρ (∆E) 2 (%) 2-level for loops. The results are shown in Figure 9. In Figure 9 (left column), the best-fit a-values and θvalues are marked with a cross. The circles represent the MC-true values. Circles and crosses lie on the dashed lines, because an uncertainty principle connects a-values to θ-values (see Section V A). Specifically, Equation 29 may be used to show, to first-order in ∆θ = θ − θ C : a∆θ = c √ 2p sin θ C = constant(41) The pulse width σ t = √ 2p is a constant derived from the waveform, implying that the product of a and ∆θ is constant. The parameters a and ∆θ are therefore inversely proportional: a ∝ ∆θ −1 . The shape of the ρ > 0.95 contour follows this inverse proportionality. The dashed lines represent Equation 41. These results suggest that a measurement of the Askaryan pulse width would constrain the cascade shape and geometry. The best-fit waveforms are shown in Figure 9 (right column). Typical correlation coefficients exceed ρ = 0.98. Table VII contains the fit results. The fit results include estimates of the lateral width parameter, l, derived from f 0 (see Section III A). Despite making the symmetric approximation to arrive at Equation 28, the fits include fractional power differences of ≈ 3%. Hadronic case. The fit procedure for the hadronic cascades was the same as the electromagnetic case, except that the range for E 0 was expanded to [1.0, 20.0] V GHz −2 . As in the on-cone procedure, the hadronic cascade energy was E C = 100 PeV. The results are shown in Figure 10. From left to right, the viewing angle, longitudinal length parameter, form-factor cutoff frequency, the energy-scaling normalization, the lateral width of the cascade, the best-fit correlation coefficient, and the relative power difference between NuRadioMC semi-analytic parameterization and the fully analytic model. The parameter means and errors in the mean are quoted in the bottom two rows. θwave (deg), θ MC (deg) awave (m), a MC (m) f 0 (GHz) E 0 (V GHz −2 ) l (cm) ρ (∆E) 2( As with the electromagnetic case, ρ is maximized and (∆E) 2 is minimized. Table VIII are in agreement with the MC values from NuRadioMC. The E 0 -values match expectations for 100 PeV cascacdes, because they are a factor of 10 higher than those of the 10 PeV electromagnetic case. The results for a, l, and f 0 , however, are not statistically different between Tables VII and VIII. Future studies will require computing the probability distributions of these parameters from large numbers of UHE-ν cascades. As a first exercise for statistical energy reconstruction from waveform parameters, assume that θ = θ C + 3.0 • is already measured. For example, θ could be determined by measuring the cutoff-frequency in the Fourier domain below 1 GHz (see Fig. 5 of [34], for example). Scanning Equation 28 over all NuRadioMC waveforms at fixed θ = θ C + 3.0 • yields Figure 11, in which the fitted avalue from each waveform is graphed versus the MC-true a-value. The a-errors in all cases are taken to be ±10 cm (± two ∆a step-sizes). A least-squares linear fit was applied to the data. The linear function fits the data, and the correlation coefficient is 0.97. The results in Figure 11 imply an energy reconstruction technique using the formulas found in Section III B. Consider the relationship between a and ln(E C /E crit ): a = c 1 ln(E C /E crit ). The fractional error in ln(E C /E crit ) is proportional to the fractional error in a: # θwave (deg), θ MC (deg) awave (m), a MC (m) f 0 (GHz) E 0 (V GHz −2 ) l (cm) ρ (∆E) 2(σ ln(EC/Ecrit) ln(E C /E crit ) = 2c 1 σ a a(42) If a reliable fit for the a-parameter is obtained from observed Askaryan waveforms, Equation 42 shows that the logarithm of the energy can be constrained. VII. CONCLUSION We have presented a fully analytic Askaryan model in the time-domain, and we have shown that it matches results generated with semi-analytic parameterizations used in NuRadioMC. Pearson correlation coefficients between the fully analytic and semi-analytic paremeterizations were found to be greater than 0.95, and typical fractional differences in total power were found to be ≈ 5%. New results and potential applications are summarized in the following sections. A. Summary of New Results The main results are summarized in Table IX. This work represents the first time the two distinct pole frequencies f 0 and f C have been used to characterize the time-domain field equations of the Askaryan effect for both θ = θ C and θ = θ C . The uncertainty principle was verified on-cone (θ = θ C ), serving as a check on the model. By fitting on-cone cascade parameters, we have shown that an analytic model matches semi-analytic predictions. The parameter reveals a potential cascade classification scheme. Next, the off-cone (θ = θ C ) field equations were derived, and again the uncertainty principle was verified. Off-cone cascade parameters were fit, and the results are in excellent agreement with semianalytic results. Fitting a-values has revealed a potential energy reconstruction. To obtain the fields on and off-cone, η < 1 was assumed. The restriction η < 1 means that Eqs. 15 and 28 must be applied to the far-field. Given that a and θ C are fixed by cascade physics and ice density, and that the relevant Askaryan bandwidth for ice is [0.1 − 1] GHz, the parameter most easily varied within η is the observer distance r. Taking ν = 0.5 GHz, n = 1.78, c = 0.3 m GHz, θ = θ C , and a = 5 m, requiring that η = 1 gives r ≥ 0.4 km. Scaling to ν = 0.25 GHz gives r ≥ 0.2 km. According to NuRadioMC [23] (Fig. 13), the r corresponding to UHE-ν at 10 18 eV ranges from 0.7-3.2 km, and 0.2 km is rare. The "acceleration argument" invoked by RB in [33] states that if r(t) points to the ICD, r(t) must be constant enough to ensure that ∆r < λ. Using the law of cosines, with two sides being r and r + ∆r, and a third being a, the criteria that (a/r) 2 1 leads to \|∆r\| ≈ a/n which is O(2) m. When in doubt about usage and event geometry, determining if (a/r) 2 1 is a good check. If the UHE-ν event is a charged-current interaction with an electromagnetic cascade far above the LPM energy for ice, a grows faster than ln(E C /E crit ) [20]. B. Utility of the Analytic Model There are at least four advantages of fully analytic Askaryan models. First, when analytic models are matched to observed data, cascade properties may be derived directly from the waveforms. Second, in large scale simulations, evaluating a fully analytic model technically provides a speed advantage over other approaches. Third, fully analytic models, combined with RF channel response, can be embedded in firmware to form a matched filter that enhances UHE-ν detection probability. Fourth, parameters in analytic models may be scaled to produce results that apply to media of different density than ice. This application is useful for understanding potential signals in the Antarctic firn, or the upper layer of snow and ice that is of lower density than the solid ice beneath it. The ability to fit cascade properties from waveforms will be a useful tool for the radio component of IceCube-Gen2. Examples of current reconstruction techniques include the forward-folding method [25] and information field theory (IFT) [26]. In particular, the longitudinal length parameter a leads to a reconstruction of ln(E C ), given knowledge of ∆θ ( Fig. 11 and Equation 42). Further, all designs for detector stations in IceCube-Gen2 radio include many distinct RF channels and one phasedarray of channels. Matching our analytic model to each channel waveform will provide a separate measurement of parameters like a and θ (see gray contours of Figures 4 and 5). The ensuing global fit should constrain the event energy and geometry. The most intriguing usage for a fully analytic Askaryan model would be to embed the model as a matched filter in detector firmware. Because cascade properties are unknown a priori, an array of matched filters could be implemented to form a matched filter bank. One example of this approach was the TARA experiment [41], which was designed to detect low-SNR cosmic ray radar echoes. This is similar to the challenge faced by IceCube-Gen2 radio: pushing the limit of low-SNR RF pulse detection in a remote setting. For example, a matched filter bank could be formed with an array of off-cone field formulas with fixed a-value and varying θ-values, which would then be convolved with the RF channel impulse response (see Section 6 of [27]). Finally, a fully analytic model enhances the ability of IceCube-Gen2 radio to identify signals that originate in the firn. At the South Pole, the RF index of refraction begins around 1.35 and does not reach the solid ice value of 1.78 until 150-200 meters [28]. There are at least two signals that could originate in the firn: UHE-ν events that create Askaryan radiation, and UHE cosmic ray cascades partially inside or fully inside the firn. The altitude of the South Pole makes the latter possible. The Askaryan radiation of the firn UHE-ν events could be modeled via appropriate density-scaling of the cascade parameters. The complete field from the original RB model [33], including the form factor F , ψ = −i exp(ikr) sin θ, and E is r E(ω, θ) = E 0 ω 2π ψ E(η, θ) F (A4) Let Equation 6 for the form factor, with σ = ω/ω CF and η = ω/ω CF , and letting E 0 be proportional to cascade energy E C : r E(ω, θ C ) = (−iω)E 0 sin(θ C )e iωr/c (1 − iω/ω C ) 1/2 (1 + (ω/ω CF ) 2 ) 3/2 (A5) Suppose ω < ω C , and ω < ω CF , such that the following approximations of the factors in the denominator are valid: (1 − iω/ω C ) 1/2 ≈ 1 − i 2 ω ω C (A6) (1 + (ω/ω CF ) 2 ) 3/2 ≈ 1 + 3 2 ω ω CF 2 (A7) Using the approximations introduces simple poles into the complex formula for the frequency-dependent electric field. Inserting the approximations in the denominator of Equation A5, we have r E(ω, θ C ) = (−iω)E 0 sin(θ C )e iωR/c 1 − i 2 ω/ω C 1 + 3 2 (ω/ω CF ) 2 (A8) The denominator can be rearranged by factoring the ω coefficients, and defining ω 0 = 2 3 ω CF . r E(ω, θ C ) = 2iω C ω 2 0 (−iω)E 0 sin(θ C )e iωr/c (2iω C + ω) (ω + iω 0 )(ω − iω 0 )(A9) LetÊ 0 = E 0 sin(θ C ), and let the retarded time be t r = t − r/c. Taking the inverse Fourier transform, using the same sign convention as RB [33] (f (t) = (2π) −1 ∞ −∞ F (ω)e −iωt dω), converts the field to the timedomain: rE(t, θ C ) =Ê 0 iω C ω 2 0 π d dt r ∞ −∞ e −iωtr (2iω C + ω) (ω + iω 0 )(ω − iω 0 ) dω (A10) 1. If t r > 0: Consider the contour comprised of the real axis and the clockwise-oriented negative infinite semi-circle. On the contour, the exponential phase factor in Equation A10 goes as exp(−iωt r ) = exp(−i(R cos φ + iR sin φ)t r ) (A11) For the semi-circle, φ ∈ [π, 2π], so sin φ < 0 and t r > 0. Exponential decay occurs and the integrand vanishes on the semi-circle for \|ω\| = R → ∞. 2. If t r < 0: Consider the contour comprised of the real axis and the counter-clockwise-oriented positive infinite semi-circle. On the contour, the exponential phase factor in Equation A10 goes again as exp(−iωt r ) = exp(−i(R cos φ + iR sin φ)t r ) (A12) For the semi-circle, φ ∈ [0, π], so sin φ > 0 and t r < 0. Exponential decay occurs and the integrand vanishes on the semi-circle for \|ω\| = R → ∞. Using cases 1 and 2, Equation A10 can be solved using the Cauchy integral formula. Beginning with t r > 0, two poles are enclosed in the semi-circle: one that originated from the coherence cutoff frequency, and the other that originated from the form factor. The Cauchy integral formula yields rE(t, θ C ) = 2Ê 0 ω C ω 2 0 d dt r e −2ωCtr i 2 (−2ω C + ω 0 )(−2ω C − ω 0 ) + e −ω0tr i 2 (−ω 0 + 2ω C )(−2ω 0 ) (A13) Define the ratio of the cutoff frequencies: = ω 0 /ω C . After evaluating the time derivatives, Equation A13 be-comes rE(t, θ C ) =Ê 0 ω 2 0 e −2ωCtr (1 − 2 )(1 + 2 ) − e −ω0tr (2)(1 − 2 ) (A14) Expanding to linear order in , assuming < 1, and recalling that ω 2 0 = 2 3 ω 2 CF : rE(t, θ C ) ≈ 1 3Ê 0 ω 2 CF 2e −2ωCtr − 1 + 2 e −ω0tr (A15) Turning to the case of t r < 0, consider integrating Equation A10 along the contour comprised of the real axis and the counter-clockwise-oriented positive infinite semi-circle. The contour encloses one pole, and the exponent ensures convergence: rE(t, θ C ) = (2πi)Ê 0 (π) −1 iω C ω 2 0 d dt r e ω0tr (2iω C + iω 0 ) (2iω 0 ) (A16) After evaluating the derivative, the expression simplifies with = ω 0 /ω C : rE(t, θ C ) = 1 2Ê 0 ω 2 0 e ω0tr 1 + 1 2 (A17) Finally, using the same first-order approximation in as the t r > 0 case: rE(t, θ C ) ≈ 1 3Ê 0 ω 2 CF 1 − 1 2 e ω0tr (A18) Collecting the t r > 0 and t r < 0 results together: rE(t, θ C ) = 1 3Ê 0 ω 2 CF 1 − 1 2 e ω0tr t r < 0 2e −2ωCtr − 1 + 1 2 e −ω0tr t r > 0 (A19) Appendix B: Details of the Off-Cone Field Equation Derivation Using Tabs. II-IV, Equation A2 reduces to E(u, x) = f (u, x)g(u, x)(1 − h(u, x)) (B1) Expanding to first-order with respect to u near (u = 1) gives E(u, x) = E(x, 1) + (u − 1)Ė(x, 1) + O(u − 1) 2 (B2) The first term is f g(1−h) evaluated at u = 1: exp(−y) ( Table IV). The second term requires the first derivative of E(u, x) with respect to u, evaluated at u = 1. Using the definition of u (Table II), the result may be written E(u, x) = e −y 1 − 1 2 jη (2y + 2q − 1)(B7) Proceding with the inverse Fourier transform of thê θ-component: rE(t, θ) = F −1 E 0 ω 2π F ψE(B8) Let η = ω/ω C , y = pω 2 (Table II). Inserting the Taylor series for E, the form factor F , and ψ = −i exp(ikr) sin θ (Sec. II), and following the same steps as the on-cone case produces 2πrE(t, θ) = E 0 ω 2 0 sin(θ) 4πiω C d dt r ∞ −∞ e −iωtr−pω 2 2iω C + 2pω 3 + (2q − 1)ω ω 2 + ω 2 0 dω (B9) Unlike the on-cone case, Equation B9 cannot be integrated with infinite semi-circle contours, because the exponential term diverges along the imaginary axis far from the origin. Let I 0 represent the constant term with respect to ω in the numerator: I 0 = ∞ −∞ e −iωtr−pω 2 (2iω C ) ω 2 + ω 2 0 dω (B10) Further, let I 1 and I 3 represent the linear and cubic terms, respectively. Completing the square in the exponent of I 0 , with ω 1 = t r /(2p), yields I 0 = 2iω C e − t 2 r 4p ∞ −∞ e −p(ω+iω1) 2 ω 2 + ω 2 0 dω (B11) Equation B11 may be re-cast as the line-broadening function, H (DLMF 7.19, [39]) common to spectroscopy applications: I 0 = 2πi ω C ω 0 e − t 2 r 4p H( √ pω 0 , i √ pω 1 )(B12) Assume that ω > ω 1 . This approximating step will be called the symmetric approximation. I 0 ≈ 2iω C e − t 2 2p ∞ −∞ e −pω 2 ω 2 + ω 2 0 dω (B13) The result for I 0 involves the complementary error function (DLMF 7.7.1, [39]): I 0 = 2iω C e − t 2 2p πω −1 0 e pω 2 0 erfc( √ pω 0 ) (B14) The integrals I 1 and I 3 are zero by symmetry, with odd integrands over (−∞, ∞). Inserting the result for I 0 into Equation B9 and evaluating the derivative finishes the problem (see Sec. V). FIG. 2 : 2(Black) Equation 6, graphed versus σ = ω/ωCF. (Gray) The two-pole approximation. Top) Equation 15 from [−4, 4] ns, with (black) ωC = 2π(1.25) GHz, ω0 = 2π(1.56) GHz, = 1.25, (gray) ωC = 2π(1.25) GHz, ω0 = 2π(0.94) GHz, = 0.75, (light gray) ωC = 2π(1.25) GHz, ω0 = 2π(0.625) GHz, = 0.5. The amplitudes of all curves are normalized to the peak of the = 1.25 (black) data. (Bottom) Same as top panel, plotted between [−1, 1] ns. represents the time-domain solution for the off-coneθ-component of the Askaryan electric field. Equation 28 is graphed in Figs. 4 and 5. In Fig. 4 (top), E(t, θ) is shown normalized to the maximum value for the angular range displayed, [θ C + 1.5 • , θ C + 5.5 • ], from t = [−5, 5] ns. Pulses with viewing angles closer to θ C have larger relative amplitudes and shorter pulse widths. Figure 4 (bottom) contains the same results, but for t = [−1.5, 1.5] ns. The pulses are symmetric and all FIG. 4 : 4E(t, θ) vs. tr (Equation 28), normalized. The viewing anlge θ is varied from θC + 1.5 • to θC + 5.5 • in steps of 0.5 • . Top: ω0/(2π) = 1.0 GHz. Bottom: Same as top, zoomed in on central region. FIG. 5 : 5Contours of E(t, θ) vs. θ vs. tr (Equation 28), normalized. The normalization is the same asFig. 4. Although the contour lines extend into the region near θC, Equation 5 is only being evaluated at ∆θ > 1.5 • (see text for details). ≈ 1 , f 0 10≈ 1 GHz, n = 1.78 for solid ice, and c = 0.3 m ns −1 (see Sec. VI A), k 0 ≈ 35 m −1 . Taking r = 1000 m, ∆θ min ≈ 1 • . Simple rules-of-thumb for the application of Equation 28 field are: ∆θ min ≥ 1 • FIG. 6 : 6Fit results: electromagnetic case, θ = θC, EC = 10 PeV. The rows correspond to NuRadioMC waveforms 1-6, 10 PeV electromagnetic cascades. (Left column) The best-fits for f0 and fC. Dashed line: = 1. Solid line: = 2. Gray contour: ρ > 0.95. Black cross: best-fit. (Right column) Best-fit waveforms. Gray: semi-analytic parameterizations from [23]. Black: Equation 15. FIG. 7 : 7The fractional difference in the sum of amplitude differences squared ((∆E) 2 ) versus correlation coefficient (ρ) for waveform 1 at EC = 10 PeV, electromagnetic case. FIG. 8 : 8Fit results: hadronic case, θ = θC, EC = 100 PeV. The six rows (from top to bottom) correspond to Nu-RadioMC waveforms 1-6, 100 PeV hadronic cascades. (Left column) The best-fits for f0 and fC. Dashed line: = 1. Solid line: = 2. Gray contour: ρ > 0.9. Black cross: best-fit. (Right column) The best-fit waveforms. Gray: semi-analytic parameterizations from 5 • , θ C + 10.0 • ] and [0.1, 10] meters, respectively, to maximize ρ. Once the best-fit values for a and θ were determined, (∆E) 2 was minimized by varying f 0 = ω 0 /(2π) and E 0 from [0.3, 3.0] GHz and [0.1, 2.0] V GHz −2 , respectively. The (θ, a) scan and the (f 0 , E 0 ) scan were each separate FIG. 9 : 9Fit results: electromagnetic case, θ = θC, EC = 10 PeV. The six rows (from top to bottom) correspond to NuRadioMC waveforms 1-6, 10 PeV electromagnetic cascades. (Left column) Best-fit θ and a-values. Crosses: best-fits. Circles: MC true values. Gray contour: ρ > 0.95. Dashed line: a versus θ from Equation 29. (Right column)The best-fit waveforms. Gray: semi-analytic parameterizations from[23]. Black: Equation28. FIG. 10 : 10Fit results: hadronic case, θ = θC, EC = 100 PeV. The six rows (from top to bottom) correspond to Nu-RadioMC waveforms 1-6, 100 PeV hadronic cascades. (Left column) Best-fit θ and a-values. Crosses: best-fits. Circles: MC true values. Gray contour: ρ > 0.95. Dashed line: a versus θ from Equation 29 (uncertainty principle). (Right column) The best-fit waveforms. Gray: semi-analytic parameterizations from [23]. Black: Equation 28. # E (u, x) = fġ +ḟ g − (f gḣ + fġh +ḟ gh) (B3) E(1, x) = fġ +ḟ g − (f gḣ + fġh +ḟ gh) \| u=1 (B4)The first-derivatives of f , g, and h, evaluated at u = 1, are given in Tab. IV. Because h(x, 1) = 0, terms proportional to h will vanish. The result iṡ E(1, x) = 1 2 e −y (2y + 2q − 1) The function n(z ) describes the total cascade excess charge, and it has a characteristic width a. The ICD has an instantaneous width much smaller than a[34].a FIG. 1: (a) Side view of the coordinate systems used in the analysis. Spherical unprimed coordinates refer to the observer. Primed cylindrical coordinates refer to J(ρ , z ). (b) Front view of the coordinate system. The instantaneous charge density (ICD) is assumed to have no φ -dependence. (c) TABLE I : IThe parameters used to build Equation 15. Fitted values in comparison to semi-analytic parameterizations are shown in Section VI. TABLE II : IIUseful variables for the derivation of the off-cone Askaryan electromagnetic field.Function Definition TABLE III : IIIUseful functions for the derivation of the off-cone Askaryan electromagnetic field. The last row contains the vector structure of theθ-component of the field. TABLE IV : IVSpecial cases of the functions defined in Table III, when u = 1. inverse Fourier transform of theθ-component gives the time-domain results, after including the expanded E(u, x):The TABLE V : VFit results: electromagnetic case, θ = θC, EC = 10 PeV. The six rows (from top to bottom) correspond to NuRadioMC waveforms 1-6, 10 PeV electromagnetic cas- cades. From left to right, the form-factor cutoff-frequency, coherence cuoff-frequency, energy-scaling normalization, lon- gitudinal length parameter, the best-fit correlation coefficient, and the relative power difference between NuRadioMC semi- analytic parameterization and the fully analytic model. The parameter means and errors in the mean are quoted in the bottom two rows. TABLE VI : VIFitresults: hadronic case, θ = θC, EC = 100 PeV. The six rows (from top to bottom) correspond to Nu- RadioMC waveforms 1-6, 100 PeV hadronic cascades. TABLE VII : VIIFit results: electromagnetic case, θ = θC, EC = 10 PeV. The six rows (from top to bottom) corre- spond to NuRadioMC waveforms 1-6, 10 PeV electromag- netic cascades. Table VIII contains the best-fit parameters, along with ρ and (∆E) 2 . Solutions with ρ ≈ 0.98 and (∆E) 2 ≈ 5 % were found. Similar to the results shown in Table VII, the results in TABLE VIII : VIIIFit results: hadronic case, θ = θC, EC = 100 PeV. The six rows (from top to bottom) correspond to NuRadioMC waveforms 1-6, 10 PeV hadronic cascades. From left to right, the viewing angle, longitudinal length parame- ter, form-factor cutoff frequency, the energy-scaling normal- ization, the lateral width of the cascade, the best-fit corre- lation coefficient, and the relative power difference between NuRadioMC semi-analytic parameterization and the fully an- alytic model. The parameter means and errors in the mean are quoted in the bottom two rows. FIG. 11: The longitudinal length parameter a derived from the Equation 28 best-fit verus the a-value derived from the cascade profile in NuRadioMC. A linear fit and correlation coefficient are shown (slope: 0.83 ± 0.05, intercept: 0.2 ± 0.2 (m), correlation coefficient = 0.97). Result Location r E(tr, θC), on-cone field (θ) Eq. 15, Sec. IV σtσν ≥ 1/(2π), on-cone Eq. 20, Sec. IV A r E(tr, θ), off-cone field (θ) Eq. 28, Sec. V σtσν ≥ 1/(2π), off-cone Eq. 31, Sec. V A On-cone EM comparison to [32] Fig. 6, Tab. V On-cone HAD comparison to [32] Fig. 8, Tab. VI Off-cone EM comparison to [32] Fig. 9, Tab. VII Off-cone HAD comparison to [32] Fig. 10, Tab. VIII TABLE IX : IXA summary of results in this work. VIII. ACKNOWLEDGEMENTSWe would like to thank our families for their support throughout the COVID-19 pandemic. We could not have completed this work without their help. We would also like to thank our colleagues for helpful discussions regarding analysis techniques. In particular, we want to thank Profs. Steve Barwick, Dave Besson, and Christian Glaser for useful discussions. Finally, we would like to thank the Whittier College Fellowships Committee, and specifically the Fletcher-Jones Fellowship Program for providing financial support for this work. This work was partially funded by the Fletcher-Jones Summer Fellowship of 2020, Whittier College Fellowships program.Appendix A: Details of the On-Cone Field Equation DerivationThe original equations for theθ-component of E are: . 0036-8075, 1311.5238Science. 3421242856The IceCube Collaboration, Science 342, 1242856 (2013), ISSN 0036-8075, 1311.5238. . M Ahlers, L Anchordoqui, M Gonzalez-Garcia, F Halzen, S Sarkar, 0927-6505Astroparticle Physics. 34M. Ahlers, L. Anchordoqui, M. Gonzalez-Garcia, F. Halzen, and S. Sarkar, Astroparticle Physics 34, 106 (2010), ISSN 0927-6505. . K Kotera, D Allard, A Olinto, 1475-7516, 1009.1382Journal of Cosmology and Astroparticle Physics. 13K. Kotera, D. 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[ "SAP-DETR: Bridging the Gap between Salient Points and Queries-Based Transformer Detector for Fast Model Convergency", "SAP-DETR: Bridging the Gap between Salient Points and Queries-Based Transformer Detector for Fast Model Convergency" ]	[ "Yang Liu [email protected] \nInstitute of Computing Technology (ICT)\nChinese Academy of Sciences\n\n\nUniversity of Chinese Academy of Sciences\n\n", "Yao Zhang [email protected] \nInstitute of Computing Technology (ICT)\nChinese Academy of Sciences\n\n\nUniversity of Chinese Academy of Sciences\n\n", "Yixin Wang [email protected] \nStanford University\n\n", "Yang Zhang [email protected] \nAI Lab\nLenovo Research\n", "Jiang Tian [email protected] \nAI Lab\nLenovo Research\n", "Zhongchao Shi \nAI Lab\nLenovo Research\n", "Jianping Fan \nAI Lab\nLenovo Research\n", "Zhiqiang He \nInstitute of Computing Technology (ICT)\nChinese Academy of Sciences\n\n\nUniversity of Chinese Academy of Sciences\n\n\nLenovo Ltd\n\n" ]	[ "Institute of Computing Technology (ICT)\nChinese Academy of Sciences\n", "University of Chinese Academy of Sciences\n", "Institute of Computing Technology (ICT)\nChinese Academy of Sciences\n", "University of Chinese Academy of Sciences\n", "Stanford University\n", "AI Lab\nLenovo Research", "AI Lab\nLenovo Research", "AI Lab\nLenovo Research", "AI Lab\nLenovo Research", "Institute of Computing Technology (ICT)\nChinese Academy of Sciences\n", "University of Chinese Academy of Sciences\n", "Lenovo Ltd\n" ]	[]	Recently, the dominant DETR-based approaches apply central-concept spatial prior to accelerating Transformer detector convergency. These methods gradually refine the reference points to the center of target objects and imbue object queries with the updated central reference information for spatially conditional attention. However, centralizing reference points may severely deteriorate queries' saliency and confuse detectors due to the indiscriminative spatial prior. To bridge the gap between the reference points of salient queries and Transformer detectors, we propose SAlient Point-based DETR (SAP-DETR) by treating object detection as a transformation from salient points to instance objects. Concretely, we explicitly initialize a query-specific reference point for each object query, gradually aggregate them into an instance object, and then predict the distance from each side of the bounding box to these points. By rapidly attending to query-specific reference regions and the conditional box edges, SAP-DETR can effectively bridge the gap between the salient point and the query-based Transformer detector with a significant convergency speed. Experimentally, SAP-DETR achieves 1.4× convergency speed with competitive performance and stably promotes the SoTA approaches by ∼1.0 AP. Based on ResNet-DC-101, SAP-DETR achieves 46.9 AP. The code will be released at https: //github.com/liuyang-ict/SAP-DETR.	10.48550/arxiv.2211.02006	[ "https://export.arxiv.org/pdf/2211.02006v2.pdf" ]	253,264,979	2211.02006	774edded0de3f7093246b368597f637cdb1282d6	SAP-DETR: Bridging the Gap between Salient Points and Queries-Based Transformer Detector for Fast Model Convergency Yang Liu [email protected] Institute of Computing Technology (ICT) Chinese Academy of Sciences University of Chinese Academy of Sciences Yao Zhang [email protected] Institute of Computing Technology (ICT) Chinese Academy of Sciences University of Chinese Academy of Sciences Yixin Wang [email protected] Stanford University Yang Zhang [email protected] AI Lab Lenovo Research Jiang Tian [email protected] AI Lab Lenovo Research Zhongchao Shi AI Lab Lenovo Research Jianping Fan AI Lab Lenovo Research Zhiqiang He Institute of Computing Technology (ICT) Chinese Academy of Sciences University of Chinese Academy of Sciences Lenovo Ltd SAP-DETR: Bridging the Gap between Salient Points and Queries-Based Transformer Detector for Fast Model Convergency Recently, the dominant DETR-based approaches apply central-concept spatial prior to accelerating Transformer detector convergency. These methods gradually refine the reference points to the center of target objects and imbue object queries with the updated central reference information for spatially conditional attention. However, centralizing reference points may severely deteriorate queries' saliency and confuse detectors due to the indiscriminative spatial prior. To bridge the gap between the reference points of salient queries and Transformer detectors, we propose SAlient Point-based DETR (SAP-DETR) by treating object detection as a transformation from salient points to instance objects. Concretely, we explicitly initialize a query-specific reference point for each object query, gradually aggregate them into an instance object, and then predict the distance from each side of the bounding box to these points. By rapidly attending to query-specific reference regions and the conditional box edges, SAP-DETR can effectively bridge the gap between the salient point and the query-based Transformer detector with a significant convergency speed. Experimentally, SAP-DETR achieves 1.4× convergency speed with competitive performance and stably promotes the SoTA approaches by ∼1.0 AP. Based on ResNet-DC-101, SAP-DETR achieves 46.9 AP. The code will be released at https: //github.com/liuyang-ict/SAP-DETR. Introduction Object detection is a fundamental task in computer vision, whose target is to recognize and localize each object from input images. In the last decade, various detec-* This work was done when working as an intern at AI Lab, Lenovo Research, Beijing, China. † Corresponding author. tors [6,11,14,18,20,22] based on Convolutional Neural Networks (CNNs), have received widespread attention and made significant progress. Recently, Carion et al. [2] proposed a new end-to-end paradigm for object detection based on the Transformer [24], called DEtection TRansformer (DETR), which treats object detection as a problem of set prediction. In DETR, a set of learnable positional encodings, namely object queries, are employed to aggregate instance features from the context image in Transformer Decoder. The predictions of queries are finally assigned to the ground truth via bipartite matching to achieve end-to-end detection. SAP-DETR (Ours Despite the promising results of DETR, its application is largely limited by considerably longer training time compared to conventional CNNs. To address this problem, many variants attempted to take a close look at query paradigm and introduced various spatial priors for model convergency and efficacy. According to the type of spatial prior, they can be categorized into implicit and explicit methods. The implicit ones [5,16,31] attempt to decouple a reference point from the object query and make use of this spatial prior to attend to the image context features efficiently. The current state-ofthe-arts (SoTAs) are dominated by the explicit ones [13,25], which suggest to instantiate a position with spatial prior for each query, i.e., explicit reference coordinates with a center point or an anchor box. These reference coordinates serve as helpful priors and enable the queries to focus on their expected regions easily. For instance, Anchor DETR [25] introduced an anchor concept (center point with different box size patterns) to formulate the query position and directly regressed the central offsets of the bounding boxes. DAB-DETR [13] further stretched the center point to a 4D anchor box concept [cx, cy, w, h] to refine proposal bonding boxes in a cascaded manner. However, instantiating the query location as a target center may severely degrade the classification accuracy and convergency speed. As illustrated in Fig. 1, there exist many plausible queries [19] with high-quality classification scores ( Fig. 1(a) within red box) and box Intersection over Union (IoU, see the redundant blue boxes in Fig. 1(b) and (c)), which only brings a slight improvement on precision rate but inevitably confuses the detector on the positive query assignments when training with bipartite matching strategy. This is because the plausible predictions are considered in negative classification loss, which severely decelerates the model convergency. As shown in Fig. 1(b) and (c), the predefined reference point of the positive query may not be the nearest one to the center of the ground truth bounding box, and the reference points tend to be centralized or marginalized (cyan arrows in Fig. 1(b)), hence losing the spatial specificity. With further insight into the one-to-one label assignment during the training process, we find that the query, whose reference point is closest to the center point, also has a high-quality IoU, but it still exists a disparity with the positive query in the classification confidence. Therefore, we argue that such a centralized spatial prior may cause degeneration of target consistency in both classification and localization tasks, which leads to inconsistent predictions. Furthermore, the mentioned central point-based variants also have difficulties in detecting occluded objects, because their queries may be assigned to the ambiguous spatial prior with overlapping centers. For example, Fig. 1(d) shows that the baseman in front of the image is detected twice while the other is totally omitted when they are largely overlapped. One solution proposed in Anchor DETR [25] is to predefine different receptive fields (similar to the scaling anchor box in YOLO [17]) for the position of each query. However, increasing the diversity of the receptive fields for each position query is unsuitable for non-overlapped targets, as it still generates massive indistinguishable predictions for one position as same as other center-based models. To bridge these gaps, in this paper, we present a novel framework for Transformer detector, called SAlient Pointbased DETR (SAP-DETR), which treats object detection as a transformation from salient points to instance objects. Instead of regressing the reference point to the target center, we define the reference point belonging to one positive query as a salient point, keep this query-specific spatial prior with a scaling amplitude, and then gradually update them to an instance object by predicting the distance from each side of the bounding box. Specifically, we tile the mesh-grid referenced points and initialize their center/corner as the query-specific reference point. To disentangle the reference sparsity as well as stabilize the training process, a movable strategy with scaling amplitude is applied for reference point adjustment, which prompts queries to consider their reference grid as the salient region to perform image context attention. By localizing each side of the bounding box layer by layer, such query-specific spatial prior enables compensation for the over-smooth/inadequacy problem during center-based detection, thereby vastly promoting model convergency speed. Inspired by [5,13,16], we also take advantage of both Gaussian spatial prior and conditional cross-attention mechanism, and then a salient point enhanced cross-attention mechanism is developed to distinguish the salient region and other conditional extreme regions from the context image features. We bridge the gap between salient points and query-based Transformer detector by speedily attending to the queryspecific region and other conditional regions. The extensive experiments have shown that SAP-DETR achieves superior convergency speed and performance. To the best of our knowledge, this is the first work to introduce the salient point based regression into end-to-end query-based Transformer detectors. Our contributions can be summarized as follows. 1) We introduce the salient point concept into query-based Transformer detectors by assigning query-specific reference points to object queries. Unlike center-based methods, we restrict the reference location and define the point of the positive query as the salient one, hence enlarging the discrepancy of query as well as reducing the redundant predictions (see Fig. 1). Thanks to the efficacy of the query-specific prior, our SAP-DETR accelerates the convergency speed greatly, achieving competitive performance with 30% fewer training epochs. The proposed movable strategy further boosts SAP-DETR to a new SoTA performance. 2) We devise a point-enhanced cross-attention mechanism to imbue query with spatial prior based on both reference point and box sides for final specific region attention. 3) Evaluation over COCO dataset has demonstrated that SAP-DETR achieves superior convergency speed and detection accuracy. Under the same training settings, SAP-DETR outperforms the SoTA approaches with a large margin. Related Work Anchor-Free Object Detectors. Classical anchor-free object detectors can be grouped into center-based and keypointbased approaches. The center-based approaches aim to localize the target objects based on the central locations [11] or predefined ROI [22]. For example, FCOS [22] treated all points within the bounding box as positive ones to predict their distances from each side ([ , t, r, b]), and a centerness score was then considered to prohibit the low-quality prediction whose point is located near the border. Compared with FCOS, we also restrict the candidate queries within the bounding box but treat only one as positive to perform end-to-end object detection via an inner matching cost. The target of keypoint-based approaches is to localize the specific object locations and assign them to the predefined keypoints of the object for box localized training. For instance, diagonal corner points were considered in Corner-Net [8], center point was further grouped into CenterNet [29], and ExtremeNet [30] added some conjectural extreme points for object localization. These works showed an impressive performance, but the complicated keypoint matching may limit their upper bound. Our SAP-DETR takes the advantage of salient point regression to focus on the distinct regions without complicated point-based supervision. Query-Based Transformer Detectors. DETR [2] pioneered a new paradigm of Transformer detector for endto-end object detection without any post-processing [1]. In DETR, a new representation, namely object query, aggregates the instance features and then yields a detection result for each instance object [15]. Following DETR, many votarists put efforts on the optimization of convergency and accuracy. Sun et al. [21] revealed that the main reason for slow convergency of DETR is attributed to the Transformer decoder, and they considered an encoder-only structure to alleviate such a problem. For in-depth understanding of the object query, one way is to generate a series of implicit spatial priors from queries to guide feature aggregation in crossattention layers. SMCA [5] applied a Gaussian-like attention map to augment the query-concerned features spatially. The reference point concept was first introduced by Deformable DETR [31], where the sampling offsets are predicted by each reference point to perform deformable cross-attention. Following such a concept, Conditional DETR [16] reformulated the attention operation and rebuilt positional queries based on the reference points to facilitate extreme region discrimination. Another way is towards position-instantiation explicitly, where this position information enables to directly conduct positional query generation. Anchor DETR [25] utilized a predefined 2D anchor point [cx, cy] to explicitly capitalize on the spatial prior during cross-attention and box regression. DAB-DETR [13] extended such a 2D concept to a 4D anchor box [cx, cy, w, h] and refined it layer-by-layer. Figure 2. Illustration of SAP-DETR. Each object query in SAP-DETR is assigned to a specific grid region and initialized by the corner/center of the grid as its reference point. A learnable 4D coordinate represents the distance from the four sides of the box to the reference point. Both reference points and box sides are served as positional encodings added/concatenated to content embeddings. All embeddings are refined to predict target objects gradually. The recent accelerating convergency methods are based on auxiliary queries for facilitating detector discrimination. DN-DETR [9] demonstrates the slow model convergency is mainly caused by the instability of bipartite matching, thus providing a denoising training to eliminate this issue. DINO [28] inherits this advance and further introduces negative queries to perform contrastive denoising. Group-DETR [4] proposes a group-wise one-to-many label assignment to match multiple positive object queries with more gradients for fast DETR convergency. The most relevant approaches to ours are Point-DETR [3] and SAM-DETR [26,27]. The former applied a point encoder for annotated point label infusion in teacher model, and the latter directly updated content embeddings by extracting salient points from image features for query-image semantic alignment. Unlike these concepts, we redefine the salient point from the perspective of the positive query's position and replace the center-concept prior with the query-specific position, thereby attending extreme regions, differentiating queries' saliency, and alleviating redundant predictions. Method We propose SAP-DETR to bridge the gap between salient points and query-based detectors. Following DETR, the extracted image features are fed into Transformer encoder after adding positional encodings, and then re-aggregated by object queries in Transformer decoder. In pursuit of the query-specific prior, we dispense a movable strategy for each query based on a fixed grid region. The query, whose reference region overlaps with ground truth objects, is allowed to predict the relative offsets from four sides of the bounding box to the points. Given the query-specific reference point and the proposal box sides, we propose salient point enhanced cross-attention mechanism to imbue query with spatial prior, thereby attending to extreme regions effectively. Additionally, we discuss two common issues in DETR-like models and address them for further improvements. Salient Points-Based Object Detection Overview. Previous methods [13,16,25] normally decompose the object query into both content and position embeddings (queries), and form a center-based anchor point/box prior on the position ones. Unlike the central concept, we tile a fixed mesh-grid region, initialize their left-top corner as the reference points with 2D coordinate r = {x, y} ∈ [0, 1] 2 , and instantiate a learnable 4D offset distance s = { , t, r, b} ∈ [0, 1] 4 from the reference point to the sides of proposal bounding box for each object query. The object query can be referred as q = {e; r, s}, where e ∈ R d is the content embedding with d dimension. Instead of regressing the center, width, and height of a bounding box, we follow FCOS [22] and directly supervise the 4D offset from the four sides of a bounding box to the reference point. The final box prediction is formulated asb = {x−ˆ ,ŷ−t,x+r,ŷ+b}. Ideally, we here fix the reference point {x,ŷ} = {x, y} (the movable update strategy is introduced in the next subsection) and only update the 4D box side prediction layer by layer. The prediction for each decoder layer can be calculated by ∆s l = BoxHead l (s l−1 , e l−1 , r l−1 ), s l = σ(σ −1 (s l−1 ) + ∆s l ), s l = Detach(ŝ l ), r l =r l = r l−1 ,b l = {r l −ŝ l [: 2],r l +ŝ l [2 :]},(1) where σ and σ −1 are the sigmoid and inverse sigmoid operation, respectively. ∆s l denotes the side offset prediction.ŝ l , r l , andb l are the predicted side distance, reference points, and box location from the l decoder layer, respectively. The BoxHead l is the prediction head following the layer-l decoder, which is independent between different decoder layers in our settings. Detach operation follows DAB-DETR [13]. During the training process, each query is only allowed to predict the bounding boxes that overlap its reference region. We adapt this rule into the one-to-one bipartite matching process via an inner matching cost L inner . Given N queries Q = {q 1 , q 2 , · · · , q N } and M ground truth objects G = {g 1 , g 2 , · · · , g M }, the L inner (g i , q j ) of each query-box pair is a step function to penalize the reference point r j of q j with value of k when r j is outside the bounding box of g i . We denote i ∈ [1, M ] and j ∈ [1, N ] as the index of query and ground truth, respectively. k can be viewed as a penalty cost, and default to 10 5 . The final permutation of the one-to-one label assignment is formulated as L inner (g i , q j ) := k rj / ∈g i , η = argmin η∈Y N N i L match + L inner ,(2) where L match is the original pair-wise matching cost consist of both classification and localization costs [2]. η ∈ Y N is a permutation of N elements for bipartite matching. Movable Reference Point. Due to the sparseness of the fixed reference point, some small and slender objects may be indistinguishable when there is no reference point inside these objects. Despite the bipartite matching forcing each object to be assigned to one object query, the positive query, whose reference point is outside the assigned bounding box, is unable to accurately regress the distance from each side by a value between 0 and 1. One straightforward solution is to adjust the locations of reference points inside the ground truth bounding boxes to ensure that each object can be detected by an inner reference point. Similar to the aforementioned box refinement, we first perform a movable reference point design to dynamically update the reference points of each query layer by layer. However, such a fullimage point regression inevitably expands the search space as vast variable determinations, causing the final reference point to be trapped in an unexpected corner of the bounding box. To reduce the training search spaces, we scale the offset amplitude of points within their specific grid regions, as illustrated in Fig. 3. Such an operation limits the range of offset values, and hence prevents a large searching space. It is implemented by applying the sigmoid activation σ and multiplying a scale factor s grid whose value equals to the height and width of one grid. The update process of the reference points is formulated as ∆r l = PointHead l (s l−1 , e l−1 , r l−1 ), ∆r l = σ(σ −1 (r l−1 − r 0 ) + ∆r l ),r l = r 0 + ∆r l · s grid , r l = Detach(r l ),(3) where ∆r l and ∆r l are the predicted offsets from r l to both r l−1 and r 0 before the sigmoid activation σ, respectively. Salient Point Enhanced Cross-Attention In cross-attention layers, existing center-based methods are limited to the attention on both center and sides of the ground truth bounding box, causing detector confusion among the queries with the same center and side attention. To this end, we expect the queries to focus on their specific regions based on the reference points, four box sides, and other conditional regions in different heads. Accordingly, we consider an improved Gaussian [5] G and conditional attention [16] A peca to enhance query specificity and spatially extreme region discrimination. The final attention map A cross is the sum of the two attentions A cross = G + A peca . Side Directed Gaussian (SDG). Similar to the movable strategy, we enforce the predicted Gaussian attention to be inside the proposal bounding box to reduce the searching space. Given a reference point r, the offset scales o ∈ [−1, 1] 2 for H heads are produced by a simple MLP with a tanh activation, and then multiply to the two sides of the proposal bounding box for head-specific point offset generation, where the direction is guided by the sign of the offset scales. The head-specific points are generated by Algorithm 1. For each head, the Gaussian-like spatial weight map G i effecting on each pixel (x, y) of context features is then formulated as G i (x, y) = exp − (x − c w,i ) 2 v 2 w,i − (y − c h,i ) 2 v 2 h,i .(4) Point Enhanced Cross-Attention (PECA). As aforementioned, the semantic class for the query is closely related to its referenced location in our SAP-DETR. To enhance the correlation between queries and their references, we concatenate the locations to the content queries after the sinusoidal positional encoding (PE). Take a close look at the conditional attention [16], we find that the linear positional embedding mostly focuses on one box side in each attention head. So we introduce a more straightforward attention mechanism, where the four side coordinates are concatenated and assigned to the corresponding head for side attention. The process of PECA is formulated as A peca = e q e k + TPE(r q )PE(r k ) + Tg(PE(r q −{ , t}, r q +{r, b}))PE(r k ) ,(5) where g is a linear layer mapping PE(4D) into PE(2D) to keep channel dimension consistency. T is a scaling matrix that follows Conditional DETR [16], and more details of T are available in Appendix C. According to the index of direction, predict head-specific point, ci = oi · s[a, b]+r; 5: end for 6: return ci, vi, ∀i = 1, ..., H SAP-DETR with Denoising Strategy To further explore the capability of our proposed SAP-DETR, we develop SAP-DN-DETR and SAP-DINO-DETR by adding the denoising auxiliary loss [9,28] into the training process. In the denoised SAP-DETR, the main difference from both DN-DETR and DINO lies in the noise design. Instead of the center point, we perform the box jittering and randomly sample a point from the intersection region between the original bounding box and the jittering one as the reference point. As the denoising strategy only serves as an auxiliary training loss increasing the training cost, the variants of denoising models are test-free whose Params and GFLOPs are the same as SAP-DETR models. Experimental Results Implementation Details We conduct the experiments on the COCO 2017 [12] object detection dataset, containing about 118K training images and 5K validation images. All models are evaluated by the standard COCO evaluation metrics. We follow the vanilla DETR [2] structure that consists of a CNN backbone, a stack of Transformer encoder-decoder layers, and two prediction heads for class label and bounding box prediction. We use ImageNet-pretrained ResNet [7] as our backbone, and report results based on the ResNet and its ×1/16-resolution extension ResNet-DC. Unlike DAB-DETR [13] sharing box and label head for each layer, we share the class head except the first layer and use an independent box head for the box regression of each layer (for more details please refer to Appendix D). As the mesh-grid initialization for reference points in SAP-DETR, we consider the number of queries N as a perfect square for uniform distribution. Unless otherwise specified, we use N = 400 queries in the experiments. Precisely, we also provide a comparison under N = 300 in Tab. 2, the standard setting in DETR-like models. We adopt two different Transformer structures for experiments where a 3-layer encoder-decoder stack is evaluated to Main Results As shown in Tab. 1 and Tab. 2, we comprehensively compare our proposed SAP-DETR with the traditional CNN detectors [18], the original DETR [2], and other DETR-like detectors [5,13,16,25,26,31] on COCO 2017 validation dataset. For in-depth analysis, we conduct the comparison in two aspects: model convergency and efficacy. Model Convergency. Compared with traditional CNN detectors, Transformer detectors are always subject to laborious training time. For example, under the same 12-epoch training scheme, Faster RCNN [18] still achieves good performance, but the mainstream DETR-like models may suffer from inadequate training and perform poorly without the help of auxiliary losses [9]. Under the 12-epoch training scheme, our proposed SAP-DETR can accelerate model convergency significantly, boosting DAB-DETR [13] by 3.9 AP and 2.7 AP on 3-layer and 6-layer encoder-decoder structures, respectively. Compared with the current SoTA, our SAP-DETR also outperforms SAM-DETR [26] by ∼1.3 AP, with reducing ∼17% parameters and ∼10% GFLOPs. Take a close look at the training process, as illustrated in Fig. 4, SAP-DETR conducts with rapid descent curves in both classification and box regression losses. Notably, there is a large margin in classification loss between ours and SoTA methods, which is benefited from the query-specific reference point, hence boosting model performance in early epochs. Model Efficacy. To analyze model efficacy, we report results on long training epochs and high-resolution features in Tab. 1. Under the 36-epoch training scheme, SAP-DETR achieves superior performance among all single-scale Transformer detectors, especially on middle and large targets. For example, SAP-DETR boosts DAB-DETR by 2.0 AP M and 4.1 AP L with 3-layer models, 1.0 AP M and 1.9 AP L with 6-layer models, which further verifies the effectiveness of our proposed salient point concept for overlapping object detection. Along with layer increase, a deficient upper-bound of SAM-DETR is exposed, with obviously lower 0.5 AP promotion compared to our 1.0 AP improvement. Persuasively, we also report the 50-epoch training results based on the 300-query setting. To align with our mesh-grid initialization strategy, we tile a 17 × 18 mesh-grid (306 queries) for each reference point initialization. Tab. 2 shows our main results and the most representative approaches with their original reported performance. Notably, SAP-DETR outperforms the current SoTAs with comparable costs based on all backbones. With low-resolution features (×1/32), it significantly boosts both middle and large object detection accuracy. Combine with Other Fast Convergency Methods. As shown in Tab. 3, we compare our SAP-DETR variants with the current fast convergency methods [4,9,28]. With such a subtle modification, our SAP-DETR (grey rows) results in a significant performance improvement compared with the original methods (white rows). Under the 12-epoch training scheme, there exist 0.5-1.9 AP improvements on DN-DETR [9] and 0.7-1.6 AP improvements on Group-DETR [4], but the promotions are slightly reduced when implemented on DINO [28]. We hypothesise that there exists the same effect between the negative query of contrastive denoising [28] and our query-salient reference point. Moreover, we observe that the performance improvements largely originate from the large object detection, especially based on ResNet-DC5 family backbones. We speculate that DETR may prefer the high-resolution features (×1/16) rather than the low-resolution ones (×1/32), and our SAP-DETR can distinguish the salient points accurately on the high-resolution, thereby taking full advantage of the large object detection. beddings improves the performance from 32.3 AP to 33.5 AP compared to baseline DAB-DETR (row 8-9). Such a query-specific spatial prior enables queries to attend to their expected region from content features (see Figures in Appendix G) and reduces the false detection rate on occluded and partial objects (see Fig. 1 1 and 2). We argue that the Gaussian-like map of SDG might be easily overlapped with PECA on small objects. Scaling Factor of Movable Strategy. We perform an ablation study on the scaling factor of the movable strategy and further investigate the effectiveness of the inner cost in Tab. 5. Notably, it is observed that there exists a conflict between the inner loss and the global search strategy, behaving a sharp drop when only reserving the inner loss. Furthermore, searching within the grid enables the detector to more attend to small objects and avoid a drastic deterioration in normal object detection. See Appendix E for more detailed analyses. Scaling Factor of SDG. We also compare our side-directed manner with the standard offset prediction method in Tab. 6. Ablation Study Based on PECA, the side-directed scaling factor may limit the detector on small object detection but significantly promote the performance on other objects. This phenomenon would be broken without the help of PECA in which a precipitous decline is emerged on all-scale object detection (row 5 in Tab. 4 vs. row 1 in Tab. 6). We hypothesise that it because the predicted reference points may be outside of the proposal boxes, or even the region of the image. Conclusion In this paper, we propose SAP-DETR for promoting model convergency by treating object detection as a transformation from the salient points to the instance objects. Our SAP-DETR explicitly initializes a query-specific reference point for each object query, gradually aggregates them into an instance object, and predicts the distance from each side of the bounding box. By speedily attending to the queryspecific region and other extreme regions from contextual image features, it thus can effectively bridge the gap between the salient points and the query-based Transformer detector. Our extensive experiments have demonstrated that SAP-DETR achieves superior model convergency speed. With the same training settings, our proposed SAP-DETR outperforms SoTA approaches with large margins. Future Work This point-based design for DETR-like models also comes with remaining issues, in particular regarding training with deformable attention, multi-scale features, and negative query design. Following current center-based methods working for similar issues, we expect future work to successfully address them for point-based design of SAP-DETR. . Table 7. Comparison of DETR-like models and our proposed SAP-DETR. Appendix A. Comparison of DETR Family Tab. 7 detailedly compares various representative properties for the DETR family. DETR [2] follows the vanilla Transformer structure and leverages the learnable positional encodings to help Transformer distinguish paralleled input queries. However, such learnable positional encodings without any spatial prior help severely affect the convergency speed of the Transformer detector. To this end, the mainstream approaches make effort to introduce different spatial prior into DETR, which can be divided into implicit and explicit methods. Specifically, the former decouples reference coordinates from the learnable positional encodings, while the latter directly sets a 2D/4D coordinate for each query and maps such low-dimensional coordinate into a high dimension positional encoding via the sinusoidal PE [24]. From the perspective of the spatial prior indoctrination, a straightforward way for object query is to predict the offset between their reference and the target bounding boxes. For example, previous approaches [5,16,25] only regress the offset of center points, while the current approaches [13,26] directly regress the 4D offset based on the reference coordinate. Another spatial prior indoctrination benefits from the redesign of the cross-attention mechanism. Deformable DETR [31], SMCA [5], and SAM-DETR [26] aggregate multiple extreme point regions from the content features by directly predicting the coordinates of these points from the object queries. Conditional DETR [16] and DAB-DETR [13] utilize a Gaussian-like positional cross-attention map to attend to distinct regions dynamically. Take a close insight at the Gaussian map, the region of box sides and center point are attended by different heads in the multi-head attention mechanism. From the perspective of the spatial prior update, the prevailing approaches [13,26] apply a cascaded way to refine the box prediction as well as update the reference spatial prior. However, all of these methods view center points as the reference spatial prior, eroding the discrimination of the positional encodings during performing the redundant prediction, thereby confusing the Transformer detector as well as leading to the slow model convergency. In our proposed SAP-DETR, such confusing reference spatial prior is replaced by the query-specific reference point. Specifically, each object query in SAP-DETR is assigned a non-overlapping fixed grid-region, which prompts queries to consider the grid area as a salient region to attend to image features and compensate for the over-smooth/inadequacy during center-based detection by localizing each side of the bounding box layer by layer. Considering the sparseness of the reference points, the movable strategy is proposed to enhance small/slender object detection. Therefore, there exists the 2D+4D reference spatial prior in the proposed SAP-DETR, and the final prediction is based on such a 6D reference coordinates ([∆x, ∆y, ∆ ∆t, ∆r, ∆b]). Taking an insight into the Conditional attention mechanism, we investigate that the highlight region is most relevant to four sides of bounding boxes, hence facilitating the final box localization. More intuitively, we devise the PECA to indicate the location of bounding box sides to object queries, where they should attend from context image features. B. Temperature Consistency in PE Following DETR, we also use the 2D sinusoidal function PE(x, y) as positional encoding. Given a position, the PE pos is calculated by PE T pos (i) = sin(pos · ω t ) i = 2t cos(pos · ω t ) i = 2t + 1, ω t = T −2t/d , t = 1, · · · , d/2,(6) where T is an adjustable temperature and i is the channel index of the positional embedding. As shown in Fig. 5, the receptive field size of the positional attention map tends to become wider with increasing temperature [13]. Before the softmax operation, the positional query-to-key similarity A in the cross-attention mechanism is computed by a dotproduct between query position PE Tq pos q and key position PE T k pos k . Clearly, the resulting positional similarity in Fig. 5(a) and (b) subjects to a Gaussian-like distribution. We fix the T1=1000，T2=1000 T1=20，T2=20 T1=10000，T2=20 T1=20，T2=10000 (a) Tq=Tk=20 (b) Tq=Tk=10 4 (c) Tq=20, Tk=10 4 (d) Tq=10 4 , Tk=20 Figure 5. Positional attention maps. Given two sequential PE of query-key pairs, we fix one PE of the query, reshape its sequential attention map for all PE of the key into original 2D image size. Consequently, there exists an offset center for each channel of the positional attention map if T k = T q . Literally, each channel of the positional attention map can be viewed as a superposition by several horizontal and vertical line masks (see Fig. 6). So it is easy to illustrate the offset center and irregular width/height of the positional attention maps as shown in Fig. 5(c) and (d). Without loss of generality, we eliminate the effect of conditional scaling transformation and fix the temperature of encoder's positional encoding to 20. As shown in Tab. 9, the reported results compare the different temperature settings based on PECA. Clearly, both point and box site positional encodings are benefit from a relative small consistent temperature, especially when concatenating with box side PE. C. Scaling Transformation for PE Revisiting Conditional Spatial Query Prediction. Given a set of content queries and their corresponding reference points, the conditional spatial query prediction adaptively maps the reference points into high-dimensional positional embeddings according to a spatial transformation generated by content queries. Let r ∈ R k denotes the 2D unnormalized reference point, e ∈ R d denotes the content query, and T ∈ R d indexes the scaling spatial transformation where d is the query dimension. Then the conditional spatial query prediction is calculated by p q = T · PE(sigmoid(r )), T = FFN(e),(9) where FFN is a feed-forward network consisting of a linear layer, a ReLU activation, and a linear layer. PE is the sinusoidal positional encoding as illustrated in Eq. (6). In Conditional DETR [16], the unnormalized reference point is either a learnable 2D coordinate or generated by its corresponding content query. Scaling Transformation in PECA. As introduced in Section Appendix B, the proposed PECA concatenates both point and box side PEs for conditional spatial cross-attention. Following the scaling transformation of Conditional DETR, we also conduct ablations on different ways of scaling transformation in PECA. The following settings are involved: • Comparing the effectiveness of scaling transformation with and without box side PE concatenation. • Comparing the effectiveness of scaling consistency in both point PE and box side PE, and then considering three types of ablation: no scaling, shared, and independent scaling transformation. • Exploiting a learnable diagonal matrix to transform the positional encoding of the key-vector, which also can be shared between point PE and box side PE. D. Independent Prediction Heads Taking a close insight into the semantic representation of these object queries, we map each query output into a 2D distribution via t-SNE [23]. As shown in Fig. 7, each dot here represents an query output from the decoder layer. It can be seen that the instance objects (blue and green dots in Fig. 7(c)-(f)) whose location at the edge/corner of the distribution are easy to distinguish from the background queries. More precisely, the instance objects, except from the first decoder layer, are at a closer distance than the semantic-close queries. Inspired by this, we employ a dedicated classification head for the first decoder layer and a shared head for the others in the auxiliary training process. Tab. 10 reports the ablation study on the 3-layer encoderdecoder decoder neck. As we can see, the detach operation generally boosts the detector performance by ∼0.3%AP, and the independent box prediction head is conducive to the Transformer detector for further improvements. Moreover, There exists a slight performance drop when using the independent classification prediction head. E. Movable Reference Points We evaluate two types of training strategies for reference points. As illustrated in Fig. 8(a), we tile the mesh-grid reference points for their initialization and set such coordinates as fixed/learnable parameters. By visualizing the learnable reference points in Fig. 8(b), their distribution are observed to be uniform within the image, similar to the learnable anchor points in Anchor DETR [25]. It indicates that the learnable reference coordinates would not be affected by properties of the target regression. We further hypothesize that there exists partial denominators between salient points and the center anchor points, to a certain extent. As introduced in Sec. 3.1, the proposed movable reference points significantly facilitate detecting small and slender objects, which are omitted caused by the sparseness of the reference point distribution. The experiments in Sec. 4 we conduct another ablation on the number of queries to verify that such a vulnerability is attribute to the query sparsity. Fig. 9 describes the performance histogram of 3-layer detectors based on both 12-epoch and 36-epoch training schemes. Without the help of the movable component, the standard SAP-DETR relatively benefits more from the query number growth compared to the counterpart. Along with query number increase, the performance gap is reduced progressively (from 1.2 AP to 0.2 AP), which further verifies our sparsity analysis and the effectiveness of the movable strategy. To further demonstrate the effectiveness of the movable strategy, the update processes of salient points are plotted in Fig. 10 and Fig. 11. Indeed, some small and slender objects can be localized well after moving the reference point within the objects. However, some queries whose reference points are located within the large objects behave an unstable matching result that the matched queries in the latter layers are inconsistent with the previous layers. Hence there exists a slight performance deterioration for large object detection after adding the movable reference points. F. Training Details and More Configurations Warm Up Training Strategy. In the early training process, the bipartite matching in Transformer detectors may appear to be fragile and instable, where the positive label are assigned to one false prediction. This phenomenon is also reported in DN-DETR [9]. Following the conventional training strategy, we conduct a warm-up step during the early training process. In our experiments, we set warm-up steps to 400 and 1000 iterations for 3-layer and 6-layer Encoder-Decoder Transformer detectors, respectively. Detailed Configurations. We list the all configurations in Tab. 11. For each number of query in Appendix E, the batch size of 8 is applied in our 3-layer SAP-DETR. G. Visualization of Attention Maps Visualization of Query-Specific Region. To understand how query-specific reference point affect on the object queries aggregation, we visualize the cross-attention map and the output bounding box for each query based on DAB-DETR and our proposed SAP-DETR in Fig. 12 to Fig. 15 Table 11. All configurations of SAP-DETR Precisely, we visualize the query-specific region in various scenes. For example, the #785 validation image with sample background and sparse instance, the #71226 validation image with complex background and different scale objects, the #1000 validation image with sophisticated instance objects, and the #3255 validation image with sophisticated small instance objects. Compared with redundant prediction and wilderness attention region in DAB-DETR, each query of SAP-DETR only has a compact attention receptive field except for the positive instance query, which benefits from the query-specific reference point and PECA attention mechansim, hence resulting in a superior convergency speed. Visualization of PECA. Fig. 16 visualizes both content and side attention generated by the proposed PECA. For each positive object query, we visualize each head attention map from the cross-attention mechanism. Then we compare them with the conditional spatial cross-attention. All models are based on ResNet-50 and 6-layer encoder-decoder structure under 50 training epochs. Intuitively, our content attention region mostly falls within the foreground content features, whereas a proportion of the head of Conditional DETR focus on the background. For the side attention, the attention maps of Conditional DETR are inaccurate, with several attention regions outside the bounding box. These inaccurate regions make it fail to locate the extremities efficiently and accurately. The visualization proves the effectiveness of PECA for extreme region attention and partial object detection. Figure 1 . 1Comparison of SAP-DETR and DAB-DETR under 36 training epochs. (a) Statistics of the query count in different classification score intervals. (b) and (c) Distribution of reference points and the visualization of the query with top-20 classification score (blue proposal bounding boxes and red reference points) in different decoder layers. (d) Visualization of bounding boxes for positive queries (blue) and ground truth (red) during training process. Figure 3 . 3Movable reference point. The reference points are initialized by the center/corner points of the mesh-grid. Based on the inner loss, only the green dashed box can be predicted by the inner points when reference points are fixed. By moving the reference points within their grid, the blue dashed boxes can be detected accurately without extended searching space. Content embedding e, reference point r and box s. Output: Head-specific points c = {(cw,i, c h,i )\|i ∈ H} and headspecific attention v = {(vw,i, v h,i) \|i ∈ H}. 1: Predict offset scale and attention scale based on content embedding, o = tanh(MLP(e)), v = MLP(e); 2: for h ← 1 ∈ H do 3: Select the index of direction guided by the sign of offset scale, {a, b} = sgn(oi) + {1, 2}, a, b ∈ {0, Figure 4 . 4Comparison of performance and training losses curves. demonstrate our lightweight model efficacy compared with the traditional CNN detectors, and a 6-layer encoder-decoder stack is aligned with previous DETR variants to investigate the performance of large model. Both are trained on two training schemes: the 12-epoch and 36-epoch schemes with a learning rate drop after 11 and 30 epochs, respectively. All models are trained on the Nvidia A100 GPUs with batch size of 16 and 8 for ResNet and ResNet-DC, respectively. For more training details, please refer to Appendix F. Effectiveness of Each Component. To offer an intuitionistic comparison of model convergency for each component, Tab. 4 reports the effectiveness of them based on the 3-layer encoder-decoder structure and 12-epoch training scheme. 1). The proposed salient point concept based on content em- Figure 6 .A 6Positional attention maps in each head. PE Tq pos q and then the center of A is calculated by PE pos q )sin(ω k t pos k )+cos(ω q t pos q )cos(ω k t pos k pos q −ω k t pos k ), pos q , pos k ∈ X = [ t pos q − ω k t pos k ) = (T k /T q ) 2t/d pos q . Figure 7 . 7Visualization of t-SNE. Both grids and slots in t-SNE represent object queries, where the green and blue color are the positive queries, corresponding to the same colored ground truth. Figure 8 . 8Distribution. 1 https:// github.com/facebookresearch/detectron2Table 1. Comparison between Transformer necks. Based on ResNet-50 backbone, all models are trained by the official source codes with their original settings and evaluated on COCO val2017. All models uses 400 queries except Anchor DETR, while Anchor DETR uses 200 queries with 2 pattern embeddings. GFLOPs and Params are measured by Detectron2 1 .Table 2. Comparison of Transformer necks with 300 queries on COCO val2017. All results are reported from their original paper. All models uses 300 queries except Anchor DETR, while Anchor DETR uses 100 queries with 3 pattern embeddings. All inference speeds are measured by a single Nvidia A100 GPU. † denotes the results are measured by ourselves.Method #Epochs #Params(M) GFLOPs AP AP 50 AP 75 AP S AP M AP L 3-Layer Encoder-Decoder Transformer Neck with ResNet-50 Backbone DETR-R50 [2] 36 33 82 15.8 28.0 15.4 5.3 16.7 24.6 Deformable DETR-R50 [31] 36 30 77 37.1 57.6 39.4 18.3 40.8 51.6 SMCA-DETR-R50 [5] 12 / 36 - - 28.8 / 37.7 48.1 / 58.7 29.9 / 40.1 13.8 / 19.4 31.3 / 40.5 41.3 / 54.8 Conditional DETR-R50 [16] 12 / 36 40 82 29.6 / 37.1 48.7 / 57.9 30.7 / 39.0 13.0 / 17.6 32.3 / 40.3 43.1 / 55.0 Anchor DETR-R50 [25] 12 / 36 31 79 30.8 / 37.6 51.1 / 58.7 31.8 / 39.7 14.3 / 18.8 34.1 / 41.5 44.3 / 53.5 DAB-DETR-R50 [13] 12 / 36 34 83 32.3 / 39.0 51.3 / 58.6 34.0 / 41.8 15.7 / 20.0 35.2 / 42.5 45.7 / 56.0 SAM-DETR-w/SMCA-R50 [26] 12 / 36 41 89 35.1 / 40.4 54.7 / 60.7 36.7 / 42.7 16.0 / 20.2 38.4 / 44.4 52.1 / 58.3 SAP-DETR-R50 (Ours) 12 / 36 36 84 36.2 / 41.2 56.2 / 61.6 37.9 / 43.4 16.4 / 21.0 39.5 / 44.5 53.8 / 60.1 6-Layer Encoder-Decoder Transformer Neck with ResNet-50 Backbone DETR-R50 [2] 36 42 89 14.0 24.4 14.0 4.2 13.7 22.5 Deformable DETR-R50 [31] 36 34 81 38.0 58.2 40.4 18.5 41.7 54.2 SMCA-DETR-R50 [5] 12 / 36 - - 32.4 / 40.1 52.3 / 61.4 34.0 / 42.8 15.5 / 20.3 34.9 / 43.3 47.7 / 57.1 Conditional DETR-R50 [16] 12 / 36 44 90 33.1 / 40.2 53.0 / 61.0 34.8 / 42.4 14.5 / 19.9 35.9 / 43.5 49.2 / 58.8 Anchor DETR-R50 [25] 12 / 36 37 85 33.7 / 39.7 54.5 / 60.5 35.1 / 41.9 15.6 / 19.9 37.3 / 43.5 49.8 / 57.3 DAB-DETR-R50 [13] 12 / 36 44 92 34.9 / 41.0 55.5 / 61.7 36.4 / 43.4 16.2 / 21.3 38.4 / 44.7 51.5 / 58.9 SAM-DETR-w/SMCA-R50 [26] 12 / 36 59 105 36.2 / 40.9 57.2 / 62.2 37.4 / 43.1 16.1 / 20.1 39.8 / 44.7 55.3 / 60.7 SAP-DETR-R50 (Ours) 12 / 36 47 94 37.5 / 42.2 58.5 / 62.7 39.2 / 44.6 17.3 / 22.6 40.6 / 45.7 55.4 / 60.8 Method #Epochs #Params(M) GFLOPs AP AP 50 AP 75 AP S AP M AP L Infer. Time(s/img) † ResNet-50 Backbone Faster RCNN-FPN-R50 [10, 18] 108 42 180 42.0 62.1 45.5 26.6 45.5 53.4 0.039 DETR-R50 [2] 500 41 86 42.0 62.4 44.2 20.5 45.8 61.1 0.040 Deformable DETR-R50 [31] 50 34 78 39.4 59.6 42.3 20.6 43.0 55.5 0.043 SMCA-DETR-R50 [5] 50 42 86 41.0 - - 21.9 44.3 59.1 0.045 Conditional DETR-R50 [16] 50 44 90 40.9 61.8 43.3 20.8 44.6 59.2 0.057 Anchor DETR-R50 [25] 50 39 85 42.1 63.1 44.9 22.3 46.2 60.0 0.050 DAB-DETR-R50 [13] 50 44 90 † 42.2 63.1 44.7 21.5 45.7 60.3 0.059 SAM-DETR-w/SMCA-R50 [26] 50 58 100 41.8 63.2 43.9 22.1 45.9 60.9 0.065 SAP-DETR-R50 (Ours) 50 47 92 43.1 63.8 45.4 22.9 47.1 62.1 0.063 ResNet-101 Backbone Faster RCNN-FPN-R101 [10, 18] 108 60 246 44.0 63.9 47.8 27.2 48.1 56.0 0.050 DETR-R101 [2] 500 60 152 43.5 63.8 46.4 21.9 48.0 61.8 0.066 Conditional DETR-R101 [16] 50 63 156 42.8 63.7 46.0 21.7 46.6 60.9 0.070 Anchor DETR-R101 [25] 50 58 150 43.5 64.3 46.6 23.2 47.7 61.4 0.068 DAB-DETR-R101 [13] 50 63 157 † 43.5 63.9 46.6 23.6 47.3 61.5 0.072 SAP-DETR-R101 (Ours) 50 67 158 44.4 64.9 47.1 24.1 48.7 63.1 0.078 DC5-ResNet-50 Backbone DETR-DC5-R50 [2] 500 41 187 43.3 63.1 45.9 22.5 47.3 61.1 0.087 Conditional DETR-DC5-R50 [16] 50 44 195 43.8 64.4 46.7 24.0 47.6 60.7 0.093 Anchor DETR-DC5-R50 [25] 50 39 151 44.2 64.7 47.5 24.7 48.2 60.6 0.069 DAB-DETR-DC5-R50 [13] 50 44 194 † 44.5 65.1 47.7 25.3 48.2 62.3 0.094 SAM-DETR-w/SMCA-DC5-R50 [26] 50 58 210 45.0 65.4 47.9 26.2 49.0 63.3 0.126 SAP-DETR-DC5-R50 (Ours) 50 47 197 46.0 65.5 48.9 26.4 50.2 62.6 0.116 DC5-ResNet-101 Backbone DETR-DC5-R101 [2] 500 60 253 44.9 64.7 47.7 23.7 49.5 62.3 0.101 Conditional DETR-DC5-R101 [16] 50 63 262 45.0 65.6 48.4 26.1 48.9 62.8 0.105 Anchor DETR-DC5-R101 [25] 50 58 227 45.1 65.7 48.8 25.8 49.4 61.6 0.083 DAB-DETR-DC5-R101 [13] 50 63 263 † 45.8 65.9 49.3 27.0 49.8 63.8 0.110 SAP-DETR-DC5-R101 (Ours) 50 67 266 46.9 66.7 50.5 27.9 51.3 64.3 0.130 Table 3 . 3Comparison with denoised methods on COCO dataset based on the 12-epoch training schedule and 300 object queries.Comment Movable Inner Loss PECA SDG AP AP 50 AP 75 AP S AP M AP L SAP-DETR (Ours) 36.2 56.2 37.9 16.4 39.5 53.8 −SDG 35.6 56.2 36.9 16.3 38.9 52.7 −PECA 34.8 55.5 36.0 15.7 37.3 52.0 −PECA & SDG 34.0 54.9 35.3 15.0 36.7 51.5 −Movable 35.2 55.4 36.8 15.8 38.5 53.8 −Inner Loss 35.9 56.3 37.4 16.2 39.3 52.5 DAB-DETR (Baseline) - - - - 32.3 51.3 34.0 15.7 35.2 45.7 +Salient Point Concept - - - - 33.5 54.3 35.1 14.3 36.5 51.0 Table 4. Ablation on each components Inner Cost (L inner ) Movable within Grid (s grid ) AP AP S AP M AP L 35.9 17.0 38.8 52.7 26.3 11.3 28.0 39.5 36.2 16.4 39.5 53.8 Table 5. Ablation on scaling factor of grid PECA Scaling Factor of SDG AP AP S AP M AP L 33.6 14.7 36.0 50.7 35.7 17.5 38.8 52.6 36.2 16.4 39.5 53.8 Table 6. Ablation on scaling factor of SDG unable to localize objects accurately, and this phenomenon always exists in large objects. 4). For salient point enhanced cross-attention, both SDG and PECA serve as the essential components, independently emerging 0.8 AP and 1.6 AP improvements compared to the standard model (rows 2-4). Interestingly, there exists an effectiveness overlap on small objects, with only 0.1 AP S improvement when adding SDG to the equipped PECA model (row Table 8 . 8Ablation study on the scaling transformation of PE.Table 9. Ablation Study on the temperature consistency of PE.Concatenate Box Side PE Temperature of PE AP AP 50 AP 75 AP S AP M AP L T k T qp T qb 20 1000 - 31.8 52.8 32.1 12.8 34.4 50.5 1000 20 - 32.1 53.0 32.5 12.7 35.2 50.6 20 20 - 32.2 53.2 32.7 12.7 35.0 51.0 20 1000 1000 32.2 52.9 32.9 12.8 35.1 51.1 1000 20 20 32.3 52.7 32.8 13.3 35.2 50.8 20 20 20 33.0 53.6 33.5 13.7 36.3 52.1 .2 demonstrate that the performance of small object detection is prompted after applying the movable strategy. Dialectically,Table 10. Ablation study on the independent prediction head.Detach Indep. Prediction Head AP AP 50 AP 75 AP S AP M AP L Head cls Head bbox 34.6 54.8 35.7 14.4 37.6 52.8 34.7 54.8 36.0 16.2 37.6 53.2 34.8 55.0 35.8 15.3 37.8 53.5 34.7 54.8 36.1 14.5 38.1 52.7 35.0 55.1 36.5 15.6 38.3 53.0 34.6 55.1 35.9 14.6 37.9 52.1 35.2 55.4 36.8 15.8 38.5 53.6 35.0 55.2 36.3 14.7 38.2 54.0 .Item Value lr 1e-4 lr backbone 1e-5 weight decay 1e-4 k pe temp 20 q point pe temp 20 q bbox pe temp 20 enc layers 3 / 6 dec layers 3 / 6 dim feedforward 2048 hidden dim 256 dropout 0.0 nheads 8 warm up 1000 batch size 4×4 Item Value mask loss 1 obj loss 1 class loss 1 bbox loss 5 giou loss 2 obj cost 2 class cost 2 class cost 2 bbox cost 5 giou cost 2 inner cost 9999 focal alpha 0.25 transformer activation relu num queries 400 DAB -12 Epochs SAM -12 Epochs SAP w/o Movable -12 Epochs (a) w/o Movable Reference Point (a) w/o Movable Reference Point DAB -12 Epochs SAM -12 Epochs SAP w/o Movable -12 Epochs(a) w/o Movable Reference Point (b) Movable Reference Point Figure 9. Comparison of performance and training losses curves between our purposed SAP-DETR and the current SOTA methods. Reference Point Output Point from Layer #0 Output Point from Layer #5 Figure 10. Movable point update for COCO validation image #3255. Initialized Reference Point Output Point from Layer #0 Output Point from Layer #5 Initialized Reference Point Output Point from Layer #0 Output Point from Layer #5 Figure 11. Movable point update for COCO validation image #14473.Visualization 30 34 36 38 40 300 400 625 900 1225 mAP Number of Queries (b) 6-Layer Encoder-Decoder 42.8 42.7 42.0 38.7 40.9 41.1 40.9 42 34.9 36.2 41.1 36.4 41.2 40.7 DAB -36 Epochs SAM -36 Epochs SAP (ours) -36 Epochs 41.0 mAP 34.0 34.2 34.4 34.6 34.8 34.1 35.0 34.8 w/ SDG w/o SDG Side Directed Gaussian mAP 30 34 36 38 40 300 400 625 900 1225 Number of Queries 40.6 40.4 40.0 39.0 38.7 40.4 40.6 40.9 42 32.3 35.1 35.2 40.9 40.0 DAB -36 Epochs SAM -36 Epochs SAP w/o Movable -36 Epochs 41.3 41.0 40.2 39.7 41.6 31.6 32.4 33.1 34.0 35.9 35.0 35.4 35.6 36.7 36.2 36.0 34.6 mAP 30 34 36 38 40 300 400 625 900 1225 Number of Queries 40.6 40.4 40.0 39.0 38.7 40.4 40.6 40.9 42 32.3 35.1 36.2 40.9 40.0 DAB -36 Epochs SAM -36 Epochs Movable SAP -36 Epochs 41.4 41.3 41.2 40.6 41.6 DAB -12 Epochs SAM -12 Epochs Movable SAP -12 Epochs 31.6 32.4 33.1 34.0 35.9 35.0 35.4 35.6 36.8 35.5 (b) Movable Reference Point 36.8 36.9 Visualization 30 34 36 38 40 300 400 625 900 1225 mAP Number of Queries (b) 6-Layer Encoder-Decoder 42.8 42.7 42.0 38.7 40.9 41.1 40.9 42 34.9 36.2 41.1 36.4 41.2 40.7 DAB -36 Epochs SAM -36 Epochs SAP (ours) -36 Epochs 41.0 mAP 34.0 34.2 34.4 34.6 34.8 34.1 35.0 34.8 w/ SDG w/o SDG Side Directed Gaussian mAP 30 34 36 38 40 300 400 625 900 1225 Number of Queries 40.6 40.4 40.0 39.0 38.7 40.4 40.6 40.9 42 32.3 35.1 35.2 40.9 40.0 DAB -36 Epochs SAM -36 Epochs SAP w/o Movable -36 Epochs 41.3 41.0 40.2 39.7 41.6 31.6 32.4 33.1 34.0 35.9 35.0 35.4 35.6 36.7 36.2 36.0 34.6 mAP 30 34 36 38 40 300 400 625 900 1225 Number of Queries 40.6 40.4 40.0 39.0 38.7 40.4 40.6 40.9 42 32.3 35.1 36.2 40.9 40.0 DAB -36 Epochs SAM -36 Epochs Movable SAP -36 Epochs 41.4 41.3 41.2 40.6 41.6 DAB -12 Epochs SAM -12 Epochs Movable SAP -12 Epochs 31.6 32.4 33.1 34.0 35.9 35.0 35.4 35.6 36.8 35.5 (b) Movable Reference Point 36.8 36.9 Initialized Figure 16. 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[]	[]	[]	[]	A PH N . S. , H eavy Ion Physi cs 5 (1997) 000{000 HProperti es ofsuperdeform ed ssi on i som ers i n the cranked rel ati vi sti c H artree-Bogol i ubov theory.A .V .A fanasjev 1;a and P.R i ng 2 1 Physi k-D epartm entder Techni schen U ni versi t at M unchen, D -85747 G archi ng,G erm any A bstract.T he rotati onaland deform ati on properti esofsuperdeform ed ssi on i som ers i n the A 240 m ass regi on have been i nvesti gated w i thi n the fram ework ofthe cranked rel ati vi sti cH artree-B ogol i ubov theory.T hedependence of the resul ts ofthe cal cul ati ons on the param etri zati on ofthe R M F Lagrangi an has been studi ed. T he rotati onalproperti es are best descri bed by the N L1 force.	10.1556/aph.13.2001.1-3.15	[ "https://export.arxiv.org/pdf/nucl-th/0006033v1.pdf" ]	14,016,645	nucl-th/0006033	719af3c4f205ce857e5aab10e1ed1d2633d35cd2	arXiv:nucl-th/0006033v1 17 Jun 2000 arXiv:nucl-th/0006033v1 17 Jun 2000Received 1 January 1996;revised version 1 January 19970231-4428/97/ $ 5. 00 c 1997 A kad em i aiK i ad o,B udapest A PH N . S. , H eavy Ion Physi cs 5 (1997) 000{000 HProperti es ofsuperdeform ed ssi on i som ers i n the cranked rel ati vi sti c H artree-Bogol i ubov theory.A .V .A fanasjev 1;a and P.R i ng 2 1 Physi k-D epartm entder Techni schen U ni versi t at M unchen, D -85747 G archi ng,G erm any A bstract.T he rotati onaland deform ati on properti esofsuperdeform ed ssi on i som ers i n the A 240 m ass regi on have been i nvesti gated w i thi n the fram ework ofthe cranked rel ati vi sti cH artree-B ogol i ubov theory.T hedependence of the resul ts ofthe cal cul ati ons on the param etri zati on ofthe R M F Lagrangi an has been studi ed. T he rotati onalproperti es are best descri bed by the N L1 force. T heregi on ofA 240 i sthe rstonew herethesuperdeform ed (SD )shapeshave been di scovered experi m ental l y [ 1]i n the ssi on i som ers i n 1962. A ccordi ng to the m easurem entsofthem om entsofi nerti a and thequadrupol em om entsthey areshape i som ersw i th the deform ati on m uch l argerthan the usualground state deform ati on. D espi te the fact that duri ng al m ost 40 years num erous experi m entale orts have been m adeforthestudy of ssi on i som ers,the experi m entaldata on thei rproperti es i n som e respects i s m ore l i m i ted than the one col l ected i n the l ast 15 years i n the regi ons ofsuperdeform ati on at hi gh spi n such as A 60;80;130;150 and 190 [ 2] . R ecent advances i n the experi m entaltechni ques l eadi ng to the observati on of hyperdeform ed (thi rd)m i ni m um [ 3{5]and vi brati onalstructuresi n theSD (second) m i ni m um [ 6{8]revi ved the i nterest to thi s m ass regi on ofsuperdeform ati on. A num ber oftheoreti cali nvesti gati ons has been perform ed i n thi s m ass regi on studyi ng di erent properti es of ssi on i som ers w i thi n the fram eworks ofthe phenom enol ogi calm acroscopi c + m i croscopi c m ethod and non-rel ati vi sti c m i croscopi c theori es based on the zero-range Skyrm e forces and ni te range G ogny forces,see R efs. [ 9{13]and references therei n. R ecentl y, som e properti es of ssi on i som ers i n 226 R a, 232 T h and 240 Pu have been studi ed i n the rel ati vi sti c m ean el d (R M F) theory by the Frankfurt group [ 14,15] . T hese i nvesti gati ons,however,have been perform ed i n the B C S approxi m ati on usi ng a schem ati c treatm entofpai ri ng i n the constant gap approxi m ati on w i th the pai ri ng param eters adjusted to the experim entaldata i n the rst m i ni m um . T he presenti nvesti gati on i sai m ed on a m ore system ati c study ofthe properti es 2 A .V .A fanasjev and P.R i ng ofSD ssi on i som ers w i thi n the fram ework ofR M F theory i n the present state of the art. T he resul ts ofthi s study w i l lbe presented i n a forthcom i ng arti cl e [ 16] . In the present contri buti on,we w i l lconcentrate on the even-even nucl eii n w hi ch rotati onalstructures have been observed so far [ 17] ,nam el y on 236;238 U and 240 Pu nucl ei . Table 1. A s a theoreti caltoolwe are usi ng the recentl y devel oped C ranked R el ati vi sti c H artree-B ogol i ubov (C R H B ) theory [ 18]w hi ch has been very successfuli n the descri pti on ofthe properti es ofSD bands i n the A 190 m ass regi on [ 19,18]and rare-earth nucl ei [ 20] . C om pared w i th previ ous rel ati vi sti c studi es i n the A 240 m ass regi on,i t has the fol l ow i ng advantages: (i) the cranked R M F equati ons are sol ved on the H artree-B ogol i ubov l evel , (ii) the ni te range D 1S G ogny force i s used i n the parti cl e-parti cl e channel ,(iii) approxi m ate parti cl e num ber projecti on i s perform ed by m eans of the Li pki n-N ogam im ethod (further A PN P(LN )), (iv) the cranki ng m odelapproxi m ati on i sem pl oyed w hi ch al l ow sto study the rotati onal properti es of ssi on i som ers and (v) thi s theory i s form ul ated i n three-di m ensi onal C artesi an coordi natesw hi ch al l ow sto study the possi bl e appearance oftri axi aldeform ati on. Si nce thi s study i s concerned w i th the properti es of the states i n the second m i ni m um , onl y re ecti on sym m etri c shapes are consi dered. T hi s approxi m ati on i s justi ed si nce octupol e deform ati on becom es i m portant onl y after the second wel l(see for exam pl e R efs. [ 11,14] ). Q uanti ty N LSH N L3 N L1 N L-Z Exp. (LN /U npr/C R M F) J (1) [ M eV 1 ] T he cal cul ated m om ents of i nerti a of the SD i som ers i n 236 U and 240 Pu are show n i n Tabl e 1 and Fi g. 1. T he change ofthe force from N LSH [ 21]vi a N L3 [ 22] and N L1 [ 23] Table 2. Experi m entaland theoreti calcharge quadrupol e m om ents ofSD ssi on i som ers. T he resul ts of the cal cul ati ons w i th the N L1 and N L3 forces are gi ven. Experi m entaldata for U and Pu i sotopes are taken from R ef. [ 17] ,w hi l e the one for 242 A m from R ef. [ 26] . T he charge quadrupol e m om ents cal cul ated w i th the forces N L1 and N L3 are com pared w i th avai l abl e experi m entaldata i n Tabl e 2. O ne shoul d note that the sm al lerror bars on the experi m entalval ues ofQ 0 gi ven for 238 U and 240 A m nucl ei shoul d be treated w i th cauti on si nce even m odern experi m ents do not provi de an accuracy ofthe absol ute Q 0 val ues better than 15% ,see di scussi on i n R ef. [ 19] . In addi ti on, w hen com pari ng the cal cul ati ons w i th experi m ent one shoul d take i nto account that (i) the Q exp 0 val ues have been obtai ned w i th di erent experi m ental techni ques [ 17] ,(ii) i t i s reasonabl e to expect that an addi ti on ofone neutron to 239 Pu w i l lnotchangeconsi derabl y theQ 0 val ueand thusQ exp 0 ( 239 Pu)coul d beused for com pari son w i th the cal cul ated Q 0 ( 240 Pu). W i th these consi derati ons i n m i nd, i t i s cl ear that the resul ts ofthe cal cul ati ons com e reasonabl y cl ose to experi m ent. In concl usi on,the cranked rel ati vi sti c H artree-B ogol i ubov theory has been appl i ed for the descri pti on ofthe rotati onaland deform ati on properti es of superdeform ed ssi on i som ers i n the A 240 m ass regi on. T hi s theory does not em pl oy any adjustabl e param eters and for the descri pti on ofpai ri ng correl ati ons uses the wel lestabl i shed D 1S G ogny force. T he study ofthe dependence ofthe resul ts on the param etri zati on ofthe R M F Lagrangi an reveal s that rotati onalproperti es are best descri bed by the N L1 force,w hi l e the uncertai nti es ofthe experi m entaldata on charge quadrupol e m om ents do not al l ow to use thi s experi m entalquanti ty for the sel ecti on ofthe best R M F force. Properti es ofsuperdeform ed ssi on . . . K i nem ati c m om ents ofi nerti a (J (1) ),charge quadrupol e (Q 0 ) and m ass hexadecupol e(Q 40 )m om entsofSD band i n 240 Pu cal cul ated atrotati onalfrequency x = 0: 01 M eV w i th di erentparam etri zati onsoftheR M F Lagrangi an.In the case oftheN L1 force,theresul tsoftheC R H B cal cul ati onsw i th and w i thoutA PN P(LN ) [ m arked as ' LN ' and ' U npr' ] and the resul ts of the C R M F cal cul ati ons w i th no pai ri ng [ m arked as ' C R M F' ] are gi ven. For other forces, onl y the resul ts of the C R H B cal cul ati ons w i th A PN P(LN ) are show n. F ig. 1 . 1to N L-Z[ 24]l eadsto the i ncreaseofthe ki nem ati c m om entsofi nerti a, charge quadrupol e and m ass hexadecupol e m om ents (Tabl e 1). T he experi m ental m om entofi nerti a ofthe SD band i n 240 Pu[ 17]i sl ocated between the resul tsofthe cal cul ati onsw i th N L3 and N L1 (seeTabl e 1).T he useofthe N LSH and N L-Z forces for the R M F Lagrangi an l eads to l arger devi ati ons from experi m ent. T he resul ts of the cal cul ati ons w i th N L3 underesti m ate the experi m entalm om ents of i nerti a by 3 5% . T hi s i s al so the case for 238 U w here experi m entaland cal cul ated (w i th Properti es ofsuperdeform ed ssi on . . .3 N L3) m om ents ofi nerti a are 153. 1 and 142. 8 M eV 1 . T he di erence between J (1) m om entsofi nerti a of 236;238 U nucl eii s3. 8 M eV 1 i n experi m entw hi l e i ti sonl y 0. 6 M eV 1 i n the cal cul ati ons w i th N L3. T he N L1 force descri bes the absol ute val ues ofthe m om entsofi nerti a and thei rdi erence i n 236;238 U betterthan N L3. In these nucl ei ,the J (1) val ues cal cul ated at x = 0: 01 M eV are 150. 5 and 154. 7 M eV 1 w hi ch agree very wel lw i th the experi m entalval ues of149. 25 and 153. 06 M eV 1 . O n the other hand,the cal cul ati ons w i th N L1 som ew hat (by 4% ) overesti m ate the experi m entalm om ents ofi nerti a i n 240 Pu (see Fi g.Experi m entaland cal cul ated ki nem ati c(J (1) )and dynam i c(J (2) )m om ents ofi nerti a ofSD ssi on i som ers i n 236 U and 240 Pu. T he notati on ofthe l i nes and sym bol s i s gi ven i n the gure. T he C R H B cal cul ati ons i ndi cate that ki nem ati c and dynam i c m om ents ofi nerti a i ncrease w i th i ncreasi ng rotati onalfrequency x (Fi g. 1). In addi ti on, the di erence between these m om entsgrow sw i th the i ncrease of x . T he experi m ental data i n 240 Pu show s such features,w hi l e they are not seen i n 236 U .T hese features are predom i nantl y due to the gradualal i gnm ent ofthe N = 8 neutrons and N = 7 protonsand a sm ooth decrease ofpai ri ng correl ati onsw i th i ncreasi ng x .T hey are si m i l ar to the ones observed i n the A 190 regi on ofsuperdeform ati on,see R efs. [ 19,18]and references quoted therei n. T he i m portance of A PN P(LN ) for the descri pti on of the rotati onal and deform ati on properti es of SD ssi on i som ers i s cl earl y seen on the exam pl e of the cal cul ati ons perform ed w i th the N L1 force (Tabl e 1). T he C R M F cal cul ati ons [ 25] w i th no pai ri ng gi ve J (1) = 189: 1 M eV 1 w hi ch i s bel ow the ri gi d body m om ent of i nerti a (J rig = 211: 7 M eV 1 ) de ned from the densi ty di stri buti on. T he i ncl usi on ofpai ri ng som ew hatdecreasesthe cal cul ated m om entofi nerti a w hi ch sti l lconsi derabl y exceedsthe experi m entalval ue.O nl y the C R H B resul tsw i th A PN P(LN )com e cl ose to experi m ent. O ne shoul d al so note thati n the case of 240 Pu the i ncl usi on of pai ri ng w i thoutA PN P(LN )hasonl y a m argi nale ecton thechargequadrupol eand m asshexadecupol e m om ents ofi nerti a (Tabl e 1). O n the contrary,A PN P(LN ) has a strong i m pacton these m om entsdecreasi ng thei rval ues.In addi ti on,A PN P(LN ) i ncreases the strength ofpai ri ng correl ati ons especi al l y for the neutron subsystem . In the C R H B theory,the pai ri ng energi esare de ned asE pairin g = 1 2 T r( )[ 18] . T he val ues ofneutron and proton pai ri ng energi es obtai ned i n the C R H B cal cul ati onsw i thoutand w i th A PN P(LN )areE pairin g = 1: 823 M eV ,E pairin g = 10: 030 M eV and E pairin g = 13: 129 M eV ,E pairin g = 13: 603 M eV ,respecti vel y. 5 A 5. V . A .acknow l edges support from the A l exander von H um bol dt Foundati on. T hi s work i s al so supported i n part by the B undesm i ni steri um f ur B i l dung und Forschung under the project 06 T M 979. a. A l exander von H um bol dt fel l ow , on l eave of absence from the Laboratory of R adi ati on Physi cs, Insti tute of Sol i d State Physi cs, U ni versi ty of Latvi a, LV 2169 Sal aspi l s,M i era str.31,Latvi a . 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[ "AIM: An Adaptive and Iterative Mechanism for Differentially Private Synthetic Data", "AIM: An Adaptive and Iterative Mechanism for Differentially Private Synthetic Data" ]	[ "Ryan Mckenna [email protected] \nUniversity of Massachusetts Amherst\nMassachusetts\n", "Brett Mullins [email protected] \nUniversity of Massachusetts Amherst\nMassachusetts\n", "Daniel Sheldon [email protected] \nUniversity of Massachusetts Amherst\nMassachusetts\n", "Gerome Miklau [email protected] \nUniversity of Massachusetts Amherst\nMassachusetts\n" ]	[ "University of Massachusetts Amherst\nMassachusetts", "University of Massachusetts Amherst\nMassachusetts", "University of Massachusetts Amherst\nMassachusetts", "University of Massachusetts Amherst\nMassachusetts" ]	[]	We propose AIM, a novel algorithm for differentially private synthetic data generation. AIM is a workload-adaptive algorithm, within the paradigm of algorithms that first selects a set of queries, then privately measures those queries, and finally generates synthetic data from the noisy measurements. It uses a set of innovative features to iteratively select the most useful measurements, reflecting both their relevance to the workload and their value in approximating the input data. We also provide analytic expressions to bound per-query error with high probability, which can be used to construct confidence intervals and inform users about the accuracy of generated data. We show empirically that AIM consistently outperforms a wide variety of existing mechanisms across a variety of experimental settings.	null	[ "https://arxiv.org/pdf/2201.12677v1.pdf" ]	246,430,835	2201.12677	df2248ea8de1ad2f6f753b07eb086e4d99c146f4	AIM: An Adaptive and Iterative Mechanism for Differentially Private Synthetic Data Ryan Mckenna [email protected] University of Massachusetts Amherst Massachusetts Brett Mullins [email protected] University of Massachusetts Amherst Massachusetts Daniel Sheldon [email protected] University of Massachusetts Amherst Massachusetts Gerome Miklau [email protected] University of Massachusetts Amherst Massachusetts AIM: An Adaptive and Iterative Mechanism for Differentially Private Synthetic Data We propose AIM, a novel algorithm for differentially private synthetic data generation. AIM is a workload-adaptive algorithm, within the paradigm of algorithms that first selects a set of queries, then privately measures those queries, and finally generates synthetic data from the noisy measurements. It uses a set of innovative features to iteratively select the most useful measurements, reflecting both their relevance to the workload and their value in approximating the input data. We also provide analytic expressions to bound per-query error with high probability, which can be used to construct confidence intervals and inform users about the accuracy of generated data. We show empirically that AIM consistently outperforms a wide variety of existing mechanisms across a variety of experimental settings. INTRODUCTION Differential privacy [15] has grown into the preferred standard for privacy protection, with significant adoption by both commercial and governmental enterprises. Many common computations on data can be performed in a differentially private manner, including aggregates, statistical summaries, and the training of a wide variety predictive models. Yet one of the most appealing uses of differential privacy is the generation of synthetic data, which is a collection of records matching the input schema, intended to be broadly representative of the source data. Differentially private synthetic data is an active area of research [1,2,5,11,12,19,25,27,29,30,43,45,46,[48][49][50][52][53][54][55] and has also been the basis for two competitions, hosted by the U.S. National Institute of Standards and Technology [40]. Private synthetic data is appealing because it fits any data processing workflow designed for the original data, and, on its face, the user may believe they can perform any computation they wish, while still enjoying the benefits of privacy protection. Unfortunately it is well-known that there are limits to the accuracy that can be provided by synthetic data, under differential privacy or any other reasonable notion of privacy [14]. As a consequence, it is important to tailor synthetic data to some class of tasks, and this is commonly done by asking the user to provide a set of queries, called the workload, to which the synthetic data can be tailored. However, as our experiments will show, existing workload-aware techniques often fail to outperform workloadagnostic mechanisms, even when evaluated specifically on their target workloads. Not only do these algorithms fail to produce accurate synthetic data, they provide no way for end-users to detect the inaccuracy. As a result, in practical terms, differentially private synthetic data generation remains an unsolved problem. In this work, we advance the state-of-the-art of differentially private synthetic data in two key ways. First, we propose a novel workload-aware mechanism that offers lower error than all competing techniques. Second, we derive analytic expressions to bound the per-query error of the mechanism with high probability. Our mechanism, AIM, follows the select-measure-generate paradigm, which can be used to describe many prior approaches. 1 Mechanisms following this paradigm first select a set of queries, then measure those queries in a differentially private way (through noise addition), and finally generate synthetic data consistent with the noisy measurements. We leverage Private-PGM [37] for the generate step, as it provides a robust and efficient method for combining the noisy measurements into a single consistent representation from which records can be sampled. The low error of AIM is primarily due to innovations in the select stage. AIM uses an iterative, greedy selection procedure, inspired by the popular MWEM algorithm for linear query answering. We define a highly effective quality score which determines the private selection of the next best marginal to measure. Through careful analysis, we define a low-sensitivity quality score that is able to take into account: (i) how well the candidate marginal is already estimated, (ii) the expected improvement measuring it can offer, (iii) the relevance of the marginal to the workload, and (iv) the available privacy budget. This novel quality score is accompanied by a host of other algorithmic techniques including adaptive selection of rounds and budget-per-round, intelligent initialization, and novel set of candidates from which to select. In conjunction with AIM, we develop new techniques to quantify uncertainty in query answers derived from the generated synthetic data. The problem of error quantification for data independent mechanisms like the Laplace or Gaussian mechanism is trivial, as they provide unbiased answers with known variance to all queries. The problem is considerably more challenging for data-dependent mechanisms like AIM, where complex post-processing is performed and only a subset of workload queries have unbiased answers. Some mechanisms, like MWEM, provide theoretical guarantees on their worst-case error, under suitable assumptions. However, this is an a priori bound on error obtained from a theoretical analysis of the mechanism under worst-case datasets. Instead, we develop an a posteriori error analysis, derived from the intermediate differentially private measurements used to produce the synthetic data. Our error estimates therefore reflect the actual execution of AIM on the input data, but do not require any additional privacy budget for their calculation. Formally, our guarantees represent one-sided confidence intervals, and we refer to them simply as "confidence bounds". To our knowledge, AIM is the only differentially private synthetic data generation mechanism that provides this kind of error quantification. This paper makes the following contributions: (1) In Section 3, we assess the prior work in the field, characterizing different approaches via key distinguishing elements and limitations, which brings clarity to a complex space. (2) In Section 4, we propose AIM, a new mechanism for synthetic data generation that is workload-aware (for workloads consisting of weighted marginals) as well as data-aware. (3) In Section 5, we derive analytic expressions to bound the perquery error of AIM with high probability. These expressions can be used to construct confidence bounds. (4) In Section 6, we conduct a comprehensive empirical evaluation, and show that AIM consistently outperforms all prior work, improving error over the next best mechanism by 1.6× on average, and up to 5.7× in some cases. BACKGROUND In this section we provide relevant background and notation on datasets, marginals, and differential privacy required to understand this work. Data, Marginals, and Workloads Data. A dataset is a multiset of records, each containing potentially sensitive information about one individual. Each record ∈ is a -tuple ( 1 , . . . , ). The domain of possible values for is denoted by Ω , which we assume is finite and has size \|Ω \| = . The full domain of possible values for is thus Ω = Ω 1 × · · · × Ω which has size = . We use D to denote the set of all possible datasets, which is equal to ∪ ∞ =0 Ω . Marginals. A marginal is a central statistic to the techniques studied in this paper, as it captures low-dimensional structure common in high-dimensional data distributions. A marginal for a set of attributes is essentially a histogram over : it is a table that counts the number of occurrences of each ∈ Ω . Definition 1 (Marginal). Let ⊆ [ ] be a subset of attributes, Ω = ∈ Ω , = \|Ω \|, and = ( ) ∈ . The marginal on is a vector ∈ R , indexed by domain elements ∈ Ω , such that each entry is a count, i.e., [ ] = ∈ 1[ = ]. We let : D → R denote the function that computes the marginal on , i.e., = ( ). In this paper, we use the term marginal query to denote the function , and marginal to denote the vector of counts = ( ). With some abuse of terminology, we will sometimes refer to the attribute subset as a marginal query as well. Workload. A workload is a collection of queries the synthetic data should preserve well. It represents the measure by which we will evaluate utility of different mechanisms. We want our mechanisms to take a workload as input, and adapt intelligently to the queries in it, providing synthetic data that is tailored to the queries of interest. In this work, we focus on the special (but common) case where the workload consists of a collection of weighted marginal queries. Our utility measure is stated in Definition 2. Definition 2 (Workload Error). A workload consists of a list of marginal queries 1 , . . . , where ⊆ [ ], together with associated weights ≥ 0. The error of a synthetic datasetˆis defined as: Error( ,ˆ) = 1 · \| \| ∑︁ =1 ( ) − (ˆ) 1 Differential privacy Differential privacy protects individuals by bounding the impact any one individual can have on the output of an algorithm. This is formalized using the notion of neighboring datasets. Two datasets , ′ ∈ D are neighbors (denoted ∼ ′ ) if ′ can be obtained from by adding or removing a single record. Definition 3 (Differential Privacy). A randomized mechanism M : D → R satisfies ( , )-differential privacy (DP) if for any neighboring datasets ∼ ′ ∈ D, and any subset of possible outputs ⊆ R, Pr[M ( ) ∈ ] ≤ exp( ) Pr[M ( ′ ) ∈ ] + . A key quantity needed to reason about the privacy of common randomized mechanisms is the sensitivity, defined below. Definition 4 (Sensitivity). Let : D → R be a vector-valued function of the input data. The 2 sensitivity of is Δ( ) = max ∼ ′ ∥ ( ) − ( ′ )∥ 2 . It is easy to verify that the 2 sensitivity of any marginal query is 1, regardless of the attributes in . This is because one individual can only contribute a count of one to a single cell of the output vector. Below we introduce the two building block mechanisms used in this work. Definition 5 (Gaussian Mechanism). Let : D → R be a vectorvalued function of the input data. The Gaussian Mechanism adds i.i.d. Gaussian noise with scale Δ( ) to each entry of ( ). That is, M ( ) = ( ) + Δ( )N (0, I), where I is a × identity matrix. Definition 6 (Exponential Mechanism). Let : D → R be quality score function defined for all ∈ R and let ≥ 0 be a real number. Then the exponential mechanism outputs a candidate ∈ R according to the following distribution: Pr[M ( ) = ] ∝ exp 2Δ · ( ) , where Δ = max ∈R Δ( ). Our algorithm is defined using zCDP, an alternate version of differential privacy definition which offers beneficial composition properties. We convert to ( , ) guarantees when necessary. Definition 7 (zero-Concentrated Differential Privacy (zCDP)). A randomized mechanism M is -zCDP if for any two neighboring datasets and ′ , and all ∈ (1, ∞), we have: (M ( ) \|\| M ( ′ )) ≤ · , where is the Rényi divergence of order . Proposition 1 (zCDP of the Gaussian Mechanism [6]). The Gaussian Mechanism satisfies 1 2 2 -zCDP. Proposition 2 (zCDP of the Exponential Mechanism [10]). The Exponential Mechanism satisfies 2 8 -zCDP. We rely on the following propositions to reason about multiple adaptive invocations of zCDP mechanisms, and the translation from zCDP to ( , )-DP. The proposition below covers 2-fold adaptive composition of zCDP mechanisms, and it can be inductively applied to obtain analogous k-fold adaptive composition guarantees. Proposition 3 (Adaptive Composition of zCDP Mechanisms [6]). Let M 1 : D → R 1 be 1 -zCDP and M 2 : D ×R 1 → R 2 be 2 -zCDP. Then the mechanism M = M 2 ( , M 1 ( )) is ( 1 + 2 )-zCDP. Proposition 4 (zCDP to DP [9]). If a mechanism M satisfies -zCDP, it also satisfies ( , )-differential privacy for all ≥ 0 and = min >1 exp ( − 1)( − ) − 1 1 − 1 . Private-PGM An important component of our approach is a tool called Private-PGM [34,35,37]. For the purposes of this paper, we will treat Private-PGM as a black box that exposes an interface for solving subproblems important to our mechanism. We briefly summarize Private-PGM and three core utilities it provides. Private-PGM consumes as input a collection of noisy marginals of the sensitive data, in the format of a list of tuples (˜, , ) for = 1, . . . , , wherẽ = ( ) + N (0, 2 I). 2 Distribution Estimation. At the heart of Private-PGM is an optimization problem to find a distributionˆthat "best explains" the noisy observations˜: ∈ argmin ∈S ∑︁ =1 1 ( ) −˜ 2 2 Here S = { \| ( ) ≥ 0 and ∈Ω ( ) = } is the set of (scaled) probability distributions over the domain Ω. 3 When˜are corrupted with i.i.d. Gaussian noise, this is exactly a maximum likelihood estimation problem [34,35,37]. In general, convex optimization over the scaled probability simplex is intractable for the high-dimensional domains we are interested in. Private-PGM overcomes this curse of dimensionality by exploiting the fact that the objective only depends on through its marginals. The key observation is that one of the minimizers of this problem is a graphical modelˆ. The parameters provide a compact representation of the distribution that we can optimize efficiently. Junction Tree Size. The time and space complexity of Private-PGM depends on the measured marginal queries in a nuanced way, the main factor being the size of the junction tree implied by the measured marginal queries [35,36]. While understanding the junction tree construction is not necessary for this paper, it is important to note that Private-PGM exposes a callable function JT-SIZE( 1 , . . . , ) that can be invoked to check how large a junction tree is. JT-SIZE is measured in megabytes, and the runtime of distribution estimation is roughly proportional to this quantity. If 2 Private-PGM is more general than this, but this is the most common setting. 3 When using unbounded DP, is sensitive and therefore we must estimate it. Algorithm 1 MWEM+PGM Input: Dataset , workload , privacy parameter , rounds Output: Synthetic DatasetÎ nitializeˆ0 = Uniform[X] = 2 √︁ / = √︁ / for = 1, . . . , do select ∈ using exponential mechanism with budget: ( ) = ∥ ( ) − (ˆ− 1 ) ∥ 1 − measure marginal on : = ( ) + N (0, 2 I) estimate data distribution using Private-PGM: = argmin ∈ ∑︁ =1 ( ) − 2 2 end for generate synthetic dataˆusing Private-PGM: returnâ rbitrary marginals are measured, JT-SIZE can grow out of control, no longer fitting in memory, and leading to unacceptable runtime. Synthetic Data Generation. Given an estimated modelˆ, Private-PGM implements a routine for generating synthetic tabular data that approximately matches the given distribution. It achieves this with a randomized rounding procedure, which is a lower variance alternative to sampling fromˆ [35]. PRIOR WORK ON SYNTHETIC DATA In this section we survey the state of the field, describing basic elements of a good synthetic data mechanism, along with novelties of more sophisticated mechanisms. We focus our attention on marginal-based approaches to differentially private synthetic data in this section, as these have generally seen the most success in practical applications. These mechanisms include PrivBayes [52], PrivBayes+PGM [37], MWEM+PGM [37], MST [35], PrivSyn [55], RAP [3], GEM [32], and PrivMRF [8]. We will review other related work in Section 8. We will begin with a formal problem statement: Problem 1 (Workload Error Minimization). Given a workload , our goal is to design an ( , )-DP synthetic data mechanism M : D → D such that the expected error defined in Definition 2 is minimized. The Select-Measure-Generate Paradigm We begin by providing a broad overview of the basic approach employed by many differentially private mechanisms for synthetic data. These mechanisms all fit naturally into the select-measuregenerate framework. This framework represents a class of mechanisms which can naturally be broken up into 3 steps: (1) select a set of queries, (2) measure those queries using a noise-addition mechanism, and (3) generate synthetic data that explains the noisy measurements well. We consider iterative mechanisms that alternate between the select and measure step to be in this class as well. Mechanisms within this class differ in their methodology for selecting queries, the noise mechanism used, and the approach to generating synthetic data from the noisy measurements. MWEM+PGM, shown in Algorithm 1, is one mechanism from this class that serves as a concrete example as well as the starting point for our improved mechanism, AIM. As the name implies, MWEM+PGM is a scalable instantiation of the well-known MWEM algorithm [22] for linear query answering, where the multiplicative weights (MW) step is replaced by a call to Private-PGM. It is a greedy, iterative mechanism for workload-aware synthetic data generation, and there are several variants. One variant is shown in Algorithm 1. The mechanism begins by initializing an estimate of the joint distribution to be uniform over the data domain. Then, it runs for rounds, and in each round it does three things: (1) selects (via the exponential mechanism) a marginal query that is poorly approximated under the current estimate, (2) measures the selected marginal using the Gaussian mechanism, and (3) estimates a new data distribution (using Private-PGM) that explains the noisy measurements well. After rounds, the estimated distribution is used to generate synthetic tabular data. In the subsequent subsections, we will characterize existing mechanisms in terms of how they approach these different aspects of the problem. Basic Elements of a Good Mechanism In this section we outline some basic criteria reasonable mechanisms should satisfy to get good performance. These recommendations primarily apply to the measure step. Measure Entire Marginals. Marginals are an appealing statistic to measure because every individual contributes a count of one to exactly one cell of the marginal. As a result, we can measure every cell of ( ) at the same privacy cost of measuring a single cell. With a few exceptions ( [3,32,48]), existing mechanisms utilize this property of marginals or can be extended to use it. The alternative of measuring a single counting query at a time sacrifices utility unnecessarily. Use Gaussian Noise. Back of the envelope calculations reveal that if the number of measurements is greater than roughly log (1/ ) + , which is often the case, then the standard deviation of the required Gaussian noise is lower than that of the Laplace noise. Many newer mechanisms recognize this and use Gaussian noise, while older mechanisms were developed with Laplace noise, but can easily be adapted to use Gaussian noise instead. Use Unbounded DP. For fixed ( , ), the required noise magnitude is lower by a factor of √ 2 when using unbounded DP (add / remove one record) over bounded DP (modify one record). This is because the 2 sensitivity of a marginal query is 1 under unbounded DP, and √ 2 under bounded DP. Some mechanisms like MST, PrivSyn, and PrivMRF use unbounded DP, while other mechanisms like RAP, GEM, and PrivBayes use bounded DP. We remark that these two different definitions of DP are qualitatively different, and because of that, the privacy parameters have different interpretations. The √ 2 difference could be recovered in bounded DP by increasing the privacy budget appropriately. Distinguishing Elements of Existing Work Beyond the basics, different mechanisms exhibit different novelties, and understanding the design considerations underlying the existing work can be enlightening. We provide a simple taxonomy of this space in Table 1 in terms of four criteria: workload-, data-, budget-, and efficiency-awareness. These characteristics primarily pertain to the select step of each mechanism. Workload-awareness. Different mechanisms select from a different set of candidate marginal queries. PrivBayes and PrivMRF, for example, select from a particular subset of -way marginals, determined from the data. Other mechanisms, like MST and PrivSyn, restrict the set of candidates to 2-way marginal queries. On the other end of the spectrum, the candidates considered by MWEM+PGM, RAP, and GEM, are exactly the marginal queries in the workload. This is appealing, since these mechanisms will not waste the privacy budget to measure marginals that are not relevant to the workload, however we show the benefit of extending the set of candidates beyond the workload. Data-awareness. Many mechanisms select marginal queries from a set of candidates based on the data, and are thus data-aware. For example, MWEM+PGM selects marginal queries using the exponential mechanism with a quality score function that depends on the data. Independent, Gaussian, and HDMM+PGM are the exceptions, as they always select the same marginal queries no matter what the underlying data distribution is. Budget-awareness. Another aspect of different mechanisms is how well do they adapt to the privacy budget available. Some mechanisms, like PrivBayes, PrivSyn, and PrivMRF recognize that we can afford to measure more (or larger) marginals when the privacy budget is sufficiently large. When the privacy budget is limited, these mechanisms recognize that fewer (and smaller) marginals should be measured instead. In contrast, the number and size of the marginals selected by mechanisms like MST, MWEM+PGM, RAP, and GEM does not depend on the privacy budget available. 4 Efficiency-awareness. Mechanisms that build on top of Private-PGM must take care when selecting measurements to ensure JT-SIZE remains sufficiently small to ensure computational tractability. Among these, PrivBayes+PGM, MST, and PrivMRF all have built-in heuristics in the selection criteria to ensure the selected marginal queries give rise to a tractable model. Gaussian, HDMM+PGM and MWEM+PGM have no such safeguards, and they can sometimes select marginal queries that lead to intractable models. In the extreme case, when the workload is all 2-way marginals, Gaussian selects all 2-way marginals, model required for Private-PGM explodes to the size of the entire domain, which is often intractable. Mechanisms that utilize different techniques for post-processing noisy marginals into synthetic data, like PrivSyn, RAP, and GEM, do not have this limitation, and are free to select from a wider collection of marginals. While these methods do not suffer from this particular limitation of Private-PGM, they have other pros and cons which were surveyed in a recent article [34]. Summary. With the exception of our new mechanism AIM, no mechanism listed in Table 1 is aware of all four factors we discussed. Mechanisms that do not have four checkmarks in Table 1 are not necessarily bad, but there are clear ways in which they can be improved. Conversely, mechanisms that have more checkmarks than other mechanisms are not necessarily better. For example, RAP has 3 checkmarks, but as we show in Section 6, it does not consistently beat Independent, which only has 1 checkmark. Other Design Considerations Beyond these four characteristics summarized in the previous section, different methods make different design decisions that are relevant to mechanism performance, but do not correspond to the four criteria discussed in the previous section. In this section, we summarize some of those additional design considerations. Selection method. Some mechanisms select marginals to measure in a batch, while other mechanisms select them iteratively. Generally speaking, iterative methods like MWEM+PGM, RAP, GEM, and PrivMRF are preferable to batch methods, because the selected marginals will capture important information about the distribution that was not effectively captured by the previously measured marginals. On the other hand, PrivBayes, MST, and PrivSyn select all the marginals before measuring any of them. It is not difficult to construct examples where a batch method like PrivSyn has suboptimal behavior. For example, suppose the data contains three perfectly correlated attributes. We can expect iterative methods to capture the distribution after measuring any two 2-way marginals. On the other hand, a batch method like PrivSyn will determine that all three 2-way marginals need to be measured. Budget split. Every mechanism in this discussion, except for PrivSyn, splits the privacy budget equally among selected marginals. This is a simple and natural thing to do, but it does not account for the fact that larger marginals have smaller counts that are less robust to noise, requiring a larger fraction of the privacy budget to answer accurately. PrivSyn provides a simple formula for dividing privacy budget among marginals of different sizes, but this approach is inherently tied to their batch selection methodology. It is much less clear how to divide the privacy budget within a mechanism that uses an iterative selection procedure. Algorithm 2 Initialize (subroutine of Algorithm 4) 1: for ∈ { ∈ + \| \| \| = 1} do 2: = + 1 ← 0 ← 3:˜= ( ) + N (0, 2 I) 4: ← + 1 2 2 5: end for 6:ˆ= argmin ∈ =1 1 ( ) −˜ 2 2 Algorithm 3 Budget annealing (subroutine of Algorithm 4) 1: if (ˆ) − (ˆ− 1 ) 1 ≤ √︁ 2/ · · then 2: +1 ← 2 · 3: +1 ← /2 4: else 5: +1 ← 6: +1 ← 7: end if 8: if ( − ) ≤ 2 1 2 2 +1 + 1 8 2 +1 then 9: +1 = √︁ 8 · (1 − ) · ( − ) 10: +1 = √︁ 1/(2 · · ( − )) 11: end if Hyperparameters. All mechanisms have some hyperparameters than can be tuned to affect the behavior of the mechanism. Mechanisms like PrivBayes, MST, PrivSyn, and PrivMRF have reasonable default values for these hyperparameters, and these mechanisms can be expected to work well out of the box. On the other hand, MWEM+PGM, RAP, and GEM have to tune the number of rounds to run, and it is not obvious how to select this a priori. While the open source implementations may include a default value, the experiments conducted in the respective papers did not use these default values, in favor of non-privately optimizing over this hyperparameter for each dataset and privacy level considered [3,32]. AIM: AN ADAPTIVE AND ITERATIVE MECHANISM FOR SYNTHETIC DATA While MWEM+PGM is a simple and intuitive algorithm, it leaves significant room for improvement. Our new mechanism, AIM, is presented in Algorithm 4. In this section, we describe the differences between MWEM+PGM and AIM, the justifications for the relevant design decisions, as well as prove the privacy of AIM. Intelligent Initialization. In Line 7 of AIM, we spend a small fraction of the privacy budget to measure 1-way marginals in the set of candidates. Estimatingˆfrom these noisy marginals gives rise to an independent model where all 1-way marginals are preserved well, and higher-order marginals can be estimated under an independence assumption. This provides a far better initialization than the default uniform distribution while requiring only a small fraction of the privacy budget. New Candidates. In Line 13 of AIM, we make two notable modifications to the candidate set that serve different purposes. Specifically, the set of candidates is a carefully chosen subset of +1 ← 0 +1 ← √︁ 8(1 − ) / 10: while < do 11: = + 1 12: ← + 1 8 2 + 1 2 2 13: = { ∈ + \| JT-SIZE( 1 , . . . , )) ≤ · MAX-SIZE} 14: select ∈ using the exponential mechanism with: ( ) = ∥ ( ) − (ˆ− 1 )∥ 1 − √︁ 2/ · · 15: measure marginal on : = ( ) + N (0, 2 I) 16: estimate data distribution using Private-PGM: = argmin ∈ ∑︁ =1 1 ( ) −˜ 2 2 17: anneal +1 and +1 using Algorithm 3 18: end while 19: generate synthetic dataˆfromˆusing Private-PGM 20: returnt he marginal queries in the downward closure of the workload. The downward closure of the workload is the set of marginal queries whose attribute sets are subsets of some marginal query in the workload, i.e., + = { \| ⊆ , ∈ }. Using the downward closure is based on the observation that marginals with many attributes have low counts, and answering them directly with a noise addition mechanism may not provide an acceptable signal to noise ratio. In these situations, it may be better to answer lower-dimensional marginals, as these tend to exhibit a better signal to noise ratio, while still being useful to estimate the higher-dimensional marginals in the workload. We filter candidates from this set that do not meet a specific model capacity requirement. Specifically, the set will only consist of candidates that, if selected, ill lead to a JT-SIZE below a prespecified limit (the default is 80 MB). This ensures that AIM will never select candidates that lead to an intractable model, and hence allows the mechanism to execute consistently with a predictable memory footprint and runtime. Better Selection Criteria. In Line 14 of AIM, we make two modifications to the quality score function for marginal query selection to better reflect the utility we expect from measuring the selected marginal. In particular, our new quality score function is ( ) = ∥ ( ) − ( −1 ) ∥ 1 − √︁ 2/ · · ,(1) which differs from MWEM+PGM's quality score function ( ) = ∥ ( ) − ( −1 ) ∥ − in two ways. First, the expression inside parentheses can be interpreted as the expected improvement in 1 error we can expect by measuring that marginal. It consists of two terms: the 1 error under the current model minus the expected 1 error if it is measured at the current noise level (Theorem 5 in Appendix B). Compared to the quality score function in MWEM+PGM, this quality score function penalizes larger marginals to a much more significant degree, since ≫ 1 in most cases. Moreover, this modification makes the selection criteria "budget-adaptive", since it recognizes that we can afford to measure larger marginals when is smaller, and we should prefer smaller marginals when is larger. Second, we give different marginal queries different weights to capture how relevant they are to the workload. In particular, we weight the quality score function for a marginal query using the formula = ∈ \| ∩ \|, as this captures the degree to which the marginal queries in the workload overlap with . In general, this weighting scheme places more weight on marginals involving more attributes. Note that now the sensitivity of is rather than 1. When applying the exponential mechanism to select a candidate, we must either use Δ = max ∈ , or invoke the generalized exponential mechanism instead, as it can handle quality score functions with varying sensitivity [39]. This quality score function exhibits an interesting trade-off: the penalty term √︁ 2/ discourages marginals with more cells, while the weight favors marginals with more attributes. However, if the inner expression is negative, then the larger weight will make it more negative, and much less likely to be selected. Adaptive Rounds and Budget Split. In Lines 12 and 17 of AIM, we introduce logic to modify the per-round privacy budget as execution progresses, and as a result, eliminate the need to provide the number of rounds up front. This makes AIM hyper-parameter free, relieving practitioners from that often overlooked burden. Specifically, we use a simple annealing procedure (Algorithm 3) that gradually increases the budget per round when an insufficient amount of information is learned at the current per-round budget. The annealing condition is activated if the difference between (ˆ) and (ˆ− 1 ) is small, which indicates that not much information was learned in the previous round. If it is satisfied, then for the select step is doubled, while for the measure step is cut in half. This check can pass for two reasons: (1) there were no good candidates (all scores are low in Equation (1)) in which case increasing will make more candidates good, and (2) there were good candidates, but they were not selected because there was too much noise in the select step, which can be remedied by increasing . The precise annealing threshold used is √︁ 2/ · · , which is the expected error of the noisy marginal, and an approximation for the expected error ofˆon marginal . When the available privacy budget is small, this condition will be activated more frequently, and as a result, AIM will run for fewer rounds. Conversely, when the available privacy budget is large, AIM will run for many rounds before this condition activates. As decreases throughout execution, quality scores generally increase, and it has the effect of "unlocking" new candidates that previously had negative quality scores. We initialize and conservatively, assuming the mechanism will be run for = 16 rounds. This is an upper bound on the number of rounds that AIM will run, but in practice the number of rounds will be much less. As in prior work [8,55], we do not split the budget equally for the select and measure step, but rather allocate 10% of the budget for the select steps, and 90% of the budget for the measure steps. This is justified by the fact that the quality function for selection is a coarser-grained aggregation than a marginal, and as a result can tolerate a larger degree of noise. Privacy Analysis. The privacy analysis of AIM utilizes the notion of a privacy filter [41], and the algorithm runs until the realized privacy budget spent matches the total privacy budget available, . To ensure that the budget is not over-spent, there is a special condition (Line 8 in Algorithm 3) that checks if the remaining budget is insufficient for two rounds at the current and parameters. If this condition is satisfied, and are set to use up all of the remaining budget in one final round of execution. Theorem 1. For any ≥ , 0 < < 1, and ≥ 0, AIM satisfies -zCDP. Proof. There are three steps in AIM that depend on the sensitive data: initialization, selection, and measurement. The initialization step satisfies 0 -zCDP for 0 = \|{ ∈ + \| \| \| = 1}\|/2 2 0 ≤ /2 2 0 = 2 /2 ≤ . For this step, all we need is that the privacy budget is not over-spent. The remainder of AIM runs until the budget is consumed. Each step of AIM involves one invocation of the exponential mechanism, and one invocation of the Gaussian mechanism. By Propositions 1 to 3, round of AIM is -zCDP for = 1 8 2 /8 + 1/2 2 . Note that at round , = =0 , and we need to show that never exceeds [41]. There are two cases to consider: the condition in Line 8 of Algorithm 3 is either true or false. If it is true, then we know after round that − ≥ 2 +1 , i.e., the remaining budget is enough to run round + 1 without over-spending the budget. If it is false, then we modify +1 and +1 to exactly use up the remaining budget. Specifically, +1 = 8(1 − ) ( − )/8 + 2 ( − )/2 = − . As a result, when the condition is true, at time + 1 is exactly , and after that iteration, the main loop of AIM terminates. The remainder of the mechanism does not access the data. □ UNCERTAINTY QUANTIFICATION In this section, we propose a solution to the uncertainty quantification problem for AIM. Our method uses information from both the noisy marginals, measured with Gaussian noise, and the marginal queries selected by the exponential mechanism. Importantly, the method does not require additional privacy budget, as it quantifies uncertainty only by analyzing the private outputs of AIM. We give guarantees for marginals in the (downward closure of the) workload, which is exactly the set of marginals the analyst cares about. We provide no guarantees for marginals outside this set, which is an area for future work. We break our analysis up into two cases: the "easy" case, where we have access to unbiased answers for a particular marginal, and the "hard" case, where we do not. In both cases, we identify an estimator for a marginal whose error we can bound with high probability. Then, we connect the error of this estimator to the error of the synthetic data by invoking the triangle inequality. The subsequent paragraphs provide more details on this approach. Proofs of all statements in this section appear in Appendix B. The Easy Case: Supported Marginal Queries. A marginal query r is "supported" whenever ⊆ for some . In this case, we can readily obtain an unbiased estimate of ( ) from , and analytically derive the variance of that estimate. If there are multiple satisfying the condition above, we have multiple estimates we can use to reduce the variance. We can combine these independent estimates to obtain a weighted average estimator: Theorem 2 (Weighted Average Estimator). Let 1 , . . . , and 1 , . . . , be as defined in Algorithm 4, and let = { 1 , . . . , }. For any ∈ + , there is an (unbiased) estimator¯= ( 1 , . . . , ) such that: ∼ N ( ( ),¯2I) where¯2 = ∑︁ =1 ⊆ 2 −1 , While this is not the only (or best) estimator to use, 5 the simplicity allows us to easily bound its error, as we show in Theorem 3. Theorem 3 (Confidence Bound). Let¯be the estimator from Theorem 2. Then, for any ≥ 0, with probability at least 1 − exp (− 2 ): ∥ ( ) −¯∥ 1 ≤ √︁ 2 log 2¯+¯√2 Note that Theorem 3 gives a guarantee on the error of¯, but we are ultimately interested in the error ofˆ. Fortunately, it easy easy to relate the two by using the triangle inequality, as shown below: Corollary 1. Letˆbe any synthetic dataset, and let¯be the estimator from Theorem 2. Then with probability at least 1 − exp (− 2 ): ( ) − (ˆ) 1 ≤ (ˆ) −¯ 1 + √︁ 2 log 2¯+¯√2 The LHS is what we are interested in bounding, and we can readily compute the RHS from the output of AIM. The RHS is a random quantity that, with the stated probability, upper bounds the error. When we plug in the realized values we get a concrete numerical bound that can be interpreted as a (one-sided) confidence interval. In general, we expect (ˆ) to be close to¯, so the error bound forˆwill not be that much larger than that of¯. 6 The Hard Case: Unsupported Marginal Queries. We now shift our attention to the hard case, providing guarantees about the error of different marginals even for unsupported marginal queries (those not selected during execution of AIM). This problem is significantly more challenging. Our key insight is that marginal queries not selected have relatively low error compared to the marginal queries that were selected. We can easily bound the error of selected queries and relate that to non-selected queries by utilizing the guarantees of the exponential mechanism. In Theorem 4 below, we provide expressions that capture the uncertainty of these marginals with respect toˆ− 1 , the iterates of AIM. 5 A better estimator would be the minimum variance linear unbiased estimator. Ding et al. [13] derive an efficient algorithm for computing this from noisy marginals. 6 From prior experience, we might expect the error ofˆto be lower than the error of [37,38], so we are paying for this difference by increasing the error bound when we might hope to save instead. Unfortunately, this intuition does not lend itself to a clear analysis that provides better guarantees. Theorem 4 (Confidence Bound). Let , , ,˜, ,ˆbe as defined in Algorithm 4, and let Δ = max ∈ . For all ∈ , with probability at least 1 − − 2 1 /2 − − 2 : ∥ ( ) − (ˆ− 1 )∥ 1 ≤ −1 + 1 √ + 2 2Δ where is equal to: (ˆ− 1 ) − 1 estimated error on + √︁ 2/ − relationship to non-selected candidates + 2Δ log (\| \|) uncertainty from exponential mech. We can readily compute from the output of AIM, and use it to provide a bound on error in the form of a one-sided confidence interval that captures the true error with high probability. While these error bounds are expressed with respect toˆ− 1 , they can readily be extended to give a guarantee with respect toˆ. Corollary 2. Letˆbe any synthetic dataset, and let be as defined in Theorem 4. Then with probability at least 1 − − 2 1 /2 − − 2 : ( ) − (ˆ) 1 ≤ (ˆ) − (ˆ− 1 ) 1 + −1 + 1 √ + 2 2Δ Again, the LHS is what we are interested in bounding, and we can compute the RHS from the output of AIM. We expectˆ− 1 to be reasonably close toˆ, especially when is larger, so this bound will often be comparable to the original bound onˆ− 1 . Putting it Together. We've provided guarantees for both supported and unsupported marginals. The guarantees for unsupported marginals also apply for supported marginals, although we generally expect them to be looser. In addition, there is one guarantee for each round of AIM. It is tempting to use the bound that provides the smallest estimate, although unfortunately doing this invalidates the bound. To ensure a valid bound, we must pick only one round, and that cannot be decided based on the value of the bound. A natural choice is to use only the last round, for three reasons: (1) is smallest and is largest in that round, (2) the error ofĝ enerally goes down with , and (3) the distance betweenˆand should be the smallest in the last round. However, there may be some marginal queries which were not in the candidate set for that round. To bound the error on these marginals, we use the last round where that marginal query was in the candidate set. EXPERIMENTS In this section we empirically evaluate AIM, comparing it to a collection of state-of-the-art mechanisms and baseline mechanisms for a variety of workloads, datasets, and privacy levels. Experimental Setup Datasets. Our evaluation includes datasets with varying size and dimensionality. We describe our exact pre-processing scheme in Appendix A, and summarize the pre-processed datasets and their characteristics in the table below. Workloads. We consider 3 workloads for each dataset, all-3way, target, and skewed. Each workload contains a collection of 3-way marginal queries. The all-3way workload contains queries for all 3-way marginals. The target workload contains queries for all 3-way marginals involving some specified target attribute. For the adult and titanic datasets, these are the income>50K attribute and the Survived attribute, as those correspond to the attributes we are trying to predict for those datasets. For the other datasets, the target attribute is chosen uniformly at random. The skewed workload contains a collection of 3-way marginal queries biased towards certain attributes and attribute combinations. In particular, each attribute is assigned a weight sampled from a squared exponential distribution. 256 triples of attributes are sampled with probability proportional to the product of their weights. This results in workloads where certain attributes appear far more frequently than others, and is intended to capture the situation where analysts focus on a small number of interesting attributes. All randomness in the construction of the workload was done with a fixed random seed, to ensure that the workloads remain the same across executions of different mechanisms and parameter settings. Mechanisms. We compare against both workload-agnostic and workload-aware mechanisms in this section. The workload-agnostic mechanisms we consider are PrivBayes+PGM, MST, PrivMRF. The workload-aware mechanisms we consider are MWEM+PGM, RAP, GEM, and AIM. We set the hyper-parameters of every mechanism to default values available in their open source implementations. We also consider baseline mechanisms: Independent and Gaussian. The former measures all 1-way marginals using the Gaussian mechanism, and generates synthetic data using an independence assumption. The latter answers all queries in the workload using the Gaussian mechanism (using the optimal privacy budget allocation described in [55]). Note that this mechanism does not generate synthetic data, only query answers. Privacy Budgets. We consider a wide range of privacy parameters, varying ∈ [0.01, 100.0] and setting = 10 −9 . The most practical regime is ∈ [0.1, 10.0], but mechanism behavior at the extremes can be enlightening so we include them as well. Evaluation. For each dataset, workload, and , we run each mechanism for 5 trials, and measure the workload error from Definition 2. We report the average workload error across the five trials, along with error bars corresponding to the minimum and maximum workload error observed across the five trials. Runtime Environment. We ran most experiments on a single core of a compute cluster with a 4 GB memory limit and a 24 hour time limit. 7 These resources were not sufficient to run PrivMRF or RAP, so we utilized different machines to run those mechanisms. PrivMRF requires a GPU to run, so we used one node a different compute cluster, which has a Nvidia GeForce RTX 2080 Ti GPU. RAP required significant memory resources, so we ran those experiments on a machine with 16 cores and 64 GB of RAM. all-3way Workload Results on the all-3way workload are shown in Figure 1. Workloadaware mechanisms are shown by solid lines, while workload-agnostic mechanisms are shown with dotted lines. From these plots, we make the following observations: (1) AIM consistently achieves competitive workload error, across all datasets and privacy regimes considered. On average, across all six datasets and nine privacy parameters, AIM improved over PrivMRF by a factor of 1.3×, MST by a factor of 8.4×, MWEM+PGM by a factor 2.1×, PrivBayes+PGM by a factor 2.6×, RAP by a factor 9.5×, and GEM by a factor 2.3×. In the most extreme cases, AIM improved over PrivMRF by a factor 3.6×, MST by a factor 118×, MWEM+PGM by a factor 16×, PrivBayes+PGM by a factor 14.7×, RAP by a factor 47.1×, and GEM by a factor 11.7×. (2) Prior to AIM, PrivMRF was consistently the best performing mechanism, even outperforming all workload-aware mechanisms. The all-3way workload is one we expect workload agnostic mechanisms like PrivMRF to perform well on, so it is 7 These experiments usually completed in well under the time limit. interesting, but not surprising that it outperforms workloadaware mechanisms in this setting. (3) Prior to AIM, the best workload-aware mechanism varied for different datasets and privacy levels: MWEM+PGM was best in 65% of settings, GEM was best in 35% of settings 8 , and RAP was best in 0% of settings. Including AIM, we observe that it is best in 85% of settings, followed by MWEM+PGM in 11% of settings and GEM in 4% of settings. Additionally, in the most interesting regime for practical deployment ( ≥ 1.0), AIM is best in 100% of settings. target Workload Results for the target workload are shown in Figure 2. For this workload, we expect workload-aware mechanisms to have a significant advantage over workload-agnostic mechanisms, since they are aware that marginals involving the target are inherently more important for this workload. From these plots, we make the following observations: (1) All three high-level findings from the previous section are supported by these figures as well. PrivMRF is not workload-aware, it is clear from their paper that every detail of the mechanism was carefully thought out to make the mechanism work well in practice, which explains it's impressive performance. While AIM did outperform PrivMRF again, the relative performance did not increase by a meaningful margin -offering a 1.4× improvement on average and a 4.6× improvement in the best case. skewed Workload Results for the skewed workload are shown in Figure 3. For this workload, we again expect workload-aware mechanisms to have a significant advantage over workload-agnostic mechanisms, since they are aware of the exact (biased) set of marginals used to judge utility. From these plots, we make the following observations: (1) All four high-level findings from the previous sections are generally supported by these figures as well, with the following interesting exception: (2) PrivMRF did not score well on salary, and while it was still generally the second best mechanism on the other datasets (again out-performing the workload-aware mechanisms in many cases), the improvement offered by AIM over PrivMRF is much larger for this workload, averaging a 2× improvement with up to a 5.7× improvement in the best case. We suspect for this setting, workload-awareness is essential to achieve strong performance. Tuning Model Capacity In Line 12 of AIM (Algorithm 4), we construct a set of candidates to consider in the current round based on an upper limit on JT-SIZE. 80 MB was chosen to match prior work, 9 but in general we can tune it as desired to strike the right accuracy / runtime trade-off. Unlike other hyper-parameters, there is no "sweet spot" for this one: setting larger model capacities should always make the mechanism perform better, at the cost of increased runtime. We demonstrate this trade-off empirically in Figure 4 (a-b). For = 0.1, 1, and 10, we considered model capacities ranging from 1.25 MB to 1.28 GB, and ran AIM on the fire dataset with the all-3way workload. Results are averaged over five trials, with error bars indicating the min/max runtime and workload error across those trials. Our main findings are listed below: (1) As expected, runtime increases with model capacity, and workload error decreases with capacity. The case = 0.1 is an exception, where both the plots level off beyond a capacity of 20 . This is because the capacity constraint is not active in this regime: AIM already favors small marginals when the available privacy budget is small by virtue of the quality score function for marginal query selection, so the model remains small even without the model capacity constraint. (2) Using the default model capacity and = 1 resulted in a 9 hour runtime. We can slightly reduce error further, by about 13%, by increasing the model capacity to 1.28 and waiting 7 days. Conversely, we can reduce the model capacity to 9 Cai et al. [8] limit the size of the largest clique in the junction tree to have at most 10 7 cells (80 MB with 8 byte floats), while we limit the overall size of the junction tree. which increases error by about 75%, but takes less than one hour. The law of diminishing returns is at play. Ultimately, the model capacity to use is a policy decision. In realworld deployments, it is certainly reasonable to spend additional computational time for even a small boost in utility. Uncertainty Quantification In this section, we demonstrate that our expressions for uncertainty quantification correctly bound the error, and evaluate how tight the bound is. For this experiment, we ran AIM on the fire dataset with the all-3way workload at = 10. In Figure 4 (c), we plot the true error of AIM on each marginal in the workload against the error bound predicted by our expressions. We set = 1.7 in Corollary 1, and 1 = 2.7, 2 = 3.7 in Corollary 2, which provides 95% confidence bounds. Our main findings are listed below: (1) For all marginals in the (downward closure of the) workload, the error bound is always greater than true error. This confirms the validity of the bound, and suggests they are safe to use in practice. Note that even if some errors were above the bounds, that would not be inconsistent with our guarantee, as at a 95% confidence level, the bound could fail to hold 5% of the time. The fact that it doesn't suggests there is some looseness in the bound. (2) The true errors and the error bounds vary considerably, ranging from 10 −4 all the way up to and beyond 1. In general, the supported marginals have both lower errors, and lower error bounds than the unsupported marginals, which is not surprising. The error bounds are also tighter for the supported marginals. The median ratio between error bound and observed error is 4.4 for supported marginals and 8.3 for unsupported marginals. Intuitively, this makes sense because we know selected marginals should have higher error than non-selected marginals, but the error of the non-selected marginal can be far below that of the selected marginal (and hence the bound), which explains the larger gap between the actual error and our predicted bound. LIMITATIONS AND OPEN PROBLEMS In this work, we have carefully studied the problem of workloadaware synthetic data generation under differential privacy, and proposed a new mechanism for this task. Our work significantly improves over prior work, although the problem remains far from solved, and there are a number of promising avenues for future work in this space. We enumerate some of the limitations of AIM below, and identify potential future research directions. Handling More General Workloads. In this work, we restricted our attention to the special-but-common case of weighted marginal query workloads. Even in this special case, there are many facets to the problem and nuances to our solution. Designing mechanisms that work for the more general class of linear queries (perhaps defined over the low-dimensional marginals) remains an important open problem. While the prior work, MWEM+PGM, RAP, and GEM can handle workloads of this form, they achieve this by selecting a single counting query in each round, rather than a full marginal query, and thus there is likely significant room for improvement. Beyond linear query workloads, other workloads of interest include more abstract objectives like machine learning efficacy and other non-linear query workloads. These metrics have been used to evaluate the quality of workload-agnostic synthetic data mechanisms, but have not been provided as input to the mechanisms themselves. In principle, if we know we want to run a given machine learning model on the synthetic dataset, we should be able to tailor the synthetic data to provide high utility on that model. Handling Mixed Data Types. In this work, we assumed the input data was discrete, and each attribute had a finite domain with a reasonably small number of possible values. Data with numerical attributes must be appropriately discretized before running AIM. The quality of the discretization could have a significant impact on the quality of the generated synthetic data. Designing mechanisms that appropriately handle mixed (categorical and numerical) data type is an important problem. There may be more to this problem than meets the eye: a new definition of a workload and utility metric may be in order, and new types of measurements and post-processing techniques may be necessary to handle numerical data. Note that some mechanisms, like GAN-based mechanisms, expect numerical data as input, and categorical data must be one-hot encoded prior to usage. While they do handle handle numerical data, their utility is often not competitive with even the simplest marginal-based mechanisms we considered in this work [44]. Utilizing Public Data. A promising avenue for future research is to design synthetic data mechanisms that incorporate public data in a principled way. There are many places in which public data can be naturally incorporated into AIM, and exploring these ideas is a promising way to boost the utility of AIM in real world settings where public data is available. Early work on this problem includes [31,32,35], but these solutions leave room for improvement. RELATED WORK In Section 3 we focused our discussion on marginal-based mechanisms in the select-measure-generate paradigm. While this is a popular approach, it is not the only way to generate differentially private synthetic data. In this section we provide a brief discussion of other methods, and a broad overview of other relevant work. One prominent approach is based on differentially private GANs. Several architectures and private learning procedures have been proposed under this paradigm [1,4,18,27,43,45,46,49,54]. Despite their popularity, we are not aware of evidence that these GAN-based mechanisms outperform even baseline marginal-based mechanisms like PrivBayes on structured tabular data. Most empirical evaluations of GAN-based mechanisms exclude PrivBayes, and the comparisons that we are aware of show the opposite effect: that marginal-based mechanisms outperform the competition [8,35,44]. GAN-based methods may be better suited for different data modalities, like image or text data. One exception is CT-GAN [51], which is an algorithm for synthetic data that does compare against PrivBayes and does outperform it in roughly 85% of the datasets and metrics they considered. However, this method does not satisfy or claim to satisfy differential privacy, and gives no formal privacy guarantee to the individuals who contribute data. Nevertheless, an empirical comparison between CT-GAN and newer methods for synthetic data, like AIM and PrivMRF, would be interesting, since these mechanisms also outperformed PrivBayes+PGM in nearly every tested situation, and PrivBayes+PGM outperforms PrivBayes most of the time as well [8,37]. Differentially private implementations of CT-GAN have been proposed, but empirical evaluations of the method suggest it is not competitive with PrivBayes [42,44]. A DATA PREPROCESSING We apply consistent preprocessing to all datasets in our empirical evaluation. There are three steps to our preprocessing procedure, described below: Attribute selection. For each dataset, we identify a set of attributes to keep. For the adult, salary, nltcs, and titanic datasets, we keep all attributes from the original data source. For the fire dataset, we drop the 15 attributes relating to incident times, since after discretization, they contain redundant information. The msnbc dataset is a streaming dataset, where each row has a different number of entries. We keep only the first 16 entries for each row. Domain identification. Usually we expect the domain to be supplied separately from the data file. For example, the IPUMS website contains comprehensive documentation about U.S. Census data products. However, for the datasets we used, no such domain file was available. Thus, we "cheat" and look at the active domain to automatically derive a domain file from the dataset. For each attribute, we identify if it is categorical or numerical. For each categorical attribute, we list the set of observed values (including null) for that attribute, which we treat as the set of possible values for that attribute. For each numerical attribute, we record the minimum and maximum observed value for that attribute. Discretization. We discretize each numerical attribute into 32 equal-width bins, using the min/max values from the domain file. This turns each numerical attribute into a categorical attribute, satisfying our assumption. ∼ N ( ( ),¯2I) where¯2 = ∑︁ =1 ⊆ 2 −1 , Proof. For each ⊇ , we observe˜∼ ( ) + N (0, 2 I). We can use this noisy marginal to obtain an unbiased estimate ( ) by marginalizing out attributes in the set \ . This requires summing up / cells, so the variance in each cell becomes 2 / . Moreover, the noise is still normally distributed, since the sum of independent normal random variables is normal. We thus have such an estimate for each satisfying ⊇ , and we can combine these independent estimates using inverse variance weighting [23], resulting in an unbiased estimator with the stated variance. For the same reason as before, the noise is still normally distributed. □ Theorem 3 (Confidence Bound). Let¯be the estimator from Theorem 2. Then, for any ≥ 0, with probability at least 1 − exp (− 2 ): ∥ ( ) −¯∥ 1 ≤ √︁ 2 log 2¯+¯√2 Proof. Noting that ( ) −¯∼ N (0, 2 I), the statement is a direct consequence of Theorem 5, below. □ Theorem 5. Let ∼ (0, 2 ) , then: E[∥ ∥ 1 ] = √︁ 2/ and Pr[∥ ∥ 1 ≥ √︁ 2 log 2 + √ 2 ] ≤ exp (− 2 ) Proof. First observe that \| \| is a sample from a half-normal distribution. Thus, E[ ] = √︁ 2/ . From the linearity of expectation, we obtain E[∥ ∥ 1 ] = √︁ 2/ , as desired. For the second statement, we begin by deriving the moment generating function of the random variable \| \|. By definition, we have: E[exp ( · \| \|)] = ∫ ∞ −∞ ( ) exp ( · \| \|) = 2 ∫ ∞ 0 ( ) exp ( · ) = 2 ∫ ∞ 0 1 √ 2 exp − 2 2 2 exp ( · ) = 1 √︂ 2 ∫ ∞ 0 exp − 2 2 2 + · = exp 2 2 2 Φ √ 2 + 1 Moreover, since ∥ ∥ 1 = =1 \| \| is a sum of i.i.d random variables, the moment generating function of ∥ ∥ 1 is: E[exp ( · ∥ ∥ 1 )] = exp 2 2 2 Φ √ 2 + 1 From the Chernoff bound, we have Pr[∥ ∥ 1 ≥ ] ≤ min ≥0 E[exp ( · ∥ ∥ 1 )] exp ( ) = min ≥0 exp 2 2 2 − Φ √ 2 + 1 ≤ min ≥0 2 exp 2 2 2 − ≤ 2 exp 2 ( / 2 ) 2 2 − ( / 2 ) = 2 exp 2 2 2 − 2 2 = 2 exp − 2 2 2 = exp − 2 2 2 + log 2 With some further manipulation of the bound, we obtain: Pr[∥ ∥ 1 ≥ √ 2 ] ≤ exp − 2 + log 2 ( = √ 2 ) Pr[∥ ∥ 1 ≥ ( + √︁ log 2) √ 2 ] ≤ exp (− 2 ) ( = + √︁ log 2) Pr[∥ ∥ 1 ≥ √︁ 2 log 2 + √ 2 ] ≤ exp (− 2 ) □ B.2 The Hard Case: Unsupported Marginals Theorem 4 (Confidence Bound). Let , , ,˜, ,ˆbe as defined in Algorithm 4, and let Δ = max ∈ . For all ∈ , with probability at least 1 − − 2 1 /2 − − 2 : ∥ ( ) − (ˆ− 1 )∥ 1 ≤ −1 + 1 √ + 2 2Δ where is equal to: (ˆ− 1 ) − 1 estimated error on + √︁ 2/ − relationship to non-selected candidates + 2Δ log (\| \|) uncertainty from exponential mech. Proof. By the guarantees of the exponential mechanism, we know that, with probability at most − 2 , for all ∈ we have: ≤ − 2Δ (log (\| \|) + 2 ) Now define = ∥ ( ) − ( −1 )∥ 1 . Plugging in = ( − √︁ 2/ ) and rearranging gives: ≥ ( − √︁ 2/ ) + 2Δ (log (\| \|) + 2 ) + √︁ 2/ From Theorem 6, with probability at most − 2 1 /2 , we have: ( −1 ) − 1 + 1 √ ≤ Combining these two facts via the union bound, along with some algebraic manipulation, yields the stated result. □ Theorem 6. Let , ∈ R and let = + where ∼ N (0, 2 ) . Pr[∥ − ∥ 1 ≤ ∥ − ∥ 1 − √ ] ≤ exp − 1 2 2 Proof. First note that \| − \| = \| − − \|, which is distributed according to a folded normal distribution with mean \| − \|. It is well known [47] that the moment generating function for this random variable is ( ), where: ( ) = exp 1 2 2 2 + \| − \| Φ(\| − \|/ + ) + exp 1 2 2 2 − \| − \| Φ(−\| − \|/ + ). Moreover, the moment generating function of ∥ − ∥ 1 is ( ) = ( ). We will begin by focusing our attention on bounding We are now ready to plug this result into the Chernoff bound, which states: Pr[∥ − ∥ 1 ≤ ] ≤ min ≥0 exp ( · ) (− ) ≤ min ≥0 exp ( · ) exp 2 2 2 − \| − \| = min ≥0 exp ( · + 2 2 2 − ∥ − ∥ 1 ) Setting = ∥ − ∥ 1 − √ gives the desired result Pr[∥ − ∥ 1 ≤ ∥ − ∥ 1 − √ ] ≤ min ≥0 exp ( · (∥ − ∥ 1 − √ ) + 2 2 2 − ∥ − ∥ 1 ) = min ≥0 exp − √ + 2 2 2 ≤ exp (− 2 /2) (set = / √ ) □ Lemma 1. Let , ≥ 0, and let Φ denote the CDF of the standard normal distribution. Then, exp 1 2 2 + Φ(− − ) ≤ exp 1 2 2 − Φ( − ) Proof. First observe that: exp 1 2 2 + Φ(− − ) = exp − 1 2 2 Φ(− − ) (− − ) exp 1 2 2 − Φ( − ) = exp − 1 2 2 Φ( − ) ( − ) Since , ≥ 0, we know that − − ≤ − . We will now argue that the function Φ( ) ( ) is monotonically increasing in , which suffices to prove the desired claim. To prove this, we will observe that this is this quantity is known as the Mills ratio [21] for the normal distribution. We know that the Mills ratio is connected to a particular expectation; specifically, if ∼ N (0, 1), then E[ \| < ] = − ( ) Φ( ) Using this interpretation, it is clear that the LHS (and hence the RHS) is monotonically increasing in . Since − ( ) Φ( ) is monotonically increasing, so is Φ( ) ( ) . □ C INTERPRETABLE ERROR RATE AND SUBSAMPLING MECHANISM In Section 6, we saw that AIM offers the best error relative to existing synthetic data mechanisms, although it is not obvious whether a given 1 error should be considered "good". This is necessary for setting the privacy parameters to strike the right privacy/utility tradeoff. We can bring more clarity to this problem by comparing AIM to a (non-private) baseline that simply resamples records from the dataset. Then, if AIM achieves the same error as resampling = 2 records, this provides a clear interpretation: that the price of privacy is losing about half the data. Due to the simplicity of this baseline, we can compute the expected workload error in closed form, without actually running the mechanism. We provide details of these calculations in the next section. Figure 5 plots the performance of AIM on each dataset, epsilon, and workload considered, measured using the fraction of samples needed for the subsampling mechanism to match the performance of AIM. These plots reveal that at = 10, the median subsampling fraction is about 0.37 for the general workload, 0.62 for the target workload, and 0.85 for the weighted workload. At = 1, these numbers are 0.13, 0.15, and 0.21, respectively. The results are comparable across five out of six datasets, with nltcs being a clear outlier. For that dataset, a subsampling fraction of 1.0 was reached by = 0.31 for all workload. This could be an indication of overfitting to the data; a possible reason for this behavior is that the domain size of the nltcs data is small compared to the number of records. mnsbc is also an outlier to a lesser extent, with worse performance than the other datasets for larger . A possible reason for this behavior is that msnbc has the most data points, so subsampling with the same fraction of points has much lower error. AIM may not be able to match that low error due to the computational constraints imposed on the model size, combined with the fact that this dataset has a large domain. C.1 Mathematical Details of Subsampling We begin by analyzing the expected workload error of the (nonprivate) mechanism that randomly samples items with replacement from . Then, we will connect that to the error of AIM, and determine the value of where the error rates match. Theorem 7 gives a closed form expression for the expected 1 error on a single marginal as a function of the number of sampled records. Theorem 7. Letˆbe the dataset obtained by sampling items with replacement from . Further, let ì = 1 ( ) and ì = ⌈ ì⌉. E 1 ( ) − 1 (ˆ) = 2 ∑︁ ∈Ω ( ) ( ) ( ) ( ) (1 − ( )) − ( )+1 Proof. The theorem statement follows directly from Lemma 4 and Lemma 3. □ Lemma 2 (Mean Deviation [16,26]). Let ∼ ( , ), then: E − = 2 (1 − ) − +1 , where = ⌈ · ⌉. Proof. This statement appears and is proved in [16,26]. □ Lemma 3 ( 1 Deviation). Let ì ∼ ( , ì), then: E[ ì − ì / 1 ] = 2 ∑︁ ( ) ( ) ( ) ( ) (1 − ( )) − ( )+1 , where ( ) = ⌈ · ( )⌉. Proof. The statement follows immediately from Lemma 2 and the fact that ( ) ∼ ( , ( )). □ Lemma 4. Letˆbe the dataset obtained by sampling items with replacement from . Then, (ˆ) ∼ ,1 ( ) Proof. The statement follows from the definition of the multinomial distribution. □ D STRUCTURAL ZEROS In this section, we describe a simple and principled method to specify and enforce structural zeros in the mechanism. These capture attribute combinations that cannot occur in the real data. Without specifying this, synthetic data mechanisms will usually generate records that violate these constraints that hold in the real data, as the process of adding noise can introduce spurious records, especially in high privacy regimes. These spurious records can be confusing for downstream analysis of the synthetic data, and can lead the analyst to distrust the quality of the data. By imposing known structural zero constraints, we can avoid this problem, while also improving the quality of the synthetic data on the workload of interest. Structural zeros, if they exist, can usually be enumerated by a domain expert. We can very naturally incorporate these into our mechanism with only one minor change to the underlying Private-PGM library. These structural zeros can be specified as input as a list of pairs ( , Z ) where Z ⊆ Ω . The first entry of the pair specifies the set of attributes relevant to the structural zeros, while the second entry enumerates the attribute combinations whose counts should all be zero. The method we propose can be used within any mechanism that builds on top of Private-PGM, and is hence more broadly useful outside the context of AIM. To understand the technical ideas in this section, please refer to the background on Private-PGM [37]. Usually Private-PGM is initialized by setting ( ) = 0 for all in the model and all ∈ Ω . This corresponds to a model where ( ) is uniform across all . Our basic observation is that by initializing Private-PGM by setting ( ) = −∞ for each ∈ , the cell of the associated marginal will be ( ) = 0, as desired. Moreover, each update within the Private-PGM estimation procedure will try to update ( ) by a finite amount, leaving it unchanged. Thus, ( ) will remain 0 during the entire estimation procedure. We conjecture that the estimation procedure solves the following modified convex optimization problem:ˆ= min ∈M ( )=0 ( ) This approach is appealing because other simple approaches that discard invalid tuples can inadvertently bias the distribution, which is undesirable. Note that for each clique in the set of structural zeros, we must include that clique in our model, which increases the size of that model. Thus, we should treat it as we would treat a clique selected by AIM. That is, when calculating JT-SIZE in line 12 of AIM, we need to include both the cliques selected in earlier iterations, as well as the cliques included in the structural zeros. D.1 Experiments In this section, we empirically evaluate this structural zeros enhancement, showing that it can reduce workload error in some cases. For this experiment, we consider the general workload on the fire dataset, and compare the performance of AIM with and without imposing structural zero constraints. This dataset contains several related attributes, like "Zipcode of Incident" and "City". While these attributes are not perfectly correlated, significant numbers of attribute combinations are impossible. We identified a total of nine attribute pairs which contain some structural zeros, and a total of 2696 structural zero constraints within these nine marginals. The results of this experiment are shown in Table 3. On average, imposing structural zeros improves the performance of the mechanism, although the improvement is not universal across all values of epsilon we tested. Nevertheless, it is still useful to impose these constraints for data quality purposes. E RUNTIME EXPERIMENTS Our primary focus in the main body of the paper was mechanism utility, as measured by the workload error. In this section we discuss the runtime of AIM, which is an important consideration when deploying it in practice. Note that we do not compare against runtime of other mechanisms here, because different mechanisms were executed in different runtime environments. Figure 6 below shows the runtime of AIM as a function of the privacy parameter. As evident from the figure, runtime increases drastically with the privacy parameter. This is not surprising because AIM is budget-aware: it knows to select larger marginals and run for more rounds when the budget is higher, which in turn leads to longer runtime. For large , the constraint on JT-SIZE is essential to allow the mechanism to terminate at all. Without it, AIM may try to select marginal queries that exceed memory resources and result in much longer runtime. For small , this constraint is not active, and could be removed without affecting the behavior of AIM. Recall that these experiments were conducted on one core of a compute cluster with 4 GB of memory and a CPU speed of 2.4 GHz. These machines were used due to the large number of experiments we needed to conduct, but in real-world scenarios we only need to run one execution of AIM, for a single dataset, workload, privacy parameter, and trial. For this, we can use machines with much better specs, which would improve the runtime significantly. In this paper, we built AIM on top of Private-PGM, leveraging prior work for the generate step of the select-measure-generate paradigm. Private-PGM is not the only method in this space, although it was the first general purpose and scalable method to our knowledge. "Relaxed Projection" [3] is another general purpose and scalable method that solves the same problem, and could be used in place of Private-PGM if desired. RAP, the main mechanism that utilizes this technique, did not perform well in our experiments. However, it is not clear from our experiments if the poor performance can be attributed to the relaxed projection algorithm, or some other algorithmic design decisions. In this section, we attempt to precisely pin down the differences between these two related methods, taking care to fix possible confounding factors. We thus consider two mechanisms: MWEM+PGM, which is defined in Algorithm 1, and MWEM+Relaxed Projection which is identical to MWEM+PGM in every way, except the call to Private-PGM is replaced with a call to the relaxed projection algorithm of Aydore et al. For this experiment, we consider the all-3way workload, and we run each algorithm for = 5, 10, . . . , 100, with five trials for each hyper-parameter setting. We average the workload error across the five trials, and report the minimum workload error across hyperparameter settings in Figure 7. Although the algorithms are conceptually very similar, MWEM+PGM consistently outperforms MWEM+Relaxed Projection, across every dataset and privacy level considered. The performance difference is modest in many cases, but significant on the fire dataset. AP-PGM [36] offers another alternative to Private-PGM for the generate step, and while it was shown to be an appealing alternative to Private-PGM in some cases, within the context of an MWEMstyle algorithm, their own experiments demonstrate the superiority of Private-PGM. Generator networks [32] offer yet another alternative to Private-PGM for the generate step. To the best of our knowledge, no direct comparison between this approach and Private-PGM has been done to date, where confounding factors are controlled for. Conceptually, this approach is most similar to the relaxed projection approach, so we conjecture the results to look similar to those shown in Figure 7. Figure 1 : 1Workload error of competing mechanisms on the all-3way workload for = 0.01, . . . , 100. ( 2 ) 2Somewhat surprisingly, PrivMRF outperforms all workloadaware mechanisms prior to AIM on this workload. This is an impressive accomplishment for PrivMRF, and clearly highlights the suboptimality of existing workload-aware mechanisms like MWEM+PGM, GEM, and RAP. Even though Figure 2 : 2Workload error of competing mechanisms on the target workload for = 0.01, . . . , 100. Figure 3 : 3Workload error of competing mechanisms on the skewed workload for = 0.01, . . . , 100. 5 Figure 4 ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grants IIS-1749854 and CNS-1954814, and by Oracle Labs, part of Oracle America, through a gift to the University of Massachusetts Amherst in support of academic research. Weighted Average Estimator). Let 1 , . . . , and 1 , . . . , be as defined in Algorithm 4, and let = { 1 , . . . , }. For any ∈ + , there is an (unbiased) estimator¯= ( 1 , . . . , ) such that: 1 below; = , = / ) Figure 5 : 5Performance of AIM as measured by the number of samples needed to match the achieved workload error. Figure 6 :F 6Runtime of AIM on the all-3way workload. PRIVATE-PGM VS. RELAXED PROJECTION Figure 7 : 7MWEM+Relaxed Projection vs. MWEM+PGM on the all-3way workload. Table 1 : 1Taxonomy of select-measure-generate mechanisms.Name Year Workload Data Budget Efficiency Aware Aware Aware Aware Independent - ✓ Gaussian - ✓ PrivBayes [52] 2014 ✓ ✓ ✓ HDMM+PGM [37] 2019 ✓ PrivBayes+PGM [37] 2019 ✓ ✓ ✓ MWEM+PGM [37] 2019 ✓ ✓ PrivSyn [55] 2020 ✓ ✓ ✓ MST [35] 2021 ✓ ✓ RAP [3] 2021 ✓ ✓ ✓ GEM [32] 2021 ✓ ✓ ✓ PrivMRF [8] 2021 ✓ ✓ ✓ AIM [This Work] 2022 ✓ ✓ ✓ ✓ Table 2 : 2Summary of datasets used in the experiments.Dataset Records Dimensions Min/Max Total Domains Domain Size adult [28] 48842 15 2-42 4 × 10 16 salary [24] 135727 9 3-501 1 × 10 13 msnbc [7] 989818 16 18 1 × 10 20 fire [40] 305119 15 2-46 4 × 10 15 nltcs [33] 21574 16 2 7 × 10 4 titanic [17] 1304 9 2-91 9 × 10 7 Table 3 : 3Error of AIM on the fire dataset, with and without imposing structural zero constraints.AIM AIM+Structural Zeros Ratio 0.010 0.613 0.542 1.130 0.031 0.303 0.263 1.151 0.100 0.141 0.153 0.924 0.316 0.087 0.077 1.124 1.000 0.052 0.053 0.979 3.162 0.044 0.045 0.964 10.00 0.038 0.032 1.170 31.62 0.029 0.026 1.149 100.0 0.025 0.025 1.004 Another common approach is based on GANs[20], however recent research[44] has shown that published GAN-based approaches rarely outperform simple baselines; therefore we do not compare with those techniques in this paper. 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[ "On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems", "On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems" ]	[ "Qinian Jin ", "Ulrich Tautenhahn " ]	[]	[]	We consider the computation of stable approximations to the exact solution x † of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods	10.1007/s00211-008-0198-y	[ "https://arxiv.org/pdf/0810.4185v1.pdf" ]	18,868,602	0810.4185	eff1698d2f96bfe89633762349d18405a1598cc3	On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems 23 Oct 2008 Qinian Jin Ulrich Tautenhahn On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems 23 Oct 2008Noname manuscript No. (will be inserted by the editor) the date of receipt and acceptance should be inserted later We consider the computation of stable approximations to the exact solution x † of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods Introduction In this paper we will consider the nonlinear inverse problems which can be formulated as the operator equations F(x) = y, (1.1) the inverse problems. Such problems arise naturally from the parameter identification in partial differential equations. Throughout this paper · and (·, ·) denote respectively the norms and inner products for both the spaces X and Y since there is no confusion. The nonlinear operator F is always assumed to be Fréchet differentiable, the Fréchet derivative of F at x ∈ D(F) is denoted as F ′ (x) and F ′ (x) * is used to denote the adjoint of F ′ (x). We assume that y is attainable, i.e. problem (1.1) has a solution x † ∈ D(F) such that F(x † ) = y. Since the right hand side is usually obtained by measurement, thus, instead of y itself, the available data is an approximation y δ satisfying y δ − y ≤ δ (1.2) with a given small noise level δ > 0. Due to the ill-posedness, the computation of a stable solution of (1.1) from y δ becomes an important issue, and the regularization techniques have to be taken into account. Many regularization methods have been considered to solve (1.1) in the last two decades. Tikhonov regularization is one of the well-known methods that has been studied extensively (see [17,11,19] and the references therein). Due to the straightforward implementation, iterative methods are also attractive for solving nonlinear inverse problems. In this paper we will consider some Newton type methods in which the iterated solutions {x δ k } are defined successively by x δ k+1 = x 0 − g α k F ′ (x δ k ) * F ′ (x δ k ) F ′ (x δ k ) * F(x δ k ) − y δ − F ′ (x δ k )(x δ k − x 0 ) ,(1.3) where x δ 0 := x 0 is an initial guess of x † , {α k } is a given sequence of numbers such that α k > 0, 1 ≤ α k α k+1 ≤ r and lim k→∞ α k = 0 (1. 4) for some constant r > 1, and g α : [0, ∞) → (−∞, ∞) is a family of piecewise continuous functions satisfying suitable structure conditions. The method (1.3) can be derived as follows. Suppose x δ k is a current iterate, then we may approximate F(x) by its linearization around x δ k , i.e. F(x) ≈ F(x δ k ) + F ′ (x δ k )(x − x δ k ). Thus, instead of (1.1), we have the approximate equation F ′ (x δ k )(x − x δ k ) = y δ − F(x δ k ). (1.5) If F ′ (x δ k ) has bounded inverse, the usual Newton method defines the next iterate by solving (1.5) for x. For nonlinear ill-posed inverse problems, however, F ′ (x δ k ) in general is not invertible. Therefore, we must use linear regularization methods to solve (1.5). There are several ways to do this step. One way is to rewrite (1.5) as F ′ (x δ k )h = y δ − F(x δ k ) + F ′ (x δ k )(x δ k − x 0 ),(1.6) where h = x − x 0 . Applying the linear regularization method defined by {g α } we may produce the regularized solution h δ k by h δ k = g α k F ′ (x δ k ) * F ′ (x δ k ) F ′ (x δ k ) * y δ − F(x δ k ) + F ′ (x δ k )(x δ k − x 0 ) . The next iterate is then defined to be x δ k+1 := x 0 + h δ k which is exactly the form (1.3). In order to use x δ k to approximate x † , we must choose the stopping index of iteration properly. Some Newton type methods that can be casted into the form (1.3) have been analyzed in [3,12,14] under a priori stopping rules, which, however, depend on the knowledge of the smoothness of x 0 − x † that is difficult to check in practice. Thus a wrong guess of the smoothness will lead to a bad choice of the stopping index, and consequently to a bad approximation to x † . Therefore, a posteriori rules, which use only quantities that arise during calculations, should be considered to choose the stopping index of iteration. One can consult [3,8,4,9,2,14] for several such rules. One widely used a posteriori stopping rule in the literature of regularization theory for illposed problems is the discrepancy principle which, in the context of the Newton method (1.3), defines the stopping index k δ to be the first integer such that F(x δ k δ ) − y δ ≤ τδ < F(x δ k ) − y δ , 0 ≤ k < k δ ,(1.7) where τ > 1 is a given number. The method (1.3) with g α (λ ) = (α + λ ) −1 together with (1.7) has been considered in [3,8]. Note that when g α (λ ) = (α + λ ) −1 , the method (1.3) is equivalent to the iteratively regularized Gauss-Newton method [1] x δ k+1 = x δ k − α k I + F ′ (x δ k ) * F ′ (x δ k ) −1 F ′ (x δ k ) * (F(x δ k ) − y δ ) + α k (x δ k − x 0 ) . (1.8) When F satisfies the condition like F ′ (x) = R(x, z)F ′ (z) + Q(x, z), I − R(x, z) ≤ C R x − z , x, z ∈ B ρ (x † ), (1.9) Q(x, z) ≤ C Q F ′ (z)(x − z) , where C R and C Q are two positive constants, for the method defined by (1.8) and (1.7) with τ being sufficiently large, it has been shown in [3,8] that if x 0 − x † satisfies the Hölder source condition x 0 − x † = (F ′ (x † ) * F ′ (x † )) ν ω (1.10) for some ω ∈ X and 0 ≤ ν ≤ 1/2, then x δ k δ − x † ≤ o(δ 2ν/(1+2ν) ); while if x 0 − x † satisfies the logarithmic source condition x 0 − x † = − log(F ′ (x † ) * F ′ (x † )) −µ ω (1.11) for some ω ∈ X and µ > 0, then x δ k δ − x † ≤ O((− ln δ ) −µ ). Unfortunately, except the above results, there is no more result available in the literature on the general method defined by (1.3) and (1.7). During the attempt of proving regularization property of the general method defined by (1.3) and (1.7), Kaltenbacher realized that the arguments in [3,8] depend heavily on the special properties of the function g α (λ ) = (α + λ ) −1 , and thus the technique therein is not applicable. Instead of the discrepancy principle (1.7), she proposed in [13] a new a posteriori stopping rule to terminate the iteration as long as max F(x δ m δ −1 ) − y δ , F(x δ m δ −1 ) + F ′ (x δ m δ −1 )(x δ m δ − x δ m δ −1 ) − y δ ≤ τδ (1.12) is satisfied for the first time, where τ > 1 is a given number. Under the condition like (1.9), it has been shown that if x 0 − x † satisfies the Hölder source condition (1.10) for some ω ∈ X and 0 ≤ ν ≤ 1/2, then there hold the order optimal convergence rates x δ m δ − x † ≤ C ν ω 1/(1+2ν) δ 2ν/(1+2ν) if {g α } satisfies some suitable structure conditions, τ is sufficiently large and ω is sufficiently small. Note that any result on (1.12) does not imply that the corresponding result holds for (1.7). Note also that k δ ≤ m δ − 1 which means that (1.12) requires more iterations to be performed. Moreover, the discrepancy principle (1.7) is simpler than the stopping rule (1.12). Considering the fact that it is widely used in practice, it is important to give further investigations on (1.7). In this paper, we will resume the study of the method defined by (1.3) and (1.7) with completely different arguments. With the help of the ideas developed in [9,19,10], we will show that, under certain conditions on {g α }, {α k } and F, the method given by (1.3) and (1.7) indeed defines a regularization method for solving (1.1) and is order optimal for each 0 < ν ≤ν − 1/2, whereν ≥ 1 denotes the qualification of the linear regularization method defined by {g α }. In particular, when x 0 − x † satisfies (1.10) for 1/2 ≤ ν ≤ν − 1/2, we will show that the order optimality of (1.3) and (1.7) even holds under merely the Lipschitz condition on F ′ . This is the main contribution of the present paper. We point out that our results are valid for any τ > 1. This less restrictive requirement on τ is important in numerical computations since the absolute error could increase with respect to τ. This paper is organized as follows. In Section 2 we will state various conditions on {g α }, {α k } and F, and then present several convergence results on the methods defined by (1.3) and (1.7). We then complete the proofs of these main results in Sections 3, 4, and 5. In Section 6, in order to indicate the applicability of our main results, we verify those conditions in Section 2 for several examples of {g α } arising from Tikhonov regularization, the iterated Tikhonov regularization, the Landweber iteration, the Lardy's method, and the asymptotic regularization. Assumptions and main results In this section we will state the main results for the method defined by (1.3) and the discrepancy principle (1.7). Since the definition of {x δ k } involves F, g α and {α k }, we need to impose various conditions on them. We start with the assumptions on g α which is always assumed to be continuous on [0, 1/2] for each α > 0. We will set r α (λ ) : = 1 − λ g α (λ ), which is called the residual function associated with g α . Assumption 1 1 (a) There are positive constants c 0 and c 1 such that 0 < r α (λ ) ≤ 1, r α (λ )λ ≤ c 0 α and 0 ≤ g α (λ ) ≤ c 1 α −1 for all α > 0 and λ ∈ [0, 1/2]; (b) r α (λ ) ≤ r β (λ ) for any 0 < α ≤ β and λ ∈ [0, 1/2]; (c) There exists a constant c 2 > 0 such that r β (λ ) − r α (λ ) ≤ c 2 λ α r β (λ ) for any 0 < α ≤ β and λ ∈ [0, 1/2]. The conditions (a) and (b) in Assumption 1 are standard in the analysis of linear regularization methods. Assumption 1(a) clearly implies 0 ≤ r α (λ )λ 1/2 ≤ c 3 α 1/2 and 0 ≤ g α (λ )λ 1/2 ≤ c 4 α −1/2 (2.1) with c 3 ≤ c 1/2 0 and c 4 ≤ c 1/2 1 . We emphasize that direct estimates on r α (λ )λ 1/2 and g α (λ )λ 1/2 could give smaller c 3 and c 4 . From Assumption 1(a) it also follows for each 0 ≤ ν ≤ 1 that r α (λ )λ ν ≤ c ν 0 α ν for all α > 0 and λ ∈ [0, 1/2]. Thus the linear regularization method defined by {g α } has qualificationν ≥ 1, where, according to [20], the qualification is defined to be the largest numberν with the property that for each 0 ≤ ν ≤ν there is a positive constant d ν such that r α (λ )λ ν ≤ d ν α ν for all α > 0 and λ ∈ [0, 1/2]. (2.2) Moreover, Assumption 1(a) implies for every µ > 0 that r α (λ )(− ln λ ) −µ ≤ min (− ln λ ) −µ , c 0 αλ −1 (− ln λ ) −µ for all 0 < α ≤ α 0 and λ ∈ [0, 1/2]. It is clear that (− ln λ ) −µ ≤ (− ln(α/(2α 0 ))) −µ for 0 ≤ λ ≤ α/(2α 0 ) . By using the fact that the function λ → c 0 αλ −1 (− ln λ ) −µ is decreasing on the interval (0, e −µ ] and is increasing on the interval [e −µ , 1), it is easy to show that there is a positive constant a µ such that c 0 αλ −1 (− ln λ ) −µ ≤ a µ (− ln(α/(2α 0 ))) −µ for α/(2α 0 ) ≤ λ ≤ 1/2. Therefore for every µ > 0 there is a positive constant b µ such that r α (λ )(− ln λ ) −µ ≤ b µ (− ln(α/(2α 0 ))) −µ (2.3) for all 0 < α ≤ α 0 and λ ∈ [0, 1/2]. This inequality will be used to derive the convergence rate when x 0 − x † satisfies the logarithmic source condition (1.11) The condition (c) in Assumption 1 seems to appear here for the first time. It is interesting to note that one can verify it for many well-known linear regularization methods. Moreover, the conditions (b) and (c) have the following important consequence. Lemma 1 Under the conditions (b) and (c) in Assumption 1, there holds [r β (A * A) − r α (A * A)]x ≤ x − r β (A * A)x + c 2 √ α Ax (2.4) for all x,x ∈ X, any 0 < α ≤ β and any bounded linear operator A : X → Y satisfying A ≤ 1/ √ 2. Proof For any 0 < α ≤ β we set p β ,α (λ ) := r β (λ ) − r α (λ ) r β (λ ) , λ ∈ [0, 1/2]. It follows from the conditions (a) and (b) in Assumption 1 that 0 ≤ p β ,α (λ ) ≤ min 1, c 2 λ α . (2.5) Therefore, for any x,x ∈ X, [r β (A * A) − r α (A * A)]x = p β ,α (A * A)r β (A * A)x ≤ p β ,α (A * A)[r β (A * A)x −x] + p β ,α (A * A)x ≤ r β (A * A)x −x + p β ,α (A * A)x . (2.6) Let {E λ } be the spectral family generated by A * A. Then it follows from (2.5) that p β ,α (A * A)x 2 = 1/2 0 p β ,α (λ ) 2 d E λx 2 ≤ c 2 2 1/2 0 λ α d E λx 2 = c 2 2 α (A * A) 1/2x 2 = c 2 2 α Ax 2 . Combining this with (2.6) gives the desired assertion. 2 For the sequence of positive numbers {α k }, we will always assume that it satisfies (1.4). Moreover, we need also the following condition on {α k } interplaying with r α . Assumption 2 There is a constant c 5 > 1 such that r α k (λ ) ≤ c 5 r α k+1 (λ ) for all k and λ ∈ [0, 1/2]. We remark that for some {g α } Assumption 2 is an immediate consequence of (1.4). However, this is not always the case; in some situations, Assumption 2 indeed imposes further conditions on {α k }. As a rough interpretation, Assumption 2 requires for any two successive iterated solutions the errors do not decrease dramatically. This may be good for the stable numerical implementations of ill-posed problems although it may require more iterations to be performed. Note that Assumption 2 implies r α k (A * A)x ≤ c 5 r α k+1 (A * A)x (2.7) for any x ∈ X and any bounded linear operator A : X → Y satisfying A ≤ 1/ √ 2. Throughout this paper, we will always assume that the nonlinear operator F : D(F) ⊂ X → Y is Fréchet differentiable such that B ρ (x † ) ⊂ D(F) for some ρ > 0 (2.8) and F ′ (x) ≤ min c 3 α 1/2 0 , β 1/2 0 , x ∈ B ρ (x † ),(2.9) where 0 < β 0 ≤ 1/2 is a number such that r α 0 (λ ) ≥ 3/4 for all λ ∈ [0, β 0 ]. Since r α 0 (0) = 1, such β 0 always exists. The scaling condition (2.9) can always be fulfilled by rescaling the norm in Y . The convergence analysis on the method defined by (1.3) and (1.7) will be divided into two cases: (i) x 0 − x † satisfies (1.10) for some ν ≥ 1/2; (ii) x 0 − x † satisfies (1.10) with 0 ≤ ν < 1/2 or (1.11) with µ > 0. Thus different structure conditions on F will be assumed in order to carry out the arguments. It is remarkable to see that for case (i) the following Lipschitz condition on F ′ is enough for our purpose. Assumption 3 There exists a constant L such that F ′ (x) − F ′ (z) ≤ L x − z (2.10) for all x, z ∈ B ρ (x † ). As the immediate consequence of Assumption 3, we have F(x) − F(z) − F ′ (z)(x − z) ≤ 1 2 L x − z 2 for all x, z ∈ B ρ (x † ). We will use this consequence frequently in this paper. During the convergence analysis of (1.3), we will meet some terms involving operators such as r α k (F ′ (x δ k ) * F ′ (x δ k )). In order to make use of the source conditions (1.10) for x 0 − x † , we need to switch these operators with r α k (F ′ (x † ) * F ′ (x † )). Thus we need the following commutator estimates involving r α and g α . Assumption 4 There is a constant c 6 > 0 such that r α (A * A) − r α (B * B) ≤ c 6 α −1/2 A − B , (2.11) [r α (A * A) − r α (B * B)] B * ≤ c 6 A − B , (2.12) A [r α (A * A) − r α (B * B)] B * ≤ c 6 α 1/2 A − B ,(2. 13) and [g α (A * A) − g α (B * B)] B * ≤ c 6 α −1 A − B (2.14) for any α > 0 and any bounded linear operators A, B : X → Y satisfying A , B ≤ 1/ √ 2. This assumption looks restrictive. However, it is interesting to note that for several important examples we indeed can verify it easily, see Section 6 for details. Moreover, in our applications, we only need Assumption 4 with A = F ′ (x) and B = F ′ (z) for x, z ∈ B ρ (x † ), which is trivially satisfied when F is linear. Now we are ready to state the first main result of this paper. (1.3) and let k δ be the first integer satisfying (1.7) with τ > 1. Let x 0 − x † satisfy (1.10) for some ω ∈ X and 1/2 ≤ ν ≤ν − 1/2. Then Theorem 1 tells us that, under merely the Lipschitz condition on F ′ , the method (1.3) together with (1.7) indeed defines an order optimal regularization method for each 1/2 ≤ ν ≤ν − 1/2; in case the regularization method defined by {g α } has infinite qualification the discrepancy principle (1.7) provides order optimal convergence rates for the full range ν ∈ [1/2, ∞). This is one of the main contribution of the present paper. ρ > 4 x 0 − x † . Let {x δ k } be defined byx δ k δ − x † ≤ C ν ω 1/(1+2ν) δ 2ν/(1+2ν) if L u ≤ η 0 , where u ∈ N (F ′ (x † ) * ) ⊥ ⊂ Y is the unique element such that x 0 − x † = F ′ (x † ) * u, η 0 > 0 We remark that under merely the Lipschitz condition on F ′ we are not able to prove the similar result as in Theorem 1 if x 0 − x † satisfies weaker source conditions, say (1.10) for some ν < 1/2. Indeed this is still an open problem in the convergence analysis of regularization methods for nonlinear ill-posed problems. In order to pursue the convergence analysis under weaker source conditions, we need stronger conditions on F than Assumption 3. The condition (1.9) has been used in [3,8] to establish the regularization property of the method defined by (1.8) and (1.7), where the special properties of g α (λ ) = (λ + α) −1 play the crucial roles. In order to study the general method (1.3) under weaker source conditions, we need the following two conditions on F. Assumption 5 There exists a positive constant K 0 such that F ′ (x) = F ′ (z)R(x, z), I − R(x, z) ≤ K 0 x − z for any x, z ∈ B ρ (x † ). Assumption 6 There exist positive constants K 1 and K 2 such that [F ′ (x) − F ′ (z)]w ≤ K 1 x − z F ′ (z)w + K 2 F ′ (z)(x − z) w for any x, z ∈ B ρ (x † ) and w ∈ X. Assumption 5 has been used widely in the literature of nonlinear ill-posed problems (see [17,11,9,19]); it can be verified for many important inverse problems. Another frequently used assumption on F is (1.9) which is indeed quite restrictive. It is clear that Assumption 6 is a direct consequence of (1.9). In order to illustrate that Assumption 6 could be weaker than (1.9), we consider the identification of the parameter c in the boundary value problem −∆ u + cu = f in Ω u = g on ∂ Ω (2.15) from the measurement of the state u, where Ω ⊂ R n , n ≤ 3, is a bounded domain with smooth boundary ∂ Ω , f ∈ L 2 (Ω ) and g ∈ H 3/2 (∂ Ω ). We assume c † ∈ L 2 (Ω ) is the sought solution. This problem reduces to solving an equation of the form (1.1) if we define the nonlinear operator F to be the parameter-to-solution mapping F : L 2 (Ω ) → L 2 (Ω ), F(c) := u(c) with u(c) ∈ H 2 (Ω ) ⊂ L 2 (Ω ) being the unique solution of (2.15). Such F is well-defined on D(F) := c ∈ L 2 (Ω ) : c −ĉ L 2 ≤ γ for someĉ ≥ 0 a.e. for some positive constant γ > 0. It is well-known that F has Fréchet derivative F ′ (c)h = −A(c) −1 (hF(c)), h ∈ L 2 (Ω ), (2.16) where A(c) : H 2 ∩ H 1 0 → L 2 is defined by A(c)u := −∆ u + cu which is an isomorphism uniformly in a ball B ρ (c † ) ⊂ D(F) around c † . Let V be the dual space of H 2 ∩H 1 0 with respect to the bilinear form ϕ, ψ = Ω ϕ(x)ψ(x)dx. Then A(c) extends to an isomorphism from L 2 (Ω ) to V . Since (2.16) implies for any c, d ∈ B ρ (c † ) and h ∈ L 2 (Ω ) F ′ (c) − F ′ (d) h = −A(c) −1 (c − d)F ′ (d)h − A(c) −1 (h(F(c) − F(d))) , and since L 1 (Ω ) embeds into V due to the restriction n ≤ 3, we have (F ′ (c) − F ′ (d))h L 2 ≤ A(c) −1 (c − d)F ′ (d)h L 2 + A(c) −1 (h(F(c) − F(d))) L 2 ≤ C (c − d)F ′ (d)h V + C h(F(c) − F(d)) V ≤ C (c − d)F ′ (d)h L 1 + C h(F(c) − F(d)) L 1 ≤ C c − d L 2 F ′ (d)h L 2 + C F(c) − F(d) L 2 h L 2 . (2.17) On the other hand, note that F(c) − F(d) = −A(d) −1 ((c − d)F(c)), by using (2.16) we obtain F(c) − F(d) − F ′ (d)(c − d) = −A(d) −1 ((c − d) (F(c) − F(d))) . Thus, by a similar argument as above, F(c) − F(d) − F ′ (d)(c − d) L 2 ≤ C c − d L 2 F(c) − F(d) L 2 . Therefore, if ρ > 0 is small enough, we have F(c) − F(d) L 2 ≤ C F ′ (d)(c − d) L 2 , which together with (2.17) verifies Assumption 6. The validity of (1.9), however, requires u(c) ≥ κ > 0 for all c ∈ B ρ (c † ), see [7]. In our next main result, Assumption 5 and Assumption 6 will be used to derive estimates related to x δ k − x † and F ′ (x † )(x δ k − x † ) respectively. Although Assumption 6 does not explore the full strength of (1.9), the plus of Assumption 5 could make our conditions stronger than (1.9) in some situations. One advantage of the use of Assumption 5 and Assumption 6, however, is that we can carry out the analysis on the discrepancy principle (1.7) for any τ > 1, in contrast to those results in [3,8] where τ is required to be sufficiently large. It is not yet clear if only one of the above two assumptions is enough for our purpose. From Assumption 6 it is easy to see that F(x) − F(z) − F ′ (z)(x − z) ≤ 1 2 (K 1 + K 2 ) x − z F ′ (z)(x − z) (2.18) and F(x) − F(z) − F ′ (z)(x − z) ≤ 3 2 (K 1 + K 2 ) x − z F ′ (x)(x − z) . (2.19) for any x, z ∈ B ρ (x † ). We still need to deal with some commutators involving r α . The structure information on F will be incorporated into such estimates. Thus, instead of Assumption 4, we need the following strengthened version. Assumption 7 (a) Under Assumption 5, there exists a positive constant c 7 such that r α F ′ (x) * F ′ (x) − r α F ′ (z) * F ′ (z) ≤ c 7 K 0 x − z (2.20) for all x, z ∈ B ρ (x † ) and all α > 0. F ′ (x) r α F ′ (x) * F ′ (x) − r α F ′ (z) * F ′ (z) ≤ c 8 (K 0 + K 1 )α 1/2 x − z + c 8 K 2 F ′ (x)(x − z) + F ′ (z)(x − z) (2.21) for all x, z ∈ B ρ (x † ) and all α > 0. Now we are ready to state the second main result in this paper which in particular says that the method (1.3) together with the discrepancy principle (1.7) defines an order optimal regularization method for each 0 < ν ≤ν − 1/2 under stronger conditions on F. We will fix a constant γ 1 > c 3 r 1/2 /(τ − 1). i , i = 0, · · · , 8, such that if (K 0 + K 1 + K 2 ) x 0 − x † ≤ η 1 then (i) If x 0 − x † satisfies the Hölder source condition (1.10) for some ω ∈ X and 0 < ν ≤ν − 1/2, then x δ k δ − x † ≤ C ν ω 1/(1+2ν) δ 2ν/(1+2ν) , (2.22) where C ν is a constant depending only on r, τ, ν and c i , (1.11) for some ω ∈ X and µ > 0, then i = 0, · · · , 8. (ii) If x 0 − x † satisfies the logarithmic source conditionx δ k δ − x † ≤ C µ ω 1 + ln δ ω −µ , (2.23) where C µ is a constant depending only on r, τ, µ, and c i , i = 0, · · · , 8. In the statements of Theorem 1 and Theorem 2, the smallness of L u and (K 0 + K 1 + K 2 ) x 0 − x † are not specified. However, during the proof of Theorem 1, we indeed will spell out all the necessary smallness conditions on L u . For simplicity of presentation, we will not spell out the smallness conditions on (K 0 + K 1 + K 2 ) x 0 − x † any more; the readers should be able to figure out such conditions without any difficulty. Note that, without any source condition on x 0 − x † , the above two theorems do not give the convergence of x δ k δ to x † . The following theorem says that x δ k δ → x † as δ → 0 provided x 0 − x † ∈ N (F ′ (x † )) ⊥ . In fact, it tells more, it says that the convergence rates can even be improved to o(δ 2ν/(1+2ν) ) if x 0 − x † satisfies (1.10) for 0 ≤ ν <ν − 1/2.ω ∈ N (F ′ (x † )) ⊥ and 1/2 ≤ ν <ν − 1/2, then x δ k δ − x † ≤ o(δ 2ν/(1+2ν) ) as δ → 0. (ii) Let all the conditions in Theorem 2 be fulfilled. If x 0 − x † satisfies (1.10) for some ω ∈ N (F ′ (x † )) ⊥ and 0 ≤ ν <ν − 1/2, then x δ k δ − x † ≤ o(δ 2ν/(1+2ν) ) as δ → 0. Theorem 1, Theorem 2 and Theorem 3 will be proved in Sections 3, 4 and 5 respectively. In the following we will give some remarks. Remark 1 A comprehensive overview on iterative regularization methods for nonlinear ill-posed problems may be found in the recent book [14]. In particular, convergence and convergence rates for the general method (1.3) are obtained in [14,Theorem 4.16] in case of a priori stopping rules under suitable nonlinearity assumptions on F. Remark 2 In [18] Tautenhahn introduced a general regularization scheme for (1.1) by defining the regularized solutions x δ α as a fixed point of the nonlinear equation x = x 0 − g α F ′ (x) * F ′ (x) F ′ (x) * F(x) − y δ − F ′ (x)(x − x 0 ) , (2.24) where α > 0 is the regularization parameter. When α is determined by a Morozov's type discrepancy principle, it was shown in [18] that the method is order optimal for each 0 < ν ≤ν/2 under certain conditions on F. We point out that the technique developed in the present paper can be used to analyze such method; indeed we can even show that, under merely the Lipschitz condition on F ′ , the method in [18] is order optimal for each 1/2 ≤ ν ≤ν − 1/2, which improves the corresponding result. Remark 3 Alternative to (1.3), one may consider the inexact Newton type methods x δ k+1 = x δ k − g α k F ′ (x δ k ) * F ′ (x δ k ) F ′ (x δ k ) * F(x δ k ) − y δ (2.25) which can be derived by applying the regularization method defined by {g α } to (1.5) with the current iterate x δ k as an initial guess. Such methods have first been studied by Hanke in [5,6] where the regularization properties of the Levenberg-Marquardt algorithm and the Newton-CG algorithm have been established without giving convergence rates when the sequence {α k } is chosen adaptively during computation and the discrepancy principle is used as a stopping rule. The general methods (2.25) have been considered later by Rieder in [15,16], where {α k } is determined by a somewhat different adaptive strategy; certain sub-optimal convergence rates have been derived when x 0 − x † satisfies (1.10) with η < ν ≤ 1/2 for some problem-dependent number 0 < η < 1/2, while it is not yet clear if the convergence can be established under weaker source conditions. The convergence analysis of (2.25) is indeed far from complete. The technique in the present paper does not work for such methods. Throughout this paper we will use {x k } to denote the iterated solutions defined by (1.3) corresponding to the noise free case. i.e. x k+1 = x 0 − g α k F ′ (x k ) * F ′ (x k ) F ′ (x k ) * F(x k ) − y − F ′ (x k )(x k − x 0 ) . (2.26) We will also use the notations A := F ′ (x † ) * F ′ (x † ), A k := F ′ (x k ) * F ′ (x k ), A δ k := F ′ (x δ k ) * F ′ (x δ k ), B := F ′ (x † )F ′ (x † ) * , B k := F ′ (x k )F ′ (x k ) * , B δ k := F ′ (x δ k )F ′ (x δ k ) * , and e k := x k − x † , e δ k := x δ k − x † . For ease of exposition, we will use C to denote a generic constant depending only on r. τ and c i , i = 0, · · · , 8, we will also use the convention Φ Ψ to mean that Φ ≤ CΨ for some generic constant C. Moreover, when we say L u (or (K 0 + K 1 + K 2 ) e 0 ) is sufficiently small we will mean that L u ≤ η (or (K 0 + K 1 + K 2 ) e 0 ≤ η) for some small positive constant η depending only on r, τ and c i , i = 0, · · · , 8. Proof of Theorem 1 In this section we will give the proof of Theorem 1. The main idea behind the proof consists of the following steps: • Show the method defined by (1.3) and (1.7) is well-defined. • Establish the stability estimate x δ k − x k δ / √ α k . This enables us to write e δ k δ e k δ + δ / √ α k δ . • Establish α k δ ≥ C ν (δ / ω ) 2/(1+2ν) under the source condition (1.10) for 1/2 ≤ ν ≤ν − 1/2. This is an easy step although it requires nontrivial arguments. • Show e k δ ≤ C ν ω 1/(1+2ν) δ 2ν/(1+2ν) , which is the hard part in the whole proof. In order to achieve this, we pick an integerk δ such that k δ ≤k δ and α¯k δ ∼ (δ / ω ) 2/(1+2ν) . Suchk δ will be proved to exist. Then we connect e k δ and e¯k δ by establishing the inequality e k δ e¯k δ + 1 α¯k δ F(x k δ ) − y + δ . (3.1) The right hand side can be easily estimated by the desired bound. • In order to establish (3.1), we need to establish the preliminary convergence rate estimate e δ k δ u 1/2 δ 1/2 when x 0 − x † = F ′ (x † ) * u for some u ∈ N (F ′ (x † ) * ) ⊥ ⊂ Y . Therefore, in order to complete the proof of Theorem 1, we need to establish various estimates. A first result on convergence rates In this subsection we will derive the convergence rate e δ k δ u 1/2 δ 1/2 under the source condition x 0 − x † = F ′ (x † ) * u, u ∈ N (F ′ (x † ) * ) ⊥ . (3.2) To this end, we introducek δ to be the first integer such that α˜k δ ≤ δ γ 0 u < α k , 0 ≤ k <k δ ,(3.3) where γ 0 is a number satisfying γ 0 > c 0 r/(τ − 1), and c 0 is the constant from Assumption 1 (a). Because of (1.4), suchk δ is well-defined. 3) and let k δ be determined by the discrepancy principle (1.7) with τ > 1. If x 0 − x † satisfies (3.2) and if L u is sufficiently small, then (i) For all 0 ≤ k ≤k δ there hold x δ k ∈ B ρ (x † ) and e δ k ≤ 2(c 3 + c 4 γ 0 )r 1/2 α 1/2 k u . (3.4) (ii) k δ ≤k δ , i.e. the discrepancy principle (1.7) is well-defined. (iii) There exists a generic constant C > 0 such that e δ k δ ≤ C u 1/2 δ 1/2 . Proof We first prove (i). Note that ρ > 4 x 0 − x † , it follows from (3.2) and (2.9) that (3.4) is trivial for k = 0. Now for any fixed integer 0 < l ≤k δ , we assume that (3.4) is true for all 0 ≤ k < l. It follows from the definition (1. 3) of {x δ k } that e δ k+1 = r α k (A δ k )e 0 − g α k (A δ k )F ′ (x δ k ) * F(x δ k ) − y δ − F ′ (x δ k )e δ k .e δ k+1 ≤ r α k (A δ k )F ′ (x δ k ) * u + r α k (A δ k )[F ′ (x † ) * − F ′ (x δ k ) * ]u + c 4 α −1/2 k F(x δ k ) − y δ − F ′ (x δ k )e δ k ≤ c 3 α 1/2 k u + L u e δ k + 1 2 c 4 L e δ k 2 α −1/2 k + c 4 δ α −1/2 k . Note that δ α −1 k ≤ γ 0 u for 0 ≤ k <k δ . Note also that α k ≤ rα k+1 by (1.4). Therefore, by using Therefore, by using ρ > 4 e 0 , we have (3.4) with k = l − 1, we obtain e δ l ≤ r 1/2 α 1/2 l   (c 3 + c 4 γ 0 ) u + L u e δ l−1 √ α l−1 + 1 2 c 4 L e δ l−1 √ α l−1 2   ≤ 2(c 3 + c 4 γ 0 )r 1/2 α 1/2 l u if L u ise δ k ≤ 3 4 ρ + c 4 γ 1/2 0 u 1/2 δ 1/2 < ρ if δ > 0 is small enough. Thus (3.4) is also true for all k = l. As l ≤k δ has been arbitrary, we have completed the proof of (i). Next we prove (ii) by showing that k δ ≤k δ . From (3.5) and (3.2) we have for 0 ≤ k <k δ that F ′ (x † )e δ k+1 − y δ + y = F ′ (x δ k )r α k (A δ k ) F ′ (x δ k ) * + F ′ (x † ) * − F ′ (x δ k ) * u + F ′ (x † ) − F ′ (x δ k ) r α k (A δ k ) F ′ (x δ k ) * + F ′ (x † ) * − F ′ (x δ k ) * u − F ′ (x † ) − F ′ (x δ k ) g α k (A δ k )F ′ (x δ k ) * F(x δ k ) − y δ − F ′ (x δ k )e δ k − g α k (B δ k )B δ k F(x δ k ) − y − F ′ (x δ k )e δ k − r α k (B δ k )(y δ − y). By using Assumption 3, Assumption 1(a), (2.1), (1.2) and (3.4), and noting that δ /α k ≤ γ 0 u , we obtain F ′ (x † )e δ k+1 − y δ + y ≤ δ + c 0 α k u + 2c 3 L u α 1/2 k e δ k + L 2 u e δ k 2 + c 4 L e δ k δ α −1/2 k + 1 2 c 4 L 2 α −1/2 k e δ k 3 + 1 2 L e δ k 2 ≤ δ + (c 0 + ε 1 ) α k u , where ε 1 = 2r 1/2 (c 3 + c 4 γ 0 )(2c 3 + c 4 γ 0 ) + 2(c 3 + c 4 γ 0 ) 2 r L u + 4 (c 3 + c 4 γ 0 ) 2 r + (c 3 + c 4 γ 0 ) 3 c 4 r 3/2 L 2 u 2 . From (1.2), (3.2) and (2.9) we have F ′ (x † )e 0 − y δ + y ≤ δ + A u ≤ δ + c 0 α 0 u . Thus, by using (1.4), F ′ (x † )e δ k − y δ + y ≤ δ + r (c 0 + ε 1 ) α k u , 0 ≤ k ≤k δ . Consequently F(x δ k δ ) − y δ ≤ F ′ (x † )e δ k δ − y δ + y + F(x δ k δ ) − y − F ′ (x † )e δ k δ ≤ δ + r (c 0 + ε 1 ) α˜k δ u + 1 2 L e δ k δ 2 ≤ δ + r c 0 + ε 1 + 2(c 3 + c 4 γ 0 ) 2 rL u α˜k δ u ≤ δ + r c 0 + ε 1 + 2(c 3 + c 4 γ 0 ) 2 rL u γ −1 0 δ ≤ τδ if L u is so small that ε 1 + 2(c 3 + c 4 γ 0 ) 2 rL u ≤ (τ − 1)γ 0 − c 0 r r . By the definition of k δ , it follows that k δ ≤k δ . Finally we are in a position to derive the convergence rate in (iii). If k δ = 0, then, by the definition of k δ , we have F(x 0 ) − y δ ≤ τδ . This together with Assumption 3 and (1.2) gives F ′ (x † )e 0 ≤ F(x 0 ) − y − F ′ (x † )e 0 + F(x 0 ) − y ≤ 1 2 L e 0 2 + (τ + 1)δ . Thus, by using (3.2), we have e 0 = (e 0 , F ′ (x † ) * u) 1/2 = (F ′ (x † )e 0 , u) 1/2 ≤ F ′ (x † )e 0 1/2 u 1/2 ≤ 1 2 L u e 0 + √ τ + 1 u 1/2 δ 1/2 . By assuming that L u ≤ 1, we obtain e δ k δ = e 0 u 1/2 δ 1/2 . Therefore we will assume k δ > 0 in the following argument. It follows from (3.5), (2.1), Assumption 3 and (3.4) that for 0 ≤ k <k δ e δ k+1 ≤ r α k (A δ k )e 0 + c 4 δ α −1/2 k + 1 2 c 4 L e δ k 2 α −1/2 k ≤ r α k (A δ k )e 0 + c 4 (γ 0 u δ ) 1/2 + (c 3 + c 4 γ 0 )c 4 r 1/2 L u e δ k .(3.7) By (3.2), (2.12) in Assumption 4, and Assumption 3 we have r α k (A δ k )e 0 − r α k (A )e 0 = [r α k (A δ k ) − r α k (A )]F ′ (x † ) * u ≤ c 6 u F ′ (x δ k ) − F ′ (x † ) ≤ c 6 L u e δ k . (3.8) Thus e δ k+1 ≤ r α k (A )e 0 + c 4 (γ 0 u δ ) 1/2 + c 6 + (c 3 + c 4 γ 0 )c 4 r 1/2 L u e δ k ≤ r α k (A )e 0 + c 4 (γ 0 u δ ) 1/2 + 1 4c 5 e δ k (3.9) if we assume further that 4c 5 c 6 + (c 3 + c 4 γ 0 )c 4 r 1/2 L u ≤ 1. (3.10) Note that (2.9) and the choice of β 0 imply r α 0 (A )e 0 ≥ 3 4 e 0 . Thus, with the help of (2.7), by induction we can conclude from (3.9) that e δ k ≤ 4 3 c 5 r α k (A )e 0 + C u 1/2 δ 1/2 , 0 ≤ k ≤k δ . This together with (3.8) and (3.10) implies e δ k ≤ 2c 5 r α k (A δ k )e 0 + C u 1/2 δ 1/2 , 0 ≤ k ≤k δ . (3.11) The combination of (3.7), (3.11) and (3.10) gives e δ k+1 ≤ 3 2 r α k (A δ k )e 0 + C u 1/2 δ 1/2 , 0 ≤ k <k δ . (3.12) We need to estimate r α k (A δ k )e 0 . By (3.2), Assumption 1(a) and Assumption 3 we have r α k (A δ k )e 0 2 = r α k (A δ k )e 0 , r α k (A δ k )F ′ (x † ) * u = r α k (A δ k )e 0 , r α k (A δ k ) F ′ (x δ k ) * + F ′ (x † ) * − F ′ (x δ k ) * u ≤ F ′ (x δ k )r α k (A δ k )e 0 u + L u e δ k r α k (A δ k )e 0 . Thus r α k (A δ k )e 0 ≤ F ′ (x δ k )r α k (A δ k )e 0 1/2 u 1/2 + L u e δ k . With the help of (3.5), (1.2), Assumption 1(a) and Assumption 3 we have F ′ (x δ k )r α k (A δ k )e 0 ≤ F ′ (x δ k )e δ k+1 + g α k (B δ k )B δ k F(x δ k ) − y δ − F ′ (x δ k )e δ k ≤ F(x δ k+1 ) − y δ + 2δ + F(x δ k+1 ) − y − F ′ (x δ k+1 )e δ k+1 + [F ′ (x δ k+1 ) − F ′ (x δ k )]e δ k+1 + F(x δ k ) − y − F ′ (x δ k )e δ k ≤ F(x δ k+1 ) − y δ + 2δ + L e δ k 2 + 2L e δ k+1 2 . Therefore r α k (A δ k )e 0 ≤ u 1/2 F(x δ k+1 ) − y δ 1/2 + √ 2 u 1/2 δ 1/2 + 2L u e δ k+1 + L u + L u e δ k . Combining this with (3.11) and (3.12) yields r α k (A δ k )e 0 ≤ u 1/2 F(x δ k+1 ) − y δ 1/2 + C u 1/2 δ 1/2 + 1 2 3 √ 2 + 4c 5 L u + 4c 5 L u r α k (A δ k )e 0 . Thus, if 3 √ 2 + 4c 5 L u + 4c 5 L u ≤ 1, we then obtain r α k (A δ k )e 0 u 1/2 F(x δ k+1 ) − y δ 1/2 + u 1/2 δ 1/2 . This together with (3.12) gives e δ k u 1/2 F(x δ k ) − y δ 1/2 + u 1/2 δ 1/2 for all 0 < k ≤k δ . Consequently, we may set k = k δ in the above inequality and use the definition of k δ to obtain e δ k δ u 1/2 δ 1/2 . 2 Stability estimates In this subsection we will consider the stability of the method (1.3) by deriving some useful estimates on x δ k − x k , where {x k } is defined by (2.26). It is easy to see that e k+1 = r α k (A k )e 0 − g α k (A k )F ′ (x k ) * F(x k ) − y − F ′ (x k )e k .(3.13) We will prove some important estimates on {x k } in Lemma 3 in the next subsection. In particular, we will show that, under the conditions in Theorem 4, x k ∈ B ρ (x † ) and e k ≤ 2c 3 r 1/2 α 1/2 k u (3.14) for all k ≥ 0 provided L u is sufficiently small. Proof For each 0 ≤ k ≤k δ we set Lemma 2 Let all the conditions in Theorem 4 and Assumption 4 hold. If L u is sufficiently small, then for all 0 ≤ k ≤k δ there hold x δ k − x k ≤ 2c 4 δ √ α k (3.15) and F(x δ k ) − F(x k ) − y δ + y ≤ (1 + ε 2 )δ ,(3.u k := F(x k ) − y − F ′ (x k )e k , u δ k := F(x δ k ) − y − F ′ (x δ k )e δ k .(3.17) It then follows from (3.5) and (3.13) that x δ k+1 − x k+1 = I 1 + I 2 + I 3 + I 4 ,(3.18) where I 1 := r α k (A δ k ) − r α k (A k ) e 0 , I 2 := g α k (A δ k )F ′ (x δ k ) * (y δ − y), I 3 := g α k (A k )F ′ (x k ) * − g α k (A δ k )F ′ (x δ k ) * u k , I 4 := g α k (A δ k )F ′ (x δ k ) * (u k − u δ k ). By using (3.2), (2.11), (2.12), Assumption 3 and (3.14) we have I 1 ≤ r α k (A δ k ) − r α k (A k ) F ′ (x † ) * − F ′ (x k ) * u + [r α k (A δ k ) − r α k (A k )]F ′ (x k ) * u ≤ c 6 L 2 u e k x δ k − x k α −1/2 k + c 6 L u x δ k − x k ≤ c 6 L u + 2c 3 r 1/2 L 2 u 2 x δ k − x k . With the help of (2.1) and (1.2) we have I 2 ≤ c 4 δ √ α k . By applying Assumption 1(a), (2.14), Assumption 3 and (3.14) we can estimate I 3 as I 3 ≤ g α k (A k )[F ′ (x δ k ) * − F ′ (x k ) * ]u k + [g α k (A k ) − g α k (A δ k )]F ′ (x δ k ) * u k ≤ (c 1 + c 6 )L u k x δ k − x k α −1 k ≤ 1 2 (c 1 + c 6 )L 2 e k 2 x δ k − x k α −1 k ≤ 2(c 1 + c 6 )c 2 3 rL 2 u 2 x δ k − x k . For the term I 4 , we have from (2.1) that I 4 ≤ c 4 √ α k u δ k − u k . By using Assumption 3, (3.4) and (3.14) one can see u k − u δ k ≤ F(x δ k ) − F(x k ) − F ′ (x k )(x δ k − x k ) + [F ′ (x δ k ) − F ′ (x k )]e δ k ≤ 1 2 L x δ k − x k 2 + L e δ k x δ k − x k ≤ 1 2 L 3 e δ k + e k x δ k − x k ≤ (4c 3 + 3c 4 γ 0 ) r 1/2 α 1/2 k L u x δ k − x k .(3.19) Therefore I 4 ≤ (4c 3 + 3c 4 γ 0 ) c 4 r 1/2 L u x δ k − x k . Thus, if L u is so small that c 6 + (4c 3 + 3c 4 γ 0 )c 4 r 1/2 L u + 2 c 3 c 6 r 1/2 + c 2 3 (c 1 + c 6 )r L 2 u 2 ≤ 1 2 , then the combination of the above estimates on I 1 , I 2 , I 3 and I 4 gives for 0 ≤ k <k δ that x δ k+1 − x k+1 ≤ c 4 δ √ α k + 1 2 x δ k − x k . This implies (3.15) immediately. Next we prove (3.16). We have from (3.18) that F ′ (x δ k )(x δ k+1 − x k+1 ) − y δ + y = F ′ (x δ k ) (I 1 + I 2 + I 3 + I 4 ) − y δ + y. (3.20) From (3.2), (2.12), (2.13), Assumption 3, (3.14) and (3.15) it follows that F ′ (x δ k )I 1 ≤ F ′ (x δ k )[r α k (A δ k ) − r α k (A k )][F ′ (x † ) * − F ′ (x k ) * ]u + F ′ (x δ k )[r α k (A δ k ) − r α k (A k )]F ′ (x k ) * u ≤ c 6 L 2 u e k x δ k − x k + c 6 L u α 1/2 k x δ k − x k ≤ 2c 4 c 6 L u + 4c 3 c 4 c 6 r 1/2 L 2 u 2 δ . By using Assumption 1(a) and (1.2) it is easy to see F ′ (x δ k )I 2 − y δ + y = r α k (B δ k )(y δ − y) ≤ δ . (3.21) In order to estimate F ′ (x δ k )I 3 , we note that F ′ (x δ k )I 3 = F ′ (x δ k ) − F ′ (x k ) g α k (A k )F ′ (x k ) * u k + r α k (B δ k ) − r α k (B k ) u k . (3.22) Thus, it follows from (2.1), Assumption 3, (2.11), (3.14) and (3.15) that F ′ (x δ k )I 3 ≤ F ′ (x δ k ) − F ′ (x k ) g α k (A k )F ′ (x k ) * u k + r α k (B δ k ) − r α k (B k ) u k ≤ (c 4 + c 6 )α −1/2 k L x δ k − x k u k ≤ 1 2 (c 4 + c 6 )α −1/2 k L 2 e k 2 x δ k − x k ≤ 4(c 4 + c 6 )c 2 3 c 4 rL 2 u 2 δ . For the term F ′ (x δ k )I 4 we have from Assumption 1(a), (3.19) and (3.15) that F ′ (x δ k )I 4 ≤ u k − u δ k ≤ 2(4c 3 + 3c 4 γ 0 )c 4 r 1/2 L u δ . Combining the above estimates, we therefore obtain F ′ (x δ k )(x δ k+1 − x k+1 ) − y δ + y ≤ (1 + ε 3 )δ , 0 ≤ k <k δ , where ε 3 :=2c 4 c 6 + (4c 3 + 3c 4 γ 0 )r 1/2 L u + 4c 3 c 4 c 6 r 1/2 + (c 4 + c 6 )c 3 r L 2 u 2 . This together with Assumption 3, (3.4), (3.15) and (1.4) implies for 0 ≤ k <k δ that F ′ (x δ k+1 )(x δ k+1 − x k+1 ) − y δ + y ≤ F ′ (x δ k )(x δ k+1 − x k+1 ) − y δ + y + L x δ k+1 − x δ k x δ k+1 − x k+1 ≤ (1 + ε 3 )δ + 2c 4 L( e δ k+1 + e δ k ) δ √ α k+1 ≤ (1 + ε 4 )δ , where ε 4 := ε 3 + 8(c 3 + c 4 γ 0 )c 4 rL u . Thus F ′ (x δ k )(x δ k − x k ) − y δ + y ≤ (1 + ε 4 )δ , 0 ≤ k ≤k δ . Therefore, noting that δ /α k ≤ rγ 0 u for 0 ≤ k ≤k δ , we have F(x δ k ) − F(x k ) − y δ + y ≤ F(x δ k ) − F(x k ) − F ′ (x δ k )(x δ k − x k ) + F ′ (x δ k )(x δ k − x k ) − y δ + y ≤ 1 2 L x δ k − x k 2 + (1 + ε 4 )δ ≤ 2c 2 4 L δ α k δ + (1 + ε 4 )δ ≤ (1 + ε 4 + 2rc 2 4 γ 0 L u )δ . The proof of (3.16) is thus complete. e k+1 − r α k (A )e 0 ≤ [r α k (A k ) − r α k (A )]F ′ (x † ) * u + c 4 √ α k F(x k ) − y − F ′ (x k )e k ≤ c 6 L u e k + c 4 2 √ α k L e k 2 .1/2 k u + c 6 L u e k + c 4 2 √ α k L e k 2 . Note that (3.2) and (2.9) imply e 0 ≤ c 3 α 1/2 0 u . By induction one can conclude the assertion (3.23) if L u is so small that 2(c 6 r 1/2 + c 3 c 4 r)L u ≤ 1. If we assume further that Lemma 4 Let all the conditions in Lemma 2 and Assumption 1(c) hold. If k δ > 0 and L u is sufficiently small, then for all k ≥ k δ we have e k δ e k + 1 √ α k F(x k δ ) − y + δ . (3.29) Proof It follows from (3.13) that x k δ − x k = [r α k δ −1 (A ) − r α k−1 (A )]e 0 + [r α k δ −1 (A k δ −1 ) − r α k δ −1 (A )]e 0 − r α k−1 (A k−1 ) − r α k−1 (A ) e 0 − g α k δ −1 (A k δ −1 )F ′ (x k δ −1 ) * F(x k δ −1 ) − y − F ′ (x k δ −1 )e k δ −1 + g α k−1 (A k−1 )F ′ (x k−1 ) * F(x k−1 ) − y − F ′ (x k−1 )e k−1 . (3.30) Thus, by using (3.2), (2.12), Assumption 3, (2.1), (3.23) and (3.27), we have x k δ − x k ≤ [r α k δ −1 (A ) − r α k−1 (A )]e 0 + c 6 L u e k−1 + e k δ −1 + c 4 2 √ α k δ −1 L e k δ −1 2 + c 4 2 √ α k−1 L e k−1 2 ≤ [r α k δ −1 (A ) − r α k−1 (A )]e 0 + 1 5c 5 e k−1 + e k δ −1 . (3.31) Since k ≥ k δ , we have α k−1 ≤ α k δ −1 . Since Assumption 1(b) and (c) hold, we may apply Lemma 1 with x = e 0 ,x = e k δ , α = α k−1 , β = α k δ −1 and A = F ′ (x † ) to obtain [r α k δ −1 (A ) − r α k−1 (A )]e 0 ≤ r α k δ −1 (A )e 0 − e k δ + c 2 √ α k−1 F ′ (x † )e k δ . Note that (3.28) implies e k δ − r α k δ −1 (A )e 0 ≤ 1 5c 5 e k δ −1 . Note also that Assumption 3 implies F ′ (x † )e k δ ≤ F(x k δ ) − y + 1 2 L e k δ 2 . Thus [r α k δ −1 (A ) − r α k−1 (A )]e 0 ≤ 1 5c 5 e k δ −1 + C √ α k F(x k δ ) − y + L e k δ 2 . Since Lemma 2, Theorem 4 and the fact k δ ≤k δ imply e k δ e δ k δ + δ √ α k δ u 1/2 δ 1/2 , we have [r α k δ −1 (A ) − r α k−1 (A )]e 0 ≤ 1 5c 5 e k δ −1 + C √ α k F(x k δ ) − y + L u δ . Combining this with (3.31) and using Lemma 3 gives x k δ − x k ≤ 4 5 e k δ + C e k + C √ α k F(x k δ ) − y + δ . This completes the proof. F ′ (x † )e k r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 (3.32) for all k ≥ 0. Proof We first use (3.13) to write F ′ (x † )e k+1 = F ′ (x † )r α k (A )e 0 + F ′ (x † ) r α k (A k ) − r α k (A ) e 0 − F ′ (x † )g α k (A k )F ′ (x k ) * F(x k ) − y − F ′ (x k )e k .F ′ (x † )e k+1 F ′ (x † )r α k (A )e 0 + L e k [r α k (A k ) − r α k (A )]F ′ (x † ) * u + F ′ (x k )[r α k (A k ) − r α k (A )]F ′ (x † ) * u + (1 + L e k α −1/2 k )L e k 2 r α k (A )A 1/2 e 0 + L 2 u e k 2 + α 1/2 k L u e k + L e k 2 r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 .(τ − 1)δ r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 (3.34) for all 0 ≤ k < k δ , Proof By using (3.16), Lemma 3 and Lemma 5, we have for 0 ≤ k < k δ that τδ ≤ F(x δ k ) − y δ ≤ F(x δ k ) − F(x k ) − y δ + y + F(x k ) − y ≤ (1 + ε 2 )δ + F ′ (x † )e k + 1 2 L e k 2 ≤ (1 + ε 2 )δ + C r α k (A )A 1/2 e 0 + Cα 1/2 k r α k (A )e 0 . Since τ > 1, by the smallness condition ε 2 ≤ (τ − 1)/2 on L u we obtain (3.34). 2 Proof of Theorem 1. If k δ = 0, then the definition of k δ implies F(x 0 ) − y δ ≤ τδ . From Theorem 4 we know that e 0 u 1/2 δ 1/2 . Thus F ′ (x † )e 0 ≤ F(x 0 ) − y − F ′ (x † )e 0 + F(x 0 ) − y δ + δ ≤ 1 2 L e 0 2 + (1 + τ)δ δ . Since e 0 = A ν ω for some 1/2 ≤ ν ≤ν − 1/2, we may use the interpolation inequality to obtain e δ k δ = e 0 = A ν ω ≤ ω 1/(1+2ν) A 1/2+ν ω 2ν/(1+2ν) = ω 1/(1+2ν) F ′ (x † )e 0 2ν/(1+2ν) ω 1/(1+2ν) δ 2ν/(1+2ν) , which gives the desired estimate. Therefore, we may assume that k δ > 0 in the remaining argument. By using e 0 = A ν ω for some 1/2 ≤ ν ≤ν − 1/2 and Lemma 6 it follows that there exists a positive constant C ν such that (τ − 1)δ < C ν α ν+1/2 k ω , 0 ≤ k < k δ . Now we define the integerk δ by α¯k δ ≤ (τ − 1)δ C ν ω 2/(1+2ν) < α k , 0 ≤ k <k δ . Then k δ ≤k δ . Thus, by using Lemma 2 and Lemma 4, we have e δ k δ e k δ + δ √ α k δ e¯k δ + F(x k δ ) − y + δ α¯k δ + δ √ α k δ . Note that Lemma 2 and the definition of k δ imply F(x k δ ) − y ≤ F(x δ k δ ) − y δ + F(x δ k δ ) − F(x k δ ) − y δ + y δ . This together with (3.24), k δ ≤k δ and r α k (A )e 0 α ν k ω then gives e δ k δ α ν k δ ω + δ √ α k δ + δ α¯k δ α ν k δ ω + δ α¯k δ . (3.35) Using the definition ofk δ and (1.4), we therefore complete the proof. 2 Proof of Theorem 2 In this section we will give the proof of Theorem 2. The essential idea is similar as in the proof of Theorem 1. Thus we need to establish similar results as those used in Section 3. However, since we do not have source representation e 0 = F ′ (x † ) * u any longer and since F satisfies different conditions, we must modify the arguments carefully. We will indicate the essential steps without spelling out all the necessary smallness conditions on (K 0 + K 1 + K 2 ) e 0 . We first introduce the integer n δ by α n δ ≤ δ γ 1 e 0 2 < α k , 0 ≤ k < n δ . (4.1) Recall that γ 1 is a constant satisfying γ 1 > c 3 r 1/2 /(τ − 1). Proof of Theorem 2. In order to complete the proof of Theorem 2, we need to establish various estimates. We will divide the arguments into several steps. Step 1. We will show that for all 0 ≤ k ≤ n δ x δ k ∈ B ρ (x † ), e δ k e 0 ,(4.2)F ′ (x † )e δ k α 1/2 k e 0 (4.3) and that k δ ≤ n δ for the integer k δ defined by the discrepancy principle (1.7) with τ > 1. To see this, we note that, for any 0 ≤ k < n δ with x δ k ∈ B ρ (x † ), (3.5) and Assumption 5 imply e δ k+1 = r α k (A δ k )e 0 − 1 0 g α k (A δ k )A δ k R(x δ k − te δ k , x δ k ) − I e δ k dt + g α k (A δ k )F ′ (x δ k ) * (y δ − y). Therefore, with the help of Assumption 1(a) and (2.1), we have e δ k+1 ≤ e 0 + 1 2 K 0 e δ k 2 + c 4 δ α −1/2 k ≤ (1 + c 4 γ 1 ) e 0 + 1 2 K 0 e δ k 2 . Thus, if 2(1 + c 4 γ 1 )K 0 e 0 ≤ 1, then, by using ρ > 2(1 + c 4 γ 1 ) e 0 and an induction argument, we can conclude e δ k ≤ 2(1 + c 4 γ 1 ) e 0 < ρ for all 0 ≤ k ≤ n δ . This establishes (4.2). Next we show (4.3). It follows from (3.5), Assumption 1(a), (1.2), (2.19) and (4.1) that for 0 ≤ k < n δ F ′ (x δ k )e δ k+1 α 1/2 k e 0 + δ + F(x δ k ) − y − F ′ (x δ k )e δ k α 1/2 k e 0 + (K 1 + K 2 ) e δ k F ′ (x † )e δ k . By Assumption 6 we have [F ′ (x † ) − F ′ (x δ k )]e δ k+1 ≤ K 1 e δ k F ′ (x † )e δ k+1 + K 2 e δ k+1 F ′ (x † )e δ k . The above two inequalities and (4.2) then imply F ′ (x † )e δ k+1 α 1/2 k e 0 + K 1 e 0 F ′ (x † )e δ k+1 + (K 1 + K 2 ) e 0 F ′ (x † )e δ k . Thus, if (K 1 + K 2 ) e 0 is sufficiently small, we can conclude (4.3) by an induction argument. As direct consequences of (4.2), (4.3) and Assumption 6 we have F ′ (x δ k )e δ k α 1/2 k e 0 , 0 ≤ k ≤ n δ (4.4) and F ′ (x δ k+1 )(x δ k+1 − x δ k ) α 1/2 k e 0 , 0 ≤ k < n δ . (4.5) In order to show k δ ≤ n δ , we note that (3.5) gives F ′ (x † )e δ k+1 − y δ + y = F ′ (x δ k )r α k (A δ k )e 0 + F ′ (x † ) − F ′ (x δ k ) r α k (A δ k )e 0 − F ′ (x † ) − F ′ (x δ k ) g α k (A δ k )F ′ (x δ k ) * F(x δ k ) − y δ − F ′ (x δ k )e δ k − g α k (B δ k )B δ k F(x δ k ) − y − F ′ (x δ k )e δ k − r α k (B δ k )(y δ − y).≤ k < n δ F ′ (x † )e δ k+1 − y δ + y ≤ δ + c 3 α 1/2 k e 0 + c 3 K 1 e 0 e δ k α 1/2 k + K 2 e 0 F ′ (x δ k )e δ k + K 1 e δ k δ + 1 2 (K 1 + K 2 ) e δ k F ′ (x δ k )e δ k + c 4 K 2 α −1/2 k F ′ (x δ k )e δ k δ + 1 2 (K 1 + K 2 ) e δ k F ′ (x δ k )e δ k + 1 2 (K 1 + K 2 ) e δ k F ′ (x δ k )e δ k ≤ δ + (c 3 + C(K 1 + K 2 ) e 0 ) α 1/2 k e 0 ≤ δ + r 1/2 (c 3 + C(K 1 + K 2 ) e 0 ) α 1/2 k+1 e 0 . Recall that γ 1 > c 3 r 1/2 /(τ − 1). Thus, with the help of (4.2), (4.3) and the definition of n δ , one can see that, if (K 1 + K 2 ) e 0 is sufficiently small, then F(x δ n δ ) − y δ ≤ F(x δ n δ ) − y − F ′ (x † )e δ n δ + F ′ (x † )e δ n δ − y δ + y ≤ δ + r 1/2 (c 3 + C(K 1 + K 2 ) e 0 )α 1/2 n δ e 0 + 1 2 (K 1 + K 2 ) e δ n δ F ′ (x † )e δ n δ ≤ δ + r 1/2 (c 3 + C(K 1 + K 2 ) e 0 )α 1/2 n δ e 0 ≤ δ + r 1/2 (c 3 + C(K 1 + K 2 ) e 0 )γ −1 1 δ ≤ τδ . This implies k δ ≤ n δ . Step 2. We will show, for the noise-free iterated solutions {x k }, that for all k ≥ 0 and for all 0 ≤ k ≤ l e k e l + 1 r α k (A )e 0 e k r α k (A )e 0 ,(4.√ α l F(x k ) − y . (4.8) In fact, from (3.13) and Assumption 5 it is easy to see that This together with (4.9) and (2.20) gives e k+1 − r α k (A k )e 0 ≤ 1 2 K 0 e k 2 .e k+1 − r α k (A )e 0 [r α k (A k ) − r α k (A )]e 0 + K 0 e k 2 K 0 e 0 e k . (4.11) Thus, by Assumption 2 and the smallness of K 0 e 0 we obtain (4.6) by induction. (4.7) is an immediate consequence of (4.11) and (4.6). In order to show (4.8), we first consider the case k > 0. Note that x k − x l has a similar expression as in (3. [r α k−1 (A ) − r α l−1 (A )]e 0 r α k−1 (A )e 0 − e k + 1 √ α l−1 F ′ (x † )e k . With the help of (2.18), (4.10), and the smallness of (K 1 + K 2 ) e 0 , we have F ′ (x † )e k ≤ F(x k ) − y + 1 2 F ′ (x † )e k . (4.13) Therefore F ′ (x † )e k ≤ 2 F(x k ) − y . This together with (4.11) and (4.7) then implies [r α k−1 (A ) − r α l−1 (A )]e 0 K 0 e 0 e k + 1 √ α l F(x k ) − y . Combining this with (4.12) gives x k − x l K 0 e 0 e k + e l + 1 √ α l F(x k ) − y which implies (4.8) if K 0 e 0 is sufficiently small. For the case k = 0, we can assume l ≥ 1. Since (4.8) is valid for k = 1, we may use (4.7) to conclude that (4.8) is also true for k = 0. Step 3. We will show for all k ≥ 0 that F ′ (x † )e k r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 . (4.14) To this end, first we may use the similar manner in deriving (4.3) to conclude F ′ (x † )e k α 1/2 k e 0 . (4.15) Note that Assumption 6 and (4.10) imply [F ′ (x † ) − F ′ (x k )]e k ≤ (K 1 + K 2 ) e k F ′ (x † )e k (K 1 + K 2 ) e 0 F ′ (x † )e k . Therefore F ′ (x k )e k F ′ (x † )e k . (4. 16) In particular this implies F ′ (x k )e k α 1/2 k e 0 .F ′ (x † )e k+1 r α k (A )A 1/2 e 0 + (K 0 + K 1 ) e 0 e k α 1/2 k + K 2 e 0 F ′ (x † )e k + F ′ (x k )e k + (K 1 + K 2 ) e k F ′ (x k )e k + K 1 (K 1 + K 2 ) e k 2 F ′ (x k )e k + K 2 (K 1 + K 2 ) e k F ′ (x k )e k 2 α −1/2 k . Thus, with the help of (4.6), (4.15), (4.16), (4.17) and (4.10), we obtain F ′ (x † )e k+1 r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 + K 2 e 0 F ′ (x † )e k . The estimates (4.14) thus follows by Assumption 2 and an induction argument if K 2 e 0 is sufficiently small. Step 4. Now we will establish some stability estimates. We will show for all 0 ≤ k ≤ n δ that x δ k − x k δ √ α k (4.18) and F(x δ k ) − F(x k ) − y δ + y ≤ (1 + C(K 0 + K 1 + K 2 ) e 0 )δ . (4.19) In order to show (4.18), we use again the decomposition (3.18) for x δ k+1 − x k+1 . We still have I 2 ≤ c 4 δ / √ α k . By using (2.20) the term I 1 can be estimated as I 1 K 0 e 0 x δ k − x k . In order to estimate I 3 , we note that Assumption 5 implies I 3 = 1 0 g α k (A k )A k − g α k (A δ k )A δ k [R(x k − te k , x k ) − I]e k dt + 1 0 g α k (A δ k )F ′ (x δ k ) * F ′ (x δ k ) − F ′ (x k ) [R(x k − te k , x k ) − I]e k dt = 1 0 r α k (A δ k ) − r α k (A k ) [R(x k − te k , x k ) − I]e k dt + 1 0 g α k (A δ k )A δ k I − R(x k , x δ k ) [R(x k − te k , x k ) − I]e k dt. Thus, by using (2.20) and (4.10), we obtain I 3 K 2 0 e k 2 x δ k − x k K 2 0 e 0 2 x δ k − x k . In order to estimate I 4 , we again use Assumption 5 to write I 4 = g α k (A δ k )F ′ (x δ k ) * F(x k ) − F(x δ k ) − F ′ (x δ k )(x k − x δ k ) + g α k (A δ k )F ′ (x δ k ) * F ′ (x δ k ) − F ′ (x k ) e k = 1 0 g α k (A δ k )A δ k R(x δ k + t(x k − x δ k ), x δ k ) − I (x k − x δ k )dt + g α k (A δ k )A δ k I − R(x k , x δ k ) e k . Hence, we may use (4.2) and (4.10) to derive that I 4 K 0 x δ k − x k 2 + K 0 e k x δ k − x k K 0 e 0 x δ k − x k . Combining the above estimates we obtain for 0 ≤ k < n δ x δ k+1 − x k+1 δ √ α k + K 0 e 0 x δ k − x k . Thus, if K 0 e 0 is sufficiently small, we can obtain (4.18) immediately. Next we show (4.19) by using (3.20). We still have (3.21). In order to estimate F ′ (x δ k )I 1 , F ′ (x δ k )I 3 and F ′ (x δ k )I 4 , we note that Assumption 6, (4.10), (4.15) and (4.18) imply [F ′ (x k ) − F ′ (x † )](x δ k − x k ) ≤ K 1 e k F ′ (x † )(x δ k − x k ) + K 2 F ′ (x † )e k x δ k − x k K 1 e 0 F ′ (x † )(x δ k − x k ) + K 2 e 0 δ , which in turn gives F ′ (x k )(x δ k − x k ) F ′ (x † )(x δ k − x k ) + δ . (4.20) Similarly, we have F ′ (x δ k )(x δ k − x k ) F ′ (x † )(x δ k − x k ) + δ .F ′ (x δ k )I 1 (K 0 + K 1 ) e 0 α 1/2 k x δ k − x k + K 2 e 0 F ′ (x δ k )(x δ k − x k ) + F ′ (x k )(x δ k − x k ) (K 0 + K 1 + K 2 ) e 0 δ + K 2 e 0 F ′ (x † )(x δ k − x k ) .F ′ (x δ k )I 3 (K 0 + K 1 ) x δ k − x k u k + α −1/2 k K 2 F ′ (x k )(x δ k − x k ) u k (K 0 + K 1 + K 2 )(K 1 + K 2 ) e 0 2 δ + K 2 (K 1 + K 2 ) e 0 2 F ′ (x † )(x δ k − x k ) . while, by using Assumption 6, (2.18), (4.2), (4.10), (4.4), (4.18), (4.20) and (4.21), F ′ (x δ k )I 4 can be estimated as F ′ (x δ k )I 4 ≤ F(x δ k ) − F(x k ) − F ′ (x k )(x δ k − x k ) + [F ′ (x δ k ) − F ′ (x k )]e δ k (K 1 + K 2 ) x δ k − x k F ′ (x k )(x δ k − x k ) + K 1 x δ k − x k F ′ (x δ k )e δ k + K 2 F ′ (x δ k )(x δ k − x k ) e δ k (K 1 + K 2 ) e 0 δ + (K 1 + K 2 ) e 0 F ′ (x † )(x δ k − x k ) . Combining the above estimates we get F ′ (x δ k )(x δ k+1 − x k+1 ) − y δ + y ≤ (1 + C(K 0 + K 1 + K 2 ) e 0 )δ + C(K 1 + K 2 ) e 0 F ′ (x † )(x δ k − x k ) . (4.22) This in particular implies F ′ (x δ k )(x δ k+1 − x k+1 ) δ + (K 1 + K 2 ) e 0 F ′ (x † )(x δ k − x k ) . On the other hand, similar to the derivation of (4.20), by Assumption 6, (4.2), (4.4) and (4.18) we have for 0 ≤ k < n δ that F ′ (x † )(x δ k+1 − x k+1 ) K 2 e 0 δ + F ′ (x δ k )(x δ k+1 − x k+1 ) . Therefore F ′ (x † )(x δ k+1 − x k+1 ) δ + (K 1 + K 2 ) e 0 F ′ (x † )(x δ k − x k ) . Thus, if (K 1 + K 2 ) e 0 is small enough, then we can conclude F ′ (x † )(x δ k − x k ) δ , 0 ≤ k ≤ n δ . (4.23) Combining this with (4.22) gives for 0 ≤ k < n δ F ′ (x δ k )(x δ k+1 − x k+1 ) − y δ + y ≤ (1 + C(K 0 + K 1 + K 2 ) e 0 ) δ . (4.24) Hence, by using (4.24), Assumption 6, (4.2), (4.5), (4.18), (4.21) and (4.23), we obtain for 0 ≤ k ≤ n δ F ′ (x δ k )(x δ k − x k ) − y δ + y ≤ (1 + C(K 0 + K 1 + K 2 ) e 0 ) δ . This together with (2.18), (4.2) and (4.10) implies (4.19). Step 5. Now we are ready to complete the proof. By using the definition of k δ , (4.19), (2.18) and (4.14) we have for 0 ≤ k < k δ τδ ≤ F(x δ k ) − y δ ≤ F(x δ k ) − F(x k ) − y δ + y + F(x k ) − y ≤ (1 + C(K 0 + K 1 + K 2 ) e 0 )δ + C F ′ (x † )e k ≤ (1 + C(K 0 + K 1 + K 2 ) e 0 ) δ + C r α k (A )A 1/2 e 0 + Cα 1/2 k r α k (A )e 0 . Since τ > 1, by assuming (K 0 + K 1 + K 2 ) e 0 is small enough, we can conclude for 0 ≤ k < k δ that (τ − 1)δ r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 .(4.25) When x 0 − x † satisfies (1.10) for some ω ∈ X and 0 < ν ≤ν − 1/2, by using (4.25), (4.8), (4.6), (4.18), (4.19) and the definition of k δ , we can employ the similar argument as in the last part of the proof of Theorem 1 to conclude (2.22). When x 0 − x † satisfies (1.11) for some ω ∈ X and µ > 0, we have from Assumption 1(a) and (2.3) that r α k (A )A 1/2 e 0 + α 1/2 k r α k (A )e 0 ≤ c 0 b 1/2 2µ + b µ α 1/2 k (− ln(α k /(2α 0 ))) −µ ω . This and (4.25) imply that there exists a constant C µ > 0 such that (τ − 1)δ < C µ α 1/2 k (− ln(α k /(2α 0 ))) −µ ω , 0 ≤ k < k δ . If we introduce the integerk δ by α 1/2 k δ − ln(αˆk δ /(2α 0 )) −µ ≤ (τ − 1)δ C µ ω < α 1/2 k (− ln(α k /(2α 0 ))) −µ , 0 ≤ k <k δ , then k δ ≤k δ . Thus, by using (4.8), (4.18), (4.19), the definition of k δ and the fact e k r α k (A )e 0 (− ln(α k /(2α 0 ))) −µ ω , we can use the similar manner in deriving (3.35) to get e δ k δ − ln(αˆk δ /(2α 0 )) −µ ω + δ αˆk δ δ αˆk δ . (4.26) By elementary argument we can show from (1.4) and the definition ofk δ that there is a constant c µ > 0 such that αˆk δ ≥ r −1 αˆk δ −1 ≥ c µ δ ω 2 1 + ln δ ω 2µ . This together with (4.26) implies the estimate (2.23). 2 Proof of Theorem 3 If x 0 = x † , then k δ = 0 and the result is trivial. Therefore, we will assume x 0 = x † . We definek δ to be the first integer such that r αk δ (A )A 1/2 e 0 + α 1/2 k δ r αk δ (A )e 0 ≤ cδ , where the constant c > 0 is chosen so that we may apply Lemma 6 or (4.25) to conclude k δ ≤k δ . By (1.4), suchk δ is clearly well-defined and is finite. Moreover, by a contradiction argument it is easy to show thatk δ → ∞ as δ → 0. e k δ + δ √ α k δ e k δ + δ αˆk δ eˆk δ + 1 αˆk δ F(x k δ ) − y + δ r αk δ (A )e 0 + δ αˆk δ δ αˆk δ . (5.2) We therefore need to derive the lower bound of αˆk δ under the conditions on e 0 . We set for each α > 0 and 0 ≤ µ ≤ν c µ (α) := 1/2 0 α −2µ r α (λ ) 2 λ 2µ d(E λ ω, ω) 1/2 , where {E λ } denotes the spectral family generated by A . It is easy to see for each 0 ≤ µ <ν that α −2µ r α (λ ) 2 λ 2µ is uniformly bounded for all α > 0 and λ ∈ [0, 1/2] and α −2µ r α (λ ) 2 λ 2µ → 0 as α → 0 for all λ ∈ (0, 1/2]. Since ω ∈ N (F ′ (x † )) ⊥ , by the dominated convergence theorem we have for each 0 ≤ µ <ν c µ (α) → 0 as α → 0. Since 0 ≤ ν <ν − 1/2, this together with (5.1) and (5.3) gives the desired conclusion. Applications In this section we will consider some specific methods defined by (1.3) by presenting several examples of {g α }. We will verify that those assumptions in Section 2 are satisfied for these examples. Example 1 We first consider the function g α given by g α (λ ) = (α + λ ) m − α m λ (α + λ ) m ,(6.1) where m ≥ 1 is a fixed integer. This function arises from the iterated Tikhonov regularization of order m for linear ill-posed problems. Note that when m = 1, the corresponding method defined by (1.3) is exactly the iteratively regularized Gauss-Newton method (1.8). It is clear that the residual function corresponding to (6.1) is r α (λ ) = α m (α + λ ) m .c 3 = 1 √ 2m − 1 2m − 1 2m m and c 4 = 1 − m + 1 m + 3 m √ m. By using the elementary inequality 1 − (1 − t) n ≤ √ nt, 0 ≤ t ≤ 1 (6.2) for any integer n ≥ 0, we have for 0 < α ≤ β and λ ≥ 0 that r β (λ ) − r α (λ ) = r β (λ ) 1 − 1 − λ /α − λ /β 1 + λ /α m ≤ m 1/2 λ α r β (λ ). Note also that g α (λ ) = α −1 ∑ m i=1 α i (α + λ ) −i . We have, by using (2.12), [g α (A * A) − g α (B * B)]B * ≤ α −1 m ∑ i=1 α i [(αI + A * A) −i − (αI + B * B) −i ]B * α −1 A − B , which verifies (2.14). Finally we verify Assumption 7 by assuming that F satisfies Assumption 5 and Assumption 6. We will use the abbreviation F ′ x := F ′ (x) for x ∈ B ρ (x † ). With the help of (6.3) with A = F ′ x and B = F ′ z , we obtain from Assumption 5 that (2.20). In order to show (2.21), we note that, for any a ∈ X and b ∈ Y satisfying a = b = 1, (6.3) implies r α (F ′ * x F ′ x ) − r α (F ′ * z F ′ z ) ≤ α m m ∑ i=1 (αI + F ′ * x F ′ x ) −i F ′ * x F ′ x [R(z, x) − I](αI + F ′ * z F ′ z ) −m−1+i + α m m ∑ i=1 (αI + F ′ * x F ′ x ) −i [I − R(x, z)] * F ′ * z F ′ z (αI + F ′ * z F ′ z ) −m−1+i ≤ α m m ∑ i=1 α −i+1 I − R(z, x) α −m−1+i + α m m ∑ i=1 α −i I − R(x, z) α −m+i I − R(z, x) + I − R(x, z) K 0 x − z which verifies(F ′ x [r α (F ′ * x F ′ x ) − r α (F ′ * z F ′ z )]a, b) ≤ α m m ∑ i=1 α −i+1 (F ′ z − F ′ x )(αI + F ′ * z F ′ z ) −m−1+i a b + α m m ∑ i=1 α −m−1/2+i (F ′ z − F ′ x )(αI + F ′ * x F ′ x ) −i F ′ * x b a . Example 2 We consider the function g α given by g α (λ ) = [1/α] ∑ i=0 (1 − λ ) i (6.4) which arises from the Landweber iteration applying to linear ill-posed problems. With such choice of g α , the method (1.3) becomes x δ k+1 = x 0 − [1/α k ] ∑ i=0 I − F ′ (x δ k ) * F ′ (x δ k ) i F ′ (x δ k ) * F(x δ k ) − y δ − F ′ (x δ k )(x δ k − x 0 ) which is equivalent to the form x δ k,0 = x 0 , x δ k,i+1 = x δ k,i − F ′ (x δ k ) * F(x δ k ) − y δ + F ′ (x δ k )(x δ k,i − x δ k ) , 0 ≤ i ≤ [1/α k ], x δ k+1 = x δ k,[1/α k ]+1 . This method has been considered in [12] and is called the Newton-Landweber iteration. Note that the corresponding residual function is Moreover, by (6.2) we have for any 0 < α ≤ β that This verifies Assumption 1(c) with c 2 = 1. It is well-known that the qualification of linear Landweber iteration isν = ∞ and (2.2) is satisfied with d ν = ν ν for each 0 ≤ ν < ∞. r α (λ ) = (1 − λ ) [1/α]+1 . In order to verify Assumption 2, we restrict the sequence {α k } to be of the form α k := 1/n k , where {n k } is a sequence of positive integers such that 0 ≤ n k+1 − n k ≤ q and lim k→∞ n k = ∞ (6.6) for some q ≥ 1. Then for λ ∈ [0, 1/2] we have r α k (λ ) = (1 − λ ) n k −n k+1 r α k+1 (λ ) ≤ 2 q r α k+1 (λ ). Thus Assumption 2 is also true. In order to verify Assumption 4, we will use some techniques from [7,12] and the following well-known estimates (I − A * A) j (A * A) ν ≤ ν ν ( j + ν) −ν , j ≥ 0, ν ≥ 0 (6.7) for any bounded linear operator A satisfying A ≤ 1. (6.8) By using (6.7) we have r α (A * A) − r α (B * B) k ∑ j=0 ( j + 1) −1/2 + (k + 1 − j) −1/2 A − B √ k A − B 1 √ α A − B . This verifies (2.11). From With the help of (6.7), we can estimate J is a constant depending only on r, τ and c i , and C ν is a positive constant depending only on r, τ, ν and c i , i = 0, · · · , 6. (b) Under Assumption 5 and Assumption 6, there exists a positive constant c 8 such that Theorem 2 2Let {g α } and {α k } satisfy Assumption 1, (1.4), Assumption 2 and Assumption 7, let ν ≥ 1 be the qualification of the linear regularization method defined by {g α }, and let F satisfy (2.8), (2.9), Assumption 5 and Assumption 6 with ρ > 2(1 + c 4 γ 1 ) x 0 − x † . Let {x δ k } be defined by (1.3) and let k δ be the first integer satisfying (1.7) with τ > 1. Then there exists a constant η 1 > 0 depending only on r, τ and c Theorem 3 3(i) Let all the conditions in Theorem 1 be fulfilled. Ifν > 1 and x † − x 0 satisfies the Hölder source condition (1.10) for some (c 3 + c 4 γ 0 )c 4 r 1/ 2 L u e δ l− 1 ≤ e 0 + c 4 302104so small that 2 r 1/2 + (c 3 + c 4 γ 0 )c 4 r L u ≤ 1. (3.6) By using (3.5), (2.1), Assumption 3, (1.2), Assumption 1(a), (3.4) with k = l − 1 and (3.6), we also obtain e δ l ≤ r α l−1 (A δ l−1 )e 0 + c 4 (c 6 + rc 4 γ 0 ) + (4c 3 + 3c 4 γ 0 )r 1/2 + 4(c 3 + c 4 γ 0 )r L u + 4c 3 c 4 c 6 r 1/2 + (c 4 + c 6 )c 3 r L 2 u 2 . 2 3. 3 23Some estimates on noise-free iterationsLemma 3 Let all the conditions in Theorem 4 be fulfilled. If L u is sufficiently small, then for all k ≥ 0 we havex k ∈ B ρ (x † )and e k ≤ 2c 3 rIf, in addition, Assumption 1(b) is satisfied, then2 3 r α k (A )e 0 ≤ e k ≤ 4 3 c 5 r α k (A )k ≤ e k+1 ≤ 2 e k .(3.25)Proof By using (3.2), (2.1), (2.12) and Assumption 3, we have from (3.13) that and (3.2) imply r α k (A )e 0 ≤ c 5c 5 c 6 6+ c 3 c 4 r 1/2 L u ≤ 1, (3.27) the combination of (3.26) and (3.23) gives e k+1 − r α k (A )e 0 ≤ c 6 + c 3 c 4 r 1/2 L u e k ≤ 1 5c 5 e k . (3.28) Note that Assumption 1(b) and α k ≤ α k−1 imply r α k (A )e 0 ≤ r α k−1 (A )e 0 . Note also that Assumption 1(a) and (2.9) imply (3.24) with k = 0. Thus, from (3.28) and (2.7) we can conclude (3.24) by an induction argument. (3.25) is an immediate consequence of (3.28) and (3.24). 2 Assume that all the conditions in Lemma 3 are satisfied. Then 0 e 0 ≤ 1, then by induction we can see that {x k } is well-defined and e k ≤ 2 e 0 for all k ≥ 0. (4.10) 30), so we may use (2.20), Assumption 5 and (4.10) to obtain x k − x l r α k−1 (A )e 0 − r α l−1 (A )e 0 + K 0 e 0 ( e k−1 + e l−1 ) + K 0 e k−1 2 + K 0 e l−1 2 [r α k−1 (A ) − r α l−1 (A )]e 0 + K 0 e 0 ( e k−1 + e l−1 ) . (4.12) By Lemma 1 with x = e 0 ,x = e k , α = α l−1 , β = α k−1 and A = F ′ (x † ), we have by using (2.21), (4.18), (4.20) and (4.21) we have Moreover, by employing (3.22), (2.20), Assumption 6, (2.18), (4.10), (4.17), (4.18) and (4.20), F ′ (x δ k )I 3 can be estimated as under the conditions of Theorem 3 (i) we use Lemma 2, Lemma 4 and (3.24), while under the conditions of Theorem 3 (ii) we use (4.18), (4.19), (4.6) and (4.8), then from the definition of k δ we have e δ k δ By elementary calculations it is easy to see that Assumption 1(a) and (b) are satisfied with c 0 = (m − 1) m−1 /m m and c 1 = m. Moreover (2.1) is satisfied with ( c 2 = m 1/2 . It is well-known that the qualification for g α is ν = m and (2.2) is satisfied withd ν = (ν/m) ν ((m − ν)/m) m−ν ≤ 1 for each 0 ≤ ν ≤ m. For the sequence {α k } satisfying (1.4), Assumption 2 is satisfied with c 5 = r m .In order to verify Assumption 4, we note thatr α (A * A) − r α (B * B) αI + A * A) −i [A * (B − A) + (B * − A * )B](αI + B * B) −m−1+i . (6.3)Thus, by using the estimates(αI + A * A) −i (A * A) µ ≤ α −i+µ for i ≥ 1 and 0 ≤ µ ≤ 1,we can verify (2.11), (2.12) and (2.13) easily. r β (λ ) − r α (λ ) = r β (λ ) 1 − (1 − λ ) [1/α]−[1/β ] ≤ λ α r β (λ ). For any α > 0, we set k := [1/α]. Let A and B be any two bounded linear operators satisfying A , B ≤ 1. Then it follows from (6.5) that r α (A * A) − r α (B * B) = k ∑ j=0 (I − A * A) j [A * (B − A) + (B * − A * )B] (I − B * B) k− j . (A(( (6.8) we also have A [r α (A * A) − r α (B * B)] B * = J 1 + J I − AA * ) j AA * (B − A)(I − B * B) k− j B * , (I − A * A) j (B * − A * )(I − BB * ) k− j BB * .In order to verify (2.13), it suffices to show J 1 (k + 1) −1/2 A − B since the estimate on J 2 is exactly the same. We writeJ 1 = J I − AA * ) j AA * (B − A)(I − B * B) k− j B * , I − AA * ) j AA * (B − A)(I − B * B) k− j B * . ( j + 1 )((( 1−1 (k + j − 1) −1/2 A − B 1 − j) −1/2 A − B (k + 1) −1/2 A − B .In order to estimate J (1) 1 , we use AA * = I − (I − AA * ) to rewrite it asJ I − AA * ) j (B − A)(I − B * B) k− j B I − AA * ) j (B − A)(I − B * B) k+1− j B * =(B − A)(I − B * B) k B * − (I − AA * ) [k/2]+1 (B − A)(I − B * B) k−[k/2] B I − AA * ) j (B − A)(I − B * B) k− j (B * B)B * .Thus, in view of (6.7), we obtainJ (1) 1 (k + 1) −1/2 A − B + (k − [k/2] + 1) −1/2 A − B + [k/2] ∑ j=1 (k − j + 1) −3/2 A − B (k + 1) −1/2 A − B . Recently we realized that (c) can be derived from (a) and (b). Thus, by using Assumption 6, we have. This verifies (2.21).The above analysis shows that Theorem 1, Theorem 2 and Theorem 3 are applicable for the method defined by (1.3) and (1.7) with g α given by (6.1). Thus we obtain the following result.Corollary 1 Let F satisfy(2.8)and(2.9), let {α k } be a sequence of numbers satisfying(1.4), and let {x δ k } be defined by(1.3)with g α given by(6.1)for some fixed integer m ≥ 1. Let k δ be the first integer satisfying (1.7) with τ > 1.(i) If F satisfies Assumption 3 and if x 0 − x † satisfies (1.10) for some ω ∈ X and 1/2 ≤ ν ≤ m − 1/2, thendepending only on r, τ and m, and C ν > 0 is a constant depending only on r, τ, m and ν.(ii) Let F satisfy Assumption 5 and Assumption 6, and let x 0 − x † ∈ N(F ′ (x † )) ⊥ . Then there exists a constant η 1 > 0 depending only on r, τ and m such that ifmoreover, when x 0 − x † satisfies (1.10) for some ω ∈ X and 0 < ν ≤ m − 1/2, thenfor some constant C ν > 0 depending only on r, τ, m and ν; while when x 0 − x † satisfies(1.11)for some ω ∈ X and µ > 0, thenfor some constant C µ depending only on r, τ, m and µ.Corollary 1 with m = 1 reproduces those convergence results in[3,8]for the iteratively regularized Gauss-Newton method (1.8) together with the discrepancy principle (1.7) under somewhat different conditions on F. Note that those results in[3,8]require τ be sufficiently large, while our result is valid for any τ > 1. This less restrictive requirement on τ is important in numerical computations since the absolute error could increase with respect to τ. Moreover, when x 0 − x † satisfies (1.10) with ν = 1/2, Corollary 1 with m = 1 improves the corresponding result in[3], since we only need the Lipschitz condition on F ′ here.Corollary 1 shows that the method defined by (1.3) and (1.7) with g α given by (6.1) is order optimal for 0 < ν ≤ m − 1/2. However, we can not expect better rate of convergence than O(δ (2m−1)/(2m) ) even if x 0 − x † satisfies (1.10) with m − 1/2 < ν ≤ m. An a posteriori stopping rule without such saturation has been studied in[9,10]for the iteratively regularized Gauss-Newton method (1.8).We thus verify(2.13). The verification of (2.12) can be done similarly.Applying the estimate (2.12), we obtainwhich verifies (2.14).Finally we verify Assumption 7 by assuming that F satisfies Assumption 5 and Assumption 6. From (6.8) and Assumption 5 it follows thatThus we may use the argument in the verification of (2.13) to concludeThis verifies (2.20). By using (6.8) and Assumption 5 we also have for any w ∈ XBy employing (6.7) it is easy to see thatWith the help of (6.7) and Assumption 6, we haveBy using the argument in the verification of (2.13) and Assumption 6 we obtainUsing Assumption 5 and the the similar argument in the verification of (2.13) we also haveCombining the above estimates we thus obtain for any w ∈ Xwhich implies (2.21). Therefore, Theorem 1, Theorem 2 and Theorem 3 are applicable for the method defined by (1.3) and (1.7) with g α given by(6.4).The similar argument as above also applies to the situation where g α is given by(1 + λ ) −i which arise from the Lardy's method for solving linear ill-posed problems.In summary, we obtain the following result.Corollary 2 Let F satisfy(2.8)and(2.9), and let {α k } be a sequence given by α k = 1/n k , where {n k } is a sequence of positive integers satisfying (6.6) for some q ≥ 1. Let {x δ k } be defined byand let k δ be the first integer satisfying (1.7) with τ > 1.(i) If F satisfies Assumption 3, and if x 0 − x † satisfies (1.10) for some ω ∈ X and ν ≥ 1/2, thenis a constant depending only on τ and q, and C ν is a constant depending only on τ, q and ν.(ii) Let F satisfy Assumption 5 and Assumption 6, and let x 0 − x † ∈ N(F ′ (x † )) ⊥ . Then there exists a constant η 1 > 0 depending only on τ and q such that if (moreover, when x 0 − x † satisfies (1.10) for some ω ∈ X and ν > 0, thenfor some constant C ν > 0 depending only on τ, q and ν; while when x 0 − x † satisfies (1.11) for some ω ∈ X and µ > 0, thenfor some constant C µ depending only on τ, q and µ.Example 3As the last example we consider the method (1.3) with g α given bywhich arises from the asymptotic regularization for linear ill-posed problems. In this method, the iterated sequence {x δ k } is equivalently defined as x δ k+1 :Note that the corresponding residual function isIt is easy to see that Assumption 1(a), (b) and (2.1) hold withBy using the inequality 1 − e −t ≤ √ t for t ≥ 0 we have for 0 < α ≤ β thatThis verifies Assumption 1(c) with c 2 = 1. It is well-known that the qualification of the linear asymptotic regularization isν = ∞ and (2.2) is satisfied with d ν = (ν/e) ν for each 0 ≤ ν < ∞.In order to verify Assumption 2, we assume that {α k } is a sequence of positive numbers satisfying 0 ≤ 1 α k+1 − 1 α k ≤ θ 0 and lim k→∞ α k = 0 (6.10)for some θ 0 > 0. Then for all λ ∈ [0, 1] we haveThus Assumption 2 is also true. In order to verify Assumption 4 and Assumption 7, we set for every integer n ≥ 1 r α,n (λ ) := 1 + λ nα −n , g α,n (λ ) := 1Note that, for each fixed α > 0, {r α,n } and {g α,n } are uniformly bounded over [0, 1], and r α,n (λ ) → r α (λ ) and g α,n (λ ) → g α (λ ) as n → ∞. By the dominated convergence theorem, we have for any bounded linear operator A with A ≤ 1 thatfor any x ∈ X, where {E λ } denotes the spectral family generated by A * A. Thus it suffices to verify Assumption 4 and Assumption 7 with g α and r α replaced by g α,n and r α,n with uniform constants c 6 , c 7 and c 8 independent of n. Let A and B be any two bounded linear operators satisfying A , B ≤ 1. We need the following inequality which says for any integer n ≥ 1 there holdsBy noting thatwe thus obtainFurthermore, by noting that g α,n (λ ) = 1 nα ∑ n i=1 r α,i (λ ), we may use (6.13) to concludeAssumption 4 is therefore verified. It remains to verify Assumption 7 with g α and r α replaced by g α,n and r α,n with uniform constants c 7 and c 8 independent of n. By using (6.12), Assumption 5 and (6.11) we haveThis implies(2.20). By using (6.12), Assumption 6 and (6.11) we also have for any a ∈ X and b ∈ Y satisfying a = b = 1 thatThis implies (2.21). Therefore, we may apply Theorem 1, Theorem 2 and Theorem 3 to conclude the following result.(2.8)and(2.9), and let {α k } be a sequence of positive numbers satisfying (6.10) for some θ 0 > 0. Let {x δ k } be defined by (1.3) with g α given by(6.9)and let k δ be the first integer satisfying (1.7) with τ > 1.Corollary 3 Let F satisfy(i) If F satisfies Assumption 3, and if x 0 − x † satisfies (1.10) for some ω ∈ X and ν ≥ 1/2, then x δ k δ = x † ; moreover, when x 0 − x † satisfies (1.10) for some ω ∈ X and ν > 0, thenfor some constant C ν > 0 depending only on τ, θ 0 , α 0 and ν; while when x 0 − x † satisfies(1.11)for some ω ∈ X and µ > 0, thenfor some constant C µ depending only on τ, θ 0 , α 0 and µ.AcknowledgementsThe authors wish to thank the referee for careful reading of the manuscript and useful comments. The problems of the convergence of the iteratively regularized Gauss-Newton method. A B Bakushinskii, Comput. Math. Math. Phys. 32A. B. 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[ "Bound states of a localized magnetic impurity in a superfluid of paired ultracold fermions", "Bound states of a localized magnetic impurity in a superfluid of paired ultracold fermions" ]	[ "Eric Vernier \nPhysics Department\nHarvard University\n02138CambridgeMassachusettsUSA\n\nDépartement de Physique\nEcole Normale Supèrieure\nParisFrance\n", "David Pekker \nPhysics Department\nHarvard University\n02138CambridgeMassachusettsUSA\n", "Martin W Zwierlein \nDepartment of Physics\nResearch Laboratory of Electronics\nMIT-Harvard Center for Ultracold Atoms\n02139CambridgeMAUSA\n", "Eugene Demler \nPhysics Department\nHarvard University\n02138CambridgeMassachusettsUSA\n" ]	[ "Physics Department\nHarvard University\n02138CambridgeMassachusettsUSA", "Département de Physique\nEcole Normale Supèrieure\nParisFrance", "Physics Department\nHarvard University\n02138CambridgeMassachusettsUSA", "Department of Physics\nResearch Laboratory of Electronics\nMIT-Harvard Center for Ultracold Atoms\n02139CambridgeMAUSA", "Physics Department\nHarvard University\n02138CambridgeMassachusettsUSA" ]	[]	We consider a localized impurity atom that interacts with a cloud of fermions in the paired state. We develop an effective scattering length description of the interaction between an impurity and a fermionic atom using their vacuum scattering length. Treating the pairing of fermions at the mean-field level, we show that the impurity atom acts like a magnetic impurity in the condensed matter context, and leads to the formation of a pair of Shiba bound states inside the superconducting gap. In addition, the impurity atom can lead to the formation of deeply bound states below the Fermi sea. 67.85.Lm, 37.10.Jk Magnetic impurities in superconductors are known not only to alter the BCS ground-state by introducing potential scattering, but also to be at the origin of the pair-breaking effect leading to elementary excitations fundamentally different than those present in pure superconductors. Their presence serves to attenuate superconductivity by formation of in-gap Shiba bound states 1 . In fact, at high concentrations magnetic impurities induce gapless superconductivity 2 . While magnetic impurities in a superconductor have often been discussed as one of the simplest models that exhibit interplay between superconductivity and magnetism, the consequences of this interplay are still not fully understood.The experimental realization of paired states in ultracold atom systems has shed new light on many problems in superconductivity. In particular it has been instrumental in prompting a better understanding of the BEC-BCS crossover 3-14 , including the case of pairing in systems with large spin imbalance 15,16 . It has also prompted a new generation of research on the subject of the dynamics of theses systems 17,18 .Combine magnetic impurities and ultracold atom systems can have rich physical consequences, some of which we explore in the present paper. In particular, introducing magnetic impurities into fermionic superfluids would help in understanding the interactions between magnetism and superfluidity, and could help to resolve long standing problems such as how superconductivity becomes gapless.Alternatively, instead of studying how the system changes in response to magnetism, one can use localized magnetic impurities as a form of local probe. As an example, in the setting of high temperature superconductors, detection of the modulation of the local density of states by a magnetic impurity via a scanning tunneling microscopy (STM) was used to great advantage to probe the nature of quasi-particle states of these materials both in the superconducting and pseudo-gap phases[19][20][21].Although STM spectroscopy is not currently possible in the ultracold atom setting Radio Frequency (RF) spectroscopy 22,23 could be used to probe the nature of the superconducting state in the vicinity of the magnetic impurity. Combining a magnetic impurity with RF spectroscopy can be used to directly probe the size of the superconducting gap eliminating the uncertainty due to effects like Hartree shifts 24,25 . Further, we envision that additional information from momentum resolved RF spectroscopy 26 in the vicinity of a magnetic impurity could provide data on the symmetry of the gap and its nodal structure.In this paper we propose and investigate theoretically a scheme for introducing localized magnetic impurities into the ultracold atom fermionic superfluid. The impurity is formed by an atom of a different species (from the species making up the superfluid) that is localized by a deep optical lattice potential. The laser frequency is chosen such that the optical lattice interacts only weakly with the two atomic species, \|↑ and \|↓ , that make up the superfluid (i.e. \|↑ and \|↓ do not become localized). The magnetic character of the impurity originates in the different interactions strengths between the impurity atom and \|↑ and \|↓ atoms, which we describe by a pair of effective scattering lengths a ↑ and a ↓ .The main input into our theory of impurity localized states is the description of the atom scattering on a localized impurity. (1) We begin by showing that, under rather general conditions, we can describe the interaction between a localized impurity and the free atoms via an effective s-wave scattering length. (2) As pointed out by Shiba, due to the sharpness of the BCS density of states, the magnetic impurity always results in the formation of a pair of localized bound states, called the Shiba states. Indeed, we find that as long as a ↑ = a ↓ , the impurity atoms always induce a pair of bound states inside the superconducting gap. (3) Interestingly, we find that the Shiba state is not related to the under-sea bound state. By the undersea bound state we mean the natural extension to the case of a filled Fermi sea of the Feshbach bound state formed between a fermion and a localized impurity in the absence of a Fermi sea when the effective scattering length is positive. Indeed, if the Feshbach bound state exists, it becomes the under-sea bound state when the Fermi sea is filled, remaining completely separate of the Shiba state. (4) We show that both the under-sea bound states and the Shiba bound states can be resolved via RF spectroscopy.This paper is organized as follows: In section I we relate the bare scattering length between a pair of atoms in vacuum to the effective scattering length when one of the atoms is localized by a parabolic confining potential. In section II we use the effective scattering lengths to find the under-sea as well as arXiv:1010.6085v1 [cond-mat.quant-gas]	10.1103/physreva.83.033619	[ "https://arxiv.org/pdf/1010.6085v1.pdf" ]	15,278,537	1010.6085	3f7190a965fccce6756aa19a40f1f9e587546c93	Bound states of a localized magnetic impurity in a superfluid of paired ultracold fermions 28 Oct 2010 Eric Vernier Physics Department Harvard University 02138CambridgeMassachusettsUSA Département de Physique Ecole Normale Supèrieure ParisFrance David Pekker Physics Department Harvard University 02138CambridgeMassachusettsUSA Martin W Zwierlein Department of Physics Research Laboratory of Electronics MIT-Harvard Center for Ultracold Atoms 02139CambridgeMAUSA Eugene Demler Physics Department Harvard University 02138CambridgeMassachusettsUSA Bound states of a localized magnetic impurity in a superfluid of paired ultracold fermions 28 Oct 2010 We consider a localized impurity atom that interacts with a cloud of fermions in the paired state. We develop an effective scattering length description of the interaction between an impurity and a fermionic atom using their vacuum scattering length. Treating the pairing of fermions at the mean-field level, we show that the impurity atom acts like a magnetic impurity in the condensed matter context, and leads to the formation of a pair of Shiba bound states inside the superconducting gap. In addition, the impurity atom can lead to the formation of deeply bound states below the Fermi sea. 67.85.Lm, 37.10.Jk Magnetic impurities in superconductors are known not only to alter the BCS ground-state by introducing potential scattering, but also to be at the origin of the pair-breaking effect leading to elementary excitations fundamentally different than those present in pure superconductors. Their presence serves to attenuate superconductivity by formation of in-gap Shiba bound states 1 . In fact, at high concentrations magnetic impurities induce gapless superconductivity 2 . While magnetic impurities in a superconductor have often been discussed as one of the simplest models that exhibit interplay between superconductivity and magnetism, the consequences of this interplay are still not fully understood.The experimental realization of paired states in ultracold atom systems has shed new light on many problems in superconductivity. In particular it has been instrumental in prompting a better understanding of the BEC-BCS crossover 3-14 , including the case of pairing in systems with large spin imbalance 15,16 . It has also prompted a new generation of research on the subject of the dynamics of theses systems 17,18 .Combine magnetic impurities and ultracold atom systems can have rich physical consequences, some of which we explore in the present paper. In particular, introducing magnetic impurities into fermionic superfluids would help in understanding the interactions between magnetism and superfluidity, and could help to resolve long standing problems such as how superconductivity becomes gapless.Alternatively, instead of studying how the system changes in response to magnetism, one can use localized magnetic impurities as a form of local probe. As an example, in the setting of high temperature superconductors, detection of the modulation of the local density of states by a magnetic impurity via a scanning tunneling microscopy (STM) was used to great advantage to probe the nature of quasi-particle states of these materials both in the superconducting and pseudo-gap phases[19][20][21].Although STM spectroscopy is not currently possible in the ultracold atom setting Radio Frequency (RF) spectroscopy 22,23 could be used to probe the nature of the superconducting state in the vicinity of the magnetic impurity. Combining a magnetic impurity with RF spectroscopy can be used to directly probe the size of the superconducting gap eliminating the uncertainty due to effects like Hartree shifts 24,25 . Further, we envision that additional information from momentum resolved RF spectroscopy 26 in the vicinity of a magnetic impurity could provide data on the symmetry of the gap and its nodal structure.In this paper we propose and investigate theoretically a scheme for introducing localized magnetic impurities into the ultracold atom fermionic superfluid. The impurity is formed by an atom of a different species (from the species making up the superfluid) that is localized by a deep optical lattice potential. The laser frequency is chosen such that the optical lattice interacts only weakly with the two atomic species, \|↑ and \|↓ , that make up the superfluid (i.e. \|↑ and \|↓ do not become localized). The magnetic character of the impurity originates in the different interactions strengths between the impurity atom and \|↑ and \|↓ atoms, which we describe by a pair of effective scattering lengths a ↑ and a ↓ .The main input into our theory of impurity localized states is the description of the atom scattering on a localized impurity. (1) We begin by showing that, under rather general conditions, we can describe the interaction between a localized impurity and the free atoms via an effective s-wave scattering length. (2) As pointed out by Shiba, due to the sharpness of the BCS density of states, the magnetic impurity always results in the formation of a pair of localized bound states, called the Shiba states. Indeed, we find that as long as a ↑ = a ↓ , the impurity atoms always induce a pair of bound states inside the superconducting gap. (3) Interestingly, we find that the Shiba state is not related to the under-sea bound state. By the undersea bound state we mean the natural extension to the case of a filled Fermi sea of the Feshbach bound state formed between a fermion and a localized impurity in the absence of a Fermi sea when the effective scattering length is positive. Indeed, if the Feshbach bound state exists, it becomes the under-sea bound state when the Fermi sea is filled, remaining completely separate of the Shiba state. (4) We show that both the under-sea bound states and the Shiba bound states can be resolved via RF spectroscopy.This paper is organized as follows: In section I we relate the bare scattering length between a pair of atoms in vacuum to the effective scattering length when one of the atoms is localized by a parabolic confining potential. In section II we use the effective scattering lengths to find the under-sea as well as arXiv:1010.6085v1 [cond-mat.quant-gas] We consider a localized impurity atom that interacts with a cloud of fermions in the paired state. We develop an effective scattering length description of the interaction between an impurity and a fermionic atom using their vacuum scattering length. Treating the pairing of fermions at the mean-field level, we show that the impurity atom acts like a magnetic impurity in the condensed matter context, and leads to the formation of a pair of Shiba bound states inside the superconducting gap. In addition, the impurity atom can lead to the formation of deeply bound states below the Fermi sea. Magnetic impurities in superconductors are known not only to alter the BCS ground-state by introducing potential scattering, but also to be at the origin of the pair-breaking effect leading to elementary excitations fundamentally different than those present in pure superconductors. Their presence serves to attenuate superconductivity by formation of in-gap Shiba bound states 1 . In fact, at high concentrations magnetic impurities induce gapless superconductivity 2 . While magnetic impurities in a superconductor have often been discussed as one of the simplest models that exhibit interplay between superconductivity and magnetism, the consequences of this interplay are still not fully understood. The experimental realization of paired states in ultracold atom systems has shed new light on many problems in superconductivity. In particular it has been instrumental in prompting a better understanding of the BEC-BCS crossover [3][4][5][6][7][8][9][10][11][12][13][14] , including the case of pairing in systems with large spin imbalance 15,16 . It has also prompted a new generation of research on the subject of the dynamics of theses systems 17,18 . Combine magnetic impurities and ultracold atom systems can have rich physical consequences, some of which we explore in the present paper. In particular, introducing magnetic impurities into fermionic superfluids would help in understanding the interactions between magnetism and superfluidity, and could help to resolve long standing problems such as how superconductivity becomes gapless. Alternatively, instead of studying how the system changes in response to magnetism, one can use localized magnetic impurities as a form of local probe. As an example, in the setting of high temperature superconductors, detection of the modulation of the local density of states by a magnetic impurity via a scanning tunneling microscopy (STM) was used to great advantage to probe the nature of quasi-particle states of these materials both in the superconducting and pseudo-gap phases [19][20][21] . Although STM spectroscopy is not currently possible in the ultracold atom setting Radio Frequency (RF) spectroscopy 22,23 could be used to probe the nature of the superconducting state in the vicinity of the magnetic impurity. Combining a magnetic impurity with RF spectroscopy can be used to directly probe the size of the superconducting gap eliminating the uncertainty due to effects like Hartree shifts 24,25 . Further, we envision that additional information from momentum resolved RF spectroscopy 26 in the vicinity of a magnetic impurity could provide data on the symmetry of the gap and its nodal structure. In this paper we propose and investigate theoretically a scheme for introducing localized magnetic impurities into the ultracold atom fermionic superfluid. The impurity is formed by an atom of a different species (from the species making up the superfluid) that is localized by a deep optical lattice potential. The laser frequency is chosen such that the optical lattice interacts only weakly with the two atomic species, \|↑ and \|↓ , that make up the superfluid (i.e. \|↑ and \|↓ do not become localized). The magnetic character of the impurity originates in the different interactions strengths between the impurity atom and \|↑ and \|↓ atoms, which we describe by a pair of effective scattering lengths a ↑ and a ↓ . The main input into our theory of impurity localized states is the description of the atom scattering on a localized impurity. (1) We begin by showing that, under rather general conditions, we can describe the interaction between a localized impurity and the free atoms via an effective s-wave scattering length. (2) As pointed out by Shiba, due to the sharpness of the BCS density of states, the magnetic impurity always results in the formation of a pair of localized bound states, called the Shiba states. Indeed, we find that as long as a ↑ = a ↓ , the impurity atoms always induce a pair of bound states inside the superconducting gap. (3) Interestingly, we find that the Shiba state is not related to the under-sea bound state. By the undersea bound state we mean the natural extension to the case of a filled Fermi sea of the Feshbach bound state formed between a fermion and a localized impurity in the absence of a Fermi sea when the effective scattering length is positive. Indeed, if the Feshbach bound state exists, it becomes the under-sea bound state when the Fermi sea is filled, remaining completely separate of the Shiba state. (4) We show that both the under-sea bound states and the Shiba bound states can be resolved via RF spectroscopy. This paper is organized as follows: In section I we relate the bare scattering length between a pair of atoms in vacuum to the effective scattering length when one of the atoms is localized by a parabolic confining potential. In section II we use the effective scattering lengths to find the under-sea as well as the Shiba bound states of the magnetic impurity. Next, we describe RF spectroscopy of the ultracold atom system with bound states in section III. We discuss possible experimental realizations and atom species that could be used in section IV, discuss the outlook in section V, and draw conclusions in section VI. I. EFECTIVE SCATTERING LENGTH The goal of this section is to show that the scattering of a fermionic atom off of a confined impurity can, under reasonable conditions, be described by a single quantity -the effective scattering length. The problem of scattering on a confined impurity was previously studied in Ref. 27 , here we review the basic arguments and summarize the results. We begin by assuming that the impurity-fermion scattering in vacuum can indeed be defined by a single scattering length for the s-wave scattering process. This condition means that the effective range r 0 of the impurity-fermion interaction potential is much smaller than the typical fermion wavelength 1/k F , and thus we can treat r 0 as being essentially zero. Since we want the fermion-impurity interaction to be tunable, we shall be primarily interested in operating in the vicinity of a wide Feshbach resonance (i.e. a resonance that meets the condition r 0 1/k F ). If the effective range condition is not satisfied for the case of a free impurity (e.g. for the case of a narrow Feshbach resonance), it will not be satisfied for the case of a localized impurity, necessitating a more complicated description of the effective scattering process. Although, we do not treat the more complicated case in the present paper, we expect that the qualitative features, including the Shiba bound states, of a system with a narrow impurity resonance will be similar to those of a system with a wide resonance. In the problem with a confined impurity we have two important energy scales: the typical kinetic energy of a scattering fermion, which in our case is set by the Fermi energy scale F , and the level spacing of the impurity atom which we label ω i . We begin by pointing out that the scattering is elastic in the regime F ω i . Further, in order for the scattering to be dominated by s-wave channel, we demand that the ground state wavefunction of the impurity must have a length scale /m i ω i that is much smaller than the wavelength of the scattering particle / √ 2m α F (here m i stands for the mass of the impurity and m α for the mass of the scattering fermion). The two conditions are identical, up to a ratio of the masses. That is we demand that max(1, m α /m i ) F ω i . Having derived the conditions for s-wave scattering we can write the resulting T-matrix for the scattering atom in the form T (ω) = 1 mα 2π 1 aα + i √ 2mω .(1) It is important to point out that since the impurity is localized the T-matrix features the fermion mass as opposed to the reduced mass µ = m −1 i + m −1 α −1 and an effective scattering length a α instead of the vacuum scattering a 0,α . In order to relate the effective scattering length to the vacuum scattering length, we must solve the scattering problem. In general the scattering problem is complicated, and requires a numerical solution. In appendix A, we state the scattering problem and derive an analytic solution for the special case of weak impurity-fermion interactions using a Born-Oppenheimer type approximation. II. IMPURITY BOUND STATES In this section, we study the conditions for the existence of impurity bound states both in the normal (single component, non-interacting Fermi gas) and in the superconducting case. Our strategy is to obtain the T-matrix for scattering off of an isolated impurity in the presence of the Fermi-sea. Having the T-matrix, we can find the energies of the bound states from its poles. Further, we can also find the spectral function of the fermions, which we shall use in the next section to compute the RF spectra. In general, we can express the effect of the impurities on the Green function of the clean system G 0 (k, ω) via an expansion in the impurity density 28 G(k, ω) = G 0 (k, ω) + n i G 0 (k, ω)T (ω)G 0 (k, ω) (2) + O(n 2 i ) where G(k, ω) is the Green function of the dirty system, T (ω) is the T-matrix, and n i is the impurity density. In this paper we shall always work in the dilute impurity limit, and thus drop terms of order O(n 2 i ) and higher. The resulting equation is represented diagrammatically in Fig. 1a. T (ω) is obtained from the Lippmann-Schwinger equation T (ω) = V + V G 0 (ω)T (ω),(3) which relates the T-matrix to the impurity-fermion interaction potential V and the momentum integrated Green function of the clean system G 0 (ω) = d 3 k (2π) 3 G 0 (k, ω).(4) The Lippmann-Schwinger equation is illustrated diagrammatically in Fig. 1b; it can be formally solved for the T-matrix by inversion T −1 (ω) = V −1 − G 0 (ω).(5) Having specified the Lippmann-Schwinger equation, we first apply it to the case of an impurity in a one component non-interacting Fermi gas. This trivial case serves as an exercise that demonstrates (1) regularization of point contact interactions, (2) properties of the T-matrix, and (3) relation between Feshbach molecules and under sea states. Having learned how to use the Lippmann-Schwinger equation in this context, we apply it to the T-matrix of the BCS state. (2) and (b) of equation (3). Thin lines represent the clean (unperturbed) Green functions (G 0 (k, ω)), the thick lines the impurity-perturbed Green function (Gk), and the dashed line the interaction of these fermions with the impurity (V ) A. Impurity in a one component non-interacting Fermi gas We first consider a fixed impurity interacting with a one component Fermi sea, the interaction being described by the scattering length a. Since the Fermi sea is non-interacting and the impurity is static, we can proceed simply by finding the one-particle eigenstates in the vicinity of the impurity potential, and filling them up to the Fermi energy. For negative a, all eigenstates are part of the continuum, and there are no localized bound states on the impurity. As a becomes positive, a single state, the Feshbach molecular state, with energy −1/(2ma 2 ) peels off the continuum and becomes localized by the impurity (henceforth, the mass of the impurity no longer features and therefore we will use m for the mass of the fermion). When we fill the Fermi sea, the Feshbach molecular state appears as an under-sea bound state. In this subsection, we show how to recover this simple picture in the T-matrix language. In the absence of impurity, the fermions are described by the following Green function 28 G(k, ω) = 1 ω − ξ k + i0 + sgn(ω) ,(6) where ξ k ≡ k − F ≡ 2 k 2 2m − F and F is the Fermi energy. In order to cancel the divergence of the integral of the Green function in the Lippmann-Schwinger equation we must use a renormalized interaction potential 29,30 1 V = 2m 4π 2 a − 2m 2 d 3 k (2π) 3 1 k 2 .(7) As described in Sec. I, because the impurity atom is confined in the expression for the interaction potential we must use the fermion mass m and the effective scattering length a instead of the reduced mass and the vacuum scattering length. The momentum integrated Green function G 0 , that enters the Lippmann-Schwinger equation, can be obtained via contour integration G 0 (ω) = d 3 k (2π) 3 1 k − i (2m) 3/2 4π √ ω + F .(8) The divergence in G 0 (ω) is perfectly canceled by the renormalized interaction to yield the T-matrix T (ω) = 1 m 2π 1 a + i 2m(ω + F ) .(9) Unsurprisingly, the T-matrix has the same form as the vacuum T-matrix, but with frequency shifted by F . This reflects the fact that energies must be measured with respect to the Fermi energy. The bound states of the system introduced by the presence of the impurity are defined by the poles of the T-matrix. We thus find that a bound state exists only for positive values of the scattering length, with an energy ω b = − F − 1 2ma 2 .(10) B. Impurity in BCS state In this subsection, we generalize the results of the previous subsection to the case of a localized impurity atom immersed in an ultracold BCS gas. We shall describe the BCS state at the mean-field level. Since BCS quasi-particles involve mixing particles and holes, it is convenient to use Nambu's 4-dimensional spinor basis 1,31,32 Ψ k =     c k↑ c k↓ c † -k↑ c † -k↓    (11) In this formalism, the BCS Hamiltonian becomes H BCS =    ξ k 0 0 −∆ 0 ξ k ∆ 0 0 ∆ −ξ k 0 −∆ 0 0 −ξ k    ,(12) where ∆ is the BCS order parameter. The BCS Green function of the clean system is G 0 (k, ω) = 1 ω − ξ k ρ 3 − ∆σ 2 ρ 2 = ω + ξ k ρ 3 + ∆σ 2 ρ 2 ω 2 − ξ 2 k − ∆ 2 ≡ 1 ω 2 − ξ 2 k − ∆ 2    ω + ξ k 0 0 −∆ 0 ω + ξ k ∆ 0 0 ∆ ω − ξ k 0 −∆ 0 0 ω − ξ k    .(13) Here, {σ 1 , σ 2 , σ 3 } and {ρ 1 , ρ 2 , ρ 3 } are two sets of Pauli matrices, the first one operating on the spin space and the second on the particle-hole space. The interaction potentials between each of the two species that make up the BCS state and the impurity atom have the same form as the interaction potential in the single component case 1 V ↑(↓) = 2m 4π 2 a ↑(↓) − 2m 2 d 3 k (2π) 3 1 k 2 .(14) Here a ↑ corresponds to the effective scattering length between a \|↑ atom and the localized impurity, while a ↓ between a \|↓ atom and the impurity. In Nambu basis, the interaction potential becomes V = 1 V1 0 0 1 V2 ⊗ ρ 3 =     1 V1 0 0 0 0 1 V2 0 0 0 0 − 1 V1 0 0 0 0 − 1 V2     . (15) Substituting G 0 (k, ω) and V into the T-matrix equation (5) we see that the four-dimensional Nambu space reduces into a pair of two-dimensional subspaces that can treated separately: the 'outer' (or 'first') subspace acts on the first and fourth Nambu components, whereas the 'inner' (or 'second') subspace acts on the second and third components. From this point, we will limit ourselves to one of them, say the first one. To use Lippmann-Schwinger equation in order to yield the T-matrix, our first step is the calculation of G 0 , which can be written as the sum of a regular part G 0 r (ω) and a diverging part, as G 0 (ω) = G 0 r (ω) − d 3 k (2π) 3 1 k ρ 3 .(16) From this definition, the regular part is G 0 r (ω) = d 3 k (2π) 3 1 ω 2 − ξ 2 k − ∆ 2 ω −∆ −∆ ω + ξ k ω 2 − ξ 2 k − ∆ 2 + 1 k 1 0 0 −1 .(17) G 0 r (ω) can be expressed as a function of two integrals I 1 (ω) = ∞ 0 κ 2 dκ ω 2 − ∆ 2 − (κ 2 − F ) 2 ,(18)I 2 (ω) = ∞ 0 dκ ω 2 − ∆ 2 − (κ 2 − F ) 2 ,(19) where we have used the notation κ = k/ √ 2m. Both integrals can be evaluated using contour integration, to give I 1 (ω) = 1 4π √ ∆ 2 − ω 2 F + i ∆ 2 − ω 2 + F − i ∆ 2 − ω 2(20)I 2 (ω) = 1 4π √ ∆ 2 − ω 2 F + i √ ∆ 2 − ω 2 + F − i √ ∆ 2 − ω 2 2 F + ∆ 2 − ω 2(21) Using the fact that F + i ∆ 2 − ω 2 + F − i ∆ 2 − ω 2 = 2 F + 2 F + ∆ 2 − ω 2 2 1/2(22) we find G 0 r (ω) = −i m 3/2 ( F + Ξ) 1/2 2π √ ω 2 − ∆ 2 sgn( (ω) (ω)) ω + ( F − Ξ) −∆ −∆ ω − ( F − Ξ) ,(23) where Ξ = 2 F + ∆ 2 − ω 2 , and the sgn function ensures that we take the correct branch of the square roots. Once more, the two diverging integrals in G 0 and V −1 cancel, and (5) yields T −1 (ω) = m 2πa ↑ 0 0 − m 2πa ↓ − G 0 r (ω)(24) Having solved the Lippmann-Schwinger equation, we can look at the properties of the resulting T-matrix. In particular, we want to consider two regimes: bound states inside the gap and bound states outside the gap. Under-sea states Aiming to recover the Feshbach molecule-like bound state that we found to exist under the Fermi sea in the case of a one component gas, we make the approximation that ∆ 0. The bound state must correspond to a frequency ω = ω b + i0 + , where ω b ≤ − F . Within this approximation, T −1 (ω) ≈ m 2πa ↑ 0 0 − m 2πa ↓ (25) − √ 2 m 3/2 2π √ − F − ω 0 0 −i √ F − ω(26) We find that the T-matrix only has a pole (i.e. det T −1 (ω) = 0) when the effective scattering length a ↑ is positive. The frequency of the pole is ω b = − F − 1 2ma 2 ↑(27) By looking at the complimentary 2 × 2 Nambu subspace, we find that another bound state exists for positive values of a ↓ with frequency ω b = − F − 1/2ma 2 ↓ . If we relax the approximation ∆ 0, we find that the under-sea state only becomes a sharp bound state in the limit ω b → −∞. If the binding energy is not very large, then the under-sea bound state can serve as a Kondo impurity. However, detailed analysis of this possibility is beyond the scope of the present article. Shiba states We now turn to the in-gap bound states predicted by Shiba, that is, \|ω\| < ∆. For weakly enough interacting BCS gases, we can make the approximation \|ω\| < ∆ F . Within this approximation, T −1 (ω) ≈ m 2πa ↑ 0 0 − m 2πa ↓ + m 3/2 √ 2 F 2π √ ∆ 2 − ω 2 ω −∆ −∆ ω(28) The form of the T-matrix in the complimentary Nambu subspace can be obtained from this one by making the substitutions ∆ → −∆, a ↑ ↔ a ↓ . The poles of the T-matrix are defined by the equation ω √ ∆ 2 − ω 2 = ± 1 + k F a ↑ k F a ↓ k F a ↓ − k F a ↑ ,(29) where the + sign corresponds to the first Nambu subspace and the − sign to the second Nambu subspace. From equation (29), we see that as long as a ↑ = a ↓ there is exactly one pole of the T-matrix in each of the two subspaces. The two poles have opposite frequencies and correspond to the two Energies of the two in-gap (Shiba) bound states as a function of 1/kF a ↑ and 1/kF a ↓ (here we use the approximation of (29), we took ∆ = 0.2 F , and ω is measured in units of F ) Shiba states. We can interpret the negative frequency pole as a bound state for the quasi-particle of the gas, and the positive frequency solution as a bound quasi-hole. In Fig. 2, we split the {1/k F a ↑ , 1/k F a ↓ } plane into two domains: the blue domain corresponds to negative pole being in the first subspace, and the white domain to the negative pole in the second subspace. The corresponding frequencies of the two Shiba states are plotted as a function of 1/k F a ↑ and 1/k F a ↓ in Fig. 3. From (29) and Fig. 3, we see that the bound states are located inside the gap only for nonzero values of a ↑ − a ↓ . This fact can be quite straightforwardly interpreted: the interaction between the Cooper pairs and the impurity can be analyzed as the sum of a 'magnetic' term proportional to a ↑ − a ↓ and a non-magnetic term proportional to a ↑ + a ↓ . The impurity can break Cooper pairs and give rise to in-gap states only when the magnetic term is finite. We note that when the non-magnetic term becomes zero, we recover the formula established by Shiba for a spin impurity in an electronic superconductor. Finally, we comment on the approximation that went into (29). In figure 4 we compare the frequency of the Shiba state (of the first Nambu subspace) calculated using (29) and numerical solution of (24). We see excellent agreement between the approximate and exact answers, which persists to surprisingly large values of gap, ∆ 0.5 F . Discussion of bound states We underline that the Shiba and under-sea bound states are not related to each other. For example if both scattering lengths are negative, but unequal, then the two Shiba states are still present while the under-sea states are not. On the other hand for two positive and unequal scattering lengths there is a pair of under-sea bound states in addition to the two Shiba states. Finally if one scattering length is positive and the other is negative then there are again two Shiba states but only one under-sea state. III. RF SPECTROSCOPY We suggest that radio-frequency spectroscopy could be a good experimental probe for reading out properties of the Shiba as well as under-sea bound states. Basic tools for understanding RF spectroscopy are given in 29 . RF spectroscopy works by converting \|↑ (or equivalently \|↓ ) atoms to a third hyperfine state labeled \|3 by irradiating the system with photons of frequency ω RF that bridges the energy difference between \|↑ and \|3 states. The bound states show up as edges in the spectra of transferred atoms when ω RF matches the bound state energy. In the following, we begin by reviewing the Fermi golden rule formula, in terms of \|↑ Green function, for the \|↑ → \|3 transition rate as a function of ω RF . Next, we apply the (2) the remaining weight is transferred to a δ-function corresponding to the under sea bound state. We note that although the part labeled "Broadening" is divergent in the impurity density expansion, its frequency integral remains finite, and the spectral function fulfills the frequency sum rule. formula first to the case of one component gas and second to the BCS case. A. General formula for the RF transition rate We assume that the Hamiltonian of the system, subject to RF drive, may be written in the form H = H gas,impurity + H 3 + H RF ,(30) where H gas,impurity describes the fermion gas and the impurity, Correction to the RF transition rate obtained for the one component gas due to the presence of impurities as a function of the drive frequency ω, with kF a = −0.5 (top) and kF a = 0.5 (bottom). The RF spectrum for the clean case is sharply peaked at ω −ω3 ∼ 0 with the width set by either trap properties and temperature. The impurities have two main effects: (1) Since momentum is no longer a good quantum number, the impurities broaden the sharp absorbtion peak at ω −ω3 ∼ 0. This broadening is composed of the depletion of the δ-function indicated by the blue arrow together with population of nearby-in-frequency states. (2) If there is a bound state, it induces an edge in the spectrum of transferred atoms followed by a broad feature indicated in pink. The broadening correction cannot be accurately captured in an expansion in impurity density. In fact at first order in impurity density we find that the correction is divergent but integrable. Therefore, in the figure we cut it off with a wavy line. While feature (1) is present independently of the sign of the scattering length, feature (2) which corresponds to the coherent part of the transition rate correction (i.e. the bound state induced part) is present only for positive scattering length. H The RF drive can be described by the Hamiltonian H RF = Ω RF d 3 k (2π) 3 (e −iωRFt c † 3,k c ↑,k + e iωRFt c † ↑,k c 3,k ),(31) where Ω RF and ω RF are the intensity and frequency of the RF drive; c † 3,k (c 3,k ) and c † ↑,k (c ↑,k ) are the creation (annihilation) operators for fermions in the \|↑ and \|3 hyperfine states. Since RF photons have a very small momentum (large wavelength) we neglect the momentum imparted on the atoms by the photons. Atoms in \|3 hyperfine state are treated as free fermions and are described by the Hamiltonian H 3 = d 3 k (2π) 3 (ω 3 + k )c † 3,k c 3,k ,(32) where ω 3 is the splitting between the \|↑ and \|3 states in vacuum. The corresponding (Matsubara) Green function for \|3 fermions is G 3 (k, iω n ) = 1 iω n − ( k + ω 3 ) . The Golden rule formula states that current from \|↑ to \|3 is 28 I(ω RF ) = 2Ω 2 D(iω n → ω RF + i0 + ) ,(34) where D(iω n ) = d 3 k (2π) 3 1 β iω1 G ↑ (k, iω 1 )G 3 (k, iω 1 + iω n ),(35) and ω 1 and ω n are fermionic and bosonic Matsubara frequencies, respectively. Our Golden rule formula gives the transition rate per unit volume. To obtain the transition rate per particle, we must divide I(ω RF ) by density [we shall use units where the density is set to k 3 F /(6π 2 ) = √ 2/(3π 2 )]. We restate the golden rule formula in the more familiar real time version I(ω RF ) = Ω 2 d 3 k (2π) 3 d 2π A ↑ (k, )A 3 (k, + ω RF )n F ( ),(36) where A σ (k, ω) = −2 G σ (k, ω + i0 + ) are the spectral functions for σ = {↑, 3} fermions, n F ( ) is the Fermi function for the ↑ fermions, and we have assumed that the 3 band is empty. Using the fact that the \|3 state is non-interacting, we can simplify this expression I(ω RF ) =Ω 2 d 3 k (2π) 3 A ↑ k, k 2 2m + ω 3 − ω RF × n F ( k 2 2m + ω 3 − ω RF ).(37) Adding the assumptions that we are working at zero temperature and the system has spherical symmetry, we can simplify the expression for the current even further I(ω RF ) = Ω 2 √ 2m(ωRF−ω3) 0 k 2 dk 2π 2 A ↑ (k, k 2 2m + ω 3 − ω RF ).(38) To apply Eq. (36) to the impurity problem, we separate the spectral function into that of the clean system A 0 (k, ω) and corrections that depend on the impurity density ∆A(k, ω) A 0 (k, ω) = A 0 (k, ω) + n i [∆A c (k, ω) + ∆A i (k, ω)] .(39) Here, we have further separated the impurity contribution ∆A(k, ω) = ∆A c (k, ω)+∆A i (k, ω) into a coherent part that corresponds to the spectral weight of impurity bound states and incoherent part that corresponds to the broadening of the continuum states by impurity scattering. We apply the same criteria to separate the RF transition rate I(ω) = I 0 (ω) + n i (∆I c (ω) + ∆I i (ω)),(40) where I 0 (ω) corresponds to the transition rate of a clean system, while ∆I c (ω) and ∆I i (ω) are the coherent and incoherent corrections due to the impurities. B. RF spectrum of a one-component gas with an impurity Suppose that the atom cloud is composed of a single, noninteracting, fermionic species in the hyperfine state \|↑ . To understand the RF induced transition rate, and how it is affected by an impurity, it is useful to begin by describing the spectral function of the \|↑ fermions. The clean spectral function has the form A 0 (k, ω) = 2πδ(ω − k 2 /2m + F ). The impurity induced corrections to this spectral function ∆A(k, ω) are plotted in Fig. 5a. These corrections move spectral weight away from the clean dispersion and can be separated into an incoherent part that corresponds to the broadening of the continuum band by impurity scattering and a coherent part that corresponds to the impurity bound states. In Fig. 5b, we plot a slice through ∆A(k, ω) at fixed k = 0.5k F . In the slice we see three main features. First, we see a negative δ-function feature, the location of which coincides with the positive δ-function in A 0 (k, ω) (feature 1a). This feature corresponds to the depletion of spectral weight from A 0 (k, ω). The spectral weight is transferred in to two regions: the under-sea bound state, which appears as a positive δ-function in ∆A(k, ω) (feature 2); the spectral weight is also transferred to the vicinity of the negative δ-function feature and corresponds to the broadening of the sharp dispersion of the clean system (feature 1b). Within our classification system, features 1a and 1b correspond to incoherent spectral weight, while feature 2 corresponds to coherent spectral weight. Finally, we point out that although feature 2 is divergent, its frequency integral is finite. Indeed, the full spectral function satisfies the frequency sum rule, which means that the corrections satisfies 0 = ∞ −∞ dω 2π ∆A(k, ω),(41) for all k. Having sorted out the spectral function we move on to the question of transition rate. Since the dispersions of the \|↑ hyperfine state and \|3 state match, the clean part of the transition rate is sharply peaked at ω = ω 3 + F and has the form I 0 (ω RF ) = Ω 2 k 3 F 3π δ([ω 3 + F ] − ω RF ).(42) At this point we pause to make several remarks. First, we remark that we have been following the notation in which the bottom of the \|↑ band is shifted to the frequency − F . Therefore, the frequency difference between the bottom of the \|↑ band and the bottom of the \|3 band isω 3 = ω 3 + F . As a result, the frequencyω 3 and not ω 3 features in the transition rate formula Eq. (42). In the ultracold atom context, it is natural to fix the "bare" splitting asω 3 instead of ω 3 , since the bottom of the \|↑ band does not move as the atom density is changed. Our second remark concerns the trapping potential. It is important to focus the RF radiation on the center of the trap in order to avoid the spatial smearing (due to shift of the Fermi energy), as discussed in Ref. 29 . Next, we come back to the effects of the impurity. For positive scattering length, there is an impurity bound state which results in a coherent correction to the transition rate ∆I c (ω RF ) = Ω 2 2 2ma 2 (ω RF − [ω 3 + F ]) − 1 ma 2 (ω RF − [ω 3 + F )]) 2 . (43) In addition to this coherent correction there is also an incoherent correction, that occurs regardless of the sign of the scattering length, and results in the broadening of the sharp transition rate of the clean state. We plot the impurity induced corrections to the transition rate in Fig. 6 for both negative and positive scattering length. For the positive scattering length case, we highlight the coherent part of the transition rate, given by Eq. (43), with pink shading. The incoherent part of the transition rate correction is composed of a negative δ-function feature (indicated by an arrow in Fig. 6) and a broad positive feature. The δ-function feature corresponds to feature 1a discussed above: the depletion of spectral weight (and thus transition rate) from the clean spectral function. On the other hand the broad positive feature corresponds to feature 1b: the broadening of the dispersion curve of the clean system. Since feature 1b is divergent, we cut it off with a wavy line. As discussed above, this divergence is a spurious consequence of the expansion in impurity density, and we do not expect to see it in experiment. C. RF spectrum of a BCS gas with an impurity In the clean BCS system, the fermion spectral function (for both species of fermions) has the form A 0,σ (k, ω) = π E k [(E k + ξ k )δ(ω − E k ) +(E k − ξ k )δ(ω + E k )] ,(44) where E k = ξ 2 k + ∆ 2 . The main feature of this spectral function is the superconducting gap in the density of states around the Fermi-surface. As before, the action of the impurity is to modify the clean Green functions and consequently the spectral functions. We begin by investigating how this spectral function is modified by the presence of the impurity atom, i.e. we compute −2 G 0 (k, ω + i0 + )T (ω + i0 + )G 0 (k, ω + i0 + ). We plot the change in the spectral function for both species of fermions induced by a magnetic impurity having k F a ↑ = 0.5 and k F a ↓ = −0.5 in Fig. 7. Similar to the case of the single component gas, we see that the impurity has two effects. it induces a Shiba state just under the Fermi energy, while for \|↓ fermions it induces a Shiba state just above the Fermi energy. In addition, as a ↑ is positive, the impurity induces an under-sea state for the \|↑ fermions that is analogous to the under-sea state of the one component gas. The RF spectrum for the clean BCS system is plotted in Fig. 8a, and the impurity induced corrections for the up and down atoms are plotted in Figs. 8b and 8c, respectively. The corrections to the RF spectrum due to the magnetic impurity are strongest for the \|↑ to \|3 transition, depicted in Fig. 8b. These consist of: (1) a dramatic filling of the gap, i.e. transitions to the left of the threshold frequency for the clean system, associated with the Shiba state below the Fermi energy; and (2) an edge in the spectrum that appears to the right of the main peak for the clean system associated with under-sea bound state. In the next three subsections we give analytical expressions for the RF spectrum of the clean system and the corrections due to under-sea and Shiba bound states. RF spectrum of the clean system Using Eqs. (36) and (44) we find that the transition rate for the clean system is I 0 (ω) = Ω 2 m 3/2 ∆ 2 (ω − ω 3 ) 2 − ∆ 2 − 2 F 2π(ω − F − ω 3 ) 5/2 .(45) We note that by dividing our expression by the particle density we recover the transition rate per particle established by Ketterle and Zwierlein 29 . We plot this transition rate in Fig. 8a. The sharp onset at low frequencies corresponds to exceeding the threshold frequency ω th = ω 3 + 2 F + ∆ 2 − F ,(46) associated with the band bottom. Under-sea states We follow the approximations of subsection II B, ω ≤ − F , ∆ ≈ 0, and use the approximate T-matrix of Eq. (26). Around the pole ω b = − F − 1 2ma 2 1 , the T-matrix takes the asymptotic form T (ω ≈ ω d ) ≈ 1 ω − ω b 2π a1m 2 0 0 0 .(47) We recognize that in the vicinity of the bound state, the singularity of the [1, 1] component of the T-matrix has the same form as the singularity of the T-matrix in the single component gas case, Eq. (9). Thus, within our approximation ∆ ≈ 0, the coherent part of the RF spectrum due to an under-sea bound state is identical to that of the single component gas, Eq. 43. This contribution is indicated by the pink shaded region on the right of Fig. 8b. Shiba states Following the assumption of subsection II B (\|ω\| < ∆ F ) and using the T-matrix of equation (28) we compute the coherent contribution to the RF transition rate from a Shiba bound state. For the coherent contribution we focus solely on the pole located at ω = −\|ω b \|, which exists in either the first or second subspace of the T-matrix depending on which domain of the { 1 a1 , 1 a2 } plane we are working, see Fig. 2. We assume that we are working at sufficiently low temperature so that only the negative frequency Shiba state is filled, and focus on the case of the negative frequency pole being in the first subspace. If it is in the second subspace, then the filled Shiba state corresponds to a \|↓ atom, and thus to detect it we must use the RF transition \|↓ → \|3 instead of \|↑ → \|3 . Around the pole ω b , the asymptotic form of the T-matrix is found to be T (ω ω b ) 2π mk F 1 k F a1 − 1 k F a2 1 k F a1 − 1 k F a2 2 + 1 + 1 k F a1k F a2 2 (48) 1 ω − ω b ω b − 1 k F a2 ∆ 2 − ω 2 b −∆ −∆ ω b + 1 k F a1 ∆ 2 − ω 2 b = 1 ω − ω b R,(49) where we define R to be the regular part of the T-matrix in the vicinity of the pole. The coherent contribution to the spectral function must come from the above pole of the T-matrix. Combining the above form of the T-matrix with the clean BCS Green function Eq. 13 and the Golden Rule formula Eq. 43 we obtain ∆I c (ω) = Ω 2 mk w π [G 0 (k w ω b ) · R · G 0 (k w , ω b )] 11 ,(50) where k w = 2m(ω + ω b − ω 3 ) and · indicates a matrix product and [] 11 indicates the [1, 1] component of the matrix. From this expression, we see that for a Shiba state the threshold frequency for RF transition is ω th = ω 3 −ω b . The coherent contribution of the Shiba state to the RF spectrum is indicated by the pink shaded region on the left of Fig. 8b. From the spectrum we see that most of the weight in the coherent part of the RF spectrum occurs at frequencies significantly higher than ω th . This is due to the fact that the Shiba state has most of its spectral weight concentrated at momenta ∼ k F . IV. EXPERIMENTAL REALIZATION In this section, we turn to the experimental realization of such a system. In a typical dilute ultracold atomic gas, the Fermi wavevector will be on the order of k F ∼ 1/4000a 0 , where a 0 is the Bohr radius. The typical order of magnitude of the scattering length, in the absence of Feshbach resonance, is given by the Van der Waals interaction a ∼ 50 − 100a 0 . In this regime, the k F (a ↑ − a ↓ ) and k F (a ↑ + a ↓ ) amplitudes always remain smaller than unity. Thus, in the absence of the resonance, the Fermion-Impurity (F-I) scattering lengths a ↑ and a ↓ have roughly the same background values. As a result, the magnetic character of the interaction is vanishingly small and thus the Shiba states are too close to the gap edges to lead to observable results. The experimental conditions shall thus be chosen such as these two scattering lengths have widely different values, that is, close to an interspecies Feshbach resonance corresponding to one of the F-I interactions. Simultaneously, we wish to stay close above the F-F Feshbach resonance in order to maintain the large negative value of the associated scattering length. In conclusion, the impurity atom must be chosen to have a Feshbach resonance with one of the fermion hyperfine levels for a magnetic field slightly superior to the F-F resonant value. In addition to the requirement for Feshbach resonances, it is necessary to be able to confine the impurity very tightly in an optical lattice, while the fermions should still be relatively free. This would favor using a light fermion and a relatively heavy impurity atom, and employing a wavelength for the optical lattice that is near-detuned with respect to the optical transition of the impurity atom One possible choice of fermion atoms are the two lowest hyperfine states of 6 Li, which have a Feshbach resonance at B 0 = 834 G. In order to achieve a BCS state, we want a "slightly superior" magnetic field, which means here that the difference between B and B 0 shall be kept within the range of the Li-Li resonance width, which is approximately ∆B ∼ 300 G. Amongst the few easily trappable bosons or fermions that could form a stable ultracold mixture with 6 Li, the boson 23 Na seems to rather well fit the above condition. Several Feshbach resonances have been observed between the 23 Na hyperfine ground state and the ground state \|1 of 6 Li, at magnetic fields close to the broad 6 Li-6 Li Feshbach resonance [Ref. 33 ]. From these data, Gaesca, Pellegrini and Côté deduce in [Ref. 34 ] the existence of further resonances between Na and 6 Li in states \|1 and \|2 between 834 and 1500 G. A complete list of predicted resonances is presented by Stan in Refs. 33,35 , along with a discussion of whether each corresponding hyperfine mixture may or may not be stable towards losses due to spin-exchange collisions. For sodium-lithium mixtures, a possibility is to use a green lattice laser at 532 nm. The effective mass for the sodium and lithium atoms as a function of lattice depth is plotted in Fig. 9a. For a lattice beam with ∼ 120 µm waist, and a potential depth of about 4 lithium recoil energies, the sodium tunneling is essentially switched off (with an effective mass of m * = 1000m), while lithium is still forming an itinerant Fermi sea (with an effective mass of m * ∼ m). Another interesting combination are the lithium-rubidium interspecies resonances that have been found in Ref. 36 . Here, there is a very interesting resonance at 882 G which is 1.3 G wide, not far from the 834 G resonance in lithium, andas required for the assumptions in the paper -on the BCS side. The advantage of using the 882 G Rb-Li resonance over any of the Na-Li resonances is technical: the 882 G Rb-Li resonance has a width of 1. Throughout we have used a ↑ kF = 0.5, a ↓ = −0.5, and ∆/ = 0.4. In (b) the coherent part of the transition rate correction, i.e. the part induced by the Shiba and the under-sea bound states is indicated by pink shading, with the peak on the left corresponding to the Shiba state and the peak on the right to the under-sea state. Similar to the case of the single component Fermi gas, the incoherent part of the transition rate correction is divergent at this order in impurity density (see Fig. 6). Therefore, the total transition rate correction which is plotted in (b) and (c) is also divergent, and we cut it off with wavy lines, as before. density, the 882 G Rb-Li resonance may lie in the BEC-BCS crossover regime as opposed to the BCS regime. We suspect that the Shiba states will continue into the crossover regime, however determining their properties requires extending our theory. Alternatively, there is a Rb-Li resonance at 1067 G, which is wide (10.6 G), but lithium is then less strongly interacting, making it more difficult to attain a superfluid. For lithium-rubidium, one could use a laser tuned to about 820nm (see Fig. 9b). As rubidium is so heavy compared to lithium, it makes for a very good localized impurity. V. OUTLOOK We suggest that the "implantation" of magnetic impurities into ultracold atom systems could lead to many exciting possibilities. As already mentioned, one class of possibilities involves leveraging the interaction of magnetism and superconductivity. This class includes the application of magnetic impurities as local probes, which is the subject of the present paper. Another possibility is to study how the pair-breaking effect of the magnetic impurities leads to the destruction of superconductivity under various conditions. In 3D one would hope to realize the transition from gap-full to gapless superconductivity. On the other hand, in 2D and 1D the pairbreaking effect of the magnetic impurities is predicted to drive the superconductor-insulator transition. Another class of possibilities involves the Kondo effect. We already see a precursor to the Kondo level in the under-sea bound state. The nature of this under-sea bound state should undergo a dramatic transformation as we turn on the Kondo effect by changing the fermion-fermion interactions from attractive to repulsive. Significantly, using an optical lattice to localize the impurity atoms naturally invites the experimental realization of the Kondo-lattice model in the setting of ultracold atoms. The Kondo-lattice model is, in turn, a stepping stone on the path of studying itinerant magnetism. One significant difficulty in seeing magnetism in the setting of ultracold atoms has been the issue of achieving sufficiently low temperature. Perhaps magnetism without an underlying lattice could be technically advantageous. That is perhaps it will be easier to achieving the Kondo temperature by avoiding the lattice induced losses that feature prominently in the quest to achieve a magnetic transition (e.g. Néel temperature) in lattice systems. VI. CONCLUSIONS We have investigated the possibility of introducing a magnetic impurity into a cloud of ultracold fermions. In particular we have focused on the realization of a localized impurity atom that is immersed in a one or two component Fermi-gas. To understand the action of the impurity atom on the fermions, we have argued that it can be described by an effective scattering length, at least for the case of a broad resonance with a sufficiently tight impurity confining potential, which we relate to the impurity-fermion scattering length in vacuum. Using the effective scattering length description, we find the effects of the impurity on the free Fermi gas as well as a two component BCS condensate. In both cases we find that if there is a positive effective scattering length then the impurity forms an "under-sea" bound state. In addition impurity scattering breaks translational invariance and thereby broadens the spectral function of the clean system. Finally, for the BCS state if the impurity-fermion scattering lengths are different then the impurity always induces a pair of Shiba bound states inside the gap of the superconductor. We demonstrate that the impurity bound states appear as additional features in Radio Frequency spectroscopy that should a b be detectable experimentally. Specifically, we suggest that 6 Li BCS condensate with 23 Na impurities could be a potentially fruitful experimental system for study of magnetic impurities. We speculate that beyond the study of bound states of dilute impurities, the same setup in combination with RF spectroscopy could be useful for study of gapless superconductivity, Kondo effect, Kondo lattices, and other problems that combine localized moments and itinerant fermions. where p i , r i and m i stand for the momentum, position, and mass of the impurity, p α , r α , and m α for the momentum, position, and mass of the scattering fermion, V i−α is a pseudo potential that describes scattering of the fermion off of the impurity in vacuum. Eq. (A1) completely defines the scattering problem. However, in general, the equation must be solved numerically. For the sake of achieving a quantitate answer, we make additional assumptions. First, we assume that the effective range of the pseudo potential V i−α is much narrower than the spa- tial extent of the harmonic oscillator ground state /mω i . Combining this assumption with the assumption that the typical collision energy scales ω i are much smaller than the characteristic resonance scale 2 µa 2 0,α , we can replace the interaction potential by a δ-function V i−α (r i − r α ) = 2πa0,α µ δ(r i − r α ). Finally, to obtain an analytical answer we make the frozen impurity orbital approximation. That is we first assume that we can write the wavefunction for the fermion and the impurity in a product form Ψ(r i , r α ) = ψ i (r i )ψ α (r α ), and then we assume that ψ i (r i ) is frozen to be the impurity ground state wavefunction. This approximation is similar in spirit to the Born-Oppenheimer approximation that a heavy fermion is moving in the field of a fast impurity, with the additional assumption that the interaction between the two is sufficiently small that the impurity wavefunction is only weakly effected by the fermion. The approximation is valid for a 0,α (µ/m i ) /m i ω i . The product wavefunction and the frozen impurity wavefunction approximations often appear in scattering theory. A particularly analogous problem where these approximations have been extensively used is the elastic scattering of a low energy electron from a hydrogen atom 37 . In this example, the role of the confinement potential is played by the electrostatic potential of the nucleus, which serves to localize the electron of the hydrogen atom. Within the frozen impurity wavefunction approximation, we define the effective potential that the fermion feels to be V eff (r) = 2πa 0,α µ \|ψ 0 (r)\| 2 (A3) where ψ 0 (r) = (m i ω i /π ) 3/4 e −mωir 2 /2 is the ground state wavefunction of the impurity. The effective scattering length, for small a 0,α , is given by a α = 2.12 a 0,α m α µ 1 + O 2 a 0,α m α √ m i ω i µ . (A4) We note that even within our simple approximation, we find "geometric" resonances that are induced by bound states of V eff . These resonances can be clearly seen in the plot of the effective scattering length as a function of a 0,α in Fig. 10. We expect that some of these resonance would survive in a more complete theory of the scattering process, and could be used in experiment to tune the "magnetism" of the impurity. PACS numbers: 67.85.-d, 67.85.Lm, 37.10.Jk FIG. 1 . 1(a) diagrammatic representation of equation FIG. 2 . 2Representation of the sign of the solutions given by the first and second subspaces as a function of 1/kF a ↑ and 1/kF a ↓ . In the blue colored domain the solution given by the first subspace is negative, while in the white colored domain the one given by the second subspace is negative.FIG. 3. FIG. 4 . 4Comparison between the approximate analytical solution (blue dashed curve) and the exact numerical solution (black curve) for the frequency (in units of F ) of the in-gap bound state (of the first Nambu subspace) as a function of 1/kF a ↑ , with ∆ = 0.2 F and 1/kF a ↓ = −0.5 FIG. 5 . 5Impurity induced correction to the spectral function ∆A(k, ω) of the one component Fermi gas for the case akF = 0.5 (white -increase of spectral weight, blue -no change, red -decrease). (a) ∆A(k, ω) as a function of momentum and frequency. The dashed white line indicates the position of the Fermi energy. ∆A(k, ω) shows a depletion of spectral weight along the clean dispersion line k 2 /2m − F (indicated by the red line), an under-sea bound state at ω = −3 F , and excess spectral weight in the vicinity of the continuum band which corresponds to the impurity induced broadening. (b) ∆A(k, ω) as a function of frequency only with momentum fixed at k = 0.5kF [slice is indicated by the green line in (a)]. The spectral function can be decomposed into three (labeled) features: (1a) a δ-function corresponding to the depletion of spectral weight along the clean dispersion line; (1b) part of the depleted weight is transferred into the vicinity of the clean dispersion line resulting in its broadening; FIG. 6. Correction to the RF transition rate obtained for the one component gas due to the presence of impurities as a function of the drive frequency ω, with kF a = −0.5 (top) and kF a = 0.5 (bottom). The RF spectrum for the clean case is sharply peaked at ω −ω3 ∼ 0 with the width set by either trap properties and temperature. The impurities have two main effects: (1) Since momentum is no longer a good quantum number, the impurities broaden the sharp absorbtion peak at ω −ω3 ∼ 0. This broadening is composed of the depletion of the δ-function indicated by the blue arrow together with population of nearby-in-frequency states. (2) If there is a bound state, it induces an edge in the spectrum of transferred atoms followed by a broad feature indicated in pink. The broadening correction cannot be accurately captured in an expansion in impurity density. In fact at first order in impurity density we find that the correction is divergent but integrable. Therefore, in the figure we cut it off with a wavy line. While feature (1) is present independently of the sign of the scattering length, feature (2) which corresponds to the coherent part of the transition rate correction (i.e. the bound state induced part) is present only for positive scattering length. First, it induces a broadening of the continuum states. Second, it induces the formation of bound states. For the \|↑ fermions FIG. 7. Impurity induced correction to the spectral function of the \|↑ atoms (left) and \|↓ atoms (right) as a function of momentum and frequency for the case a ↑ kF = 0.5, a ↓ = −0.5, and ∆/ = 0.4 (white -increase of spectral weight, blue -no change, red -decrease). The dashed white line indicates the position of the Fermi energy. Both A n i ,↑ (k, ω) and A n i ,↓ (k, ω) show a depletion of spectral weight along the dispersion curve of the clean system indicated by the red line. A n i ,↑ (k, ω)shows an under-sea bound state at ω ≈ −3 F as well as a Shiba state at ω ≈ −0.13 F , while A n i ,↓ (k, ω) shows only a Shiba state at ω ≈ 0.13 F . In addition, there is spectral weight in the vicinity of the dispersion curve of the clean system which corresponds to impurity induced broadening. FIG. 8 . 83 G while the Na-Li resonances have widths of ∼ 300 mG. However, depending on the Li (a) RF transition rate for BCS state as a function of the drive frequency ω. Corrections to the transition rate for the \|↑ atoms (b) and \|↓ atoms (c). (d) Total transition rate (clean+corrections) for 10% concentration of impurities, with divergences smoothed out. FIG. 9 . 9(a) Effective mass m * of lithium and sodium atoms in a 532 nm lattice as a function of the lattice depth (measured in lithium recoil energies). Using a potential depth of ∼ 4 ER,Li it is possible to localize the sodium atoms (that serve as impurities) while lithium atoms remain itinerant. (b) Effective mass for lithium and rubidium atoms in a 820 nm lattice. Lattice depths between about 0.5 and 4 ER,Li can be used to localize the Rb atoms while Li remains itinerant. FIG. 10 . 10Effective scattering length a that describes the scattering of a free fermion of mass mα on an impurity of mass mi localized in a harmonic potential of frequency ωi as a function of fermionimpurity atom interaction strength (scattering length in vacuum a0,α) computed in the Frozen impurity approximation. For small a0,α, a depends linearly on a0,α, Eq. (A4). However, for large a we see a deviation from linear law. For large negative a it is possible to form bound states of the fermion, which result in Feshbach resonances at (mαa0,α/µ) 2miωi/π ≈ {−1.5, −9, ...}. 3 describes the Fermions in the \|3 hyperfine state, and H RF describes the action of the RF radiation. In writing H in this form, we make the standard assumption that fermions in the \|↑ and \|↓ hyperfine states do not interact with fermions in the \|3 hyperfine state except through the action of H RF . Our goal is to calculate the RF current (i.e. the transfer rate of atoms from state \|↑ to state \|3 ) that is induced by H RF , which we do in second order perturbation theory (Fermi golden rule). ACKNOWLEDGMENTSThe authors acknowledge support from a grant from the Army Research Office with funding from the DARPA OLE program, CUA, NSF Grant No. DMR-07-05472 and PHY-06-53514, AFOSR-MURI, the AFOSR Young Investigator Program, the ARO-MURI on Atomtronics, and the Alfred P. Sloan Foundation.Appendix A: Scattering ProblemIn this appendix, we state the scattering problem for the case of a confined impurity, and provide an analytic solution under special circumstances. We describe the scattering of a single fermion off of the trapped impurity by the Hamiltonian . H Shiba, Prog. Theor. 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[]	[ "\nDepartment of Electrical and Electronic Engineering\nUniversity of Peradeniya\n20400PeradeniyaSri Lanka\n", "\nDepartment of Computer Science\nUniversity of Maryland\n20742College ParkMDUSA\n", "\nDepartment of Informatics\nTechnical University of Munich\n85748GarchingGermany\n", "\nSchool of Engineering\nCardiff University\nCF24 3AACardiffWalesUK\n", "\nDepartment of Community Medicine\nUniversity of Peradeniya\n20400PeradeniyaSri Lanka\n" ]	[ "Department of Electrical and Electronic Engineering\nUniversity of Peradeniya\n20400PeradeniyaSri Lanka", "Department of Computer Science\nUniversity of Maryland\n20742College ParkMDUSA", "Department of Informatics\nTechnical University of Munich\n85748GarchingGermany", "School of Engineering\nCardiff University\nCF24 3AACardiffWalesUK", "Department of Community Medicine\nUniversity of Peradeniya\n20400PeradeniyaSri Lanka" ]	[]	Citation: Jayatilaka, G.; Hassan, J.; Sritharan, S.; Senanayaka, J.B.; Weligampola, H.; Godaliyadda, R.; Ekanayake, P.; Herath, V.; Ekanayake, J.; Dharmaratne, S. Holistic	10.3390/app12178428	[ "https://export.arxiv.org/pdf/2112.06428v4.pdf" ]	245,123,809	2112.06428	3f45292af49421eb520f0c6ce5ea0409ffbeddd6	Department of Electrical and Electronic Engineering University of Peradeniya 20400PeradeniyaSri Lanka Department of Computer Science University of Maryland 20742College ParkMDUSA Department of Informatics Technical University of Munich 85748GarchingGermany School of Engineering Cardiff University CF24 3AACardiffWalesUK Department of Community Medicine University of Peradeniya 20400PeradeniyaSri Lanka Citation: Jayatilaka, G.; Hassan, J.; Sritharan, S.; Senanayaka, J.B.; Weligampola, H.; Godaliyadda, R.; Ekanayake, P.; Herath, V.; Ekanayake, J.; Dharmaratne, S. Holistic Introduction COVID-19 is a viral infection that causes a wide range of complications, primarily in the respiratory system [1] along with other systems [2,3]. As per current statistics, even though the virus has a comparatively small case mortality rate, it has amassed a massive fatality count due to its high infectiousness. The World Health Organization (WHO) estimates that the virus has infected around 338 million people and claimed more than 5.72 million lives as of December 2021. Despite the availability of effective vaccines against virus spread, medical complications, and mortality, complete global vaccination coverage is still far overdue. Furthermore, emerging variants cast some non-trivial obstacles to vaccine efficiency [4][5][6][7]. Therefore, mitigating the spread of the disease through social distancing, mask wearing, hand washing, sanitizing, and other practices of hygiene still remains indispensable by and large [8,9] to restore normalcy whilst ensuring the safety of health. People, being a social species, tend to exhibit group behaviors frequently. Therefore, even the most mindful persons may violate social distancing protocols occasionally [10,11]. Even arXiv:2112.06428v4 [cs.CV] 17 Aug 2022 such occasional violation of social distancing protocols may garner a risk of contracting COVID-19 depending on the proximity or duration of the violation [12,13]. Conversely, monitoring such violations of social distancing protocols (i.e., proximity, duration as well as the intensity of sudden events such as maskless cough or sneezing) provide vital tools for contact tracing, monitoring, and eventually pandemic control. In essence, observing social distancing protocol violations is a task with many caveats. Thus, automating this manual process needs meticulous analysis [14]. The main two avenues of research have been (a) intrusive solutions where people are actively contributing to the measurement (by handheld devices, etc.) and (b) non-intrusive solutions with zero burden on the people (which could be deployed to any situation irrespective of who is being monitored). The first type (intrusive techniques) requires a signal to be transmitted by the people being tracked; i.e., methods of this type require an active beacon by each tracked person. Such a wearable device based on an oscillating magnetic field for proximity sensing to monitor social distancing to prevent COVID-19 has been presented in [15]. This system was shown to be more robust than Bluetooth-sensing devices [16], especially in determining the distance threshold limit. However, it is practically difficult to deploy a solution of this type in a public space in a real-world situation. Thus, a non-intrusive solution is preferable for large-scale deployment in public spaces as the people who are being tracked are done so passively. Research in non-intrusive techniques to monitor social distancing has led to a large body of work utilizing computer vision techniques. The major sub-tasks in those approaches are the detection and tracking of people, and the state of the art for these sub-tasks is now primarily dominated by convolution neural networks (CNNs). Most recent applications combine YOLO [17] and deepSORT [18] to form powerful tools which can achieve object detection and tracking in real time, and it is used to tackle object recognition problems in different scenarios such as license plate recognition [19], road marking detection [20], pedestrian detection [21], agricultural production [22], etc. The work in [23] is an example of a CNN framework built on the aforementioned detection and localization algorithms to detect people, calculate the Euclidean distance between them and spot social distancing violations. A similar approach using YOLOv3 is performed in [24,25] for birds-eye view (overhead) camera footage. However, such overhead viewpoints are not practically deployable in public settings. An SSD-based model is presented in [26], which also performs person detection and social distancing violation identification. The performance is compared for each of the deep learning models Faster RCNN, SSD, and YOLO. Reference [27] utilizes the YOLOv4 model for people detection in low light instances to enforce social distancing measures. In [28], a spatio-temporal trajectory-based social distancing measurement and analysis method is proposed. This problem has been further examined in [29][30][31]. While various solutions proposed in the literature strive to assess the adherence to social distancing protocols, they fall short of incorporating factors such as mask wearing, which is critical to the current COVID-19 pandemic. The presence or absence of a mask on a person greatly affects the efficacy of the social distancing protocols [32]. Similarly, interperson interactions such as hugs, kisses, and handshakes are more severe concerns than mere distancing amongst individuals [33,34] as far as the person-to-person spreading of COVID-19 is concerned. The detection of mask-wearing [35][36][37][38][39] as well as the detection of dyadic interactions [40][41][42] has been explored in computer vision as isolated and distinct problems. However, to the best of the knowledge of the authors, those factors have not been incorporated into a unified and holistic solution for detecting violations of social distancing protocols in the literature Table 1. Ignoring such factors vastly undermines the robustness of vision-based techniques to tackle the social distancing problem of COVID-19. Table 1. Different social distancing measures and the handling availability in our proposed system. Social Distancing Measure Specifics Handled in Our System Physical distancing [43] Singapore (1 m), South Korea (1.4 m) Mask wearing [44] Practiced in most of the countries Close contacts [45] Handshakes, hugging, etc. Hygiene practices [44,46] Washing hands, sanitizing, etc. Restricted gathering [44,47] Indoor gatherings In this light, the system proposed in this paper analyzes the spatial and temporal interactions manifested over multiple frames. A single frame was analyzed to recognize how people adhere to social distancing measures such as keeping proper distance, mask wearing, and handshake interactions. The risk of spreading COVID-19 increases when an individual interacts with multiple people and the nature of the interaction. On the other hand, if a certain set of people are in a "bubble" and they remain so until the end of observation, there is no change in the risk of spreading COVID-19. This temporal analysis of identifying bio-bubbles is also included in our proposed model. In this paper, the design, implementation, and testing of a complete end-to-end system comprising of a framework to fit in different computer vision and deep learning-based techniques, a representation to store the output of the deep learning models, and an interpretation technique to evaluate the threat level of a given scene are discussed. The key contributions of this paper are as follows: • A deep learning-based system to monitor social distancing violations and COVID-19 threat parameters. The system can utilize multiple computer vision modules to extract different information from the video sequence such as the number of people, their location, their physical interactions, and whether they wear masks. • A temporal graph representation to structurally store the information extracted by the computer vision modules. In this representation, people are represented by nodes with timevarying properties for their location and behavior. The edges between people represent the interactions and social groups. • A methodology to interpret the graph and quantify the threat level in every scene based on primary and secondary threat parameters such as individual behavior, proximity, and group dynamics extracted from the graph representation. Proposed Solution This section explains the graph-based computer vision framework proposed to quantify the risk of COVID-19 transmission in various public scenarios. The input video feed from closed circuit television (CCTV) footage is first used to extract key information such as people, handshake interactions, and face masks through computer vision models. The proposed system then quantifies the risk of transmission of COVID-19 by encoding the extracted information into a temporal graph and interpreting it using a function for the threat of transmission developed in this paper. An overview of the proposed system is depicted in Figure 1. The system takes a video stream V in (t) as the input, where t denotes the frame number. The video stream is considered to be captured from a CCTV system camera mounted at the desired vantage point with a known frame rate. V in (t) is a three-dimensional matrix with the dimensions H × W × 3, where H and W denote the frame's height and width, respectively. The video feed, V in (t), was passed into a series of functions F i ; i ∈ {p, d, g, h, m}. Each F i processes a video frame and produces different information as J i (t) = F i V in (t)(1) where J i (t) denotes an output, such as the locations of people, handshake interactions, or the presence of face masks. While the functions F i s process individual frames, processing a sequence of frames is required to analyze this information across time. Therefore, a collection of trackers F i was employed to track the above-mentioned detections provided by F i s over time as [S i (t), J i (t)] = F i V in (t), J i (t), S i (t − 1)(2) where S i (t) is the state and J i (t) is the tracking interpretations based on the sequential information. The list of functions utilized to obtain spatial information necessary for detecting and localizing persons, interactions, and face masks follows: 1. People detection (F p ) and tracking (F p ). 2. Distance estimation (F d ) and group identification (F g ). 3. Identifying and localizing physical interaction (handshakes) (F h ). 4. Mask detection (F m ). The information retrieved by the aforementioned functions, which is critical for calculating the social distancing violation measure, was encoded in a graph G = (V, E). Sections 2.1-2.4 define the functionality of each system component that works together to populate the graph G, while Section 2.5 offers a full explanation of the data contained in the graph. Finally, graph G was interpreted in the manner described in Section 2.6 in order to provide actionable insights based on the threat level analysis of the analyzed video. For ease of understanding, the notations used in this work are listed in a table in Abbreviations. People Detection and Tracking This section discusses the proposed framework's people detection and tracking models. The people in the scene were detected using the F p detection model and then tracked over different frames using theF p tracking model. The detection model used for this purpose provides a bounding box for the person's position, whilst the tracking model assigns each person a unique ID and tracks them through time. The detection model provides a time-varying vector containing information on people's spatial location. It is defined as J p (t) = {bb p1 (t), bb p2 (t), . . . , bb pk (t), . . . , bb pn (t)}, where n is the number of bounding boxes and bb pk (t) = (u, v, r, h, c p ) is a five-tuple that represents the bounding box representing a person at time t. In bb pk (t), variables u and v represent the two-dimensional coordinates of the bounding box's center, r represents the bounding box's aspect ratio, h represents the bounding box's height, and c p represents the detection's confidence level, as shown in Figure 2. The tracker assigns an ID and updates bounding box information based on previous and current data. The output of the tracker is defined as J p (t) = {bbi p1 (t), bbi p2 (t), . . . , bbi pk (t), . . . , bbi pn (t)}, where bbi pk (t) = (u, v, r, h, c p , i) is a six-tuple representing updated bounding box information with assigned ID, i, for kth person. Given its robustness and real-time prediction capabilities, the YOLO network [48] for people detection (F p ) and the DeepSORT algorithm [18] for tracking (F p ) were used in this paper. DeepSORT by itself handles minor occlusions for people by the Kalman filtering mechanism. We implement another level of interpolation to handle the missed detections on top of this. Given an image, the YOLO network predicts the bounding boxes of many predefined object classes that are present in a scene. Following that, the output is created by applying non-max suppression [49] and filtering the bounding boxes belonging to people. The DeepSORT algorithm then assigns indices, J p , to these detected bounding boxes using the Mahalanobis distance and the cosine similarity of the deep appearance descriptors. The publicly available weights trained using the COCO dataset [50] were used to initialize the weights of the YOLO model, whereas the weights trained using the MOT dataset [51] were used to initialize the Deep-SORT model. Distance Estimation This section discusses the method for estimating the distance between identified individuals. The distance between people was estimated in three steps: first, by identifying the people's standing locations in the video, then by performing perspective transform and finally by measuring their Euclidean distance [52]. First, the standing locations of the people s (i,t) (denoted by thick black dots in Figure 3) were determined using the bounding box data as follows, s (i,t) = (u, v + 0.5h).(3) The standing locations were then transformed via perspective transform from an overhead wall mount camera viewpoint to a two-dimensional bird's eye viewpoint. The required transformation matrix M T was obtained as follows, R = M T R R R T = M T RR T R R T (RR T ) −1 = M T (RR T )(RR T ) −1 M T = R R T (RR T ) −1 (4) where the R values are 2 × 4 matrices that contain the coordinates of four reference points in the video frame (refer blue trapezoid in Figure 3-left) and the corresponding coordinates of those four points in the two-dimensional plane. This two-dimensional plane is referred to as the "floor plane" (refer Figure 3-right). The projections were performed as, f loorLocation (i,t) = M T s (i,t)(5) where s (i,t) are the input coordinates from (3) and f loorLocation (i,t) are the output coordinates on the floor plane. Finally, the distances between each pair of people i and j in frame t were calculated as dist (i,j,t) = \|\| f loorLocation (i,t) − f loorLocation (j,t) \|\|(6) Since the detected bounding boxes of people cannot be directly used to estimate distances between people due to the overhead camera viewing angle, the estimation is performed after perspective transform. The transform is performed based on the following assumptions. These assumptions hold for most of the scenes with a CCTV camera. 1. All the people are on the same plane. 2. The camera is not a fisheye-like camera. 3. The camera is placed at an overhead level. Group Identification The group identification model discussed in this section utilizes the people detection, tracking and distance estimation models introduced in Sections 2.1 and 2.2. This was achieved by two algorithms F g and F g . F g was run on the information from individual frames, while F g analyzed the results from F g across time to properly deduce which people fall into groups based on sustained physical proximity. Given a frame V in (t), a matrix M d (t) called the distance matrix is created based on the calculated distances between people. The affinity matrix M a (t) was then calculated as follows, M a = exp (−αM d )(7) where α is an input parameter that is used to introduce the camera and scene pair to a scale. This parameter acts as a threshold for the closeness of people in the 2D projected plane prior to clustering. Then, clustering was performed on M a to split the people into clusters. clusters = spectral_clustering(M a )(8) According to the group identification model, a person is considered to be a member of a group if they are close to at least one member of the group. While conventional clustering algorithms attempt to minimize the distance between individual elements and the cluster center, this is not how humans behave. As a result, this result was obtained using spectral clustering of affinity matrices [53]. Human behavior, on the other hand, cannot be analyzed in terms of discrete frames. As a result, a temporal analysis of the clusters was performed to determine the actual groups of people using a time threshold τ. The primary idea is that a group is detected only if it persists for a specified time period τ. People P i ∈ P were being clustered from a video frame at time t as follows, cluster_id(P i , t) ←− spectral_clustering(M a (t)) (9) cluster_id(P i , t) = cluster_id(P j , t) if ∃ t 0 s.t t 0 ≤ t ≤ t 0 + τ(10) where P i and P j were considered to be in the same social group as per Equation (10). Social distancing violations between the people in the same social group was ignored in the proposed system as justified in Section 1. For cases involving a few people, a simplified algorithm based on naive thresholding of interpersonal distance violation occurrences was used instead of spectral clustering. When spectral clustering was used, τ was picked so that a group should be in proximity for 10 s. For the thresholding case, people spending upwards of 20% of the time in proximity were considered groups. Mask Detection This section describes the model used to detect the presence/absence of masks. The framework's mask recognition stage entails identifying and tracking the presence (or absence) of masks. The model used for this purpose computes the bounding box of the face as well as the degree of confidence in the presence of a mask. As with prior object detection models, this model outputs a time-varying vector representing the spatial localization information for faces as J m (t) = {bb m1 (t), bb m2 (t), . . . , bb mk (t), . . . , bb mn (t)}, where n is the number of face bounding boxes at time t and bb mk (t) = (u, v, r, h, cm) is a five-tuple representation of the bounding box encompassing a detected face at time t. The variables u, v, r, and h have the same definitions as those in Section 2.1. The confidence measure c m = (c mask , c nomask ) is a two-tuple in which c mask ∈ [0, 1] indicates the probability of the presence of a mask and c nomask ∈ [0, 1] indicates the absence. Similar to Section 2.1, the tracking model returns a vector of the same size J m (t) containing tracked bounding boxes for each t. Similar to Section 2.1, the YOLO network was utilized for mask detection and the Deep-SORT algorithm was utilized for tracking the masks across frames. The YOLO model was first initialized with the pre-trained COCO weights and then fine-tuned using the images from the Moxa3k dataset [35] as well as the UT and UOP datasets, which were labeled for mask detection. The DeepSORT model used the weights trained using the MOT dataset [54] for initialization. The DeepSORT algorithm handles minor occlusions of masks. However, another layer of interpolation (such as for people detection) was not implemented because the algorithm is supposed to detect when people remove masks (i.e., people cannot disappear while masks can). Graph Representation The information extracted using different models in Sections 2.1-2.4 need to be combined to provide meaningful insights into the threat level of the given scene. This is accomplished by encoding the data into a graph structure. This section describes how the graph structure is modeled using the different outputs from the models for interpretation. The information retrieved from the video is stored as a time-varying graph G(t) given by G(t) = V(t), E(t)(11) and V(t) = {v 1 (t), v 2 (t), . . . , v n (t)}(12)E(t) = {e 1,1 (t), e 1,2 (t), . . . , e i,j (t), . . . , e n,n (t)}(13) where V(t) is the set of vertices and E(t) is the set of edges at time t. Each person P i is denoted by a vertex v i (t) which contains the features representing the person extracted from the video as time-varying vertex parameters. The vertex v i (t) is given by v i (t) = [ location i (t), mask i (t), group i (t) ](14) where location i (t) = (x i (t), y i (t)) is a two-tuple that represents the position of the person P i at time t obtained through perspective transform to a bird's-eye view position on a 2D plane (refer to Section 2.2). mask i (t) = c m is two-tuple, which shows the confidence level that a person P i is wearing a mask at time t. This information is extracted from bb mi (t) depending on the index ID mi (t) (refer to Section 2.4). group i (t) is a matrix that represents the probability that two people belong to the same group (refer to Section 2.3). The edge e i,j (t) is a binary value (0/1) that represents the presence (denoted by 1) or absence (denoted by 0) of an interaction between person P i and P j at time t detected using [39]. E(t) is stored as a sparsely filled adjacency matrix with null values for instances where interactions are not detected. A visual example of a frame and its constructed graph is shown in Fig. 4. Threat Quantification The information extracted from the models described in the proposed system in Sections 2.1-2.5 needs to be processed from the created temporal graph in order to provide a quantifiable metric that denotes the risk of transmission for the given scene/frame. In this section, the derivation of the threat level function which quantifies the threat of the given frame is described in detail. Table 2 contains a list of the parameters that contribute to the spread of COVID-19. The parameters are divided into two categories: primary and secondary parameters, which will be discussed further in this section using the threat level function. First, we calculate the threat level contribution of each pair of people in the frame at time t as described in (16). Then, we find the threat level of the particular frame as per (15). T(t) = ∑ (v 1 ,v 2 ∈V) T v 1 ,v 2 (t)(15)T v 1 ,v 2 (t) = ∑ p i ∈P p i (v 1 , v 2 ) × ∏ q j ∈Q j − q j (v 1 , v 2 )(16) P = {p h , p d } is the set of parameters that directly attributes to the transmission of COVID-19 from one person to another. This includes the distance between people and the handshake interactions. As the distance between people (people coming close) and their interactions (handshakes) play a primary role in the COVID-19 virus transmission, these values were first considered as the primary parameters P. The probability of two people shaking hands p h and the probability of them coming extremely close p d were represented as scalar values in the range [0, 1], where 1 represents a high probability of occurrence (for the distance probability, 1m is used as the threshold distance for being extremely close in this study). Q = {q m , q g } is the set of secondary parameters which are relevant only when two people are in close proximity, and in such a case, these parameters can increase or decrease the probability of COVID-19 transmission accordingly. This includes whether people are wearing masks, since two people not wearing masks is irrelevant if they are far apart, and whether the persons belong to the same group. First, the mask-wearing probability q m was used to quantify the effect of masks in transmission. Furthermore, people belonging to the same group (q g ) have a similar effect on transmission, since it is assumed that the disease spread between them does not increase depending on what is happening in the video frame (it is more likely they were both infected or not, even before coming into the frame). The values of q j are in the range [0, 1]. j ≥ 1 is used as a tuneable parameter that dictates the influence of a particular parameter q j on the overall threat level. A higher j value gives a lower significance to the corresponding q j in calculating the total threat T(t). Because the influence of various factors varies depending on variations and different pandemics, the j option can be used to change the influence of various parameters, and new parameters can be added to the threat-level equation based on consultations with appropriate authorities. By substituting the parameters and setting m = 2.0, g = 1.0, the equation was rewritten as follows, T v 1 ,v 2 (t) = (p h + p d )(2.0 − q m ) 1.0 − q g(17) When analyzing the threat equation in Equation (16), it can be noted that when the secondary parameter probabilities decrease (i.e., q j ), the effect of the multiplicative term ( j − q j ) is higher. This implies that the effects of the primary parameters p j to the threat of the given scene are compounded when the two persons have worsening secondary parameters (i.e., are not wearing masks or when they are of different groups). It can also be observed that (17) does not carry any terms with the p d p h product. This could be intuitively understood because shaking hands requires them to be physically close, and thus, incorporating this term is redundant. While (17) is tuned for the implemented system, the generic form (16) can incorporate any number of parameters being extracted from a video scene. Evaluation In this section, we discuss the methodology used to evaluate the system. The proposed solution was executed on a chosen set of datasets as the input, and the results were evaluated using different metrics. The following subsections describe the datasets, the metrics, and the evaluation execution process in detail. Datasets Existing public datasets such as MOT [51,54,55] and UT-interaction [56] were chosen to evaluate the performance of the individual components of the system. However, there are no existing datasets to perform a holistic analysis. Thus, in order to analyze this, a new dataset was created from the University of Peradeniya premises, which is referred to as the UOP dataset. The multiple object tracking (MOT) datasets are a set of image sequences with annotations for people localization and people IDs. Three datasets [51,54,55] were used to evaluate the capability of an algorithm to uniquely identify and track a person through a video. The University of Texas-Interaction (UTI) [56,57] dataset comprises twenty video sequences of human interactions in two or four-people settings. The actions in the dataset include handshake, punch, point, kick, push and hug where each video spans roughly 1 min. The UOP dataset [39] is a collection of ten video sequences that were collected from the University of Peradeniya premises by enacting a scene with human interactions such as handshakes, close contacts, and grouped conversations. These videos were recorded by a wall-mounted CCTV camera in the university corridor and waiting area. The video consists of either four or five persons, with each video spanning 1 min. The ground truth for this dataset was annotated manually for training and evaluation. Evaluation Metrics The outputs were evaluated on the given datasets based on the metrics average precision (AP) and the mean average precision (mAP). mAP is the key metric used in evaluating detector performance in prominent object detection tasks such as the PASCAL VOC challenge [58], COCO detection challenge [50] and the Google Open Images competition [59]. The average precision (AP) is the precision value averaged across different recall values between 0 and 1 [60]. This was computed as the area under the curve (AUC) of the precision vs. recall curve, which was plotted as a function of the confidence threshold of detection with a fixed intersection over union (IoU) for the bounding box threshold [61]. Model Evaluation People Detection The people detection component used here is the YOLO network, which is a wellestablished detector. Hence, no modifications were introduced to this segment of the detector. The YOLOv4 model which was used here is extensively compared in terms of frame rate and mAP in [40]. Group Identification The group identification component was evaluated using the existing MOT datasets. Since the ground truth for the datasets considered in this work do not contain the group annotated information, an alternative methodology was required for evaluation. For this purpose, a visual inspection of frames was used to determine if two individuals belonged to the same group in a given frame. Mask Detection The mask detection component requires localized information about masks. Thus, the UTinteraction dataset was re-annotated. However, this dataset only consists of unmasked faces, and as such, the annotated UOP dataset was used together with the UT-interaction dataset to train and evaluate the mask detection component. The 17 videos from the UT-interaction dataset and the five videos from the UOP dataset were used for training. The dataset was annotated with the two class information: namely, masked and unmasked faces in frames, where the faces were visible and the presence of masks can be interpreted by a human. The mask detection model was evaluated using both the AP and mAP measures. First, the model's ability to localize the faces was determined by measuring the AP of the localization component of the models disregarding the class labels. Next, the performance of the model in terms of both the localization and the accuracy was determined by the mean average precision (mAP) value. Note that since both the classes correspond to the same object (i.e., faces), this two-metric evaluation process helps us identify the specific shortcomings of the model considered. For instance, a high AP and a low mAP show poor mask detection (classification), whereas a high accuracy and low mAP denote poor face localization. Threat Level Assessment (End-to-End System) The threat level quantification algorithm was tested on the three datasets mentioned earlier. Since there is no publicly available ground truth for videos for this parameter, the results of the algorithm were evaluated by comparison with expert human input. For this purpose, 462 samples of frame pairs from video sequences were chosen. The system was then evaluated by observing the increment/decrement of the inferred threat level T(t) and comparing the results with the expert human input. The performance of the full system is evaluated using accuracy, precision, and recall. The expert responses were obtained by showing a pair of frames and asking if the threat of COVID-19 spread has increased or decreased from the first frame to the second. Since a high disparity in identifying the impact of COVID-19 spread can exist amongst human experts in certain instances, ground truth cannot be established for such pairs of frames. To identify such instances, a thresholding minimum majority required to establish ground truth was set as 70%, and all frame pairs with a higher disparity (i.e., less than 70% majority) for any given choice were removed. In the evaluation conducted, five such frame pairs were identified and removed. One such frame pair is shown in Figure 5 to conclude this factor. As it can be observed, it is difficult to assess the change in threat for COVID-19 spread across these two frames. Results and Discussion The proposed system was implemented using the Python programming language alongside and Tensorflow and OpenCV libraries. The system is deployed on a high-performance computing server with NVIDIA GPU. The output of each component of the system as well as the final output of the entire system are discussed below. People Detection and Tracking The results shown in Figure 6 are indicative of the performance of the human detection and tracking segment of the proposed system. The first row shows a sequence of frames where people are detected properly and tracked with unique IDs. However, the model fails to perform satisfactorily in specific scenarios. The bottom row gives examples of the cases in which the model can fail. From left, (1) a person is not being identified because of occlusion, (2) the identified bounding box is smaller than the person due to occlusion, and (3) a person is going undetected due to the lack of contrast with the background. The model has an mAP = 65%. As observed in Figure 6, a given frame from the output consists of multiple markings. The blue quadrilateral on the ground is the reference used for perspective transformation. The people detected are identified by uniquely colored rectangular bounding boxes. The location of each person in the 2D plane is marked using a black dot on the bottom edge of the respective bounding box. The threat level for the given frame is numerically displayed. Further details of the relevant markings will be discussed in the subsequent sections. Distance Estimation A scene consisting of four people from the UTI dataset is considered in Figure 7 to show how the distance between people contributes to the threat level. The distance between people is given by the distance activity matrix shown beside each frame in Figure 7. Each element (square) in the activity matrix denotes the proximity between the person IDs corresponding to the row and column indices. The color changes to a warmer shade (yellow) when the people are closer, and it becomes a colder shade (blue) when they are farther away. Considering the frames in Figure 7, the person ID 2 and 3 can be observed to be closer in the second frame than in the first frame. This causes a higher contribution to the threat level between them in the second frame and a lower contribution to the threat level in the first frame. This is seen in the distance activity matrix by the blue shade turning to cyan, indicating closer proximity between those persons. The reader's attention is drawn to the threat level shown in each frame. As it can be observed, when the distance activity matrix lightens up, the threat level has also risen. The errors in people detection ( Figure 6) can propagate to the distance estimation. While people going fully undetected is usually handled by interpolation, predicted bounding boxes becoming cropped due to the occlusion of feet leads to a faulty prediction for where the person's feet are. Therefore, the calculation of distance between people becomes erroneous. Group Identification The results for a few frames for the group identification model are shown in Figure 8. An example from the UTI dataset and Oxford towncenter datasets is shown here. The frames with the persons detected are shown on the left and the group activity matrices showing the group characteristics are shown on the right. If two people are of the same group, the group activity matrix element corresponding to the row and column of the IDs of these two persons is shown in yellow, and otherwise, it is shown in blue. The people of the same group are also joined by a white line in the original frame to show this. Figure 9 shows the performance of the system in detecting the presence/absence of masks. One example from the UTI, UOP, and Moxa3K datasets is shown. Overall the system performs well while dealing with high-resolution images (Moxa3K). However, as the resolution drops (UTI/UOP), the efficacy reduces drastically. This can be observed in Table 3, which lists the numerical evaluation metrics (AP and mAP) for localization on different datasets. Another prominent failure case is when the proposed system is unable to detect the presence or the absence of masks when people face away from cameras. Mask Detection Threat Level Assessment (End-to-End System) To evaluate the proposed system performance, the threat level metric provided for each frame of a given scene is evaluated across multiple frames. The successful output of this value is evaluated by the full system for both datasets UTI (Figures 10-12) and Oxford ( Figure 13). It should be noted that it is not the absolute values of the threat level that are significant but the increments or decrements between the frames. Considering Figures 10-12, it can be observed that the threat level increases from top to bottom frames as 14.7, 16.9 and 20.0. From the first frame to the second frame ( Figure 10 to Figure 11), we can see the distance activity matrix brightening in the right top and left bottom edges. This is due to the close proximity of persons ID 1 and 4. This leads to an increase in the threat level of the frame by 16.9 − 14.7 = 2.2. Similarly, when looking at the first and third frames ( Figure 10 to Figure 12), this time, the interaction activity matrix brightens up in the third frame due to the handshake interaction in this frame. This also leads to an increase in threat level, which is by 20.0 − 14.7 = 5.3. It is also clearly observed in the threat activity matrix for the third frame in Figure 12, where the center squares brighten up to show a significant threat between persons 2 and 3. This increment (of 5.3) in the threat level is higher than the previous comparison (of 2.2) in Figure 10, and Figure 11 since the handshake interaction poses a higher threat than proximity alone. The same can be observed by comparing the second and third frames. A simpler situation is analyzed in Figure 13. Here, there are only two people belonging to the same group, and they are present in the video throughout the time. However, there are no physical interactions such as shaking hands. Therefore, the only parameter that dictates the threat level is the number of people and their interpersonal distances in each frame. When analyzing Figure 13, the people in the first frame are moving away from each other until the second frame. This is why the threat level goes down from 95.0 to 46.0 from the first frame to the second. In the third frame, new people come into the frame, and they move closer to each other. Therefore, an increase in the threat level of 105.3 is observed. However, this dataset does not contain a rich set of scenes to evaluate all components of the proposed system. 14.7 Full System Evaluation The performance of the full system in comparison to human expert responses is provided in Table 4 in terms of accuracy, precision, and recall. The complete system is evaluated on both the UTI and UOP datasets. However, it should be noted that the UTI dataset does not contain anyone wearing masks, and hence, the mask detection component does not contribute to the threat calculation here. It can be noted that the system performance is not biased toward either dataset and is able to generalize with considerable accuracy of nearly 76%. A few of the notable failure cases of the system are shown in Figures 14-17, where the threat level predicted was contrary to human expert opinion. Out of the four cases shown here, three of them failed to evaluate the proper threat value due to a failure in one of the components in the system pipeline. In Figure 14, the person indicated by the purple arrow was not detected by the person detection model due to occlusion. Similarly, in Figure 15, the two individuals hugging are detected as a single person. Since it is the proximity of the three individuals in Figure 14 and the hugging individuals in Figure 15 that pose a high threat to COVID-19 spread, the system fails to reflect this, deviating from the expert opinion. In Figure 16, the high proximity of the individuals in the first frame results in a high threat value for the first frame. However, the handshake interaction model fails to detect the interaction in the second frame, hence leading to a lower threat level output by the system and hence failing to identify the increase in threat for COVID-19 spread. In the case of Figure 17, since the system design was not accounted for incidents such as a pushing action as in the second frame, the system provides a higher threat value for the first frame contrary to human expert opinion. However, there were a few rare cases where in retrospect, the system output was more plausible or instances where the failure of the system was unexplained. Considering Figure 18, the ground truth from human expert opinion was that the threat level decreases, which is explained by the handshake interaction in the first frame, which is a serious violation of social distancing protocols. However, the system output for threat value increases significantly in the second frame as a new person is identified in the far left. Since an increase in the number of people and the closer proximity of this new person in a given space should also be accounted for, this leads to the increased threat value predicted by the system. Meanwhile, Figure 19 is an instance where the system output states the threat of COVID-19 spread has increased, whereas human expert opinion is on the contrary. This deviation by the system is an edge case where the deviation is unexplained. Conclusions An end-to-end solution utilizing CCTV footage to provide a practical and versatile mechanism for monitoring crowds to identify possible instances of spreading COVID-19 is proposed in this paper. The proposed system detects people, their locations, their individual actions (wearing masks), and their dyadic interactions (handshakes) using deep learning-based video interpretation. This information is stored in temporal graphs to enable further insights such as identifying social groups. Finally, these temporal graphs undergo a more holistic interpretation using rule-based algorithms. This analysis uncovers the individual's contributions to the spread of COVID-19 (not wearing masks) and pairwise contributions (handshakes, staying close to others) on a per frame basis. These results are brought together to calculate the total threat levels in frames. Finally, these outputs are examined against expert human opinion. Consistent accuracies over 75% across all datasets could be considered a strong indication of the robust performance of the proposed system. Furthermore, this unified framework allows for the future incorporation of possible other future measures for curtailing the spread of COVID-19 or any other epidemics impacting the health and safety of society. Therefore, this proposed framework may be strengthened by incorporating additional COVID-19 specific features, and it could be adapted and adopted for similar other scenarios as well that may benefit from video or CCTV-based non-intrusive observations. The proposed system evaluation is limited by the availability of datasets. While it was tested on existing datasets and a newly collected dataset, testing this on a wider range of diverse scenes is required. Moreover, threat level estimation is performed at the frame level. As a result, the contribution of the time of interactions such as approaching close to each other and shaking hands is not taken into account in our system. Future work could improve the system by processing the time series of the threat level (individual contributors such as interactions as well as the total score) by techniques such as moving averages and filtering. Due to the widespread use of CCTV cameras, the proposed system is applicable to a wide variety of real-world scenarios. The decreasing performance-to-cost ratio of computing hardware has enabled even small organizations to acquire the system. The release of the codebase as free and open source software (FOSS) can accelerate both third-party deployment and solution improvement. However, concerns about privacy, bias, and fairness in conducting analytics on people's CCTV footage should be addressed on an individual basis in accordance with the rules and regulations of individual organizations and countries. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study for all stages including dataset creation and system evaluation. Data Availability Statement: The data collecetd in this study could be requested from J.E. via email ([email protected]). The requests will be handled in a case by case basis. Acknowledgments: GPU computing resources were provided by NVIDIA to the Embedded Systems and Computer Architecture Laboratory (ESCAL), University of Peradeniya through the GPU donation program. This research was made possible through the University of Peradeniya research infrastructure funded by the residents of Sri Lanka through their contribution to public education. The comments and suggestions from the anonymous reviewers has been instrumental in revising this paper. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Abbreviations Notations and description. Notation Definition V in (t) Input video feed F p ,F p People detection and tracking F d Distance estimation F g ,F g Group identification and tracking F h Identifying and localizing physical interaction (handshakes) F m , F m Mask detection and tracking J i (t) Output of model F i S i (t) State information bb pk (t), bbi pk (t) Bounding box encompassing person k at time t and bounding box encompassing person k at time t which is being tracked with their unique index bb hk (t), bbi hk (t) Bounding box encompassing handshake interaction k at time t and bounding box encompassing handshake interaction k at time t which is being tracked with their unique index. bb mk (t), bbi mk (t) Bounding box encompassing the face of person k at time t and bounding box encompassing face of person k at time t which is being tracked with their unique index u, v The 2D coordinates of the center of the bounding box h, r The height and aspect ratio of the bounding box R, R The coordinates of the reference points in the video frame and two-dimensional floor plane, respectively M T Transformation matrix for the perspective transform from CCTV perspective to floor plane s (i,t) Standing location of person i at time t in the CCTV perspective f loorLocation (i,t) Standing location of person i at time t in the floor plane dist (i,j,t) Distance between a pair of people i and j at time t P i Person i in the frame G(t) Graph at time t V(t) Vertices of graph G at time t given by {v 1 (t), v 2 (t), . . . , v n (t)}, each vertex corresponding to person P i with the vertex parameters embedded E(t) Edges of graph G at time t given by {e 1,1 (t), e 1,2 (t), . . . , e i,j (t), . . . , e n,n (t)}, where e i,j is the edge between person(vertex) i and j T(t) Threat level of frame at time t P = {p d , p h } Primary parameters-set of parameters that have a direct attribute to COVID-19 transmission Q = {q g , q m } Secondary parameters-set of parameters that are relevant to COVID-19 transmission when two individuals are in close proximity j Tuneable parameter dictating influence of parameter q j on overall threat level. Figure 1 . 1A high-level overview of the proposed system. Figure 2 . 2The parameters for the bounding boxes. Figure 3 . 3Perspective transformation. The (right) frame is a visualization of how a camera-captured scene (left) is projected to the 'floor plane' after perspective transformation. The trapezoidal floor is being transformed into a square. Figure 4 . 4Graph representation figure. (a) Bounding boxes for people and handshake; (b) Corresponding graph representation. Figure 5 . 5Example frames that were removed from full system evaluation due to disparity in human expert responses. Figure 6 . 6Results of people detection. (Top row)-cases where the people detection model is successful. (Bottom row)-instances where people detection is erroneous. Undetected people are marked by the purple oval. The green bounding box (marked by the purple arrow) does not span the full person. Figure 7 . 7Distance estimation results. Figure 8 .Figure 9 . 89Group identification. (Left): video frames where groups of people are denoted by white lines connecting individuals. (Right): group activity matrices showing people belonging to the same group by yellow and else blue. Mask detection detection examples. (a) UTI dataset; (b) UOP dataset; (c) Moxa3K dataset. Figure 10 . 9 Figure 11 . 0 Figure 12 .Figure 13 . 1091101213Full system result of UTI interaction dataset at t 1 .16.Full system result of UTI interaction dataset at t 2 .20.Full system result of UTI interaction dataset at t 3 . Full system result of oxford dataset. Figure 14 . 14Failure case 1 threat level interpretations. System output for threat-Decreases, Human expert opinion on threat-Increases. Figure 15 . 15Failure case 2 threat level interpretations. System output for threat-Increases, Human expert opinion on threat-Decreases. Figure 16 . 16Failure case 3 threat level interpretations. System threat evaluation output-Decreases, Human expert opinion output-Increases. Figure 17 . 17Failure case 4 threat level interpretations. System threat evaluation output-Decreases, Human expert opinion output-Increases. Figure 18 . 18Edge case 1 threat level interpretations. System threat evaluation output-Increases, Human expert opinion output-Decreases. Figure 19 . 19Edge case 2 threat level interpretations. System threat evaluation output-Increases, Human expert opinion output-Decreases. Author Contributions: Conceptualization, G.J., J.H., S.S., R.G., P.E., V.H. J.E. and S.D.; Data curation, G.J., J.H. and S.S.; Formal analysis, G.J., J.H. and S.S.; Funding acquisition, R.G., P.E., V.H. J.E and S.D.; Methodology, G.J., J.H. and S.S.; Project administration, R.G., P.E., V.H. and J.E.; Software, G.J., J.H. and S.S.; Supervision, R.G., P.E., V.H., J.E. and S.D.; Validation, J.B.S., H.W.; Visualization, G.J., J.H., S.S., J.B.S., H.W.; Writing-original draft, G.J., J.H. and S.S.; Writing-review & editing, G.J., J.H., S.S., J.B.S., H.W., R.G., P.E., V.H. and J.E. All authors have read and agreed to the published version of the manuscript. Funding: This work is funded by (1) International Development Research Centre (IDRC), Canada through grant number 109586-001 and (2) Lewis Power, Singapore. The APC was funded by IDRC as well. Institutional Review Board Statement: Not applicable. Table 2 . 2Parameters used in threat quantification.Set Notation Description P p d Distance between people p h Handshake interactions between people Q q g People belonging to the same group q m People wearing masks Table 3 . 3Performance metrics of the mask detection.Dataset AP/mAP /% UT-interaction (Unmasked) 29.30 UOP (Masked) 41.47 Moxa3K 81.04 Table 4 . 4Full system performance.Test Accuracy Precision Recall UTI dataset 75% 75% 75% UOP dataset 76% 85% 79% Overall 76% 81% 77% A comparative study on the clinical features of coronavirus 2019 (COVID-19) pneumonia with other pneumonias. D Zhao, F Yao, L Wang, L Zheng, Y Gao, J Ye, F Guo, H Zhao, R Gao, Clin. Infect. 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[ "Osculating properties of decomposable scrolls", "Osculating properties of decomposable scrolls" ]	[ "Antonio Lanteri \nDipartimento di Matematica \"F. Enriques\"\nUniversità degli Studi di Milano\nVia C. Saldini, 50I-20133MilanoItalia\n\nFirst address\n\n\nFirst address\n\n", "Raquel Mallavibarrena \nDepartamento de Algebra\nFacultad de Matemáticas\nUniversidad Complutense de Madrid\nCiudad UniversitariaE-28040MadridSpain\n\nSecond address\n\n\nSecond address\n\n" ]	[ "Dipartimento di Matematica \"F. Enriques\"\nUniversità degli Studi di Milano\nVia C. Saldini, 50I-20133MilanoItalia", "First address\n", "First address\n", "Departamento de Algebra\nFacultad de Matemáticas\nUniversidad Complutense de Madrid\nCiudad UniversitariaE-28040MadridSpain", "Second address\n", "Second address\n" ]	[]	Osculating spaces of decomposable scrolls (of any genus and not necessarily normal) are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus -deriving from flexes-are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls.	10.1002/mana.200610834	[ "https://arxiv.org/pdf/0711.3759v1.pdf" ]	17,767,269	0711.3759	fe47f3af1c288542b00b82b0e163605b4d6b7c80	Osculating properties of decomposable scrolls 23 Nov 2007 Antonio Lanteri Dipartimento di Matematica "F. Enriques" Università degli Studi di Milano Via C. Saldini, 50I-20133MilanoItalia First address First address Raquel Mallavibarrena Departamento de Algebra Facultad de Matemáticas Universidad Complutense de Madrid Ciudad UniversitariaE-28040MadridSpain Second address Second address Osculating properties of decomposable scrolls 23 Nov 2007mn header will be provided by the publisherScroll (non-normal)osculating spaceinflectional locus(higher) discriminant locus MSC (2000) Primary: 14F05, 14N05Secondary: 14J26, 14J40, 14C20, 53A20 Osculating spaces of decomposable scrolls (of any genus and not necessarily normal) are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus -deriving from flexes-are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls. Introduction The inflectional behavior of a projective variety belongs to its extrinsic geometry. In particular, flexes can appear on projective manifolds under (isomorphic) projections. Though this observation is obvious, it seems that several projective manifolds have been extensively investigated from the point of view of their osculatory behavior only in the linearly normal case. This is true e.g., for rational scrolls of any dimension [9] and also for elliptic surface scrolls [7]. In this paper we mainly consider decomposable scrolls, not necessarily linearly normally embedded, and we study their inflectional behavior. Decomposable scrolls X ⊂ P N , whose construction generalizes that of rational normal scrolls, are generated by n curves C i (i = 1, . . . , n) isomorphic each other, lying in linearly independent linear subspaces generating the whole P N (see Section 1). They are very well suited to investigate their k-th inflectional loci Φ k (X). We do that developing systematically the local description used in [9] and [7], and in Sections 1 and 2 we succeed to describe several properties of Φ k (X), relating them to the inflectional loci of the generating sections C i . In particular, restricting to the case of rational non-normal scrolls our approach allows us to produce in Section 3 a new series of counterexamples to the even dimensional part of a conjecture of Piene and Tai [10]. While the odd dimensional part of this conjecture has been proved several years ago [10], [3], the even dimensional part is false for certain linearly normal scrolls, as shown by the first author [6]. However we want to stress that the new counterexamples exhibited here are rational scrolls, though, of course, not linearly normal. We also characterize these examples in the framework of decomposable scrolls (Theorem 3.4). This adds some information in order to correct the even dimensional part of the conjecture. Let X be a decomposable scroll. While describing Φ k (X) for k > 2 involves inflectional loci of lower order, the description becomes very easy for k = 2. In particular, we show that for a decomposable scroll X, Φ 2 (X) can have only two types of irreducible components. Let G be any such a component. Then, either G is a sub-fibre of a fibre of X, or X is rational, some curve C i is a line, and G is a sub-scroll of X given by a Segre product (Proposition 4.2). This precise description of Φ 2 (X) allows us to study in Section 4 the second discriminant locus of a decomposable scroll X ⊂ P N . This is the Zariski closed subset D of P N ∨ parameterizing all hyperplane sections of X admitting a triple point. The main component of D is the second dual variety of X, which parameterizes osculating hyperplanes to X at general points and their limits. But when X has flexes, extra components D G of D arise, coming from the irreducible components G of Φ 2 (X). Our study of Φ 2 (X) allows us to describe these components: either D G is a linear space or it is a 1-dimensional family of linear spaces. In particular, we show that D G is a scroll if and only if X is a rational normal scroll generated by some lines plus conics and/or twisted cubics (Example 4.3 and Proposition 4.4). Moreover, we characterize rational normal scrolls generated by some lines plus some conics as the decomposable scrolls admitting an irreducible component D G of D which is a rational normal scroll (Theorem 4.7). In Section 5, we come to surface scrolls, not necessarily linearly normal, regardless the fact they are decomposable or not. Here the techniques developed in the previous sections fail. We discuss two points arising from [6]. a) Indecomposable elliptic surface scrolls of invariant −1 have been studied in [6]. By adapting the approach used there, we investigate those of invariant 0, providing a description of their flexes in terms of base points of suitable linear systems related to the one giving the embedding (Proposition 5.1). b) The lowest dimension of any osculating space to a surface scroll is 3, as shown in [6]. Moreover, Example 3.2 shows that any higher order osculating space can have very low dimension at some points. Here we find sufficient conditions to grant that all k-th osculating spaces of a surface scroll have dimension ≥ k + 1. They are formulated in terms of the (relatively) good properties of the linear system giving rise to the embedding (Theorem 5.2). Notation and background We work over the field of complex numbers. Let X be a smooth projective variety of dimension n ≥ 1. If L is a line bundle on X we denote by \|W \| the (not necessarily complete) linear system defined by a vector subspace W ⊆ H 0 (X, L). Suppose that \|W \| is very ample, i. e., the map defined by W is an embedding ϕ W : X ֒→ P(W ) = P N . Then L = ϕ * W (O P N (1)). In this case, frequently we look at the pair (X, W ), or at the triplet (X, L, W ), in place of the non-degenerate embedded variety ϕ W (X) ⊂ P N and sometimes we do not distinguish between X and its image. For any integer k ≥ 0 let J k L be the k-th jet bundle of L. For every x ∈ X we denote by j (X,W ) k,x : W → (J k L) x the homomorphism associating to every section σ ∈ W its k-th jet evaluated at x. When the subspace W we are dealing with is clear from the context, or the discussion involves a single pair (X, W ), we simply write j X k,x or j k,x respectively, instead of j (X,W ) k,x . Recall that j k,x (σ) is represented in local coordinates by the Taylor expansion of σ at x, truncated after the order k. So, if \|W \| is very ample, the k-th osculating subspace to X at a point x ∈ X is defined as Osc k x (X) := P(Imj (X,W ) k,x ). Identifying P N with P(W ) (the set of codimension 1 vector subspaces of W ) we see that Osc k x (X) is a linear subspace of P N . To avoid that it fills up the whole ambient space we assume that N is large enough. For instance, to discuss osculation for surfaces, i. e., k = n = 2, a reasonable assumption is that N ≥ 6 or even 5, depending on the regularity of the surface we are dealing with. Recalling that rk(J k L) = k+n : W → (J k L) x attains its maximum, say s(k) + 1. The k-th inflectional locus of (X, W ) is defined by Φ k (X) = X \ U. So x ∈ Φ k (X) if and only if dim Osc k x (X) < s(k). By flex we simply mean a point in Φ 2 (X), while a higher flex is a point of Φ k (X) with k > 2. We say that X is uninflected to mean that Φ 2 (X) = ∅. Of course Φ h (X) ⊆ Φ k (X) for h ≤ k. Let n = 1. If N < k, then clearly Φ k (X) = X. However, if N ≥ k then Φ k (X) X (e. g., see [1, p. 37, Ex C-2]). In particular, Φ N (X) = ∅ if and only if X is a rational normal curve [1,p. 39,. Now let x ∈ U. A hyperplane H ∈ P N ∨ is said to be k-th osculating to X at x if H ⊇ Osc k x (X). Then the k-th dual variety X ∨ k of (X, W ) is defined as the closure in P N ∨ of the locus parameterizing all k-th osculating hyperplanes to X at points of U. By scroll we mean an embedded smooth projective variety Y ⊂ P N of dimension n ≥ 1 endowed with a morphism π : Y → C over a smooth curve C such that (f, O P N (1)\| f ) = (P n−1 , O P n−1 (1)) for every fibre f of π, or the corresponding pair (X, W ) with \|W \| very ample, such that Y = ϕ W (X). Of course X = C if n = 1. We need to fix some more notation. mn header will be provided by the publisher 3 Let (X, W ) be a scroll. As is known, for any k ≥ 2 we have a strict inequality dim Osc k x (X) < k+n n − 1 at every point x ∈ X. In fact, there are local coordinates (u, v 2 , . . . , v n ) around every point x ∈ X such that the homogeneous coordinates x i (i = 0, . . . , N ) of the points of the variety locally can be written as x i = a i (u) + n j=2 v j b ij (u), where a i and b ij are holomorphic functions of u. Since every section σ ∈ W is a linear combination σ = N i=0 λ i x i we thus see that the second derivatives σ vj v h vanish at every point. Then dim Osc 2 x (X) ≤ 2n, and differentiating further up to the order k we see that dim Osc k x (X) ≤ nk for every x ∈ X. Finally, we set F e = P(O P 1 ⊕ O P 1 (−e)) to denote the Segre-Hirzebruch surface of invariant e (e ≥ 0). Then, as in [5, p. 372], C 0 stands for a section of minimal self-intersection and f for a fibre. Decomposable scrolls and their flexes The situation we consider for the most part of this paper is inspired by that in [9] and [7,Sec. 2]. Let C be a smooth curve of genus g. For i = 1, . . . , n let L i be a very ample line bundle on C and let V i ⊆ H 0 (C, L i ) be a vector subspace such that \|V i \| gives rise to an embedding ϕ i : C → P ri = P(V i ). Set C i = ϕ i (C). Let V = ⊕ n i=1 V i , E = ⊕ n i=1 L i and consider the projective bundle P = P(E). By identifying V with a vector subspace of H 0 (P, L), where L is the tautological line bundle on P , we get an embedding ϕ : P → P N = P(V ). We set X = ϕ(P ). According to [7, p. 151] we say that X is the decomposable scroll generated by C 1 , . . . , C n . For a point p ∈ C, let p i = ϕ i (p) ∈ C i . Geometrically, X is generated by the linear spaces f p := p 1 , . . . , p n ∼ = P n−1 as the point p varies on C; note that all the linear spans C i = P ri of the C i 's are skew each other and generate the whole ambient space P N . Let t be a local parameter on C such that ϕ i (p) = (x 0 (0), . . . , x ri (0)) corresponds to t = 0. Locally, around p, the homomorphism j Ci k : V i → J k L i is represented by the matrix M i k (t) =       x 0 (t) x 1 (t) . . . x ri (t) x ′ 0 (t) x ′ 1 (t) . . . x ′ ri (t) . . . . . . . . . . . . x (k) 0 (t) x (k) 1 (t) . . . x (k) ri (t)       . The linear space spanned by the row vectors of the matrix M i k (0) defines the k-th osculating space to C i at p i . Note that, if k > r i , then every k-th osculating space to C i is the whole space P ri = P(V i ). Now let λ 1 , . . . , λ n denote homogeneous coordinates corresponding to a local trivialization of E around p, and, for λ n = 0, set v i = λ i /λ n . Then (t, v 1 , . . . , v n−1 ) provide local coordinates on X at a point x ∈ f p \ p 1 , . . . , p n−1 . Writing down the parametric equations for X around f p we can easily get the matrix M X Let x ∈ f p \ p 1 , . . . , p s , . . . , p n , where denotes suppression. Up to reordering the C i 's, there is no restriction if we suppose that s = n, hence the matrix representing j X k is that given by Lemma 1.1. Sometimes, however, it is convenient to order the C i 's according to some criterion (e. g., in such a way that r 1 ≤ r 2 ≤ · · · ≤ r n ). In this case, we can write x = u 1 p 1 + · · · + u s−1 p s−1 + p s + u s p s+1 + · · · + u n−1 p n , and then, with respect to the local coordinates (t, u 1 , . . . , u n−1 ) the matrix representing j X k near x is the following  k (t, v 1 , . . . , v n−1 ) representing j X k : V → J k L near x. Set M i k−1 = M i min{k−1,ri} . Lemma 1.1 [7, p. 152] We have M X k (t, v 1 , . . . , v n−1 ) =            v 1 M 1 k v 2 M 2 k . . . v n−1 M n−1 k M n k M 1 k−1 0 . . . 0 0 0 M 2 k−1 . . . 0 0 . . . . . . . . . . . . . . 0 0 . . . M n−1 k−1 0 0 0 . . . 0 0            (t) .                u 1 M 1 k . . . u s−1 M s−1 k M s k u s M s+1 k . . . u n−1 M n k M 1 k−1 . . . 0 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . 0 . . . M s−1 k−1 0 0 . . . 0 0 . . . 0 0 0 . . . 0 0 . . . 0 0 M s+1 k−1 . . . 0 . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 0 0 . . . M n k−1                  (t) . We say that two matrices A and B of type m × n are row equivalent if the vector subspace of C n spanned by the rows of A is the same as that spanned by the rows of B. Here is an immediate application. Theorem 1.2 Let X be a decomposable scroll generated by C 1 , . . . , C n and let Φ 2 (X) be its inflectional locus. (1) The following three conditions are equivalent: (i) f p \ n i=1 p 1 , . . . , p i , . . . , p n ∩ Φ 2 (X) = ∅; (ii) p i is a flex of C i for every i = 1, . . . , n; (iii) f p ⊆ Φ 2 (X). (2) p i ∈ Φ 2 (X) if and only if it is a flex of C i . (3) Let x ∈ Φ 2 (X): if x ∈ f p \ p 1 , . . . , p s , . . . , p n then p s is a flex of C s . P r o o f. To prove (1) it is enough to show that (i) ⇒ (ii) ⇒ (iii) . Let x ∈ f p \ p 1 , . . . , p n−1 , so that we can write x = v 1 p 1 + . . . v n−1 p n−1 + p n . Then x ∈ Φ 2 (X) if and only if j X 2,x : V → (J 2 L) x has rank < 2n + 1. Note that rk M i 1 (t) = 2 and rk M i 2 (t) ≥ 2(1) for every i and for every t. Then Lemma 1.1 shows that rk M X 2 (0, v 1 , . . . , v n−1 ) < 2n + 1 if and only if both rk M n 2 (0) = 2 and rk (v 1 M 1 2 v 2 M 2 2 . . . v n−1 M n−1 2 )(0) = 2. The former condition says that j 2,p : V n → (J 2 L n ) p has rank 2, while by (1) the latter one is equivalent to saying that either v i = 0 or rk M i 2 (0) = 2 for every i = 1, . . . , n − 1. In conclusion we have that j 2,p : V i → (J 2 L i ) p has rank 2 for i = n and for every i such that v i = 0. So, if x ∈ Φ 2 (X) is a general point as in (i), we get (ii). On the other hand, if (ii) holds, then we see that f p \ p 1 , . . . , p n−1 , hence its closure f p , lies in Φ 2 (X). So (1) is proved. Moreover, the above argument proves the "only if" part of (2) when x = p n , and (3) in the special case s = n. As to the "if" part of (2), note that if x = p n then the matrix M X 2 (0, 0, . . . , 0) of Lemma 1.1 has the mn header will be provided by the publisher 5 following special form:            0 0 . . . 0 M n 2 M 1 1 0 . . . 0 0 0 M 2 1 . . . 0 0 . . . . . . . . . . . . . . 0 0 . . . M n−1 1 0 0 0 . . . 0 0            . So, if p n is a flex of C n we get rk M X 2 (0, 0, . . . , 0) = 2n, since M n 2 (0) has rank 2. Now, let x be as in (3); so we can write x = u 1 p 1 + . . . u s−1 p s−1 + p s + u s p s+1 + · · · + u n−1 p n . Then one can easily see that the matrix representing j X 2 : V → J 2 L near x is                  u 1 M 1 2 . . . u s−1 M s−1 2 M s 2 u s M s+1 2 . . . u n−1 M n 2 M 1 1 . . . 0 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . . 0 . . . M s−1 1 0 0 . . . 0 0 . . . 0 0 0 . . . 0 0 . . . 0 0 M s+1 1 . . . 0 . . . . . . . . . . . . . . . . . . . . . . 0 . . . 0 0 0 . . . M n 1                  (t) . Thus the same argument as above works and shows that since x ∈ Φ 2 (X), p s must be a flex of C s . This completes the proof of (3) and (2). The same argument proving Theorem 1.2 says more. Proposition 1.4 For any x ∈ Φ 2 (X) we have Osc 2 x (X) = Osc 1 p1 (C 1 ), . . . , Osc 2 ps (C s ), . . . , Osc 1 pn (C n ) for some s, where, p s ∈ Φ 2 (C s ). Moreover, Osc 2 x (X) is the same linear P 2n−1 for all x ∈ Φ 2 (X) ∩ f p . P r o o f. First, suppose that x ∈ p 1 , . . . , p n−1 . Then x = v 1 p 1 + · · · + v n−1 p n−1 + p n . As x ∈ Φ 2 (X), the first block of rows in the matrix M X 2 (0, v 1 , . . . , v n−1 ) appearing in Lemma 1.1 for k = 2 has rank 2. In particular, p n ∈ Φ 2 (C n ) by Theorem 1. 2(3). Moreover, either v i = 0 or v i M i 2 is row equivalent to M i 1 . Hence M X 2 is row equivalent to the matrix            0 0 . . . 0 M n 2 M 1 1 0 . . . 0 0 0 M 2 1 . . . 0 0 . . . . . . . . . . . . . . 0 0 . . . M n−1 1 0 0 0 . . . 0 0            . This means exactly that Osc 2 x (X) = Osc 1 p1 (C 1 ), . . . , Osc 1 pn−1 (C n−1 ), Osc 2 pn (C n ) . Next, suppose that x ∈ p 1 , . . . , p n−1 \ p 1 , . . . , p n−2 . Then, x = v 1 p 1 + · · · + v n−2 p n−2 + p n−1 can also be written as x = u 1 p 1 + · · · + u s−1 p s−1 + p s + u s p s+1 + · · · + u n−1 p n , as done after Lemma 1.1, with s = n − 1 and u n−1 = 0. Then look at the matrix appearing after Lemma 1.1 in the present situation:         u 1 M 1 2 . . . u n−2 M n−2 2 M n−1 2 0 M 1 1 . . . 0 0 0 . . . . . . . . . . . . . . 0 . . . M n−2 1 0 0 0 . . . 0 0 M n 1         (0) . If x ∈ Φ 2 (X) , arguing as before we see that p n−1 ∈ Φ 2 (C n−1 ) and this matrix is row equivalent to         0 . . . 0 M n−1 2 0 M 1 1 . . . 0 0 0 . . . . . . . . . . . . . . 0 . . . M n−2 1 0 0 0 . . . 0 0 M n 1         (0) . This means that Osc 2 x (X) = Osc 1 p1 (C 1 ), . . . , Osc 1 pn−2 (C n−2 ), Osc 2 pn−1 (C n−1 ), Osc 1 pn (C n ) . Now, let s ≤ n−2. By repeating the argument for x ∈ p 1 , . . . , p s \ p 1 , . . . , p s−1 , we see that p s ∈ Φ 2 (C s ) and Osc 2 x (X) is the linear span of Osc 2 ps (C s ) and the spaces Osc 1 pi (C i ) for i = s. This proves the first assertion. Now, note that all Osc 1 pi (C i ) are lines. Moreover, as we have shown, p s is a flex for C s , hence Osc 2 ps (C s ) is also a line. Thus, for any x ∈ X, Osc 2 x (X) is the linear space generated by the n tangent lines to C i at p i for i = 1, . . . , n. Note that they generate a P 2n−1 . It turns out that Osc 2 x (X) is the same P 2n−1 for all x ∈ Φ 2 (X) ∩ f p . Higher flexes and fibres Let X be a decomposable scroll over a smooth curve C generated by C 1 , . . . , C n as in Section 1, and let f p = p 1 , . . . , p n be the fibre over p ∈ C. In this section we explore some connections between the higher inflectional loci Φ k (X) and the fibres of X. Remark 2.1 We have Osc k ps (X) = Osc k−1 p1 (C 1 ), . . . , Osc k ps (C s ), . . . , Osc k−1 pn (C n )(2) for any s = 1, . . . , n (the only k-th osculating space on the right hand is that at p s ). In particular, if p s ∈ Φ k (C s ), then p s ∈ Φ k (X). P r o o f. Up to reordering the curves we can suppose that s = n. Then the matrix representing j X k,ps is, according to Lemma 1.1,         0 . . . 0 M n k M 1 k−1 . . . 0 0 . . . . . . . . . . . . 0 . . . M n−1 k−1 0 0 . . . 0 0         .(3) This proves the first assertion. Note that all linear spaces appearing on the right hand of (2) are skew each other. mn header will be provided by the publisher 7 Then the second assertion follows from the inequality: dim Osc k ps (X) ≤ (n − 1)(k − 1) + dim Osc k ps (C s ) + (n − 1) < (n − 1)k + k = nk.(4) As to the converse, if p s ∈ Φ k (X), we cannot claim that p s ∈ Φ k (C s ) if k > 2. However, we have Remark 2.2 If p s ∈ Φ k (X), then either p s ∈ Φ k (C s ), or p j ∈ Φ k−1 (C j ) for some j = s. Proposition 2.3 Let p i ∈ Φ k (C i ) for i = 1, . . . , s, . . . , n, where denotes suppression. Then for every point x ∈ f p \ p 1 , . . . , p s , . . . , p n we have Osc k x (X) = Osc k−1 p1 (C 1 ), . . . , Osc k ps (C s ), . . . , Osc k−1 pn (C n ) (the only k-th osculating space on the right hand is that at p s ). P r o o f. Up to reordering we can suppose that s = n. Due to the assumption, we have Osc k pi (C i ) = Osc k−1 pi (C i ) for i = 1, . . . , n − 1. This means that the two matrices M i k and M i k−1 are row equivalent for i = 1, . . . , n − 1. Now look at the matrix M of Lemma 1.1. By subtracting suitable linear combinations of the subsequent rows from the first block of rows we see that M is row equivalent to the matrix (3). This proves the assertion. The same argument proves more. Actually, assume that p ij ∈ Φ k (C ij ) for j = 1, . . . , s and set Λ = p i1 , . . . , p is . Up to reordering we can suppose that (i 1 , . . . , i s−1 , i s ) = (1, . . . , s − 1, n). Then for any By subtracting suitable linear combinations of the subsequent rows of M from the first block we see that M is row equivalent to the matrix in (3). Now, since also p n ∈ Φ k (C n ) we have rk M n k < k + 1 and then the same computation done to prove Remark 2.1 holds at x, giving dim Osc k x (X) < nk. Thus Λ \ p 1 , . . . , p s−1 ⊆ Φ k (X). On the other hand Φ k (X) ∩ f p is a Zariski closed subset, hence Λ ⊆ Φ k (X). Now suppose that (i 1 , . . . , i s ) = (1, . . . , s), with s ≤ n − 1 and p n ∈ Φ k (C n ). Then rk M n k = k + 1, and the same argument as above applied to any point x ∈ p 1 , . . . p s , p n \ Λ shows that x ∈ Λ \ p 1 , . . . , p s−1 we can write x = v 1 p 1 + . . . v s−1 p s−1 + p n .dim Osc k x (X) = n−1 i=1 rk M i k−1 + (k + 1) − 1. In particular, if p i ∈ Φ k−1 (C i ) for i = 1, . . . , s, then all the first n − 1 summands are equal to k, hence dim Osc k x (X) = nk, and so x ∈ Φ k (X). This proves the following Proposition 2.4 If p ij ∈ Φ k (C ij ) for j = 1, . . . , s ≤ n, then p i1 , . . . , p is ⊆ Φ k (X). Moreover, if p ij ∈ Φ k (C ij ) \ Φ k−1 (C ij ) for j = 1, . . . , s ≤ n − 1 and p ij ∈ Φ k (C ij ) for j = s + 1, . . . , n, then Φ k (X) ∩ f p = p i1 , . . . , p is . Corollary 2.5 i) If p i ∈ Φ k (C i ) for every i = 1, . . . , n, then f p ⊆ Φ k (X). ii) If f p ⊆ Φ k (X), then p i ∈ Φ k (C i ) for some i. P r o o f. i) is obvious; ii) follows from Remark 2.2, taking into account the inclusion Φ k−1 (C j ) ⊆ Φ k (C j ) In particular, if f p ⊆ Φ k (X) and k > 2, we see that not necessarily p i ∈ Φ k (C i ) for all i's. For n = 2 we can be more explicit. a) p i ∈ Φ k−1 (C i ) for some i, or b) p i ∈ Φ k (C i ) for i = 1, 2. P r o o f. Let f p ⊆ Φ k (X) . By Corollary 2.5, ii), up to reordering, we can suppose that p 1 ∈ Φ k (C 1 ). Then, by Proposition 2.3, for every x ∈ f p \ {p 1 }, we have Osc k x (X) = Osc k−1 p1 (C 1 ), Osc k p2 (C 2 ) .(5) Hence dim Osc k x (X) = dim Osc k−1 p1 (C 1 ) + dim Osc k p2 (C 2 ) + 1.(6) Since x ∈ Φ k (X) this shows that either p 1 ∈ Φ k−1 (C 1 ), case a), or p 2 ∈ Φ k (C 2 ), case b). To prove the converse, in both cases a) and b), up to renaming, we can assume that p 1 ∈ Φ k (C 1 ). Then Proposition 2.3 gives again (5) for any x ∈ f p \ {p 1 } and then (6) shows that dim Osc k x (X) ≤ k − 2 + k + 1 in case a), k − 1 + (k − 1) + 1 in case b). Hence f p \ {p 1 } ⊆ Φ k (X) in both cases, and then, taking the closure, we get f p ⊆ Φ k (X). Theorem 2.7 Let n = 2. Suppose that x ∈ Φ k (X) and let f p be the fibre of X containing x. i) If x = p 1 , p 2 , then f p ⊆ Φ k (X); ii) if x = p i , then either f p ⊆ Φ k (X) or p i ∈ Φ k (C i ). P r o o f. If x = p 1 , then we can write x = vp 1 + p 2 . According to Lemma 1.1, j X k,x is represented by the following matrix M = vM 1 k M 2 k M 1 k−1 0 . Since x ∈ Φ k (X), M has rank ≤ 2k. This implies either α) rk M 1 k−1 < k − 1, i. e., p 1 ∈ Φ k−1 (C 1 ), or β) rk vM 1 k M 2 k < k. Condition β) in turn implies both rk M 1 k < k and rk M 2 k < k. The latter condition means that p 2 ∈ Φ k (C 2 ), while the former is equivalent to saying that either v = 0, or rk M 1 k < k. In other words, either x = p 2 or p 1 ∈ Φ k (C 1 ). In conclusion, if x = p 1 , p 2 then either α) p 1 ∈ Φ k−1 (C 1 ), or β) p i ∈ Φ k (C i ) for i = 1, 2. In both cases f p ⊆ Φ k (X) by Proposition 2.6. This proves i). Now let x = p i . By Remark 2.2 either p i ∈ Φ k (C i ) or p j ∈ Φ k−1 (C j ) for j = i. But in the latter case Proposition 2.6 says that f p ⊆ Φ k (X) again. This proves ii). mn header will be provided by the publisher 9 Corollary 2.8 Let n = 2. Then Φ k (X) = ∅ if and only if Φ k (C i ) = ∅ for i = 1, 2. P r o o f. If p i ∈ Φ k (C i ) for some i, we know that p i ∈ Φ k (X) by Remark 2.2. This proves the "only if part". To prove the "if" part suppose, by contradiction, that x ∈ Φ k (X), and let f p be the fibre of X through x. By Theorem 2.7 either f p ⊆ Φ k (X) or p i ∈ Φ k (C i ) for some i. In both cases, taking into account Proposition 2.6, we see that Φ k (C i ) = ∅ for some i. But this is a contradiction. Now suppose that r 1 ≤ r 2 , where C i = P ri . If r 1 < k − 1, then we have dim(Osc k−1 y (C 1 )) ≤ r 1 < k − 1 for every y ∈ C 1 . In other words, Φ k−1 (C 1 ) = C 1 and therefore Φ k (X) = X by Proposition 2.6. Let r 1 ≥ k−1. If r 2 < k (i. e., r 1 = r 2 = k − 1), then Φ k (C i ) = C i for i = 1, 2, hence Φ k (X) = X again, by Proposition 2.6. If r 1 = k − 1 but r 2 ≥ k then Φ k (C 1 ) = C 1 but Φ k (C 2 ) C 2 (e. g., see [1, p. 37, Ex. C-2]); so Φ k (X) contains every fibre of X passing through a point of either Φ k−1 (C 1 ) or Φ k (C 2 ). Taking into account also Remark 2.1, we thus get the following Corollary 2.9 Let n = 2 and suppose that r 1 ≤ r 2 . If Φ k−1 (C 1 ) = ∅ but Φ k (C 1 ) = C 1 , then Φ k (X) consists of C 1 plus the fibres containing a point of Φ k (C 2 ). Surface scrolls: examples and applications In this section we focus on the surface case (n = 2). Let X ⊂ P N be a decomposable surface scroll as in Section 1. We present some examples concerned with the dimension that Osc k x (X) can have at some point x, and with the structure of Φ 2 (X), focusing in particular on the case of non-normal rational scrolls. First it is useful to recall the situation for normal rational scrolls. Example 3.1 Notation as in Section 1; let C = P 1 , E = O P 1 (r 1 ) ⊕ O P 1 (r 2 ) , with 1 ≤ r 1 ≤ r 2 , and let X ⊂ P N be the image of P(E) in the embedding given by complete linear system associated with the tautological line bundle L. Note that N = r 1 + r 2 + 1. Let p = (t 0 : t 1 ) ∈ P 1 and set t = t 1 /t 0 (or t 0 /t 1 ). At any point x ∈ X \ C 1 we can use local coordinates (t, v) to write x = vp 1 + p 2 on the fibre f p ; then, according to Lemma 1.1, the homomorphism j X k : H 0 (X, L) → J k L is represented near x by the matrix M X k (t, v) =   vM 1 k M 2 k M 1 k−1 0 0 0   (t) . Note that rk M 2 k (t) = min{k + 1, r 2 + 1}. Moreover rk M 1 k−1 (t) = rk(M 1 k−1 ) = k if k − 1 ≤ r 1 , rk(M 1 r1 ) = r 1 + 1 otherwise. It follows that rk(j X k,x ) = rk M X k (t, v) = rk M 2 k (t) + rk M 1 k−1 (t) = min{k + 1, r 2 + 1} + min{k, r 1 + 1}.(7) Therefore dim Osc k x (X) =      2k if k ≤ r 1 + 1, k + r 1 + 1 if r 1 + 1 ≤ k ≤ r 2 , r 1 + r 2 + 1 if k ≥ r 2 .(8) Note that at any point x ∈ X \ C 1 the dimension of the k-th osculating space can be strictly smaller than 2k. This is obvious when N < 2k, but it can happen also for k ≤ N −1 2 , e. g., for a very unbalanced rational normal scroll (i. e., with invariant e := r 2 − r 1 very large). In fact, for k ≤ N −1 2 we have dim Osc k x (X) < 2k if r 1 + 1 < k from (8). This means N + 1 − e < 2k ≤ N − 1, hence e ≥ 3 is enough. For k ≥ 2, (8) also shows that k + 2 ≤ dim Osc k x (X) ≤ 2k at any point x ∈ X \ C 1 . In particular, letting k = 2 we see that dim Osc 2 x (X) = 4 for any x ∈ X \ C 1 . We want to stress that X was linearly normal in the example above. Here is an enlightening example showing how small the dimension of Osc k x (X) can be at some point x, for any k, when we drop linear normality. Example 3.2 Fix integers k ≥ 2 and r = r 2 ≥ 3. Let C = P 1 , E = L 1 ⊕ L 2 , where L 1 = O P 1 (1), L 2 = O P 1 (k + r − 1), and let V 1 = H 0 (P 1 , L 1 ) = t 0 , t 1 , V 2 = t k+r−1 0 , t k+r−2 0 t 1 , t r−2 0 t k+1 1 , . . . , t 0 t k+r−2 1 , t k+r−1 1 . Note that ϕ 2 : P 1 → P r defines an embedding, which is not linearly normal, since dim V 2 = r + 1 < h 0 (L 2 ). Then X ⊂ P r+2 , defined as in Section 1, is a rational non-normal scroll. Let L be the hyperplane bundle and let V ⊂ H 0 (X, L) be the subspace giving rise to the embedding. Note that at the point p ∈ P 1 , corresponding to (t 0 : t 1 ) = (1 : 0) we have \|V 2 − 2p\| = · · · = \|V 2 − (k + 1)p\|. This means that for every h, (2 ≤ h ≤ k) the homomorphism j C2 h,p : V 2 → (J k L 2 ) p has a 2-dimensional image (isomorphic to (J 1 L 2 ) p ), i. e., rk(j C2 k,p ) = 2. On the other hand, at every point q ∈ P 1 it is obvious that rk(j C1 h,q ) = 2 for any h ≥ 1. Now, let x ∈ f p . If x ∈ f p \ {p 2 }, Proposition 2.3 shows that Osc h x (X) = Osc h p1 (C 1 ), Osc h−1 p2 (C 2 ) for any h = 2, . . . , k. On the other hand, Remark 2.1 tells us that Osc h p2 (X) = Osc h−1 p1 (C 1 ), Osc h p2 (C 2 ) for any h ≥ 2. In both cases Osc h x (X) is the linear span of two skew lines, namely C 1 and Osc 1 p2 (C 2 ); hence Osc k x (X) = Osc k−1 x (X) = · · · = Osc 2 x (X) for every k ≥ 3, at every point x ∈ f p . In particular, dim Osc k x (S) = 3 for all k ≥ 2 at any point x ∈ f p . We recall that if X ∈ P N is any scroll of dimension n, then dim Osc 2 x (X) ≥ n + 1 [2] (see also [6] for n = 2). Example 3.3 Let C = P 1 , L 1 = O P 1 (m) with m ≥ 2, V 1 = H 0 (P 1 , L 1 ), and consider L 2 = O P 1 (d) with d ≥ m + 2. The vector space H 0 (P 1 , L 2 ) defines an embedding of C as a rational normal curve Γ ⊂ P d . Projecting Γ from a general linear space T of dimension d − m − 2 to a P m+1 we get an embedding. Let V 2 = V (T ) be the vector subspace of H 0 (P 1 , L 2 ) corresponding to this embedding. Let C i be the image of C in the embedding defined by V i , i = 1, 2, and in the space P 2m+2 = P(V 1 ⊕ V 2 ) consider the decomposable rational scroll X generated by C 1 and C 2 . Note that X = F d−m . We claim that Φ m (X) = ∅. Of course Φ m (C 1 ) = ∅. Let O Γ be the m-th osculating developable of Γ (i. e., the variety generated by the linear spaces Osc m x (Γ), as x varies on Γ). Note that dim(O Γ ) = m+1, hence T ∩O Γ = ∅ for a general T . Since no osculating space Osc m x (Γ) meets the center of projection T , we conclude that Φ m (C 2 ) = ∅. Then the claim follows from Corollary 2.8. Let us recall the following conjecture of Piene-Tai [10]. Let S ⊂ P N (N ≥ 5) be a non-degenerate smooth projective surface such that dim Osc k x (S) ≤ 2k for all points x ∈ S and for every k, with equality for k = [ N −1 2 ], where [ ] stands for the greatest integer function. (i) If N is odd, then S is the balanced rational normal scroll of degree N − 1 (i. e., S is F 0 embedded by \|C 0 + [ N −1 2 ]f \|). (ii) If N is even, then S is the semibalanced rational normal scroll of degree N − 1 (i. e., S is F 1 embedded by \|C 0 + ([ N −1 2 ] + 1)f \|). Part (i) of this conjecture is true, as proved in [3], while part (ii) is not (see [6, Theorem A and comment after Corollary 2.3]). Example 3.3 provides a new series of counterexamples to the even dimensional part of the conjecture. We want to stress that all these scrolls are decomposable, while those appearing in [6, Theorem A] are not, all being isomorphic to the elliptic P 1 -bundle of invariant −1. Moreover, we have the following characterization, which provides more information in order to correct the conjecture. Theorem 3.4 Let X ⊂ P 2m+2 (m ≥ 2) be a decomposable scroll with n = 2 such that Φ m (X) = ∅. Then either X is the semibalanced rational normal scroll of degree m + 1, or X is of the type described in Example 3.3. P r o o f. By Corollary 2.8 it must be Φ m (C i ) = ∅, for i = 1, 2. In particular, C 1 cannot be a line, hence r 1 = dim C 1 ≥ 2. We can assume that r 1 ≤ r 2 and then from r 1 + r 2 + 1 = 2m + 2 we get that r 1 ≤ m. As Φ m (C 1 ) = ∅, this implies that r 1 = m, and then r 2 = m + 1. So C 1 is a rational normal curve of degree m in P m while C 2 is either the rational normal curve of degree m + 1 in P m+1 or any other rational non-normal curve of some degree d ≥ m + 2 in P m+1 . In the former case X is the semibalanced rational normal scroll. In the latter, C 2 is obtained by projecting a rational normal curve of degree d in P d to P m+1 from a general center as in Example 3.3. The examples in the next part of this section are concerned with Φ 2 (X). First we would like to stress that for the cubic scroll X ⊂ P 4 the inflectional locus Φ 2 (X) consists exactly of the generating line C 1 . In fact this is the only semi-balanced rational normal scroll which is not uninflected. As to quartic rational normal scrolls in P 5 the situation is also well known [11]. Let us note that the one isomorphic to F 0 is uninflected according to Corollary 1.3, being generated by two conics C 1 , C 2 . On the other hand, the one isomorphic to F 2 is generated by a line C 1 and a rational normal cubic C 2 , which has no flexes. Hence, according to Theorem 1.2(1) its inflectional locus Φ 2 consists exactly of C 1 . Example 3.5 We consider quintic non-normal rational scrolls in P 5 . Let X be as in Example 3.3, with m = 1 and d = 4. According to Theorem 1.2(1), the inflectional locus Φ 2 (X) consists of the line C 1 and the fibres passing through the flexes of the non-normal quartic rational curve C 2 . Now the center of projection T is a point. If T ∈ O Γ , then C 2 has no flexes, as we said, and then Φ 2 (X) = C 1 . On the other hand, if c ∈ O Γ , then C 2 has ǫ flexes, where ǫ is the number of osculating planes to Γ passing through T . According to the enumerative formula counting the weighted number of 2-osculating lines and 3-osculating planes to C 2 [8, Theorem 3.2] we can see that ǫ = 1 or 2. Depending on this, Φ 2 (X) consists of C 1 plus one or two fibres. Example 3.6 In the same vein we can construct non-normal rational scrolls having a finite inflectional locus. Let C, L 2 , V 2 be as in the previous example but now put L 1 = O P 1 (2) and V 1 = H 0 (P 1 , L 1 ). Again let C i be the image of C in the embedding defined by V i , i = 1, 2, and in the space P 6 = P(V 1 ⊕ V 2 ) consider the decomposable sextic rational scroll X generated by C 1 and C 2 . Now C 1 is a conic, hence it has no flexes. On the other hand C 2 has ǫ = 1 or 2 flexes provided that the projection of Γ giving rise to C 2 is made from a center T ∈ O Γ . Therefore, according to Theorem 1.2 ((1) and (2)), Φ 2 (X) consists of one or two points (the flexes of C 2 ). Example 3.7 Let C be a smooth curve of genus 1, L 1 = O C (3p), for some point p ∈ C, V 1 = H 0 (C, L 1 ). Then C 1 = ϕ 1 (C) is a smooth plane cubic having exactly 9 flexes, one of which is p 1 := ϕ 1 (p). Now let L 2 be a line bundle of degree 5 on C. The vector space H 0 (C, L 2 ) defines an embedding of C in P 4 whose image, say Γ, is a quintic normal elliptic curve: then Γ has no flexes (but 25 hyperflexes). Projecting Γ from a point c ∈ P 4 \ Sec(Γ) to a P 3 we get an embedding; let V 2 = V (c) be the corresponding vector subspace of H 0 (C, L 2 ) and let C 2 be the image of C in the embedding ϕ 2 : C → P 3 defined by V 2 . In the space P 6 = P(V 1 ⊕ V 2 ) consider the decomposable elliptic scroll X generated by C 1 and C 2 . It can happen that p 2 := ϕ 2 (p) is a flex of C 2 or not. According to Theorem 1.2(1), in the former case the whole fibre f p is in the inflectional locus Φ 2 (X), while in the latter we have Φ 2 (X) ∩ f p = {p 1 }. Note that if c is general enough, then C 2 has no flexes and therefore X has only 9 flexes: those of C 1 . The second discriminant locus of decomposable scrolls Let (X, L, W ) be as at the beginning of Section 0, and let U ⊆ X be the Zariski dense open subset of X where j (X,W ) k,x : W → (J k L) x attains the maximum rank s(k) + 1. If x ∈ U, the fact that H ∈ \|W \| is a k-th osculating hyperplane to X at x is equivalent to the fact that H = (σ) 0 , where σ ∈ W and j k,x (σ) = 0. Equivalently, this means that H ∈ \|W − (k + 1)x\|, i. e., the hyperplane section cut out by H on X has a point of multiplicity ≥ (k + 1) at x. Note however that if x ∈ U and H ∈ \|W − (k + 1)x)\|, this does not necessarily mean that H ∈ X ∨ k . Actually H ∈ X ∨ k if and only if H is a limit of k-th osculating hyperplanes to X at points of U. On the other hand we can consider the k-th discriminant locus D k (X, W ) of (X, W ), which is defined as the image of J := {(x, H) ∈ X × \|W \| \| H ∈ \|W − (k + 1)x\|} via the second projection of X × \|W \|. It parameterizes all hyperplane sections of X ⊂ P N admitting a singular point of multiplicity ≥ k + 1; of course D k (X, W ) ⊇ X ∨ k with equality if and only if U = X, i. e., if and only if dim Osc k x (X) = s(k) for every x ∈ X. In general D k (X, W ) contains some extra components coming from the irreducible components of Φ k (X). The discussion above says that D k (X, W ) = X ∨ k if and only if Φ k (X) = ∅. From this point of view, the characterization of balanced rational normal surface scrolls due to Ballico, Piene and Tai [3], mentioned after Example 3.3, can be rephrased as follows. Proposition 4.1 Let X ⊂ P N be any smooth surface, where N = 2m + 1 ≥ 5. Then D m (X, W ) = X ∨ m if and only if X = P 1 × P 1 and W = H 0 (P 1 × P 1 , O P 1 ×P 1 (1, m)). Now, let X ⊂ P N = P(V ) be a decomposable scroll as in Section 1. For simplicity we identify X with the corresponding abstract projective bundle P . So, we denote by D 2 (X, V ) the second discriminant locus of (P, L, V ). Its main component is the second dual variety X ∨ 2 of X. Note that if X is not linearly normal then X ∨ 2 corresponds to a suitable linear section of the second dual variety of the linearly normal scroll giving rise to X via the projection to P N . Here, relying on the results of Sections 1 and 2, we want to describe the extra components of D 2 (X, V ). Of course we assume that Φ 2 (X) = ∅. As a first thing we need to describe the irreducible components of Φ 2 (X). Proposition 4.2 Let X ⊂ P N be a decomposable scroll as in Section 1, generated by C 1 , . . . , C n , and assume that Φ 2 (X) = X. Let G be an irreducible component of Φ 2 (X). Then, up to reordering the curves C i 's, either (1) G = p 1 , . . . , p s ⊆ f p = p 1 , . . . , p n , or (2) X is rational and G = C 1 × p 1 , . . . , p s is the image of P 1 × P s−1 via the Segre embedding. Moreover, Osc 2 x (X) = Osc 2 p1 (C 1 ), Osc 1 p2 (C 2 ), . . . , Osc 1 pn (C n ) for all x ∈ G in case (1) and for all x ∈ G ∩ f p in case (2). In particular, dim Osc 2 x (X) = 2n − 1 for any x ∈ G in both cases. P r o o f. As Φ 2 (X) = ∅ it follows from Theorem 1.2(3) that Φ 2 (C i ) = ∅ for some i. If Φ 2 (C i ) = C i for every i = 1, . . . , n, then we get an irreducible component as in case (1). Actually, up to reordering the curves, we can assume that p i ∈ Φ 2 (C i ) for i = 1, . . . , s. Then G := p 1 , . . . , p s ⊆ Φ 2 (X) by Proposition 2.4. Moreover, mn header will be provided by the publisher 13 since Φ 2 (C i ) is a finite set for every i, we conclude that G is an irreducible component of Φ 2 (X). Now suppose that Φ 2 (C i ) = C i for some i. Then up to reordering the curves we can assume that Φ 2 (C i ) = C i for i = 1, . . . , s and Φ 2 (C i ) = C i for i > s. This implies that C i is a line for i = 1, . . . , s and a rational curve of higher degree for i > s. In particular C = P 1 , i. e., X is a rational scroll. Moreover G p := p 1 , . . . , p s ⊆ Φ 2 (X) by Proposition 2.4, for every p ∈ P 1 . Let G := p∈P 1 G p . Then G is the sub-scroll of X generated by the lines C 1 , . . . C s . In other words, G is P(O P 1 (1) ⊕s ) = P 1 × P s−1 embedded in the linear span of C 1 , . . . , C s via the Segre embedding. This gives case (2) and there are no further possibilities. The last assertions follow from Proposition 1.4, since G ⊆ Φ 2 (X). Of course it may happen that a fibre G p of an irreducible component of Φ 2 (X) of type (2) is contained in a larger component of Φ 2 (X) of type (1). This happens if C j has a flex at the point p j for some j > s. Now let us consider the second discriminant locus: for simplicity we set D = D 2 (X, V ) and denote by D G the component of D arising from an irreducible component G of Φ 2 (X). Then D G = {H ∈ P N ∨ \| H ⊇ Osc 2 x (X) for any x ∈ G}. Let G be an irreducible component of Φ 2 (X). We say that G is of type (1) or (2) according to the cases of Proposition 4.2. Let G be of type (1). Then, recalling that Osc 2 x (X) is a fixed P 2n−1 for all x ∈ G, we conclude that the component D G is a linear P N −2n . Now suppose that G is of type (2). By Proposition 4.2, Osc 2 x (X) is a fixed linear space T p := P 2n−1 for x ∈ G ∩ f p . So, letting p vary on P 1 we see that D G = p∈P 1 {H ∈ P N ∨ \| H ⊇ T p }. We can think of T p as Osc 2 p1 (X). Note that if Φ 2 (X) = X, the tangent line to C n varies as p varies on C. Hence for points x, y ∈ G, lying on general distinct fibres f p , f q of X we have Osc 2 x (X) = Osc 2 p1 (X) = Osc 2 q1 (X) = Osc 2 y (X). It follows that dim(D G ) = N − 2n + 1, D G being a family of P N −2n parameterized by P 1 . More precisely, we can describe the structure of D G in this way. Consider the incidence correspondence P = {(p 1 , H) ∈ C 1 × P N ∨ \| H ⊇ Osc 2 p1 (X)}. Note that P is a P N −2n -bundle over C 1 = P 1 via the first projection of C 1 × P N ∨ , since Osc 2 p1 (X) is a P 2n−1 for any p 1 ∈ C 1 . Then D G = π(P), where π is the second projection of C 1 × P N ∨ . Example 4.3 Let X be a decomposable scroll as in Section 1, generated by lines C 1 , . . . , C n−1 and by a nondegenerate rational curve C n ⊂ P r , r = r n ≥ 3, of degree d. Then d ≥ 3 and X ⊂ P N , where N = 2n − 2 + r. According to Proposition 4.2, G = C 1 × P n−2 , Segre embedded in P 2n−3 = C 1 , . . . C n−1 . Let p, q be any two distinct points of C = P 1 . The tangent lines to C n at p n and q n generate at most a P 3 . Therefore dim C 1 , . . . , C n−1 , Osc 1 pn (C n ), Osc 1 qn (C n ) ≤ 2n + 1. Recall that the linear space above is just the linear span Osc 2 p1 (X), Osc 2 q1 (X) by Proposition 4.2. So, if r ≥ 4, for any two distinct points p, q ∈ C there exists a hyperplane H of P N containing both Osc 2 p1 (X) and Osc 2 q1 (X). Note that any such a hyperplane corresponds to a singular point of D G : actually, π\| −1 P (H) = {(p 1 , H), (q 1 , H)}. In particular, if r ≥ 4, then Sing(D G ) contains the P 2 parameterizing the double symmetric product of C n with itself. Now let r = 3. If d ≥ 4, then C n is not normal. Hence it is the image of a rational normal curve C ⊂ P d of degree d via a projection from a general linear space T of dimension d − 4. Any two tangents to C span a P 3 , so, one sees by a dimension count that in the dual space P d∨ there is a one dimensional family of hyperplanes of P d containing two tangent lines to C and T . Projecting to P 3 they provide infinitely many pairs of coplanar tangent lines to C n (see also [5,Remark 5.2]). This being a closed condition, implies that for any p n ∈ C n there exists some other point q n ∈ C n such that the two tangent lines Osc 1 pn (C n ) and Osc 1 qn (C n ) are coplanar. For such a pair of points, dim C 1 , . . . , C n−1 , Osc 1 pn (C n ), Osc 1 qn (C n ) = 2n. Hence H := Osc 2 p1 (X), Osc 2 q1 (X) is a hyperplane of P 2n+1 giving rise to a singular point of D G . Finally, let r = 3 = d. Then C n is a twisted cubic. Being a rational normal curve, we know that any two distinct tangent lines to C n do not meet. Thus, for any two distinct points p, q ∈ C we have that Osc 2 p1 (X), Osc 2 q1 (X) = C 1 , . . . , C n−1 , Osc 1 pn (C n ), Osc 1 qn (C n ) = C 1 , . . . , C n is the whole P 2n+1 . In fact, in this case D G is a scroll over C (see Proposition 4.4 below). What we said in Example 4.3 when either r ≥ 4 or r = 3 and d ≥ 4 holds, "a fortiori", if C 1 , . . . , C s are lines (s ≥ 1) and for some i = s + 1, . . . , n either r i ≥ 4 or r i = 3 and deg C i ≥ 4. Actually, also in this case there are hyperplanes H of P N containing both Osc 2 p1 (X) and Osc 2 q1 (X), for distinct points p, q of C, and any such hyperplane gives rise to a singular point of D G . From now on in this Section, we assume that 1 = deg C 1 = · · · = deg C s < deg C s+1 ≤ · · · ≤ deg C n . Example 4.3 shows that D G is not a scroll if deg C n ≥ 4. On the other hand, we can prove the following Proposition 4.4 Let X be a decomposable scroll generated by C 1 , . . . , C n , where C i is a line for i = 1, . . . , s and 2 ≤ deg C s+1 ≤ · · · ≤ deg C n ≤ 3. Let G be the sub-scroll of X generated by C 1 , . . . , C s . Then D G is a rational scroll. We need to point out some facts. Remark 4.5 Let Y be the decomposable scroll generated by C s+1 , . . . , C n , and let P M be its linear span in P N . We denote by Σ the minimal sub-scroll of Y generated by the curves C s+1 , . . . , C n−1 and by F any fibre of Y . Note that Σ ∩ C n = ∅, while Σ ∩ F is a hyperplane of F . We have Pic(Y ) ∼ = Z 2 and we can choose as generators the classes of Σ and F . Then: (i) any hyperplane of P M cuts Y along a divisor D linearly equivalent to Σ + bF , for some integer b > 0. In particular, since Σ does not meet C n we see that deg C n = DC n = (Σ + bF )C n = b.(9) (ii) For any hyperplane H of P N not containing P M set h := H ∩ P M . If h contains Osc 1 pi (C i ) for every i = s + 1, . . . , n then h cuts Y along a divisor of the form D = 2F p + R where F p = p s+1 , . . . , p n and R is an effective divisor linearly equivalent to Σ + βF , with β ≥ 0. Indeed, the tangent space to Y at p i is Osc 1 pi (Y ) = p s+1 , . . . , Osc 1 pi (C i ), . . . , p n for i = s + 1, . . . , n, by Remark 2.1 with k = 1. Hence h is tangent to Y at all points p s+1 , . . . , p n . Since they are linearly independent, this says that h is tangent to Y along the whole fibre F p . Thus the divisor D cut out by h on Y is singular at all points of F p , hence the summand 2F p appears in the expression of D as positive linear combination of its irreducible components. Now we can prove Proposition 4.4. P r o o f. As the fibres of P are mapped linearly into P N ∨ by π, it is enough to show that the bundle projection of P induces a morphism D G → C 1 . To do that we prove that π is bijective, i. e., for any H ∈ D G , the fibre π\| −1 P (H) consist of a single element. Equivalently, for any pair of distinct points p, q ∈ C, there is no hyperplane H ⊂ P N containing both Osc 2 p1 (X) and Osc 2 q1 (X). Set R p = Osc 1 ps+1 (C s+1 ), . . . , Osc 1 pn (C n ) , R q = Osc 1 qs+1 (C s+1 ), . . . , Osc 1 qn (C n ) . By Proposition 4.2 we know that Osc 2 p1 (X) = C 1 , . . . , C s , R p , Osc 2 q1 (X) = C 1 , . . . , C s , R q . Thus the assertion follows once we show that R p , R q = P M . By contradiction, suppose that there is a hyperplane h of P M containing both R p and R q . Then, according to Remark 4.5 (ii), h cuts Y along a divisor D = 2F p + 2F q + R, with R linearly equivalent to Σ + βF , for some integer β ≥ 0. Then, dotting with C n and recalling (9), we get deg C n = DC n = 4 + β ≥ 4, a contradiction. A further property of D G is that it is degenerate in P N ∨ . In fact, for any x ∈ G, Osc 2 x (X) contains the lines C 1 , . . . , C s , and hence their linear span Λ := C 1 , . . . , C s which is a P 2s+1 . By duality, this means that D G is contained in the linear subspace P N −2s ⊂ P N ∨ parameterizing the hyperplanes containing Λ. Moreover, D G = P N −2s . Next we want to determine the degree of D G when G is of type (2). To do that, recall that C i = P ri , and let d i = deg C i . Proposition 4.6 Let G be an irreducible component of Φ 2 (X) of type (2). Then deg D G = 2 n i=s+1 (d i −1). P r o o f. Since G is of type (2) we know that d 1 = · · · = d s = 1 and d i ≥ 2 for i > s. Also r 1 = · · · = r s = 1 and r i ≥ 2 for i > s. Recalling that N = n i=1 r i + n − 1, we note that dim D G = N − 2n + 1 = n i=1 r i − n = n i=s+1 (r i − 1). So deg D G is the number of elements of D G contained in a linear system S ⊂ \|V \| defined by n i=s+1 (r i − 1) linear conditions, general enough. Choose r i − 1 general points in each P ri for i = s + 1, . . . , n and call Z i ⊂ P ri the linear subspace they generate. Let S be the linear system of hyperplanes of P N defined by the condition of passing through all these points. Let H ∈ D G be a hyperplane of P N not containing P ri for a given i, s + 1 ≤ i ≤ n. Then h i := H ∩ P ri is a hyperplane of P ri tangent to the curve C i . More precisely, if H ⊃ Osc 2 x (X) and x ∈ f p , then Osc 2 x (X) ⊃ Osc 1 pi (C i ), and so h i is tangent to C i at p i . On the other hand, if our H is also in S, then, in particular, h i contains Z i . Conversely, suppose that h i is a hyperplane of P ri containing Z i and tangent to C i at a point p i , and set H := C 1 , . . . , C i−1 , h i , C i+1 , . . . , C n . Clearly H is a hyperplane of P N . Moreover, H ∈ D G , because H contains all C j for j = i and also Osc 1 pi (C i ). Furthermore, H ∈ S since H ⊃ Z i for every i = s + 1, . . . , n. It thus follows that deg D G = n i=s+1 b i , where b i is the number of hyperplanes of P ri containing Z i , that are tangent to C i . To compute b i note that dim Z i = r i − 2, so Z i is the axis of a pencil of hyperplanes of P ri . The number of hyperplanes in this pencil that are tangent to C i is that of the ramification points of the morphism C i → P 1 defined by the projection of C i from Z i . Thus the Riemann-Hurwitz formula tells us that b i = 2(d i − 1) and this concludes the proof. Relying on the above results we get the following characterization. Theorem 4.7 Let X ⊂ P N be a decomposable scroll generated by C 1 , . . . , C n , and let d i = deg C i , for i = 1, . . . , n. Suppose that G is an irreducible component of type (2) of Φ 2 (X). Then D G is a rational normal scroll if and only if, up to reordering the curves, d 1 = · · · = d s = 1 and d s+1 = · · · = d n = 2 for some s ≥ 1. P r o o f. As G is of type (2), we can assume that d 1 = · · · = d s = 1 for some s ≥ 1 and d i ≥ 2 for i ≥ s + 1, by Proposition 4.2. As we noted, D G has dimension N − 2n + 1 and is non-degenerate in P N −2s . Thus, recalling Proposition 4.6, the inequality deg D G ≥ codimD G + 1 becomes 2 n i=s+1 (d i − 1) ≥ 2(n − s).(10) Note that this is an equality if and only if d i = 2 for i = s + 1 . . . , n. So, if D G is a rational normal scroll, then (11) holds. On the other hand, if (11) holds then we know that X is a rational scroll, by Proposition 4.4, and then equality in (10) says that it is normal. A general lower bound In [6, Theorem A] it is shown that the highest inflectional locus of an indecomposable linearly normal elliptic scroll of invariant e = −1 is empty. By adapting the argument used in [6] we can locate the highest inflectional locus of an elliptic indecomposable scroll of invariant e = 0, which is linearly normally embedded. Let C be a smooth curve of genus 1 and let S = P(E), where E is the holomorphic rank-2 vector bundle on C defined by the non-split extension 0 → O C → E → O C → 0.(12) Let π : S → C be the ruling projection and denote by C 0 the tautological section on S. Let δ ∈ Div(C) be a divisor of degree deg δ = m + 1 ≥ 3 and set L := O S (C 0 + π * δ). Note that L is very ample, because deg δ ≥ e + 3 [4, Ex. 2.12(b), p. 385] and the morphism given by \|L\| embeds S as a linearly normal scroll of degree 2m + 2 in P N , where N = 2m + 1 (note that m = N −1 2 = [ N −1 2 ]). Let x ∈ S. By [6, (1.0 m )] we have dim Osc m x (S) = N − 1 − dim(\|L − (m + 1)x\|) = 2m − dim(\|L − (m + 1)x\|).(13) On the other hand, by [6, Remark 1.2] we know that \|L − (m + 1)x\| = mf x + \|L − mf x − x\|,(14) where f x is the fibre through x. Now, twisting (12) by O C (δ − mπ(x)) we immediately see that h 0 (L − mf x ) = h 0 (E(δ − mπ(x)) = 2h 0 (O C (δ − mπ(x)) = 2. Hence (15) gives dim(\|L − mf x − x\|) = 0 if and only if L ⊗ O S (−mf x ) is spanned at x and taking into account (14) and (13) we get dim Osc m x (S) = 2m if and only if L ⊗ O S (−mf x ) is spanned at x. This proves the following Proposition 5.1 Let S ⊂ P 2m+1 be a linearly normal surface scroll over an elliptic curve C, defined by an indecomposable vector bundle as in (12), and let L be the hyperplane bundle. Then x ∈ Φ m (S) if and only if the line bundle L ⊗ O S (−mf x ) is not spanned at x, where f x is the fibre of S through x. Now let S ⊂ P N be any surface scroll. Though S can be not decomposable, according to Example 3.2 it seems natural to ask whether, under some assumption, we can get a global lower bound for the dimension of Osc k x (S), i. e., a lower bound holding at every point x ∈ S, bigger than 3. We determine such a lower bound, depending on k, under certain assumptions on the linear system (not necessarily complete) giving rise to the embedding. In the following we use the same notation as in [6]. dim Osc k x (S) ≥ k + 2 for k ≥ 3. Note that this global lower bound is the same holding for rational normal scrolls, as shown by Example 3.1. P r o o f. First let us prove, by induction, that dim Osc k x (S) ≥ k + 1 for any k ≥ 2.(16) For k = 2 this comes from [6, Theorem B] (noting that N ≥ 4 is enough in the proof). So let k ≥ 3 and set L = L − (k − 2)f x and \|W \| = \|V − (k − 2)f x \|. Note that \|W \| is very ample by assumption and that S embedded by \|W \| is also a scroll. Actually, for every fibre f of S we have If ii) holds, then L = O(1, k − 1), hence h 0 (L) = 2k, which implies that Lf = (L − (k − 2)f x )f = Lf = 1.N = dim(\|V \|) ≤ dim(\|L\|) = 2k − 1, but this contradicts our assumption that N ≥ 2k. Therefore condition i) holds. Now suppose, by contradiction, that (16) is not true, i. e., dim Osc k x (S) ≤ k. Because \|V − (k − 3)f x \| is very ample, by induction we know that dim Osc k−1 x (S) ≥ k. So, due to the obvious inclusion Osc k x (S) ⊇ Osc k−1 x (S) we conclude that Osc k x (S) = Osc k−1 x (S). Equivalently, this says that \|V − (k + 1)x\| = \|V − kx\|. This in turn, according to [6, Remark (1. 2)], implies the equality dim(\|V − kf x − x\|) = dim(\|V − (k − 1)f x − x\|). Hence f x ⊆ Bs(\|V − (k − 1)f x − x\|) = Bs(\|W − f x − x\|) = ϕ −1 x (ϕ x (x) ). But this contradicts condition i). To conclude the proof we show that equality cannot occur in (16) for k ≥ 3. First of all, since \|V − tf x \| is very ample for all t ≤ k − 2, by applying [6, Remark (1.7)] inductively we see that dim(\|W \|) = dim(\|V \|) − 2(k − 2) = N − 2(k − 2).(17) Note that Finally, combining (17) with (18) and assuming equality in (16) gives k ≤ 2. This completes the proof. Enumeration of Hamiltonian paths in a graph Let A = (a ij ) be the adjacency matrix of graph G. The corresponding Kirchhoff matrix K = (k ij ) is obtained from A by replacing in −A each diagonal entry by the degree of its corresponding vertex; i.e., the ith diagonal entry is identified with the degree of the ith vertex. It is well known that det K(i\|i) = the number of spanning trees of G, i = 1, . . . , n where K(i\|i) is the ith principal submatrix of K. \det\mathbf{K}(i\|i)=\text{ the number of spanning trees of $G$}, Let C i(j) be the set of graphs obtained from G by attaching edge (v i v j ) to each spanning tree of G. Denote by C i = j C i(j) . It is obvious that the collection of Hamiltonian cycles is a subset of C i . Note that the cardinality of C i is k ii det K(i\|i). Let X = {x 1 , . . . ,x n }. $\wh X=\{\hat x_1,\dots,\hat x_n\}$ Define multiplication for the elements of X bŷ x ixj =x jxi ,x 2 i = 0, i, j = 1, . . . , n.(2) Letk ij = k ijxj andk ij = − j =ik ij . Then the number of Hamiltonian cycles H c is given by the relation [8] n j=1x j H c = 1 2k ij det K(i\|i), i = 1, . . . , n.(3) The task here is to express (3) in a form free of anyx i , i = 1, . . . , n. The result also leads to the resolution of enumeration of Hamiltonian paths in a graph. * Corresponding author: e-mail: [email protected], Phone: +00 999 999 999, Fax: +00 999 999 999 mn data will be provided by the publisher It is well known that the enumeration of Hamiltonian cycles and paths in a complete graph K n and in a complete bipartite graph K n1n2 can only be found from first combinatorial principles [4]. One wonders if there exists a formula which can be used very efficiently to produce K n and K n1n2 . Recently, using Lagrangian methods, Goulden and Jackson have shown that H c can be expressed in terms of the determinant and permanent of the adjacency matrix [3]. However, the formula of Goulden and Jackson determines neither K n nor K n1n2 effectively. In this paper, using an algebraic method, we parametrize the adjacency matrix. The resulting formula also involves the determinant and permanent, but it can easily be applied to K n and K n1n2 . In addition, we eliminate the permanent from H c and show that H c can be represented by a determinantal function of multivariables, each variable with domain {0, 1}. Furthermore, we show that H c can be written by number of spanning trees of subgraphs. Finally, we apply the formulas to a complete multigraph K n1...np . The conditions a ij = a ji , i, j = 1, . . . , n, are not required in this paper. All formulas can be extended to a digraph simply by multiplying H c by 2. Main Theorem Notation For p, q ∈ P and n ∈ ω we write (q, n) ≤ (p, n) if q ≤ p and A q,n = A p,n . \begin{notation} For $p,q\in P$ and $n\in\omega$ ... \end{notation} Let B = (b ij ) be an n × n matrix. Let n = {1, . . . , n}. Using the properties of (2), it is readily seen that Lemma 3.1 i∈n j∈n b ijxi = i∈nx i per B (4) where per B is the permanent of B. Let Y = {ŷ 1 , . . . ,ŷ n }. Define multiplication for the elements of Y bŷ y iŷj +ŷ jŷi = 0, i, j = 1, . . . , n. Then, it follows that Lemma 3.2 i∈n j∈n b ijŷj = i∈nŷ i det B.(6) Note that all basic properties of determinants are direct consequences of Lemma 3.2. Write j∈n b ijŷj = j∈n b (λ) ijŷ j + (b ii − λ i )ŷ iŷ (7) where b (λ) ii = λ i , b (λ) ij = b ij , i = j.(8) Let B (λ) = (b (λ) ij ). By (6) and (7), it is straightforward to show the following result: Theorem 3.3 det B = n l=0 I l ⊆n i∈I l (b ii − λ i ) det B (λ) (I l \|I l ),(9) where I l = {i 1 , . . . , i l } and B (λ) (I l \|I l ) is the principal submatrix obtained from B (λ) by deleting its i 1 , . . . , i l rows and columns. mn header will be provided by the publisher 5 Remark 3.4 Let M be an n × n matrix. The convention M(n\|n) = 1 has been used in (9) and hereafter. Before proceeding with our discussion, we pause to note that Theorem 3.3 yields immediately a fundamental formula which can be used to compute the coefficients of a characteristic polynomial [9]: Corollary 3.5 Write det(B − xI) = n l=0 (−1) l b l x l . Then b l = I l ⊆n det B(I l \|I l ).(10) Let K(t, t 1 , . . . , t n ) =     D 1 t −a 12 t 2 . . . −a 1n t n −a 21 t 1 D 2 t . . . −a 2n t. . −a n1 t 1 −a n2 t 2 . . . D n t     ,(11) \begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\ -a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\ \hdotsfor [2]{4}\\ -a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix} where D i = j∈n a ij t j , i = 1, . . . , n.(12) Set D(t 1 , . . . , t n ) = δ δt det K(t, t 1 , . . . , t n )\| t=1 . Then D(t 1 , . . . , t n ) = i∈n D i det K(t = 1, t 1 , . . . , t n ; i\|i), where K(t = 1, t 1 , . . . , t n ; i\|i) is the ith principal submatrix of K(t = 1, t 1 , . . . , t n ). Theorem 3.3 leads to det K(t 1 , t 1 , . . . , t n ) = I∈n (−1) \|I\| t n−\|I\| i∈I t i j∈I (D j + λ j t j ) det A (λt) (I\|I).(14) Note that det K(t = 1, t 1 , . . . , t n ) = I∈n (−1) \|I\| i∈I t i j∈I (D j + λ j t j ) det A (λ) (I\|I) = 0.(15) Let t i =x i , i = 1, . . . , n. H c = 1 2n n l=0 (−1) l D l ,(17) where D l = I l ⊆n D(t 1 , . . . , t n )2\| ti= n 0, if i∈I l 1, otherwise , i=1,...,n .(18) Application We consider here the applications of Theorems 5.2 and 5.3 to a complete multipartite graph K n1...np . It can be shown that the number of spanning trees of K n1...np may be written T = n p−2 p i=1 (n − n i ) ni−1(19) where n = n 1 + · · · + n p . It follows from Theorems 5.2 and 5.3 that H c = 1 2n n l=0 (−1) l (n − l) p−2 l1+···+lp=l p i=1 n i l i · [(n − l) − (n i − l i )] ni−li · (n − l) 2 − p j=1 (n i − l i ) 2 .(21) ... \binom{n_i}{l _i}\\ and H c = 1 2 n−1 l=0 (−1) l (n − l) p−2 l1+···+lp=l p i=1 n i l i · [(n − l) − (n i − l i )] ni−li 1 − l p n p [(n − l) − (n p − l p )].(22) The enumeration of H c in a K n1···np graph can also be carried out by Theorem 7.5 or 7.6 together with the algebraic method of (2). Some elegant representations may be obtained. For example, H c in a K n1n2n3 graph may be written H c = n 1 ! n 2 ! n 3 ! n 1 + n 2 + n 3 i n 1 i n 2 n 3 − n 1 + i n 3 n 3 − n 2 + i + n 1 − 1 i n 2 − 1 n 3 − n 1 + i n 3 − 1 n 3 − n 2 + i .(23) mn header will be provided by the publisher 7 Secret Key Exchanges Modern cryptography is fundamentally concerned with the problem of secure private communication. A Secret Key Exchange is a protocol where Alice and Bob, having no secret information in common to start, are able to agree on a common secret key, conversing over a public channel. The notion of a Secret Key Exchange protocol was first introduced in the seminal paper of Diffie and Hellman [1]. [1] presented a concrete implementation of a Secret Key Exchange protocol, dependent on a specific assumption (a variant on the discrete log), specially tailored to yield Secret Key Exchange. Secret Key Exchange is of course trivial if trapdoor permutations exist. However, there is no known implementation based on a weaker general assumption. The concept of an informationally one-way function was introduced in [5]. We give only an informal definition here: In the non-uniform setting [5] show that these are not weaker than one-way functions: Theorem 5.2 ([5] (non-uniform)) The existence of informationally one-way functions implies the existence of one-way functions. We will stick to the convention introduced above of saying "non-uniform" before the theorem statement when the theorem makes use of non-uniformity. It should be understood that if nothing is said then the result holds for both the uniform and the non-uniform models. It now follows from Theorem 5.2 that Theorem 5.3 (non-uniform) Weak SKE implies the existence of a one-way function. More recently, the polynomial-time, interior point algorithms for linear programming have been extended to the case of convex quadratic programs [11,13], certain linear complementarity problems [7,10], and the nonlinear complementarity problem [6]. The connection between these algorithms and the classical Newton method for nonlinear equations is well explained in [7]. Review We begin our discussion with the following definition: Definition 6.1 A function H : ℜ n → ℜ n is said to be B-differentiable at the point z if (i) H is Lipschitz continuous in a neighborhood of z, and (ii) there exists a positive homogeneous function BH(z) : ℜ n → ℜ n , called the B-derivative of H at z, such that lim v→0 H(z + v) − H(z) − BH(z)v v = 0. The function H is B-differentiable in set S if it is B-differentiable at every point in S. The B-derivative BH(z) is said to be strong if lim (v,v ′ )→(0,0) H(z + v) − H(z + v ′ ) − BH(z)(v − v ′ ) v − v ′ = 0. Lemma 6.2 There exists a smooth function ψ 0 (z) defined for \|z\| > 1 − 2a satisfying the following properties: (i) ψ 0 (z) is bounded above and below by positive constants c 1 ≤ ψ 0 (z) ≤ c 2 . (ii) If \|z\| > 1, then ψ 0 (z) = 1. (iii) For all z in the domain of ψ 0 , ∆ 0 ln ψ 0 ≥ 0. (iv) If 1 − 2a < \|z\| < 1 − a, then ∆ 0 ln ψ 0 ≥ c 3 > 0. P r o o f. We choose ψ 0 (z) to be a radial function depending only on r = \|z\|. Let h(r) ≥ 0 be a suitable smooth function satisfying h(r) ≥ c 3 for 1 − 2a < \|z\| < 1 − a, and h(r) = 0 for \|z\| > 1 − a 2 . The radial Laplacian ∆ 0 ln ψ 0 (r) = d 2 dr 2 + 1 r d dr ln ψ 0 (r) has smooth coefficients for r > 1−2a. Therefore, we may apply the existence and uniqueness theory for ordinary differential equations. Simply let ln ψ 0 (r) be the solution of the differential equation d 2 dr 2 + 1 r d dr ln ψ 0 (r) = h(r) with initial conditions given by ln ψ 0 (1) = 0 and ln ψ ′ 0 (1) = 0. Next, let D ν be a finite collection of pairwise disjoint disks, all of which are contained in the unit disk centered at the origin in C. We assume that D ν = {z \| \|z − z ν \| < δ}. Suppose that D ν (a) denotes the smaller concentric disk D ν (a) = {z \| \|z − z ν \| ≤ (1 − 2a)δ}. We define a smooth weight function Φ 0 (z) for z ∈ C − ν D ν (a) by setting Φ 0 (z) = 1 when z / ∈ ν D ν and Φ 0 (z) = ψ 0 ((z − z ν )/δ) when z is an element of D ν . It follows from Lemma 6.2 that Φ 0 satisfies the properties: (i) Φ 0 (z) is bounded above and below by positive constants (2) and (3) c 1 ≤ Φ 0 (z) ≤ c 2 . (ii) ∆ 0 ln Φ 0 ≥ 0 for all z ∈ C − ν D ν (a), the domain where the function Φ 0 is defined. (iii) ∆ 0 ln Φ 0 ≥ c 3 δ −2 when (1 − 2a)δ < \|z − z ν \| < (1 − a)δ. Let A ν denote the annulus A ν = {(1 − 2a)δ < \|z − z ν \| < (1 − a)δ}, and set A = ν A ν . The propertiesof Φ 0 may be summarized as ∆ 0 ln Φ 0 ≥ c 3 δ −2 χ A , where χ A is the characteristic function of A. Suppose that α is a nonnegative real constant. We apply Proposition 3.6 with Φ(z) = Φ 0 (z)e α\|z\| 2 . If u ∈ C ∞ 0 (R 2 − ν D ν (a)), assume that D is a bounded domain containing the support of u and A ⊂ D ⊂ R 2 − ν D ν (a). A calculation gives D ∂u 2 Φ 0 (z)e α\|z\| 2 ≥ c 4 α D \|u\| 2 Φ 0 e α\|z\| 2 + c 5 δ −2 A \|u\| 2 Φ 0 e α\|z\| 2 . The boundedness, property (1) of Φ 0 , then yields D ∂u 2 e α\|z\| 2 ≥ c 6 α D \|u\| 2 e α\|z\| 2 + c 7 δ −2 A \|u\| 2 e α\|z\| 2 . Let B(X) be the set of blocks of Λ X and let b(X) = \|B(X)\|. If φ ∈ Q X then φ is constant on the blocks of Λ X . P X = {φ ∈ M \| Λ φ = Λ X }, Q X = {φ ∈ M \| Λ φ ≥ Λ X }.(24)If Λ φ ≥ Λ X then Λ φ = Λ Y for some Y ≥ X so that Q X = Y ≥X P Y . Thus by Möbius inversion \|P Y \| = X≥Y µ(Y, X) \|Q X \| . Thus there is a bijection from Q X to W B(X) . In particular \|Q X \| = w b(X) . Next note that b(X) = dim X. We see this by choosing a basis for X consisting of vectors v k defined by v k i = 1 if i ∈ Λ k , 0 otherwise.χ(A, t) = B⊆A (−1) \|B\| t dim T (B) . In order to compute R ′′ recall the definition of S(X, Y ) from Lemma 3. 1. Since H ∈ B, A H ⊆ B. Thus if T (B) = Y then B ∈ S(H, Y ). Let L ′′ = L(A ′′ ). Then R ′′ = H∈B⊆A (−1) \|B\| t dim T (B) = Y ∈L ′′ B∈S(H,Y ) (−1) \|B\| t dim Y = − Y ∈L ′′ B∈S(H,Y ) (−1) \|B−AH \| t dim Y = − Y ∈L ′′ µ(H, Y )t dim Y = −χ(A ′′ , t).(A1) = xyz(x − z)(x + z)(y − z)(y + z) Fig. 2 Q(A2) = xyz(x + y + z)(x + y − z)(x − y + z)(x − y − z) The Poincaré polynomial of an arrangement will appear repeatedly in these notes. It will be shown to equal the Poincaré polynomial of the graded algebras which we are going to associate with A. It is also the Poincaré polynomial of the complement M (A) for a complex arrangement. Here we prove that the Poincaré polynomial is the chamber counting function for a real arrangement. The complement M (A) is a disjoint union of chambers M (A) = C∈Cham(A) C. The number of chambers is determined by the Poincaré polynomial as follows. Theorem 6.7 Let A R be a real arrangement. Then \|Cham(A R )\| = π(A R , 1). P r o o f. We check the properties required in Corollary 6.6: (i) follows from π(Φ l , t) = 1, and (ii) is a consequence of Corollary 3.5. Theorem 6.8 Let φ be a protocol for a random pair (X, Y ). If one of σ φ (x ′ , y) and σ φ (x, y ′ ) is a prefix of the other and (x, y) ∈ S X,Y , then σ j (x ′ , y) ∞ j=1 = σ j (x, y) ∞ j=1 = σ j (x, y ′ ) ∞ j=1 . P r o o f. We show by induction on i that σ j (x ′ , y) i j=1 = σ j (x, y) i j=1 = σ j (x, y ′ ) i j=1 . mn header will be provided by the publisher 11 The induction hypothesis holds vacuously for i = 0. Assume it holds for i − 1, in particular [σ j (x ′ , y)] i−1 j=1 = [σ j (x, y ′ )] i−1 j=1 . Then one of [σ j (x ′ , y)] ∞ j=i and [σ j (x, y ′ )] ∞ j=i is a prefix of the other which implies that one of σ i (x ′ , y) and σ i (x, y ′ ) is a prefix of the other. If the ith message is transmitted by P X then, by the separatetransmissions property and the induction hypothesis, σ i (x, y) = σ i (x, y ′ ), hence one of σ i (x, y) and σ i (x ′ , y) is a prefix of the other. By the implicit-termination property, neither σ i (x, y) nor σ i (x ′ , y) can be a proper prefix of the other, hence they must be the same and σ i (x ′ , y) = σ i (x, y) = σ i (x, y ′ ). If the ith message is transmitted by P Y then, symmetrically, σ i (x, y) = σ i (x ′ , y) by the induction hypothesis and the separate-transmissions property, and, then, σ i (x, y) = σ i (x, y ′ ) by the implicit-termination property, proving the induction step. If φ is a protocol for (X, Y ), and (x, y), (x ′ , y) are distinct inputs in S X,Y , then, by the correct-decision property, σ j (x, y) ∞ j=1 = σ j (x ′ , y) ∞ j=1 . Equation (25) defined P Y 's ambiguity set S X\|Y (y) to be the set of possible X values when Y = y. The last corollary implies that for all y ∈ S Y , the multiset 1 of codewords {σ φ (x, y) : x ∈ S X\|Y (y)} is prefix free. One-Way Complexitŷ C 1 (X\|Y ), the one-way complexity of a random pair (X, Y ), is the number of bits P X must transmit in the worst case when P Y is not permitted to transmit any feedback messages. Starting with S X,Y , the support set of (X, Y ), we define G(X\|Y ), the characteristic hypergraph of (X, Y ), and show that C 1 (X\|Y ) = ⌈ log χ(G(X\|Y ))⌉ . Let (X, Y ) be a random pair. For each y in S Y , the support set of Y , Equation (25) defined S X\|Y (y) to be the set of possible x values when Y = y. The characteristic hypergraph G(X\|Y ) of (X, Y ) has S X as its vertex set and the hyperedge S X\|Y (y) for each y ∈ S Y . We can now prove a continuity theorem. for every x ∈ Ω\S u . Let f : R m → R k be a Lipschitz continuous function such that f (0) = 0, and let v = f (u) : Ω → R k . Then v ∈ BV (Ω; R k ) and Jv = (f (u + ) − f (u − )) ⊗ ν u · H n−1 Su .(27) In addition, for Du -almost every x ∈ Ω the restriction of the function f to T u x is differentiable atũ(x) and Dv = ∇( f \| T u x )(ũ) Du Du · Du .(28) Before proving the theorem, we state without proof three elementary remarks which will be useful in the sequel. \|Dv\| (B) ≤ K \|Du\| (B) ∀B ∈ B(Ω),(29) where K > 0 is the Lipschitz constant of f . By (13) and by the approximation result quoted in §3, it is possible to find a sequence (u h ) ⊂ C 1 (Ω; R m ) converging to u in L 1 (Ω; R m ) and such that lim h→+∞ Ω \|∇u h \| dx = \|Du\| (Ω). The functions v h = f (u h ) are locally Lipschitz continuous in Ω, and the definition of differential implies that \|∇v h \| ≤ K \|∇u h \| almost everywhere in Ω. The lower semicontinuity of the total variation and (13) yield \|Dv\| (Ω) ≤ lim inf h→+∞ \|Dv h \| (Ω) = lim inf h→+∞ Ω \|∇v h \| dx ≤ K lim inf h→+∞ Ω \|∇u h \| dx = K \|Du\| (Ω).(30) Since f (0) = 0, we have also Ω \|v\| dx ≤ K Ω \|u\| dx; therefore u ∈ BV (Ω; R k ). Repeating the same argument for every open set A ⊂ Ω, we get (29) for every B ∈ B(Ω), because \|Dv\|, \|Du\| are Radon measures. To prove Lemma 6.2, first we observe that S v ⊂ S u ,ṽ(x) = f (ũ(x)) ∀x ∈ Ω\S u .(31) In fact, for every ε > 0 we have {y ∈ B ρ (x) : \|v(y) − f (ũ(x))\| > ε} ⊂ {y ∈ B ρ (x) : \|u(y) −ũ(x)\| > ε/K}, hence lim ρ→0 + \|{y ∈ B ρ (x) : \|v(y) − f (ũ(x))\| > ε}\| ρ n = 0 whenever x ∈ Ω\S u . By a similar argument, if x ∈ S u is a point such that there exists a triplet (u + , u − , ν u ) satisfying (14), (15), then (v + (x) − v − (x)) ⊗ ν v = (f (u + (x)) − f (u − (x))) ⊗ ν u if x ∈ S v and f (u − (x)) = f (u + (x)) if x ∈ S u \S v . Hence, by (1.8) we get Jv(B) = B∩Sv (v + − v − ) ⊗ ν v dH n−1 = B∩Sv (f (u + ) − f (u − )) ⊗ ν u dH n−1 = B∩Su (f (u + ) − f (u − )) ⊗ ν u dH n−1 and Lemma 6.2 is proved. To prove (31), it is not restrictive to assume that k = 1. Moreover, to simplify our notation, from now on we shall assume that Ω = R n . The proof of (31) is divided into two steps. In the first step we prove the statement in the one-dimensional case (n = 1), using Theorem 5.3. In the second step we achieve the general result using Theorem 7.1. Step 1 Assume that n = 1. Since S u is at most countable, (7) yields that Dv (S u \S v ) = 0, so that (19) and (21) Du -almost everywhere in R. It is well known (see, for instance, [12, 2.5.16]) that every one-dimensional function of bounded variation w has a unique left continuous representative, i.e., a functionŵ such thatŵ = w almost everywhere and lim s→t −ŵ(s) =ŵ(t) for every t ∈ R. These conditions implŷ u(t) = Du(]−∞, t[),v(t) = Dv(]−∞, t[) ∀t ∈ R (32) andv (t) = f (û(t)) ∀t ∈ R.(33) Let t ∈ R be such that Du ([t, s[) > 0 for every s > t and assume that the limits in (22) exist. By (23) and (24) we getv (s) −v(t) Du ([t, s[) = f (û(s)) − f (û(t)) Du ([t, s[) = f (û(s)) − f (û(t) + Du Du (t) Du ([t, s[)) Du ([t, s[) + f (û(t) + Du Du (t) Du ([t, s[)) − f (û(t)) Du ([t, s[) for every s > t. Using the Lipschitz condition on f we find v(s) −v(t) Du ([t, s[) − f (û(t) + Du Du (t) Du ([t, s[)) − f (û(t)) Du ([t, s[) ≤ K û(s) −û(t) Du ([t, s[) − Du Du (t) . By (29), the function s → Du ([t, s[) is continuous and converges to 0 as s ↓ t. Therefore Remark 7.2 and the previous inequality imply Dv Du (t) = lim h→0 + f (û(t) + h Du Du (t)) − f (û(t)) h Du -a.e. in R. By (22),û(x) =ũ(x) for every x ∈ R\S u ; moreover, applying the same argument to the functions u ′ (t) = u(−t), v ′ (t) = f (u ′ (t)) = v(−t), we get Dv Du (t) = lim h→0 f (ũ(t) + h Du Du (t)) − f (ũ(t)) h Du -a.e. in R and our statement is proved. Step 2 Let us consider now the general case n > 1. Let ν ∈ R n be such that \|ν\| = 1, and let π ν = {y ∈ R n : y, ν = 0}. In the following, we shall identify R n with π ν × R, and we shall denote by y the variable ranging in π ν and by t the variable ranging in R. By the just proven one-dimensional result, and by Theorem 3.3, we get lim h→0 f (ũ(y + tν) + h Du y Du y (t)) − f (ũ(y + tν)) h = Dv y Du y (t) Du y -a.e. in R for H n−1 -almost every y ∈ π ν . We claim that Du, ν Du, ν (y + tν) = Du y Du y (t) Du y -a.e. in R(34) for H n−1 -almost every y ∈ π ν . In fact, by (16) and (18) and (24) follows from (13). By the same argument it is possible to prove that Dv, ν Du, ν (y + tν) = Dv y Du y (t) Du y -a.e. in R(35) for H n−1 -almost every y ∈ π ν . By (24) and (25) we get lim h→0 f (ũ(y + tν) + h Du, ν Du, ν (y + tν)) − f (ũ(y + tν)) h = Dv, ν Du, ν (y + tν) for H n−1 -almost every y ∈ π ν , and using again (14), (15) we get lim h→0 f (ũ(x) + h Du, ν Du, ν (x)) − f (ũ(x)) h = Dv, ν Du, ν(x) mn header will be provided by the publisher 15 Du, ν -a.e. in R n . Since the function Du, ν / Du is strictly positive Du, ν -almost everywhere, we obtain also lim h→0 f (ũ(x) + h Du, ν Du (x) Du, ν Du, ν (x)) − f (ũ(x)) h = Du, ν Du (x) Dv, ν Du, ν (x) Du, ν -almost everywhere in R n . Finally, since and since both sides of (33) are zero Du -almost everywhere on Du, ν -negligible sets, we conclude that lim h→0 f  ũ (x) + h Du Du (x), ν   − f (ũ(x)) h = Dv Du (x), ν , Du -a.e. in R n . Since ν is arbitrary, by Remarks 7.3 and 7.4 the restriction of f to the affine space T u x is differentiable atũ(x) for Du -almost every x ∈ R n and (26) holds. It follows from (13), (14), and (15) that D(t 1 , . . . , t n ) = I∈n (−1) \|I\|−1 \|I\| i∈I t i j∈I (D j + λ j t j ) det A (λ) (I\|I).(36) Let t i =x i , i = 1, . . . , n. Lemma 1 leads to D(x 1 , . . . ,x n ) = i∈nx i I∈n (−1) \|I\|−1 \|I\| per A (λ) (I\|I) det A (λ) (I\|I).(37) By (3), (13), and (37), we have the following result: Theorem 7.5 H c = 1 2n n l=1 l(−1) l−1 A (λ) l ,(38) where A (λ) l = I l ⊆n per A (λ) (I l \|I l ) det A (λ) (I l \|I l ), \|I l \| = l.(39) It is worth noting that A (λ) l of (39) is similar to the coefficients b l of the characteristic polynomial of (10). It is well known in graph theory that the coefficients b l can be expressed as a sum over certain subgraphs. It is interesting to see whether A l , λ = 0, structural properties of a graph. We may call (38) a parametric representation of H c . In computation, the parameter λ i plays very important roles. The choice of the parameter usually depends on the properties of the given graph. For a complete graph K n , let λ i = 1, i = 1, . . . , n. It follows from (39) that A (1) l = n!, if l = 1 0, otherwise.(40) By (38) H c = 1 2 (n − 1)!.(41) For a complete bipartite graph K n1n2 , let λ i = 0, i = 1, . . . , n. By (39), A l = −n 1 !n 2 !δ n1n2 , if l = 2 0, otherwise .(42) Theorem 7.5 leads to H c = 1 n 1 + n 2 n 1 !n 2 !δ n1n2 .(43) Now, we consider an asymmetrical approach. Theorem 3.3 leads to det K(t = 1, t 1 , . . . , t n ; l\|l) = I⊆n−{l} (−1) \|I\| i∈I t i j∈I (D j + λ j t j ) det A (λ) (I ∪ {l}\|I ∪ {l}). (44) By (3) and (16) we have the following asymmetrical result: Theorem 7.6 H c = 1 2 I⊆n−{l} (−1) \|I\| per A (λ) (I\|I) det A (λ) (I ∪ {l}\|I ∪ {l})(45) which reduces to Goulden-Jackson's formula when λ i = 0, i = 1, . . . , n [9]. Various font features of the amsmath package 8.1 Bold versions of special symbols In the amsmath package \boldsymbol is used for getting individual bold math symbols and bold Greek letters-everything in math except for letters of the Latin alphabet, where you'd use \mathbf. For example, A_\infty + \pi A_0 \sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+} \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}} looks like this: A ∞ + πA 0 ∼ A ∞ + πA 0 mn header will be provided by the publisher "Poor man's bold" If a bold version of a particular symbol doesn't exist in the available fonts, then \boldsymbol can't be used to make that symbol bold. At the present time, this means that \boldsymbol can't be used with symbols from the msam and msbm fonts, among others. In some cases, poor man's bold (\pmb) can be used instead of \boldsymbol: ∂x ∂y ∂y ∂z \[\frac{\partial x}{\partial y} \pmb{\bigg\vert} \frac{\partial y}{\partial z}\] So-called "large operator" symbols such as and require an additional command, \mathop, to produce proper spacing and limits when \pmb is used. For further details see The T E Xbook. i<B i odd κ κF (r i ) i<B i odd κ κ(r i ) \[\sum_{\substack{i<B\\\text{$i$ odd}}} \prod_\kappa \kappa F(r_i)\qquad \mathop{\pmb{\sum}}_{\substack{i<B\\\text{$i$ odd}}} \mathop{\pmb{\prod}}_\kappa \kappa(r_i) \] 9 Compound symbols and other features 9.1 Multiple integral signs \iint, \iiint, and \iiiint give multiple integral signs with the spacing between them nicely adjusted, in both text and display style. \idotsint gives two integral signs with dots between them. A f (x, y) dx dy A f (x, y, z) dx dy dz (46) A f (w, x, y, z) dw dx dy dz · · · A f (x 1 , . . . , x k )(47) Over and under arrows Some extra over and under arrow operations are provided in the amsmath package. (Basic L A T E X provides \overrightarrow and \overleftarrow). − −−−−− → ψ δ (t)E t h = ψ δ (t)E t h − −−−−− → ← −−−−− − ψ δ (t)E t h = ψ δ (t)E t h ← −−−−− − ←−−−−→ ψ δ (t)E t h = ψ δ (t)E t h ←−−−−→ Dots Normally you need only type \dots for ellipsis dots in a math formula. The main exception is when the dots fall at the end of the formula; then you need to specify one of \dotsc (series dots, after a comma), \dotsb (binary dots, for binary relations or operators), \dotsm (multiplication dots), or \dotsi (dots after an integral). For example, the input Then we have the series $A_1,A_2,\dotsc$, the regional sum $A_1+A_2+\dotsb$, the orthogonal product $A_1A_2\dotsm$, and the infinite integral \[\int_{A_1}\int_{A_2}\dotsi\]. produces Then we have the series A 1 , A 2 , . . . , the regional sum A 1 + A 2 + · · · , the orthogonal product A 1 A 2 · · · , and the infinite integral A1 A2 · · · Accents in math Double accents: HČTÁG˙ḊDBB V \[\Hat{\Hat{H}}\quad\Check{\Check{C}}\quad \Tilde{\Tilde{T}}\quad\Acute{\Acute{A}}\quad \Grave{\Grave{G}}\quad\Dot{\Dot{D}}\quad \Ddot{\Ddot{D}}\quad\Breve{\Breve{B}}\quad \Bar{\Bar{B}}\quad\Vec{\Vec{V}}\] This double accent operation is complicated and tends to slow down the processing of a L A T E X file. Dot accents \dddot and \ddddot are available to produce triple and quadruple dot accents in addition to the \dot and \ddot accents already available in L A T E X: ... Q .... R \[\dddot{Q}\qquad\ddddot{R}\] Roots In the amsmath package \leftroot and \uproot allow you to adjust the position of the root index of a radical: \sqrt[\leftroot{-2}\uproot{2}\beta]{k} gives good positioning of the β: β √ k Boxed formulas The command \boxed puts a box around its argument, like \fbox except that the contents are in math mode: \boxed{W_t-F\subseteq V(P_i)\subseteq W_t} W t − F ⊆ V (P i ) ⊆ W t . Extensible arrows \xleftarrow and \xrightarrow produce arrows that extend automatically to accommodate unusually wide subscripts or superscripts. The text of the subscript or superscript are given as an optional resp. mandatory argument: Example: 0 α ← − ζ F × △[n − 1] ∂0α(b) − −−− → E ∂0b \[0 \xleftarrow[\zeta]{\alpha} F\times\triangle[n-1] \xrightarrow{\partial_0\alpha(b)} Eˆ{\partial_0b}\] 9.9 \overset, \underset, and \sideset Examples: * X X * a X b \[\overset{ * }{X}\qquad\underset{ * }{X}\qquad \overset{a}{\underset{b}{X}}\] The command \sideset is for a rather special purpose: putting symbols at the subscript and superscript corners of a large operator symbol such as or , without affecting the placement of limits. Examples: * * * * k ′ 0≤i≤m E i βx \[\sideset{_ ˆ }{_ ˆ }\prod_k\qquad \sideset{}{'}\sum_{0\le i\le m} E_i\beta x \] The \text command The main use of the command \text is for words or phrases in a display: y = y ′ if and only if y ′ k = δ k y τ (k) \[\mathbf{y}=\mathbf{y}'\quad\text{if and only if}\quad y'_k=\delta_k y_{\tau(k)}\] Operator names The more common math functions such as log, sin, and lim have predefined control sequences: \log, \sin, \lim. The amsmath package provides \DeclareMathOperator and \DeclareMathOperator * for producing new function names that will have the same typographical treatment. Examples: The following special operator names are predefined in the amsmath package: \varlimsup, \varliminf, \varinjlim, and \varprojlim. Here's what they look like in use: f ∞ =lim n→∞ Q(u n , u n − u # ) ≤ 0 (48) lim n→∞ \|a n+1 \| / \|a n \| = 0 (49) lim − → (m λ i ·) * ≤ 0 (50) lim ← − p∈S(A) A p ≤ 0 (51) \begin{align} &\varlimsup_{n\rightarrow\infty} \mathcal{Q}(u_n,u_n-uˆ{\#})\le0\\ &\varliminf_{n\rightarrow\infty} \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0\\ &\varinjlim (m_iˆ\lambda\cdot)ˆ\le0\\ &\varprojlim_{p\in S(A)}A_p\le0 \end{align} \mod and its relatives The commands \mod and \pod are variants of \pmod preferred by some authors; \mod omits the parentheses, whereas \pod omits the 'mod' and retains the parentheses. Examples: x ≡ y + 1 (mod m 2 ) (52) x ≡ y + 1 mod m 2 (53) x ≡ y + 1 (m 2 ) (54) \begin{align} x&\equiv y+1\pmod{mˆ2}\\ x&\equiv y+1\mod{mˆ2}\\ x&\equiv y+1\pod{mˆ2} \end{align} Fractions and related constructions The usual notation for binomials is similar to the fraction concept, so it has a similar command \binom with two arguments. Example: γ∈ΓC I γ = 2 k − k 1 2 k−1 + k 2 2 k−2 + · · · + (−1) l k l 2 k−l + · · · + (−1) k = (2 − 1) k = 1(55) Continued fractions The continued fraction 1 √ 2 + 1 √ 2 + 1 √ 2 + 1 √ 2 + 1 √ 2 + · · ·(59) can be obtained by typing \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+\dotsb }}}}} Left or right placement of any of the numerators is accomplished by using \cfrac [l] or \cfrac[r] instead of \cfrac. Smash In amsmath there are optional arguments t and b for the plain T E X command \smash, because sometimes it is advantageous to be able to 'smash' only the top or only the bottom of something while retaining the natural depth or height. In the formula X j = (1/ √ λ j )X ′ j \smash[b] has been used to limit the size of the radical symbol. $X_j=(1/\sqrt{\smash[b]{\lambda_j}})X_j'$ Without the use of \smash[b] the formula would have appeared thus: X j = (1/ λ j )X ′ j , with the radical extending to encompass the depth of the subscript j. The 'cases' environment 'Cases' constructions like the following can be produced using the cases environment. P r−j = 0 if r − j is odd, r! (−1) (r−j)/2 if r − j is even.(60) \begin{equation} P_{r-j}= \begin{cases} 0& \text{if $r-j$ is odd},\\ r!\,(-1)ˆ{(r-j)/2}& \text{if $r-j$ is even}. \end{cases} \end{equation} Notice the use of \text and the embedded math. Matrix Here are samples of the matrix environments, \matrix, \pmatrix, \bmatrix, \Bmatrix, \vmatrix and \Vmatrix: To produce a small matrix suitable for use in text, use the smallmatrix environment. ϑ ̺ ϕ ̟ ϑ ̺ ϕ ̟ ϑ ̺ ϕ ̟ ϑ ̺ ϕ ̟ ϑ ̺ ϕ ̟ ϑ ̺ ϕ ̟(61) \begin{math} \bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr) \end{math} To show the effect of the matrix on the surrounding lines of a paragraph, we put it here: a b c d and follow it with enough text to ensure that there will be at least one full line below the matrix. \hdotsfor{number} produces a row of dots in a matrix spanning the given number of columns: W (Φ) = ϕ (ϕ 1 , ε 1 ) 0 . . . 0 ϕk n2 (ϕ 2 , ε 1 ) ϕ (ϕ 2 , ε 2 ) . . . . . . ϕk n n−1 (ϕ n , ε n−1 ) ϕ (ϕ n , ε n ) \[W(\Phi)= \begin{Vmatrix} \dfrac\varphi{(\varphi_1,\varepsilon_1)}&0&\dots&0\\ \dfrac{\varphi k_{n2}}{(\varphi_2,\varepsilon_1)}& \dfrac\varphi{(\varphi_2,\varepsilon_2)}&\dots&0\\ \hdotsfor{5}\\ \dfrac{\varphi k_{n1}}{(\varphi_n,\varepsilon_1)}& \dfrac{\varphi k_{n2}}{(\varphi_n,\varepsilon_2)}&\dots& \dfrac{\varphi k_{n\,n-1}}{(\varphi_n,\varepsilon_{n-1})}& \dfrac{\varphi}{(\varphi_n,\varepsilon_n)} \end{Vmatrix}\] The spacing of the dots can be varied through use of a square-bracket option, for example, \hdotsfor [1.5]{3}. The number in square brackets will be used as a multiplier; the normal value is 1. The \substack command The \substack command can be used to produce a multiline subscript or superscript: for example \sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j) produces a two-line subscript underneath the sum: 0≤i≤m 0<j<n P (i, j)(62) A slightly more generalized form is the subarray environment which allows you to specify that each line should be left-aligned instead of centered, as here: 0≤i≤m 0<j<n P (i, j)(63) \sum_{\begin{subarray}{l} 0\le i\le m\\ 0<j<n \end{subarray}} P(i,j) Big-g-g delimiters Here are some big delimiters, first in \normalsize: E y tε 0 L x,y x (s) ϕ(x) ds \[\biggl(\mathbf{E}_{y} \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds \biggr) \] and now in \Large size: E y t ε 0 L x,y x (s) ϕ(x) ds {\Large \[\biggl(\mathbf{E}_{y} \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds \biggr) \]} A Examples of multiple-line equation structures Note: Starting on this page, vertical rules are added at the margins so that the positioning of various display elements with respect to the margins can be seen more clearly. A.1 Split The split environment is not an independent environment but should be used inside something else such as equation or align. If there is not enough room for it, the equation number for a split will be shifted to the previous line, when equation numbers are on the left; the number shifts down to the next line when numbers are on the right. f h,ε (x, y) = εE x,y tε 0 L x,yε(εu) ϕ(x) du = h L x,z ϕ(x)ρ x (dz) + h 1 t ε E y tε 0 L x,y x (s) ϕ(x) ds − t ε L x,z ϕ(x)ρ x (dz) + 1 t ε E y tε 0 L x,y x (s) ϕ(x) ds − E x,y tε 0 L x,yε(εs) ϕ(x) ds = h L x ϕ(x) + hθ ε (x, y),(64)f h,ε (x, y) = εE x,y tε 0 L x,yε(εu) ϕ(x) du = h L x,z ϕ(x)ρ x (dz) + h 1 t ε E y tε 0 L x,y x (s) ϕ(x) ds − t ε L x,z ϕ(x)ρ x (dz) + 1 t ε E y tε 0 L x,y x (s) ϕ(x) ds − E x,y tε 0 L x,yε(εs) ϕ(x) ds = h L x ϕ(x) + hθ ε (x, y), Some text after to test the below-display spacing. If the option centertags is included in the options list of the amsmath package, the equation numbers for split environments will be centered vertically on the height of the split: \|I 2 \| = T 0 ψ(t) u(a, t) − a γ(t) dθ k(θ, t) θ a c(ξ)u t (ξ, t) dξ dt ≤ C 6 f Ω S −1,0 a,− W 2 (Ω, Γ l ) \|u\| • → W e A 2 (Ω; Γ r , T ) .(65) Some text after to test the below-display spacing. 28 Sh. First Author, Sh. Second Author, and Sh. Third Author: Short Title Use of split within align: \|I 1 \| = Ω gRu dΩ ≤ C 3 Ω x a g(ξ, t) dξ 2 dΩ 1/2 × Ω u 2 x + 1 k x a cu t dξ 2 cΩ 1/2 ≤ C 4 f S −1,0 a,− W 2 (Ω, Γ l ) \|u\| • → W e A 2 (Ω; Γ r , T ) .(66)\|I 2 \| = T 0 ψ(t) u(a, t) − a γ(t) dθ k(θ, t) θ a c(ξ)u t (ξ, t) dξ dt ≤ C 6 f Ω S −1,0 a,− W 2 (Ω, Γ l ) \|u\| • → W e A 2 (Ω; Γ r , T ) .(67) Some text after to test the below-display spacing. Unnumbered align, with a number on the second split: \|I 1 \| = Ω gRu dΩ ≤ C 3 Ω x a g(ξ, t) dξ 2 dΩ 1/2 × Ω u 2 x + 1 k x a cu t dξ 2 cΩ 1/2 ≤ C 4 f S −1,0 a,− W 2 (Ω, Γ l ) \|u\| • → W e A 2 (Ω; Γ r , T ) . \|I 2 \| = T 0 ψ(t) u(a, t) − a γ(t) dθ k(θ, t) θ a c(ξ)u t (ξ, t) dξ dt ≤ C 6 f Ω S −1,0 a,− W 2 (Ω, Γ l ) \|u\| • → W e A 2 (Ω; Γ r , TNumbered version: b a b a [f (x) 2 g(y) 2 + f (y) 2 g(x) 2 ] − 2f (x)g(x)f (y)g(y) dx dy = b a g(y) 2 b a f 2 + f (y) 2 b a g 2 − 2f (y)g(y) b a f g dy (68) To test the use of \label and \ref, we refer to the number of this equation here: (68). \begin{multline}\label{eq:E} \int_aˆb\biggl\{\int_aˆb[f(x)ˆ2g(y)ˆ2+f(y)ˆ2g(x)ˆ2] -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ =\int_aˆb\biggl\{g(y)ˆ2\int_aˆbfˆ2+f(y)ˆ2 \int_aˆb gˆ2-2f(y)g(y)\int_aˆb fg\biggr\}\,dy \end{multline} Unnumbered version: b a b a [f (x) 2 g(y) 2 + f (y) 2 g(x) 2 ] − 2f (x)g(x)f (y)g(y) dx dy = b a g(y) 2 b a f 2 + f (y) 2 b a g 2 − 2f (y)g(y) b a f g dy Some text after to test the below-display spacing. \begin{multline } \int_aˆb\biggl\{\int_aˆb[f(x)ˆ2g(y)ˆ2+f(y)ˆ2g(x)ˆ2] -2f(x)g(x)f(y)g(y)\,dx\biggr\}\,dy \\ =\int_aˆb\biggl\{g(y)ˆ2\int_aˆbfˆ2+f(y)ˆ2 \int_aˆb gˆ2-2f(y)g(y)\int_aˆb fg\biggr\}\,dy \end{multline * } A.4 Align Numbered version: γ x (t) = (cos tu + sin tx, v), (72) γ y (t) = (u, cos tv + sin ty), γ z (t) = cos tu + α β sin tv, − β α sin tu + cos tv . Some text after to test the below-display spacing. \begin{align} \gamma_x(t)&=(\cos tu+\sin tx,v),\\ \gamma_y(t)&=(u,\cos tv+\sin ty),\\ \gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, -\frac\beta\alpha\sin tu+\cos tv\right). \end{align} Unnumbered version: γ x (t) = (cos tu + sin tx, v), γ y (t) = (u, cos tv + sin ty), γ z (t) = cos tu + α β sin tv, − β α sin tu + cos tv . Some text after to test the below-display spacing. \begin{align * } \gamma_x(t)&=(\cos tu+\sin tx,v),\\ \gamma_y(t)&=(u,\cos tv+\sin ty),\\ \gamma_z(t)&=\left(\cos tu+\frac\alpha\beta\sin tv, -\frac\beta\alpha\sin tu+\cos tv\right). \end{align * } A variation: x = y by (84)(75) x ′ = y ′ by (85) (76) x + x ′ = y + y ′ by Axiom 1. Some text after to test the below-display spacing. A.5 Align and split within gather When using the align environment within the gather environment, one or the other, or both, should be unnumbered (using the * form); numbering both the outer and inner environment would cause a conflict. Automatically numbered gather with split and align * : ϕ(x, z) = z − γ 10 x − γ mn x m z n = z − M r −1 x − M r −(m+n) x m z n (78) ζ 0 = (ξ 0 ) 2 , ζ 1 = ξ 0 ξ 1 , ζ 2 = (ξ 1 ) 2 , Here the split environment gets a number from the outer gather environment; numbers for individual lines of the align * are suppressed because of the star. The * -ed form of gather with the non-* -ed form of align. ϕ(x, z) = z − γ 10 x − γ mn x m z n = z − M r −1 x − M r −(m+n) x m z n ζ 0 = (ξ 0 ) 2 ,(79)ζ 1 = ξ 0 ξ 1 ,(80)ζ 2 = (ξ 1 ) 2 ,(81)V i = v i − q i v j , X i = x i − q i x j , U i = u i , for i = j;(82)V j = v j , X j = x j , U j u j + i =j q i u i .(83) Some text after to test the below-display spacing. V i = v i − q i v j , X i = x i − q i x j , U i = u i , for i = j; V j = v j , X j = x j , U j u j + i =j q i u i . Some text after to test the below-display spacing. The most common use for alignat is for things like x = y by (66) (84) x ′ = y ′ by (82) (85) x + x ′ = y + y ′ by Axiom 1. Some text after to test the below-display spacing. Required packages This class requires the standard L A T E X packages calc, sidecap, and caption2 and the A M S-L A T E X packages. 1 New documentclass options separatedheads (default): This gives the normal section, subsection and subsubsection headings with white space above and below the heading. embeddedheads: With the embeddedheads option all section headings on all levels will be typeset as runin headings without numbering like the standard LaTeX paragraph. If the numbering shall remain one has to set \setcounter{secnumdepth}{3} explicitly in the preamble of the document. autolastpage (default): The pagenumber of the last page will automatically be determined by the classfile. In the \pagespan{}{} command only the first entry should be entered. If the second entry is also given it will be ignored. The pagenumber of the lastpage will be used in the running head of the first page. In order to get correct results the document has to be run through L A T E X at least twice. noautolastpage: The value of the second argument of the \pagespan{}{} command will be printed as the last pagenumber. referee: Prints the document with a larger amount of interline whitespace. Floating objects -figures and tables We have two different table environments: table and vchtable. The same holds true for figure: figure and vchfigure. The vch-types including their captions (vchcaption) are typically leftindented by an amount equal to the indentation of mathematical formulas. For the caption layout the caption2.sty package is preloaded. Additionally the sidecap package of the L A T E X-distribution will be loaded with the option "rightcaption" by the w-art class. This package defines the SCfigure and SCTable environment for figures and tables with captions on one side. Tables The L A T E X code for Table 2 1 If these packages are not part of your installation you may download them from the nearest CTAN server. mn header will be provided by the publisher 5 Figures The L A T E X code for Fig. 1 The L A T E X code for Fig. 2 The L A T E X code for Fig. 5a SCfigure and SCtable environments The SCfigure and SCTable environment may be used as provided and described in the documentation of the sidecap package. So a typical SCfigure environment would look as follows: In order to print a figure and a table side by side the \setfloattype command is introduced. The L A T E X code for Fig. 7 and Table 4 Test of math environments Equations are always left-aligned. Therefore the option fleqn is used for the documentclass command by default. Note that fleqn does not work with unnumbered displayed equations written as $$ Ax =b $$, so please use \[ Ax=b \] or an equation* or gather* environment instead. By default the equations are consecutively numbered. This may be changed by putting the following command inside the preamble For more mathematical commands and environments please refer to the document -tma.tex and the documentation of the A M S classes. Some predefined theorem like environments Some predefined theorem like environments may be used by loading the package w-thm.sty. This package will load by itself the package amsthm.sty. So it will be easy to define new theorem-and definition-like environments. For further details refer to the documentation of the amsthm.sty package. P r o o f. This is a proof. Definition 3.3 This is a definition. Proposition 3.4 This is a proposition. Lemma 3.5 This is a lemma. Corollary 3.6 This is a corollary. Example 3.7 This is an example. Remark 3.8 This is a remark. Definition of new theorem like environments Because w-thm.sty uses amsthm.sty the definition of new theorem like environments will be done in the same manner as in the amsthm package. The definition of \theoremstyle{plain} \newtheorem{criterion}{Criterion} \theoremstyle{definition} \newtheorem{condition}[theorem]{Condition} inside the preamble of the document will give the following envirenments. Criterion 1 This is a Criterion. Condition 3.9 This is a Condition. If the name of a predefined environment has to be changed it can be done by e.g. typing \renewcommand{\definitionname}{Definitions} after the \begin{document} command. n − 1 1we have dim Osc k x (X) ≤ min{N, k+n n − 1}. Let U ⊆ X be theZariski dense open subset where the rank of the homomorphism j (X,W ) k,x Corollary 1. 3 3X is uninflected if and only if C 1 , . . . , C n are uninflected. Arguing as in the proof of Proposition 2.3 we have that the matrices M i k and M i k−1 are row equivalent for i = 1, . . . , s − 1. Now look at the matrix M of Lemma 1.1, representing j k,x . The first block of rows of M is Let n = 2 and k ≥ 2. Then f p ⊆ Φ k (X) if and only if either Note that the line bundle L ⊗ O S (−mf x ) = O S (C 0 + π (δ − mπ(x)) is not necessarily spanned, because deg(δ − mπ(x)) = m + 1 − m = 1 < e + 2 [4, Ex. 2.12(a), p. 385]. We have that dim(\|L − mf x − x\|) = dim(\|L − mf x \|) − 1 if and only if L ⊗ O S (−mf x ) is spanned at x. (15) Theorem 5. 2 2Let S ⊂ P N be a surface scroll embedded by \|V \|, where V ⊆ H 0 (S, L), L = O P N (1)\| S , and suppose that N ≥ 2k. Let x ∈ S and denote by f x the fibre of S through x. If \|V − tf x \| is very ample for every non-negative integer t ≤ k − 2, then Thus [ 6 , 6Lemma (1.4) and Lemma (1.5)] imply the following facts. The linear system \|W − f x \| is base-point free and if ϕ x : S → P M denotes the associated morphism then one of the following conditions holds: i) every fibre of ϕ x intersects any fibre f of S at a finite set, ii) (S, L) = (P 1 × P 1 , O P 1 ×P 1 (1, 1)) and W = H 0 (S, L). dim(\|V − (k + 1)x\|) = dim(\|W − 3x\|) by [6, Remark (1.2)]. Due to (16) and the assumption N ≥ 2k, S is embedded by \|W \| as a scroll in a projective space of dimension ≥ 4; hence \|W − 3x\| = \|W − 2x\| by [6, Theorem B]. Since \|W \| is very ample this says that dim(\|W − 3x\|) < dim(\|W \|) − 3. Thus, recalling the equality dim(\|V − (k + 1)x\|) + dim Osc k x (S) = N − 1, we get dim Osc k x (S) > N + 2 − dim(\|W \|). paper contains examples of various features from A M S-L A T E X. Definition 5. 1 1A polynomial time computable function f = {f k } is informationally one-way if there is no probabilistic polynomial time algorithm which (with probability of the form 1 − k −e for some e > 0) returns on input y ∈ {0, 1} k a random element of f −1 (y). ( 25 ) 25Corollary 6.4 Let (A, A ′ , A ′′ ) be a triple of arrangements. Then π(A, t) = π(A ′ , t) + tπ(A ′′ , t). Definition 6.5 Let (A, A ′ , A ′′ ) be a triple with respect to the hyperplane H ∈ A. Call H a separator if T (A) ∈ L(A ′ ). Corollary 6. 6 6Let (A, A ′ , A ′′ ) be a triple with respect to H ∈ A. ( i ) iIf H is a separator then µ(A) = −µ(A ′′ ) and hence \|µ(A)\| = \|µ(A ′′ )\| . ( ii ) iiIf H is not a separator then µ(A) = µ(A ′ ) − µ(A ′′ ) and \|µ(A)\| = \|µ(A ′ )\| + \|µ(A ′′ )\| . P r o o f. It follows from Theorem 5.2 that π(A, t) has leading term (−1) r(A) µ(A)t r(A) .The conclusion follows by comparing coefficients of the leading terms on both sides of the equation in Corollary 6.4. If H is a separator then r(A ′ ) < r(A) and there is no contribution from π(A ′ , t). Fig. 1 Q(A1) = xyz(x − z)(x + z)(y − z)(y + z) Theorem 7. 1 1Let Ω ⊂ R n be an open set, let u ∈ BV (Ω; R m ), and letT u x = y ∈ R m : y =ũ(x) + Du \|Du\| (x), z for some z ∈ R n(26) Remark 7. 2 2Let ω : ]0, +∞[ → ]0, +∞[ be a continuous function such that ω(t) → 0 as t → 0. Then lim h→0 + g(ω(h)) = L ⇔ lim h→0 + g(h) = L for any function g : ]0, +∞[ → R. for every z ∈ Q n and that L is a linear function of z. Then g is differentiable at 0.Remark 7.4 Let A : R n → R m be a linear function, and let f : R m → R be a function. Then the restriction of f to the range of A is differentiable at 0 if and only if f (A) : R n → R is differentiable at 0 and ∇( f \| Im(A) )(0)A = ∇(f (A))(0). P r o o f. We begin by showing that v ∈ BV (Ω; R k ) and imply that Dv = Dv + Jv is the Radon-Nikodým decomposition of Dv in absolutely continuous and singular part with respect to Du . By Theorem 5. ·ν) · Du y dH n−1 (y) -a.e. in R n \overleftarrow{\psi_\delta(t) E_t h}& =\underleftarrow{\psi_\delta(t) E_t h}\\ \overleftrightarrow{\psi_\delta(t) E_t h}& =\underleftrightarrow{\psi_\delta(t) E_t h} \end{align * } These all scale properly in subscript sizes: 0 . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ϕk n1 (ϕ n , ε 1 ) ϕk n2 (ϕ n , ε 2 ) \begin{alignat}{2} x& =y && \qquad \text {by (\ref{eq:A})}\label{eq:C}\\ x'& = y' && \qquad \text {by (\ref{eq:B})}\label{eq:D}\\ x+x' & = y+y' && \qquad \text {by Axiom 1.} \end{alignat} 4 Sh. First Author, Sh. Second Author, and Sh. Third Author: Short Title Fig. 1 1The usual figure environment. It may be used for figures spanning the whole page width. Fig. 3 3Two figures side by side with different numbers.Fig. 4This is the second picture. Fig. 5 5Two figures with one number. The figures are referred to as a) and b). where ¡relwidth¿ (optional) is the caption width relative to the width of the figure or table. A large value (e.g., 50) reserves the maximum width that is possible.And ¡float¿ (optional) is like the floating position parameter of the original table/figure environments. Default is [tbp]. The alignment rules are: • Figures and tables on top of a page should be top aligned with the caption. • Figures and tables on bottom of a page should be bottom aligned with the caption.The L A T E X code forFig. 6is \begin{SCfigure}[4][htb] \includegraphics[width=.3\textwidth]{empty.eps}% \caption{Caption of a SCfigure figure. These captions are always bottom aligned.} \label{fig:6} \end{SCfigure} Fig. 7 7Figure and table side by side. This is the picture. \numberwithin{equation}{section} The latex math display environment \[ . . . \] \esssup and \meas would be defined in the document preamble asess sup x∈R n \|f (x)\| \[\norm{f}_\infty= \esssup_{x\in Rˆn}\abs{f(x)}\] meas 1 {u ∈ R 1 + : f * (u) > α} = meas n {x ∈ R n : \|f (x)\| ≥ α} ∀α > 0. \[\meas_1\{u\in R_+ˆ1\colon fˆ(u)>\alpha\} =\meas_n\{x\in Rˆn\colon \abs{f(x)}\geq\alpha\} \quad \forall\alpha>0.\] \DeclareMathOperator {\esssup}{ess\,sup} \DeclareMathOperator{\meas}{meas} Some text after to test the below-display spacing.Sh. First Author, Sh. Second Author, and Sh. Third Author: Short Title Unnumbered version:\begin{equation} \begin{split} f_{h,\varepsilon}(x,y) &=\varepsilon\mathbf{E}_{x,y}\int_0ˆ{t_\varepsilon} L_{x,y_\varepsilon(\varepsilon u)}\varphi(x)\,du\\ &= h\int L_{x,z}\varphi(x)\rho_x(dz)\\ &\quad+h\biggl[\frac{1}{t_\varepsilon}\biggl(\mathbf{E}_{y} \int_0ˆ{t_\varepsilon}L_{x,yˆx(s)}\varphi(x)\,ds -t_\varepsilon\int L_{x,z}\varphi(x)\rho_x(dz)\biggr)\\ &\phantom{{=}+h\biggl[}+\frac{1}{t_\varepsilon} \biggl(\mathbf{E}_{y}\int_0ˆ{t_\varepsilon}L_{x,yˆx(s)} \varphi(x)\,ds -\mathbf{E}_{x,y}\int_0ˆ{t_\varepsilon} L_{x,y_\varepsilon(\varepsilon s)} \varphi(x)\,ds\biggr)\biggr]\\ &=h\wh{L}_x\varphi(x)+h\theta_\varepsilon(x,y), \end{split} \end{equation} 26 mn header will be provided by the publisher 29\begin{align} \begin{split}\abs{I_1} &=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ &\le C_3\left[\int_\Omega\left(\int_{a}ˆx g(\xi,t)\,d\xi\right)ˆ2d\Omega\right]ˆ{1/2}\\ &\quad\times \left[\int_\Omega\left\{uˆ2_x+\frac{1}{k} \left(\int_{a}ˆx cu_t\,d\xi\right)ˆ2\right\} c\Omega\right]ˆ{1/2}\\ &\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}ˆ{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert. \end{split}\label{eq:A}\\ \begin{split}\abs{I_2}&=\left\lvert \int_{0}ˆT \psi(t)\left\{u(a,t) -\int_{\gamma(t)}ˆa\frac{d\theta}{k(\theta,t)} \int_{a}ˆ\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ &\le C_6\left\lvert \left\lvert f\int_\Omega \left\lvert \wt{S}ˆ{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert. \end{split} \end{align} ) .Some text after to test the below-display spacing.(67 ′ ) \begin{align * } \begin{split}\abs{I_1}&=\left\lvert \int_\Omega gRu\,d\Omega\right\rvert\\ &\le C_3\left[\int_\Omega\left(\int_{a}ˆx g(\xi,t)\,d\xi\right)ˆ2d\Omega\right]ˆ{1/2}\\ &\phantom{=}\times \left[\int_\Omega\left\{uˆ2_x+\frac{1}{k} \left(\int_{a}ˆx cu_t\,d\xi\right)ˆ2\right\} c\Omega\right]ˆ{1/2}\\ &\le C_4\left\lvert \left\lvert f\left\lvert \wt{S}ˆ{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert. \end{split}\\ \begin{split}\abs{I_2}&=\left\lvert \int_{0}ˆT \psi(t)\left\{u(a,t) -\int_{\gamma(t)}ˆa\frac{d\theta}{k(\theta,t)} \int_{a}ˆ\theta c(\xi)u_t(\xi,t)\,d\xi\right\}dt\right\rvert\\ &\le C_6\left\lvert \left\lvert f\int_\Omega \left\lvert \wt{S}ˆ{-1,0}_{a,-} W_2(\Omega,\Gamma_l)\right\rvert\right\rvert \left\lvert \abs{u}\overset{\circ}\to W_2ˆ{\wt{A}} (\Omega;\Gamma_r,T)\right\rvert\right\rvert. \end{split}\tag{\theequation$'$} \end{align * } A.2 Multline \begin{align} x& =y && \text {by (\ref{eq:C})}\\ x'& = y' && \text {by (\ref{eq:D})}\\ x+x' & = y+y' && \text {by Axiom 1.} \end{align} mn header will be provided by the publisher33 Some text after to test the below-display spacing.\begin{gather * } \begin{split} \varphi(x,z) &=z-\gamma_{10}x-\gamma_{mn}xˆmzˆn\\ &=z-Mrˆ{-1}x-Mrˆ{-(m+n)}xˆmzˆn \end{split}\\[6pt] \begin{align} \zetaˆ0&=(\xiˆ0)ˆ2,\\ \zetaˆ1 &=\xiˆ0\xiˆ1,\\ \zetaˆ2 &=(\xiˆ1)ˆ2, \end{align} \end{gather * } A.6 Alignat Numbered version: \begin{alignat}{3} V_i & =v_i -q_i v_j, & \qquad X_i & = x_i -q_i x_j, & \qquad U_i & = u_i, \qquad \text{for $i\ne j$;}\label{eq:B}\\ V_j & = v_j, & \qquad X_j & = x_j, & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. \end{alignat} Unnumbered version: \begin{alignat * }3 V_i & =v_i -q_i v_j, & \qquad X_i & = x_i -q_i x_j, & \qquad U_i & = u_i, \qquad \text{for $i\ne j$;} \\ V_j & = v_j, & \qquad X_j & = x_j, & \qquad U_j & u_j + \sum_{i\ne j} q_i u_i. \end{alignat * } mn header will be provided by the publisher 35 is \begin{table}[htb] \caption{The caption inside a table environment.} \label{tab:2}\renewcommand{\arraystretch}{1.5} \begin{tabular}{lll} \hline Description 1 & Description 2 & Description \\ \hline Row 1, Col 1 & Row 1, Col 2 & Row 1, Col 3 \\ Row 2, Col 1 & Row 2, Col 2 & Row 2, Col 3 \\ \hline \end{tabular} \end{table} The L A T E X code for Table 3 (a vchtable) is \begin{vchtable}[htb] \vchcaption{The caption inside a vchtable environment.} \label{tab:3}\renewcommand{\arraystretch}{1.5} \begin{tabular}{lll} \hline Description 1 & Description 2 & Description \\ \hline Table 2 2The caption inside a table environment. Description 1 Description 2 Description Row 1, Col 1 Row 1, Col 2 Row 1, Col 3 Row 2, Col 1 Row 2, Col 2 Row 2, Col 3 Row 1, Col 1 & Row 1, Col 2 & Row 1, Col 3 \\ Row 2, Col 1 & Row 2, Col 2 & Row 2, Col 3 \\ \hline \end{tabular} \end{vchtable} Table 3 3The caption inside a vchtable environment. Description 1 Description 2 Description Row 1, Col 1 Row 1, Col 2 Row 1, Col 3 Row 2, Col 1 Row 2, Col 2 Row 2, Col 3 is \begin{figure}[htb] \includegraphics[width=\textwidth, height=2cm]{empty.eps} \caption{The usual figure environment. It ...} \label{fig:1} \end{figure} (a vchfigure) is \begin{vchfigure}[htb] \includegraphics[width=.5\textwidth]{empty.eps} \vchcaption{A vchfigure environment with a vchcaption. Figure and caption are leftindented.} \label{fig:2} \end{vchfigure} The L A T E X code for Fig. 3 and 4 is \begin{figure}[htb] \begin{minipage}[t]{.45\textwidth} \includegraphics[width=\textwidth]{empty.eps} \caption{Two figures side by side with different numbers.} \label{fig:3} Sh. First Author, Sh. Second Author, and Sh. Third Author: Short Title Fig. 2 A vchfigure environment with a vchcaption. Figure and caption are leftindented. \end{minipage} \hfil \begin{minipage}[t]{.45\textwidth} \includegraphics[width=\textwidth]{empty.eps} \caption{This is the second picture.} \label{fig:4} \end{minipage} \end{figure}6 and b is \begin{figure}[htb] \includegraphics[width=.45\textwidth]{empty.eps}˜a) \hfil \includegraphics[width=.45\textwidth]{empty.eps}˜b) \caption{Two figures with one number. The figures are referred to as a) and b).} \label{fig:5} \end{figure} \begin{SCfigure}[<relwidth>][<float>] \includegraphics[<options>]{filename.eps}% \caption{Caption of a SCfigure.} \label{fig:x1} % Give a unique label \end{SCfigure} is \begin{figure}[htb] \begin{minipage}{.45\textwidth} \includegraphics[width=\textwidth]{empty.eps} \caption{Figure and table side by side. This is the picture.} \label{fig:8} \end{minipage} Sh. First Author, Sh. Second Author, and Sh. Third Author: Short TitleFig. 6 Caption of a SCfigure figure. These captions are always bottom aligned.8 \hfil \begin{minipage}{.45\textwidth} \setfloattype{table} \caption{This is the table. ...} \label{tab:4} \renewcommand{\arraystretch}{1.5} \begin{tabular}{lll} ... \end{tabular} \end{minipage} \end{figure} Table 4 4This is the table. Picture and table are both numbered independendly. Description 1 Description 2 Description Row 1, Col 1 Row 1, Col 2 Row 1, Col 3 Row 2, Col 1 Row 2, Col 2 Row 2, Col 3 Table 5 5Some predefined theorem like environments. Theorem 3.1 This is a theorem. Theorem 3.2 Another theorem.environment caption theoremstyle thm, theorem Theorem theorem prop, proposition Proposition theorem lem, lemma Lemma theorem cor, corollary Corollary theorem axiom Axiom theorem defs, defn, definition Definition definition example Example definition rem, remark Remark definition notation Notation definition Antonio Lanteri and Raquel Mallavibarrena: Osculating properties of decomposable scrolls A multiset allows multiplicity of elements. Hence, {0, 01, 01} is prefix free as a set, but not as a multiset. Sh. First Author, Sh. Second Author, and Sh. Third Author: Short Title Acknowledgements During the preparation of this paper the first author has been supported by the MUR of the Italian Government in the framework of the PRIN "Geometry on Algebraic Varieties", and the second author by the projects BFM2003-03917/MATE (Spanish Ministry of Education) and Santander/UCM PR27/05-13876. The first author would also like to thank the GNSAGA-INDAM and Azione Integrata Italia-Spagna IT200 for support received at an early stage of this research. Both authors are grateful to the University of Milan for financial support.Acknowledgements An acknowledgement may be placed at the end of the article.The style of the following references should be used in all documents. [1] xxx.ReferencesAcknowledgements An acknowledgement may be placed at the end of the article.The style of the following references should be used in all documents.This is an example input file. Comparing it with the output it generates can show you how to produce a simple document of your own.IntroductionThe class file w-art.cls represents an adaptation of the L A T E X 2 ε -standard class file article.cls and the A M S class file amsart.cls with the size option 10pt to the specific requirements of journal production at WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. It can be used through the L A T E X-command\documentclass[<abbr>,fleqn, other options]{w-art}where <abbr> is an abbreviation of the journal name. Greek symbolsw-greek.styTable 3: Boldface variants of slanted greek letters α α α \pmb{\upalpha} θ θ θ \pmb{\uptheta} ο ο ο \pmb{\upo} τ τ τ \pmb{\uptau} β β β \pmb{\upbeta} ϑ ϑ ϑ \pmb{\upvartheta} π π π \pmb{\uppi} υ υ υ \pmb{\upupsilon} γ γ γ \pmb{\upgamma} ι ι ι \pmb{\upiota} ϖ ϖ ϖ \pmb{\upvarpi} φ φ φ \pmb{\upphi} δ δ δ \pmb{\updelta} κ κ κ \pmb{\upkappa} ρ ρ ρ \pmb{\uprho} ϕ ϕ ϕ \pmb{\upvarphi} ε ε ε \pmb{\upepsilon} λ λ λ \pmb{\uplambda} ̺ ̺ ̺ \pmb{\varrho} χ χ χ \pmb{\upchi} ε ε ε \pmb{\varepsilon} µ µ µ \pmb{\upmu} σ σ σ \pmb{\upsigma} ψ ψ ψ \pmb{\uppsi} ζ ζ ζ \pmb{\upzeta} ν ν ν \pmb{\upnu} ς ς ς \pmb{\upvarsigma} ω ω ω \pmb{\upomega} η η η \pmb{\upeta} ξ ξ ξ \pmb{\upxi} E Arbarello, M Cornalba, Ph, J Griffiths, Harris, Geometry of Algebraic Curves. New YorkSpringerIE. Arbarello, M. Cornalba, Ph. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. I (Springer, New York, 1985). On the osculatory behaviour of higher dimensional projective varieties. E Ballico, C Fontanari, Collect. Math. 55E. Ballico and C. Fontanari, On the osculatory behaviour of higher dimensional projective varieties, Collect. Math. 55, 229-236 (2004). A characterization of balanced rational normal surface scrolls in terms of their osculating spaces. E Ballico, R Piene, H Tai, Math. Scand. IIE. Ballico, R. Piene, and H. Tai, A characterization of balanced rational normal surface scrolls in terms of their osculating spaces, II, Math. Scand. 70, 204-206 (1992). R Hartshorne, Algebraic Geometry. New YorkSpringerR. Hartshorne, Algebraic Geometry (Springer, New York, 1977). Plane projections of a smooth space curve, Parameter Spaces. T Johnsen, Banach Center Publications36T. Johnsen, Plane projections of a smooth space curve, Parameter Spaces, Banach Center Publications, vol 36 (1996) pp. 89-110 . On the osculatory behavior of surface scrolls. A Lanteri, Le Matematiche (Catania). 55A. Lanteri, On the osculatory behavior of surface scrolls, Le Matematiche (Catania) 55, 447-458 (2000). Duality for elliptic normal surface scrolls, Enumerative Algebraic Geometry. R Mallavibarrena, R Piene, Contemporary Mathematics. S. L. Kleiman and A. Thorup123Amer. Math. SocR. Mallavibarrena and R. Piene, Duality for elliptic normal surface scrolls, Enumerative Algebraic Geometry, edited by S. L. Kleiman and A. Thorup, Contemporary Mathematics, vol. 123, (Amer. Math. Soc., 1991), pp. 149-160. Numerical characters of a curve in projective n-space, Real and Complex Singularities. R Piene, P. Holm (Sijthoff and NoordhoffGroningenR. Piene, Numerical characters of a curve in projective n-space, Real and Complex Singularities, edited by P. Holm (Sijthoff and Noordhoff, Groningen, 1978), pp. 475-496. Duality for rational normal scrolls. R Piene, G Sacchiero, Comm. Algebra. 12R. Piene and G. Sacchiero, Duality for rational normal scrolls, Comm. Algebra 12, 1041-1066 (1984). A characterization of balanced rational normal scrolls in terms of their osculating spaces, Enumerative Geometry. R Piene, H S Tai, Proc. Sitges. SitgesSpringer1436edited by S. Xambo-DescampsR. Piene and H. S. Tai, A characterization of balanced rational normal scrolls in terms of their osculating spaces, Enumerative Geometry, Proc. Sitges, 1987 edited by S. Xambo-Descamps, Lecture Notes in Math., vol 1436, (Springer, 1990), pp. 215-224. The osculatory behavior of surfaces in P 5. T Shifrin, Pacif. J. Math. 123T. Shifrin, The osculatory behavior of surfaces in P 5 , Pacif. J. Math. 123, 227-256 (1986). Gather Numbered version with \notag on the second line: D(a, r) ≡ {z ∈ C : \|z − a\| < r}. A.3 Gather Numbered version with \notag on the second line: D(a, r) ≡ {z ∈ C : \|z − a\| < r}, (69) = ≡ {z ∈ C : ℑz, ℑa, \|z − a\| < r}, c(e, θ, r) ≡ {(x, y) ∈ C : \|x − e\| < y tan θ, 0 < y < r}. seg(a, r) ≡ {z ∈ C : ℑz = ℑa, \|z − a\| < r}, c(e, θ, r) ≡ {(x, y) ∈ C : \|x − e\| < y tan θ, 0 < y < r}, (70) . C(e, Θ, R) ≡ E∈e C, e, θ, r)C(E, θ, r) ≡ e∈E c(e, θ, r). \\ \seg(a,r)\equiv\{z\in\mathbf{C}\colon \Im z= \Im a,\ \abs{z-a}<r\},\notag\\ c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} \colon \abs{x-e}<y\tan\theta. \begin{gather} D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\}. \ 0<y<r\},\\ C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r)\begin{gather} D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ \seg(a,r)\equiv\{z\in\mathbf{C}\colon \Im z= \Im a,\ \abs{z-a}<r\},\notag\\ c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). \end{gather} Unnumbered version. \end{gather} Unnumbered version. D(a, R) ≡ {z ∈ C, \|z − a\| < r}, seg(a, r) ≡ {z ∈ C : ℑz = ℑa, \|z − a\| < r}, c(e, θ, r) ≡ {(x, y) ∈ C : \|x − e\| <. y tan θ, 0 < y < r}, C(E, θ, r) ≡ e∈E c(e, θ, r)D(a, r) ≡ {z ∈ C : \|z − a\| < r}, seg(a, r) ≡ {z ∈ C : ℑz = ℑa, \|z − a\| < r}, c(e, θ, r) ≡ {(x, y) ∈ C : \|x − e\| < y tan θ, 0 < y < r}, C(E, θ, r) ≡ e∈E c(e, θ, r). D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ \seg (a,r)\equiv\{z\in\mathbf{C}\colon \Im z= \Im a,\ \abs{z-a}<r\},\\ c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} \colon \abs{x-e}<y\tan\theta,\ 0<y<r\}. D(a,r)\equiv\{z\in\mathbf{C}\colon \abs{z-a}<r\},\\ \seg (a,r)\equiv\{z\in\mathbf{C}\colon \Im z= \Im a,\ \abs{z-a}<r\},\\ c(e,\theta,r)\equiv\{(x,y)\in\mathbf{C} \colon \abs{x-e}<y\tan\theta,\ 0<y<r\},\\ C(e \theta, r)\equiv\bigcup_{e\in E}c(e,\theta,r). C(E,\theta,r)\equiv\bigcup_{e\in E}c(e,\theta,r). New directions in cryptography. W Diffie, E Hellman, IEEE Transactions on Information Theory. 225W. Diffie and E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory 22 (1976), no. 5, 644-654. Cichon's diagram, 1983/1984, presented at the Séminaire Initiationà l'Analyse. D H Fremlin, ; G Choquet, M Rogalski, J Raymond, Parisat the Université Pierre et Marie Curie23e annéeD. H. Fremlin, Cichon's diagram, 1983/1984, presented at the Séminaire Initiationà l'Analyse, G. Choquet, M. Rogal- ski, J. Saint Raymond, at the Université Pierre et Marie Curie, Paris, 23e année. The enumeration of directed closed Euler trails and directed Hamiltonian circuits by Langrangian methods. I P Goulden, D M Jackson, European J. Combin. 2I. P. Goulden and D. M. Jackson, The enumeration of directed closed Euler trails and directed Hamiltonian circuits by Langrangian methods, European J. Combin. 2 (1981), 131-212. F Harary, E M Palmer, Graphical enumeration. Academic PressF. Harary and E. M. Palmer, Graphical enumeration, Academic Press, 1973. Pseudo-random generation from one-way functions. R Impagliazzo, L Levin, M Luby, Proc. 21st STOC. 21st STOCNew YorkACMR. Impagliazzo, L. Levin, and M. Luby, Pseudo-random generation from one-way functions, Proc. 21st STOC (1989), ACM, New York, pp. 12-24. A new continuation method for complementarity problems with uniform pfunctions. M Kojima, S Mizuno, A Yoshise, Dept. of Information Sciences. Tech. Report B-194Tokyo Inst. of TechnologyM. Kojima, S. Mizuno, and A. Yoshise, A new continuation method for complementarity problems with uniform p- functions, Tech. Report B-194, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information Sciences. A polynomial-time algorithm for a class of linear complementarity problems. Dept. of Information Sciences. Tech. Report B-193Tokyo Inst. of Technology, A polynomial-time algorithm for a class of linear complementarity problems, Tech. Report B-193, Tokyo Inst. of Technology, Tokyo, 1987, Dept. of Information Sciences. On operator and formal sum methods for graph enumeration problems. C J Liu, Yutze Chow, SIAM J. Algorithms Discrete Methods. 5C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algorithms Discrete Methods 5 (1984), 384-438. A survey of matrix theory and matrix inequalities. M Marcus, H Minc, Complementary Series in Math. 14M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Complementary Series in Math. 14 (1964), 21-48. Practical polynomial time algorithms for linear complementarity problems. S Mizuno, A Yoshise, T Kikuchi, 13Tokyo Inst. of Technology. Dept. of Industrial Engineering and ManagementTech. ReportS. Mizuno, A. Yoshise, and T. Kikuchi, Practical polynomial time algorithms for linear complementarity problems, Tech. Report 13, Tokyo Inst. of Technology, Tokyo, April 1988, Dept. of Industrial Engineering and Management. Interior path following primal-dual algorithms, part II: Quadratic programming. R D Monteiro, I Adler, Working paper, Dept. of Industrial Engineering and Operations ResearchR. D. Monteiro and I. Adler, Interior path following primal-dual algorithms, part II: Quadratic programming, August 1987, Working paper, Dept. of Industrial Engineering and Operations Research. Singular integrals and differentiability properties of functions. E M Stein, Princeton Univ. PressPrinceton, N.J.E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. Interior algorithms for linear, quadratic and linearly constrained convex programming. Y Ye, Stanford Univ., Palo Alto, Calif. ; Dept. of Engineering-Economic SystemsPh.D. thesis. unpublished. Long title L. First Author * 1 , L. Second Author * * 1,2 , and L. Third Author * * * 2Y. 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[ "Document-aware Positional Encoding and Linguistic-guided Encoding for Abstractive Multi-document Summarization", "Document-aware Positional Encoding and Linguistic-guided Encoding for Abstractive Multi-document Summarization", "Document-aware Positional Encoding and Linguistic-guided Encoding for Abstractive Multi-document Summarization", "Document-aware Positional Encoding and Linguistic-guided Encoding for Abstractive Multi-document Summarization" ]	[ "Congbo Ma [email protected] ", "Au ", "Wei Emma Zhang [email protected] ", "Au ", "Pitawelayalage Dasun [email protected] ", "Dileepa Pitawela ", "Au ", "Yutong Qu [email protected] ", "Au ", "Haojie Zhuang [email protected] ", "Au ", "Hu Wang [email protected] ", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "Congbo Ma [email protected] ", "Au ", "Wei Emma Zhang [email protected] ", "Au ", "Pitawelayalage Dasun [email protected] ", "Dileepa Pitawela ", "Au ", "Yutong Qu [email protected] ", "Au ", "Haojie Zhuang [email protected] ", "Au ", "Hu Wang [email protected] ", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n", "\nThe University of Adelaide Adelaide\nAustralia\n" ]	[ "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia", "The University of Adelaide Adelaide\nAustralia" ]	[]	One key challenge in multi-document summarization is to capture the relations among input documents that distinguish between single document summarization (SDS) and multi-document summarization (MDS). Few existing MDS works address this issue. One effective way is to encode document positional information to assist models in capturing cross-document relations. However, existing MDS models, such as Transformerbased models, only consider token-level positional information. Moreover, these models fail to capture sentences' linguistic structure, which inevitably causes confusions in the generated summaries. Therefore, in this paper, we propose documentaware positional encoding and linguistic-guided encoding that can be fused with Transformer architecture for MDS. For document-aware positional encoding, we introduce a general protocol to guide the selection of document encoding functions. For linguistic-guided encoding, we propose to embed syntactic dependency relations into the dependency relation mask with a simple but effective non-linear encoding learner for feature learning. Extensive experiments show the proposed model can generate summaries with high quality.	10.1109/ijcnn55064.2022.9892026	[ "https://export.arxiv.org/pdf/2209.05929v1.pdf" ]	252,211,797	2209.05929	8de285b767bd733a09fde193b2fe4122395992e4	Document-aware Positional Encoding and Linguistic-guided Encoding for Abstractive Multi-document Summarization Congbo Ma [email protected] Au Wei Emma Zhang [email protected] Au Pitawelayalage Dasun [email protected] Dileepa Pitawela Au Yutong Qu [email protected] Au Haojie Zhuang [email protected] Au Hu Wang [email protected] The University of Adelaide Adelaide Australia The University of Adelaide Adelaide Australia The University of Adelaide Adelaide Australia The University of Adelaide Adelaide Australia The University of Adelaide Adelaide Australia The University of Adelaide Adelaide Australia Document-aware Positional Encoding and Linguistic-guided Encoding for Abstractive Multi-document Summarization Index Terms-Multi-document summarizationDeep neural networkDocument positionLinguistic knowledge One key challenge in multi-document summarization is to capture the relations among input documents that distinguish between single document summarization (SDS) and multi-document summarization (MDS). Few existing MDS works address this issue. One effective way is to encode document positional information to assist models in capturing cross-document relations. However, existing MDS models, such as Transformerbased models, only consider token-level positional information. Moreover, these models fail to capture sentences' linguistic structure, which inevitably causes confusions in the generated summaries. Therefore, in this paper, we propose documentaware positional encoding and linguistic-guided encoding that can be fused with Transformer architecture for MDS. For document-aware positional encoding, we introduce a general protocol to guide the selection of document encoding functions. For linguistic-guided encoding, we propose to embed syntactic dependency relations into the dependency relation mask with a simple but effective non-linear encoding learner for feature learning. Extensive experiments show the proposed model can generate summaries with high quality. Abstract-One key challenge in multi-document summarization is to capture the relations among input documents that distinguish between single document summarization (SDS) and multi-document summarization (MDS). Few existing MDS works address this issue. One effective way is to encode document positional information to assist models in capturing cross-document relations. However, existing MDS models, such as Transformerbased models, only consider token-level positional information. Moreover, these models fail to capture sentences' linguistic structure, which inevitably causes confusions in the generated summaries. Therefore, in this paper, we propose documentaware positional encoding and linguistic-guided encoding that can be fused with Transformer architecture for MDS. For document-aware positional encoding, we introduce a general protocol to guide the selection of document encoding functions. For linguistic-guided encoding, we propose to embed syntactic dependency relations into the dependency relation mask with a simple but effective non-linear encoding learner for feature learning. Extensive experiments show the proposed model can generate summaries with high quality. Index Terms-Multi-document summarization, Deep neural network, Document position, Linguistic knowledge I. INTRODUCTION Multi-document summarization (MDS) aims at generating fluent and informative summaries from a set of topic-related documents. Similar to single document summarization (SDS), the process of summary generation in MDS can be divided into two types: extractive summarization and abstractive summarization. Abstractive summarization requires the model to have profound natural language understanding, based on which the generated summaries could be formed by new words, phrases, or sentences that do not exist in the original documents [1]; while extractive summarization models select existing sentences from the original documents. Such that, in principle, summaries generated by abstractive summarization models can have higher readability, as well as conciseness, than extractive summarization models [2]. Recent years have witnessed an increasing number of neural network models applied in MDS due to the rapid improvement of computational power [3]. Transformer is a popular one among them. It is based on a self-attention mechanism and has natural advantages for capturing cross-token relations. Liu et al. [4] proposed a Flat Transformer-based decoderonly sequence transduction to generate the Wikipedia articles. Besides Flat Transformer, Hierarchical Transformerbased models [5]- [7] utilize multiple encoders to embed the hierarchical relations among the source documents. To inject positional information for textual sequences, token-level positional encoding has been considered before data being pumped into encoders and decoders of these Transformerbased models. However, token-level positional encoding is not sufficient to capture document-level positional information. Missing document-level positional encoding significantly prevents models from detecting cross-document relationships. In addition to the aforementioned issue, there is another issue that obstructs the performance of Transformer-based MDS models: Transformer can explicitly compute relations between every token via a self-attention mechanism. However, this mechanism lacks explicit syntactic support that will cause the content irrelevance and deviation problem for the generated summaries [8]. Dependency parsing represents the grammatical structure between each pair of words and it has been widely used in a variety of natural language processing tasks to help the model retain the syntactic structure [9], [10]. When it comes to document summarization, according to Hirao et al. [11], no matter how the word order changes from the source documents to generated summaries, the dependency structures will keep consistent in most cases. Incorporating dependency structures into summarization models is crucial to retain the correct logics from source documents. To solve the above-mentioned two problems, in this paper, we propose an encoding mechanism combining documentaware positional encoding and linguistic-guided encoding for abstractive MDS. Figure 1 illustrates a general overview of the proposed method. We construct a document-aware positional encoding protocol to guide the encoding process and the selection of document-level positional encoding functions. Like most of the Transformer-based models, we add documentaware positional encoding with the input token embedding at the bottoms of the encoder stacks. Furthermore, a novel linguistic-guided encoding is introduced to incorporate the dependency relation mask, containing pair-wise dependency relations, into the Transformer-based multi-head attention. The proposed linguistic-guided encoding method allows the model to better understand the relationship between each pair of words, and retains the correct dependency structure as well as grammatical associations when generating the summaries. We highlight our contributions as follows: • We propose an effective and informative encoding mechanism to encode the multi-document positional information and the dependency structure for MDS tasks. We further propose a general protocol to guide the selection of document encoding functions. • We compare the proposed model with multiple competitive baselines. The results demonstrate that models equipped with the proposed encoding mechanism receive superior performances over the comparing models. We conduct an ablation study to assess the contribution of different encoding methods. • Extensive analysis on various settings of the documentaware positional encoding and linguistic-guided encoding are provided. These results help researchers understand the intuitiveness of the proposed model and could serve as an informative reference to the MDS research community. II. RELATED WORK Abstractive Multi-document Summarization. Abstractive MDS has been an active area of the natural language processing community in recent years. Yang et al. [12] augmented the Transformer architecture [4] by encoding multiple documents hierarchically. Zhang et al. [13] tried to tackle the MDS problem by utilizing a hierarchical encoder-decoder framework [14] with a PageRank [15] based attention module. Fabbri et al. [16] introduced an end-to-end Hierarchical maximal margin relevance-Attention Pointer-generator (Hi-MAP) model, which expanded the existing pointer-generator network into a hierarchical fashion. It incorporates the hidden-state-based maximal margin relevance module with sentence-level representations to generate abstractive summaries. Li et al. [6] followed the encoder-decoder architecture with graph representations to gather rich cross-document relationships while encoding process. However, these models do not take dependency relations into account, which can assist summarization models in fetching the grammatical structure of a sentence within source documents. Song et al. [17] developed a shift-reduce dependency parsing system to guide summary generation by "SHIFT" operation and pairwise dependency arc addition by "REDUCE" operation. It transformed source sequences into summary sequences in the linearized parse tree form. Jin et al. [8] constructed semantic dependency graphs by utilizing the off-the-shelf semantic dependency parser [18]. Nevertheless, little consideration paid to syntactic dependency information in the MDS area. In this paper, we propose a linguistic-guided encoding to incorporate dependency knowledge with a strong dependency learner for better attention. Positional Encoding for Transformer. Due to considering each token separately, Transformer does not keep internal sequential or order information. However, in natural language, tokens with incorrect order result in different meanings or incorrect grammar. Sequential information is crucial for language models. Vaswani et al. [19] appended position encoding, representing the relative or absolute token position information, to the token embeddings. Sukhbaatar et al. [20] embraced the relative position encoding assigned by a predefined piecewise function. Wang et al. [21] extended word embedding vectors to continuous word functions with absolute global positions as independent variables to model the smooth shift among sequential positions of words. However, these models only consider token-level positional encoding in the Transformer architecture but fail to be aware of documentlevel positional information, which will cause models to fail to identify different source documents. Therefore, in this paper, we propose to encode document-level information into positional encoding. It enables the model to perceive the token with explicit document positions for easier model optimization. Moreover, we further propose a general protocol to guide the selection of document encoding functions. III. METHODOLOGY In this work, we incorporate two types of encodings into Transformer-based abstractive MDS model: i) documentaware positional encoding considers document positional information; ii) linguistic-guided encoding incorporate dependency information into summarization process. The encodings will be introduced based on the following problem formulation and notations: given a set of q documents D = d 1 , d 2 , ..., d q on the same topic, the task of MDS is to generate a concise and informative summary Sum distilling knowledge from D. Let t k i denotes the i-th token in the k-th document d k (k = 1, 2, ..., q) in D. e k i represents the token embedding assigned to t k i by the Transformer model. A. Document-aware Positional Encoding For the token t k i from the source documents, the token positional encoding P os k tokeni and document positional encoding P os k doci can be represented as: P os k token i = f token (i) P os k doc i = f doc (k)(1) where f token and f doc are encoding functions for token positional encoding and document positional encoding respectively. Different from the token positional embedding that considers the order of tokens, document positional embedding does not require the document order information as the order does not affect the MDS tasks. In order to find a proper f doc , we design a protocol with three considerations: (1) The encoding of each document should be unique. The purpose is to distinguish the documents and trace the source document for the tokens. We adopt the sin function as document positional encoding function. Many other functions satisfying the document positional encoding protocol. We discuss their performances in Section V-B. The final positional encoding P os k i for token t k i combines the token-level and document-level positional encoding by a linear combination: P os k i = αP os k doc i + P os k token i(2) where P os k doc i = Stack dim token (P os k doc i ) where Stack dim token (·) is to repeat P os k doci for dim token times to have the same dimension with P os k tokeni . Then the overall input representations to the Transformer-based model are obtained by simply adding the token embedding and its corresponding positional encoding: Figure 2 illustrates the process of proposed document-aware positional encoding. Given a set of documents (containing q documents), the document positional encoding combines with token positional encoding to form the document-aware positional encoding, which later serves as part of the input to the encoder of Transformer. E k i = P os k i + e k i(4) B. Linguistic-guided Encoding In order to retain the dependency structures and distill the generated summaries in a better manner, we propose a novel linguistic-guided encoding method to encode the informative dependency relations into the Transformer-based model. Specifically, for each sentence of source documents, an external dependency parser [22] is adopted to extract the grammatical structure containing dependency relations between head words and corresponding dependent words. We construct the three-order tensor Dep to place the dependency relations (the tokens discussed below are all from the same document, so the superscript k is omitted). The specific dependency relations dep ij ∈ Dep can be defined as below: dep ij = v rel ti tj 0 ti tj(5) where v rel ∈ R N * 1 is the one-hot vector of dependency relations between token t i and t j . There are a variety of dependency relations between paired words in dependency parsing and N represents the total number of these dependencies. t i t j indicates there is a dependency relation for t i and t j , while t i t j represents no existing dependency between the two tokens. To encode these dependency relations into the Transformer-based models, we first transfer the dependency tensor into a dependency encoding weight through a two-layers encoding function: mij = F depEnc (depij)(6) where F depEnc contains two linear transformations and one LeakReLU non-linear mapping in between: F depEnc (x) = Linear • LeakyReLU • Linear(x)(7) where • represents the concatenation of multiple subfunctions. In general, we discover that the complexity of designing the encoding function for dependency information is crucial for model optimization. A too-naive encoding function may lack the ability to embed the information well enough; while an encoding function with overly strong fitting abilities results in a slow training process and may cause failures in transforming the dependencies in an easy-optimizable manner. Figure 3 shows the process of the transformation from dependency relation tensor Dep to dependency relation mask M ij . Each fiber of the dependency relation tensor represents a one-hot vector for a specific dependency relation. Only the corresponding element of the one-hot vector has the value (highlight in red). The dependency relation weight m ij is joined with the multi-head attention from source documents to generate syntactic-rich features in the following manner: M HAtt(ti, tj, mij) = j Aij · Vj(8) where Aij = Mij Aij + Aij (9) Aij = sof tmax Qi T Kj √ dim (10) Mij = Stack h(mij)(11) where Q i , K j , V j ∈ R h * d k * 1 are corresponding key, query, value for token t i and t j . dim is the dimension of the key, query and value. h is the number of attention heads. Both dim and h are fixed values that we followed the original settings in Transformer. In order to fuse dependency relation weight m ij into dependency relation mask M ij , function Stack h(·) is to repeat p ij on the dimension of head to have the same size with Att ij ∈ R h * 1 * 1 . denotes the element-wise Hadamard product. Then two layer-normalization operations are applied to get the output vector of the current encoder or decoder layer for the token t i . IV. EXPERIMENTAL SETTINGS A. Datasets Multi-News Dataset [16] is a large-scale English dataset containing various topics in news domain. It includes 56,216 document-summary pairs and it is further scattered with the ratio 8:1:1 for training, validation, and test respectively. Each document set contains 2 to 10 documents with a total length of 2103.49 words. The average length of the golden summaries is 263.66. Multi-XScience Dataset [23] is a large-scale English dataset and it contains 40,528 document-summary pairs collected from scientific articles. The task of the Multi-XScience dataset is to generate the related work section of a target scientific paper based on the abstract of the same target paper and the abstracts of the articles it refers to. The dataset contains 30,369 training, 5,066 validation and 5,093 testing data. Samples have an average input length of 778 tokens and an average length of 116 tokens on the summary. B. Implementation Details To have a fair comparison, we keep all the experimental settings consistent throughout all experiments. In our Transformer-based model, eight encoder layers and decoder layers are adopted. The Biaffine parser [22] is used for generating dependency relations among the source documents. Our model adopts 45 dependency relations. We use Adam optimizer (β1=0.9 and β2=0.998) for model parameter optimization. The initial learning rate of the model is set to 1 × 10 −3 and 0.1 dropout rate is set for both the encoder and decoder. The trade-off hyper-parameter α is set to 0.1. In the training phase, the first 8 × 10 3 steps are trained for warming up and the models are trained with a multi-step learning rate reduction strategy. In the experiments, the model accumulates gradients and updates once every four iterations. The minimum and maximum lengths of the generated summaries are set to 200 and 300 words for the Multi-News dataset, while 110 and 300 words for the Multi-XScience dataset. C. Baselines and Metrics We compare our proposed method with the following strong baselines: LexRank [24] computes textual unit salience based on the eigenvector centrality algorithm using heuristic features in the similarity graph-based sentence representations. TextRank [25] leverages the graph-based ranking formula, deciding on the importance of a text unit representative within a graph built for information extraction. SummPip [26] constructs sentence graphs by incorporating both linguistic knowledge and deep neural representations. Maximal Marginal Relevance (MMR) [27] combines query relevance and information novelty from source documents, benefiting summarization in reducing redundancy while remaining the most salient information. Bidirectional recurrent neural network (BRNN) superimposes two RNNs of opposing directions on the same output according to RNN states. Transformer [19] follows an encoder-decoder structure based on attention mechanism, which has been extensively utilized in a wide range of natural language processing tasks 1 . CopyTransformer restricts abstractive summarizer to copy tokens from source documents. Pointer-Generator (PG) [28] equips with the coverage mechanism between the pointer network and the standard sequence-to-sequence attention model. Hierarchical MMR-Attention Pointer-generator (Hi-MAP) model [16] integrates sentence representatives with hidden-state-based MMR into a standard pointer-generator network, an end-to-end model for abstract summarization. Hierarchical Transformer (HT) [5] captures relationships across multiple paragraphs via the hierarchical Transformer encoders and flat Transformer decoders 2 . D. Automatic Evaluation Metrics We evaluate the models by using ROUGE scores [29] and BERTScore [30]. Unigram and bigram overlap (ROUGE-1 and ROUGE-2 scores) are adopted to indicate the literal quality of generated summaries. ROUGE-SU score is a unigrambased co-occurrence statistic, bringing out the soft skip bigram by computing both the skip-bigram and unigram. ROUGE F1 scores are considered in our work 3 . BERTScore is an automatic language evaluation metric for text generation based on contextual token embeddings of the pre-trained BERT [31]. We mark ROUGE-1, ROUGE-2, ROUGE-SU and BERTScore as "R-1, "R-2", "R-SU" and "BS" in this paper. V. EXPERIMENTAL RESULTS A. Overall Performance In this section, we compare our proposed model with several strong baselines and list the comparison results in Table I (Multi-News) and Table II (Multi-XScience). The results of our proposed model on the Multi-News dataset show the best overall results on both ROUGE scores and BERTScore. To give a fair comparison, we rerun all the baseline models. 3 The scores are computed with ROUGE-1.5.5 script with option "-c 95 -2 -1 -U -r 1000 -n 4 -w 1.2 -a -m" It is observed that our model performs particularly well on R-SU than other models. It gains 1.06 improvement to the second best, Hi-MAP. Given that R-SU takes more skipbigram plus unigram-based co-occurrence statistics into account, it contains additional comprehensive information to evaluate the models. The BERTScore on different models shows relative marginal differences. However, our proposed model still achieves the best among all the evaluate models, which indicates our proposed model can generate high-quality summaries in a semantic level. We also evaluate our proposed models based on the Multi-XScience datasets. Comparing the Transformer baseline models and our model with documentaware positional encoding and linguistic guided encoding, we observe that these two encodings help to improve the performance by 2.59 on R-1, 1.07 on R-2 and 1.36 on R-SU. The results on the Multi-XScience dataset show that our model performs better than most of the models. Our proposed model does not achieve the best results on all evaluation metrics because the proposed model is based on the Transformer models which are dataset sensitive. This means the Transformer-basd models do not always work well on all the MDS datasets. This phenomenon can also be found in the paper [7], [26] and [2]. In these paper, the Transformerbased model (CopyTransformer) shows poor results on DUC-2004 dataset 4 although it works well on the Multi-News dataset. A potential reason is Multi-XScience and DUC-2004 datasets have higher novel n-grams score than Multi-News dataset [16], [23]. For example, paper [23] reported that the proportion novel of unigrams/bigrams/trigrams/4grams in the golden summaries of the B. Ablation Study To better understand the contribution of document-aware positional encoding and linguistic-guided encoding techniques to overall model performance individually, we conduct an ablation study on the proposed model on both Multi-News and Multi-XScience datasets. methods perform considerably better than the model without them. This is due to (1) document-aware positional encoding has the capability of capturing cross-document information in MDS; (2) with linguistic-guided encoding, dependency relations within the source documents are well preserved, enabling the summarization model to effectively learn a much more faithful syntactic structure than that working on the model without it. C. Encoding Strategies In addition to the model performance evaluation, we report our findings on different encoding functions and the ways to incorporate the encoding. (1) Document-aware Positional Encoding Strategies. We evaluate the contribution of different document positional encoding functions. All these functions satisfy the proposed protocol described in Section III-A. The experience results are shown in the upper part of Table IV. sin function helps the MDS model achieve the best ROUGE score and the combination of sin and cos produce similar results. However, cos function greatly reduces the model performance. The reason could be related to the document number in a document set of Multi-News dataset. Most of the document sets contain two documents in the Multi-News dataset. When applying cos on two documents, the value differences for the two encodings is smaller than what the sin function provides, which means cos has less distinguishing ability than sin. This may result in lower model performance for MDS tasks. Additionally, we also try to adjust α in Equation (2). Results are shown in the lower part of Table IV. We test the model performance on validation set when α = 0.1, 0.5, 1 and observe model perform best when α = 0.1. Therefore, we fix this hyper-parameter to 0.1 and report the final results on the test set. (2) Document-aware Positional Encoding Protocol. To verify the proposed three considerations of document encoding functions, we select some other functions except sin and cos, and the functions are not satisfy the conditions proposed in III-A for experiments. We also randomly assign values to document positional encoding to verify the effectiveness of our chosen function. The results are shown in Table V. We observe that the performance are not well when (1) the document The sequence of dependency relation list in the first methods is constructed according to the sequence of the occurrence of dependency relations in the source documents. We select the top-8 dependents from the official core dependents of clausal predicates 5 to build the relation lists for "Arithmetic sequence (core)". "Arithmetic sequence (root)" is to assign the largest weight to the root word since the dependency relation "root" is proven to be the most important token in the syntax dependency tree [10]. "One-hot (one layer)" means the one-hot representation of dependencies with only one linear transformation between the dependency relation tensor and the dependency relation mask. The "One-hot (one layer)" model performs substantially poorer than the one-hot encoding model with non-linear function F depEnc . It is because non-linearity enlarges the learning capability of encoding functions significantly. The"One-hot (F depEnc )" represents our final model. The F depEnc function can outperform all arithmetic sequence models since it delegates the construction of dependency relations to a non-linear learner. It enables the model to learn the importance of gradient descent directly. From another point of view, besides the addition of the linguistic-guided encoding on keys and queries within self-attention, we also tried to add the encoding on values. However, model performance dropped greatly. We hypothesize the reason is that keys and queries are adopted to calculate attention, but values are the final receptors of attention. Small changes in values will have a large influence on the model optimization process. D. Human Evaluation Apart from automatic evaluation, we conduct a human evaluation to assess the quality of the generated summaries on three aspects: text fluency checks whether the summary is natural, well-formed, and both syntactically and semantically correct; conciseness assesses whether the summary is concise and without repeated or useless information; informativeness examines whether the summary keeps the salient information from the source documents. We randomly sample 10 examples from the Multi-News dataset [16]. Three experienced researcher are invited to score summaries (from 4 models) on the above aspects. The score range is 1-5 (1 means very bad; 5 means very good). The final scores for each model are averaged across different examples and raters. The results are listed in Table VII. The text fluency score of our model is 3.13, which is higher than 2.50 of Transformer, 2.60 of CopyTransformer, and 3.07 of Hi-Map, which means the summaries generated by our model are more natural and wellformed. In terms of the score of informativeness, our model achieves 3.10 and is higher than the second-best model (Hi-Map) by 0.23, indicating our model is better at capturing the most important information from different sources. Moreover, the generated summaries by our model are more concise and better at reducing redundant information, which could be concluded by the conciseness score. E. Case Study Table VIII presents the generated summaries from four models: Transformer, CopyTransformer, Hi-Map, and our models. In this example, the Transformer model only captures "james savage has been swimming off the san francisco bay area" (in red) but takes the age wrong. It should be 9 in fact. Besides, Transformer model also generates something that are not supported in the source document (in orange). For the CopyTransformer, the salient information (in green) is in the generated summary. However, this model also outputs unsupported text (in orange). The Hi-Map model misses some key information (e.g. the red highlight in the source document). In contrast, the summary generated by our proposed model keeps the significant information and shows content consistent with the source documents. It could demonstrate that our model equipped with the proposed informative encoding mechanism could generate summaries more accurately than the other comparing models. VI. CONCLUSION In this paper, we propose to incorporate document-aware positional encoding and linguistic-guided encoding for abstractive multi-document summarization. We conduct extensive experiments on two benchmark datasets and the results demonstrate the superior performance of the proposed two encoding methods. The analysis of various settings of the document-aware positional encoding and linguistic-guided encoding can help researchers understand the intuitiveness of the proposed model and could serve as an informative reference to the MDS research community. In the future, we would like to explore different ways to capture cross-document relations to further improve the quality of generated summaries. Source Docs a nine-year-old boy from los banos has completed quite the journey on tuesday , as he swam through the san francisco bay , all the way to alcatraz island and back . james savage is hoping to be the youngest swimmer on record to make the swim . the title is currently held by a 10-year-old . it ' s been quite an emotion day in san francisco , but it ended with smiles and cheers as james walked on shore , after hours in the bay . james began the swim at 7 : 00 a.m. , and it took him a little over two hours to complete the feat . he struggled for a big out in the open water , about 30 minutes into the swim . james said the waves were hitting him , and the current was too strong , making it difficult for him to fight them . in the end , however , with help from his coach and a promise from his father , james found his second wind , and was off . " i wanted to turn back and they kept yelling , ' i believe i you you can do it ' , " said james . " we kind of pushed each other . finally got him going . smooth sailing all the way ......james savage , a 9-year-old boy from los banos , calif...... Transformer james savage has been swimming off the san francisco bay area since he was 14 years old , but he ' s not the only one who ' s been in the water for more than a century . " i ' ve never seen anything like this before , " he tells nbc bay area . " it ' s a great thing to do . i ' m not going to be able to do it . " he says he ' ll be swimming in the bay of san francisco ' s bay in san francisco when he ' d swim to the beach , and he ' s 〈blank〉〈blank〉〈blank〉...... CopyTrans james savage , a 9-year-old boy who completed the tiring trek last year , was able to set a record as the youngest swimmer to make the journey to the island and back on june 14 , reports nbc bay area ...... james tells ksee . " he ' s not going to do anything about it , " he says , adding that his second wind was off the rails . " it ' s really hard to get to the bottom of the road , " savage says of his father . " it was just a slap on the wrist . it was a slap in the water . it ' s 〈blank〉〈blank〉〈blank〉...... Hi-Map james savage is hoping to be the youngest swimmer on record to make the swim to the former prison , reports the san francisco chronicle . " i wanted to turn back and they kept yelling , ' i believe i you you can do it ' , " james told ksee . " we kind of pushed each other . finally got him going . smooth sailing all the way , " he says . " i 〈blank〉〈blank〉〈blank〉...... Ours james savage , a 9-year-old boy from los banos , calif. , has completed the swim at 7 : 〈unk〉and 57 pounds , was able to set a record as the youngest swimmer to make the journey to the alcatraz island and back on june 14 , nbc news reports. " i wanted to turn back and they kept yelling , ' i believe i you can do it , ' " james tells ksee . " we kind of pushed each other . finally got him going . smooth sailing all the way , " ...... Fig. 1 . 1The framework of our proposed document-aware positional encoding and linguistic-guided encoding. Document-aware positional encoding serves as part of the input of the encoder; the proposed dependency relation mask will be incorporated with multi-head attention. Fig. 2 . 2The proposed document-aware positional encoding. It contains a document-level positional encoding and a token-level positional encoding. The selection of document positional encoding functions is according to our proposed protocol. ( 2 ) 2The values of encoding should be bonded. It will inevitably introduce large bias to certain documents if the encoding values are not bonded. (3) The values of encoding can not be remarkably larger than the value of token positional encoding. It will overwhelm the values of the token positional encoding if the document encoding values are too large, which impedes the model optimization process. Fig. 3 . 3The transformation of dependency relation mask (right) from dependency relation tensor (left). positional encoding of each document is the same (SameEncoding); (2) the values of document positional encoding are not bonded (y = x, y = 2x, y = 5x, y = 10x); and (3) the values of document positional encoding are remarkable larger than the values of token positional encoding (y = 10x); (4) randomly assign values to document positional encoding (Random).(3) Linguistic-guided Encoding Strategies. There are 45 dependency relations existing in the Biaffine parser. Some dependency relations have a great influence on the generated summaries; and vice versa. This section discusses how to encode these various relations into multi-head attention mechanism by considering their importance. The performance of different linguistic-guided encoding methods is shown inTable VI. The importance of dependency relations in the first three methods are manually set and the following two are automatically learned. "Arithmetic sequence" represents a sequence with the values of 1, N −1/N, N −2/N, ..., 1/N , which means the dependency relations at the top of the list have a larger weight. N denotes the number of dependency relations in total. TABLE I PERFORMANCE ICOMPARISON ON THE MULTI-NEWS DATASET. WE RERUN ALL THE BASELINE MODELS UNDER THE SAME SETTINGS. "COPYTRANS" REPRESENTS COPYTRANSFORMER. THE BEST RESULTS FOR EACH COLUMN ARE IN BOLD.TABLE II PERFORMANCE COMPARISON ON THE MULTI-XSCIENCE DATASET. WE RERUN ALL THE BASELINE MODELS UNDER THE SAME SETTINGS. "COPYTRANS" REPRESENTS COPYTRANSFORMER. THE BEST RESULTS FOR EACH COLUMN ARE IN BOLD.Models R-1 R-2 R-SU BS LexRank 37.92 13.10 12.51 0.83 TextRank 39.02 14.54 13.08 0.83 SummPip 42.29 13.29 16.16 0.84 MMR 42.12 13.19 15.63 0.84 BRNN 38.36 13.55 14.65 0.83 Transformer 25.82 5.84 6.91 0.80 CopyTrans 42.98 14.48 16.91 0.84 PG 34.13 11.01 11.58 0.83 Hi-MAP 42.98 14.85 16.93 0.83 HT 36.09 12.64 12.55 0.84 Ours 44.35 15.04 17.97 0.85 Models R-1 R-2 R-SU BS LexRank 31.31 5.85 9.13 0.83 TextRank 31.15 5.71 9.07 0.84 SummPip 29.66 5.54 8.11 0.82 MMR 30.04 4.46 8.15 0.83 BRNN 27.95 5.78 8.43 0.83 Transformer 28.34 4.99 8.21 0.82 CopyTrans 26.92 4.92 7.50 0.83 PG 30.30 5.02 9.04 0.84 Hi-MAP 30.41 5.85 9.13 0.81 HT 25.31 4.23 6.64 0.83 Ours 30.93 6.06 9.57 0.84 TABLE III ABLATION IIISTUDY OF OUR MODEL ON MULTI-NEWS AND MULTI-XSCIENCE DATASET. "DOC-POS EN" AND "DEPEN EN" STAND FOR DOCUMENT-AWARE POSITIONAL ENCODING AND LINGUISTIC-GUIDED ENCODING.Dataset Model Variants R-1 R-2 R-SU Multi- w/o doc-pos en 44.16 15.06 17.74 News w/o depen en 43.73 14.86 17.37 Full Model 44.35 15.04 17.97 Multi- w/o doc-pos en 28.81 5.53 8.56 XScience w/o depen en 29.69 5.62 8.86 Full Models 30.93 6.06 9.57 Table III presents the results. The experiments confirm that the proposed two encoding TABLE IV PERFORMANCE IVOF OUR MODEL USING DIFFERENT DOCUMENT POSITIONAL ENCODING STRATEGIES. THE STRATEGIES INCLUDE DIFFERENT ENCODING FUNCTIONS (UPPER) AND DIFFERENT DOCUMENT POSITIONAL ENCODING WEIGHTS(LOWER). ITER(A, B) MEANS TO USE FUNCTIONS A AND B ALTERNATELY. VALUES OBTAINED FROM THE VALIDATION SET BASED ON THE MULTI-NEWS DATASET.Models R-1 R-2 R-SU sin(x) 43.80 14.74 17.59 cos(x) 42.82 14.49 16.71 iter(sin(x), cos(x)) 43.56 14.43 17.38 iter(sin(0.1x), cos(0.1x)) 43.66 14.52 17. 47 α= 0.1 44.11 14.81 17.74 α= 0.5 43.68 14.54 17.45 α= 1 43.80 14.74 17.59 TABLE V PERFORMANCE VOF MODELS WITH FUNCTIONS THAT DO NOT MEET THE DOCUMENT POSITIONAL ENCODING PROTOCOL. VALUES OBTAINED FROM THE VALIDATION SET BASED ON THE MULTI-NEWS DATASET.Models R-1 R-2 R-SU SameEncoding 42.82 14.28 16.63 y = x 42.25 14.08 16.25 y = 2x 42.57 14.06 16.60 y = 5x 40.56 12.06 15.13 y = 10x 38.94 11.50 14.24 Random 43.19 14.67 16.87 TABLE VI PERFORMANCE OF OUR MODEL BASED ON DIFFERENT LINGUISTIC-GUIDED ENCODING METHODS. VALUES OBTAINED FROM THE VALIDATION SET BASED ON THE MULTI-NEWS DATASET. Models R-1 R-2 R-SU Arithmetic sequence 43.71 14.54 17.43 Arithmetic sequence (core) 43.79 14.57 17.47 Arithmetic sequence (root) 43.89 14.64 17.55 One-hot (one layer) 43.15 14.40 17.03 One-hot (F depEnc ) 44.11 14.81 17.74 Added on values 42.41 13.89 16.47 TABLE VII HUMAN VIIEVALUATION RESULTS ON THE MULTI-NEWS DATASET. THE BEST RESULTS FOR EACH COLUMN ARE IN BOLD. "COPYTRANS" REPRESENTS COPYTRANSFORMER.Models Fluency Informativeness Conciseness Transformer 2.50 1.97 2.50 CopyTrans 2.60 2.60 2.83 Hi-MAP 3.07 2.87 2.97 Ours 3.13 3.10 3.20 TABLE VIII GENERATED VIIISUMMARIES OF DIFFERENT MODELS GIVEN THE SAME SOURCE DOCUMENTS. "COPYTRANS" REPRESENTS COPYTRANSFORMER. DIFFERENT COLORS REPRESENT DIFFERENT THOUGHT GROUPS. 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[ "https://github.com/Alex-Fabbri/Multi-News/tree/master/code/OpenNMT-py-baselines2" ]
[ "Spin Squeezing via One-Axis Twisting with Coherent Light", "Spin Squeezing via One-Axis Twisting with Coherent Light" ]	[ "M Takeuchi \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n", "S Ichihara \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n", "T Takano \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n", "M Kumakura \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n\nCREST\nJST, 4-1-8 Honcho Kawaguchi332-0012SaitamaJapan\n", "T Yabuzaki \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n", "Y Takahashi \nDepartment of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan\n\nCREST\nJST, 4-1-8 Honcho Kawaguchi332-0012SaitamaJapan\n" ]	[ "Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan", "Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan", "Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan", "Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan", "CREST\nJST, 4-1-8 Honcho Kawaguchi332-0012SaitamaJapan", "Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan", "Department of Physics\nGraduate School of Science\nKyoto University\n606-8502KyotoJapan", "CREST\nJST, 4-1-8 Honcho Kawaguchi332-0012SaitamaJapan" ]	[]	We propose a new method of spin squeezing of atomic spin, based on the interactions between atoms and off-resonant light which are known as paramagnetic Faraday rotation and fictitious magnetic field of light. Since the projection process, squeezed light, or special interactions among the atoms are not required in this method, it can be widely applied to many systems. The attainable range of the squeezing parameter is ζ S −2/5 , where S is the total spin, which is limited by additional fluctuations imposed by coherent light and the spherical nature of the spin distribution.PACS numbers: 03.67. Mn, 42.50.Lc Squeezed spin state (SSS) is one of the non-classical states in collective spin system. In SSS, the quantum uncertainty of the spins along an axis orthogonal to the mean spin vector ∆ S 2 ⊥ is suppressed below the standard quantum limit (SQL) such as ∆ S 2 ⊥ < \| S \|/2, where S is the mean spin vector, due to an entanglement formation among the individual spins. The degree of the squeezing is usually evaluated by the squeezing parameter ζ ≡ 2 ∆ S 2 ⊥ /\| S \|, in terms of the variance to average ratio[1].For the last several years, SSS has been extensively interested in not only for precision measurement of a spin component[2,3,4], but also for the application to the quantum infomation[5,6,7]. There have been many proposals and experiments to realize the spin squeezing of atoms. They can be put into three categories as follows : (i) Quantum non-demolition (QND) measurement of spin via paramagnetic Faraday rotation and spin squeezing by quantum projection[5,8,9,10,11,12,13]. The QND measurement has been already performed by some groups for the electronic ground states of atom[5,11,13], and the squeezed parameter has reached about ζ ∼ 0.7 for S ∼ 4 × 10 7 [11], and ζ ∼ 0.1 for S ∼ 10 11[13]. Since the projection causes the squeezing in this method, the degree of the squeezing will be finally determined by the performance of the detector. (ii) Quantum-state transfer from squeezed light to spin[6,7,14,15,16,17]. One type is based on the complete absorption of squeezed vacuum, and has been experimentally demonstrated for the electronic exited states of atom (ζ ∼ 0.97 for S ∼ 5 × 10 7 )[14]. Another type is based on the stimulated Raman adiabatic passage[6,7,16,17]. Since the squeezed light is the source of spin squeezing in these methods, the degree of the squeezing will be finally determined by the quality of the squeezed light. (iii) Special systems to induce nonlinear interactions among the individual spins such as Bose-Einstein condensates[18,19,20], cold atoms in optical lattice[21], atoms in optical cavity[22,23,24]. They are not easy to prepare and difficult to operate after squeezed.In this paper, we propose a new method to realize the spin squeezing, which can not be put into any of the three categories. Our method does not rely on the projection by the measurement, use of squeezed light, and the specialities of the systems. Instead, the new method only requires a coherent light pulse and a few linear optics, so it can be widely applied to many systems. It should be noted that a recent electronic archive by K.Hammerer et al.[25]includes another proposal of an unconditional spin squeezing with coherent light.Our method is based on the interaction between atoms and off-resonant light, whose interaction Hamiltomian takes a form [9]where α is a real constant, and the z-axis is set parallel to the wave vector of the light. S is the summation over the individual spin, which obeys the usual commutation relation of angular momenta [S, S] = iS. J is quantum-mechanical Stokes vector of light, which also obeys the usual commutation relation of angular momenta [J, J] = iJ. For a light pulse with the duration T propagating in free space, J can be written aswhere a ± is the annihilation operators of σ ± circular polarization mode, respectively[26]. The interaction of Eq.(1) represents the addition of the phase difference for σ ± light, which causes the rotation of the polarization plane for linear polarization at the angular frequency αS z /2, known as paramagnetic Faraday rotation. It also represents the spin rotation around the z-axis at the angular frequency αJ z , known as fictitious magnetic field of light[27]. If we are able to apply a light pulse whose J z is proportional to S z as a fitctitious magnetic field, the collective spin will nonlinearly rotate at angular frequencies proportional to S z , whose evolution will be similar to one-axis twisting[1]. This is the basic idea of our proposal.To design such an interaction, we propose a system illustrated inFig.1. Initially a light pulse \|ψ J is lin-	10.1103/physrevlett.94.023003	[ "https://arxiv.org/pdf/quant-ph/0410132v1.pdf" ]	204,932,799	quant-ph/0410132	abc1fb7d0be21935968bc7cdcb53efba580b973a	Spin Squeezing via One-Axis Twisting with Coherent Light arXiv:quant-ph/0410132v1 18 Oct 2004 (Dated: July 2, 2018) M Takeuchi Department of Physics Graduate School of Science Kyoto University 606-8502KyotoJapan S Ichihara Department of Physics Graduate School of Science Kyoto University 606-8502KyotoJapan T Takano Department of Physics Graduate School of Science Kyoto University 606-8502KyotoJapan M Kumakura Department of Physics Graduate School of Science Kyoto University 606-8502KyotoJapan CREST JST, 4-1-8 Honcho Kawaguchi332-0012SaitamaJapan T Yabuzaki Department of Physics Graduate School of Science Kyoto University 606-8502KyotoJapan Y Takahashi Department of Physics Graduate School of Science Kyoto University 606-8502KyotoJapan CREST JST, 4-1-8 Honcho Kawaguchi332-0012SaitamaJapan Spin Squeezing via One-Axis Twisting with Coherent Light arXiv:quant-ph/0410132v1 18 Oct 2004 (Dated: July 2, 2018) We propose a new method of spin squeezing of atomic spin, based on the interactions between atoms and off-resonant light which are known as paramagnetic Faraday rotation and fictitious magnetic field of light. Since the projection process, squeezed light, or special interactions among the atoms are not required in this method, it can be widely applied to many systems. The attainable range of the squeezing parameter is ζ S −2/5 , where S is the total spin, which is limited by additional fluctuations imposed by coherent light and the spherical nature of the spin distribution.PACS numbers: 03.67. Mn, 42.50.Lc Squeezed spin state (SSS) is one of the non-classical states in collective spin system. In SSS, the quantum uncertainty of the spins along an axis orthogonal to the mean spin vector ∆ S 2 ⊥ is suppressed below the standard quantum limit (SQL) such as ∆ S 2 ⊥ < \| S \|/2, where S is the mean spin vector, due to an entanglement formation among the individual spins. The degree of the squeezing is usually evaluated by the squeezing parameter ζ ≡ 2 ∆ S 2 ⊥ /\| S \|, in terms of the variance to average ratio[1].For the last several years, SSS has been extensively interested in not only for precision measurement of a spin component[2,3,4], but also for the application to the quantum infomation[5,6,7]. There have been many proposals and experiments to realize the spin squeezing of atoms. They can be put into three categories as follows : (i) Quantum non-demolition (QND) measurement of spin via paramagnetic Faraday rotation and spin squeezing by quantum projection[5,8,9,10,11,12,13]. The QND measurement has been already performed by some groups for the electronic ground states of atom[5,11,13], and the squeezed parameter has reached about ζ ∼ 0.7 for S ∼ 4 × 10 7 [11], and ζ ∼ 0.1 for S ∼ 10 11[13]. Since the projection causes the squeezing in this method, the degree of the squeezing will be finally determined by the performance of the detector. (ii) Quantum-state transfer from squeezed light to spin[6,7,14,15,16,17]. One type is based on the complete absorption of squeezed vacuum, and has been experimentally demonstrated for the electronic exited states of atom (ζ ∼ 0.97 for S ∼ 5 × 10 7 )[14]. Another type is based on the stimulated Raman adiabatic passage[6,7,16,17]. Since the squeezed light is the source of spin squeezing in these methods, the degree of the squeezing will be finally determined by the quality of the squeezed light. (iii) Special systems to induce nonlinear interactions among the individual spins such as Bose-Einstein condensates[18,19,20], cold atoms in optical lattice[21], atoms in optical cavity[22,23,24]. They are not easy to prepare and difficult to operate after squeezed.In this paper, we propose a new method to realize the spin squeezing, which can not be put into any of the three categories. Our method does not rely on the projection by the measurement, use of squeezed light, and the specialities of the systems. Instead, the new method only requires a coherent light pulse and a few linear optics, so it can be widely applied to many systems. It should be noted that a recent electronic archive by K.Hammerer et al.[25]includes another proposal of an unconditional spin squeezing with coherent light.Our method is based on the interaction between atoms and off-resonant light, whose interaction Hamiltomian takes a form [9]where α is a real constant, and the z-axis is set parallel to the wave vector of the light. S is the summation over the individual spin, which obeys the usual commutation relation of angular momenta [S, S] = iS. J is quantum-mechanical Stokes vector of light, which also obeys the usual commutation relation of angular momenta [J, J] = iJ. For a light pulse with the duration T propagating in free space, J can be written aswhere a ± is the annihilation operators of σ ± circular polarization mode, respectively[26]. The interaction of Eq.(1) represents the addition of the phase difference for σ ± light, which causes the rotation of the polarization plane for linear polarization at the angular frequency αS z /2, known as paramagnetic Faraday rotation. It also represents the spin rotation around the z-axis at the angular frequency αJ z , known as fictitious magnetic field of light[27]. If we are able to apply a light pulse whose J z is proportional to S z as a fitctitious magnetic field, the collective spin will nonlinearly rotate at angular frequencies proportional to S z , whose evolution will be similar to one-axis twisting[1]. This is the basic idea of our proposal.To design such an interaction, we propose a system illustrated inFig.1. Initially a light pulse \|ψ J is lin- We propose a new method of spin squeezing of atomic spin, based on the interactions between atoms and off-resonant light which are known as paramagnetic Faraday rotation and fictitious magnetic field of light. Since the projection process, squeezed light, or special interactions among the atoms are not required in this method, it can be widely applied to many systems. The attainable range of the squeezing parameter is ζ S −2/5 , where S is the total spin, which is limited by additional fluctuations imposed by coherent light and the spherical nature of the spin distribution. Squeezed spin state (SSS) is one of the non-classical states in collective spin system. In SSS, the quantum uncertainty of the spins along an axis orthogonal to the mean spin vector ∆ S 2 ⊥ is suppressed below the standard quantum limit (SQL) such as ∆ S 2 ⊥ < \| S \|/2, where S is the mean spin vector, due to an entanglement formation among the individual spins. The degree of the squeezing is usually evaluated by the squeezing parameter ζ ≡ 2 ∆ S 2 ⊥ /\| S \|, in terms of the variance to average ratio [1]. For the last several years, SSS has been extensively interested in not only for precision measurement of a spin component [2,3,4], but also for the application to the quantum infomation [5,6,7]. There have been many proposals and experiments to realize the spin squeezing of atoms. They can be put into three categories as follows : (i) Quantum non-demolition (QND) measurement of spin via paramagnetic Faraday rotation and spin squeezing by quantum projection [5,8,9,10,11,12,13]. The QND measurement has been already performed by some groups for the electronic ground states of atom [5,11,13], and the squeezed parameter has reached about ζ ∼ 0.7 for S ∼ 4 × 10 7 [11], and ζ ∼ 0.1 for S ∼ 10 11 [13]. Since the projection causes the squeezing in this method, the degree of the squeezing will be finally determined by the performance of the detector. (ii) Quantum-state transfer from squeezed light to spin [6,7,14,15,16,17]. One type is based on the complete absorption of squeezed vacuum, and has been experimentally demonstrated for the electronic exited states of atom (ζ ∼ 0.97 for S ∼ 5 × 10 7 ) [14]. Another type is based on the stimulated Raman adiabatic passage [6,7,16,17]. Since the squeezed light is the source of spin squeezing in these methods, the degree of the squeezing will be finally determined by the quality of the squeezed light. (iii) Special systems to induce nonlinear interactions among the individual spins such as Bose-Einstein condensates [18,19,20], cold atoms in optical lattice [21], atoms in optical cavity [22,23,24]. They are not easy to prepare and difficult to operate after squeezed. In this paper, we propose a new method to realize the spin squeezing, which can not be put into any of the three categories. Our method does not rely on the projection by the measurement, use of squeezed light, and the specialities of the systems. Instead, the new method only requires a coherent light pulse and a few linear optics, so it can be widely applied to many systems. It should be noted that a recent electronic archive by K.Hammerer et al. [25] includes another proposal of an unconditional spin squeezing with coherent light. Our method is based on the interaction between atoms and off-resonant light, whose interaction Hamiltomian takes a form [9] H = αJ z S z ,(1) where α is a real constant, and the z-axis is set parallel to the wave vector of the light. S is the summation over the individual spin, which obeys the usual commutation relation of angular momenta [S, S] = iS. J is quantum-mechanical Stokes vector of light, which also obeys the usual commutation relation of angular momenta [J, J] = iJ. For a light pulse with the duration T propagating in free space, J can be written as J x ≡ 1 2 T 0 (a † + a − +a † − a + )dt, J y ≡ 1 2i T 0 (a † + a − −a † − a + )dt, J z ≡ 1 2 T 0 (a † + a + − a † − a − )dt, where a ± is the annihilation operators of σ ± circular polarization mode, respectively [26]. The interaction of Eq.(1) represents the addition of the phase difference for σ ± light, which causes the rotation of the polarization plane for linear polarization at the angular frequency αS z /2, known as paramagnetic Faraday rotation. It also represents the spin rotation around the z-axis at the angular frequency αJ z , known as fictitious magnetic field of light [27]. If we are able to apply a light pulse whose J z is proportional to S z as a fitctitious magnetic field, the collective spin will nonlinearly rotate at angular frequencies proportional to S z , whose evolution will be similar to one-axis twisting [1]. This is the basic idea of our proposal. To design such an interaction, we propose a system illustrated in Fig early polarized along the x-axis and contains 2J(≫ 1) photons as an average. Atoms \|ψ S are spin-polarized along the x-axis and contains total spin S. The light is weakly focused to match the atomic ensemble [26]. The averages of the Stokes components is then J x = J, J y = J z = 0, and the averages of the collective spin vector is S x = S, S y = S z = 0. Since the light pulse is a strong coherent state, we can approximate the commutation relation as [J y , J z ] = iJ [26]. Firstly, a light pulse passes through the atoms and the polarization plane is then rotated. We call it "the first interaction", whose interaction time is labelled as t 1 . The Stokes vector becomes J (FI) = e it1H Je −it1H , whose y component is approximately written as J (FI) y ≃ J y + αt 1 JS z , for αt 1 S z ≪ 1. Since the average of the J (FI) y becomes ψ J \|J (FI) y \|ψ J = αt 1 JS z , we can say that the information of S z is copied and held on J (FI) y as a Faraday rotation angle. We note that S z is conserved because the interaction of Eq.(1) satisfies the back-action evasion (BAE) condition of [S z , H] = 0. Secondly, the pulse passes through twice the λ/8 wave plate by the totally retroreflecting mirror. As a result, λ/4 phase difference is induced between the two orthogonal modes of linear polarization. We call it "the local operation" for the light. The Stokes vector becomes J (LO) = e i(π/2)Jx J (FI) e −i(π/2)Jx , whose z component is J (LO) z = J (FI) y . We can say that the infor- mation of S z is shifted from J (FI) y to J (LO) z , converting the angle of the polarization plane to the photon number difference of the σ ± modes. Thus, the required light is achieved whose J z is approximately proportional to S z . Finally, the pulse passes through the atomic ensemble again. We call it "the second interaction", whose interaction time is labelled as t 2 . The interaction Hamiltonian of the second interacton can be roughly written as H (SI) ∼ αJ (LO) z S z ∝ S 2 z , which takes a form similar to the one-axis twisting Hamiltonian χS 2 z [1]. Thus, we can expect that the spin state becomes SSS after the second interaction. Next, we derive the density operator of the spin after the second interaction to calculate the properties of the spin state obtained by this method. The initial density operator of the whole system can be written as ρ SJ ≡ ρ S ⊗ \|ψ J ψ J \|, where ρ S = \|ψ S ψ S \|. After the second interaction, it becomes ρ SJ ≡ U ρ SJ U † where U = e −it2H e −i(π/2)Jx e −it1H . The reduced density operator representing the spin state after the second interaction ρ S can be written as ρ S = Tr J ( ρ SJ ), where Tr J is the partial trace for the light. For convenience, we consider the set of eigenstates for S 2 and S z , say \|S, M , where σ MM ′ = e −µ ′ (M−M ′ ) 2 /2 e −iµ(M 2 −M ′2 )/2 ,(2) where we have set µ ≡ (αt 1 )(αt 2 )J and µ ′ ≡ ((αt 1 ) 2 + (αt 2 ) 2 )J/2. If t 1 = t 2 then µ = µ ′ . Since the atoms \|ψ S are polarized along the x-axis, the matrix elements of ρ S can be written as S, M \|ρ S \|S, M ′ = 1 2 2S 2S S + M 1/2 2S S + M ′ 1/2 .(4) In the following discussions, we use the experessions of Eq.(3) and Eq.(4). We note that the ideal one-axis twisted state [1] corresponds to the case of µ ′ = 0. To know how uncertainties evolve, we calculate the quasiprobability distributions (QPD), which is defined as Q(θ, φ) = θ, φ\| ρ S \|θ, φ , where \|θ, φ ≡ e −iφSz e −iθSy \|S, S is a spin state polarized along the direction whose polar and azimuth angles are θ and φ, respectively [1]. The results of the calculations in the case of S = 20 are shown in Fig.2 for the initial spin state (a), the spin state after the first interaction (b), and that after the second (c). The initial spin state is isotropically distributed along the x-direction as is shown in Fig.2(a). After the first interaction, the distribution is a little broadened along the y direction, as Fig.2(b) indicates. This is explained by additional fluctuation imposed by coherent light. In fact, the y components after the first interaction is approximately written as S (FI) y = e it1H S y e −it1H ≃ S y + αt 1 J z S x for αt 1 J z ≪ 1. Since S z is the BAE variable, the distribution along the z direction does not change at all. After the second interaction, the distribution looks twisted around the zaxis and squeezed along the z ′ -axis, as Fig.2(c) indicates. Although not clear from the figure, the distribution is also broadened along the y-axis as in the case after the first interaction. In fact, the y component after the second interaction is roughly written as S y ∼ e it2H (SI) S (FI) y e −it2H (SI) ∼ S y +µS z S x +(αt 1 J z +αt 2 J y )S x . By these additional fluctuations imposed by coherent light, the spin state after the second interaction is different from the ideal one-axis twisted state [1]. The additional fluctuations would be reduced by use of a polarization squeezed light pulse whose squeezed component is t 1 J z + t 2 J y , approaching the ideal one-axis twisting interaction of µ ′ → 0. We mention that the additional fluctuations by light in the method of Ref. [25] are imposed both on the y and z components, while the z component is squeezed. Therefore, the squeezing parameter does not become small in that scheme. From Eq.(3) and Eq.(4), we can derive the averages and variances of the spin components. The averages can be calculated as S x = Se −µ ′ /2 cos 2S−1 (µ/2) and S y = S z = 0, where S represents the spin operator after the second interaction. They indicates that the orientation of the mean spin vector remains (π/2, 0) direction or the x-axis, as is shown in Fig.2(c). To characterize a elliptical distribution around the x-axis, we define the minor and major axis, say z ′ and y ′ , respectively, as is shown in Fig.2(c), so that the variances of whose components ∆ S 2 z ′ and ∆ S 2 y ′ give the minimum and maximum on the y − z plane, respectively. The variances can be calculated as ∆ S 2 x = S 2 − S x 2 −S(S −1/2)A/2 and ∆ S 2 y ′ z ′ = S 2 + S 2 S − 1/2 2 A ± A 2 + B 2 ,(5) where we have set A = 1 − e −2µ ′ cos 2S−2 µ, and B = 4e −µ ′ /2 sin(µ/2) cos 2S−2 (µ/2). We can also calculate δ, which is an angle between the directions of the z ′ -and z-axes, or the y ′ -and y-axes, as is shown in Fig.2(c), and obtain δ = arctan(B/A)/2. For S ≫ 1 and S −1 ≪ µ ∼ µ ′ ≪ S −1/2 , we find the approximate value of the variance of the z ′ component ∆ S 2 z ′ ≃ S 2 γ ′ γ 2 + γ ′ + 2 3 β 2 ,(6) where we have set γ = Sµ/2, γ ′ = Sµ ′ /2 and β = Sµ 2 /4. Also we find S x ≃ S(1 − β). To examine the dependence on the interaction strength αt 1 , αt 2 and the input photon number 2J, we plot the variances of the y ′ components and the z ′ components as a function of µ(= µ ′ ) in Fig.3(a). We also plot the approximate value for the z ′ components written as Eq.(6). It is clearly known that the variance of the z ′ component is reduced for small µ, minimized at an optimal value of µ, and becomes large for large µ. It means that too strong interaction or too large photon number deteriorates the squeezing. This is explained by the spherical nature of the spin distribution, and in fact, the variance of the y ′ component is almost saturated at the largest value of S 2 /2 for large µ, which was entirely ignored in the analysis in Ref. [25]. As a typical value of µ, we introduce µ half as the value of µ to attain ∆ S 2 z ′ = S/4, the half variance of the SQL. We also introduce µ min as the value to attain the minimum of the ∆ S 2 z ′ . We plot the numerical solutions of µ half and µ min in Fig.3 (b) for the case of µ = µ ′ . One can see that both µ half and µ min become small as S increases but they obey different power laws. From Eq.(6), we find µ half ≃ 2S −1 and µ min ≃ 2(3/2) 1/5 S −3/5 . We show these approximate solutions in Fig.3 (b), which are in good agreement with the numerical ones. We also find that the squeezing parameter at µ = µ ′ = µ min becomes ζ min ≃ (2/3) 1/5 S −2/5 . We note that it is slightly worse than (1/3) 1/3 S −2/3 , which is the squeezing parameter for the ideal one-axis twisting, due to the additional fluctuations imposed by coherent light, as is mentioned above. Finally, we discuss the feasibility of our method. In the following, we consider the case that the shape of the light pulse is a square wave with its peak power P and pulse duration T . As in Ref. [26], we assume ∆ ≫ Γ, Ω ≪ ∆, and rT ≪ (Sµ) −1 ≪ 1, where ∆ represents the detuning from the resonance frequency, Γ the full natural linewidth at half maximum of the transition, Ω the Rabi frequency, and r the photon scattering rate [28]. After some calculations, we obtain µ = µ ′ = rT σ 0 /(2πw 2 ), where w represents the beam waist and σ 0 the photonabsorption cross section of an atom, which can be written as σ 0 = 3λ 2 0 /(2π) with the resonance wavelength λ 0 . We note that Ω, r and µ are exactly the same as 2g 2N p /(cT ), 4ε a /T , and κ 2 /N a in Ref. [26], respectively. The condition to obtain µ ≥ µ half , or ζ ≤ 1/2, can be rewritten as d 0 8(rT ) −1 , where d 0 = 2Sσ 0 /(πw 2 ) is the optical depth. We also note that this condition is the same as κ √ 2, similar to that of QND measurement [26]. Such a condition has been satisfied in several systems, such as atoms in a cell [5], laser cooled and trapped atom, and so on. The feasibility of our scheme also comes from the simple experimental setup depicted in Fig.1, which is also the great advantage over another scheme in Ref. [25]. This suggests efficient squeezing can be realized by the current technologies. As one ideal example, we consider a ytterbium atoms ( 171 Yb) in optical trap [29,30], which contains S = 4 × 10 6 . The atom collision and the precession due to the stray magnetic field, which causes the transverse relaxation, are well surpressed because it is ultracold fermion and has only nuclear spin 1/2 whose gyromagnetic ratio is about three orders smaller than paramagnetic atoms like alkali metal. From the parameters given in Ref. [30], w = 3µm [30], λ 0 = 399nm and Γ = 2π × 29MHz, the light pulse of µ = 5.4 × 10 −6 , for example, is obtained by setting ∆ = 2π × 24GHz, P = 17nW, T = 0.24ms, which satisfies the assumptions ∆ ≫ Γ, Ω = 2π × 21MHz ≪ ∆, rT = 4.0 × 10 −3 ≪ (Sµ) −1 = 4.6 × 10 −2 ≪ 1. and J = 4.0 × 10 6 ≫ 1. In this case the squeezing parameter becomes ζ = 0.08. We note that the decay constant of the atom number in optical trap is about 4s [31], which is so longer than the the pulse duration T that we can treat the total spin S as a constant. We also note that the length of the atom distribution shuould be adjusted to L ∼ 70µm to satisfy the condition of πw 2 /(λ 0 L) ∼ 1 [26], which is easy for atoms in optical trap of crossed configuration. To avoid the interference between the three steps of the first interaction, the local operation and the second interaction within one pulse, we can use a pluse train, each duration of which is so short that the three steps are separable and the repetition rate is so slow that the pulse number travelling in the path is at most one. Since the each matrix element S, M \|ρ S \|S, M ′ would evolve like a geometric progression whose common ratio is σ MM ′ for the every pulse as Eq.(2) indicates, and σ MM ′ is the power of the mean photon number of the each pulse as Eq. (3) indicates, we can say that the same SSS would be obtained as long as the total mean photon number passed through the atomic ensemble are equal. PACS numbers: 03.67.Mn, 32.80.-t, 42.50.Lc . 1 .FIG. 1 : 11Initially a light pulse \|ψ J is lin-System of our proposal. A linearly polarized light pulse passes through an atomic ensemble and the polarization plane is rotated. The rotation angle is proporitonal to Sz and converted to the circular polarization components after passing through the λ/8 plate twice. When the pulse passes through the atomic ensemble again, the pulse induce a nonlinear rotation to the atomic ensemble around the z-axis as a fictitious magnetic field. See text for details. S 2 2\|S, M = S(S + 1)\|S, M and S z \|S, M = M \|S, M . The matrix elements take a form S, M \| ρ S \|S, M ′ = σ MM ′ S, M \|ρ S \|S, M ′ , FIG. 2 : 2State evolutions expressed as the quasiprobability distribution for S = 20. The value of QPD for (θ, φ) direction is represented by the gray scale on the unit sphere, which is normalized by the maximum value. (a) The initial spin state. (b) The spin state after the first interaction, where we have set (αt1) 2 J/2 = 0.1 and t2 = 0, in other words, µ = 0 and µ ′ = 0.1. (c) The spin state after the second interaction, where we have set (αt1) 2 J/2 = (αt2) 2 J/2 = 0.1. in other words, µ = µ ′ = 0.2. 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[ "Thermodynamic bounds on the ultra-and infra-affinity of Hsp70 for its substrates", "Thermodynamic bounds on the ultra-and infra-affinity of Hsp70 for its substrates" ]	[ "Basile Nguyen \nII. Institut für Theoretische Physik\nUniversität Stuttgart\nStuttgartGermany\n\nInstitute of Physics\nSchool of Basic Science and Institute of Bioengineering\nSchool of Life Sciences\nLaboratory of Statistical Biophysics\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nLausanneSwitzerland\n", "David Hartich \nII. Institut für Theoretische Physik\nUniversität Stuttgart\nStuttgartGermany\n", "Udo Seifert \nII. Institut für Theoretische Physik\nUniversität Stuttgart\nStuttgartGermany\n", "Paolo De ", "Los Rios \nInstitute of Physics\nSchool of Basic Science and Institute of Bioengineering\nSchool of Life Sciences\nLaboratory of Statistical Biophysics\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nLausanneSwitzerland\n" ]	[ "II. Institut für Theoretische Physik\nUniversität Stuttgart\nStuttgartGermany", "Institute of Physics\nSchool of Basic Science and Institute of Bioengineering\nSchool of Life Sciences\nLaboratory of Statistical Biophysics\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nLausanneSwitzerland", "II. Institut für Theoretische Physik\nUniversität Stuttgart\nStuttgartGermany", "II. Institut für Theoretische Physik\nUniversität Stuttgart\nStuttgartGermany", "Institute of Physics\nSchool of Basic Science and Institute of Bioengineering\nSchool of Life Sciences\nLaboratory of Statistical Biophysics\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nLausanneSwitzerland" ]	[]	The 70 kDa Heat Shock Proteins Hsp70 have several essential functions in living systems, such as protecting cells against protein aggregation, assisting protein folding, remodeling protein complexes and driving the translocation into organelles. These functions require high affinity for non-specific amino-acid sequences that are ubiquitous in proteins. It has been recently shown that this high affinity, called ultra-affinity, depends on a process driven out of equilibrium by ATP hydrolysis. Here we establish the thermodynamic bounds for ultra-affinity, and further show that the same reaction scheme can in principle be used both to strengthen and to weaken affinities (leading in this case to infra-affinity). We show that cofactors are essential to achieve affinity beyond the equilibrium range. Finally, biological implications are discussed.	10.1016/j.bpj.2017.06.010	[ "https://arxiv.org/pdf/1702.01649v2.pdf" ]	8,929,552	1702.01649	8cb9580ade1ae73401719fa6405ac233b11a3f6b	Thermodynamic bounds on the ultra-and infra-affinity of Hsp70 for its substrates Basile Nguyen II. Institut für Theoretische Physik Universität Stuttgart StuttgartGermany Institute of Physics School of Basic Science and Institute of Bioengineering School of Life Sciences Laboratory of Statistical Biophysics École Polytechnique Fédérale de Lausanne (EPFL) LausanneSwitzerland David Hartich II. Institut für Theoretische Physik Universität Stuttgart StuttgartGermany Udo Seifert II. Institut für Theoretische Physik Universität Stuttgart StuttgartGermany Paolo De Los Rios Institute of Physics School of Basic Science and Institute of Bioengineering School of Life Sciences Laboratory of Statistical Biophysics École Polytechnique Fédérale de Lausanne (EPFL) LausanneSwitzerland Thermodynamic bounds on the ultra-and infra-affinity of Hsp70 for its substrates 1 The 70 kDa Heat Shock Proteins Hsp70 have several essential functions in living systems, such as protecting cells against protein aggregation, assisting protein folding, remodeling protein complexes and driving the translocation into organelles. These functions require high affinity for non-specific amino-acid sequences that are ubiquitous in proteins. It has been recently shown that this high affinity, called ultra-affinity, depends on a process driven out of equilibrium by ATP hydrolysis. Here we establish the thermodynamic bounds for ultra-affinity, and further show that the same reaction scheme can in principle be used both to strengthen and to weaken affinities (leading in this case to infra-affinity). We show that cofactors are essential to achieve affinity beyond the equilibrium range. Finally, biological implications are discussed. INTRODUCTION Most proteins must fold into specific three-dimensional structures (native states) to be functional and take part in cellular processes. During, and right after, translation, newly synthesized polypeptides are not yet fully folded. As a consequence, they still expose hydrophobic surfaces, that could lead to inter-protein interaction and cytotoxic aggregation (1). Furthermore, mutations or environmental cues, such as heat-shock or oxidative stress, can destabilize native proteins, leading to their unfolding, misfolding and potential aggregation. In cells, the protein quality control system acts to maximize the reliability of protein folding, and to clear proteins that cannot be driven back to their native state (2). Defects in protein quality control are associated with age-related diseases such as type II diabetes, heart diseases, specific cancers and, most notably, neurodegenerative disorders (e.g Alzheimer's or Parkinson's diseases) (3). Chaperones proteins are key players in protein quality control, and are present in all organisms. Their broadly recognized role is to assist the folding process, and minimize protein aggregation. Intriguingly, the action of most chaperones stringently depends on ATP hydrolysis, although in most cases its precise role has not been fully understood. Central among proteins is the 70 kDa Heat Shock Protein (Hsp70). Hsp70 is possibly the most versatile of the chaperones and takes part in disparate functions beyond quality control. It drives the translocation of hundreds of different proteins into mitochondria and the endoplasmic reticulum, disassembles functional oligomers and facilitates protein translation, among others (2,4). In order to be functional, Hsp70s must be able to strongly bind to a diverse array of amino-acid sequence. The structure of Hsp70 comprises two domains: the nucleotide binding domain (NBD), where ATP or ADP are lodged, and the substrate binding domain (SBD), which is made of two halves and is responsible for the interactions between Hsp70s and their substrates (5), see Fig. 1 for an illustration. In the ATP-bound state, the two halves of the SBD are preferably docked onto the NBD ("open" conformation, Fig. 1), whereas in the ADP bound state the two halves of the SBD preferably detach from the NBD and bind to each other, forming a "closed" clamp (which remains linked to the NBD by a flexible linker). Note that since binding and unbinding rates are significant in the ADP bound state, spontaneous opening and closing occurs (6). The spontaneous ATPase rate of Hsp70 is very low (10 −4 − 10 −3 s −1 ), but is greatly accelerated (up to 1 s −1 ) upon substrate binding and an associated, mandatory, J-domain containing protein (7). Upon contact, thus, Hsp70 latches onto the substrate after rapid ATP hydrolysis, entrapping it into the closed clamp (Fig. 1). Remarkably, the measured substrate affinity of the ATPbound, open conformation is only slightly lower than the substrate affinity of the closed ADP-bound conformation (K ATP D > K ADP D , see Refs. (8,9) or Table 1 for experimental values). Experiments, though, have shown that substrate binding occurs mainly in the ATP state rather than ADP state, despite the latter being the state characterized by the smallest dissociation constant (10,11). It had been proposed that this effect was inherently due to the non-equilibrium, ATP-consuming, nature of Hsp70s. Recently, this enhanced affinity (dubbed ultra-affinity) was linked to the kinetic properties of the ATP-bound and ADP-bound states (12). The substrate binding and unbinding rates are faster for ATP-bound Hsp70s, because the SBD is preferably open and easily accessible, than for ADP-bound Hsp70s, whose SBD is preferably closed and thus difficult to bind to, but also difficult to unbind from (see Fig. 1). Due to an excess of ATP in living cells and in the vast majority of experiments, most free Hsp70 molecules are bound to ATP. As a consequence, substrate binding occurs mostly in the ATP-bound Hsp70, with the fastest binding rate k ATP + . ATP hydrolysis rapidly follows, with the closure of the SBD on the substrate. Substrate unbinding takes place then with the smallest dissociation rate, k ADP − . The effective non-equilibrium dissociation constant can thus be lowered down to K eff = k ADP − /k ATP + , which is not related to the individual dissociation constants of the ATP-bound and ADP-bound states. It can be smaller than the dissociation constant attainable without hydrolysis, which is the average of the dissociation constants of the two nucleotide-bound states, weighted by their respective populations. Ultra-affinity is a remarkable principle, which allows Hsp70s to bind very effectively to a broad, non-specific, array of amino-acid sequences. Experiments show that Hsp70s bind to their substrates with different intrinsic affinities (8). When energy from ATP hydrolysis is available, Hsp70s can bind to their substrates with a higher affinity which might be further enhanced during stresses. We provide new insight into this selectivity mechanism in response to stress. Furthermore, we show that Hsp70 can achieve such a selectivity depending on stress level, which relates to heat shock experiments that show changes in nucleotide levels (13)(14)(15) and changes in the activity of cofactors (16,17) during stress. A careful analysis of ultra-affinity, though, reveals that the energy budget of the process should also be taken into account. In this work, we consider a thermodynamic description of the Hsp70 system. Specifically, we characterize the relation between affinity and energy consumption by computing the thermodynamic bounds of ultra-affinity. Moreover, we show that it is possible to obtain the opposite of ultra-affinity, namely infra-affinity, that is an affinity which is lower than what would be possible at equilibrium. Note that the Hsp70 system shares many similarities with kinetic proofreading (18)(19)(20)(21) where error reduction is achieved with a chemical force driving the system out of equilibrium (22)(23)(24)(25)(26). A similar ultra-sensitive response was found for the E. coli chemotaxis system (27) where the increase of sensitivity by a non-equilibrium driving force was described and compared with kinetic proofreading (28). METHODS: Local detailed balance We model the Hsp70 system by a canonical four state system from (12), see Fig. 1 for an illustration. The Hsp70 system can be in an open ATP state (H ATP , S · H ATP ) or in a closed ADP state (H ADP , S · H ADP ), where "S" labels the presence of a substrate. Chemical forces arising from an ATP hydrolysis cycle drive the system out of equilibrium, which allow Hsp70 to tune its affinity to substrates. To better understand the benefit of such chemical forces, we Figure 1: Canonical model of the Hsp70 cycle. Horizontal rates correspond to binding/unbinding of a substrate with Hsp70's two states. The ADP-state has a lower dissociation constant and slow binding kinetics k ADP ± . The ATP-state has a higher dissociation constant and fast binding kinetics (k ATP ± > k ADP ± ). Vertical rates correspond to hydrolysis ω ± and nucleotide exchange κ ± reactions. Specifically, ω + is a release of P i and κ + is an exchange of ADP with ATP, both of which are emphasized by thick arrows. [S]k ADP + k ADP − [S]k ATP + k ATP − ω S + ω S − κ S − κ S + ω + ω − κ − κ + have to explain the local detailed balance relation (29), which connects the dynamics of single reactions with the laws of thermodynamic. We introduce the local detailed balance relation by considering the individual reactions corresponding to substrate binding and unbinding, which is illustrated in Fig. 1 by horizontal transitions. The binding and unbinding of the substrate to and from the chaperones corresponds to the reaction H X + S [S]k X + −−− −−− k X − S · H X ,(1) where X = ATP, ADP indicates the state of the heat shock protein and [S]k X + , k X − are transition rates. At temperature T , the transition rates must satisfy the local detailed balance relation (29), which for X = ATP, ADP reads k B T ln [S]k X + k X − = F H X − F S·H X + µ S ,(2) where F H X is the free energy of state H X , F S·H X the free energy of state S · H X , µ S the chemical potential of the substrate, and k B Boltzmann's constant. Experimental measurements of binding rates for the Hsp70 system can be found in Table 1. The vertical transitions in Fig. 1 involve the consumption of chemical energy due to ATP hydrolysis allowing the system to outperform equilibrium chaperone systems. More precisely, the vertical transition in Fig. 1 correspond to a hydrolysis reaction H ATP ω + − − − − ω − H ADP + P i ,(3) and a nucleotide exchange reaction Table 1: Experimental transition rates for the Hsp70 system and dissociation constants K ADP H ADP + ATP κ + − − − − κ − H ATP + ADP,(4)D = k ADP − /k ADP + , K ATP D = k ATP − /k ATP + . where κ ± and ω ± are transition rates in the absence of a substrate. Analogously, in the presence of a substrate the transition rates are denoted by an additional superscript "S", see κ S ± and ω S ± in Fig. 1. Note that in reality, the nucleotide exchange is a two-step reaction involving unbinding of ADP (ATP) and binding of ATP (ADP). Nevertheless, Eq. 4 is an effective relation which is equivalent as nucleotide binding is very fast and nucleotide affinity with Hsp70 is high. Performing one step in "+"-direction of reaction 3 and then one step in "+"-direction of reaction 4 does not change the state of the Hsp70 system, whereas it turns one ATP into an ADP and P i . Such a complete cycle consumes a chemical energy (work) ∆µ ≡ µ ATP − µ ADP − µ P i = ∆µ 0 + k B T ln [ATP] [ADP][P i ] ,(5) where µ X is the chemical potential of species X = ATP, ADP, P i . The last equality in Eq. 5 is the approximation for an ideal solution, where [X] is the concentration of species X = ATP, ADP, P i , and ∆µ 0 a reference value. Equilibrium corresponds to ∆µ = 0, whereas under physiological conditions an excess of ATP is maintained that implies ∆µ > 0. For such a cycle the local detailed balance relation implies β∆µ = ln κ + ω + κ − ω − = ln κ S + ω S + κ S − ω S − ,(6) where β = 1/(k B T ) is the inverse thermal energy. This relation connects the kinetics of hydrolysis Eq. 3 and nucleotide exchange Eq. 4 to the chemical driving force ∆µ from Eq. 5. Along a complete ATP hydrolysis cycle, this chemical energy ∆µ is dissipated in the environment. Moreover, a constant chemical driving force ∆µ > 0 (supply of ATP) drives the chaperone system into a non-equilibrium steady state, leaving more room to tune the system compared to a system with an equilibrium Boltzmann distribution, where ∆µ = 0. RESULTS We are interested in a thermodynamic relation between energy consumption and Hsp70's affinity for its substrates. We consider the effective dissociation constant K eff which measures how well Hsp70s can bind to their substrates. It is defined as K eff = [S][Hsp70]/[Hsp70 · S](7) where [S] is the concentration of free substrate, [Hsp70] is the concentration of free Hsp70s and [Hsp70·S] is the concentration of substrates bound to Hsp70 (see Appendix A for the full expression). In equilibrium, the dissociation constant is a linear combination of the ADP and ATP dissociation constants, therefore, K ADP D ≤ K eq eff ≤ K ATP D . Under physiological conditions, an excess of ATP is maintained which induces a positive chemical force (∆µ > 0). Under these non-equilibrium conditions, it has been found that K eff < K ADP D can be achieved, which is called ultraaffinity (12). We provide a thermodynamic description of ultra-affinity and show that this system can also achieve infra-affinity for its substrates. Figure 2: Thermodynamic bounds on the effective dissociation constant based on experimental binding rates as listed in Table 1. In equilibrium, the dissociation constant is bounded by the ADP and ATP dissociation constant K ADP,ATP eff . Ultra-affinity corresponds to K eff < K ADP D and infra-affinity to K eff > K ADP D , these regimes can be achieved only with a non-equilibrium driving force. In addition, infra-affinity can be achieved only with a minimum driving force ∆µ > (1/β) ln(K ATP D /K ADP D ). The dissociation constant can be optimally tuned close to the ultra-affinity limit (k ADP − /k ATP + ) and the infra-affinity limit (k ATP − /k ADP + ), provided sufficient driving force ∆µ. The black arrow shows the effect of stress which can trigger ultra-affinity in response to experimental observations (13)(14)(15)(16)(17) as explained in the main text. The normal system H normal has the following parameters K D ADP K D ATP k - ADP /k + ATP k - ATP /k + ADP e -βΔμ K D ADP e βΔμ K D ADP 1 β ln ( K D ATP /K D ADP ) Ultra-(ω + = 10 −4 s −1 , ω S + = 0.01s −1 , κ + = 27s −1 , κ − = 0.5s −1 and κ S + = 0.003s −1 ) . The other rates are computed using local detailed balance, see Eqs. 9-11 in Appendix A. We first consider ultra-affinity qualitatively as explained in (12). In the case of Hsp70, the substrate binding and unbinding kinetics is faster in the ATP state as it is in the ADP state (see Table 1). The slow substrate binding and unbinding kinetics in the ADP state is indicated by horizontal dashed arrows in Fig. 1. For ultra-affinity, when a substrate is bound to Hsp70 it should ideally switch to the closed S · H ADP configuration to benefit from slow substrate unbinding k ADP − , whereas in the absence of the substrate Hsp70 should ideally switch to the open H ATP configuration to benefit from fast substrate binding k ATP + . Infra-affinity on the other hand requires an opposite switching behavior. When a substrate is bound to Hsp70 it should ideally switch to the open S · H ATP configuration to benefit from fast substrate unbinding k ATP − , whereas in the absence of the substrate it should ideally switch to the closed H ADP configuration to benefit from slow substrate binding k ADP + . Note that with a finite budget of chemical energy ∆µ, such an ideal switching behavior cannot be perfectly realized as shown in the following. An optimization of the effective dissociation constant K eff , while keeping the thermodynamic constraint Eqs. 2 and 6 allow us to derive bounds for K eff as shown in Fig. 2 (see Appendix A for the derivation). For a given energy budget ∆µ and substrate kinetics k ATP ± , k ADP ± (see Table 1 for experimental values), we optimize K eff with respect to the hydrolysis rates ω ± , ω S ± and nucleotide exchange rates κ ± , κ S ± . The driving force ∆µ then determines the maximum decrease of dissociation constant allowing K eff < K ADP D . First, e −β∆µ K ADP D ≤ K eff provides a simple lower bound. Second, k ADP − /k ATP + ≤ K eff provides another simple lower bound, which is relevant in the limit of infinite driving (∆µ → ∞). It corresponds to the ideal case where binding occurs only in the open ATP state and unbinding in the closed ADP state. Both lower bounds on the effective dissociation constant are shown as dashed purple lines in Fig. 2. Finally, a minimization of the effective dissociation constant K eff by varying the kinetic parameters while satisfying the energetic constraints leads to an analytic lower bound K min eff . The analytical expression and derivation of K min eff is presented in the Appendix A. Note that in the case of high substrate concentration, the effective dissociation constant K eff is bounded by the equilibrium dissociation constants K ADP,ATP D . Infra-affinity (K eff > K ATP D ), in contrast with ultra-affinity, requires investing a minimum free energy difference ∆µ > (1/β) ln(K ATP D /K ADP D ) which must work against an equilibrium bias that arises from the allosteric interaction induced by different dissociation constants K ADP, ATP D (see Fig. 2). Hence, the binding and unbinding kinetics of Hsp70 from Table 1 favors ultra-affinity. For ∆µ > (1/β) ln(K ATP D /K ADP D ), however, infraaffinity can be achieved, where a simple upper bound on the effective dissociation constant K eff is given by min e β∆µ K ADP D , k ATP − /k ADP + which is indicated by two dashed red lines in Fig. 2. A more detailed calculation, as shown in Appendix A, leads to an upper bound on infra-affinity K max eff , which is obtained from a maximization of the effective dissociation constant K eff while keeping the energetic constraint fixed. Unlike ultra-affinity, the upper bound k ATP − /k ADP + corresponds to the case, where binding occurs only in the closed ADP state and unbinding only in the open ATP state. Most remarkably, a similar concept was found in kinetic proofreading, where it was called "anti-proofreading" (26). This kinetic limit uses the non-equilibrium features to lower the discrimination between substrates. In this regime, the non-equilibrium force must also overcome a critical equilibrium bias. The cofactors of Hsp70 are needed to reach affinity beyond the equilibrium range. In the case of ultra-affinity, our optimization showed that, first, hydrolysis must be much faster than nucleotide exchange in the substrate-bound state, whereas nucleotide exchange must be much faster than hydrolysis in the absence of a substrate. Second, these reactions must be much faster than substrate binding and unbinding kinetics. These two key requirements are necessary to optimally tune the effective dissociation constant beyond the equilibrium restrictions. Most remarkably, J-proteins and nucleotide exchange factors (NEFs) have a similar role in the Hsp70 system (7). First, J-proteins bind to a specific sequence of amino acids present in non-native (misfolded) proteins and catalyze the hydrolysis reaction by four order of magnitudes over the basal rate (10). The second key elements are the NEFs. They have high affinity for the H ADP state and catalyze the dissociation of ADP (34). NEFs should ideally only boost nucleotide exchange without bound substrate. Nevertheless, J-proteins catalyze hydrolysis much stronger than NEFs catalyze nucleotide exchange in the presence of a substrate, thus, favoring hydrolysis over nucleotide exchange in that case. In the absence of a substrate, hydrolysis is slow (not catalyzed), whereas nucleotide exchange is catalyzed by NEFs (see Fig. 3). Experimental observations on the Hsp70 system, thus, match the kinetics requirements found during our optimization. Hsp70 can tune its affinity depending on the stress level. Using our model, we can show the effect of stress on the affinity induced by the response of cofactors and the change in nucleotide levels measured during heat-shock experiments. During heat-shock experiments, where the temperature is increased from 37 • to 45 • , the effect of J-proteins on hydrolysis becomes stronger (≈ 100ω S ± ) and the effect of NEFs becomes weaker (≈ 0.1κ ± , 0.1κ S ± ) (16,17). Moreover, the ATP concentration decreases and the ADP concentration increases leading to a weaker driving force ∆µ (13-15). We consider a system tuned in the equilibrium range at ∆µ = 10, which is represented at H normal in Fig. 2. We model the response of cofactors to stress by tuning the hydrolysis in the substrate-bound state by a factor 100 (rates ω S ± ) and decreasing the nucleotide exchange rates by a factor 10 (rates κ ± , κ S ± ). In addition, we model the decrease of the ATP ([ATP] stress ≈ 0.4[ATP] normal ) leading to a decrease of the rates κ + , κ S + by a factor e −1 , and the increase of ADP ([ADP] stress ≈ 7[ADP] normal ) leading to an increase of the rates κ − , κ S − by a factor e 2 . We obtain a stressed system at H stress in Fig. 2 which can achieve ultra-affinity at ∆µ = 7 despite a weaker non-equilibrium driving. This example shows that ultra-affinity can dynamically be turned on and off. J-domain containing proteins have specific substrate preferences, necessary to recruit Hsp70s on the targets that need their intervention. Since ultra-affinity is triggered by ATP hydrolysis, which in turn is stringently accelerated by the interaction with the J-domain (see thick red arrows ω M ± in Fig. 3A), ultra-affinity only works on the substrates that have been selected for Hsp70 by J-proteins. In the case of protein folding, energy consumption enhances the affinity of Hsp70s only toward non-native (misfolded, unfolded and aggregated) proteins, because the associated J-proteins have a low affinity for native polypeeptides. After hydrolysis, J-proteins are expelled and unfolding can occur in the ADP state (7,35). After unfolding, the protein is then rejected through nucleotide exchange and unbinds from the ATP state (2,4). The specificity of J-proteins is thus central for efficient protein refolding and allows binding with high affinity and rejecting the substrate after unfolding. Ultra-affinity can only be achieved when reactions induce cycling in the counter clockwise direction as shown in the lower left panel in Fig. 3A (see also discussion from Appendix A). For a partially folded substrate X, nucleotide exchange reactions do not induce directed cycling, which allows the substrate to be rejected with a higher effective dissociation constant K eff . In an ideal refolding scheme, partially folded substrate should be rejected with infra-affinity, where reactions should induce cycling in the opposite direction of ultra-affinity. In the case of Hsp70, this would require a strongly catalyzed hydrolysis reaction in the absence of substrate. Hence, due to the allosteric barrier ∆µ > (1/β) ln(K ATP D /K ADP D ), as shown in Fig. 2, it may be quite challenging to realize infra-affinity for the Hsp70 system. Small GTPases, however, may provide a quite similar kinetic scheme for which infra-affinity will be biologically more relevant. Small GTPases are molecular switches which are key regulators in many cellular processes (36). They are GTP-driven machines going trough a GTP hydrolysis cycle similar to Hsp70. Small GTPases also work with two cofactors: GTP hydrolysis is boosted by GTPase activating proteins (GAPs) and GTP exchange is catalyzed by Guanine nucleotide exchange factors (GEFs) (37,38), see Fig. 3B. They rely on allosteric regulation, where the active GTP-state binds more tightly to its effector (substrate) than the inactive GDP-state contrary to Hsp70 (39)(40)(41)(42). Surprisingly, bound complexes have a short lifetime (43). Infra-affinity with fast binding kinetics can optimize the transmission of signal with fast activation and efficient release to further molecules (44,45). In Fig. 3B, we show the kinetic requirements to achieve infra-affinity in small GTPases. First, nucleotide exchange must be much faster than hydrolysis in the substrate-bound state and hydrolysis must be much faster than nucleotide exchange in the absence of a substrate. Second, these reactions must be much faster than substrate binding and unbinding kinetics. Most remarkably, GAPs and GEFs fulfills these conditions for small GTPases. GEFs catalyze the dissociation of GDP similar to NEFs, which results in faster nucleotide exchange κ ± , κ E ± reactions. Contrary to Hsp70, GAPs boost hydrolysis by binding to small GTPases as an effector (substrate), therefore, resulting in faster hydrolysis ω ± reactions (37). Second, these reactions are strongly catalyzed (38,46). Experimental observations of small GTPases, thus, match the kinetic requirements for infra-affinity found during our optimization; see Appendix B for more details. DISCUSSION AND SUMMARY In this work, we have assessed the thermodynamic bounds of ultra-affinity, namely, how effectively the energy available from ATP hydrolysis can be converted into the non-equilibrium enhanced affinity of Hsp70 for its substrates. Ultra-affinity can prevent aggregation and improve the refolding efficiency. Most substrates bound to Hsp70 are protected and do not aggregate (47). In addition, substrates bound to Hsp70 are unfolded rather than misfolded compared to free specimens (48,49). The unfolding process could be due to an interaction between Hsp70 and its substrate in the closed ADP-state (50). Therefore, this unfolding activity coupled with ultra-affinity could ultimately help to shift the energy landscape to favor the refolding of the substrate after release in an unfolded, refoldable species. Recent single-molecule experiments show new insights into the role of Hsp70 in protein folding (51). For instance, they show that Hsp70 also protects partially folded structures against aggregation in addition to misfolded substrates. Moreover, they find that Hsp70 can both stabilize and destabilize native structures depending on the nucleotide concentrations. It would be interesting to add protein dynamics to our model (as sketched in Fig. 3A) and investigate refolding strategies under different stress levels. The relation between stress and affinity could be further investigated in experiments. In Fig. 1, we show that changes in nucleotide levels (13)(14)(15) and temperature dependent cofactors (16,17) can lead to an increase of affinity with a smaller driving force ∆µ. Using our framework, we explain how these two responses to stress lead to a higher affinity for substrates (see Fig. 3). It would be interesting to make in vitro experiments investigating the relation between the driving force ∆µ, the individual ADP and ATP levels together with the effective affinity. H ADP M·H ADP M·H ATP H ATP [M]k ADP + k ADP − [M]k ATP + k ATP − ω M + ω M − κ M − κ M + ω + ω − κ − κ + X·H ADP H ADP X·H ATP H ATP [X]k ADP + k ADP − [X]k ATP + k ATP − ω X + ω X − κ X − κ X + ω + ω − κ − κ + G GDP E·G ADP E·G GTP G GTP [E]k GDP + k GDP − [E]k GTP + k GTP − ω E + ω E − κ E − κ E + ω + ω − κ − κ + A B Hsp70 Small GTPases Ultra-affinity Infra-affinity Adjusting the nucleotide levels accordingly, increasing the concentration of J-proteins and decreasing the concentration of NEFs should trigger an ultra-affinity response in Hsp70, thus, mimicking a heat-shock without any temperature change. We found that infra-affinity is quite difficult to realize with Hsp70 due to an allosteric barrier. For small GTPases, however, we show how infra-affinity can be essential for efficient molecular switches. While the link between infra-affinity and small GTPases has not been experimentally observed yet, the function of small GTPases can benefit from infra-affinity. We have shown under which conditions small GTPases could achieve fast binding and unbinding with an effector coupled with low affinity (see Fig. 3B and Appendix B). Our analysis was based on biological properties of cofactors and the fact that the GTP state has a higher affinity to effectors than the GDP state. Our framework shows that infra-affinity could only be achieved if the binding and unbinding kinetics in the GTP state are faster than the GDP state (k GTP ± > k GDP ± ) which has not been confirmed yet experimentally to our knowledge. Measuring slow binding and unbinding rates in a low affinity state may be difficult during experiments. Nevertheless, if an experiment directly measured an effective dissociation constant K eff lower than the equilibrium range, our framework could be used to give insights into the the binding and unbinding kinetics in the GDP state. The framework proposed in (12) and exploited here might also be applicable to other non-equilibrium systems relying on allosteric regulation. Specifically, other chaperones system such as Hsp90s, Hsp100s or the GroEL-GroES system (2,4), which have been proposed to exhibit non trivial non-equilibrium properties (52) could benefit from our general scheme and give us more insights into the role of energy consumption in these systems. Figure 4: Four state model from Fig. 1 with total transition rates α 1 , . . . , α 4 including both hydrolysis and nucleotide exchange reactions. H ADP S · H ADP H ATP S · H ATP [S]k ADP + k ADP − [S]k ATP + k ATP − α 1 α 2 α 3 α 4 Appendix A: Derivation of the thermodynamic bounds We introduce a simple binding model in Fig. 4, where we consider the total rates α 1 = ω + + κ − , α 2 = ω − + κ + , α 3 = ω S + + κ S − , and α 4 = ω S − + κ S + , which include both hydrolysis ω ± and nucleotide exchange κ ± reactions. We define the coarse-grained affinity A ≡ ln α 2 α 3 k ADP − k ATP + α 1 α 4 k ADP + k ATP − = ln K ADP D (ω − + κ + ) ω S + + κ S − K ATP D (ω + + κ − ) ω S − + κ S + ,(8) where we have identified the total transition rates in the second step. Note that this coarse-grained affinity A should not be confused with the chemical affinity which is the inverse of the effective dissociation constant K eff . A positive sign of A indicates a cycling in the counter clockwise direction. Most importantly, A = 0 can be attained only if the system is driven out of equilibrium. The local detailed balance relation from Eqs. 2 and 6 impose the following constraints on the transition rates ω + κ + ω − κ − = ω S + κ S + ω S − κ S − = e β∆µ ,(9)ω + κ S + K ATP D ω − κ S − K ADP D = e β∆µ ,(10)ω S + κ + K ADP D ω S − κ − K ATP D = e β∆µ ,(11) where β = 1/(k B T ) is the inverse thermal energy. From Eqs. 9-11, the coarse-grained affinity 8 is bound between − β∆µ ≤ A ≤ β∆µ.(12) Thereby, a maximization over κ ± , κ S ± , ω ± , ω S ± while keeping Eqs. 9-11 fixed is equivalent to a maximization over the coarse-grained rates α 1 , . . . , α 4 while satisfying Eq. 12. Denoting the steady state probabilities by P i , where i labels the states i = H ATP , H ADP , S · H ATP , S · H ADP , allows us to write the effective dissociation constant in the form K eff = P off P on [S] ≡ P H ADP + P H ATP P S·H ADP + P S·H ATP [S],(13) which is the inverse of the effective affinity for substrates. We calculate the stationary probability distribution with standard methods (53,54) and obtain K eff = (α 1 + α 2 )(α 3 k ADP − + α 4 k ATP − ) + B (α 3 + α 4 )(α 1 k ADP + + α 2 k ATP + ) + C ,(14) where B = (α 1 + α 2 )k ADP − k ATP − + (α 3 k ADP − k ATP + + α 4 k ADP + k ATP − )[S] and C = α 1 k ATP − k ADP + + α 2 k ADP − k ATP + + (α 3 + α 4 )k ADP + k ATP + [S]. The terms B and C are linear functions of the total transition rates α 1 , . . . , α 4 which do not allow the dissociation constant to be controlled beyond the equilibrium restrictions, since min[K ADP D , K ATP D ] ≤ B/C ≤ max[K ADP D , K ATP D ]. Note that in the limit of high substrate concentration, the effective dissociation constant Eq. 14 is also restricted to the equilibrium range. Any dissociation constant can be written in the form K eff = pk ADP − + (1 − p)k ATP − qk ADP + + (1 − q)k ATP +(15) where p, q are positive weight parameters satisfying 0 ≤ p, q ≤ 1 and e −β∆µ ≤ p(1 − q) (1 − p)q k ADP − k ATP + k ADP + k ATP − ≤ e β∆µ ,(16) which follows from Eqs. 8 and 12. The maximal attainable range for the dissociation constant is given by K min D ≤ K eff ≤ K max D(17) where K min D (K max D ) is the minimum (maximum) of Eq. 15 with respect to p, q within the allowed range from Eq. 16. The effective dissociation constant K eff is best controlled if the vertical transitions α 1 , . . . , α 4 are much faster than the transition rates involving substrate binding and unbinding, where p ≈ α 3 /(α 3 + α 4 ) and q ≈ α 1 /(α 1 + α 2 ). Note that K eff saturates to its lower (upper) limit in Eq. 17 if the weight parameters p, q are chosen such that the expression in Eq. 16 equals e β∆µ (e −β∆µ ), i.e., Eq. 16 must saturate as well. Specifically, for Hsp70 with the binding and unbinding kinetics from Table 1, we obtain the following bounds. For ∆µ ≥ 0 the minimal dissociation constant is given by K min D = K ADP D K ATP D k ATP + − k ADP + e β∆µ 2 √ Θ + k ATP − e β∆µ − k ADP − e β∆µ k ADP + − k ATP + + √ Θ + k ADP + k ATP − e β∆µ − k ADP − k ATP + (e β∆µ − 1) − √ Θ + ,(18) where Θ + = k ADP − − k ATP − k ADP + − k ATP + −1 + e β∆µ e β∆µ − K ADP D K ATP D e β∆µ k ADP + k ATP − .(19) Note that the system is optimally tuned for ultra-affinity when A = β∆µ (i.e., p, q maximize Eq. 16). Therefore, the lowest dissociation constant requires hydrolysis to be much faster than nucleotide exchange in the substrate-bound state, whereas in the absence of a substrate nucleotide exchange must be much faster than hydrolysis. Notably, infra-affinity requires a large enough chemical potential ∆µ > (1/β) ln(K ATP D /K ADP D ), in which case the maximal dissociation constant is given by However, small chemical potentials 0 ≤ β∆µ ≤ ln(K ATP D /K ADP D ) do not allow infra-affinity, since K max D = K ATP D = k ATP − /k ATP + . Note that the system is optimally tuned for infra-affinity when A = −β∆µ (i.e., p, q minimize Eq. 16). Therefore, the maximal dissociation constant requires nucleotide exchange to be much faster than hydrolysis in the substrate-bound state, whereas in the absence of a substrate hydrolysis must be much faster than nucleotide exchange. K max D = K ADP D K ATP D k ADP + − k ATP + e β∆µ 2 √ Θ − k ADP − e β∆µ − k ATP − e β∆µ k ADP + − k ATP + + √ Θ − k ADP − k ATP + e β∆µ − k ADP + k ATP − (e β∆µ − 1) + √ Θ − ,(20) Appendix B: Analysis of Small GTPases We provide in this section a detailed analysis of small GTPases. Small GTPases go through a GTP hydrolysis cycle similar to Hsp70. Our four state model Fig. 4 introduced previously can be extended to small GTPases by changing ATP(ADP) to GTP(GDP). Small GTPases have fast binding kinetics and high affinity to substrate in their active GTP state. In the inactive GTP state, they have low affinity for substrate and slow binding kinetics (40)(41)(42). Experiments have measured the binding and unbinding rates in the GTP state (40), however, only estimation of binding kinetics in the GDP state are available at the current time. We show here under which kinetic conditions small GTPases can be tuned for infra-affinity. Small GTPases work with cofactors similar to Hsp70. Therefore, we can tune the effective dissociation constant K eff by varying the the hydrolysis ω ± and nucleotide exchange rate for a given energy budget ∆µ. Similar to Appendix A, we extremize Eq. 22 with respect to p, q within the allowed range from e −β∆µ ≤ p(1 − q) (1 − p)q k GDP − k GTP + k GDP + k GTP − ≤ e β∆µ ,(22) which is adopted from Eq. 16, where the kinetic parameters from Fig. 1 are replaced by the ones from Fig. 3B. We obtain bounds for K eff as shown in Figure 3 : 3The role of cofactors in Hsp70s and small GTPases. (A) The effect of Hsp70's cofactors depends on the state of the substrate. For misfolded substrate (left panel), J-proteins binds to misfolded substrate and strongly catalyze hydrolysis ω M ± reactions (7, 10). Nucleotide exchange factors (NEFs) catalyze the nucleotide exchange κ ± , κ M,X ± reactions. Ultra-affinity requires reaction cycling in the counter clockwise direction as shown in the lower left panel. For partially folded substrate (middle panel), J-proteins do not interact with them. However, the mechanism of NEFs does not depend on the state of substrate (34). (B) Small GTPases have cofactors which promote infra-affinity. Guanine nucleotide exchange factors (GEFs) are similar to NEFs, they facilitate GDP dissociation which boost nucleotide exchange κ ± , κ E ± reactions. Contrary to J-proteins, GTPase activating proteins (GAPs) compete with substrate effector to bind small GTPases and catalyze the hydrolysis ω ± reactions (37). In addition, we have K GDP GTP − /k GTP + . Based on these observations, infra-affinity can only be achieved when k GDP ± < k GTP ± (see Appendix B and Fig. 5). Figure 5 : 5whereΘ − = k ADP − − k ATP − k ADP + − k ATP + e β∆µ − 1 e β∆µ − K ATP D K ADP D e β∆µ k ADP − k ATP + .Thermodynamic bounds on the effective dissociation constant for small GTPases based on experimental binding rates. In the GTP state k GTP+ = 4.3s −1 µM −1 , k GTP − = 6.4s −1 where K GDP D = 1.5µM(40). To achieve infra-affinity, the binding and unbinding kinetics must be slow, we choose rate k GDP+ = 10 −4 s −1 µM −1 and k GDP − = 10 −2 s −1 such that K GDP D ≈ 10 2 -10 3 K GTP D and matches experimental estimation (40-42). Ultra-affinity can be achieved only with a minimum driving force ∆µ > (1/β) ln(K GDP D /K GTP D ). Fig. 5 . 5Similar to Hsp70, simple lower and upper bounds are given respectively by min e β∆µ K GDP D , k GTP − /k GDP + and max e β∆µ K GDP D , k GTP − /k GDP + . 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[ "the double-trace spectrum of super gluons", "the double-trace spectrum of super gluons", "the double-trace spectrum of super gluons", "the double-trace spectrum of super gluons" ]	[ "J M Drummond \nSchool of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom\n", "R Glew \nSchool of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom\n", "M Santagata \nSchool of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom\n", "J M Drummond \nSchool of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom\n", "R Glew \nSchool of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom\n", "M Santagata \nSchool of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom\n" ]	[ "School of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom", "School of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom", "School of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom", "School of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom", "School of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom", "School of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJHighfieldUnited Kingdom" ]	[ "BCJ relations in AdS 5 × S", "BCJ relations in AdS 5 × S" ]	We revisit the four-point function of super gluons in AdS5 × S 3 in the spirit of the large p formalism and show how the integrand of a generalised Mellin transform satisfies various non-trivial properties such as U (1) decoupling identity, BCJ relations and colour-kinematic duality, in a way that directly mirrors the analogous relations in flat space. We unmix the spectrum of double-trace operators at large N and find all anomalous dimensions at leading order. The anomalous dimensions follow a very simple pattern, resembling those of other theories with hidden conformal symmetries.	10.1103/physrevd.107.l081901	[ "https://export.arxiv.org/pdf/2202.09837v1.pdf" ]	247,012,044	2202.09837	5e4f76dcfcf783e1b33775ca3f63c7133fea815f	the double-trace spectrum of super gluons 20 Feb 2022 J M Drummond School of Physics and Astronomy University of Southampton SO17 1BJHighfieldUnited Kingdom R Glew School of Physics and Astronomy University of Southampton SO17 1BJHighfieldUnited Kingdom M Santagata School of Physics and Astronomy University of Southampton SO17 1BJHighfieldUnited Kingdom the double-trace spectrum of super gluons BCJ relations in AdS 5 × S 320 Feb 2022 We revisit the four-point function of super gluons in AdS5 × S 3 in the spirit of the large p formalism and show how the integrand of a generalised Mellin transform satisfies various non-trivial properties such as U (1) decoupling identity, BCJ relations and colour-kinematic duality, in a way that directly mirrors the analogous relations in flat space. We unmix the spectrum of double-trace operators at large N and find all anomalous dimensions at leading order. The anomalous dimensions follow a very simple pattern, resembling those of other theories with hidden conformal symmetries. Introduction Understanding properties of (quantum) gravity theories and their relation to gauge theories is a primary goal in modern physics. Most of these relations are obscured in a lagrangian formulation, and seem to manifest all their majesty only through observables such as scattering amplitudes. The study of scattering amplitudes has led to a series of impressive and deep results, for example BCJ dualities [1] and double-copy constructions [2] in various theories, both at tree and loop-level (for a recent review, see [3]). The upshot is that there seems to be an underlying common structure between gravity and gauge theories, yet to be fully understood. While most of these efforts have related to flat space, mainly because of the difficulties in performing such computations in curved backgrounds, a recent series of papers have begun the exploration of these properties in AdS backgrounds, both in Mellin space [4][5][6][7], as well as in position [8,9] and momentum space [10,11]. Many of these developments have use bootstrap methods that have been highly successful in studying supergravity in AdS [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], as well as string corrections [27][28][29][30][31][32][33][34][35][36][37][38][39]. In particular, in [5], AdS versions of colour-kinematic and double copy relations have been found. In this letter we further explore these relations by focusing on the four-point function of half-BPS operators dual to the scattering of four super gluons in AdS 5 × S 3 , first computed in [4]. In common with [5], we will focus on the 'reduced' Mellin amplitude (which manifests the supersymmetry of the theory) and cast this in a form which makes colour-kinematics and BCJ relations manifest by mirroring directly the form of the flat space amplitude. At leading order, the theory enjoys a hidden 8d conformal symmetry that nicely repackages all Kaluza-Klein modes into a simple reduced Mellin amplitude M p . In notation inspired by the 'large p' limit of [35] it takes the form M p = n s c s s + 1 + n t c t t + 1 + n u c u u + 1 , where the kinematic (n) and colour (c) numerators obey the same Jacobi type relations. We will see that the as-sociated colour-ordered amplitudes also satisfy BCJ relations for all Kaluza-Klein modes. These take the form, where M p (1,2,3,4) are the colour-ordered amplitudes. The fact that these relations directly mirror their flat space counterparts is related to the existence of the 8d conformal symmetry. In the second part of the letter, we unmix the spectrum of double-trace operators exchanged in the OPE and compute all the anomalous dimensions at leading order. These CFT data are an important part of the bootstrap program for computing one-loop correlators beyond the lowest KK mode correlator. As shown in e.g. [6,14,15,22], the CFT data of the double-trace operators can be used to build the leading discontinuities of the correlator and then from them one can construct the full amplitude with the help of crossing symmetry. In common with many other cases where the AdS theory exhibits a hidden conformal symmetry, we find that the anomalous dimensions are given by a strikingly simple formula whose form we sketch here, η ± τ = − 2 N δ (2) h,j δ (2) h,j (l ± 8d + 1) 4 .(3) Here the 'effective' spin l 8d and the quantity δ (2) h,j are functions of twist τ , spin l and SU (2) × SU (2) labels [ab] of the double-trace operators and we will give their specific form later on. The results are reminiscent of previous computations in other backgrounds [19,40,42], and suggest that the hidden conformal symmetry, unavoidably, plays a primary role in constraining the data of these SCFTs. AdS5 × S 3 Mellin transform and the large p formalism The AdS 5 × S 3 background arises in two basic stringy setups. One can either consider a stack of N D3branes probing F-theory 7-brane singularities or a stack of N F D7-branes wrapping an AdS 5 × S 3 subspace in the AdS 5 × S 5 geometry of a stack of N D3-branes. In both cases, the system preserves 8 supercharges, therefore the dual CFT is a 4d N = 2 theory with flavour group G F , which we will keep generic because it is mostly irrelevant for the details considered in this paper. The low-energy degrees of freedom are those of a N = 1 vector multiplet which transforms in the adjoint of G F . Upon reducing on the sphere, it provides an infinite tower of Kaluza-Klein modes organised in different multiplets. In the dual CFT, the super primaries of these multiplets are half-BPS scalar operators of the form O Ia1a2...ap;ā1ā2...āp−2 p . Here I is the colour index, p is the scaling dimension of the operator, a 1 , . . . , a p are symmetrised SU (2) R Rsymmetry indices and similarlyā i are indices of an additional SU (2) L flavour group; these last two groups realise the isometry group of the sphere S 3 . In practice, and as usual in these contexts, is convenient to contract the indices with auxiliary bosonic two-component vectors η andη to keep track of the SU (2) R × SU (2) L indices: O I p ≡ O I;a1a2...ap;ā1ā2...āp−2 p η a1 . . . η apηā1 . . .ηā p−2 . (4) In this paper we consider the amplitude of four super gluons, which we denote by G I1I2I3I4 p (x i , η i ,η i ) ≡ O I1 p1 O I2 p2 O I3 p3 O I4 p4 .(5) A crucial point is that, in these theories, the strength of the self-gluon coupling is larger than the coupling of gluons to gravitons [4]. In light of this, one can perform an expansion in 1/N in which gravity is 1/N suppressed. Schematically, we have G I1I2I3I4 p = G I1I2I3I4 disc, p + 1 N G I1I2I3I4 tree-gluon, p + · · ·(6) The first 'disconnected' term is a sum over products of two-point functions and takes the form of (generalised) free theory. In terms of OPE data it contains the leading order contributions to the three-point functions of the external operators with exchanged two-particle operators. We will refer to the second term as the 'tree-level' amplitude. The correlator is subject to constraints due to superconformal symmetry. In particular, the superconformal Ward identites [43] allow us to split it into two parts, each separately respecting crossing symmetry, G tree-gluon, p = G 0, p + P I A p .(7) The term G 0, p contains all contributions due to protected multiplets at this order in 1/N . The second term contains all the logarithmic terms which arise due to two-particle operators receiving anomalous dimensions. It contains certain kinematic factors P and I, due to bosonic and femionic symmetries respectively. First, let us define the propagator via g ij = y 2 ij x 2 ij(8) where y 2 ij = η i η j η iηj with η i η j = η ia η jb ǫ ab and similarly η iηj =η iāηjb ǫāb. We also introduce cross-ratios via Note that we can write the y,ȳ variables in terms of the η andη variables as y = η 1 η 2 η 3 η 4 η 1 η 3 η 2 η 4 ȳ = η 1η2 η 3η4 η 1η3 η 2η4 .(9) The kinematic factors are then given by P ≡ g ks 12 g kt 14 g ku 24 g 13 g 24 p3 η 1η3 2 η 2η4 2 , I = (x − y)(x − y) ,(10) where k s = p1+p2−p3−p4 2 , k t = p1+p4−p2−p3 2 , k u = p2+p4−p3−p1 2 . Note that, due to the presence of the factor I = (x − y)(x − y), the remaining function A I1I2I3I4 p has the same degree in y,ȳ. Moreover, since A I1I2I3I4 p is symmetric under y,ȳ exchange, we can write it as a function ofŨ ,Ṽ as well as U and V and the charges p. The function A I1I2I3I4 p admits a very compact and natural representation, that extends the well known Mellin transform [44,45] to the compact space. The transform makes manifest the so-called large p limit [35] -where p here refers to the charges -and it was found to be very useful in the context of AdS 5 × S 5 [35,37] and AdS 3 × S 3 backgrounds [40]. In our conventions the generalised Mellin transform M is defined via A I1I2I3I4 p = − dsdt dsdt U s V tŨsṼt Γ M I1I2I3I4 p (11) where M I1I2I3I4 p ≡ M I1I2I3I4 p (s, t,s,t). The kernel Γ is factorised into AdS 5 and S 3 contributions and takes the form Γ = S Γ s Γ t Γ u with S = π 2 (−)t(−)ũ sin(πt ) sin(πũ) , Γ s = Γ[−s]Γ[−s+ks] Γ[1+s]Γ[1+s+ks](12) and Γ t , Γ u defined similarly. Note that the Mellin variables obey the relations, s + t + u = −p 3 − 1,s +t +ũ = p 3 − 2 ,(13) which may be used to eliminate u andũ. Note also that the amplitude A I1I2I3I4 p is polynomial inŨ andṼ . In fact, the integral overs,t can be turned into a discrete sum over a certain domain that in our case is given by T = {s ≥ max(0, −k s ),t,ũ ≥ 0} .(14) The contour integral in s and t requires a little care and we will return to this point in the next section. The double integral (11), when combined with the amplitude M p given in the next section, precisely coincides with the result given in [4]. This generalised AdS 5 × S 3 Mellin transform is quite useful because, as shown in [35], in the large p limit, the integrals localise on a classical saddle point. The authors show that the computation matches with that of four geodesics shooting from the boundary and meeting in a common bulk point at which the particles scatter as if they were in flat space. At the saddle point, the 'boldface' variables s = s +s, t = t +t, u = u +ũ, s + t + u = −3 (15) become proportional to the flat space Mandelstam variables. This explains why, for large p, the Mellin amplitude M I1I2I3I4 p is fixed by the flat space S-matrix with the Mandelstam variables replaced by the bold-face variables s, t, u. Moreover, as we will see, the integrand M I1I2I3I4 p satisfies BCJ and double-copy relations, directly analogous to the flat space relations, incorporating all Kaluza-Klein modes. BCJ and colour-kinematics in AdS5 × S 3 Let us consider the field theory amplitude computed in [4] within this formalism. As in [5] we consider the reduced Mellin amplitude M p . In the colour-factor basis, the amplitude M p takes the following very simple form when written in terms of the bold-face variables, M I1I2I3I4 p = n s c s s + 1 + n t c t t + 1 + n u c u u + 1 .(16) Here we have n s = 1 3 1 t + 1 − 1 u + 1 , c s = f I1I2J f I3I4J , n t = 1 3 1 u + 1 − 1 s + 1 , c t = f I1I4J f I2I3J , n u = 1 3 1 s + 1 − 1 t + 1 , c u = f I1I3J f I2I4J . (17) As described above, the large p limit ensures that the amplitude reduces to the flat amplitude with the Mandelstam replaced by bold-face variables M I1I2I3I4 p − −−−−−−− → s,t,s,t, p→∞ V I1I2I3I4 YM (s, t), V I1I2I3I4 YM (s, t) = n s c s s + n t c t t + n u c u u , n s = 1 3 1 t − 1 u , . . .(18) where V I1I2I3I4 YM is the field theory gluon amplitude in flat space, see e.g. [46]. Note that this limit somewhat restores the symmetry between AdS and S; in this sense it is a generalisation of the usual flat space limit in which only the AdS (Mellin) variables s, t are taken to be large. In principle, away from large p, nothing would prevent the amplitude to depend on s,s, · · · separately. However, from (16) we see that in fact the full amplitude M is just a function of the bold face variables. This fact is a consequence of a hidden 8d conformal symmetry of the amplitude. This symmetry allows one to promote the correlator M I1I2I3I4 2222 to a generating function for correlators with arbitrary charges p. Then, M I1I2I3I4 p follows from 'covariantising' M I1I2I3I4 2222 : M I1I2I3I4 2222 (s, t) 8d-symm − −−−− → M I1I2I3I4 p = M I1I2I3I4 2222 (s, t). These features are entirely analogous to AdS 3 × S 3 [40,[47][48][49] and AdS 5 × S 5 [20,35] backgrounds where the dynamics is also controlled by hidden conformal symmetries. In other words, A I1I2I3I4 p is generated from A I1I2I3I4 2222 upon acting with a differential operator which takes a very simple form. In fact, we can give a general formula of the operator that interpolates between the three cases. Parametrising the space as AdS θ1+1 ×S θ2+1 , the amplitude A for general p is generated from the one with the lowest charges p = (qqqq) with q = θ1 2 via a differential operator, A I1I2I3I4 p = D θ1,θ2 p U θ1+θ2 2 A I1I2I3I4 qqqq , q = θ1 2 . (19) The operator D θ1,θ2 p takes the following form D θ1,θ2 p = U − θ1+θ2 2 s,t Ũ U s Ṽ V tD θ1,θ2 p,s,t(20) wherê D θ1,θ2 p,s,t = a={0,ks} (U∂U +1− θ1+θ2 2 −s−a)s+a (−) a (s+a)! (21) × b={0,kt} (V ∂V +1−t−b)t +b (−) b (t+b)! c={0,ku} (U∂U +V ∂V )ũ+c (ũ+c)! and we turned the sphere integral into a sum restricted to the domain T . The operator transforms the gamma functions of A I1I2I3I4 qqqq into those of A I1I2I3I4 p and replaces s, t, u with s, t, u, as it can be easily checked using (a consequence of) Euler's reflection identity. In fact, as observed above, with the AdS 5 × S 3 background, our variables obey s + t + u = −3. Therefore the the Mellin amplitude M is literally the same function as the flat space amplitude with the Mandelstam variables s, t, u replaced by the shifted bold face variables (s + 1), (t + 1), (u + 1). It follows immediately that all the relations obeyed by the flat space amplitudes also apply to M. Note that it is not trivial that this holds; for example, the analogous relation for AdS 5 ×S 5 is s+t+u = −4 [35]. As an example of the properties obeyed by M we have that n s + n t + n u = 0 , c s + c t + c u = 0 ,(22) which gives an AdS version of the colour-kinematic duality, which was already observed in [5]. Note that (22) captures this duality for all Kaluza-Klein modes. This duality is intimately connected with the so-called BCJ relations between colour-ordered amplitudes. Recall that the full colour-dressed amplitude is: M I1I2I3I4 p = P(2,3,4) Tr T I1 T I2 T I3 T I4 M p (1, 2, 3, 4)(23) where the partial amplitudes M p (1, 2, 3, 4) are the colour-ordered amplitudes and P(2, 3, 4) are the permutations of points (2,3,4). The translation from one basis to another is: c s = Tr T I1 T I2 T I3 T I4 + Tr T I1 T I4 T I3 T I2 − Tr T I1 T I2 T I4 T I3 − Tr T I1 T I3 T I4 T I2 , c t = Tr T I1 T I4 T I2 T I3 + Tr T I1 T I3 T I2 T I4 − Tr T I1 T I4 T I3 T I2 − Tr T I1 T I2 T I3 T I4 , c u = Tr T I1 T I3 T I4 T I2 + Tr T I1 T I2 T I4 T I3 − Tr T I1 T I3 T I2 T I4 − Tr T I1 T I4 T I2 T I3 . (24) The colour-ordered amplitudes then read as follows, where we used the on-shell relation s + t + u = −3. We stress again that the relations (27) Having introduced the colour-ordered amplitudes, let us return to the issue of the contour in the Mellin integral (11). It should be noted that the presence of poles at s = −1, t = −1 and u = −1 is potentially a problem for the contour of integration. In fact, since s + t + u = −3, the simultaneous presence of these poles leaves no region in the real s, t plane for the contour to pass through, while separating left moving and right moving sequences of poles in the Mellin integrand. Thus the same property which leads to the direct analogy with the flat space amplitudes also leads to a subtlety in returning to position space from Mellin space. For the colour ordered amplitudes, one does not have all three poles present simultaneously. Thus we propose that the correct definition for the contour is tied to the colour-ordering and we define analogously a colour-ordered correlator, A(1, 2, 3, 4) = − dsdt dsdt U s V tŨsṼt Γ M(1, 2, 3, 4) , The contour can now be taken to lie slightly below s = −1 and t = −1. Note then that this introduces a subtlety in interpreting the BCJ relations (27) back in position space, since the left and right hand sides of these equations are to be integrated over slightly different contours. To conclude, let us point out that there is also an AdS version of the double-copy prescription [5]. Replacing colour with kinematic factors we get M I1I2I3I4 p − −−− → ci→ni n 2 s s + 1 + n 2 t t + 1 + n 2 u u + 1 (28) = 1 (s + 1)(t + 1)(u + 1) ∝ M SUGRA p . This is nothing but the SUGRA amplitude in AdS 5 × S 5 [12] rewritten in the large p formalism [35], upon reinterpreting s, t, u as the N = 4 variables, i.e. subject to the constraint u = −s − t − 4. Note also that, similarly to flat space [3], we can use BCJ and colour-kinematic duality to derive an AdS version of the KLT relations: Long disconnected free theory The rest of the letter will be devoted to investigate the structure of the anomalous dimensions of the doubletrace operators exchanged in the OPE at large N . In order to do so, we need two ingredients: the superconformal block decomposition of disconnected generalised free theory and that of the log U discontinuity of the tree-level correlator. The anomalous dimensions are then nothing but the eigenvalues of a certain matrix built out of the block coefficients of these two decompositions. On top of the above mentioned (usual) technology, we also have to deal with the non-trivial flavour structure of the amplitude. However, since all of this just amounts to considering certain symmetric or antisymmetric combinations built out of the correlator, we postpone the discussion on flavour structures to the end of next section. A more detailed discussion can be found in [6]. Let us begin with disconnected free theory. The only correlators with non-zero disconnected contributions are with pairwise equal charges and their spacetime dependence can be computed by performing simple Wick contractions. We have G I1I2I3I4 disc,pqpq =δ I1I2 δ I3I4 δ pq g p 12 g p 34 η 1η2 2 η 3η4 2 + δ I1I3 δ I2I4 g p 13 g p 24 η 1η3 2 η 2η4 2 u-channel + δ I1I4 δ I2I3 δ pq g p 14 g p 23 η 1η4 2 η 2η3 2 t-channel .(30) However, due to the non-trivial colour structure of the amplitude, only representations with a definite parity under t ↔ u exchange enter the OPE. In practice, we need to decompose the following combinations of diagrams G ± disc,pqpq = δ pq g p 14 g p 23 η 1η4 2 η 2η3 2 ± g p 13 g p 24 η 1η3 2 η 2η4 2 (31) Now, following [43], we first extract the unprotected contribution and then decompose it in long superblocks [52], whose form is given in the appendix. The block decomposition reads G ± disc,pqpq long = τ L ± τ L τ ,(32) where L τ are the long superblocks. We find that the coefficients take a particularly simple form, L ± τ = − ±1 + (−1) a+l δ pq (p − 1)(q − 1) A h AhB j Bjδ .(33) Here the A and B factors are given by A h = Γ(h + p−q 2 )Γ(h − p−q 2 )Γ(h + p+q 2 − 1) Γ(2h − 1)Γ(h − p+q 2 + 1) ,(34)B j = Γ(2 − 2j) Γ(1 − j + p−q 2 )Γ(1 − j − p−q 2 ) × 1 Γ( p+q 2 + j − 1)Γ( p+q 2 − j) , while δ is given by δ = δ (2) h,j − δ (2) h,j δ (2) h,j δ (2) h,j , δ (2) h,j = (h − j)(h + j − 1). (35) Here, h,h and j,j label, respectively, the conformal and internal representations. We can also express them in terms of the more common quantum labels τ = (τ, b, l, a) h = τ 2 + 1 + l,h = τ 2 , j = − b 2 − a,j = − b 2 ,(36) where τ, l are twist and spin, and b, a can be seen as the analogues of twist and spin on the sphere. Note the different ways the two internal SU (2) factors enter the coefficients. On the one hand, SU (2) L only comes in through the function Bj. On the other hand, the decomposition under the R-symmetry group SU (2) R produces also the function δ and, in particular the combination δ (2) h,j δ (2) h,j . This object is the eigenvalue of a Casimir operator operator acting on the blocks, D 4 U p43 2Ũ 2− p43 2 (x −x)G τ,l H b,a = δ (2) h,j δ (2) h,j U p43 2Ũ 2− p43 2 (x −x)G τ,l H b,a .(37) Here the differential operator D 4 is given by D 4 = (D + x − D − y )(D + x − D − y ), D ± x F ± h (x) = h(h − 1)F ± h (x), where D ± x is [51], (38) and the functions G τ,l , H b,a , F ± h , that appear in the long superblocks, are defined in the appendix. Note that F − j (ȳ) is a spectator in (37). The presence of δ D ± x = x 2 ∂ x (1 − x)∂ x ± (p 12 + p 34 )x 2 ∂ x − p 12 p 34 x.(2) h,j δ (2) h,j suggests that the hidden symmetry in free theory is realised not on the correlator of the O p but on a correlator of superconformal descendants of O p , obtained by action of the Casimir. A more detailed discussion can be found in [20] for AdS 5 × S 5 and in [42] for AdS 2 × S 2 background, where the logic is exactly the same. In these last two cases, D 4 is replaced by D 8 and D 2 , respectively. In the rest of the section we would like to highlight some features common to various AdS θ1+1 × S θ2+1 backgrounds. To start with, the coefficients of long disconnected free theory are very similar in all these theories, (c.f. formulas in [40]). In fact, upon shifting (p, q) → (p + 1, q + 1) in (33), they are the same when written in h−type variables, except for the function δ which depends on the theory, 1 δ = δ (4) h,h,j,j δ (4) h,h,j,j δ (4) h,h,j,j + δ (4) h,h,j,j , AdS 3 × S 3 , 1 δ = δ (4) h,h,j,j δ (4) h,h,j,j δ (4) h,h,j,j − δ (4) h,h,j,j , AdS 5 × S 5 , 1 δ = δ (2) h,j δ (2) h,j δ (2) h,j − δ (2) h,j , AdS 5 × S 3 .(39) where δ (4) h,h,j,j ≡ δ (2) h,j δ (2) h,j . Note also that the dictionary between h labels and τ labels depends on the theory. By borrowing the results from [40], we can write down the general dictionary interpolating between the three backgrounds h = τ + θ 2 2 + l,h = τ + θ 2 − θ 1 2 + 1, j = − b + θ 2 2 − a + 1,j = − b 2 .(40) The existence of such formulas for disconnected graphs interpolating between different theories turns out to be a particular case of a more general formula for all freetheory diagrams which can be proved through a Cauchy identity [41]. Anomalous dimensions and residual degeneracy We will not give too many details of the computation, which can be found in [16,19] for the similar AdS 5 × S 5 case; analogous computations in AdS 3 × S 3 can be found in [40]. The main difference with the N = 4 case is that here double-trace operators have a flavour structure. Because of this, there will be two types of anomalous dimensions, those of operators exchanged in symmetric or antisymmetric channels. At large N , the operators acquiring anomalous dimensions are of the schematic form, O ± pq = P ± I1I2 O I1 p ∂ l 1 2 (τ −p−q) O I2 q(41) where P ij is an appropriate projector that projects onto symmetric or antisymmetric representations of the gauge group exchanged in the OPE. For any given quantum numbers τ = (τ, b, l, a), the number of operators exchanged in the OPE can be represented with the number of pairs (pq) filling a rectangle [19], R τ := (p, q) : p = i + \|a\| + 1 + r q = i + a + 1 + b − r , for i = 1, . . . , (t − 1) r = 0, . . . , (µ − 1) .(42) The rectangle R τ consists of d = µ(t − 1) allowed lattice points where t ≡ (τ − b) 2 − (a + \|a\|) 2 , µ ≡ b+a−\|a\|+2 2 a + l even, b+a−\|a\|+1 2 a + l odd. The picture below shows an example with µ = 4, t = 9. p q A B C D A = (\|a\| + 2, a + b + 2) B = (\|a\| + 1 + µ, a + b + 3 − µ) C = (\|a\| + µ + t − 1, a + b + 1 + t − µ) D = (\|a\| + t, a + b + t) This representation turns out to be particularly useful when we take into account 1/N corrections. In fact, operators on the same vertical line will continue to be degenerate at this order. To see this, let us consider the OPE at genus zero. This is best cast in a matrix form [16]. First, arrange a d × d matrix of correlators δ p1p3 δ p2p4 G ± disc, p long + 1 N P(x − y)(x − y)A ± p(43) with the pairs (p 1 , p 2 ) and (p 3 , p 4 ) running over the same R τ . Here, we denote by A ± p the inverse Mellin transform of the following Mellin amplitudes, M ± p = 1 2 M p (1, 2, 3, 4) ± M p (1, 3, 4, 2) = 1 2 1 s + 1 1 t + 1 ± 1 u + 1 .(44) The OPE equations then read C ± τ C ± τ T = L ± τ , C ± τ η ± τ C ± τ T = M ± τ .(45) Here L ± τ is a (diagonal) matrix of CPW coefficients of disconnected free theory defined by (32), while M ± τ is a matrix of CPW coefficients of the log U discontinuity of A ± p P(x − y)(x − y)A ± p log U = τ M ± τ L τ .(46) Finally, η ± τ is a diagonal matrix of anomalous dimensions and C ± τ = O p O q K ± rs is a matrix of three-point functions with two half-BPS and one double-trace operator. Here, we denote with K ± rs the true two-particle operator in interacting theory, that differs by O ± pq , precisely because there is mixing. Note that, since A ± p can be written as a function ofŨ andṼ , the SU (2) L × SU (2) R representations contributing to M ± τ can be reorganised into SO (4) representations, while this is not so for the disconnected contribution L ± τ . It is simple to show, with some linear algebra, that the anomalous dimensions are the eigenvalues of the matrix M τ L ± τ −1 . By computing them for various quantum numbers, we find that the anomalous dimensions follow a very simple pattern, η ± τ = − 2 N δ (2) h,j δ (2) h,j (l ± 8d + 1) 4(47) where l 8d is l ± 8d = l + 2(p − 2) + 1 ∓ (−1) a+l 2 − \|a\| ,(48) and can be interpreted as a sort of effective 8d spin, the definition being dictated by the partial wave decomposition of the flat amplitude in 8d [20]. Note that (47) only depends on p, not q, or in other words, operators on the same vertical line in the rectangle will acquire the same anomalous dimensions. We stress again that these are the anomalous dimensions associated to the double-trace operators exchanged in the amplitudes M ± p : the gauge group enters the anomalous dimensions only through an overall constant which does not play any significant role in the computation. Finally, note that in AdS 5 × S 5 the anomalous dimensions read [19] η τ = − 2 N 2 δ (4) h,h,j,j δ (4) h,h,j,j (l 10d + 1) 6 ,(49) where we recall that δ (4) h,h,j,j δ (4) h,h,j,j ≡ δ (2) h,j δ (2) h,j δ (2) h,j δ (2) h,j . Note that the numerator is doubled with respect to the AdS 5 × S 3 case, as a consequence of the fact that supersymmetry is also doubled. Finally, let us also point out that the object δ (2) h,j appearing ubiquitously is, perhaps with no much surprise, nothing but the anomalous dimension of the two-derivative sector in AdS 2 × S 2 [42]. We conclude the section by commenting on the flavour structure of the correlator. One way to deal with it is to decompose t, u channel flavour structures (of both disconnected and tree-level correlators) in a basis of representations appearing in the tensor product of two adjoint representations in the s channel. We then read off the coefficients associated to each flavour structure which are of the form G I1I2I3I4 a ∝ G I1I2I3I4 t + G I1I2I3I4 u a ∈ symm G I1I2I3I4 a ∝ G I1I2I3I4 t − G I1I2I3I4 u a ∈ anti where a runs over all symmetric (antisymmetric) representations in adj ⊗ adj with the proportionality coefficient depending on the specific group as well as the exchanged representation. Examples of such coefficients are given in [6]. The unmixing procedure can then be consistently carried for each a separately. For the symmetric (antisymmetric) representations the relevant doubletrace operators exchanged are of the type O + pq (O − pq ) with the respective anomalous dimensions proportional to η + τ (η − τ ). Actually, the only antisymmetric representation exchanged is the adjoint itself. Outlook and conclusions In the first part of this letter we have discussed colourkinematics and BCJ relations between colour-ordered amplitudes of super gluons in AdS 5 × S 3 , by making use of the large p formalism [35]. We believe this formalism makes clearer the direct parallel with the flat space versions of these relations and that they hold for all Kaluza-Klein modes. This, in turn, shows that, like in flat space, there is a precise relation between colour-kinematic duality and BCJ relations. In the second part of the paper we have computed the anomalous dimensions in the large N limit. As a consequence of the 8d hidden conformal symmetry, and in common with the analogous problems in AdS 5 × S 5 and AdS 3 × S 3 , the anomalous dimensions turn out to have a residual degeneracy which is nicely captured by the vertical columns of the rectangular lattice R τ described in (42) and below. We were also able to give a number of formulae which interpolate results on the spectrum between different backgrounds of different dimensions. These results open a lot of exciting possibilities. Firstly, as we mentioned already in the introduction, the knowledge of the anomalous dimensions can be of use in bootstrapping loop corrections, beyond the lowest charge correlator studied in [6], and help in further exploring whether some features of the double-copy relations persist beyond tree level. Moreover, following the procedure described in [50], one can imagine treating the theory of gluons as an effective model and introduce higher order D n F 4 interactions, analogous to the higher curvature corrections present for gravitons in e.g. AdS 5 ×S 5 . Much like the curvature corrections responsible for completing the Virasoro-Shapiro amplitude in AdS 5 × S 5 [34,37], such terms will induce a splitting of the residual degeneracy in the anomalous dimensions. Finally, the computation of open and closed string amplitudes in AdS might give a clue on how KLT and world-sheet monodromy relations work in a curved spacetime. by the ERC Consolidator grant 648630 IQFT. RG is supported by an STFC studentship. MS is supported by a Mayflower studentship from the University of Southampton. Appendix: superconformal blocks We quickly review here the superconformal block technology needed in this letter. The long superconformal blocks [52], which capture the the non-protected multiplets exchanged in the OPE, are the simplest. They are the product of ordinary conformal and internal blocks for both SU (2) factors. In our notation they take the form L τ = P(x − y)(x − y) Ũ U p3 G τ,l (x,x)H b,a (y,ȳ) ,(51) where G τ,l (x,x) = (−1) l (x −x)U p43 2 F + τ 2 +1+l (x)F + τ 2 (x) − F + τ 2 (x)F + τ 2 +1+l (x) , H b,a (y,ȳ) = 1 U 2− p43 2 F − − b 2 −a (y)F − − b 2 (ȳ),(52) with F ± h (x) = x h 2 F 1 h ∓ p12 2 , h ∓ p43 2 , 2h (x).(53) Here, G τ,l (x,x) are the standard 4d conformal blocks (up to a shift by 2 in the twist τ ) and H b,a (y,ȳ) are the internal blocks. The latter are the product of two SU (2) spherical harmonics, one corresponding to the Rsymmetry group SU (2) R and the other corresponding to the flavour group SU (2) L . Finally, τ, l are, respectively, twist and spin, and b, a label the different representation of SO(4) and can be viewed as the analogues of twist and spin on the sphere. Notice that the internal blocks are not invariant under y ↔ȳ exchange. In fact, the two SU (2) have a different nature; in paricular, notice that free theory is also not invariant under y ↔ȳ. This means that, unlike other theories like AdS 5 ×S 5 and AdS 5 ×S 3 , the decomposition is extended to spherical harmonics with label a < 0. In particular, for given charges p i , we decompose a function in spherical harmonics labelled by two quantum numbers that we denote by [ab]. The values of a run over the following set: relations obeyed by the flat space colour-ordered amplitudes obviously also hold here. In particular, an npoint function satisfies cyclicity, reflection, Kleiss-Kuijf relations and the U (1) decoupling identity that reduce the number of independent colour-ordered amplitudes to (n−2)!. These last two relations coincide for a four-point function. From (25), we can see that an analogous U (1) decoupling identity holds here: further relations, known as BCJ relations, that reduce the number of independent colour-ordered amplitudes to (n − 3)!,(t + 1)M p (1, 2, 3, 4) = (u + 1)M p (1, 3, 4, 2) , (s + 1)M p (1, 2, 3, 4) = (u + 1)M p (1, 4, 2, 3) , (t + 1)M p (1, 4, 2, 3) = (s + 1) M p (1, 3, 4, 2) , −κ p ≤ a ≤ κ p where κ p = min(p 1 + p 2 , p 3 + p 4 ) − p 43 − capture the appearance of BCJ relations in AdS for all Kaluza-Klein modes. Such relations are manifest at level of the reduced Mellin amplitude while they do not hold, at least directly, for the full Mellin amplitude[7]. It is an interesting open question how such relations might extend to higher point amplitudes in AdS and what the role of a reduced Mellin amplitude might be in this regard. AcknowledgementsWe thank F. Aprile, P. Heslop, K. Rigatos and X. Zhou for providing important feedback and comments on the manuscript. MS thanks D. Bufalini, H. Paul and S. 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[ "Effective one-band models for the 1D cuprate Ba 2−x Sr x CuO 3+δ", "Effective one-band models for the 1D cuprate Ba 2−x Sr x CuO 3+δ" ]	[ "A E Feiguin ", "Christian Helman ", "S C De Bariloche ", "Argentina A A Aligia ", "\nPhysics Department\nCentro Atómico Bariloche and Instituto Balseiro\nNortheastern University\n02115BostonMAUSA\n", "\nInstituto de Nanociencia y Nanotecnología CNEA-CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro\nCNEA\n8400, 8400BarilocheGAIDIArgentina\n" ]	[ "Physics Department\nCentro Atómico Bariloche and Instituto Balseiro\nNortheastern University\n02115BostonMAUSA", "Instituto de Nanociencia y Nanotecnología CNEA-CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro\nCNEA\n8400, 8400BarilocheGAIDIArgentina" ]	[]	We consider a multiband Hubbard model Hm for Cu and O orbitals in Ba2−xSrxCuO 3+δ similar to the tree-band model for two-dimensional (2D) cuprates. The hopping parameters are obtained from maximally localized Wannier functions derived from ab initio calculations. Using the cell perturbation method, we derive both a generalized t − J model HtJ and a one-band Hubbard model HH to describe the low-energy physics of the system. HtJ has the advantage of having a smaller relevant Hilbert space, facilitating numerical calculations, while additional terms should be included in HH to accurately describe the multi-band physics of Hm. Using HtJ and DMRG, we calculate the wave-vector resolved photoemission and discuss the relevant features in comparison with recent experiments. In agreement with previous calculations, we find that the addition of an attractive nearest-neighbor interaction of the order of the nearest-neighbor hopping shifts weight from the 3kF to the holon-folding branch. Kinetic effects also contribute to this process.	null	[ "https://export.arxiv.org/pdf/2303.11905v1.pdf" ]	257,636,990	2303.11905	216c83cb9385464d959d2cd63a2c9ee4fa1ef214	Effective one-band models for the 1D cuprate Ba 2−x Sr x CuO 3+δ A E Feiguin Christian Helman S C De Bariloche Argentina A A Aligia Physics Department Centro Atómico Bariloche and Instituto Balseiro Northeastern University 02115BostonMAUSA Instituto de Nanociencia y Nanotecnología CNEA-CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro CNEA 8400, 8400BarilocheGAIDIArgentina Effective one-band models for the 1D cuprate Ba 2−x Sr x CuO 3+δ We consider a multiband Hubbard model Hm for Cu and O orbitals in Ba2−xSrxCuO 3+δ similar to the tree-band model for two-dimensional (2D) cuprates. The hopping parameters are obtained from maximally localized Wannier functions derived from ab initio calculations. Using the cell perturbation method, we derive both a generalized t − J model HtJ and a one-band Hubbard model HH to describe the low-energy physics of the system. HtJ has the advantage of having a smaller relevant Hilbert space, facilitating numerical calculations, while additional terms should be included in HH to accurately describe the multi-band physics of Hm. Using HtJ and DMRG, we calculate the wave-vector resolved photoemission and discuss the relevant features in comparison with recent experiments. In agreement with previous calculations, we find that the addition of an attractive nearest-neighbor interaction of the order of the nearest-neighbor hopping shifts weight from the 3kF to the holon-folding branch. Kinetic effects also contribute to this process. I. INTRODUCTION After more than three decades from the discovery of high-T c superconductivity, the pairing mechanism is still not fully understood, although it is believed that it is related to spin fluctuations originating from the effective Cu-Cu superexchange J, and complicated by the existence of several phases competing with superconductivity 1-4 . However, there is consensus that for energies below an energy scale of the order of 1 eV, the physics of the two-dimensional (2D) superconducting cuprates is described by the three-band Hubbard model 5,6 H 3b , which contains the 3d x 2 −y 2 orbitals of Cu and the 2p σ orbitals of O 7 . More recently, one-dimensional (1D) cuprates have attracted a great deal of attention, in particular because numerical techniques in 1D are more powerful and also fieldtheoretical methods like bosonization can be used [8][9][10][11][12][13][14][15][16][17][18][19][20] . Neudert et al. have studied experimentally and theoretically the distribution of holes in the 1D cuprate Sr 2 CuO 3 8 . The authors discuss several multiband models and the effect of several terms. Recently, angleresolved photoemission experiments have been carried out in a related doped compound Ba 2−x Sr x CuO 3+δ and analyzed on the basis of a one-band Hubbard model with parameters chosen ad hock 14 . The need to add nearestneighbor attraction or phonons to fit the experiment has been suggested 14,17,19,20 . Li et al. studied a four-band and a one-band Hubbard model and noted that the latter lacks the electron-hole asymmetry observed in resonant inelastic x-ray scattering experiments 16 . The questions we want to address in this work are: 1) which is the appropriate multiband Hubbard model H m to describe Ba 2−x Sr x CuO 3+δ ? 2) What are the physical values of the parameters? 3) To what extent can this model be represented by simpler one-band ones? Due to the large Hilbert space of H m , different lowenergy reduction procedures have been used to obtain simpler effective Hamiltonians for the 2D cuprates [21][22][23][24][25][26][27] . Most of them are based on projections of H m onto the low-energy space of Zhang-Rice singlets (ZRS) 28 . In spite of some controversy remaining about the validity of this approach [29][30][31][32][33][34] , the resulting effective models seem to describe well the physics of the 2D cuprates. However, the effect of excited states above the ZRS, often neglected, can have an important role 35 . For example, if one considers the Hubbard model as an approximation to H m (as done in Ref. 14), it is known that it leads, in secondorder in the hopping t, to a term which in one dimension (1D) takes the form H t = t iσ c † i+2σ c † i+1σ c i+1σ c iσ − c † i+2σ n i+1σ c iσ + H.c. ,(1)with t = t 2 /U > 0, where n iσ = c † iσ c iσ andσ = −σ. This term is an effective repulsion and inhibits superconductivity in 1D, while as expected, it favors superconductivity if the sign is changed 36,37 . Interestingly, some derivations of the generalized t−J model for 2D cuprates suggest that t can be negative for some parameters of H m 27,38 , and a very small term t = −t/20 can have a dramatic effect favoring d-wave superconductivity 39 . Even if the realistic t is positive, it is expected to be smaller than that derived from the Hubbard model and might explain why studies of the superconductivity in the one-band Hubbard model conclude that part of the pairing interaction is missing 4 . arXiv:2303.11905v1 [cond-mat.str-el] 21 Mar 2023 Therefore, a discussion on the appropriate model to describe the 1D cuprates and, in particular, Ba 2−x Sr x CuO 3+δ seems necessary. In this work we calculate the hopping parameters of the multiband model for this compound and use this information to derive simpler one-band models. The paper is organized as follows. In Section II we explain the multiband Hubbard model H m and derive its hopping parameters using maximally localized Wannier functions (MLWF). In Section III we describe the resulting generalized t − J model obtained from H m by a low-energy reduction procedure explained briefly in the appendix. In Section IV we explain the corresponding results for the one-band Hubbard model. In Section V we calculate the photoemission spectrum using the time-dependent density-matrix renormalizationgroup method [40][41][42][43] and we compare it with previous experimental and theoretical results. Section VI contains a summary and discussion. II. THE MULTIBAND HUBBARD MODEL We use the following form of the Hamiltonian H m = U d i d † i↑ d i↑ d † i↓ d i↓ + iσ { Cu d † iσ d iσ + O 2 δ p † i+δσ p i+δσ + ap O γ p † i+γσ p i+γσ +[d † iσ (t x pd δ p i+δσ + t y pd γ p i+δγ ) −t pp δγ p † i+δσ p i+γσ + H.c.]},(2) where d † iσ (p † jσ ) creates a hole with spin σ at Cu (O) site i (j). We choose the chain direction as x (a in Fig. 1) and δ = ±ax/2 denote the vectors that connect a Cu atom with their nearest O atoms in the chain direction, where a is the Cu-Cu distance that we take as 1 in what follows. γ has a similar meaning for the apical O atoms, displaced from the chain in the y direction (c in Fig. 1). The relevant O orbitals are those pointing towards their nearest Cu atoms. To simplify the form of the Hamiltonian, we have changed the signs of half of the orbitals in such a way that sign of the hopping terms do not depend on direction and t x pd , t y pd , t pp > 0 44 . In comparison with previous approaches 8, 16 , two terms are missing: the intratomic O repulsion U p and the interatomic Cu-O repulsion U pd . Although the former is rather sizeable (U p ∼ 4 eV has been estimated in 2D cuprates 45 ), we find that it has very little influence on the parameters of the one-band models because of the low probability of double hole occupancy at the O sites. The value of U pd is difficult to determine from spectroscopic measurements 46 and its effect on different quantities can be absorbed in other parameters 8 . We obtain a better agreement with the measured photoemission spectra assuming U pd = 0. We also take U d = 10 eV and ∆ = O − Cu = 3.5 eV, from calculations in the 2D cuprates 45 and ap O − O = −0.4 eV was determined as the value that leads to a ratio 1.225 between the occupancy of apical and chain O atoms, very similar to determined experimentally in Sr 2 CuO 3 8 . The values of the hopping parameters t x pd = 1.10 eV, t y pd = 1.04 eV, and t pp = 0.60 eV were determined from density functional theory (DFT) calculations along with the MLWF method. For the DFT calculations, we use the QUANTUM ESPRESSO code 47,48 , with the GGA approximation for the exchange and correlation potential and PAW-type pseudopotentials. The energy cut for the plane waves is 80 Ry, and the mesh used in reciprocal space is 15 × 15 × 5. The unit cell is an orthorhombic structure with lattice parameters a = 3.85Å, b = 4.17Å and c = 13.18Å, and contains two formula units, see Fig. 1. We consider the spin unpolarized case and obtain the bands shown in Fig. 2. The MLWF procedure involves band fitting of the DFT results, as shown in blue in Fig. 2. The energy window selected to project the Wannier orbital is between 3.75 eV and 10.75 eV, and the orbitals are centered in Cu and O atoms with d and p character, respectively. Other convergence parameters are also successfully evaluated, as suggested in Ref. 49. Finally, the hopping parameters are extracted from the Hamiltonian expressed in the Wannier basis. III. THE GENERALIZED t − J MODEL Using the cell-perturbation method 24,25 , with appropriate modifications for this 1D compound, we find that the system can be described with the following general- ized t − J model H tJ = −t iσ c † iσ c i+1σ + H.c. −t 2 iσ c † iσ c i+2σ + H.c. + i (JS i · S i+1 + V n i n i+1 ) + H t ,(3) with t 2 = t/5. Minor terms of magnitude below 0.04 eV were neglected. The parameters of the model are given by analytical expressions in terms of the eigenstates and energies of a cell Hamiltonian, that are obtained after solving a 6×6 matrix and two 3×3 matrices. A summary of the method is included in the appendix. For the parameters of the multiband model described above, we obtain t = 0.443 eV, J = 0.314 eV, V = −0.143 eV, and t = 0.068 eV. Interestingly, our values for J and t without adjustable parameters are similar to those corresponding to the Hubbard model chosen to explain the experiments by Chen et al. 14 . The larger values of t and J compared to the 2D cuprates (for example t = 0.37 meV, J = 0.15 meV for T-CuO 32 ) are expected due to the larger overlap between the normalized O orbitals δ p i+δσ that hybridize with the Cu for nearest-neighbor Cu positions. This leads to a larger overlap between non-orthogonal ZRS 27,50 and to a larger extension of the orthogonal oxygen Wannier functions centered at the Cu sites (see appendix). For Sr 2 CuO 3 , the reported values of J ∼ 0.24 meV 8-12 are also larger than those of 2D cuprates. The resulting value of t is somewhat smaller than that used in Ref. 14 but is is compensated by the hopping to second nearest neighbors. The fact that the nearest-neighbor attraction V is larger than J/4 as expected for the mapping from the Hubbard to the t − J model is due to the contribution of excited local triplets absent in the Hubbard model. For other parameters of H m , in particular increasing the ratio t pp /t pd and the difference between O and Cu on-site energies, t changes sign as expected from calculations in 2D cuprates 27,38 . For example increasing t pp to 1 eV and both on-site energy differences to 7 eV (unrealistic for Ba 2−x Sr x CuO 3+δ but near to the values expected for nickelates), we obtain t = 0.531 eV, J = 0.105 eV, V = −0.096 eV, and t = −0.019 eV, due to the increasing relative importance of excited triplets. IV. THE EFFECTIVE ONE-BAND HUBBARD MODEL The generalized t − J model discussed above describes the movement of ZRS (two-hole states) in a chain of singly occupied cells. If the cells with no holes are also considered (because for example one is interested in larger energy scales), one can also derive a one-band Hubbard-like model using the cell perturbation method. A simple version of this model has the form 21-23 H H = −t iσ c † iσ c i+1σ + H.c. [t AA (1 − n iσ )(1 − n i+1σ ) +t BB n iσ n i+1σ + t AB (n iσ + n i+1σ − 2n iσ n i+1σ )] +U i n i↑ n i↓ .(4) As for H tJ , we map the ZRS into empty states of H H . Then t AA coincides with t of H tJ . From the mapping procedure we obtain t AA = 0.443 eV, t AB = 0.421 eV, t BB = 0.369 eV, and U = 2.083 eV. The model is electron-hole symmetric if and only if t AA = t BB , while the photoemission of H m is asymmetric in general 16 . Another shortcoming of H H is that if the model is reduced to a generalized t − J one by eliminating double occupied sites, the effective J = 4t 2 AB /U = 0.376 eV and t = J/4 = 0.094eV are overestimated with respect to the values obtained in the previous section: J = 0.314 eV, and t = 0.068 eV. Instead, the NN attraction −J/4 is underestimated (V = −0.143 eV above). This is due to the neglect of the triplets in H H (see appendix and Ref. 38). Therefore H tJ is more realistic to describe the photoemission spectrum of hole doped 1D cuprates, unless additional terms are added to H H . V. PHOTOEMISSION SPECTRUM OF HtJ A. Photoemission intensity as a function of wave vector While the effective Hamiltonian H tJ is enough to accurately describe the energy spectrum of H m at low energies, this is not the case for the spectral intensity since one needs to map the operators for the creation of Cu and O holes in H m to the corresponding ones of the effective low-energy Hamiltonian that one uses 51,52 In the 2D cuprates, it has been found by numerical diagonalization of small clusters, that the photoemission intensity due to O atoms at low energies can be well approximated by the expression 52 I O = 1.22 × Z(k)[sin 2 (k x /2) + sin 2 (k y /2)],(5) where Z(k) is the quasiparticle weight of the generalized t − J model. The dependence on wave vector can be understood from the fact that at k = 0, the O states which point towards their nearest Cu atoms are odd under reflection through the planes perpendicular to the orbitals, while the low-energy orbitals that form the ZRS are even under those reflections. Comparison of this expression to experiment is very good 30 . A variational treatment of a spin-fermion model for the cuprates also leads to a vanishing weight at k x = k y = 0 29 . A similar dependence is expected in the 1D case due to the contribution of the O orbitals along the chain, which increases the relative weight for k x ∼ π/2. To estimate the relative weight due to these orbitals, we have calculated the probability of creating a hole (e ik p † i+δ − p † i−δ )/ √ 2 (the minus sign is due to the choice of phases in H m ) in a singly occupied cell leaving a ZRS, and also the corresponding result for apical O and Cu. In addition, the observed total intensity depends on the cross sections f for Cu and O, which in turn depend sensitively on the frequency of the radiation used. The ratio of cross sections for the reported energy (65 eV) of the x-ray beam (available at https://vuo.elettra.eu/) is f Cu /f O = 3.077. Using this result and the above mentioned probabilities for the parameters of H m described in the previous Section, we obtain that the wave vector dependence of the photoemission intensity can be written as I(k) ∼ Z(k)[A + B sin 2 (k/2)],(6) where Z(k) is the quasiparticle weight of the generalized t − J model, and A = 1.097 and B = 0.205. B. Numerical results We calculated the photoemission spectrum of the generalized t − J model using time-dependent densitymatrix renormalization group (tDMRG) 40,41,43,53 . The simulation yields the single particle two-time correlator G(x, t) = i c † σ (x, t)c σ (L/2, 0) . This is Fourier transformed to frequency and momentum, allowing one to retrieve the spectral function as A(k, ω) = −ImG(k, ω)/π. The method has been extensively described elsewhere 43,53 and we hereby mention some (standard) technical aspects. Simulations are carried out using a time-targeting scheme with a Krylov expansion of the evolution operator 42 . Since open boundary conditions are enforced and the chain length is even, the correlations in real space are symmetrized with respect to the "midpoint" x = L/2. In order to reduce boundary effects we convolve G(x, t) with a function that decays smoothly to zero at the ends of the chain and at long times, automatically introducing an artificial broadening. The function of choice is the so-called "Hann window" (1 + cos (xπ/σ))/2, where σ is the window width. We apply this window to the first and last quarter of the chain. We study systems of length L = 80 sites, and N = 74 electrons, corresponding to a 7.5% doping, using m = 600 DMRG states (guaranteeing a truncation error below 10 −6 ), a time step δt = 0.05 and a maximum time σ = t max = 20. This density is chosen to maximize the holon folding (hf) effect discussed in Ref. 14. Due to the one-dimensionality, the low-energy physics of the models discussed here falls into the universality class of Luttinger liquid theory [54][55][56][57] . Accordingly, excitations are not full-fledged Landau quasi-particles and the spectrum displays edge singularities instead of Lorentzians. In addition, and most remarkably, they realize the phenomenon known as spin-charge separation, with independent charge and spin excitations that propagate with different velocities and characteristic energy scales: the spinon bandwidth is determined by J, while the holon bandwidth by the hopping t. In the photoemission spectrum, these excitations appear as separate branches between −k F and k F , with the spinon branch looking like an arc connecting the two points. Unlike non interacting systems, the photoemission spectrum extends beyond \|k\| > k F due to momentum transfer between spinons and holons: An electron with energy (k) can fractionalize into spin and charge excitations such that s (q) + c (k − q) = (k), leading to a high energy continuum and additional branches leaking out from k = ±k F and k = ±3k F 58-60 (the first one is referred-to as the holon-folding band in Ref. 14 and as "shadow bands" in Ref. 61). Results for the generalized t − J model [Eq. (3)] are shown in Fig. 3(a) for the physical parameters corresponding to Ba 2−x Sr x CuO 3+δ , as discussed above. To understand the contributions of the different terms we also considered the cases without second-neighbor hopping in Fig. 3(b), and without correlated hopping (t = 0), Fig. 3(c). In all these curves, the spectral density has been rescaled according to Eq. (6). As a reference, we also show results for the extended Hubbard chain with U = 8t and second neighbor attraction V = −t (t = 0.6eV ). We notice that the Hubbard chain has more spectral weight concentrated on the holon branches, while in the t − J model it is more distributed in the continuum and even in the continuation of the holon bands at high energies. In addition, we observe that the t − J model has a larger spinon velocity with a wider spinon branch, and a larger charge velocity with a holon band ∼ 20% wider than the one for the Hubbard model (we measure the holon bandwidth as the distance between the Fermi energy and the crossing of the two holon branches at k = 0). Since the value of J remains unchanged, we attribute these effects to kinematic sources (the extra hopping terms). In order to compare to previous attempts to interpret the experimental observations, we analyze momentum distribution curves at fixed frequency values, plotted in Fig. 4, corresponding to the yellow dashed lines in Fig. 3. We include results for the extended Hubbard model with U/t = 8, V = −t and V = 0, that agree very well with similar previous calculations 17, 19 . Note that the parameters of this Hubbard model are not those that correspond to the mapping discussed in section IV, but the chosen value of U gives rise to an effective J similar to the correct one. In this figure it is easier to observe the signatures of the k F and 3k F holon branches at high momentum, which are quite faint in the color density plot and are highlighted here with arrows. One can also appreciate the qualitative differences between the extended Hubbard model with V = −t and the other cases. In particular, by comparing to the standard Hubbard model with V = 0 we notice a transfer of weight from the edge of the continuum (3k F -band) to the holon-folding band, which in Ref. 14 is attributed to a phonon induced attraction. On the other hand, the t − J model realizes a more prominent feature at 3k F and the hf-band, and the continuum contains markedly more spectral weight in the sidebands than the Hubbard model. We also include results for the generalized t − J model with V = −t and we observe results practically identical to those for the extended Hubbard model with attraction. Our results indicate that there are both kinetic as well as many-body effects that affect the relative spectral weight concentrated in the sidebands: While t 2 and t shift weight from the center toward the folding and 3k F bands, the attraction V shifts weight toward the center and from the 3k F band into the holon-folding (hf) one. However, comparing with the cases of vanishing t 2 and t , we see that these terms also have an effect of shifting weight from the 3k F to the hf band but of smaller magnitude. VI. SUMMARY AND DISCUSSION We have started our description of CuO 3 chains of Ba 2−x Sr x CuO 3+δ from a four-band model (with one relevant orbital per Cu or O atom). The hopping parameters of the model were obtained using maximally localized Wannier functions. Extending the cell-perturbation method used for CuO 2 planes of the superconducting cuprates to this one-dimensional compound, we derive simpler one-band models that are more amenable to numerical techniques due to the smaller Hilbert space. In order to account for the effect of excited triplets, the oneband Hubbard model should be supplemented by other terms not usually considered. In addition the hopping term depends on the occupancy of the sites involved. For energies below the value of the effective Coulomb repulsion U , it is more convenient to use the generalized t − J model. We have calculated the photoemission spectrum of this model using time-dependent density-matrix renormalization group. The results are in semiquantitative agreement with experiment. We obtain that the hopping to second nearest-neighbors and the three-site term H t have a moderate effect in shifting weight from the 3k F peak to the holon-folding branch, but a nearest-neighbor attraction has a stronger effect. For energies below U and if only either electron or hole doping is of interest, a Hubbard model with an artificially enlarged U that leads to the correct value of the effective nearest-neighbor exchange J, shows a photoemission spectrum very similar to the corresponding results for the generalized t − J model. Here we summarize the application of the cellperturbation method 24,25 , to the case of the onedimensional compound. The Cu orbitals at each site are hybridized with symmetric linear combinations of O orbitals of the form (dropping for the moment the spin subscripts) a † i = p † i+γ + p † i−γ √ 2 , q † i = p † i+δ + p † i−δ ,(A1) To change the basis of the q † i to orthonormal Wannier functions 28 , we Fourier transform q † k = 1 √ N l e −ikl q † i = 2 cos(kδ) √ N j e −ikj p † j ,(A2) where the sum over l(j) runs over all Cu(O) sites. The operators π † k = 1 2\| cos(kδ)\| q † k ,(A3) satisfy {π † k1 , π k2 } = δ k1,k2 . Transforming to real space one obtains the Wannier O orbitals centered at the Cu sites π † l = 1 √ N k e ikl π † k = j A(l − j)p † j , A(j) = 1 N k e ikj sgn [cos(kδ)] = (−1) j−1/2 jπ . (A4) Changing the basis of the O orbitals, the hopping terms in the Hamiltonian Eq. (2) (those proportional to t x pd , t y pd , t pp ) that act inside each cell that includes a Cu site and the O Wannier functions centered at the same site, becomes H intra hop = iσ [d † iσ (V x π iσ + V y a iσ ) + V O π † iσ a iσ + H.c.], V x = 2A(1/2)t x pd , V y = √ 2t y pd , V O = −2 √ 2A(1/2)t pp ,(A5) while the remaining part of the hopping takes the form H inter hop = iσ l =0 B l [π † i+lσ (t x pd d iσ − √ 2t pp a iσ ) + H.c.], B l = A(l + 1/2) + A(l − 1/2).(A6) The on-site terms of H 3b retain the same form. This part and H intra hop is solved exactly in the subspaces of one and two holes. For one hole and given spin, one has a 3×3 matrix, and we denote as E 1 the lowest energy in this subspace. For two holes and neglecting U p there is a 6×6 matrix for the singlet states and a 3 × 3 matrix for each spin projection of the triplet states . The ground state of the subspace of singlets with energy E s is identified as the Zhang-Rice singlet (ZRS) 28 and mapped into an empty site in the effective generalized t−J model. The Coulomb repulsion in the effective Hubbard model is U = E s −2E 1 . An advantage of the cell perturbation method is that most of the hopping terms are included in H intra hop and including exactly in these matrices. The rest of the hopping H inter hop is treated in perturbation theory. The firstorder correction gives rise to effective hopping at different distances. An important difference with the twodimensional case is that the larger overlap between linear combinations of the original O orbitals centered at a Cu site [as the q † i in Eq. (A1)] leads to larger effective hoppings and to a slower decay with distance. Note that the ratio of second nearest-neighbor (NN) hopping to the first NN one is t 2/ /t 1 = −B 2/ /B 1 = 1/5 (the minus sign comes from restoring the original signs of half of the orbitals, which have been changed to simplify H 3b ) and for third NN t 3/ /t 1 = B 3/ /B 1 = 3/35 (these ratios change if corrections due to U p and U pd are included). In the effective Hubbard model, there are actually three different hopping terms depending on the occupancy of the sites involved, while in the generalized t − J model, only the one related with the exchange of Zhang-Rice singlets with singly occupied sites is important. The most important second-order corrections in H inter hop lead to a superexchange J and nearest-neighbor attraction −V in the generalized t − J model. For example, one of these second-order processes leads to an effective spin-flip process between a state with one hole with spin ↑ at site i and another with spin ↓ at site i + 1, and another one with the spins interchanged, through an intermediate state with no holes at site i and two holes at site i + 1. While the states with one hole correspond to the ground state of the above mentioned 3 × 3 matrix, the two-hole part of the intermediate states include all singlet and triplet states of the corresponding 6 × 6 and 3 × 3 matrices. This is an important difference with the Hubbard model, because in the latter, only the ground state of the 6 × 6 matrix of singlets is included in the effective exchange J H = 4t 2 AB /U and the triplets are neglected, leading to an overestimation of J H , because the contribution of the triplets is negative. The next important second-order corrections in H inter hop correspond to three-site terms. They lead for example to an effective mixing between states with one hole at sites i and i + 1 and a ZRS at site i + 2 and states with a ZRS at site i and one hole at sites i + 1 and i + 2. As before, performing second-order perturbation theory in the Hubbard model includes only a few of these contributions. The different terms can be expressed analytically in terms of the eigenstates and eigenenergies of the matrices of the local cell mentioned above. The expressions are lengthy and are not reproduced here. FIG. 1 . 1Unit cell of BaCuO3. The gray/blue/green ball are Ba/Cu/O respectively. The lattice parameters are a = 3.85Å, b = 4.17Å and c = 13.18Å. The CuO chains are along the a direction with distance among them of 4.14Å. FIG. 2 . 2Band structure for the BaCuO3 obtained for unpolarized DFT calculation. In blue the bands from the MLWF procedure superposed with DFT ones. FIG. 3 . 3Photoemission spectrum of (a) the generalized t − J, Eq. 3; (b) same but setting the second-neighbor hopping to zero; (c) t = 0. Panel (d) shows results for the extended Hubbard model with t = 0.6, U/t = 8; V /t = −1 for comparison. FIG. 4 . 4Momentum distribution curves: cuts along the fixed frequency dashed lines inFig. 3. The features corresponding to the holon-folding (hf) and 3kF bands are highlighted by arrows. The curves are ordered from bottom to top and shifted 0.1 up in intensity from the previous one for clarity. ACKNOWLEDGMENTS We thank Alberto de la Torre and Giorgio Levy for information regarding Cu and O cross sections for photoionization. We enjoyed fruitful discussions with Alberto Nocera, Steven Johnston and Yao Wang.AAA acknowledges financial support provided by PICT 2017-2726 and PICT 2018-01546 of the ANPCyT, Argentina. AEF acknowledges support from the U.S. Department of Energy, Office of Basic Energy Sciences under grant No. DE-SC0014407. 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[ "Nuclear fission reaction simulations in compact stars", "Nuclear fission reaction simulations in compact stars" ]	[ "Alex Deibel ", "M E Caplan ", "C J Horowitz ", "\nDepartment of Astronomy\nDepartment of Physics\nIndiana University\n47405BloomingtonINUSA\n", "\nCenter for Exploration of Energy and Matter and Department of Physics\nIllinois State University\n61790NormalILUSA\n", "\nIndiana University\n47405BloomingtonINUSA\n" ]	[ "Department of Astronomy\nDepartment of Physics\nIndiana University\n47405BloomingtonINUSA", "Center for Exploration of Energy and Matter and Department of Physics\nIllinois State University\n61790NormalILUSA", "Indiana University\n47405BloomingtonINUSA" ]	[]	Type-Ia supernovae are powerful stellar explosions that provide important distance indicators in cosmology. Recently, we proposed a new SN Ia mechanism that involves a nuclear fission chainreaction in an isolated white dwarf [PRL 126, 1311010]. Here we perform novel reaction network simulations of the actinide-rich first solids in a cooling white dwarf. The network includes neutroncapture and fission reactions on a range of U and Th isotopes with various possible values for 235 U enrichment. We find, for modest 235 U enrichments, neutron-capture on 238 U and 232 Th can breed additional fissile nuclei so that a significant fraction of all U and Th nuclei may fission during the chain-reaction. Finally, we compute the energy release from the fission chain-reaction for various uranium enrichments; a novel result that is a necessary input for thermal diffusion simulations of carbon ignition. *	10.1103/physrevc.106.045803	[ "https://export.arxiv.org/pdf/2109.14714v2.pdf" ]	238,226,958	2109.14714	a9a3fc5aa9e760c492fd8b0ef28b66285669fb7b	Nuclear fission reaction simulations in compact stars Alex Deibel M E Caplan C J Horowitz Department of Astronomy Department of Physics Indiana University 47405BloomingtonINUSA Center for Exploration of Energy and Matter and Department of Physics Illinois State University 61790NormalILUSA Indiana University 47405BloomingtonINUSA Nuclear fission reaction simulations in compact stars (Dated: September 29, 2022) Type-Ia supernovae are powerful stellar explosions that provide important distance indicators in cosmology. Recently, we proposed a new SN Ia mechanism that involves a nuclear fission chainreaction in an isolated white dwarf [PRL 126, 1311010]. Here we perform novel reaction network simulations of the actinide-rich first solids in a cooling white dwarf. The network includes neutroncapture and fission reactions on a range of U and Th isotopes with various possible values for 235 U enrichment. We find, for modest 235 U enrichments, neutron-capture on 238 U and 232 Th can breed additional fissile nuclei so that a significant fraction of all U and Th nuclei may fission during the chain-reaction. Finally, we compute the energy release from the fission chain-reaction for various uranium enrichments; a novel result that is a necessary input for thermal diffusion simulations of carbon ignition. * I. INTRODUCTION Type-Ia supernovae (SN Ia) are widely used distance indicators in cosmology [1][2][3], but significant tension remains between the Hubble constant determined from SN Ia and the value determined from other data [4][5][6]. Despite the importance of SN Ia for cosmology, their progenitor systems and explosion mechanism are still somewhat uncertain. Traditionally, SN Ia are thought to involve the thermonuclear explosion of a C/O white dwarf (WD) in a binary system. Here the companion is either a conventional star (single-degenerate mechanism) or another WD (double-degenerate mechanism) [7][8][9]. Recently we proposed an additional SN Ia mechanism that may occur in isolated WDs [10,11] wherein the cooling WD core rapidly precipitates a fission-critical uranium crystal within 30 s. If a fission chain-reaction proceeds in the crystal, it is unknown, a priori, the fraction of uranium consumed or the resulting energy release. In the present paper we perform nuclear reaction network simulations of fission chain-reactions in compact stars. Our goal is to determine the fraction of fissile fuel consumed during a fission chain-reaction and the resulting energy release and how this depends on the uranium enrichment f 5 -the fraction of all uranium that is 235 U. This is important input for thermal diffusion simulations of carbon ignition in an isolated WD. We present these simulations in a separate paper [12] where we find that carbon ignition is likely, at high densities. Our simulations are novel, apparently the first such calculations for a compact star. To provide context, we briefly review fission chain-reactions in conventional nuclear reactors and nuclear weapons in Sec. II. Our reaction network formalism is described in Sec. III, results presented in Sec. IV and we conclude in Sec. VI. II. CONTEXT A fission reaction in a compact star is a unique hybrid between a nuclear reactor and a nuclear weapon. Like a nuclear weapon the chain-reaction is expected to proceed extremely rapidly. A WD is degenerate, however, and the temperature can rise without a large increase in pressure. As a result, the system does not rapidly disassemble as in a nuclear weapon. This allows time -as in a nuclear reactor -for fertile isotopes such as 238 U or 232 Th to capture neutrons and breed additional fissile material. Many conventional nuclear reactors slow neutrons to (terrestrial) thermal energies to take advantage of the large fission cross section of 235 U at low energies. In the WD core, however, the temperature is of order ∼ 1 keV and the 235 U fission cross section is much smaller. Even if plenty of light nuclei are present to moderate the neutrons, the neutron energy will only be reduced to the ∼ 1 keV ambient temperature. Therefore, unlike a terrestrial nuclear reactor, a stellar system can not take advantage of the large low-energy 235 U cross section. Terrestrial nuclear weapons, on the other hand, disassemble extremely rapidly because of the large energy release. This necessitates using fast neutrons in the chainreaction because slow neutrons simply take too long to cause additional fissions. By the time a slow neutron arrives to cause another fission the fuel may have been blown apart. This reliance on fast neutrons requires the use of highly enriched uranium or plutonium in a nuclear weapon. The necessary uranium enrichment may be reduced if it is possible for the chain-reaction to breed additional fissile nuclei. For example 238 U in a nuclear reactor can capture a neutron to become 239 U that in turn beta decays twice to produce 239 Pu. Therefore, a fission chainreaction in a star could breed some of its nuclear fuel as the reaction progresses. We now begin our study by discussing the composition of the first solids to form as a WD cools. Next, we list the fission and neutron-capture reactions that are included in our network simulations and present the results for composition and energy release as a function of time. We end with a discussion of sensitivity to uranium enrichment and to the 238 U fission cross section. We conclude that for modest enrichments a large fraction of the available fuel is expected to fission producing a large energy release that could ignite carbon burning. III. FORMALISM Initial abundances: The composition of the first solids to form as material in a WD just starts to crystallize has been studied using free energy models and with molecular dynamics simulations [10,11]. The material is U and Th rich since these elements have the highest charge Z. In addition some Pb is present because the solar system abundance of Pb is 100 times that of U. The initial abundance in nuclei per baryon Y i = n i /n b are listed in Table I. Here n i is the number density of species i and n b is the baryon density. In addition to heavy nuclei, some C and O may be present in the first solids. Elastic scattering from the light C and O nuclei can lower the energy of fission neutrons. At this time the amount of C and O is uncertain and may be zero. For simplicity in this first study, we assume there is no C and O present. As a result there will be little moderation of the initial neutron energies. The fission spectrum has a most probable energy near ≈ 1 MeV. In this paper we simply assume all neutrons have an energy of 1 MeV and evaluate all cross sections at this energy. This assumption of mono-energetic neutrons greatly simplifies thermally averaged reaction rates that are proportional to the cross section times the relative velocity, σ(E)v ≈ σ(1 MeV)v 0 ,(1) with v 0 the velocity of a 1 MeV neutron. Furthermore, this thermal average is independent of temperature. We consider neutron-capture (n, γ) and neutroninduced fission reactions on the U and Th isotopes. We use 1 MeV cross sections from the ENDF 2011 data set available at the National Nuclear Data Center [13], see Table II. Our reaction network has 232 Th, 233 Th, and 234 Th isotopes and U isotopes from 235 U to 242 U. In addition we have neutrons and fission fragments. We do not distinguish different possible fission fragments and simply assume that any fission will produce two fragments. Our network has a total of 13 species consisting of 3 Th isotopes, 8 U isotopes, n and fission fragments. There are simple equations for the change in abundance dY i /dt from a given reaction [14]. For example the change in neutron abundance Y n from the fission of nucleus A of abundance Y A is dY n dt = (ν − 1)n b σ f v 0 Y n Y A .(2) Here each fission produces an average number of neutrons ν (see Table II, and one neutron was absorbed to cause the fission). The fission also increases the abundance of fission fragments Y ff where we include a factor of two for the two fragments, dY ff dt = 2n b σ f v 0 Y n Y A .(3) Likewise neutron absorption on nucleus A decreases its abundance and increases the abundance of nucleus A + 1, dY A+1 dt = − dY A dt = n b σ n,γ v 0 Y n Y A .(4) We sum terms with these forms over all of the (n, γ) and fission reactions in Table II. The initial abundance of 235 U is f 5 Y u with Y u the uranium abundance from Table I and f 5 the uranium enrichment. Likewise the 238 U abundance is Y u (1−f 5 ). Finally for the initial neutron abundance one can use any small seed value. The cases explored are listed in Table III. The energy released per fission is about 200 MeV, or 100 MeV per fission fragment. Therefore the heating rate TABLE III. Four cases of enrichment f5, fission cross section of 238 U, and neutron absorption cross section of fission fragments σ ff . Also listed are the initial n abundance Yn, the fission heating S, the fraction of U and Th that fission, and the final temperature T f . in MeV per baryon per time iṡ Case f5 σ f ( 238 U) σ ff Yn S U Th T f (b) (b) (MeV/A) % % 10 9 K AS = (100 MeV) dY ff /dt,(5) and the total energy released by a time t final in MeV per baryon is S = (100 MeV) Y ff (t final ) .(6) The large fission energy release will raise the temperature of the system. We assume the reaction proceeds at constant pressure. Previously we calculated the heat capacity at constant pressure and obtained the final temperature T f [11], T f ≈ 8 F S 5π 2 Y e 1/2 .(7) Here F is the electron Fermi energy and Y e is the electron fraction, see Table I. IV. RESULTS A. Abundance evolution We now present results for three cases of enrichment and 238 U fission cross section as listed in Table III. In all cases the chain-reaction proceeds rapidly, in less than 10 −11 s, as shown in Fig. 1. This is because of the high density of the system, the large neutron cross sections, and the high velocity of 1 MeV neutrons. In general, the reaction proceeds in two stages. In the first stage neutrons from 235 U fission transform or breed some 238 U and 232 Th nuclei into more easily fissionable 239 U and 233 Th. In the second stage, or breeder reaction, most of the original U and a significant fraction of the Th fission. In Case A we use f 5 = 0.14 and the unmodified 1 MeV fission cross section for 238 U of 0.014 b. One needs an enrichment of at least f 5 ≈ 0.12 for the system to be critical. If f 5 is smaller than that no chain-reaction will take place. If f 5 is only slightly larger than 0.12 we expect the chain-reaction to burn a small amount of 235 U until the system becomes sub-critical and the chain reaction stops. This will only release a small amount of fission heating. In Fig. 1 (a) we show results for Y i versus time for Case A. The system fissions ≈ 9.6, % of the total U. This includes over half of the original 235 U and only a small amount of 238 U. Only a small amount of Th fissions near ≈ 0.4 %. We see that the neutron abundance rises exponentially with time until enough 235 U has been burned so that the system is no longer critical. After that Y n decreases as the remaining neutrons are captured. Results are very sensitive to the small 238 U fission cross section. This is 0.014 b at 1 MeV but rises rapidly at higher energies. A detailed calculation averaging the energy-dependent cross section over the neutron fission spectrum may give a larger value. Alternatively, as the temperature of the medium rises, 238 U nuclei will occupy a range of excited states and these may have higher fission cross sections for 1 MeV neutrons. For example, Zhu and Pei calculate that the spontaneous fission half life of 240 Pu decreases by 12 orders of magnitude as the temperature is increased from 0 MeV to 0.1 MeV [15]. In Case B we use σ f = 0.04 b instead of 0.014 b for σ f ( 238 U). The results in Fig. 1 (b) show dramatic differences from Case A. The reaction now proceeds in two stages. First mostly 235 U fissions. This releases enough neutrons so the n capture converts both 232 Th and 238 U into odd A nuclei with significant fission cross sections. In the second or breeder reaction stage these nuclei fission. As a result fully 95% of the U and 58% of the Th fissions. Note that in a terrestrial nuclear reactor there is time for 239 U to decay to 239 Pu. Here there is not enough time and 239 U instead can be used directly as a fuel. Alternatively, even if the 238 U cross section is only 0.014 b, one can obtain a breeder reaction stage by modestly increasing f 5 . Case C has σ f = 0.014 b but uses an enrichment of f 5 = 0.20 instead of 0.14. The results in Fig. 1 panel (c) show two well-separated reaction stages and the fission of a large fraction of available nuclei similar to Case B. We have assumed that the fission fragments are essentially inert. While there may not be time for beta-decay, the fission fragments could capture neutrons. Many fission fragments have (n, γ) cross sections for 1 MeV neutrons of order ∼ 0.01 b [13]. Therefore to explore this we simply assign all fission fragments a 0.01 b capture cross section for Case D, see Table III. This case is otherwise identical to Case B. Note that we are not keeping track of the identity of each fission fragment so when a fragment captures a neutron its identity does not change. Therefore, neutron-capture on fission fragments simply acts as a neutron sink. Figure 1 panel (d) shows that capture on fission fragments somewhat reduces the abundance of neutrons. Indeed Case B has a finite abundance of neutrons remaining. This artificial result reflects the limitations of the reaction network. In Case D all neutrons are eventually captured. The reduction in neutrons slows the production of fission fragments somewhat. However by the time the reaction is over, the total number of fission fragments and therefore the total fission energy released is only Table III. slightly smaller in Case D with capture than originally in Case B. B. Heat release and final temperature The heating rate for the different scenarios is shown in Fig. 2 and the total heating is plotted in Fig. 3 and listed in Table III. The final temperature is plotted in Fig. 4 for a range of f 5 values. There is a minimum value of f 5 for the system to be critical. Below this value there is almost no fission heating and T f is small. Next there is a modest range of f 5 values where significant 235 U fissions but little 238 U or Th fissions. Here T f is between ≈ 1-2×10 9 K. Finally there is a sharp transition when there are enough neutrons for an essentially complete breeder reaction stage that fissions most of the U and 58% of the Th. Note that the breeder reaction stage leads to a large total energy release of ≈ 0.36 MeV/nucleon as listed in Table III; this leads to T f ≈ 6×10 9 K. As the 238 U fission cross section increases, the necessary f 5 for the breeder reaction decreases. In general, the breeder reaction uses all available fuel resulting in a nearly complete burn. According to Fig. 6 of Timmes and Woosley [16] a final temperature of T f ≈ 6 × 10 9 K for a 5 mg mass may be hot enough to ignite carbon burning. However, hydrodynamical simulations should be performed to explicitly verify that the energy release S heats the system enough to start carbon burning, which we reserve for future work. We note that the presence of Pb in Table I significantly increases the heat capacity without increasing the fission energy released. If the amount of Pb were less (or absent) the system would reach higher temperatures. Table III. FIG. 3. Total fission heating per baryon S(t) versus time for the four cases in Table III. C. Cross section and enrichment sensitivity To explore sensitivity to input parameters we have performed large numbers of reaction network simulations. The frames in Fig. 4 were prepared by computing a grid of networks to find the final temperature as a function of f 5 , σ f ( 238 U), and σ ff . In Fig. 4a and 4b we compare the final temperature when excluding neutron-captures on the fission fragments (i.e., σ ff = 0.0 b) and assuming a modest σ ff = 0.01 b as an average neutron-capture cross section, respectively. The grid is computed between f 5 = 0.08 and f 5 = 0.30 at a resolution of ∆f 5 = 0.01, and between σ f ( 238 U) = 0.00 b and 0.075 b with a resolution of ∆σ f ( 238 U) = 0.005 b, for a total of 368 network calculations. The cases A, B, C, and D from Table III are marked in Fig. 4. In Fig. 4a, where no neutron-captures occur on the fission fragments, there is a sharp transition between the incomplete burn or 'fizzle' behavior at low f 5 and the complete burn at high f 5 where the final temperature is T f 5 × 10 9 K. For enrichments f 5 0.25 the transition between incomplete/complete burns depends on σ f ( 238 U), requiring larger σ f ( 238 U) for a complete burn at lower enrichment f 5 . For enrichments f 5 0.25 a nonzero σ f ( 238 U) always results in a complete burn. In Fig. 4b the fission fragments act as a neutron sink with a cross section of σ ff = 0.01 b. In this case, the burning transition shifts to higher f 5 . However, at nonzero σ f ( 238 U) we still find a robust burn of the 238 U and Th. In Fig. 4c we explore the sensitivity to the fission fragments as a neutron sink assuming constant σ f ( 238 U)=0.014 b. We use a resolution of ∆σ ff = 0.002 b; the resolution in f 5 is the same as above, for a total of 483 networks. For low values of σ ff (i.e. σ ff σ f ( 238 U)), we find that complete burns are still readily achieved for f 5 0.20. It is only at σ ff σ f ( 238 U) that the fission fragments begin to 'outcompete' the 238 U for neutrons and quench the burning. Thus, even with some degree of 'poisoning' due to neutron captures onto fission fragments, or other impurities, the system may still undergo a complete burn so long as σ f ( 238 U) (and f 5 ) is sufficiently high. In Fig. 5 we show the total percentage of Th and U that fissions in the grid of networks computed in Fig. 4. It is clear that barely any Th (top) burns in networks that fizzle at low f 5 (Fig. 5a), and that the conditions required for a complete burn Th burning have a sharp transition. Neutron captures on the fission fragments can suppress the total Th fraction that fission (Figs. 5b and 5c), and the sharp turn-on is shifted to slightly greater f 5 . When we compare to the U fraction that burns (bottom) in the networks that fizzle, we see that most of the 235 U burns, but barely more than f 5 , so the heating is largely due to a 235 U burning which then stalls before igniting the breeder stage. We conclude that the final temperatures observed in Fig. 4 can be explained by a steady increase in 235 U burning with increasing f 5 in the fizzling regime, with complete burns achieved after a very sharp turn-on which burns the Th and 238 U in a breeder stage. V. DISCUSSION In this section we discuss limitations in our reaction network and then carbon ignition. A. Limitations of reaction network We now explore possible limitations in our reaction network. First we have only included (n, γ) and fission reactions. These reactions proceed very rapidly on a timescale of τ ∼ 10 −15 s and neglecting beta-decay and (γ, n) should be a good first approximation. We have ne- glected (n, 2n) reactions because these should be unimportant except at high neutron energies above 1 MeV. In future work we will examine temperature-dependent cross sections and any new reaction pathways that may result. Our reaction network assumes 1 MeV neutrons. This is a reasonable first approximation to the fission spectrum as long as the amount of light nuclei such as C or O is small. If light nuclei are present, then nuclear recoil following elastic scattering will reduce the neutron energies. In future work we will explore sensitivity of the reaction network to the neutron spectrum. Our reaction network is somewhat incomplete and does not include reactions for very neutron-rich Th or U isotopes. For U we include reactions on isotopes up to 241 U. The omission of reactions on 242 U or heavier isotopes is not expected to be important because most of the U fissions and only a tiny fraction captures enough neutrons to reach 242 U. For Th we only include reactions on 232 Th and 233 Th. For Cases B, C and D a significant fraction of the original Th is converted to 234 Th (which is stable in our network). Including reactions on heavier Th isotopes could lead to the fission of more Th. In particular, the neutroninduced fission of 235 Th could be a significant addition to our reaction network. There may not be data for this neutron-rich isotope, however, and further progress using our reaction network likely will require theoretical rates for very neutron-rich isotopes. We assumed a constant baryon density in Sec. III Eqs. 2,3, and 4. In reality, the system will expand slightly because of the large fission energy release. However this decrease in density is only about 25% because the electrons are degenerate [11]. This will slightly slow down the rate of all neutron reactions and therefore the chain reaction will take slightly longer to complete. We have neglected cooling from heat conduction and neutrino emission. This should be a good approximation during the fission reaction because the reaction rates are so high. The fission heating rate in Fig. 2 is consistent with the estimated rate in Ref. [11]. The rate of cooling from heat conduction via the large thermal conductivity of the degenerate electrons is estimated in Ref. [11] to be two to three orders of magnitude lower than the heating rate in Fig. 2. Therefore heat conduction is unimportant during the fission reaction. B. Carbon ignition and explosive yield We now consider carbon ignition. After the fission chain reaction, the system may be so hot that selfpropagating thermonuclear carbon burning is initiated. Such a scenario is unstudied and future hydrodynamical simulations will be needed. In the meantime, however, we can examine the fission-scenario through analogy with terrestrial nuclear weapons. First for context, we discuss ignition of hydrogen isotopes in a terrestrial nuclear weapon. The Classical Super was the original idea to use heat from an atomic bomb to start fusion in a deuterium, tritium mixture. It is said that losses from thermal radiation will help quench thermonuclear fusion in the Classical Super. Instead, radiation implosion can be used to compress the hydrogen fuel first. This compression increases the energy density from hydrogen fusion without increasing the radiation losses. In a WD, the carbon fuel is already at a very high density. Therefore radiation losses are likely unimportant and our system may avoid this problem with the Classical Super. Because of the high initial density there may be no need for radiation implosion to compress the system further. It is interesting to compare the explosive yields in our system to those of nuclear weapons. Previously we had estimated the initial mass of the uranium rich crystal to be about 5 mg [10,11]. In Cases B, C, and D we find a significant fraction of the U and Th in this 5 mg mass fissions. This will release energy equivalent to about 50 kg of TNT. The yield is much lower than the approximately 15 kiloton yield of the first atomic bombs because the 5 mg mass of our system is much less than the multi kg core masses of conventional fission weapons. Note that the efficiency of our system may be much higher, with nearly all of the U and Th fissioning, compared to the few % efficiency of an atomic bomb. Nevertheless, because the mass and critical mass are so much smaller, our fission yield is almost a million times smaller. Therefore, if carbon burning is not initiated, the fission chain reaction may have very little effect on the star. The situation is dramatically different if carbon burning is initiated. The thermonuclear burning of a significant fraction of C in a WD will release energy comparable to a SN Ia. This corresponds to an explosive yield of almost 10 29 megatons (MT)! Thus, we propose using a 50 kg yield fission primary to ignite a 10 29 MT fusion secondary. The temperature required for carbon ignition depends on the ignition scenario including the system size, density and ignition timescale. Our system has a density near 10 8 g cm −3 and a total mass of order 5 mg. The original work of Timmes and Woosley may be the most directly relevant previous calculation of ignition temperature for our conditions [16]. They consider ignition in a C/O liquid where they instantaneously replace a small mass of C/O by carbon burning ashes and assume the temperature of this ash has been raised to T i . They find an ignition temperature of T i ≈ 5 × 10 9 K. If the final temperature T f > T i a flame may propagate in the surrounding C/O liquid. If T f < T i the system will cool via heat conduction without initiating carbon burning. This ignition temperature of 5 × 10 9 K should be verified with future hydrodynamic simulations. VI. CONCLUSIONS In this paper we have performed novel reaction network simulations of fission chain-reactions in a cooling WD. The first solids to form when material in a WD just starts to crystallize are expected to be U and Th rich because of their high charges. These solids may support a fission chain-reaction if the uranium enrichment f 5 is high enough > 0. 12. We find that the reaction proceeds very quickly (within ≈ 10 −11 s) because the density is high and the neutron cross sections are large. In general, the reaction proceeds in two stages. In the first stage neutrons from 235 U fission transform or breed some 238 U and 232 Th nuclei into more easily fissionable 239 U and 233 Th. In the second, or breeder reaction, stage most of the original U and a significant fraction of the Th fission. These reaction stages release ≈ 0.36 MeV/nucleon and raise the final temperature to T f ≈ 6 × 10 9 K. This is important input for our thermal diffusion simulations where we find that carbon ignition is likely at high densities [12]. Acknowledgements: We thank Ezra Booker, Con- FIG. 1 . 1Abundance per baryon Yi versus time for Case A in panel (a), B in (b) and C in (c). Panel (d) shows abundance per baryon of neutrons (black curves) and fission fragments (red curves) vs time. Results are shown for Case B without neutron capture on fission fragments (dotted) and for Case D with capture (solid), see FIG. 2 . 2Heating rate per baryonṠ(t) versus time for the four cases in FIG. 4 .FIG. 5 . 45Final temperature as a function of f5 and σ f ( 238 U) using constant σ ff = 0 (left) and 0.01 b (center). Right panel shows the final temperature as a function of f5 and σ ff at constant σ f ( 238 U) = 0.014 b. We compute 368 networks in (a) and (b) and 483 networks in (c); the jagged regions are an artifact of the finite resolution of the grid. Scenarios A, B, C, and D fromTable IIIare indicated in the panels. The Red region approximately reaches carbon ignition[16]. Percentage of Th that fissions (top) and total fraction of U that fissions (bottom) for the same simulations as inFig. 4. TABLE I . IInitial abundances Yi, electron fraction Ye, and baryon number density n b .TABLE II. Cross sections for neutron absorption (n, γ) and fission reactions for 1 MeV neutrons on Th and U isotopes. Also listed is the average number of neutrons emitted per fissionν. Data from [13].U Th Pb Ye n b (cm −3 ) 1.3 × 10 −3 9.51 × 10 −4 2.28 × 10 −3 0.391 6.02 × 10 31 Isotope σn,γ (b) σ f (b)ν 232 Th 0.14 0.0013 2.18 233 Th 0.068 0.094 2.69 235 U 0.11 1.20 2.53 236 U 0.17 0.36 2.49 237 U 0.082 0.68 2.57 238 U 0.13 0.014 2.69 239 U 0.097 0.38 2.91 240 U 0.086 0.007 2.69 241 U 0.17 0.24 2.88 stantine Deliyannis, Erika Holmbeck, Wendell Misch, Matthew Mumpower, Witek Nazarewicz, Catherine Pilachowski, Tomasz Plewa, and Rebecca Surman for helpful discussions. The work of CJH was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. This research was supported in part by the US Department of Energy Office of Science Office of Nuclear Physics grants DE-FG02-87ER40365 and DE-SC0018083 (NU-CLEI SCIDAC). . 0 0 8 . 0 0 1 6 . 0 0 2 4 . 0 0 3 2 . 0 0 4 0 . 0 0 4 8 . 0 0 5 6 . 0 0 . 0 0 8 . 0 0 1 6 . 0 0 2 4 . 0 0 3 2 . 0 0 4 0 . 0 0 4 8 . 0 0 5 6 . 0 0 . T M C Abbott, S Allam, P Andersen, C Angus, J Asorey, A Avelino, S Avila, B A Bassett, K Bechtol, G M Bernstein, 10.3847/2041-8213/ab04faThe Astrophysical Journal. 87230T. M. C. Abbott, S. Allam, P. Andersen, C. Angus, J. Asorey, A. Avelino, S. Avila, B. A. Bassett, K. Bechtol, G. M. Bernstein, and et al., The Astrophysical Journal 872, L30 (2019). . D A Howell, 10.1038/ncomms1344Nature Communications. 2350D. A. Howell, Nature Communications 2, 350 (2011). Type ia supernovae and cosmology. M Sullivan, 10.1007/978-3-642-10598-2_2Lectures on Cosmology: Accelerated Expansion of the Universe. G. WolschinBerlin Heidelberg; Berlin, HeidelbergSpringerM. Sullivan, "Type ia supernovae and cosmology," in Lec- tures on Cosmology: Accelerated Expansion of the Uni- verse, edited by G. Wolschin (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010) pp. 59-97. . A G Riess, L M Macri, S L Hoffmann, D Scolnic, S Casertano, A V Filippenko, B E Tucker, M J Reid, D O Jones, J M Silverman, R Chornock, P Challis, W Yuan, P J Brown, R J Foley, 10.3847/0004-637X/826/1/56arXiv:1604.01424Astrophys. J. 826astro-ph.COA. G. Riess, L. M. Macri, S. L. Hoffmann, D. Scolnic, S. Casertano, A. V. Filippenko, B. E. Tucker, M. J. Reid, D. O. Jones, J. M. Silverman, R. Chornock, P. Challis, W. Yuan, P. J. Brown, and R. J. Foley, Astrophys. J. 826, 56 (2016), arXiv:1604.01424 [astro-ph.CO]. . A G Riess, S Casertano, W Yuan, J B Bowers, L Macri, J C Zinn, D Scolnic, 10.3847/2041-8213/abdbafThe Astrophysical Journal. 9086A. G. Riess, S. Casertano, W. Yuan, J. B. Bowers, L. Macri, J. C. Zinn, and D. Scolnic, The Astrophys- ical Journal 908, L6 (2021). . E Di Valentino, O Mena, S Pan, L Visinelli, W Yang, A Melchiorri, D F Mota, A G Riess, J Silk, 10.1088/1361-6382/ac086dClassical and Quantum Gravity. E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, Classical and Quantum Gravity (2021), 10.1088/1361- 6382/ac086d. . B Wang, Z Han, 10.1016/j.newar.2012.04.001arXiv:1204.1155astro-ph.SRNew Astronomy Reviews. 56B. Wang and Z. Han, New Astronomy Reviews 56, 122 (2012), arXiv:1204.1155 [astro-ph.SR]. Towards an understanding of type ia supernovae from a synthesis of theory and observations. W Hillebrandt, M Kromer, F K Röpke, A J Ruiter, arXiv:1302.6420astro-ph.COW. Hillebrandt, M. Kromer, F. K. Röpke, and A. J. Ruiter, "Towards an understanding of type ia supernovae from a synthesis of theory and observations," (2013), arXiv:1302.6420 [astro-ph.CO]. . P Ruiz-Lapuente, 10.1016/j.newar.2014.08.002New Astronomy Reviews. 62P. Ruiz-Lapuente, New Astronomy Reviews 62-63, 15 (2014). . C J Horowitz, M E Caplan, 10.1103/PhysRevLett.126.131101Phys. Rev. Lett. 126131101C. J. Horowitz and M. E. Caplan, Phys. Rev. Lett. 126, 131101 (2021). . C J Horowitz, M E Caplan, Arxiv 2107.03568C. J. Horowitz and M. E. Caplan, Arxiv 2107.03568 (2021). . C J Horowitz, arXiv:2208.00053C. J. Horowitz, arXiv:2208.00053 (2022). . J Lippuner, L F Roberts, 10.3847/1538-4365/aa94cbThe Astrophysical Journal Supplement Series. 23318J. Lippuner and L. F. Roberts, The Astrophysical Jour- nal Supplement Series 233, 18 (2017). . Y Zhu, J C Pei, 10.1103/PhysRevC.94.024329Phys. Rev. C. 9424329Y. Zhu and J. C. Pei, Phys. Rev. C 94, 024329 (2016). . F X Timmes, S E Woosley, 10.1086/171746Astrophys. J. 396649F. X. Timmes and S. E. 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[ "Cross-layer Application-aware Power/Energy Management for Extreme Scale Science [Position paper]", "Cross-layer Application-aware Power/Energy Management for Extreme Scale Science [Position paper]" ]	[ "Ivan Rodero \nRutgers Discovery Informatics Institute\nDepartment of Electrical and Computer Engineering\nNSF Cloud and Autonomic Computing Center\nRutgers University\n\n", "Manish Parashar [email protected] \nRutgers Discovery Informatics Institute\nDepartment of Electrical and Computer Engineering\nNSF Cloud and Autonomic Computing Center\nRutgers University\n\n" ]	[ "Rutgers Discovery Informatics Institute\nDepartment of Electrical and Computer Engineering\nNSF Cloud and Autonomic Computing Center\nRutgers University\n", "Rutgers Discovery Informatics Institute\nDepartment of Electrical and Computer Engineering\nNSF Cloud and Autonomic Computing Center\nRutgers University\n" ]	[]	MotivationHigh Performance Computing (HPC) has evolved over the past decades into increasingly complex and powerful systems. Current HPC systems consume several MWs of power, enough to power small towns, and are in fact soon approaching the limits of the power available to them. Estimates are with the given current technology, achieving exascale will require hundreds of MW, which is not feasible from multiple perspectives. Architecture and technology researchers are aggressively addressing this; however as past history is shown, innovation at these levels are not sufficient and have to be accompanied with innovations at higher levels (algorithms, programming, runtime, OS) to achieve the multiple orders of magnitude reduction -i.e., a comprehensive cross-layer and application-aware strategy is required. Furthermore, energy/power-efficiency has to be addressed in combination with quality of solutions, performance and reliability and other objectives and appropriate tradeoffs are required.	null	[ "https://arxiv.org/pdf/1304.2840v1.pdf" ]	42,061,281	1304.2840	14860fc83458bcc59b7870d25c44556ab26b2313	Cross-layer Application-aware Power/Energy Management for Extreme Scale Science [Position paper] Ivan Rodero Rutgers Discovery Informatics Institute Department of Electrical and Computer Engineering NSF Cloud and Autonomic Computing Center Rutgers University Manish Parashar [email protected] Rutgers Discovery Informatics Institute Department of Electrical and Computer Engineering NSF Cloud and Autonomic Computing Center Rutgers University Cross-layer Application-aware Power/Energy Management for Extreme Scale Science [Position paper] MotivationHigh Performance Computing (HPC) has evolved over the past decades into increasingly complex and powerful systems. Current HPC systems consume several MWs of power, enough to power small towns, and are in fact soon approaching the limits of the power available to them. Estimates are with the given current technology, achieving exascale will require hundreds of MW, which is not feasible from multiple perspectives. Architecture and technology researchers are aggressively addressing this; however as past history is shown, innovation at these levels are not sufficient and have to be accompanied with innovations at higher levels (algorithms, programming, runtime, OS) to achieve the multiple orders of magnitude reduction -i.e., a comprehensive cross-layer and application-aware strategy is required. Furthermore, energy/power-efficiency has to be addressed in combination with quality of solutions, performance and reliability and other objectives and appropriate tradeoffs are required. Challenges addressed The main goal is to expand runtime energy/power management at multiple levels and how address this more effectively in a cross-layer manner and tradeoff energy/power with the quality of the solution. Current approaches try to optimize energy efficiency at different levels, such as, runtime or component-based power management; however, the optimization goals of these approaches may be conflicting. Also, current data management approaches typically operate on centralized data repositories and cannot handle the extreme rates of data generation and distribution scales. In these thrusts, we will explore autonomic mechanisms to manage energy efficiency through dynamic cross-layer adaptations in an integrated and holistic manner as illustrated in Figure 1. Hardware' Network' The main challenges of such a cross-layer approach can be summarized as follows: (1) understanding the application patterns and how keeping the application in the loop can facilitate performing effective optimizations and adaptations at different levels in a cross-layer approach; (2) studying the implications of these adaptations to other dimensions such as performance, data analysis/management, and methods to effectively use all levels of the memory hierarchy [1]; and (3) understanding the control plane and the mechanisms that can be used to implement and orchestrate such a management approach. Maturity Our vision is revolving on a cross-layer approach to perform effective optimizations/adaptations. We have studied and modeled performance/energy tradeoffs of power management techniques at different levels and designed and evaluated model-driven autonomic optimizations and adaptations, including applicationcentric aggressive power management of data center's resources for HPC workloads [6], multi-channel DRAM power management [2] and application-aware cross-layer power management for PGAS applications on many-core platforms [3]. We have also explored energy-efficient and thermal-aware autonomic management of HPC workloads [5,7], and created innovative testbeds to conduct experimental evaluation and explored cutting-edge platforms such as the Intel SCC many-core system. Uniqueness The proposed approach address energy efficiency keeping the application/workload in the loop, which facilitates performing more effective optimizations and adaptations, for example, by using language extensions to improve runtime decisions. A key aspect is providing these adaptations along multiple dimensions, for example, adapting the resources to the applications (e.g., using low power modes to conform to a power budget) or adapting the application to the resources (e.g., reducing the precision of the computation or changing convergence values in order to adapt the accuracy/quality of the solution to guarantee completion time in a given time frame). Novelty The core idea of our approach is having a control plane along multiple levels including OS mechanisms and interfaces that can be used to optimize multiple tradeoffs and perform adaptations. They can be automatically orchestrated at runtime, in a cross-layer way to respond to the heterogeneity and dynamics, both of the applications and the infrastructure. Our approach is based on model-based power/energy management; however, in contrast to existing approaches, we propose end-to-end optimizations and optimizations that integrate the application with the runtime (e.g., using language extensions). An example of this approach is to reactively and proactively manage and optimize adaptive simulations (e.g., SAMR applications [8]) by detecting conditions under which the parameters affecting the application deviate from their acceptable behavior or operation (e.g., simulation performance may be degrade severely due to low available low-latency memory and increased data transfer over the network) and determining the appropriate application reconfiguration strategy and the resources required to repartition work (e.g., amount of memory of each level of the hierarchy) to optimize the simulation performance and energy efficiency. Applicability The overall cross-layer approach can be applied to other relevant areas, such as, data processing, transfer and management, and the design and usage modes of deep memory hierarchies, in addition to energy efficiency. The proposed approach could also influence the design or adoption of advanced hardware (e.g., exposing sufficient power controls and low power modes for different subsystems), how applications are developed and data analyzed, and how the scientists interact with applications data. An example of how cross-layer optimizations can be applied to data management is transferring data over the network only when analyzing data (generated during simulation) in dedicated nodes is more energy/performance efficient than doing the analysis in-situ where the data is generated. Energy/power optimizations can be enabled, for example, using low power modes in the dedicated nodes when the data is in-transit. At the same time, deciding when/where to move/analyze the data can be influenced by application constraints or the application can be adapted to the resources based on energy/performance tradeoffs (e.g., accuracy of the solution or frequency of data analysis within a power budget). Effort The effort to effectively explore the proposed approach can be decomposed into different stages: (1) characterize relevant compute-and data-intensive applications and HPC benchmarks and explore performance/power tradeoffs to define models, (2) develop mechanisms, strategies, application extensions and usage modes to implement model-driven optimizations/adaptations, (3) enable cross-layer interactions and integrate them with the runtime system, and (4) explore our approach at larger scale using simulation [4] considering non-standard hardware configurations, hardware and software co-design and how they impact the applications and system. Figure 1 : 1Conceptual view of the proposed cross-layer approach The Opportunities and Challenges of Exascale Computing. S Ashby, Summary Report of the Advanced Scientific Computing Advisory Committee (ASCAC) Subcommittee. S. Ashby et al., "The Opportunities and Challenges of Exascale Computing", Summary Report of the Advanced Scientific Computing Advisory Committee (ASCAC) Subcommittee, 2010. Adaptive Memory Power Management Techniques for HPC Workloads. K Elangovan, I Rodero, M Parashar, F Guim, I Hernandez, 18th IEEE International Conference on High Performance Computing (HiPC). K. Elangovan, I. Rodero, M. Parashar, F. Guim, I. Hernandez, "Adaptive Memory Power Management Techniques for HPC Workloads", 18th IEEE International Conference on High Performance Computing (HiPC), 2011. Exploring Cross-layer Power Management for PGAS Applications on the SCC Platform. M Gamell, I Rodero, M Parashar, R Muralidhar, 21st International ACM Symposium on High-Performance Parallel and Distributed Computing (HPDC). M. Gamell, I. Rodero, M. Parashar, R. Muralidhar, "Exploring Cross-layer Power Management for PGAS Applications on the SCC Platform", 21st International ACM Symposium on High-Performance Parallel and Distributed Computing (HPDC), pp. 235-256, 2012. Using simulation to design extreme scale applications and architectures: programming model exploration. C L Janssen, H Adalsteinsson, J P Kenny, SIGMETRICS Perform. 38C.L. Janssen, H. Adalsteinsson, and J.P. Kenny, "Using simulation to design extreme scale applications and architectures: programming model exploration," SIGMETRICS Perform. Eval. Rev., Volume 38, Issue 4, pp. 4-8, 2011. Energy-Efficient Application-Aware Online Provisioning for Virtualized Clouds and Data Centers. I Rodero, J Jaramillo, A Quiroz, M Parashar, F Guim, S Poole, IEEE International Green Computing Conference (IGCC). I. Rodero, J. Jaramillo, A. Quiroz, M. Parashar, F. Guim, S. Poole, "Energy-Efficient Application- Aware Online Provisioning for Virtualized Clouds and Data Centers", IEEE International Green Computing Conference (IGCC), pp. 31-45, 2010. Investigating the Potential of Application-Centric Aggressive Power Management for HPC Workloads. I Rodero, S Chandra, M Parashar, R Muralidhar, H Seshadri, S Poole, 17th IEEE International Conference on High Performance Computing (HiPC). I. Rodero, S. Chandra, M. Parashar, R. Muralidhar, H. Seshadri, S. Poole, "Investigating the Potential of Application-Centric Aggressive Power Management for HPC Workloads", 17th IEEE International Conference on High Performance Computing (HiPC), 2010 Energy-efficient Thermalaware Autonomic Management of Virtualized HPC Cloud Infrastructure. I Rodero, H Viswanathan, E K Lee, M Gamell, D Pompili, M Parashar, Journal of Grid Computing. 103I. Rodero, H. Viswanathan, E.K. Lee, M. Gamell, D. Pompili, M. Parashar, "Energy-efficient Thermal- aware Autonomic Management of Virtualized HPC Cloud Infrastructure", Journal of Grid Computing, Volume 10, Issue 3, pp. 447-473, 2012. Autonomic Proactive Runtime Partitioning Strategies for SAMR Applications. Y Zhang, J Yang, S Hariri, S Chandra, M Parashar, 18th International Parallel and Distributed Processing Symposium (IPDPS) -Workshop 10. Y. Zhang, J. Yang, S. Hariri, S. Chandra, M. Parashar, "Autonomic Proactive Runtime Partitioning Strategies for SAMR Applications", in 18th International Parallel and Distributed Processing Symposium (IPDPS) -Workshop 10, 2004.	[]
[ "The optical spectroscopy of extraterrestrial molecules", "The optical spectroscopy of extraterrestrial molecules" ]	[ "T W Schmidt \nSchool of Chemistry\nUniversity of Sydney\n2006NSWAustralia\n", "R G Sharp \nAnglo-Australian Observatory\nPO Box 2961710EppingNSWAustralia\n" ]	[ "School of Chemistry\nUniversity of Sydney\n2006NSWAustralia", "Anglo-Australian Observatory\nPO Box 2961710EppingNSWAustralia" ]	[]	The ongoing quest to identify molecules in the interstellar medium by their electronic spectra in the visible region is reviewed. Identification of molecular absorption is described in the context of the elucidation of the carriers of the unidentified diffuse interstellar bands while molecular emission is discussed with reference to the unidentified Red Rectangle bands. The experimental techniques employed in undertaking studies on the optical spectroscopy of extraterrestrial molecules are described and critiqued in the context of their application.	10.1071/ch04269	[ "https://export.arxiv.org/pdf/astro-ph/0501180v1.pdf" ]	16,161,166	astro-ph/0501180	705d21c4a86d1e1513542f348ea9bcbb4f159c55	The optical spectroscopy of extraterrestrial molecules 11 Jan 2005 T W Schmidt School of Chemistry University of Sydney 2006NSWAustralia R G Sharp Anglo-Australian Observatory PO Box 2961710EppingNSWAustralia The optical spectroscopy of extraterrestrial molecules 11 Jan 2005 The ongoing quest to identify molecules in the interstellar medium by their electronic spectra in the visible region is reviewed. Identification of molecular absorption is described in the context of the elucidation of the carriers of the unidentified diffuse interstellar bands while molecular emission is discussed with reference to the unidentified Red Rectangle bands. The experimental techniques employed in undertaking studies on the optical spectroscopy of extraterrestrial molecules are described and critiqued in the context of their application. INTRODUCTION As the last embers of a red giant star die down, it undergoes a series of expansions and contractions, puffing away the outer layers of the star, resulting in the expulsion of its carbon rich atmosphere into the cosmos. As the central stellar core contracts under gravity into a white dwarf, the atmosphere evolves into a nascent proto-planetary nebula, rich in carbon, oxygen, nitrogen (Fig. 1). Such a dignified end to the life of an intermediate mass star, such as our own Sun is in sharp contrast to the violent end encountered by higher mass stars 1 which end their lives in supernova explosions. A supernova is triggered when the collapsing stellar core is too massive to be supported, against gravitational collapse, by electron degeneracy pressure (i.e. the Pauli exclusion principle, as is the case in a white dwarf star). The resulting violent nuclear explosion heats the surrounding interstellar medium, through a variety of mechanisms, to temperatures in excess of 10 6 − 10 8 K, sufficiently hot to generate X-ray emission. Elements from all over the periodic table are ejected into the cosmos: the atoms to be later incorporated into molecular clouds and future solar systems. All elements heavier than Fe are formed in supernovae. The chemistry of interstellar space is different to that performed in a conical flask. It is slow, it is driven by ion-molecule and neutral-radical reactions in the gas phase 1 and on the surfaces of dust grains, and it is highly exotic. The harsh radiation field in interstellar environments is also of great importance. There are a number of chemical models of interstellar space. Of note are the "UMIST gas-phase chemical network" of Millar and co-workers in Manchester 2 and the "New Standard Model" of Herbst and co-workers in Columbus (NSM) 3. Both these models use complicated networks of kinetic equations to model the chemistry of various interstellar environments. However, these models may only be tested by spectroscopic observation of the relative abundances of interstellar molecules, which is a field 1 The mass limit above which electron degeneracy pressure cannot support a stellar core against the relentless crushing force of gravity was first derived by S. Chandrasekhar, for which he was later awarded a Nobel prize. Above 1.44 Solar masses a stellar core will collapse to a neutron star or black hole, resulting in a supernova explosion rather than the formation of planetary nebular. Precursor stars with masses in excess of 1.44 Solar masses may still avoid ending their lives as supernova if enough mass is lost from the star, during it's late evolutionary stages, to prevent the remaining core mass exceeding the Chandrasekhar limit. unto itself. Also, the numerous (thousands of) rate constants are being constantly updated with better experimental and ab initio results. To understand the chemistry of interstellar clouds one must begin by first identifying the molecules therein. It is a great challenge posed by Nature to remotely identify the menagerie of molecules extant in the interstellar medium (ISM). Planetary nebulae are gas clouds surrounding stars typically hundreds of light years away. They take their name from their appearance when imaged in small telescopes, whereby they resemble gas giant planets from our solar system such as Uranus or Neptune. There is no association between planetary nebular and planets beyond this appearance. Molecules are identified in the interstellar regions by their spectroscopic signatures in the millimetre, infrared and optical regions of the electromagnetic spectrum. While it is the millimetre region which has most greatly illuminated our understanding of the structures of interstellar molecules, this technique is blind to a family of molecules of interest: those without permanent dipole moments. For this reason, the UMIST and NSM models concentrate on reproducing the observed abundances of polar molecules 2, 3. An up-to-date list of molecules known to exist in the interstellar regions is given in Table 1 5. In the following paragraphs, work concerning the identification of extraterrestrial molecules in various wavelength regimes is outlined. The astronomical facilities exemplified are done so in an Australian context, where possible, and so are not necessarily indicative of those facilities globally. 1.1. The millimetre-wave region. Molecules are heated by gravitational collapse, converting potential energy to kinetic energy which is distributed among the degrees of freedom of the constituent species. This energy can be radiated back into space by molecules as visible, infrared or millimetre-wave radiation. Molecules with permanent dipole moments, upon relaxation, radiate in the millimetre-wave region of the spectrum by cascading down the ladder of energy levels which describe molecular rotation. This radiation N 2 H + N 2 O NaCN OCS SO 2 c-SiC 2 CO 2 NH 2 H + 3 AlNC SiCN SiNC H 2 D + HD + 2 4 c-C 3 H l-C 3 H C 3 N C 3 O C 3 S C 2 H 2 CH 2 D + ? HCCN HCNH + HNCO HNCS CH 3 HOCO + H 2 CO H 2 CN H 2 CS H 3 O + NH 3 SiC 3 5 C 5 C 4 H C 4 Si l-C 3 H 2 c-C 3 H 2 CH 2 CN CH 4 HC 3 N HC 2 NC HCOOH H 2 CNH H 2 C 2 O H 2 NCN HNC 3 SiH 4 H 2 COH + 6 C 5 H C 5 O C 2 H 4 CH 3 CN CH 3 NC CH 3 OH CH 3 SH HC 3 NH + HC 2 CHO HCONH 2 l -HC 4 H? l-H 2 C 4 C 5 N 7 C 6 H CH 2 CHCN CH 3 C 2 H HC 5 N CH 3 CHO NH 2 CH 3 c-C 2 H 4 O CH 2 CHOH 8 CH 3 C 3 N HCOOCH 3 CH 3 COOH C 7 H H 2 C 6 CH 2 OHCHO l -HC 6 H? CH 2 CHCHO? 9 CH 3 C 4 H CH 3 CH 2 CN (CH 3 ) 2 O CH 3 CH 2 OH HC 7 N C 8 H 10 CH 3 C 5 N? (CH 3 ) 2 CO NH 2 CH 2 COOH? (CH 2 OH) 2 ? CH 3 CH 2 CHO 11 HC 9 N 12 C 6 H 6 ? 13 HC 11 N is collected and analyzed to produce a forest of sharp, well-defined spectral lines. These lines are matched to rotational spectra observed in laboratory experiments and in doing so the extraterrestrial species are identified. Molecules with larger dipole moments are easier to observe by this technique and as such asymmetric carbon chains are a dominant motif in the list of known molecules from the interstellar regions. Millimetre-wave spectroscopy is performed on extraterrestrial objects by so-called radio telescopes. Excellent examples of this type of instrument are the 64 m Parkes radio telescope, the 22 m Mopra telescope, and the Australia Telescope Compact Array, all administered by the Australia Telescope National Facility of the CSIRO 6 ( Fig. 2). There are many groups who undertake laboratory experiments to which astronomical observations may be compared. In the laboratory, rotational spectroscopy is performed by a fourier-transform technique whereby the rotational spectrum is obtained in a similar fashion to the free-induction decay well known in the field of nuclear magnetic resonance. The group of Thaddeus and co-workers have discovered over fifty molecules of astrophysical relevance 7. 1.2. The infrared region. The infrared region features emission corresponding to vibrational relaxation of specific functional groups and bonds comprising interstellar molecules. Of note is the 3.3 µm emission lines which are thought to originate from polycyclic aromatic hydrocarbons (PAHs), such as naphthalene, anthracene and phenanthrene. Spectroscopy in the infrared is difficult, however, due to the interference of sources of background infrared emission (sky and telescope). Some regions of the infrared are "off-limits" to astronomers due to absorption of radiation by water in the atmosphere. Emission in the far infrared has been used to identify interstellar C 3 23 and C 5 22. Due to technical difficulties associated with ground-based observations in the infrared, last year (2003), a new satellite was launched by NASA. This new instrument, SPITZER, (Fig. 2) is capable of performing infrared spectroscopy in the wavelength region spanning 3-180 µm at various levels of resolution. The ground-based Michelle (Mid-Infrared Echelle spectrograph) mounted on Gemini North, Hawaii, is capable of spectroscopy in the 7-26 µm region with resolution of 30000 2 . The Anglo-Australian Telescope (AAT) at Coonabarabran is capable of spectroscopy in the 0.9-2.5 µm range with resolution of 2400 10. In the laboratory, infrared spectroscopy is routinely performed using a fourier-transform technique (FTIR). However, this technique is less sensitive than tunable diode laser spectroscopy (TDL). TDL spectroscopy as an absorption technique has been applied to the IR spectroscopy of many carbon chains and rings of astrophysical relevance. Of particular note is the Cologne Carbon Cluster experiment 11 of Winnewisser and co-workers. FIG. 2 Astronomical instruments used to observe extraterrestrial molecules. Top to bottom: Parkes radio-telescope; Australia Telescope Compact Array; Spitzer; Hubble Space Telescope; AAT at Coonabarabran; Gemini North (CFH in background). 1.3. The optical region. It is in the optical regions of the electromagnetic spectrum (300-950 nm) that there is much to be done. Despite the optical region being the part of the electromagnetic spectrum originally accessed by astronomers, there have been scarce new identifications of interstellar molecules by their electronic spectra. Examples of astronomical instruments available in Australia that access this region for the purposes of optical spectroscopy are the ultra-high resolution facility (UHRF) (R ∼ 900, 000) and the University College London Coudé Echelle Spectrograph (UCLES) on the AAT 10. Facilities with Australian access include the Gemini Multi-Object Spectrograph (GMOS) on Gemini North (Hawaii) and South (Chile), which is a low resolution optical spectrograph. The Bench-mounted High Resolution Optical Spectrograph (bHROS) on Gemini South will be in routine operation shortly. Molecules without permanent dipole moments may not be observed by millimetre wave spectroscopy, and those without infrared active vibrational modes cannot be observed by spectroscopy in that region. For instance, C 2 may only be observed by optical spectroscopy. For these reasons, spectroscopy in the optical region offers possibilities for observing new interstellar molecules. This review focusses on the research being undertaken in the field of the identification of new interstellar molecules through their electronic spectra in the near-infrared, visible and ultraviolet spectrum. The article is arranged as follows. Firstly, the absorption of starlight by molecules is discussed and astronomical and laboratory work in this area is reviewed in the context of the unidentified "Diffuse Interstellar Bands" (DIBs). Molecular emission is discussed in relation to the Red Rectangle paradigm and ongoing work in this area. The experimental techniques and results involved in performing interstellar spectroscopy in the laboratory are reviewed in the context of their application. Finally, state-of-the-art experiments are described and suggestions are made as to the future directions of the field. MOLECULAR ABSORPTION -THE DIFFUSE INTERSTELLAR BANDS The spectrum of light from many stars is well described to first order by black-body radiation theory: the hotter the star, the bluer its spectrum. So-called "reddened" stars are those for which the blue part of the spectrum is attenuated by scattering. The scattering takes place in the interstellar clouds, along the line of sight to the background star. The passage of the starlight though such a cloud can leave the signature of the DIBs imprinted on the stellar spectrum. With starlight as a "white-light" source, the interstellar cloud as a sample, and with spectrographs on Earth, one has all the essential elements of a benchtop spectrophotometer (albeit slightly larger). Molecular clouds vary in their density. The denser clouds greatly diminish the brightness of the background star thereby also sheltering the interior of the cloud from harmful deep ultraviolet radiation. The denser clouds naturally have higher abundances of molecular species, which are measured by "column density". Column density is an effective number of molecules in a column of space between the observer and the light source, typically quoted as having a 1 cm 2 cross-section. The spectra of many diatomic molecules have been known for decades 12. Diatomic species, such as CH, C 2 , CN, and CH + , have been detected toward a number of clouds. Their column densities are regarded as standards with which chemical models of interstellar clouds may be compared. Less is known about absorption in the optical region by polyatomic molecules. In the spectra taken towards diffuse clouds, there are approximately 300 absorption features of unknown origin. These features, collectively known as the "Diffuse Interstellar Bands" (DIBs), vary in width from 0.1-3 nm and cover the entire visible and near IR regions. There is a vast body of phenomenology pertaining to the identity of the DIB carriers, which has been reviewed elsewhere 13. There are various hypotheses as to which types of transitions are responsible for the DIBs 14. These are outlined below. 2.1. The carbon chain hypothesis. Carbon chains were first proposed as the carriers of the DIBs by Douglas 15. That the DIB absorbers are carbon chains is predicated upon the observation that carbon chains can exhibit very strong transitions in the visible region, and that they are known to exist in molecular clouds 16. The list in Table 1 demonstrates that carbon chains are a widely exhibited motif of the known interstellar molecules. The carbon chains observed by millimetre-wave astronomy are necessarily strongly polar (e.g. H − C ≡ C − C ≡ C − C ≡ C − C ≡ C − C ≡ C − C ≡ N) yet it is expected that the bare carbon chains (C n ) will also be present in molecular clouds. It is hypothesised that some or all of the DIB absorbers are carbon chain molecules. The hypothesis is easy to prove, at least in principle. One must measure the absorption spectra of target molecules, in the gas phase, under isolated conditions, and compare these to astronomical spectra. An example of a positive identification of an interstellar carbon chain by this method is the observation of C 3 in interstellar clouds 17. C 3 had been previously identified in the laboratory and its spectrum at 405 nm was well known from laser-induced fluorescence spectroscopy 18 and comets 19. A survey towards ζ-Ophiuchi, 20-Aquilae and ζ-Persei revealed absorption by the A 1 Π u ← X 1 Σ + g transition of C 3 17. The rotational profile was fitted to 80 K and the column density was determined to be 1 − 2 × 10 12 cm −2 (Fig. 3). Since then, C 3 has been observed towards many stars with column densities reported up to 10 13 cm −2 . A strong relationship between the column densities of C 2 and C 3 with N (C 2 )/N (C 3 ) ≃ 40 has been reported 20. So far, searches for C 4 and C 5 by optical spectroscopy have been unsuccessful 21. Upper bounds have been placed on column densities of C 4 and C 5 towards ζ-Ophiuchi of 10 13 cm −2 and 2×10 11 cm −2 respectively. The value of N (C 4 ) relies on a calculated oscillator strength and is therefore less certain. It should be pointed out that C 5 and C 3 have been observed by high resolution IR absorption spectroscopy in carbon-rich nebulae 22, 23. Carbon chains, bare, monohydrogenated or dihydrogenated are expected or have been shown to exist in the ISM. Whether they are the DIB absorbers can be rigorously tested by a combination of laboratory spectroscopy and observation. A large part of the work reviewed here was performed in Basel, Switzerland, in the group of J.P. Maier. 2.1.1. Resonant 2-Colour 2-Photon Ionization (R2C2PI) spectroscopy. The spectra of dihydrogenated chains, HC n H, were first observed in solution and in solid matrices. The gas-phase spectra of the even series, HC 2n H, were recorded only recently. The spectra of HC 2n H (n = 8−13) were obtained by Resonant 2-Colour 2-Photon Ionization (R2C2PI) spectroscopy in a molecular beam produced by a discharge of diacetylene in argon 24. R2C2PI spectroscopy is a type of Resonance-Enhanced Multi-Photon Ionization (REMPI) spectroscopy 25 (see Fig. 4). In such an experiment, the gas-phase supersonically cooled products of the discharge (containing C n H m species with n ≥ m) are irradiated by two laser beams. The first is scanned in wavelength. If a photon from the first laser is absorbed, then absorption from the second laser will overcome the ionization potential of the molecule and nascent ions will be produced. The ions are accelerated towards a detector, arriving at a time-of-flight (TOF) characteristic of the mass to charge ratio of the ion (generally speaking only singly charged molecules are observed). This experiment yielded the spectra of the even chains up to HC 26 H, none of which were observed to lie in the visible region (the longest chain, HC 26 H, absorbs at 340 nm). The reason for this behaviour is bond length alternation. The hydrogen end-caps of the chain induce triple-bond/single-bond alternation which slowly decays toward the molecular centre. The overall effect is that of the chain exhibiting a bandgap, which slows the movement of the absorption positions to lower energies as the chain length is increased 26. Very weak, forbidden, bands of the HC 2n H series have been observed in the visible region, yet these are not of relevance to astrophysical studies due to their low oscillator strengths 26. FIG. 4 In R2C2PI spectroscopy, supersonically cooled molecules are ionized in two steps by photons of two colours (top-left) and then mass-analyzed. The energy of λ 2 is not energetic enough to ionize the ground state molecule in a one photon process, so appreciable ion signal is only observed when λ 1 is resonant with an excited state of the molecule. As the wavelength λ 1 is scanned, an excitation spectrum is produced. The mass spectrum of a hydrocarbon discharge appears as indicated in the top-right of the figure. When λ 1 is resonant, a mass peak will be enhanced by orders of magnitude in strength. Cavity Ringdown Spectroscopy (CRDS). Absorption spectra of the odd chains HC 2n+1 H (n = 3 − 6) were observed in the visible region by cavity ringdown spectroscopy (CRDS) 27. CRDS can be performed using pulsed lasers 29,30 or continuous wave lasers 28. The principle is as follows (see Fig. 5). molecules are expanded into a vacuum chamber such that they cool supersonically. In experiments related to carbon chains, the molecules expanded into the vacuum are the products of a hydrocarbon discharge, much like the R2C2PI experiments outlined above 28. The free jet expands in an optical cavity defined by two highly reflective mirrors mounted on either side of the vacuum chamber. A laser beam is injected into the cavity (tuned to support the particular wavelength). The decay of the light pulse in the cavity, as observed by a photodetector mounted exterior to the vacuum chamber, can be related to the reflectivity of the mirrors, and absorption of laser light by some molecular species extant in the free jet expansion. Scanning the laser beam produces spectra which must be analysed by the rotational structure, for there is no mass information about the absorber (unlike R2C2PI spectroscopy). Nevertheless, the spectra of HC 2n+1 H (n = 3 − 6) in the visible region were recorded in this way 27. These spectra were not found to match any DIBs. A simple free electron model of the electronic structure showed that the oscillator strength for these visible transitions increased with chain length. High-level MRCI calculations and CASSCF calculations did not support this assertion 31-33. It was found by computation that the excited states of the HC 2n+1 H series could be described by an admixture of two determinants. The determinants combined in even and odd combination to produce a lower energy excited state which carried very little oscillator strength and a higher energy state which carried a very large oscillator strength. The lower energyÃ states, represented by absorption of HC 2n+1 H (n = 3 − 6) in the visible region were thus found to be irrelevant to the DIB problem, on account of their vanishing oscillator strengths. This realization spurred a search for theB states by R2C2PI spectroscopy. Two bands were observed, for HC 13 H and HC 19 H, both lying in the UV region 33. It is concluded that dihydrogenated chains cannot be responsible for DIBs unless they are considerably longer than HC 19 H (it is estimated that HC 30 H will absorb in the visible region). Due to their strong dipole moment, the monohydrogenated chains, C 2n H, have been observed in the ISM by rotational spectroscopy 7. The optical spectra of monohydrogenated chains have been observed in the visible region by CRDS and R2C2PI spectroscopy 34, 35. That the spectra did not match any known DIBs places upper limits on their column densities of ≈ 10 12 cm −2 . The optical transitions observed for the C 2n H chains do not have large oscillator strengths. In general, polarization of the carbon chain reduces the overlap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital, thereby reducing the possible oscillator strength of the transition. Of the bare carbon chains, C 3 and C 5 have been observed in the interstellar medium 17, 22, 23. The absorption by C 3 towards ζ-Oph is due to a relatively weak perpendicular transition, first observed in fluorescence in comet tails. There is a much stronger parallel transition of C 3 in the vacuum ultraviolet region 38. This strong parallel transition is seen analogously in all C 2n+1 chains. The oscillator strength grows with the size of the chain and the wavelength shifts linearly (as opposed to HC 2n H which exhibits a band gap). The absorption spectra of C 7 to C 21 have been observed in a neon matrix 36, 37. Absorptions due to the strong parallel transition all lie in the visible region yet gas phase spectra, with which astronomical observations may be compared, have so far proved elusive. It is the authors' opinion that a carbon cluster source coupled to the R2C2PI technique represents the best chance of observing these spectra in the laboratory. Gas-phase spectra of the even carbon chains have also proved elusive. Condensed phase spectra of several members of the C 2n series have been observed 39, 40. The transitions observed often lie in the visible region and thus these molecules cannot be ruled out as the DIB carriers. Definitive proof, one way or the other, will come with unambiguous gas-phase spectra. Open shell carbon chains are not the best candidates as DIB absorbers, as their transitions are highly mixed, either distributing oscillator strength across many transitions or shifting the oscillator strength into the UV for medium length chains. Of the closed shell chains, there are HC 2n H, HC + 2n+1 , and C 2n+1 . As noted above, the polyyne series, HC 2n H, suffers from a non-linear dependence of absorption wavelength with chain size. The gas-phase spectrum of HC 26 H lies in the near UV and only much longer chains will begin to absorb in the visible (2n ∼ 40). Of the HC + 2n+1 cations, not much is known. Their spectra should mimic the C 2n+1 neutrals however the transitions will not be quite so strong since the overlap between the highest occupied and the lowest unoccupied molecular orbital is reduced in the polar molecules. 2.1.3. Photodetachment spectroscopy. It would be remiss not to discuss the possibility of carbon chain anions in the ISM. Indeed, carbon chains have been found to possess very high electron affinities 41. In particular, polar carbon chains may be efficient at electron capture due to the existence of dipole-bound states 42, 43. The spectrum of C − 7 was observed to match very closely several DIBs 44, 45. More precise observation showed that the match was not exact 46, and as such C − 7 was not responsible for any DIBs. The spectra of anions are recorded in the gas phase is a similar manner to R2C2PI spectra. The set-up is illustrated in Fig. 6. Anions are produced in a hydrocarbon discharge and accelerated in a time of flight (TOF) tube to the laser interaction region. The mass-selected anion bunch is intercepted by a pair of laser beams which cooperatively excite the anion and subsequently photodetach an electron. The nascent neutral molecules are oblivious to the ion mirror which reflects the remaining anions. The neutrals impacting onto the MCP at 2.7 keV induce a signal due to production of secondary electrons. The neutral signal as a function of the first laser pulse yields the excitation spectrum of the anion. While the possibilities are numerous, the outstanding candidates as carbon chain carriers of the DIBs are the odd-numbered carbon chains. It is expected that this hypothesis will be tested within the next few years. The examples of carbon chains terminated with heteratoms 47, 48 are too numerous to review here. However, they have been observed in molecular clouds (see Table 1) and are likely important species in interstellar chemistry 49. In time, some of these will be detected in the interstellar medium by optical spectroscopy. The PAH hypothesis. Polycyclic aromatic hydrocarbons (PAHS) are a class of molecule characterized by conjoined "benzene ring" moieties. They may be thought of as fragments of graphite with hydrogens bound to the edge. Examples of PAHs are naphthalene, anthracene (see Fig. 9). The presence of PAHs in the ISM was first proposed in the context of their possible role in the form of the UV extinction curve 50. The suggestion that PAHs might be the carriers of the DIBs came later 51, 52, with the realization that in order that smaller PAHs absorb in the visible they should be in cationic form 54. With the "coming of age" of Mid-IR astronomy, particularly results from the European Space Agency (ESA) Infrared Space Observatory (ISO) mission such as the identification of interstellar benzene 53, PAH molecules have been interpreted as the natural carriers for the observed Mid-IR emission band 55. Indeed, recent work has seen the first indications that the Mid-IR emission features can be well fitted by composites of emission from numerous PAH species 56. Unfortunately, the composite nature of the Mid-IR bands makes this region of the spectrum poor for identification of specific PAH molecules. The promising possibilities of using Far-IR emission bands (vibrational frequencies associated with the bending of the skeletal structure of molecules) to uniquely identify PAH molecules has been discussed 57. Further support for the PAH origin of DIBs comes from their spectral stability (with respect to environmental variations in the excitation spectrum). While the Mid-IR PAH features are observed to vary greatly in structure and intensity ratios between observations, the DIBs are observed to be surprisingly uniform in structure, with only the relative intensity of different bands altering with sight line, presumably an effect of differing abundances of species along different sight line. It has been demonstrated that environmental effects are significant in altering the emission profiles of the Mid-IR PAH features 56. This can account for the large variations seen in the observed Mid-IR emission and the discrepant intensity ratios when compared with laboratory observations. However, it has been shown that this is not the case for visible light PAH transitions 58. They show that PAH absorption features would be observed to be stable, at the level of current observations, over an extreme range of environmental conditions. Circumstantial evidence in favor of larger molecular carbon species is reported 59 whereby a potential "carbon-crisis" is highlighted: insufficient carbon is identified in ISM to support the proposed build up of carbon rich dust grains. High molecular weight PAH molecules could act as repositories for carbon. It has been estimated that up to 20% of cosmic carbon in the Galaxy is contained in PAHs 60 yet more likely it is less than this. The current status of possible matches between DIBs and laboratory PAH features has been reviewed 61. With the presence of PAH molecules in the ISM now widely accepted, the search for transitions in the visible region which could give rise to the DIBs is a logical next step. This is a problem for laboratory astrochemistry. Since most neutral PAHs amenable to spectroscopic study absorb only in the UV, research has concentrated on measuring the spectra of the PAH cations in the visible region. Until recently, only matrix isolation spectra of PAH cations were available. The first gas phase spectrum of a PAH cation was measured by resonance enhanced dissociation spectroscopy of naphthalene cation, Np + 62. The spectrum produced was noisy and not definitive. It was not until 1999 that the absorption spectrum of the naphthalene cation (the smallest PAH) was recorded 63, 64. Two bands were recorded by CRDS FIG. 6 An illustration of the apparatus used to produce the spectra of carbon chain anions. Anions are produced in a discharge and then accelerated in a time-of-flight (TOF) tube. The mass-selected ion bunch is intercepted by a pair of laser pulses which coorperatively neutralize the anion. The neutral signal as a function of λ 1 is the photodetachment spectrum. (vide supra) in a pulsed discharge source. Following this, the entire spectrum of the Np + cation was produced using an action spectroscopy technique pioneered earlier in work on benzene analogs 65. Np + was clustered with argon "spectator atoms". The cluster mass distribution as measured by time-of-flight mass spectrometry was observed to change when energy was absorbed by the cationic chromophore. Essentially, the spectator atom is evaporated when the Np + chromophore absorbs a visible photon, and the mass change is recorded. The "action" as a function of wavelength yields a proxy excitation spectrum. This technique is powerful however suffers from the spectator atoms inducing slight changes in the electronic chromophore. The change (0.1 nm at 648 nm) is enough to exclude these spectra from being directly comparable to astronomical observation. More recently, acephenene and naphthalene cations have been studied in more detail by CRDS 66. The technique which yielded the first PAH excitation spectrum, namely resonance-enhanced multiphoton dissociation (REMPD) has undergone a small revival. The development of ion-traps had led to the ability to irradiate a population of ions with multiple photons, thereby accumulating signal before detection. Mass-selected ions are trapped in the gas phase and irradiated with several laser shots. Mass spectrometry of the fragments reveals acetylene loss as an indicator of the ions having absorbed energy. The probability of the ion having absorbed enough to break carbon-carbon bonds is greatly enhanced if the first photon may be absorbed in such a way as to place the ion in an excited state. In this way, warm gas phase spectra of naphthalene and anthracene have been recorded 67. Mounting an ion trap on a cold head is one way of alleviating the problems of temperature on the spectrum 68. At the time of writing, not one PAH, neutral or cationic, has been identified in the ISM by optical spectroscopy. Indeed, not one DIB carrier has been positively identified. While it is certain that PAHs exist in the ISM, it is unclear whether they should be dominantly ionized or neutral. As PAHs are likely to have large electron affinities, there is also scope for the existence of PAH anions in the ISM. Indeed, the electron affinities of some carbon-based molecules exceed the ionization energy of alkaline earth atoms. As such, in an environment where one finds sodium atoms, one may also find carbonaceous anions. It is also unclear as to whether the PAHs should be wholly intact. One conclusion that may be drawn from the observed PAH cation spectra is that they are most probably not responsible for narrow DIBs, because the PAH cation bands observed thus far are all very broad (≈ 10Å), presumably due to lifetime effects associated with internal conversion processes. The Possibilities regarding extensions of the PAH hypothesis will be discussed below. 2.3. Something completely different?. There have been many suggestions over the years as to the identity of the DIB absorbers. Those suggestions taken most seriously, namely carbon chains and PAHs, have been discussed above. The remaining candidates are plentiful, and only need be tested in the laboratory. The possible existence of buckminsterfullerenes in space was first suggested by Kroto 69. That C 60 in particular is so symmetrical and so stable lends credence to the hypothesis. However, the strongest absorptions of C 60 occur in the UV and as such this molecule is not responsible for any DIBs 70. Following the publication of the matrix isolation spectrum of the C + 60 cation 71, a search was carried out towards a number of stars which revealed two new DIBs. The DIBs were found to have a spacing and absorption wavelength consistent with the observed spectrum of C + 60 in an argon matrix. While the assignment of the DIBs observed near 9500Å to C + 60 seems entirely reasonable, definitive proof can only come by the laboratory gas phase spectrum of C + 60 , which has so far proved elusive. C 2+ 60 may also exist in appreciable concentrations in the ISM, since the second ionization potential of C 60 is extraordinarily low (11.4 eV) 73. The column density of H 2 in molecular clouds is approximately 10 6 times greater than that of the most abundant polyatomic carbon species. This implies that for a given oscillator strength of a transition, only one millionth of the H 2 present need be in a particular state to effect the same absorption as other proposed carriers of the DIBs. Indeed, there exist inter-Rydberg transitions of H 2 calculated to match DIBs very well 74, 75. Experimental observations of the inter-Rydberg transitions have revealed intriguing properties of H 2 such as "outer well" states with W-shaped potential energy curves 76, 77. Another theory put forward attributes the DIBs to Rydberg matter 78, aggregations of excited atoms and molecules bonded through electrostatic interactions of Rydberg electrons. The existence of interstellar diamonds was suggested in 1969 79. They have since been shown to exist as nanometre sized crystallites in carbonaceous meteorites 80. Given the ability for defects and surface effects to produce colour centres, nanodiamonds present themselves as possible candidates as the carriers of some of the DIBs. MOLECULAR EMISSION -THE RED RECTANGLE The search for molecules in Space by optical spectroscopy may be performed by the molecules' absorption or emission of visible and UV light. The search for molecules by absorption is intrinsically linked to the search for the DIB carriers. In the case of C 3 , the molecule was first observed by emission in comet tails, subsequently in the laboratory and then finally by absorption in interstellar clouds 17. The "Red Rectangle" (see Fig 7) is a biconical proto-planetary nebula. In the case of the Red Rectangle, the core is a binary system. One of the stars has come to the end of its life and has started puffing off its atmosphere to leave behind a white dwarf. The other component is a helium white dwarf which died previously in a similar manner. It will have been more massive, leading to an earlier evolution. FIG. 7 The "Red Rectangle", a nearby proto-planetary nebula, is a carbon-rich object in which there are unidentified emitters, thought to be molecular 4 (Image reproduced with permission of the authors of reference 4.) Much of its mass will most likely have been accreted onto the other star, rather than puffed off in a previous nebula. The system will evolve into a binary white dwarf system. The unusual geometry of the nebula is not entirely understood 81. The Red Rectangle exhibits a "bipolar flow" which carries mass away from the central stars into the interstellar medium. It is suggested that the central stars give rise to a pair of jets that precess about one another (like a spinning top). The nebula's emission is also unusual: it displays a number of unidentified emission lines, the so-called Red Rectangle Bands (RRBs) 82. These bands, occuring in the visible region are speculated to be due to unusual carbon containing molecules. There is also suggestion that at least one of the features is related to a DIB 83. Thus, identification of the carrier of the band may be a "foot in the door" to the identification of the DIB carriers. This feature is illustrated in Fig. 8. The advantage of looking for molecules in emission is that the species necessarily fluoresce, or phosphoresce. In the case of fluorescence, the carriers may be observed by laser induced fluorescence (LIF) spectroscopy, so long as the species can be created in the laboratory. One possible problem is that the LIF spectrum of a hydrocarbon discharge may be too rich to positively identify individual species. However, there has been much progress in the last five years in diagnosing the products of a hydrocarbon discharge by gas-phase spectroscopy. It is now possible that much of the fluorescence can be assigned. The remainder will belong to new molecules. Identification of mass-unresolved spectra will come about by a combination of ab initio theory, isotopic studies, and rotational structure. In this way a molecular carrier at 443 nm was very recently identified as C 5 H 5 84. It has been suggested that some of the RRBs are due toã 3 Π u →X 1 Σ + g phosphorescence 85 of C 3 (which has never been observed in the gas phase). Indeed, CO is seen in the Red Rectangle due to its phosphorescence: the so-called Cameron Bands (between 1850 and 2600Å). If the species corresponding to the molecular carrier of the RRBs is produced in a hydrocarbon discharge, but the emission observed is due to phosphorescence, then it is likely that the molecules will pass out of the light collection region and into the vacuum pump before emitting a detectable number of photons (phosphorescence typically occurs on the ms time-scale, and molecular beams move at about 1mm/µs). An experiment designed to circumvent this problem is described in section 4. In Fig. 8, the observed spectrum of one of the RRBs is displayed alongside a simulated spectrum performed by the authors. The seemingly convincing simulation was performed with A ′′ = 0.84 cm −1 and A ′ = 0.76 cm −1 . Such a change of rotational constant (10%) upon excitation is unusual yet not unheard of. One class of molecule with A constants very sensitive to excited state are the carbenes. These molecules possess lone-pair electrons which, when excited, bring about large changes in geometry and thus rotational constant 86. A rotational constant in the range given is slightly unusual. It is too large to be due to three collinear second period atoms, so must be accounted for by an effective diatomic (or some other slightly non-collinear structure 87). Candidates include radical molecules such as those observed in discharges by R2C2PI spectroscopy 88,90 and rotational spectroscopy 89. Of these, C 7 H 3 88 has a structure which has calculated rotational constants in the ground state consistent with the observed spectrum. UNDER CONSTRUCTION: WHERE TO FROM HERE? The identification of extraterrestrial molecules in the optical region can occur in two ways. Either the spectrum is recorded firstly in the laboratory and subsequently in an extraterrestrial object, or the absorption or emission line is observed by astronomy and subsequently in the laboratory. Neither approach has been particularly successful. A search for C 5 21, which we know to exist in the ISM 22, at optical wavelengths, was unsuccessful. It was concluded that the column density was only one order of magnitude too low for optical detection. However, a search for lines which might match C + 60 turned up two promising features 72. Unfortunately, the gas-phase optical spectrum of C + 60 is unknown and thus this identification required confirmation. As described above, there are hundreds of unidentified absorption and emission features in astronomical spectra. Identifying these is a job for laboratory spectroscopy. Many avenues have been explored, including a host of carbon chain species and PAH cations. New experiments, presently under construction, are described below. FIG. 9 Structures of cations thought to exist in the interstella medium. clockwise from top-left: naphthalenylium cation, phenanthrenylium cation, anthracenylium cation and buckminsterfullerenylium cation (C + 60 ). Spectroscopy of exotic cations. Cation spectroscopy is difficult. They possess a much higher density of states than neutral species and as such often have efficient internal conversion pathways. As a consequence, only small cations fluoresce. LIF is thus of limited applicability. Direct absorption measurements by CRDS are possible. However, the species must have a density in the free-jet expansion above a threshold limit for detection. This technique is also mass-unresolved and thus identification of a band carrier is often not straightforward. R2C2PI spectroscopy is currently only applied to neutral species, but in principle could be applied to cations if the mass-to-charge ratio can be changed in a resonant process. Since a single ion can be detected, this technique does not suffer from the problems of sensitivity which plagues CRDS. Cations are difficult to doubly ionize. As they are already charged, removal of an extra electron is approximately twice as difficult as the first. The most amenable example may be C + 60 . The ionization potential of C 60 is 7.62 eV, and that of C + 60 is 11.4 eV (109 nm)73. These photon energies required are only just becoming convenient. Another problem is the internal conversion mechanisms which preclude LIF from being applied to cations. A consequence of internal conversion is that the double ionization step must occur from the electronic ground state. Signal will be very sensitive to the photon energy of the second laser pulse. One unexplored direction is the implementation of ultrafast lasers for ionization (τ F W HM ≈ 100fs). The ionization laser need not be high-resolution (and necessarily are not due to the time-energy uncertainty principle). One advantage of ultrafast laser pulses is that their wavelength can be changed by non-linear optical techniques with high efficiency (due to the high peak-power). Thus, deep UV wavelengths may be accessed more easily than with nanosecond laser pulses. The ionization step may also be effected by multiphoton processes, which might be called resonant 2-colour multi-photon double-ionization spectroscopy (R2CMPDI). One problem of C + 60 , and for large cations in general, is that they are difficult to place into the gas phase and they are difficult to cool to temperatures comparable to the interstellar medium. One solution is to trap the ions and cool them with a buffer gas 68. In this way, it is possible to load an ion trap with mass-selected C + 60 or another large cationic species and then cool to 5 K or higher with a helium buffer. Spectroscopy is then performed in the trap. This may be done either by R2CMPDI, or by REMPD as in Ref. 67. Mass selection prior to trapping opens up the possibility of performing spectroscopy on derivatives of PAH cations. It has been observed by one of the authors (TWS), that nascent hydrocarbon cations produced in a R2C2PI experiment will readily absorb photons of energy ≈ 6 eV and shed hydrogen atoms. In this way, the signal observed for C 9 H + 3 also yielded the same resonance enhanced ion signal at the masses for C 9 H + 2 , C 9 H + and C + 9 90. It is thus likely that, in the interstellar medium, PAH cations will absorb UV photons and shed hydrogen atoms. This process will be in equilibrium with a hydrogen capture process (ion-atom reaction) and it is possible that the derivativized population of PAH cations and neutrals will be significant. It is worth noting that a mono-dehydrogenated PAH neutral will have a π-electronic structure similar to its cation and will likely absorb in the visible. Obvious candidates for these studies are naphthalenylium and anthracenylium cations and the corresponding neutrals. Phosphorescence spectroscopy. Fluorescence spectroscopy is performed by observing the emission, by molecules, of photons at the point of laser-molecule interaction. Where the emission lifetime is much longer than ns-µs, the emission cannot be observed in this way. For this reason, relatively little is known of laser-induced phosphorescence (LIP) spectroscopy, and indeed about forbidden transitions in exotic molecules. It has been speculated that the RRBs may be due to phosphorescence 85. While it is unlikely that forbidden transitions play any part in the DIBs, if a molecule is formed in a triplet state by some reactive mechanism, the radiative lifetime is irrelevant: molecules experience long delays between collisions in the rarified environments of molecular clouds and nebulae (10 to 10,000 s). One way to observe phosphorescence in the laboratory is to excite a forbidden transition in the gasphase with a powerful laser, then freeze the triplet excited molecules onto a substrate at 5 K. The ensuing phosphorescence is then detected at leisure as the molecules are now frozen into a matrix of the carrier gas (e.g. argon). Phosphorescence as a function of laser wavelength yields a gas-phase phosphorescence spectrum which may be compared to astronomical spectra. This technique has already been applied to the spectroscopy of benzaldehyde 91. CONCLUDING REMARKS Models of interstellar chemistry are not only tested by predicting observations of column densities by millimetre-wave spectroscopy, but must also predict abundances of species without permanent dipole moments to which millimetre-wave spectroscopy is blind. The identification of molecules in the optical region of the electromagnetic spectrum requires high resolution astronomical observation coupled with sophisticated laboratory experiments. While astronomical observations have uncovered hundreds of unidentified and presumably molecular absorption and emission features, and laboratory spectroscopy has produced cold, gas-phase spectra of hundreds of candidate carriers, there is as of yet not one certain match between a DIB or RRB and a laboratory spectrum. C 3 has been observed in molecular clouds, but so far C 5 has been elusive in the optical region. As far as identification of the DIBs is concerned, the outstanding candidates within the frame of the carbon chain hypothesis are the odd numbered pure carbon clusters. These chains possess strong transitions which increase linearly with the size of the chain. They are also known to absorb in the visible region. Other candidates yet to be tested are the iso-electronic monohydrogenated carbon chain cations. These are only now being studied in the condensed phase. Gas-phase spectroscopy of cations is a field under development. The coming years should see some progress. An outstanding yet unobserved spectrum is that of C + 60 in the gas-phase. This spectrum, when obtained, will confirm whether or not this cation is abundant in the interstellar medium. Modeling its formation should be a great challenge for theoretical astrochemists. The spectroscopy of PAHs in the laboratory is ongoing business. Very few gas-phase spectra of PAH cations and derivatives have been observed. More studies are needed before an informed opinion can be formed on the importance of PAHs with respect to the DIBs. One of the great challenges facing these studies is the methodology with which the spectra of cations may be observed in the gas-phase. The RRBs remain unidentified. It is clear that the carrier emits, and as such it will be observed in the laboratory by observation of its laser-induced emission. The challenge to the experimentalist is to build the laser-induced fluorescence/phosphorescence apparatus and find a way of making the presumably exotic carrier in situ. The carrier must be abundant in the Red Rectangle nebula and thus should be produced in a discharge of the right precursor mixture. Whether the carrier possesses a heteroatom remains to be seen. The identification of molecules in the interstellar medium is an on-going quest. The coming decade should see the identification of several of the DIBs, or if not, then certainly the gas-phase spectra of troublesome cations will be obtained. On this quest, physical chemists and astronomers walk together in an example of cooperation and collaboration between two seemingly different fields of scientific endeavour. FIG. 1 1Planetary nebulae as imaged by the Hubble Space Telescope and the Anglo-Australian Telescope. FIG. 3 3The absorption spectrum of C 3 observed by Maier et al towards ζ-Ophiuchi as compared to a simulation from known line positions at 80 K. (Adapted from Ref.17) FIG. 5 5In cavity ringdown spectroscopy (CRDS), supersonically cooled molecules are injected into an optical cavity in a vacuum chamber. The decay profile of the laser pulse as observed by a photodetector exterior to the cavity is modulated by absorption by molecular species. The modulation of the decay profile as a function of laser wavelength yields CRDS spectra. FIG. 8 8A high resolution portion of the extended red emission of the Red Rectangle as compared to a simulation. TABLE 1 A 1list of molecules identified in the interstellar regions 5. Carbon chains as long as HC 11 N have been observed in molecular clouds by millimetre-wave spectroscopy. 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[ "Cosmic antiprotons as a probe for supersymmetric dark matter?", "Cosmic antiprotons as a probe for supersymmetric dark matter?" ]	[ "Lars Bergström \nDepartment of Physics\nStockholm University\nBox 6730SE-113 85StockholmSweden\n", "Joakim Edsjö \nDepartment of Physics\nStockholm University\nBox 6730SE-113 85StockholmSweden\n", "Piero Ullio \nDepartment of Physics\nStockholm University\nBox 6730SE-113 85StockholmSweden\n" ]	[ "Department of Physics\nStockholm University\nBox 6730SE-113 85StockholmSweden", "Department of Physics\nStockholm University\nBox 6730SE-113 85StockholmSweden", "Department of Physics\nStockholm University\nBox 6730SE-113 85StockholmSweden" ]	[]	The flux of cosmic ray antiprotons from neutralino annihilations in the galactic halo is computed for a large sample of models in the MSSM (the Minimal Supersymmetric extension of the Standard Model). We also revisit the problem of estimating the background of low-energy cosmic ray induced secondary antiprotons, taking into account their subsequent interactions (and energy loss) and the presence of nuclei in the interstellar matter.We consider a two-zone diffusion model, with and without a galactic wind. We find that, given the uncertainties in the background predictions, there is no need for a primary (exotic) component to explain present data. However, allowing for a signal by playing with the uncertainties in the background estimate, we discuss the characteristic features of the supersymmetric models which give a satisfactory description of the data. We point out that in some cases the optimal kinetic energy to search for a signal from supersymmetric dark matter is above several GeV, rather than the traditional sub-GeV region.The large astrophysical uncertainties involved do not, one the other hand, allow the exclusion of any of the MSSM models we consider, on the basis of data.We present besides numerical results also convenient parameterizations of the antiproton yields of all 'basic' two-body final states. We also give examples of the yield and differential energy spectrum for a set of supersymmetric models with high rates.We also remark that it is difficult to put a limit on the antiproton lifetime from present measurements, since the injection of antiprotons from neutralino annihilation can compensate the loss from decay. *	10.1086/307975	[ "https://export.arxiv.org/pdf/astro-ph/9902012v1.pdf" ]	6,549,801	astro-ph/9902012	d237c031e6059fd725564907b69002c680637d24	Cosmic antiprotons as a probe for supersymmetric dark matter? 1 Feb 1999 February 1, 1999 Lars Bergström Department of Physics Stockholm University Box 6730SE-113 85StockholmSweden Joakim Edsjö Department of Physics Stockholm University Box 6730SE-113 85StockholmSweden Piero Ullio Department of Physics Stockholm University Box 6730SE-113 85StockholmSweden Cosmic antiprotons as a probe for supersymmetric dark matter? 1 Feb 1999 February 1, 19991 The flux of cosmic ray antiprotons from neutralino annihilations in the galactic halo is computed for a large sample of models in the MSSM (the Minimal Supersymmetric extension of the Standard Model). We also revisit the problem of estimating the background of low-energy cosmic ray induced secondary antiprotons, taking into account their subsequent interactions (and energy loss) and the presence of nuclei in the interstellar matter.We consider a two-zone diffusion model, with and without a galactic wind. We find that, given the uncertainties in the background predictions, there is no need for a primary (exotic) component to explain present data. However, allowing for a signal by playing with the uncertainties in the background estimate, we discuss the characteristic features of the supersymmetric models which give a satisfactory description of the data. We point out that in some cases the optimal kinetic energy to search for a signal from supersymmetric dark matter is above several GeV, rather than the traditional sub-GeV region.The large astrophysical uncertainties involved do not, one the other hand, allow the exclusion of any of the MSSM models we consider, on the basis of data.We present besides numerical results also convenient parameterizations of the antiproton yields of all 'basic' two-body final states. We also give examples of the yield and differential energy spectrum for a set of supersymmetric models with high rates.We also remark that it is difficult to put a limit on the antiproton lifetime from present measurements, since the injection of antiprotons from neutralino annihilation can compensate the loss from decay. * INTRODUCTION The mystery of the dark matter in the Universe remains unsolved. Among the most plausible candidates are Weakly Interacting Massive Particles (WIMPs), of which the supersymmetric neutralino is a favourite candidate from the point of view of particle physics. The neutralino arises naturally in supersymmetric extensions of the standard model, and has the attractive feature of giving a relic density which in large regions of parameter space is adequate to explain cosmological dark matter. We will in this paper consider the neutralino as a dark matter candidate within the Minimal Supersymmetric extension of the Standard Model (MSSM). For a thorough review of neutralino dark matter, see Jungman, Kamionkowski & Griest (1996). Neutralino dark matter can be and is searched for in several ways: directly through detection of nuclear recoils and/or ionization in direct detection experiments, and indirectly through searches for their annihilation products from annihilation in the Earth or Sun (for neutrinos) and the galactic halo. In this paper, we discuss the detection prospects of antiprotons from neutralino annihilation in the galactic halo. As antimatter seems not to exist in large quantities in the observable Universe, including our own Galaxy, any contribution to the cosmic ray generated antimatter flux (besides antiprotons also positrons) from exotic sources may in principle be a good signature for such sources. Since neutralinos are constrained by supersymmetry to be Majorana fermions they are their own antiparticles and therefore the final state in their annihilations in the halo will contain equal amounts of matter and antimatter (given the particle physics constraints on CP violating couplings and in particular on baryon number violation). The excess of particles would drown in the background of particles from astrophysical sources, but there is a chance that antiparticles from this new primary source could be detectable. This issue has recently come into new focus thanks to upcoming space experiments like Pamela (Adriani & al. 1995) and Ams (Ahlen & al. 1994) with increased sensitivity to the cosmic antimatter flux. Cosmic ray induced secondary antiprotons are generated mainly through pp →p + X collisions of cosmic ray protons with interstellar matter. For kinematical reasons they are born with a non-zero momentum. The strategy to search for exotic signals has thus been to investigate the low-energy region since, e.g., a neutralino-induced component does not drop as fast at low energies. However, as we will see, this ideal picture is blurred to a large extent by a 'tertiary' component caused by scattering with energy loss of the secondary antiprotons. Also, heavier nuclei in the interstellar medium target (primarily helium) cause a significant antiproton flux at low energy. In particular, it is known that in proton-nucleus collisions antiprotons may be produced well below the nominal pp energy threshold. In addition, low energy particles have difficulties entering the heliosphere which makes the connection of the measured fluxes to the insterstellar ones dependent on a not completely known correction due to this solar modulation (which follows the 11-year solar cycle). Using reasonable parameters for the computation of all these effects, we will show that we are able to explain in a satisfactory way the present experimental data on cosmic ray antiprotons without the need of a primary component. This is seen in Fig. 1, where our computed reference distribution for the background is compared to the recent data from the Bess experiment (Bess 95: Matsunaga & al. 1998;Bess 97: Orito 1998). The satisfactory agreement can be compared to the conclusions of a recent analysis (Bottino & al. 1998), where the need for an exotic component was more apparent. The main cause of this difference lies in our improved treatment of energy loss during propagation and our inclusion of the helium component of the interstellar medium. From Fig. 1 one can also see that the more recent Bess data (Orito 1998) indicate a lower flux at low energy than earlier data. If reacceleration effects were included, the need for an exotic component would be even less. It is, however, also evident from Fig. 1 that the statistical sample of antiprotons presently available is very limited, so that a new primary component can not yet be ruled out with high significance even if the propagation parameters were known. By varying all parameters in this complex astrophysical problem, the room for an exotic contribution can, as we will see, be made quite large, so it is certainly worthwhile to investigate what a favoured dark matter candidate like the MSSM neutralino may yield in terms of a signal. It is apparent from Fig. 1 that the low-energy tail of cosmic ray induced antiprotons does not fall very rapidly with decreasing kinetic energy, so this particular region of phase-space may not yield as a nice signature for a dark matter neutralino as previously thought. Therefore, we will also discuss the cases (mainly for high-mass neutralinos) when the optimal kinetic energy for finding an exotic contribution is above several GeV. Unfortunately, it will turn out that the rates in that energy region are not large enough to cause a spectral distortion, unless the background of secondary antiprotons is considerably smaller and/or the signal is considerably larger than our canonical results show. This lack of spectral features in the neutralino-induced antiproton flux causes severe fundamental limitations for this indirect method of detecting supersymmetric dark matter. The idea of a dark matter-induced component in the cosmic antiproton spectrum has a long history. An early report (Buffington & al. 1981) of anomalous excess of cosmic ray antiprotons at low energies led to the suggestion that annihilation of relic neutralinos could be the source. Calculations of fluxes have since then been performed with different degrees of sophistication, ranging from order of magnitude estimates (Silk & Srednicki 1984), analytical expressions (Stecker, Rudaz & Walsh 1985) to results from Monte-Carlo simulations (Ellis & al. 1988;Stecker & Tylka 1989;Bottino & al. 1995;Bottino & al. 1997). The latter method has also been used together withp production from neutralino annihilation in a minimal supergravity scheme (Diehl & al. 1995). In this work, we use the Lund Monte Carlo Pythia 6.115 (Sjöstrand 1994) to simulate the energy spectrum of antiprotons from neutralino annihilation. We have used large numerical tables for our computations, but for convenience we also present useful parametrisations of thep fluxes from various annihilation channels. In the following Sections, we will describe the MSSM model we use, describe the Monte Carlo simulations, discuss the antiproton propagation model, discuss the background fluxes and the uncertainties in both the background and the signal. Finally, we will show and discuss our results. L. Bergström, J. Edsjö and P. Ullio, 1999 Solar Modulated, φ F = 500 MV (Matsunaga & al. 1998;Orito 1998). Parameter µ M 2 tan β m A m 0 A b /m 0 A t /m 0 Unit GeV GeV 1 GeV GeV 1 1 Min -50000 -50000 1.0 0 100 -3 -3 Max 50000 50000 60.0 10000 30000 3 3 Table 1 The ranges of parameter values used in our scans of the MSSM parameter space. Note that several special scans aimed at interesting regions of the parameter space has been performed. In total we have generated approximately 116 000 models which obey all accelerator constraints. Of these, about 41000 have a relic density in the range 0.025 < Ω χ h 2 < 1. DEFINITION OF THE SUPERSYMMETRIC MODEL We work in the Minimal Supersymmetric Standard Model (MSSM). In general, the MSSM has many free parameters, but with some reasonable assumptions we can reduce the number of parameters to the Higgsino mass parameter µ, the gaugino mass parameter M 2 , the ratio of the Higgs vacuum expectation values tan β, the mass of the CP -odd Higgs boson m A (or m H 0 3 ), the scalar mass parameter m 0 and the trilinear soft SUSY-breaking parameters A b and A t for the third generation. In particular, we do not impose any restrictions from supergravity other than gaugino mass unification, which relates the other gaugino mass parameter M 1 to M 2 . (We remind that one of the most attractive features of the MSSM is that, unlike the non-supersymmetric Standard Model, it is compatible with gauge coupling unification given the current data on the running of low-energy gauge couplings.) For a more detailed definition of the parameters and a full set of Feynman rules we refer to (Edsjö & Gondolo 1997;Edsjö 1997). The lightest stable supersymmetric particle is in most models the lightest neutralino (which we will henceforth just call 'the neutralino', χ), which is a superposition of the superpartners of the gauge and Higgs fields, χ ≡χ 0 1 = N 11B + N 12W 3 + N 13H 0 1 + N 14H 0 2 .(1) It is convenient to define the gaugino fraction of the neutralino, Z g = \|N 11 \| 2 + \|N 12 \| 2 .(2) For the masses of the neutralinos and charginos we use the one-loop corrections from the literature (Drees & al. 1997;Pierce & Papadopoulos 1994;Pierce & Papadopoulos 1994a;Lahanas, Tamvakis & Tracas 1994) and for the Higgs boson masses we use the leading logarithmic two-loop radiative corrections, calculated within the effective potential approach given by Carena & al. (1995). We make extensive scans of the model parameter space, some general and some specialized to interesting regions. In total we make 22 different scans of the parameter space. The scans are done randomly and are mostly distributed logarithmically in the mass parameters and in tan β. For some scans the logarithmic scan in µ and M 2 has been replaced by a logarithmic scan in the more physical parameters m χ and Z g /(1 − Z g ) where m χ is the neutralino mass. Combining all the scans, the overall limits of the seven MSSM parameters we use are given in Table 1. We check each model to see if it is excluded by the most recent accelerator constraints, of which the most important ones are the LEP bounds (Carr 1998) on the lightest chargino mass, m χ + 1 > 91 GeV , \|m χ + 1 − m χ 0 1 \| > 4 GeV 85 GeV , otherwise(3) and on the lightest Higgs boson mass m H 0 2 (which range from 72.2-88.0 GeV depending on sin(β − α) with α being a mixing angle in the Higgs sector) and the constraints from b → sγ (Ammar et al. 1993;Alam et al. 1995). The new higher-precision measurement from CLEO (Glenn & al. 1998) gives a slightly smaller range for that process than the one we have allowed; we have checked, however, that this causes no major changes in the properties related top yield for the allowed models. For each allowed model we compute the relic density of neutralinos Ω χ h 2 , where Ω χ is the density in units of the critical density and h is the present Hubble constant in units of 100 km s −1 Mpc −1 . We use the formalism of Gondolo & Gelmini (1991) for resonant annihilations, threshold effects, and finite widths of unstable particles and we include all two-body tree-level annihilation channels of neutralinos. We also include the so-called coannihilation processes according to the results of Edsjö & Gondolo (1997) in the relic density calculation. Present observations favor h = 0.6 ± 0.1, and a total matter density Ω M = 0.3 ± 0.1, of which baryons may contribute 0.02 to 0.08 (see, e.g., Schramm & Turner 1998). Not to be overly restrictive, we accept Ω χ h 2 in the range from 0.025 to 1 as cosmologically interesting. The lower bound is somewhat arbitrary as there may be several different components of non-baryonic dark matter, but we demand that neutralinos are at least as abundant as required to make up the dark halos of galaxies. In principle, neutralinos with Ω χ h 2 < 0.025 would still be relic particles, but only making up a small fraction of the dark matter of the Universe. We will consider models with Ω χ h 2 < 0.025 only when discussing the dependence of the signal on Ω χ h 2 . Table 2 The parameters fitted to the antiproton distributions for neutralino masses 50-5000 GeV. The parameters a 23 , a 24 , a 33 and a 34 are always zero and not given in the table. The parameterized distributions are given by Eqs. (4)-(5). It may also be of interest to consider specifically models which naturally give a value of Ω χ h 2 which is close to the present 'best fit' value. We will therefore present some Figures where the range between 0.1 and 0.2 for Ω χ h 2 is shown by special symbols. Since in general a small relic density implies a large annihilation cross section and vice versa, this tends to cut out a large fraction of the models with otherwise observable rates -a fact not always highlighted in previous analyses. ANTIPROTON PRODUCTION BY NEUTRALINO ANNIHILATION Introduction Neutralinos are Majorana fermions and will annihilate with each other in the halo producing leptons, quarks, gluons, gauge bosons and Higgs bosons. The quarks, gauge bosons and Higgs bosons will decay and/or form jets that will give rise to antiprotons (and antineutrons which decay shortly to antiprotons). At tree level the relevant final states forp production will be qq, ℓl, W + W − , Z 0 Z 0 , W + H − , ZH 0 1 , ZH 0 2 , H 0 1 H 0 3 and H 0 2 H 0 3 . We will include all the heavier quarks (c, b and t), gauge bosons and Higgs boson final states in our analysis. In addition, we will include the Zγ (Ullio and the 2 gluon (Drees & al. 1994; final states which occur at one loop-level. Note that for the antiproton-rich 2-gluon final state the improved and corrected formulas given in the second reference generally imply a lower branching ratio than those of the former reference which has been used in several previous analyses. The hadronization for all final states (including gluons) is simulated with the well-known particle physics Lund Monte Carlo program Pythia 6.115 (Sjöstrand 1994), which is used extensively at accelerators in simulations of jet production at the full energy range which we need to consider here. A word of caution should be raised, however, that antiproton data is not very abundant, in particular not at the lowest antiproton lab energies which tend to dominate our signal. Therefore an uncertainty in normalization, probably of the order of a factor 2, cannot be excluded at least in the low energy region. Simulations To get the energy distribution of antiprotons for each of the final states listed in the previous subsection we generate the final states cc, bb, tt, W + W − , Z 0 Z 0 and gg and let them decay/hadronize according to Pythia 6.115. We do not need to include lighter quarks since the branching ratios to these are negligible. The annihilation channels containing Higgs bosons need not be simulated separately since they decay to the other particles for which we do simulate. They are then let to decay in flight and the spectrum from the decay products are boosted and averaged over the decay angles. During the simulations, antineutrons are let to decay since we would otherwise underestimate the flux by a factor of two. We have performed simulations for the neutralino masses m χ = 10, 25, 50, 80.3, 91.2, 100, 150, 176, 200, 250, 350, 500, 750, 1000, 1500, 2000, 3000 and 5000 GeV and for intermediate masses an interpolation is used. For each mass and annihilation channel, 2.5 × 10 5 events have been simulated. For easier use, we have also parameterized the antiproton distributions for the 'basic' annihilation channels given above. It is for this purpose more convenient to define x = Tp/m χ , with Tp being the kinetic energy of the antiproton, as the independent variable. A suitable parameterization is then given by dN dx = (p 1 x p3 + p 2 \|log 10 x\| p4 ) −1(4) where the parameters p i depend on both annihilation channel and the neutralino mass. The latter dependence is parameterized as p i (m χ ) = a i1 m ai2 χ + a i3 m ai4 χ −1(5) The values of the a ij for the different annihilation channels are given in Table 2. These parameterizations are valid for neutralino masses in the range 50-5000 GeV. The error in the most relevant regions, i.e. the low-energy tail to most of the high-energy slope is usually less than 20%. At worst (for tt), it can be up to 50% in isolated regions. This should be compared with the uncertainties in Pythia which probably can be up to a factor of 2. In Fig. 2 we give as an example the distributions for a 100 GeV neutralino annnihilating into bb and a 1000 GeV neutralino annihilating into W + W − . Note that the above parameterizations are only given for the reader's convenience -in our calculations we use the results of the simulations directly. For convenience we also show in Table 3 the individual branching ratios of the main modes for a set of models with high antiproton yield but different mass, and gaugino content. The Antiproton Source Function The source function Q χ p gives the number of antiprotons per unit time, energy and volume element produced in annihilation of neutralinos locally in space. It is given by Q χ p (T, x ) = (σ ann v) ρ χ ( x ) m χ 2 f dN f dT B f (6) where T is thep kinetic energy. For a given annihilation channel f , B f and dN f /dT are, respectively, the branching ratio and the fragmentation function, and (σ ann v) is the annihilation rate at v = 0 (which is very good approximation since the velocity of the neutralinos in the halo is so low). As dark matter neutralinos annihilate in pairs, the source function is proportional to the square of the neutralino number density n χ = ρ χ /m χ . Assuming that most of the dark matter in the Galaxy is made up of neutralinos and that these are smoothly distributed in the halo, one can directly relate the neutralino number density to the dark matter density profile in the galactic halo ρ. Given a generic parametrization of ρ, we fix: ρχ( x ) ≡ ρ( x) = ρ 0 r 0 \| x\| γ 1 + (r 0 /a) α 1 + (\| x\|/a) α (β−γ)/α(7) where ρ 0 is the value of the local halo density, r 0 is the galactocentric distance of the Sun and a is some length scale; we assume ρ 0 = 0.3 GeVcm −3 and r 0 = 8.5 kpc. In the actual computation we will mainly restrict ourselves to the case in which the dark matter density profile is described by a modified isothermal distribution, (α, β, γ) = (2, 2, 0), mentioning what changes are expected in case more cuspy profiles, which are favoured by results in N -body simulations of hierarchical clustering, are considered. We will in particular consider the example of the Navarro et al. profile (Navarro, Frenk & White 1996), (α, β, γ) = (1, 2, 1). Although we are here focusing on the case of a smooth distribution of dark matter particles in the halo, an extension to a clumpy distribution is potentially interesting as well (Bergström, Edsjö, Gondolo & Ullio 1999;. Given thep distributions calculated in the previous section, we can now get the source function for any given annihilation channel. Table 2. Br(tt) − − 0.11 − 0.65 − − Br(W + W − ) − − 0.024 0.87 < 10 −5 − < 10 −5 Br(Z 0 Z 0 ) − − 0.020 0.12 < 10 −5 − − Br(gg) 0.00060 0.0028 0.00017 0.0016 0.0028 0.00054 0.00043 Br(γγ) 3.7 × 10 −7 3.9 × 10 −6 6.5 × 10 −5 1.8 × 10 −4 2.3 × 10 −6 9.6 × 10 −7 1.2 × 10 −6 Br(Zγ) 5.8 × 10 −13 5.8 × 10 −6 1.7 × 10 −4 6.5 × 10 −4 2.3 × 10 −5 6.3 × 10 −10 2.1 × 10 −6 Ωχh 2 0.025 0.028 0.032 0.025 0.050 0.034 0.034 Φp 4.6 × 10 −2 1.6 × 10 −2 8.5 × 10 −4 8.8 × 10 −3 4.0 × 10 −3 2.1 × 10 −2 1.1 × 10 −2 Φ µ, Earth 3.8 × 10 −1 1.0 × 10 0 2.4 × 10 −5 1.1 × 10 −2 2.5 × 10 −7 7.5 × 10 1 2.3 × 10 1 Φµ, Sun 7.0 × 10 −2 2.6 × 10 0 4.1 × 10 −1 2.7 × 10 3 8.7 × 10 −1 1.3 × 10 0 1.8 × 10 1 σSI 2.9 × 10 −7 4.0 × 10 −7 9.6 × 10 −10 1.2 × 10 −8 1.2 × 10 −10 1.2 × 10 −6 2.3 × 10 −6 Φ e + 5.0 × 10 −8 2.9 × 10 −8 3.2 × 10 −8 2.6 × 10 −8 2.0 × 10 −8 3.1 × 10 −8 2.6 × 10 −8 Φcont. γ 8.1 × 10 −8 3.5 × 10 −8 5.9 × 10 −9 1.4 × 10 −8 1.1 × 10 −8 4.2 × 10 −8 2.6 × 10 −8 Table 3 Example of models giving high antiproton rates. All masses are given in GeV; the annihilation rate is given in 10 −24 cm 3 s −1 ; the solar modulatedp flux at 0.35 GeV is in m −2 s −1 sr −1 sr −1 ; the neutrino flux from the Earth and the Sun is with a threshold of 25 GeV and in units of km −2 yr −1 ; the spin-independent cross section is in pb; the solar modulated positron flux is in one of the HEAT bins (8.9-14.8 GeV) and in units of cm −2 s −1 sr −1 GeV −1 ; the continuum gamma flux is for high Galactic latitudes and is given in cm −2 s −1 sr −1 , integrated above 1 GeV. The rates and fluxes are calculated as given in (Bergström & Gondolo 1996;Bergström, Edsjö, Gondolo & Ullio 1999;Baltz & Edsjö 1999;Bergström, Ullio & Buckley 1998;Bergström, Edsjö & Gondolo 1998). PROPAGATION MODEL In the absence of a well established theory to describe the interactions of charged particles with the magnetic field of the Galaxy and the interstellar medium, the propagation of cosmic rays has generally been treated by postulating a semiempirical model and fitting the necessary set of unknown parameters to available data. A common approach is to use a diffusion approximation defined by a transport equation and an appropriate choice of boundary conditions (see e.g. Berezinskii & al. 1990;Gaisser 1990 and references therein). In most diffusion models the form of the terms present in the diffusion equation is a compromise between physical insight and the possibility of an analytical solution. Only recently more realistic models have been studied by applying numerical solutions (Strong & Moskalenko 1998) or Monte Carlo simulations (Porter & Protheroe 1997). Our analysis is focused on comparing the characteristics of cosmic-ray antiproton signals of different origin: secondary antiprotons produced in cosmic-ray interactions and, eventually, a primary flux from neutralino annihilations. We want in particular to examine the dependence of the relative strength and spectral signatures on the diffusion model and the distribution of particle dark matter in the galactic halo. As both of them are not well constrained, we believe that an analytic solution of a reasonable physical model will be sufficient to provide most of the information needed on the behaviour of the two types of signals. We choose to describe the propagation of cosmic rays in the Galaxy by a transport equation of the diffusion type as written by Ginzburg and Syrovatskii (1964) (see also Berezinskii & al. 1990;Gaisser 1990). In the case of a stationary solution, the number density N of a stable cosmic ray species whose distribution of sources is defined by the function of energy and space Q(E, x), is given by: ∂N (E, x) ∂t = 0 = ∇ · (D(R, x) ∇N (E, x)) − ∇ · ( u( x) N (E, x)) − p(E, x) N (E, x) + Q(E, x) .(8) Here and below we try to keep the notation as general as possible. Although our goal is to compute N for antiprotons, in order to determine the source function for the secondary flux we will have to obtain the spatial density distribution for protons as well. On the right hand side of Eq. (8) the first term implements the diffusion approximation for a given diffusion coefficient D, generally assumed to be a function of rigidity R, while the second term describes a large-scale convective motion of velocity u. The third term is added to take into account losses of cosmic rays due to to collisions with the interstellar matter. It is a very good approximation to include in this term only the interactions with interstellar hydrogen (on the other hand, we will point out below that heavier elements, in particular helium, cannot be neglected when computing the source function for secondary antiprotons); in this case p is given by: p(E, x) = n H ( x) v(E) σ in cr p (E)(9) where n H is the hydrogen number density in the Galaxy, v is the velocity of the cosmic ray particle considered 'cr', while σ in cr p is the inelastic cross section for cr-proton collisions. In Eq. (8) we have neglected continuous energy losses; this will be included in an implicit form when considering secondary antiprotons. We will briefly mention in the conclusions the possibility that antiprotons have a finite lifetime τ . To take this effect into account the term −(1/τ ) N (E, x) should be added on the right hand side of Eq. (8), and this corresponds to shifting p to p + 1/τ in all the equations below. We now have to specify the parameters introduced and the boundary conditions. We mainly follow the approach of Ginzburg, Khazan & Ptuskin (1980), given also in Berezinskii & al. 1990 and analogous to that of Webber, Lee & Gupta 1992, Chardonnet & al. 1996, and Bottino & al. 1998. The main feature is that the propagation region is assumed to have a cylindrical symmetry: the Galaxy is split into two parts, a disk of radius R h and height 2 · h g , where most of the interstellar gas is confined, and a halo of height 2 · h h and the same radius. We assume that the diffusion coefficient is isotropic with possibly two different values in the disk and in the halo, reflecting the fact that in the disk there may be a larger random component of the magnetic fields. We then have a spatial dependence: D( x) = D(z) = D g θ(h g − \|z\|) + D h θ(\|z\| − h g ) .(10) Regarding the rigidity dependence, fits to cosmic ray data in models which do not include reacceleration effects indicate that D scales as R 0.6 (Webber, Lee & Gupta 1992; Strong & Moskalenko 1998) with a cutoff below some rigidity R 0 . We consider the same functional form as in Chardonnet &: D l (R) = D 0 l 1 + R R 0 0.6(11) where l = g, h. We will briefly discuss below what changes are expected in case reacceleration is included, without making numerical predictions. The convective term has been introduced in Eq. (8) to describe the effect of particle motion against the wind of cosmic rays leaving the disk. We will therefore not consider any convection in the radial direction, assuming instead a galactic wind of velocity u( x) = (0, 0, u(z))(12) where u(z) = sign(z) u h θ(\|z\| − h g ) .(13) An analytic solution is possible also in the case of a linearly increasing wind (Ullio 1999). The last parameter we have to specify is the distribution of gas in the Galaxy: for convenience we assume that this has the very simple z dependence n H ( x) = n H (z) = n H g θ(h g − \|z\|) + n H h θ(\|z\| − h g )(14) where n h ≪ n g (in practice we will take n h = 0) and an average in the radial direction is performed. Finally, as we eventually want to treat the case of clumpy neutralino dark matter (Ullio 1999) we will not assume any symmetry in the source function Q( x) (note that to apply the results of this Section to sources with a cylindrical symmetry, as for instance Eq. (6) with the assumption in Eq. (7), it is sufficient to set everywhere the index k equal to 0). As boundary condition, it is usually assumed that cosmic rays can escape freely at the border of the propagation region, i.e. N (R h , z) = N (r, h h ) = N (r, −h h ) = 0(15) as the density of cosmic rays is assumed to be negligibly small in the intergalactic space. To check whether this hypothesis holds even in the case of a source from dark matter annihilations, we will compare the flux of outgoing antiprotons with the one entering the diffusion region due to sources in 'free space'. For references about other possible choices of boundary conditions see Berezinskii & al. 1990. The cylindrical symmetry and the free escape at the boundaries makes it possible to solve the transport equation expanding the number density distribution N in a Fourier-Bessel series: N (r, z, θ) = ∞ k=0 ∞ s=1 J k ν k s r R h · M k s (z) cos(kθ) +M k s (z) sin(kθ)(16) which automatically satisfies the boundary condition at r = R h , ν k s being the s-th zero of J k (the Bessel function of the first kind and of order k). In the same way the source function can be expanded as: Q(r, z, θ) = ∞ k=0 ∞ s=1 J k ν k s r R h · Q k s (z) cos(kθ) +Q k s (z) sin(kθ) (17) where Q k s (z) = 2 R h 2 J k+1 2 (ν k s ) R h 0 dr ′ r ′ J k ν k s r ′ R h 1 α k π π −π dθ ′ cos(kθ ′ ) Q(r ′ , z, θ ′ ) .(18) In the equation above α 0 = 2, while α k = 1 for k ≥ 1; it is not necessary to specify the coefficients of the terms in sin(kθ) as we fix the coordinate system such that θ = 0 at our location and we are only interested in computing fluxes for this value of θ. Inserting the two expansions in Eq. (8), we can derive the equation relevant for the propagation in the z direction: ∂ ∂z D(z) ∂ ∂z M k s (z) − D(z) ν k s R h 2 M k s (z) − ∂ ∂z u(z) M k s (z) − p(z)M k s (z) + Q k s (z) = 0 .(19) The solution of this equation is straightforward: it can be easily derived writing a solution separately for h g < z < h h , −h g < z < h g and −h h < z < −h g , and then imposing the boundary conditions at z = ±h h , i.e. Eq. (15), and the continuity of the number density and of the flux, that is of M k s (z) and [−D(z)∂/∂z M k s (z) + u(z) M k s (z)], at z = ±h g . For −h g ≤ z ≤ h g the solution is given by: M k s (z) = M k s (0) cosh(λ ks g z) − 1 D g λ ks g z 0 dz ′ sinh λ ks g (z − z ′ ) Q k s (z ′ ) (20) where M k s (0) = 1 cosh(λ ks g h g ) I H sinh λ ks h (h h − h g ) + D h I GS D g λ ks g γ h + λ ks h coth λ ks h (h h − h g ) + I GC × D g λ ks g tanh λ ks g h g + D h γ h + D h λ ks h coth λ ks h (h h − h g ) −1(21) and we have introduced the following set of definitions: λ ks g = ν k s R h 2 + n H g vσ in cr p D g , λ ks h = ν k s R h 2 + n H h vσ in cr p D h + γ h 2 , γ h = u h 2 D h(22) and I H = h h hg dz ′ sinh λ ks h (h h − z ′ ) exp (γ h (h g − z ′ )) · Q k s (z ′ ) + Q k s (−z ′ ) 2 I GS = hg 0 dz ′ sinh λ ks g (h g − z ′ ) · Q k s (z ′ ) + Q k s (−z ′ ) 2 I GC = hg 0 dz ′ cosh λ ks g (h g − z ′ ) · Q k s (z ′ ) + Q k s (−z ′ ) 2 .(23) For \|z\| > h g the solution is analogous, but less compact and will not be reproduced here. It is more useful to give explicitly M k s (0) in case of a source which is constant in the disk and negligible in the halo; in this case M k s (0) = 1 D g λ ks g 2 1 − D h γ h + λ ks h coth λ ks h (h h − h g ) D g λ ks g sinh λ ks g h g + D h cosh λ ks g h g γ h + λ ks h coth λ ks h (h h − h g ) Q k s (0) ≡ M * (s, k) Q k s (0) .(24) It is easy to check that in the limit u h → 0, Eq. (24) correctly reduces to the result quoted in Berezinskii & al. 1990, Chapter 3, Section 3. Also Eq. (21) corresponds to the number density found in Chardonnet & in the limits h g → 0, D h → D g , u h → 0 and for a source function symmetric with respect to the z axis. Before applying these results to compute cosmic ray antiproton fluxes, let us pause for a moment. Most commonly, data on cosmic rays have been treated within the framework of the leaky box approximation. This is, to a certain extent, a simplified version of the diffusion model, where it is assumed that diffusion takes place rapidly. As was noticed for instance in Berezinskii & al. 1990, the path length distribution function of particles for a diffusion model with sources in the disk is very close to the exponential form characteristic of the leaky box treatment. For the purpose of computing secondary antiprotons, which are mainly generated in the gaseous disk, we expect it to be essentially equivalent to write a diffusion equation and fit its parameters to existing data on cosmic ray nuclei, or to derive from the same data in the simple leaky box scenario the grammage as a function of rigidity and use it to compute the secondary antiproton flux. In this sense we do expect to find an antiproton secondary spectrum which is analogous to the results of several papers in which this has been calculated in the leaky box approximation (except that in some of these papers not all the relevant effects have been included). On the other hand, in the case of neutralino sources which are more homogeneously distributed extending through the full halo, it is unlikely that the effective average matter density particles have gone through can be of the same form as for sources located only in the disk. We will try to analyze in some detail this dependence on the geometry of the source, and we believe that this will give a real improvement with respect to the leaky box approximation. SOLAR MODULATION A further complication when comparing predictions of a theoretical model with data on cosmic rays taken at Earth is given by the solar modulation effect. During their propagation from the interstellar medium through the solar system, charged particles are affected by the solar wind and tend to lose energy. The net result of the modulation is a shift in energy between the interstellar spectrum and the spectrum at the Earth and a substantial depletion of particles with non-relativistic energies. The simplest way to describe the phenomenon is the analytical force-field approximation by Gleeson & Axford (1967;1968) for a spherically symmetric model. The prescription of this effective treatment is that, given an interstellar flux at the heliospheric boundary, dΦ b /dT b , the flux at the Earth is related to this by dΦ ⊕ dT ⊕ (T ⊕ ) = p 2 ⊕ p 2 b dΦ b dT b (T b )(25) where the energy at the heliospheric boundary is given by E b = E ⊕ + \|Ze\|φ F(26) and p ⊗ and p b are the momenta at the Earth and the heliospheric boundary respectively. Here e is the absolute value of the electron charge and Z the particle charge in units of e (e.g. Z = −1 for antiprotons). An alternative approach is to solve numerically the propagation equation of the spherically symmetric model (Fisk 1971): the solar modulation parameter one has to introduce with this method roughly corresponds to φ F as given above. When computing solar modulated antiproton fluxes, the two treatments seem not to be completely equivalent in the low energy regime (the reader may check for instance Fig. 4 in (Labrador & Mewaldt 1997) against Fig. 8 in (Bottino & al. 1998)); keeping this in mind, we will anyway implement the force field approximation, avoiding the problem of having to solve a partial differential equation for each of our supersymmetric models. We just mention here that the picture can be much more complicated: non-spherical propagation models which take into account the polarity of the solar magnetic field have been studied as well (e.g. Webber & Potgieter 1989). In this case the solar modulation effects on particles and antiparticles can be quite different. If one would translate this into the force field effective treatment, one should use different values for the modulation parameter for protons and antiprotons (in contrast with the standard procedure of assigning to antiprotons the value found from proton flux measurements). The relation between these two would be very model dependent. To compare with the two sets of Bess measurements, which are both near solar minimum, we choose φ F = 500 MV, in reasonable agreement with what the Bess collaboration uses in their analysis. Otherwise, we will focus on the interstellar fluxes which are not affected by these uncertainties. BACKGROUND ESTIMATES General considerations Secondary antiprotons are produced in cosmic ray collisions with the interstellar gas. Looking at the composition of incident and target particles it is easy to guess that the main contribution to thep flux is given by cosmic ray protons colliding with interstellar hydrogen atoms. Because of baryon number conservation, the minimal p + p →p + X reaction has three protons in the final state; in the rest frame of the target hydrogen atom, the energy threshold for an incident proton to produce an antiproton is therefore E p = 7m p . Due to this feature, the energy distribution of the producedp shows a sharp peak at a few GeV and a steep fall-off at lower energies. It is reasonable to expect that in this low energy region reactions involving heavier nuclei, both as targets and projectiles, may play some role: they imply in fact different kinematics and the spectrum of the producedp need not fall as fast as for low energy pp collisions. We have verified that the interaction of primary protons with interstellar helium is indeed a relevant process, while all others can be safely neglected as their contributions add up to below a few per cent of the total at any energy (this is essentially the same conclusion as that of Simon, Molnar & Roesler 1998, although our approach is slightly different, as we will point out below). We assume therefore that the source function for secondary antiprotons has the following form (the factor of 2 accounts for antiprotons produced by antineutron decays): Qp( x, E) = 2 · 4π ∞ E thresh dE ′ dσ pH→p dE (E, E ′ ) n H ( x) + dσ pHe→p dE (E, E ′ ) n He ( x) Φ p ( x, E ′ ) .(27) In this formula Φ p ( x, E ′ ) is the primary proton cosmic ray flux at the position x in the Galaxy and for the energy E ′ , n He is the helium number density which we assume to be 7% of n H (Garcia- Munoz & al. 1987) and have the same spatial dependence, while dσ/dE (E, E ′ ) stands for the differential cross section for the production of an antiproton with energy E for an incident proton of energy E ′ , in the two processes considered. For pH collisions we implement the standard parametrization for the differential cross section introduced in Tan & Ng 1983; as already mentioned the energy threshold E thresh for this process is E ′ = 7m p . Antiproton production in collisions with nuclei A much smaller set of data is available in the case of antiproton production in proton collisions with heavier elements. In particular, it is difficult to estimate the effects of production below the nominal energy threshold for p + p →p + X, which is known to occur in hadron-nucleus collisions. Recently, several experiments have shown substantial sub-thresholdp production for deuterium, helium, carbon and copper targets (Chiba & al. 1993;Schröter & al. 1993). Possible collective effects allow the abundant low-energy cosmic rays to produce antiprotons below threshold. On their way out of the nucleus, the produced antiprotons may also suffer inelastic losses which slow them down, creating a potentially important component in the low-energy cosmic ray-inducedp spectrum. (Note that helium and heavier nuclei in the cosmic rays also give different kinematics for produced antiprotons. However, this gives an extra contribution at higherp energies, and is therefore not important for our study.) These sub-threshold effects have been modeled by Sibirtsev & al. 1997, where the limited data set available can be described by a transport equation solved by a Monte Carlo technique. Here we follow a much simplified approach, which describes the C and Cu data displayed in Sibirtsev & al. 1997 reasonably well and which we apply to p + He →p + X. We find that the collective effects can be mimicked by a shift in the incident proton energy: E in → E eff = E in + 0.6(E thresh − E in )θ(E thresh − E in ) + 1.1 GeV,(28) where E thresh is the nominal threshold (7m p ) forp production in pp collisions. The energy loss due to inelastic rescattering can be approximated by decreasing the energy of the outgoing antiproton (using pp kinematics) by 1 GeV. The yield of antiprotons per collision is taken to scale with the total pA cross section, parametrised according to Letaw, Silberberg & Tsao 1983. In this way, we have a parametrisation which is asymptotically correct at high energies and which also fits the subthreshold data. However, we are unable to assess the accuracy of this treatment for the problem at hand, believing it to be at the 50 % level, but acknowledging the need for improved data and theoretical modeling of this seldom discussed problem. Primary proton flux The last step to make before implementing Eq. (27) is to determine the primary proton flux Φ p ( x, E ′ ). The formalism introduced in Section 4 is suitable for this purpose once we specify the source function for primary protons. It is generally believed that supernova remnants are the main sources of cosmic rays. Nevertheless the gradient of Φ p as a function of the distance from the galactic centre which is obtained from the observed distribution of supernovae or the related pulsar distribution is not consistent with models for gamma-ray emission (see Strong & Moskalenko 1998, and references therein). We take advantage of the phenomenological approach of Strong & Moskalenko 1998, where a generic form for the radial distribution of cosmic-ray sources was considered and its parameters fitted to EGRET gamma-ray data (Eq. (6) and Fig. 12 in Strong & Moskalenko 1998). We therefore assume that the primary proton source, in cylindrical coordinates, is of the form: Q p (E, x) =q(E) q( x) =q(E) r r 0 0.5 exp − r − r 0 r 0 θ(h g − \|z\|)(29) where r 0 = 8.5 kpc is our galactocentric distance, and we have assumed that the energy spectrum of emitted protons is the same everywhere in the Galaxy. The functionq(E), which we may interpret as a normalization factor, can be rewritten, after propagation, in terms of the local proton flux Φ p (r 0 , E) which has been measured in several experiments. It was argued in the past that the spread among different experimental determinations of Φ p (r 0 , E) introduces one of the main factors of uncertainty in the prediction for the secondary antiproton flux (see for instance Gaisser & Schaefer 1992). The recent measurements by the Imax (Menn & al. 1997) and Caprice (Boezio & al. 1999) collaborations are in better agreement with each other. In Bottino & al. 1998, a fit of the data of these two experiments with a single power law in energy or rigidity was made, using the force-field method to take solar modulation into account. The fits in rigidity (see also Boezio & al. 1999) show a steeper fall-off than those in energy; however, this may not be the case if a break in the spectrum at low rigidities is assumed instead (Ormes & Protheroe 1983). As in the next Section the background at high energies will be important, we prefer to conservatively consider fits with a power law in energy. From Eq. (1) and Table 1 in Bottino & al. 1998, Φ p (r 0 , E) = A E 2 − m p 2 E E 1 GeV −α(30) with A = 12300 ± 3000 and α = 2.67 ± 0.06 for the Imax data, and A = 19600 ± 3000 and α = 2.85 ± 0.04 for the Caprice data. Interstellar secondary antiproton flux We are now ready to give the formula for the interstellar secondary antiproton flux at our galactocentric distance. We find: Φp(r 0 , E) = 1 4 π vp(E) Np(r 0 , E) = = 2 vp(E) ∞ s=1 J 0 ν 0 s r 0 R h Mp(s, E) I R (s) ∞ E thresh dE ′ dσi dE (E, E ′ ) n i Φ p (r 0 , E ′ ) M p (s, E ′ ) ∞ s ′ =1 J 0 ν 0 s ′ r0 R h M p (s ′ , E ′ ) I R (s ′ )(31) where the repeated index i stands for the sum over hydrogen and helium, and we have introduced the notation: I R (s) = 2 R h 2 J 1 2 (ν 0 s ) R h 0 dr ′ r ′ J 0 ν 0 s r ′ R h q(r ′ )(32) where T and E are in units of GeV. To derive Eq. (31) we have assumed that for z < h g the approximation Φ p (z) ≃ Φ p (z = 0) is valid. This is generally a very good approximation, as for most choices of the parameters in the propagation model Φ p is nearly constant in the disk and rapidly decreasing in the halo (see for instance Fig. 3.10 in Berezinskii & al. 1990). Only in extreme cases can Φ p (z = h g ) be 10% lower than Φ p (z = 0) and the correction to the result in Eq. (31), always below few per cent, can be obtained keeping track of the full z dependence in J p (use Eqs. (20) and (21), all numerical integrals in Eq. (23) can still be performed analytically; the result follows easily). In Bottino & al. 1998, it was suggested that it is a good approximation to assume that the energy spectrum for the protons after propagation is roughly independent of location in the Galaxy. This hypothesis simplifies the computation (for us, it is especially needed to compute numerically the tertiary contribution described below); Eq. (31) reduces to: Φp(r 0 , E) = 2 vp(E) ∞ E thresh dE ′ dσi dE (E, E ′ ) n i Φ p (r 0 , E ′ ) ∞ s ′ =1 J 0 ν 0 s ′ r0 R h M p (s ′ ,Ê) I R (s ′ ) ∞ s=1 J 0 ν 0 s r 0 R h Mp(s, E) M p (s,Ê) I R (s)(35) whereÊ is an arbitrary normalization energy. As there are some indications that the energy spectrum may indeed be steeper far away from the sources, because of the energy dependence in the propagation coefficient (Mori 1997), we compare in one case Eq. (35) against Eq. (31) to check if the simplification in any way changes the result. For the set of parameters as in example in Fig. 1, which we will soon discuss, we find that Eq. (35) gives a very slight overestimate of Eq. (31), below 3% for interstellar antiproton kinetic energies up to 1 GeV, a maximal 5.5% overestimate at 3 GeV, while for higher energies the difference decreases again and is below 4% at 50 GeV (we remark however that we have not tuned our propagation model to reproduce the effect in Mori 1997 so we cannot claim that this effect is not relevant). Tertiary antiprotons In Eq. (8) we have not introduced any energy-changing term. We now include energy losses for secondary antiprotons due to scattering processes during their propagation in the Galaxy. The main effect is due to non-annihilation inelastic interactions of antiprotons with interstellar protons, giving lower energy antiprotons in the final state. Actually, the Interstellar fluxes total p-p secondary p-He secondary tertiary L. Bergström, J. Edsjö and P. Ullio, 1999 Fig. 3.-The interstellar antipron flux and the contribution from secondary and tertiary antiprotons. The uncertainty due to the parametrization of the primary proton spectrum is also given as the shaded band. The solid line corresponds to the same set of parameters as in Fig. 1. energy distribution after the non-annihilation interaction is not well known as there are no direct measurements; the usual assumption (Tan & Ng 1982) is that the distribution is similar to the final state proton in pp inelastic (non-diffractive) interactions, i.e., a rather flat distribution in kinetic energy between zero and the kinetic energy of the incident antiproton. One may think that elastic scattering processes are relevant as well but available data show that the cross section is dominated by the forward peak with very small energy transfer (Eisenhandler & al. 1976;Brückner & al. 1986), and hence with a marginal net effect from our point of view. We include therefore only non-annihilation processes considering a 'tertiary' source function generated by inelastically scattered secondary antiprotons in the form: Q tert p ( x, E) = 4 π n H ( x)   ∞ E σ non−ann pp (E ′ ) T ′ Ip( x, E ′ )dE ′ − σ non−ann pp (E)Ip( x, E)  (36) where σ non−ann pp is obtained as the difference between the total inelastic cross section Eq. (34) and the inelastic annihilation cross section: where for lower energies we have used the parametrization in Tan & Ng 1982, while in the high energy range we apply the approximation given in Protheroe 1981. Both this parametrization and those needed above have been checked against a compilation of more recent data (Caso & al. 1998). The second term in Eq. (36) takes into account antiprotons which are depleted from the energy E and which we propagate as a negative flux; it actually gives an effect that is less than few per cent at any energy and is not needed in our formalism. As was done for Eq. (27), it is straightforward to write a Fourier-Bessel expansion for Q tert p and then compute Φ tert p which has to be summed to Φp to get the final expression for the background interstellar antiproton flux. Numerical results Coming to the actual numerical predictions for the background flux of antiprotons, we base our choice of parameters in the propagation model on previous work in which diffusion models analogous to the one described in Section 4 were used to fit data on cosmic-ray nuclei, such as ratios of secondaries to primaries and of radioactive nuclei to their stable counterparts. Actually slightly discrepant results are present in the literature, partially reflecting the fact that it is not easy to find a propagation model which is consistent with the whole set of existing data. We consider here and in the following Section, when describing the signal from neutralino annihilations, three different scenarios. We only keep the following parameters fixed: n H g = 1 cm −3 , n H h = 0, h g = 0.1 kpc and R h = 20 kpc. The first three are the standard values which are inferred from direct observation. The last one, which in the literature is taken sometimes as small as 15 kpc, (1992). Their conclusion is that a thin halo is preferred, with height h h ∈ (1.1, 3.8) kpc and D ≃ (6±4)·10 27 cm 2 s −1 at the rigidity R = 1 GV. The flux shown in Fig. 1 is obtained in this scenario, setting h h = 3 kpc, D 0 = 6 · 10 27 cm 2 s −1 and R 0 = 3 GV, choosing the proton flux at the Earth as the medium value in the fit of IMAX data, i.e. with A = 12300 and α = 2.67, and taking into account solar modulation with the force field method with φ F = 500 MV as suggested by the analysis of the Bess collaboration. There is no consensus in the literature on the value of the solar modulation parameter at solar minimum, nevertheless the spectrum does not change dramatically if a slightly different value for φ F is assumed. For instance, φ F = 400 MV gives about a 7% increase at the kinetic energy T = 0.2 GeV and about an 8% increase at the maximum. For φ F = 600 MV the effect is reversed and we find that the flux is lower by roughly the same percentages as in the previous case. As will become clear in the following, the background antiproton flux shown in Fig. 1 is only an example of the possibility of a good fit to the data; we keep it as reference case to compare with. In Fig. 3, we show for the same parameters the interstellar antiproton flux versus kinetic energy T , plotting also its three main components: the secondary antiproton flux due to pp collisions, the contribution from pHe scattering processes and the tertiary component due to energy loss. As can be seen the first contribution is dominant at the maximum and at high energies, while the other two are important in the low energy region. We take advantage of Fig. 3 to show another feature that is common for all choices of the propagation parameters, the uncertainty due to the interstellar proton flux. The band around our reference antiproton flux is the envelope of the predictions obtained by using the uncertainty in the proton flux (Bottino & al. 1998): the upper bound is given choosing the fit of IMAX data with A = 15300 and α = 2.61 (average values +1σ and −1σ respectively), while the lower bound below T = 2.5 GeV is obtained from the IMAX data fit with A = 9300 and α = 2.73 and above 2.5 GeV from the CAPRICE data fit with A = 16300 and α = 2.89 (average values −1σ and +1σ respectively, actually these two spectra are nearly overlapping at all energies). Coming back to the uncertainty in the choice of the propagation parameters in the Webber-Lee-Gupta scenario, if we now pick the average value for the halo height h h = 2 kpc and vary the diffusion coefficient in the suggested interval, D 0 ≃ (3 − 7) · 10 27 cm 2 s −1 for R 0 = 3 GV, we find that the the flux at intermediate energies increases by up to about 30% for the smallest value of the diffusion coefficient, while it decreases with a slightly higher percentage for the highest value of D 0 . This is represented by the band in Fig. 4, region (a) (both in this Figure and in Fig. 6 below we defined fractional differences as (Φ − Φ R )/Φ R , with Φ R being the reference value, i.e. in this case the flux shown as a solid line in Fig. 3). If we assume on the other hand that D 0 and h h are linearly related, a degeneracy which may indeed not be resolved by available data, and fix D 0 = 2.5(h h /kpc) · 10 27 cm 2 s −1 varying h h between 1.1 and 3.8 kpc we get a band of very small width, below a few percent (solid and dashed lines at about −20% in Fig. 4, region (a)). b) In Strong & Moskalenko 1998, a scenario is favoured with a thicker halo, with h h ∈ (4, 12) kpc, in case no convection is assumed. Their treatment of propagation is less close to our model than the previous case, so the way we translate their typical choice of parameters into our picture is less safe but should give at least the right qualitative behaviour. We sketch the thick halo scenario taking h h ∈ (4, 12) kpc, D 0 h = 2.5(h h /kpc) · 10 27 cm 2 s −1 , D 0 g = 6 · 10 27 cm 2 s −1 and R 0 = 1 GV, a choice consistent with the results in Ginzburg, Khazan & Ptuskin (1980), where the propagation model we have chosen was first considered. Comparing to our reference case we find a band of 20% width around an average suppression of the flux of about 50%, where the least severe suppression is given by the smallest halo considered and the maximal suppression corresponds to the large halo h h = 12 kpc (solid and dashed lines respectively in Fig. 4 region (b)). c) As a third scenario we allow for the presence of a galactic wind driving cosmic rays out of the galactic disk. Self consistent models for the propagation of cosmic rays in magnetohydrodynamic flows have been studied recently in detail (Zirakashvili & al. 1996;. The much simpler approach we take here is intended to compare qualitatively the effects of the wind on the background antiproton flux and on the signal from neutralino annihilations. Considering again the model in the previous scenario with h h = 4 kpc (solid curve in Fig. 4 region (c)), we take as an example the case of v h = 10 km s −1 (lower dashed line) and v h = 20 km s −1 (lower dash-dotted line). As can be seen, in perfect analogy with the case of the solar wind, the galactic wind alters the spectrum at low energies (up to 30% in the example we are considering) while the effect gets smaller and smaller for more energetic particles. If in analogy with the parameter choice of Strong and Moskalenko we suppose that there is some scaling between v h and D 0 h , for instance a simple linear scaling D 0 h = 2.5(h h /kpc)[(40 km s −1 −v h )/40 km s −1 ]·10 27 cm 2 s −1 , we find that at intermediate energies the flux is nearly unchanged (upper dashed and dash-dotted lines). Kinetic Energy, Tp (GeV) Effects of changing diffusion parameters We will not combine the uncertainty bands we have just derived and make a definite statement about the uncertainty on the prediction of the cosmic antiproton background. To be able to do that on a firmer basis we should compare the predictions of our propagation model directly against the whole set of data on cosmic ray nuclei and this is beyond the aim of the paper. We stress again that this Section was mainly intended to show that the most recent data on cosmic ray antiprotons can be fitted by the background flux for some natural choice of the diffusion parameters. On the other hand we find that the prediction for the background could be lower as well, leaving room for an antiproton flux generated by an exotic source, possibly dark matter neutralinos. Reacceleration It seems very plausible that cosmic rays are reaccelerated by a Fermi type of acceleration by stochastic magnetic fields during propagation. This has been treated, e.g., in Seo & Ptuskin (1994);Heinbach & Simon (1995) and Simon & Heinbach (1996). There is also a possibility that cosmic rays get reaccelerated by weak shock waves from supernova remnants (Letaw, Silberberg & Tsao 1993). We will here focus on the former process, usually called diffusive reacceleration since it can be treated as a diffusion in momentum space. In Heinbach & Simon (1995) it was shown that data on low energy cosmic rays are compatible with the predictions of models without diffusive reacceleration only if the mean path length variation with energy shows a sharp break around 1-2 GeV. In models which include reaccelaretion effects on the other hand, depending on reacceleration strength, a path length distribution that is a simple power law for all energies may be considered. This is theoretically appealing since this from observations derived form agrees well with that expected from Kolmogorov turbulence. The result in Heinbach & Simon (1995) is confirmed in the analysis by Strong & Moskalenko (1998), who conclude as well that the reacceleration scheme allows a more natural choice for the parameters in the propagation model. Even though reacceleration implies that the average energy increases as the cosmic rays propagate through the galaxy, there is a general smearing of the injected spectrum meaning that there is also a 'leakage' of cosmic rays from higher energies to lower. This might be important for antiprotons where the injected spectrum drops below the maximum at few GeV. In Simon & Heinbach (1996) it was found that this leakage could substantially increase the secondaryp spectrum below 1 GeV, with the flux at a few hundred MeV that hardly can be lower than 1/3 of the value at around 1 GeV. We cannot compare directly with our analysis as we are not applying the same primary proton spectrum and propagation model, but still we can conclude that including reacceleration in the propagation model might add up to the effects of the proton-nuclei interactions and of the tertiary component in flattening the antiproton spectrum at low energies, and making it even more problematic to separate an exotic signal from the background in this region of the antiproton spectrum. L. Bergström, J. Edsjö and P. Ullio, 1999 Fig. 5.-The value of C prop for the same choice of diffusion parameters as in Fig. 1 and an isothermal sphere distribution of dark matter with a = 3.5 kpc (solid line), compared to the value for the same diffusion parameters and a Navarro et al. profile with a = 9 kpc (dotted line). The band gives the range of values of C prop in the thin halo scenario (with h h = 2 kpc) as described in the text. SIGNAL FROM NEUTRALINO ANNIHILATION General discussion With the source function introduced in Eq. (6), the antiproton flux from neutralino annihilations in the galactic halo is readily obtained from the formulas derived in Section 4. It is given by: Φp(r 0 , T ) = 1 4 π vp(T ) Np(r 0 , T ) = 1 4 π vp(T ) ∞ s=1 J 0 ν 0 s r 0 R h M 0 s (0)(38) where M 0 s (0) is obtained from Eq. (21) with σ in cr p = σ in p p in Eq. (22) and Q 0 s (z) in Eq. (23) given by Q 0 s (z) = 2 R h 2 J 1 2 (ν 0 s ) R h 0 dr ′ r ′ J 0 ν 0 s r ′ R h Q χ p (r ′ , z) .(39) It is possible to separate in the expression for the signal the part which depends on the MSSM parameter space from the terms which are related only to the distribution of sources in the propagation region and to the propagation model itself. We introduce the definition: Φp(r 0 , T ) ≡ (σ ann v) f dN f dT B f ρ 0 mχ 2 C prop (T ) .(40) The quantity C prop , which can be obtained explicitly by comparing Eq. (38) with Eq. (40), has the dimension of length divided by solid angle and is analogous to the coefficient defined in Eq. (46) of Bottino & al. 1998; note however that in Eq. (40) we have factorized the value of the local halo density ρ 0 rather than some reference density. Uncertainties related to propagation Having factorized out in Eq. (40) the dependence of the signal on the choice of the dark matter candidate, we analyse first how sensitive the result is to the set of parameters which define both the location of the sources and the propagation of the produced antiprotons. We fix a reference configuration selecting for the propagation model, in analogy to the analysis of the background flux, the same parameters as in the example in Fig. 1 and Fig. 3 L. Bergström, J. Edsjö and P. Ullio, 1999 Fig. 6.-The changes of C prop when the diffusion zone height is changed within (a) the thin halo scenario derived from Webber, Lee and Gupta model and (b) the thick halo scenario inspired by the Moskalenko and Strong analysis. In (c) the effects of a convective wind are sketched: in (c1) the diffusion coefficient is kept fixed while in (c2) it is linearly scaled with the galactic wind. and R 0 = 3 GV), while as a reference dark matter density profile we choose a modified isothermal distribution, Eq. (7) with (α, β, γ) = (2, 2, 0), and with an intermediate value for the length scale, a = 3.5 kpc. In Fig. 5 we plot the value of C prop for this reference case (solid curve) versus the antiproton kinetic energy. C prop is increasing in the low energy range as it contains the kinematic factor vp(T ) and at the same time we have assumed that the diffusion coefficient is roughly constant at low rigidities (see Eq. (11)). The coefficient C prop then reaches a maximum at about T ≃ 2 GeV, while at higher energies it decreases as a consequence of the 0.6 power law increase in the diffusion coefficient. First we analyse how the result changes if as the dark matter density distribution we consider the Navarro et al. profile (Navarro, Frenk & White 1996) which is singular towards the galactic centre, Eq. (7) with (α, β, γ) = (1, 2, 1). Choosing a = 9 kpc, we obtain (see the dotted line in Fig. 5) roughly a 33% increase in C prop (and therefore in the signal from neutralino annihilations) at any T, while the more cuspy profile with a = 3.5 kpc gives a result which is more than twice the reference value. This is rather surprising because even though the singularity in the profile induces a sharp enhancement in the neutralino number density and therefore in the strength of the source, this cusp is at the galactic centre rather far away from the solar system. It is commonly believed that in the diffusion regime the local sources are the most relevant, but at least in the propagation model we are considering, this is not true: for the Navarro et al. profile with a = 9 kpc, 23% of the signal is given by sources contained in a spherical region of 1 kpc around the galactic centre; the percentage increases to 42% for a = 3.5 kpc while it is as low as 1% for the isothermal sphere profile. We conclude that non-local sources may give a significant contribution provided their strength is much enhanced with respect to the local ones. This effect should be considered in more detail when considering a clumpy scenario (Ullio 1999) for which it may be even more relevant. Fig. 5 contains another piece of information. Going back to the case where the dark matter density profile is described by an isothermal sphere, we have varied the parameters which define the propagation model, going through the same three scenarios described when discussing the background. The band in the figure is given by fixing h h = 2 kpc and varying the diffusion coefficient in the interval D 0 ≃ (3 − 7) · 10 27 cm 2 s −1 , and corresponds to the band shown in part (a) of Fig. 4 (again the highest value of D 0 gives the lowest value for the flux). Unlike the latter, the band in Fig. 5 does not overlap the reference value (solid line): for the signal from neutralino annihilations the decrease in the height of the propagation zone from 3 kpc to 2 kpc is not compensated by the decrease in the central value for the diffusion coefficient. This gives a first hint on how sensitive the dark matter signal is to the choice of the value of the height of the diffusion zone. The same effect is studied in the upper part of Fig. 6, fixing the kinetic energy to T = 1 GeV, varying h h and linearly relating D 0 to h h , as introduced in the previous Section. According to the Webber-Lee-Gupta scenario, h h is constrained to be between 1.1 and 3.8 kpc: the degeneracy we found for the background flux (the nearly overlapping solid and dashed line in part (a) of Fig. 4) is completely removed for the signal from neutralino annihilations, going from an 80% suppression to a 20% increase compared to the reference value, for minimal and maximal h h respectively. Changing the diffusion zone height modifies the number of sources which contribute to the flux, as sources which are outside the diffusion box are not included in the model. Therefore it is not surprising that a very thin diffusion zone gives a suppressed signal while larger values for h h enhance it. In the same way, in the thick halo scenario without convection, inferred from the analysis of Strong and Moskalenko, the suppression band found for the background flux (part (b) of Fig. 4) becomes a much wider band for which the signal flux is increased instead (solid line in part (b) of Fig. 6). The enhancement with the diffusion zone height is flattened out at high values of h h as the new sources we include are further and further away from the observer and moreover the density profile falls at large galactocentric distances. While the effects we have considered so far give roughly the same result for any value of the antiproton kinetic energy, the effect of convection in the z direction is clearly energy dependent. In part (c1) of Fig. 6 we plot C prop as a function of the galactic wind speed, for h h = 4 kpc, D 0 h = 10 28 cm 2 s −1 , D 0 g = 6 · 10 27 cm 2 s −1 and R 0 = 1 GV (we consider here as reference value the one obtained with this set of parameters and u h = 0). In part (c2), as we did for the background, D 0 h is taken to be related linearly to the value of the wind speed. In both parts five different kinetic energies (given in the Figure in GeV) have been considered. As can be seen going back to part (c) of Fig. 4, the effect of convection is greater on the signal from neutralino annihilation than on the background flux. Especially at low energies, it is not well compensated by the linear scaling of the diffusion coefficient. For the background flux the effect for introducing a galactic wind is to drive antiprotons more quickly from the disk, where they are generated, to the border of the diffusion zone where they are lost. This effect can be balanced by lowering the diffusion coefficient, that is, assuming that diffusion takes place less efficiently. The sources of the signal are on the other hand distributed over the whole diffusion box; setting with the galactic wind a preferred direction of propagation lowers the probability for an antiproton generated relatively far away from the disk to reach our location. The effect is not compensated by the rescaling of D 0 h , at least not for the rescaling needed for cosmic ray species generated in the disk. The last check we perform regards the role played by the antiprotons which are produced outside the propagation region. Our solution to the diffusion equation has been derived under the hypothesis that the number density of the considered cosmic ray species is zero at the boundary of the diffusion zone. This is not strictly true for the signal from neutralino annihilations. One possibility of verifying what kind of corrections might be needed in this case is to compare the antiproton flux leaving the diffusion zone with the flux injected by external sources. We restrict the analysis to exchanges at the boundary z = ±h h , as the effect is much suppressed in the radial direction, being generally R h >> h h . The outgoing flux can be computed by keeping track of the full dependence on z of the number density, as this flux is related to the gradient of the number density at the boundaries; one derives a rather lengthy expression which we do not reproduce here but which follows in a straightforward way. For the injected flux we use the very simple picture of propagation in free space, summing contributions over the line of sight. For the modified isothermal sphere profile and a Bergström, J. Edsjö and P. Ullio, 1999 Fig. 8.-The interstellar antiproton flux at 1 GeV versus the relic density. This is the only figure where models having Ω χ h 2 < 0.025 are shown. diffusion zone height of 3 kpc we find that the ingoing flux is about one third of the outgoing flux for very small radial coordinates, while they become roughly equal at r = 8 kpc and at larger radii the injected flux becomes prevailing; the total number of antiprotons per second that penetrate the diffusion box is about 70% of those that leave it. As can be understood, this fraction is smaller if we consider instead the Navarro et al. profile or if we pick a higher value for the diffusion zone height, and might turn into a rather large number for very thin haloes. To give a precise numerical estimate for the effect one should add in the propagation model a third zone above \|z\| = h h . However, taking into account all the other uncertainties that enter in the prediction for the signal, we do not consider this worthwhile at present. We believe that a safe assumption is that the signal from neutralino annihilations has not been underestimated due to this effect by more that a factor of 2 in the most extreme cases. Antiprotons from specific MSSM models The antiproton spectrum from neutralino annihilation have been calculated for all the different MSSM models given in Table 1. In Figs. 7-11 we show our main results. We use our canonical parameters for propagation and the isothermal sphere model for the halo profile. Interstellar fluxes In Fig. 7 (a) we show the predicted interstellar antiproton flux at a kinetic energy of 1 GeV (i.e., without corrections for solar modulation) versus the neutralino mass. We clearly see the trend that the flux goes down with the mass of the neutralino. The reason for this is that the number density of neutralinos goes down as the mass increases, for a given interval of the dark matter mass density. Since n χ = ρ χ /m χ , and the annihilation rate scales as n 2 χ the suppression increases rapidly with mass. At the lower mass end, the present accelerator limits preclude a neutralino in the MSSM below a few tens of GeV. The low-flux models at low masses will be discussed in connection to Fig. 8 below. The points in the figure are coded with different symbols for different composition of the neutralino. We define models with 0 < Z g < 0.01 as Higgsino-like, 0.01 < Z g < 0.99 as mixed and 0.99 < Z g < 1 as gaugino-like. As can be seen, most of the models with high rates are either gaugino-like or mixed, except at masses greater than several hundred GeV, where also Higgsino-like models can be important. (In fact, there is also a small mass window around 80 GeV where Higgsinos may be relevant.) In Fig. 7 (b), we show the ratio of the interstellar flux at 0.5, 3, 5, and 10 GeV to the flux at 1 GeV displayed in (a) for the same set of models (but without coding the composition). It is seen that for the higher mass models, it can be more advantageous to study the flux at higher kinetic energies. In Fig. 8 we show the same fluxes as in Fig. 7 (a), but versus the relic density, Ω χ h 2 . There is a very clear trend that the highest flux is obtained when Ω χ h 2 is close to the lowest acceptable relic density. The reason for this is that if the annihilation cross section is increased, the flux of antiprotons increases, but since the relic density Ω χ h 2 is approximately inversely proportional to the annihilation cross section, Ω χ h 2 decreases and hence the strong correlation. The correlation L. Bergström, J. Edsjö and P. Ullio, 1999 Fig. 9.-The flux of antiprotons from neutralino annihilation at the optimal kinetic energy, T opt , versus T opt . T opt is defined as the energy at which Φ signal /Φ background is highest and if the spectrum has more than one optimum, the highest two have been included in the plot. The models have been coded according to the neutralino mass in GeV. is not perfect however, since it is the thermally averaged cross section at a temperature of about m χ /20 that determines Ω χ h 2 , whereas the annihilation in the halo to a very good approximation occurs at rest (the speeds are typically ∼ 0.001c). This also exlplains some features in Fig. 7 (a). In the mass range between 40 and 60 GeV there exist models which give exceedingly small rates, but also some which give the highest of all rates. This large spread reflects peculiarities near the Z 0 (and neutral Higgs) resonances and the W + W − threshold. For the low-flux models around 40-60 GeV, the annihilation cross section at rest is very small, but either a resonance or threshold can be reached through thermal motion in the early Universe and the relic density is reduced to our selected range 0.025 < Ω χ h 2 < 1. As shown by Chen & Kamionkowski (1998), three-body final states can be important (not too far) below the W + W − and the tt thresholds and this could enhance the signal for these low-rate models. We also have some models around 130 GeV that give high fluxes. In this case it is the other way around, the masses are just so that we are on the H 0 3 -resonance for the non-relativistic speeds in the halo, but the thermal average in the early Universe gives a lower annihilation cross section and hence the relic density is increased to our desired range 0.025 < Ω χ h 2 < 1. The behaviour at 130 GeV is just accidental, it could happen at any important resonance. In fact, we found only one high-flux model around 130 GeV in our 'normal' scans, and performed a small scan varying the parameters slightly around this model. The relic density was essentially unchanged, but the antiproton flux showed large variations depending on if we were below, on or above the resonance. In Fig. 8 we also show models with a value of Ωh 2 lower than our required limit of 0.025. In principle, one could accept these models at the expense of introducing other components of dark matter. To be consistent, one should then rescale the local dark matter density in the form of neutralinos by some unknown factor. In the lack of better procedures, one usually employs a linear rescaling ρ χ = (Ω χ h 2 /0.025) · ρ DM . Since the annihilation rate is quadratic in the number density, this rescaling factor enters squared in the predictedp rate, something which is clearly visible in Fig. 8. We are now interested in finding out if there are any special features of the antiproton spectra from neutralino annihilation which distinguish these spectra from the background. We have already mentioned that the window at low energies may not be as good as previously thought. We will here investigate other features and energy regions of the spectrum to see if there are good signatures of a neutralino contribution to the flux. One thing that might differ is the slope at different energies. The background is expected to have a rising trend at low energies, reaching a maximum between 1 and 2 GeV (see Fig. 3) and a slope of around −3 at high energies. On the other hand, the high-rate models tend to be decreasing at 1 GeV (see Fig. 11 (a)). This may cause a shift of the maximum of the summed spectrum (signal plus background) to a lower energy, which is a possible signature. We next investigate if there is an optimal energy at which Φ signal /Φ background has a maximum (for this purpose we will use the reference background given in Fig. 3). In Fig. 9 we show the flux at these optimal energies, T opt , versus T opt . We now see that we have two classes of models. One class which have highest signal to noise below 0.5 GeV (i.e. inaccessible in the solar system due to the solar modulation) and one which have highest signal to noise at 10-30 GeV. For this first class of models, we note that there exist a proposal of an extra-solar space probe (Wells, Moiseev & Ormes 1998) which would avoid the solar modulation problem and is thus an attractive possibility for this field. However, these models have high rates in the range 0.5-1 GeV as well, even though it would be even more advantagous to go to lower energies. The second class of models are much less affected by solar modulation and also give reasonably high fluxes. In Fig. 11 (a) we show some examples of spectra. All of these have optimal energies in the low-energy region, but e.g. spectrum 3 has an optimum at high energy as well. Solar modulated fluxes We now turn to the solar modulated fluxes and will compare with the Bess 97 measurements, which we recall are in very good agreement with our estimate for the background flux. We will compare the fluxes in two of the Bess energy bins, the one at 0.35 GeV and the one where the measured flux is the highest. In Fig. 10 we show the solar modulated fluxes versus the neutralino mass. We see the same general trend as for the interstellar fluxes, Fig. 7, but we also see that there are many models with fluxes above the Bess measurements. However, this conclusion depends strongly on which range one allows for the neutralino relic density. In Fig. 10 we have coded the symbols according to the relic density interval. As can be seen, essentially all models which are in the Bess measurement band have a relic density Ω χ h 2 < 0.1. If we instead require 0.1 ∼ < Ω χ h 2 ∼ < 0.2 the rates are never higher than the measured flux. This points to a weakness of this indirect method of detecting supersymmetric dark matter: once the predicted rate is lower than the presently measured flux, the sensitivity to an exotic component is lost. This is because of the lack of a distinct signature which could differentiate between the signal and the background. Alternative indirect search methods, like gamma rays from the halo (see e.g. Bergström, Ullio & Buckley 1998), or neutrinos from the Sun or the Earth (see e.g. Bergström, Edsjö & Gondolo 1998) have the added virtue of giving both a directional and a spectral signature which can be used to improve the signal to background ratio well beyond the limits of present-day measurements. The highest values for the fluxes in Fig. 10 (a) are 4 times higher than the Bess measurement. However, the uncertainty coming from the local halo density alone is larger than this. Given the total mass of the galaxy, and restricting to our choice of halo profile (isothermal sphere with a = 3.5 kpc), we find a minimal local halo density of 0.14 GeV/cm 3 which would correspond to a flux reduction of a factor of 4.6. To that one should add the uncertainties of the Monte Carlo simulations (up to a factor of 2) and the halo profile, the propagation model and solar modulation. For this reason it is presently not possible to exclude any supersymmetric model on the basis of antiproton measurements alone. Example of models In Table 3 we show 7 MSSM models that all give highp fluxes. These models have acceptable relic densities, cover a large mass range and have varying composition (and obey present accelerator bounds). In Fig. 11 (a) the predicted differentialp flux is shown for the 7 models. They show maxima occurring at lower energies than for our canonical background. At higher energies, the trend is that the slope of the flux decreases as the neutralino L. Bergström, J. Edsjö and P. Ullio, 1999 Solar Modulated, φ F = 500 MV total background signal Fig. 11.-(a) Antiproton spectra for all 7 models appearing in Table 3. (b) Example of a composite spectrum consisting of our reference backgroundp flux ( Fig. 1) reduced by 24 % with the addition of the predicted flux from annihilating dark matter neutralinos of MSSM model number 5 in Table 3. mass increases. Model number 3 corresponds to a heavy neutralino and its spectrum is significantly less steep than the background. If such a spectrum is enhanced, for instance by changing the dark matter density distribution, we would get a bump in the spectrum above 10 GeV. In Table 3 we also show the annihilation rate and the most important branching ratios. With the help of the results in the earlier Sections, the antiproton flux from neutralino annihilation can be derived. The difference between the parameterizations given in Section 3.2 and the full simulation results (that we have used) is typically less than 20%. Also note that the branching ratio to gg is never important for our high-rate models (not only the ones in the table, but all high-rate models). This is not in agreement with the results found by . The reason for this difference is that we use the improved gg annihilation cross section of . The table also contains an indication of the rates for other detection methods. The neutrino-induced muon flux in neutrino telescopes does not show a strong correlation with thep flux, and it is possible to find models that give either low or high rates. Current limits are about 10 3 -10 4 muons km −2 yr −1 . We also give the spin-indepedent neutralino-nucleon cross section (Bergström & Gondolo 1996), which should be compared with the current limits that are of the order of 10 −5 pb. These show a better, but not perfect, correlation with thep fluxes. The correlation is even stronger between thē p flux and both the e + flux and the γ flux with continuum energy spectrum. Both of these do not decrease as much with neutralino mass as the antiproton flux does, however. For more details, see Baltz & Edsjö (1999) and Bergström, Edsjö, Gondolo & Ullio 1999. The cross section for annihilation into monochromatic γs (through γγ and Zγ) are uncorrelated with thep flux. Finally, in Fig. 11 (b) we show an example of a hypothetical composite spectrum which consists of our canonical background flux decreased by 24 % (obtained e.g. by decreasing the primary proton flux by 1σ), and the signal for model 5 in Table 3. We can obtain a nice fit to the Bess data, but as noted before, there are no special features in the spectrum that allow us to distinguish between this case and the case of no signal. DISCUSSION AND CONCLUSIONS We have seen that there is room, but no need, for a signal in the measured antiproton fluxes. We have also seen that the optimal energy to look for when searching for antiprotons is either below the solar modulation cut-off or at higher energies than currently measured. However, there are no special spectral features in the signal spectra compared to the background, unless the signal is enhanced and one looks at higher energies (above 10 GeV). We have stressed the somewhat disappointing fact that since the present measurements by the Bess collaboration already exclude a much higherp flux at low energies than what is predicted through standard cosmic-ray production processes, an exotic signal could be drowned in this background. Even if it is not, the similar shape of signal and background spectra will make it extremely hard to claim an exotic detection even with a precision measurement, given the large uncertainties in the predicted background flux (at least a factor of a few, up to ten in a conservative approach). We note that some of the uncertainties may be reduced if an extrasolar probe aimed at low-energy detection would be launched, a possibility that has been recently proposed (Wells, Moiseev & Ormes 1998). Although it is tempting to conclude that what has been measured by the Bess experiment is the standard cosmic-ray induced background flux of antiprotons, one should keep in mind that it could, on the contrary, be almost entirely due to an exotic source like neutralino annihilation. Since this possibility cannot be excluded (at least until the problem of the dark matter in the Galactic halo has been solved), one has to be cautious about using the measured antiproton flux to deduce properties of antiproton propagation and, as has recently been done (Geer & Kennedy 1998), the antiproton lifetime. We have checked that, using one of our high-mass neutralino models and a clumpy distribution of dark matter in the halo, we can get an excellent fit to the Bess data for antiproton lifetimes as low as 10 5 years, clearly violating the claimed lower bound of Geer & Kennedy (1998). (For details, see Ullio (1999).) Fig. 1 . 1-The background of antiprotons solar modulated with φ F = 500 MV. The Bess 95 and 97 data are also shown L.Bergström, J. Edsjö and P. Ullio, 1999 Tp dN / dT-p Fig. 2.-Examples of the fits to the antiproton distributions from neutralino annihilation, dN/dT = (1/m χ )dN/dx with dN/dx given by Eqs. (4)-5) and and M p (s, E) = M * (s, k = 0) setting the total inelastic cross section appropriate for pp collisions σ incr p = σ in p p , while Mp(s, E) = M * (s, k = 0) with σ in cr p = σ in p p .The parametrizations for both of these cross sections were given inTan00262·T −(17.9+13.8 ln T +4.41 ln 2 T ) mb , 0.3 ≤ T < 3 GeV 32.2 · [1 + 0.0273 ln(E/200)] mb , 3 ≤ T < 200 GeV 32.2 · 1 + 0.0273 ln(E/200) + 0.01 (ln(E/200)) 2 mb , T ≥ 200 GeV (33) and σ in p p (T ) = 24.7 · 1 + 0.584 T −0.115 + 0.856 T −0.566 mb , T ≥ 0.05 GeV 1 + 0.0115 T −0.774 − 0.948 T 0.0151 mb , T < 15.5 GeV 36 T −0.5 mb , T ≥ 15.5 GeV Fig. 4 . 4-The effects of changing the diffusion parameters are shown. In (a) the thin halo scenario, in (b) the thick halo scenario, while in (c) a convective halo is considered. See the text for further details. and which in Strong & Moskalenko (1998) is set equal to 30 kpc, does not play a major role and different choices lead to nearly equivalent results. The three scenarios are: a) In the case D 0 g = D 0 h = D 0 our propagation model is fairly close to the one considered by Webber, Lee and Gupta (h h = 3 kpc, D 0 = 6 · 10 27 cm 2 s −1 Fig. 7 . 7-In (a) the interstellar antiproton flux at 1 GeV is shown and in (b) the ratio of the flux at different kinetic energies to that at 1 GeV is shown. To make the figure clearer (and avoid showing artifacts of sampling frequency) the figure is binned. We also show with different symbols in which bins there are models which are gaugino-like, mixed and Higgsino-like. Fig. 10 . 10-The solar modulated antiproton fluxes at (a) 0.35 GeV and (b) 1.76 GeV compared with Bess 97. The models have been coded according to their relic density, Ω χ h 2 . 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[ "Apache Point Observatory", "Apache Point Observatory" ]	[ "Peregrine M Mcgehee [email protected] \nLANSCE-8\nLos Alamos National Laboratory\nH820, 87545Los AlamosMS, NM\n\nDepartment of Astronomy\nNew Mexico State University\nMSC 4500Box 3000188003Las CrucesNM\n", "J Allyn Smith \nLos Alamos National Laboratory\nISR-4D448, 87545Los AlamosMS, NM\n\nDepartment of Physics & Astronomy\nUniversity of Wyoming\n1000 E. University Blvd82071LaramieWY\n", "Arne A Henden \nUniversities Space Research Association/US Naval Observatory\nP.O. Box 114986002Flagstaff Station, FlagstaffAZ\n", "Michael W Richmond \nDepartment of Physics\nRochester Institute of Technology\n85 Lomb Memorial Drive14623RochesterNY\n", "Gillian R Knapp \nPrinceton University Observatory\n08544PrincetonNJ\n", "Douglas P Finkbeiner \nPrinceton University Observatory\n08544PrincetonNJ\n", "Željko Ivezić \nPrinceton University Observatory\n08544PrincetonNJ\n", "J Brinkmann " ]	[ "LANSCE-8\nLos Alamos National Laboratory\nH820, 87545Los AlamosMS, NM", "Department of Astronomy\nNew Mexico State University\nMSC 4500Box 3000188003Las CrucesNM", "Los Alamos National Laboratory\nISR-4D448, 87545Los AlamosMS, NM", "Department of Physics & Astronomy\nUniversity of Wyoming\n1000 E. University Blvd82071LaramieWY", "Universities Space Research Association/US Naval Observatory\nP.O. Box 114986002Flagstaff Station, FlagstaffAZ", "Department of Physics\nRochester Institute of Technology\n85 Lomb Memorial Drive14623RochesterNY", "Princeton University Observatory\n08544PrincetonNJ", "Princeton University Observatory\n08544PrincetonNJ", "Princeton University Observatory\n08544PrincetonNJ" ]	[]	We present Sloan Digital Sky Survey and United States Naval Observatory observations of the V1647 Ori protostar and surrounding field near NGC 2068. V1647 Ori, the likely driving source for HH 23, brightened significantly in November 2003. Analysis of SDSS imaging acquired in November 1998 and February 2002 during the quiescent state, recent USNO photometry, and published 2MASS and Gemini data shows that the color changes associated with brightening suggest an EXor outburst rather than a simple dust clearing event.	10.1086/425069	[ "https://arxiv.org/pdf/astro-ph/0408308v1.pdf" ]	119,099,155	astro-ph/0408308	7b218c1f2ead67bf5113e3133f064ac7493e9e07	Apache Point Observatory 2001 Peregrine M Mcgehee [email protected] LANSCE-8 Los Alamos National Laboratory H820, 87545Los AlamosMS, NM Department of Astronomy New Mexico State University MSC 4500Box 3000188003Las CrucesNM J Allyn Smith Los Alamos National Laboratory ISR-4D448, 87545Los AlamosMS, NM Department of Physics & Astronomy University of Wyoming 1000 E. University Blvd82071LaramieWY Arne A Henden Universities Space Research Association/US Naval Observatory P.O. Box 114986002Flagstaff Station, FlagstaffAZ Michael W Richmond Department of Physics Rochester Institute of Technology 85 Lomb Memorial Drive14623RochesterNY Gillian R Knapp Princeton University Observatory 08544PrincetonNJ Douglas P Finkbeiner Princeton University Observatory 08544PrincetonNJ Željko Ivezić Princeton University Observatory 08544PrincetonNJ J Brinkmann Apache Point Observatory Apache Point Road, Sunspot, NM 8834920018Subject headings: stars: formation -stars: pre-main-sequence -stars: circumstel- lar matter -stars: individual (V1647 Ori, IRAS 05436-0007) We present Sloan Digital Sky Survey and United States Naval Observatory observations of the V1647 Ori protostar and surrounding field near NGC 2068. V1647 Ori, the likely driving source for HH 23, brightened significantly in November 2003. Analysis of SDSS imaging acquired in November 1998 and February 2002 during the quiescent state, recent USNO photometry, and published 2MASS and Gemini data shows that the color changes associated with brightening suggest an EXor outburst rather than a simple dust clearing event. Introduction In January 2004, J.W. McNeil discovered a new reflection nebula in the dark cloud Lynds 1630 near M 78 (McNeil 2004) This object, now known as McNeil's Nebula, is apparently associated with an EXor-type eruption (Reipurth & Aspin 2004) of the embedded protostar V1647 Ori. EXors belong to the class of pre-main-sequence optical variables (Herbig 1977). They are classical T Tau stars which undergo irregular outbursts in the optical/UV of several magnitudes, named for the prototype EX Lup (Herbig et al. 2001). These outbursts are interpreted as episodes of substantial mass transfer resulting from instabilities in the accretion disk; they are present very early in the evolution of a protostar, as shown by the detection of EXor outbursts from deeply embedded Class 1 protostars in the Serpens star formation region (Hodapp et al. 1996). Clark (1991) first identified V1647 Ori as the young stellar object IRAS 05436-0007 on the basis of its IRAS colors. I-band and [SII] narrow-band imaging of the region by Eislöffel & Mundt (1997) revealed a faint I band source at the position of the IRAS object and reflection nebulosity extending to the north, identifying V1647 Ori as the likely driver for HH 23, located 170 arcsec north of the star. The bolometric flux of the source derived from IRAS and sub-millimeter photometry by Lis, Menten, & Zylka (1999) yields a luminosity of 2.7 L ⊙ and an inferred molecular gas mass of 0.4 M ⊙ assuming a distance of 400 pc to the Orion star-forming complex (Anthony-Twarog 1982). Estimates of the extinction towards V1647 Ori, A V = 11 m -15 m , are found to be similar from photometry taken during the quiescent phase (Abraham et al. 2004) and during the eruptive phase (Reipurth & Aspin 2004;Vacca et al. 2004;Briceño et al. 2004;Andrews, Rothberg & Simon 2004). Thus, it is not clear whether the appearance of McNeil's Nebula is due only to the eruption of V1647 Ori or to the eruption plus additional clearing of obscuring circumstellar dust. In this paper, we examine this question using pre-eruption multiband optical and near infrared data from the Sloan Digital Sky Survey (SDSS) and the Two Micron All Sky Survey (2MASS) compared with post-eruption data in the SDSS and 2MASS bands observed at the United States Naval Observatory. Observations We detect the protostar at four epochs of Sloan Digital Sky Survey (SDSS) imaging as a point source (SDSS J054613.14-000604.1) coincident with the 2MASS K-band position (α 2000 =05 h 46 m 13.1 s , δ 2000 =-00 o 06 ′ 05 ′′ ). The SDSS observations consist of two pairs of scans acquired in November 1998 and February 2002. Figure 1 shows an SDSS composite image made with the riz filters in which the protostar and the faint nebulosity to the north can be seen. A technical summary of the SDSS is given by York et al. (2000). The SDSS imaging camera is described by Gunn et al. (1998). The Early Data Release and the Data Release One are described by Stoughton et al. (2002) and Abazajian et al. (2003). The former includes an extensive discussion of the data outputs and software. Pier et al. (2003) describe the astrometric calibration of the survey and the network of primary photometric standard stars is described by Smith et al. (2002). The photometric system itself is defined by Fukugita et al (1996), and the system which monitors the site photometricity by Hogg et al. (2001). Abazajian et al. (2003) discuss the differences between the native SDSS 2.5m ugriz system and the u ′ g ′ r ′ i ′ z ′ standard star system defined on the USNO 1.0 m (Smith et al. 2002). The SDSS low Galactic latitude data which includes the Orion equatorial imaging used in this work are described by Finkbeiner et al. (2004). The U.S. Naval Observatory Flagstaff Station 1.0 m and 1.55 m telescopes were used to obtain eruptive phase BV RIz ′ and JHK photometry of V1647 Ori. For BV RIz ′ , frames were taken and flatfielded using twilight sky flats. DAOPHOT PSF fitting as implemented in IRAF was used to obtain photometric measures of the target since there is a bright part of the nebula only a few arcsec distant. For each dataset, ensemble differential photometry was performed using a set of secondary standard stars calibrated on two photometric nights with the USNO 1.0 m telescope. The z ′ measures were relative to z ′ secondary standard stars calibrated by the SDSS PT telescope (Hogg et al. 2001). For JHK, we used 2MASS stars in the field, eliminating obvious variables, to calibrate the data. We used a standard K filter for 2004 February 11, but used a K ′ filter for 2004 April 12. Results The pre-eruptive and recent optical and near-infrared photometry of V1647 Ori are summarized in Tables 1, 2, and 3. The quiescent and eruptive phase spectral energy distributions are shown as νF ν in Figure 2. We examine the evolution of reddening invariant colors from the quiescent to the eruptive states to constrain the underlying physical processes. These colors take the generic form C xyz = (x − y) − (y − z) * E(x − y)/E(y − z) where x, y, and z are the observed magnitudes in each passband. A color change is computed as ∆C xyz , where color changes having ∆C xyz statistically distinct from 0 indicate changes in the spectral energy distribution (SED) not consistent with pure dust clearing. The conversion from E(B − V) to extinction in each band follows Schlegel, Finkbeiner, & Davis (1998) and Finkbeiner et al.(2004), in preparation The reddening invariant colors measured before and during the eruption are listed in Table 4 for both R V = 3.1 and R V = 5.5, the latter appropriate for the larger dust grains found in star formation regions. R V , defined as A V /E(B − V), is the ratio of the general to selective extinction. Characteristic grain sizes are inferred to increase from 0.17 µm to 0.21 µm as R V ranges from 3.1 to 5.5 assuming a mix of silicate and carbonaceous populations (Weingartner & Draine 2001). In the coldest portions of the molecular clouds additional grain species such as refractory and volatile organics, olivine, water ice, orthopyroxene, trolite, and metallic iron are expected to contribute to the opacity (Pollack et al. 1994). Vacca et al. (2004) detect absorption due to water ice at 3.1 µm in the near-IR outburst spectrum. Study of the selective extinction in background stars has revealed that R V is not constant within the interiors of dark molecular clouds. Strafella et al. (2001) find that for the Bok globule CB 107 R V increases inwards reaching a value of 6.5 at the core. Given the complex nature of the V1647 Ori environment including outburst cavities, accretion disks, and remnant envelopes it is likely that R V can vary on small spatial scales. Determination of Extinction In the foregoing analysis we choose to characterize the intervening dust in terms of the SDSS z band extinction and R V . Since the shape of the extinction curve is dependent upon the grain size distribution for λ < 0.9µm (Cardelli, Clayton & Mathis 1989) the traditional A V and E(B − V) quantities are sensitive to both the dust column density and R V . By adopting reference passbands longward of 0.9 µm we remove the dependency on the form of the dust law although R V must still be considered in the optical and UV. Alternate passbands that are free of R V effects include the near-IR J and Johnson-Cousins I, the latter used by Weingartner & Draine (2001) who adopted A I /N(HI) = 2.5 × 10 −22 cm 2 for a standard gas-to-dust ratio. Following Abraham et al. (2004) and Reipurth & Aspin (2004) we assume that the quiescent phase near-IR colors are that of an embedded low-mass Classical T Tauri and obtain E(J − H) = 1.43 m by dereddening onto the Classical T Tauri locus of Meyer, Calvet, & Hillenbrand (1997). This value of E(J − H) corresponds to A J = 3.39 m (A z = 6.40 m ) given λ eff (J) = 1.25µm and λ eff (H) = 1.65µm and application of the Cardelli, Clayton & Mathis (1989) methodology. Changes in the Spectral Energy Distribution In order to analyze the general differences between the quiescent and eruptive appearance we combine sets of observations to form representative values. The quiescent state is taken as the October 7 (2MASS J, H, K) and the 1998 November 17 (SDSS r, i, z) observations. The optical photometry for the eruptive state is obtained from the 2004 February 14 Gemini r and i data (Reipurth & Aspin 2004), and the 2004 February 26 USNO z band observations. The J, H, and K data for the eruptive state are from the 2004 February 11 USNO observations. Table 5 presents the reddening invariant colors in the quiescent and eruptive states for R V = 3.1 and R V = 5.5. As evidenced by the greater than 5σ change in all colors except for C riz , which has a marginal r band detection in the quiescent state, it is clear that the increase in emission is not consistent with a dust clearing event and thus we conclude that an intrinsic change occurred to the SED of V1647 Ori. In Figure 3 we compare the observed magnitude variations against ∆A z = −2 m and −4 m dust clearing events and, as concluded above based on reddening invariant colors, find they are not explainable in terms of a simple diminishing of the line of sight extinction. The Quiescent Phase From observations of Class II protostars (Classical T Tauris) we would expect that the veiling continuum due to magnetospheric accretion will be present in the SDSS u and g bands but weakening into the redder r, i and z bands (Calvet & Gullbring 1998). In a similar fashion, the thermal emission from the inner portion of the circumstellar disk should be bright in the H and K bands but diminishing into the bluer J band (Meyer, Calvet, & Hillenbrand 1997). For all but the mostly heavily veiled stars the i and z bands will only be affected by extinction. None of the reddening invariant colors can be used to give the intrinsic spectral type. Young (1 Myr) T Tauri stars exhibit spectral types between M0 and M4 for masses of 0.1 to 1.0 M ⊙ ( Baraffe et al. 1998). The use of the Meyer, Calvet, & Hillenbrand (1997) near-IR Classical T Tauri locus by Abraham et al. (2004) and Reipurth & Aspin (2004) implicitly assumes a similar spectral type as this locus is based on observations of K7/M0 stars in the Taurus star formation region. The i − z colors for these early to mid M spectral types range from 0.38 to 0.80 with considerable scatter on the order of several tenths of magnitudes (West et al. 2004). Given the observed i − z (2.01±0.06), dereddening to these intrinsic colors requires A z = 3.1 m to 4.2 m for R V = 3.1 and 4.3 m to 5.7 m for R V = 5.5. Dereddening using the estimated A z ∼ 6.4 and R V = 3.1 results in an intrinsic i − z of -0.47, which is too blue for the stellar locus. Assumption of R V = 5.5 yields an i − z of 0.20, corresponding to a late K spectral type (Finlator et al. 2000). If V1647 Ori is indeed a low mass protostar with a late K or M spectral type than either R V is high or significant i band veiling is present in the quiescent state. In Figure 4 we compare the observed quiescent phase SED against an M0 photosphere seen under an extinction of A z =6.4 magnitudes. Excess emission is seen in the J, H, and K bands that could indicate the presence of a circumstellar disk. The Eruptive Phase The optical spectra acquired during the outburst by Walter et al. (2004) lack the TiO molecular absorption bands characteristic of late K and M stars. Walter et al. (2004) attribute this to either an early photospheric spectral type or overwhelming veiling. We adopt the latter interpretation given the apparent change to the intrinsic SED and the large brightness increase in the optical (Figure 3). If the new component to the SED is due to an EXor outburst then we expect to see emission from the high temperature (6000 -8000 K) inner disk (Bell et al. 1995). In Figure 5 we show the observed flux (νF ν ) increase in comparison with a 7000 K blackbody reddened by A z = 3.2 m and 6.4 m for R V = 3.1 and 5.5. In common with Walter et al. (2004) we see that a single EXor-like high temperature component can not reproduce the observed excess in both the optical and the near-IR. For the partial dust clearing events suggested by Reipurth & Aspin (2004) and Walter et al. (2004) we find that due to the increase in optical depth towards the blue the additional flux due to a new source must fall more rapidly at the shorter wavelengths. The observed change in J − H is -0.61 m , which if entirely due to dust clearing requires ∆A z = -3.2 m . The intrinsic flux increase required in this case is shown in Figure 5 and would require either a lower temperature for the new source or a significantly higher extinction than anticipated. Spectral indices are commonly defined by the ratio of the flux of the feature (F s ) and that of the nearby pseudo-continuum (F c ), so the intrinsic measure of the feature strength is I 0 = F s /F c (Gizis 1997). If a veiling continuum is present, defined by F v = rF c , then the measured spectral index, I, is I = F s + rF c F c + rF c = I 0 1 + r + r 1 + r . (1) Figure 6 shows the observed depression of the line relative to the continuum level (1 −I) on a logarithmic scale as a function of the veiling, r, for unveiled indices of I 0 = 0.0, 0.1, to 0.9. From Figure 3 we see that the flux in the i band has increased by ∆m = -5 or a factor of ∼100 which means the TiO features will be visible as less than a 1% fluctuation in the observed continuum. Conclusions The observed photometric variations are not consistent with a simple dust clearing event as evidenced by changes in reddening invariant indices. We infer that the process of eruption involves changes to both the optical and near-IR SED. Application of an inferred pre-outburst extinction of A z = 6.4 m suggests a late K spectral type if the r − i and i − z colors are minimally affected by the veiling continuum arising from magnetospheric accretion shocks and that R V has a value expected for a star formation region. An early to mid M spectral type is possible if the i band includes an excess emission of 0.2 m to 0.6 m , corresponding to a veiling between 0.2 and 0.7. We interpret the quiescent phase of V1647 Ori as an embedded low mass Classical T Tauri. Comparison with a reddened M0 photosphere shows a near-IR excess that is presumably due to thermal emission from a normal circumstellar disk. The outburst SED is dominated by the new component. We see that a single blackbody having the ∼7000 K temperature expected for an EXor inner disk can not simultaneously reproduce both the optical and near-IR portions of the SED. It is possible, as suggested by Reipurth & Aspin (2004), that a partial dust clearing event occurred in combination with an intrinsic brightening. We note that a reduction of the line of sight extinction would preferentially lessen the required flux increase at shorter wavelengths. This steepening of the curve shown in Figure 5 would increase the difficulty of fitting the high temperature (B spectral type) component observed by Briceño et al. (2004). Further study of V1647 Ori planned using the facilities at USNO, Apache Point Observatory, and the Spitzer Space Telescope may clarify the nature and evolutionary state of this young star. Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Finally we thank the referee, Colin Aspin, for his helpful comments. c Nσ is the ratio of color change to the quadrature sum of measurement errors. Fig. 1.-SDSS pre-eruption riz band mosaic image of V1647 Ori. The location of V1647 Ori is marked on this 2x2 binned mosaic which maps the SDSS r, i, and z bands onto blue, green, and red. Herbig-Haro objects, such as the large HH 24 complex (bottom center), are seen in blue due to the Hα emission appearing in the r band. The image is roughly 10 arcminutes on a side and is displayed using a negated grayscale. North is up and east is to the left. The faint emission immediately north of V1647 Ori significantly brightens during the eruptive phase where it is seen as McNeil's Nebula. The diffuse r band emission due to HH 23, which may be driven by V1647 Ori, is highlighted by the box in the upper center. Color R V = 3.1 R V = 5.5 C riz (r − i) − 0.987 * (i − z) (r − i) − 1.002 * (i − z) C izJ (i − z) − 0.824 * (z − J) (i − z) − 0.605 * (z − J) C zJH (z − J) − 2.443 * (J − H) (z − J) − 2.443 * (J − H) C JHK (J − H) − 1.563 * (H − K) (J − H) − 1.563 * (H − K) Facilities: SDSS, USNO, Gemini-N, 2MASS. Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 . 23456A color rendition of this image is available in the electronic version of the journal. -Quiescent and Eruptive Spectral Energy Distributions. The optical and near-IR SEDs are shown for the quiescent state (squares) and during eruption (circles). Data acquired by the SDSS and at the USNO are indicated by filled symbols. The open squares are the 2MASS JHK observations and the open circles are the r and i band eruptive phase measurements of Reipurth & Aspin (2004). -Changes in Spectral Energy Distributions. The observed magnitude differences (squares) are compared against pure dust clearing events for ∆A z = -2 m (circles) and ∆A z = -4 m (triangles) and for R V = 3.1 (solid) or R V = 5.5 (open) dust laws. -Modeling the Quiescent Spectral Energy Distribution. Observed preeruption SED compared with an M0 photosphere (3800 K) seen under a line of sight reddening of A z = 6.4 m for R V = 3.1 (solid line) and R V = 5.5 (dotted line). Excess emission is evident longward of 1.2 µm (J). -Modeling the Eruptive Component. This figure compares the observed flux increase (filled squares) against a 7000 K blackbody viewed under (R V =3.1 (solid), 5.5 (dotted)) extinctions of A z = 3.2 m and 6.4 m , scaled to match the observed change in the z band flux. The triangles show the required intrinsic flux increase in the case of a ∆A z = −3.2 m partial dust clearing for R V = 3.1 (filled) and 5.5 (open). -Effect of Veiling on Spectral Features. This figure plots the observed depression of an absorption line relative to the continuum level (1 − I) on a logarithmic scale as a function of the veiling, r, for unveiled indices of I 0 = 0.0, 0.1, to 0.9. The dashed lines indicate the eruptive phase i band veiling of 100 and a relative depression level of 1%. Table 1 . 1riz Photometry of V1647 Ori b The USNO 1.55m observations used standard K on 2004 February 11 and K ′ on 2004 April 12, both placed on the 2MASS zeropoint system, but probably with transformation differences.Table 4. Reddening Invariant ColorsDate r i z Telescope 1998 Nov 17 23.04 ± 0.22 20.81 ± 0.05 18.80 ± 0.04 SDSS 1998 Nov 28 24.00 ± 0.80 21.03 ± 0.09 19.19 ± 0.08 SDSS 2002 Feb 07 22.69 ± 0.29 20.77 ± 0.08 18.77 ± 0.06 SDSS 2002 Feb 09 23.88 ± 0.73 21.33 ± 0.11 19.33 ± 0.07 SDSS 2004 Feb 14 17.4 15.6 . . . Gemini a 2004 Feb 26 . . . . . . 14.60 ± 0.06 USNO 1.55m 2004 Apr 16 . . . . . . 14.79 ± 0.06 USNO 1.55m 2004 Apr 27 . . . . . . 14.72 ± 0.09 USNO 1.55m Table 5 . 5Reddening Invariant Color ChangesR V = 3.1 R V = 5.5 Color Quiescent a Eruptive b Nσ c Quiescent a Eruptive b Nσ c aThe quiescent state is defined by the 1998 October 7 2MASS and 1998 November 17 SDSS data. The eruptive phase data are from the 2004 February 14 Gemini r and i, the 2004 February 26 USNO z, and the 2004 February 11 USNO J, H, and K measurements.C riz 0.25±0.23 0.81±0.10 2.2 0.22±0.23 0.80±0.11 2.3 C izJ -1.34±0.08 -2.14±0.09 -6.7 -0.45±0.07 -1.31±0.09 -7.7 C zJH -2.24±0.12 -0.98±0.07 9.4 -2.24±0.12 -0.98±0.07 9.4 C JHK -0.37±0.07 0.23±0.03 8.0 -0.37±0.07 0.23±0.03 8.0 b a FromReipurth & Aspin (2004). http://arxiv.org/ps/astro-ph/0408308v1 . K Abazajian, AJ. 1262081Abazajian, K., et al. 2003, AJ, 126, 2081 . P Abraham, A&A. 41939Abraham, P. et al. 2004, A&A, 419L, 39 . 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[ "A model-independent analysis of final-state interactions inB 0 d/s → J/ψππ", "A model-independent analysis of final-state interactions inB 0 d/s → J/ψππ" ]	[ "J T Daub [email protected] \nHelmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n", "C Hanhart [email protected] ", "B Kubis [email protected] \nHelmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany\n", "\nInstitut für Kernphysik\nInstitute for Advanced Simulation\nJülich Center for Hadron Physics\nForschungszentrum Jülich\nD-52425JülichGermany\n" ]	[ "Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics\nUniversität Bonn\nD-53115BonnGermany", "Institut für Kernphysik\nInstitute for Advanced Simulation\nJülich Center for Hadron Physics\nForschungszentrum Jülich\nD-52425JülichGermany" ]	[]	Exploiting B-meson decays for Standard Model tests and beyond requires a precise understanding of the strong final-state interactions that can be provided modelindependently by means of dispersion theory. This formalism allows one to deduce the universal pion-pion final-state interactions from the accurately known ππ phase shifts and, in the scalar sector, a coupled-channel treatment with the kaon-antikaon system. In this work an analysis of the decaysB 0 d → J/ψπ + π − andB 0 s → J/ψπ + π − is presented. We find very good agreement with the data up to 1.05 GeV in the ππ invariant mass, with a number of parameters reduced significantly compared to a phenomenological analysis. In addition, the phases of the amplitudes are correct by construction, a crucial feature for many CP violation measurements in heavy-meson decays.	10.1007/jhep02(2016)009	[ "https://arxiv.org/pdf/1508.06841v2.pdf" ]	119,197,216	1508.06841	d690076c2054c9c1ccbdf572944a1fafa5801259	A model-independent analysis of final-state interactions inB 0 d/s → J/ψππ 28 Jan 2016 J T Daub [email protected] Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany C Hanhart [email protected] B Kubis [email protected] Helmholtz-Institut für Strahlen-und Kernphysik (Theorie) and Bethe Center for Theoretical Physics Universität Bonn D-53115BonnGermany Institut für Kernphysik Institute for Advanced Simulation Jülich Center for Hadron Physics Forschungszentrum Jülich D-52425JülichGermany A model-independent analysis of final-state interactions inB 0 d/s → J/ψππ 28 Jan 2016 Exploiting B-meson decays for Standard Model tests and beyond requires a precise understanding of the strong final-state interactions that can be provided modelindependently by means of dispersion theory. This formalism allows one to deduce the universal pion-pion final-state interactions from the accurately known ππ phase shifts and, in the scalar sector, a coupled-channel treatment with the kaon-antikaon system. In this work an analysis of the decaysB 0 d → J/ψπ + π − andB 0 s → J/ψπ + π − is presented. We find very good agreement with the data up to 1.05 GeV in the ππ invariant mass, with a number of parameters reduced significantly compared to a phenomenological analysis. In addition, the phases of the amplitudes are correct by construction, a crucial feature for many CP violation measurements in heavy-meson decays. Introduction B-meson decays can be exploited for Standard Model tests and beyond, in particular to determine the Cabibbo-Kobayashi-Maskawa (CKM) couplings and to study CP violation. For a theoretical description of many of these decays, it is mandatory to understand the strong final-state interactions in terms of amplitude analysis techniques [1], with tight control over the magnitudes and phase motions of the various partial waves involved. For example, the decays B → f 0 (980)K S and B → φ(1020)K S are explored for an experimental determination of the CP asymmetry sin 2β [2][3][4][5], β being one of the angles of the unitarity triangle, which requires precise knowledge of the strange and non-strange scalar form factors that we discuss in this article. We focus on the decaysB 0 d → J/ψπ + π − and B 0 s → J/ψπ + π − , measured by the LHCb collaboration [6,7]. The tree-level process of the weak decay into J/ψ and a qq pair is depicted in Fig. 1 (exemplarily for theB 0 s decay). These analyses complement former related studies ofB 0 d andB 0 s decays by the BaBar [8], Belle [9], CDF [10], and D0 [11] Collaborations as well as older LHCb results [12,13]. Universality of final-state interactions dictates that the hadronization into pions and the rescattering effects in the π + π − system for S-and P -waves are closely related to the scalar Figure 1. TheB 0 s → J/ψπ + π − diagram to leading order via W − exchange. The hadronization into pions (S-wave dominated) proceeds through the pion strange scalar form factor Γ s π (s). In the case of theB 0 d → J/ψπ + π − decay, with s ↔ d, the pions are generated out of a non-strange scalar source, i.e. Γ s π (s) is replaced by the pion non-strange scalar form factor Γ n π (s) for S-wave and by the vector form factor for P -wave pions. and vector pion form factors, respectively. We describe these form factors using dispersion theory, using Omnès (or Muskhelishvili-Omnès) representations. In doing so we exploit the fact that LHCb found no obvious structures in the J/ψπ + invariant mass distribution, suggesting that left-hand-cut contributions in the π + π − system due to the crossed-channel J/ψπ + interaction are small and can be neglected. The advantage of the dispersive framework is that all constraints imposed by analyticity (i.e., causality) and unitarity (probability conservation) are fulfilled by construction. Further, it is a model-independent approach, so we do not have to specify any contributing resonances or conceivable non-resonant backgrounds. For the vector form factor a singlechannel (elastic) treatment works very well below 1 GeV. In the scalar sector the strong coupling of two S-wave pions to KK near 1 GeV due to the f 0 (980) resonance, causing a sharp onset of the KK inelasticity, necessitates a coupled-channel treatment. Therefore a two-channel Muskhelishvili-Omnès problem is solved. This two-channel approach breaks down at energies where inelasticities caused by 4π states become important, we are thus not able to cover the complete phase space, but restrict ourselves to the low-energy range √ s ≤ 1.05 GeV. In Ref. [6] theB 0 d decay is described by six resonances in the π + π − channel, f 0 (500), ρ(770), ω(782), ρ(1450), ρ(1700), and f 2 (1270), which are modeled by Breit-Wigner functions. This parametrization of especially the f 0 (500) meson is somewhat precarious, as the broad bump structure of this scalar resonance is not well described by a Breit-Wigner shape. As demonstrated for the first time in the context of B decays in Ref. [14], it should be replaced by the corresponding scalar form factor. In the present work this idea is extended and rigorously applied using form factors derived from dispersion theory. In particular, there is no need to parametrize any resonance, since the input required to describe the final-state interactions is taken from known phase shifts, and therefore the f 0 (500) appears naturally in the non-strange scalar form factor. TheB 0 s decay, described in the experimental analysis by five resonances, f 0 (980), f 0 (1500), f 0 (1790), f 2 (1270), and f ′ 2 (1525) (Solution I) or with an additional non-resonant contribution (Solution II), dominantly occurs in an S-wave state [7], while the P -wave is shown to be negligible. Given the almost puress source the pions are generated from, this decay shows great promise to provide insight into the strange scalar form factor. The idea of such a "scalar-source model", where an S-wave pion pair is generated out of a quark-antiquark pair and the final-state interactions are described by the scalar form factor, is also used in Ref. [15] for the description of theB 0 s andB 0 d decays into the scalar resonances f 0 (980) and f 0 (500), respectively. It was employed earlier e.g. in analyses of the decay of the J/ψ into a vector meson (ω or φ) and a pair of pseudoscalars (ππ or KK) [16,17]. In these references the strong-interaction part is described by a chiral unitary theory including coupled channels, which yields a dynamical generation of the scalar mesons. In contrast to the present study, the very precise information available on pion-pion [18][19][20][21] and pion-kaon [22] phase shifts is not strictly implemented there. Related studies using the chiral unitary approach are performed in Ref. [23], where the J/ψ-vectormeson final state is analyzed, and in Ref. [24], which includes resonances beyond 1 GeV. In contrast to models of dynamical resonance generation, the scalar resonances are considered as qq or tetraquark states in Ref. [25]. Other theoretical approaches employ light-cone QCD sum rules to describe the form factors [26]. Progress on the short-distance level is made in Ref. [27], where the factorization formulae (which we treat in a naive way) are improved in a perturbative QCD framework. This manuscript is organized as follows. In Sec. 2, we review the construction of the transversity amplitudes and partial waves, after sketching the kinematics. We provide explicit expressions that relate the theoretical quantities to the angular moments determined in experiment. Section 3 is focused on the Omnès formalism. The fits to the LHCb data, using theB 0 d/s → J/ψπ + π − angular moment distributions, are discussed in Sec. 4, where we use several configurations with and without D-wave corrections to study the impact of certain corrections to our fits. We also predict the S-wave amplitude for the related B 0 s → J/ψK + K − decay. The paper ends with a summary and an outlook in Sec. 5. Some technical details are relegated to the appendices. Kinematics, decay rate, and angular moments In this section we derive the decay rate and angular moments for theB 0 d → J/ψπ + π − decay mode in terms of partial-wave amplitudes up to D-waves, employing the transversity formalism of Ref. [28]. The formalism works analogously for theB 0 s decay. Kinematics The kinematics of the decayB 0 d/s (p B ) → J/ψ(p ψ )π + (p 1 )π − (p 2 ) (J/ψ → µ + µ − ) can be described by four variables: • the invariant dimeson mass squared, s = (p 1 + p 2 ) 2 , and three helicity angles, see Fig. 2, • θ J/ψ , the angle between the µ + in the J/ψ rest frame (Σ J/ψ ) and the J/ψ in theB 0 • θ π , the angle between the π + in the π + π − center-of-mass frame Σ ππ and the dipion line-of-flight in Σ B ; d/s rest frame (Σ B ); B 0 d/s π + π − plane µ + µ − plane π + π − µ + µ − φ θ π θ J/ψ π + π − J/ψ • φ, the angle between the dipion and the dimuon planes, where the latter originate from the decay of the J/ψ. The three-momenta of either of the two pions in the dipion center-of-mass system (p π ) and of the J/ψ in theB 0 d/s rest frame (p ψ ) are given by \|p π \| = λ 1/2 (s, M 2 π , M 2 π ) 2 √ s ≡ σ π √ s 2 , \|p ψ \| = λ 1/2 (s, m 2 ψ , m 2 B ) 2m B ≡ X m B , (2.1) with the Källén function λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc. We define the two remaining Mandelstam variables as t = (p B − p 1 ) 2 and u = (p B − p 2 ) 2 , (2.2) where the difference of these two determines the scattering angle θ π , t − u = −2p ψ (p 1 − p 2 ) = −2σ π X cos θ π . (2.3) Further, we introduce two additional vectors as combinations of the above four-momenta, P µ = p µ 1 + p µ 2 , Q µ = p µ 1 − p µ 2 . (2.4) Matrix element To calculate the matrix element we make use of the effective Hamiltonian that governs the b → ccd transition [29], H eff = G F √ 2 V cb V * cd [C 1 (µ)O 1 (µ) + C 2 (µ)O 2 (µ)] + . . . ,(2.5) where the C i are Wilson coefficients and the O i local current-current operators O 1 =c k γ µ (1 − γ 5 )b ldl γ µ (1 − γ 5 )c k =c k γ µ (1 − γ 5 )c kdl γ µ (1 − γ 5 )b l , O 2 =c k γ µ (1 − γ 5 )b kdl γ µ (1 − γ 5 )c l =c k γ µ (1 − γ 5 )c ldl γ µ (1 − γ 5 )b k ,(2.6) with k, l being color indices. In the second step the quark operators are regrouped by means of a Fierz rearrangement. The ellipses in H eff denote operators beyond tree-level, including penguin topologies. V cb and V cd are the CKM matrix elements for c → b and c → d (where V cd is to be replaced by V cs for theB 0 s decay), and G F = 1.166365 × 10 −5 GeV −2 is the Fermi constant. Under the assumption that the final-state interaction between the J/ψ and the pions is negligible (no obvious structures are found in the J/ψπ channel experimentally [6,7], and a close-to-zero J/ψπ scattering length a J/ψπ = −0.01(1) fm results from lattice calculations [30]) a factorization approach appears to be justified. Note that on the quark level this naive factorization ansatz may be spoiled [31,32], for instance due to (large) penguin contributions that we have neglected in Eq. (2.5) [33,34]. However, a more complicated structure of the source term does not conflict with our approach: any factorization limitations due to color structures do not concern the hadronic final-state interaction, for which the short-distance factorizations are sufficient but not mandatory. All we use is the fact that the B decays provide cleanqq sources of much shorter range than that of the final-state interaction. In our approach, any deviations from clean point sources would be parametrized by derivatives of the source term. An excellent fit to the data even without those correction terms is a proof that with respect to the final-state interactions the sources can be regarded as point-like. We express the matrix elements of the four-quark operators by two independent hadronic currents, valid if the cc system produced by the hadronization of the virtual W − is well separated from the spectator quark system. For the decay of theB 0 d meson considered here the matrix element is, in analogy to theB 0 s expression given in Ref. [35], written as M f i = G F √ 2 V cb V * cd a eff (µ) π + (p 1 )π − (p 2 )\|dγ µ (1 − γ 5 )b\|B 0 d/s (p B ) M ππ µ × J/ψ(p ψ , ǫ)\|cγ µ c\|0 M cc µ , M ππ µ = M ψ P µ (0) X F 0 + Q µ ( ) √ s F − ip µ ⊥ √ s F ⊥ , M cc µ = f ψ M ψ ǫ * µ (p ψ , λ),(2.7) with a eff = C 1 (µ)+C 2 (µ)/N c +. . . , the ellipses denoting combinations of Wilson coefficients due to penguin diagrams, we have not taken into account explicitly. The scale (µ) dependence of the Wilson coefficients is cancelled by the scale dependence of the hadronic matrix elements, cf. Sec. 3; µ is chosen to be of order O(m B ), such that heavier particles, in particular the W , are integrated out. The current that creates the J/ψ from the vacuum is related to the decay constant f ψ . The matrix element containing the pions is given by the three transversity form factors F 0 , F , and F ⊥ , corresponding to the orthogonal basis of momentum vectors [28] p µ ψ ,p µ (⊥) = ǫ µαβγ X (p ψ ) α P β Q γ , Q µ ( ) = Q µ − (P · p ψ )(Q · p ψ ) X 2 P µ + s(Q · p ψ ) X 2 p µ ψ . (2.8) We define ǫ µνρσ such that ǫ 0123 = −ǫ 0123 = +1. -5 - The partial-wave expansions of the transversity form factors read 1 F 0 (s, θ π ) = ℓ √ 2ℓ + 1 F (ℓ) 0 (s)P ℓ (cos θ π ) = F (S) 0 (s) + √ 3 cos θ π F (P ) 0 (s) + √ 5 2 3 cos 2 θ π − 1 F (D) 0 (s) + . . . , F ,⊥ (s, θ π ) = ℓ √ 2ℓ + 1 ℓ(ℓ + 1) F (ℓ) ,⊥ (s)P ′ ℓ (cos θ π ) = 3 2 F (P ) ,⊥ (s) + √ 5 cos θ π F (D) ,⊥ (s) + . . . , (2.9) where the ellipses denote waves larger than D-waves. In Appendix A the relation to the helicity form factors is briefly sketched, which have a well-known partial-wave expansion. Decay rate and angular moments When comparing the angular moments to the experimental data we have to deal with flavoraveraged expressions due to the B 0 -B 0 mixing and take into account the CP -conjugated amplitudes (the B 0 d decay mode) as well. Since the interfering term between the amplitudes is negligibly small [6], the decay rate can be written as the sum of the decay rates for the directB 0 d and the mixed CP -conjugated B 0 d mode, d 2 Γ B 0 d → J/ψπ + π − d √ s d cos θ π ≈ d 2 Γ (direct) d √ s d cos θ π + d 2 Γ B 0 d → J/ψπ + π − d √ s d cos θ π . (2.10) Note that this neglect is less justified when applying the formulae to theB 0 s decay rate. In the analysis of Ref. [7] an interference term is added to Eq. (2.10). However, in Sec. 4.2 we find that it is sufficient to take into account S-waves. In that case the interference term does not affect the fit procedure and merely generates a tiny shift of the resulting fit parameter (the normalization c s 0 ). In this section we provide expressions for one particular mode. The CP -related amplitude can be deduced straightforwardly by multiplying the transversity partial-wave amplitudes with CP eigenvalues as outlined in detail below (cf. the discussion around Eq. (2.15)). The differential decay rate is given by d 2 Γ d √ s d cos θ π = G 2 F \|V cb \| 2 \|V cd \| 2 f 2 ψ M 2 ψ Xσ π √ s 4(4π) 3 m 3 B × F (S) 0 (s) + √ 3 cos θ π F (P ) 0 (s) + √ 5 2 3 cos 2 θ π − 1 F (D) 0 (s) 2 + 3 2 σ 2 π sin 2 θ π F (P ) + √ 5 cos θ π F (D) (s) 2 + F (P ) ⊥ + √ 5 cos θ π F (D) ⊥ (s) 2 , (2.11) see Appendix A for details. By weighting this decay rate by spherical harmonic functions Y 0 l (cos θ π ), we define the angular moments Y 0 l (s) = 1 −1 d 2 Γ d √ s d cos θ π Y 0 l (cos θ π )d cos θ π . (2.12) With the orthogonality property 1 −1 Y 0 i (cos θ π )Y 0 j (cos θ π )d cos θ π = δ ij 2π , (2.13) we obtain √ 4π Y 0 0 = G 2 F \|V cb \| 2 \|V cd \| 2 f 2 ψ M 2 ψ Xσ π √ s 2(4π) 3 m 3 B F (S) 0 2 + F (P ) 0 2 + F (D) 0 2 + σ 2 π F (P ) 2 + F (P ) ⊥ 2 + F (D) 2 + F (D) ⊥ 2 , √ 4π Y 0 1 = G 2 F \|V cb \| 2 \|V cd \| 2 f 2 ψ M 2 ψ Xσ π √ s (4π) 3 m 3 B Re F (S) 0 F (P ) * 0 + 2 √ 5 Re F (P ) 0 F (D) * 0 + 3 5 σ 2 π Re F (P ) ⊥ F (D) * ⊥ + Re F (P ) F (D) * , √ 4π Y 0 2 = G 2 F \|V cb \| 2 \|V cd \| 2 f 2 ψ M 2 ψ Xσ π √ s (4π) 3 m 3 B Re F (S) 0 F (D) * 0 + 1 √ 5 F (P ) 0 2 + √ 5 7 F (D) 0 2 − σ 2 π 2 √ 5 F (P ) 2 + F (P ) ⊥ 2 + σ 2 π √ 5 14 F (D) 2 + F (D) ⊥ 2 , (2.14) where Y 0 0 corresponds to the event distribution, Y 0 1 describes the interference between S-and P -wave as well as P -and D-wave amplitudes, and Y 0 2 contains P -wave, D-wave, and S-D-wave interference contributions. The corresponding expressions for the CP -conjugated modes are related to the above equations by certain sign changes due to the CP eigenvalues η CP = ±1 in the definitions of the transversity partial-wave amplitudes, as already mentioned in the beginning of this section. We declare the amplitudes F (ℓ) τ to describe the B 0 d decay, then the correspondinḡ B 0 d decay amplitudes are given byF (ℓ) τ = η CP F (ℓ) τ ,(2.15) with η CP = +1 for the τ = 0, P -waves and the τ =⊥ D-wave, and η CP = −1 otherwise. Consequently the angular moments Y 0 0 and Y 0 2 are unchanged under CP conjugation, while the conjugated moment Y 0 1 has opposite sign, such that when considering flavoraveraged quantities and summing over the B 0 d andB 0 d contributions, Y 0 1 vanishes. In the following we thus consider Y 0 0 and Y 0 2 only. Omnès formalism We describe the S-and P -wave amplitudes using dispersion theory. This approach allows us to treat the pion-pion rescattering effects in a model-independent way, based on the fundamental principles of unitarity and analyticity: the partial waves are analytic functions in the whole s-plane except for a branch-cut structure dictated by unitarity. In the following we deal with the functions f I ℓ (s) (referring to isospin I and angular momentum ℓ) that possess a right-hand cut starting at the pion-pion threshold s thr = 4M 2 π and are analytic elsewhere, i.e. we do not consider any left-hand-cut or pole structure related to crossing symmetry. This is justified from the observation that there are practically no structures observed for the crossed J/ψπ + channel in the region of interest [6]. Considering two-pion intermediate states only, Watson's theorem holds, i.e. the phase of the partial wave is given by the elastic pion-pion phase shift [36], and the discontinuity across the cut can be written as discf I ℓ (s) = f I ℓ (s + iǫ) − f I ℓ (s − iǫ) = 2iσ π f I ℓ (s) t I ℓ (s) * = f I ℓ (s)e −iδ I ℓ sin δ I ℓ . (3.1) A solution of this unitarity relation can be constructed analytically, setting (compare Ref. [37]) f I ℓ (s) = P (s)Ω I ℓ (s), (3.2) where P (s) is a polynomial not fixed by unitarity, and the Omnès function Ω I ℓ (s) is entirely determined by the phase shift δ I ℓ (s) [38], Ω I ℓ (s) = exp s π ∞ s thr δ I ℓ (s ′ ) s ′ (s ′ − s − iǫ) ds ′ ,(3.3) with Ω I ℓ (0) = 1 and Ω I ℓ (s) = 0 ∀ s. (3.4) The P -wave amplitudes can be well described in the elastic approximation up to energies of roughly 1 GeV. 2 The simplest possible application is the pion vector form factor F V π (s), 0\|j µ em (0)\|π + (p 1 )π − (p 2 ) = (p 2 − p 1 ) µ F V π (s), j µ em = 2 3ū γ µ u − 1 3d γ µ d, (3.5) which obeys a representation like (3.2) with a linear polynomial P F V π (s) = 1 + αs, α ≈ 0.1 GeV −2 [39] up to √ s ≈ 1 GeV, with the exception of a small energy region around the ω resonance that couples to the two-pion channel via isospin-violating interactions. In this context it is important to note that the electromagnetic current j µ em , introduced in Eq. (3.5), can be decomposed as j µ em = 1 2 ūγ µ u −dγ µ d + 1 6 ūγ µ u +dγ µ d . (3.6) Thus it contains with the first term an isovector and with the second term an isoscalar component. The latter couples directly to the ω, whose decay into π + π − is suppressed by isospin, but enhanced by a small energy denominator (i.e., the small width of the ω), hence leading to a clearly observable effect in the pion form factor [40][41][42]. Theoretically, this effect is correctly taken into account by the replacement [43][44][45] P F V π (s)Ω 1 1 (s) −→ P F V π (s)Ω 1 1 (s) 1 + κ em s M 2 ω − iM ω Γ ω − s . (3.7) Note that in case of the ω the use of a Breit-Wigner parametrization is appropriate since the ω pole is located far above the relevant decay thresholds and since Γ ω = 8.5 MeV is very small. A fit of the form factor parametrization introduced in Eq. (3.7) to the KLOE data [41] yields κ em ≈ 1.8 × 10 −3 . This fixes the strength of the so-called ρ-ω mixing amplitude phenomenologically. The isospin-violating coupling κ em is of the usual size, however, near the ω peak its smallness is balanced by the factor M ω /Γ ω ≈ 90 from the ω propagator, giving rise to an isospin-violating correction as large as 15% on the amplitude level, corresponding to 30% in observables due to interference with the leading term. Note also that the ρ-ω mixing amplitude has been pointed out to significantly enhance certain CP -violating asymmetries in hadronic B-meson decays [46]. The effect of the ω on theB 0 d → J/ψπ + π − decay can be related straightforwardly to that on the pion vector form factor. To see this observe that the source term for the ππ system isdd at tree level, see Fig. 1, such that the isospin decomposition of the corresponding vector current readsd γ µ d = − 1 2 ūγ µ u −dγ µ d + 1 2 ūγ µ u +dγ µ d . (3.8) Comparison to Eq. (3.6) shows that the relative strength of the isoscalar component differs from the electromagnetic current by a factor of −3, such that we will fix the ρ-ω mixing contribution in analogy to Eq. (3.7), but with the replacement κ em → κ = −3κ em ≈ −5.4 × 10 −3 . Notice that this is in contrast with the experimental analysis [6], where the ω contribution is fitted with free coupling constants. The (elastic) single-channel treatment, introduced in the beginning of this section, cannot be used in the S-wave case: there are strong inelastic effects in the region around 1 GeV due to the opening of the KK channel, coinciding with the f 0 (980) resonance, which affects the phase of the scalar pion form factors (see e.g. the discussion in Ref. [47]). Thus the Omnès problem has to be generalized, with the Watson theorem fulfilled in the elastic region and inelastic effects included above the KK threshold. This leads to the two-channel Muskhelishvili-Omnès equations that intertwine the pion and kaon form factors, defined as π + (p 1 )π − (p 2 ) \|qq\| 0 = B q Γ q π (s), K + (p 1 )K − (p 2 ) \|qq\| 0 = B q Γ q K (s),(3.9) where the quark flavors may be eitherqq = (ūu+dd)/2 for the light quarks, with the superscript q = n denoting the corresponding scalar form factor, orqq =ss for strange quarks (with superscript q = s). Furthermore, B n = M 2 π /(m u + m d ), B s = (2M 2 K − M 2 π )/(2m s ) . Note that the form factors Γ q π,K (s) are invariant under the QCD renormalization group, while the hadronic matrix elements are not due to the scale dependence inherent in the factors B q . This in turn allows for the cancellation of the scale dependence in the Wilson coefficients introduced in the effective Hamiltonian of Sec. 2. Appealing to the tree-level diagram of Fig. 1, we expect the non-strange scalar form factors to contribute dominantly in theB 0 d decay, while the strange ones should feature mainly in the corresponding decay of theB 0 s . As discussed in detail below, these expectations are confirmed by the data analysis. The Muskhelishvili-Omnès formalism is briefly reviewed in Appendix B. It requires three input functions: in addition to the ππ phase shift already necessary in the elastic case, modulus and phase of the ππ → KK S-wave amplitude also need to be known. Our main solution is based on the Roy equation analysis by the Bern group [20,21] for the ππ phase shift, the modulus of the ππ → KK S-wave as obtained from the solution of Roy-Steiner equations for πK scattering performed in Orsay [22], and its phase from partial-wave analyses [48,49]. Alternatively, we employ the T -matrix constructed by Dai and Pennington (DP) in Ref. [50]: here, a coupled-channel K-matrix parametrization is fitted to ππ data [51][52][53][54][55], and the Madrid-Kraków Roy-equation analysis [19] is used as input; furthermore, the KK threshold region is improved by fitting also to Dalitz plot analyses of D + s → π + π − π + [56] and D + s → K + K − π + [57] by the BaBar Collaboration. In addition, the channel coupling manifests itself through the fact that even in the simplest case, corresponding to the polynomial of Eq. (3.2) reducing to a constant, the scalar form factors depend on two such constants, corresponding to the form factor normalizations for both pion and kaon. In contrast to the single-channel case, here the shape of the resulting form factors depends on the relative size of these two normalization constants; on the other hand, once this relative strength is fixed, it relates the final states ππ and KK to each other unambiguously. We will make use of this additional predictiveness in Sec. 4.3. In order to apply this formalism to the transversity partial waves we have to construct partial waves f (ℓ) τ (s) that are free of kinematical singularities, i.e. represented by functions whose only non-analytic behavior is related to unitarity. In Appendix A the hadronic matrix element is introduced (using the basis of the momenta p µ ψ , P µ , and Q µ , Eq. (2.4)) in terms of the form factors A i and V, Eq. (A.1), and related to the transversity basis, Eq. (A.5). Given that the form factors A i and V are regular, Eq. (A.5) implies that there are additional factors of X, σ π , and √ s introduced into the transversity form factors, which give rise to artificial branch cuts in the unphysical region. To avoid those, we write the partial waves as F (S) 0 (s) = Xf (S) 0 (s), F (P ) 0 (s) = σ π f (P ) 0 (s), F (P ) (s) = √ sf (P ) (s), F (P ) ⊥ (s) = √ sXf (P ) ⊥ (s),(3.10) where the f For the S-wave, we a priori allow for contributions of both non-strange (n) and strange (s) scalar form factors. The coefficients of the polynomials P (ℓ) τ (s) are to be determined from a fit to the efficiency-corrected and background-subtracted LHCb data, in particular to the angular moments Y 0 0 and Y 0 2 . Basically we assume the various polynomials to be well approximated by constants. However, to study the impact of a linear correction at a later stage, we also consider linear polynomials P (S,n) 0 = b n 0 (1 + b ′n 0 s) and P (P ) τ = a τ (1 + a ′ τ s) for the non-strange S-wave and the P -wave amplitudes, respectively. The strange S-wave contribution is expected to be very small (in the LHCb analysis ofB 0 d → J/ψπ + π − the f 0 (980) meson is not seen), but tested in the fits. On the contrary, theB 0 s → J/ψπ + π − distribution is dominated by the f 0 (980) resonance, described by a constant polynomial times Omnès function, P (S,s) 0 = c s 0 , while there is no structure in the f 0 (500) region reported by LHCb. Thus in that case the non-strange S-wave amplitude is assumed to be negligible, to be confirmed in the fits. Although the first D-wave resonance seen is the f 2 (1270), it may affect also the region below √ s ≈ 1 GeV due to its finite width, Γ f 2 = 185.1 +2.9 −2.4 MeV [58]. Therefore we also test its influence on the fit. The D-waves could be treated in the same dispersive way as S-and P -waves, but this would increase the number of free parameters in our fits to the LHCb data. As the effect of D-wave corrections is rather small, we avoid introducing additional fit parameters and take over the amplitudes (with fixed couplings) used in the LHCb analysis, where the f 2 (1270) resonance is modeled by a Breit-Wigner shape. Since the data are given in arbitrary units, we collect all prefactors in normalizations that we subsume into the fit parameters (and into the transversity coefficients α f 2 τ that we extract from the LHCb fit results). Writing Y 0 i in terms of Omnès functions for S-and P -waves, supplemented by the D-wave resonance contribution, yields √ 4π Y 0 0 = Xσ π √ s X 2 b n 0 (1 + b ′n 0 s)Γ n π (s) + c s 0 Γ s π (s) 2 + σ 2 π Ω 1 1 (s) 2 a 0 (1 + a ′ 0 s) 2 + s a (1 + a ′ s) 2 + sX 2 a ⊥ (1 + a ′ ⊥ s) 2 + τ =0,⊥, α f 2 τ e iφ f 2 τ A (τ ) f 2 (s) 2 , √ 4π Y 0 2 = Xσ π √ s 2Re X b n 0 (1 + b ′n 0 s)Γ n π (s) + c s 0 Γ s π (s) α f 2 0 e iφ f 2 0 A (0) f 2 (s) * + σ 2 π √ 5 Ω 1 1 (s) 2 2 a 0 (1 + a ′ 0 s) 2 − s a (1 + a ′ s) 2 − sX 2 a ⊥ (1 + a ′ ⊥ s) 2 + √ 5 7 2 α f 2 0 e iφ f 2 0 A (0) f 2 (s) 2 + τ = ,⊥ α f 2 τ e iφ f 2 τ A (τ ) f 2 (s) 2 . (3.12) For details concerning the definition of the Breit-Wigner amplitudes A and f (P ) (a 0 , a ). (We find that including the τ =⊥ P -wave amplitude practically does not change the χ 2 , i.e. a ⊥ is a redundant parameter.) In the basic fit only S-and P -waves are considered. Beyond that, we study the relevance of certain corrections: in FIT II we use again the same three parameters as in FIT I, but in addition we include the D-wave contributions, fixed to their strengths as determined by LHCb. To further improve FIT II, supplemental linear terms (b ′ 0 , a ′ 0 , a ′ -cf. Eq. (3.12)) are allowed in FIT III. Performing FIT III we find that two of the slope parameters, the linear non-strange S-wave term (b ′ 0 ) and the τ = P -wave slope (a ′ ), yield no significant improvement of the fits; their values are compatible with zero within uncertainties. We thus fix them to zero, and in FIT III only the four parameters b n 0 , a 0 , a , and a ′ 0 are varied. Furthermore, the effect of an inclusion of a strange S-wave component is tested. Its strength is found to be compatible with zero, justifying its omission. Note that the scalar pion form factors depend on the normalizations of both the pion and kaon form factors. While the normalizations in the case of the pion form factor are known quite precisely, there are considerable uncertainties for the kaon form factor normalizations, having an impact on the shapes of both pion form factors, see Appendix B. The non-strange kaon normalization Γ n K (0) is limited to the range (0.4 . . . 0.6). In our fits we fix the value to Γ n K (0) = 0.5, which is compatible with the current algebra result. The effect from a variation of Γ n K (0) in the allowed interval shows up only in the second decimal place of the χ 2 /ndf. The fitted coefficients and the resulting χ 2 /ndf, referring to Eq. (3.12), are listed in Table 1. The large uncertainties can be traced back to the correlations between the fit parameters, especially present in FIT III. For a comparison to the LHCb fit, we insert their fit results (best model) into our definition of the χ 2 . In more specific terms this means that we do not compare to the χ 2 published in Ref. [6], for which the full energy range up to √ s = 2.1 GeV is fitted with 34 parameters and the data of all angular moments Y 0 i for i = 0, . . . , 5 are included, but we calculate the χ 2 in the region we use in our fits, i.e. including data up to √ s = 1.02 GeV and the angular moments Y 0 0 and Y 0 2 only. We obtain χ 2 LHCb /ndf = 2.08. In this limited energy range the Breit-Wigner description, including the f 0 (500), ρ(770) and ω(782), requires 14 fit constants, while we have three (FIT I, II) or four (FIT III) free parameters and find χ 2 /ndf = 2.0 (FIT I), χ 2 /ndf = 1.5 (FIT II) and χ 2 /ndf = 1.3 (FIT III). The calculated angular moments for the three fit models in comparison to the data are shown in Fig. 3. Probably the most striking feature of our solution is the pronounced effect of the ω that leads to the higher peak in Fig. 3. As mentioned above, this isospin-violating contribution is fixed completely from an analysis of the pion vector form factor, however, its appearance here is utterly different, since the coupling strength is multiplied by a factor of −3. This not only enhances the impact of the ω on the amplitude level to about 50%, but also implies that the change in phase of the signal is visible a lot more clearly: while in case of the vector form factor the ω amplitude leads to an enhancement on the ρ-peak and some depletion on the right wing, forming a moderate distortion of the line shape, here we obtain a depletion on the ρ-peak accompanied by an enhancement on the right wing. While the current data do not show the ω peak clearly, a small shape variation due to the ρ-ω interference is better seen in Ref. [33], where a finer binning is used. The ρ-ω mixing strength obtained from a fit in that reference is consistent with the strength we obtain in a parameter-free manner. Nonetheless, improved experimental data are called for, since an experimental confirmation of the ω effect onB 0 d → J/ψπ + π − would allow one to establish that theB 0 d decay indeed provides a rather cleandd source. A key feature of the formalism employed here is its correct description of the S-wave. Figure 4 shows the comparison of the S-wave amplitude strength of the LHCb Breit-Wigner parametrization with the ones obtained in FIT I-III, as well as the comparison of the corresponding phases. In the elastic region, the phase of the non-strange scalar form factor δ Γ n = arg(Γ n π ) coincides with the ππ phase shift δ 0 0 that we use as input for the Omnès matrix, in accordance with Watson's theorem. Right above the KK threshold, δ Γ n drops quickly, which causes the dip in the region of the f 0 (980), visible in the modulus of the amplitudes as well as the non-Breit-Wigner bump structure in the f 0 (500) region. We find that the phase due to a Breit-Wigner parametrization largely differs from the dispersive solution, indicating that parametrizations of such kind are not well suited for studies of CP violation in heavy-meson decays. χ 2 /ndf \|b n 0 \| \|a 0 \| \|a \| a ′ 0 Note that in the analysis of Ref. [33] the f 0 (500) is modeled not by a Breit-Wigner function, but by the theoretically better motivated parametrization of Ref. [59]. In this work, higher resonances are included by multiplying S-matrix elements. While this procedure preserves unitarity, it produces terms at odds with any microscopic description of the coupled ππ-KK system. As such also this approach introduces uncontrolled theoretical uncertainties into the analysis. The only stringently model-independent way to include hadronic final-state interactions is via dispersion theory. 4.2B 0 s → J/ψπ + π − TheB 0 s → J/ψπ + π − distribution in the region up to roughly 1 GeV is clearly dominated by the f 0 (980). We therefore describe the data with the strange S-wave component only, using a constant subtraction polynomial (c s 0 ). The only non-zero contribution to the fit thus comes from Y 0 0 . Fitting the data up to √ s = 1.05 (1.02) GeV yields χ 2 /ndf = 2.2 (1.8) and c s 0 = 16.8±0.4 (16.8±0.4). In analogy to theB 0 d decay we also perform the fit including the D-wave parametrization of the LHCb analysis. This yields an additional non-zero contribution to Y 0 2 due to the S-D-wave interference, which is fitted simultaneously with Y 0 0 . Further, the influence of a linear subtraction polynomial for the strange S-wave is tested. However, none of these corrections exhibits a considerable improvement. In the LHCb analysis the full energy range, √ s ≤ 2.1 GeV, is fitted with 22 (24) parameters for Solution I (II). Confining to the region we examine in our fit and considering the f 0 (980) resonance only, the number of fit parameters reduces to four (six), and we calculate χ 2 LHCb /ndf = 0.76 (0.82), when using our definition of the χ 2 . Figure 5. Left panel: Y 0 0 fitted using the strange S-wave with constant subtraction polynomial for two different phase inputs (red, solid: B+O input [20][21][22], green, dotted: DP input [50], based on the Madrid-Kraków analysis [19]). Right panel: comparison of the phase of the strange scalar pion form factor for the B+O (blue, dashed) and DP (red, solid) input, respectively, with the S-wave phase extracted from the LHCb analysis (Solution I and II, shown with error bands). The strange scalar form factor, or the f 0 (980) peak in the dispersive formalism, depends crucially on the ππ → KK S-wave transition amplitude, which is not as accurately known as elastic ππ scattering (and even contains subtleties as non-negligible isospin breaking effects due to the different thresholds of charged and neutral kaons, see e.g. Ref. [60]). As there are no error bands available for the Omnès matrix (or the various input quantities), to estimate the theoretical uncertainty we use and compare the fits resulting from the two different coupled-channel T -matrices described in Sec. 3. A minimization of the χ 2 using the modified Omnès solution based on Ref. [50] yields χ 2 /ndf = 3.4 (2.4) and c s 0 = 18.3 ± 0.5 (18.2 ± 0.5). 3 The resulting Y 0 0 curves for both fits, using the phase input from the Bern [20,21] and Orsay [22] groups (B+O), as well the one of Ref. [50] (DP), are presented in Fig. 5. Furthermore we show the phase shifts and the phases of the strange form factor for both phase inputs and compare to the LHCb phase due to Solution II (with f 0 (980) and a non-resonant S-wave contribution) as well as Solution I (f 0 (980) parametrization only). While the latter phase has a negative slope for s 1 GeV, which does not agree with the known phase shift, the phase extracted in Solution II is remarkably close to both the Bern and Madrid phase motions. 4.3B 0 s → J/ψK + K − S-wave prediction Having obtained theB 0 s → J/ψπ + π − fit parameters, we can make a prediction for thē B 0 s → J/ψK + K − S-wave amplitudes, using the relation between the ππ and the KK final states provided by the coupled-channel formalism, cf. Appendix B. 4 3 A similar procedure for theB 0 d decay has a rather small effect since the S-wave is not dominant in that case, and the difference of the P -wave phase of Refs. [19][20][21] is quite small (the S-or P -wave phase modification yields, in the most perceptible cases, a 4% correction of the χ 2 ). 4 In the case of theB 0 d → J/ψK + K − decay [61], the prediction of the S-wave does not work in such a direct way due to the I = 1 S-wave contribution (with a prominent a0(980) resonance) in addition to f0 In particular an understanding of the S-wave background to the prominent φ(1020) is of interest. In the LHCb analysis [62], the f 0 (980) as well as a non-resonant S-wave content is reported within a mass window of ±12 MeV around the φ(1020), which contribute an S-wave fraction of (1.1 ± 0.1 +0.2 −0.1 )%-consistent with former measurements from LHCb, CDF, and ATLAS [63][64][65], as well as theoretical estimates [66]. We calculate the S-wave fraction in the same mass interval ±12 MeV around the φ(1020) mass adopting the LHCb Breit-Wigner parametrization for the φ(1020), but using the predicted S-wave for the J/ψK + K − final state. Naively, this S-wave can be obtained by replacing the pion scalar form factor and all pion masses and momenta by the respective kaon quantities and taking the resulting fit parameters from the pion case. However, the fit result depends on the normalization of theB 0 s → J/ψπ + π − distribution. Hence, taking over the pion fit results for such a prediction requires a proper normalization of both decay channels relative to each other. To achieve this, we use the absolute branching fractions [58] B B 0 s → J/ψK + K − = (7.9 ± 0.7) × 10 −4 , B B 0 s → J/ψπ + π − = (2.12 ± 0.19) × 10 −4 , and define normalization constants N {π,K} = B(B 0 s → J/ψ{π + π − , K + K − }) N (B 0 s → J/ψ{π + π − , K + K − }) , (4.1) where N (B 0 s → J/ψ{π + π − , K + K − }) = √ 4π Y 0 0 B 0 s → J/ψ{π + π − , K + K − } d √ s (4.2) is the total number of events. 5 The S-wave contribution to the φ(1020) peak region is given by R S/φ ≡ N π m φ +12 MeV m φ −12 MeV X 3 σ K √ s \|c s 0 Γ s K ( √ s)\| 2 d √ s m φ +12 MeV m φ −12 MeV √ 4π Ỹ 0 0 B0 s → J/ψK + K − d √ s , (4.3) where we can approximate the (normalized) angular moment in the region of interest by the S-wave and the φ(1020) contribution, √ 4π Ỹ 0 0 B 0 s → J/ψK + K − \| √ s−m φ \| 12 MeV ≈ Xσ K √ s X 2 N π c s 0 Γ s K ( √ s) 2 + N K τ α φ τ A (τ ) φ (s) 2 . (4.4) Using the B+O input, we obtain R S/φ = 1.1%, in agreement with the LHCb result. However, there is a notable uncertainty due to the estimated ambiguity in the phase input in the region of the f 0 (980) resonance discussed in Sec. 4.2. Using the DP phase instead of the B+O phase input yields a fraction of 1.95%. resonances in the I = 0 S-wave. 5 For theB 0 s → J/ψK + K − decay [62] no data for the efficiency-corrected angular moments are available. We therefore extract the strength of the φ(1020) Breit-Wigner amplitude from the published expected signal yield Nexp and use N (B 0 s → J/ψK + K − ) = B(B 0 s → J/ψK + K − )/Nexp. Summary and outlook In this article, we have described the strong-interaction part of theB 0 d → J/ψπ + π − and B 0 s → J/ψπ + π − decays by means of dispersively constructed scalar and vector pion form factors. This formalism respects all constraints from analyticity and unitarity. The nonstrange and strange scalar form factors are calculated from a two-channel Muskhelishvili-Omnès formalism that requires the pion-pion elastic S-wave phase shift as well as modulus and phase of the corresponding ππ → KK amplitude as input. For the vector form factor, an elastic Omnès representation based solely on the pion-pion P -wave phase shift is sufficient, supplemented by an enhanced isospin-breaking contribution of ρ-ω mixing, which can be fixed from data on e + e − → π + π − . For energies √ s ≤ 1.02 GeV, a minimal description of all S-and P -waves (constructed in a form free of kinematical singularities) as the corresponding form factors, multiplied by real constants, has been shown to be sufficient. Allowing for subtraction polynomials with linear s-dependence leads to a slightly improved fit quality solely in the case of one P -wave component, with a slope still compatible with zero within uncertainties. In particular considering the S-wave slope as a free fit parameter (as opposed to fixing it to zero) only yields a minimal improvement of the χ 2 . In accordance with expectations from the underlying tree-level decay mechanism, below the onset of D-wave contributions that become important with the f 2 (1270), only the non-strange scalar and the vector form factors feature in theB 0 d decay, while the strange scalar form factor determines theB 0 s S-wave. The overall fit quality in the energy range considered is at least as good as in the phenomenological fits by the LHCb collaboration [6,7], where Breit-Wigner resonances and non-resonant background terms were used. However, since the dispersive analysis allows one to use input from other sources, our analysis calls for a much smaller number of parameters to be determined from the data. In addition, a comparison of theB 0 d Swave obtained from the dispersive analysis with the one deduced from the LHCb analysis shows drastic differences in both modulus and phase: it is well-known that the f 0 (500) does not have a Breit-Wigner shape, and therefore such parametrizations should be avoidedespecially when it comes to studies of CP violation that need a reliable treatment of the phases induced by the hadronic final-state interactions [14]. The LHCb analysis of thē B 0 s S-wave uses a Flatté parametrization of the f 0 (980), solely (corresponding to their Solution I) or combined with a non-resonant background (Solution II). Only Solution II yields a phase that is close to the phase of the strange scalar form factor, and approximately compatible with Watson's final-state interaction theorem in the elastic region. Finally we have made a prediction for theB 0 s → J/ψK + K − S-wave, which is related to the corresponding π + π − final state through channel coupling. Only the results of the fit to the π + π − final state are required to predict an S-wave fraction below the φ(1020) resonance of about 1.1%, in agreement with the findings by the LHCb collaboration. We have not attempted a corresponding prediction for theB 0 d → J/ψK + K − S-wave, since this has an isovector component (corresponding e.g. to the a 0 (980) resonance). This would have to be described by a coupled-channel treatment of the πη and KK S-waves [67]. To extend our description of the form factors to higher energies, eventually covering most of the energy range accessible inB 0 d/s → J/ψπ + π − , inelastic channels with corresponding higher resonances have to be taken into account. Here, a formalism recently developed for the vector form factor [45] that correctly implements the analytic structure and unitarity, reduces to the Omnès representation in the elastic regime, but maps smoothly onto an isobar-model picture at higher energies should be extended to the scalar sector. Even an extraction of the scalar form factors from these high-precision LHCb data sets seems feasible, and should be pursued in the future. as well as those to the set {A i , V}, F ⊥ = − √ sXV, F = √ sA 2 , F 0 = X 2M ψ 2A 1 + σ π cos θ π (P · p ψ ) X A 2 , F t = P · Q M ψ A 1 + σ π X cos θ π 2XM ψ (P · p ψ )(P · Q) − sM 2 ψ A 2 + M ψ A 3 . (A.5) The unphysical time component F t does not contribute. We expand the remaining three form factors H 0,± in partial waves. The latter relation is of particular interest when defining partial waves that are free of kinematical singularities and zeros, see Sec. 3. The partial-wave expansion of the helicity amplitudes reads H λ (s) = ℓ √ 2ℓ + 1H (ℓ) λ (s)d ℓ λ0 (θ π )e λiφ , (A.6) where the d ℓ λλ ′ are the small Wigner-d functions. Using d ℓ 00 (θ π ) = P ℓ (cos θ π ), d ℓ 10 (θ π ) = −d ℓ −10 (θ π ) = − sin θ π ℓ(ℓ + 1) P ′ ℓ (cos θ π ), (A.7) we see that the zero-component H 0 (s) is expanded in terms of Legendre polynomials P ℓ (cos θ π ) and thus contains all S-, P -, and D-wave contributions, while the H ± (s) partialwave expansions, proceeding in derivatives of the Legendre polynomials P ′ ℓ (cos θ π ), start with the P -wave amplitudes, i.e. ± (s) = ∓ σπ √ 2 F (ℓ) (s) ± F (ℓ) ⊥ (s) , we arrive at the partial-wave expansion of the transversity form factors given in Eq. (2.9) in the main text. In order to calculate the differential decay rate we sum over the squared helicity amplitudes, M 2 = G 2 F 2 \|V cb \| 2 \|V cq \| 2 f 2 ψ M 2 ψ \|H 0 \| 2 + \|H + \| 2 + \|H − \| 2 (q = {d, s}) (A.9) and integrate over the invariant three-particle phase space, which is given by dΦ (3) = Xσ π 4(4π) 2 m 2 B ds d cos θ π dφ. (A.10) Neglecting waves larger than D-waves and integrating over φ we arrive at Eq. (2.11). B Coupled-channel Omnès formalism We briefly discuss the coupled-channel derivation of the scalar pion and kaon form factors and Σ(s) = diag σ π (s)Θ(s − 4M 2 π ), σ K (s)Θ(s − 4M 2 K ) , with σ i (s) = 1 − 4M 2 i /s 1/2 and Θ(.) denoting the Heaviside function. There are three input functions entering the Tmatrix, the ππ S-wave isoscalar phase shift δ 0 0 (s) and the ππ → KK S-wave amplitude g 0 0 (s) = \|g 0 0 (s)\|exp(iψ 0 0 (s)) with modulus and phase. The modulus \|g 0 0 (s)\| is related to the inelasticity parameter η 0 0 (s) by η 0 0 (s) = 1 − 4σ π (s)σ K (s)\|g 0 0 (s)\| 2 Θ(s − 4M 2 K ). (B.3) Writing the two-dimensional dispersion integral over the discontinuity (B.1) leads to a system of coupled Muskhelishvili-Omnès equations, Γ(s) = 1 π ∞ 4M 2 π T 0 * 0 (s ′ )Σ(s ′ )Γ(s ′ ) s ′ − s − iǫ ds ′ . (B.4) A solution can be constructed introducing a two-dimensional Omnès matrix, which is connected to the form factors by means of a multiplication with a vector containing the normalizations Γ π (0) and Γ K (0) [68], Γ π (s) where Γ π,K (s) represents both strange and non-strange form factors, Γ s π,K (s) and Γ n π,K (s), which differ merely in their respective normalizations. Thus the problem reduces to finding a matrix Ω(s) that fulfills which has to be solved numerically [68][69][70][71]. To ensure an adequate asymptotic behavior, we exploit the correlation between the high-energy behavior of the Omnès solution and the sum of the eigen phase shifts δ I ℓ (s) [69], where m is the number of channels that are treated in the formalism. According to the Feynman-Hellmann theorem, the form factors for zero momentum are related to the corresponding Goldstone boson masses, which at next-to-leading order in the Figure 6 shows the results obtained for the modulus of the pion form factor (see also Ref. [74]). The sensitivity due to the uncertainty in the kaon form factor normalization is illustrated by the uncertainty bands. The strange form factor exhibits a peak around 1 GeV, which is produced by the f 0 (980) resonance. On the contrary in the pion nonstrange form factor the σ meson appears as a broad bump (notice the non-Breit-Wigner shape) around 500 MeV. Figure 2 . 2Definition of the kinematical variables forB 0 d/s → J/ψπ + π − . (τ ) f 2 2(s), τ = 0, , ⊥, see Ref.[6]. , Eq. (3.12), simultaneously. Taking up the discussion of Sec. 3, our basic fit, FIT I, includes three fit parameters (to be compared to 14 free parameters in the Breit-Wigner parametrization used in the LHCb analysis, see below): the normalization factors for the S-wave (b n 0 ) and for two P -waves f Figure 3 . 3Y 0 0 (left) simultaneously fitted with Y 0 2 (right), using 3 parameters without D-wave contribution (FIT I, red, solid), and improving step by step by adding a Breit-Wigner-parametrized D-wave contribution (FIT II, blue, dashed) and by allowing for 4 free parameters, also supplemented by the D-wave contribution (FIT III, green, dotted). Figure 4 . 4Comparison of the S-wave amplitude strength and phase obtained in the LHCb and in our fits, respectively. In the left panel the S-wave part of the decay rate for the three fit configurations FIT I-III is depicted together with the LHCb outcome. The right panel shows the phases of the non-strange scalar form factor δ Γ n (equal to the ππ S-wave phase shift δ 0 0 below the KK threshold) compared to the S-wave phase δ f0 extracted from the LHCb analysis. ± (s) e ±iφ + . . . , (A.8) where the ellipses denote F -waves and larger. Equivalently, due to Eq. (A.4) and using H (ℓ) (I = 0, ℓ = 0). The two-channel unitarity relation reads disc Γ(s) = 2iT 0 * 0 (s)Σ(s)Γ(s), (B.1)where the two-dimensional vector Γ(s) contains the pion and kaon scalar s ′ )Σ(s ′ )Ω(s ′ ) s ′ − s − iǫ ds ′ , Ω(0) = ½, (B.6) Figure 6 . 6Modulus of the scalar pion non-strange (left panel) and strange (right panel) form factors, depicted for three different normalizations inside the allowed range, illustrated by the uncertainty band.chiral expansion in terms of quark masses depend on certain low-energy constants. These are determined in lattice simulations with N f = 2 + 1 dynamical flavors at a running scale µ = 770MeV[72], limiting the form factor normalizations to the ranges6 Though we expect the D-and higher waves to be small and therefore describe only S-and P -waves in the Omnès formalism, we present the formulae including the D-wave contribution, as we will study their impact at a later stage. In the following we will suppress the isospin indices as Bose symmetry demands the S-waves to be isoscalar, while the P -waves are restricted to I = 1. Similar ranges, with slightly increased values in the case of the kaon form factor normalizations, are found in simulations with N f = 2 + 1 + 1 dynamical flavors[73]. AcknowledgmentsWe would like to thank the LHCb collaboration for the invitation to the Amplitude Analysis Workshop where this work was initiated, and in particular Tim Gershon, Jonas Rademacker, Sheldon Stone, and Liming Zhang for useful discussions. We are furthermore grateful to Mike Pennington for providing us with the coupled-channel T -matrix parametrization of Ref.[50]. Financial support by DFG and NSFC through funds provided to the Sino-German CRC 110 "Symmetries and the Emergence of Structure in QCD" is gratefully acknowledged.A Form factors and partial-wave expansionIn the standard basis of momenta p ψ , P µ , and Q µ , Eq. (2.4), the matrix element describing the hadronic part of theB 0 d/s decay is given by four dimensionless form factors, three axial (A i ) and one vector (V),In Sec. 2.2 we use a different (orthogonal) basis of momentum vectors, p µ ψ ,p µ (⊥) , and Q µ ( ) , see Eq. (2.8), corresponding to the orthonormal basis of polarization vectors of the J/ψ meson[28],This allows us to describe the matrix element M µ ππ in terms of the transversity form factors, Eq. 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[ "Transferring Inter-Class Correlation", "Transferring Inter-Class Correlation" ]	[ "Hui Wen \nSchool of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina\n", "Yue Wu \nSchool of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina\n", "Chenming Yang [email protected] \nNational Key Lab on Communication\nUniversity of Electronic Science and Technology of China\nChina\n", "Jingjing Li \nSchool of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina\n", "Yue Zhu \nNVIDIA Semiconductor Technology (Shanghai) Co., Ltd\nShanghaiChina\n", "Xu Jiang \nSchool of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina\n", "Hancong Duan [email protected] \nSchool of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina\n" ]	[ "School of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina", "School of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina", "National Key Lab on Communication\nUniversity of Electronic Science and Technology of China\nChina", "School of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina", "NVIDIA Semiconductor Technology (Shanghai) Co., Ltd\nShanghaiChina", "School of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina", "School of Computer Science and Engineering\nUniversity of Electronic Science and Technology of China\nChina" ]	[]	The Teacher-Student (T-S) framework is widely utilized in the classification tasks, through which the performance of one neural network (the student) can be improved by transferring knowledge from another trained neural network (the teacher). Since the transferring knowledge is related to the network capacities and structures between the teacher and the student, how to define efficient knowledge remains an open question. To address this issue, we design a novel transferring knowledge, the Self-Attention based Inter-Class Correlation (ICC) map in the output layer, and propose our T-S framework, Inter-Class Correlation Transfer (ICCT). Notably, the analysis of our ICCT illustrates that the student combines with its own belief comprehensively adopting the teachers ICC map, rather than curtly mimic the teacher. Since the ICC map can impose effective regularization, the knowledge from the teacher, which has a higher, equal, or lower capacity than the student can bring the benefit for the student training process. Experimental results on the CIFAR-10, CIFAR-100, and ILSVRC2012 image classification datasets demonstrate that our ICCT can improve the student's performance and outperform other state-of-theart T-S frameworks in T-S application scenarios with different network capacities and structures.	null	[ "https://arxiv.org/pdf/2008.10444v1.pdf" ]	221,266,771	2008.10444	7c3e32fcfc7b24b1ae89412c4ecbb7da8a8d30fe	Transferring Inter-Class Correlation Hui Wen School of Computer Science and Engineering University of Electronic Science and Technology of China China Yue Wu School of Computer Science and Engineering University of Electronic Science and Technology of China China Chenming Yang [email protected] National Key Lab on Communication University of Electronic Science and Technology of China China Jingjing Li School of Computer Science and Engineering University of Electronic Science and Technology of China China Yue Zhu NVIDIA Semiconductor Technology (Shanghai) Co., Ltd ShanghaiChina Xu Jiang School of Computer Science and Engineering University of Electronic Science and Technology of China China Hancong Duan [email protected] School of Computer Science and Engineering University of Electronic Science and Technology of China China Transferring Inter-Class Correlation The Teacher-Student (T-S) framework is widely utilized in the classification tasks, through which the performance of one neural network (the student) can be improved by transferring knowledge from another trained neural network (the teacher). Since the transferring knowledge is related to the network capacities and structures between the teacher and the student, how to define efficient knowledge remains an open question. To address this issue, we design a novel transferring knowledge, the Self-Attention based Inter-Class Correlation (ICC) map in the output layer, and propose our T-S framework, Inter-Class Correlation Transfer (ICCT). Notably, the analysis of our ICCT illustrates that the student combines with its own belief comprehensively adopting the teachers ICC map, rather than curtly mimic the teacher. Since the ICC map can impose effective regularization, the knowledge from the teacher, which has a higher, equal, or lower capacity than the student can bring the benefit for the student training process. Experimental results on the CIFAR-10, CIFAR-100, and ILSVRC2012 image classification datasets demonstrate that our ICCT can improve the student's performance and outperform other state-of-theart T-S frameworks in T-S application scenarios with different network capacities and structures. I. INTRODUCTION With the development of deep neural networks, considerable attractive approaches have been made to tackle challenging tasks. Though numerous studies have been conducted to improve the model performance, the improvement which only relying on enlarging the number of parameters [1], [2] or searching for more complex network structures [3], [4] becomes increasingly difficult. To overcome the difficulty, a supervised learning framework -Teacher-Student (T-S) framework -was proposed to promote the generalization accuracy of some neural network (the student) by bringing in transferring knowledge from another neural network (the teacher) [5]. Since the generalization accuracy can be regarded as the model capacity [6], we utilize the "Cap T " and "Cap S " to represent the capacity of the student and the teacher, respectively. Furthermore, we employ the relationship between "Cap T " and "Cap S " to divide the T-S framework's application scenario. In Figure 1, we illustrate the three capacity-based T-S application scenarios. The T-S framework was popular in the model compression area [7], as for its capability of improving the performance of the low-capacity student with few-parameters by a highcapacity teacher with large-parameters (Cap T > Cap S ) [8]- [10]. Recently, the T-S framework was proved to be also effective in improving the student performance in other capacitybased application scenarios, Cap T = Cap S [11], [12] and Cap T < Cap S [12], [13], in which the T-S framework excavates the potential generalization ability of the student. Since T-S framework transfers knowledge from the teacher to the student, one of the critical issues is how to define the knowledge. Hinton et al. [8] first proposed the concept of Knowledge Distillation (KD) to guide the low-capacity student by a trained high-capacity teacher's softened output (Cap T > Cap S ), controlled by a hyperparameter called "temperature". The transferring knowledge is only related to the categories of classification, and therefore does not constrain the network structure between the teacher and the student. Furlanello et al. [11], Yang et al. [13], and Yuan et al. [12] further exploited the intrinsic mechanism of the original KD and extended its capacity-based application scenarios where the student has equal (Cap T = Cap S ) and higher capacity than the teacher (Cap T < Cap S ). Different from the knowledge definition in KD, another kind of transferring knowledge was defined in hidden layers. For instance, Zagoruyko and Komodakis [14] defined the knowledge as the spatial attention map of the hidden-layers outputs. Yim et al. [15] adopted the correlation matrix of two layers' feature maps as the knowledge. Tung and Mori [10] and Hou et al. [16] defined the knowledge by the similarity activation maps from hidden layers. The advantage of these hidden-layers is that they can optimize students step by step. However, since the feature maps of the student and the teacher are required to have the same shape [14]- [16], these T-S frameworks with hidden-layers knowledge have to convert the hidden features. Due to the missing of beneficial information during conversion processing, the performance of these frameworks is restricted by network structures between the teacher and the student. Furthermore, the knowledge from hidden layers has only been explored in the Cap T > Cap S scenario, and it is still unclear whether they are available in the Cap T = Cap S or Cap T < Cap S scenario in the image classification task. In this paper, we define knowledge based on a new perspective, named Inter-Class Correlation (ICC), for classification tasks. Our intuitive motivation is that it is usually difficult to distinguish two similar classes if we can only obtain the Cap T = Cap S , Cap T < Cap S ) . For clarity, we adopt the structure of "MLP" as the demonstration. The purple is applied to mark the teacher, and the yellow is applied to mark the student. information of these two classes. However, if we can compare them with other dissimilar classes and form the ICC map, this task could be easier. Figure 2 shows an example of the ICC map in the CIFAR-10 dataset. Though the "Truck" and the "Automobile" are similar, the similarity between the "Truck" and other classes (like the "Cat") can be different from the similarity between the "Automobile" and the other classes (like the "Cat"). Because the correlation of classes is only related to the images in the dataset, it is reasonable to require that the student and the teacher have similar ICC map corresponding with the same input image. To form the ICC map of knowledge, we employ the popular Self-Attention mechanism [17] on every two "logit" of the output layer. We then use the Self-Attention based ICC map as the transferring knowledge and propose a novel T-S framework, named Inter-Class Correlation Transfer (ICCT). Revealed by the analyzed gradients, the operating the student of our ICCT does not just mimic the teacher in that class. Instead, the student gathers all the information related to this class in the ICC map and adds its own belief about this class. We call this "comprehensive" learning mode, which makes it reasonable to expect our ICCT to be applied in all three capacity-based application scenarios. The ICC map is only related to classification outputs, like KD [8], rather than the hidden features [14]- [16]. Therefore, our ICCT eliminates the restrictions of network structure on the output shape of the hidden layer and avoids the loss of beneficial information from the teacher. Notably, our ICC does not introduce hyperparameters, like "temperature", which could change the original outputs. To validate it, we design extensive experiments in three capacity-based T-S application scenarios Cap T > Cap S , Cap T = Cap S , and Cap T < Cap S , in which the structure of the teacher and student networks is unrestricted. The experiments are conducted on three standard image classification datasets, CIFAR-10, CIFAR-100 [18], and ILSVRC2012 [19], which have incremental complexity. Experimental results demonstrate that our ICCT is not only able to improve the student's performance but also outperform KD and hidden-layers based T-S frameworks consistently, regardless of the application scenarios based on different network capacities and structures. The main contributions of this paper are summarized as follows: • We design a novel T-S framework named Inter-Class Correlation Transfer (ICCT), which uses Self-Attention based Inter-Class Correlation as the transferring knowledge. Our ICCT eliminates the restrictions of network structure and avoids the loss of beneficial information from the teacher. • We analyze the gradients of the ICC map and compare them with KD. The ICCT updates the student network in a "comprehensive" mode, which means the student can comprehensively study teacher's knowledge based on its own belief, rather than completely mimicking the teacher. Benefited from the effective regularization of ICC, our ICCT can be flexibly implemented in different capacitybased T-S application scenarios. • Experiments are conducted on three standard image classification datasets in three capacity-based application scenarios with different network structures to show the outstanding performance of our framework. II. PRELIMINARY A. Inter-Class Correlation The Inter-Class Correlation has been well studied in clustering, measuring the relationship between categories among clusters [20]. Benefited from the researches in clustering, studies have been conducted to improve the classification methods with the help of the Inter-Class Correlation. In [21], [22], co-occurrence correlation was utilized to enforce object recognition in images. Wen et al. [20] proposed the "center loss" as the loss function to get inter-class dispensation and intra-class compactness in face recognition. Wu et al. [23] jointly learned feature relationships and exploited the interclass relationships for improving video classification performance. Fig. 2. Examples of pairwise Inter-Class Correlation in the CIFAR-10 dataset. "TA" means the pairwise ICC between class "Truck" and "Automobile," and others are the same. Though the "Truck" and the "Automobile" are hard to distinguish, the "TC" and "TB" is different from "AC" and "AB." In recent years, another kind of correlation called Self-Attention has been attractive, since the success of "Transformer" [17] in natural language processing. The Self-Attention mechanism is a variant of the original attention mechanism, which is widely used in the Encoder-Decoder framework [24]. Unlike the original attention mechanism extracting the correlations between the input and the output, the Self-Attention mechanism try to capture the correlations among different output neurons. There are various operating modes to implement the Self-Attention mechanism, like dot product [17], multiplicative [25], and additive [26]. To be noticed, the Self-Attention mechanism is served as a kind of intra-class correlations in natural language processing by treating the output as a whole. However, in the classification task, the output neurons correspond with different classes, which means that the Self-Attention mechanism measures the Inter-Class Correlation. III. INTER-CLASS CORRELATION TRANSFER In this section, we present our Self-Attention based T-S framework, called Inter-Class Correlation Transfer (ICCT), which defines the ICC map as the transferring knowledge. We first introduce the architecture of the T-S framework and then provide a Self-Attention based implementation of the ICC knowledge. Based on the gradients, we finally explain how our ICCT works, which we called "comprehensive" mode. A. Teacher-Student Framework In the supervised image classification task, given a training set D = {(x, y)\|(x, y) ∈ X × Y}, where x and y are images and labels, the goal is to train a model f (x) : → X × Y, which is able to generalize to unseen data. In this paper, we qualify the model to be a neural network f (x; θ θ θ) and learn the parameters θ θ θ via Empirical Risk Minimization (ERM). The training process is to minimize the loss function: θ θ θ * = arg min θ θ θ L label (y, f (x; θ θ θ)),(1) where L label is some function to measure similarity. Suppose that there are N categories in total, and the output y = f (x; θ θ θ) = (y 1 , . . . , y N ) is generated by: y i = e zi N j=1 e zj ,(2) where z = (z 1 , . . . , z N ) is the input of the last softmax classifier and named by "logits." Due to the unbalance between model capacity and data space, this training process often limits the ability of the trained model generalizing to unseen data. The T-S framework was proposed to tackle this problem. In the T-S framework, the target model is treated as the student model f (x; θ θ θ S ). Suppose we have got a trained teacher model f (x; θ θ θ T ). We then need to define the knowledge transferred from f (x; θ θ θ T ) to f (x; θ θ θ S ). One of the general definition of the knowledge is the one in Knowledge Distillation (KD) [8], which gets a softened the softmax distribution q by distilling the logits z: q i = e zi/M N j=1 e zj /M ,(3) where M is the soften hyperparameter, called the "temperature." Consequently, the loss function of the student model is: θ θ θ * S = arg min θ θ θ S (L label (y, f (x; θ θ θ S )) + λ KD L KD (q S , q T )),(4) where L KD is a similarity measure function, and λ KD controls the proportion of the two parts of loss. Previous researches have demonstrated its ability to promote the generalization ability of the student by adding the teacher's knowledge in the student's loss function [8]. Other hidden-layers based knowledge very likely restricts the output shape of the hidden layers, which is contrary to our demands. So we omit the details of the hidden-layers based knowledge without losing integrity. B. Self-Attention Based Inter-Class Correlation As illustrated in Section I, the ICC map can help the model to make better decisions in the image classification task. To better utilize the information contained in the outputs, we propose a new definition of the transferring knowledge -Inter-Class Correlation (ICC) map, which contains the fine-grained relationship between every two classes in a mini-batch and represents a specific pattern of data. The implementation of the ICC map is vital in our framework. To simplify the computation and explain it clearly, we adopt the dot-product Self-Attention mechanism [17]. The ICC map is calculated as follows. Firstly, compute the unnormalized Self-Attention matrix by dot-product on "logits" of the s-th sample in a mini-batch, A s = [a s ij ] = z s z s ∈ R N ×N . Then A s should be normalized to make it comparable within different models and thus can Finally, the matrices of all the samples should be averaged to measure the ICC in the mini-batch, A = 1 b b s=1Ã s ,(6) where b is the batch size. An example of the ICC maps of four neural networks with the same input is shown in Figure 3. Apparently, the shades of the rows/columns in each matrix are very different, revealing the difference of the ICC between different classes. The four matrices of four neural networks are similar, which is consistent with our illustration that the ICC is highly related to the input. Noteworthy, the original Self-Attention mechanism [17] has components "key," "query," and "value." In the ICC map, the "key" and "query" are the "logits" on the one hand, which are just like the original mechanism. On the other hand, the "value" can be seemed as "1", which means that each class is equally treated. Similar to other T-S frameworks, the hypothesis of our ICCT is that if an input image has a specific ICC map in the teacher network, it should have a similar ICC map in the student network. Compared with KD, the ICC mapÃ does not introduce additional hyperparameter "temperature" as KD in Eq. (3) The reason is that we regard the ICC map as a representation of the average Inter-Class Correlation, which is determined by the samples in the mini-batch. If we add the "temperature" like KD, we change the correlations manually by reducing the higher correlations and amplifying the lower correlations. The purpose of our ICCT is not to justify the ICC artificially, but to measure it as accurate as possible. After defining the ICC map as the transferring knowledge, we implement it in the T-S framework to form our ICCT. Given an input mini-batch of b images, we compute the ICC maps of the studentÃ S and the teacherÃ T according to the corresponding "logits." The two matrices encode the average ICC of the input mini-batch. The loss function of our ICCT is: L ICCT = L label (Y, f (X; θ θ θ S )) + λ ICC L ICC (Ã T ,Ã S ), (7) where Y = (y 1 , . . . , y b ) and X = (x 1 , . . . , x b ) are the labels and inputs in the mini-batch; λ ICC is a hyperparameter to balance the influence of labels and ICC maps. In our experiments, we set L label to be the cross-entropy function and L ICC to be the KL-divergence between the teacher and The student. The framework diagram is shown in Figure 4. C. Gradients of the ICCT In this section, we analyze the gradients with respect to the "logits" to interpret how our ICCT works. Suppose that there are N classes in the dataset and b samples in each mini-batch. In our ICCT, the gradient of L ICC with respect to the "logits" are shown in Eq. (8): ∂L ICC ∂z s k,S = 2 b b s=1 N i=1 z s i,S (ã s ik,S −ã s ik,T ),(8) whereã s ik,S andã s ik,T are the corresponding element of the student's and the teacher's ICC maps in the s-th sample. The details of calculating the gradients are presented in the supplementary material. To illustrate the different operating mode of ICCT, we compare its gradient with that of KD: ∂L KD ∂z s k,S = 1 bM b s=1 (q s k,S − q s k,T ) = 1 bM b s=1 ( e (z s k,S /M ) N j=1 e (z s j,S /M ) − e (z s k,T /M ) N j=1 e (z s j,T /M ) ).(9) Eq. (9) reveals that the student's each "logit" z s k,S is updated independently by just mimicking the teacher's "logit" z s k,T . Each class is treated equally when minimizing the loss function. In our ICCT, the student's each "logit" z s k,S is updated by gathering all the correlations between class k and all the classes. This could bring about more information to the student. Moreover, due to the existence of z s i,S in Eq. (8), the student does not equally treat all the classes. Instead, the student can decide which class should be learned more, Fig. 4. ICCT Framework. This figure shows the training process of the student in our ICCT. The purple is applied to mark the teacher, and the yellow is applied to mark the student. Mini-batch SA SA SOFTMAX label label ICC ICC ICCT ICCT Teacher Student T Z S Z S A S A T A T A according to its own outputs. z s i,S acts as an belief weight for the i-th class. When z s i,S is large, the student believes that class i is possible to be the ground truth and tends to learn more about the gap in this class between the teacher and the student. Otherwise, the student learns less about the gaps in the relatively less critical classes. In such a way, the student can comprehensively study teachers knowledge based on its own belief. We called this operating mode as "comprehensive" mode, though which the student can learn smarter and may obtain better generalization capability. D. Differences With Previous Approaches Our ICCT defines the transferring knowledge at the output layer, similar to the traditional KD 3. Instead of using soften outputs as KD, we employ the Self-Attention mechanism to extract Inter-Class Correlation to form the knowledge. Other hidden-layers based T-S frameworks [10], [14], [16] define the transferring knowledge at the hidden layers or blocks. To be noticed, [10], [16] recently proposed activationbased T-S frameworks, which also utilize the concept of correlations. However, there is a key difference between our ICCT and these two works. In [10], [16], the knowledge is defined by the similarity between two inputs' activation maps. They hypothesize that if two inputs produce highly similar activations in the teacher network, it is beneficial to guide the student network towards a configuration that also results in the two inputs producing highly similar activations in the student. However, in our ICCT, knowledge is the averaged Inter-Class Correlation in a mini-batch. Our hypothesis is that the correlations between classes should be determined by the data. The ICC maps of the student and the teacher are just two estimations about the correlations. Our ICCT is to guide the student's estimation close to the teacher's estimation. Moreover, [10] mainly concentrated in the scenario Cap T > Cap S for image classification, and [16] considers the scenario Cap T = Cap S for lane detection. Our ICCT is also designed for the image classification task, but we consider more general scenarios Cap T > Cap S , Cap T = Cap S , and Cap T < Cap S . The Cap T = Cap S , and Cap T < Cap S can further validate our hypothesis about the Inter-Class Correlation and benefit for promoting the performance of the student when we do not have trained high-capacity teachers. IV. EXPERIMENTAL EVALUATION A. Experimental Settings In this section, experiments are conducted to validate the efficiency of our ICCT in classification tasks, in which the median error of test over the ten standard splits is reported. The experiments implement on three common datasets, CIFAR-10, CIFAR-100 [18], and large-scale dataset ILSVRC2012 [19], and the complexity of these data increases successively. To demonstrate the superiority of our ICCT, we compare the performance of commonly used frameworks Knowledge Distillation (KD) [8], logit-transfer (LT) [27], attention transfer (AT) [14], and Similarity-Preserving (SP) [10], Tf-KD [12] with our framework. 1) Experimental Networks: To further illustrate that our ICCT is not constrained by the network structure, we utilize five kinds of networks with heterogeneous structures. We start with a tiny toy model, a convolutional neural network with 5 layers (CNN-5). We perform three 3 × 3 convolutional layers, each followed by batch normalization, max pooling, and ReLU activation. The first convolutional layer has 32 output channels, while the following two convolutional layers have 64 and 128 output channels, respectively. After that, two consecutive fully-connected layers are added. We then choose four mainstream classification models, ResNet [6], ResNeXt [28], WideResNet [29], and MobileNetV2 [30]. For the sake of brevity, we abbreviate "WideResNet" as "WRN." Distillation between different types of the model structure has also been considered. Based on the above five network structures, we design four kinds of pairs between the student and the teacher: heterogeneity of architecture, heterogeneity of depth, heterogeneity of width, and heterogeneity of depth&width. 2) Experimental Scenarios: To show our ICCT can flexibly work in capacity-based application scenarios, we split the T-S application scenarios based on the network capacities between the student and the teacher. We design three categories of T-S capacity-based application scenarios: Cap T > Cap S , Cap T = Cap S , and Cap T < Cap S , where "T" and "S" denote the teacher and the student, respectively. Specifically, we design experiments in generations in Cap T = Cap S , which means that the student in one generation is to be the teacher in the next generation. For the competitors, KD and LT are tested in Cap T > Cap S , Cap T = Cap S , and Cap T < Cap S . Notably, in Cap T = Cap S , the KD is employed without an ensemble, which is the same as the sequential version in [11]. Besides, in Cap T < Cap S , we implement KD by adding secondary information as the implementation in [13]. According to [10], [14], AT and SP are both scenariospecialization approaches for Cap T > Cap S . Meanwhile, the Tf-KD [12] is specific to the scenario of Cap T = Cap S . B. CIFAR-10 and CIFAR-100 We first evaluate our framework on the CIFAR-10 and CIFAR-100 datasets [18], which consists of 60,000 32x32 RGB images in uniformly distributed 10 and 100 classes, with each class having an equal number of images. 1) Network Baselines: Based on the four mainstream classification models, we select nine baseline models including CNN-5, WRN-16-2, WRN-28-2, MobileNetV2, ResNet-18, WRN-16-10, ResNeXt-29, WRN-28-10, ResNet-101. The "ResNeXt-29" is short for "ResNeXt-29(8 × 64d)." The parameters of baseline networks on the CIFAR-100 are shown in Table I. 2) Training Settings: For datasets CIFAR-10 and CIFAR-100, We applied the standard horizontal flip and random crop data augmentation. The training protocols of CNN-5 and residual networks are not the same. For CNN-5, we train it by Adam with batch size 256 (140 epochs; an initial learning rate of 0.001, decayed by a factor of 0.2 at epochs 40, 80, and 120). For residual networks, We use the standard Stochastic Gradient Descent (SGD) with a weight decay of 0.0001 and a Nesterov momentum of 0.9 (200 epochs; an initial learning rate of 0.1, decayed by a factor of 0.2 at epochs 60, 120, and 160). For KD, we set M = 4 on the CIFAR-10, following the experiments [8], [10], [12], and M = 10 on the CIFAR-100. The weights of the knowledge transfer loss are obtained by heldout validation as in [10] on a subset of the training set (CIFAR-10/100: λ ICC = 1500/1800, λ KD &λ T f −KD = 300/800, λ LT = 80/150, λ AT = 1000/400 and λ SP = 1600/550). 3) Results and Discussions: Following the capacity settings, we show the corresponding experimental results in the CIFAR-10 and CIFAR-100 datasets. The part of the experimental results of CIFAR-10 is shown in the appendix. • Cap T > Cap S . In Table II, the first seven T-S pairs are composed of the high-capacity teacher with large parameters and the lowcapacity student with small parameters for the approach of model compression. The last pair in this table is the highcapacity teacher with small parameters and the low-capacity student with large parameters, which is more suitable for model optimization. Therefore, we can observe from this table, in our ICCT, no matter how many times the parameters of the teacher network differ from those of the student network, the transferring information of the teacher can always help the student improve the capability. As shown in II, our ICCT can be applied in all the heterogenous T-S pairs. However, due to the restriction of network structure in AT, AT can not handle the heterogeneous teacher and student, which are so diverse, e.g., the student of CNN-5 with the teacher of ResNet-18. Meanwhile, the missing information of transferring information from hidden layers, e.g., the missing information of channel dims in AT, the missing information of spatial dims and spatial dims in "THE ERROR RATE OF THE BASELINE OF THE STUDENT" AND "THE ERROR RATE OF THE BASELINE OF THE TEACHER," SP, introduces a negative impact on learning results. This table illustrates that our ICCT can systematically improve the performance of the students in the CIFAR-100 dataset. Moreover, the error rates of students trained by ICCT are lower than other competitors regardless of the compression rate and different types of network architecture, showing the superiority of our framework. Furthermore, due to defining the knowledge at the output layer, our ICCT is easy to extend by adding other hidden-layers based knowledge transfer techniques and obtained preferable performance. • Cap T = Cap S . In this scenario, the student is taught by itself in generations. As shown in Table III, the students in our ICCT perform better than themselves in all generations. Similar to the phenomenon in [11], training models with our ICCT for multiple generations leads to inconsistent but positive improvements, which saturates after a few generations. This improvement demonstrates that the Self-Attention based Inter-Class Correlation is useful. • Cap T < Cap S . In this scenario, the student is taught by a low-capacity teacher. Generally speaking, a low-capacity teacher can not teach the high-capacity student well. However, the results in Table IV show that the high-capacity model can also learn something useful from the low-capacity model. Though the low-capacity teacher transfers a gentler supervision signal, the Inter-Class Correlation from the teacher can serve as a generic regularization method and reduce the risk of overfitting for the high-capacity student. To further analyze the effectiveness of our ICCT in three kinds of capacity-based scenarios, we conduct twodimensional visualization of the feature map distribution after the fully connected layer of Resnet-18. Following CIFAR-10 experiments, we can clearly observe from Figure 5, compared with all students in our ICCT, the feature map distribution of the baseline is significantly disordered. In contrast, our ICCT devotes to cluster the feature map by reducing the inner-class distance and expanding the inter-class distance. In the three scenarios, the student taught by the teachers with high-capacity, learns more compact and discriminative representations. Besides, there are encouraging signs that even the low-capacity teacher still has certain help for the student to obtain the better capability of feature understanding. C. ILSVRC2012 We now investigate the more complex ILSVRC2012 dataset [19]. The ILSVRC2012 dataset is a popular subset of the ImageNet database [31], which contains 1.3 million highresolution images and 1,000 classes in total. 1) Network Baselines: To handle the ILSVRC2012 dataset, which is more complicated than CIFAR-10 and CIFAR-100, we choose the models with higher capacity and leave alone the CNN-5 model. The 2) Training Settings: Due to the complex of the ILSVRC2012 dataset, a series of data-augmentation techniques are applied, including cropping, rescaling, and randomly mirroring the image. For all networks, The SGD with a weight decay of 0.0001 and a Nesterov momentum of 0.9 is used with batch size 256 (90 epochs; an initial learning rate of 0.1, decayed by the cosine annealing). For KD, we fix M = 5, following the experiments in [8], [12], [13]. The weights of the knowledge transfer loss are obtained as in [10] on a subset of the training set (λ ICC = 2000, λ KD &λ T f −KD = 350, λ LT = 100, λ AT = 120 and λ SP = 850). 3) Results and Discussions: The results are also presented in three capacity-based application scenarios. • Cap T > Cap S . Due to the complexity of the task, though the advantage of the ICCT in the ILSVRC2012 dataset is not as much as that in the CIFAR-10 and CIFAR-100 datasets, the ICCT still outperforms other competitors. The possible reason is that the teacher is also unable to completely handle the complex task, resulting in the teacher's ICC containing less useful information. • Cap T = Cap S . As shown in Table VI, the improvement from our ICCT is also inconsistent from generation to generation, which is similar to previous CIFAR-10 and CIFAR-100 experiments. The third generation of ResNet-18 in our ICCT presents the test rate of 28.25%, which is very close to the same student taught by ResNet-50 28.19%. This small gap demonstrates the potential of training neural networks in generations. • Cap T < Cap S . We can observe similar results in Table VII as those in the CIFAR-100 experiments. The ICCT can bring in relatively little but consistent progress in different students and teachers. Simultaneously, our ICCT still outperforms other competitors. This experiment proves that a low-capacity teacher can still improve the performance of high-capacity students through the ICCT in complicated tasks. V. RELATED WORK A. Teacher-Student Framework The goal of the T-S framework is training a student model by learning from the knowledge of the teacher model to achieve better generalizing accuracy than being trained directly. The performance of the T-S framework is sensitive to how knowledge is defined. Some early researches defined the knowledge at the output layer by extracting something useful in the outputs. Hinton et al. [8] first proposed the Knowledge Distillation (KD) by introducing the concept of "dark knowledge," which is the softened output of the teacher's softmax distribution with a soften hyperparameter "temperature." KD can be regarded as a generalization of previous work [27] by setting the "temperature" coefficient large enough. After that, some studies turned to define the knowledge in the hidden layers, aiming at teaching the student step by step. Zagoruyko and Komodakis [14] forced the student to match the attention map of the teacher (norm across the channel dimension in each spatial location) at the end of each residual stage. Czarnecki et al. [32] tried to minimize the difference between the teacher and student derivatives of the loss concerning the input. Yim et al. [15] designed a correlation matrix of two hidden layers' feature maps to be the knowledge and employed Knowledge Distillation (KD) in the final layer. Tung and Mori [10] and Hou et al. [16] defined the knowledge by the similarity activation maps in hidden layers. Excepting [10], the output shape of student's hidden layers should be similar to the teachers', considering the complexity of hidden features. In the image classification task, all the above frameworks were implemented in the application scenario of Cap T > Cap S . Recently, the potential of the T-S framework has been further explored. Based on the model capacity of the teacher (Cap T ) and the student (Cap S ), the capacity-based application scenarios of T-S frameworks could be divided into three categories: (1) Cap T > Cap S ; (2) Cap T = Cap S ; (3) Cap T < Cap S . Furlanello et al. [11] and Yuan et al. [12] considered the scenario Cap T = Cap S and demonstrated the capability of KD to improve the performance of a network by itself in generations. Yang et al. [13] and Yuan et al. [12] researched the more difficult scenario Cap T < Cap S and find that the student's accuracy could also be improved by secondary information and KD in generations, although the teacher has a lower capacity than the student. Another important question is how to transfer the knowledge between students and teachers. Inspired by generative adversarial networks, Wang et al. [33] introduced an additional discriminator network to the original T-S framework, and jointly trained the student, teacher, and discriminator networks. Other ensemble-based variants, such as Lan et al. [34], utilized the techniques of ensemble learning to train a teacher together with multiple students. VI. CONCLUSION AND FUTURE WORK In this paper, we study the problem of improving the performance of a student neural network by a teacher neural network in the classification task. To tackle this problem, we propose a new Teacher-Student framework named Inter-Class Correlation Transfer (ICCT), in which the transferring knowledge is defined as the ICC map implemented by the Self-Attention mechanism. ICC map eliminates the restrictions of network structure between the teacher and the student, and avoids the loss of beneficial transferring information. Our ICCT works in a "comprehensive" mode, which means the student can jointly utilize the fine-grained pairwise correlations from the teacher and add its own beliefs. Since the "comprehensive" mode provides an effective regularization, our ICCT can be flexibly implemented in all capacity-based T-S application scenarios. Experiments validate the availability of our ICCT and show that it is superior to existing frameworks in all capacity-based scenarios, Cap T > Cap S , Cap T = Cap S , and Cap T < Cap S , with different network structures, regardless of the complexity of the task. Moreover, due to defining the knowledge at the output layer, our ICCT is easy to extend by adding other hidden-layers based knowledge transfer techniques. In the future, we will study other forms of ICC besides Self-Attention and extend the pairwise correlations to tripartite or more correlations. We believe the performance of ICCT can be further promoted with additional Inter-Class Correlation. Furthermore, we will consider the inter-sample correlations to verify whether they can be useful to transfer. This may be helpful to the research of the random crop strategies. Fig. 3 . 3The Self-Attention based ICC maps of the four networks with a same input image of the CIFAR-10 dataset. The structure of networks will explain in Section IV. The row/column in the matrix contain all the pairwise ICC between a specific class and each class. Brighter colors indicate lower Inter-Class Correlation extents. be transferred. As with the normalization in training single neural network, we choose the softmax function to generate the normalized formÃ s = [ã s ij ] ∈ R N ×N , Fig. 5 . 5Distributed visualization of Feature maps. Feature maps come from the four training methods for resnet-18 on the CIFAR-10. Figuer (A) represents the feature map visualization of the baseline network. Figure (B) represents the feature map visualization of the student, which is trained by our ICCT in the Cap T < Cap S scenario. Figure (C) represents the feature map visualization of the student, trained by our ICCT in the Cap T = Cap S scenario. Figure (D) represents the feature map visualization of the student, trained by our ICCT in the Cap T > Cap S scenario. baselines contain ResNet-18, ResNet-50, ResNeXt-101 (64 × 4d), MobileNetV2, WRN-18-1, WRN-18-2, WRN-34-1, WRN-34-2 We abbreviate ResNeXt-101 (64 × 4d) as ResNeXt-101. arXiv:2008.10444v1 [cs.CV] 11 Aug 2020T S Cap Cap < T S Cap Cap > = T S Cap Cap Teacher Student Fig. 1. Three capacity-based application scenarios (Cap T > Cap S , TABLE I PARAMETERSTABLE II TEST IIIOF NETWORKS. ERROR (%) IN CAPACITY-BASED APPLICATION SCENARIO Cap T > Cap S ON THE CIFAR-100. "S(B)" AND "T(B)" ARE SHORT FOR "THE ERROR RATE OF THE BASELINE OF THE STUDENT" AND "THE ERROR RATE OF THE BASELINE OF THE TEACHER," RESPECTIVELY. IN CAPACITY-BASED APPLICATION SCENARIO Cap T = Cap S ON THE CIFAR-100.Networks CNN-5 WRN-16-2 WRN-28-2 MobileNetV2 ResNet-18 WRN-16-10 ResNeXt-29 WRN-28-10 ResNet-101 Parameters 0.36M 0.70M 1.48M 3.42M 11.17M 17.12M 34.43M 36.50M 42.51M "NA" MEANS "NOT AVAILABLE." Heterogeneity S T PR S(B) T(B) KD LT AT SP ICCT ICCT+AT ICCT+SP Architecture CNN-5 ResNeXt-29 1:95.64 39.66 17.70 37.93 38.16 NA 37.71 37.35 37.02 36.87 MobileNetV2 ResNeXt-29 1:10.07 34.82 17.70 33.78 33.95 33.26 32.81 31.84 31.43 31.12 ResNet-18 ResNeXt-29 1:3.08 24.34 17.70 23.35 23.32 23.17 22.98 22.32 21.94 21.45 Depth ResNet-18 ResNet-101 1:3.81 24.34 22.19 23.52 23.59 23.43 23.19 22.41 22.15 21.76 WRN-16-10 WRN-28-10 1:2.13 21.56 20.48 21.10 21.22 20.97 20.84 20.53 20.36 20.05 Width WRN-28-2 WRN-28-10 1:24.66 25.28 20.48 24.73 24.82 24.56 24.27 23.61 23.36 23.14 Depth & Width WRN-16-2 WRN-28-10 1:52.14 27.29 20.48 26.84 26.92 26.58 26.27 25.66 25.42 25.23 Architecture ResNet-101 ResNeXt-29 1:0.81 22.19 17.70 21.66 21.72 21.48 21.30 20.78 20.61 20.54 TABLE III TEST ERROR (%) Model Generations KD LT Tf-KD ICCT ResNet-18 Baseline 24.34 24.34 24.34 24.34 Gen #1 23.97 23.83 23.15 22.90 Gen #2 23.62 23.96 23.32 22.97 Gen #3 23.59 23.68 23.49 22.74 Gen #4 23.84 23.63 23.22 23.08 ResNet-101 Baseline 22.19 22.19 22.19 22.19 Gen #1 21.98 21.83 21.06 20.86 Gen #2 21.80 21.86 20.73 20.95 Gen #3 21.76 21.81 20.87 20.84 Gen #4 21.91 22.04 21.45 20.87 MobileNetV2 Baseline 34.82 34.82 34.82 34.82 Gen #1 34.65 34.04 33.74 33.25 Gen #2 34.12 34.15 33.68 32.07 Gen #3 33.90 34.28 32.87 32.64 Gen #4 34.18 34.47 33.02 32.92 TABLE RESPECTIVELY.Heterogeneity S T S(B) T(B) KD LT ICCT Architecture ResNeXt-29 CNN-5 17.70 39.66 17.58 17.61 17.48 ResNeXt-29 MobileNetV2 17.70 34.82 17.49 17.55 17.41 ResNeXt-29 ResNet-18 17.70 24.34 17.43 17.46 17.33 Depth ResNet-101 ResNet-18 22.19 24.34 21.96 22.07 21.23 WRN-28-10 WRN-16-10 20.48 21.56 20.28 20.35 20.07 Width WRN-28-10 WRN-28-2 20.48 25.28 20.32 20.38 20.15 Depth & Width WRN-28-10 WRN-16-2 20.48 27.29 20.37 20.41 20.26 TABLE V TEST VERROR (%) IN CAPACITY-BASED APPLICATION SCENARIO Cap T > Cap S ON THE ILSVRC2012. "S(B)" AND "T(B)" ARE SHORT FOR"THE ERROR RATE OF THE BASELINE OF THE STUDENT" AND "THE ERROR RATE OF THE BASELINE OF THE TEACHER," RESPECTIVELY.Heterogeneity S T S(B) T(B) KD LT AT SP ICCT ICCT+AT ICCT+SP Architecture ResNet-18 ResNeXt-101 29.12 20.40 28.74 28.94 28. 31 28.02 27.66 27.52 27.41 MobileNetV2 ResNeXt-101 28.13 20.40 27.90 27.97 27.43 27.12 26.74 26.63 26.45 ResNet-50 ResNeXt-101 22.71 20.40 22.16 22.32 21.87 21.75 21.22 21.03 20.86 Depth ResNet-18 ResNet-50 29.12 22.71 28.78 28.94 28.65 28.42 28.19 27.86 27.73 WRN-18-2 WRN-34-2 25.56 23.35 25.28 25.30 25.14 25.03 24.81 24.62 24.47 Width WRN-34-1 WRN-34-2 26.74 23.35 26.37 26.45 26.32 26.18 25.86 25.79 25.56 Depth & Width WRN-18-1 WRN-34-2 30.38 23.35 29.96 30.08 29.71 29.57 29.32 29.25 29.04 TABLE VI TEST ERROR (%) IN CAPACITY-BASED APPLICATION SCENARIO Cap T = Cap S ON THE ILSVRC2012. Name Generation KD LT Tf-KD ICCT ResNet-18 Baseline 29.12 29.12 29.12 29.12 Gen #1 29.03 29.08 28.84 28.91 Gen #2 28.94 29.13 28.75 28.66 Gen #3 28.91 28.96 28.83 28.25 Gen #4 28.97 29.04 28.72 28.41 MobileNetV2 Baseline 28.13 28.13 28.13 28.13 Gen #1 27.97 28.15 27.89 26.86 Gen #2 28.03 28.02 27.54 26.77 Gen #3 27.94 28.06 26.97 27.50 Gen #4 27.96 28.03 27.15 26.91 TABLE VII TEST VIIERROR (%) IN CAPACITY-BASED APPLICATION SCENARIO Cap T < Cap S ON THE ILSVRC2012. "S(B)" AND "T(B)" ARE SHORT FOR"THE ERROR RATE OF THE BASELINE OF THE STUDENT" AND "THE ERROR RATE OF THE BASELINE OF THE TEACHER," RESPECTIVELY.Heterogeneity S T S(B) T(B) KD LT ICCT Architecture ResNeXt-101 ResNet-18 20.40 29.12 20.26 20.34 20.15 ResNeXt-101 MobileNetV2 20.40 28.13 20.21 20.28 20.03 ResNeXt-101 resnet-50 20.40 22.71 20.16 20.25 19.97 Depth resnet-50 ResNet-18 22.71 29.12 22.44 22.56 22.10 WRN-34-2 WRN-18-2 23.35 25.56 23.08 23.14 22.95 Width WRN-34-2 WRN-34-1 23.35 26.74 23.19 23.21 23.13 Depth & Width WRN-34-2 WRN-18-1 23.35 30.38 23.22 23.24 23.18 Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. N Shazeer, A Mirhoseini, K Maziarz, A Davis, Q Le, G Hinton, J Dean, Proceedings of the International Conference on Learning Representations. the International Conference on Learning RepresentationsN. Shazeer, A. Mirhoseini, K. Maziarz, A. Davis, Q. Le, G. Hinton, and J. 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[ "Beta-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes from deformed quasiparticle random-phase approximation", "Beta-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes from deformed quasiparticle random-phase approximation" ]	[ "P Sarriguren \nInstituto de Estructura de la Materia\nIEM-CSIC\nSerrano 123E-28006MadridSpain\n" ]	[ "Instituto de Estructura de la Materia\nIEM-CSIC\nSerrano 123E-28006MadridSpain" ]	[]	β-decay properties of even and odd-A neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes involved in the astrophysical rapid neutron capture process are studied within a deformed proton-neutron quasiparticle random-phase approximation. The underlying mean field is described self-consistently from deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual interactions in the particle-hole and particle-particle channels are also included in the formalism. The isotopic evolution of the various nuclear equilibrium shapes and the corresponding charge radii are investigated in all the isotopic chains. The energy distributions of the Gamow-Teller strength as well as the β-decay half-lives are discussed and compared with the available experimental information. It is shown that nuclear deformation plays a significant role in the description of the decay properties in this mass region. Reliable predictions of the strength distributions are essential to evaluate decay rates in astrophysical scenarios. PACS numbers: 21.60.Jz, 23.40.Hc, 27.60.+j, 26.30.-k J π exp ( 95 Sr) = 1/2+ J π exp ( 97 Sr) = 1/2+ J π exp ( 99 Sr) = 3/2+ J π exp ( 101 Sr) = (5/2-) FIG. 5: (Color online) Same as in Fig. 2, but for Sr isotopes. Experimental charge radii are from [59].	10.1103/physrevc.91.044304	[ "https://arxiv.org/pdf/1504.01640v1.pdf" ]	73,699,215	1504.01640	5893ae6a37f2c471329ad0da6d551c2c37673a77	Beta-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes from deformed quasiparticle random-phase approximation 7 Apr 2015 (Dated: April 8, 2015) P Sarriguren Instituto de Estructura de la Materia IEM-CSIC Serrano 123E-28006MadridSpain Beta-decay properties of neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes from deformed quasiparticle random-phase approximation 7 Apr 2015 (Dated: April 8, 2015) β-decay properties of even and odd-A neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes involved in the astrophysical rapid neutron capture process are studied within a deformed proton-neutron quasiparticle random-phase approximation. The underlying mean field is described self-consistently from deformed Skyrme Hartree-Fock calculations with pairing correlations. Residual interactions in the particle-hole and particle-particle channels are also included in the formalism. The isotopic evolution of the various nuclear equilibrium shapes and the corresponding charge radii are investigated in all the isotopic chains. The energy distributions of the Gamow-Teller strength as well as the β-decay half-lives are discussed and compared with the available experimental information. It is shown that nuclear deformation plays a significant role in the description of the decay properties in this mass region. Reliable predictions of the strength distributions are essential to evaluate decay rates in astrophysical scenarios. PACS numbers: 21.60.Jz, 23.40.Hc, 27.60.+j, 26.30.-k J π exp ( 95 Sr) = 1/2+ J π exp ( 97 Sr) = 1/2+ J π exp ( 99 Sr) = 3/2+ J π exp ( 101 Sr) = (5/2-) FIG. 5: (Color online) Same as in Fig. 2, but for Sr isotopes. Experimental charge radii are from [59]. I. INTRODUCTION The rapid structural changes occurring in the ground state and low-lying collective excited states of neutronrich nuclei in the mass region A = 80 − 128 have been extensively studied both theoretical and experimentally (see e.g. [1,2] and references therein). From the theoretical side, the equilibrium nuclear shapes in this mass region have been shown to suffer rapid changes as a function of the number of nucleons with competing spherical, axially symmetric prolate and oblate, and triaxial shapes at close energies. Both relativistic [3,4] and nonrelativistic [5][6][7][8][9] approaches agree in the general description of the nuclear structural evolution in this mass region, which is supported experimentally by spectroscopic studies [10][11][12], 2 + lifetime measurements [13][14][15] and quadrupole moments for rotational bands [15], as well as by laser spectroscopy measurements [16]. However, the nuclear structure richness is not the only attractive feature characterizing these nuclei. Another remarkable property of nuclei in this mass region is that they are involved in the astrophysical rapid neutron capture process (r process), which is considered to be one of the main nucleosynthesis mechanisms leading to the production of heavy neutron-rich nuclei in the universe [17,18]. The r-process nucleosynthesis involves many neutron-rich unstable isotopes, whose neutron capture rates, masses, and β-decay half-lives (T 1/2 ) are crucial quantities to understand the possible r-process paths, the isotopic abundances, and the time scales of the process [18][19][20]. Although much progress has been done measuring masses (see for example the Jyväskylä mass database [21]) and half-lives [22][23][24], most of the nuclear properties of relevance for the r process are experimentally unknown * Electronic address: [email protected] due to their extremely low production yields in the laboratory. Therefore, reliable nuclear physics models are required to simulate properly the r process. The quasiparticle random-phase approximation (QRPA) is considered a well suited model to describe medium-mass open-shell nuclear properties and specifically β-decay properties. QRPA calculations for neutron-rich nuclei have been carried out within different spherical formalisms, such as Hartree-Fock-Bogoliubov (HFB) [25], continuum QRPA with density functionals [26], and relativistic mean field approaches [27]. However, the mass region we are dealing with requires nuclear deformation as a relevant degree of freedom to characterize the nuclear structure involved in the calculation of the β-strength functions. The deformed QRPA formalism was developed in Refs. [28][29][30][31], using phenomenological mean fields. A Tamm-Dancoff approximation with Sk3 interaction was also implemented in Ref. [32]. More recently, deformed QRPA calculations using deformed Woods-Saxon potentials and realistic CD-Bonn residual forces have been performed in [33,34]. First-forbidden transitions were also considered in those references, showing that their effect in this mass region can be neglected. Various self-consistent deformed QRPA calculations to describe the β-decay properties, either with Skyrme [35] or Gogny [36] interactions are also available in the literature. In Refs. [37,38] the decay properties of neutronrich Zr and Mo isotopes were studied within a deformed proton-neutron QRPA based on a self-consistent Hartree-Fock (HF) mean field formalism with Skyrme interactions and pairing correlations in BCS approximation. Residual spin-isospin interactions were also included in the particle-hole and particle-particle channels [39,40]. In this work this study is extended to the neighboring regions including even and odd-A neutron-rich Ge, Se, Kr, Sr, Ru, and Pd isotopes. These calculations are timely because they address a mass region which is at the bor-derline of present experimental capabilities for measuring half-lives at MSU and RIKEN [22][23][24]. In addition, theoretical calculations can be tested with the available experimental information on half-lives providing simultaneously predictions for the underlying Gamow-Teller strength distributions and for the half-lives of more exotic nuclei not yet measured. Finally, this more comprehensive study allows one to judge better the extent to which the method is able to describe the decay properties of nuclei in a wider mass region that includes spherical, well deformed, and weakly deformed transitional isotopes, as well as isotopes exhibiting shape coexistence. Therefore, the theoretical method will be tested over a rich set of different nuclear structures that will reveal the limitations of the model. The paper is organized as follows. In Sec. II a review of the theoretical formalism used is introduced. Section III contains the results obtained for the potential energy curves (PEC), Gamow-Teller (GT) strength distributions, and β-decay half-lives, which are compared with the experimental data. Section IV summarizes the main conclusions. II. THEORETICAL FORMALISM A summary of the theoretical framework used in this paper to describe the β-decay properties in neutron-rich isotopes is shown in this section. More details of the formalism can be found elsewhere [39,40]. The method starts from a self-consistent calculation of the mean field by means of a deformed Skyrme Hartree-Fock procedure with pairing correlations in BCS approximation. Singleparticle energies, wave functions, and occupation amplitudes are generated from this mean field. The Skyrme interaction SLy4 [41] is used as a representative of modern Skyrme forces. It has been very successful describing nuclear properties all along the nuclear chart and has been extensively studied [6,7,42]. The solution of the HF equation, assuming time reversal and axial symmetry, is found by using the formalism developed in Ref. [43]. The single-particle wave functions are expanded in terms of the eigenstates of an axially symmetric harmonic oscillator in cylindrical coordinates, using twelve major shells. The pairing gap parameters for protons and neutrons in the BCS approximation are determined phenomenologically from the odd-even mass differences [44]. In a further step, constrained HF calculations with a quadratic constraint are performed to construct the PECs, analyzing the nuclear binding energies in terms of the quadrupole deformation parameter β. Calculations for GT strengths are performed subsequently for the various minima in the energy curves indicating the equilibrium shapes of each nucleus. Since decays connecting different shapes are disfavored, similar shapes are assumed for the ground state of the parent nucleus and for all populated states in the daughter nucleus. The validity of this assumption was discussed for example in Refs. [28,30]. To describe GT transitions, a separable spin-isospin residual interaction in the particle-hole (ph) and particleparticle (pp) channels is added to the mean field and treated in a deformed proton-neutron QRPA [28-32, 39, 40, 45]. An optimum set of coupling strengths could be chosen following a case by case fitting procedure and one will finally get different answers depending on the nucleus, shape, and Skyrme force. However, since the purpose here is to test the ability of QRPA to account for the GT strength distributions in this mass region with as few free parameters as possible, the same coupling strengths are used for all the nuclei considered in this paper, which are taken from previous works [37,38]. We use χ ph GT = 0.15 MeV and κ pp GT = 0.03 MeV for the residual interaction in the ph and pp channels, respectively. The sensitivity of the GT strength distributions to the various ingredients contributing to the deformed QRPA calculations, namely to the nucleon-nucleon effective force, to deformation, to pairing correlations, and to residual interactions, have been investigated in the past [39,40,[46][47][48]. In this work the most reasonable choices found in those references are used. Summarizing the various sensitivities, the conclusion is that the main features of the GT strength distributions are in general very robust against the Skyrme force used, showing some more sensitivity in the spherical cases, where the location of the single-particle energies is more critical to determine the excitation energies of the GT transitions. Deformation has been shown to be an important issue to describe the profiles of the GT strength distributions. First, because the degeneracy of the spherical shells is broken making the GT strength distributions more fragmented than the corresponding spherical ones. Secondly, because the energy levels of deformed orbitals cross each other in a way that depends on the magnitude of the quadrupole deformation as well as on the oblate or prolate character. This level crossing may lead in some instances to sizable differences in the GT profiles, a fact that has been exploited to learn about the nuclear shape from the measured β-decay pattern [49][50][51]. Pairing correlations are also important to describe nuclei out of closed shells. Their influence on the GT profiles was studied in Ref. [40], concluding that the main effect is to decrease slightly the strength at low energies and to create new transitions, mainly at high energies, that are forbidden in the absence of such correlations. The effect of the ph and pp residual interactions is also well known. The repulsive ph interaction redistributes the GT strength by shifting it to higher excitation energies causing a displacement of the GT resonance. It also reduces somewhat the total strength. The attractive pp interaction moves the strength to lower energies. Its effect on the GT resonance is in general negligible, but nevertheless, the changes induced in the low-energy region are of great relevance in the calculation of the βdecay half-lives, which are only sensitive to the strength contained in the energy region below the Q-energy window. The GT transition amplitudes in the intrinsic frame connecting the ground state \|0 + of an even-even nucleus to one phonon states in the daughter nucleus \|ω K (K = 0, 1) are found to be ω K \|σ K t ± \|0 = ∓M ωK ± ,(1) where M ωK − = πν (q πν X ωK πν +q πν Y ωK πν ) ,(2) M ωK + = πν (q πν X ωK πν + q πν Y ωK πν ) ,(3) withq πν = u ν v π Σ νπ K , q πν = v ν u π Σ νπ K ,(4) in terms of the occupation amplitudes for neutrons and protons v ν,π (u 2 ν,π = 1 − v 2 ν,π ) and the matrix elements of the spin operator, Σ νπ K = ν \|σ K \| π , connecting proton and neutron single-particle states, as they come out from the HF+BCS calculation. X ωK πν and Y ωK πν are the forward and backward amplitudes of the QRPA phonon operator, respectively. Once the intrinsic amplitudes in Eq. (1) are calculated, the GT strength B ω (GT ± ) in the laboratory system for a transition (5) in [g 2 A /4π] units. To obtain this expression, the initial and final states in the laboratory frame have been expressed in terms of the intrinsic states using the Bohr-Mottelson factorization [52]. I i K i (0 + 0) → I f K f (1 + K) can be obtained as B ω (GT ± ) = ωK ω K=0 σ 0 t ± 0 2 δ(ω K=0 − ω) +2 ω K=1 σ 1 t ± 0 2 δ(ω K=1 − ω) , The specific treatment of odd-A systems has been described [31,47] by considering two types of GT contributions. One type is due to phonon excitations in which the odd nucleon acts only as a spectator. The transition amplitudes in the intrinsic frame are in this case basically the same as in the even-even case, but with the blocked spectator excluded from the calculation. The other type of transitions involves the odd nucleon and is treated perturbatively by taking into account phonon correlations to first order in the quasiparticle transitions. The excitation energies of the GT states with respect to the ground state in the daughter nuclei have been discussed in Ref. [47] for both types of transitions in terms of the QRPA phonon energy and the quasiparticle energies. The β-decay half-life is obtained by summing all the allowed transition strengths to states in the daughter nucleus with excitation energies lying below the correspond- r c (fm) Ru (a) (b) 5/2- 5/2+ 7/2- 3/2+ 5/2- 7/2- 9/2- 9/2- 7/2- 1/2+ 7/2-ing Q-energy, Q β ≡ Q β − = M (A, Z)− M (A, Z + 1)− m e , written in terms of the nuclear masses M (A, Z) and the electron mass (m e ), and weighted with the phase space factors f (Z, Q β − E ex ), T −1 1/2 = (g A /g V ) 2 eff D 0<Eex<Q β f (Z, Q β − E ex ) B(GT, E ex ) ,(6) with D = 6200 s and (g A /g V ) eff = 0.77(g A /g V ) free , where 0.77 is a standard quenching factor. The same quenching factor is included in all the figures shown later for the GT strength distributions. The bare results can be recovered by scaling the results in this paper for B(GT ) and T 1/2 with the square of this quenching factor. The Fermi integral f (Z, Q β −E ex ) is computed numerically for each value of the energy including screening and finite size effects, as explained in Ref. [53], f β ± (Z, W 0 ) = W0 1 pW (W 0 − W ) 2 λ ± (Z, W )dW , (7) with λ ± (Z, W ) = 2(1 + γ)(2pR) −2(1−γ) e ∓πy \|Γ(γ + iy)\| 2 [Γ(2γ + 1)] 2 ,(8) where γ = 1 − (αZ) 2 ; y = αZW/p ; α is the fine structure constant and R the nuclear radius. W is the total energy of the β particle, W 0 is the total energy available in m e c 2 units, and p = √ W 2 − 1 is the momentum in m e c units. III. RESULTS AND DISCUSSION In this section I present first the PECs in the isotopic chains studied. Quadrupole deformation parameters as well as charge r.m.s. radii (r c ) are analyzed as a function of the mass number. Then, energy distributions of the GT strength corresponding to the local minima in the PECs are calculated. Finally, half-lives are evaluated and compared with the experiment. A. Structural isotopic evolution In Fig. 1 the PECs, i.e., the energies relative to that of the ground state, are plotted as a function of the quadrupole deformation β for the neutron-rich Ge, Se, Kr, Sr, Zr, Mo, Ru, and Pd isotopes. The results correspond to the SLy4 interaction. The isotopes covered in this study include middle-shell nuclei with proton numbers between shell closures Z = 28 and Z = 50, namely Z = 32, 34,36,38,40,42,44,46 and neutron numbers between shell closures N = 50 (as in 82 Ge) and N = 82 (as in the heaviest 128 Pd). In most of the isotopic chains one can see the appearance of several equilibrium nuclear shapes, whose relative energies change with the number of neutrons. In Ge isotopes, prolate shapes that are ground states in the lighter isotopes are found with the only exception of 82 Ge, where a spherical shape is found in accordance with its N = 50 semi-magic character. At N = 58, 60 ( 90,92 Ge) oblate and prolate shapes are practically degenerate in energy and oblate shapes become ground states for heavier isotopes. The case of Se isotopes is similar with oblate and prolate minima all along the isotopic chain. The lighter (heavier) isotopes have prolate (oblate) ground states with transitional isotopes around N = 58, 60 ( 92,94 Se). In this case the energy barriers are more pronounced than in the case of Ge isotopes. Kr isotopes show competing shapes in the lighter isotopes that become oblate at N = 58, 60 ( 94,96 Kr) and then turn into prolate shapes beyond 98 Kr. In the heavier isotopes, as in the case of Se isotopes, shape coexistence is found with very well developed oblate and prolate minima separated with high energy barriers. Sr isotopes show a transition from oblate at N = 58 ( 96 Sr) to prolate at N = 60 ( 98 Sr) with a two minima structure for heavier isotopes. The cases of Zr and Mo isotopes were discussed in Refs. [37,38]. Both oblate and prolate minima are observed in the lighter isotopes of Zr and Mo with prolate ground states. Whereas the prolate shape remains ground state in most of the heavier Zr isotopes, oblate shapes are lower in energy for the heavier Mo isotopes. Finally, Ru and Pd isotopes show oblate and prolate minima in the lighter isotopes and a gradual transition into spherical shapes as one approaches the shell closure at N = 82. In summary, a large diversity of nuclear structures are found in this mass region, from spherical to well deformed shapes, passing through soft transitional nuclei and even possible shape-coexistence structures. This rich variety of shapes represents a challenge to any theoretical model trying to describe them in a unified manner. In the next subsections the results are compared with the available experimental data, which are restricted at present to βdecay half-lives. Then, the theoretical approach will be tested against this information and the limitations of the model will be established. These results are in qualitative agreement with similar calculations obtained in this mass region from different theoretical approaches, including macroscopicmicroscopic methods based on liquid drop models with shell corrections [54,55], relativistic mean fields [56], as well as nonrelativistic calculations with Skyrme [5] and Gogny [8,9,57] interactions. Thus, a consistent theoretical description emerges, which is supported by the still scarce experimental information available [2,[10][11][12][13][14][15][16]58]. The isotopic evolution can be better appreciated in Figs. 2-7, where quadrupole deformations β (a) and r.m.s. charge radii r c (b) of the various energy minima are plotted as a function of the mass number A. The deformation corresponding to the ground state for each isotope is encircled. Also shown in these figures for odd-A isotopes, are the spin and parity (J π ) of the different shapes and the experimental assignments [44]. The experimental assignments based on systematics estimated from trends in neighboring nuclides have not been included. In Fig. 2 for Ge isotopes one can see clearly the shape transition at A = 90 − 92 (N = 58 − 60) from prolate shapes with β ≈ 0.2 to oblate shapes with β ≈ −0.2. Charge r.m.s. radii have not been measured in these isotopes, but it is expected from these calculations a very smooth behavior given that the magnitude of β in the prolate and oblate sectors are very similar. Fig. 3 shows the analogous results for Se isotopes. In this case one can see the transition from prolate (β ≈ 0.2) to oblate (β ≈ −0.2) at A = 92 (N = 58). The prolate shape grows in the heavier isotopes (β ≈ 0.3), but they are never ground states and then, the expected increasing in the charge radii is smooth. Kr isotopes in Fig. 4 show first a shape transition from prolate (β ≈ 0.15) to oblate (β ≈ −0.25) and a subsequent transition from oblate to prolate (β ≈ 0.35) shapes. The radii are sensitive to this transitions, although the measured radii [59] seem to favored prolate shapes in the lighter isotopes. Sr isotopes in Fig. 5 show a clear transition from oblate to strong prolate (β ≈ 0.4) deformations at A = 96 − 98 (N = 58 − 60). This shape transition is well correlated with the trend change observed in the charge radii that shows a sizable jump between 96 Sr and 98 Sr both theoretical and experimentally [59]. In the case of Ru (Pd) isotopes shown in Fig. 6 (7), one can see a smooth transition from deformed oblate (prolate) solutions in the lighter isotopes to spherical shapes in the heavier ones. This change is felt in the trends of the radii, but no experiments are yet available to compare with. Spins and parities in odd-A isotopes can be compared with their experimental assignments. In the Ge isotopes the calculations agree reasonably well with the assignments taking into account that oblate and prolate shapes are very close in energy and that a 1/2 + isomer is observed experimentally in 83 Ge at 248 keV. In the lighter Se isotopes, 1/2 + and 3/2 + states are obtained, whereas experimental assignments are (5/2 + ). In both isotopes, 5/2 + states very close in energy to the ground states are also obtained, although somewhat above. Similarly, in the lighter Kr isotopes the experimental assignments are obtained very close in energy to the ground states, although slightly above. On the other hand, a 7/2 + isomer is experimentally observed in 93 Kr at 355 keV that corresponds to the ground state here. Sr isotopes exhibit a nice agreement. The measured spin and parities of ground states in 95,99,101 Sr correspond to the prolate calculations. A (7/2 + ) state is also observed experimentally in 95 Sr at 56 keV. In 97 Sr the observed 1/2 + ground state appears as an excited state. It is also worth noting that the prolate ground state (3/2 − ) for this isotope is observed experimentally at 645 keV. In the case of Ru isotopes the measured J π are difficult to reproduce. They are found in the calculations, but not as ground states. On the other hand, the negative parity 7/2 − states found in the calculations are also seen experimentally at low energies. In particular, an isomeric state (7/2 − ) at an undetermined energy has been seen in 113 Ru. Finally, in Pd isotopes the negative parity isomers, which are oblate in this description, are reproduced in the calculations, but not the ground states. B. Gamow-Teller strength distributions In the next figures, the energy distributions of the GT strength corresponding to the various deformed equilibrium shapes are shown for each isotopic chain. The results are obtained from QRPA with the force SLy4 with pairing correlations and with residual interactions with the parameters written in Sec. II. The GT strength is plotted versus the excitation energy of the daughter nucleus with a quenching factor 0.77. Zr and Mo isotopes were already studied in Refs. [37,38] and are not repeated here. increases with N , as it is expected to fulfill the Ikeda sum rule. The various shapes produce quite similar GT strength distributions on a global scale. Nevertheless, the small differences among the various shapes at the low energy tails (below the Q β ) of the GT strength distributions that can be appreciated because of the logarithmic scale, lead to sizable effects in the β-decay half-lives. Unfortunately, comparison with experiment is still not possible for the GT strength distributions, the measured half-lives will be compared to the calculations in the next subsection. Comparison with calculated GT distributions from other theoretical approaches is also restricted to the few cases where these results have been published [33,35]. In Refs. [33,60] the authors performed QRPA calculations with deformed Woods-Saxon potentials and realistic CD-Bonn residual forces using the G-matrix formalism and compared these results with the results obtained from separable forces. While in Ref. [33] the comparison between the results obtained from realistic or separable residual interactions is restricted to the halflives, in Ref. [60] the authors compared those results in the context of two-neutrino double-beta decay, conclud- ing that both approaches, realistic and separable, lead to similar results. On the other hand, in Ref. [35] the Skyrme force SLy4 was used to generate the mean field as it is done in this work. The residual interaction in the ph channel was self-consistently introduced and not reduced to a separable form. Finally the pp residual interaction was written as a contact force with a coupling strength fitted to reproduce the half-life in 100 Zr. The GT strength distributions in neutron-rich Zr isotopes obtained from this approach were compared with the corresponding distributions obtained with separable forces in Figs. 5-6 in Ref. [35]. From this comparison one can conclude that in many aspects the main characteristics of the consistent force are maintained by a separable force with a much lower computational cost. The comparison of the half-lives shows also a remarkable agreement between both approaches. In the next figures, Figs. 14-19, one can see in more detail the accumulated GT strength in the energy region below the corresponding Q β energy of each isotope, which is the relevant energy range for the calculation of the half-lives. The vertical solid (dashed) arrows show the Q β (S n ) energies, taken from experiment [44]. In these figures the sensitivity of these distributions to deformation can be appreciated and one can understand that measurements of the GT strength distribution from β-decay can be, in particular cases, an additional source of information about the nuclear deformation, as it was shown in Refs. [49][50][51]. The GT strength distribution in odd-A isotopes is found to be displaced to higher energies (typically about 2-3 MeV) with respect to the even-even case. The shift corresponds roughly to the breaking of a neutron pair and therefore it amounts to about twice the neutron pairing gap. Below this energy only transitions involving the odd nucleon are possible. The energy distribution of the GT strength is fundamental to constrain the underlying nuclear structure. For a theoretical model, it represents a more demanding test than just reproducing half-lives or total GT strengths that are integral quantities obtained from these strength distributions properly weighted with phase factors (see Eq. 6). These quantities might be reproduced even with wrong strength distributions. This is of especial importance in astrophysical scenarios of high densities and tem- peratures that cannot be reproduced in the laboratory. Given that the phase factors in the stellar medium are different from those in the laboratory, the stellar half-lives become dependent on the electron distribution in the stellar plasma that eventually may block the β-particle emission [61]. Therefore, to describe properly the decay rates under extreme conditions of density and temperature, it is not sufficient to reproduce the half-lives in the laboratory. One needs, in addition, to have a reliable description of the GT strength distributions [62,63]. C. Beta-decay Half-lives The calculation of the half-lives in Eq. (6) involves knowledge of the GT strength distribution and of the β energies (Q β − E ex ), which are evaluated by using Q β values obtained from the mass differences between parent and daughter nuclei obtained from SLy4 with a zerorange pairing force and Lipkin-Nogami obtained from the code HFBTHO [64]. In Figs dots, open dots stand for experimental values from systematics) [23,44] are compared with the theoretical results obtained with the prolate, oblate, and spherical equilibrium shapes, for the various isotopic chains. In Fig. 20 one can see the half-lives for Ge isotopes. The lighter isotopes are not well reproduced, being largely overestimated. This point will be discussed later. The half-lives obtained from oblate shapes are larger than the corresponding prolate ones. This feature is correlated with the GT strength contained below the Q β energy in Figs. 8 and 14. Prolate shapes, which are closer to experiment, are also the ground states in this range of masses according to the calculations (see Fig. 2). For heavier isotopes, the half-lives for oblate and prolate shapes are very similar. In the case of Se isotopes in Fig. 21, the calculations also overestimate the half-lives of the lighter isotopes, but the agreement with experiment is in this case much better. In the middle region the experimental half-lives, which are taken from systematics, are reasonably well reproduced. The half-lives of heavier isotopes exhibit a rather flat behavior. Half-lives of Kr isotopes are shown in Fig. 22. As in the previous figures, the half-lives from the oblate shapes are larger than the prolate ones in the lighter Kr isotopes, but the situation is reversed at 94 Kr. This is again nicely correlated with the GT strength at low excitation energies shown in Fig. 16. In general, the half-lives in the middle region are well described. This is also true for Sr and Ru isotopes in Figs. 23 and 24, respectively, where the trends observed experimentally are well reproduced, except for the lighter Sr isotopes that are clearly underestimated and the heavier Ru isotopes, where the data from systematics fall down faster than the calculations. Finally, in the case of Pd isotopes, shown in Fig. 25, the calculations underestimate (overestimate) the measured half-lives in the lighter (heavier) isotopes. All in all, the agreement with experiment is reasonable, especially in the middle regions. These regions contain in general well deformed nuclei, where the present approach is more suitable. On the other hand, weakly deformed transitional isotopes, such as light Ge and Se isotopes and heavy Ru and Pd isotopes are not so well described. Furthermore, in the light isotopes of all the isotopic chains, which are closer to the valley of stability, the half-lives are larger because of the small Q β energies involved. In these cases the half-lives are determined exclusively by the very low energy tail of the GT strength distribution contained in the narrow window below Q β . Therefore, tiny variations in the description of the GT strength distribution in the low-lying energy region can drive sizable effects in the half-lives. Of course it is also important to describe the half-lives of the long-lived iso- Fig. 20, but for Kr isotopes. Experimental half-lives are from [23,44]. topes, but their significance to constrain the GT strength distribution is minor since the half-lives are insensitive to most of this distribution. Half-lives for neutron-rich Kr, Sr, Zr, and Mo isotopes calculated from self-consistent deformed QRPA calculations with the Gogny D1M interaction and experimental values of Q β [36] agree with the results in this work within the uncertainties of the calculations. The agreement is also very reasonable between the calculated half-lives and those obtained from deformed QRPA calculations using deformed Woods-Saxon potentials to generate the mean field and complemented with realistic CD-Bonn residual forces [33,34]. The agreement is also good with the results in Ref. [35] using the Skyrme force SLy4 with consistent residual interactions in the ph channel as mentioned earlier. Fig. 7 in that reference displays this comparison. It is also worth noticing that the worst agreement with experiment occurs in the light Ge isotopes, as well as in heavy Pd isotopes. In these cases the calculations overestimate the experiment leaving room for contributions coming from first forbidden (FF) transitions. One can understand from simple qualitative arguments that the role of FF transitions is expected to be more important in lighter Ge and in Pd isotopes. Thus, for 32 negative-parity spherical shells. On the other hand the neutrons in 80−94 Ge isotopes occupy orbitals belonging to the 1g 9/2 , 2d 5/2 and 1g 7/2 positive-parity spherical shells. Therefore, in the β-decay one neutron in a positive-parity state is transformed into a proton that would sit in a negative-parity state, thus suppressing GT and favoring FF transitions in the low-lying transitions. This is particularly true for the lighter Ge isotopes. In the heavier ones, other neutron states with negative parity (1h 11/2 ) have to be considered because of deformation effects. The same argument can be applied to the lighter Se, Kr, and Sr isotopes, but in these cases proton states from positive parity (1g 9/2 ) are closer in energy and would participate in the decay favoring GT transitions. The situation is different in the case of Ru and Pd isotopes. Now the available proton states for the decay are of positive parity (1g 9/2 ), whereas most of the last occupied neutrons belong to negative-parity states (1h 11/2 ), thus favoring FF transitions. According to calculations [26,33] of the FF transitions in this mass region, minor effects are expected from them. Nevertheless, it would be very interesting in the future to study systematically the FF contributions in all the isotopes in this mass region. Another feature observed in the present calculations is the existence of some odd-even staggering effect in the calculated half-lives, which is not observed experimentally. This effect is particularly evident in Ru and Pd isotopes. There are not many calculations involving simultaneously even-even and odd-A isotopes, but some of them exhibit some sort of staggering effect as well [33]. The appearance of this effect in the half-lives suggests some deficiency in the model that might be related to the determination of ground-state energies in the odd-A systems [65]. Unfortunately, there are more sources of uncertainty related to the odd-A systems that should be considered as well [66], such as the spin and parity assignments, the blocking procedures or the treatment of the 1qp excitations involving the odd nucleon. This issue will be the subject of a future investigation in this direction. It is also interesting to look for the simultaneous appearance of structural effects that eventually can appear in different observables. One example can be seen in the evolution of the experimental half-lives with the number of neutrons in the isotopic chains. At some points one observes discontinuities in the general trends of behavior, such as in the mass regions 90,92 Se, 92,94 Kr, 96,98 Sr, and 118,120 Ru. These experimental findings on the half-lives are correlated with the shape transitions in Figs. 3-6 predicted in the model. One cannot state firmly that these sharp changes in the behavior of the half-lives are signatures of shape transitions, but certainly this correlation cannot be discarded given that a change of the deformation in the nuclear system involves a structural change to whom the half-lives are also sensitive. Finally, the impact of deformation on the decay properties can be better appreciated in a systematic comparison of the half-lives calculated with both the spherical approximation and the deformation that corresponds to the minimum of the PEC for each isotope. Then, Fig. 26 shows the ratios of the calculated and experimental half-lives for two sets of data corresponding to a spherical calculation (open dots) and to a deformed calculation (solid dots) at the self-consistent deformation that gives the minimum of the PECs. These ratios are plotted as a function of the experimental half-lives (a) and as a function of the quadrupole deformation at the minimum of the PECs (b). To increase the size of the sample, besides the isotopes considered in this work with measured half-lives, I have also included the set of Zr and Mo neutron-rich isotopes studied in Ref. [38] with measured half-lives. In the upper panel of Fig. 26 (a) one can see how deformation improves the description of the half-lives. Practically all the full black dots are contained within the horizontal lines defining the region of one order of magnitude agreement. On the other hand, the results from the spherical calculation are more spread out with larger discrepancy with experiment. One can also see that the results are better in both spherical and deformed calculations for shorter half-lives, whereas the results for larger half-lives show sizable deviations. The latter correspond to isotopes close to the valley of stability with small Q β -values, where the half-lives are only sensitive to the small portion of the GT strength distribution at low excitation energies below Q β . In the lower panel (b) one can see the results from a different point of view and it can be studied whether deformation improves the results evenly in the whole range of deformations or whether its effect is stronger at large deformations. Three regions of accumulation of results can be distinguished. Two of them correspond to well deformed nuclei located at β ≈ −0.2 and β ≈ 0.35. In these regions the deformed calculations clearly improve the results from the spherical ones that show a tendency to underestimate the experiment. The other region corresponds to 0 < β < 0.2 values, where nuclei are softer or transitional and the deformed formalism should be improved. In this region the results are more scattered than in the well deformed regions, but the deformed calculations show deviations that rarely exceed one order of magnitude, still representing an improvement over the spherical results. In order to have a quantitative estimation of the quality of the various calculations, following the analysis made in Ref. [29], the logarithms of the ratios of the calculated and experimental half-lives are introduced through the quantities r = log 10 T 1/2 (calc) T 1/2 (exp) . Then, the average position of the points, M r , the standard deviation, σ r , and the total error, Σ r , are defined as M r = 1 n n i=1 r i ; σ r = 1 n n i=1 (r i − M r ) 2 1/2 ; Σ r = 1 n n i=1 (r i ) 2 1/2 ,(10) and their corresponding factors M 10 r = 10 Mr , σ 10 r = 10 σr , and Σ 10 r = 10 Σr . The analysis of the results shown in IV. CONCLUSIONS A microscopic approach based on a deformed QRPA calculation on top of a self-consistent mean field obtained with the SLy4 Skyrme interaction has been used to study the nuclear structure and the decay properties of even and odd neutron-rich isotopes in the mass region A ≈ 80 − 130. The nuclear model and interaction have been successfully tested in the past providing good agreement with the available experimental information on bulk properties all along the nuclear chart. Decay properties in different mass regions have been well reproduced as well. The structural isotopic evolution has been studied from their PECs. Depending on the isotopic chain, a large variety of nuclear shapes is found, including spherical shapes, well developed deformed shapes, and transitional soft shapes. Charge radii have been also investigated, showing the connection between a discontinuous behavior in the isotopic trend with a shape transition and comparing the results with the available measurements from laser spectroscopy. Then, Gamow-Teller strength distributions and β-decay half-lives have been computed for the equilibrium shapes. The isotopic evolution of the GT strength distributions exhibits some typical features, such as GT resonances increasing in energy and strength as the number of neutrons increases. Effects of deformation are hard to see on a global scale, but they become apparent in the low excitation energy below Q β energies, a region that determines the half-lives. Half-lives have been calculated using Q β energies calculated with the force SLy4. In general, a reasonable agreement with experiment is obtained, especially in the short-lived nuclei of Ge, Se, Kr, Sr, and Ru isotopes. The results are comparable to other calculations using different approaches for the mean field and/or residual interactions. Special difficulties are found to describe properly the half-lives of the lighter Ge isotopes and the Pd isotopes. These are examples of transitional nuclei where the nuclear structure is more involved and the concept of a well defined shape might not be meaningful. A systematic comparison of the ratios of the calculated and experimental half-lives has been done using both spherical and deformed calculations, showing that the inclusion of deformation improves significantly the description of the decay properties. Experimental information on the energy distribution of the GT strength is a valuable piece of knowledge about nuclear structure in this mass region. The study of these distributions is within the current experimental capabilities in the case of the lighter isotopes considered in this work. Here, I have presented theoretical predictions for them based on microscopic calculations. Similarly, measuring the half-lives of the heavier isotopes will be highly beneficial to model the r process and to constrain theoretical nuclear models. This possibility is also open within present capabilities at RIKEN. FIG. 6 :FIG. 7 : 67(Color online) Same as inFig. 2, but for Ru isotopes. (Color online) Same as inFig. 2, but for Pd isotopes. FIG. 8 : 8(Color online) QRPA-SLy4 Gamow-Teller strength distributions for Ge isotopes as a function of the excitation energy in the daughter nucleus. The calculations correspond to the various equilibrium deformations found in the PECs. FIG. 9 : 9(Color online) Same as inFig. 8, but for Se isotopes. FIG. 10 : 10(Color online) Same as inFig. 8, but for Kr isotopes. FIG. 11 : 11(Color online) Same as inFig. 8, but for Sr isotopes. FIG . 12: (Color online) Same as inFig. 8, but for Ru isotopes. FIG . 13: (Color online) Same as inFig. 8, but for Pd isotopes. 14: (Color online) QRPA-SLy4 accumulated GT strengths in Ge isotopes calculated for the various equilibrium shapes. Q β and Sn energies are shown by solid and dashed vertical arrows, respectively. FIG. 15 : 15(Color online) Same as inFig. 14, but for Se isotopes. Figures 8 - 813 contain the results for Ge, Se, Kr, Sr, Ru, and Pd isotopes. The energy distributions of the individual GT strengths corresponding to the ground state shapes are shown, together with continuous distributions for the ground state shapes as well as for the other possible shapes, obtained by folding the strength with 1 MeV width Breit-Wigner functions. Q β values are shown with vertical arrows. In both cases, even and odd isotopes, the Q β values increase with the number of neutrons in each isotopic chain and the values in the odd-A isotopes (Z, N + 1) are about 2-3 MeV larger than the values in the neighbor even-even isotopes (Z,N ). The general structure of the GT distributions is characterized by the existence of a GT resonance, which is placed at increasing excitation energy as the number of neutrons N increases in a given isotopic chain. The total GT strength also 16: (Color online) Same as inFig. 14, but for Kr isotopes. . 17: (Color online) Same as inFig. 14, but for Sr isotopes. . 18: (Color online) Same as inFig. 14, but for Ru isotopes. 19: (Color online) Same as inFig. 14, but for Pd isotopes. . 20-25 the measured β-decay half-lives (solid . 20: (Color online) Measured β-decay half-lives for Ge isotopes compared to theoretical QRPA-SLy4 results calculated from different shapes. Circles are experimental values (open circles are experimental values from systematics)[44]. . 21: (Color online) Same as inFig. 20, but for Se isotopes. . 22: (Color online) Same as in . 23: (Color online) Same as inFig. 20, but for Sr isotopes. Experimental half-lives are from[23,44]. . 24: (Color online) Same as inFig. 20, but for Ru isotopes. . 25: (Color online) Same as inFig. 20, but for Pd isotopes. FIG. 26: (Color online) Ratio of calculated to experimental β-decay half-lives for two sets of calculations, with the spherical approximation (open dots) and with the deformation that corresponds to the minimum of the PECs (solid dots). The ratios are plotted as a function of the experimental half-lives (a) and as a function of the quadrupole deformation at the minimum of the PECs (b). Fig. 26 involving n = 81 nuclei leads to the values M 10 r = 1.105, σ 10 r = 10.21, and Σ 10 r = 10.24 in the spherical case and M 10 r = 0.937, σ 10 r = 3.09, and Σ 10 r = 3.09 in the deformed one, showing clearly the improvement achieved with the deformed formalism. FIG. 1: Potential energy curves for even-even neutron-rich Ge, Se, Kr, Sr, Zr, Mo, Ru, and Pd isotopes obtained from constrained HF+BCS calculations with the Skyrme force SLy4.Ge 32 Se 34 Kr 36 Sr 38 Zr 40 Mo 42 Ru 44 Pd 46 80 82 84 86 88 90 92 94 86 90 A=94 100 104 110 114 116 112 106 102 96 92 88 90 94 98 104 108 114 118 120 116 110 106 100 96 92 94 98 102 108 112 118 122 124 120 114 110 104 100 96 98 102 106 112 116 122 126 128 124 118 114 108 104 100 -0.2 -0.1 0 0.1 0.2 β prolate oblate g.s. 80 82 84 86 88 90 92 94 A 4.1 4.15 4.2 4.25 4.3 r c (fm) Ge (a) (b) 9/2+ 1/2+ 3/2+ 1/2+ 3/2+ 3/2+ 1/2+ 5/2+ 1/2+ 3/2+ 7/2+ 3/2+ 1/2+ J π exp ( 81 Ge) = (9/2+) J π exp ( 83 Ge) = (5/2+) J π exp ( 85 Ge) = (3/2+, 5/2+) FIG. 2: (Color online) Isotopic evolution of the quadrupole deformation parameter β (a) and charge radius (b) corre- sponding to the energy minima obtained from the Skyrme interaction SLy4 for Ge isotopes. Ground state results are encircled. -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 β prolate oblate g.s. 86 88 90 92 94 96 98 100 A 4.15 4.2 4.25 4.3 4.35 r c (fm) Se (a) (b) 3/2+ 1/2+ 3/2+ 5/2+ 1/2+ 5/2+ 1/2+ 1/2+ 3/2+ 7/2+ 3/2+ 3/2+ 1/2+ 5/2+ J π exp ( 87 Se) = (5/2+) J π exp ( 89 Se) = (5/2+) FIG. 3: (Color online) Same as in Fig. 2, but for Se isotopes. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 β 90 92 94 96 98 100 102 104 A 4.2 4.25 4.3 4.35 4.4 4.45 r c (fm) prolate oblate g.s. exp Kr (a) (b) 1/2+ 3/2+ 5/2+ 3/2+ 5/2- 5/2-5/2+ 3/2+ 7/2+ 11/2-3/2+ 1/2+ 5/2+ 9/2- J π exp ( 91 Kr)=5/2(+) J π exp ( 93 Kr)=1/2+ J π exp ( 95 Kr)=1/2(+) J π exp ( 97 Kr)=(3/2+) FIG. 4: (Color online) Same as in Fig. 2, but for Kr isotopes. 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[ "NEIGHBORLY INSCRIBED POLYTOPES AND DELAUNAY TRIANGULATIONS", "NEIGHBORLY INSCRIBED POLYTOPES AND DELAUNAY TRIANGULATIONS" ]	[ "Bernd Gonska \nInstitut für Mathematik\nFreie Universität Berlin\nArnimallee 214195BerlinGermany\n", "Arnau Padrol \nInstitut für Mathematik\nFreie Universität Berlin\nArnimallee 214195BerlinGermany\n" ]	[ "Institut für Mathematik\nFreie Universität Berlin\nArnimallee 214195BerlinGermany", "Institut für Mathematik\nFreie Universität Berlin\nArnimallee 214195BerlinGermany" ]	[]	We prove that there are superexponentially many combinatorially distinct d-dimensional neighborly Delaunay triangulations on n points. These are the first examples of neighborly Delaunay triangulations that cannot be obtained via a stereographic projection of an inscribed cyclic polytope, and provide the current best lower bound for the number of combinatorial types of Delaunay triangulations. To prove this bound we combine recent results on constructions for neighborly and inscribable polytopes to obtain a very simple explicit technique to generate a rich family of inscribable neighborly polytopes, and hence of point configurations with neighborly Delaunay triangulations.	10.1515/advgeom-2015-0045	[ "https://arxiv.org/pdf/1308.5798v1.pdf" ]	14,234,330	1308.5798	81335106f3bb4489af39272910a5416e1aa4f2b2	NEIGHBORLY INSCRIBED POLYTOPES AND DELAUNAY TRIANGULATIONS Bernd Gonska Institut für Mathematik Freie Universität Berlin Arnimallee 214195BerlinGermany Arnau Padrol Institut für Mathematik Freie Universität Berlin Arnimallee 214195BerlinGermany NEIGHBORLY INSCRIBED POLYTOPES AND DELAUNAY TRIANGULATIONS We prove that there are superexponentially many combinatorially distinct d-dimensional neighborly Delaunay triangulations on n points. These are the first examples of neighborly Delaunay triangulations that cannot be obtained via a stereographic projection of an inscribed cyclic polytope, and provide the current best lower bound for the number of combinatorial types of Delaunay triangulations. To prove this bound we combine recent results on constructions for neighborly and inscribable polytopes to obtain a very simple explicit technique to generate a rich family of inscribable neighborly polytopes, and hence of point configurations with neighborly Delaunay triangulations. Introduction Delaunay triangulations and Voronoi diagrams are one of the most important objects in computational geometry. For example, they are used for nearest-neighbors search, pattern matching, clustering, mesh generation for the finite-element method and surface reconstruction. Understanding their combinatorial complexity (in terms of the number of faces) becomes crucial for these applications. This subject has received a lot of recent attention both for low dimensional configurations [4,5,12,13,14] and for those with arbitrary dimension [2,3,7,11]. Seidel [22,23] proved an upper bound theorem for the complexity of Delaunay triangulations in terms of neighborly polytopes. Indeed, Brown observed in 1979 [8] that using a stereographic projection one can identify the combinatorial types of Delaunay triangulations in R d with those of inscribable (d + 1)-dimensional polytopes. Therefore, by McMullen's Upper Bound Theorem [19], the complexity of Voronoi diagrams is bounded by that of neighborly polytopes. The existence of inscribed neighborly polytopes was already known by Carathéodory in 1911, when he presented an inscribed realization of the cyclic polytope [9]. Every instance of inscribed neighborly polytope found since then has been combinatorially equivalent to the cyclic polytope. In this paper we provide a very simple construction (Theorem 5.6 and Construction 6.1) for higher dimensional inscribable neighborly polytopes (and hence Delaunay triangulations and Voronoi diagrams). In short, the vertex set of an initial neighborly polytope in R d is lifted to a neighborly Delaunay triangulation in R d+1 , which can then be lifted to an inscribable neighborly polytope in R d+2 . This technique generates a surprisingly rich family of combinatorial types of neighborly inscribable polytopes/Delaunay triangulations, and shows that, in combinatorial terms, inscribable neighborly polytopes are more frequent than what one might have thought. Indeed, we show in Theorem 6.6 that inp(n, d), the number of different labeled combinatorial types of inscribable neighborly d-polytopes with n vertices, is at least inp(n, d) ≥ n d 2 n(1+o(1)) (as n → ∞ with d fixed). This contrasts with the fact that pol(n, d), the number of different labeled combinatorial types of d-polytopes with n vertices, is not larger than pol(n, d) ≤ n d d 2 n(1+o(1)) when n d → ∞ (see [1] and [17]). This means that inp(n, d) is at least of the same order of magnitude as d pol(n, d) when d is fixed and n → ∞. Previous work. A polytope is called inscribable if it is combinatorially equivalent to a polytope that has all its vertices on a sphere. The question of inscribability was raised for the fist time by Steiner in 1832 [25], who asked whether all 3-dimensional polytopes are inscribable. A negative answer was given by Steinitz in 1928 [26]. For example, the Triakis tetrahedron is not inscribable. The existence of inscribable neighborly polytopes has been known since their discovery by Carathéodory [9], who found a realization of cyclic polytopes with all their vertices on a sphere. While many inscribed realizations of the cyclic polytope have been found (c.f. [15], [16], [18] and [23]), a proof for the existence of other combinatorial types of neighborly polytopes inscribed on the sphere has been elusive. Without the constraint of inscribability, Grünbaum [18] found the first examples of non-cyclic neighborly polytopes. Even more, Shemer used a "sewing construction" to prove in 1982 that the number of combinatorial types of neighborly d-polytopes with n vertices is of order n n 2 (1+o(1)) [24]. His bound was recently improved by the second author in [20,21], by proposing a new construction for neighborly polytopes that contains Shemer's family. When d is even, the number of different combinatorial types of labeled neighborly d-polytopes with n vertices obtained with this construction is at least (n − 1) ( n−1 2 ) 2 (n − d − 1) ( n−d−1 2 ) 2 d ( d 2 ) 2 e 3 n−d−1 2 d 2 , which implies that, regardless of the dimension, the number of labeled neighborly d-polytopes with n vertices is greater than (1) n d 2 n(1+o(1)) as n → ∞. Even if this method cannot generate all neighborly polytopes, (1) is currently also the best lower bound for the number of combinatorial types of labeled d-polytopes with n vertices. Our main contribution in this paper is to show that all these neighborly polytopes are inscribable. To this end, we revisit the construction in [20,21] providing a new direct proof that avoids the use of Gale duality and oriented matroid theory needed in the original proof. Thanks to this new proof we are able to show that these polytopes are inscribable by using a technique developed by the first author in [15,16] to construct inscribable cyclic polytopes. This implies that the lower bound (1) is also valid for the number of neighborly Delaunay triangulations, and begs the question of whether every neighborly polytope is inscribable. Preliminaries 2.1. Polytopes and point configurations. Let A = {a 1 , . . . , a n } be a point configuration in R d whose points are labeled by {1, . . . , n}. We will say that A is in general position if no d+1 points lie in a common hyperplane and no d + 2 points lie on a common sphere. The convex hull of A is a polytope P := conv(A) ⊂ R d and the intersection of P with a supporting hyperplane is a face of P . Faces of dimensions 0 and d − 1 are called vertices and facets, respectively. If all facets of P are simplices, then P is called simplicial and if A is in general position, then conv(A) is always simplicial. A face F of P is called an equatorial face if the projection π that forgets the last coordinate maps F onto a face of π(P ). If A is in sufficient general position, then the dimensions of F and π(F ) coincide for every equatorial face F . In particular, in this case there cannot be equatorial facets. A face F of a polytope P is visible from a point p / ∈ P if there is a point x ∈ relint(F ) such that the segment [x, p] intersects P only at x. We will usually assume that A is in convex position, i.e. it coincides with vert(P ), the set of vertices of P . In particular, each face F of P can be identified with the set of labels of the points a i ∈ F . The face lattice of P is then a poset of subsets of {1, . . . , n}. In this context, two vertex-labeled polytopes are combinatorially equivalent if their face lattices coincide. A polytope P is called k-neighborly if every subset of k vertices of P forms a face of P . No d-dimensional polytope other than the simplex can be k-neighborly for any k > d 2 , which motivates the definition of neighbory polytopes as those that are d 2 -neighborly. McMullen's Upper Bound Theorem [19] states that the number of idimensional faces of a d-polytope P with n vertices is maximized by simplicial neighborly polytopes, for all i. A canonical example of neighborly polytope is the cyclic polytope, C n (d), obtained as the convex hull of n points in any d-order curve in R d [27]. For example, the moment curve γ : t → (t, t 2 , . . . , t d ) is a d-order curve. In even dimensions, the trigonometric moment curve τ : t → (sin(t), cos(t), sin(2t), cos(2t), . . . , sin( d 2 t), cos( d 2 t) ) is a d-order curve, providing an inscribed realization of the cyclic polytope. Triangulations. A triangulation of a point configuration A is a collection T of simplices called cells such that • vert(S) ⊆ A for every S ∈ T . • If S ∈ T and F is a face of S, then F ∈ T . • S∈T S = conv(A). • If S, S ∈ T and S = S then relint(S) ∩ relint(S ) = ∅. Just as with polytopes, we can define the face lattice of a triangulation and consider their combinatorial equivalence. If A is in general position, this condition always defines a unique triangulation of A. We will be particularly interested in neighborly triangulations, which will provide us with extreme examples in terms of number of faces. Definition 2.2. A triangulation T of a point configuration A ⊂ R d is neighborly if conv(S) is a cell of T for each subset S ⊂ A of size \|S\| = d+1 2 . Liftings and triangulations 3.1. Lexicographic liftings. The main tool for our construction are lexicographic liftings, which are a way to derive (d + 1)-dimensional point configurations from d-dimensional point configurations. Definition 3.1. Let A = {a 1 , . . . , a n } be a configuration of n labeled points in R d . We say that a configuration A = { a 1 , . . . , a n } of n labeled points in R d+1 is a Delaunay lexicographic lifting (or just a D-lifting) of A (with respect to the order induced by the labeling) if a i = (a i , h i ) ∈ R d+1 for each 1 ≤ i ≤ n, for some collection of heights h i ∈ R that fulfill: We omit the proof of this lemma, which follows from the definition. 3.2. Lexicographic triangulations. The combinatorics of D-liftings are easily explained in terms of lexicographic triangulations. We refer to [10, Section 4.3] for a detailed presentation, and we will only present here the parts that will be directly useful for us. . . , a n } be a point configuration in convex position and let T be a triangulation of the point configuration A \ a i . Then there is a triangulation T of A whose cells are (i) h 1 = · · · = h d+1 = 0, (ii) \|h d+2 \| > 0, and (iii) for each i > d + 2, \|h i \| > 0 is large enough so that if h i > 0 (resp. h i < 0) then a i is above (resp. below) H for every hyperplane H spanned by d + 1 points of { a 1 , . . . , a i−1 }. (iv) for each i > d + 2, a i isT := T ∪ {conv(B ∪ a i ) \| B ∈ T and is visible from a i } . Moreover, T is the only triangulation of A that contains T . Proof. The proof is by induction on n. If n = d + 2 then both triangulations consist on the single simplex spanned by A. Otherwise, let T be the Delaunay triangulation of A \ a n . By induction hypothesis, this is the placing triangulation of A \ a n . Moroever, since the lifting fulfills condition (iv), every Delaunay cell of T is still a Delaunay cell of T . In particular, T refines T . Since by Lemma 3.4 the triangulation that refines T is unique, T must be the placing triangulation of A. Theorem 4.2 can be deduced from the "Gale sewing" technique presented in [20] (cf. [21,Theorem 4.5]). However, the original proof of the theorem exploits Gale duality and oriented matroid theory, while this primal proof is elementary. Moreover, this is the setting that will eventually allow us to prove inscribability in Theorem 5.6. A = {a 1 , . . . , a n } be a configuration of n ≥ d + 2 labeled points in convex general position in R d and let A be a D-lifting of A. Theorem 4.2. Let If conv(A) is k-neighborly (as a polytope), then conv( A) is k-neighborly (as a polytope) and the placing triangulation of A is (k + 1)-neighborly (as a triangulation). Proof. The proof of the first claim (conv( A) is k-neighborly) is straightforward. Indeed, since A is a projection of A, every face of conv(A) is a projection of an equatorial face of conv( A). In particular, since every subset of k-points of A is a face of conv(A), the corresponding points must also form an equatorial face of conv( A). The second claim is proved by induction on n, and it is trivial when n = d + 2. For n > d + 2, let T be the placing triangulation of A and T the corresponding placing triangulation of A \ a n . Now fix a subset S of A of size k + 1. If a n / ∈ S, then S forms a cell of T by induction hypothesis, and hence of T . Otherwise, if a n ∈ S, then S = S \ a n must be an equatorial face of A \ a n . By Lemma 3.3, S is visible from a n and hence by the definition of the placing triangulation, S must be a cell of T . The combination of these two theorems directly proves our main result (see also Remark 6.4). Theorem 4.3. Let A = {a 1 , . . . , a n } be a configuration of n ≥ d + 2 labeled points in convex general position in R d such that conv(A) is k-neighborly, and let A be a D-lifting of A. Then conv( A) is a k-neighborly polytope with vertex set A and its Delaunay triangulation is a (k + 1)-neighborly triangulation. Polytopes We can easily adapt Theorem 4.2 to obtain a statement in terms of neighborly polytopes instead of neighborly triangulations. It suffices to do another suitable D-lifting. The construction will start on a neighborly polytope and increase its dimension by 2. A first approach would be to start with a k-neighborly configuration A ⊂ R d of n points, then make a D-lifting of A to obtain A, and finally a positive D-lifting of A to obtain A . Then A is easily seen to be a (d+2)-dimensional (k + 1)-neighborly configuration of n points. However, we will do a slight variation of this method to obtain a (d + 2)dimensional configuration of n + 2 points. The reason is that, in a D-lifting A of A, the position of the point a n in A does not affect the combinatorics of conv( A). Hence, we can always add an "extra point" to play the role of this last point. To this end, we will extend the concept of D-lifting to that of pointed D-lifting, which could be understood as doing a D-lifting and then placing the extra point so high that plays the role of a point at infinity. This will become convenient to translate from triangulations to polytopes, and to estimate with more precision the number of combinatorial types obtained. 5.1. Pointed lexicographic liftings. As usual, let A = {a 1 , . . . , a n } be a configuration of n labeled points in R d . A configuration A[ p] = { a 1 , . . . , a n , p} of n + 1 labeled points in R d+1 is a pointed Delaunay lexicographic lifting of A (with respect to the order induced by the labels) if p has label n + 1 and there is some p ∈ R d such that A[ p] is a D-lifting of A = A ∪ p and such that the height for p is positive. We call p the apex of the D-lifting. In short, to do a pointed lifting, a point p is added to A and is lifted in the positive direction as the last element of the configuration. ] of the Gale dual of A (see [6,Section 7.2]). This provides the link with the results presented in [20] and [21]. Polytopes from triangulations. To recover inscribed polytopes from Delaunay triangulations, we need the following classical result from Brown [8] (cf. [15,Proposition 0.3.13]). Lemma 5.3. Let A = {a 1 , . . . , a n } be a configuration of n points in R d in general position, and let D(A) be its Delaunay triangulation. Then there is an inscribable simplicial (d+1)-polytope P A with n+1 vertices {å 1 , . . . ,å n ,p} whose faces are: (i) {å i 1 , . . . ,å i k } if {a i 1 , . . . , a i k } is a cell of D(A) and (ii) {p,å i 1 , . . . ,å i k } if {a i 1 , . . . , a i k } is a face of conv(A). An inscribed realization of P A can be found by inverting a stereographic projection. Observation 5.4. This lemma implies a bijection between labeled combinatorial types of inscribable d-polytopes with n vertices and (d − 1)-dimensional Delaunay triangulations of n − 1 points. To go from the face lattice of an inscribable polytope to the face lattice of a Delaunay triangulation, simply delete all faces containing the point with the largest label. This is clearly a bijection. We omit the proof of the next lemma, which only needs a combinatorial description of the face lattice of a positive pointed lifting (cf. [10,Lemma 4.3.4]). Lemma 5.5. If the Delaunay triangulation of A coincides with its placing triangulation, then the polytope P A of Lemma 5.3 is combinatorially equivalent to a positive pointed D-lifting of A. As a direct consequence of the combination of Lemma 5.3 and Theorem 4.3, we obtain the following result. Observe how the strategy is to start with a k-neighborly d-polytope, lift it to a (k + 1)-neighborly (d + 1)triangulation, and lift it again (with a positive lifting with the same order) to a (k + 1)-neighborly (d + 1)-polytope. because conv( A[ p]) is k-neighborly. Lower bounds Theorem 5.6 provides the following method to construct many inscribable neighborly (d ≥ 2)-polytopes with n vertices starting with an arbitrary polygon or 3-polytope. Remark 6.2. We deferred the discussion about the order the points for the D-lifting until now. However, the relabeling step in Construction 6.1 is crucial, since it is the choice of these permutations what produces the variety of combinatorial types. It is also important to remark that the choice of the permutation is only done once every two dimensions, since the lifting with apex p and the lifting with apex q need to follow the same order or otherwise Theorem 5.6 does not hold. Remark 6.3. Construction 6.1 relies on Delaunay lexicographic liftings, which in their definition depend on some h i 's being "large enough". One might wonder how feasible, computationally, is to find these h i s. On the one hand, it is not hard to find an upper bound for the minimal valid h i that depends only on d and n if we assume, for example, that the a i 's have integer entries whose absolute value is bounded by K. Indeed, the conditions required in Definition 3.1 only depend on certain determinants being positive. However, if we apply this method to construct our point configurations from scratch, we will end up with points having extremely large coordinates. It remains open which are the minimal coordinates needed to realize these configurations. Can they be realized with polynomially large coordinates? In lower dimensions this kind of questions has already been considered (see [12] and [13]). Remark 6.4. Theorem 4.3 provides neighborly Delaunay triangulations that are stereographic projections of neighborly polytopes from vertices. However, they do not attain the upper bound of Theorem 2.3, because their convex hull is not a simplex. To obtain such triangulations, we need neighborly polytopes with an inscribed realization that can be stacked on a facet such that the result is still inscribed (see [23] and [15] [ p]), apply a stellar subdivision to the the first simplex directly after it appears. What we then get is the vertex projection of an inscribed neighborly polytope that has been stacked with a point on its circumsphere. We can then apply Brown's projection from the stacked point to get the desired triangulation with a maximal number of faces. 6.1. The bounds. It remains to discuss how many different labeled combinatorial types of inscribable neighborly polytopes (and of neighborly Delaunay triangulations) can be obtained with Construction 6.1. These bounds were obtained in [20] using a construction that can be seen to be equivalent. We will only sketch the main ideas for the originial proof, which is based on oriented matroids, and refer to [20] and [21,Chapter 5] for details. These bounds concentrate on the case of even-dimensional neighborly polytopes (see Remark 6.7). The main ingredient for them is the following lemma, which ignores the variability provided by the signs of the h i 's and only focuses on positive liftings. Stronger bounds are discussed in [21,Chapter 5]. Then there are at least (n+1)! (d+2)! different elements in P. (with respect to the same ordering). Finally the permutation σ −1 is applied on the labels of A so that the orginal labeling of A is preserved. The next step in the proof of this lemma in [21,Proposition 5.4] is to show that, if we restrict to the case when both D-liftings are positive, then the n − d − 2 first elements of σ can be recovered from the combinatorics of P , the convex hull of A [ p] [ q], and that 2 different cases can be distinguished for the last d + 2 elements. A final step is that, when conv(A) is neighborly, if q and the face lattice of P are fixed, then there are at most two points of A [ p] that could be p. Hence the label of p can also be chosen almost arbitrarely, adding a factor of n+1 2 to the number of combinatorial types. Applying recursively this lemma one obtains the following bound, which is estimated using Euler-Maclaurin approximation. Remark 6.7. For the odd-dimensional case, observe that any pyramid over an even dimensional inscribable neighborly polytope is again an inscribable neighborly polytope, which shows that the bound (1) also applies for odd dimensional configurations. Remark 6.8. Although the use of labeled types is needed to prove these bounds, observe that we can easily recover bounds for non-labeled combinatorial types just by dividing by n!. The bounds obtained this way are still superexponential and comparable to (1). Definition 2. 1 . 1The Delaunay triangulation D(A) of a point configuration A ⊂ R d in general position is the triangulation that consists of all cells defined by the empty circumsphere condition: S ∈ D(A) if and only if there exists a (d − 1)-sphere that passes through all the vertices of S and all other points of A lie outside this sphere. Theorem 2.3 (Upper Bound Theorem for balls [10, Corollary 2.6.5]). A triangulation T of a point configuration A has the maximal number of d-cells among all triangulations of n points in R d if and only if T is neighborly and conv(A) is a d-simplex. not contained in any of the circumspheres of any simplex spanned by d + 2 points of { a 1 , . . . , a i−1 }. If h i ≥ 0 for every 1 ≤ i ≤ n, the lexicographic lifting is called positive. Remark 3 . 2 . 32If A is in general position, then any D-lifting A of A is also in general position. Furthermore, if A is in convex position, so is A. Lemma 3 . 3 . 33For any point configuration A, and for any D-lifting A, every equatorial face of conv( A \ a n ) is visible from a n . Figure 1 . 1A D-lifting { a 1 , . . . , a 5 } of the point configuration {a 1 , . . . , a 5 }. . Let A = {a 1 , . Definition 3 . 5 . 35The triangulation T in the previous lemma is called a refinement of T obtained by placing a i .The placing triangulation of A (with respect to the order induced by the labels) is then the triangulation T := T n obtained iteratively as follows: T 1 is the trivial triangulation of {a 1 } and T i is the triangulation of {a 1 , . . . , a i } obtained by placing a i on T i−1 .4. The constructionOur construction is based on the following theorems, which show how certain triangulations of certain D-liftings are always Delaunay (Theorem 4.1) and neighborly (Theorem 4.2).The proof of Theorem 4.1 is inspired in [15, Proposition 1.3.1] and[16, Proposition 17], where a similar argument is used to prove that the cyclic polytope is inscribable. Theorem 4. 1 . 1Let A = {a 1 , . . . , a n } be a configuration of n ≥ d + 2 labeled points in general position in R d and let A be a D-lifting of A. Then the Delaunay triangulation T := D( A) coincides with the placing triangulation of A. Observation 5 . 1 . 51Let A[ p] be a pointed D-lifting of A. Then the faces of conv( A[ p]) that contain the apex p are the join of p with every face of conv(A). Observe that Theorems 4.1, 4.2 and 4.3 still hold if we replace the Dlifting by a pointed D-lifting with apex p. Indeed, the proof of Theorem 4.2 only uses the neighborliness of A \ a n , and those of Theorems 4.1 and 4.3 follow directly. Remark 5.2. The pointed D-lifting of A with heights h i corresponds to the lexicographic extension by [a − sign(hn) n , . . . , a − sign(h d+2 ) d+2 Theorem 5 . 6 . 56Let A be a d-dimensional configuration of n ≥ d + 2 points in convex general position such that conv(A) is k-neighborly, let A[ p] be a pointed D-lifting of A and let A [ p][ q] be a positive pointed D-lifting of A[ p].Thenconv( A [ p][ q]) is an inscribable (k + 1)-neighborly (d + 2)-polytope. In particular, if conv(A) is neighborly, so is conv( A [ p][ q]).Proof. Inscribability follows from Lemma 5.5, which says that conv( A [ p][ q])is combinatorially equivalent to the inscribable polytope P A of Lemma 5.3.For k-neighborliness, observe that every subset of k + 1 points of A[ p] is a face of its placing triangulation, and hence a face of conv( A [ p][ q]) by point (i) of Lemma 5.3. And every set of k + 1 points of A [ p][ q] cointaining q is a face of conv( A [ p][ q]), by point (ii) of Lemma 5.3 Construction 6 . 1 .. 61To construct an inscribable neighborly d-polytope with n vertices P and a (d − 1)-dimensional set of n − 1 points with a neighborly Delaunay triangulation T : 1. Set n 0 := n − 2 Let P 0 be any simplicial d 0 -polytope with n 0 vertices.3. P 0 is neighborly because d 0 ∈ {2, 3}. 4. Set A 0 = vert(P 0 ). 5. For i from 1 to dChoose a permutation σ ∈ S n i−1 and relabel the points of A i−1 with a j → a σ(j) . 2.2. Let A i−1 [ p] be a pointed D-lifting of A i−1 . 3.3. Set A i := A i−1 [ p][ q]be a pointed positive D-lifting of A i−1 [ p]. 4.4. Set n i := n i−1 + 2 and d i := d i−1 + 2. 5.5. By Theorem 5.6, P i := conv(A i ) is a neighborly d i -polytope with n i vertices. 6. P := P d−2 2 is an inscribable neighborly d-polytope with n vertices by Theorem 5.6. 7. T := D( A d−4 2 [ p]) is a neighborly Delaunay triangulation of a (d − 1)dimensional set of n − 1 points by Theorem 4.3. Lemma 6.5 ([21, Proposition 5.4],[21, Lemma 5.9]). Let A be a configuration of n > d + 2 labeled points in convex general position in R d , for d ≥ 2 even, such that conv(A) is neighborly.Denote by P the set of different labeled combinatorial types of polytopes P that fulfill:• P = conv( A [ p][ q])is obtained by doing a couple of positive D-liftings (as in Theorem 5.6) from a initial relabeling of A, and • the labeled combinatorial type of conv(A) can be recovered from that of conv( A [ p][ q]). Theorem 6 . 6 . 66For even d, inp(n, d), the number of different labeled combinatorial types of inscribable neighborly d-polytopes with n vertices, fulfills inp(n, 5.4, the same bound holds for the number of labeled combinatorial types of neighborly Delaunay triangulations of configurations of n − 1 points in R d−1 . ). 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[ "COHOMOLOGY AND THE CONTROLLING ALGEBRA OF CROSSED HOMOMORPHISMS ON 3-LIE ALGEBRAS", "COHOMOLOGY AND THE CONTROLLING ALGEBRA OF CROSSED HOMOMORPHISMS ON 3-LIE ALGEBRAS" ]	[ "Shuai Hou ", "Meiyan Hu ", "ANDLina Song ", "Yanqiu Zhou " ]	[]	[]	In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an L ∞ -algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted L ∞ -algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.	null	[ "https://export.arxiv.org/pdf/2212.02729v1.pdf" ]	254,275,039	2212.02729	b8d2011ed20b25397b4cb6d9f5af9e82653e56a2	COHOMOLOGY AND THE CONTROLLING ALGEBRA OF CROSSED HOMOMORPHISMS ON 3-LIE ALGEBRAS 6 Dec 2022 Shuai Hou Meiyan Hu ANDLina Song Yanqiu Zhou COHOMOLOGY AND THE CONTROLLING ALGEBRA OF CROSSED HOMOMORPHISMS ON 3-LIE ALGEBRAS 6 Dec 2022 In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a Lie algebra to the semidirect product Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an L ∞ -algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted L ∞ -algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras. Introduction The notion of 3-Lie algebras and more generally, n-Lie algebras (also called Filippov algebras) was introduced in [11]. See the review article [7,26] for more details. The n-Lie algebra is the algebraic structure corresponding to Nambu mechanics [28]. In recent years, 3-Lie algebras have been widely studied and applied in the fields of mathematics and physics, especially in string theory and M2-branes [2,12,20]. For example, metric 3-Lie algebras play a significant role in the basic model of Bagger-Lambert-Gustavsson theory [8,9], the supersymmetric Yang-Mills theory can be studied by a special structure of 3-Lie algebras [15], and in [5], Basu and Harvey suggested to replace the Lie algebra appearing in the Nahm equation by a 3-Lie algebra for the lifted Nahm equations. The notion of a crossed homomorphism of on Lie algebras was introduced by Lue in [25]. A crossed homomorphism is also called a relative difference operator or differential operator of weight 1 with respect to the adjoint representation [17,18,24]. Crossed homomorphisms are related to post-Lie algebras and can be used to study the integration of post-Lie algebras [27]. In [30], using the crossed homomorphisms on Lie algebras, they studied the relationship between the category of weak representations of Lie-Rinehart algebras and the monoidal category of representations of Lie algebras of Cartan type. They also introduced the cohomology theory of crossed homomorphisms on Lie algebras and studied linear deformations of crossed homomorphisms. In [22], the authors studied the controlling algebra of relative difference Lie algebras and defined the cohomology of difference Lie algebras with coefficients in arbitrary representations. Crossed homomorphisms on Hopf algebras and Cartier-Kostant-Milnor-Moore theorem for difference Hopf algebras were studied in [19]. The research on the deformation theory of algebraic structures began with the seminal work of Gerstenhaber for associative algebras [13]. Next, Nijenhuis and Richardson extended the study of deformation theory to Lie algebra [29]. In [10,26], the deformation problem of n-Lie algebras and 3-Lie algebras were studied respectively. See the review [16] for more details. Recently, the deformations of certain operators, e.g. morphisms, relative Rota-Baxter operators on 3-Lie algebras have been deeply studied [1,33]. Actually, an invertible linear map is a differential operator if and only its inverse is a (relative) Rota-Baxter operator on 3-Lie algebras [4,3]. The purpose of this paper is to study crossed homomorphisms on 3-Lie algebras, with particular interests in the cohomology and deformation theories. The crossed homomorphisms introduced in this paper are closely related to relative Rota-Baxter operators of weight 1 on 3-Lie algebras introduced in [21]. More precisely, the inverse of an invertible crossed homomorphism is a relative Rota-Baxter operators of weight 1, which generalizes the classical relations between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on Lie algebras, and thus justifies its correctness. A crossed homomorphism gives rise to a new representation, and the corresponding cohomology of 3-Lie algebras is taken to be the cohomology of the crossed homomorphism. As expected, the second cohomology group classifies infinitesimal deformations of the crossed homomorphism. Furthermore, we use Voronov's higher derived brackets to construct an L ∞ -algebra whose Maurer-Cartan elements are crossed homomorphisms. Consequently, we obtain the L ∞ -algebra governing deformations of a crossed homomorphism. Note that in the Lie algebra context, it is a differential graded Lie algebra governing deformations of a crossed homomorphism on Lie algebras. While for 3-Lie algebras, it is indeed an L ∞ -algebra with nontrivial l 3 governing deformations of a crossed homomorphism, which is totally different from the case of Lie algebras. The paper is organized as follows. In Section 2, we introduce the notion of crossed homomorphisms on 3-Lie algebras and illustrate the relationship between crossed homomorphisms and relative Rota-Baxter operator of weight 1. In Section 3, we establish the cohomology theory of crossed homomorphisms on 3-Lie algebras. We use the cohomology theory of crossed homomorphisms that we established to classify infinitesimal deformations of crossed homomorphisms. In Section 4, we construct an L ∞ -algebra whose Maurer-Cartan elements are precisely crossed homomorphisms on 3-Lie algebras. We also using Getzler's twisted L ∞ -algebra theory to characterize deformations of crossed homomorphisms on 3-Lie algebras. In this paper, we work over an algebraically closed filed K of characteristic 0. Acknowledgements. This research is supported by NSFC (12001226). Crossed homomorphisms on 3-Lie algebras In this section, we introduce the notion of crossed homomorphisms on 3-Lie algebras, and find that there is a close relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Definition 2.1. ( [11]) A 3-Lie algebra is a vector space g together with a skew-symmetric linear map [·, ·, ·] g : ∧ 3 g → g, such that for x i ∈ g, 1 ≤ i ≤ 5, the following Fundamental Identity holds: [x 1 , x 2 , [x 3 , x 4 , x 5 ] g ] g = [[x 1 , x 2 , x 3 ] g , x 4 , x 5 ] g + [x 3 , [x 1 , x 2 , x 4 ] g , x 5 ] g + [x 3 , x 4 , [x 1 , x 2 , x 5 ] g ] g .(1)For x 1 , x 2 ∈ g, define ad x 1 ,x 2 ∈ gl(g) by ad x 1 ,x 2 x := [x 1 , x 2 , x] g , ∀x ∈ g. Then ad x 1 ,x 2 is a derivation, i.e. ad x 1 ,x 2 [x 3 , x 4 , x 5 ] g = [ad x 1 ,x 2 x 3 , x 4 , x 5 ] g + [x 3 , ad x 1 ,x 2 x 4 , x 5 ] g + [x 3 , x 4 , ad x 1 ,x 2 x 5 ] g . Definition 2.2. ([23]) A representation of a 3-Lie algebra (g, [·, ·, ·] g ) on a vector space V is a linear map: ρ : ∧ 2 g → gl(V), such that for all x 1 , x 2 , x 3 , x 4 ∈ g, the following equalities hold: ρ(x 1 , x 2 )ρ(x 3 , x 4 ) = ρ([x 1 , x 2 , x 3 ] g , x 4 ) + ρ(x 3 , [x 1 , x 2 , x 4 ] g ) + ρ(x 3 , x 4 )ρ(x 1 , x 2 ), (2) ρ(x 1 , [x 2 , x 3 , x 4 ] g ) = ρ(x 3 , x 4 )ρ(x 1 , x 2 ) − ρ(x 2 , x 4 )ρ(x 1 , x 3 ) + ρ(x 2 , x 3 )ρ(x 1 , x 4 ).(3) Let (g, [·, ·, ·] g ) be a 3-Lie algebra. The linear map ad : ∧ 2 g → gl(g) defines a representation of the 3-Lie algebra g on itself, which is called the adjoint representation of g. Definition 2.3. ( [11]) Let (g, [·, ·, ·] g ) be a 3-Lie algebra. Then the subalgebra [g, g, g] g is called the derived algebra of g, and denoted by g 1 . The subspace C(g) = {x ∈ g \| [x, y, z] g = 0, ∀y, z ∈ g} is called the center of g. Definition 2.4. [21] Let (g, [·, ·, ·] g ) and (h, [·, ·, ·] h ) be two 3-Lie algebras. Let ρ : ∧ 2 g → gl(h) be a representation of the 3-Lie algebra g on the vector space h. If for all x, y ∈ g, u, v, w ∈ h, ρ(x, y)u ∈ C(h), (4) ρ(x, y)[u, v, w] h = 0,(5) then ρ is called an action of g on h. We denote an action by (h; ρ). Proposition 2.5. [21] Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). There is a 3-Lie algebra structure on g ⊕ h, defined by [x + u, y + v, z + w] ρ = [x, y, z] g + ρ(x, y)w + ρ(y, z)u + ρ(z, x)v + [u, v, w] h ,(6) for all x, y, z ∈ g, u, v, w ∈ h. This 3-Lie algebra is called the semidirect product of the 3-Lie algebra g and the 3-Lie algebra h with respect to the action ρ, and denoted by g ⋉ ρ h. Next we give the notion of crossed homomorphisms on 3-Lie algebras. Definition 2.6. Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). A linear map H : g → h is called a crossed homomorphism with respect to the action ρ if H[x, y, z] g = ρ(x, y)(Hz) + ρ(y, z)(Hx) + ρ(z, x)(Hy) + [Hx, Hy, Hz] h , ∀x, y, z ∈ g. Remark 2.7. If the action ρ of g on h is zero, then any crossed homomorphism from g to h is nothing but a 3-Lie algebra homomorphism. If h is commutative, then any crossed homomorphism from g to h is simply a derivation from g to h with respect to the representation (h; ρ). In the Lie algebra context, crossed homomorphisms play important roles in the study of representations of Lie algebras of Cartan type. An essential ingredient in the whole theory is that a crossed homomorphism H : g → h induces a homomorphism from the Lie algebra g to the semidirect product Lie algebra g ⋉ h ([30, Theorem 2.7]). Now for crossed homomorphisms on 3-Lie algebras, we still have this characterization. Theorem 2.8. Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). Then a linear map H : g → h is a crossed homomorphism from g to h if and only if the map φ H : g → g ⋉ ρ h defined by φ H (x) := (x, Hx), ∀x ∈ g, (8) is a 3-Lie algebra homomorphism. Proof. For all x, y, z ∈ g, we have φ H [x, y, z] g = ([x, y, z] g , H[x, y, z] g ); [φ H (x), φ H (y), φ H (z)] ρ = ([x, y, z] g , ρ(x, y)(Hz) + ρ(y, z)(Hx) + ρ(z, x)(Hy) + [Hx, Hy, Hz] h ). Thus, φ H [x, y, z] g = [φ H (x), φ H (y), φ H (z)] ρ if and only if H is a crossed homomorphism from g to h with respect to the action ρ. Example 2.9. Let (g, [·, ·, ·] g ) be a 4-dimensional 3-Lie algebra with a basis {e 1 , e 2 , e 3 , e 4 } and the nonzero multiplication is given by [e 2 , e 3 , e 4 ] g = e 1 . The center of g is the subspace generated by {e 1 }. It is obvious that the adjoint representation In particular, ad : ∧ 2 g → gl(g) is an action of g on itself. For a matrix               a 11 a                 H(e 1 ) = 0, H(e 2 ) = e 2 , H(e 3 ) = e 3 , H(e 4 ) = −e 4 , is a crossed homomorphism from g to g with respect to the adjoint action ad. Definition 2.10. Let H and H ′ be two crossed homomorphisms from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. A homomorphism from H to H ′ consists of 3-Lie algebra homomorphisms ψ g : g → g and ψ h : h → h such that ψ h • H = H ′ • ψ g , (9) ψ h (ρ(x, y)u) = ρ(ψ g (x), ψ g (y))(ψ h (u)), ∀x, y ∈ g, u ∈ h.(10) In particular, if both ψ g and ψ h are invertible, (ψ g , ψ h ) is called an isomorphism from H to H ′ . Lemma 2.11. Let H : g → h be a crossed homomorphism from g to h with respect to an action ρ. Let ψ g : g → g and ψ h : h → h be 3-Lie algebra isomorphisms such that (10) holds. Then ψ −1 h • H • ψ g is a crossed homomorphism from g to h with respect to the action ρ. Proof. For all x, y, z ∈ g, we have (ψ −1 h • H • ψ g )[x, y, z] g = ψ −1 h ρ(ψ g (x), ψ g (y))H(ψ g (z)) + ρ(ψ g (y), ψ g (z))H(ψ g (x)) + ρ(ψ g (z), ψ g (x))H(ψ g (y)) +[H(ψ g (x)), H(ψ g (y)), H(ψ g (z))] h = ρ(x, y)(ψ −1 h • H • ψ g (z)) + ρ(y, z)(ψ −1 h • H • ψ g (x)) + ρ(z, x)(ψ −1 h • H • ψ g (y)) +[ψ −1 h • H • ψ g (x), ψ −1 h • H • ψ g (y), ψ −1 h • H • ψ g (z)] h , which implies that ψ −1 h • H • ψ g is a crossed homomorphism. At the end of this section, we establish the relationship between crossed homomorphisms and relative Rota-Baxter operators of weight 1 on 3-Lie algebras. Recall from [21] that a linear map T : h → g is called a relative Rota-Baxter operator of weight λ ∈ K from a 3-Lie algebra h to a 3-Lie algebra g with respect to an action ρ if [T u, T v, T w] g = T ρ(T u, T v)w + ρ(T v, T w)u + ρ(T w, T u)v + λ[u, v, w] h ,(11) for all u, v, w ∈ h. Proposition 2.12. Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). An invertible linear map H : g → h is a crossed homomorphism from the 3-Lie algebra g to the 3-Lie algebra h with respect to the action ρ if and only if H −1 is a relative Rota-Baxter operator of weight 1 from the 3-Lie algebra h to the 3-Lie algebra g with respect to the action ρ. Proof. If an invertible linear map H : g → h is a crossed homomorphism, then for u 1 , u 2 , u 3 ∈ h, by (7), we have [H −1 (u 1 ), H −1 (u 2 ), H −1 (u 3 )] g =H −1 (H[H −1 (u 1 ), H −1 (u 2 ), H −1 (u 3 )] g ) =H −1 ρ(H −1 (u 1 ), H −1 (u 2 ))(u 3 ) + ρ(H −1 (u 2 ), H −1 (u 3 ))(u 1 ) + ρ(H −1 (u 3 ), H −1 (u 1 ))(u 2 ) + [u 1 , u 2 , u 3 ] h . Therefore, H −1 is a relative Rota-Baxter operator of weight 1. Conversely, if H −1 be a relative Rota-Baxter operator of weight 1. For all x 1 , x 2 , x 3 ∈ g, assume x i = H −1 (u i ), 1 ≤ i ≤ 3, for u i ∈ h. By (11), we have H[x 1 , x 2 , x 3 ] g =H[H −1 (u 1 ), H −1 (u 2 ), H −1 (u 3 )] g =H(H −1 (ρ(H −1 (u 1 ), H −1 (u 2 ))(u 3 ) + ρ(H −1 (u 2 ), H −1 (u 3 ))(u 1 ) + ρ(H −1 (u 3 ), H −1 (u 1 ))(u 2 ) + [u 1 , u 2 , u 3 ] h )) =ρ(x 1 , x 2 )H(x 3 ) + ρ(x 2 , x 3 )H(x 1 ) + ρ(x 3 , x 1 )H(x 2 ) + [Hx 1 , Hx 2 , Hx 3 ] h . So H is a crossed homomorphism. Cohomologies of crossed homomorphisms on 3-Lie algebras In this section, we define the cohomology of crossed homomorphisms on 3-Lie algebras, and we use the second cohomology group to study infinitesimal deformations of crossed homomorphisms. 3.1. Cohomologies of crossed homomorphisms. First, we recall the cohomologies theory of 3-Lie algebras. Let (V; ρ) be a representation of a 3-Lie algebra (g, [·, ·, ·] g ). Denote by C n 3Lie (g; V) := Hom(∧ 2 g ⊗ · · · ⊗ ∧ 2 g (n−1) ∧g, V), (n ≥ 1), which is the space of n-cochains. The coboundary operator d : C n 3Lie (g; V) → C n+1 3Lie (g; V) is defined by (d f )(X 1 , · · · , X n , x n+1 ) = 1≤ j<k≤n (−1) j f (X 1 , · · · ,X j , · · · , X k−1 , [x j , y j , x k ] g ∧ y k +x k ∧ [x j , y j , y k ] g , X k+1 , · · · , X n , x n+1 ) + n j=1 (−1) j f (X 1 , · · · ,X j , · · · , X n , [x j , y j , x n+1 ] g ) + n j=1 (−1) j+1 ρ(x j , y j ) f (X 1 , · · · ,X j , · · · , X n , x n+1 ) +(−1) n+1 ρ(y n , x n+1 ) f (X 1 , · · · , X n−1 , x n ) + ρ(x n+1 , x n ) f (X 1 , · · · , X n−1 , y n ) , for all X i = x i ∧ y i ∈ ∧ 2 g, i = 1, 2, · · · , n and x n+1 ∈ g. It was proved in [6,32] that d • d = 0. Thus, (⊕ +∞ n=1 C n 3Lie (g; V), d) is a cochain complex. Definition 3.1. The cohomology of the 3-Lie algebra g with coefficients in V is the cohomology of the cochain complex (⊕ +∞ n=1 C n 3Lie (g; V), d). Denote by Z n 3Lie (g; V) and B n 3Lie (g; V) the set of n-cocycles and the set of n-coboundaries, respectively. The n-th cohomology group is defined by H n 3Lie (g; V) = Z n 3Lie (g; V)/B n 3Lie (g; V). Lemma 3.2. Let H be a crossed homomorphism from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. Define ρ H : ∧ 2 g → gl(h) by ρ H (x, y)u := ρ(x, y)u + [Hx, Hy, u] h , ∀x, y ∈ g, u ∈ h.(12) Then ρ H is a representation of g on h. Proof. By a direct calculation using (1)- (7), for all x i ∈ g, 1 ≤ i ≤ 4, u ∈ h, we have ρ H (x 1 , x 2 )ρ H (x 3 , x 4 ) − ρ H (x 3 , x 4 )ρ H (x 1 , x 2 ) −ρ H ([x 1 , x 2 , x 3 ] g , x 4 ) + ρ H ([x 1 , x 2 , x 4 ] g , x 3 ) (u) = ρ(x 1 , x 2 )ρ(x 3 , x 4 )u + [Hx 1 , Hx 2 , ρ(x 3 , x 4 )u] h + ρ(x 1 , x 2 )[Hx 3 , Hx 4 , u] h +[Hx 1 , Hx 2 , [Hx 3 , Hx 4 , u] h ] h − ρ(x 3 , x 4 )ρ(x 1 , x 2 )u − [Hx 3 , Hx 4 , ρ(x 1 , x 2 )u] h −ρ(x 3 , x 4 )[Hx 1 , Hx 2 , u] h − [Hx 3 , Hx 4 , [Hx 1 , Hx 2 , u] h ] h − ρ([x 1 , x 2 , x 3 ] g , x 4 )u −[H[x 1 , x 2 , x 3 ] g , Hx 4 , u] h + ρ([x 1 , x 2 , x 4 ] g , x 3 )u + [H[x 1 , x 2 , x 4 ] g , Hx 3 , u] h = 0, and ρ H ([x 1 , x 2 , x 3 ] g , x 4 ) − ρ H (x 1 , x 2 )ρ H (x 3 , x 4 ) −ρ H (x 2 , x 3 )ρ H (x 1 , x 4 ) − ρ H (x 3 , x 1 )ρ H (x 2 , x 4 ) (u) = ρ([x 1 , x 2 , x 3 ] g , x 4 )u + [H[x 1 , x 2 , x 3 ] h , Hx 4 , u] h − ρ(x 1 , x 2 )ρ(x 3 , x 4 )u −[Hx 1 , Hx 2 , ρ(x 3 , x 4 )u] h − ρ(x 1 , x 2 )[Hx 3 , Hx 4 , u] h − [Hx 1 , Hx 2 , [Hx 3 , Hx 4 , u] h ] h −ρ(x 2 , x 3 )ρ(x 1 , x 4 )u − [Hx 2 , Hx 3 , ρ(x 1 , x 4 )u] h − ρ(x 2 , x 3 )[Hx 1 , Hx 4 , u] h −[Hx 2 , Hx 3 , [Hx 1 , Hx 4 , u] h ] h − ρ(x 3 , x 1 )ρ(x 2 , x 4 )u − [Hx 3 , Hx 1 , ρ(x 2 , x 4 )u] h −ρ(x 3 , x 1 )[Hx 2 , Hx 4 , u] h − [Hx 3 , Hx 1 , [Hx 2 , Hx 4 , u] h ] h = 0. Therefore, we deduce that (h; ρ H ) is a representation of the 3-Lie algebra (g, [·, ·, ·] g ). Let d ρ H : C n 3Lie (g; h) → C n+1 3Lie (g; h), (n ≥ 1) be the corresponding coboundary operator of the 3-Lie algebra (g, [·, ·, ·] g ) with coefficients in the representation (h; ρ H ). More precisely, for all f ∈ Hom(∧ 2 g ⊗ · · · ⊗ ∧ 2 g (n−1) ∧g, h), X i = x i ∧ y i ∈ ∧ 2 g, i = 1, 2, · · · , n and x n+1 ∈ g, we have (d ρ H f )(X 1 , · · · , X n , x n+1 ) = 1≤i<k≤n (−1) i f (X 1 , · · · ,X i , · · · , X k−1 , [x i , y i , x k ] g ∧ y k +x k ∧ [x i , y i , y k ] g , X k+1 , · · · , X n , x n+1 ) + n i=1 (−1) i f (X 1 , · · · ,X i , · · · , X n , [x i , y i , x n+1 ] g ) + n i=1 (−1) i+1 ρ(x i , y i ) f (X 1 , · · · ,X i , · · · , X n , x n+1 ) + n i=1 (−1) i+1 [Hx i , Hy i , f (X 1 , · · · ,X i , · · · , X n , x n+1 )] h +(−1) n+1 ρ(y n , x n+1 ) f (X 1 , · · · , X n−1 , x n ) + [Hy n , Hx n+1 , f (X 1 , · · · , X n−1 , x n )] h +(−1) n+1 ρ(x n+1 , x n ) f (X 1 , · · · , X n−1 , y n ) + [Hx n+1 , Hx n , f (X 1 , · · · , X n−1 , y n )] h . It is obvious that f ∈ Hom(g; h) is closed if and only if f ([x 1 , x 2 , x 3 ] g ) = ρ(x 1 , x 2 ) f (x 3 ) + [Hx 1 , Hx 2 , f (x 3 )] h + ρ(x 2 , x 3 ) f (x 1 ) +[Hx 2 , Hx 3 , f (x 1 )] h + ρ(x 3 , x 1 ) f (x 2 ) + [Hx 3 , Hx 1 , f (x 2 )] h , ∀x 1 , x 2 , x 3 ∈ g. Define δ : ∧ 2 g → Hom(g, h) by Proof. For all x 1 , x 2 , x 3 ∈ g, by (1)- (7), we have δ(X)z = ρ(y, z)H(x) + ρ(z, x)H(y) + [Hx, Hy, Hz] h , ∀X = x ∧ y ∈ ∧ 2 g, z ∈ g.(d ρ H δ(X))(x 1 , x 2 , x 3 ) = ρ(x 1 , x 2 )δ(X)(x 3 ) + ρ(x 2 , x 3 )δ(X)(x 1 ) + ρ(x 3 , x 1 )δ(X)(x 2 ) − δ(X)([x 1 , x 2 , x 3 ] g ) +[Hx 1 , Hx 2 , δ(X)x 3 ] h + [δ(X)x 1 ,, x 2 , x 3 ] g ] h − ρ(y, [x 1 , x 2 , x 3 ] g )(Hx) − ρ([x 1 , x 2 , x 3 ] g , x)(Hy) = 0. Thus, we deduce that d ρ H δ(X) = 0. The proof is finished. We now give the cohomology of crossed homomorphisms on 3-Lie algebras. Let H be a crossed homomorphism from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. Define the set of n-cochains by C n H (g; h) = C n−1 3Lie (g; h), n ≥ 2, g ∧ g, n = 1. At the end of this subsection, we show that certain homomorphisms between crossed homomorphisms induce homomorphisms between the corresponding cohomology groups. Let H and H ′ be two crossed homomorphisms from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. Let (ψ g , ψ h ) be a homomorphism from H to H ′ in which ψ g is invertible. Define a map p : C n H (g; h) → C n H ′ (g; h) by p(ω)(X 1 , · · · , X n−2 , x n−1 ) = ψ h ω ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (x n−1 ) , for all ω ∈ C n H (g; h), X i = x i ∧ y i ∈ ∧ 2 g, i = 1, 2, · · · , n − 2 and x n−1 ∈ g. Theorem 3.5. With above notations, p is a cochain map from the cochain complex ( ∞ ⊕ n=2 C n H (g; h), d ρ H ) to the cochain complex ( ∞ ⊕ n=2 C n H ′ (g; h), d ρ H ′ ). Consequently, it induces a homomorphism p * from the cohomology group H n H (g; h) to H n H ′ (g; h). Proof. For all ω ∈ C n H (g; h), by (9)-(10) and (13)-(15), we have d ρ H ′ (p(ω))(X 1 , · · · , X n−1 , x n ) = 1≤i<k≤n−1 (−1) i p(ω)(X 1 , · · · ,X i , · · · , X k−1 , [x i , y i , x k ] g ∧ y k + x k ∧ [x i , y i , y k ] g , X k+1 , · · · , X n−1 , x n ) + n−1 i=1 (−1) i p(ω)(X 1 , · · · ,X i , · · · , X n , [x i , y i , x n ] g ) + n−1 i=1 (−1) i+1 ρ(x i , y i )p(ω)(X 1 , · · · ,X i , · · · , X n−1 , x n ) + n−1 i=1 (−1) i+1 [H ′ x i , H ′ y i , p(ω)(X 1 , · · · ,X i , · · · , X n−1 , x n )] h + (−1) n ρ(y n−1 , x n )p(ω)(X 1 , · · · , X n−2 , x n−1 ) + [H ′ y n−1 , H ′ x n , p(ω)(X 1 , · · · , X n−2 , x n−1 )] h + (−1) n ρ(x n , x n−1 )p(ω)(X 1 , · · · , X n−2 , y n−1 ) (17) that + [H ′ x n , H ′ x n−1 , p(ω)(X 1 , · · · , X n−2 , y n−1 )] h = 1≤i<k≤n−1 (−1) i ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · ,ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x k−1 ) ∧ ψ −1 g (y k−1 ), ψ −1 g ([x i , y i , x k ] g ) ∧ ψ −1 g (y k ) + ψ −1 g (x k ) ∧ ψ −1 g ([x i , y i , y k ] g ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n )) + n−1 i=1 (−1) i ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · ,ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g ([x i , y i , x n ] g )) + n−1 i=1 (−1) i+1 ρ(x i , y i )ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n )) + n−1 i=1 (−1) i+1 [H ′ x i , H ′ y i , ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n )) ] h + (−1) n ρ(y n−1 , x n )ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (x n−1 )) + [H ′ y n−1 , H ′ x n , ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (x n−1 )) ] h + (−1) n ρ(x n , x n−1 )ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (y n−1 )) + [H ′ x n , H ′ x n−1 , ψ h (ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (y n−1 )) ] h = 1≤i<k≤n−1 (−1) i ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · ,ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x k−1 ) ∧ ψ −1 g (y k−1 ), ([ψ −1 g (x i ), ψ −1 g (y i ), ψ −1 g (x k )] g ) ∧ ψ −1 g (y k ) + ψ −1 g (x k ) ∧ ([ψ −1 g (x i ), ψ −1 g (y i ), ψ −1 g (y k )] g ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n )) + n−1 i=1 (−1) i ψ h ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · ,ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ([ψ −1 g (x i ), ψ −1 g (y i ), ψ −1 g (x n )] g )) + n−1 i=1 (−1) i+1 ψ h ρ(ψ −1 g (x i ), ψ −1 g (y i )) ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · ,ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n )) + n−1 i=1 (−1) i+1 ψ h [H(ψ −1 g (x i )), H(ψ −1 g (y i )), ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x i ) ∧ψ −1 g (y i ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n ))] h + (−1) n ψ h ρ(ψ −1 g (y n−1 ), ψ −1 g (x n ))ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (x n−1 )) + [Hψ −1 g (y n−1 ), Hψ −1 g (x n ), ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (x n−1 ))] h + (−1) n ψ h ρ(ψ −1 g (x n ), ψ −1 g (x n−1 ))ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (y n−1 )) + [Hψ −1 g (x n ), Hψ −1 g (x n−1 ), ω(ψ −1 g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−2 ) ∧ ψ −1 g (y n−2 ), ψ −1 g (y n−1 ))] hH 1 (z) = H 2 (z) + (∂X)(z), ∀z ∈ g, which implies that H 1 and H 2 are in the same cohomology class. Maurer-Cartan characterization of crossed homomorphisms on 3-Lie algebras In this section, we construct a suitable L ∞ -algebra, which characterize crossed homomorphisms on 3-Lie algebras as Maurer-Cartan elements. Then we construct a twisted L ∞ -algebra that controls deformations of crossed homomorphisms. Definition 4.1. An L ∞ -algebra is a Z-graded vector space g = ⊕ k∈Z g k equipped with a collection (k ≥ 1) of linear maps l k : ⊗ k g → g of degree 1 with the property that, for any homogeneous elements x 1 , · · · , x n ∈ g, we have (i) (graded symmetry) for every σ ∈ S n , l n (x σ(1) , · · · , x σ(n−1) , x σ(n) ) = ε(σ)l n (x 1 , · · · , x n−1 , x n ), (ii) (generalized Jacobi Identity) for all n ≥ 1, n i=1 σ∈S (i,n−i) ε(σ)l n−i+1 (l i (x σ(1) , · · · , x σ(i) ), x σ(i+1) , · · · , x σ(n) ) = 0. Let α be a Maurer-Cartan element of an L ∞ -algebra (g, {l i } +∞ i=1 ). For all k ≥ 1 and x 1 , · · · , x k ∈ g, define a series of linear maps l α k : ⊗ k g → g of degree 1 by l α k (x 1 , · · · , x k ) = +∞ n=0 1 n! l n+k {α, · · · , α n , x 1 , · · · , x k }.(19)Theorem 4.3. ([14]) With the above notations, (g, {l α i } +∞ i=1 ) is an L ∞ -algebra, obtained from the L ∞ -algebra (g, {l i } +∞ i=1 ) by twisting with the Maurer-Cartan element α. Moreover, α + α ′ is a Maurer-Cartan element of (g, {l i } +∞ i=1 ) if and only if α ′ is a Maurer-Cartan element of the twisted L ∞ -algebra (g, {l α i } +∞ i=1 ) . In [34], Th. Voronov developed the theory of higher derived brackets, which is a useful tool to construct explicit L ∞ -algebras. • F is an abelian graded Lie subalgebra of (L, [·, ·]), • P : L → L is a projection, that is P • P = P, whose image is F and kernel is a graded Lie subalgebra of (L, [·, ·]), • ∆ is an element in ker(P) 1 such that [∆, ∆] = 0. Theorem 4.5. ([34]) Let (L, F, P, ∆) be a V-data. Then (F, {l k } +∞ k=1 ) is an L ∞ -algebra, where l k (a 1 , · · · , a k ) = P [· · · [[ k ∆, a 1 ], a 2 ] , · · · , a k ], for homogeneous a 1 , · · · , a k ∈ F. (20) We call {l k } +∞ k=1 the higher derived brackets of the V-data (L, F, P, ∆). Let g be a vector space. We consider the graded vector space C * (g, g) = ⊕ n≥0 C n (g, g) = ⊕ n≥0 Hom(∧ 2 g ⊗ · · · ⊗ ∧ 2 g n ∧g, g). Theorem 4.6. ( [31]) The graded vector space C * (g, g) equipped with the graded commutator bracket [P, Q] 3Lie = P•Q − (−1) pq Q•P, ∀ P ∈ C p (g, g), Q ∈ C q (g, g),(21) is a graded Lie algebra, where P•Q ∈ C p+q (g, g) is defined by (1) , · · · , X σ(k−1) , x k+q ∧ Q X σ(k) , · · · , X σ(k+q−1) , y k+q , X k+q+1 , · · · , X p+q , x + σ∈S(p,q) (−1) pq (−1) σ P X σ(1) , · · · , X σ(p) , Q X σ(p+1) , · · · , X σ(p+q−1) , X σ(p+q) , x , for all X i = x i ∧ y i ∈ ∧ 2 g, i = 1, 2, · · · , p + q and x ∈ g. (P•Q)(X 1 , · · · , X p+q , x) = p k=1 (−1) (k−1)q σ∈S(k−1,q) (−1) σ P X σ(1) , · · · , X σ(k−1) , Q X σ(k) , · · · , X σ(k+q−1) , x k+q ∧ y k+q , X k+q+1 , · · · , X p+q , x + p k=1 (−1) (k−1)q σ∈S(k−1,q) (−1) σ P X σ Moreover, µ : ∧ 3 g −→ g is a 3-Lie bracket if and only if [µ, µ] 3Lie = 0, i.e. µ is a Maurer-Cartan element of the graded Lie algebra (C * (g, g), [·, ·] 3Lie ). Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). For convenience, we use π : ∧ 3 g → g to indicate the 3-Lie bracket [·, ·, ·] g and µ : ∧ 3 h → h to indicate the 3-Lie bracket [·, ·, ·] h . In the sequel, we use π + ρ + µ to denote the element in Hom(∧ 3 (g ⊕ h), g ⊕ h) given by (22) (π + ρ + µ)(x + u, y + v, z + w) = [x, y, z] g + ρ(x, y)w + ρ(y, z)u + ρ(z, x)v + [u, v, w] h , for all x, y, z ∈ g, u, v, w ∈ h. Note that the right hand side is exactly the semidirect product 3-Lie algebra structure given in Proposition 2.5. Therefore by Theorem 4.6, we have [π + ρ + µ, π + ρ + µ] 3Lie = 0. Proof. It follows from straightforward computations. Proposition 4.8. Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). Then we have a V-data (L, F, P, ∆) as follows: • the graded Lie algebra (L, [·, ·]) is given by (C * (g ⊕ h, g ⊕ h), [·, ·] 3Lie ); • the abelian graded Lie subalgebra F is given by F = C * (g, h) = ⊕ n≥0 C n (g, h) = ⊕ n≥0 Hom(∧ 2 g ⊗ · · · ⊗ ∧ 2 g n ∧g, h); • P : L → L is the projection onto the subspace F; • ∆ = π + ρ + µ. Consequently, we obtain an L ∞ -algebra (C * (g, h), l 1 , l 3 ), where l 1 (P) = [π + ρ + µ, P] 3Lie , l 3 (P, Q, R) = [[[π + ρ + µ, P] 3Lie , Q] 3Lie , R] 3Lie , for all P ∈ C m (g, h), Q ∈ C n (g, h) and R ∈ C k (g, h). Proof. By Theorem 4.5, (F, {l k } ∞ k=1 ) is an L ∞ -algebra, where l k is given by (20). It is obvious that ∆ = π + ρ + µ ∈ ker(P) 1 . For all P ∈ C m (g, h), Q ∈ C n (g, h) and R ∈ C k (g, h), by Lemma 4.7, we have [[π + ρ + µ, P] 3Lie , Q] 3Lie ∈ ker(P), which implies that l 2 = 0. Similarly, we have l k = 0, when k ≥ 4. Therefore, the graded vector space C * (g, h) is an L ∞ -algebra with nontrivial l 1 , l 3 , and other maps are trivial. Theorem 4.9. Let ρ : ∧ 2 g → gl(h) be an action of a 3-Lie algebra (g, [·, ·, ·] g ) on a 3-Lie algebra (h, [·, ·, ·] h ). Then Maurer-Cartan elements of the L ∞ -algebra (C * (g, h), l 1 , l 3 ) are precisely crossed homomorphisms from the 3-Lie algebra (g, [·, ·, ·] g ) to the 3-Lie algebra (h, [·, ·, ·] h ) with respect to the action ρ. Proof. It is straightforward to deduce that [π + ρ + µ, H] 3Lie (x, y, z) = ρ(y, z)(Hx) + ρ(z, x)(Hy) + ρ(x, y)(Hz) − Hπ(x, y, z); [[[π + ρ + µ, H] 3Lie , H] 3Lie , H] 3Lie (x, y, z) = 6µ (Hx, Hy, Hz). Let H be a Maurer-Cartan element of the L ∞ -algebra (C * (g, h), l 1 , l 3 ). We have +∞ n=1 1 n! l n (H, · · · , H)(x, y, z) = µ(Hx, Hy, Hz) + ρ(y, z)(Hx) + ρ(z, x)(Hy) + ρ(x, y)(Hz) − Hπ(x, y, z) = 0, which implies that H is a crossed homomorphism from the 3-Lie algebra (g, [·, ·, ·] g ) to the 3-Lie algebra (h, [·, ·, ·] h ) with respect to the action ρ. = [π + ρ + µ,+ n i=1 (−1) n−1 (−1) i−1 (π + ρ + µ)(X i , f (X 1 , · · · ,X i , · · · , X n , x n+1 )) −(−1) n−1 n−1 k=1 k i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X k , (π + ρ + µ)(X i , x k+1 ) ∧ y k+1 , X k+2 , · · · , X n , x n+1 −(−1) n−1 n−1 k=1 k i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X k , x k+1 ∧ (π + ρ + µ)(X i , y k+1 ), X k+2 , · · · , X n , x n+1 −(−1) n−1 n i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X n , (π + ρ + µ)(X i , x n+1 ) = ρ(y n , x n+1 ) f (X 1 , · · · , X n−1 , x n ) + ρ(x n+1 , x n ) f (X 1 , · · · , X n−1 , y n ) + n i=1 (−1) n−1 (−1) i−1 ρ(x i , y i ) f (, X 1 , · · · ,, X i , · · · , , X n , x n+1 ) −(−1) n−1 n−1 k=1 k i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X k , π(x i , y i , x k+1 ) ∧ y k+1 , X k+2 , · · · , X n , x n+1 −(−1) n−1 n−1 k=1 k i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X k , x k+1 ∧ π(x i , y i , y k+1 ), X k+2 , · · · , X n , x n+1 −(−1) n−1 n i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X n , π(x i , y i , x n+1 ) . By Lemma 4.7, we have l 3 (H, H, f )(X 1 , · · · , X n , x n+1 ) = [[[π + ρ + µ, H] 3Lie , H] 3Lie , f ] 3Lie (X 1 , · · · , X n , x n+1 ) = [[π + ρ + µ, H] 3Lie , H] 3Lie f (X 1 , · · · , X n−1 , x n ) ∧ y n , x n+1 +[[π + ρ + µ, H] 3Lie , H] 3Lie x n ∧ f (X 1 , · · · , X n−1 , y n ), x n+1 + n i=1 (−1) n−1 (−1) i−1 [[π + ρ + µ, H] 3Lie , H] 3Lie X i , f (X 1 , · · · ,X i , · · · , X n , x n+1 ) y k+1 ), X k+2 , · · · , X n , x n+1 −(−1) n−1 n i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X n , [[π + ρ + µ, H] 3Lie , H] 3Lie (X i , x n+1 ) = 2 [ f (X 1 , · · · , X n−1 , x n ), Hy n , Hx n+1 ] h + [Hx n , f (X 1 , · · · , X n−1 , y n ), Hx n+1 ] h + n i=1 (−1) n−1 (−1) i−1 [Hx i , Hy i , f (X 1 , · · · ,X i , X n , x n+1 )] h COHOMOLOGY AND THE CONTROLLING ALGEBRA OF CROSSED HOMOMORPHISMS ON 3-LIE ALGEBRAS 17 −(−1) n−1 n−1 k=1 k i=1 (−1) i+1 f X 1 , · · · ,X i , · · · , X k , [[π + ρ + µ, H] 3Lie , H] 3Lie (X i , x k+1 ) ∧ y k+1 +x k+1 ∧ [[π + ρ + µ, H] 3Lie , H] 3Lie (X i , Thus, we deduce that d H f = (−1) n−1 l 1 ( f ) + 1 2 l 3 (H, H, f ) , that is d H f = (−1) n−1 l H 1 f . Proposition 3. 3 . 3Let H be a crossed homomorphism from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. Then δ(X) is a 1-cocycle of the 3-Lie algebra (g, [·, ·, ·] g ) with coefficients in (h; ρ H ). Definition 4. 2 . 2A Maurer-Cartan element of an L ∞ -algebra (g = ⊕ k∈Z g k , {l i } +∞ i=1 ) is an element α ∈ g 0 satisfying the Maurer-Cartan equation +∞ n=1 1 n! l n (α, · · · , α) = 0. Definition 4.4. ([34]) A V-data consists of a quadruple (L, F, P, ∆), where • (L, [·, ·]) is a graded Lie algebra, Lemma 4.7. Let H : g → h be a crossed homomorphism from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. For all x, y, z ∈ g, u, v, w ∈ h, we have[[π + ρ + µ,H] 3Lie , H] 3Lie (x + u, y + v, z + w) = 2 [Hx, Hy, w] h + [Hx, v, Hz] h + [u, Hy, Hz] h H] 3Lie (x, y, z) + 1 3! [[[π + ρ + µ, H] 3Lie , H] 3Lie , H] 3Lie (x, y, z) Now we are ready to give the main result in this section. g (x 1 ) ∧ ψ −1 g (y 1 ), · · · , ψ −1 g (x n−1 ) ∧ ψ −1 g (y n−1 ), ψ −1 g (x n ) =p(d ρ H ω)(X 1 , · · · , X n−1 , x n ), where X i = x i ∧ y i ∈ ∧ 2 g, i = 1, 2, · · · , n − 1 and x n ∈ g. Thus p is a cochain map, and induces a homomorphism p * from the cohomology group H n H (g; h) to H n H ′ (g; h).3.2.Infinitesimal deformations of crossed homomorphisms. In this section, we use the established cohomology theory to characterize infinitesimal deformations of crossed homomorphisms on 3-Lie algebras. Let (g, [·, ·, ·] g ) be a 3-Lie algebra over K and K[t] be the polynomial ring in one variable t., where the 3-Lie algebra structure is defined byIn the sequel, all the vector spaces are finite dimensional vector spaces over K and we denoteDefinition 3.6. Let H : g → h be a crossed homomorphism from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. Let H : g → h be a linear map. If H t = H + tH is still a crossed homomorphism modulo t 2 , they we say that H generates an infinitesimal deformation of the crossed homomorphism H. . Definition 3.7. Let H be a crossed homomorphism from a 3-Lie algebra (g, [·, ·, ·] g ) to a 3-Lie algebra (h, [·, ·, ·] h ) with respect to an action ρ. Two one-parameter infinitesimal deformationsIn particular, an infinitesimal deformation H t = H + tH 1 of a crossed homomorphism H is said to be trivial if there exists X ∈ g ∧ g such that (Id g + tad X , Id h + tρ(X)) is a homomorphism modulo t 2 from H t to H.Let (Id g + tad X , Id h + tρ(X)) be a homomorphism modulo t 2 from H 1 t to H 2 t . By (9), we get, (Id h + tρ(X))(H + tH 1 )(z) = (H + tH 2 )(Id g + tad X )(z), ∀X = x ∧ y ∈ ∧ 2 g, z ∈ g.Proposition 4.10. Let H be a crossed homomorphism from a 3-Lie algebra g to a 3-Lie algebra h with respect to an action ρ. Then C * (g, h) carries a twisted L ∞ -algebra structure as following:l H 2 (P, Q) = l 3 (H, P, Q),(24)l H 3 (P, Q, R) = l 3 (P, Q, R),where P ∈ C m (g, h), Q ∈ C n (g, h) and R ∈ C k (g, h).Proof. Since H is a Maurer-Cartan element of the L ∞ -algebra (C * (g, h), l 1 , l 3 ), by Theorem 4.3, we have the conclusions.The above L ∞ -algebra controls deformations of crossed homomorphisms on 3-Lie algebras.Proof. By Theorem 4.9, H + H ′ is a crossed homomorphism if and only ifApplying l 1 (H) + 1 3! l 3 (H, H, H) = 0, the above condition is equivalent towhich implies that H ′ is a Maurer-Cartan element of the twisted L ∞ -algebra (C * (g, h), l H 1 , l H 2 , l H 3 ). Next we give the relationship between the coboundary operator d ρ H and the differential l H 1 defined by (23) using the Maurer-Cartan element H of the L ∞ -algebra (C * (g, h), l 1 , l 3 ).Theorem 4.12. 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[ "Electroweak Skyrmions in the HEFT", "Electroweak Skyrmions in the HEFT" ]	[ "Juan Carlos Criado [email protected] \nInstitute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDH1 3LEDurhamUK\n", "Valentin V Khoze \nInstitute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDH1 3LEDurhamUK\n", "Michael Spannowsky [email protected] \nInstitute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDH1 3LEDurhamUK\n" ]	[ "Institute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDH1 3LEDurhamUK", "Institute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDH1 3LEDurhamUK", "Institute for Particle Physics Phenomenology\nDepartment of Physics\nDurham University\nDH1 3LEDurhamUK" ]	[]	We study the existence of skyrmions in the presence of all the electroweak degrees of freedom, including a dynamical Higgs boson, with the electroweak symmetry being non-linearly realized in the scalar sector. For this, we use the formulation of the Higgs Effective Field Theory (HEFT). In contrast with the linear realization, a well-defined winding number exists in HEFT for all scalar field configurations. We classify the effective operators that can potentially stabilize the skyrmions and numerically find the region in parameter spaces that support them. We do so by minimizing the static energy functional using neural networks. This method allows us to obtain the minimal-energy path connecting the vacuum to the skyrmion configuration and calculate its mass and radius. Since skyrmions are not expected to be produced at colliders, we explore the experimental and theoretical bounds on the operators that generate them. Finally, we briefly consider the possibility of skyrmions being dark matter candidates.	10.1007/jhep12(2021)026	[ "https://arxiv.org/pdf/2109.01596v1.pdf" ]	237,416,598	2109.01596	0f2b8afa0c420d6a5b5fa90aa97a048573e3a9f6	Electroweak Skyrmions in the HEFT Juan Carlos Criado [email protected] Institute for Particle Physics Phenomenology Department of Physics Durham University DH1 3LEDurhamUK Valentin V Khoze Institute for Particle Physics Phenomenology Department of Physics Durham University DH1 3LEDurhamUK Michael Spannowsky [email protected] Institute for Particle Physics Phenomenology Department of Physics Durham University DH1 3LEDurhamUK Electroweak Skyrmions in the HEFT We study the existence of skyrmions in the presence of all the electroweak degrees of freedom, including a dynamical Higgs boson, with the electroweak symmetry being non-linearly realized in the scalar sector. For this, we use the formulation of the Higgs Effective Field Theory (HEFT). In contrast with the linear realization, a well-defined winding number exists in HEFT for all scalar field configurations. We classify the effective operators that can potentially stabilize the skyrmions and numerically find the region in parameter spaces that support them. We do so by minimizing the static energy functional using neural networks. This method allows us to obtain the minimal-energy path connecting the vacuum to the skyrmion configuration and calculate its mass and radius. Since skyrmions are not expected to be produced at colliders, we explore the experimental and theoretical bounds on the operators that generate them. Finally, we briefly consider the possibility of skyrmions being dark matter candidates. Introduction Skyrmions are extended field configurations that behave as new particle degrees of freedom. Initially, they were proposed as a description of baryons within an Effective Field Theory (EFT) description of strong interactions containing only the pion fields [1][2][3]. Since the pions can be viewed as pseudo-Goldstone bosons arising from the breaking of an SU (2) symmetry, the original setting can be directly applied to the electroweak sector, in the limit in which the Higgs field is infinitely massive, and the gauge bosons are decoupled, so that only the SU (2) would-be Goldstone bosons are present [4]. Electroweak skyrmions have been shown to survive under certain conditions in more realistic settings, in which these limits are partially removed, which can destroy the topological protection they enjoyed in the first place [5]. The purpose of ref. [6] and of this work is to consider skyrmions in the full electroweak theory, including the effects of both the gauge fields and a dynamical Higgs boson. As in the original Skyrme setting, the existence of skyrmions in the Standard Model (SM) Lagrangian is forbidden by Derrick's theorem [7], but they can be stabilized by including higher-order effective operators. Since the discovery of the Higgs [8,9], two effective descriptions of the electroweak sector have emerged: the Standard Model EFT (SMEFT), in which the scalars furnish a linear representation of the electroweak symmetry group; and the Higgs Effective Field Theory (HEFT), in which the realization of this symmetry is non-linear. The SMEFT version is studied in ref. [6]. In this paper, we focus on the HEFT framework, which we find to be better suited for the description of skyrmions because of the non-trivial topology of its scalar sector. In section 2, we briefly introduce the relevant sector of the HEFT, discuss the differences with the SMEFT and with the approximations that have been previously taken, and introduce the topological numbers that characterize the topology of its field configurations. In section 3, we study the existence of skyrmions numerically in the presence of the different combinations of HEFT operators. In section 4, we consider the phenomenological consequences of skyrmions and the operators that generate them. This allows us to obtain constraints on the parameter space, in which we include positivity bounds. We summarize our conclusions in section 5. Theory The relevant degrees of freedom for skyrmions in the electroweak sector are the SU (2) gauge bosons W a µ , the would-be Goldstone bosons G a , and the Higgs boson h. We neglect the effects of the U (1) Y gauge sector. The Higgs is invariant under SU (2) gauge transformations, while the Goldstones are collected in a non-linear representation U = exp iσ a G a √ 2v ∈ SU (2), (2.1) with no relation to the Higgs singlet field h. 1 We write the effective Lagrangian as L = i F i (h/v)Q i , F i (η) = ∞ n=0 c i,n η n , (2.2) where Λ is the HEFT cut-off scale; the Q i are monomials in W µν , U , h and their covariant derivatives, with h appearing only through its derivatives. That is, schematically Q i ∼ h H i (W µν ) W i U U i D D i ,(2.3) where H i , W i , D i are, respectively, the number of Higgs fields, field-strength tensors, and covariant derivatives contained in Q i . In this setting, D i corresponds to the general chiral dimension [10]. We adopt a power counting based on the chiral dimension, in which each c i,n coefficient is of order Λ 2−D i , multiplied by the necessary power of v for the coefficient to have the correct energy dimensions. Thus, terms with higher chiral dimensions are suppressed by higher powers of v/Λ. We keep terms with chiral dimension 4, and impose custodial symmetry, which is needed for configurations in the spherical ansatz to give spherically symmetric contributions to the energy, as described in section 3.1. A list of all relevant operators Q i is given in table 1, partially following the notation of ref. [11], where angle brackets · denote a trace and L µ = iU D µ U † . 1 This is to be contrasted with the more restrictive linear realization where h(x) and U (x) are assembled into the Higgs doublet φ = 1 √ 2 (v + h) U 0 1 . Name Operator Radial energy density ρ i in spherical ansatz The relevant sector of the SM Lagrangian is given by the chiral dimension 2 operators, with Q 1 1 − r 2 e 2 v 4 Q h ∂ µ h∂ µ h r 2 2 (η ) 2 Q U D µ U † D µ U 2 v 2 f 2 1 + f 2 2 + r 2 2 b 2 Q Xh2 W µν W µν −8e 2 (f 1 − 2bf 2 ) 2 + (f 2 + (2f 1 − 1)b) 2 + 2 r 2 (f 2 1 + f 2 2 − f 1 ) 2 Q Xh5 µνρσ W µν W ρσ 0 Q XU 8 i W µν [L µ , L ν ] 16e 2 2r 2 (f 2 1 + f 2 2 )(f 2 1 + f 2 2 − f 1 + 2r 2 b 2 ) − br 2 (f 2 f 1 − f 1 f 2 + bf 1 ) Q XU 11 i µνρσ W µν [L ρ , L σ ] 0 Q D1 L µ L µ 2 − 4e 2 r 2 2(f 2 1 + f 2 2 ) + r 2 b 2 2 Q D2 L µ L ν L µ L ν − 4e 2 r 2 2(f 2 1 + f 2 2 ) 2 + r 4 b 4 Q D7 L µ L µ ∂ ν h∂ ν h −e 2 v 2 (η ) 2 2(f 2 1 + f 2 2 ) + r 2 b 2 Q D8 L µ L ν ∂ µ h∂ ν h −e 2 v 2 (η ) 2 r 2 b 2 Q D11 (∂ µ h∂ µ h) 2 − e 2 v 4 4 (η ) 4 r 2F 1 (h/v) = V (h) = λv 4 (h/v) 2 + (h/v) 3 + 1 4 (h/v) 4 = λ v 2 h 2 + vh 3 + h 4 4 , (2.4) F h (h/v) = 1 2 , F Xh2 (h/v) = − 1 2g 2 , F U (h/v) = v 2 4 1 + h v 2 . (2.5) Deviations from the SM are encoded in modifications of any of the F i (h). Derrick's theorem forbids the existence of solitons in the SM. A necessary condition for them to exist is that higher-derivative terms are present. The original term proposed by Skyrme [1] to stabilize skyrmions can be written in the HEFT Lagrangian as L Sk = − 1 16e 2 (Q D1 − Q D2 ), (2.6) that is, setting F D1 (h/v) = −F D2 (h/v) = − 1/(16e 2 ). In the chiral dimension powercounting, the size of the coefficient is given by e ∼ Λ/(4v). The theory L SM + L Sk is thus a candidate for the stabilization of skyrmions. Two limits of it have been previously studied in the literature: A. Frozen Higgs. This corresponds to m h → ∞, which implies that the Higgs is set to its vev h = 0 everywhere. B. No gauge fields. This is obtained when the SU (2) vanishes, g → 0. In this limit, the coefficient of the W µν W µν term becomes large, and the gauge fields are forced to approach a pure gauge configuration in order to minimize the energy. One can gauge them away. The only degrees of freedom left are the Goldstone bosons and the Higgs. Taking both limits leads to a theory with only the Goldstone bosons as dynamical degrees of freedom, which has been studied in, e.g. ref. [3]. Limit A has been considered in ref. [5], while limit B has been considered in ref. [12]. In any of these limits, and in the full theory, the Skyrme term can be generalized by allowing other linear combinations of the Q D1 and Q D2 operators. This has been done in the case where both limits are taken, in ref. [4], and in limit B, in ref. [13]. In ref. [6] skyrmions were studied in the full theory without assuming any of the two limits above. This was done within the SMEFT framework, in which the electroweak symmetry is realized linearly. The purpose of the present paper is to continue this program in the non-linear realization. Ultimately, the existence of skyrmions turns out to be much harder to prove in the SMEFT than in the HEFT, as discussed below. Ref. [14], which appeared during the preparation of this work, has a similar scope. In limit B, the theory contains stable field configurations separated from the vacuum by an infinite energy barrier. This fact can be understood from a topological point of view. To have finite energy, the scalar fields must satisfy the following boundary conditions: lim \|x\|→∞ h(x) = 0, lim \|x\|→∞ U (x) = 1 2×2 , (2.7) which means that all directions towards infinity can be identified with a single point, effectively compactifying space into S 3 . Thus, the fields can be viewed as a S 3 → R × S 3 mapping. We can then define a topological charge, the winding number for the U : S 3 → S 3 part of the mapping: n U = 1 24π 2 ijk d 3 x L i L j L k . (2.8) This is a homotopy invariant of U , and therefore it can never change with smooth time evolution. However, this number is only well defined when the target space of the scalar sector has the topology R×S 3 . This is true generically both in the full HEFT and in limit B, but it ceases to be in the particular case of the SMEFT, in which the Lagrangian becomes independent of U (x) when h(x) = −v (see footnote 1 above). One can then identify all points with this value of h, turning the scalar manifold into R 4 ∼ = C 2 . The scalar degrees of freedom are thus collected into a SU (2) doublet φ. In general, the topology of the static configurations of φ cannot be characterized in terms of the number n U since U is only defined through φ = 1 √ 2 (v + h) U · (0 1) T when φ = 0 everywhere. 2 One can recover a welldefined n U in the SMEFT by taking the frozen Higgs limit A, which disallows h(x) = −v and forces the scalars to be in the submanifold S 3 . As already noted earlier, we will not follow this route in this paper and will instead use the HEFT formulation of the theory where h and U are independent and without imposing limits A and B. The inclusion of gauge fields destroys the topological protection of n U = 0 configurations from decaying into the vacuum. However, the W ij W ij term in the energy induces a finite-energy barrier between configurations in which W µ is a pure gauge, W i = U∂ i U † , W 0 = 0, possibly making them metastable. In order to describe this, we use the Chern-Simons number n CS = 1 16π 2 ijk d 3 x W i W jk + 2i 3 W i W j W k . (2.9) For a pure-gauge W i = U∂ i U † , n CS is the integer winding number of the gauge transfor- mation U(x) : S 3 → S 3 . A skyrmion is a field configuration for which n U and n CS differ by (approximately 3 ) one unit. We thus define the skyrmion number as n Sk = n U − n CS . (2.10) While n U and n CS are not gauge invariant, n Sk is, because n U and n CS change by the same integer under a large gauge transformation. An anti-skyrmion is similarly a configuration where n Sk −1, and multi-skyrmions have \|n Sk \| > 1. A CP transformation changes the sign of the skyrmion number. Skyrmion configurations and energy landscape The energy functional in the spherical ansatz We parametrize the space of static configurations of the fields W a µ , U and h in the W 0 = 0 gauge by means of 4 real functions of one variable: f 1 , f 2 , b and η. We do so by further imposing the unitary gauge U (x) = 1 2×2 and the spherical ansatz: W i (x) = veτ a ija n j f 1 (r) r + (δ ia − n i n a ) f 2 (r) r + n i n a b(r) , h(x) = v √ 2 η(r), (3.1) where τ a are the Pauli matrices, n i = x i /\|x\|, r = ve\|x\|, and e is a parameter we will adjust as a function of Wilson coefficients. The energy density in this ansatz is spherically symmetric when all interactions are invariant under custodial symmetry. 4 We can then write the energy as E = 4πv e ∞ 0 dr i F i (η)ρ i , (3.2) where the contributions ρ i to the radial energy density of each Q i operator are given in table 1. Requiring that the energy is finite and that the fields are regular at the origin gives rise to the following boundary conditions: f 1 (0) = f 1 (0) = f 2 (0) = f 2 (0) − b(0) = η (0) = 0, (3.3) f 1 (∞) = f 2 (∞) = b(∞) = η(∞) = 0. (3.4) Since we have fixed the unitary gauge, the skyrmion number is just n Sk = −n CS . For convenience, we define which agrees with n CS at integer values. Thus, skyrmion and anti-skyrmions will be found at n W −1 and n W 1, respectively. CP symmetry, which takes one into the other, is given here by n W = i 24π 2 ijk d 3 x W i W j W k = 2 π ∞ 0 dr b(f 2 1 + f 2 2 ),(3.f 1 → f 1 , f 2 → −f 2 , b → −b, η → η. All the operators we consider are invariant under this transformation. This is because the two operators that violate CP vanish for static field configurations. Thus, the static-configuration energy functional is invariant under n W → −n W . The Skyrme term We focus first on the case in which − c D1,0 = c D2,0 ≡ 1 16e 2 ,(3.6) with the rest of non-SM coefficients in the HEFT Lagrangian being set to zero. The last equality is to be understood as fixing the free parameter e of the ansatz. This corresponds to the original Skyrme term, given in eq. (2.6). The total energy functional is given by E = 4πv e ∞ 0 dr ρ SM + (f 2 1 + f 2 2 ) b 2 + f 2 1 + f 2 2 2r 2 ,(3.7) where ρ SM is the contribution from the SM. We shall now describe the field configurations and energy landscape that arise in this setting. We study them using the method described in appendix A. We display two example configurations for e = 1.8 and different values of n W in figure 1. In figure 2, we show the minimal energy as a function of n W , for different values of e. For e > e crit 0.9, we find a finite-energy barrier separating the skyrmion, with n W 1, and the vacuum at n W = 0. This barrier disappears below e crit . Thus, the skyrmion solution exists only when e > e crit and is a metastable configuration. The energy E of the local minimum is the skyrmion mass. We find that the normalized energy eM Sk /(4πv) is approximately constant, with a value of 3.3 at e = e crit , and a limiting In figure 3, we show this behaviour and compare it to the case in which no gauge fields are present, labelled limit B in section 2. The curves are similar for large e. This is to be expected since a large value of e makes the W µν W µν term dominant, with similar effects as taking g → 0, which is limit B. However, some differences arise at small e. Just above e crit , the mass of the skyrmion in the full theory is slightly lower than in limit B. This is because the n W = 1 is no longer topologically fixed in the full theory, and so n W can move to another value with lower energy. For e < e crit , skyrmions become unstable in the full theory, but nothing changes in limit B, as they are still topologically protected. For the height of the barrier, the energy of the local maximum near n W = 1/2, we find that E barrier \| e=e crit 11 TeV, and E barrier \| e→∞ 10 TeV. (3.10) We also define the radius of the skyrmion R Sk by averaging over the n W density as R 2 Sk = i 24π 2 ijk d 3 x \|x\| 2 W i W j W k = 2 π(ve) 2 dr r 2 b (f 2 1 + f 2 2 ). (3.11) We find that R Sk \| e=e crit 1.4 ve , R Sk \| e→∞ 1.9 ve ,(3. Skyrmion stabilisation from other operators in HEFT We consider here the possibility that skyrmions are stabilized by some operator from table 1 other than Q D1 − Q D2 . Some of these operators can be discarded for this purpose from general considerations: Q 1 , Q h and Q U , by Derrick's theorem; and all operators containing a field-strength tensor can also be neglected since they vanish when the gauge fields are set to a pure gauge configuration. There are five remaining operators that can contribute: the Q Di in table 1. We consider now turning on one c Di,n coefficient at a time while fixing the others to zero. We find that none of them are capable of stabilizing skyrmions except for c D1,0 and c D2,0 . Indeed, for all the others, their radial energy density ρ i is multiplied by some monomial in η or η . One can then take η = 0 everywhere, which implies F i (η)ρ i = 0, and then skyrmions become unstable by Derrick's theorem. We have checked this numerically in several examples. It remains to study the skyrmions generated by c D1,0 and c D2,0 . It turns out that both individually, as well as some of their linear combinations, generate meta-stable skyrmions. We parametrize the space of linear combinations with two parameters e and θ, with the former to be used as the corresponding parameter in the spherical ansatz: c D1,0 = √ 2 16e 2 cos θ, c D2,0 = √ 2 16e 2 sin θ. (3.13) The Skyrme term is recovered for θ = 3π/4. In terms of these parameters, the non-SM contribution to the radial energy reads c D1,0 ρ D1 + c D2,0 ρ D2 = − cos θ 4r 2 2(f 2 1 + f 2 2 )r 2 b 2 + 2(2 + tan θ)(f 2 1 + f 2 2 ) 2 + (1 + tan θ)r 4 b 4 (3. 14) This is positive everywhere if and only if cos θ ≤ 0 and tan θ ≥ −1, or, equivalently 3π/4 ≤ θ ≤ 3π/2. Numerically, we find that skyrmions are stabilized in a slightly wider range: 5 0.71π θ min ≤ θ ≤ θ max 1.6π, (3.15) for e > e crit (θ), where e crit (θ) is a θ-dependent critical value of e, that we show on the left panel in figure 4. The skyrmion mass also depends on θ for constant e, with M Sk = 0 at The radius is similarly given by R Sk (20 TeV −1 ) · [tan(θ max )c D1,0 − c D2,0 ] 1/2 . (3.17) The condition e > e crit (θ) is just a θ-independent upper bound on the skyrmion mass M Sk < 11 TeV. The region where skyrmions exist in the (c D1,0 , c D2,0 ) plane is thus determined by c D2,0 < tan(θ min )c D1,0 , 0 < tan(θ max )c D1,0 − c D2,0 0.13. (3.18) Although the rest of the c i,n coefficients are not enough by themselves to stabilize skyrmions, they may have effects in the configurations generated by c D1,0 and c D2,0 . Figure 7 shows the contribution of each Q i to the energy density in the configuration with θ = 3π/4 and e = 1.8. The contributions from the operators not included in the generation of the configuration are negligible compared to the energy. This means that whenever the c i,n coefficients are chosen so that their contribution is positive, they will not change the skyrmion configuration in a significant way. However, they might be chosen so that their contribution to the energy is arbitrarily negative, destabilizing the skyrmion. We find numerically that this happens when c D8,0 = 1, for example. 2.5 V (η) 1 2 Q h v 2 4 (1 + η) 2 Q U − 1 2g 2 Q Xh2 − 1 16e 2 Q D1 − 1 16e 2 Q D20.0150 − 1 16e 2 Q XU 8 − 1 16e 2 v 2 Q D7 − 1 16e 2 v 2 Q D8 − 1 16e 2 v 4 Q D11 Phenomenology Collider signals The process of electroweak skyrmion production is similar to the electroweak instanton, as it is a B + L violating transition over a barrier of a few TeV. As such, it is expected to be exponentially suppressed, even at energies above the potential barrier [15,16]. Thus, it is unlikely that this process will take place at colliders. However, one can indirectly study the existence of skyrmions through other effects of the operators that generate them. The two skyrmion-stabilizing operators Q D1 and Q D2 induce an anomalous quartic gauge coupling (aQGC) while preserving the SM triple gauge coupling. Most LHC searches for aQGC [17][18][19][20][21][22][23] use a parametrization in terms of dimension-8 SMEFT operators which was first proposed in ref. [24]. This set of operators was corrected in ref. [25] by introducing missing operators and removing redundant ones in order for them to form a basis. The space of operators with four covariant derivatives was shown to have dimension 3. However, the experimental searches with the strongest constraint on this space [21,22] give their results in terms of only two operators, coming from an incomplete set of ref. [24]: L S = f S0 Λ 4 (D µ φ † D ν φ)(D µ φ † D ν φ) + f S1 Λ 4 (D µ φ † D µ φ)(D ν φ † D ν φ) (4.1) Therefore, their results cannot be used in general to constrain the full 3-dimensional space of Wilson coefficients. Only when the measured final state uniquely selects one aQCG vertex (W W W W , W W ZZ or ZZZZ) can the results in the incomplete set be translated into the complete EFT basis, as shown in ref. [26]. Following this reference, we obtain limits over c D1,0 and c D2,0 (denoted α 5 and α 4 there) from the 95% CL limits over f S0 and f S1 found in ref. [22] individually for W W and W Z production at √ s = 13 TeV and Ldt = 137 fb −1 . W W production comes from the W W W W vertex, and the limits and from CMS [21], using data from W Z production (red) and W W production (green). Transparent shaded blue region: excluded by positivity bounds. Color-gradient region: allowed values of the coefficients for the existence of skyrmions, from the numerical calculations in this work. The coloring represents the skyrmion mass. Solid black-line perimeter encloses the triangle of allowed values of the coefficients that support a skyrmion. conversion are given by −2.7 × 10 −3 ≤ 2c D1,0 + c D2,0 = v 4 f S1 8Λ 4 ≤ 2.9 × 10 −3 , (4.2) −8.2 × 10 −3 ≤ c D2,0 = v 4 f S0 8Λ 4 ≤ 8.9 × 10 −3 ,(4.3) whereas W Z production comes from the W W ZZ vertex, they are −1.3 × 10 −3 ≤ c D1,0 = v 4 f S1 16Λ 4 ≤ 1.3 × 10 −3 , (4.4) −1.9 × 10 −3 ≤ c D2,0 = v 4 f S0 16Λ 4 ≤ 1.9 × 10 −3 . (4.5) We show these limits in figure 8. We point out that the experimental bounds in ref. [27] are presented in terms of a basis for 2-dimensional custodial-invariant subspace of the 3-dimensional space of qQGC operators containing only covariant derivatives, and thus directly translatable into our setting. However, they are weaker than the ones we have obtained, and they are not shown in figure 8. Positivity bounds The space of Wilson coefficients can also be constrained theoretically by imposing general principles such as unitarity, locality and causality. The bounds obtained in this way are known as positivity bounds [28], and can be interpreted as necessary conditions for the existence of a UV completion to the EFT in question. In the HEFT, causality implies that [29][30][31][32] c D1,0 + c D2,0 > 0, c D2,0 > 0. (4.6) These inequalities also arise in the chiral Lagrangian without gauge bosons [33]. The region excluded by them is shown in blue in figure 8. It follows that skyrmions can only exist in the angular region θ min ≤ θ < 3π/4. Combining this fact with the experimental limits gives an upper bound on the mass of the skyrmion: M Sk 1.6 TeV. (4.7) Dark matter Similarly to skyrmion production, skyrmion decay is a B + L-violating process which expected to be exponentially suppressed. The skyrmion lifetime is thus likely longer than the age of the universe, opening the possibility of skyrmions being Dark Matter (DM) candidates. We use the following order-of-magnitude estimate of the freeze-out skyrmion density [6] Ω Sk h 2 3 × 10 −27 cm 3 s −1 σ ann v , σ ann πR 2 Sk , v 1/2. (4.8) Requiring that the skyrmion density is at most the total DM density Ω Sk h 2 0.1 results in a lower limit for the skyrmion mass M Sk 60 GeV (4.9) This limit would be saturated if skyrmions formed all of the DM. Conclusions We have studied the skyrmion configurations that arise in the HEFT. We have found that a meta-stable configuration with skyrmion number close to one exists whenever the coefficients c D1,0 and c D2,0 lie on the strip c D2,0 ≤ tan(θ min )c D1,0 , 0 < tan(θ max )c D1,0 − c D2,0 0.13. The mass of this skyrmion is given by M Sk = (30 TeV) · [tan(θ max )c D1,0 − c D2,0 ] 1/2 . It is separated from the trivial vacuum by an energy barrier of about 11 TeV. This value also represents the maximal theoretical M Sk , as above it the barrier disappears. Since skyrmions are unlikely to be created at colliders, we have focused on the experimental signals of the operators that stabilize them. LHC searches for aQCG put bounds of order 10 −3 on both coefficients. Combining these bounds with positivity constraints, we have found that the allowed parameter space for skyrmions in the triangle 1 < c D2,0 /c D1,0 ≤ tan(θ min ), −1.3 × 10 −3 ≤ c D1,0 ≤ 0. (5.1) This allowed us to obtain a stronger upper bound on the mass of the skyrmion, of about 1.6 TeV. Skyrmions are also expected to be long-lived, so they contribute to the DM density. By assuming that their abundance is generated by the freeze-out mechanism and adopting a simple approximation for the skyrmion annihilation cross-section, we have computed an order-of-magnitude lower bound on the skyrmion mass, of 60 GeV. A Numerical method We use a neural network to model the function taking r to R(r) = (f 1 (r), f 2 (r), b(r), η(r)). We choose an architecture with sigmoid σ(x) = (1 + e −x ) −1 activation functions and two hidden layers, each having 5 units. That is, we parameterize with the weights ω i being sufficiently high. In practice, we set ω BC = 10 3 and ω n W = 10 4 . To perform the numerical minimization of L, the training of the network, we use the functional-minimization features of Elvet [34,35]. This procedure allows us to find the minimal energy E for a fixed n W . In order to find the local minimum near n W = 1, which is the skyrmion, we first train the network with n W = 1. Once the minimum under this condition is reached, we resume the training with this condition removed by setting ω n W = 0. Since the network is already close to the local minimum, the training cannot overcome the finite barrier, so it can only take the network into the skyrmion configuration. R(r) = A 3 • σ • A 2 • σ • A 1 (r) (A.1) where A 1 : R 1 → R 5 , A 2 : R 5 → R 5 , We check that the configurations found in this way are compatible with Derrick's argument. We first split the energy as E = i E i , E i = 4πv e ∞ 0 drF i (η)ρ i . (A.3) In terms of the functions defined here, a spatial scale transformation is given as r → λr, b → b/λ. If one applies such transformation to a local minimum, the energy must satisfy: In order to find the critical value e crit , we note that for e > e crit the height of the local minimum must be lower than the height of the barrier, so at e = e crit they must become of the same height. For small e, the barrier is found at around n W = 0.4 and the local minimum is close to n W = 0.8. We can thus obtain e crit approximately as the value of e that minimizes \|E(n W = 0.4) − E(n W = 0.8)\|. We search for this value in steps of 0.05. 0 = dE dλ λ=1 = 3E 1 + (E h + E U ) − i E i ,(A. 5) 3 Figure 1 . 31Due to metastability. 4 Indeed, if one takes any M ∈ SU (2) and R its representation as a spatial rotation, one has M Wi(x)M † = Rij Wj(R −1 x), so invariance under spatial rotations and under custodial symmetry become equivalent. Minimal energy configurations for e = 1.2, c D1,0 = −c D2,0 = 1/(16e 2 ), and n W = 0.4 (left) or n W = 0.8 (right). Figure 2 . 2Minimal energy as a function of n W , for c D1,0 = −c D2,0 = 1/(16e 2 ) and different values of e. The finite-energy disappears around e = 0.9. value of 3 as e → ∞, so the skyrmion mass is given by M Sk \| e e crit 41v e , M Sk \| e→∞ 38v e . (3.8) The maximum of M Sk is reached at e = e crit : M Sk ≤ M Sk \| e=e crit 11 TeV. (3.9) Figure 3 . 3Skyrmion mass M Sk as a function of e for the full theory and for limit B. Figure 4 .Figure 5 . 45Left: e crit as a function of θ. Right: skyrmion mass M Sk as a function of θ, for e = Minimal energy as a function of n W , for e = 1.8 and θ = 3π/4, π, 3π/2. Figure 6 . 6Skyrmion mass M Sk as a function of c D1,0 and c D2,0 . θ = θ max . We show this on the right panel of figure 4. The normalized mass eM Sk /(4πv) has little variation with e, as it happened for θ = 3π/4. In figure 5, we display the energy profile for e = 1.8 > max θ e crit (θ), and different values of θ. Finally, in figure 6 we show the region of (c D1,0 , c D2,0 ) space where meta-stable skyrmions exist, and the values the masses of the skyrmions inside it, which are given approximately by M Sk (30 TeV) · [tan(θ max )c D1,0 − c D2,0 ] 1/2 . (3.16) Figure 7 . 7Radial energy densities ρ i in the skyrmion configuration with θ = 3π/4 and e = 1.8. The operators in the left plot are included in the calculation of the skyrmion configuration. The ones in the right plot are computed once this configuration is obtained and fixed. Figure 8 . 8Dashed lines: 95% CL limits on c D1,0 and c D2,0 at √ s = 13 TeV, dtL = 35.9 fb −1 Table 1 . 1Custodial-invariant operators Q i containing the Higgs only through derivatives, of order up to Λ 0 , together with their contribution to the radial energy density ρ i in the spherical ansatz, defined in eq. (3.2). and A 3 : R 5 → R 4 are affine transformations. The problem is to adjust the parameters of A 1 , A 2 and A 3 to minimize the energy E[R] while satisfying the boundary conditions BC[R] = 0 and fixing n W [R] to some value n W,0 . We reformulate this problem as minimizing the functional L[R] = E[R] + ω BC BC[R] 2 + ω n W (n W [R] − n W,0 ) 2 , (A.2) 4 ) 4where i runs over rest of operators in table 1. Numerically, we get1 E dE dλ λ=1 < 1%. (A.5) Even if φ = 0 only at an isolated point p, U becomes a mapping S 3 − {p} ∼ = R 3 → S 3 , and all such mappings are homotopically equivalent. 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[ "Temporal Knowledge Consistency for Unsupervised Visual Representation Learning", "Temporal Knowledge Consistency for Unsupervised Visual Representation Learning" ]	[ "Weixin Feng [email protected] \nMegvii Technology\n\n", "Yuanjiang Wang [email protected] ", "† Lihua [email protected] ", "Ma ", "Ye Yuan [email protected] ", "Chi Zhang [email protected] ", "\nBeijing University of Posts and Telecommunications\n\n" ]	[ "Megvii Technology\n", "Beijing University of Posts and Telecommunications\n" ]	[]	The instance discrimination paradigm has become dominant in unsupervised learning. It always adopts a teacherstudent framework, in which the teacher provides embedded knowledge as a supervision signal for the student. The student learns meaningful representations by enforcing instance spatial consistency with the views from the teacher. However, the outputs of the teacher can vary dramatically on the same instance during different training stages, introducing unexpected noise and leading to catastrophic forgetting caused by inconsistent objectives. In this paper, we first integrate instance temporal consistency into current instance discrimination paradigms, and propose a novel and strong algorithm named Temporal Knowledge Consistency (TKC). Specifically, our TKC dynamically ensembles the knowledge of temporal teachers and adaptively selects useful information according to its importance to learning instance temporal consistency. Experimental result shows that TKC can learn better visual representations on both ResNet and AlexNet on linear evaluation protocol while transfer well to downstream tasks. All experiments suggest the good effectiveness and generalization of our method. Code will be made available.	10.1109/iccv48922.2021.01001	[ "https://arxiv.org/pdf/2108.10668v1.pdf" ]	237,278,120	2108.10668	3b759f66c0aac8266c8418a8c22abdbad5565453	Temporal Knowledge Consistency for Unsupervised Visual Representation Learning Weixin Feng [email protected] Megvii Technology Yuanjiang Wang [email protected] † Lihua [email protected] Ma Ye Yuan [email protected] Chi Zhang [email protected] Beijing University of Posts and Telecommunications Temporal Knowledge Consistency for Unsupervised Visual Representation Learning The instance discrimination paradigm has become dominant in unsupervised learning. It always adopts a teacherstudent framework, in which the teacher provides embedded knowledge as a supervision signal for the student. The student learns meaningful representations by enforcing instance spatial consistency with the views from the teacher. However, the outputs of the teacher can vary dramatically on the same instance during different training stages, introducing unexpected noise and leading to catastrophic forgetting caused by inconsistent objectives. In this paper, we first integrate instance temporal consistency into current instance discrimination paradigms, and propose a novel and strong algorithm named Temporal Knowledge Consistency (TKC). Specifically, our TKC dynamically ensembles the knowledge of temporal teachers and adaptively selects useful information according to its importance to learning instance temporal consistency. Experimental result shows that TKC can learn better visual representations on both ResNet and AlexNet on linear evaluation protocol while transfer well to downstream tasks. All experiments suggest the good effectiveness and generalization of our method. Code will be made available. Introduction The rise of Deep Convolutional Neural Networks (DCNN) [24,29,47] has led to significant success in computer vision benchmarks [9,14,34]. The excellent performance of supervised DCNN always relies on a large quantity of manually labeled data, which is costly to collect [19,53]. Unsupervised representation learning has been attracted more and more interest, for it can learn a good Iterations(k) proportion(%) Figure 1. Mainstream unsupervised methods adopt the teacherstudent framework, where the teacher is an EMA ensemble of previous student encoders. This figure illustrates the proportion of previous students in the teacher with respect to training steps. The red curve shows that the EMA teacher ensembles the previous encoders by a predesigned factor α, where only alomst encoders in the very close steps are ensembled. Our TKC (the green curve) reuses the early models and adaptively learns the importance ω for each of them, thus leads to temporal consistent representations. representation without human annotations. These methods are generally to manually design a pretext task to learn representations, such as image in-painting [43], colorization [10,62,31,32], rotate predicting [19,6,15] and clustering [2,64,4]. All these pretext tasks are based on specific domain knowledge, which has poor generation on various downstream tasks. Recently, instance discrimination [53,22,5,20,37] paradigm has led to remarkable progress in unsupervised representation learning and even surpasses the supervised pre-training on extensive downstream tasks [37,22]. The instance discrimination paradigm treats each sample itself as its own category and trains the CNN to sep-arate all the different samples from each other. The current paradigm can be formulated as a teacher-student framework enforcing the instance spatial consistency of two networks, which are the student network and the EMA teacher network [7,20,5]. The instance spatial consistency constrains the similarity of different spatial views from the same instance, and its ultimate goal is to learn instancediscriminative and spatial-invariant representations. One of the key points in these instance discrimination works is the EMA teacher. For instance, MoCo [22] uses the EMA teacher to output consistent negative samples for the student; BYOL [20] trains a student to mimic the representations from the EMA teacher; SimCLR [5] maintains a realtime EMA teacher of the student. However, we argue that the current EMA teacher is suboptimal as illustrated in Fig. 1: (1) the EMA teacher only ensembles the rare knowledge of recent encoders by a handcraft proportion, which means that it only concentrates on instance spatial consistency while the instance temporal consistency is ignored. As a consequence, the outputs of the same sample can vary dramatically among different training stages, which can introduce unexpected noise and finally lead to catastrophic forgetting [35,63]. (2) The EMA manner can't leverage the importance of different encoders. It assumes that the outputs of later models are largely more important than the earlier ones, despite that the benefits of previous epochs have been observed in previous works [30,63]. In this paper, we integrate instance temporal consistency into the instance discrimination paradigm and propose a novel and strong algorithm, namely Temporal Knowledge Consistency(TKC), which contains the temporal teacher and the knowledge transformer. Specifically, temporal teacher supplies instance temporal consistency via introducing the temporal knowledge from previous models. And the knowledge transformer dynamically learns the importance of different temporal teachers, then adaptively ensembles the useful information according to their importance, to generate instance temporal consistency objective. In addition, we provide a computation-economical implementation, which can provide temporal knowledge without preserving multiple previous models. Our experimental results on different tasks and benchmarks have demonstrated that TKC can learn a better visual representation with excellent transferability and scalability. Concretely, we achieve state-of-the-art performance on ResNet and AlexNet backbones on linear evaluation protocol. Moreover, we evaluate representations learned by TKC on many downstream tasks and architectures. All results suggest the effectiveness of TKC. Overall, the main contributions in this work include: • We are the first to integrate instance temporal consistency into the current EMA teacher in the instance dis-crimination paradigm. • We propose a novel and strong algorithm, named Temporal Knowledge Consistency (TKC), which can dynamically ensemble the knowledge from different temporal teachers. • Extensive experiments are conducted on several benchmarks and architectures, which shows the superior performance on mainstream benchmarks and the scalability of TKC. Related Works Unsupervised Pretext Tasks. Unsupervised representation learning aims to learn meaningful representations from large amounts of data samples via constructing a wide range of pretext tasks without human labels. These pretext tasks usually vary in different forms. Among them, one family of these typical pretext works are generative-based which rely on auto-encoder [45] or GAN [36,13], such as colorization [10,62,31,32] and image in-painting [43]. And the others are discriminative-based, like predicting rotation or augmentation [19,6,15] of the image, solving jigsaw puzzles [39], locating relative patch [11,12], ordering video frames [16,56,51], matching corresponding audio [42,18,41,28], and clustering [2,3,64,60,1,54,58,57,17]. All the pretext methods are based on specific domain knowledge, fail to generalize to different downstream tasks. Recent progress in unsupervised representation learning mainly benefits from instance discrimination and attracts widespread attention from researchers. Instance Discrimination. Instance discrimination methods [53,22,7,5,20,40,15,37] have dominated the unsupervised learning field in the few years, which treat each sample itself as its own category and train the CNN to separate all the different samples from each other. This paradigm commonly includes a teacher model to provide a supervised signal, and a student model to learn the embedded knowledge from the former. Wu et al. [53] is the first to propose instance discrimination in unsupervised learning, which regards the student model in the last epoch as the teacher model. It learns meaningful representations by means of the classic InfoNCE loss [21,40] and the target generated by the teacher. MoCo [22,7] takes the EMA ensemble of the student as the teacher model to provide consistent and robust objectives, and brings a breakthrough by solving the knowledge out-of-date problem with the help of the EMA teacher. It also maintains a queue of negative samples and keeps them fresh. SimCLR [5] builds symmetrical architecture between the student and the teacher, while uses stronger data augmentation to enforce network to learn instance spatial consistency. BYOL [20] also implements the teacher with the EMA ensemble of the student, and makes use of L2 loss to pull the embedding features of positives pairs while removing explicitly negative samples. Our TKC explicitly integrates instance temporal consistency into the instance discrimination paradigm, making the targets generated by the teacher more accurate and stable. Temporal Knowledge. Temporal knowledge is widely used in both semi-supervised learning and optimization. In the field of semi-supervised learning, plenty of the proposed works adopt the EMA ensemble to take advantage of the knowledge of the previous training stage, to learn the time-consistent representations. Temporal Ensemble [30] ensembles the output of different epochs to yield better predictions. Mean teacher [48] instead ensembles the previous student as a teacher to prevent incorrect target and outweighs the cost of misclassification. Tian et al. [63] points out the catastrophic forgetting problems in semi-supervised, and solves it by measuring the time consistency of samples and filtering the inconsistent ones. In the field of optimization, temporal knowledge is integrated by different advanced optimization strategies during training. SGD only uses the gradient computed by a mini-batch to backpropagate, which is noisy and inaccurate. Momentum [44] and NAG [38] instead use the gradient in a mini-batch by the momentum of the gradient to accelerate the convergence of model training and suppress shocks. Adam [27] is another momentum updating strategy which further introduces the second momentum to leverage different channels. All these works use temporal knowledge to reduce noise and accelerate convergence. Method In this section, we first point out the limitation of the current EMA teacher in Sec 3.1. Secondly, we propose temporal teacher to improve it in Sec 3.2. Thirdly, we introduce the knowledge transformer to dynamically leverage the importance of different models in Sec 3.3. Then we propose a temporal loss to learn instance temporal consistency in Sec 3.4. At last, we describe our overall framework and the algorithm in Sec 3.5. Limitation of EMA teacher Instance discrimination paradigm always involves two encoders, the teacher encoder T and the student encoder S. For a training sample x, the augmentation from augmentation distribution T is applied twice to obtain two augmented sample x 0 , x n . The teacher output r T n = T (x n ) as the target to provide instance spatial knowledge. The student network takes the other sample x 0 then outputs r S 0 = T (x 0 ), and learn knowledge by constrainting its similarity with r T n . In this teacher-student framework, the teacher encoder has the same architecture with the student, and its parameters are updated by an exponential moving average (EMA) of the models: T n+1 = αT n + (1 − α)S n(1) where n is the training step, α is to control the updating speed of the teacher. We name the teacher as EMA teacher. In current training step n + 1, the teacher is ensembled by the last teacher T n with ratio α, and the last student S n with ratio 1 − α. The last teacher T n is also an ensemble of previous students. In order to explore the temporal knowledge in EMA teacher, we expand T n in Eq 1 as following: T n+1 = (1 − α) · n−1 m=0 (α m S n−m ) + α n · T 1 ≈ (1 − α) · [S n + αS n−1 + ... + α n S 0 ](2) where S m means the student model at step m, α is the updating factor. In Eq 2, we can find out that current teacher T n+1 is an ensemble of a sequence of student S from step 0 to step n. However, we note that the EMA teacher can only preserve the knowledge from the latest encoders. On the one hand, as m goes to infinity m → ∞, the weight of student S n−m approaches 0, for α is lower than 1. When training MoCo [22] on ImageNet, only student models within an epoch can provide the knowledge, as illustrated in Fig. 1. This knowledge from only near steps is insufficient, which can cause the dramatically changes among different training stages and prevent the student to learn instance temporal consistency. On the other hand, the strategy of EMA is also too simple. It assumes that the importance of earlier models is decreased exponentially with time, even though the earlier models can provide useful information to mitigate the catastrophic forgetting. In a summary, these two flaws prevent the instance discrimination paradigm from making full use of temporal knowledge and learning instance temporal consistency. Temporal Teachers EMA teacher in Eq.2 only attaches importance to recent models. However, the output of these models is smooth and similar due to the low learning rate and momentum optimizer. As a consequence, they fail to supply instance temporal consistency to lighten the dramatic changes of the models, which can easily lead to training failure and catastrophic forgetting. We claim that jointly utilizing the knowledge from previous models can provide a more consistent and robust target. To achieve that, we propose to take out the previous models, which have few proportions in the EMA teacher, to build our temporal teacher. Then we make full use of them to alleviate catastrophic forgetting and learn instance temporal consistency in the instance discrimination paradigm. We explicitly preserve a group of previous teachers as temporal teacher to reuse the knowledge from previous encoders. To formulate our proposal, we use T n to denote ... " ($) ' ( "+ ($) x ". "+ ($) " . ($) ". ($) ($) ... ... x "+ Temporal Teacher Knowledge Transformer x " x ' Figure 2. The overall framework of our TKC. For each training image x, TKC generate a target from the EMA teacher, and h targets from the temporal teachers. Temporal teachers is a set of encoders from previous training stages. The knowledge transformer is appended behind the temporal teacher to dynamically leverage their importance. Every teacher in TKC frameworks can provide a supervised signal, which then feeds into the temporal loss and backward to update the student and the knowledge transformer. The green dotted line means backpropagate. current EMA teacher, and {T n−1 , T n−2 , ...} to represent the temporal teachers. The lower subscript means earlier. Each of these teachers is saved for each s training step, including the knowledge mainly from this training step. Note that the subscript n means different with the superscript in Eq.1. The distance between T j−1 and T j−2 is s training steps. The teacher far away from now is too out-of-date, whose knowledge can be inconsistent and noisy for the current teacher. Hence we only preserve adjacent teachers as temporal teacher, while throwing away the previous one. We use h to represent the number of temporal teachers, and denote the temporal teachers by {T n−1 , T n−2 , ..., T n−h }. We illustrate the temporal teacher in the brown dotted box in Fig.2. For a sample x in the training set, we apply h times data augmentations from the augmentation distribution T , to obtain x j , j ∈ [n − h, n − 1]. The temporal teachers are set stop-gradient, and take the augmented views as input to yield representations z T j = T j (x j ), j ∈ [n − h, n − 1] as the temporal predicting target. The subscript j of z T j indicate that the target is corresponding to the teacher T j . In the implementation, we propose a more efficient way to achieve temporal teacher. Instead of getting the target from previous teachers, we preserve the representations of all the training data in the previous h stages in a memory named history bank. For each training sample x, we can get z T j from the history bank instead of from the teacher T j . History bank is an approximate implementation of the temporal teachers, for both of them can provide temporal knowledge. In this way, the computational cost is largely reduced and the additional GPU memory allocation is negligible. The detail can be seen in the supplementary material. Knowledge Transformer In EMA teacher, the weights of different ensemble models are decreased over time exponentially. However, the importance of different models may not be in line with the EMA rule. In this section, we propose to dynamically predict the importance of the temporal teachers's knowledge by knowledge transformer. The knowledge transformer is illustrated in the blue box in Fig.2. It takes the rough target z T j , j ∈ [n−1, n−h] from the teacher T j as input, and then transfer the knowledge of them to leverage their importance. The formulation is as follows: r T j = K j (z T j )(3) where r T j denotes the target after leveraging the importance, which has thrown the harmful information from it and only preserves temporal consistent knowledge. This strategy can adaptively learn and adjust the importance of the temporal teachers in the early or later encoders, which is better than coupled it to the handcraft proportion in the EMA teacher. In the implementation, we use an MLP with one hidden layer to transfer the knowledge for each temporal teacher. During training, the knowledge transformer is training simultaneously with the student. Temporal Loss Algorithm 1 Temporal knowledge consistency input: S(·), T n (·), K(·) hyperparameters: α, h, s 1: for each sample x do 2: draw h+2 augmentations 3: # the original models 4: r S 0 = S(x 0 ) 5: r T n = T n (x n ) 6: for all j ∈ {n − 1, n − h} do 7: # temporal teacher 8: z T j = T j (x j ) 9: # knowledge transformer 10: r T j = K j (z T j ) 11: end for 12: # temporal loss 13: compute the loss in Eq. 4 14: backward to update S and K 15: update the T n by Eq. 1 16: end for 17: return S(·) Different from previous works that only maximize the mutual information (MI) of the student output r S 0 and the target r T n from the EMA teacher, we propose to combine maximal the MI between r S 0 and each r T j , j ∈ [n−h, n−1]. This is in the intuitive that we hope the student can synchronously learn instance temporal consistency from temporal knowledge. Our objective is as follows: L tem = max r S 0 (I(r S 0 ; r T n )+ n−1 j=n−h I(r S 0 ; r T j )) (4) The first term maximizes the MI in the current phase, like previous works do [40,25,22,20], which can only learn spatial consistent representations between different views. The second term maximizes the MI with previous knowledge to encourage the temporal consistency representations between different training stages, to mitigate the oscillation and catastrophic forgetting. Because the mutual information is notoriously hard to estimate, we instead maximizing the lower bound of it by the InfoNCE [40,22,5]: L (1) tem = n j=n−h −log sim(r S 0 ·r T j ) sim(r S 0 ·r T j )+ r − j sim(r S 0 ·r − j )(5) where r − j presents the representation of other samples from the same teacher T j , and sim(r S 0 · r T j ) means their cosine similarity as following: sim(r S 0 · r T j ) = exp(r S 0 · r T j /τ )(6) where τ is temperature coefficient. In Eq. 5, the term j = n estimates the MI with the current target, the other terms estimate the MI with temporal targets. InfoNCE is relied on the negative samples to estimate the probability distributions. Furthermore, our methods can also work on the methods without negative samples like BYOL [20]. We minimize the L2 distance to maximize the MI for these works: L (2) tem = n j=n−h \|\|r S 0 − r T j \|\| 2(7) 3. Overall Framework As previous works do, TKC also introduce a student S(·) amd an EMA teacher T n (·). For a training sample x from the data distribute, we obtain r S 0 from S and r T n from T n . To learn consistent knowledge, we also get targets from the temporal teachers as z T j , j ∈ [n − h, n − 1]. These targets should transport to the knowledge transformer to filter important knowledge as r T j , j ∈ [n − h, n − 1]. Then all the representations are fed into the temporal loss in Eq. 4. During training, all the teachers are set stop-gradient. The loss will be back-propagated to update the student S and the knowledge transformer K j , j ∈ [n − h, n − 1]. Algorithm 1 summarizes the algorithmic flow of the TKC procedure. Experiments In this section, we evaluate the quality of feature representation learned by our proposed TKC on several unsupervised benchmarks. We first follow standard linear evaluation protocol to assess the learned representations on Im-ageNet [9]. Then we transfer the pre-trained features to different downstream tasks, including object detection, instance segmentation, and semi-supervised classification. Finally, we perform a set of analysis studies to give an intuition of its performance. For brief-expression, all the experiments are based on MoCo v2 [7] framework and ResNet-50 [24] backbone unless otherwise stated. Evaluation on Linear Classification We implement our TKC based on MoCo v2, which is composed of a standard ResNet-50 [24] backbone and an MLP layer in the teacher-student framework. And the number of temporal teachers h is set to 2. We train TKC model on 8 NVidia-1080ti GPUs with a mini-batch size of 256 and set α as 0.999, τ as 0.2. Moreover, we set the base learning rate lr as 0.3, weight decay as 0.0001, and introduce a warm-up stage in the first 10 epochs, where linearly increase the learning rate from 0.01 to 0.03. All other hyperparameters, training settings on pretext task and linear eval- uation are strictly kept aligned with the implementations in [7]. Table 1 summaries the top-1 and top-5 accuracy of our method. We report our results for different epochs pretrained and also list top-performing methods. TKC improves MoCo v2 by 1.5 % on 200 epochs results, which indicates that temporal teachers can provide more accurate targets to learn consistent representations. Our results are also superiors to previous works on different pretext tasks, including all other instance discrimination paradigms. This demonstrates that temporal knowledge can benefit from stable training while mitigates the effect of catastrophic forgetting. In order to verify the scalability of TKC, we respectively conduct our TKC on BYOL [20] baseline, and AlexNet [29] backbone. The number of teachers h is changed to 3 for AlexNet. Specifically, We use a PyTorch implementation of BYOL in Momentum 2 Teacher [33] as BYOL baseline and train the model for 100 epochs with 128 batch size on 8 Nvidia-1080ti GPUs. As for AlexNet, we adopt the implementation in Deep Clustering [2], where we train the network with a mini-batch of 1024 on 4 NVidia-1080ti GPUs, and the learning rate is initialized by 0.24 with a cosine decay schedule for 200 epochs. More detail can be seen in the supplementary material. For AlexNet, as shown in Table 3, TKC achieves stateof-the-art top-1 accuracy on conv1 to conv4, which outperforms all self-supervised methods on this track. Despite that TKC from conv5 underperforms Rot-decouple [15] by 0.3%, our best result is from conv4, which surpasses the best of Rot-decouple by 1.9%. The results show that TKC is also a leading method on AlexNet linear classification benchmark. We notice that TKC has more improvement on AlexNet than ResNet-50. This might be because the dropout layer in AlexNet can provide various temporal knowledge, which could be more effective in learning instance temporal consistency. Transfer to Downstream Tasks The primary goal of self-supervised learning is to learn good representations that transfer well on downstream tasks. In this subsection, we transfer the representations of 200 epoch TKC to three benchmarks: object detection, instance segmentation, and semi-supervised learning. We show that TKC learns better transferable representations on all three downstream tasks. Object Detection. We both transfer to VOC [14] and COCO [34] dataset to evaluate our representations. As for Pascal VOC, We use Faster R-CNN [46] with ResNet50 backbone as the detector. We fine-tune the candidate pretrained model for 48k iterations with a min-batch size of 8 on Pascal VOC [14] training set. The learning rate is initialized from 0.001 and then decayed at 36k and 44k iterations. The weight decay is set to 0.0001, and training image scales range between 480 to 800. We use AP 50 , AP , AP 75 as evaluation metric on VOC test2007 set. For COCO [34] We also evaluate TKC on detection in another way. We freeze the Faster R-CNN backbone and only train from the detection head to challenge it. This is somewhat like linear classification. Table 6 shows the results on VOC dataset. For both AP , AP 50 and AP 75 , TKC surpass MoCo v2 baseline by more than 1.0%, and also surpass the supervised counterpart. Table 5 shows that on COCO dataset, the improvement is even more than 5.5 %. Train on the frozen backbone can better reflect the pre-trained model's representation because that the trained head is more dependent on what the pretext task learns. The results on frozen backbone show that TKC does learn better semantics representations. The training detail can be seen in the supplementary material. Instance Segmentation. We evaluate the instance segmentation on the COCO dataset, following the same setting as COCO detection. Table 7 shows both the results by finetune and freezing. The finetune results gains 0.5% AP 75 on MoCo v2 baseline, indicates the temporal consistency can better locate the target to improve the IOU of instances. Moreover, the gain is further expanded to 1.9 % when only train the segmentation head. We note in this way AP 5 0 is improved by 3.0%, show that TKC can also learn better representations on a simple task. Table 8. Semi-supervised Learning on ImageNet. We finetune the model with 1% and 10% labels. Center-crop top-5 accuracy is reported to compare with previous methods. † indicates that the score is from this work [55]. ‡ means that we implement under the same strategy using the officially released pre-trained model. Semi-supervised Learning. We then evaluate the utility of TKC in a data-efficient setting by performing semisupervised learning on ImageNet. In this benchmark, We follow the experimental setup of [5,37]. The dataset is sampled of 1% and 10% from the labeled ImageNet-1k training data in a class-balanced way. We finetune the TKC pre-trained model on these two labeled subsets and validate it on the whole ImageNet validation data. In order to compare with previous works, we report the top-5 accuracy. The supervised baseline from [59] is trained only using 1% and 10% labels, with a stronger architecture of ResNet50-v2, trained for 1000 epochs. Table 8 shows that our TKC surpasses all the previous methods trained for 200 epochs. When only 1% of data is labeled, TKC surpasses our MoCo v2 baseline by a large margin of 9.6%, indicating that the temporal knowledge is more beneficial when lacking labeled data. In addition, the mainstream semi-supervised learning methods adopt a consistent regularization to learn smooth manifold. The intuition in this field is similar to us, where they consider that consistent representation between similar samples can bring up accuracy classification boundary. Similarly, TKC also encourages consistent representations between different training stages to get a smoother manifold. The significant improvement on semi-supervised benchmarks shows we indeed learn temporal consistent representations. Analysis Ablation study. Our method introduces two new hyperparameters, the step interval of each teacher s, and amount of teachers h. We use s as the steps among an epoch and do not tune it. For h, we take an ablation on the AlexNet backbone. In Table 9, the first column h = 1 means only the EMA teacher is used, which is an implementation of MoCo v2. We see that the temporal teacher can boost accuracy from 39.9 to 42.2 when introduced only one temporal teacher. TKC achieves the best performance when maintaining three teachers, for this setting can acquire the most temporal knowledge to stable the target. When increasing h even more, the result is declined unexpectedly. This might be because when involving the too old teachers, their representations are changed too much. It is hard to learn consistently with these teachers. Nonetheless, this confirms our motivation again that the inconsistency between different training stages has alleviated convergence. Table 9. Ablation study on the effect of teacher numbers. Convergence comparison. In Section 3.4, We consider that TKC can combine maximize the mutual information with the target from the different stages, and therefore will enforce the network to learn temporal consistent representations and mitigate the catastrophic forgetting. To confirm our proposal, we use a kNN classifier to validate the model performance during training. As shown in Fig. 3, TKC has a lower accuracy in the earlier training, which is because that the model is more inconsistent and noisy in the earlier stage, resulting in a big difference between the temporal teacher and the current teacher. This difference prevents TKC from providing consistent signals. However, the TKC catches up MoCo v2 from the middle stage and finally surpasses it for 4.6% at the end of the training, which indicates that TKC can stably provide a consistent signal from the middle training. This consistent signal can guide a more accurate training direct and accelerate convergence. Fig. 3 shows that the TKC at 160 epochs meets the accuracy of fully trained MoCo, reducing 80% training time by mitigating catastrophic forgetting. Conclusion We summarize the existing instance discrimination methods into a teacher-student framework and note that the teacher can only provide instance spatial consistency. However, the output of the same instance can vary dramatically between different epochs when only spatial consistency is involved. We instead present a novel and strong method named Temporal Knowledge Consistency (TKC), which integrates the knowledge from previous teachers to improve the model's robustness and prevent possible catastrophic forgetting. TKC contains three modules. The temporal teacher introduces the instance temporal consistency from previous models, the knowledge transformer leverages the knowledge of these teachers, and the temporal loss reduces the MI between the student and the temporal teacher. Temporal teacher is an orthogonal improvement for different instance discrimination methods. Our experimental results show that TKC can improve different frameworks MoCo, BYOL, and architectures ResNet-50, AlexNet. It also provides transferable representations on downstream tasks such as object detection, instance segmentation, and semi-supervised learning. Moreover, we hope our study can draw much attention to solve the unstable in unsupervised learning and search for effective ways to generate stable output with no labels. A. Implementation Details A.1. Linear Classification. For TKC on ResNet-50 [24], we freeze the ResNet-50 backbone and train a linear classifier after the frozen features from the global pooling layer. We use the student network as a pre-trained model. The classifier is trained for 100 epochs, with initialized learning rate lr = 30. We set momentum as 0.9, weight decay as 0, and decay the learning rate by 0.1 at the 60th epoch and 80th epoch. The batch size is set to 256 on 8 NVidia-1080ti GPUs. A.2. Based on BYOL Pretext Training. In the paper, we have implemented an experiment based on BYOL [20] to show that TKC can improve different methods (in Table 2). We use the BYOL baseline based on a pytorch implementation in Momentum 2 teacher [33]. Their code is publicly available at https:// github.com/zengarden/momentum2-teacher. We use momentum SGD with momentum 0.9 and weight decay 1e-4. We train both the BYOL baseline and our TKC for 100 epochs, the basic learning rate is 0.05. We use a warm-up stage at the beginning of training for 10 epochs, and then cosine decays the learning rate. The batch size is 256 on 8 NVidia-1080ti GPUs. The data augmentation and the architecture are the same as the original paper [20], except that we use batch normalization instead of SyncBN. The MLP projection head consists of two linear layers, with a batch norm layer and a ReLU layer between them. Our TKC+BYOL shares the same setting with the baseline, we set h as 3, and also use a symmetrized loss. Linear Classification. The setup of linear classification is also following the reproduction in [33]. We fetch out the teacher encoder and freeze its backbone. Then we train a classifier consisting of a linear layer and a batch norm layer, following the global average pooling layer in the backbone. We train for 5 epochs. This reproduction is not strictly reimplemented the results in BYOL [20], thus we do not compare this result with other methods. The results in Table 2 show that TKC has good scalability, and can improve different methods. A.3. Based on AlexNet Pretext Training. We adopt the AlexNet [29] implementation in Deep Clustering [2] as the backbone, and additionally append a two-layers MLP behind it, following MoCo v2 [7]. We train the model on 4 NVidia-1080ti GPUs with 1024 batch size. The learning rate in initialized as 0.24, and cosine decayed for 200 epochs. We set τ as 0.2, α as 0.9, following MoCo v2 [7]. Differently, we set the number of negative samples K as 8192 to accelerate training. Linear Classification. In this section, we freeze the backbone of the student model and train a classifier containing a linear layer with 1000 output dimensions after the backbone for 100 epochs. The initial learning rate is set as 0.01, and decayed by 0.1 at the 60 th epoch and 80 th epoch. A.4. Linear Detection and Segmentation Object Detection on PASCAL VOC. We have performed linear detection on PASCAL VOC in Table 5 to show that TKC learns better representations for object detection. We use Faster R-CNN [46] detector based on Detectron2 [52]. The backbone ends with conv5 stage and is set frozen. The training hyperparameters are kept consistent with finetuning, except the backbone is frozen. We train it for 48k iterations on VOC07+12 trainval. The initial learning rate is 0.02 and decayed at 36k and 44k iterations. The warmup stage lasts for 200 iterations. Object Detection and Instance Segmentation on COCO. We have also implemented linear detection on COCO. We use Mask R-CNN [23] detector based on Detectron2 [52], and synchronously train the object detection head and the instance segmentation head following [22], meanwhile we freeze the backbone. We also use a 2x scheduler the same as finetuning, where we train it for 180k iterations. The size of the shorter side is in [640,800] pixels during training and is fixed as 800 at inference. We use 8 GPUs and 16 batchsize. B. Architecture of History Bank We use history bank as a more effective implementation of the temporal teacher. Fig 4 illustrates the architecture and mechanism of history bank. The history bank is a matrix with size of D rows and h columns. D is the train set, h is the number of temporal teachers. A row of history bank stores the features from the same image but different teachers, while a column of history bank stores the ones from the same teacher but different images. This matrix can be saved at CPU memory, with no need to allocate the GPU memory. We illustrate the history bank by the blue cube in Fig 4. For a sample x, we first get the feature r S 0 from the student model, and the feature r T n from the EMA teacher model. Then we fetch out all the features from the same image x from the history bank as z T n−h ∼ z T n−1 , as shown in the middle in the Fig 4. For the implementation based on MoCo, we also fetch out the negative samples from the history bank. For each z T j j ∈ [n − h, n − 1], the corresponding negative features are randomly selected from the same column in the history bank as r − j . For the implementation based on BYOL, there is no need of the negative samples. History bank is an effective implementation of temporal teacher, the training procedure is the same as in the paper. C. Computational Cost We compare the computational cost of the two methods to show the efficiency of TKC. TKC use the history bank to approximate the temporal teachers. History bank stores the features of the recent epochs in CPU memory, and only part of them corresponding to the current batch will be dumped to GPU memory. This optimization can avoid duplicate forwarding, and the features from history bank are the same as forwarding the image into the temporal teachers. TKC has not too many additional costs thanks to history bank. As shown in Table 10, TKC has similar memory allocation with MoCo v2, which indicates that TKC has no special requirements for the capacity of machines. The time cost is higher than MoCo v2 for 47 %, which is mainly from the matrix multiplications between temporal features r T j , j ∈ [n − h, n − 1] and r S 0 , the knowledge transformer, and the data transport between memory and GPU memory. D. Further analysis D.1. More Experiment on AlexNet Backbone In this section, we implement MoCo v2 based on AlexNet [29] backbone to show that the temporal knowledge introduced by TKC can improve instance discrimina- D.2. Relation to No EMA Methods Some recent works [8] claim that the EMA encoder is not necessary to prevent model collapse. However, their works have no conflict with our work. SimSiam has shown that the stop gradient but not the EMA encoder is the key to prevent model collapse, but it also admits that the EMA encoder can improve accuracy (in the last paragraph in Section 2). Table 4 in [8] reveal that SimSiam with EMA encoder (BYOL) surpasses it by 3.0% for 800 epochs training, which shows that EMA encoder is important to learn good representations. However, the EMA encoder is not good enough, for it can not learn the temporal consistency between different training stages, as shown in our works. We note that all the reproductions in [8] applies the symmetrized loss. Section 4.6 in [8] shows that the symmetrized loss can boost the accuracy for 3%, and the computational cost has also doubled. So comparing TKC which is asymmetric to the symmetric methods [20,8] is unfair. Our improvement based on BYOL shows that the temporal knowledge is orthogonal to the symmetric loss. We consider providing the result of symmetric TKC in the next version and compare it with the symmetric methods. D.3. Does TKC improve the consistency? In this section, we visualize the inconsistency during training and indicate that TKC can improve the stability during pretext training. We randomly select some images and compare the stability of each sample between TKC and MoCo v2 baseline. Firstly, We define stability of a sample as the cosine similarity between current teacher output r T n and the counterpart in the last epoch z T n−1 . The formulation is: Figure 5. Comparison of stability between two methods in some randomly selected samples. The red curve represents MoCo v2, and the green curve represents TKC. The curves demonstrate that TKC can lead a consistent training and yield better representations. stable(x) = r T n · z T n−1(8) Then we randomly select some samples from the training set and compute the stability of these samples respectively during the whole training procedure. We compute the stability for both MoCo v2 [7] baseline and TKC. Each figure in Fig 5 represents the stability of the same sample in different methods, the red curve represents MoCo v2, the green curve represents TKC. As shown in Fig 5: (1) The output of the teacher model can dramatically vary even in the later stage during training. The stability of the samples usually gets down below 0.8 shows that the training target is inconsistent. Also, the stability is changed a lot in different epochs. These phenomena have confirmed our hypothesis that the targets from the teacher are noisy and inconsistent. (2) The stability of TKC is totally better than MoCo, where the green curve is higher than the red curve as a whole, which shows that the temporal knowledge from our method can lead to a more consistent training procedure, and improve the quality of the teacher's output. In this section, we conduct ablation studies on different structures of the knowledge transformer, as shown in the table 12. All models are trained on ResNet50 for 100 epochs. In our work, we use an MLP to implement the knowledge transformer. This MLP consists of a linear layer with output dimension 256 followed by a ReLU nonlinearity and a final linear layer with output size 256. This structure can reach an accuracy of 65.91. We observe that increasing the layer of MLP can better extract the importance of different teachers to achieve better performance. The 4-layer MLP can further improve the top-1 accuracy to 66.31. And a design of bottleneck MLP can also boost the performance, where we change the hidden size from 256 to 4096 to obtain a bottleneck structure. This structure can boost the result to 66.21 with less additional computational cost. These results show that the temporal teacher depends on the knowledge transformer to leverage the importance of different teachers. Using more complex structures as attention may further improve the performance. We will explore it in future work. E. Difference with Related Works Some previous works also involve information from previous periods. MoCo [22] and Temporal Ensembling [30] both use the samples from previous training. In this section, we will clarify the difference between TKC and their works in both motivation and methodology. Difference with MoCo v2. MoCo [22] believe that a large and consistent group of negative samples is critical for contrastive learning, and use the EMA encoder to construct a large and consistent negative bank. They think that the negative samples from previous training stages are harmful, and only use the negative samples which are near in time. In our work, we notice that the outputs of the teacher can vary dramatically on the same sample during different training stages, which can introduce unexpected noise and lead to catastrophic forgetting caused by inconsistent objectives. We believe that the knowledge from previous stages is essential to learn the instance temporal consistency and stable the position of the teacher's outputs in the latent space. Empirically results show that the output of temporal teachers can provide the temporal knowledge and gain the performance. Note that our negative bank is all consistent [22], for we use the negative samples from the same teacher to compute the temporal loss in Eq 4. Difference with Temporal Ensembling. Temporal Ensembling [30] is a semi-supervised learning method that ensemble the output of the same sample from previous epochs as the predicting target. Their work is different from ours in this aspect: (1) Temporal Ensembling relies heavily on dropout regularization to obtain various outputs in different epochs to yield a more accurate target. TKC also works well with networks without dropout layer [24], for TKC can restrict consistency between different epochs. (2) Temporal Ensembling also uses an exponential moving average to ensemble the output from different epochs, which can't leverage the importance of different outputs. On the contrary, TKC preserves the temporal teacher independently and uses knowledge transformer to dynamically learn their importance. Figure 3 . 3Comparison of validation accuracy between MoCo v2 and TKC. The top-1 accuracy is from a kNN classifier. Figure 4 . 4The architecture of history bank. History bank is an effective implementation of the temporal teacher. A row in the history bank stores the features from the same image, a column in the history bank stores the features from the same teacher. The green cubes indicate positive features, the grey cubes indicate negative features. Table 2 2shows our results on BYOL[20] baseline. We find that TKC can bootstrap BYOL for 2.3%, which shows that temporal knowledge can also benefit different instance discrimination methods via maximizing the mutual information for temporal targets. The results of BYOL are incompatible with the official ones because we use an unofficial reproduction of BYOL. We conduct this experiment only to prove that TKC can improve different instance discrimination methods. For more details about this reproduction, please refer to the supplementary material.Method conv1 conv2 conv3 conv4 conv5 Random 11.6 17.1 16.9 16.3 14.1 Supervised 19.3 36.3 44.2 48.3 50.5 Jigsaw [39] 19.2 30.1 34.7 33.9 28.3 Rotation [19] 18.8 31.7 38.7 38.2 36.5 DeepCluster [2] 12.9 29.2 38.2 39.8 36.1 NPID [53] 16.8 26.5 31.8 34.1 35.6 AET [61] 19.2 32.8 40.6 39.7 37.7 LA [64] 14.9 30.1 35.7 39.4 40.2 ODC [60] 19.6 32.8 40.4 41.4 37.3 Rot-Decouple [15] 19.3 33.3 40.8 41.8 44.3 TKC 20.3(+1.1) 34.2(+0.9) 42.6(+1.8) 46.2(+4.4) 44.0 Table 3 . 3Top-1 accuracy under the linear classification protocol on ImageNet with the AlexNet backbone. We fine-tune a fc layer from the top of different layers. on the test stage. All hyper-parameters of this finetuning protocol are consistent with the MoCo v2 baseline.As shown inTable 4, our TKC achieves 81.8 AP on the PASCAL VOC dataset, which outperforms all pre-trained models from competitors include the supervised ones. Our TKC shows consistent improvement for both AP 50 , AP , AP 75 , which shows that TKC indeed learns more consistent and transferable representations than MoCo v2. The upper part of the table 5 shows the results on COCO, TKC as well surpass the MoCo v2 AP 75 by 0.5% . The results on these two datasets indicated that comprehensive temporal knowledge can lead to transferable representations and learn better representations on different scenes and tasks.dataset, we train a Mask R-CNN [23] to learn the object detection and instance segmentation tasks pre-train AP 50 AP AP 75 random-init 60.2 33.8 33.1 supervised 81.3 53.5 58.8 NPID++ [53] 79.1 52.3 56.9 PIRL [37] 80.7 54.0 59.7 MoCo v2 [22] 81.5 55.9 62.6 TKC 81.8(+0.3) 56.5(+0.6) 62.8(+0.2) Table 4. Object detection fine-tuned on PASCAL VOC with Faster-RCNN. pre-train AP 50 AP AP 75 supervised 59.8 40.2 43.8 fine-tune MoCo v2 60.0 40.1 43.4 TKC 60.1(+0.1) 40.4(+0.3) 43.9(+0.5) supervised 54.3 34.3 36.5 freeze MoCo v2 48.1 29.2 30.8 TKC 54.2(+6.1) 34.7(+5.5) 37.1(+6.3) Table 5. Object detection on COCO. The detection framework is Mask R-CNN. We report the results for both fine-tune and freezing the backbone. synchronously. We train it for 180k iterations and decay the learning rate by 0.1 at 120k and 160k iterations. The input image size is between 640 and 800 on the training stage and 800 pre-train AP 50 AP AP 75 random-init 24.6 11.6 9.7 supervised 80.2 51.4 55.5 MoCo v2 [22] 79.0 51.7 56.2 TKC 80.9(+1.9) 52.7(+1.0) 57.6(+1.4) Table 6. Object detection on PASCAL VOC by freezing the back- bone and only training the detection head of Faster-RCNN. Table 7. Instance segmentation on COCO. The detection framework is Mask R-CNN. We report the results for both fine-tune and freezing the backbone.dataset pre-train AP 50 AP AP 75 supervised 56.7 34.9 37.1 fine-tune MoCo v2 56.8 35.0 37.2 TKC 56.8 35.2(+0.2) 37.7(+0.5) supervised 51.1 30.6 31.7 freeze MoCo v2 48.1 29.2 30.8 TKC 51.1(+3.0) 30.9(+1.7) 32.7(+1.9) Method Model Epochs Label fraction 1% 10% Supervised R50v2 48.4 80.4 NPID [53] R50 200 39.2 77.4 PIRL [37] R50 800 57.2 83.8 MoCo v1[22] † R50 200 61.3 84.0 SimCLR [5] † R50 200 64.5 82.6 MoCo v2 [7] ‡ R50 200 61.7 84.6 TKC R50 200 72.1(+10.4) 86.2(+1.6) Table 10. Computational cost. We report the time the GPU memory cost of our method and MoCo v2 baseline.Method GPU batchsize GPU·Time/Epoch memory/GPU MoCo v2 8×2080ti 256 3.4h 4.9G TKC 8×2080ti 256 5.0h 5.0G Table 11 . 11Comparison with MoCo v2 baseline on AlexNet.tion methods on the different backbone. InTable 3, we only compare TKC with SOTA methods, here we supplement the result of MoCo v2. The MoCo v2 baseline follows the same setup with TKC. 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[ "ASYMPTOTIC LINKING OF VOLUME-PRESERVING ACTIONS OF R k", "ASYMPTOTIC LINKING OF VOLUME-PRESERVING ACTIONS OF R k" ]	[ "José Luis ", "Lizarbe Chira ", "Paul A Schweitzer ", "S J " ]	[]	[]	We extend V. Arnold's work on asymptotic linking for two volume preserving flows on a domain in R 3 and S 3 to volume preserving actions of R k and R ℓ on certain domains in R n and also to linking of a volume preserving action of R k with a closed oriented singular ℓ-dimensional submanifold in R n , where n = k + ℓ + 1.	null	[ "https://export.arxiv.org/pdf/2008.01823v5.pdf" ]	220,968,884	2008.01823	41b616fbcfbe086e317c23df6e42c82fe25633a3	ASYMPTOTIC LINKING OF VOLUME-PRESERVING ACTIONS OF R k 19 Dec 2022 José Luis Lizarbe Chira Paul A Schweitzer S J ASYMPTOTIC LINKING OF VOLUME-PRESERVING ACTIONS OF R k 19 Dec 2022 We extend V. Arnold's work on asymptotic linking for two volume preserving flows on a domain in R 3 and S 3 to volume preserving actions of R k and R ℓ on certain domains in R n and also to linking of a volume preserving action of R k with a closed oriented singular ℓ-dimensional submanifold in R n , where n = k + ℓ + 1. Introduction V.I. Arnold, in his paper "The asymptotic Hopf invariant and its applications" [1] published in 1986 (also see [2,6,15,4]), considered a compact domain Ω in R 3 or S 3 with a smooth boundary and trivial homology and two divergence free vector fields X and Y in Ω tangent to the boundary ∂Ω. He defined an asymptotic linking invariant lk(X, Y ) that measures the average linking of trajectories of X with those of Y , and another invariant I(X, Y ) = Ω α ∧ dβ, where dα = i X ω and dβ = i Y ω (interior products with the volume form ω on Ω), and showed that lk(X, Y ) = I(X, Y ). We extend these results to volume-preserving actions Φ and Ψ of R k and R ℓ on a compact convex domain Ω with smooth boundary in R n , where Φ and Ψ are tangent to ∂Ω and k + ℓ = n − 1. The first author thanks the CAPES postdoc program 2015, Brazil for support. Arnol'd defines the invariant lk(X, Y ) as follows. For p ∈ Ω and T > 0, let ϑ X (p, T ) = {φ X t (p)\|0 ≤ t ≤ T } be the segment of orbit beginning at p and continuing for a time T , and letθ X (p, T ) be this curve closed by adding a short path in Ω from φ X T (p) to p. Defineθ Y (q, S) similarly. The asymptotic linking invariant of X and Y is lk(X, Y ) = Ω×Ωl k(p, q) wherel k(p, q) = lim S,T →∞ 1 ST lk(θ X (p, T ),θ Y (q, S)). Then lk(X, Y ) is well-defined, since lk(θ X (p, T ),θ Y (q, S)) is defined and the limit exists for almost all (p, q) ∈ Ω × Ω, and furthermore the functionlk(p, q) is in L 1 (Ω × Ω) [15]. The way that Arnol'd closes up partial orbits with short curves was used earlier on by Schwartzman to define asymptotic cycles for a continuous flow φ on a compact polyhedron X [10]. Let ϑ φ (p, T ) be the partial orbit from p ∈ X to φ T (p), let ϑ φ (p, T ) be a (possibly singular) loop formed by adding a short curve, and let [θ φ (p, T )] ∈ H 1 (X; R) be its first real homology class. Then the p asymptotic cycle is the limit A p = lim 1/t[θ φ (p, T )] ∈ H 1 (M, R) which exists for almost all points p ∈ X, as described in a geometric interpretation ( [10], p. 275). Schwartzman's proof is quite different, since he uses homomorphisms from the cohomology to R to define A p . If the short curves are chosen in a measurable fashion for a normalized invariant measure µ, then the µ asymptotic cycle is defined to be the integral A µ = X A p dµ ∈ H 1 (X; R), the average of the cycles A p . In [11], Schwartzman also defines asymptotic cycles for a smooth action of R k on a compact smooth manifold M n . This asymptotic cycle could also be defined by capping off the boundary of a partial orbit by a small (possibly singular) manifold, if that can be done in a measurable way, as in the present paper, though this is not carried out in [11]. In §2 we define an asymptotic linking invariant lk(Φ, Ψ) which measures the degree of linking between orbits of the actions Φ and Ψ and another invariant I(Φ, Ψ) defined in terms of differential forms. Our main result, Theorem 2 (proven in §11), states that lk(Φ, Ψ) = I(Φ, Ψ). Analogous results are given for the asymptotic linking of the action Φ with a closed oriented ℓ-dimensional submanifold N (Theorem 4, proven in §10). We use extensions of the gradient, curl, and divergence to multivectors in higher dimensions that are presented in §4, and in §7 an extension to higher dimensions of the classical Biot-Savart formula that gives an inverse for the curl of a divergencefree vector field on a compact domain R 3 . A version of the ergodic theorem due to Tempelman [13] that is used in the proofs is given in §5. As an application, we show that our invariant gives a lower bound for the energy of an action in §12. Examples in which the invariant is non-trivial are given in the last section, §13. These results are taken from the doctoral thesis [9] of the first author, under the direction of the second author at the Pontifical Catholic University of Rio de Janeiro (PUC-Rio). Some similar results were obtained by García-Compéan and Santos-Silva in [5]. It would be interesting to extend these results to S n and other Riemannian manifolds and also to linking of R k -actions with leaves of foliations endowed with an invariant transverse volume form (see [7]). Definitions and statements of results Throughout the paper M is an oriented Riemannian n-dimensional manifold and Ω ⊂ M is a compact convex domain with smooth boundary ∂Ω. In the main results of this paper, M will be R n with the standard metric, but many of the details are valid more generally. We consider a smooth (C ∞ ) action Φ : R k × Ω → Ω, of the k-dimensional real vector space R k on Ω. Then Φ is defined by k vector fields tangent to ∂Ω, X 1 , X 2 , . . . , X k , whose corresponding flows φ 1 , φ 2 , . . . , φ k commute with each other, so that for t = (t 1 , . . . , t k ) ∈ R k and x ∈ Ω, Φ(t, x) = φ 1 (t 1 , φ 2 (t 2 , . . . , φ k (t k , x), . . . )). In other words, if we set Φ t = Φ(t, ·) and φ i ti = φ i (t i , ·) for each i, then Φ t = φ 1 t1 • · · · • φ k t k . As usual, φ i is related to X i by the identity ∂ ∂t φ i (t, x) = X i (φ i (t, x)) and the commutation of φ i and φ j is equivalent to the vanishing of the Lie bracket [X i , X j ]. Definition 1. A (smooth) action Φ : R k × Ω → Ω on Ω is conservative if it is volume-preserving (i.e., for each t ∈ R k , Φ t : Ω → Ω preserves the Riemannian volume form on M ) and the generating vector fields X i are tangent to the boundary ∂Ω. Let Φ : R k × Ω → Ω and Ψ : R ℓ × Ω → Ω be conservative actions on Ω, k + ℓ + 1 = n. Let X = X 1 ∧· · · ∧X k and Y = Y 1 ∧· · · ∧Y ℓ be the exterior products of the k vector fields that generate the action Φ and the ℓ vector fields that generate Ψ, and let ω be the volume form on Ω. Denote the differential forms of degree r on Ω (resp., the forms that vanish on ∂Ω) by E r (Ω) (resp., E r (Ω, ∂Ω)). Since Ω is convex, their deRham cohomology groups H * (Ω; R) and H * (Ω, ∂Ω; R) vanish for 0 < r < n. The differential forms i X ω ∈ E ℓ+1 (Ω, ∂Ω) and i Y ω ∈ E k+1 (Ω, ∂Ω) given by the interior products with X and Y vanish on the boundary ∂Ω since X and Y are tangent to the boundary, and these forms are closed since the actions are volume-preserving. Since Ω is convex, they are exact, so there exist differential forms α ∈ E ℓ (Ω, ∂Ω) and β ∈ E k (Ω, ∂Ω) of degrees ℓ and k, respectively, such that dα = i X ω and dβ = i Y ω. Then we define the invariant I(Φ, Ψ) = Ω α ∧ dβ, which obviously does not depend on the choice of β. Since d(α ∧ β) = dα ∧ β + (−1) ℓ α ∧ dβ and both α and β vanish on ∂Ω, Stokes' theorem gives the following result. Lemma 1. This invariant satisfies I(Φ, Ψ) = (−1) ℓ+1 Ω dα ∧ β = (−1) (ℓ+1)(k+1) I(Ψ, Φ). Hence it depends only on the actions Φ and Ψ, and not on the choice of the differential forms α and β. We shall define an asymptotic linking number lk(Φ, Ψ) that measures the degree of linking between orbits of Φ and Ψ. For sets T ⊂ R k and Y ⊂ Ω we set Φ(T, Y ) = {Φ(t, y) \| t ∈ T, y ∈ Y }. Let T k be the set of k-rectangles T = [0, T 1 ] × · · · × [0, T k ], (T 1 , . . . , T k ) ∈ R k + where R k + is the space of k-tuples of non-negative real numbers, and fix a point p ∈ Ω. Then we let θ Φ (p, T ) be the closed oriented singular k-manifold in the domain Ω θ Φ (p, T ) = Φ(T, p) ∪ σ(p, T ) where σ(p, T ) = Φ(∂T, p) * p (1) is the cone composed of the geodesic segments joining each point of Φ(∂T, p) to p. We construct the closed oriented singular ℓ-manifold θ Ψ (q, S) = Ψ(S, q)∪σ ′ (q, S) in like manner, replacing T by S = [0, S 1 ]×· · ·×[0, S ℓ ] ∈ T ℓ for some (S 1 , . . . , S ℓ ) ∈ R ℓ + , Φ by Ψ, and p by another point q = p. For fixed T and S, since the sum of the dimensions of θ Φ (p, T ) and θ Φ (q, S) is n − 1, the following lemma holds. It will be proved in §9. Lemma 2. Fix T ∈ T k and S ∈ T ℓ . Then for almost every pair (p, q) ∈ Ω × Ω the singular manifolds θ Φ (p, T ) and θ Ψ (q, S)) are disjoint and therefore lk(θ Φ (p, T ), θ Ψ (q, S)) is defined. The set D(Φ, Ψ) = {(p, T, q, S) ∈ Ω × T k × Ω × T ℓ \| θ Φ (p, T ) ∩ θ Ψ (q, S) = ∅}, where the compact sets θ Φ (p, T ) and θ Ψ (q, S)) are disjoint, is clearly open, and since it has full measure, it must be dense, so we have: Corollary 1. D(Φ, Ψ) is an open dense set in Ω × T k × Ω × T ℓ . It follows from the Lemma that the function lk T,S (p, q) := 1 λ k (T )λ ℓ (S) lk(θ Φ (p, T ), θ Ψ (q, S)) is defined for almost all pairs (p, q) ∈ Ω × Ω, where λ k (T ) = T 1 · · · T k and λ ℓ (S) = S 1 · · · S ℓ are the Lebesgue measures on R k and R ℓ . The following theorem, proved in §11, affirms that this function is in L 1 (Ω × Ω) and permits us to define the linking index for the orbits of Φ and Ψ. We write T, S → ∞ to signify that min{T 1 , . . . T k , S 1 , . . . , S ℓ } → ∞. Theorem 1. Suppose that Ω is a compact convex domain in R n . Let Φ : R k × Ω → Ω and Ψ : R ℓ × Ω → Ω be conservative actions with k + ℓ + 1 = n. Then 1. The limit function lim T,S→∞ lk T,S exists as a function in L 1 (Ω × Ω), i.e., there is an integrable function lk Φ,Ψ : Ω × Ω → R defined almost everywhere such that lim T,S→∞ Ω Ω \|lk T,S (p, q) − lk Φ,Ψ (p, q)\| dpdq = 0. The integral Ω Ω lk Φ,Ψ (p, q)dpdq is independent of the choice of the distinct points p and q. Then the asymptotic linking number of Φ and Ψ is defined to be lk(Φ, Ψ) := Ω Ω lk Φ,Ψ (p, q)dpdq Our main theorem is the following. Linking of an action with a submanifold. There is a similar theory for asymptotic linking between a (smooth) conservative action Φ : R k × Ω → Ω and a closed oriented singular ℓ-submanifold N ⊂ Ω, where as above Ω is a compact convex domain in n-dimensional Euclidean space and n = k + ℓ + 1. As before, let α be an ℓ-form on Ω satisfying dα = i X ω where the vector fields X 1 , X 2 , . . . , X k generate the action Φ, X = X 1 ∧ · · · ∧ X k and let ω be the volume form on Ω. Then we define I(Φ, N ) = N α.(2) By analogy to the previous case of two actions, we can also define an asymptotic linking number between the action Φ and N . As before, let θ Φ (p, T ) = Φ(T, p) ∪ σ(p, T ) with the apex of the cone at p ∈ Ω \ N . The proof of the following Lemma is analogous to the proof of Lemma 2 and will also be given in §9. Lemma 3. Fix T ∈ T k and let N ′ be a compact oriented singular ℓ-submanifold N ′ ⊂ Ω, possibly with boundary. Then for almost every point p ∈ Ω, θ Φ (p, T )∩N ′ = ∅. Hence when N ′ = N , 1 λ k (T ) lk(θ Φ (p, T ), N ) is defined for almost all p ∈ Ω. Furthermore, the limit as T → ∞ exists in L 1 (Ω), and the integral is well-defined: Theorem 3. Let Φ : R k × Ω → Ω be a conservative action on a compact convex domain Ω in R n and let N ⊂ Ω be a smooth closed oriented ℓ-manifold, with k + ℓ + 1 = n. Then 1. The limit function lk Φ,N (p) := lim T →∞ 1 λ k (T ) lk(θ Φ (p, T ), N ) exists as a function in L 1 (Ω), i.e., there is an integrable function lk Φ,N : Ω → R defined almost everywhere such that lim T →∞ Ω \| 1 λ k (T ) lk(θ Φ (p, T ), N ) − lk Φ,N (p)\| dp = 0. 2. The integral Ω lk Φ,N (p)dp is independent of the choice of the point p. Then we define the asymptotic linking number of Φ and N to be lk(Φ, N ) := Ω lk Φ,N (p) dp. Theorem 1 follows from Proposition 9 in §11, the proof of Theorem 2 is given at the end of §11, Theorem 3 follows from Proposition 7 in §10, and the proof of Theorem 4 is given in §10. Higher Dimensional Vector Algebra We recall vector algebra on an oriented Riemannian n-dimensional manifold with metric g. Let E x,r = ∧ r T x M be the rth exterior power of the tangent space T x M at x ∈ M , with exterior multiplication ∧ : E x,r × E x,s → E x,r+s . The elements of E x,r are called r-vectors or multivectors. Recall that the Hodge operator * : E x,r → E x,n−r is defined for any positive orthonormal basis e 1 , . . . , e n of T x M = E x,1 by setting * (e i1 ∧ · · · ∧ e ir ) = e j1 ∧ · · · ∧ e jn−r , if (i 1 , . . . , i r , j 1 , . . . , j n−r ) is a positive permutation of (1, . . . , n), and extending over E x,r by linearity and antisymmetry. Then * • * = (−1) r(n−r) id : E r → E r .(3) The inner product given by the Riemannian metric < , > on T x M defines an inner product on E x,r ; for decomposable multivectors u = u 1 ∧ · · · ∧ u r and v 1 ∧ · · ·∧v r , u ·v = det(< u i , v j >). This inner product extends to an R-bilinear product (4) · : E x,r × E x,s → E x,s−r , (u, v) → u · v = * (u ∧ * v), and there is also a generalization to R n of the classical cross product on R 3 (5) × : E x,r × E x,s → E x,n−r−s , (u, v) → u × v = * (u ∧ v), In particular, u · v = u × * v. Proposition 1. Let u ∈ E x,r , v ∈ E x,s , and w ∈ E x,m be multivectors. (1) u × (v × w) = u · (v ∧ w). (2) If r + s + m = n, then (u × v) · w = * (u ∧ v ∧ w). Proof. 1. u · (v ∧ w) = * (u ∧ * (v ∧ w)) = * (u ∧ (v × w)) = u × (v × w). 2. Note that u × v and w are both in E x,m , so (u × v) · w ∈ E x,0 = R and (u×v)·w = w·(u×v). Now w·(u×v) = * (w∧ * * (u∧v)) = (−1) m(r+s) * (w∧u∧v) = * (u ∧ v ∧ w). It follows from item 2 of the preceding Proposition that if the vectors u, v, and w are decomposable, say u = v 1 ∧· · ·∧v r , v = v s+1 ∧· · ·∧v r+s , and w = v r+s+1 ∧· · ·∧v n with v i = j a ij e j for a positive orthonormal basis e 1 , . . . , e n , then (6) (u × v) · w = det(a ij ). Example 1. As usual, a multi-index I is an ordered subset I = (i 1 , . . . , i k ) of {1, . . . , n} with i 1 < i 2 < · · · < i k , and we set e I = e i1 ∧ · · · ∧ e i k and \|I\| = k. Proposition 2. For vectors u, v 1 , . . . , v k ∈ T x M n , we have u · (v 1 ∧ · · · ∧ v k ) = (−1) (k−1)(n−k) k i=1 (−1) i−1 (u · v i ) v 1 ∧ . . . v i · · · ∧ v k . Proof. Using (2) of Example 1 with e I = e ji and e J = e j1 ∧· · ·∧e j k with 1 ≤ i ≤ k, we have \|K\| = k − 1 and so (7) e ji · (e j1 ∧ · · · ∧ e j k ) = (−1) (k−1)(n−k)+i−1 e j1 ∧ . . . e ji · · · ∧ e j k . Note that expanding u = n i=1 u i e i and v = v 1 ∧ · · · ∧ v k with v i = n ji=1 v iji e ji we obtain u · v = n j1,...,j k =1 k i=1 u ji (v 1j1 . . . v kj k )(e ji · (e j1 ∧ · · · ∧ e j k )). so the desired formula follows by substituting (7) and reassembling the terms u and v 1 , . . . , v k . Example 2. For vectors u, v, w in R n , by the definition of the product × and Proposition 2, u × (v × w) = u · (v ∧ w) = (−1) n [(u · v)w − (u · w)v]. In particular, in R 3 we have the well-known formula u × (v × w) = (u · w)v − (u · v)w. Extensions of Gradient, Curl, and Divergence. Let E k = E k (M ) be the space of smooth k-vector fields on a Riemannian manifold M , and let E k = E k (M ) be the dual space of differential k-forms. The inner product (U, V ) → U · V on E k determines an isomorphism (8) j : E k → E k , j(U )(V ) = U · V. The interior product i : Proof. Consider X = e I where I = (i 1 , . . . , i k ) and J = (j 1 , . . . , j ℓ ) are ordered multiindices such that (i 1 , . . . , i k , j 1 , . . . , j ℓ ) is a positive permutation of (1, . . . , n), and let η 1 , . . . , η n be the basis dual to a local positive orthonormal basis e 1 . . . , e n . Then i eI ω = η j1 ∧ · · · ∧ η j ℓ = j(e J ) = j( * e I ) since e J = * e I . The lemma follows since every X ∈ E k is a linear combination of the elements e I . E k × E r → E r−k , (X, α) → i X α, is defined i X α(Y ) = α(X ∧ Y ) for Y ∈ E r−k . The duality between E k e E k will be expressed using the isomorphism j. For example, the gradient operator ∇, defined ∇f = j −1 (df ) for a smooth function f on M , can be extended to a linear operator ∇ : E k → E k+1 , ∇X = j −1 dj(X). We can also extend the curl and divergence to operators rot : E k → E ℓ and div : E k → E k−1 by setting (9) rot(X) = (−1) (k+1)ℓ * (∇X) and (10) div(X) = (−1) (k+1)ℓ * ∇( * X) where we always set ℓ = n − k − 1. On R 3 these definitions coincide with the classical definitions of curl and divergence for vector fields. For the rest of this section we suppose that M = R n with the canonical basis {e 1 , . . . , e n } and the dual basis {dx 1 , . . . dx n }. For a k-vector field of the form X = f e i1 ∧ · · · ∧ e i k where f is a smooth function it is easy to check that j(f e i1 ∧ · · · ∧ e i k ) = f dx i1 ∧ · · · ∧ dx i k , ∇X = (∇f ) ∧ e i1 ∧ . . . e i k , and (11) div(X) = (−1) k k s=1 (−1) s ∂f ∂x is e i1 ∧ . . . e is · · · ∧ e i k . Recall that a vector field U = n i=1 u i e i on R n acts on a function f by setting U (f ) = < U, ∇f > = n i=1 u i ∂f ∂xi . The action of U on a vector field V = n i=1 v i e i is defined by setting U (V ) = n i=1 U (v i )e i = n i,j=1 u j ∂v i ∂x j e i so the Lie bracket can be written [U, V ] = U (V ) − V (U ). Proposition 3. Let V = V 1 ∧ · · · ∧ V k be the exterior product of vector fields V 1 , . . . , V k on R n . Then div(V ) = (−1) k k i=1 (−1) i div(V i ) V 1 ∧ · · · V i · · · ∧ V k +(−1) k 1≤i<j≤k (−1) i+j [V i , V j ] ∧ V 1 ∧ · · · ∧ V i ∧ · · · ∧ V j ∧ · · · ∧ V k where [V i , V j ] is the Lie bracket. Proof. Note that this is a dual version of the well-known formula for the exterior derivative of a product of 1-forms evaluated on vector fields. Let (12) V i = n ℓ=1 v i ℓ e ℓ for every i, so expanding V we have (13) V = n ℓ1,...,ℓ k =1 v 1 ℓ1 . . . v k ℓ k e ℓ1 ∧ · · · ∧ e ℓ k . Then by (11) div (V ) = (−1) k k i=1 n ℓ1,...,ℓ k =1 (−1) i ∂(v 1 ℓ1 . . . v k ℓ k ) ∂x ℓi e ℓ1 ∧ · · · ∧ e ℓi ∧ . . . e ℓ k = (−1) k k i,j=1 n ℓ1,...,ℓ k =1 (−1) i ∂v j ℓj ∂x ℓi v 1 ℓ1 . . . v j ℓj . . . v k ℓ k e ℓ1 ∧ · · · ∧ e ℓi ∧ . . . e ℓ k . Since div (V i ) = n ℓi=1 ∂v i j ∂x ℓ i , the terms with i = j give (−1) k k i=1 (−1) i div(V i ) V 1 ∧ · · · V i · · · ∧ V k while the remaining terms give the second sum in the proposition; in fact, if I ab with a < b is the sum of the terms with (i, j) = (a, b) and (i, j) = (b, a), then I ab = (−1) k n ℓa,ℓ b =1 (−1) a+b v a ℓa ∂v b ℓ b ∂x ℓa e ℓ b − v b ℓ b ∂v a ℓa ∂x ℓ b e ℓa ∧ ∧ (v 1 ℓ1 . . . v a ℓa . . . v b ℓ b . . . v k ℓ k )e ℓ1 ∧ . . . e ℓa . . . e ℓ b · · · ∧ e ℓ k = (−1) k+a+b [V a , V b ] ∧ V 1 ∧ . . . V a . . . V b · · · ∧ V k since n ℓa,ℓ b =1 v a ℓa ∂v b ℓ b ∂x ℓa e ℓ b − v b ℓ b ∂v a ℓa ∂x ℓ b e ℓa = [V a , V b ]. Example 3. If U and V are vector fields in R n , then by Proposition 3 and the definitions of rot and ×, rot(U × V ) = div(U ∧ V ) = (div(V ))U − (div(U ))V − [U, V ]. Proposition 4. Let ω be the positive unit volume form. Given a k-vector field U ∈ E k (Ω) and a k-form α ∈ E k (Ω) with 0 ≤ k ≤ n, we have: α(U )ω = α ∧ i U ω,(14)dj(U ) = i rot(U) ω.(15) Proof. If U = e i1 ∧ · · · ∧ e i k with i 1 < · · · < i k and α = dx j1 ∧ · · · ∧ dx j k with j 1 < · · · < j k , then α(U ) = 0 if and only if the sequences (i 1 , . . . , i k ) and (j 1 , . . . , j k ) coincide, and then α(U )ω = ω = α ∧ i U ω. If the two sequences do not coincide, then both sides vanish. By expanding any U and α and using linearity, we conclude that the equation (14) holds in general. Next, dj(U ) = j(∇U ) = j((−1) (k+1)(n−k) * * ∇U ) = j( * rot(U )) which is equal to i rot(U) ω by Lemma 4, thus proving (15). The Ergodic Theorem for actions of R k In this section we present Theorem 5, a special case of Tempelman's version of the Ergodic Theorem [13] (also see [14]), for volume-preserving actions of R k . This result is an essential step in showing that the asymptotic linking invariant is well-defined. Let M be a compact Riemannian manifold (possibly with boundary) with Riemannian volume form µ and let Φ : R k × M → M be a conservative action of R k on M . Let L 1 (M ) denote the space of measurable real functions f : M → R such that M \|f \|dµ < ∞. Consider a sequence of k-rectangles T n := [0, T 1 n ] × · · · × [0, T k n ], n ∈ N with each T i n > 0, such that for each i (1 ≤ i ≤ k) lim n→∞ T i n = ∞. For a function f ∈ L 1 (M ), define a sequence of means f n ∈ L 1 (M ), n ∈ N, by setting f n (p) := 1 λ(T n ) t∈Tn f (Φ t (p))dλ(t) = 1 T 1 n T 2 n . . . T k n T k n 0 T k−1 n 0 · · · T 1 n 0 f (Φ (t1,...t k ) (p))dt 1 dt 2 . . . dt k where λ is the Lebesgue measure on R k and t = (t 1 , . . . t k ). The following theorem is a special case of Theorem 6.2 of Tempelman [13] and also of Theorem 3.3 of Lindenstrauss [8]. Of course, uniqueness of f is understood in the sense of L 1 , i.e., two such functionsf agree outside of a set of measure zero. Lindenstrauss' Theorem 3.3 implies this theorem since R k is an amenable group and {T n } is a tempered Følner sequence. Outline of the Proof. First we observe that for a fixed sequence {T n } of krectangles the set of f ∈ L 1 (M ) for which the Theorem holds is a closed vector subspace of L 1 (M ). Then the essential idea is Tempelman's decomposition of L 1 (M ) into invariant functions and functions with zero mean (Theorem 5.1 of [13]). Let W be the vector subspace of L 1 (M ) generated by functions h − h • Φ t where h = χ A is the characteristic function of a measurable set A and t ∈ R k , and let W be its closure in L 1 (M ). One shows that the conclusions of the Theorem hold for f = h − h • Φ t , if h is the characteristic function of a measurable set A in Ω, and consequently for every f ∈ W . By approximation, the same is true for all f ∈ W . On the other hand, let I ⊂ L 1 (M ) be the set of invariant functions where f ∈ L 1 (M ) is invariant if there exists a measurable set A with µ(M \ A) = 0 such that for every x ∈ A and t = (t 1 , . . . , t k ) ∈ R k we have f (Φ t (x)) = f (x). For every invariant function f it is clear that f n = f , so it is easy to see that the conclusions of the Theorem hold for every f ∈ I by setting f = f . Since by Theorem 5.1 of [13] every function f ∈ L 1 (M ) can be uniquely represented as a sum f = f 1 + f 2 with f 1 ∈ I and f 2 ∈ W , the Theorem holds for every f ∈ L 1 (M ). The Generalized Gauss Divergence Theorem for a Multivector Field In this section, Ω is a compact domain with smooth boundary in R n . We define the integral of a k-vector field X = 1≤i1<···<i k ≤n f i1...i k e i1 ∧ · · · ∧ e i k ∈ E k (Ω) to be the k-vector Ω X ω := 1≤i1<···<i k ≤n Ω f i1...i k ω e i1 ∧ · · · ∧ e i k ∈ E k (Ω) (16) where ω is the unit volume form. Using this definition of the integral, we can extend the Gauss divergence theorem to k-vector fields on Ω with k > 1. Theorem 6. (Generalized Gauss Divergence Theorem for a Multivector Field) If V ∈ E k (Ω), then Ω div(V ) ω = (−1) (k+1)ℓ ∂Ω N · V dA , where N is the unit normal vector field pointing outwards along ∂Ω, N · V is the extended dot product (4), ω and dA are the positive unit volume forms on Ω and ∂Ω, and ℓ = n − k − 1. Proof. Since every element of E k (Ω) is a sum of decomposable ones, it suffices to prove the proposition for a decomposable k-vector V = V 1 ∧ · · · ∧ V k where V i is given by (12). Then from (13) and (11) we get div(V ) = (−1) k k i=1 n ℓ1,... ℓi...,ℓ k =1 (−1) i div(v 1 ℓ1 . . . v i ℓi . . . v k ℓ k V i )e ℓ1 ∧ . . . e ℓi · · · ∧ e ℓ k since div(v 1 ℓ1 . . . v i ℓi . . . v k ℓ k V i ) = n ℓi=1 ∂(v 1 ℓ1 . . . v k ℓ k ) ∂x ℓi . By Stokes' Theorem we have Ω div v 1 ℓ1 . . . v i ℓi . . . v k ℓ k V i ω = ∂Ω v 1 ℓ1 . . . v i ℓi . . . v k ℓ k < N, V i > dA so Ω div(V ) ω = (−1) k k i=1 (−1) i ∂Ω < N, V i > V 1 ∧ . . . V i · · · ∧ V k dA = ∂Ω (−1) k k i=1 (−1) i < N, V i > V 1 ∧ . . . V i · · · ∧ V k dA = (−1) (k+1)ℓ ∂Ω N · V 1 ∧ · · · ∧ V k dA using V j = n ℓj =1 v j ℓj e ℓj and Proposition 2. Corollary 2. Set Ω − x = {u − x ∈ R n \| u ∈ Ω}. For a k-vector field V (x, u) on R n × R n we have div x Ω−x V (x, u)du = −(−1) (k+1)ℓ ∂Ω−x N · V (x, u)dA(u) + Ω−x div x V (x, u)du. Proof. By the change of variables v = u + x Ω−x V (x, u)du = Ω V (x, v − x)dv, so div x Ω−x V (x, u)du = div 1,x Ω V (x, v − x)dv + div 2,x Ω V (x, v − x)dv, where the notation indicates that the divergence is calculated with respect to the first or second occurrence of the variable x. Now div 1,x Ω V (x, v − x)dv = Ω div 1,x V (x, v − x)dv = Ω−x div x V (x, u)du by reversing the change of variables. On the other hand, if we introduce a new variable z = x to separate the two arguments of V , div 2,x Ω V (x, v − x)dv = div x Ω V (z, v − x)dv = −div v Ω V (z, v − x)dv = −(−1) (k+1)ℓ ∂Ω N · V (z, v − x)dA(v) = −(−1) (k+1)ℓ ∂Ω−x N · V (x, u)dA(u) by Theorem 6, reversing the change of variables. Adding the last two expressions gives the desired result. Extension of the Biot-Savart Formula We now give an extension of the Biot-Savart formula to higher dimensions. For a smooth divergence-free vector field V that is tangent to the boundary on a bounded domain Ω in R 3 , it is well known that the Biot-Savart formula BS(V )(x) = −1 4π (x − y) × V (y) \|\|x − y\|\| 3 dy gives a right inverse for the curl, i.e., rot(BS(V )) = V (e.g., see [3] §5). We generalize this result to R n since it will be used in our proofs. Let Ω be a bounded domain with smooth boundary ∂Ω in R n and consider k commuting vector fields V 1 , . . . , V k on Ω that are divergence-free and tangent to ∂Ω, 1 ≤ k < n, with ℓ = n − k − 1. They generate an action of R k on Ω. Let V = V 1 ∧ · · · ∧ V k be the exterior product of the vector fields V i . Theorem 7. For x ∈ Ω, the ℓ-vector field (17) BS(V )(x) = (−1) k a n Ω (x − y) \|\|x − y\|\| n × V (y)dy, where a n is the (n − 1)-volume of the unit sphere in R n and we use the standard Lebesgue measure dy on R n , satisfies rot(BS(V ))(x) = V (x). Proof. Note that the integral is well defined since the pole along the singular set has order n − 1. We prove the theorem for x ∈Ω, to avoid the problem of a singularity of order n − 1 when we integrate along ∂Ω. It will follow by continuity that the theorem holds for every x ∈ Ω. By the change of variables u = y − x on Ω − x, we have BS(V )(x) = (−1) k+1 a n Ω−x u \|\|u\|\| n × V (u + x)du. Since rot(u × v) = div(u ∧ v), from Corollary 2 we get I :=rot(BS(V ))(x) = (−1) k+1 a n div x Ω−x u \|\|u\|\| n ∧ V (u + x)du = I 1 + I 2 where I 1 = (−1) k+1 a n Ω−x div x u \|\|u\|\| n ∧ V (u + x) du and I 2 = − (−1) k+1+k(ℓ+1) a n ∂Ω−x N · ( u \|\|u\|\| n ∧ V (u + x))dA(u). Applying Proposition 3 and the facts that div(V i ) = 0, [V i , V j ] = 0, and u \|\|u\|\| n does not depend on the variable x, we have I 1 = 1 a n Ω−x k i=1 (−1) i u \|\|u\|\| n x V i (u + x) V 1 ∧ . . . V i · · · ∧ V k (u + x) du, where u \|\|u\|\| n x V i is the action of the vector field u \|\|u\|\| n on V i (u + x) with derivatives in the variable x. Expanding the V i 's by (12), using the definition of the integral (16), and avoiding the singularity at u = 0, we can write I 1 = −1 a n lim ǫ→0 n j1,...,j k =1 Ω ′ u \|\|u\|\| n , ∇ x (v 1 j1 . . . v k j k )(u + x) du e j1 ∧ · · · ∧ e j k where Ω ′ = (Ω − x) \ {\|\|u\|\| ≤ ǫ} and e ji ∧ e j1 . . . e ji . . . e j k = (−1) i−1 e j1 . . . e j k . Now ∇ x (v 1 j1 . . . v k j k )(u + x) = ∇ u (v 1 j1 . . . v k j k )(u + x), so, for ǫ > 0 so small that {\|\|u\|\| ≤ ǫ} ⊂Ω, the integral I(ǫ) := Ω ′ u \|\|u\|\| n , ∇ x (v 1 j1 . . . v k j k )(u + x) du can be written as I(ǫ) = Ω ′ u \|\|u\|\| n , ∇ u (v 1 j1 . . . v k j k )(u + x) du = Ω ′ div u (v 1 j1 . . . v k j k u \|\|u\|\| n ) − v 1 j1 . . . v k j k div u ( u \|\|u\|\| n ) du = Ω ′ div u (v 1 j1 . . . v k j k u \|\|u\|\| n )du = ∂Ω−x < N, u \|\|u\|\| n > v 1 j1 . . . v k j k dA(u) − {\|\|u\|\|=ǫ} 1 ǫ n−1 v 1 j1 . . . v k j k (u + x) dA(u) by Theorem 6, since div u ( u \|\|u\|\| n ) = 0 on R n . Thus lim ǫ→0 I(ǫ) = ∂Ω−x < N, u \|\|u\|\| n > v 1 j1 . . . v k j k dA(u) − a n v 1 j1 . . . v k j k (x), so (18) I 1 = − 1 a n ∂Ω−x < N, u \|\|u\|\| n > V 1 ∧· · · ∧V k (u + x) dA(u)+ V 1 ∧· · · ∧V k (x). Next, returning to (18), we get I 2 = (−1) kℓ a n ∂Ω−x N · ( u \|\|u\|\| n ∧ V (u + x))dA(u) = 1 a n ∂Ω−x < N, u \|\|u\|\| n > V (u + x))dA(u). by Proposition 2, since the V i 's are tangent to ∂Ω − x. Adding the last result to (18) we obtain the desired conclusion, I = rot(BS(V ))(x) = I 1 + I 2 = V (x). Corollary 3. Let Ω be convex with unit volume form ω and let V ∈ E k (Ω) be as above. Then dj(BS(V )) = i V ω and(19)I(Φ, Ψ) = j(BS(X)) ∧ dβ (20) Proof. By (15) and Theorem 7, djBS(V ) = i rot(BS(V )) ω = i V ω. proving (19). By Lemma 1 I(Φ, Ψ) = Ω α ∧ dβ is independent of α, provided that dα = i X ω. Then by (19) with V = X I(Φ, Ψ) = j(BS(X)) ∧ dβ. Linking of submanifolds In order to study the asymptotic linking invariant we recall the linking of singular submanifolds in R n . Let N and N ′ be closed, oriented, possibly singular, disjoint submanifolds of R n of dimensions k and ℓ, where we always suppose that n = k + ℓ + 1. Then the linking number lk(N, N ′ ) of N and N ′ can be defined as follows. Let C be a compact oriented singular k + 1-dimensional manifold in R n with ∂C = N . By a small deformation of C, if necessary, we may suppose that C is transverse to N ′ and only intersects it in non-singular points of N ′ . Then the linking number of N and N ′ is defined to be lk(N, N ′ ) := p ε p where the sum is taken over all points p ∈ C ∩ N ′ , with ε p = +1 if the orientation of C × N ′ coincides with that of R n or −1 if the orientations are opposite. It is well known that this linking number is symmetric, does not depend on the choice of C, and can also be calculated as lk(N, N ′ ) = deg(f : N × N ′ → S n−1 ) where f (p, q) := q − p q − p is the normalized vector pointing from p ∈ N to q ∈ N ′ and deg(f ) is the degree of the mapping f relative to the orientations of N , N ′ , and S n−1 . If N and N ′ are disjoint images of smooth maps g :N → R n and g ′ :N ′ → R n , then the linking number can be calculated by lk(N, N ′ ) = 1 a n N ×N ′f * (σ) (21) wheref = f • (g × g ′ ) and a n = S n−1 σ is the volume form on S n−1 . In order to prove the next proposition, we observe that if (t 1 , , t 2 , . . . , t k ) are local coordinates in N , then the volume form dη on N can be written in these coordinates as (22) dη = ∂ ∂t 1 ∧ · · · ∧ ∂ ∂t k dt 1 dt 2 . . . dt k . and similarly for the volume form dη ′ on N ′ with local coordinates s 1 , . . . , s ℓ . Proposition 5. If N and N ′ are disjoint immersed closed oriented submanifolds in R n , then the linking number lk(N, N ′ ) can be calculated by the formula (23) lk(N, N ′ ) = (−1) k a n p∈N q∈N ′ (q − p) × U (p) · U ′ (q) \|\|q − p\|\| n dη(p)dη ′ (q) where U (p) is a unit k-vector on N at p and U ′ (q) is a unit ℓ-vector on N ′ at q and η and η ′ are the volume measures in N and N ′ . Furthermore, this formula holds if N and N ′ are the disjoint images of smooth manifoldsN andN ′ under smooth singular maps g :N → R n and g ′ :N ′ → R n , since the images of the singular sets (where U (p) = 0 or U ′ (q) = 0) have measure zero on N and N ′ , by Sard's Theorem. Proof. Note that the volume form σ = n i=1 (−1) i−1 x i dx 1 . . . dx i . . . dx n on S n−1 can be written σ = i Y dx 1 . . . dx n , where Y = n i=1 x i e i is the position vector in S n−1 . Then, since dx 1 . . . dx n (Z) = * Z for any Z ∈ Λ n (R n ), σ(v 2 ∧ v 3 ∧ · · · ∧ v n ) = i Y dx 1 . . . dx n )(v 2 ∧ · · · ∧ v n ) = dx 1 . . . dx n (Y ∧ v 2 ∧ · · · ∧ v n ) = * (Y ∧ v 2 ∧ · · · ∧ v n ).(24) On the other hand, using local coordinates (t 1 , . . . , t k , s 1 , . . . , s ℓ ) in N × N ′ , since f (p, q) = q−p \|\|q−p\|\| andf = f • (g × g ′ ), we have ∂f ∂t i (p, q) = −1 \|\|q − p\|\| ∂ ∂t i (p) + 1 \|\|q − p\|\| ti (q − p), ∂f ∂s j (p, q) = 1 \|\|q − p\|\| ∂ ∂s j (q) + 1 \|\|q − p\|\| sj (q − p). Setting ∂ ∂t = ∂ ∂t1 ∧ · · · ∧ ∂ ∂t k and ∂ ∂s = ∂ ∂s1 ∧ · · · ∧ ∂ ∂s ℓ we get ∂f ∂t ∧ ∂f ∂s = ∂f ∂t 1 ∧ · · · ∧ ∂f ∂t k ∧ ∂f ∂s 1 ∧ · · · ∧ ∂f ∂s ℓ = (−1) k \|\|q − p\|\| k+ℓ ∂ ∂t ∧ ∂ ∂s + W ∧ (q − p)(25) where W is a (k +ℓ−1)-vector. Thus, at the point (p, q) in N ×N ′ that corresponds to the point q−p \|\|q−p\|\| ∈ S n−1 , using local coordinates and k + ℓ + 1 = n we get f * (σ)(p, q) = σ( ∂f ∂t ∧ ∂f ∂s )dt 1 . . . dt k ds 1 . . . ds ℓ = dx 1 . . . dx n q − p \|\|q − p\|\| ∧ ∂f ∂t ∧ ∂f ∂s dt 1 . . . dt k ds 1 . . . ds ℓ = (−1) k \|\|q − p\|\| n dx 1 . . . dx n (q − p) ∧ ∂ ∂t ∧ ∂ ∂s dt 1 . . . dt k ds 1 . . . ds ℓ by (25) = (−1) k \|\|q − p\|\| n * (q − p) ∧ ∂ ∂t ∧ ∂ ∂s dt 1 . . . dt k ds 1 . . . ds ℓ by (24) = (−1) k \|\|q − p\|\| n * (q − p) ∧ \|\| ∂ ∂t \|\|U ) ∧ (\|\| ∂ ∂s \|\|U ′ dt 1 . . . dt k ds 1 . . . ds ℓ = (−1) k \|\|q − p\|\| n * (q − p) ∧ U (p) ∧ U ′ (q) dη(p)dη ′ (q) by (22) = (−1) k \|\|q − p\|\| n * (−1) (k+1)ℓ U ′ (q) ∧ (q − p) ∧ U (p) dη(p)dη ′ (q) = (−1) k \|\|q − p\|\| n * U ′ (q) ∧ * * [(q − p) ∧ U (p)] dη(p)dη ′ (q) by (3) = (−1) k \|\|q − p\|\| n * U ′ (q) ∧ * [(q − p) × U (p)] dη(p)dη ′ (q) by (5) = (−1) k \|\|q − p\|\| n U ′ (q) · (q − p) × U (p) dη(p)dη ′ (q) by (4) = (−1) k \|\|q − p\|\| n (q − p) × U (p) · U ′ (q)dη(p)dη ′ (q) since U ′ (q) and (q − p) × U (p) are in the same dimension ℓ so the dot product commutes. Thus by (21) lk(N, N ′ ) = 1 a n p∈N q∈N ′f * (σ)(p, q) = (−1) k a n p∈N q∈N ′ (q − p) × U (p) · U ′ (q) \|\|q − p\|\| n dη(p)dη ′ (q).lk(N, N ′ ) = −1 4π t0 0 s0 0 (α ′ (s) − α(t)) ×α(t) ·α ′ (s) \|\|α ′ (s) − α(t)\|\| 3 dtds. A double differential form L(x, y) on R n × R n of bidegree (k, ℓ), k + ℓ = n − 1, is called a linking form if whenever N = g(N ) and N ′ = g ′ (N ′ ) are disjoint images of smooth singular maps g :N → R n and g ′ :N ′ → R n , whereN andN ′ are closed oriented manifolds of dimensions k and ℓ, then we have (26) L = L(x, y) = (−1) k a n lk(N, N ′ ) = N N ′ L.(y − x) × U (x) · U ′ (y) \|\|y − x\|\| n dη(x)dη ′ (y) is a linking form on R n × R n , where U (x) is a unit k-vector on N at x, U ′ (y) is a unit ℓ-vector on N ′ at y, and η and η ′ are the volume measures in N and N ′ . This is evident from Proposition 5. Proofs of Lemmas 2 and 3 As in §2, consider two volume-preserving actions Φ : R k × Ω → Ω and Ψ : R ℓ × Ω → Ω on a compact convex domain Ω in a Riemannian n-manifold M tangent to the (smooth) boundary ∂Ω, n = k + ℓ + 1. Recall that T k is the set of k-rectangles T = [0, T 1 ] × · · · × [0, T k ] ⊂ R k for (T 1 , . . . , T k ) ∈ R k + . Fix points p,q ∈ Ω,p =q, and consider the geodesic cones σ(p, T ), (p, T ) ∈ Ω × T k , and σ ′ (q, S), (q, S) ∈ Ω × T ℓ , with apicesp andq, as defined in (1). We now prove Lemma 2. Proof of Lemma 2. We must show that for every T ∈ T k and S ∈ T ℓ the set X = {(p, q) ∈ Ω × Ω \| θ Φ (p, T ) ∩ θ Φ (q, S) = ∅} has measure zero in Ω × Ω. Set A q = Φ(−T, Ψ(S, q)), B q = Φ(−T, σ ′ (q, S)), B ′ p = Ψ(−S, σ(p, T )), and C p = {q ∈ Ω \| σ(p, T ) ∩ σ ′ (q, S) = ∅}. Note that for any set K ⊂ Ω and p ∈ Ω, p ∈ Φ(−T, K) ⇐⇒ Φ(T, p) ∩ K = ∅. Consequently p ∈ A q ⇐⇒ Φ(T, p) ∩ Ψ(S, q) = ∅, p ∈ B q ⇐⇒ Φ(T, p) ∩ σ ′ (q, S) = ∅, and q ∈ B ′ p ⇐⇒ Ψ(S, q) ∩ σ(p, T ) = σ(p, T )∅. Since θ Φ (p, T ) = Φ(T, p) ∪ σ(p, T ) and similarly for θ Ψ (q, S), it follows that X = q∈Ω ((A q ∪ B q ) × {q}) ∪ p∈Ω ({p} × (B ′ q ∪ C p )). Each of the sets A p , B p , and B ′ q is a singular compact (n − 1)-dimensional submanifold with open dense complement in Ω, and therefore has measure zero in Ω. Next we shall show that if p = q the set C p has measure zero in Ω. Let N be the cone consisting of straight segments beginning at q, passing through a point of σ(p, T ), and ending at a point of ∂Ω. Let N be the closure of the component of N \ σ(p, T ) that does not contain the point q. Now σ ′ (q, S) meets σ(p, T ) if and only if Ψ(∂S, q) meets N . Thus C p = Ψ(−∂S, N ), which is a compact singular manifold (the product of the image of the union of the 2ℓ faces of S with N ) of dimension (ℓ − 1) + (k + 1) = n − 1, so it has measure zero. Note that each of the sets ∪ q (A q × {q}), ∪ q (B q × {q}), ∪ p ({p} × B ′ p ) , and ∪ p ({p} × C p ) is closed and therefore measurable in Ω × Ω. Hence the function f : Ω × Ω → {0, 1}, defined by setting f (p, q) = 1 if p ∈ A q and 0 otherwise, is measurable. Since A q has measure zero in Ω for almost all q ∈ Ω, and therefore Ω f (p, q)dp = 0 for almost all q, Fubini's theorem shows that Ω Ω f (p, q)dpdq = 0, which means that the set ∪ q (A q × {q}) has measure zero in Ω × Ω. Parallel arguments show that the sets ∪ q (B q × {q}), ∪ p ({p} × B ′ p ), and ∪ p ({p} × C q ) also have measure zero, so their union X has measure zero in Ω × Ω, as claimed. Proof of Lemma 3. The proof is similar to the last proof. We must show that for every T ∈ T k the set Y = {p ∈ Ω \| θ Φ (p, T ) ∩ N ′ = ∅} has measure zero in Ω. Observe that Y = A∪C where A = Φ(T −1 , N ′ ) and C = {p ∈ Ω \| σ(p, T )∩N ′ ) = ∅}. Let B be the cone consisting of segments beginning at p, passing through a point of N ′ , and ending at a point of ∂Ω. Let B be the closure of the component of B \ N ′ that does not contain the point p. As in the previous proof, we find that C = Ψ(−∂T, N ′ ), and then A, C, and their union Y have measure zero in Ω. Asymptotic linking of an action and a submanifold Consider a volume-preserving action Φ : R k ×Ω → Ω tangent to the boundary on a compact convex domain Ω ⊂ R n with smooth boundary and let N ⊂ Ω be a closed singular ℓ-dimensional oriented submanifold of Ω, with k + ℓ = n − 1. As before, T k is the set of k-rectangles T = [0, T 1 ] × · · · × [0, T k ] ⊂ R k for (T 1 , . . . , T k ) ∈ R k + , p ∈ Ω \ N is fixed, and X = X 1 ∧ · · · ∧ X k generates Φ. According to Lemma 3, for every T ∈ T k the sets σ(p, T ) defined in (1) are disjoint from N for almost all p ∈ Ω. The invariant I(Φ, N ) = N α with dα = i X ω was defined in (2). Lemma 5. This invariant satisfies I(Φ, N ) = N jBS(X) and does not depend on the choice of α. To show that h T is integrable and that the limit converges to zero, we parametrize σ(p, T ) by setting Proof. By (19) djBS(X) = i X ω = dα so d(α − jBS(X)) = 0. Since Ω is convex, α − jBS(X) isT i = [0, T 1 ] × · · · [0, T i ] · · · × [0, T k ] and ∂ iδ T = [0, T 1 ] × · · · × {t iδ } × · · · × [0, T k ](27) where t i0 = 0 and t i1 = T i are the extremities of the interval [0, T i ]. Then ∂T = ∪ k i=1 ∪ 1 δ=0 ∂ iδ T and Φ(∂T, p) = ∪ k i=1 ∪ 1 δ=0 Φ(∂ iδ T, p), so σ(p, T ) = ∪ k i=1 ∪ 1 δ=0 σ iδ (p, T ), where σ iδ (p, T ) is the cone with base Φ(∂ iδ T, p) and apex p. It suffices to prove the proposition using each σ iδ (p, T ) in place of their union σ(p, T ). Parametrize σ iδ (p, T ) by σ p (r, t i ) = (1 − r)Φ(t iδ , p) + r p, (r, t i ) ∈ [0, 1] × ∂ iδ T,(28) where t i = (t 1 , . . . , t i−1 , t i+1 , . . . , t k ) and t iδ = (t 1 , . . . , t i−1 , t iδ , t i+1 , . . . , t k ). Then ∂σp ∂r (r, t i ) = p − Φ(t iδ , p) and ∂σ p ∂t j (r, t i ) = (1 − r)X i (Φ(t iδ , p)), where X i = X 1 ∧ · · · X i · · · ∧ X k . Hence, setting \|T \| = T 1 · · · T k and h iδ T (p) = 1 \|T \| x∈σ iδ (p,T ) y∈N L(x, y), where L(x, y) is the linking form (26), we have \|h iδ T (p)\| ≤ 1 \|T \| x∈σ iδ (p,T ) y∈N \|L(x, y)\| = 1 \|T \| r∈[0,1] t i ∈T i y∈N L(r, t i , y, p)drdt i dη(y) (29) where L(r, t i , y, p) = [(y − σ p (r, t i )) × ∂σp ∂r∂t i (r, t i )] · U (y) \|\|y − σ p (r, t i )\|\| n ,(30)∂σ p ∂r∂t i (r, t i ) = ∂σ p ∂r ∧ ∂σ p ∂t 1 ∧ · · · ∂σ p ∂t i · · · ∧ ∂σ p ∂t k , U (y) is the unit ℓ-vector in ∧ ℓ (T y (N )), and dη(y) is the volume measure on N . This lemma will be proven at the end of this section. We use it now to show that h iδ T ∈ L 1 (Ω). In fact, by (29), p∈Ω \|h iδ T (p)\| ≤ 1 \|T \| p∈Ω r∈[0,1] t i ∈T i y∈N L(r, t i , y, p)drdt i dη(y) dλ(p) = 1 \|T \| t i ∈T i y∈N r∈[0,1] p∈Ω L(r, t i , y, p)drdλ(p) dt i dη(y) ≤ W i \|T \| t i ∈T i dt i y∈N dη(y) = W i Vol(N )T 1 · · · T i · · · T k \|T \| = W i Vol(N ) T i . so h iδ T ∈ L 1 (Ω) and lim T →∞ p∈Ω \|h T (p)\|dλ(p) = 0. Proof of Lemma 6. Using σ p (r, t i ) and its derivatives, ∂σ p ∂r∂t i (r, t i ) = ∂σ p ∂r ∧ ∂σ p ∂t 1 ∧ · · · ∂σ p ∂t i · · · ∧ ∂σ p ∂t k = (1 − r) k−1 [Φ(t iδ , p) − p] ∧ X 1 (Φ(t iδ , p)) ∧ · · · X i (Φ(t iδ , p)) · · · ∧ X k (Φ(t iδ , p)) = (1 − r) k−1 [Φ(t iδ , p) − p] ∧ X i (Φ(t iδ , p)) where X i = X 1 ∧ · · · X i · · · ∧ X k . Note that (1 − r) k−1 ≤ 1, \|Φ(t iδ , p) − p\| is less than or equal to the diameter D of Ω, there is a constant B such that \|\|X i (p)\|\| ≤ B for all p ∈ Ω, and \|\|U (y)\|\| = 1, so by (30) we have L(r, t i , y, p) ≤ \|\| ∂σp ∂r∂t i (r, t i )\|\| \|\|U (y)\|\| \|\|σ p (r, t i ) − y\|\| n−1 ≤ (1 − r) k−1 \|\|Φ(t iδ , p) − p\|\| \|\|X i (Φ(t iδ , p))\|\| \|\|U (y)\|\|) \|\|σ p (r, t i ) − y\|\| n−1 ≤ DB \|\|(1 − r)Φ(t iδ , p) + r p − y\|\| n−1 .\|\|(1 − r)Φ(t iδ , p) + r p − y\|\| ≥ d/2, where d is the distance from p to N . Then r∈[1−ǫ,1] p∈Ω DB \|\|(1 − r)Φ(t iδ , p) + r p − y\|\| n−1 drdλ(p) ≤ r∈[1−ǫ,1] p∈Ω DB (d/2) n−1 drdλ(p) = 2 n−1 DBǫ d n−1 . On the other hand, for r ∈ [0, 1 − ǫ], Φ(t iδ , ·) = Φ t iδ is a volume-preserving diffeomorphism of Ω, so we can make the substitution p ′ = Φ(t iδ , p) and get r∈[0,1−ǫ] p∈Ω DB \|\|(1 − r)Φ(t iδ , p) + r p − y\|\| n−1 drdλ(p) = r∈[0,1−ǫ] p ′ ∈Ω DB \|\|(1 − r)p ′ + r p − y\|\| n−1 drdλ(p ′ ). Now for each r we let p r = (1 − r)p ′ + r p. Then dλ(p r ) = (1 − r) n dλ(p ′ ) and Ω is replaced by by Ω r ⊂ Ω (a contraction moving towards p), so r∈[0,1−ǫ] p ′ ∈Ω DB \|\|(1 − r)p ′ + r p − y\|\| n−1 drdλ(p ′ ) = r∈[0,1−ǫ] 1 (1 − r) n pr∈Ωr DB \|\|p r − y\|\| n−1 drdλ(p r ) ≤ r∈[0,1−ǫ] 1 ǫ n pr∈Ω DB \|\|p r − y\|\| n−1 drdλ(p r ) ≤ r∈[0,1−ǫ] DBΓ ǫ n dr = DBΓ(1 − ǫ) ǫ n since 1−r ≥ ǫ and Ω r ⊂ Ω, by the following lemma, which holds since the singularity at q has order n − 1, and that is less than the dimension n. Lemma 7. There is a constant Γ such that the function g(q) = Ω\{q} 1 \|\|p − q\|\| n−1 dλ(p) satisfies \|g(q)\| ≤ Γ for all q ∈ Ω. Combining the last two results with (31), we get r∈[0,1] p∈Ω L(r, t i , y, p)drdλ(p) ≤ 2 n−1 DBǫ d n−1 + DBΓ(1 − ǫ) ǫ n =: W i . Since θ Φ (p, T ) and N are disjoint for almost all (p, T ) ∈ Ω × T k , the linking number lk(θ Φ (p, T ), N ) is defined on an open dense set. Then we have 1 T 1 · · · T k σ(p,T ) N L = 0. (33) Let g(p) = (−1) k a n y∈N (y − p) × X(p) · U (y) \|\|y − p\|\| n dη(y) where U is the positive unit ℓ-form on N . The function g is smooth on Ω \ N . Then \|g(p)\| ≤ 1 a n y∈N \|\|y − p\|\|\|\|X(p)\|\|\|\|U (y)\|\| \|\|y − p\|\| n dη(y). Let K be an upper bound for \|\|X(p)\|\|, p ∈ Ω. Since \|\|U (y)\|\| = 1, \|g(p)\| ≤ K a n y∈N 1 \|\|y − p\|\| n−1 dη(y). By Fubini's Theorem p∈Ω \|g(p)\|dλ(p) ≤ K a n y∈N p∈Ω 1 \|\|y − p\|\| n−1 dλ(p)dη(y) ≤ KΓ a n N dη = KΓV ol(N ) a n so g ∈ L 1 (Ω). On the other hand, note that x∈Φ(T,p) y∈N L(x, y) = = T1 0 · · · T k 0 y∈N (y − Φ(t, p)) × X(Φ(t, p)) · U (y) \|\|y − Φ(t, p)\|\| n dη(y)dt = T1 0 · · · T k 0 g(Φ(t, p))dt. Thus, by (32), (33), and the Ergodic Theorem, since g ∈ L 1 (Ω), the limit lim T →∞ 1 T 1 · · · T k lk(θ Φ (p, T ), N ) = lim T →∞ 1 T 1 · · · T k T1 0 · · · T k Asymptotic linking of two actions In this section, we assume that M = R n , so Ω is a compact convex region with smooth boundary in R n and consider volume-preserving actions Φ and Ψ of R k and R ℓ that are tangent to the boundary on Ω, k + ℓ = n − 1, as in §2. Recall that D(Φ, Ψ) ⊂ Ω × T k × Ω × T ℓ is the dense open set of points (p, T, q, S) for which θ Φ (p, T ) and θ Ψ (q, S)) are disjoint. Proposition 8. The following conditions are satisfied: (1) The functions (p, T ) → θ Φ (p, T ) and (q, S) → θ Ψ (q, S) are continuous functions on Ω × R k . Furthermore, the function θΦ(p,T ) θΨ(q,S) L(p, q) is continuous on D(Φ, Ψ) and therefore measurable. (2) The limits lim T,S→∞ 1 λ k (T )λ ℓ (S) p∈Ω q∈Ω Ap Bq L(p, q) dpdq = 0,(34) where we set (A p , B q ) equal to (Φ(T, p), σ ′ (q, S)), (σ(p, T ), Ψ(S, q)), and (σ(p, T ), σ ′ (q, S)), exist, and all three limits are zero. Proof of (2). As before, T, S → ∞ means that min(T 1 , . . . , T k , S 1 , . . . , S ℓ ) → ∞. When the compact sets A p and B q are disjoint, it is clear that the integral Ap Bq L(p, q) converges, but it is not evident that the integral in (34) converges, although the integrand is measurable. First, consider A p = Φ(T, p) and B q = σ ′ (q, S). We decompose σ ′ (q, S) = ∪σ ′ jε (q, S) analogous to the decomposition (27) of σ(p, T ) with the parametrization (28). Let s j0 = 0 and s j1 = S j be the extremities of the interval [0, S j ]. Note that Ψ(∂S, q) is the union of 2ℓ sets, Ψ(∂S, q) = ∪ ℓ j=1 ∪ 1 ε=0 Ψ(∂ jε S, q), ε ∈ {0, 1}, where ∂ jε S = [0, S 1 ], × · · · × {s jε } × · · · × [0, S ℓ ], so the singular submanifold σ ′ (q, S) = ∪ ℓ j=1 ∪ 1 ε=0 σ ′ jε (q, S)(35) where σ ′ jε (q, S) is the cone joining Ψ(∂ jε S, q) to the vertex q. We shall prove the Proposition for B = σ ′ jε (q, S) instead of σ ′ (q, S); then the same proof works for the other components of σ ′ (q, S). Let S j = [0, S 1 ], × . . . [0, S j ] · · · × [0, S ℓ ]. To each point s j = (s 1 , . . . , s j−1 , s j+1 , . . . , s ℓ ) ∈ S j we naturally associate the point s jε = (s 1 , . . . , s j−1 , s jε , s j+1 , . . . , s ℓ ) ∈ ∂ jε S. We use the parametrizations x p (t) = Φ(t, p), t ∈ T, of Φ(T, p) and y q (u, s j ) = (1 − u)Ψ(s jε , q) + u q, (u, s j ) ∈ [0, 1] × S j , of σ ′ jε (q, S). Note that ∂xp ∂ti = X i , ∂yq ∂s j = (1 − u)Y j and ∂yq ∂u = q − Ψ(s jε , q). Since Ω is compact, there is a constant C that is a common upper bound for \|\|X(p)\|\| = \|\|X 1 ∧· · ·∧X k (p)\|\|, \|Y j (q)\|\| = \|\|Y 1 ∧. . . Y j · · ·∧Y ℓ (q)\|\| and for \| q −Ψ(s jε , q)\|, p, q ∈ Ω. Recall that for multivectors \|\|(u × v) · w\|\| ≤ \|\|u\|\|\|\|v\|\|\|\|w\|\|. Then Ap B L(p, q) = Φ(T,p) σ ′ jε (q,S) L ≤ Φ(T,p) σ ′ jε (q,S) \|L\|, but using (26), dη(x) = dλ(t), and dη ′ (y) = ( q − Ψ(s jε , q))dλ(s j )du, \|L\| ≤ 1 a n \|\|x p (t) − y q (u, s j )\|\|\|\|X(Φ(t, p)\|\|\|\|Y j (y q (u, s j ))\|\| \|\|x p (t) − y q (u, s j )\|\| n dη ′ (y)dη(x) ≤ C ′ 1 0 1 \|\|x p (t) − y q (u, s j )\|\| n−1 dλ(s j )dudλ(t), where C ′ = C 3 /a n , so Ap B L ≤ C ′ T 1 0 S j 1 \|\|x p (t) − y q (u, s j )\|\| n−1 dλ(s j )dudλ(t). Integrating \| Ap B L\| on Ω × Ω we have p∈Ω q∈Ω Ap B L dλ(p)dλ(q) ≤ C ′ p∈Ω q∈Ω T 1 0 Sj 1 \|\|x p (t) − y q (u, s j )\|\| n−1 dλ(s j )dudλ(t)dλ(p)dλ(q) ≤ C ′ T 1 0 Sj q∈Ω p∈Ω 1 \|\|Φ(t, p) − y q (h, s j )\|\| n−1 dλ(p) dλ(q)dλ(s j )dudλ(t) by Fubini's Theorem, since we shall see that the last integral converges. Since the action Φ t preserves the volume, if we set Φ(t, p) = p ′ , the measure dλ(p ′ ) coincides with dλ(p), and the last integral becomes C ′ T 1 0 Sj q∈Ω p∈Ω 1 \|\|p ′ − y q (u, s j )\|\| n−1 dλ(p ′ ) dλ(q)dλ(s j )dudλ(t). (36) Lemma 7 shows that this integral coverges. Then, working backwards, it follows that all the previous integrals in this proof also converge. The integral (36) is less than or equal to C ′ T 1 0 S j q∈Ω Γdλ(q)dλ(s j )dudλ(t) ≤ C ′ ΓV ol(Ω)V ol(T )V ol([0, 1])V ol(S j ) = C ′ ΓV ol(Ω)T 1 . . . T k S 1 . . . S j . . . S ℓ . In the limit we have 0 ≤ lim T,S→∞ 1 T 1 . . . T k S 1 . . . S ℓ p∈Ω q∈Ω Ap B L \|dλ(p)dλ(q) ≤ lim T1,...,T k ,S1,...,S ℓ →∞ C ′ ΓV ol(Ω) S j = 0, so (34) holds for A p = Φ(T, p) and B q = B = σ ′ jε (q, S). Thus the limit vanishes for Φ(T, p) and σ ′ (q, S) and similarly for the case A p = σ(p, T ) and B q = Ψ(S, q). For the case when A p = σ(p, T ) and B q = σ ′ (q, S), we use the decompositions (27) of σ(p, T ) and (35) of σ ′ (q, S) and the parametrizations x p = σ p (r, t i ) = (1 − r)Φ(t iδ , p) + r p, (r, t i ) ∈ [0, 1] × T, and y q = σ ′ q (u, s j ) = (1 − u)Φ(t jε , q) + u q, (u, s j ) ∈ [0, 1] × T, of σ iδ (p, T ) and σ ′ jε (q, S) , with t i , t iδ , s j and s jε as before. Then we have \|L(x p , y q )\| ≤ 1 a n \|\|x p − y q \|\|\|\|X(x p )\|\|\|\|Y j (y q )\|\| \|\|x p − y q \|\| n dη ′ (y q )dη(x p ) ≤ C \|\|x p − y q \|\| n−1 dλ(s j )dudλ(t), where Ca n is an upper bound for \|X(p)\| \|Y (q)\|. It suffices to show that the limit of L = 1 \|S\| \|T \| p∈Ω q∈Ω xp∈σ iδ (p,T ) yq∈σ ′ jε (q,S) C \|\|x p − y q \|\| n−1 dλ(s j )dudλ(t)dpdq converges to zero as S, T → ∞. We shall do this in three cases. Case 1. r, u ∈ [1 − ǫ, 1], where ǫ > 0 is such that \|\|y q − x p \|\| ≥ d/2 when u, r ∈ [1 − ǫ, 1] and d is the distance from p to q. Such an ǫ exists since x p → p and y q → q as r, u → 1. In this case C \|\|x p − y q \|\| n−1 ≤ 2 d n−1 , the volume D of Ω is finite, \|T \| −1 Vol(σ iδ (p, T )) ≤ 1/T i , and \|S\| −1 Vol(σ ′ jε (q, S)) ≤ 1/S j so the limit of L is zero. Case 2. r ∈ [0, 1 − ǫ]. L ′ = p∈Ω q∈Ω yq∈σ ′ jε (q,S) xp∈σ iδ (p,T ) 1 \|\|x p − y q \|\| n−1 dy q drdt i dpdq = t i ∈T i 1−ǫ r=0 1 \|\|(1 − r)Φ(t iδ , p) + r p − y q \|\| n−1 dy q dudt i dpdq Then Φ(t iδ , ·) = Φ t iδ is a volume-preserving diffeomorphism of Ω, so we can make the substitution p ′ = Φ(t iδ , p) and get L ′ = r∈[0,1−ǫ] p∈Ω 1 \|\|(1 − r)Φ(t iδ , p) + r p − y q \|\| n−1 dy q dt i dλ(p)dqdrdt i = r∈[0,1−ǫ] p ′ ∈Ω 1 \|\|(1 − r)p ′ + r p − y q \|\| n−1 dy q dt i dλ(p ′ )dqdrdt i . For each r we let p r = (1 − r)p ′ + r p. Then dλ(p r ) = (1 − r) n dλ(p ′ ) and Ω is replaced by by Ω r ⊂ Ω (a contraction moving towards p), so L ′ = r∈[0,1−ǫ] 1 (1 − r) n pr ∈Ωr 1 \|\|p r − y\|\| n−1 drdλ(p r ) ≤ r∈[0,1−ǫ] 1 ǫ n pr∈Ω 1 \|\|p r − y\|\| n−1 drdλ(p r ) ≤ r∈[0,1−ǫ] Γ ǫ n dr = Γ(1 − ǫ) ǫ n by Lemma 7, since 1 − r ≥ ǫ and Ω r ⊂ Ω. Now the volume of Ω is finite, Vol(σ iδ (p, T )) ≤ \|T \|/T i , and Vol(σ ′ jε (q, S)) ≤ \|S\|/S j , so it follows that lim S,T →∞ L = 0. Case 3. u ∈ [0, 1 − ǫ] . This case is exactly parallel to Case 2, with p and q interchanged, so it is omitted. There is an overlap in the three cases, but all values of (r, u) ∈ [0, 1] × [0, 1] are covered. Then for almost all (p, T ) ∈ Ω × T k and (q, S) ∈ Ω × T ℓ , θ Φ (p, T ) and θ ′ Ψ (q, S) are disjoint and the linking number lk(θ Φ (p, T ), θ ′ Ψ (q, S)) is defined. Proposition 9. The limit lk(p, q) = lim T1,...,T k ,S1,...,S ℓ →∞ 1 T 1 . . . T k S 1 . . . S ℓ lk(θ Φ (p, T ), θ ′ Ψ (q, S)) (37) exists as an integrable L 1 -function on Ω × Ω and does not depend on the choice of the pointsp andq. Proof. Calculating the linking number using the linking form (26), it suffices to integrate over the sets Φ(p, T ) and Ψ(q, S), since by Proposition 8 the limits of the integrals over the other three sets vanish, i.e., lim T,S→∞ 1 λ k (T )λ ℓ (S) lk(θ Φ (p, T ), θ ′ Ψ (q, S)) = lim T,S→∞ 1 λ k (T )λ ℓ (S) Φ(T,p) Ψ(S,q)) L.(38) As before, X = X 1 ∧ · · · ∧ X k and Y = Y 1 ∧ · · · ∧ Y ℓ are the exterior products of the vector fields that generate the actions of Φ and Ψ, respectively. Define the function f : Ω × Ω → R by (39) f (p, q) := (−1) k [(q − p) × X(p)] · Y (q) a n \|\|q − p\|\| n . For every (p, q) we have \|f (p, q)\| ≤ \|\|X(p)\|\| \|\|Y (q)\|\| a n \|\|q − p\|\| n−1 ≤ K a n \|\|q − p\|\| n−1 (40) where K is an upper bound for \|\|X(p)\|\| \|\|Y (q)\|\|, p, q ∈ Ω. Now, by Lemma 8, g(p, q) = 1/\|\|q − p\|\| n−1 is an integrable function in Ω × Ω, since Then by (40) we get (p,q)∈Ω×Ω \|f (p, q)\|dλ(p)dλ(q) ≤ ΓKVol(Ω) a n so f ∈ L 1 (Ω × Ω). To calculate Φ(T,p) Ψ(S,q)) L we use the natural parametrizationsp = x p (t) = Φ t (p) = Φ(t, p) andq = y q (s) = Ψ s (q) = Ψ(s, q) induced by the actions Φ and Ψ on Φ(T, p) and Ψ(S, q). Then ∂xp ∂ti (t) = X i (Φ t (p)), i = 1, . . . , k, and ∂yq ∂sj (s) = Y j (Ψ s (q)), j = 1, . . . , ℓ. Let ∂xp ∂t = ∂xp ∂t1 ∧ · · · ∧ ∂xp ∂t k (t) and ∂yq ∂s (s) = ∂yq ∂s1 ∧ · · · ∧ ∂yq ∂s ℓ (s), so ∂x p ∂t (t) = X 1 ∧ · · · ∧ X k (Φ t (p)) = X(Φ t (p)) and ∂y q ∂s (s) = Y 1 ∧ · · · ∧ Y ℓ (Ψ s (q) = Y (Ψ s (q)). Let U (p) and U ′ (q) denote the unit k-and ℓ-vectors atp ∈ Φ(T, p) andq ∈ Ψ(S, q), respectively. Then by (26) (−1) k a n Φ(T,p) Ψ(S,q)) L = p∈Φ(T,p) q∈Ψ(S,q) [(q −p) × U (p)] · U ′ (q) \|\|q −p\|\| n dη(p)dη(q) = t∈T s∈S [(y q (s) − x p (t)) × U (x p (t))] · U ′ (y q (s)) \|\|y q (s) − x p (t)\|\| n \|\| ∂y q ∂s (s)\|\|ds \|\| ∂x p ∂t (t)\|\|dt = T S [(Ψ s (q) − Φ t (p)) × (\|\|X(Φ t (p))\|\|U (Φ t (p))] · (\|\|Y (Ψ s (q))\|\|U ′ (Ψ s (q)) \|\|Ψ s (q) − Φ t (p)\|\| n dsdt = t∈T s∈S [(Ψ s (q) − Φ t (p)) × X(Φ t (p))] · Y (Ψ s (q)) \|\|Ψ s (q) − Φ t (p)\|\| n dsdt = t∈T s∈S f (Φ t (p), Ψ s (q))dsdt = t∈T s∈S f (Θ (t,s) (p, q))ds 1 . . . ds ℓ dt 1 . . . dt k = T1 0 · · · T k 1 S1 0 · · · S1 0 · · · S ℓ 0 f (Θ t1,...,t k ,s1,...,s ℓ (p, q))ds 1 . . . ds ℓ dt 1 . . . dt k , where Θ = Φ × Ψ is the product action of R k+ℓ in Ω × Ω defined by setting Θ (t,s) (p, q) = (Φ t (p), Ψ s (q)). Then the Ergodic Theorem, Theorem 5, applied to the action Θ, shows that the limit lim T1,...T k ,S1,...S ℓ →∞ 1 T 1 . . . T k S 1 . . . S ℓ Φ(T,p) Ψ(S,q)) L converges and defines a function lk ∈ L 1 (Ω × Ω), lk(p, q) = lim T1,...T k ,S1,...S ℓ →∞ 1 S1 0 · · · S1 0 · · · S ℓ 0 f (Θ (t,s) (p, q))dt 1 . . . dt k ds 1 . . . ds ℓ so we get (p,q)∈Ω×Ω lk(p, q)dp × dq = (p,q)∈Ω×Ω f (p, q)dp × dq. Then (38) shows that this function satisfies (37). Clearly it does not depend on the choices of p and q. As a consequence of this Proposition, we can define the asymptotic linking invariant to be lk(Φ, Ψ) = p∈Ω q∈Ω lk(p, q)dη(p)dη(q), and then Theorem 2 states that lk(Φ, Ψ) = I(Φ, Ψ). Proof of Theorem 2. With the volume forms ω, dη(p), and dη(q) on Ω, we have lk(Φ, Ψ) = Ω×Ω lk(p, q)dη(p)dη(q) = Ω×Ω f (p, q)dη(p)dη(q) = (−1) k a n p∈Ω q∈Ω q − p \|\|q − p\|\| n × X(p) · Y (q)dη(p)dη(q) by (39) = q∈Ω (−1) k a n p∈Ω q − p \|\|q − p\|\| n × X(p)dη(p) · Y (q)dη(q) by Fubini's Theorem, and then, by the Biot-Savart formula (17), the definition of j, (14), and Corollary 3, this is equal to Ω (BS(X) · Y )ω = Ω jBS(X)(Y )ω = Ω jBS(X) ∧ i Y ω = Ω jBS(X) ∧ dβ = I(Φ, Ψ). A lower bound for the energy of an action We remark that in the case when Φ = Ψ and n = 2k + 1, the invariant lk(Φ, Φ) = I(Φ, Φ) is a lower bound for the energy of the generating k-vector X. Definition 2. Let Φ be a conservative k-action on Ω and let X be the k-vector field that generates Φ. The energy of the k-action Φ is defined to be the value of the integral E(Φ) = \|\|X\|\| 2 = p∈Ω X(p) · X(p)dλ(p) = p∈Ω \|\|X(p)\|\| 2 dλ(p). Note that we can decrease the energy of Φ by conjugating Φ by volume-preserving diffeomorphisms. Can we make it arbitrarily close to zero? The following result gives a negative answer to this question. Theorem 8. There exists a constant C > 0 depending only on Ω such that C −1 \|lk(Φ, Φ)\| ≤ E(Φ). Proof. By Corollary 2, (7), and the definition of j, lk(Φ, Φ) = Ω jBS(X) ∧ dα = Ω jBS(X) ∧ i X dλ = Ω jBS(X)(X)dα = Ω BS(X) · Xdλ. By the Cauchy-Schwarz inequality \|lk(Φ, Φ)\| = \| < BS(X), X > \| ≤ \|\|BS(X)\|\| \|\|X\|\|. (41) Furthermore BS(X)(p) = q∈Ω (p − q) × X(q) \|\|p − q\|\| 2k+1 dλ(q) so \|\|BS(X)(p)\|\| ≤ q∈Ω \|\|(p − q) × X(q)\|\| \|\|p − q\|\| 2k+1 dλ(q) ≤ q∈Ω (\|\|X(q)\|\| \|\|p − q\|\| 2k dλ(q)) = q∈Ω \|\|X(q)\|\| \|\|p − q\|\| k 1 \|\|p − q\|\| k dλ(q)) ≤ q∈Ω \|\|X(q)\|\| 2 \|\|p − q\|\| 2k dλ(q) 1/2 q∈Ω 1 \|\|p − q\|\| 2k λ(q) We can decrease the energy of Φ by volume-preserving diffeomorphisms, but these diffeomorphisms do not change the value of the asymptotic linking number lk(Φ, Φ), so by (43) Γ −1 lk(Φ, Φ) is the desired lower bound for the energy of Φ. Examples Example 4. For every pair of integers k, ℓ ≥ 1, k + ℓ + 1 = n, and every t ∈ R, there are conservative actions Φ of R k and Ψ of R ℓ on the unit closed ball D n ⊂ R n such that lk(Φ, Ψ) = I(Φ, Ψ) = t. The construction uses several lemmas. Proof. Given M and N , by a translation we may assume that their images lie in the positive half space x 1 > 0 ⊂ R n ⊂ R n+1 . Let P be the (n − 1)-plane in R n perpendicular to the x 1 -axis, and rotate M around P to get M × S 1 ⊂ R n+1 . Clearly M × S 1 is disjoint from N . If we let Σ ⊂ R n be a compact singular (k + 1)-manifold tranverse to N such that ∂Σ = M , then lk(M, N ) = Int(Σ, N ). By rotating Σ around P we obtain Σ × S 1 , whose boundary is M × S 1 . Then Int(Σ × S 1 , N ) = Int(Σ, N ), and therefore the linking number is the same. In case N is an affine ℓ-plane a similar argument works, taking P to be an affine plane parallel to N . Lemma 9. There exist disjoint embeddings of T k × D ℓ+1 and T ℓ × D k+1 in D n , where T k and T ℓ are tori of dimensions k and ℓ, such that lk(T k × 0, T ℓ × 0) = 1. Proof. Begin with disjoint smooth embeddings of two circles M and N in R 3 such that lk(M, N ) = 1. Applying Lemma 8 repeatedly, switching the roles of M and N , gives disjoint embeddings of T k and T ℓ in R n with intersection number 1. Since the normal bundles are trivial we can extend the embeddings to disjoint embeddings of T k × D ℓ+1 and T ℓ × k + 1. then a homothety will move these sets into D n . Lemma 10. Let T r act on T r × D s by the product action on the first factor and identity on the second factor. For any smooth volume form ω on T r × D s there is a smooth isotopy h t of T r × D s taking each factor T r × {y} to itself such that h 0 = id and h * 1 (ω) is T r -invariant. Proof. Here we need a slightly modified form of Moser's Theorem [12] acting on each orbit. We use the standard coordinates (x, y) = (x 1 , . . . , x k , y 1 , . . . , y s ) and the standard Euclidean volume form ω * = dx ∧ dy on T r × D s to simplify the notation. Let f 0 : T r × D s → R be the (unique) non-vanishing smooth function such that ω = f 0 ω * and define f 1 : D s → R by setting f 1 (y) = T k ×{y} f 0 (x, y)dx. Note that α * y = f 0 (y)dx is T k -invariant. Now the volume forms α y = f 1 (x, y)dx and α * y have the same integral T k α y dx = T k α * y dx, so there exists a smooth function f : T r × D s → R such that α = f α * and we can apply Moser's proof [12] on each factor T k × {y}. Following Moser, we may suppose that there is a positive ǫ such that \|f (y) − 1\| < ǫ for every y by expressing any positive function f as a sum of functions close to 1. We use the same cover of T k by open cubes U 0 , U 1 , . . . , U m and the same functions η i (k = 1, . . . , m), independent of y. Then it is straightorward to check that Moser's isotopies of each T k × {y} fit together to give a smooth isotopy of T r × D s transforming each α y into α * y . This isotopy also transforms ω into f 1 ω * , which is invariant under the action of T k . Construction of the Example. Take W = T k × D ℓ+1 ⊔ T ℓ × D k+1 embedded in D n by Lemma 9, where k + ℓ + 1 = n. The compact Lie groups T k and T ℓ act on W , T k acting on T k × D ℓ+1 by multiplication on the first factor and trivially on T ℓ × D k+1 , and analogously for the action of T ℓ . By Lemma 10, we may conjugate the action of T k on T k × D ℓ+1 by a diffeomorphism isotopic to the identity so that it preserves the Euclidean volume form, and similarly for the action of T ℓ . Lift the actions of T k and T ℓ to volume preserving actions φ : R k × (T k × D ℓ+1 ) → T k × D ℓ+1 and ψ : R ℓ × (T ℓ × D k+1 ) → T ℓ × D k+1 . Let W ǫ = T k × D ℓ+1 0 ∪ T ℓ × D k+1 0 be a smaller invariant neighborhood of T k ∪ T ℓ and let λ : W → [0, 1] be constant on the orbits with the values 1 on W ǫ and 0 on D n \W . Then let Φ(t, z) = φ(λ(z)t, z) for z in the ǫ-neighborhood of T k ×D ℓ+1 and identity elsewhere, while Ψ(t, z) = ψ(λ(z)t, z) on the ǫ-neighborhood of T ℓ × D k+1 and identity elsewhere. Thus Φ and Ψ are commuting conservative actions of R k and R ℓ on D n . The linking number of the orbits T k × {y} and T ℓ × {z} are lk(T k × y, T ℓ × z) = 1 for y ∈ D ℓ+1 0 and z ∈ D k+1 0 . Now it is easy to check that the linking number lk(Φ, Ψ) > 0 since for points p ∈ D ℓ+1 0 and q ∈ D k+1 0 and for T = [0, 2rπ] k and S = [0, 2sπ] ℓ , lk(θ Φ (p, T), θ Ψ (q, S) = r k s ℓ since for these rectangles T and S the cones σ(p, T ) and σ ′ (q, S) are empty. When we normalize by dividing by (2rπ) k · (2sπ) ℓ we get the constant (2π) −(k+ℓ) , which is therefore the value of the limit for orbits in W ǫ as r, s → ∞. Other points p, q contribute positively, so we get lk(Φ, Ψ) > 0. To get a negative value it suffices to change one of the orientations. Finally by multiplying t ∈ R k by s we multiply the asymptotic linking number by s k and thus we can obtain all real numbers as values of lk(Φ, Ψ). Example 5. Given a closed connected oriented submanifold N ℓ embedded in D n and a real number t, by a similar construction we can find a conservative action Φ of R k on D n , k = n − ℓ − 1, such that lk(Φ, N) = t. Here the construction is similar to the previous example. By applying Lemma 8 repeatedly we can obtain T k ⊂ R n \ P , where P is an affine ℓ-plane, such that the linking number is lk(T k , P) = 1. Now locally the smooth embedding of N in D n is diffeomorphic to the embedding of P in R n , so we can find a small torus T k ⊂ D k \ N such that lk(T k , N) = 1. The rest of the construction proceeds as in Example 4. Theorem 2 . 2Under the hypotheses of Theorem 1, the asymptotic linking number and the invariant I(Φ, Ψ) coincide, i.e., lk(Φ, Ψ) = I(Φ, Ψ). Theorem 4 . 4Under the hypotheses if Theorem 3, the asymptotic linking number and the invariant I(Φ, N ) coincide, i.e., lk(Φ, N ) = I(Φ, N ). ( 1 ) 1For (ordered) multi-indices I and J we have e I × e J = e K if I ∩ J = ∅, K = {1, . . . , n} \ (I ∪ J), and the ordered union I ∪ J ∪ K is a positive permutation of (1, . . . , n); but e I × e J = 0 if I ∩ J = ∅. (2) In addition, e I · e J = (−1) \|K\|(n−\|J\|) e K if I ⊂ J, K = J \ I, and the ordered union I ∪ K is a positive permutation of J; furthermore, e I · e J vanishes if I ⊂ J. Lemma 4 . 4Let ω be the positive unit volume form on M . Then i X ω = j( * X). Theorem 5 . 5(Ergodic Mean Theorem) There is a unique function f in L 1 (M ) to which the sequence {f n } n∈N converges almost everywhere, i.e., lim n→∞ M \|f n − f \| dµ = 0. Furthermore, f is independent of the choice of the sequence {T n } n∈N and satisfies Remark 1 . 1(See, e.g.,[3]) In dimension 3, when N and N ′ are curves parametrized by arclength by α : [0, t 0 ] → N and α ′ : [0, s 0 ] → N ′ , the formula (23) becomes the well-known Gauss linking number formula exact and there exists a form θ such that dθ = α − jBS(X). ThenI(Φ, N ) − N jBS(X) = N α − N jBS(X) = N dθ = ∂N θ = 0 since ∂N = ∅.Clearly N jBS(X) does not depend on α. Proposition 6. The following conditions are satisfied:(1) The sets σ(p, T ) vary measurably in the sense that for every T ∈ T k there is a function h T : Ω → R defined byh T (p) = 1 T 1 . . . T k x∈σ(p,T ) y∈N L(x, y),and h T ∈ L 1 (Ω), i.e., Ω \|h T (p)\|dη(p) < ∞.(2) The family of functions {h T } converges to zero in L 1 (Ω), i.e., lim T1,...,T k →∞ Ω \|h T (p)\|dη(p) = 0.Proof. To prove (1), let Y T := {p ∈ Ω \| σ(p, T ) ∩ N = ∅} and note that h T (p) = (T 1 · · · T k ) −1 σ(p,T ) N L(x, y) is defined and varies continuously on the dense open set Ω \ Y T , where the compact sets σ(p, T ) and N are disjoint. Then since Y T has measure zero, h T is measurable in Ω. Lemma 6 . 6There exists a constant W i > 0 such that for all t i ∈ T i and y ∈ N r∈[0,1] p∈Ω L(r, t i , y, p)drdλ(p) ≤ W i where dλ(p) is the euclidean measure on Ω. \|\|(1 − r)Φ(t iδ , p) + r p − y\|\| n−1 drdλ(p).(31)Now for p / ∈ N there exists ǫ > 0 such that for all y ∈ N and r ∈ [1 − ǫ, 1] Proposition 7 . 7The limitlk Φ,N (p) = lim T1,...,T k →∞ 1 T 1 . . . T k lk(θ Φ (p, T ), N )exists as an integrable L 1 -function on Ω and does not depend on the choice of the pointp ∈ Ω \ N .Proof.lk(θ Φ (p, T ), N ) = defines an L 1 function lk Φ,N (p) on Ω that satisfies p∈Ω lk Φ,N (p)dλ(p) = p∈Ω g(p)dλ(p) and does not depend on the choice of p. Then we define the asymptotic linking invariant to be lk(Φ, N ) = Ω lk Φ,N (p) dη and prove Theorem 4, which states that lk(Φ, N ) = I(Φ, N ). − p) × X(p) \|\|y − p\|\| n dλ(p) · U (y)dη(y) by Fubini's Theorem, so by (17) and the definition of the isomorphism j lk(Φ, N ) = N BS(X) · U dη = N jBS(X)(U )dη. Then since U is a unit ℓ-vector and dη is a unit ℓ-form, Lemma 5 shows that lk(Φ, N ) = N jBS(X) = I(Φ, N ). Proof. ( 1 ) 1Since the actions are continuous and line segments depend continuously on their extremities, it is clear that the functions (p, T ) → θ Φ (p, T ) and (q, S) → θ Ψ (q, S) are continuous, and so the function θΦ(p,T ) θΨ(q,S) L(p, q) is continuous and measurable on the dense open set D(Φ, Ψ). Γdλ(p) = ΓVol(Ω). 1 T1...T k S1...S ℓ Φ(T,p) Ψ(S,q)) inequality in (41) we get \|lk(Φ, Φ)\| ≤ \|\|BS(X)\|\| \|\|X\|\| ≤ Γ\|\|X\|\| 2 \| = ΓE(Φ). Lemma 8 . 8Given disjoint smooth embeddings of closed oriented manifolds M, N , of dimensions k and ℓ in R n , there exist disjoint smooth embeddings M ×S 1 , N ⊂ R n+1 such that lk(M × S 1 , N ) = lk(M, N ). The same holds if N is an affine ℓ-space disjoint from M . The asymptotic Hopf invariant and its applications. V , Sel. Math. Sov. 5V. Arnol'd, The asymptotic Hopf invariant and its applications, Sel. Math. Sov. 5 (1986), 327-354. V , B Khesin, Topological Methods in Hydrodynamics. SpringerV. Arnol'd and B. Khesin, Topological Methods in Hydrodynamics (Springer 1998). The Biot-Savart operator for application to knot theory, fluid dynamics and plasma physics. J Cantarella, D De Turck, H Gluck, Jour. of Math. Physics. J. Cantarella, D. De Turck, and H. Gluck, The Biot-Savart operator for application to knot theory, fluid dynamics and plasma physics, Jour. of Math. Physics, 42 (2001), 876-904. Average linking numbers. G Contreras, R Iturriaga, Erg. Th. Dyn. Syst. 19Contreras, G. and Iturriaga, R., Average linking numbers, Erg. Th. Dyn. Syst. 19 (1999), 1425-1435. H García-Compeán, R Santos-Silva, arXiv:0908.3218Link invariants for flows in higher dimensions. hep-thGarcía-Compeán, H. and Santos-Silva, R. Link invariants for flows in higher dimensions, arXiv:0908.3218 [hep-th] Topological fluid dynamics. B Khesin, Notices Amer. Math. Soc. 52Khesin, B., Topological fluid dynamics, Notices Amer. Math. Soc. 52 (2005), 9-19. Linking number of measured foliations. D Kotschick, T Vogel, Erg. Th. Dyn Sys. 23Kotschick, D. and Vogel, T., Linking number of measured foliations, Erg. Th. Dyn Sys. 23 (2003), 541-558. Pointwise theorems for amenable groups. E Lindenstrauss, Inventiones Math. 146Lindenstrauss, E., Pointwise theorems for amenable groups, Inventiones Math. 146 (2001), 259-295. Indices de enlaçamento assintótico para ações de R k em variedades Riemannianas compactas, doctoral thesis. J L Lizarbe Chira, Pontifícia Universidade Católica do Rio de JaneiroLizarbe Chira, J.L., Indices de enlaçamento assintótico para ações de R k em variedades Riemannianas compactas, doctoral thesis, Pontifícia Universidade Católica do Rio de Janeiro, 2006, available on the internet at: http://www.mat.puc-rio.br/∼paul/tesechira.pdf. Asymptotic cycles. S Schwartzman, Annals of Math. 66Schwartzman, S., Asymptotic cycles, Annals of Math. 66 (1957), 279-284. Smooth actions of R n. S Schwartzman, Proc. Amer. Math. Soc. 134Schwartzman, S., Smooth actions of R n , Proc. Amer. Math. Soc. 134 (2005), 379-384. On the volume elements on a manifold. J Moser, Trans. Amer. Math. Soc. 120Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. Ergodic theorems for general dynamical systems. A A Tempelman, Sov. Math. Dokl. 8Tempelman, A.A., Ergodic theorems for general dynamical systems, Sov. Math. Dokl. 8 (1967), 1213-1216. Ergodic theorems for group actions. A A Tempelman, SpringerTempelman, A.A., Ergodic theorems for group actions (Springer 1992). On the asymptotic linking number. T Vogel, Proc. Amer. Math. Soc. 131Vogel, T., On the asymptotic linking number, Proc. Amer. Math. Soc. 131 (2003), 2289-2298.	[]
[ "Automorphisms of Chevalley groups over commutative rings", "Automorphisms of Chevalley groups over commutative rings" ]	[ "E I Bunina \nBar Ilan University\n\n" ]	[ "Bar Ilan University\n" ]	[]	In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i. e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.	null	[ "https://export.arxiv.org/pdf/2304.13447v2.pdf" ]	258,331,798	2304.13447	f34b4a06ea9f1daea5434680595eb3dad7f702af	Automorphisms of Chevalley groups over commutative rings 3 Jun 2023 E I Bunina Bar Ilan University Automorphisms of Chevalley groups over commutative rings 3 Jun 2023 In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i. e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description. Introduction Automorphisms and isomorphisms of classical linear groups Automorphisms and isomorphisms of linear groups are studied by mathematicians from the beginning of XX century. First papers on automorphisms and isomorphisms of linear groups appeared already in the beginning of the 20th century. In particular, in the paper by Schreier and van der Warden [74] they described all automorphisms of the group PSL n (n 3) over an arbitrary field. Later on, Hua [49] generalized this method and applied it to the description of automorphisms of symplectic groups over a field of characteristic = 2. Diedonne [38] (1951) and Rickart [72] (1950) introduced the involution method, and described automorphisms of the group GL n (n 3) over a skew field, and then also of unitary and symplectic groups over skew fields of characteristic = 2 [73]. The first step towards the description of automorphisms of classical groups over rings was made by Hua and Reiner [48]. They dealt with the case GL n (Z). This result was extended to non-commutative principal ideal domains by Landin and Reiner in [57] and by Yan Shi-jian in [77]. The methods of the papers mentioned above were based mostly on studying involutions in the corresponding linear groups. O'Meara in 1976 invented very different (geometrical) method, which did not use involutions. By its aid, O"Meara described automorphisms of the group GL n (n 3) over domains [64] and automorphisms of symplectic groups of a special form over fields (so-called groups rich in transvections) [65]. Independently, Yan Shi jian in [77] described automorphisms of the group E n (R), n 3, where R is a domain of characteristic = 2 using the involution method. In the paper [62] Pomfret and MacDonald studied automorphisms of the groups GL n , n 3, over a commutative local ring with 1/2. Further on, Waterhouse in [96] obtained a description of automorphisms of the group GL n , n 3, over arbitrary commutative rings with 1/2. In 1982 Petechuk [66] described automorphisms of the groups GL n , SL n (n 4) over arbitrary commutative rings. If n = 3, then automorphisms of linear groups are not always standard [68]. They are standard either if in a ring 2 is invertible, or if a ring is a domain, or it is a semisimple ring. McQueen and McDonald in [63] obtained the description of automorphisms of the groups Sp n , n 6 over commutative local rings with 1/2. Continuing research in this direction, in 1980 Petechuk in [69] studied automorphisms of symplectic groups over arbitrary commutative local rings. In 1982 he extended description of automorphisms to the case Sp n (R), n 6, over arbitrary commutative ring R, using the localization method, see [70]. Isomorphisms of the groups GL n (R) and GL m (S) over arbitrary associative rings with 1/2 for n, m 3 were described in 1981 by Golubchik and Mikhalev [43] and independently by Zelmanov [101]. In 1997 Golubchik described isomorphisms between these groups for n, m 4, over arbitrary associative rings with 1 [44]. In 1983 Golubchik and Mikhalev in [42] studied isomorphisms of unitary linear groups over arbitrary associative rings with 1/2, with some conditions for the dimension of the group and the rank of the form. For the case when n = 2k and the hyperbolic rank of the form Q is maximal, the automorphism of U n (R, Q), k 3, were independently classified in 1985 by Zelmanov, see [101]. Automorphisms and isomorphisms of Chevalley groups In 50-th years of the previous century Chevalley, Steinberg and others introduced the concept of Chevalley groups over commutative rings. The foundations of the theory of Chevalley groups have been laid in the papers of Chevalley, Tits, Borel, Weil, Grothendieck, Demazure, Stenberg, etc. In 1956-1958 Chevalley obtained a classification of semisimple algebraic groups over algebraically closed fields. Later on, Chevalley showed that all semisimple groups over an algebraically closed field are actually defined under Z, or, in other words, are obtained as a result of expanding to an arbitrary ring of some group scheme defined over Z. These group schemes are called Chevalley-Demazure schemes. The groups of points of Chevalley-Demazure schemes over commutative rings are called Chevalley groups. Chevalley groups include classical linear groups (special linear SL , special orthogonal SO , symplectic Sp , spinor Spin , and also projective groups connected with them) over commutative rings. Finite simple groups of Lie type are the central quotients of Chevalley groups. Isomorphisms and automorphisms of Chevalley groups over different classes of rings were were intensively studied. The description of isomorphisms of Chevalley groups over fields was obtained by Steinberg [82] for the finite case and by Humphreys [50] for the infinite one. Many papers are devoted to description of automorphisms of Chevalley groups over commutative rings. We can mention here the papers of Borel-Tits [12], Carter-Chen Yu [27], Chen Yu [28]- [32], Abe [1], Klyachko [56]. Usually complete description of automorphisms of Chevalley groups means standardity of all these automorphisms, that is, all automorphisms are compositions of some simple and well-described types of automorphisms: inner automorphisms, automorphisms induced by ring automorphisms, etc. Abe in [1] proved the standardity of automorphisms for Noetherian rings with 1/2, which could help to close the question of automorphisms of Chevalley groups over arbitrary commutative rings with 1/2. However, in considering the case of adjoint elementary groups has a gap, which cannot be eliminated by the methods of this article. The cases when the ring contains a lot of invertible integers (in some sense) are completely clarified in the paper of Klyachko [56]. In the paper [15] Bunina proved that automorphisms of adjoint elementary Chevalley groups with root systems A l , D l , E l , l 2, over local rings with invertible 2 can be represented as the composition of ring automorphism and an automorphism-conjugation (by automorphismconjugation we call conjugation of elements of a Chevalley group in the adjoint representation by some matrix from the normalizer of this group in GL (V )). By the similar token it was proved in [17] that every automorphism of an arbitrary Chevalley (or its arbitrary subgroup) group is standard, i. e., it is a composition of ring, inner, central and graph automorphisms. In the same paper it was obtained the theorem describing the normalizer of Chevalley groups in their adjoint representation, which also holds for local rings without 1/2. In the series of papers [19], [16], [18], [20], [25] the similar methods made it possible to obtain the standardity of all automorphisms of Chevalley groups G(Φ, R) where Φ = F 4 , B l , l 3, R is a local ring and 1/2 ∈ R, or Φ = G 2 and 1/2, 1/3 ∈ R. The same is true for Φ = A l , D l E l , G 2 , l 2, R is a local ring and 1/2 / ∈ R. As we already mentioned the case C l (symplectic linear groups and projective symplectic linear groups) was considered in the papers of Petechuk and Golubchik-Mikhalev (even for non-commutative rings). The non-standard automorphisms are described by Steinberg in [81] for the cases of Chevalley groups of types B 2 and F 4 over fields of characteristic 2 and of type G 2 over fields of characteristic 3. For fields of characteristic 2 also there exists an isomorphism between Chevalley groups of types B l and C l , l 3. In [68] Petechuk described (non-standard) automorphisms of Chevalley groups of the type A 2 over local rings without 1/2. Therefore the cases of Chevalley groups of the types A 2 , B l , C l , F 4 over rings without 1/2 and of the type G 2 over rings without 1/3 require separate consideration. In the paper [21] Bunina used the localization method and ideas of Petechuk and generalized the description of automorphisms of Chevalley groups over local rings to adjoint Chevalley groups over arbitrary commutative rings. In the paper [22] the isomorphisms between these Chevalley groups were described. In this paper we extend the result of [21] to arbitrary Chevalley groups over rings. The paper is organized as follows. Section 2 deals with definitions and formulation of the Main Theorem. The proof of the Main Theorem for elementary case is situated in Section 3. The next Section 4 is devoted to the proof of the Main Theorem in the general case. 2 Definitions and main theorem. Root systems and semisimple Lie algebras We fix an indecomposable root system Φ of the rank ℓ > 1, with the system of simple roots ∆, the set of positive (negative) roots Φ + (Φ − ), and the Weil group W . Recall that any two roots of the same length are conjugate under the action of the Weil group. Let \|Φ + \| = m. More detailed texts about root systems and their properties can be found in the books [51], [13]. Recall also that for α, β ∈ Φ α, β = 2 (α, β) (β, β) . Suppose now that we have a semisimple complex Lie algebra L with the Cartan subalgebra H (more details about semisimple Lie algebras can be found, for instance, in the book [51]). Lie algebra L has a decomposition L = H ⊕ α =0 L α , L α := {x ∈ L \| [h, x] = α(h)x for every h ∈ H}, and if L α = 0, then dim L α = 1, all nonzero α ∈ H such that L α = 0, form some root system Φ. The root system Φ and the semisimple Lie algebra L over C uniquely (up to automorphism) define each other. On the Lie algebra L we can introduce a bilinear Killing form κ(x, y) = tr ( ad x ad y), that is non-degenerated on H. Therefore we can identify the spaces H and H * . We can choose a basis {h 1 , . . . , h l } in H and for every α ∈ Φ elements x α ∈ L α so that {h i ; x α } is a basis in L and for every two elements of this basis their commutator is an integral linear combination of the elements of the same basis. This basis is called a Chevalley basis. Elementary Chevalley groups Introduce now elementary Chevalley groups (see [81]). Let L be a semisimple Lie algebra (over C) with a root system Φ, π : L → gl(V ) be its finitely dimensional faithful representation (of dimension n). If H is a Cartan subalgebra of L, then a functional λ ∈ H * is called a weight of a given representation, if there exists a nonzero vector v ∈ V (that is called a weight vector ) such that for any h ∈ H π(h)v = λ(h)v. In the space V in the Chevalley basis all operators π(x α ) k /k! for k ∈ N are written as integral (nilpotent) matrices. An integral matrix also can be considered as a matrix over an arbitrary commutative ring with 1. Let R be such a ring. Consider matrices n × n over R, matrices π(x α ) k /k! for α ∈ Φ, k ∈ N are included in M n (R). Now consider automorphisms of the free module R n of the form exp(tx α ) = x α (t) = 1 + tπ(x α ) + t 2 π(x α ) 2 /2 + · · · + t k π(x α ) k /k! + . . . Since all matrices π(x α ) are nilpotent, we have that this series is finite. Automorphisms x α (t) are called elementary root elements. The subgroup in Aut (R n ), generated by all x α (t), α ∈ Φ, t ∈ R, is called an elementary Chevalley group (notation: E π (Φ, R)). In elementary Chevalley group we can introduce the following important elements and subgroups: • w α (t) = x α (t)x −α (−t −1 )x α (t), α ∈ Φ, t ∈ R * ; • h α (t) = w α (t)w α (1) −1 ; • N is generated by all w α (t), α ∈ Φ, t ∈ R * ; • H is generated by all h α (t), α ∈ Φ, t ∈ R * ; • The subgroup U = U(Φ, R) of the Chevalley group G(Φ, R) (resp. E(Φ, R)) is generated by elements x α (t), α ∈ Φ + , t ∈ R, the subgroup V = V (Φ, R) is generated by elements x −α (t), α ∈ Φ + t ∈ R. The action of x α (t) on the Chevalley basis is described in [26], [90]. It is known that the group N is a normalizer of H in elementary Chevalley group, the quotient group N/H is isomorphic to the Weil group W (Φ). All weights of a given representation (by addition) generate a lattice (free Abelian group, where every Z-basis is also a C-basis in H * ), that is called the weight lattice Λ π . Elementary Chevalley groups are defined not even by a representation of the Chevalley groups, but just by its weight lattice. More precisely, up to an abstract isomorphism an elementary Chevalley group is completely defined by a root system Φ, a commutative ring R with 1 and a weight lattice Λ π . Among all lattices we can mark two: the lattice corresponding to the adjoint representation, it is generated by all roots (the root lattice Λ ad ) and the lattice generated by all weights of all reperesentations (the lattice of weights Λ sc ). For every faithful reperesentation π we have the inclusion Λ ad ⊆ Λ π ⊆ Λ sc . Respectively, we have the adjoint and simply connected elementary Chevalley groups. Every elementary Chevalley group satisfies the following relations: (R1) ∀α ∈ Φ ∀t, u ∈ R x α (t)x α (u) = x α (t + u); (R2) ∀α, β ∈ Φ ∀t, u ∈ R α + β = 0 ⇒ [x α (t), x β (u)] = x α (t)x β (u)x α (−t)x β (−u) = x iα+jβ (c ij t i u j ), where i, j are integers, product is taken by all roots iα + jβ, taken in some fixed order; c ij are integer numbers not depending on t and u, but depending on α and β and the order of roots in the product. (R3) ∀α ∈ Φ w α = w α (1); (R4) ∀α, β ∈ Φ ∀t ∈ R * w α h β (t)w −1 α = h wα(β) (t); (R5) ∀α, β ∈ Φ ∀t ∈ R * w α x β (t)w −1 α = x wα(β) (ct), where c = c(α, β) = ±1; (R6) ∀α, β ∈ Φ ∀t ∈ R * ∀u ∈ R h α (t)x β (u)h α (t) −1 = x β (t β,α u). For a given α ∈ Φ by X α we denote the subgroup {x α (t) \| t ∈ R}. Chevalley groups Introduce now Chevalley groups (see [81], [33], [11], [26], [36], [88], [90], and references therein). Consider semisimple linear algebraic groups over algebraically closed fields. These are precisely elementary Chevalley groups E π (Φ, K) (see. [81], § 5). All these groups are defined in SL n (K) as common set of zeros of polynomials of matrix entries a ij with integer coefficients (for example, in the case of the root system C ℓ and the universal representation we have n = 2l and the polynomials from the condition (a ij )Q(a ji ) − Q = 0, where Q is a matrix of the symplectic form). It is clear now that multiplication and taking inverse element are defined by polynomials with integer coefficients. Therefore, these polynomials can be considered as polynomials over an arbitrary commutative ring with a unit. Let some elementary Chevalley group E over C be defined in SL n (C) by polynomials p 1 (a ij ), . . . , p m (a ij ). For a commutative ring R with a unit let us consider the group G(R) = {(a ij ) ∈ SL n (R) \| p 1 (a ij ) = 0, . . . , p m (a ij ) = 0}, where p 1 (. . . ), . . . p m (. . . ) are polynomials having the same coefficients as p 1 (. . . ), . . . , p m (. . . ), but considered over R. This group is called the Chevalley group G π (Φ, R) of the type Φ over the ring R, and for every algebraically closed field K it coincides with the elementary Chevalley group. In more advanced terms a Chevalley group G(Φ, R) is the value of the Chevalley-Demazure group scheme, see [?]. The subgroup of diagonal (in the standard basis of weight vectors) matrices of the Chevalley group G π (Φ, R) is called the standard maximal torus of G π (Φ, R) and it is denoted by T π (Φ, R). This group is isomorphic to Hom(Λ π , R * ). Let us denote by h(χ) the elements of the torus T π (Φ, R), corresponding to the homomorphism χ ∈ Hom(Λ(π), R * ). In particular, h α (u) = h(χ α,u ) (u ∈ R * , α ∈ Φ), where χ α,u : λ → u λ,α (λ ∈ Λ π ). Connection between Chevalley groups and their elementary subgroups Connection between Chevalley groups and corresponding elementary subgroups is an important problem in the structure theory of Chevalley groups over rings. For elementary Chevalley groups there exists a convenient system of generators x α (ξ), α ∈ Φ, ξ ∈ R, and all relations between these generators are well-known. For general Chevalley groups it is not always true. If R is an algebraically closed field, then G π (Φ, R) = E π (Φ, R) for any representation π. This equality is not true even for the case of fields, which are not algebraically closed. However if G is a simply connected Chevalley group and the ring R is semilocal (i.e., contains only finite number of maximal ideals), then we have the condition G sc (Φ, R) = E sc (Φ, R). [60], [2], [79], [6]. If, however, π is arbitrary and R is semilocal, then: [2], [6], [60]), and the elements h(χ) are connected with elementary generators by the formula G π (Φ, R) = E π (Φ, R)T π (Φ, R)] (seeh(χ)x β (ξ)h(χ) −1 = x β (χ(β)ξ). (1) Remark 1. Since χ ∈ Hom (Λ(π), R * ), if we know the values of χ on some set of roots which generate all roots (for example, on some basis of Φ), then we know χ(β) for all β ∈ Φ and respectively all x β (ξ) h(χ) for all β ∈ Φ and ξ ∈ R * . Therefore (in particular) if for all roots β from some generating set of Φ we have [x β (1), h(χ)] = 1, then h(χ) ∈ Z(E π (Φ, R)) and hence h(χ) ∈ Z(G π (Φ, R)). We will use this observation in the next section many times. If Φ is an irreducible root system of a rank ℓ 2, then E(Φ, R) is always normal and even characteristic in G(Φ, R) (see [86], [47]). In the case of semilocal rings it is easy to show that [G(Φ, R), G(Φ, R)] = E(Φ, R). except the cases Φ = B 2 , G 2 , R = F 2 . In the case ℓ = 1 the subgroup of elementary matrices E 2 (R) = E sc (A 1 , R) is not necessarily normal in the special linear group SL 2 (R) = G sc (A 1 , R) (see [35], [85], [83]). In the general case the difference between G π (Φ, R) and E π (Φ, R) is measured by K 1 -functor. Standard automorphisms of Chevalley groups Define four types of automorphisms of a Chevalley group G π (Φ, R), we call them standard. Central automorphisms. Let C G (R) be a center of G π (Φ, R), τ : G π (Φ, R) → C G (R) be some homomorphism of groups. Then the mapping x → τ (x)x from G π (Φ, R) onto itself is an automorphism of G π (Φ, R), denoted by τ . It is called a central automorphism of the group G π (Φ, R). Ring automorphisms. Let ρ : R → R be an automorphism of the ring R. The mapping (a i,j ) → (ρ(a i,j )) from G π (Φ, R) onto itself is an automorphism of the group G π (Φ, R), denoted by the same letter ρ. It is called a ring automorphism of the group G π (Φ, R). Note that for all α ∈ Φ and t ∈ R an element x α (t) is mapped to x α (ρ(t)). Inner automorphisms. Let S be some ring containing R, g be an element of G π (Φ, S), that normalizes the subgroup G π (Φ, R). Then the mapping x → gxg −1 is an automorphism of the group G π (Φ, R), denoted by i g . It is called an inner automorphism, induced by the element g ∈ G π (Φ, S). If g ∈ G π (Φ, R), then we call i g a strictly inner automorphism. Graph automorphisms. Let δ be an automorphism of the root system Φ such that δ∆ = ∆. Then there exists a unique automorphisms of G π (Φ, R) (we denote it by the same letter δ) such that for every α ∈ Φ and t ∈ R an element x α (t) is mapped to x δ(α) (ε(α)t), where ε(α) = ±1 for all α ∈ Φ and ε(α) = 1 for all α ∈ ∆. Now suppose that δ 1 , . . . , δ k are all different graph automorphisms for the given root system (for the systems E 7 , E 8 , B l , C l , F 4 , G 2 there can be just identical automorphism, for the systems A l , D l , l = 4, E 6 there are two such automorphisms, for the system D 4 there are six automorphisms). Suppose that we have a system of orthogonal idempotents of the ring R: {ε 1 , . . . , ε k \| ε 1 + · · · + ε k = 1, ∀i = j ε i ε j = 0}. Then the mapping Λ ε 1 ,...,ε k := ε 1 δ 1 + · · · + ε k δ k of the Chevalley group onto itself is an automorphism, called a graph automorphism of the Chevalley group G π (Φ, R). Similarly we can define four types of automorphisms of the elementary subgroup E π (Φ, R). An automorphism σ of the group G π (Φ, R) (or E π (Φ, R)) is called standard if it is a composition of automorphisms of these introduced four types. In [21] the following theorem was proved: Theorem 1. Let G = G ad (Φ, R) be an adjoint Chevalley group (or its elementary subgroup (E ad (Φ, R))) of rank > 1, R be a commutative ring with 1. Suppose that for Φ = A 2 , B l , C l or F 4 we have 1/2 ∈ R, for Φ = G 2 we have 1/2, 1/3 ∈ R. Then every automorphism of the group G is standard and the inner automorphism in the composition is strictly inner. Our goal is to prove the following theorem: Theorem 2. Let G = G π (Φ, R) be a Chevalley group (or its elementary subgroup E π (Φ, R))) of rank > 1, R be a commutative ring with 1. Suppose that for Φ = A 2 , B l , C l or F 4 we have 1/2 ∈ R, for Φ = G 2 we have 1/2, 1/3 ∈ R. Then every automorphism of the group G is standard. Proof of the main theorem for elementary Chevalley groups and subgroups 3.1 Localization of rings and modules; injection of a ring into the product of its localizations. Definition 1. Let R be a commutative ring. A subset Y ⊂ R is called multiplicatively closed in R, if 1 ∈ Y and Y is closed under multiplication. Introduce an equivalence relation ∼ on the set of pairs R × Y as follows: a s ∼ b t ⇐⇒ ∃u ∈ Y : (at − bs)u = 0. By a s we denote the whole equivalence class of the pair (a, s), by Y −1 R we denote the set of all equivalence classes. On the set S −1 R we can introduce the ring structure by a s + b t = at + bs st , a s · b t = ab st . Definition 2. The ring Y −1 R is called the ring of fractions of R with respect to Y . Let p be a prime ideal of R. Then the set Y = R\p is multiplicatively closed (it is equivalent to the definition of the prime ideal). We will denote the ring of fractions Y −1 R in this case by R p . The elements a s , a ∈ p, form an ideal M in R p . If b t / ∈ M, then b ∈ Y , therefore b t is invertible in R p . Consequently the ideal M consists of all non-invertible elements of the ring R p , i. e., M is the greatest ideal of this ring, so R p is a local ring. The process of passing from R to R p is called localization at p. Proof for E π (Φ, R). Suppose that G = G π (Φ, R) or G = E π (Φ, R) is a Chevalley group (or its elementary subgroup), where Φ is an indecomposable root system of rank > 1, R is an arbitrary commutative ring (with 1/2 in the case Φ = A 2 , F 4 , B l , C l and with 1/2 and 1/3 in the case Φ = G 2 ). Suppose that ϕ ∈ Aut (G). Since the subgroup E π (Φ, R) is characteristic in G π (Φ, R), then ϕ induces the automorphism ϕ ∈ Aut (E π (Φ, R)) (we denote it by the same letter). The elementary adjoint Chevalley group E ad (Φ, R) is the quotient group of our initial elementary Chevalley group E π (Φ, R) by its center Z = Z(E π (Φ, R)). Therefore the automorphism ϕ induces an automorphism ϕ of the adjoint Chevalley group E ad (Φ, R). By Theorem 1 ϕ is the composition of a graph automorphism Λ ε 1 ,...,ε k , where ε 1 , . . . , ε k ∈ R, a ring automorphism ρ, induced by ρ ∈ Aut R, and the strictly inner automorphism i g , induced by some g ∈ G ad (Φ, R). Central automorphism is identical in the decomposition of ϕ, since the center of any adjoint Chevalley group is trivial. Since ε 1 , . . . , ε k ∈ R and for any δ i ∈ Aut ∆ and for any representation π of the corresponding Lie algebra there exists the corresponding graph automorphism δ i ∈ Aut (G π (Φ, R)), then there exists a graph automorphism Λ ε 1 ,...,ε k ∈ Aut (E π (Φ, R)) such that the induced automorphism of the group E ad (Φ, R) is precisely Λ ε 1 ,...,ε k . Also taking the ring automorphism ρ ∈ Aut (G π (Φ, R)) we see that the induced automorphism of E ad (Φ, R) is precisely ρ. Therefore if we take ϕ 1 = Λ −1 •ρ −1 •ϕ, then we obtain an automorphism of the group G (and in any cases of the group/subgroup E π (Φ, R)) which induces the strictly inner automorphism i g on E ad (Φ, R). We always assume that R is a subring of the ring S = m R m = i∈κ R i , where every R i is a local ring, therefore G π (Φ, R) ⊆ G π (Φ, S) = i∈κ G π (Φ, R i ) and E π (Φ, R) ⊆ i∈κ E π (Φ, R i ). Note that since every R i is local, then we have G π (Φ, R) = T π (Φ, R)E π (Φ, R) and therefore i∈κ G π (Φ, R i ) = i∈κ T π (Φ, R i )E π (Φ, R i ). Suppose now that g = i∈κ g i , where g i ∈ G ad (Φ, R i ). Let us consider one i ∈ κ, where g i ∈ T ad (Φ, R i )E ad (Φ, R i ), i. e., g i = t i · x i , where t i ∈ T ad (Φ, R i ), x i ∈ E ad (Φ, R i ). Since x i is a product of elementary unipotents over the ring R i , then we can take x i ∈ E π (Φ, R i ), that is the same product of the same elementary unipotents and its image under factorization of E π (Φ, R i ) by its center is precisely x i . Now let us consider the element t i ∈ T ad (Φ, R i ). This element corresponds to some homomorphism χ i ∈ Hom (Λ( ad ), R * i ) and acts on any x α (s) ∈ E ad (Φ, R i ) as t i x α (s)t −1 i = x α (χ i (α) · s). If t i / ∈ H ad (Φ, R i ), then we can extend the ring R i up to a ring S i so that there exists h i ∈ H ad (Φ, S i ) with the same action on all elementary uniponents x α (s) as our t i . The ring S i is an algebraic extension of R i , in which there exist several new roots k √ λ for a finite number of λ ∈ R * i . This S i can be obtained from R i by the standard procedure S i ∼ = R i [y]/(y k − λ). Note that S i is not necessarily local. Now since R i ⊆ S i , then S ⊆ i∈κ S i = S and R ⊆ S ⊆ S. We see that for every i ∈ κ the torus element t i acts on all x α (s), s ∈ S i as h i ∈ H ad (Φ, S i ), therefore the element y i = h i · x i acts on all x α (s), s ∈ S i as the initial g i . Consequently the element y := i∈κ y i ∈ E ad (Φ, S) acts on all x α (s), s ∈ S as the initial g. Therefore we have y ∈ E ad (Φ, S) such that i y \| E ad (Φ, S) = i g \| E ad (Φ, S) . In particular, i y \| E ad (Φ,R) = i g \| E ad (Φ,R) . Let us take y ∈ E π (Φ, S) such that its image under factorization of E π (Φ, S) by its center is precisely y. Now we can take ϕ 2 = i y −1 • ϕ 1 , it will be an isomorphism between E π (Φ, R) and the subgroup of E π (Φ, S) such that under factorization by the center of E π (Φ, S) we obtain the identical automorphism ϕ 2 of the group E ad (Φ, R). Now let us analyze the mapping ϕ 2 . Since ϕ 2 is identical, then ∀α ∈ Φ ∀s ∈ R ϕ 2 (x α (s)) = z α,s x α (s), where z α,s ∈ Z(E π (Φ, S)). If α is either any root of the systems A l , l 2, D l , l 4, E l , l = 6, 7, 8, F 4 , or any long root of the systems G 2 , B l , l 3, or any short root of the systems C l , l 3, then α can be represented as α = β + γ, where {±β, ±γ, ±α} ∼ = A 2 . In this case x α (s) = [x β (s), x γ (1)], therefore z α,s x α (s) = ϕ 2 (x α (s)) = [ϕ 2 (x β (s)), ϕ 2 (x γ (1))] = [z β,s x β (s), z γ,1 x γ (1)] = [x β (s), x γ (1)] = x α (s). Consequently, z α,s = 1 for all s ∈ R. For the root system G 2 all Chevalley groups are adjoint and so we do not need to prove Theorem 1 for this root system. For the root system B 2 if α is a long simple root and β is a short simple root, then Φ + = {α, β, α + β, α + 2β}, where α + β is short and α + 2β is long and [x α (t), x β (u)] = x α+β (±tu)x α+2β (±tu 2 ), [x α+β (t), x β (u)] = x α+2β (±2tu). (see [81], Lemma 33). Since for the root system B 2 we require 1/2 ∈ R, then [x α+β (s), x β (1/2)] = x α+2β (±s) and by the same arguments as above z γ,s = 1 for all long roots γ and all s ∈ R. Then x α+β (±s)x α+2β (±s) = [x α (s), x β (1)] = ϕ 2 ([x α (s), x β (1)]) = = ϕ 2 (x α+β (±s)x α+2β (±s)) = z α+β,±s x α+β (±s)x α+2β (±s), thus z γ,s = 1 also for all short roots γ ∈ B 2 . Therefore for B 2 for all α ∈ Φ and all s ∈ R the mapping ϕ 2 is an identical automorphism of E π (Φ, R). Since any root γ of the root system B l or C l , l 3, can be embedded to some root system isomorphic to B 2 , and in this case we also require 1/2 ∈ R, then for these root systems also z γ,s = 1 for all s ∈ R * and ϕ 2 is an identical automorphism of E π (Φ, R). Therefore for all cases under consideration ϕ 2 \| Eπ(Φ,R) = i y −1 • Λ −1 • ρ −1 • ϕ\| Eπ(Φ,R) = id Eπ(φ,R) , so ϕ\| Eπ(Φ,R) = ρ • Λ • i y \| Eπ(Φ,R) , where y ∈ E π (Φ, S) ∩ N(E π (Φ, R)), Λ is a graph automorphism of the groups G π (Φ, R) and E π (Φ, R) and ρ is a ring automorphism of the groups G π (Φ, R) and E π (Φ, R). Thus, for G = E π (Φ, R) the main theorem (Theorem 2) is proved. 4 Proof of the main theorem for the groups G π (ϕ, R) Let now G = G π (Φ, R). Initially the mapping ϕ was an automorphism of the group G. The mapping ϕ 1 from the previous section was the composition of ϕ and graph and ring automorphisms of the group G, i. e., also an automorphism of G. After that ϕ 2 (from the previous section) is the composition of ϕ 1 and the conjugation of G by some element y ∈ E π (Φ, S), where R ⊂ S. We know that y normalizes E π (Φ, R) and we want to show that in our case y normalizes also our full Chevalley group G. Note that for the simply-connected Chevalley group of the type E 6 Luzgarev and Vavilov in [58] proved that the normalizers of the Chevalley group and its elementary subgroup coincide. Then in [59] they proved the same theorem for the root system E 7 . Since all other exceptional Chevalley groups are adjoint, we only need to show the coincidence of normalizers for nonadjoint classical Chevalley groups, but our method will cover all the cases. Lemma 1. Under assumptions of Theorem 2 the elements x α (1), α ∈ Φ, by addition, multiplication and multiplication by elements from R generate the Lie algebra π(L R (Φ)) ⊂ M N (R), where N is the dimension of the representation π. Proof. For the adjoint Chevalley groups this lemma was proved in [21]. Therefore we will not repeat the proof for the root system G 2 (since it is always adjoint). If the root system differs from G 2 and 1/2 ∈ R, then x α (1) = E + π(X α ) + π(X α ) 2 /2, therefore π(X α ) = x α (1) − E − (x α (1) − E) 2 /2, and π(L R (Φ)) = π(X α ) \| α ∈ Φ R . Suppose now that we deal with systems A l , (l 3), D l , E l , 1/2 / ∈ R. For all these systems and non-adjoint representations π we have π(X α ) 2 = 0 for all α ∈ Φ, therefore π(X α ) = x α (1) − E. The lemma is proved. From Lemma 1 we see that the conjugation by y maps the Lie algebra π(L(Φ) R ) onto itself. Lemma 2. Under assumptions of Theorem 2 the Lie algebra π(L R (Φ)) together with the unity matrix E by addition, multiplication and multiplication by elements from R generate the matrix ring M N (R), where N is the dimension of the representation π. Proof. For all adjoint Lie algebras under consideration this fact was proved in the papers [15], [16], [18], [19], [20]. For classical representations of classical Lie algebras the proof is clear and direct: 1. If we have the root system A l and the standard representation, then π(X e i −e j ) = E ij , π(X e i −e j )π(X e j −e i ) = E ii , M l+1 (R) = E ij \| 1 i, j l + 1 R . 2. The Lie algebra of the type C l in its universal representation has 2l-dimensional linear space and the basis {E ii − E l+i,l+i ; E ij − E l+j,l+i ; E i,l+i ; E l+i,i ; E i,l+j + E j,l+i ; E l+i,j + E l+j,i \| 1 i = j l}. Multiplying E ij − E l+j,l+i by E j,l+j , we get all E i,l+j for all 1 i, j l. Multiplying E l+i,i by E ij − E l+j,l+i , we obtain E l+i,j for all 1 i, j l. It is clear that after that we have all E ij , 1 i, j l, and therefore the whole matrix ring M 2l (R). 3. For the root system D l the standard representation gives the algebra so 2l , where in 2l-dimensional space the basis is {E ii − E l+i,l+i ; E ij − E l+j,l+i ; E i,l+j − E j,l+i ; E i+l,j − E j+l,i \| 1 i = j l}. Since for i = j we have (E ii −E l+i,l+i ) · (E ij −E l+j,l+i ) = E ij , then the whole matrix ring M 2l (R) is generated by this Lie algebra. All other representations are described by Plotkin,Semenov and Vavilov in [71] as microweight representations with the help of so-called weight diagrams. Weight diagram is a labeled graph, its vertices correspond (bijectively) to the weights λ ∈ Λ(π). The vertices corresponding to λ, µ ∈ Λ(π), are joined by a bond marked α i ∈ ∆ (or simply i) if and only if their difference λ − µ = α i is a simple root. The diagrams are usually drawn in such way that the marks on the opposite (parallel) sides of a parallelogram are equal and at least one of them is usually omitted. All weights are numbered in any order and give the basis of our representation π. If we want to find π(X α i ), i = 1, . . . , l, then we need to find all bonds marked by i, and if they join the vertices (γ 1 , γ 1 + α i ), . . . , (γ k , γ k + α i ), then π(X α i ) = ±E γ 1 ,γ 1 +α i ± · · · ± E γ k ,γ k +α i , π(X −α i ) = ±E γ 1 +α i ,γ 1 ± · · · ± E γ k +α i ,γ k . It is clear that if we take an element π(X α i ) · π(X α j ), then it is a sum of ±E γ,γ ′ , where there exists a path from the weight γ to γ ′ of the length 2 marked by the sequence (i, j). Similarly, if we take an element π(X α i 1 ) × · · · × π(X α i k ), then it is a sum of ±E γ,γ ′ , where there exists a path from the weight γ to γ ′ of the length k marked by the sequence (i 1 , . . . , i k ). Our goal is to generate all matrix units E γ 1 ,γ 2 , where γ 1 , γ 2 ∈ Λ(π). Since all weight diagrams are connected, it is sufficient to generate all matrix units E γ,γ+α i and E γ+α i ,γ , where α i ∈ ∆, γ, γ + α i ∈ Λ(π). The general idea how to do it is the following: for any γ ∈ Λ(π) and any α i 0 ∈ ∆ such that γ + α i 0 ∈ Λ(π) we find γ ′ ∈ Λ(π) such that: (1) there exists a path (i 0 , i 1 , . . . , i k ) from γ to γ ′ ; (2) in our weight diagram there is no other path (i 0 , i 1 , . . . , i k ); (3) the path (i 1 , . . . , i k ) exists only from γ + α i to γ ′ . Then π(X α i 0 )π(X α i 1 ) . . . π(X α i k ) = ±E γ,γ ′ and π(X −α i k ) . . . π(X −α i 1 ) = ±E γ ′ ,γ+α i 0 and therefore E γ, γ+α i 0 = E γ,γ ′ E γ ′ ,γ+α i 0 . It is almost clear that such γ ′ and unique paths always exist, we will just show one diagram as an example. If we take the case A 7 with the weight ω 2 , the representation is 28-dimensional. Let us find a path which gives E γ 1 ,γ 2 . Since the path (1, 3) is unique in the diagram, then the path (2, 1, 3) is also unique and we have E γ 1 ,γ 2 = (π(X α 2 )π(X α 1 )π(X α 3 )) · (π(X −α 3 )π(X −α 1 ). Figure 1: A 7 , ω 2 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ If we want to generate, for example, E γ 4 ,γ 6 , then the suitable path is (4, 1, 5), since the path (1, 5) is unique in the diagram. Looking at the picture it is easy to find the suitable path for any pair of neighboring vertices. Therefore the lemma is proved for all the cases. Since yπ(L(Φ) R )y −1 = π(L(Φ) R and π(L(Φ) R generates the whole matrix ring M N (R), then yM N (R)y −1 = M N (R). Therefore yG π (Φ, R)y −1 ⊆ SL N (R). From the other side, since y ∈ G π (Φ, S), then yG π (Φ, R)y −1 ⊆ G π (Φ, S). Since G π (Φ, S) ∩ SL N (R) is (by definition) the Chevalley group G π (Φ, R), then y normalizes G. Now we know that ϕ 2 is an automorphism of G = G π (Φ, R), identical on the elementary subgroup E = E π (Φ, R). Let us take some g ∈ G and x 1 ∈ E and let gx 1 g −1 = x 2 ∈ E. Then ϕ 2 (g)ϕ 2 (x 1 )ϕ 2 (g) −1 = ϕ 2 (x 2 ) =⇒ ϕ 2 (g)x 1 ϕ 2 (g) −1 = x 2 , therefore ϕ 2 (g)x 1 ϕ 2 (g) −1 = gx 1 g −1 =⇒ (g −1 ϕ 2 (g))x 1 (g −1 ϕ 2 (g)) −1 = x 1 , so g −1 ϕ 2 (g) ∈ C G (E). By the main theorem from [5] C G (E) = Z(G), therefore ϕ 2 (g) = c g · g, c g ∈ Z(G) for all g ∈ G. Whence ϕ 2 is a central automorphism of G and the initial ϕ is the composition of graph, ring, inner and central automorphisms, i. .e, ϕ is standard. The theorem is proved. Some applications: isomorphisms and model theory of Chevalley groups Standard description of automorphisms of Chevalley groups allows to describe and classify Chevalley groups up to different type of equivalencies and also to study model-theoretic properties. Theorem 3. Let G 1 = G π 1 (Φ 1 , R 1 ) and G 2 = G π 2 (Φ 2 , R 2 ) be two Chevalley groups of ranks > 1, R 1 , R 2 be commutative rings with 1. Suppose that for Φ 1 = A 2 , B l , C l or F 4 we have 1/2 ∈ R 1 , for Φ 1 = G 2 we have 1/2, 1/3 ∈ R 1 . Then every isomorphism between the groups G 1 and G 2 is standard: it is a composition of inner, diagram and central automorphisms of G 1 and ring isomorphism between G 1 and G 2 . Proof. The proof is identical to the proof of Theorem 9 from [22]. One needs to replace the references to Theorem 1 in the proof by those to Theorem 2. Remark 2. The result of Theorem 3 is valid with respect to elementary Chevalley groups E π 1 (Φ 1 , R 1 ) and E π 2 (Φ 2 , R 2 ) as well. Corollary 1 (classification of Chevalley groups up to isomorphism). Under conditions from Theorem 3 two Chevalley groups G 1 and G 2 (elementary Chevalley groups, respectively) are isomorphic if and only if they have the same root systems Φ 1 and Φ 2 , same weight lattices Λ π 1 and Λ π 2 and isomorphic rings R 1 and R 2 . Proof. If G 1 ∼ = G 2 , then there exists an isomorphism ϕ : G 1 → G 2 , which is composition of a ring isomorphism ρ : G 1 → G 2 and some automorphism ψ ∈ Aut G 1 (according to Theorem 3). Therefore there exists a ring isomorphism between G 1 and G 2 , i. e., G 1 and G 2 have the same root systems, weight lattices and isomorphic rings. Another application of Theorem 3 is classification of Chevalley groups up to elementary equivalence (for adjoint Chevalley groups it was done in [22]). Definition 3. Two algebraic systems M 1 and M 2 of the same language L are called elementarily equivalent, if their first order theories coincide. Theorem 4 (Keisler-Shelah Isomorphism theorem, [76], [54]). Two models M 1 and M 2 of the same language are elementarily equivalent if and only if there exists an ultrafilter F such that F M 1 ∼ = F M 2 . Corollary 2 (classification of Chevalley groups up to elementary equivalence). Under conditions from Theorem 3 two Chevalley groups G 1 and G 2 (elementary Chevalley groups, respectively) are elementarily equivalent if and only if they have the same root systems Φ 1 and Φ 2 , same weight lattices Λ π 1 and Λ π 2 and elementarily equivalent rings R 1 and R 2 . Proof. By Theorem 4 the groups G 1 and G 2 are elementarily equivalent if and only if for some ultrafilter F their ultrapowers are isomorphic. Since F G π (Φ, R) ∼ = G π (Φ, F R), the latter is equivalent to G π 1 (Φ 1 , F R 1 ) ∼ = G π 2 (Φ 2 , F R 2 ) ⇐⇒      Λ π 1 = Λ π 2 , Φ 1 = Φ 2 , F R 1 ∼ = F R 2 , ⇐⇒      Λ π 1 = Λ π 2 , Φ 1 = Φ 2 , R 1 ≡ R 2 , what was required. Two last corollaries almost finalize classification of Chevalley groups over commutative rings up to isomorphisms and elementary equivalence. However, there are still open questions concerning the relations of Chevalley groups with model theory. In the recent work of D. Segal and K. Tent [75] the question of bi-interpretability of Chevalley groups over integral domains was considered (see [75] and [55] for the definition of biinterpretability): Theorem 5 ( [75]). Let G(R) = G π (Φ, R) be a Chevalley group of rank at least two, and let R be an integral domain. Then R and G(R) are bi-interpretable provided either (1) G is adjoint, or (2) G(R) has finite elementary width, assuming in case Φ = E 6 , E 7 , E 8 , or F 4 that R has at least two units. In the paper [23] regular bi-interpretabilty of Chevalley groups over local rings was obtained. This result used the ideas from [75] along with description of isomorphisms between Chevalley groups over local rings. It has also been proved that the class of Chevalley groups over local rings is elementarily definable: any group that is elementarily equivalent to some Chevalley group over a local ring is also a Chevalley group (of the same type) over a local ring (see [23]). 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[ "Secure and Robust Authentication for DC MicroGrids based on Power Talk Communication", "Secure and Robust Authentication for DC MicroGrids based on Power Talk Communication" ]	[ "Marko Angjelichinoski \nDepartment of Electronic Systems\nAalborg University\nDenmark\n", "Pietro Danzi \nDepartment of Electronic Systems\nAalborg University\nDenmark\n", "Čedomir Stefanović \nDepartment of Electronic Systems\nAalborg University\nDenmark\n", "Petar Popovski [email protected] \nDepartment of Electronic Systems\nAalborg University\nDenmark\n" ]	[ "Department of Electronic Systems\nAalborg University\nDenmark", "Department of Electronic Systems\nAalborg University\nDenmark", "Department of Electronic Systems\nAalborg University\nDenmark", "Department of Electronic Systems\nAalborg University\nDenmark" ]	[]	We propose a novel framework for secure and reliable authentication of Distributed Energy Resources to the centralized secondary/tertiary control system of a DC MicroGrid (MG), networked using the IEEE 802.11 wireless interface. The key idea is to perform the authentication using power talka powerline communication technique executed by the primary control loops of the power electronic converters. In addition, the scheme also promotes direct and active participation of the control system in the authentication process, a feature not commonly encountered in current networked control systems for MicroGrids. The PLECS R -based simulations verifies the viability of the proposed solution.	10.1109/icc.2017.7996878	[ "https://arxiv.org/pdf/1612.07213v2.pdf" ]	691,640	1612.07213	fe6da38fc017cab868f2a419c5acb8a5fc345944	Secure and Robust Authentication for DC MicroGrids based on Power Talk Communication Marko Angjelichinoski Department of Electronic Systems Aalborg University Denmark Pietro Danzi Department of Electronic Systems Aalborg University Denmark Čedomir Stefanović Department of Electronic Systems Aalborg University Denmark Petar Popovski [email protected] Department of Electronic Systems Aalborg University Denmark Secure and Robust Authentication for DC MicroGrids based on Power Talk Communication We propose a novel framework for secure and reliable authentication of Distributed Energy Resources to the centralized secondary/tertiary control system of a DC MicroGrid (MG), networked using the IEEE 802.11 wireless interface. The key idea is to perform the authentication using power talka powerline communication technique executed by the primary control loops of the power electronic converters. In addition, the scheme also promotes direct and active participation of the control system in the authentication process, a feature not commonly encountered in current networked control systems for MicroGrids. The PLECS R -based simulations verifies the viability of the proposed solution. access to the communication system and thus compromise MG operation. So far, there has been very little effort to design secure control architecture in which the control system actively participates in authentication and access control. This paper proposes novel, robust networked control architecture for secure authentication of DER units, that employs recent powerline communication (PLC) technique termed power talk, and runs the initial handshake over the powerlines. Power talk [6], [7], [8] is an in-band, low rate PLC solution, designed originally for self-sustainable DC MGs that do not rely on external communication systems. It modulates the information into subtle deviations of the parameters of the primary droop control of the DERs, which translate in information-carrying deviations of the steady state voltage of the DC distribution system. Using power talk, the handshake becomes effectively invisible for the conventional attacker, as the attacker needs to physically access the grid in order to perform the attack, which is significantly more complicated and often impossible (e.g., MG systems for applications guarded by a safety perimeter). Furthermore, the proposed scheme promotes active participation of the MG control system in the authentication procedure, as it is embedded within the primary control loops, without the use of any additional, external hardware, requiring only software modifications of the power electronic converters. The paper is organized as follows. Section II reviews the state-of-the art security procedures used in IEEE 802.11 and discusses their potentially weak points, with the emphasis on the handshake procedure. Section III introduces the hierarchical MG control architecture, discusses possible cyber-attacks and introduces the basics of power talk. Section IV presents the secure authentication procedure. Section V verifies the viability of the advocated solution through PLECS R simulation of a realistic low voltage DC MG. Section VI concludes the paper. II. CRYPTOGRAPHIC HANDSHAKE OF IEEE 802.11 SYSTEMS The IEEE 802.11 standard includes advanced security mechanisms to ensure the confidentiality and integrity of the communication channel. The most recent security specification is IEEE 802.11i [9], whose security aspects are summarized as follows. The standard ensures the confidentiality and integrity of the packets by means of the CCMP protocol, that is based on the Advanced Encryption Standard (AES) cipher. A shared key, named Pairwise Transient Key (PTK), guarantees the encryption between Station (STA) and Access Point (AP). A second shared key, the Group Transient Key (GTK), is used for multicast and broadcast traffic. The STA is provided with the keys after having proved to be authorized, i.e., to know the right Pairwise Master Key (PMK). The PMK can be derived from a Pre-Shared Key (PSK) or by means of advanced authentication mechanisms that involve an Authentication Server. In any case, STAs have to indirectly prove the knowledge of the PMK to the AP; this problem has been resolved by means of a cryptographic handshake [10]. The handshake consists of four messages, each requiring an Acknowledgement (ACK). The protocol steps, depicted in Fig. 1, can be summarized as: (h 1 ) AP→STA containing the ANonce and its own MAC address enabling the STA to compute the PTK, (h 2 ) STA→AP containing the CNonce and its own MAC address, enabling the AP to compute the PTK, (h 3 ) AP→STA containing the GTK, and, (h 4 ) STA→AP confirmation of the reception. Besides the basic dictionary attack [11], particularly threatening attack on the handshake is when a malicious attacker impersonates the STA and sends invalid packets leading to unsuccessful authentication [10] and disabling the STA from accessing the communication resources. This situation may result in poor regulation of the MG system, leading to performance degradation and instability as the DER is prevented from communicating and, thus, participating in the control and optimization, c.f. [12]. III. CONTROL IN DC MICROGRIDS A. Multiple-Bus DC MicroGrid Architecture A DC MG is a collection of DERs and loads, connected to distribution infrastructure that comprises set of buses interconnected via distribution lines. The total number of DERs is denoted with U , and they are indexed in the set U = {0, ..., U − 1}. They use power electronic converters to interface the distribution system, and their voltage and current (i.e., power) outputs are locally regulated via several control channels of different bandwidths. The total demand of the aggregate load is denoted with d. B. Hierarchical Control The control architecture of the MG is summarized in Fig. 2. The control plane is organized in a hierarchy, comprising fast decentralized primary control, and slow centralized secondary and tertiary control. 1 In standard implementations, the primary control uses only local measurements as feedback, i.e., it does not require any exchange of information among remote units. On the other hand, the feedback loop of the secondary/tertiary control is closed via external communication system; thereby, each power electronic converter is assumed to be pre-equipped with IEEE 802.11 modem, see Fig. 2. We assume: (i) the communication is centralized, i.e., there is a single AP and all DERs are associated with it, and (ii) the central secondary/tertiary controller (CC) is collocated with the AP in the same physical unit, see Fig. 2. Thus, besides handling the secondary/tertiary control processes, the same unit is also in charge of the authentication of DERs to the communication network. Without loss of generality, assume that the secondary/tertiary CC and AP reside in DER 0. 1) Primary control: A common primary control configuration is in the form of Voltage Source Converter (VSC). VSC DERs regulate the electrical parameters and balance the supply-demand to guarantee stability, based on local output measurements, using the following law [1], [2], [13]: v u = x u − r u i u , u ∈ U, where v u and i u are the bus output voltage and current, x u is the reference voltage and r u is the virtual resistance. This implementation is known as decentralized droop control. The value v n serves as input reference to the inner primary control loops, that operate with frequency ν, equal to the sampling frequency of the converters' ADC, see Fig. 2. The droop controller controls x u and r u , where x u determines the voltage rating of the system, while the r u determines the load sharing among different DERs. In practice, the value of the virtual resistance is set to enable proportional load sharing [2], [3]. Another primary control architecture is in the form of Current Source Converter (CSC). CSC units do not participate in output voltage regulation and are usually operated at their individual maximum efficiency point using the Maximum Power Point Tracking (MPPT) algorithm [2]. 2) Secondary control: It is well known that under decentralized primary droop control, the bus voltages vary with changes in the load demand and the power generation capacities of the renewable DERs [2], [3]. Moreover, the load is not ideally shared among different DERs due to mismatched line admittances [13]. In this context, the role of the secondary control is to alleviate the drawbacks of the decentralized droop control and restore the bus voltages to a predefined and optimized global reference (determined by the tertiary control) and to foster proper load sharing. This is achieved by adding correction offsets to the reference voltage control parameter of the local droop controller, see Fig. 2. We assume that the secondary control is centralized and implemented as follows: 1) DER u ∈ U, periodically (e.g., every 5 milliseconds) sends a short message to the CC with payload [ṽ u ,ĩ u ] withṽ u andĩ u denoting the local measurement of the output bus voltage and current; 2) the CC computes the voltage restoration and proportional load sharing offsets δx v and δx c u using Proportional-Integral (PI) loops; 3) the CC sends unicast packet with payload [δx v , δx c u ] to DER u, u ∈ U; 4) upon receiving the packet, each DER uses δx v , δx c u to correct the local droop controllers: v u = x u + δx v + δx c u − r u i u , u ∈ U. After the load/generation change, the offsets δx v and δx c u converge to stable values and remain fixed until the next fluctuation. The modified droop control law provides a global voltage regulation reference and fosters proportional load sharing. Note that only VSC DERs participate in secondary control voltage and current sharing regulation. 3) Tertiary control: The tertiary control runs with significantly lower frequency than the secondary control (e.g., every 5 − 30 minutes [14]), optimizing the performance of the system and generating the optimal references for the lower control levels. The tertiary control objective depends on the specific application. In our system, a generic centralized tertiary control is implemented as follows: 1) DER u ∈ U periodically sends a message with generic payload z u , u ∈ U to the CC; 2) the CC solves the application-specific optimization problem and generates the optimized global/local control parameters; 3) the optimal parameters are sent to the DERs in a message with payload q u , u ∈ U; 4) the DERs follow the received CC directives and reset the local control parameters which remain valid until new ones are received. C. Attacks on the control system via attacks on the wireless interface To motivate the development of our framework, we take a closer look into two potential attacks on the secondary control. Consider DC MG in which small subset of DERs are operated in VSC mode, while the rest are operated in CSC mode, see the right-hand side of Fig. 3. Such primary control architectures are typical in state-of-the-art installations, where the individual renewable DERs (e.g., solar panels) are operated at their respective maximum efficiency points [1], [2]. A known problem in such systems is the limited ability of the controlling VSC DERs to restore and stabilize the voltage around the global reference after frequent dramatic changes in the renewable power capacity of the CSC DERs; this decreases the overall system efficiency and might lead to instability [2]. One possible solution is to configure the DERs with dual mode capability, i.e., to switch between CSC and VSC modes transparently [2]. Specifically, whenever the grid voltage crosses predefined voltage thresholds, portion of the CSC DERs switch to VSC primary control mode and ask the CC to join the secondary control. To gain access to the wireless communication resources, they are initially requested to authenticate. An attacker might attack the control system by 1) attacking the handshake and disabling the authentication, which prevents the newly switched VSC DERs from joining the secondary control, see the left-hand side of Fig. 3, and 2) jamming the wireless channel of one or several VSC DERs, including the CC, that participate in secondary control [12]. Both types of attacks result in poor voltage regulation, leading to performance degradation and potential instabilities. The solution presented in the paper addresses the first type of attack (Fig. 3), by implementing the handshake over parallel powerline communication interface based on power talk, which is described in Section III-D. We also note that the power talk interface can be easily adapted to address the second type of attack, by allowing jammed units to send an alarm to the CC. The CC can then act appropriately, e.g., by sending directives to the jammed VSCs DERs via power talk to switch back to CSC mode, and asking non-jammed CSCs to switch to VSC mode and join the secondary control. Finally, the power talk interface can be also used as a safe and secure channel over which the VSC DERs can re-elect new CC in case the wireless interface of the current CC is under attack. Addressing these aspects is part of ongoing work. D. Power Talk Power talk is implemented on primary droop control level, and requires the secondary control to be switched off during its operation [6], [7], [8]. VSC DER u ∈ U modulates information into the values of the local reference voltage x u and virtual resistance r u droop control parameters, thus inducing disturbances of the output voltages. At the same time, droop controlled DERs also observe the steady state bus voltage response. We say that the inputs to the power talk multiple access channel are x u and r u , u ∈ U, while the output observed by DER k = u is represented through the disturbances of the output bus voltage. The power talk channel is non-linear and requires full knowledge of the configuration of the system, which makes it particularly challenging for implementation [6], [8]. However, the main challenge and impairment stems from the requirement for turning off the secondary control; this makes the steady state voltage susceptible to random load variations which alter the output voltage of the DERs in unpredictable manner. Large portion of the work on power talk consist in designing viable strategies to mitigate the effect of random load changes in various communication scenario, such as one-way, broadcast, all-to-all full duplex, etc. [6], [7], [8]. As detailed in the Section IV, here we employ a basic variant, suitable for one-way communication where only one transceiver pair is active at a time. IV. SECURE AND ROBUST DER AUTHENTICATION PROTOCOL BASED ON POWER TALK COMMUNICATION Here we describe the scheme for secure and reliable power talk based authentication of a DER to the external communication infrastructure. We label an "incoming" DER with U , and we assume that it is connected to the MG via primary control and does not participate in upper level control. DER U wishes to join the set U, i.e., to actively participate in the regulation and optimization of the MG. Prior to this, DER U should be authenticated and granted permission to join the wireless networking. The authentication is performed by the CC; the control architecture of the DERs engaged in the handshake is summarized in Fig. 2. Fig. 4 depicts the time organization of the external wireless and power talk interfaces; note all DERs physically connected to the system are synchronized. 2 The time axis of the wireless interface is divided into secondary and tertiary control periods of durations denoted with T sc , T tc , corresponding to the sampling frequencies of the secondary and tertiary controllers, respectively. Note that T sc T tc . A. Protocol Organization The power talk interface is based on a periodically repeating pool of D ≥ 1 consecutive secondary control periods in which the secondary control is turned off, see Fig. 4. 3 We name the periodic pool as power talk association request channel (PTARCh), with duration T sc D, occurring with frequency L T tc . The PTARCh is used by any "incoming" VSC DER that wants to join the upper level control, to send initial association request to the CC. After receiving association request, the CC keeps the secondary control off and allocates additional, ondemand pool of R ≥ 1, consecutive secondary control periods for execution of the actual handshake, establishing the ondemand power talk handshake channel (PTHaCh), see Fig. 4. Specifically, DER U , after physically connecting to the MG through primary control and synchronizing with the power talk interface, switches to VSC mode and waits until the next occurrence of the PTARCh. Then, it sends association request to the CC. The CC, after receiving the request successfully, first sends an ACK and then the response message to to DER U . DER U acknowledges the response which triggers the CC to allocate additional power talk resources for the rest of the handshake, namely the exchange of messages h 1 to h 4 , see Fig. 1, which happens in the PTHaCh channel of duration T sc R, see Fig. 4. We assume that the duration of a single secondary control period can be expressed as T sc = ST pt , where T pt denotes the duration of a single power talk slot and S ≥ 1 is an integer. 4 The described protocol relies on switching off the secondary control for limited period of time, during which the system becomes susceptible to voltage deviations, required by the power talk (see subsection III-D). However, this also makes the steady state voltage susceptible to load variations, representing the main communication impairment of the power talk interface, as described next. B. Specifics of the Power Talk PHY interface We employ binary power talk introduced in [6]. Fig. 2 gives an overview of the main functional blocks residing in the CC and DER U when engaged in handshake via power talk. Label the power talk transmitter in a power talk slot with i and the receiver with j; when i = U then j = 0 and vice versa. The handshake messages, represented as binary strings are mapped into binary stream of reference voltage deviations in the transmitters' droop control loop, as follows: 0 ↔ x i − γ, 1 ↔ x i + γ, The reference voltage deviation amplitude γ satisfies γ xi 1, see [6] for discussion on how to choose γ. The deviation ±γ 3 The secondary control is turned off to enable establishment of the power talk channel; during the off periods, the CC does not send secondary control information and the DER u uses the last available δx v , δx c u . 4 The duration T pt complies with the control bandwidth of the inner primary control loops, such that the system reaches steady state in a power talk slot. Typically S ≤ 10. leads to deviations of the output voltage of the receiving DER: 0 ↔ v j − ∆v j (0), 1 ↔ v j + ∆v j (1), where v j , corresponding to γ = 0. The receiver, in each power talk slot collects ν(T pts − τ ) noisy steady state output voltage samples, denoted withṽ j [n], then, it compares their average to the threshold v j , and makes decision on the information bits: if 1 νT pt nṽ j [n] > v j , decide 1, if 1 νT pt nṽ j [n] < v j , decide 0. Notice that τ denotes the transient interval in which the bus reaches a steady state. The above detection scheme is challenged by two major impairments: (i) the susceptibility of the primary control level to sporadic load variations during the periods in which the secondary control is off, and (ii) sampling noise of the converters' ADC. The impact of load variations on power talk has been extensively studied [6], [8]. Specifically, in the above scheme, a load change invalidates the detection threshold v j , which might lead to burst of bit errors. A simple strategy to deal with this impairment is to employ load change detection, in parallel with power talk symbol transmission/detection, in both the transmitter and the receiver [6]. 5 After detecting load change, the transmission is paused, M "blank" power talk slots, i.e., slots with γ = 0 are inserted and the communicating DERs to measure the new steady state output voltage level, which is used as the new detection threshold. The viability of this technique is verified in Section V. Note that under this strategy, each load change increases the time necessary to complete the handshake. We introduce the average handshake completion time denoted with µ, defined as the average time needed to complete the handshake, after the DER accesses the PTARCh. We model the load changing process as a Poisson process with arrival rate λ [8]. Then, the following expression for µ can be derived [8]: µ = (D + R)ST pt (1 − e −λ + e λM ). The expression shows that when the rate λ is significantly lower than the power talk signaling rate, the dominant contributing factor to µ is the total number of bits (D + R)S of the handshake messages. The sampling noise of the converter follows Gaussian distribution [15], i.e.,ṽ j [n] ∼ N (v j + ∆v j (b), σ 2 ) [8]. The bit error rate (BER) can be calculated as: P e = 1 − 1 2 erf ∆v j (1) σ √ 2 − 1 2 erf ∆v j (0) σ √ 2 . An appropriate error correction code can be employed to protect the handshake messages against noise-related errors. V. RESULTS We simulate a simple DC MG using PLECS R . There are U = 6 DER units, connected to a single bus in parallel via distribution lines with resistances 0.2 Ω. The aggregate resistive load of 1.5 Ω is also connected in parallel to the bus. We use T sc = 5 ms and T tc = 5 min. The reference voltages of the VSC DERs and the global grid voltage reference are equal and set to x u = v = 48 V. The virtual resistances of all VSC DERs are set to 0.2 Ω and are kept fixed. Given the above system parameters, the transient response time of a step change of the reference voltage is estimated to be τ = 2.35 ms. The wireless interface is IEEE 802.11n. Using Wireshark and a compatible Wi-Fi interface, we capture the handshake messages and map their binary versions directly onto the power talk interface. The length of the messages in bytes is 114 for the Association Request, 93 for the Association Response, 177 for h1, 177 for h2, 211 for h3, 155 for h4 and 32 for ACKs. The ADC sampling noise standard deviation is denoted with η = 8.58 · 10 −2 . Thus, the standard deviation of the power talk samples in each power talk slot can be calculated as σ = η √ ν(T pt −τ ) . We first demonstrate and verify the technical feasibility of the proposed authentication scheme. We use uncoded power talk modulation, i.e., we directly map the handshake binary representations of the messages to the power talk interface. The CC receives the association request packet from DER U in the PTARCh starting at t = 3 s. Fig. 5 output currents of the DERs and grid voltage during and after the handshake. Before authorization, it is clear that DER U , being connected to the MG only through primary control, does not participate into secondary control regulation and its output current is not contributing proportionally to the load. After the handshake ends and the secondary control becomes switched on, we observe that the output current of DER U quickly becomes aligned with the output currents of the other DERs, verifying that the authentication was successful and that DER U joined the secondary control. Fig. 6 illustrates a handshake during which a load change occurs, in t = 15 s. After resetting the detection threshold, the handshake resumes. Upon successful authorization to DER U and turning the secondary control on, we observe that the load is proportionally shared and the voltage is restored to its global reference. Taking into account that the load changes rather infrequently compared to the signaling rate of power talk [6], [8], sporadic load changes in the power talk channels can be efficiently mitigated using the detection reset strategy with negligible impact on the overall duration of the power talk periods. Moreover, the impact of any load change that occurs during the void PTARChs, i.e., when no DER sends association request, will be quickly eliminated after the secondary control is switched on, restoring the voltage to its global reference and fostering proportional current sharing. Figs. 5 and 6 also illustrate that the handshake can take up several seconds to complete via power talk which is a result of the fundamental fact that power talk is a narrowband communication technique. During the handshake period, if a load change occurs, the operating point will be suboptimal as the system is governed only by the primary control. However, we note that, as a proof of concept, the figures are derived using the actual handshake messages from IEEE 802.11 standard, which is a high-rate wireless interface with the symbol durations that are only fractions of microsecond. If the structure and length of the handshake messages and possibly the complete handshake procedure were redesigned to match the features of the power talk interface, the duration of the procedure could be significantly shortened. This would also significantly reduce the fraction of time during which the system is susceptible to load variations. Finally, Fig. 7 depicts the BER as a function of the reference voltage perturbation amplitudes γ and the duration of the power talk slot T pt . Clearly, the impact of the sampling noise becomes negligible as γ and T pt increase. Remarkably, already with γ = 0.01 (i.e., 0.02% of v ) volts and T pt = 0.01 s, the BER falls below 10 −7 , implying that the power talk interface can be used in conjunction with simple error correction codes with negligible redundancy and low complexity. VI. CONCLUSION If the MG control is simply placed on top of an existing communication technology, it will also inevitably inherit the corresponding security threats. According to this paradigm, adopted by previous works related to MG security, the system should include countermeasures against any known security vulnerability, to enable reliable and secure operation of the control system. In this paper we show that if the communication is secured using a non-conventional channel, i.e., the grid itself, it becomes safe against traditional cyber-attacks. In particular, we modeled and demonstrated an authentication scheme for IEEE 802.11 systems based on power talk. Its robustness stems from the fact that the initial handshake can be observed and altered by only being physically connected to the grid. Moreover, power talk can play a pivotal role in the overall security, e.g., by reporting communication outages of the primary channel and by distributing its encryption keys. These topics are included in our ongoing work. ACKNOWLEDGMENT The work presented in this paper was supported in part by EU, under grant agreement no. 607774 "ADVANTAGE". Fig. 1 : 1Representation of the four-way-handshake. Fig. 2 : 2Control diagram and flow of information in DC MG with centralized secondary/tertiary control and secure and reliable power talk based DER authentication. The red blocks are the newly added software components. Fig. 3 : 3Attacking the secondary control level via disabling initial handshake. Fig. 4 : 4Time organization of the proposed power talk-based DER authentication scheme. Fig. 5 : 5Realization of an authentication process. 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[ "BOUNDING THE PHOTON MASS WITH SOLAR PROBES", "BOUNDING THE PHOTON MASS WITH SOLAR PROBES" ]	[ "V Alan Kostelecký \nPhysics Department\nIndiana University Bloomington\n47405INU.S.A\n", "Michael Martin Nieto \nTheory Division Los Alamos National Laboratory\nUniversity of California Los Alamos\n87545NMU.S.A\n" ]	[ "Physics Department\nIndiana University Bloomington\n47405INU.S.A", "Theory Division Los Alamos National Laboratory\nUniversity of California Los Alamos\n87545NMU.S.A" ]	[]	Proposed close-encounter solar probes (Vulcan and Solar Probe) are planned to have highly eccentric orbits, with a perihelion of about 4R S and an inclination close to 90 • out of the plane of the ecliptic. We show this could allow at least an order-of-magnitude improvement in the present directly measured limit on the photon mass.	10.1016/0370-2693(93)91597-g	[ "https://export.arxiv.org/pdf/hep-ph/9304250v1.pdf" ]	119,090,896	hep-ph/9304250	2efec9df5b8b6ed7beea3a0014dccae30fb3a1a8	BOUNDING THE PHOTON MASS WITH SOLAR PROBES Apr 1993 February 1993 V Alan Kostelecký Physics Department Indiana University Bloomington 47405INU.S.A Michael Martin Nieto Theory Division Los Alamos National Laboratory University of California Los Alamos 87545NMU.S.A BOUNDING THE PHOTON MASS WITH SOLAR PROBES Apr 1993 February 1993arXiv:hep-ph/9304250v1 13 Proposed close-encounter solar probes (Vulcan and Solar Probe) are planned to have highly eccentric orbits, with a perihelion of about 4R S and an inclination close to 90 • out of the plane of the ecliptic. We show this could allow at least an order-of-magnitude improvement in the present directly measured limit on the photon mass. If the photon has a mass m γ , the Maxwell equations are replaced with the Proca equations [1]. In Gaussian units, these are ∂ µ F µν + µ 2 A ν = 4π c J ν ,(1) where F µν = ∂ µ A ν − ∂ ν A µ , the parameter µ = m γ c/h is the inverse Compton wavelength of the photon, and the four-vector potential A µ = (φ, A) is comprised, as usual, of the electric scalar potential φ and the magnetic vector potential A. For detailed discussions of these equations and their implications, see Refs. [2]. Current conservation, ∂ µ J µ = 0, enforces the Lorentz condition ∂ µ A µ = 0 and implies the breaking of gauge invariance, which under certain circumstances might be realized by spontaneous symmetry breaking (see Ref. [3] and references therein). Alternatively, a small photon mass may be generated in the context of string theory [4]. This paper describes a possible order-of-magnitude improvement in the present directly measured bound on the photon mass. For indirect limits, the reader is referred to the discussion and citations contained in [2]. The best laboratory limit on µ comes from extending the concentric-sphere test of Coulomb's law. The basic idea involves two concentric conducting spherical shells of radii a and b < a with a potential V applied to the outer shell. In general, in the presence of a nonzero photon mass, the electrostatic potential φ becomes a solution to the modified Poisson equation (∇ 2 − µ 2 )φ(x) = −4πρ(x) ,(2) subject to appropriate boundary conditions. Writing r = \|x\|, the solution between the concentric spheres is φ(x) = V a r sinh µr sinh µa .(3) The potential difference between the two spheres is no longer zero. For µa ≪ 1, φ(a) − φ(b) φ(a) ≃ 1 6 µ 2 (a 2 − b 2 ) + O[(µa) 4 ] .(4) Using a multi-shell application of this method, Ref. [5] obtained the bound µ ≤ 6 × 10 −10 cm −1 ≡ 10 −14 eV ≡ 2 × 10 −47 g . Note that in Eq. (4) the lowest-order physical effects of a nonzero photon mass appear at order (µL) 2 , where L is a length scale. This result is general [2], and it implies that an improved bound can be obtained either by a more precise measurement or by using a larger system. In particular, it suggests a consideration of the effects of a nonzero photon mass on magnetic field lines. The field from a point magnetic dipole D = Dẑ at the origin is, in polar coordinates withr ·ẑ = cos θ, B(r, θ) = D e −µr r 3 1 + µr + 1 3 µ 2 r 2 (3r cos θ −ẑ) − 2 3 µ 2 r 2ẑ .(6) This shows that the presence of a nonzero photon mass rescales the usual dipole field and introduces an additional term. On a sphere surrounding the dipole, the latter appears as a constant field antiparallel to the dipole. In 1943, Schrödinger suggested [6] using the earth's dipole field to limit µ. A modern analysis using satellite and ground-based observations provides a conservative bound of [7] µ ≤ 10 −10 cm −1 . This is an improvement over the best laboratory bound because the increased size of the earth dominates over the reduced precision obtainable in measuring the field. More generally, suppose a magnetic dipole is found to have strength S on the equator of a sphere of radius R centered about the dipole. If the smallest measurable field is δ and if no constant antiparallel field is observed, then Eq. (6) provides a bound on the photon mass of µ ∼ < 3δ 2S 1 R .(8) This shows that an increase of a factor of 10 in the length scale of an experiment is equivalent to an improvement of a factor of 100 in the sensitivity. The best bounds on the photon mass can therefore be obtained by going to larger systems. Consider, for example, the planet Jupiter with radius R J ≃ 7 × 10 4 km. Typical magnetometers aboard interplanetary probes have sensitivities δ/S ≃ 10 −4 , so repeated measurements along a trajectory at about 10R J could in principle yield a bound of µ ∼ < 10 −13 cm −1 . In practice, there are substantial complications from the nature of the jovian magnetosphere (see, for example, [8,9] In addition to Pioneer 10, there were three other Jupiter flyby missions in the 1970s: Pioneer 11, and Voyagers 1 and 2. Of these, the best trajectory for our purposes was that of Pioneer 11, which attained relatively high jovigraphic latitudes and a perijove of 1.6R J . The point is that to extract a good bound from Eq. (6) a clean separation of the usual dipole-type field from the additional term is useful, and this requires information about the field at latitudes away from the equator. Multipole fits to the planetary field using the newer data obtained could generate an improvement in the bound (9) by a factor of perhaps two. A further small improvement might eventually be made by incorporating multipole fits to data from the 1992 Ulysses encounter with Jupiter, for which the outbound pass attained relatively high southern latitudes [11]. Another possibility in the future is the Galileo probe [12], due to arrive at Jupiter in 1995. The orbiter portion of this craft will make several passes at distances ranging from an initial perijove of about 4R J to varying apojoves of about (100 ± 25)R J , before entering the 'tail-petal' orbit with apojove of about 150R J . However, the alignment of the orbits and their extension beyond the inner magnetosphere make unlikely a further improvement of much more than a factor of two in the bound on µ. The Sun provides another interesting opportunity to decrease further the upper limit on µ. Its radius is R S ≃ 7 × 10 5 km ≃ 10R J , which means any experiments would involve much larger length scales and so would access smaller values of µ. However, the solar magnetic field is more complex and less well understood than that of Jupiter (see, for example, [14]). There are additional complications stemming from such features as sunspots and coronal holes. Even the basic effects of solar rotation such as the Archimedes spiral [15] remain a subject of active research [16]. Moreover, the dipole field itself is time-dependent, changing strength, tilt, and even orientation with the solar cycle. Energetics have so far limited solar probe trajectories to within a few degrees latitude of the ecliptic, for which the dipole field is likely to dominate only well within the closest perihelia of about 65R S achieved by the Helios 1 and 2 missions. An improved bound on µ could, however, be obtained by a solar polar mission. A probe with a trajectory inclined near 90 • to the ecliptic offers the advantage of better separations between the dipole and the added terms in Eq. (6) and between analogous terms for higher-order multipoles. It also implies passage over the polar region, where it is likely that the magnetosphere is simpler because the solar wind and the rotation rate are reduced. The solar dipole moment at solar minimum is believed to be about (3 G) R 3 S , to within a factor of three. Given that current weak field magnetometers are sensitive to fields of order 10 −7 G, a probe would need to have a perihelion of no less than one or two A.U. if it is even to detect the solar dipole moment. One solar polar mission, Ulysses, is currently underway [13]. In 1994, the Ulysses explorer will reach the south polar region of the Sun in an orbit at an inclination of about 80 • to the ecliptic. It will spend about eight months above 70 • heliographic latitude and will pass over the poles at distances of about 2 A.U. However, given the complications likely to be present in the magnetosphere, even over the poles, placing a bound on µ probably requires a perihelion an order of magnitude smaller than this, within about 20R S . During the first decade of the next century, close-approach solar probes are envisioned both by NASA (Solar Probe) [17] and by ESA (Vulcan) [18,19]. These missions are planned to have highly eccentric orbits, with a perihelion limited to about 4R S by modern heat-shielding technology. To maximize heliographic latitude coverage, the orbits should be inclined at about 90 • to the ecliptic. Antenna-pointing requirements suggest a period commensurate with half an earth year. As an example, consider an operational orbit with a period of one earth year, an aphelion of 1.981 A.U., and an eccentricity 0.981 [20]. Then, the distances from the probe to the center of the Sun would be approximately 6R S , 8R S , and 12R S , at 20 • before, directly over, and 20 • past closest-polar approach, respectively. With a trajectory of this sort, the anticipated quiet conditions near solar minimum and the broad latitude coverage of the orbit should obviate difficulties arising from the local swamping of the solar dipole field by the fields due to coronal holes and sunspots. With a precision δ/S ≃ 10 −4 and data at a distance of about 10R S , Eq. (8) suggests that a bound as low as µ ≤ 10 −14 cm −1 might in principle be obtained. It is certainly plausible that repeated orbits could allow an improved photon-mass bound of µ ≤ 10 −12 cm −1 . We thank many colleagues for their helpful comments and observations. They . Only within the inner magnetosphere (up to about 10R J ) does the planetary field dominate over effects from external currents and the solar wind. Even within this region, there are contributions from external multipoles. Moreover, the planetary field itself has significant quadrupole and octupole moments and is tilted with respect to the axis of rotation. These and other complications prevent the attainment of the ideal bound suggested by Eq. (8). Using data from the Pioneer 10 flyby of Jupiter and fitting higher multipoles to the field in the inner magnetosphere, Ref. [10] derived a conservative bound of µ ≤ 2 × 10 −11 cm −1 . include: A.N. Cox, W.C. Feldman, H. Hill, J.G. Hills, T. Hoeksema, H.R. Johnson, J.R. 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[ "Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds", "Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds" ]	[ "Dominic Joyce " ]	[]	[]	Let X be a Calabi-Yau 3-fold, T = D b (coh(X)) the derived category of coherent sheaves on X, and Stab(T ) the complex manifold of Bridgeland stability conditions on T . It is conjectured that one can define invariants J α (Z, P) ∈ Q for (Z, P) ∈ Stab(T ) and α ∈ K(T ) generalizing Donaldson-Thomas invariants, which 'count' (Z, P)-semistable (complexes of) coherent sheaves on X, and whose transformation law under change of (Z, P) is known.This paper explains how to combine such invariants J α (Z, P), if they exist, into a family of holomorphic generating functions F α : Stab(T ) → C for α ∈ K(T ). Surprisingly, requiring the F α to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T ) with values in an infinite-dimensional Lie algebra L.The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.	10.2140/gt.2007.11.667	[ "https://export.arxiv.org/pdf/hep-th/0607039v1.pdf" ]	18,481,537	hep-th/0607039	6d54becdb6c18bccbcc67c49b3dda78f53f43115	Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds Jul 2006 Dominic Joyce Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds Jul 2006arXiv:hep-th/0607039v1 6 Let X be a Calabi-Yau 3-fold, T = D b (coh(X)) the derived category of coherent sheaves on X, and Stab(T ) the complex manifold of Bridgeland stability conditions on T . It is conjectured that one can define invariants J α (Z, P) ∈ Q for (Z, P) ∈ Stab(T ) and α ∈ K(T ) generalizing Donaldson-Thomas invariants, which 'count' (Z, P)-semistable (complexes of) coherent sheaves on X, and whose transformation law under change of (Z, P) is known.This paper explains how to combine such invariants J α (Z, P), if they exist, into a family of holomorphic generating functions F α : Stab(T ) → C for α ∈ K(T ). Surprisingly, requiring the F α to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T ) with values in an infinite-dimensional Lie algebra L.The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics. Introduction To set the scene we start with an analogy, which is explained by McDuff and Salamon [17]. If (M, ω) is a compact symplectic manifold, one can define the Gromov-Witten invariants Φ A (α, β, γ) of M . It is natural to encode these in a holomorphic generating function S : H ev (M, C) → C called the Gromov-Witten potential, given by a (formal) power series with coefficients the Φ A (α, β, γ). Identities on the Φ A (α, β, γ) imply that S satisfies a p.d.e., the WDVV equation. This p.d.e. can be interpreted as the flatness of a 1-parameter family of connections defined using S, which make H ev (M, C) into a Frobenius manifold. The goal of this paper (which we do not achieve) is to tell a story with many similar features. Let X be a Calabi-Yau 3-fold, coh(X) the abelian category of coherent sheaves on X, and T = D b (coh(X)) its bounded derived category. Let K(T ) be the image of the Chern character map K 0 (T ) → H even (X, Q), a lattice of finite rank. Define the Euler formχ : K(T ) × K(T ) → Z by (1) Thenχ is biadditive, antisymmetric, and nondegenerate. Following Bridgeland [3] one can define stability conditions (Z, P) on the triangulated category T , consisting of a group homomorphism Z : K(T ) → C called the central charge, and extra data P encoding the (Z, P)-semistable objects in T . The family of stability conditions Stab(T ) is a finite-dimensional complex manifold, with the map (Z, P) → Z a local biholomorphism Stab(T ) → Hom(K(T ), C). In String Theory terms, the 'stringy Kähler moduli space' of X should be thought of as a complex Lagrangian submanifold of Stab(T ), the subset of stability conditions represented by Super Conformal Field Theories. We would like to define invariants J α (Z, P) ∈ Q 'counting' (Z, P)-semistable objects in each class α ∈ K(T ) \ {0}, so roughly counting semistable sheaves. In the final version of the theory these should be extensions of Donaldson-Thomas invariants [6,20] and invariant under deformations of X, but for the present we may make do with the author's 'motivic' invariants defined in [15] for the abelian category case, which are not invariant under deformations of X. The important thing about the invariants J α (Z, P) is that their transformation laws under change of stability condition are known completely, and described in the abelian case in [15]. Basically J α (Z, P) is a locally constant function of (Z, P), except that when (Z, P) crosses a locus in Stab(T ) where α = α 1 + · · · + α n for α k ∈ K(T ) and Z(α 1 ), . . . , Z(α n ) all have the same phase e iφ in C \ {0}, then J α (Z, P) jumps by a multiple of J α1 (Z, P) · · · J αn (Z, P). This paper studies the problem of how best to combine such invariants J α (Z, P) into generating functions which should be continuous, holomorphic functions of (Z, P) on Stab(T ), a bit like the Gromov-Witten potential. In fact we shall define a function F α : Stab(T ) → C for each α ∈ K(T ) \ {0}, given by (α i , α j ) ,(2) where F n : (C × ) n → C are some functions to be determined, and C × = C \ {0}. Here the sum over graphs comes from the transformation laws (28) below for the J α (Z, P), determined in the abelian case in [15, §6.5]. Let us admit at once that there are two very major issues about (2) that this paper does not even attempt to solve, which is why the goals of the paper are not achieved. The first is that we do not define the invariants J α (Z, P). In the abelian category case A = coh(X), for Gieseker type stability conditions (τ, T, ), we do define and study such invariants J α (τ ) in [15]. But the extension to Bridgeland stability conditions on D b (coh(X)) still requires a lot of work. The second issue is the convergence of the infinite sum (2), and of other infinite sums below. I am not at all confident about this: it may be that (2) does not converge at all, or does so only in special limiting corners of Stab(T ), and I am not going to conjecture that (2) or other sums converge. Instead, we shall simply treat our sums as convergent. This means that the results of this paper are rigorous and the sums known to converge only in rather restricted situations: working with abelian categories A rather than triangulated categories T , and imposing finiteness conditions on A that do not hold for coherent sheaves A = coh(X), but do work for categories of quiver representations A = mod-KQ. The question we do actually answer in this paper is the following. Suppose for the moment that (2) converges in as strong a sense as necessary. What are the conditions on the functions F n (z 1 , . . . , z n ) for F α to be both continuous and holomorphic? Since the J α (Z, P) are not continuous in (Z, P), to make F α continuous the F n must have discontinuities chosen so that the jumps in J α (Z, P) and F n exactly cancel. The simplest example of this is that F n (z 1 , . . . , z n ) must jump by F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ) across the real hypersurface z l+1 /z l ∈ (0, ∞) in (C × ) n . We shall show that the condition that F α be holomorphic and continuous, plus a few extra assumptions on the symmetry and growth of the F n and the normalization F 1 ≡ (2πi) −1 , actually determine the F n uniquely. Furthermore, on the open subset of (C × ) n where F n is continuous it satisfies the p.d.e. dF n (z 1 , . . . , z n ) = n−1 k=1 F k (z 1 , . . . , z k )F n−k (z k+1 , . . . , z n ) · dz k+1 + · · · + dz n z k+1 + · · · + z n − dz 1 + · · · + dz k z 1 + · · · + z k . This in turn implies that the generating functions F α satisfy the p.d.e. dF α (Z, P) = − β,γ∈K(T )\{0}:α=β+γχ (β, γ)F β (Z, P)F γ (Z, P) d(Z(β)) Z(β) . (4) It seems remarkable that simply requiring the F α to be holomorphic and continuous implies they must satisfy the p.d.e. (4), which has appeared moreor-less out of nowhere. In the Gromov-Witten case the generating function S also satisfies a p.d.e., the WDVV equation. Note however that the WDVV equation holds because of identities upon Gromov-Witten invariants, but in our case (4) holds because of any identities not on the J α (Z, P) for fixed (Z, P), but rather because of identities on how the J α (Z, P) transform as (Z, P) changes. Just as the WDVV equation implies the flatness of a connection constructed using the Gromov-Witten potential, so we can interpret (4) in terms of flat connections. Define L to be the C-Lie algebra with basis formal symbols c α for α ∈ K(T ), and Lie bracket [c α , c β ] =χ(α, β)c α+β . Ignoring questions of convergence, define an L-valued connection matrix Γ on Stab(T ) by Γ(Z, P) = α∈K(T )\{0} F α (Z, P) c α ⊗ d(Z(α)) Z(α)(5) Then (4) implies that Γ is flat, that is, the curvature R Γ = dΓ+ 1 2 Γ∧Γ ≡ 0. But we do not expect that dΓ ≡ 0 and Γ ∧ Γ ≡ 0 as happens in the Gromov-Witten case, so we do not have a 1-parameter family of flat connections and a Frobenius manifold type structure. All this cries out for an explanation, but I do not have one. However, I am convinced that the explanation should be sought in String Theory, and that underlying this is some new piece of physics to do with Mirror Symmetry, just as the context of the derived category D b (coh(X)) of coherent sheaves on X is the core of the Homological Mirror Symmetry programme of Kontsevich [16]. So I am posting this paper on the hep-th archive to bring it to the attention of String Theorists, and I invite any physicists with ideas on its interpretation to please let me know. Two possible pointers towards an interpretation are discussed in §6. Firstly, ignoring convergence issues, we show that in the Calabi-Yau 3-fold triangulated category case the connection Γ above induces a flat connection on T Stab(T ), which is in fact the Levi-Civita connection of a flat holomorphic metric g C on Stab(T ), provided g C is nondegenerate. Secondly, again ignoring convergence issues, for λ ∈ C × and fixed a, b ∈ Z define a (0, 1)-form on Stab(T ) by Φ λ (Z, P) = α∈K(T )\{0} λ a e λ b Z(α) F α (Z, P) d(Z(α)) Z(α) . Then (4) implies an equation in (0, 2)-forms on Stab(T ): ∂ Φ λ (Z, P) īj = − 1 2 λ −a−2b (χ) ij ∂Φ λ (Z, P) iī ∂Φ λ (Z, P) jj ,(7) using complex tensor index notation, where (χ) ij is the (2, 0) part ofχ. This is a little similar to the holomorphic anomaly equation of Bershadsky et al. [1,2]. Here is a brief description of the paper. Despite this introduction we mostly work neither with Calabi-Yau 3-folds, nor with triangulated categories. Instead, we work with abelian categories A such as quiver representations mod-KQ, and slope stability conditions (µ, R, ) determined by a morphism Z : K(A) → C. Then we can use the author's series [12][13][14][15] on invariants counting µ-semistable objects in abelian categories; the facts we need are summarized in §2. Section 3 studies generating functions f α generalizing F α in (2), in the abelian category setting, and expressed in terms of Lie algebras L following [12][13][14][15]. In §3.1 we find conditions on F n for these f α to be holomorphic and continuous, including some conditions from the triangulated category case, and show that with a few extra assumptions any such functions F n are unique. In §3.2 we guess a p.d.e. generalizing (4) for the f α to satisfy, deduce that it implies (3), and use (3) to construct a family of functions F n by induction on n. Then §3. 3 shows that these F n constructed using (3) satisfy all the conditions of §3.1, and so are unique. Section 4 discusses L-valued flat connections Γ as above, and §5 the extension to triangulated categories. Finally, §6 explains how the ideas work out for Calabi-Yau 3-folds. Acknowledgements. I would like to thank Philip Candelas, Calin Lazaroiu, Balázs Szendrői, Richard Thomas, and especially Tom Bridgeland for useful conversations. I was supported by an EPSRC Advanced Research Fellowship whilst writing this paper. Background material The author has written six long, complicated papers [10][11][12][13][14][15] developing a framework for studying stability conditions (τ, T, ) on an abelian category A, and interesting invariants counting τ -semistable objects in A, and the transformation laws of these invariants under change of stability condition. Sections 2.1-2.2 explain only the minimum necessary for this paper; for much more detail, see [10][11][12][13][14][15]. Section 2.3 discusses the extension to triangulated categories. 2.1 The general set-up of [12][13][14][15] We start with a very brief summary of selected parts of the author's series [12][13][14][15] (B, C)× Ext j (A, B) → Ext i+j (A, C) bilinear for i, j, i + j = 0 or 1. Let K(A) be the quotient of the Grothendieck group K 0 (A) by some fixed subgroup. Suppose that if A ∈ A with [A] = 0 in K(A) then A ∼ = 0. To define moduli stacks of objects or configurations in A, we need some extra data, to tell us about algebraic families of objects and morphisms in A, parametrized by a base scheme U . We encode this extra data as a stack in exact categories F A on the category of K-schemes Sch K , made into a site with theétale topology. The K, A, K(A), F A must satisfy some complex additional conditions [12, Assumptions 7.1 & 8.1], which we do not give. In [12, §9- §10] we define data A, K(A), F A satisfying Assumption 2.1 in some large classes of examples, including the abelian category coh(X) of coherent sheaves on a projective K-scheme X, and the following: Example 2.2. A quiver Q is a finite directed graph. That is, Q is a quadruple (Q 0 , Q 1 , b, e), where Q 0 is a finite set of vertices, Q 1 is a finite set of arrows, and b, e : Q 1 → Q 0 are maps giving the beginning and end of each arrow. A representation (V, ρ) of Q consists of finite-dimensional K-vector spaces V v for each v ∈ Q 0 , and linear maps ρ a : V b(a) → V e(a) for each a ∈ Q 1 . A morphism of representations φ : (V, ρ) → (W, σ) consists of K-linear maps φ v : V v → W v for all v ∈ Q 0 with φ e(a) • ρ a = σ a • φ b(a) for all a ∈ Q 1 . Write mod-KQ for the abelian category of representations of Q. It is of finite length. Write N Q0 and Z Q0 for the sets of maps Q 0 → N and Q 0 → Z, where N = {0, 1, 2, . . .} ⊂ Z. Define the dimension vector dim(V, ρ) ∈ N Q0 ⊂ Z Q0 of (V, ρ) ∈ mod-KQ by dim(V, ρ) : v → dim K V v . This induces a surjective group homomorphism dim : K 0 (mod-KQ) → Z Q0 . Define K(mod-KQ) to be the quotient of K 0 (mod-KQ) by the kernel of dim . Then K(mod-KQ) ∼ = Z Q0 , and for simplicity we identify K(mod-KQ) and Z Q0 , so that for (V, ρ) ∈ mod-KQ the class [(V, ρ)] in K(mod-KQ) is dim(V, ρ). As in [12,Ex. 10.5] we can define a stack in exact categories F mod-KQ so that A = mod-KQ, K(mod-KQ), F mod-KQ satisfy Assumption 2.1. We will need the following notation [12,Def. 7.3], [14,Def. 3.8]: Definition 2.3. We work in the situation of Assumption 2.1. Define A set of A-data is a triple (I, , κ) such that (I, ) is a finite partially ordered set (poset) and κ : I → C(A) a map. In this paper we will be interested only in the case when is a total order, so that (I, ) is uniquely isomorphic to ({1, . . . , n}, ) for n = \|I\|. We extend κ to the set of subsets of I by defining κ(J) = j∈J κ(j). Then κ(J) ∈ C(A) for all ∅ = J ⊆ I, as C(A) is closed under addition. (8). Suppose (T, ) is a totally ordered set, and τ : C(A) → T a map. We call (τ, T, ) a stability condition on A if whenever α, β, γ ∈ C(A) with β = α + γ then either τ (α) < τ (β) < τ (γ), or τ (α) > τ (β) > τ (γ), or τ (α) = τ (β) = τ (γ). We call (τ, T, ) a weak stability condition on A if whenever α, β, γ ∈ C(A) with β = α + γ then either τ (α) τ (β) τ (γ), or τ (α) τ (β) τ (γ). Definition 2.5. Let (τ, T, ) be a weak stability condition on A, K(A) as above. Then we say that a nonzero object U in A is C(A) = [U ] ∈ K(A) : U ∈ A, U ∼ = 0 ⊂ K(A),(8)(i) τ -semistable if for all S ⊂ U with S ∼ = 0, U we have τ ([S]) τ ([U/S]); (ii) τ -stable if for all S ⊂ U with S ∼ = 0, U we have τ ([S]) < τ ([U/S]); and (iii) τ -unstable if it is not τ -semistable. Definition 2.6. Let Assumption 2.1 hold and (τ, T, ) be a weak stability condition on A. For α ∈ C(A) define Obj α ss (τ ) = [U ] ∈ Obj α A (K) : U is τ -semistable ⊂ Obj A (K). Write δ α ss (τ ) : Obj A (K) → {0, 1} for its characteristic function. We call (τ, T, ) a permissible weak stability condition if: (i) A is τ -artinian, that is, there are no chains of subobjects · · · ⊂ A 2 ⊂ A 1 ⊂ U in A with A n+1 = A n and τ ([A n+1 ]) τ ([A n /An+1 ]) for all n; and (ii) Obj α ss (τ ) is a constructible set in Obj A for all α ∈ C(A), using the theory of constructible sets and functions on Artin K-stacks developed in [10]. Examples of (weak) stability conditions on A = mod-KQ and A = coh(X) are given in [14, §4.3- §4.4]. Most of them are permissible. Here is [14,Ex. 4.14]. Example 2.7. Let Assumption 2.1 hold, and c, r : K(A) → R be group homomorphisms with r(α) > 0 for all α ∈ C(A). Define µ : C(A) → R by µ(α) = c(α)/r(α) for α ∈ C(A). Then µ is called a slope function on K(A), and (µ, R, ) is a stability condition on A. It will be useful later to re-express this as follows. Define the central charge Z : K(A) → C by Z(α) = −c(α) + ir(α). The name will be explained in §2.3. Then Z ∈ Hom K(A), C is a group homomorphism, and maps C(A) to the upper half plane H = {x + iy : x ∈ R, y > 0} in C. For α ∈ C(A), the argument arg •Z(α) lies in (0, π), and clearly µ(α) = − cot • arg •Z(α), where cot is the cotangent function. So (µ, R, ) can be recovered from Z. Since − cot : (0, π) → R is strictly increasing, it fixes orders in R. Thus (arg •Z, R, ) is an equivalent stability condition to (µ, R, ), that is, U ∈ A is µ-(semi)stable if and only if it is arg •Z-(semi)stable. Write Stab(A) = Z ∈ Hom(K(A), C) : Z(C(A)) ⊂ H, and the stability condition (µ, R, ) defined by Z is permissible .(9) In the cases we are interested in Stab(A) is an open subset of the complex vector space Hom(K(A), C), and so is a complex manifold. Such stability conditions can be defined on all the quiver examples of [12, §10], and they are automatically permissible by [14,Cor. 4.13]. In Example 2.2, as K(A) = Z Q0 and C(A) = N Q0 \ {0} we may write c, r as c(α) = v∈Q0 c v (dimα)(v) and r(α) = v∈Q0 r v (dim α)(v), where c v ∈ R and r v ∈ (0, ∞) for all v ∈ Q 0 . Thus Stab(A) = H Q0 ⊂ C Q0 . The usual notion of slope stability on A = coh(X) for X a smooth projective curve is a slight generalization of the above. We take c([U ]) to be the degree and r([U ]) the rank of U ∈ coh(X). But then for α ∈ C(A) coming from a torsion sheaf U we have r(α) = 0 and c(α) > 0, so we must allow µ to take values in (−∞, +∞], with µ(α) = +∞ if r(α) = 0. Here [14,Th. 4.4] is a useful property of weak stability conditions. We call 0 = A 0 ⊂ · · · ⊂ A n = U in Theorem 2.8 the Harder-Narasimhan filtration of U . Theorem 2.8. Let (τ, T, ) be a weak stability condition on an abelian category A. Suppose A is noetherian and τ -artinian. Then each U ∈ A admits a unique filtration 0 = A 0 ⊂ · · · ⊂ A n = U for n 0, such that S k = A k /A k−1 is τsemistable for k = 1, . . . , n, and τ ([ S 1 ]) > τ ([S 2 ]) > · · · > τ ([S n ]). A framework for discussing counting invariants Given A, K(A), F A satisfying Assumption 2.1 and weak stability conditions (τ, T, ), (τ ,T , ) on A, the final paper [15] in the series was mostly concerned with defining interesting invariants I α ss (τ ), J α (τ ), . . . which 'count' τ -semistable objects in class α for all α ∈ C(A), and computing the transformation laws which these invariants satisfy under changing from (τ, T, ) to (τ ,T , ). These different invariants all share a common structure, involving an algebra and a Lie algebra. We will now abstract this structure (which was not done in [15]) and express the various invariants of [15] as examples of this structure. We will need the following notation, from [15,Def.s 4.2,4.4 & 5.1]. In our first two definitions, S, U ( * , τ,τ ) are called transformation coefficients, and are combinatorial factors appearing in transformation laws from (τ, T, ) to (τ ,T , ). Definition 2.9. Let Assumption 2.1 hold, (τ, T, ), (τ ,T , ) be weak stability conditions on A, and ({1, . . . , n}, , κ) be A-data. If for all i = 1, . . . , n − 1 we have either (a) τ • κ(i) τ • κ(i + 1) andτ • κ({1, . . . , i}) >τ • κ({i + 1, . . . , n}) or (b) τ • κ(i) > τ • κ(i + 1) andτ • κ({1, . . . , i}) τ • κ({i + 1, . . . , n}), then define S({1, . . . , n}, , κ, τ,τ) = (−1) r , where r is the number of i = 1, . . . , n − 1 satisfying (a). Otherwise define S({1, . . . , n}, , κ, τ,τ) = 0. If (I, , κ) is A-data with a total order, there is a unique bijection φ : {1, . . . , n} → I with n = \|I\| and φ * ( ) = , and ({1, . . . , n}, , κ • φ) is A-data. Define S(I, , κ, τ,τ ) = S({1, . . . , n}, , κ • φ, τ,τ ). b) = κ(ψ −1 (b)). Define µ : {1, . . . , l} → C(A) by µ(a) = λ(ξ −1 (a)). Then τ • κ ≡ τ • λ • µ : I → T andτ • µ ≡τ (α) l a=1 S(ξ −1 ({a}), , λ, τ,τ )· (−1) l−1 l · m b=1 1 \|ψ −1 (b)\|! .(10) If (I, , κ) is A-data with a total order, there is a unique bijection φ : We say the change from (τ, T, ) to (τ ,T , ) is globally finite if this holds for C = Obj α A (K) (which is not constructible, in general) for all α ∈ C(A). Since any constructible C ⊆ Obj A (K) is contained in a finite union of Obj α A (K), globally finite implies locally finite. The following encapsulates the structure common to most of the invariants studied in [15], with some oversimplifications we discuss in Remark 2.13. Assumption 2.12. Let Assumption 2.1 hold. Suppose we are given a C-algebra H with identity 1 and multiplication * (which is associative, but not in general commutative), with a decomposition into C-vector subspaces H = α∈C(A) H α , such that 1 ∈ H 0 and H α * H β ⊆ H α+β for all α, β ∈C(A). Suppose we are given a C- Lie subalgebra L of H with Lie bracket [f, g] = f * g − g * f , with a decomposition into C-vector subspaces L = α∈C(A) L α such that L α ⊆ H α and [L α , L β ] ⊆ L α+β for all α, β ∈ C(A). Whenever (τ, T, ) is a permissible weak stability condition on A, let there be given elements δ α (τ ) ∈ H α and ǫ α (τ ) ∈ L α for all α ∈ C(A). These satisfy ǫ α (τ ) = A-data ({1, . . . , n}, , κ) : κ({1, . . . , n}) = α, τ • κ ≡ τ (α) (−1) n−1 n δ κ(1) (τ ) * δ κ(2) (τ ) * · · · * δ κ(n) (τ ), (11) δ α (τ ) = A-data ({1, . . . , n}, , κ) : κ({1, . . . , n}) = α, τ • κ ≡ τ (α) 1 n! ǫ κ(1) (τ ) * ǫ κ(2) (τ ) * · · · * ǫ κ(n) (τ ),(12) where there are only finitely many nonzero terms in each sum. If (τ, T, ), (τ ,T , ) are permissible weak stability conditions on A and the change from (τ, T, ) to (τ ,T , ) is globally finite, for all α ∈ C(A) we have δ κ(1) (τ ) * δ κ(2) (τ ) * · · · * δ κ(n) (τ ) = δ α (τ ),(13) A-data ({1, . . . , n}, , κ) : κ({1, . . . , n}) = α U ({1, . . . , n}, , κ, τ,τ)· ǫ κ(1) (τ ) * ǫ κ(2) (τ ) * · · · * ǫ κ(n) (τ ) = ǫ α (τ ),(14) where there are only finitely many nonzero terms in each sum. Equation (14) may be rewritten: total orders on I. ǫ α (τ ) =Write I = {i1, . . . , in}, i1 i2 · · · in U (I, , κ, τ,τ )· ǫ κ(i1) (τ ) * · · · * ǫ κ(in) (τ ) .(15) The term [· · · ] in (15) is a finite Q-linear combination of multiple commutators of ǫ κ(i) for i ∈ I, and so it lies in the Lie algebra L, not just the algebra H. Thus (14) and (15) can be regarded as identities in L rather than H. Remark 2.13. (a) δ α (τ ) is an invariant of the moduli space Obj α ss (τ ) of τsemistable objects in class α, which 'counts' such τ -semistable objects. Usually it is of the form δ α (τ ) = Φ(δ α ss (τ )), where δ α ss (τ ) ∈ CF(Obj A ) is the characteristic function of Obj α ss (τ ), CF(Obj A ) is the vector space of constructible functions on the Artin K-stack Obj A as in [10], and Φ : CF(Obj A ) → H is a linear map with special multiplicative properties. (b) The ǫ α (τ ) are an alternative set of generators to the δ α (τ ). Here (12) is the inverse of (11), and given (10)- (12), equations (13) and (14) are equivalent. Thus, the main nontrivial claim about the ǫ α (τ ) is that they lie in the Lie algebra L, which may be much smaller than H. Roughly speaking, the ǫ α (τ ) count τsemistable objects S in class α weighted by a rational number depending on the factorization of S into τ -stables, which is 1 if U is τ -stable. If S is decomposable this weight is 0, so ǫ α (τ ) counts only indecomposable τ -semistables. The Lie algebra L is the part of H 'supported on indecomposables'. (c) In [15] we worked with (Lie) algebras over Q, not C. But here we complexify, as we shall be discussing holomorphic functions into H, L. (d) In parts of [15], equations (13)- (15) are only proved under an extra assumption, the existence of a third weak stability condition (τ ,T , ) compatible with (τ, T, ), (τ ,T , ) in certain ways. But we will not worry about this. (e) In parts of [15] we relax the assumption that (τ, T, ), (τ ,T , ) are permissible (taking them instead to be τ -artinian, or essentially permissible), and we allow the change from (τ, T, ) to (τ ,T , ) to be locally finite rather than globally finite. Then equations (11)- (15) need no longer have only finitely many nonzero terms, and they are interpreted using a notion of convergence in H. (f ) The δ α (τ ), ǫ α (τ ) are only the simplest of the invariants studied in [15] we could call them 'one point invariants', as they depend on only one class α ∈ C(A). We also considered systems of 'n point invariants' depending on n classes α ∈ C(A), which will not enter this paper. One thing that makes the one point invariants special is that their transformation laws (13)- (14) depend only on other one point invariants, not on n point invariants for all n 1. The next six examples explain how various results in [15] fit into the framework of Assumption 2.12. Example 2.14. Let Assumption 2.1 hold with K of characteristic zero. Take H = CF(Obj A ) ⊗ Q C, the vector space of C-valued constructible functions on Obj A , and H α the subspace of functions supported on Obj α A . The multiplication * on H, studied at length in [13], has the following approximate form: for V ∈ A, (f * g)([V ]) is the 'integral' over all short exact sequences 0 → U → V → W → 0 in A of f ([U ])g([W ]) , with respect to a measure defined using the Euler characteristic of constructible subsets of K-stacks. The identity is 1 = δ [0] , the characteristic function of [0] ∈ Obj A (K). The Lie subalgebra L is CF ind (Obj A ) ⊗ Q C, functions supported on points [U ] for U ∈ A indecomposable, and δ α (τ ) = δ α ss (τ ), the characteristic function of Obj α ss (τ ). Then [13][14][15] show Assumption 2.12 holds, except that (13)- (15) are only proved under extra conditions as in Remark 2.13(d) above. We can also replace H = CF(Obj A ) ⊗ Q C and L = CF ind (Obj A ) ⊗ Q C by the much smaller (Lie) subalgebras H to τ ⊗ Q C, L to τ ⊗ Q C of [14, §7], since by [15, §5] these are very often independent of the choice of permissible weak stability condition (τ, T, ) used to define them. Example 2.15. Let Assumption 2.1 hold. Take H = SF al (Obj A ) ⊗ Q C, the algebra of stack functions on Obj A with algebra stabilizers defined in [13, §5], using the theory of stack functions from [11], a universal generalization of constructible functions. Let L = SF ind al (Obj A ) ⊗ Q C, the subspace of H supported on 'virtual indecomposables', and let H α , L α be the subspaces of H, L supported on Obj α A . Set δ α (τ ) =δ α ss (τ ), in the notation of [14]. Then [13][14][15] show Assumption 2.12 holds, but with (13)- (15) only proved under extra conditions. This also works with SF al (Obj A ) replaced by one of the 'twisted stack function' spacesSF al (Obj A , Υ, Λ),SF al (Obj A , Υ, Λ • ),SF al (Obj A , Θ, Ω) of [13]. We can also replace H = SF al (Obj A ) ⊗ Q C and L = SF ind al (Obj A ) ⊗ Q C by the much smaller (Lie) subalgebrasH to τ ⊗ Q C,L to τ ⊗ Q C of [14, §8], since by [15, §5] these are very often independent of the choice of permissible weak stability condition (τ, T, ) used to define them. dim K Hom(U, V ) − dim K Ext 1 (U, V ) = χ [U ], [V ] for all U, V ∈ A. (16) This happens when A = coh(X) with X a smooth projective curve, and for A = mod-KQ in Example 2.2 with χ given by the Ringel form χ(α, β) = v∈Q0 α(v)β(v) − a∈Q1 α(b(a))β(e(a)) for α, β ∈ Z Q0 .(17) Define Λ = C(z), the algebra of rational functions p(z)/q(z) for polynomials p, q with coefficients in C and q = 0, and define a special element ℓ = z 2 in Λ. Define Λ • to be the subalgebra of p(z)/q(z) in Λ for which z ± 1 do not divide q. The facts we need about Λ, Λ • are that the virtual Poincaré polynomial P (X; z) of a K-variety X takes values in Λ • ⊂ Λ, and ℓ = P (K; z) for K the affine line, and ℓ and ℓ k + ℓ k−1 + · · · + 1 are invertible in Λ • , and ℓ − 1 is invertible in Λ. Let a α for α ∈C(A) be formal symbols, and define H = A(A, Λ, χ) as in [13, §6.2] to be the Λ-module with basis {a α : α ∈C(A)}, with the obvious notions of addition and multiplication by C. Define a multiplication * on H by i∈I λ i a αi * j∈J µ j a βj = i∈I j∈J λ i µ j ℓ −χ(βj ,αi) a αi+βj .(18) Then H is a C-algebra, with identity a 0 . Define H α = Λ · a α for α ∈C(A). Define L α = (ℓ − 1) −1 Λ • · a α ⊂ H α for α ∈ C(A), and L = α∈C(A) L α . Then L is a Lie subalgebra of H, as (ℓ −χ(β,α) − ℓ −χ(α,β) )/(ℓ − 1) ∈ Λ • . For (τ, T, ) a permissible weak stability condition on A and α ∈ C(A), define δ α (τ ) = I α ss (τ )a α , where I α ss (τ ) is the virtual Poincaré function of Obj α ss (τ ), as defined in [11, §4.2], where we regard Obj α ss (τ ) as a finite type open Ksubstack with affine geometric stabilizers in the Artin K-stack Obj α A . Define ǫ α (τ ) by (11). Then ǫ α (τ ) ∈ H α , so we can write ǫ α (τ ) = (ℓ − 1) −1 J α (τ )a α for J α (τ ) ∈ Λ. We show in [15,Th. 6.8] that J α (τ ) ∈ Λ • , so ǫ α (τ ) ∈ L α . Then [13, §6.2] and [15, §6.2] show that Assumption 2.12 holds in its entirety when A = mod-KQ, and with extra conditions as in Remark 2.13(d) above in general. It also holds with Λ replaced by other commutative C-algebras, and virtual Poincaré polynomials replaced by other Λ-valued 'motivic invariants' Υ of K-varieties with ℓ = Υ(K); for details see [11,13,15]. Example 2.17. Let K be an algebraically closed field and X a smooth projective surface over K with K −1 X numerically effective (nef ). Take A = coh(X) with data K(A), F A satisfying Assumption 2.1 as in [12,Ex. 9.1]. Then there is a biadditive χ : K(A) × K(A) → Z such that for all U, V ∈ A we have dim K Hom(U, V ) − dim K Ext 1 (U, V ) + dim K Ext 2 (U, V ) = χ [U ], [V ] . (19) Define Λ, H, * , H α as in Example 2.16, but set L α = H α for α ∈ C(A) and L = α∈C(A) L α . Then in [15, §6.4] , for a class of weak stability conditions (τ, T, ) on A based on Gieseker stability, we define invariants I α ss (τ ),J α (τ ) ∈ Λ such that Assumption 2.12 holds with δ α (τ ) = I α ss (τ )a α and ǫ α (τ ) = (ℓ − 1) −1J α (τ )a α . But we do not prove thatJ α (τ ) ∈ Λ • , which is why we modify the definitions of L α , L. Examples 2.16 and 2.17 illustrate the relationship between 'invariants' I α ss (τ ), J α (τ ),J α (τ ) which 'count' τ -semistables in class α, and our (Lie) algebra approach. In this case, the transformation laws (13)- (14) for δ α (τ ), ǫ α (τ ) are equivalent to the following laws for I α ss (τ ), J α (τ ), from [15, Th. 6.8]: I α ss (τ ) = A-data ({1, . . . , n}, , κ) : κ({1, . . . , n}) = α S({1, . . . , n}, , κ, τ,τ) · ℓ − 1 i<j n χ(κ(j),κ(i)) · n i=1 I κ(i) ss (τ ),(20)J α (τ ) = A-data ({1, . . . , n}, , κ) : κ({1, . . . , n}) = α U ({1, . . . , n}, , κ, τ,τ) · ℓ − 1 i<j n χ(κ(j),κ(i)) · (ℓ − 1) 1−n n i=1 J κ(i) (τ ).(21) Observe that (13)- (14) are simpler than (20)- (21), since the powers of ℓ in (20)- (21) are packaged in the multiplication * in H. This is more pronounced in our next two examples, where the formulae for * are much more complicated, so the transformation laws for invariants are too. One moral is that working in the framework of Assumption 2.12 is simpler than working with systems of invariants, which is why we have adopted it. . Define H α = [I,κ]:κ(I)=α Λ · b [I,κ] . Define a multiplication * on H by b [I,κ] * b [J,λ] = eq. classes [K, µ] b [K,µ] · (ℓ − 1) \|K\|−\|I\|−\|J\| \| Aut(K, µ)\| · (22) iso. classes of finite sets L (−1) \|L\|−\|K\| \|L\|! φ : I → L, ψ : J → L and θ : L → K: φ∐ψ surjective, µ(k) = κ((θ • φ) −1 (k))+ λ((θ • ψ) −1 (k)), k ∈ K k∈K (\|θ −1 (k)\|−1)! i∈I, j∈J: φ(i)=ψ(j) ℓ −χ(λ(j),κ(i)) , extended Λ-bilinearly. Then H is a C-algebra with identity b [∅,∅] . For α ∈ C(A) define b α = b [{1},α ′ ] where α ′ (1) = α, define L α = Λ • · b α and L = α∈C(A) L α . Equation (22) yields [b α , b β ] = ℓ −χ(β,α) − ℓ −χ(α,β) ℓ − 1 b α+β ,(23) and 16. We then define δ α (τ ) by (12), giving a much more complicated answer than in Example 2.16. From [13, §6.3] and [15, §6] it follows that Assumption 2.12 holds in its entirety when A = mod-KQ, and with extra conditions as in Remark 2.13(d) above in general. (ℓ −χ(β,α) − ℓ −χ(α,β) )/(ℓ − 1) ∈ Λ • , so L is a Lie subalgebra of H. If (τ, T, ) is a permissible weak stability condition we put ǫ α (τ ) = J α (τ )b α for the same J α (τ ) ∈ Λ • as in Example 2.dim K Hom(U, V ) − dim K Ext 1 (U, V ) − dim K Hom(V, U ) − dim K Ext 1 (V, U ) =χ [U ], [V ] for all U, V ∈ A.(24) Note that (16) implies (24) withχ(α, β) = χ(α, β) − χ(β, α), so this holds for A = mod-KQ and A = coh(X) for X a smooth projective curve. But we also show in [13, §6.6] using Serre duality that (24) holds when A = coh(X) for X a Calabi-Yau 3-fold over K. c [I,κ] * c [J,λ] = eq. classes [K, µ] c [K,µ] · 1 \| Aut(K, µ)\| η : I → K, ζ : J → K: µ(k) = κ(η −1 (k)) + λ(ζ −1 (k)) simply-connected directed graphs Γ: vertices I ∐ J, edges i • → j •, i ∈ I, j ∈ J, conn. components η −1 (k) ∐ ζ −1 (k), k ∈ K edges i • → j • in Γ 1 2χ (κ(i), λ(j)) ,(25)extended C-bilinearly. Then H is a C-algebra with identity c [∅,∅] . For α ∈ C(A) define c α = c [{1},α ′ ] where α ′ (1) = α, define L α = C · c α and L = α∈C(A) L α . Equation (25) yields [c α , c β ] =χ(α, β) c α+β , so L is a Lie subalgebra. Then [15, §6.5] defines invariants J α (τ ) ∈ Q for α ∈ C(A) , such that if we set ǫ α (τ ) = J α (τ ) c α and define δ α (τ ) by (12) then Assumption 2.12 holds, with extra conditions as in Remark 2.13(d) above. These invariants J α (τ ) are defined using the Euler characteristic of constructible sets in Artin K-stacks, in a rather subtle way. As Euler characteristic and virtual Poincaré polynomials are related by χ(X) = P (X; −1), these are specializations of the virtual Poincaré polynomial invariants of Examples 2.16 and 2.18. Note that we cannot define the δ α (τ ) directly, but only reconstruct them from the ǫ α (τ ). In the notation of Example 2.15, this is because ǫ α (τ ) is defined using a Lie algebra morphism Ψ : SF ind al (Obj A ) → L which does not extend to an algebra morphism Ψ : SF al (Obj A ) → H, so we cannot define δ α (τ ) = Ψ(δ α ss (τ )) as we might hope. The above also holds with C replaced by other commutative C-algebras Ω, and Euler characteristics replaced by other Ω-valued 'motivic invariants' Θ of K-varieties with Θ(K) = 1; for details see [11,13,15]. To rewrite (14) as a transformation law for the J α (τ ) we need to compute c α1 * · · · * c αn in H. Actually it is enough to know the projection of this to L. As in [15, §6.5], calculation shows this is given by: c α1 * · · · * c αn = terms in c [I,κ] , \|I\| > 1,(26)+ 1 2 n−1 connected, simply-connected digraphs Γ: vertices {1, . . . , n}, edge i • → j • implies i < j edges i • → j • in Γχ (α i , α j ) c α1+···+αn . Here a digraph is a directed graph. Let (τ, T, ), (τ ,T , ) be weak stability conditions on A, Γ be a connected, simply-connected digraph with finite vertex set I, and κ : I → C(A). Define V (I, Γ, κ, τ,τ ) ∈ Q by V (I, Γ, κ, τ,τ ) = 1 2 \|I\|−1 \|I\|! total orders on I: edge i • → j • in Γ implies i j U (I, , κ, τ,τ ).(27) Then using ǫ α (τ ) = J α (τ ) c α and (26), it turns out [15,Th. 6.28] that (14) is equivalent to J α (τ ) = iso. classes of finite sets I κ:I→C(A): κ(I)=α connected, simply-connected digraphs Γ, vertices I V (I, Γ, κ, τ,τ ) · edges i • → j • in Γ χ(κ(i), κ(j)) · i∈I J κ(i) (τ ).(28) Example 2.19 is the reason why the title of the paper involves Calabi-Yau 3-folds, why we believe that the ideas of this paper have to do with Mirror Symmetry and String Theory, and why we want to bring them to the attention of String Theorists in particular so that they may explain them in physical terms. In brief, the point is this. In [15, §6.5], as the culmination of a great deal of work in [10][11][12][13][14][15], the author defined invariants J α (τ ) ∈ Q 'counting' τ -semistable sheaves in class α ∈ K(A) on a Calabi-Yau 3-fold X, which transform according to a complicated transformation law (28) under change of weak stability condition, reminiscent of Feynman diagrams. The author expects that some related invariants which extend Donaldson-Thomas invariants and transform according to the same law (28) should be important in String Theory, perhaps counting numbers of branes or BPS states. For conjectures on this see [15, §6.5]. This paper will study natural ways of combining these invariants in holomorphic generating functions; the author expects that these generating functions, and the equations they satisfy, should also be significant in String Theory. Comments on the extension to triangulated categories The series [12][13][14][15] studied only abelian categories, such as the coherent sheaves coh(X) on a projective K-scheme X. But for applications to String Theory and Mirror Symmetry, the whole programme should be extended to triangulated categories, such as the bounded derived category D b (coh(X)) of coherent sheaves on X. The issues involved in this are discussed in [15, §7]. For a recent survey on derived categories of coherent sheaves on Calabi-Yau m-folds, see Bridgeland [4]. One justification for this is Kontsevich's Homological Mirror Symmetry proposal [16], which explains Mirror Symmetry of Calabi-Yau 3-folds X,X as an equivalence between D b (coh(X)) and the derived Fukaya category D b (F (X)) of X. This relates the complex algebraic geometry of X, encoded in D b (coh(X)), to the symplectic geometry ofX, encoded in D b (F (X)). Building on Kontsevich's ideas, triangulated categories of branes have appeared in String Theory in the work of Douglas, Aspinwall, Diaconescu, Lazaroiu and others. The following notion of stability condition on a triangulated category, due to Bridgeland [3, §1.1], will be important in this programme. For background on triangulated categories, see Gelfand and Manin [7]. Definition 2.20. Let T be a triangulated category, and K(T ) the quotient of its Grothendieck group K 0 (T ) by some fixed subgroup. For instance, if T is of finite type over a field K we can take K(T ) to be the numerical Grothendieck group K num (T ) as in [3, §1.3], and then Bridgeland calls the resulting stability conditions numerical stability conditions. A stability condition (Z, P) on T consists of a group homomorphism Z : K(T ) → C called the central charge, and full subcategories P(φ) ⊂ T for each φ ∈ R of semistable objects with phase φ, satisfying: (a) If S ∈ P(φ) then Z([S]) = m([S])e iπφ for some m([S]) ∈ (0, ∞); (b) for all t ∈ R, P(t + 1) = P(t)[1]; (c) if t 1 > t 2 and S j ∈ P(t j ) for j = 1, 2 then Hom T (S 1 , S 2 ) = 0; and (d) for 0 = U ∈ T there is a finite sequence t 1 > t 2 > · · · > t n in R and a collection of distinguished triangles with S j ∈ P(t j ) for all j: 0 = A 0 G G A 1 G G Ð Ð Ò Ò Ò Ò Ò A 2 G G Ð Ð Ò Ò Ò Ò Ò · · · G G A n−1 G G A n = U. { { w w w w w w S 1 i i i S 2 ``S n f f f(29) This is the generalization to triangulated categories of the slope function stability conditions of Example 2.7. In both cases we have a central charge homomorphism Z : K(A) → C or Z : K(T ) → C, and semistability can be expressed in terms of arg •Z. In the abelian case arg •Z takes a unique value in (0, π), but in the triangulated case one has to choose a value of arg •Z and lift phases from R/2πZ to R. This need to choose phases is why in the abelian case Z determines the stability condition, but in the triangulated case we also need extra data P. Equation (29) is the analogue of Theorem 2.8, since both decompose an arbitrary object U ∈ A or T into semistable objects S 1 , . . . , S n with phases satisfying µ( S 1 ) > · · · > µ(S n ) or t 1 > · · · > t n . There is also a generalized notion of stability condition on T due to Gorodentscev et al. [9], not involving a central charge, which is closer in spirit to Definition 2.4 above. But we will not use it. Here is Bridgeland's main result [3, Th. 1.2], slightly rewritten: (Z, P) → Z is a local homeomorphism Σ → V Σ . When V Σ is finite-dimensional, which happens automatically when K(T ) has finite rank, Σ can be given the structure of a complex manifold uniquely so that (Z, P) → Z is a local biholomorphism Σ → V Σ . Bridgeland's stability conditions were motivated by Douglas' work on Pistability, and are natural objects in String Theory. Suppose we wish to define some kind of generating function f α encoding invariants 'counting' (Z, P)semistable objects in class α in T . These invariants will depend on (Z, P), so the generating function f α should be a function on Stab(T ) (and perhaps in other variables as well). Now Theorem 2.21 shows Stab(T ) is a complex manifold, so it makes sense to require f α to be a holomorphic function on Stab(T ). We can also try to make f α continuous, despite the fact that the invariants it encodes will change discontinuously over real hypersurfaces in Stab(T ). This problem also makes sense in the abelian setting of Example 2.7, where we can try to define a generating function f α which is a continuous, holomorphic function on the complex manifold Stab(A) of (9). In fact most of the rigorous part of the paper is about Example 2.7, but we have done it in a way that the author expects will generalize to the triangulated case when (if ever) the extension of [12][13][14][15] to triangulated categories has been worked out. Then we have a complex manifold Stab(A) of central charges Z, each of which defines a permissible stability condition (µ, R, ). For this µ we have invariants δ α (µ) ∈ H α and ǫ α (µ) ∈ L α for all α ∈ C(A). Regarded as functions of Z, these δ α (µ), ǫ α (µ) change discontinuously across real hypersurfaces in Stab(A) where arg •Z(β) = arg •Z(γ) for β, γ ∈ C(A) according to the transformation laws (13)-(14), and away from such hypersurfaces are locally constant. Holomorphic generating functions For α ∈ C(A) we shall consider a generating function f α : Stab(A) → H α of the following form, where (µ, R, ) is the stability condition induced by Z: f α (Z) = n 1, α1,...,αn∈C(A): α1+···+αn=α F n Z(α 1 ), . . . , Z(α n ) ǫ α1 (µ) * ǫ α2 (µ) * · · · * ǫ αn (µ). (30) We explain why we chose this form, and what conditions the F n must satisfy: Remark 3.1. (a) The general form of (30) is modelled on (11)- (14) above. The functions F n should map (C × ) n → C, where C × = C \ {0}. For the abelian category case of Example 2.7 we have Z(α) ∈ H = {x + iy : x ∈ R, y > 0} for α ∈ C(A), so it would be enough to define F n only on H n . However, for the extension to the triangulated category case discussed in §2.3 we must allow Z(α k ) ∈ C × , which is why we chose the domain (C × ) n . (b) We require that the functions F n satisfy F n (λz 1 , . . . , λz n ) = F n (z 1 , . . . , z n ) for all λ, z 1 , . . . , z n ∈ C × . The reason is easiest to explain in the triangulated category case. Let T , K(T ) and (Z, P) be as in Definition 2.20, and let r > 0 and ψ ∈ R. Define a new stability condition ( Z ′ , P ′ ) on T by Z ′ = re iψ Z and P ′ (φ) = P(φ − ψ/π). This gives an action of (0, ∞)×R on Stab(T ), which does not change the sets of (Z, P)-semistable objects, but only their phases φ. So we expect that in an appropriate extension of Assumption 2.12 to the triangulated case, the invariants δ α (Z, P), ǫ α (Z, P) 'counting' (Z, P)-semistable objects in class α should be also unchanged by this action. Therefore we can try and make f α and each term in (30) invariant under Z → re iψ Z, which is equivalent to (31). We can make a similar argument in the abelian case Example 2.7, but we have to restrict to re iψ such that re iψ Z C(A) ⊂ H, which makes the argument less persuasive. Requiring f α and F n instead to be homogeneous of degree d ∈ Z, so that F n (λz 1 , . . . , λz n ) = λ d F n (z 1 , . . . , z n ) for all λ, z k , is equivalent to replacing f α (Z) by Z(α) d f α (Z). So we lose nothing by restricting to d = 0. (c) Equation (31) implies that F 1 is constant, say F 1 ≡ c. For λ ∈ C × we may replace f α , F n , c by λf α , λF n , λc without changing whether f α is holomorphic or continuous, so all nonzero choices of c are equivalent. We shall take F 1 ≡ (2πi) −1 ,(32) as this simplifies formulae in §3.2 and the rest of the paper. Think of (30) as saying f α (Z) = c ǫ α (µ)+'higher order terms'. If ǫ α (µ) is an invariant 'counting' µ-semistables in class α, then so is f α (Z), to leading order. But ǫ α (µ) changes discontinuously with Z, whereas f α (Z) includes higher order correction terms which smooth out these changes and make f α continuous. (d) Following equations (14) and (15), we may rewrite (30) as f α (Z) = iso classes of finite sets I 1 \|I\|! κ:I→C(A): κ(I)=α total orders on I. Write I = {i1, . . . , in}, i1 i2 · · · in F \|I\| (Z • κ(i 1 ), . . . , Z • κ(i n ))· ǫ κ(i1) (µ) * · · · * ǫ κ(in) (µ) .(33) As for (15), we shall require the functions F n to have the property that the term [· · · ] in (33) is a finite C-linear combination of multiple commutators of ǫ κ(i) for i ∈ I, and so it lies in the Lie algebra L, not just the algebra H. Thus (30) and (33) make sense in L, and f α actually maps Stab(A) → L α . This is why we choose to write (30) in terms of the ǫ α (µ) rather than the δ α (µ). By substituting (11) into (30) we get another equation of the same form for f α , but with δ αi (µ) instead of ǫ αi (µ), and different functions F n . But using the ǫ αi (µ) means we can work in L rather than H, which is a great simplification if L is much smaller than H. This happens in Example 2.19, our motivating Calabi-Yau 3-fold example, when L α = C · c α so f α is really just a holomorphic function, but H α is in general infinite-dimensional. Now for \|I\| > 1, if a C-linear combination of products of ǫ κ(i) (µ) for i ∈ I is a sum of multiple commutators, it is easy to see that the sum of the coefficients of the products must be zero in C. Thus, a necessary condition for [· · · ] in (33) to be a linear combination of multiple commutators is that σ∈Sn F n (z σ(1) , . . . , z σ(n) ) = 0 for all n > 1 and (z 1 , . . . , z n ) ∈ (C × ) n , (34) where S n is the symmetric group of permutations σ of {1, . . . , n}. (e) We require that f α be a continuous and holomorphic function on Stab(A). These translate to conditions on the functions F n . In §3.1 we will compute the conditions on F n for f α to be continuous; it turns out that across real hypersurfaces arg z l = arg z l+1 , F n must jump by expressions in F k for k < n. For f α to be holomorphic, it is enough that the F n be holomorphic wherever they are continuous. Thus, F n is a branch of a multivalued holomorphic function, except along arg z l = arg z l+1 where it jumps discontinuously from one branch to another; but the discontinuities in ǫ α (µ) and F n (· · · ) cancel out to make f α continuous. A simple comparison is a branch of log z on C × , cut along (0, ∞). (f ) We shall ensure uniqueness of the F n by imposing a growth condition: F n (z 1 , . . . , z n ) = o \|z k \| −1 as z k → 0 with z l fixed, l = k, for all k. (35) This may assist the convergence of (30) in situations when the sum is infinite. For the moment we impose the following extra condition. It implies there are only finitely many possibilities for n and α 1 , . . . , α n in (30), and so avoids problems with infinite sums and convergence. It holds for all the quiver examples A = mod-KQ, . . . of [12, §10], but not for A = coh(X) when dim X > 0. In the rest of the section we construct functions F n satisfying the requirements of Remark 3.1, and show that they are unique, and satisfy interesting partial differential equations. Conditions on the functions Then using (14) with α i ,μ, µ in place of α, τ,τ respectively to express ǫ αi (µ) in (30) in terms of ǫ κ(j) (μ), using (36) and rewriting, we find that f α (Z) = n 1, α1,...,αn∈C(A): α1+···+αn=α ǫ α1 (μ) * ǫ κ(2) (μ) * · · · * ǫ αn (μ)· (37) m=1,...,n, 0=a0<a1< ···<am=n F m Z(α a0+1 +· · ·+α a1 ), . . . , Z(α am−1+1 +· · ·+α am ) m k=1 u a k −a k−1 Z (α a k−1 +1 ), . . . ,Z(α a k ); Z(α a k−1 +1 ), . . . , Z(α a k ) . We get this by decomposing α = β 1 + · · · + β m in (30), and then decomposing β k = α a k−1 +1 +· · ·+α a k as part of an expression (14) for ǫ β k (µ), for k = 1, . . . , m. We rewrite the bottom line [· · · ] of (37) using a function G n : H 2n → C by f α (Z) = n 1, α1,...,αn∈C(A): α1+···+αn=α G n Z(α 1 ), . . . , Z(α n );Z(α 1 ), . . . ,Z(α n ) · ǫ α1 (μ) * ǫ α2 (μ) * · · · * ǫ αn (μ),(38) where G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) = m=1,...,n, 0=a0<a1<···<am=n F m (z a0+1 + · · ·+ z a1 , . . . , z am−1+1 + · · · + z am ) m k=1 u a k −a k−1 (z a k−1 +1 , . . . ,z a k ; z a k−1 +1 , . . . , z a k ).(39) In (38), the terms ǫ α1 (μ) * · · · * ǫ αn (μ) andZ(α 1 ), . . . ,Z(α n ) are constants independent of Z. Thus it is clear that f α is continuous, or holomorphic, provided the function (z 1 , . . . , z n ) → G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) is continuous, or holomorphic, for each fixed (z 1 , . . . ,z n ) ∈ H n . We can now substitute (14) with α i , µ,μ in place of α, τ,τ respectively to express ǫ αi (μ) in (38) in terms of ǫ κ(j) (µ). Rewriting gives an expression for f α (Z) as a linear combination of ǫ α1 (µ) * · · · * ǫ αn (µ), as in (30). In fact the coefficients of ǫ α1 (µ) * · · · * ǫ αn (µ) in the two expressions must agree; we can prove this either by using [14,Ex. 7.10], in which the ǫ α1 (µ) * · · · * ǫ αn (µ) are linearly independent in H for all α 1 , . . . , α n satisfying some conditions, or by using combinatorial properties of the coefficients U (· · · ) from [15,Th. 4.8]. Equating the two writes F n Z(α 1 ), . . . , Z(α n ) in terms of the functions G m and u k . Since Z(α k ),Z(α k ) can take arbitrary values in H, we deduce an expression for F n (z 1 , . . . , z n ) when z k ,z k ∈ H, the inverse of (39): F n (z 1 , . . . , z n ) = m=1,...,n, 0=a0<a1<···<am=n G m (z a0+1 + · · ·+ z a1 , . . . , z am−1+1 + · · · + z am ; z a0+1 + · · ·+z a1 , . . . ,z am−1+1 + · · · +z am )· m k=1 u a k −a k−1 (z a k−1 +1 , . . . , z a k ;z a k−1 +1 , . . . ,z a k ). Note that although we want F n to map (C × ) n → C, for the moment (40) is defined only when z k ,z k ∈ H, since we have so far defined u n and G n only on H 2n , not on (C × ) 2n . Note too that (40) holds for arbitraryz 1 , . . . ,z n ∈ H. Here are some conclusions so far. , and let some functions F n : (C × ) n → C be given. Then a sufficient condition for the function f α of (30) to be continuous, or holomorphic, is that for fixed (z 1 , . . . ,z n ) ∈ H n the function (z 1 , . . . , z n ) → G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) is continuous, or holomorphic. This holds for some (z 1 , . . . ,z n ) ∈ H n if and only if it holds for all (z 1 , . . . ,z n ). This condition is also necessary, for all values of n occurring in (30), if the terms ǫ α1 (µ) * ǫ α2 (µ) * · · · * ǫ αn (µ) occurring in To go further we must understand the functions s n , u n (z 1 , . . . , z n ;z 1 , . . . ,z n ) better. From Example 2.7 and Definitions 2.9 and 2.10, we see that these depend on whether the inequalities arg(z a + · · · + z b ) > arg(z b+1 + · · · + z c ) and arg(z a +· · ·+z b ) > arg(z b+1 +· · ·+z c ) hold for each choice of 1 a b < c n, choosing arg(· · · ) uniquely in (0, π) as '· · · ' lies in H. For each (z 1 , . . . ,z n ) ∈ H n , define N (z1,...,zn) = (z 1 , . . . , z n ) ∈ H n : if arg(z a +· · ·+z b ) > arg(z b+1 +· · ·+z c ) then arg(z a +· · ·+z b ) > arg(z b+1 +· · ·+z c ), for all 1 a b < c n . Then N (z1,...,zn) is an open subset of H n , as it is defined by strict inequalities, and (z 1 , . . . ,z n ) ∈ N (z1,...,zn) . As in [14,Def. 4.10] we say that (μ, R, ) dominates (µ, R, ) ifμ(α) >μ(β) implies µ(α) > µ(β) for all α, β ∈ C(A). From [15, §5.2], this implies that S({1, . . . , n}, , κ, µ,μ) = 1, µ • κ(1) > · · · > µ • κ(n),μ • κ ≡μ(α), 0, otherwise,(42) for all A-data ({1, . . . , n}, , κ) with κ({1, . . . , n}) = α. If (z 1 , . . . , z n ) ∈ N (z1,...,zn) then the same argument shows that s n (z 1 , . . . , z n ;z 1 , . . . ,z n ) =    1, arg(z 1 ) > · · · > arg(z n ) and arg(z k ) = arg(z 1 + · · · +z n ), all k, 0, otherwise, arg(za) = arg(z1 + · · · +zn), all a, ψ(a) = ψ(b) implies arg(za) = arg(z b ), ψ(a) < ψ(b) and ξ •ψ(a) = ξ •ψ(b) imply arg(za) > arg(z b ) (−1) l−1 l · m c=1 1 \|ψ −1 (c)\|! ,(44) and u n (z 1 , . . . , z n ;z 1 , . . . ,z n ) = 0 if arg(z k ) = arg(z 1 + · · · +z n ), some k. We have been working with z k ,z k ∈ H, since s n , u n , G n are, so far, defined only on H 2n . We shall now restate the conditions of Proposition 3.3 for f α to be continuous, or holomorphic, in a way which makes sense for z k ,z k ∈ C × . a k , k=1,...,m G m (z a0+1 +· · ·+z a1 , . . . , z am−1+1 +· · ·+z am ; z a0+1 +· · ·+z a1 , . . . ,z am−1+1 +· · ·+z am )· m k=1 1 l k m k a k − a k−1 surjective ψ k : {a k−1 + 1, . . . , a k } → {1, . . . , m k } and ξ k : {1, . . . , m k } → {1, . . . , l k }: a b implies ψ k (a) ψ k (b), a b implies ξ k (a) ξ k (b), ψ k (a) = ψ k (b) implies arg(za) = arg(z b ), ψ k (a) < ψ k (b) and ξ k •ψ k (a) = ξ k •ψ k (b) imply arg(za) > arg(z b ), taking arg(za), arg(z b ) ∈ (c k − π/2, c k + π/2) (−1) l k −1 l k · m k c=1 1 \|ψ −1 k (c)\|! ,(45) where G m (· · · ) are some functions defined on the subsets of (C × ) 2m required by (45), such that the maps (z 1 , . . . , z m ) → G m (z 1 , . . . , z m ;z 1 , . . . ,z m ) are continuous (for f α to be continuous), and holomorphic (for f α to be holomorphic), in their domains. Here are some remarks on this: • From (44), for (z 1 , . . . , z n ) in an open neighbourhood of (z 1 , . . . ,z n ) we see that the term u a k −a k−1 (z a k−1 +1 , . . . , z a k ;z a k−1 +1 , . . . ,z a k ) in (40) is zero unless arg(z a ) = c k for all a k−1 < a a k and some c k . We have put this in as a restriction in the first line of (45), expressing it asz a ∈ e ic k (0, ∞) rather than arg(z a ) = c k because of the multivalued nature of arg. • We put in an extra condition Re(z kz −1 k ) > 0 for all k when (z 1 , . . . , z n ) ∈ N (z1,...,zn) . The main point of this is in (45) we have that Re e −ic k z a > 0 for a k−1 < a a k , so Re e −ic k (z a k−1 +1 + · · · + z a k ) > 0, and thus the argument z a k−1 +1 + · · · + z a k in G m (· · · ) in (45) is nonzero. That is, (45) only needs G m to be defined on a subset of (C × ) 2m . We also use this condition in the second line, as when a k−1 < a, b a k we can choose arg(z a ), arg(z b ) uniquely in (c k − π/2, c k + π/2). • When restricted to H n , Condition 3.4 is equivalent to the conditions of Proposition 3.3 for f α to be continuous, or holomorphic, as the arguments above show. But Condition 3.4 also makes sense on (C × ) n , where we want F n to be defined for the extension to the triangulated category case. Calculations by the author indicate that Condition 3.4 is the correct extension to the triangulated case. • The point of restricting to neighbourhoods N (z1,...,zn) is partly because there we can use the formula (44), but mostly because we do not have a meaningful extension of u n from H 2n to all of (C × ) 2n , so that (39) and (40) do not make sense. But for any fixed (z 1 , . . . ,z n ) we can use (44) to define u n (z 1 , . . . , z n ;z 1 , . . . ,z n ) for (z 1 , . . . , z n ) sufficiently close to (z 1 , . . . ,z n ), and this is the basis of Condition 3.4. Now suppose that (z 1 , . . . ,z n ) ∈ (C × ) n withz k+1 /z k / ∈ (0, ∞) for all 1 k < n, and let (z 1 , . . . , z n ) ∈ N (z1,...,zn) . Then in the first sum in (45) we cannot have a k−1 a k − 2 for any k, as thenz a k ,z a k −1 ∈ e ic k (0, ∞), contradicting z a k /z a k −1 / ∈ (0, ∞). Thus the only term in the first sum is m = n and a k = k for 0 k n, so the only terms in the second line are l k = m k = 1 and {a k } ψ k −→ {1} ξ k −→ {1} , and (40) reduces to F n (z 1 , . . . , z n ) = G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) ifz k+1 /z k / ∈ (0, ∞) for all k and (z 1 , . . . , z n ) ∈ N (z1,...,zn) . Thus Condition 3.4 requires F n to be continuous, or holomorphic, on N (z1,...,zn) , an open neighbourhood of (z 1 , . . . ,z n ). So we deduce: Proposition 3.5. Condition 3.4 implies that the function F n must be continuous, and holomorphic, on the set (z 1 , . . . , z n ) ∈ (C × ) n : z k+1 /z k / ∈ (0, ∞) for all 1 k < n .(47) Similarly, suppose (z 1 , . . . ,z n ) ∈ (C × ) n withz l+1 /z l ∈ (0, ∞) for some 1 l < n, andz k+1 /z k / ∈ (0, ∞) for all 1 k < n, k = l. Then in the first sum there are two terms, m = n and a k = k for 0 k n as before, and m = n − 1 and a k = k for 0 k < l, a k = k + 1 for l k < n. Rewriting arg(z l ) > arg(z l+1 ) as Im(z l+1 /z l ) < 0, and so on, we find (45) reduces to F n (z 1 , . . . , z n ) = G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) +G n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ; z 1 , . . . ,z l−1 ,z l +z l+1 ,z l+2 , . . . ,z n ) ·      1 2 , Im(z l+1 /z l ) < 0, 0, Im(z l+1 /z l ) = 0, − 1 2 , Im(z l+1 /z l ) > 0. By (46) this G n−1 (· · · ) is F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ), giving: Proposition 3.6. Condition 3.4 implies that if (z 1 , . . . ,z n ) ∈ (C × ) n with z l+1 /z l ∈ (0, ∞) for some 1 l < n, andz k+1 /z k / ∈ (0, ∞) for all 1 k < n with k = l, then for (z 1 , . . . , z n ) in an open neighbourhood of (z 1 , . . . ,z n ) in (C × ) n , the following function is continuous, and holomorphic: F n (z 1 , . . . , z n )− F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ) ·      1 2 , Im(z l+1 /z l ) < 0, 0, Im(z l+1 /z l ) = 0, − 1 2 , Im(z l+1 /z l ) > 0.(48) To summarize: away from the real hypersurfaces z l+1 /z l ∈ (0, ∞) in (C × ) n for 1 l < n, the F n must be continuous and holomorphic. As we cross the hypersurface z l+1 /z l ∈ (0, ∞) at a generic point, F n (z 1 , . . . , z n ) jumps by F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ), with the value on the hypersurface being the average of the limiting values from either side. Where several of the hypersurfaces z l+1 /z l ∈ (0, ∞) intersect, F n (z 1 , . . . , z n ) satisfies a more complicated condition. Roughly speaking, this says that several different sectors of (47) come together where the hypersurfaces intersect, and on the intersection F n should be a weighted average of the limiting values in each of these sectors. We now show these conditions determine the functions F n uniquely, provided they exist at all. Proof. Suppose F n and F ′ n for n 1 are two families of functions satisfying all the conditions, using functions G m , G ′ m respectively in Condition 3.4. We shall show that F n ≡ F ′ n for all n, by induction on n. We have F 1 ≡ F ′ 1 ≡ (2πi) −1 by (32). So let n 2, and suppose by induction that F k ≡ F ′ k for all k < n. By Condition 3.4 and induction on k this implies that G k = G ′ k for k < n. So taking the difference of (40) for F n and F ′ n gives f (z 1 , . . . , z n ) = F n (z 1 , . . . , z n ) − F ′ n (z 1 , . . . , z n ) = G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) − G ′ n (z 1 , . . . , z n ;z 1 , . . . ,z n ) in an open neighbourhood of (z 1 , . . . ,z n ). As G n , G ′ n are continuous and holomorphic in the z k , we see f : (C × ) n → C is holomorphic. By (31), f is the pullback of a holomorphic functionf : [z 1 , . . . , z n ] ∈ CP n−1 : z k = 0, k = 1, . . . , n → C. Taking the difference of (35) for F n , F ′ n gives \|f \| = o(\|z k \| −1 ) near points in just one hypersurface z k = 0 in CP n−1 . So by standard results in complex analysis,f extends holomorphically over these parts of CP n−1 , and so is defined except on intersections of two or more hypersurfaces z k = 0 in CP n−1 . By Hartog's theoremf extends holomorphically to all of CP n−1 , and so is constant. Since n > 1, equation (34) gives σ∈Snf [z σ(1) , . . . , z σ(n) ] = 0 for z k ∈ C × , forcingf ≡ 0. Hence f ≡ 0 and F n ≡ F ′ n . The theorem follows by induction. Note that we actually prove slightly more than the theorem says: any functions F 1 , . . . , F n satisfying the conditions up to n are unique. Partial differential equations satisfied by f α and F n We are trying to construct a family of holomorphic generating functions f α : Stab(A) → L α for α ∈ C(A). Clearly, it would be interesting if this family satisfied some nontrivial partial differential equations. We are now going to guess a p.d.e. for the f α to satisfy, and deduce a p.d.e. for the F n . We will then use this p.d.e. to construct the functions F n that we want by induction on n. In §3.3 we will prove they satisfy Remark 3.1 and Condition 3.4. Theorem 3.7 then shows these F n are unique. This means that the p.d.e.s that we shall guess for the f α and F n are actually implied by the general assumptions Remark 3.1 and Condition 3.4, which seems very surprising, as these imposed no differential equations other than being holomorphic. One possible conclusion is that our p.d.e.s are not simply something the author made up, but are really present in the underlying geometry and combinatorics, and have some meaning of their own. To guess the p.d.e. we start by determining the function F 2 . Equation (31) implies we may write F 2 (z 1 , z 2 ) = f (z 2 /z 1 ) for some f : C × → C, and then Propositions 3.5 and 3.6 and (32) imply that f is holomorphic in C \ [0, ∞) with the following continuous over (0, ∞): f (z) − 1 2πi ·      1 2 , Im(z) < 0, 0, Im(z) = 0, − 1 2 , Im(z) > 0, Since log z cut along (0, ∞) jumps by 2πi across (0, ∞), the obvious answer is f (z) = (2πi) −2 log z +C for some constant C, where we define log z on C\[0, ∞) such that Im log z ∈ (0, 2π). But equation (34) reduces to f (z) + f (z −1 ) ≡ 0, which holds provided C = −πi/(2πi) 2 . This suggests that F 2 (z 1 , z 2 ) =    1 (2πi) 2 log(z 2 /z 1 ) − πi , z 2 /z 1 / ∈ (0, ∞), 1 (2πi) 2 log(z 2 /z 1 ), z 2 /z 1 ∈ (0, ∞),(49) where log z is defined so that Im log z ∈ [0, 2π). It is now easy to check that these F 1 , F 2 satisfy Condition 3.4 and (31), (32), (34), (35) up to n = 2, so Theorem 3.7 shows (49) is the unique function F 2 which does this. Let us now consider a simple situation in which we are interested only in classes β, γ, β + γ in C(A), and β, γ cannot be written as δ + ǫ for δ, ǫ ∈ C(A), and the only ways to write β + γ = δ + ǫ for δ, ǫ ∈ C(A) are δ, ǫ = β, γ or δ, ǫ = γ, β. In this case, from (30), (32) and (49) we see that when Z ∈ Stab(A) with Z(γ)/Z(β) / ∈ (0, ∞) we have f β (Z) = 1 2πi ǫ β (µ), f γ (Z) = 1 2πi ǫ γ (µ), f β+γ (Z) = 1 2πi ǫ β+γ (µ)+ 1 (2πi) 2 log Z(γ) Z(β) −πi ǫ β (µ) * ǫ γ (µ)−ǫ γ (µ) * ǫ β (µ) . These satisfy the p.d.e. on Stab(A), at least for Z(γ)/Z(β) / ∈ (0, ∞): df β+γ (Z) = f β (Z) * f γ (Z) − f γ (Z) * f β (Z) ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) .(50) Here f β+γ is an L-valued function on Stab(A), so df β+γ is an L-valued 1-form, that is, a section of L ⊗ C T * C Stab(A). Also Z(γ), Z(β) are complex functions on Stab(A), so d(Z(γ))/Z(γ), d(Z(β))/Z(β) are complex 1-forms on Stab(A), and tensoring over C with f β (Z) * f γ (Z) − f γ (Z) * f β (Z) also gives an L-valued 1-form on Stab(A). Note that ǫ β (µ), ǫ γ (µ), ǫ β+γ (µ) are locally constant in Z away from Z(γ)/Z(β) ∈ (0, ∞), so there are no terms from differentiating them. Also, by construction f β , f γ , f β+γ are continuous and holomorphic over the hypersurface Z(γ)/Z(β) ∈ (0, ∞), so by continuity (50) holds there too. We now guess that the generating functions f α of (30) should satisfy the p.d.e., for all α ∈ C(A): df α (Z) = β,γ∈C(A):α=β+γ f β (Z) * f γ (Z) ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) = β,γ∈C(A):α=β+γ 1 2 [f β (Z), f γ (Z)] ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) = − β,γ∈C(A):α=β+γ [f β (Z), f γ (Z)] ⊗ d(Z(β)) Z(β) ,(51) where the three lines are equivalent, noting that we may exchange β, γ. In the simple case above this reduces to (50) when α = β + γ. We can now explain our choice of constant F 1 ≡ (2πi) −1 in (32). The 2πi comes from the jumping of log z over (0, ∞) as above. If we had instead set F 1 ≡ c for some c ∈ C, then f α , F n and the right hand side of (49) would be multiplied by 2πi c, and the right hand sides of (50)-(51), and (69) below, would be multiplied by (2πi c) −1 . We picked c = (2πi) −1 to eliminate constant factors in the p.d.e. (51) and flat connection (69), and so simplify our equations. For (51) to hold, it is clearly necessary that the right hand side should be closed. We check this by applying 'd' to it and using (51), giving: f β (Z) * f γ (Z) ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) = ǫ,δ∈C(A):α=ǫ+δ df ǫ (Z) * f δ (Z) ∧ d(Z(δ)) Z(δ) − d(Z(ǫ)) Z(ǫ) + β,ǫ∈C(A):α=β+ǫ f β (Z) * df ǫ (Z) ∧ d(Z(ǫ)) Z(ǫ) − d(Z(β)) Z(β) (52) = β,γ,δ∈C(A):α=β+γ+δ f β (Z) * f γ (Z) * f δ (Z) ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) ∧ d(Z(δ)) Z(δ) − d(Z(β+γ)) Z(β+γ) + d(Z(δ)) Z(δ) − d(Z(γ)) Z(γ) ∧ d(Z(γ+δ)) Z(γ+δ) − d(Z(β)) Z(β) = 0. Here expanding the first line gives β,γ df β * f γ ∧ (· · · ) + β,γ f β * df γ ∧ (· · · ), as the final 1-form is closed. These two terms appear in the second and third lines, with β, γ relabelled as ǫ, δ in the second line and β, ǫ in the third. The fourth and fifth lines substitute (51) into the second and third lines, with ǫ in place of α for the second line and ǫ, γ, δ in place of α, β, γ for the third line. The final step holds as the 2-form [· · · ] on the fourth and fifth lines is zero. Thus equation (51) has the attractive property that it implies its own consistency condition; that is, the condition for (51) to be locally solvable for f α is equation (51) for β, γ. We express (51) in terms of the functions F n and G n . and let some functions F n : (C × ) n → C be given. Then a sufficient condition for the functions f α of (30) to satisfy (51), is that for fixed (z 1 , . . . ,z n ) ∈ H n the functions (z 1 , . . . , z n ) → G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) of §3.1 should satisfy dG n (z 1 , . . . , z n ;z 1 , . . . ,z n ) = n−1 k=1 G k (z 1 , . . . , z k ;z 1 , . . . ,z k ) · G n−k (z k+1 , . . . , z n ;z k+1 , . . . ,z n ) dz k+1 +· · ·+dz n z k+1 +· · ·+z n − dz 1 +· · ·+dz k z 1 +· · ·+z k dF n (z 1 , . . . , z n ) = n−1 k=1 F k (z 1 , . . . , z k )F n−k (z k+1 , . . . , z n ) · dz k+1 + · · · + dz n z k+1 + · · · + z n − dz 1 + · · · + dz k z 1 + · · · + z k .(54) Proof. For the first part, substitute (38) in for f α , f β and f γ in the top line of (51). Then both sides can be rewritten as a sum over α 1 , . . . , α n ∈ C(A) with α 1 + · · · + α n = α of ǫ α1 (μ) * · · · * ǫ αn (μ) tensored with complex 1-forms. Equating the complex 1-form coefficients of ǫ α1 (μ) * · · · * ǫ αn (μ) gives (53), evaluated at z a = Z(α a ) andz a =Z(α a ), where the G k term comes from f β in (51) with β = α 1 + · · · + α k , and the G n−k term comes from f γ in (51) with γ = α k+1 + · · · + α n . The first two paragraphs follow, by the same arguments used to prove Proposition 3.3. For the final part, let Condition 3.4 hold. If (z 1 , . . . ,z n ) ∈ (C × ) n with z k+1 /z k / ∈ (0, ∞) for all 1 k < n and (z 1 , . . . , z n ) ∈ N (z1,...,zn) , then the proof of Proposition 3.5 shows that F n (z 1 , . . . , z n ) = G n (z 1 , . . . , z n ;z 1 , . . . ,z n ), F k (z 1 , . . . , z k ) = G k (z 1 , . . . , z k ;z 1 , . . . ,z k ) and F n−k (z k+1 , . . . , z n ) = G n−k (z k+1 , . . . , z n ;z k+1 , . . . ,z n ). Thus (53) is equivalent to (54) in N (z1,...,zn) , so (53) implies (54) in the domain (47). For the reverse implication, suppose Condition 3.4 holds and (54) holds in (47). Then the argument above shows (53) holds for (z 1 , . . . , z n ), (z 1 , . . . ,z n ) in (47) with (z 1 , . . . , z n ) ∈ N (z1,...,zn) . But whether (53) holds or not is unaffected by small changes in (z 1 , . . . ,z n ), and (47) is dense in (C × ) n . Thus, (53) holds for all (z 1 , . . . ,z n ) ∈ (C × ) n and (z 1 , . . . , z n ) ∈ N (z1,...,zn) with (z 1 , . . . , z n ) in (47). Now the functions (z 1 , . . . , z m ) → G m (z 1 , . . . , z m ;z 1 , . . . ,z m ) are continuous and holomorphic by Condition 3.4, so as (47) is open and dense we see that (53) must hold for all (z 1 , . . . , z n ) ∈ N (z1,...,zn) , by continuity. The proof of Proposition 3.8 conceals a subtlety. One might think that for generic Z ∈ Stab(A), all terms (Z(α 1 ), . . . , Z(α n )) occurring in (30) will lie in the open dense domain (47), so that (54) on (47) implies (51) for generic Z in the obvious way, and so (51) must hold for all Z by continuity. However, this is false. For example, if α 1 = α 2 then Z(α 2 )/Z(α 1 ) ≡ 1 ∈ (0, ∞), so (Z(α 1 ), . . . , Z(α n )) does not lie in (47) for any Z ∈ Stab(A). So assuming (54) on (47) apparently tells us nothing about how such terms contribute to (51). Because of this, for (51) to hold when f α in (30) includes terms with dependencies such as α 1 = α 2 , we need F n to satisfy not just (54) on (47), but other more complicated conditions on the real hypersurfaces z k+1 /z k ∈ (0, ∞) as well. The point of the proof is that these other conditions are implied by (54) on (47) and Condition 3.4, as we can express the conditions in terms of the G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) and use the fact that they are continuous and holomorphic in (z 1 , . . . , z n ) over the hypersurfaces z k+1 /z k ∈ (0, ∞). Equations (53) and (54) apparently have poles on the hypersurfaces z 1 + · · · + z k = 0 and z k+1 + · · · + z n = 0. So we would expect solutions G n , F n to have log-type singularities along these hypersurfaces; in particular, this suggests that there should not be single-valued solutions on the domain (47). In fact this is false, and single-valued, nonsingular solutions can exist across these hypersurfaces. The next proposition explains why. Proposition 3.9. For n 2 the following is a nonempty, connected set in C n : (z 1 , . . . , z n ) ∈ (C × ) n : z k+1 /z k / ∈ (0, ∞) for k = 1, . . . , n − 1 and z 1 + · · · + z n = 0 . If F n satisfies (54) on the domain (47) then F n ≡ C n on (55) for some C n ∈ C. If F n also satisfies (34) as in Remark 3.1 then C n = 0. In this case we have F n (z 1 , . . . , z n ) = (z 1 + · · · + z n )H n (z 1 , . . . , z n ) for a holomorphic function H n defined on the domain (47), including where z 1 + · · · + z n ≡ 0. Using these we rewrite (54) as dF n (z 1 , . . . , z n ) = n−1 k=1 H k (z 1 , . . . , z k )H n−k (z k+1 , . . . , z n ) · (z 1 +· · ·+z k )(dz k+1 +· · ·+dz n )−(z k+1 +· · ·+z n )(dz 1 +· · ·+dz k ) . (56) Note that (56) has no poles on (47). Proof. Let (z 1 , . . . , z n ), (z ′ 1 , . . . , z ′ n ) lie in (55). We shall construct a path between them in (55), showing (55) is connected. It is easy to see that w ∈ C :(z 1 , . . . , z n−2 , w, z n−1 + z n − w) lies in (55) = C \ {x(z n−1 + z n ) : x ∈ [0, 1]} ∪ {xz n−2 : x ∈ [0, ∞)} ,(57)w ∈ C :(z ′ 1 , . . . , z ′ n−2 , w, z ′ n−1 + z ′ n − w) lies in (55) = C \ {x(z ′ n−1 + z ′ n ) : x ∈ [0, 1]} ∪ {xz ′ n−2 : x ∈ [0, ∞)} ,(58) which are both connected subsets of C, containing z n−1 and z ′ n−1 respectively. Choose some w 0 in both (57) and (58) with \|w 0 \| ≫ \|z k \|, \|z ′ k \| for all k = 1, . . . , n. Choose paths between z n−1 and w 0 in (57), and between z ′ n−1 and w 0 in (58). These induce paths in (55) between (z 1 , . . . , z n ) and (z 1 , . . . , z n−2 , w 0 , z n−1 + z n − w 0 ), and between (z ′ 1 , . . . , z ′ n ) and (z ′ 1 , . . . , z ′ n−2 , w 0 , z ′ n−1 + z ′ n − w 0 ). It remains to find a path in (55) between (z 1 , . . . , z n−2 , w 0 , z n−1 +z n −w 0 ) and (z ′ 1 , . . . , z ′ n−2 , w 0 , z ′ n−1 + z ′ n − w 0 ). To do this, choose a path (x 1 (t), . . . , x n−2 (t)) for t ∈ [0, 1] between (z 1 , . . . , z n−2 ) and (z ′ 1 , . . . , z ′ n−2 ) in (y 1 , . . . , y n−2 ) ∈ (C × ) n−2 : y k+1 /y k / ∈ (0, ∞) for k = 1, . . . , n − 3 and y n−2 /w 0 / ∈ (0, ∞) , which is possible as using y k+1 /y k , y n−2 /w 0 as coordinates we see this domain is homeomorphic to (C \ [0, ∞)) n−2 , and thus is connected. Making w 0 larger if necessary, we can also assume that \|w 0 \| ≫ \|x k (t)\| for all k = 1, . . . , n − 2 and t ∈ [0, 1]. Then it is easy to see that the path ( (55). Therefore (55) is connected. For the second part, observe that on the hypersurface z 1 + · · · + z n = 0 we have z k+1 +· · ·+z n ≡ −(z 1 +· · ·+z k ), so the 1-form [· · · ] in (54) restricts to zero on z 1 + · · · + z n = 0. Thus the restriction of dF n to (55) is zero, so F n ≡ C n on (55) for some C n ∈ C, as (55) is connected. For generic (z 1 , . . . , z n ) in (55) all the permutations (z σ(1) , . . . , z σ(n) ) for σ ∈ S n lie in (55) as well, so (34) becomes n!C n = 0, giving C n = 0. Thus, the holomorphic function F n is zero along the nonsingular hypersurface z 1 + · · · + z n = 0 in its domain (47). Properties of holomorphic functions imply that F n (z 1 , . . . , z n ) = (z 1 + · · · + z n )H n (z 1 , . . . , z n ) for a unique holomorphic function H n on (47). Equation (56) is immediate. x 1 (t), . . . , x n−2 (t), w 0 , −x 1 (t) − · · · − x n−2 (t) − w 0 ) for t ∈ [0, 1] links (z 1 , . . . , z n−2 , w 0 , z n−1 + z n − w 0 ) and (z ′ 1 , . . . , z ′ n−2 , w 0 , z ′ n−1 + z ′ n − w 0 ) in Suppose we are given some holomorphic functions F n on the domains (47) satisfying (54). Analytically continue the F n to multivalued, singular holomorphic functionsF n on (C × ) n , still satisfying (54). The argument above shows thatF n is locally constant along z 1 + · · · + z n = 0, but it can take a different value on each sheet. So (54) can have genuine poles along z 1 + · · · + z k = 0 and z k+1 + · · · + z n = 0 whenF k ,F n−k are nonzero constants. ThusF n will have log-like singularities along z 1 + · · · + z k = 0 and z k+1 + · · · + z n = 0, and more generally singularities along z a + · · · + z b = 0 for 1 a b n with (a, b) = (1, n). One moral is that our functions F n manage to be single-valued and nonsingular on (47) for very special reasons, and their analytic continuations have much worse singularities and branching behaviour. We now construct functions F n for n 1 satisfying (54), by induction on n. Proposition 3.10. There exists a unique series of holomorphic functions F n for n 1 defined on the domain (47) with F 1 ≡ (2πi) −1 , such that F n satisfies (54) and is zero on (55). Also F n (z 1 , . . . , z n ) ≡ (z 1 + · · · + z n )H n (z 1 , . . . , z n ) for a unique holomorphic function H n defined on (47), and (56) holds. Proof. Suppose by induction that for some m 2 we have constructed unique holomorphic functions F n , H n on the domains (47) for n = 1, . . . , m−1 satisfying all the conditions of the proposition for n < m. For m = 2 this is trivial, as we must have F 1 (z) = (2πi) −1 and H 1 (z) = (2πiz) −1 . We will construct F m , H m , and show they are unique. Equations (54) and (56) for n = m give equivalent expressions for dF m on (47), with (56) being manifestly holomorphic on all of (47). Write α m for the right hand side of (54) or (56), so that α m is a holomorphic 1-form on (47), and we need to construct F m with dF m = α m . Following (52), we can compute dα m by applying d to the r.h.s. of (54) for n = m, and using (54) for n < m (which holds by induction) to substitute in for dF k and dF n−k . Then everything cancels giving dα m = 0, so α m is a closed 1-form. Although (47) is not simply-connected, it is the pullback to C m \ {0} of [z 1 , . . . , z m ] ∈ CP m−1 : z k = 0 and z k+1 /z k / ∈ (0, ∞) for all k ,(59) which is homeomorphic to C \ [0, ∞) m−1 , and so is simply-connected. Now α m is the pull-back of a 1-form on (59), which is closed as α m is closed, and so is exact as (59) is simply-connected. Therefore α m is an exact holomorphic 1-form on (47), so there exists a holomorphic function F m on (47) with dF m = α m , which is unique up to addition of a constant, as (47) is connected. To choose the constant, note that the restriction of α m to the connected set (55) is zero as in Proposition 3.9, so requiring F m to be zero on (55) fixes F m uniquely. Since F m is zero along the nonsingular hypersurface z 1 + · · · + z m = 0, by properties of holomorphic functions there is a unique holomorphic function H m on (47) with F m (z 1 , . . . , z m ) ≡ (z 1 + · · · + z m )H m (z 1 , . . . , z m ). These F m , H m satisfy all the conditions for n = m, and the proposition follows by induction. Reconciling the approaches of §3.1 and §3.2 So far we have given two quite different approaches to the functions F n used to define f α in (30). In §3.1 we found conditions on the F n on (C × ) n for the f α to be continuous and holomorphic, and showed such F n would be unique if they existed. In §3.2 we found different conditions on the F n on a subdomain (47) of (C × ) n for the f α to satisfy the p.d.e. (54), neglecting the question of whether f α would be continuous for these F n , and constructed unique F n satisfying these second conditions. There seems no a priori reason why these two sets of conditions on F n should be compatible, but we now prove that they are. That is, we show that the F n on (47) constructed in Proposition 3.10 extend uniquely to (C × ) n so as to satisfy Remark 3.1 and Condition 3.4. First we show, in effect, that Proposition 3.6 holds for the F n of Proposition 3.10. For F n as in Proposition 3.10, define a function D l,n for 1 l < n by D l,n : (z 1 , . . . ,z n ) ∈ (C × ) n :z l+1 /z l ∈ (0, ∞), These limits exist and give a continuous function D l,n , since the proof in Proposition 3.10 that the F n are continuous and holomorphic in their domains extends locally from either side over the hypersurface z l+1 /z l ∈ (0, ∞). The next result would follow from (48) if we knew Proposition 3.6 applied. z k+1 /z k / ∈ (0, ∞) for 1 k < n, k = l −→ C,(60) Proposition 3.11. We have D l,n (z 1 , . . . , z n ) ≡ F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ) on the domain of D l,n , where F n−1 is as in Proposition 3.10. Proof. Taking the difference of the limits of (54) from both sides of the hypersurface z l+1 /z l ∈ (0, ∞) gives an equation in 1-forms on the domain of D l,n : dD l,n (z 1 , . . . , z n ) = (62) l−1 k=1 F k (z 1 , . . . , z k )D l−k,n−k (z k+1 , . . . , z n ) · dz k+1 +···+dzn z k+1 +···+zn − dz1+···+dz k z1+···+z k + n−1 k=l+1 D l,k (z 1 , . . . , z k )F n−k (z k+1 , . . . , z n ) · dz k+1 +···+dzn z k+1 +···+zn − dz1+···+dz k z1+···+z k . Here if (z 1 , . . . , z n ) lies in the domain of D l,n and k < l then F k is defined and continuous at (z 1 , . . . , z k ) but F n−k is not defined (nor continuous) at (z k+1 , . . . , z n ), so the difference in limits of F k (z 1 , . . . , z k )F n−k (z k+1 , . . . , z n ) in (54) is F k (z 1 , . . . , z k )D l−k,n−k (z k+1 , . . . , z n ), giving the first term in (62). Similarly, k > l gives the second term. There is no term k = l in (62), since F l and F n−l are both defined and continuous at (z 1 , . . . , z l ) and (z l+1 , . . . , z n ) respectively, so the limits from each side of z l+1 /z l ∈ (0, ∞) cancel. As F n (z 1 , . . . , z n ) = 0 when z 1 + · · · + z n = 0, if (z 1 , . . . ,z n ) lies in the domain of D l,n withz 1 + · · · +z n = 0 then both limits in (61) are zero, as (z 1 , . . . ,z n ) is the limit of points (z 1 , . . . , z n ) with z 1 + · · · + z n = 0 from both sides of z l+1 /z l ∈ (0, ∞). Thus D l,n (z 1 , . . . , z n ) = 0 if z 1 + · · · + z n = 0. Also, it is easy to verify from Proposition 3.10 that F 2 is given by (49) in its domain, so from properties of logs we see that D 1,2 (z 1 , z 2 ) ≡ (2πi) −1 ≡ F 1 (z 1 + z 2 ).(64) Suppose by induction on m that D l,n (z 1 , . . . , z n ) ≡ F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ) whenever 1 l < n m, for some m 2. The first case m = 2 is (64). Let n = m + 1 and 1 l < n. Then comparing (54) and (62) and using the inductive hypothesis shows that dD l,n (z 1 , . . . , z n ) ≡ dF n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ). Thus D l,n (z 1 , . . . , z n ) − F n−1 (z 1 , . . . , z l−1 , z l + z l+1 , z l+2 , . . . , z n ) is constant on the domain of D l,n . But this domain is connected and contains (z 1 , . . . , z n ) with z 1 + · · · + z n = 0 as n 3, and both D l,n (· · · ) and F n−1 (· · · ) are zero at such points by Proposition 3.10 and (63). So the constant is zero, proving the inductive step and the proposition. Using this we extend the F n of Proposition 3.10 so that Condition 3.4 holds. Theorem 3.12. The functions F n of Proposition 3.10, defined on the domain (47), can be extended uniquely to F n : (C × ) n → C satisfying Condition 3.4. Proof. The idea of the proof is that by induction on n we shall construct functions G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) that for each fixed (z 1 , . . . ,z n ) are continuous and holomorphic and satisfy (53) in (z 1 , . . . , z n ) on N (z1,...,zn) , such that (45) holds with F n as in Proposition 3.10 whenever (z 1 , . . . , z n ) lies in the intersection of (47) and N (z1,...,zn) . We then extend F n uniquely from (47) to (C × ) n by requiring (45) to hold on all of N (z1,...,zn) , for all (z 1 , . . . ,z n ). Suppose by induction that for some p 2 and for all n < p we have found open neighbourhoods N (z1,...,zn) of (z 1 , . . . ,z n ) in (C × ) n for all (z 1 , . . . ,z n ) ∈ (C × ) n , and functions G n , and extensions of F n in Proposition 3.10 to (C × ) n , such that Condition 3.4 holds for n < p and the G n satisfy (53), and G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) = 0 if z 1 + · · · + z n = 0. The first case p = 2 is trivial, taking F 1 ≡ (2πi) −1 ≡ G 1 and N (z1) = C × . We shall now construct open neighbourhoods N (z1,...,zp) , the function G p , and an extension of F p , satisfying all the conditions. Choose a connected, simply-connected open neighbourhood N (z1,...,zp) of each (z 1 , . . . ,z p ) in (C × ) p , such that (z 1 , . . . , z p ) ∈ N (z1,...,zp) implies that (a) if 1 m < p, 0 = a 0 < a 1 < · · · < a m = p and c 1 , . . . , c m ∈ [0, 2π) with z a ∈ e ic k (0, ∞) for a k−1 < a a k , then (z a0+1 + · · · + z a1 , . . . , z am−1+1 + · · · + z am ) ∈ N (za 0 +1+···+za 1 ,...,za m−1 +1+···+zam ) , and (b) if 1 k < p then (z 1 , . . . , z k ) ∈ N (z1,...,z k ) and (z k+1 , . . . , z p ) ∈ N (z k+1 ,...,zp) . This is satisfied if N (z1,...,zp) is a small enough open ball about (z 1 , . . . ,z p ). The point is that (a) ensures that all the terms in (45) with n = p and m < p are well-defined when (z 1 , . . . , z p ) ∈ N (z1,...,zp) , and (b) ensures that the right hand side of (53) for n = p is well-defined when (z 1 , . . . , z p ) ∈ N (z1,...,zp) . Now regard (z 1 , . . . ,z p ) as fixed, and consider equation (53) with n = p for (z 1 , . . . , z p ) ∈ N (z1,...,zp) . The left hand side dG p (· · · ) has not yet been defined. The right hand side involves G k for k < p, which by induction are defined on their domains and satisfy (53). The choice of N (z1,...,zp) implies the r.h.s. is a 1-form defined on N (z1,...,zp) , and taking d and using (53) for n < p we find this 1-form is closed, as for dF m in the proof of Proposition 3.10. Also, as for (52) and for F n in Proposition 3.9, the inductive assumption G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) = 0 if z 1 + · · · + z n = 0 ensures that the terms (z 1 + · · · + z k ) −1 , (z k+1 + · · · + z n ) −1 in (53) do not induce singularities. This proves that the right hand side of (53) for n = p is a well-defined, closed, holomorphic, nonsingular 1-form on the connected, simply-connected domain N (z1,...,zp) . Hence there exists a holomorphic function (z 1 , . . . , z p ) → G p (z 1 , . . . , z p ;z 1 , . . . ,z p ) on N (z1,...,zp) , unique up to addition of a constant, such that (53) holds. Here is how we fix the constant. Recall that so far F p has been defined on the open dense domain (47) in Proposition 3.10, and F n for n < p has been defined on all of (C × ) n . Thus, every term in (45) with n = p is now defined on the intersection of (47) and N (z1,...,zp) ; note that the only term on the r.h.s. of (45) with m = n = p is G p (z 1 , . . . , z p ;z 1 , . . . ,z p ). This intersection is also open and nonempty, as N (z1,...,zp) is nonempty and (47) is dense. We claim that there is a unique function G p satisfying (53) such that (45) holds on the intersection of (47) and N (z1,...,zp) . To see this, note that by Proposition 3.8, equations (53) and (54) are equivalent when (45) holds. Thus, for any choice of G p satisfying (53), applying d to both sides of (45) gives the same thing, so the difference between the left and right hand sides of (45) is locally constant on the intersection of N (z1,...,zp) and (47). Fix a connected component C of this intersection. Then we can choose G p uniquely such that (45) holds on this connected component, and on every other connected component the difference between the left and right hand sides of (45) is constant. Suppose C ′ , C ′′ are connected components of the intersection of N (z1,...,zp) and (47) which meet along the real hypersurface z l+1 /z l ∈ (0, ∞) for 1 l < p. (That is, the closures of C ′ , C ′′ must contain a nonempty open subset of this hypersurface). Then Proposition 3.11 computes how much F p jumps across this hypersurface, which by Proposition 3.6 follows from the condition for G p to be continuous across the hypersurface. It is not difficult to deduce that the difference between the left and right hand sides of (45) must take the same constant value on C ′ and C ′′ . Since this value is 0 on one component C, and as N (z1,...,zp) is open and connected we can get from C to any other component C ′ by crossing hypersurfaces z l+1 /z l ∈ (0, ∞) one after the other, the constant is zero for every C ′ . This proves the claim. We have now defined the functions G p . If (z 1 , . . . , z p ) lies in the intersection of (47) and N (z1,...,zp) with z 1 + · · · + z p = 0 then (45) holds at (z 1 , . . . , z p ). There is a term G p (z 1 , . . . , z p ;z 1 , . . . ,z p ) on the right hand side, and every other term is zero by F p (z 1 , . . . , z p ) = 0 when z 1 + · · · + z p = 0 and the inductive hypothesis. Hence G p (z 1 , . . . , z p ;z 1 , . . . ,z p ) = 0. By continuity this extends to all (z 1 , . . . , z p ) in N (z1,...,zp) with z 1 + · · · + z p = 0, as we have to prove. By construction, (45) holds on the intersection of N (z1,...,zp) and the subset (47) where F p is already defined by Proposition 3.10. We now extend F p to (C × ) p by requiring F p to satisfy (47) with n = p on each domain N (z1,...,zp) . Since the N (z1,...,zp) cover (C × ) p this defines F p uniquely, but we must check that given (z 1 , . . . ,z p ) and (ẑ 1 , . . . ,ẑ p ), equation (47) for n = p gives the same answer for F p (z 1 , . . . , z p ) with thez k andẑ k on the intersection N (z1,...,zp) ∩ N (ẑ1,...,ẑp) . This holds for the same reason that the conditions of Proposition 3.3 hold for some (z 1 , . . . ,z n ) if and only if they hold for all (z 1 , . . . ,z n ). The point is that the condition for f α to be continuous is that we can write F p in the form (47) near (z 1 , . . . ,z p ) for G k continuous in (z 1 , . . . , z k ), and these continuity conditions for (z 1 , . . . ,z p ), (ẑ 1 , . . . ,ẑ p ) must be equivalent in the overlap N (z1,...,zp) ∩N (ẑ1,...,ẑp) . We are using (45) to determine how to extend F p from (47) to (C × ) p in a way that makes the f α continuous, and these continuity conditions are independent of the choice of (z 1 , . . . ,z p ) or (ẑ 1 , . . . ,ẑ p ). Thus F p is well defined and satisfies (45). This completes the inductive step, and the proof of Theorem 3.12. Our next three results verify the remaining conditions of Remark 3.1. Theorem 3.13. For n 1, define A n to be the free C-algebra with generators e 1 , . . . , e n and multiplication * , and L n to be the free Lie subalgebra of A n generated by e 1 , . . . , e n under the Lie bracket [f, g] = f * g − g * f . Then for any (z 1 , . . . , z n ) ∈ (C × ) n the following expression lies in L n : σ∈Sn F n (z σ(1) , z σ(2) , . . . , z σ(n) ) e σ(1) * e σ(2) * · · · * e σ(n) , where the F n are as in Theorem 3.12 and S n is the symmetric group. Also (34) holds, and f α in (30) maps Stab(A) → L α , as in Remark 3.1(d). Proof. We shall first prove the first part of the theorem on the domain (z 1 , . . . , z n ) ∈ (C × ) n : z k /z l ∈ (0, ∞) for all 1 k < l n . The point of this is that if (z 1 , . . . , z n ) lies in (66) then (z σ(1) , . . . , z σ(n) ) lies in the domain (47) where F n is holomorphic and satisfies (54) for all σ ∈ S n . Suppose by induction that for some m 2 and all n < m, the expression (65) lies in L n for all (z 1 , . . . , z n ) in (66). Write P m for (65) with n = m, regarded as a holomorphic function from (66) to H m . Then we have dP m (z 1 , . . . , z n ) = σ∈Sm m−1 k=1 F k (z σ(1) , . . . , z σ(k) ) e σ(1) * · · · * e σ(k) * F m−k (z σ(k+1) , . . . , z σ(m) )e σ(k+1) * · · · * e σ(m) ⊗ dz σ(k+1) +···+dz σ(m) z σ(k+1) +···+z σ(m) − dz σ(1) +···+dz σ(k) z σ(1) +···+z σ(k) = 1 2 σ∈Sm m−1 k=1 F k (z σ(1) , . . . , z σ(k) ) e σ(1) * · · · * e σ(k) , F m−k (z σ(k+1) , . . . , z σ(m) ) e σ(k+1) * · · · * e σ(m) ⊗ dz σ(k+1) +···+dz σ(m) z σ(k+1) +···+z σ(m) − dz σ(1) +···+dz σ(k) z σ(1) +···+z σ(k) (67) = 1 2 σ∈Sm m−1 k=1 1 k!(m−k)! τ ∈S k F k (z σ•τ (1) , . . . , z σ•τ (k) )e σ•τ (1) * · · · * e σ•τ (k) , υ∈S m−k F m−k (z σ(k+υ(1)) , . . . , z σ(k+υ(m−k)) )e σ(k+υ(1)) * · · · * e σ(k+υ(m−k)) ⊗ dz σ(k+1) +···+dz σ(m) z σ(k+1) +···+z σ(m) − dz σ(1) +···+dz σ(k) z σ(1) +···+z σ(k) . Here the second line is immediate from (54). The third line is the average of two copies of the second, one copy as it stands, the other relabelled with m − k in place of k and indices σ(k + 1), . . . , σ(m), σ(1), . . . , σ(k) in place of σ(1), . . . , σ(k), σ(k +1), . . . , σ(m) respectively; this is valid because of the sum over σ ∈ S m . The fourth and final line uses the fact that symmetrizing over S m on 1, . . . , m is equivalent to first symmetrizing over S k on 1, . . . , k and S m−k on k + 1, . . . , m, with factors 1/k!(m − k)!, and then symmetrizing over S m . By the inductive hypothesis, as k, m − k < m, the terms τ ∈S k · · · and just an H m -valued 1-form. As m 2 it is easy to show that each connected component of (66) with n = m contains a point (z 1 , . . . , z m ) with z 1 +· · ·+z m = 0. At this point F m (z σ(1) , . . . , z σ(m) ) = 0 for all σ ∈ S m , so P m (z 1 , . . . , z m ) = 0, which lies in L m . Thus dP m is an L m -valued 1-form and P m (z 1 , . . . , z m ) lies in L m at one point in each connected component of (66), so P m (z 1 , . . . , z m ) lies in L m at every point of (66), completing the inductive step. It remains to extend this from (66) to (C × ) n . We do this using an argument similar to Theorem 3.12, and facts about the coefficients U (· · · ) from [15, §5]. The relationships between the functions F n , G n given in (39) and (40) were derived by using the change of stability condition formula (14) to transform between ǫ α (µ) and ǫ β (μ). By [15,Th. 5.4], equation (14) can be rewritten as in (15) with the term [· · · ] a sum of multiple commutators of ǫ κ(i) (τ ) for i ∈ I, so that it lies in L α rather than just H α . Suppose the open neighbourhoods N (z1,...,zn) in Theorem 3.12 are chosen so that (z 1 , . . . , z n ) ∈ N (z1,...,zn) if and only if (z σ(1) , . . . , z σ(n) ) ∈ N (z σ(1) ,...,z σ(n) ) for all σ ∈ S n . As we can take the N (z1,...,zn) to be sufficiently small open balls about (z 1 , . . . ,z n ), this is clearly possible. Then since the changes (39)-(40) between F n , G n come from Lie algebra transformations, one can show that (65) lies in L n for all n and (z 1 , . . . , z n ) ∈ (C × ) n if and only if the expression σ∈Sn G n (z σ(1) , . . . , z σ(n) ;z σ(1) , . . . ,z σ(n) ) e σ(1) * · · · * e σ(n) Lemma 3.14. If 1 k n and z 1 , . . . , z k−1 , z k+1 , . . . , z n are fixed in C × , the function F n of Theorem 3.12 satisfies F n (z 1 , . . . , z n ) C(1 + \| log z k \|) n−1 for all z k ∈ C × , for some C > 0 depending on k, n and z 1 , . . . , z k−1 , z k+1 , . . . , z n . Proof. For n = 1, 2 the lemma follows from (32) and (49). On the domains (47), equation (54) gives an expression for ∂F n /∂z k in terms of F l for l < n, and it is easy to use this and induction on n to prove the lemma on (47). To extend from (47) to (C × ) n , we can observe that for (z 1 , . . . , z n ) in the complement of (47) in (C × ) n , F n (z 1 , . . . , z n ) is a weighted average of the limits of F n (z ′ 1 , . . . , z ′ n ) as (z ′ 1 , . . . , z ′ n ) → (z 1 , . . . , z n ), for (z ′ 1 , . . . , z ′ n ) in the various sectors of (47) meeting at (z 1 , . . . , z n ). In particular, F n (z 1 , . . . , z n ) lies in the convex hull in C of these limits, so estimates on \|F n \| on (47) imply the same estimates on (C × ) n . Proof. Condition 3.4 holds by Theorem 3.12. Given λ ∈ C × we note that all conditions on F n , G n are preserved by replacing F n (z 1 , . . . , z n ) by F n (λz 1 , . . . , λz n ) and G n (z 1 , . . . , z n ;z 1 , . . . ,z n ) by G n (λz 1 , . . . , λz n ; λz 1 , . . . , λz n ). Thus, since these conditions determine F n , G n uniquely (31) must hold. Equation (32) holds by definition, and (34) and (35) follow from Theorem 3.13 and Lemma 3.14. Remark 3.16. It is an obvious question whether the functions F n constructed above can be written in terms of known special functions. Tom Bridgeland has found a very nice answer to this, which will be published in [5]. It involves the hyperlogarithms of Goncharov [8, §2], a kind of polylogarithm, which are defined by iterated integrals and satisfy a p.d.e. reminiscent of (54). Bridgeland shows that F n (z 1 , . . . , z n ) may be written on the domain (47) as an explicit sum over rooted trees with n leaves of a product over vertices of the tree of a hyperlogarithm whose arguments are various sums of z 1 , . . . , z n , and a constant factor. This is interesting, as polylogarithms and hyperlogarithms have many links to other branches of mathematics such as number theory, Hodge theory and motives, and the author wonders whether the ideas of this paper will also have such links. Flat connections We now explain how to define a holomorphic L-valued connection Γ on Stab(A) using the generating functions f α , which the p.d.e. (51) implies is flat. Our formulae involve infinite sums over all α ∈ C(A), so we need a notion of convergence of infinite sums in L, that is, a topology on L. This also clarifies the meaning of the infinite direct sum L = α∈C(A) L α in Assumption 2.12, since we can take L to be the set of convergent sums α∈C(A) l α with l α ∈ L α . Here are simple definitions of convergence and the direct sum which go well with Assumption 3.2, and ensure the formulae below converge in this case. If Assumption 3.2 does not hold, choosing a topology on L to make the formulae below converge may be difficult or impossible; in this case the sum (30) defining f α may not converge either. Also, we must consider whether the Lie bracket [ , ] is defined on all of L × L, and whether it commutes with limits. Definition 4.1. In Assumption 2.12, by the direct sum L = α∈C(A) L α we mean simply that L is the infinite Cartesian product of the spaces L α . That is, elements of L are just arbitrary families (l α ) α∈C(A) with l α ∈ L α , with no restriction on how many l α are zero, and no other 'smallness conditions' on the l α . Write Π α : L → L α for the obvious projection. A possibly infinite sum i∈I l i in L is called convergent if for each α ∈ C(A) there are only finitely many i ∈ I with Π α (l i ) nonzero. The limit l = (l α ) α∈C(A) in L is defined uniquely by taking l α to be the sum of the nonzero Π α (l i ). That is, i∈I l i = l if i∈I Π α (l i ) = Π α (l) in L α for all α ∈ C(A), where the second sum is well-defined as it has only finitely many nonzero terms. The direct sum H = ᾱ∈C(A) H α and convergence of sums in H are defined in the same way. If Assumption 3.2 holds, it is easy to see that the Lie brackets [ , ] : L α ×L β → L α+β extend to a unique Lie bracket [ , ] : L × L → L which commutes with limits. Otherwise, the Lie bracket of two convergent sums can be a nonconvergent sum, so [ , ] can only be defined on a subspace of L × L. In the situation of §3, define a section Γ in C ∞ L ⊗ T * C Stab(A) by Γ(Z) = α∈C(A) f α (Z) ⊗ d(Z(α)) Z(α) .(69) This infinite sum is convergent in the sense of Definition 4.1, extended to L ⊗ T * C Stab(A) in the obvious way, since for each α ∈ C(A) there is only one term in the sum with Π α (· · · ) nonzero. Then Γ is a connection matrix for a holomorphic connection on the trivial complex Lie algebra bundle L × Stab(A) over Stab(A). By standard differential geometry, the curvature of this connection is the section R Γ of the vector bundle L ⊗ Λ 2 T * C Stab(A) over Stab(A) given by R Γ = dΓ + 1 2 Γ ∧ Γ.(70) To form Γ∧Γ ∈ C ∞ L⊗Λ 2 T * C Stab(A) from Γ⊗Γ ∈ C ∞ (L⊗T * C Stab(A)) 2 , we both project L⊗L → L using the Lie bracket [ , ] on L, and project T * C Stab(A)⊗ T * C Stab(A) → Λ 2 T * C Stab(A) using the wedge product ∧. Combining (51), (69) and (70) we find that R Γ = α∈C(A) df α (Z) ∧ d(Z(α)) Z(α) + 1 2 β,γ∈C(A) [f β (Z), f γ (Z)]⊗ d(Z(β)) Z(β) ∧ d(Z(γ)) Z(γ) = α,β,γ∈C(A): β+γ=α 1 2 [f β (Z), f γ (Z)] ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) ∧ d(Z(α)) Z(α) + β,γ∈C(A) 1 2 [f β (Z), f γ (Z)] ⊗ d(Z(β)) Z(β) ∧ d(Z(γ)) Z(γ) (71) = β,γ∈C(A) 1 2 [f β (Z), f γ (Z)] ⊗ d(Z(γ)) Z(γ) − d(Z(β)) Z(β) ∧ d(Z(β))+d(Z(γ)) Z(β)+Z(γ) + d(Z(β)) Z(β) ∧ d(Z(γ)) Z(γ) = 0,s(Z) = α∈C(A) f α (Z),(72) which converges as in Definition 4.1. Then from (51) and (69) we see that Let P : L → C be smooth and invariant under ad(L), that is, dP (x)·[x, y] = 0 for all x, y ∈ L. Then ∇ ad(Γ) s = 0 implies that P (s) is constant on Stab(A). For example, if ρ : L → End(V ) is a representation of L on a finite-dimensional C-vector space V then P (x) = det ρ(x) − λ id V has these properties, so the characteristic polynomial of ρ(s(Z)) is constant on Stab(A). ∇ ad(Γ) s = α∈C(A) df α (Z) + β∈C(A) f β (Z), γ∈C(A) f γ (z) ⊗ d(Z(β)) Z(β)(73) In general, for s as in (72) the eigenvalues of s(Z) in any representation of L should be constant on Stab(A). However, the author does not expect this construction to be useful with the topology on L in Definition 4.1, as it seems likely that the only finite-dimensional representations for such infinitedimensional L will be nilpotent, and so have zero eigenvalues anyway. The author feels that the topology on L given in Definition 4.1 is rather trivial, and that if the ideas of this section do have interesting applications in classes of examples it will be with a more complex topology on L appropriate to the examples. Then the convergence and validity of equations (69)-(73) would become conjectures to be (dis)proved in these examples, depending on asymptotic properties of the f α for large α. Extending all this to triangulated categories For generic Z this amounts to summing over α ∈ K(T ) \ {0}. Sums over α ∈ C(A) involving f α (Z) need a more subtle approach we describe below. We now explain two neat coincidences meaning that arguments in §3 still work with this replacement, although one might have expected them to fail. First, note that if (Z, P) ∈ Stab(T ) and α ∈ K(T ) with Z(α) = 0 then we must have δ α (Z, P) = ǫ α (Z, P) = 0. This is because δ α (Z, P), ǫ α (Z, P) are constructed from (Z, P)-semistable objects in class α, but there are no such objects if Z(α) = 0 by Definition 2.20. We also expect δ α (Z ′ , P ′ ) = ǫ α (Z ′ , P ′ ) = 0 for (Z ′ , P ′ ) in a small open neighbourhood of (Z, P) in Stab(T ). This means that omitting terms in ǫ αi (Z, P) in (30) when Z(α i ) = 0 does not cause discontinuities on the hypersurface Z(α i ) = 0 in Stab(T ), since the omitted terms are zero near there anyway. Second, note that f α (Z, P) = 0 when Z(α) = 0, since (30) now involves terms in α 1 , . . . , α n with Z(α k ) = 0 but Z(α 1 ) + · · · + Z(α n ) = Z(α) = 0, so F n (Z(α 1 ), . . . , Z(α n )) = 0, and every term in (30) is zero. However, for α = 0 we do not expect f α (Z ′ , P ′ ) ≡ 0 for (Z ′ , P ′ ) near (Z, P). So in sums such as (69) involving f α (Z)/Z(α), giving 0/0 when Z(α) = 0, it is not right to just omit α when Z(α) = 0, for α = 0. Instead, since f α (Z, P) is holomorphic and zero when Z(α) = 0, as for the functions H n in §3.2, the holomorphic function h α (Z, P) = f α (Z, P)/Z(α) on Z(α) = 0 extends uniquely over Z(α) = 0, so in (30), (32), (34) we replace terms f α (Z, P)/Z(α) by h α (Z, P). Convergence of sums. Once we replace sums over α ∈ C(A) by sums over α ∈ K(T ) with Z(α) = 0, most of the equations in §3- §4 become infinite sums, and the question of whether they converge at all in any sense becomes acute. There seems to be no triangulated analogue of Assumption 3.2 that makes the sums finite, nor can the author find any way to make the sums converge in a formal power series sense. Here are two comments which may help. Firstly, suppose the Lie algebra L is nilpotent. That is, define ideals L = L 1 ⊃ L 2 ⊃ · · · by L 1 = L, L n+1 = [L, L n ], and suppose n 1 L n = {0}. Then Theorem 3.13 implies that the sum of terms with fixed n in (30) lie in L n . Hence, projecting (30) to L/L k eliminates all terms with n k. If we use a notion of convergence such that a sum converges in L if its projections to L/L k converge for all k 1, then we only have to show the sum (30) for n < k converges in L/L k , which may be easier. Secondly, even if the sum (30) defining f α does not make sense, the p.d.e. (51) upon the f α might still converge in the triangulated case, as it is a much simpler sum. For example, if g = n + ⊕ h ⊕ n − is a Kac-Moody Lie algebra, it is known [13, §4.9] how to use Ringel-Hall algebras of abelian categories of quiver representations A = mod-KQ to realize H = U (n + ) and L = n + in examples, and people have hoped to use triangulated categories to obtain H = U (g) and L = g. If we could do this with g a finite-dimensional semisimple Lie algebra, then α, β, γ in (51) would take values in the set of roots of g, with L α being the root space g α , and (51) would become a finite sum, so trivially convergent. However, (30) would still be an infinite sum. Intuitively, what is going on is as follows. The functions F n are related to certain finite-dimensional nilpotent Lie algebras N n for n 1. In a similar way to Ramakrishnan [18] for the higher logarithms ln k , one can use the F k for k n to write down a nontrivial flat holomorphic N n -valued connection on (z 1 , . . . , z n ) ∈ C n : z a + · · · + z b = 0 for all 1 a b n, (a, b) = (1, n) . In the Ringel-Hall case H = U (n + ), L = n + above, equation (30) is about building the nilpotent Lie algebra n + , and the flat n + -valued connection Γ of §4, out of the standard family of nilpotent Lie algebras N n , and standard flat N n -valued connections. However, it may not be possible to build semisimple Lie algebras g and their flat connections from standard nilpotent building blocks N n , which is why (30) may not converge. But (51) has to do with general Lie algebras, not just nilpotent Lie algebras, and so may make sense in a more general setting. The Calabi-Yau 3-fold case Finally we discuss and elaborate the ideas of §3- §5 in the Calabi-Yau 3-fold case of Example 2.19. We use the notation of this example and §3- §5 throughout. 6.1 Holomorphic functions F α , H α and their p.d.e.s We begin with the abelian category case. Since f α maps Stab(A) → L α by Theorem 3.13 and L α = C · c α we may write f α = F α c α for a holomorphic function F α : Stab(A) → C, for α ∈ C(A). Also ǫ α (µ) = J α (µ)c α for J α (µ) ∈ Q, so combining (26) and (30) we find that where µ is the slope function associated to Z. We also have F α ≡ Z(α)H α for a holomorphic function H α : Stab(A) → C given by and the flat connection Γ of (69) is Γ(Z) = α∈C(A) F α (Z) c α ⊗ d(Z(α)) Z(α) = α∈C(A) H α (Z) c α ⊗ d(Z(α)).(77) In the triangulated category case we replace F α , H α (Z) and J α (µ) by F α , H α , J α (Z, P), and replace sums over C(A) by sums over K(T ) \ {0} in (74)-(77), and also omit terms involving α i with Z(α i ) = 0 in (74)-(75). In the triangulated case, the Lie algebra L is L = c α : α ∈ K(T ) C , with [c α , c β ] =χ(α, β)c α+β . Suppose K(T ) is a lattice of finite rank, and χ : K(T ) × K(T ) → Z is nondegenerate. Then we can interpret L as a Lie algebra of complex functions on the real torus T T = Hom(K(T ), R)/ Hom(K(T ), Z) by identifying c α with the function Thus the map c α → C α induces an injective Lie algebra morphism from L to a Lie algebra of complex functions on T T with Poisson bracket { , }. It is not clear which class of functions on T T we should consider. For instance, smooth functions C ∞ (T T ) C or real analytic functions C ω (T T ) C both give well behaved Lie algebras of functions on T T . These also come with natural topologies, and so yield notions of convergence of infinite sums in L, as discussed in §4. However, the author expects that these notions of convergence will be too strict to make the sums of §3- §5 converge in interesting examples, and some much weaker convergence criterion than smoothness or real analyticity is required. A flat connection on T Stab(T ) in the triangulated case In the triangulated category case, the invariants J α (Z, P) ∈ Q 'counting' (Z, P)semistable objects in class α ∈ K(T ) should satisfy J −α (Z, P) = J α (Z, P), since the translation operator [+1] induces a bijection between (Z, P)-semistable objects in classes α and −α. Thus we expect F −α ≡ F α for all α ∈ K(T ) \ {0}. Hence Γ in (77) is actually an L ′ -valued connection, where L ′ = c α + c −α : α ∈ K(T ) C is a Lie subalgebra of L with [c α + c −α , c β + c −β ] =χ(α, β) (c α+β + c −α−β ) − (c α−β + c −α+β ) . Regarded as a Lie algebra of functions on T T , the functions in L ′ are invariant under −1 : T T → T T acting by x+Hom(K(T ), Z) → −x+Hom(K(T ), Z), so the Hamiltonian vector fields of functions in L ′ all vanish at 0 ∈ T T . Therefore they have a Lie algebra action on T 0 T T ∼ = Hom(K(T ), C), and on its dual K(T )⊗ Z C. That is, we have found a Lie algebra representation ρ : L ′ → End K(T ) ⊗ Z C , which is given explicitly on the generators c α + c −α of L ′ by ρ(c α + c −α ) : γ −→ 2χ(α, γ)α. Comparing (78) and (79) shows ρ is a Lie algebra morphism. Note that ρ does not extend to a Lie algebra morphism L → End K(T ) ⊗ Z C . Now there is a natural isomorphism K(T ) ⊗ Z C ∼ = T * Stab(T ). Thus in the Calabi-Yau 3-fold triangulated category case, if all the relevant sums converge in End K(T ) ⊗ Z C (which seems rather unlikely), then applying ρ to the flat connection Γ of §4 induces a flat connection ∇ ρ(Γ) on the tangent and cotangent bundles T Stab(T ), T * Stab(T ) of Stab(T ). This connection is easily seen to be torsion-free: the connection on T Stab(T ) is a sum over α ∈ K(T )\{0} of a term linear in α⊗α⊗α, and the torsion vanishes because of a symmetry in exchanging two copies of α. It also preserves the symplectic form on Stab(T ) induced bȳ χ. Integrating ∇ ρ(Γ) should give new, interesting flat local coordinate systems on Stab(T ). Ignoring convergence issues, define a section g C of S 2 T * Stab(T ) by g C (Z, P) = α∈K(T )\{0} F α (Z, P) dZ(α) ⊗ dZ(α). In a calculation related to (73), differentiating using ∇ ρ(Γ) gives (β, γ)F β (Z, P)F γ (Z, P) d(Z(β)) Z(β) ⊗ − dZ(β)+dZ(γ) ⊗ dZ(β)+dZ(γ) +dZ(β)⊗ dZ(γ)+dZ(γ)⊗ dZ(β) = − β,γ∈K(T )\{0} χ(β, γ)F β (Z, P)F γ (Z, P) d(Z(β)) Z(β) ⊗ dZ(β) ⊗ dZ(β)+ dZ(γ) ⊗ dZ(γ) = 0. Here the second line applies ρ(Γ) to g C , where we replace α in the sum (77) defining Γ by β, and α in the sum (80) defining g C by γ, and use the fact that ρ(c β + c −β ) dZ(γ) ⊗ dZ(γ) = 2χ(β, γ) dZ(β) ⊗ dZ(γ) + dZ(γ) ⊗ dZ(β) . The third and fourth lines of (81) substitute (76) into the first line and set α = β + γ, and for the final step we note that as F −γ (Z, P) = F γ (Z, P), pairing terms in the fifth line with β, γ and β, −γ shows that everything cancels. Suppose now that g C is a nondegenerate section of S 2 T * Stab(T ). (If it is nondegenerate at one point in Stab(T ) it is degenerate everywhere in this connected component, as it is constant under ∇ ρ(Γ) by (81).) Then g C is a holomorphic metric on Stab(T ). Since ∇ ρ(Γ) is torsion-free with ∇ ρ(Γ) g C = 0, we see that ∇ ρ(Γ) is the Levi-Civita connection of g C , and thus g C is flat as ∇ ρ(Γ) is flat. Note that Frobenius manifolds also have flat holomorphic metrics. A variant of the holomorphic anomaly equation Several people have commented to the author that the p.d.e. (54) on F n resembles the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa [1,2], which is interpreted by Witten [23]. This equation is [2, eq. (3.6)] ∂īF g = 1 2Cījk e 2K G jj G kk ∂ j ∂ k F g−1 + g−1 r=1 ∂ j F r ∂ k F g−r ,(82) which can be repackaged as a linear equation on exp ∞ g=1 λ 2g−2 F g . It is beyond the author's competence to properly explain (82). Very roughly, F g is a complex-valued generating function which 'counts' numbers of genus g holomorphic curves in a Calabi-Yau 3-fold X -just as our generating functions F α 'count' coherent sheaves on X. It is not holomorphic, but is nearly so, in that (82) expresses∂F g in terms of ∂F r for r < g. For λ ∈ C × and fixed a, b ∈ Z define a (0, 1)-form on Stab(A) by Φ λ (Z) = α∈C(A) λ a e λ b Z(α) H α (Z) d(Z(α)). The idea here is that we have taken the complex conjugate of (77), and then replaced the Lie algebra element c α by the holomorphic function λ a e λ b Z(α) . In the abelian category case, as Im Z(α) > 0 for α ∈ C(A), if Im(λ b ) ≫ 0 then e λ b Z(α) is small, and it seems plausible that (83) may actually converge. In the triangulated case, when C(A) in (83) is replaced by K(T ) \ {0}, convergence seems less likely. The (1, 1)-form ∂Φ λ and the (0, 2)-form∂Φ λ on Stab(A) are given by ∂Φ λ (Z) = λ a+b α∈C(A) e λ b Z(α) H α (Z) d(Z(α)) ∧ d(Z(α)), ∂Φ λ (Z) = − 1 2 λ a β,γ∈C(A) e λ b Z(β) H β (Z) e λ b Z(γ) H γ (Z) · χ(β, γ) d(Z(β)) ∧ d(Z(γ)), where in (85) we have used H α ≡ Z(α) −1 F α and substituted in (76). Using index notation for complex tensors as in (82), so that i, j are type (1, 0) tensor indices andī,j type (0, 1) tensor indices, we see these satisfy ∂ Φ λ (Z) īj = − 1 2 λ −a−2b (χ) ij ∂Φ λ (Z) iī ∂Φ λ (Z) jj .(86) Here (χ) ij is the (2, 0) part ofχ, regarded as a constant tensor in Λ 2 T Stab(A) = Λ 2 Hom(K(A), C). Equation (86) is formally similar to the p.d.e. satisfied by W λ = ∞ g=1 λ 2g−2 F g in the holomorphic anomaly case above, of the form ∂W λ = λ 2 linear term in ∂ 2 W λ + ∂W λ ⊗ ∂W λ . k∈Z (−1) k dim Hom T (U, V [k]) =χ([U ], [V ]) for all U, V ∈ T . F α (Z, P) = n 1, α1,...,αn∈K(T )\{0}: α1 + · · · + αn = α, Z(α k ) = 0 all k F n Z(α 1 ), . . . , Z(α n ) andC(A) = C(A) ∪ {0}. That is, C(A) is the set of classes in K(A) of nonzero objects U ∈ A, andC(A) the set of classes of objects in A. We think of C(A) as the 'positive cone' andC(A) as the 'closed positive cone' in K(A). In Example 2.2 we haveC(A) = N Q0 and C(A) = N Q0 \ {0}. Definition 2 . 10 . 210Let Assumption 2.1 hold, (τ, T, ), (τ ,T , ) be weak stability conditions on A, and ({1, . . . , n}, , κ) be A-data. Define U ({1, . . . , n}, , κ, τ,τ) = 1 l m n surjective ψ : {1, . . . , n} → {1, . . . , m} and ξ : {1, . . . , m} → {1, . . . , l}: i j implies ψ(i) ψ(j), i j implies ξ(i) ξ(j). Define λ : {1, . . . , m} → C(A) by λ( {1, . . . , n} → I with n = \|I\| and φ * ( ) = , and ({1, . . . , n}, , κ • φ) is A-data. Define U (I, , κ, τ,τ ) = U ({1, . . . , n}, , κ • φ, τ,τ ). Definition 2 . 11 . 211Let Assumption 2.1 hold and (τ, T, ), (τ ,T , ) be weak stability conditions on A. We say the change from (τ, T, ) to (τ ,T , ) is locally finite if for all constructible C ⊆ Obj A (K), there are only finitely many sets of A-data ({1, . . . , n}, , κ) for which S({1, . . . , n}, , κ, τ,τ) = 0 and C ∩ σ({1, . . . , n}) * M ss ({1, . . . , n}, , κ, τ ) A = ∅. A-data ({1, . . . , n}, , κ) : κ({1, . . . , n}) = α S({1, . . . , n}, , κ, τ,τ)· Example 2 . 16 . 216Let Assumption 2.1 hold and χ : K(A) × K(A) → Z be biadditive and satisfy Example 2 . 18 . 218Let Assumption 2.1 hold and χ : K(A) × K(A) → Z be biadditive and satisfy (16), and let Λ, Λ • , ℓ be as in Example 2.16. Consider pairs (I, κ) with I a finite set and κ : I → C(A) a map. Define an equivalence relation '≈' on such (I, κ) by (I, κ) ≈ (I ′ , κ ′ ) if there exists a bijection i : I → I ′ with κ ′ • i = κ. Write [I, κ] for the ≈-equivalence class of (I, κ). Introduce formal symbols b [I,κ] for all such equivalence classes [I, κ].As in[13, §6.3], let H = B(A, Λ, χ) be the Λ-module with basis the b[I,κ] Example 2 . 19 . 219Let Assumption 2.1 hold andχ : K(A) × K(A) → Z be antisymmetric and biadditive and satisfy As in Example 2.18, introduce symbols c [I,κ] for all equivalence classes [I, κ], and let H = C(A, Ω, 1 2χ ) be the C-vector space with basis the c [I,κ] . Define H α = [I,κ]:κ(I)=α C · c [I,κ] . Define a multiplication * on H by Theorem 2 . 21 . 221Let T be a triangulated category and K(T ) as in Definition 2.20. Write Stab(T ) for the set of stability conditions (Z, P) on T . Then Stab(T ) has a natural, Hausdorff topology. Let Σ be a connected component of Stab(T ). Then there is a complex vector subspace V Σ in Hom(K(T ), C) with a well-defined linear topology such that the map Σ → Hom(K(T ), C) given by Consider the following situation. Let Assumptions 2.1 and 2.12 hold for A, with K(A) of finite rank, and suppose Stab(A) in Example 2.7 is a nonempty open subset of Hom(K(A), C), and so a complex manifold. This works for all the quiver examples A = mod-KQ, nil-KQ, mod-KQ/I, nil-KQ/I, mod-A of [12, §10], with H, L, . . . chosen as in one of Examples 2.14-2.19. Assumption 3 . 2 . 32In the situation of Assumption 2.1, for each α ∈ C(A) there are only finitely pairs β, γ ∈ C(A) with α = β + γ. F n for f α to be continuous Let Assumptions 2.1, 2.12 and 3.2 hold for A, with K(A) of finite rank, and suppose Stab(A) in Example 2.7 is a nonempty open subset of Hom(K(A), C), and so a complex manifold. Let Z,Z ∈ Stab(A), with associated stability conditions (µ, R, ) and (μ, R, ). We think of Z as varying in Stab(A) andZ as a fixed base point.We need some notation for the coefficients S, U ({1, . . . , n}, , κ, µ,μ) of §2.1. They depend on the 2n complex numbers Z • κ(k),Z • κ(k) for k = 1, . . . , n in H = {x + iy : x ∈ R, y > 0}, and the definition makes sense for any 2n elements of H. Thus there are unique functions s n , u n : H 2n → Q written s n , u n (z 1 , . . . , z n ;z 1 , . . . ,z n ) such that S({1, . . . , n}, , κ, µ,μ) = s n (Z •κ(1), . . . , Z •κ(n);Z •κ(1), . . . ,Z •κ(n)), U ({1, . . . , n}, , κ, µ,μ) = u n (Z •κ(1), . . . , Z •κ(n);Z •κ(1), . . . ,Z •κ(n)). Proposition 3. 3 . 3Suppose Assumptions 2.1, 2.12 and 3.2 hold for A with K(A) of finite rank and Stab(A) is a nonempty open subset of Hom(K(A), C) (30) are linearly independent in H. This happens in the examples of [14, Ex. 7.10], for arbitrarily large n. since the conditions in (41) play the same role asμ(α) >μ(β) implies µ(α) > µ(β) does in (42). From(10) we deduce that if (z 1 , . . . , z n ) ∈ N (z1,...,zn) then u n (z 1 , . . . , z n ;z 1 , . . . ,z n ) =1 l m n surjective ψ : {1, . . . , n} → {1, . . . , m} and ξ : {1, . . . , m} → {1, . . . , l}: a b implies ψ(a) ψ(b), a b implies ξ(a) ξ(b), Condition 3. 4 . 4Let some functions F n : (C × ) n → C be given for n 1. For all n 1 and (z 1 , . . . ,z n ) ∈ (C × ) n there should exist an open neighbourhood N (z1,...,zn) of (z 1 , . . . ,z n ) in (C × ) n , such that if (z 1 , . . . , z n ) ∈ N (z1,...,zn) > 0 for k = 1, . . . , n. For (z 1 , . . . , z n ) ∈ N (z1,...,zn) we must have F n (z 1 , . . . , z n ) = m=1,...,n, 0=a0<a1<···<am=n and c1, . . . , cm ∈ [0, 2π) : za∈e ic k (0,∞), a k−1 <a Theorem 3 . 7 . 37There exists at most one family of functions F n : (C × ) n → C satisfying Condition 3.4 and equations (31), (32), (34), (35) of Remark 3.1. Proposition 3 . 8 . 38Suppose Assumptions 2.1, 2.12 and 3.2 hold for A with K(A) of finite rank and Stab(A) is a nonempty open subset of Hom(K(A), C), ( 53 ) 53in H n . This holds for some (z 1 , . . . ,z n ) in H n if and only if it holds for all (z 1 , . . . ,z n ) in H n .This condition is also necessary, for all values of n occurring in (30), if the terms ǫ α1 (µ) * ǫ α2 (µ) * · · · * ǫ αn (µ) occurring in(30)are linearly independent in H. This happens in the examples of [14, Ex. 7.10], for arbitrarily large n. Now suppose Condition 3.4 holds. Then equation (53) holds for (z 1 , . . . , z n ) in N (z1,...,zn) for all n 1 and all fixed (z 1 , . . . ,z n ) ∈ (C × ) n if and only if the following p.d.e. holds on the domain (47) for all n 1: D l,n (z 1 , . . . ,z n ) = lim (z1, . . . , zn) → (z1, . . . ,zn) : (z1, . . . , zn) lies in (47), Im(z l+1 /z l ) < 0 F n (z 1 , . . . , z n ) − lim (z1, . . . , zn) → (z1, . . . ,zn) : (z1, . . . , zn) lies in (47), Im(z l+1 /z l ) > 0 F n (z 1 , . . . , z n ). (61) lies in L n for all (z 1 , . . . ,z n ) ∈ (C × ) n and (z 1 , . . . , z n ) ∈ N (z1,...,zn) .In fact one can prove more than this. For m 1, write:( * m ) Suppose (65) lies in L n for all n < m and (z 1 , . . . , z n ) ∈ (C × ) n and (68) lies in L n for all n < m, (z 1 , . . . ,z n ) ∈ (C × ) n and (z 1 , . . . , z n ) ∈ N (z1,...,zn) .One can show that if ( * m ) holds, (z 1 , . . . ,z m ) ∈ (C × ) m and (z 1 , . . . , z m ) ∈ N (z1,...,zm) , then (65) with n = m and this (z 1 , . . . , z m ) lies in L m if and only if (68) with n = m and these (z 1 , . . . , z m ), (z 1 , . . . ,z m ) lies in L m . The point is that (68) is (65) plus sums of multiple commutators of terms we know lie in L m by our assumptions for n < m, and vice versa. Suppose by induction that ( * m ) holds for some m 1. When m = 1 this is vacuous. Let (z 1 , . . . ,z m ) ∈ (C × ) m and (z 1 , . . . , z m ) ∈ N (z1,...,zm) with (z 1 , . . . , z m ) in (66) for m = n. Then (65) with n = m and this (z 1 , . . . , z m ) lies in L m by the proof above, so (68) with n = m and these (z 1 , . . . , z m ), (z 1 , . . . ,z m ) lies in L m . As L m is closed and G m (z 1 , . . . , z m ;z 1 , . . . ,z m ) is continuous in (z 1 , . . . , z m ) and the intersection of N (z1,...,zm) with (66) for m = n is dense in N (z1,...,zm) , taking limits shows (68) lies in L m for any (z 1 , . . . , z m ) ∈ N (z1,...,zm) . As this holds for all (z 1 , . . . ,z m ) ∈ (C × ) m , equation (65) lies in L m for all (z 1 , . . . , z m ) ∈ (C × ) m . Hence by induction ( * m ) holds for all m 1, which proves the first part of the theorem. The remaining two parts follow as in Remark 3.1(d). Corollary 3 . 15 . 315The functions F n of Theorem 3.12 satisfy Condition 3.4 and equations (31), (32), (34) and (35) of Remark 3.1. Thus by Theorem 3.7 they are the unique functions F n satisfying the conditions of §3.1. since the term [· · · ] in the last line is zero. Thus Γ is a flat connection. If Assumption 3.2 holds, these calculations are all valid as infinite convergent sums in the sense of Definition 4.1.If ρ : L → End(V ) is a representation of the Lie algebra L on a complex vector space V then Γ induces a flat connection ∇ ρ(Γ) on the trivial vectorbundle V ×Stab(A) over Stab(A), with connection 1-form ρ(Γ) in C ∞ End(V )⊗ T * C Stab(A) . If s : Stab(A) → V is a smooth section of this bundle then ∇ ρ(Γ) s = ds + ρ(Γ) · s in C ∞ V ⊗ T * C Stab(A) .In particular, as the tangent bundle T Stab(A) is naturally isomorphic to the trivial vector bundle Hom(K(A), C) × Stab(A), if L has a representation ρ on Hom(K(A), C) then ∇ ρ(Γ) is a flat connection on T Stab(A). We will see in §6 that this should happen in the triangulated category extension of the Calabi-Yau 3-fold invariants in Example 2.19.Take V to be L and ρ the adjoint representation ad : L → End(L). Define a section s : Stab(A) → L by s in (72) is a constant section of L × Stab(A). If Assumption 3.2 holds then (73) is valid as infinite convergent sums in the sense of Definition 4.1. F n Z(α 1 ), . . . , Z(α n ) i , α j ) , α1,...,αn∈C(A): α1+···+αn=αH n Z(α 1 ), . . . , Z(α n ) i , α j ) . , γ)H β (Z)H γ (Z)Z(γ)d(Z(β)), C α : Hom(K(T ), R)/ Hom(K(T ), Z) → C, C α : x+Hom(K(T ), Z) → e 2πix(α) . Now (2πi) −2χ induces a section of Λ 2 T (T T ) ⊗ R C yielding a Poisson bracket { , } on smooth complex functions on T T , with {C α , C β } =χ(α, β)C α+β . ∇ ρ(Γ) g C = α∈K(T )\{0} dF α (Z, P) ⊗ dZ(α) ⊗ dZ(α)+ (81) β,γ∈K(T )\{0}χ(β, γ)F β (Z, P)F γ (Z, P) Then [12, §7] defines moduli stacks Obj A of objects in A, and Obj α A of objects in A with class α in K(A), for each α ∈C(A). They are Artin K-stacks, locally of finite type, with Obj α A an open and closed K-substack of Obj A . The underlying geometric spaces Obj A (K), Obj α A (K) are the sets of isomorphism classes of objects U in A, with [U ] = α for Obj α A (K). In [14, §4] we study (weak ) stability conditions on A, generalizing Rudakov [19]. The next three definitions are taken from [14, Def.s 4.1-4.3, 4.6 & 4.7]. Definition 2.4. Let Assumption 2.1 hold and C(A) be as in υ∈S m−k · · · in the final line of (67) lie in L k with generators e σ(1) , . . . , e σ(k) and L m−k with generators e σ(k+1) , . . . , e σ(m) respectively, so they and their commutator in (67) lie in L m . Hence dP m is an L m -valued 1-form on (66), not Note too that there are no convergence issues for (86), it always makes sense as an equation on (0, 1)-forms Φ λ on Stab(A) or Stab(T ). The author has no idea whether all this is relevant to String Theory. programme cannot yet be rigorously extended from abelian categories A to triangulated categories T , because the material of [12-15] on which it rests has not yet been extended. Some remarks on the issues involved are given in [15, §7programme cannot yet be rigorously extended from abelian categories A to triangulated categories T , because the material of [12-15] on which it rests has not yet been extended. Some remarks on the issues involved are given in [15, §7]. DH(T ) under strong finiteness conditions on T , and [22, §3.3.3] an 'absolute Hall algebra' H abs (T ) under weaker conditions. It seems likely that the right way to construct examples of data satisfying a triangulated version of Assumption 2.12 is to use an algebra morphism Φ : DH(T ) or H abs (T ) → H. Also, [22] provides the tools needed to form moduli Artin ∞-stacks of objects and configurations in triangulated categories with dg-enhancement, which is the main ingredient needed to extend. The work of Bertrand Toën. 21,22] is likely to be useful here. In particular. 12-15] to the triangulated case. Here are some issues in extending the ideas of this paper to the triangulated caseThe work of Bertrand Toën [21,22] is likely to be useful here. In particular, [21] defines a 'derived Hall algebra' DH(T ) under strong finiteness conditions on T , and [22, §3.3.3] an 'absolute Hall algebra' H abs (T ) under weaker conditions. It seems likely that the right way to construct examples of data satisfying a triangulated version of Assumption 2.12 is to use an algebra morphism Φ : DH(T ) or H abs (T ) → H. Also, [22] provides the tools needed to form moduli Artin ∞-stacks of objects and configurations in triangulated categories with dg-enhancement, which is the main ingredient needed to extend [12-15] to the triangulated case. Here are some issues in extending the ideas of this paper to the triangulated case. Lifting phases from R/2πiZ to R. The δ α (τ ), ǫ α (τ ) of Assumption 2.12 are constructed from 'characteristic functions' of τ -semistable objects in A in class α ∈ C(A). Now in Bridgeland's stability conditions (Z, P) on a triangulated category T , Definition 2.20, the (Z, P)-semistable objects in class α ∈ K(T )Lifting phases from R/2πiZ to R. The δ α (τ ), ǫ α (τ ) of Assumption 2.12 are constructed from 'characteristic functions' of τ -semistable objects in A in class α ∈ C(A). Now in Bridgeland's stability conditions (Z, P) on a triangulated category T , Definition 2.20, the (Z, P)-semistable objects in class α ∈ K(T ) That is, if we write Z(α) = re iπφ for φ ∈ R, then the (Z, P)-semistable objects in class α with phase φ are the objects U in P(φ). ∈ C ×, with class α ∈ K(T∈ C × . That is, if we write Z(α) = re iπφ for φ ∈ R, then the (Z, P)-semistable objects in class α with phase φ are the objects U in P(φ) with class α ∈ K(T ). It is natural to ask whether the triangulated analogues δ α (Z, P), ǫ α (Z, P) should also depend on a choice of phase φ for Z(α). The author's view is that for the purposes of this paper, they should not depend on choice of phase. Effectively this means working in a Hall-type algebra in which the translation squared operator [+2] is the identity. The reason is that if δ α (Z, P), ǫ α (Z, P) depended on phase then f α (Z, P) should also depend on a choice of phase for Z(α), and F n (z 1. C n rather than. C × ) n . Allowing this would invalidate nearly all of §3. In particular, the uniqueness result Theorem 3.7 would fail, and the p.d.e. (51) would no longer make sense, as for a given choice of phase for Z(α) there does not seem to be a natural way to choose phases for Z(β), Z(γ) in the sumReplacing φ by φ + 2n for n ∈ Z replaces P(φ) by P(φ + 2n) = P(φ)[2n], so replaces objects U by U [2n], that is, applying the translation functor to the power 2n. Note that replacing U by U [2n] fixes the class α of U in K(T ). It is natural to ask whether the triangulated analogues δ α (Z, P), ǫ α (Z, P) should also depend on a choice of phase φ for Z(α). The author's view is that for the purposes of this paper, they should not depend on choice of phase. Effectively this means working in a Hall-type algebra in which the translation squared operator [+2] is the identity. The reason is that if δ α (Z, P), ǫ α (Z, P) depended on phase then f α (Z, P) should also depend on a choice of phase for Z(α), and F n (z 1 , . . . , z n ) on choices of phase for z 1 , . . . , z n . That is, F n should be a function of (log z 1 , . . . , log z n ) ∈ C n rather than (z 1 , . . . , z n ) ∈ (C × ) n . Allowing this would invalidate nearly all of §3. In particular, the uniqueness result Theorem 3.7 would fail, and the p.d.e. (51) would no longer make sense, as for a given choice of phase for Z(α) there does not seem to be a natural way to choose phases for Z(β), Z(γ) in the sum. as for nonzero T every element of K(T ) will be represented by a nonzero object. However, the sums over α ∈ C(A) in §3 do not make sense when replaced by α ∈ K(T ), because of problems when Z(α) = 0. For instance, F n (Z(α 1 ), . . . , Z(α n )) in (30) is undefined if any Z(α k ) = 0, and (51) is undefined if any Z(β) = 0 or Z(γ) = 0. The author proposes that the right answer is to replace sums over α ∈ C(A) in §3 involving ǫ α (µ), such as (30). Z(α) = 0 . What should be the analogue of the positive cone C(A) in a triangulated category T ? Replacing A by T in (8) will give C(T ) = K(T ). Replacing C(A) by α ∈ K(Tby sums over all α ∈ K(T ) with Z(α) = 0Replacing C(A) by α ∈ K(T ) : Z(α) = 0 . What should be the analogue of the positive cone C(A) in a triangulated category T ? 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[ "Certifiably Optimal Mutual Localization with Anonymous Bearing Measurements", "Certifiably Optimal Mutual Localization with Anonymous Bearing Measurements" ]	[ "Yingjian Wang ", "Xiangyong Wen ", "Longji Yin ", "Chao Xu ", "Yanjun Cao ", "Fei Gao " ]	[]	[]	Mutual localization is essential for coordination and cooperation in multi-robot systems. Previous works have tackled this problem by assuming available correspondences between measurements and received odometry estimations, which are difficult to acquire, especially for unified robot teams. Furthermore, most local optimization methods ask for initial guesses and are sensitive to their quality. In this paper, we present a certifiably optimal algorithm that uses only anonymous bearing measurements to formulate a novel mixed-integer quadratically constrained quadratic problem (MIQCQP). Then, we relax the original nonconvex problem into a semidefinite programming (SDP) problem and obtain a certifiably global optimum using with off-the-shelf solvers. As a result, our method can determine bearing-pose correspondences and furthermore recover the initial relative poses between robots under a certain condition. We compare the performance with local optimization methods on extensive simulations under different noise levels to show our advantage in global optimality and robustness. Realworld experiments are conducted to show the practicality and robustness.	10.1109/lra.2022.3190079	[ "https://arxiv.org/pdf/2203.09312v1.pdf" ]	247,519,178	2203.09312	c61070196a9a6f5d223ba1078410dd95cedb7c34	Certifiably Optimal Mutual Localization with Anonymous Bearing Measurements Yingjian Wang Xiangyong Wen Longji Yin Chao Xu Yanjun Cao Fei Gao Certifiably Optimal Mutual Localization with Anonymous Bearing Measurements Mutual localization is essential for coordination and cooperation in multi-robot systems. Previous works have tackled this problem by assuming available correspondences between measurements and received odometry estimations, which are difficult to acquire, especially for unified robot teams. Furthermore, most local optimization methods ask for initial guesses and are sensitive to their quality. In this paper, we present a certifiably optimal algorithm that uses only anonymous bearing measurements to formulate a novel mixed-integer quadratically constrained quadratic problem (MIQCQP). Then, we relax the original nonconvex problem into a semidefinite programming (SDP) problem and obtain a certifiably global optimum using with off-the-shelf solvers. As a result, our method can determine bearing-pose correspondences and furthermore recover the initial relative poses between robots under a certain condition. We compare the performance with local optimization methods on extensive simulations under different noise levels to show our advantage in global optimality and robustness. Realworld experiments are conducted to show the practicality and robustness. I. INTRODUCTION Recently, due to the inherent advantage, multi-robot systems have received increasing attention in many applications, such as formation control [1], exploration [2], search and rescue and surveillance. To execute each subtask correctly and complete the full task collaboratively, robots in a team are expected to be located in a common reference frame. However, this requirement is not satisfied in wild environments like underground caves where global coordinate systems are not available. Launching robots in a predetermined relative pose is another solution. However, it is obviously time-consuming and prone to failure in large-scale environments. To bridge this gap, self-localization using onboard sensors and relative pose recovery are irreplaceable in multirobot systems. There are majorly two ways to estimate the initial relative transformations between robots in a team. They are map-based localization which relies on exchanging environment features, and mutual localization which depends on robot-to-robot measurements. Most research focuses on the map-based relative pose recovery method, which can be easily adapted from loop-closing modules of existing simultaneous localization and mapping (SLAM) systems. However, it requires robots to observe the same scene and send observed environment information to others, leading to degeneration in the environments with many similar or texture-less scenes. 1 Overview of our proposed method which can obtain certifiably optimal solution for mutual localization problem with anonymous bearing measurements. Our result can be used for map fusion in multi-robot monocular SLAM and coordinate alignment in multi-robot tasks. Our study focuses on mutual localization using bearing measurements, which only utilize detected robots' 2D coordinates in the observer's image and observed robots' estimated odometry. Compared to map-based localization, it is less influenced by environments and needs less bandwidth. Despite its appeals, as we do not rely on any specialized devices, like visual tags or external sensors, data association between the visual detection and robot identifications in a team of unified robots is challenging. For this problem, existing works take the similar paradigm of establishing data association firstly and then recovering the relative pose with extra sensors such as IMU. Distinctly, in our paper, we introduce binary variables representing the data association relationships and mix them with multiple SO(3) variables representing the relative poses between observer and observed robots, formulating a mixed-integer problem. Furthermore, we rewrite it as a non-convex MIQCQP problem and employ tight convex relaxation to obtain a SDP problem. Thanks to its convexity, we obtain a certifiably globally optimal solution to our formulated problem. Moreover, we also provide a condition, under which our approach avoids local minima in noise-free cases. Complete algorithm is demonstrated in Fig.1. Extensive experiments on synthetic real-world datasets show the robustness of our method under different levels of noise. Our contributions in this paper are: 1) We provide an innovative formulation which jointly solves data association and relative poses in a MIQCQP problem. To the best of our knowledge, there is no such work in mutual localization. 2) We propose an algorithm for the non-convex MIQCQP problem, which adopts semidefinite relaxation (SDR) to make it convex. Furthermore, we provide a condition to guarantee the tightness of the relaxation. 3) We conduct sufficient simulation and real-world experiments to validate the practicality and robustness of our proposed method. 4) We release the implementation of our method in MAT-LAB and C++ for the reference of our community. II. RELATED WORKS A. Relative Pose Estimation There are mainly two ways to solve multi-robot relative pose estimation (RPE) problems: interloop detection based methods and mutual observation based methods. Most interloop detection based methods, including centralized [3,4] and decentralized architectures [5,6], firstly determine whether the robots in a team visited the same places using loop detection technique [7], then conduct the relative pose recovery. However, interloop detection-based methods require significant computation and bandwidth and have poor performance in environments with many similar scenes. Most mutual observation-based methods employ robotto-robot range or bearing measurements to recover relative poses. Early work [8]- [10] take extended Kalman filter (EKF) as nonlinear estimator using prior identified range measurements. Zhou [11] provides a set of 14 minimal analytical solutions that cover any combination of range and bearing measurements. However, their proposed algorithm has poor performance under noise because it only uses minimal measurements. Besides, all the above works assume that correspondence between measurement and estimated poses is known, which is not common in practical applications. Cognetti [12] and Franchi [13] solve mutual localization problem with particle filters (PF) using anonymous measurements. Indelman [14] and Dong [15] formulate a multi-robot pose graph problem and utilize the expectation-maximization (EM) approach to estimate initial relative poses between robots. However, it is well known that PF and EM all require extensive computation. Nguyen [16] adapts the coupled probabilistic data association filter to estimate relative pose with vision sensor and IMU. In [17], Jang proposes an alternating minimization algorithm to optimize relative poses in multirobot monocular SLAM. However, these local optimization methods are sensitive to initial values and cannot work with multiple bearing measurements in one image. Compared with the above work, our proposed method solves correspondence and relative poses together without extra sensor inputs. B. Certifiably Global Optimization Recently, based on semidefinite relaxation and advanced optimization theory, the research community has developed certifiably optimal non-minimal solvers for many computer vision and robotics problems that are non-convex and NPhard. In [18], Carlone uses Lagrangian duality to verify the optimality of candidate solution of pose graph optimization (PGO). Exploiting the strong duality of PGO, SE-Sync [19] and Cartan-Sync [20] obtain the optimal solution of PGO under acceptable noise. In [21], point registration with outliers is formulated as a QCQP by binary cloning, relaxed using SDR, and finally globally optimized by adding redundant constraints. Besides, SDR is also leveraged in 3d registration [22], camera pose estimation [23,24], extrinsic calibration [25] and so on. All of these problems involve optimization over SO(3) or SE(3) variables and add orthogonality constraints to make the convex relaxation tight. In this paper, our solution procedure is similar to [21]. Differently, we keep binary variables and introduce binary constraint and correspondence constraint to formulate a MIQCQP problem. As far as we know, our proposed algorithm is the first method that can obtain a globally optimal solution for the mutual localization problem using anonymous measurements. III. FORMULATION OF RELATIVE POSE ESTIMATION In this section, we formulate the RPE problem with anonymous measurements as a QCQP problem. Firstly, we define a loop error for mutual localization of one observed robot case in Sec.III-A. Then in Sec.III-B, we extend the error to multiple observed robots case, introduce binary variables for data association, and formulate the optimization as a mixed-integer programming problem. Finally, we marginalize distance variables, define auxiliary variables, and derive a QCQP problemin Sec.III-C. A. Loop Error for One Observed Robot In this subsection, we consider two robots, observer robot A and observed robot B, moving along two 3D trajectories. Their camera coordinates frame at time j are denoted by {A j } and {B j }, where j ∈ J, J is the timestamp collection. Robot A observes feature of robot B at time j and gets the bearing measurement b B j in frame {A j }. Assuming B be rigid body, the inner bias B P between the feature and camera on B are time-invariant, i.e., B P = Bj P . Then Aj P , the feature coordinate in frame {A j }, can be given by Aj P = D B j b B j = R Aj Bj Bj P + t Aj Bj = R Aj Bj B P + t Aj Bj , (1) where D B j is the distance between A's camera and the observed feature. R Aj Bj and t Aj Bj denote the relative rotation and translation between {A j } and {B j }. For simplicity, we set D j = D B j and b j = b B j in two robots' case. And for each time j, we have R A1Aj Aj P + t A1Aj = s AB R AB (R B1Bj B P + t B1Bj ) + t AB ,(2) where s AB denotes the scale ratio between local maps of A and B, and {s AB R AB , t AB } is the corresponding relative pose. After subtraction between Equ. (2) of j 1 , j 2 ∈ J, we eliminate variable t AB and derive the loop error: e AB j1j2 = R A1Aj 2 b j2 D j2 − R A1Aj 1 b j1 D j1 + t Aj 1 Aj 2 − (R AB R Bj 1 Bj 2 BP + s AB R AB t Bj 1 Bj 2 ),(3) where t Xj 1 Xj 2 = t X1Xj 2 −t X1Xj 1 , X ∈ {A, B} , R Bj 1 Bj 2 = R B1Bj 2 − R B1Bj 1 and BP = B P/s AB . If s AB R AB and BP are recovered, B P can be determined solely. This expression is found in [17]. In this paper, we reformulate it in a linear expression which will be used to get a quadratic cost in Sec.III-B. Firstly we define the following variables: r s . = vec(s AB R AB ) ∈ R 9×1 , r p . = vec( BP T ⊗ R AB ) ∈ R 27×1 ,(4) where ⊗ is the Kronecker product, vec(M ) is the vectorization (applied column-wise) of matrix M . Then we introduce an additional variable y and constraint y 2 = 1 to define x AB j1j2 . = [ r T s , r T p , y, D j1 , D j2 ] T ∈ R (9+27+1+2)×1 .(5) Then the loop error of the edge {j 1 , j 2 } is rewritten as e AB j1j2 = [ t T Bj 1 Bj 2 ⊗ I, vec( R Bj 1 Bj 2 ) T ⊗ I, − t Aj 1 Aj 2 , R A1Aj 1 b j1 , −R A1Aj 2 b j2 ]x AB j1j2 ,(6) The derivation of Equ.(6) from Equ. (3) is given in supplementary material. B. Mutual Localization with Anonymous Measurements In this section, we extend the above loop error to the case with N observed robots. When the amount of observed robot increases to N ≥ 2, the correct correspondence of bearing measurement sequence b X = {b X j } j∈J and estimated pose trajectory T Y = {R Yj , t Yj } j∈J is hard to provide. Here X, Y are indexes of measurement sequence and estimated trajectory respectively. Recovering the correspondence of a set of measurement sequences and a set of trajectories is called anonymity recovery problem. To solve it, we introduce binary variables Θ = {θ XY } X,Y ∈ [1,N ] , in which the binary constraint (θ XY = {0, 1}) indicates whether the X th bearing measurement corresponds to the Y th trajectory (θ XY = 1) or not (θ XY = 0). And the correspondece constraints can be written as X θ XY = 1, Y θ XY = 1, ∀X, Y ∈ [1, N ].(7) The above constraints are to guarantee that the measurement sequences and the estimated trajectories have one-to-one correspondence. We use the binary variables to rewrite the loop error Equ. (3) for the X th measurement as follow e X j1j2 = R A1Aj 2 b X j2 D X j2 − R A1Aj 1 b X j1 D X j1 + t Aj 1 Aj 2 − N Y =1 θ XY (R AY R Yj 1 Yj 2 YP + s AY R AY t Yj 1 Yj 2 ).(8)P . = [s AY , YP T ] T . Then we define extra variables Y P X . = θ XY Y P. Furthermore, we define the following variables, r XY . = vec( Y P T X ⊗ R AY ) ∈ R 36×1 ,(9)r X . = vstack({r XY } N Y =1 ) ∈ R 36N ×1 ,(10)D X . = vstack({D X j } j∈J ) ∈ R n×1 ,(11)r . = vstack({r X } N X=1 ) ∈ R 36N 2 ×1 ,(12)D . = vstack({D X } N X=1 ) ∈ R nN ×1 ,(13) x . = [ r T , y, D T ] T ∈ R (36N 2 +1+nN )×1 .(14) where the notation vstack(G) stacks all variable in G vertically and n is the number of measurements. We use variable x to rewrite Euq. (8) in linear form as e X j1j2 = c X j1j2 x, X ∈ [1, N ]. Detailed formulation of c X j1j2 is given in supplementary material. Then the error of each measurement sequence is used to formulate a nonconvex least-square problem Problem 3.1 (Original Problem): x * =arg min x X∈[1,N ] {j 1 ,j 2 }∈J (e X j1j2 ) T w X j1j2 e X j1j2 =arg min x X∈[1,N ] {j 1 ,j 2 }∈J x T (c X j1j2 ) T w X j1j2 c X j1j2 x =arg min x x T ( X∈[1,N ] {j 1 ,j 2 }∈J (c X j1j2 ) T w X j1j2 c X j1j2 ) :=C x s.t.D X j > 0, s AY > 0, R AY ∈ SO(3), X θ XY = 1, Y θ XY = 1, θ XY ∈ {0, 1},(15) where w X j1j2 is the measurement confidence parameter. The structure of x and C is shown in Fig.2. Note that C is a Gram matrix, so it is positive semidefinite and symmetric. C. Marginalization and Auxiliary Variables In this subsection, we following the procedures in [25] to marginalize the distance variables using Schur Complement. We write cost matrix C as C = CD ,D CD ,D C D,D C D,D ,(16) where the subindex D stands for the set of indexes corresponding to the distance variables or not (subindexD). Then we eliminate distance variables D and obtain Problem 3.2 (Marginalized Problem): z * = arg min z z TC z s.t. s AX > 0, R AY ∈ SO(3),(17)X θ XY = 1, Y θ XY = 1, θ XY ∈ {0, 1}, where z = [r T , y] T andC = C/C D,D = CD ,D − CD ,D C −1 D,D C D,D . Note that after marginalization, the number of involved variables is solely related to N . In contrast, exiting local optimization methods' computation is not only related to N but also the number of measurements. In our formulation, for each variable r XY = vec(θ XY Y P ⊗ R AY ), which involves θ XY and Y P = [ s AY , YP T ] T , we generalize the SO(3) constraints in [23] for our formulation as follow The variable µ is of the form µ = θ XY h Y , where h Y could be the term s AY , Y P (1) , Y P (2) or Y P (3) . In actual, constraints (19) and (20) are redundant, and we will study the effectiveness of adding them in Sec. V. (µR AY ) T (µR AY ) = µ 2 I,(18)(µR AY )(µR AY ) T = µ 2 I,(19)(µR AY ) (i) × (µR AY ) (j) = µ(µR AY ) (k) ,(20) However, it is still intractable to directly optimize the current problem. Since there is no direct variable corresponding to θ XY h Y in decision variable z, the above constraints can not be explicitly formulated into quadratic constraints in term of z. Similarly, since z neither includes θ XY , the binary constraint θ XY ∈ {0, 1}, which can be written as θ 2 XY − θ XY = 0, and the correspondence constraints all can not be constructed with decision variable z. To address above issues, the key of next step is to introduce auxiliary variables, although they are not involved in cost function directly. According to the above analysis, we need to add auxiliary variables to represent θ XY h Y and θ XY . Besides, it is necessary to link the auxiliary variables for θ XY h Y and θ XY to actual decision variable z by adding variables representing h Y and vec(h Y R AY ) and equality relationship constraints θ XY h Y = θ XY h Y ,(21)vec(θ XY h Y R AY ) = θ XY vec(h Y R AY ).(22) where the underlines denote independent variables. Summarize all necessary auxiliary variables as follows 1) Lifted Rotation Variable: Y . = vec( Y P T ⊗ R AY ) ∈ R 36×1 ,(23). = vstack({ Y } N Y =1 ) ∈ R 36N ×1 .(24) 2) Binary Variable: ϕ θ ϕ X θ . = vstack({θ XY } N Y =1 ) ∈ R N ×1 ,(25)ϕ θ . = vstack({ϕ X θ } N X=1 ) ∈ R N 2 ×1 .(26) 3) Scale Ratio and Inner Bias Variable: ϕ h ϕ h . = vstack({ Y P} N X=1 ) ∈ R 4N ×1 .(27) 4) Lifted Scale Ratio and Inner Bias Variable: ϕ µ ϕ X µ . = vstack({ Y P X } N Y =1 ) ∈ R 4N ×1 ,(28)ϕ µ . = vstack({ϕ X µ } N X=1 ) ∈ R 4N 2 ×1 .(29) Now we define the final decision variablē z . = [z T , T , ϕ T θ , ϕ T h , ϕ T µ ] T ,f * = min zz T Q 0z s.t.z T Q iz = g i , i = 1, .., m,(30) where Q 0 = C 0 dz×da 0 da×dz 0 da×da . d z and d a are dimensions of z and auxiliary variables respectively. However, the formulated non-convex QCQP is still nontrivial to solve. In next section, we provide a complete algorithm using SDR to get the global optimal solution of Problem 3.3. IV. CERTIFIABLY GLOBAL OPTIMIZATION BY SEMIDEFINITE RELAXATION In this section, we will firstly apply semidefinite relaxation to Problem 3.3 in Sec.IV-A. Then we recover data correspondence and relative poses from the solution of the SDP problem in Sec.IV-B. Lastly, we provide a condition under which the zero-duality-gap and one-rank-solution can be guaranteed in noise-free cases in Sec.IV-C. A. Semidefinite Relaxation and Dual Problem As stated above, Problem 3.3 is non-convex. Fortunately, it can be relaxed to a convex SDP, known as Shor's relaxation. By introducing matrix variable Z . =zz T , we havē z T Q iz = tr (z T Q iz ) = tr (Q izz T ) = tr (Q i Z),(31) where tr(M ) is the trace of matrix M . Together with Equ.(31) and dropping the constraint of rank (Z) = 1, we obtain the following problem. f * primal = min Z tr(Q 0 Z) s.t.Z 0, tr(Q i Z) = g i , i = 1, ..., m,(32) which is convex and can be solved by off-shelf solvers using primal-dual interior point method. Its dual problem is Problem 4.2 (Dual SDP): f * dual = max λ g T λ s.t.Q(λ) = Q 0 − i λ i Q i 0, i = 1, ..., m,(33) where g = [g 1 , ..., g m ] T , λ = [λ 1 , ..., λ m ] T . Once Z * , the solution of Problem 4.1, is obtained, we denote the part of Z * that corresponds to variable z as Z * . = Z * [1:36N 2 ,1:36N 2 ] . Moreover, if zero-duality-gap (f * primal = f * dual ) and one-rank-solution (rank (Z * ) = 1) hold, we can obtain the global optimal solution z * of Problem 3.2 as described in Sec. IV-B. Actually, both the above conditions are satisfied in noise-free cases, which is proved in Sec. IV-C. B. Recovery from the tight SDP solution Given Z * , we need to recover the optimal correspondences and relative poses. According to the one-rank-solution (rank(Z * ) = 1), we firstly deploy a rank-one decomposition to obtain z * ∈ R 36N 2 ×1 . Denoting r * XY ∈ R 36×1 as slices of z * corresponding to variable r XY , we define M X [12,3]), where mat(v, [r, c]) means reshape the vector v to one r × c matrix by colfirst order. Note that M X AY is either zero matrix or non-zero matrix due to the binary variable θ XY . So we set = 10 −5 and take \|\|M X AY \|\| 2 > to indicate that the X th measurement corresponds to the Y th estimated trajectory. AY . = θ * XY Y P * T ⊗ R * AY = mat(r * XY , Then for each M X AY whose corresponding θ XY > , we recover the scale ratio and relative rotation S * AY := s * AY R * AY = M X AY [1:3,1:3] ,(34)s * AY = 3 det(S * AY ), R * AY = S * AY /s * AY ,(35) and inner bias Y P * similarly. Recall that we have marginalized the distance variable D in Sec.III-C, we now recover the optimal D * as D * (r * ) = −C −1 D,D C D,D r * .(36) Furthermore, the optimal relative translation t * AY is recovered using D * as follow t * AY = j∈J (t A1Aj + R A1Aj (D Y * j b Y j )− (37) s * AY R * AY (R Y1Yj Y P * + t Y1Yj )). C. Tightness of Semidefinite Relaxation In this subsection, we aim to prove that there are zeroduality-gap and one-rank-solution in noise-free cases. Firstly, we introduce a lemma and a corank-one condition. Lemma 4.1: If C ∈ R n×n be positive semidefinite and x T Cx = 0 for a vector x, then Cx = 0. Definition 1: For Problem 3.1, the corank-one condition holds if the number of independent measurements n and the number of observed robots N , satisfy that n ≥ 18N + 2, where independent measurements mean that {c X j1j2 } are linearly independent vectors. Based in this condition, we have Lemma 4.2: Assume that the corank-one condition holds, then the cost matrix C is semidefinite and has corank one in noise-free cases. Furthermore, after Schur Compliment,C is also semidefinite and has corank one. The detailed proof of above two lemmas can be seen in supplementary material. Then we apply Lemma 2.1 in [26] to our problem and introduce the following proposition. Proposition 1: If bearing measurements are noise-free, their is zero-duality-gap between Problem 3.3 and Problem 4.2. Furthermore, once the corank-one condition is satisfied and given the solution Z * of Problem 4.1, we have rank(Z * ) = 1, and its rank-one decomposition z * is the global optimal minimum of Problem 3.3. Proof: Letz = [z T ,˜ T ,φ θ T ,φ p T ,φ µ T ] T be a feasible point in Problem 3.3 wherez,˜ ,φ θ ,φ p ,φ µ are all ground truth. Letλ = 0 be a feasible point in Problem 4.2. Then the zero-duality-gap is guaranteed since the below three conditions needed in Lemma 2.1 in [26] are satisfied: (i) Primal feasibility. In noise-free cases, the ground truth certainly satisfy constraints in Problem 3.3. (ii) Dual feasibility. Q(λ) = Q 0 − m i=1λ i Q i = Q 0 = C 0 dz×da 0 da×dz 0 da×da 0. (iii) Lagrangian multiplier. Sincez is ground truth,z T Q 0z equals to the optimal cost in Problem 3.1, which equals to 0. Recall that Q 0 is semidefinite according to Lemma V. EXPERIMENTS In this section, we firstly confirm the optimality and efficiency of our method by comparing it against the alternating minimization (AM) [17] and the Levenberg-Marquardt (LM) methods. Next, to present the robustness of our method, we compare its performances under different levels of noise. Then, we show the results of our method with different robot number and noise. Finally, we apply our algorithm in realworld, using estimated odometry from different sources. A. Experiments on Synthetic Data To simulate bearing measurement, we generate random trajectories for multiple robots. An example simulated environment is shown in Fig. 3. Robots trace circular routes around different centers over a common landscape consisting of multiple random sinusoidal functions. All trajectories have the same length. Then, for observer robot A and observed robot Y , we use their global poses to generate noisy bearing measurement as follow where N (1, σ) is Gaussian distribution with standard deviation σ . Then, we take the first pose of each trajectory as the local world frame and obtain each robot's local poses, which will be shared with other robots for estimation. 1) Optimality and Runtime: Given a certain initial value of relative rotation matrix R, both AM and LM can converge to a local minimum, with error distributions shown in Fig. 4. In this figure, each cell denotes the L 2 -norm error of the estimated relative pose. Fig. 4 states, the optimization converges to local minimums if the distance between the initial values and ground truth is large. b Y j = R −1 Aj (t Yj + N (1, σ)R Yj Y P − t Aj ).(38) Then we compare our method with these two methods in optimality and efficiency for four problems: RPE without scale ratio and inner bias (RPE-only), RPE with scale ratio (RPE-S), RPE with inner bias (RPE-B), and RPE with scale ratio and inner bias (RPE-SB). For each problem, we conduct 1000 experiments using different measurements. The left figure of Fig. 5 shows that for all problems, our method can always obtain the optimal solution, while both AM and LM are trapped in local minimums with random initial values. For efficiency, since our formulation fixes the number of variables by marginalizing the distance variables D (see Sec.III-C), its computing time is only related to the number of observed robots. In contrast, the number of variables in local optimization methods AM and LM increase with measurement number. The right figure of Fig. 5, which presents the mean runtime with 200 bearing measurements, show that our method solves all problems faster. 2) Robustness: To evaluate the robustness of our method and the effectiveness of the redundant constraints, we add Fig. 4. Error heatmaps for two local optimization methods over a uniform sampled initial value R, which is generated by rotating ground truth with roll (x-axis) and pitch (y-axis) from −π and π. Green region denotes range of initial values in which these approach will drop into local minimums. Benchmark results between our method and local optimization algorithms for different problems. different levels of noise into simulated measurements. We compare two versions of our method, the default version (D) and the augmented version which is added redundant rotation constraint (D+R). As the left plot of Fig. 6 shows, for each noise level, the augmented version (D+R) recover an exact minimizer of the primal problem. However, the default version (D) does not obtain the one-rank solution under extreme noise (σ ≥ 0.4). Fig. 8 presents the performance of our method (D and D+R) and several local optimization methods. AM and LM utilize random rotation as the initial value, and AM (GT) and LM (GT) use the ground truth instead. Each figure represents 100 random trials on simulated data with different noise levels σ. As Fig. 8 shows, our method is consistently more accurate compared to AM and LM and has comparable performance with AM (GT). Besides, we observe that under extreme noise (σ = 0.5), our method still performs accurately. Furthermore, we conduct experiments with multiple observed robots under noise. The right figure of Fig.6 presents the trend of rank(Z) when the noise level increases. Although as the number of robots increases, the zero-duality-gap is easily influenced by noise, our method always obtains one-rank solution with a common noise level (σ < 0.4). Moreover, we present the error distribution of obtained solutions in Fig. 9. This figure states: (1) Increasing number of robots does not influence the estimation error majorly. (2) Although there is no one-rank-solution under extreme noise, the result of one-rank decomposition has comparable accuracy with AM (GT). 3) Scalability: In our method, the number of variables is related to the squared number of observed robots. Fig. 10 shows the runtime of our method with different numbers of robots. According to the result, our method has an acceptable runtime in real multi-robot applications when the robot number is limited. B. Real-world Experiments In real-world experiments, we use motion capture and VIO for odometry estimations and AprilTag for bearing measurements, with ground truth provided by vicon motion capture. Table V-B summarizes the results of comparison between our method and others. The experiments that use Vicon (300 measurements) is labeled "Vicon+AprilTag", and using VINS [27](100 measurements) is labeled "VINS+AprilTag". Under each configuration, we conduct experiments with two and three robots. The ground truth correspondence comes from AprilTag. In experiments with three robots, AM (C+GT) optimizes with ground truth correspondence, while AM (w/o C+GT) does not. The results are in Fig.(11). Comparison of error distribution of our method with different number of robots under different levels of noise. As Table V-B shows, AM and AM (w/o C+GT) all converge to small cost but obtain egregiously large L 2 -norm error, which indicates that they are trapped in a local minimum. Compared with them, our globally optimal approach obtains the minimum cost in all experiments with most or secondly most small error. Note that, due to noise from odometry estimation and bearing measurements, obtaining the minimum cost does not mean obtaining the most accurate estimation. For runtime, the table shows that our algorithm has constant runtime which is independent to the number of measurements. It also indicates that our algorithm is suitable for bootstrapping other algorithms that use relative poses as initialization. Finally, we apply our algorithm in multi-robot map fusion as Fig.7 shows. In this experiment, each robot's local map comes from feature-based monocular SLAM. Compared with AM that traps into local minimum and fails to fuse maps, our result fuses robots' maps correctly without any initialization, while AM (GT) needs. VI. CONCLUSIONS AND FUTURE WORK In this paper, we proposed a certifiably globally optimal algorithm for mutual localization problems with anonymous bearing measurements. With our method, we can obtain bearing-pose correspondences and relative poses between robots together. Furthermore, we provide a necessary condition for optimality guarantee and conduct extensive experiments to present the optimality and robustness compared with local optimization methods. In the future, we aim to explore the noise tolerance threshold of our method to provide a more powerful guarantee for application. Top: Two robots experiment with a observer (robot 0) and a observed robot (robot 1). Bottom: Three robots experiment with a observer robot (robot 0) and two observed robots (robot 1 and 2). Fig. 1. Overview of our proposed method which can obtain certifiably optimal solution for mutual localization problem with anonymous bearing measurements. Our result can be used for map fusion in multi-robot monocular SLAM and coordinate alignment in multi-robot tasks. Problem 4 . 1 ( 41Primal SDP): . 4.2, so Q(λ)z = 0 is obtained based on Lemma. 4.1. Furthermore, suppose Z * is the optimal solution of Problem 4.1. Then Z * = 0 since at least one g i = 0. By complementary slackness, tr(Q(λ)Z * ) = tr( C Z * 0 dz×da 0 da×dz 0 da×da ) = 0, so tr(CZ * ) = 0. And sinceC and Z * are both positive semidefinite, rank(C)+rank(Z * ) <= N . So, if corank(C) = 1, rank(Z) * = 1. Moreover, its rank-one decompositionz is the unique optimum of Problem 3.3. Fig. 3 . 3Three random trajectories in a simulated environment. Fig. 5. Benchmark results between our method and local optimization algorithms for different problems. Fig . 6. (Left) Comparison between method w/ and w/o redundant constraints (Right) Comparison with different number of robots. (solid line: mean; shaded area: 1-sigma standard deviation). Fig. 7 .Fig. 8 . 78Map fusion results using feature maps from two robots, which launch at different place and observe each other when they rendezvous. Comparison of error distribution between different methods. The top colorbar presents colors corresponding to different error range. In each subfigure, each bar denotes the percentage of error range and the black line represents the mean error. Fig. 9. Comparison of error distribution of our method with different number of robots under different levels of noise. Fig. 10 . 10Runtime comparison using Matlab / C++ with different number of robots. (solid line: mean; shaded area: 1-sigma standard deviation) Fig. 11. Estimated trajectories and ground truth in real-world experiments. State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou, 310027, China.2 Huzhou Institute of Zhejiang University, Huzhou, 313000, China. E-mails:{yj wang, fgaoaa}@zju.edu.cn. Fig. 2. Structure of decision variable x and cost matrix C in Problem 3.1. Now we convert the mixed-integer expression to a linear form. Firstly, we denote the parameters that need to be estimated for robot Y as Y36N 2 r 11 36N 2 nN nN r 1N r N1 r NN y D 1 D N 36N 36N n n and use it to formulate all constraints in quadratic termsz T Q iz = g i , i ∈ [1, m],where m is the number of constraints. For detailed derivation of Q i , we refer readers to supplementary material. Now we obtain Problem 3.3 (QCQP Probem): TABLE I IREAL-WORLD EXPERIMENTS RESULTSL 2 Error Scene #Robots Method Cost Trans. (m) Rot. Runtime (ms) Ours (D+R) 0.0018 0.24 0.063 343.3 AM 0.198 2.52 2.83 1391.5 2 AM (GT) 0.132 0.323 0.087 660.5 Ours (D+R) 0.0006 0.092 0.0688 603.3 AM (C) 0.727 2.506 2.809 1289.1 AM (C+GT) 0.082 0.0305 0.0305 632.7 VICON+ AprilTag 3 AM (w/o C+GT) 0.569 1.53 0.290 901.5 Ours (D+R) 0.553 0.429 0.0716 180.1 AM 0.667 2.597 2.823 312.3 2 AM (GT) 0.650 0.601 0.159 120.2 Ours (D+R) 0.101 0.943 0.205 603.3 AM (C) 0.423 1.62 2.82 512.3 AM (C+GT) 0.337 0.773 0.118 131.1 VINS+ AprilTag 3 AM (w/o C+GT) 1.308 7.24 2.82 305.6 Distributed swarm trajectory optimization for formation flight in dense environments. L Quan, L Yin, C Xu, F Gao, arXiv:2109.07682arXiv preprintL. Quan, L. Yin, C. Xu, and F. Gao, "Distributed swarm trajectory op- timization for formation flight in dense environments," arXiv preprint arXiv:2109.07682, 2021. Meeting-merging-mission: A multi-robot coordinate framework for large-scale communication-limited exploration. 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[ "Published as a conference paper at ICLR 2023 LEARNING LABEL ENCODINGS FOR DEEP REGRES- SION", "Published as a conference paper at ICLR 2023 LEARNING LABEL ENCODINGS FOR DEEP REGRES- SION" ]	[ "Deval Shah [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada\n", "& Tor \nDepartment of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada\n", "M Aamodt [email protected] \nDepartment of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada\n" ]	[ "Department of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada", "Department of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada", "Department of Electrical and Computer Engineering\nUniversity of British Columbia\nVancouverBCCanada" ]	[]	Deep regression networks are widely used to tackle the problem of predicting a continuous value for a given input. Task-specialized approaches for training regression networks have shown significant improvement over generic approaches, such as direct regression. More recently, a generic approach based on regression by binary classification using binary-encoded labels has shown significant improvement over direct regression. The space of label encodings for regression is large. Lacking heretofore have been automated approaches to find a good label encoding for a given application. This paper introduces Regularized Label Encoding Learning (RLEL) for end-to-end training of an entire network and its label encoding. RLEL provides a generic approach for tackling regression. Underlying RLEL is our observation that the search space of label encodings can be constrained and efficiently explored by using a continuous search space of real-valued label encodings combined with a regularization function designed to encourage encodings with certain properties. These properties balance the probability of classification error in individual bits against error correction capability. Label encodings found by RLEL result in lower or comparable errors to manually designed label encodings. Applying RLEL results in 10.9% and 12.4% improvement in Mean Absolute Error (MAE) over direct regression and multiclass classification, respectively. Our evaluation demonstrates that RLEL can be combined with off-the-shelf feature extractors and is suitable across different architectures, datasets, and tasks. Code is available at https://github.com/ubc-aamodt-group/RLEL_regression.Reproducibility:We have provided details on training hyperparameters, experimental setup, and network architectures in Appendix A.3. Code is available at https://github.com/ ubc-aamodt-group/RLEL_regression. We have provided the training and inference code with trained models.Code of Ethics: Autonomous robotics and vehicles are major applications of deep regression networks. Thus improvement of regression tasks can accelerate the progress of these fields, which may lead to some negative societal impacts such as loss of jobs, privacy, and ethical concerns.REFERENCESErin L. Allwein, Robert E. Schapire, and Yoram Singer. Reducing multiclass to binary: A unifying approach for margin classifiers. consistent ordinal regression for neural networks with application to age estimation. A novel bandit-based approach to hyperparameter optimization. J. Mach. Learn. Res., 18(1): 6765-6816, jan 2017. ISSN 1532-4435. William Libaw and Leonard Craig. A photoelectric decimal-coded shaft digitizer. Electronic . Ordinal regression with multiple output CNN for age estimation.	10.48550/arxiv.2303.02273	[ "https://export.arxiv.org/pdf/2303.02273v1.pdf" ]	257,365,673	2303.02273	d785bded41e43ea695466d917059e75b8562c671	Published as a conference paper at ICLR 2023 LEARNING LABEL ENCODINGS FOR DEEP REGRES- SION Deval Shah [email protected] Department of Electrical and Computer Engineering University of British Columbia VancouverBCCanada & Tor Department of Electrical and Computer Engineering University of British Columbia VancouverBCCanada M Aamodt [email protected] Department of Electrical and Computer Engineering University of British Columbia VancouverBCCanada Published as a conference paper at ICLR 2023 LEARNING LABEL ENCODINGS FOR DEEP REGRES- SION Deep regression networks are widely used to tackle the problem of predicting a continuous value for a given input. Task-specialized approaches for training regression networks have shown significant improvement over generic approaches, such as direct regression. More recently, a generic approach based on regression by binary classification using binary-encoded labels has shown significant improvement over direct regression. The space of label encodings for regression is large. Lacking heretofore have been automated approaches to find a good label encoding for a given application. This paper introduces Regularized Label Encoding Learning (RLEL) for end-to-end training of an entire network and its label encoding. RLEL provides a generic approach for tackling regression. Underlying RLEL is our observation that the search space of label encodings can be constrained and efficiently explored by using a continuous search space of real-valued label encodings combined with a regularization function designed to encourage encodings with certain properties. These properties balance the probability of classification error in individual bits against error correction capability. Label encodings found by RLEL result in lower or comparable errors to manually designed label encodings. Applying RLEL results in 10.9% and 12.4% improvement in Mean Absolute Error (MAE) over direct regression and multiclass classification, respectively. Our evaluation demonstrates that RLEL can be combined with off-the-shelf feature extractors and is suitable across different architectures, datasets, and tasks. Code is available at https://github.com/ubc-aamodt-group/RLEL_regression.Reproducibility:We have provided details on training hyperparameters, experimental setup, and network architectures in Appendix A.3. Code is available at https://github.com/ ubc-aamodt-group/RLEL_regression. We have provided the training and inference code with trained models.Code of Ethics: Autonomous robotics and vehicles are major applications of deep regression networks. Thus improvement of regression tasks can accelerate the progress of these fields, which may lead to some negative societal impacts such as loss of jobs, privacy, and ethical concerns.REFERENCESErin L. Allwein, Robert E. Schapire, and Yoram Singer. Reducing multiclass to binary: A unifying approach for margin classifiers. consistent ordinal regression for neural networks with application to age estimation. A novel bandit-based approach to hyperparameter optimization. J. Mach. Learn. Res., 18(1): 6765-6816, jan 2017. ISSN 1532-4435. William Libaw and Leonard Craig. A photoelectric decimal-coded shaft digitizer. Electronic . Ordinal regression with multiple output CNN for age estimation. INTRODUCTION Deep regression is an important problem with applications in several fields, including robotics and autonomous vehicles. Recently, neural radiance fields (NeRF) regression networks have shown promising results in novel view synthesis, 3D reconstruction, and scene representation (Liu et al., 2020;Yu et al., 2021). However, a typical generic approach to direct regression, in which the network is trained by minimizing the mean squared or absolute error between targets and predictions, performs poorly compared to task-specialized approaches (Yang et al., 2018;Ruiz et al., 2018;Niu et al., 2016;Fu et al., 2018). Recently, generic approaches based on regression by binary classification have shown significant improvement over direct regression using custom-designed label encodings (Shah et al., 2022). In this approach, a real-valued label is quantized and converted to an M -bit binary code, and these binary-encoded labels are used to train M binary classifiers. In the prediction phase, the output code of classifiers is converted to real-valued prediction using a decoding function. Moreover, binary-encoded labels have been proposed for ordinal regression (Li & Lin, 2006;Niu et al., 2016) and multiclass classification (Allwein et al., 2001;Cissé et al., 2012). The use of binary-encoded labels for regression has multiple advantages. Additionally, predicting a set of values (e.g., classifiers' output) instead of one value (direct regression) introduces ensemble diversity, which improves accuracy (Song et al., 2021). Furthermore, encoded labels introduce redundancy in the label presentation, which improves error correcting capability and accuracy (Dietterich & Bakiri, 1995). Finding suitable label encoding for a given problem is challenging due to the vast design space. Related work on ordinal regression has primarily leveraged unary codes (Li & Lin, 2006;Niu et al., 2016;Fu et al., 2018). Different approaches for label encoding design, including autoencoder, random (Table 2), respectively. (a) Regularizer R1 encourages the distance between learned encodings to be proportional to the difference between corresponding label values. (b) Regularizer R2 reduces the number of bit transitions per bit, reducing the complexity of decision boundaries to be learned by binary classifiers. Here blue and white colors represent 1 and 0, respectively. search, and simulated annealing, have been proposed to design suitable encoding for multiclass classification (Cissé et al., 2012;Dietterich & Bakiri, 1995;Song et al., 2021). However, these encodings perform relatively poorly for regression due to differences in task objectives (Section 2). More recently, Shah et al. (2022) analyzed and proposed properties of suitable encodings for regression. They empirically demonstrated the effectiveness of manually designed encodings guided by these properties. While establishing the benefits of exploring the space of label encodings for a given task, they did not provide an automated approach to do so. In this work, we propose Regularized Label Encoding Learning (RLEL), an end-to-end approach to train the network and label encoding together. Binary-encoded labels have discrete search space. This work proposes to relax the assumption of using discrete search space for label encodings. Label encoding design can be approached by regularized search through a continuous space of real-valued label encodings, enabling the use of continuous optimization approaches. Such a formulation enables end-to-end learning of the network parameters and label encoding. We propose two regularization functions to encourage certain properties in the learned label encoding during training. Specifically, while operating on real-valued label encoding, the regularization functions employed by RLEL are designed to encourage properties previously identified as being helpful for binary-valued label encodings (Shah et al., 2022). The first property encourages the distance between learned encoded labels to be proportional to the difference between corresponding label values, which reduces the regression error. Further, each bit of label encoding can be considered a binary classifier. The second property proposes to reduce the complexity of a binary classifier's decision boundary by reducing the number of bit transitions (0 → 1 and 1 → 0 transitions in the classifier's target over the range of labels) in the corresponding bit in binarized label encoding. Figure 1 demonstrates the effect of proposed regularizers on the learned label encodings and regression errors. Figure 1a plots the L1 distance between learned encodings for different target labels versus the difference between corresponding label values. The L1 distance between encodings for distant targets is low without the regularizer. In contrast, the proposed regularizer encourages the learned label encoding to follow the first design property. Figure 1b plots learned label encoding (binarized representation for clarity). Each row represents encoding for a target value, and each column represents a classifier's output over the range of target labels. The use of regularizer R2 reduces the number of bit-transitions (i.e., 1 → 0 and 0 → 1 transitions in a column) to enforce the second design property and consequently reduces the regression error. We demonstrate that the regularization approach employed by RLEL encourages the desired properties in the label encodings. We evaluate the proposed approach on 11 benchmarks, covering diverse datasets, network architectures, and regression tasks, such as head pose estimation, facial landmark detection, age estimation, and autonomous driving. Label encodings found by RLEL result in lower or comparable errors to manually designed codes and outperform generic encoding design approaches (Gamal et al., 1987;Cissé et al., 2012;Shah et al., 2022). Further, RLEL results in lower error than direct regression and multiclass classification by 10.9% and 12.4%, respectively, and even outperforms several task-specialized approaches. We make the following contributions to this work: (d) Error distribution Figure 2: (a-c) Examples of label encodings. Each row represents the binary-encoded target label for a given target. (d) represents the error probability distribution of the classifier-1 for different target values in Johnson encoding. Green lines represent the bit transitions of a classifier. • We provide an efficient label encoding design approach by combining regularizers with continuous search space of label encodings. • We analyze properties of suitable encodings in the continuous search space and propose regularization functions for end-to-end learning of network parameters and label encoding. • We evaluate the proposed approach on 11 benchmarks and show significant improvement over different encoding design methods and generic regression approaches. BACKGROUND AND RELATED WORK This section summarizes relevant background information on regression by binary classification approach and different code design approaches. Task-specific regression approaches are summarized in Appendix A.3. However, a generic regression approach applicable to all tasks is desirable. REGRESSION BY BINARY CLASSIFICATION A regression problem can be converted to a set of binary classification subproblems. Prior works proposed to use N binary classifiers for scaled and quantized target labels ∈ {1, 2, ..., N } (Niu et al., 2016;Fu et al., 2018). Here, classifier-k's target output is 1 if the target label is greater than k, else 0. Figure 2a represents the target output of binary classifiers for this setup. Shah et al. (2022) proposed Binary-encoded Labels (BEL), a generalized framework for regression by binary classification. In the proposed approach, a real-valued target label is quantized and converted to a binary code B of length M using an encoding function E. M binary classifiers are trained using binary-encoded target labels B ∈ {0, 1} M . During inference, the output of binary classifiers, i.e., predicted code, is converted to the real-valued prediction using a decoding function D. Using encoded labels introduces error-correction capability, i.e., tolerance to classification error. Hamming distance between two codes (number of differing bits) gives a measure of error-correction capability. Error-correcting codes, such as Hadamard codes (Figure 2b), have been proposed to encode labels in multiclass classification (Dietterich & Bakiri, 1995;Verma & Swami, 2019). BEL showed that such codes are not suitable for regression due to differences in task objectives and classifiers' error probability distribution, and proposed properties of suitable codes for regression. The first property suggests a trade-off between classification errors and error correction properties. Each classifier learns a decision boundary for bit transitions from 1 → 0 and 0 → 1 in the classifier's target bit over the numeric range of labels (green lines within a column in Figure 2). For example, in Johnson encoding (Libaw & Craig, 1953) (Figure 2c), the classifier for bit B 3 learns two decision boundaries for bit transitions in intervals (2, 3) and (6, 7). The number of intervals for which the classifier has to learn a separate decision boundary increases with bit transitions, increasing its complexity. Hadamard codes have excellent error-correction properties but have several bit transitions ( Figure 2b); this increases the complexity of a classifier's decision boundary and reduces its classification accuracy compared to unary and Johnson codes (Figure 2a and Figure 2c). Rahaman et al. (2019) introduced the term spectral bias and demonstrated that neural networks prioritize learning low-frequency functions (i.e., lower local fluctuations). The spectral bias of neural networks provides insights into accuracy improvement with the reduction in the number of bit transitions. Second, Hamming distance between two codes should increase with the difference between corresponding label values to reduce the probability of making large absolute errors between predicted and target labels. The probability that erroneous predicted code for label X will be decoded as Y decreases as the hamming distance between codes for values X and Y increases. Thus the above rule reduces the regression error. Last, the encoding design should also consider the error probability of classifiers. BEL shows that the error probability of classifiers is not uniform for regression and increases near bit transitions, as shown for classifier B 1 in Figure 2d. Here, the probability of predicting 8 for target label 1 is very low, as the bit differing between corresponding codes (B 1 ) has a very low classification error probability. BEL shows that this property can be exploited to design better codes for regression. These three factors significantly affect the suitability of encodings. BEL demonstrates that simple codes sampled based on these properties, such as unary or Johnson code, result in lower errors than widely used error-correcting Hadamard code. ENCODING DESIGN Encoding design is a well-studied problem with applications in several fields. Iterative approaches, such as simulated annealing or random walk, have been proposed for code design (Dietterich & Bakiri, 1995;Song et al., 2021). However, iterative approaches are computationally expensive as each iteration requires full/partial training of the network to measure the error for sample encodings. Works on multiclass classification using binary classifiers have demonstrated the effectiveness of error-correcting codes such as Hadamard or random codes (Verma & Swami, 2019;Dietterich & Bakiri, 1995). Cissé et al. (2012) proposed an autoencoder-based approach to design compact codes for multiclass classification problems with a large number of classes. However, these approaches do not consider the task objective and classifiers' nonuniform error probability distribution for regression. Deep hashing approaches aim to find binary hashing codes for given inputs such that the hashing codes preserve the similarities in the inputs space (Luo et al., 2022;Wang et al., 2018;Xia et al., 2014;Jin et al., 2019;Liu et al., 2016). Deep supervised hashing approaches use the label information to design the loss function. In deep hashing, loss functions are designed to decrease the hamming distance between binary codes for similar images (e.g., same label). In contrast, label encoding design for regression aims to reduce the error between decoded output codes and target labels. Further, deep hashing approaches are designed for classification datasets and do not account for the nonuniform error probability distribution of classifiers observed in regression. As shown in prior work (Shah et al., 2022), classifiers' nonuniform probability significantly affects the design of suitable codes for regression. Thus, a naive adaptation of deep hashing approaches for regression problems performs poorly compared to codes designed by the proposed approach RLEL (Section A.1.5). REGULARIZED LABEL ENCODING LEARNING Regression aims to minimize the error between target labels y i and predictionsŷ i for a set of training samples i. In regression by binary classification, the network learns M -bit binary-encoded labels B i ∈ {0, 1} M . During inference, the predicted codeB i is decoded to a real-valued labelŷ i . We propose to relax label encodings' search space from a discrete ({0, 1} M ) to a continuous space (R M ), enabling the use of traditional continuous optimization methods. We propose regularizers to enable efficient search through this space. This work automates the search for label encoding using an end-to-end training approach that learns the network parameters and label encoding together. This section explains the proposed label encoding learning approach RLEL. First, we explain the regression by binary classification formulation used in this work for end-to-end training of network parameters and label encoding. Further, we introduce properties of suitable label encodings in continuous space. Lastly, we explain the proposed regularizers and loss function that accelerate the search for label encodings by encouraging learned label encoding to exhibit the proposed properties. LABEL ENCODING LEARNING Preliminaries: Figure 3 represents the formulation used in this work for label encoding learning. x i and y i represent the input and the real-valued target label for sample i, respectively. We assume y i ∈ [1, N ] for simplicity as the real-valued targets with any arbitrary numeric range can be scaled and shifted to this range. Q i ∈ {1, 2, ..., N } represents the quantized target label. The input x i is Output correlation vector of length N . HereĈ j i gives a measure of the probability that predicted label value is equal to j D Decoding matrix that converts the predicted encodings to a correlation vectorĈi passed through a feature extractor and fully connected (FC) layers to generate the predicted encodinĝ Z i ∈ R M 1 . Here, an FC layer of size θ (θ < M ) is added between the feature vector and output code. This layer reduces the number of parameters in FC layers and improves accuracy, as shown by previous work (Shah et al., 2022). Each neuron of the output code is a binary classifier, and the magnitudeẐ k i gives a measure of the confidence of the classifier-k (Allwein et al., 2001). The output code and a decoding matrix D ∈ R M ×N are multiplied 2 , and the output is passed through a softmax function to give a correlation vectorĈ i ∈ R N 3 , where the value ofĈ k i represents the probability that the predicted labelŷ i = k. This correlation vector is then converted to a real-valued prediction by taking the expected value 4 . Table 1 summarizes the notations used in this work. Prior works use custom-designed label encoding in this formulation. For example, Shah et al. (2022) proposed a series of suitable label encodings B i = E(Q i ) ∈ {0, 1} M . The network can be trained using binary cross-entropy loss between B i andẐ i , and these encodings are used as columns of the decoding matrix (D :,i = E(i)). However, it is desirable to automatically find suitable encodings B i and decoding matrix D without searching through a set of hand-designed encodings. The search space of binary label encodings is discrete and hence challenging to search using traditional continuous optimization methods (Darabi et al., 2018). Hence, we relax the assumption of binarized label encodings and use a continuous search space. This relaxation, coupled with the proposed formulation, enables the use of traditional optimizers to learn label encoding and the decoding matrix D with the entire network by optimizing the loss between targets and prediction. Let S n represent the set of training samples with quantized target Q i = n, and E ∈ R N ×M represent a label encoding matrix, where each row E n,: is the encoding for target Q i = n. E is defined as: E n,: = 1 \|S n \| i⊂SnẐ i (1) However, training the network solely with the loss betweenŷ i and y i does not constrain the search space of label encodings (E). In regression, the label encoding (E) significantly impacts the accuracy, and label encodings that follow specific properties result in lower error (Shah et al., 2022). The following section explains desirable characteristics of output codes for regression and how these properties can be encouraged in learned label encoding using regularization functions. LABEL ENCODING LEARNING WITH REGULARIZERS Section 2.1 summarizes the properties of suitable binary label encodings for regression proposed by prior works. These properties constrain the vast search space of label encodings. We further propose two properties applicable to real-valued label encodings to narrow its search space. R1 -Distance between encodings: A binary classifier's real-valued output represents its confidence (i.e., error probability). The L1 distance between real-valued predicted encodings gives more weight to classifiers that are more confident (i.e., higher value). Thus, by considering the L1 distance between real-valued codes instead of the hamming distance between binary codes, we can combine the second and third design properties of binary label encodings (Section 2.1) into a single rule for real-valued label encodings. This gives the first regularization rule: L1 distance between encodings for two labels should increase with the difference between two labels, i.e., \|\|E i,: − E j,: \|\| 1 ∝ \|i − j\|. R2 -Regularizing bit transitions: The number of bit transitions in a bit-position of label encoding gives a measure of the binary classifier's decision boundary's complexity. There are no 0 → 1 or 1 → 0 transitions in real-valued label encodings. Thus, we approximate the number of bit transitions by measuring the L1 distance between encodings for adjacent label values Q i = n and Q i = n + 1. The number of bit transitions for real-valued label encoding E can be approximated as: M i=1 N −1 j=1 \|E j,i − E j+1,i \|(2) This leads to the second regularization rule: The L1 distance between encodings for adjacent target label values should be regularized to find a balance between the complexity of the decision boundary and the error-correction capability of designed codes for a given benchmark. LOSS FUNCTION FORMULATION We propose two regularizers applicable to learned label encoding (E) to limit its search space. E is measured from the output codesẐ i over the complete training dataset (Equation 1). However, deep neural networks are trained using mini-batches, where each batch consists of K training examples sampled randomly from the (typically shuffled) training set. We extend the proposed regularization rules to apply to a minibatch-based loss function. R1: Regularizer R1 can be approximated as the following for a batch with K training examples: L 1 = K i=1 K j=1 max(0, 2 × \|y i − y j \| − \|\|Ẑ i −Ẑ j \|\| 1 )(3) The above regularization considers K 2 pairs in a minibatch of K samples, and penalizes a pair of training samples i and j if L1 distance between encodingsẐ i andẐ j is less than twice the difference between corresponding label values. The scaling parameter is set to two as it encourages at least one bit difference between two binary codes. This encourages the L1 distance between encodings to be approximately proportional to the difference between corresponding target values. R2: Regularizer R2 minimizes the L1 distance between encodings of adjacent label values. In a randomly formed minibatch consisting of only a subset of training examples, adjacent target labels might not be present. Hence it is nontrivial to apply this regularizer to the label encoding. However, we find that imposing this regularizer on the decoding matrix also helps with regularizing the bit transitions in the learned label encoding and can be added to the loss function irrespective of the batch formulation approach. Theorerical analysis and empirical verification for this approach are provided in Section A.1.7 and Section 4.4. Equation 4 represents the proposed regularizer. L 2 = M i=1 N −1 j=1 \|D i,j − D i,j+1 \|(4) Loss Function: We use the cross-entropy loss betweenĈ i and soft target labels. Here, each bit of label encoding resembles a binary classifier. However, identifying the predicted label corresponding to the multi-bit label encoding can be treated as a multiclass classification problem. Soft target labels are probability distributions generated using the distance between different classes. Soft target labels can be used with cross-entropy loss and have shown improvement over typical classification loss between the correlation vectorĈ i and quantized target label Q i or regression loss between the expected predictionŷ i and target label y i for ordinal regression (Díaz & Marathe, 2019). We use and 4) can be written as: L = K i=1 CE(Ĉ i , φ(y i ))+α M i=1 N −1 j=1 \|D i,j −D i,j+1 \|+β K i=1 K j=1 max(0, 2×\|y i −y j \|−\|\|Ẑ i −Ẑ j \|\| 1 ), where φ j (y i ) = e −\|j−yi\| N n=1 e −\|n−yi\|(5) Here, the first term is the loss between target and predicted labels. φ i represents the target probability distribution generated from target y i . The second and third terms are for regularizer R1 (Equation 3) and regularizer R2 (Equation 4), respectively. A trade-off exists between the proposed desirable properties of label encodings: Encouraging one design property comes at the cost of relaxing constraints imposed by other design properties. As demonstrated by Shah et al. (2022), finding the right balance between these properties for a given benchmark is crucial to finding the best label encoding for a given problem. Thus, these design properties can be naturally applied as regularizers, and the search for balance between different properties can be seen as tuning the regularization parameters α and β. EVALUATION This section first provides the experimental setup used to evaluate the proposed approach, then we compare RLEL with different label encoding design methods. We also compare RLEL with different regression approaches to demonstrate its effectiveness as a generic regression approach. Last, we provide an ablation study to show the impact of proposed regularizers. Table 2 summarizes the regression tasks, feature extractors architecture (Figure 3), and datasets for benchmarks used for evaluation. Selected benchmarks cover different tasks, datasets, and network architectures and have been used by prior works on regression due to the complexity of the task (Díaz & Marathe, 2019;Shah et al., 2022). We also evaluated RLEL on facial landmark detection tasks with smaller datasets to demonstrate its generalization capability. In this setup, a subset of training samples is used for training, whereas the complete test dataset is used to measure the test error. EXPERIMENTAL SETUP Landmark-free 2D head pose estimation (LFH) takes a 2D image as input and directly finds the pose of a human head with three angles: yaw, pitch, and roll. The facial landmark detection task focuses on finding (x, y) coordinates of key points in a face image. The age estimation task is used to find a person's age from the given face image. In end-to-end autonomous driving, the car's steering wheel's angle is to be predicted for a given image of the road. Normalized Mean Error (NME) or Mean Absolute Error (MAE) with respect to raw real-valued labels are used as the evaluation metrics. We compare with other encoding design approaches, including simulated annealing, autoencoder (summarized in Appendix A.2), and manually designed codes (Shah et al., 2022). We also compare RLEL with generic regression approaches, such as direct regression and multiclass classification. For direct regression, L1 or L2 loss functions with L2 regularization are used. Label value scaling (hyperparameter) is used to change the numeric range of labels. For multiclass classification, we use cross-entropy loss between the softmax output and target labels. The feature extractor and regressor are trained end-to-end for all approaches. The feature extractor architecture, data augmentation, and the number of training iterations are kept uniform across different approaches for a given benchmark. There is no notable difference between the training time for all approaches. The training dataset is divided into 70% training and 30% validation sets for tuning hyperparameters. The network is trained using the full dataset after hyperparameter tuning. We use the same values for quantization levels as prior work (Shah et al., 2022). An average of five training runs with an error margin of 95% confidence interval is reported. Appendix A.3 provides details on datasets, training parameters, related work (task-specific approaches), and evaluation metrics. Table 3 compares different encoding design approaches. RLEL results in lower error than simulated annealing and autoencoder-based approaches for most benchmarks. Both approaches are widely used for code design. However, for regression tasks, the suitability of label encoding depends upon the problem, including the task, network architecture, and dataset (Shah et al., 2022). Simulated annealing or autoencoder-based approaches do not optimize the encodings end-to-end with the regression problem., resulting in higher error. Furthermore, the gap between the error of learned label encoding with and without regularizers (RLEL and LEL) increases for smaller datasets, which suggests that RLEL-learned codes generalize better. COMPARISON OF RLEL WITH ENCODING DESIGN APPROACHES RLEL can not be used with binary-cross entropy loss for training. We observe that for some benchmarks, the autoencoder-based approach outperforms (e.g., LFH1) as it can be used with binary-cross entropy loss. The main objective of RLEL is to automatically learn label encoding that can reach the accuracies of manually designed codes (BEL), as using such codes is time and resource-consuming. Hyperparameter search for RLEL can be performed by off-the-shelf hypermeter tuners/libraries without manual efforts (Li et al., 2017;Falkner et al., 2018). In contrast, hand-designed codes need human intervention to design codes. Also, multiple training runs are still required to find suitable codes for a given benchmark from a set of hand-designed codes. As shown in Table 3, RLEL results in lower or comparable errors to hand-designed codes. COMPARISON OF RLEL WITH REGRESSION APPROACHES RLEL is a generic regression approach that focuses on regression by binary classification and proposes a label encoding learning approach. We compare RLEL with other generic regression approaches, including direct regression and multiclass classification as shown in Table 4. RLEL problem formulation introduces more fully-connected layers after the feature extractor; hence, we also perform an ablation study on increasing the number of fully connected layers in Appendix A.1.4. RLEL consistently lowers the error compared to direct regression and multiclass classification with 10.9% and 12.4% improvement on average. Figure 1a demonstrates that the use of regularizer R1 encourages the L1 distance between encodings to be proportional to the difference between target values. The second regularizer R2 is introduced to regularize the number of bit transitions in encodings. As mentioned in Section 3.3, we apply the regularization on the decoding matrix as it is nontrivial to apply this regularization on the output codes for randomly formed batches. Table 5 summarizes the effect of α (i.e., the weight of R2) on the number of bit transitions in the decoding matrix and binarized/real-valued label encoding. The second column is the number of bit transitions in the decoding matrix (Equation 4), which is used as the regularization function. The third and fourth columns are the total number of bit transitions in binarized and real-valued encodings (Equation 2). The table shows that the proposed regularizer on the decoding matrix also encourages fewer bit transitions in the label encoding. Figure 1b shows the impact of regularizer R2 on learned binarized label encoding. ABLATION STUDY As pointed out by prior works (Shah et al., 2022), there is a trade-off between the error probability and error correction capability of classifiers for regression. Hence, depending upon the benchmarks, more bit transitions can be added as the advantage of increased error correction outweighs the increase in classification error. We observe a similar trend, where adding R2 does not improve error for some benchmarks (FLD1_s, FLD2_s), as it constrains the number of bit transitions. CONCLUSION This work proposes an end-to-end approach, Regularized Label Encoding Learning, to learn label encodings for regression by binary classification setup. We propose a combination of continuous approximation of binarized label encodings and regularization functions. This combination enables an efficient and automated search of suitable label encoding for a given benchmark using traditional continuous optimization approaches. The proposed regularization functions encourage label encoding learning with properties suitable for regression, and the learned label encodings generalize better, specifically for smaller datasets. Label encodings designed by the proposed approach outperform simulated annealing-and autoencoder-designed label encodings by 12.6% and 2.1%, respectively. RLEL-designed codes show lower or comparable errors to hand-designed codes. RLEL reduces error on average by 10.9% and 12.4% over direct regression and multiclass classification. We proposed regularization function R1 to encourage the L1 distance between encodings to be proportional to the difference between corresponding label values. Figure 4a and Figure 4b represent the L1 distance between pairs of learned encodings for FLD1_s benchmark without and with 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 regularization, respectively. The X-axis and Y -axis represent the label values. Here, some columns and rows are replaced by white lines, as these label values are not present in the training dataset. The data point at coordinates (i, j) represent the L1 distance between encodings for label i and j, i.e., \|\|E i,: − E j,: \|\| 1 . For example, in Figure 4a, the L1 distance between encodings for label values 0 and 97 is ∼ 120 (light-blue coloured point at coordinate (0, 97)). In Figure 4b, the L1 distance between encodings for label values 4 and 96 is ∼ 170 (red coloured point at coordinate (4, 96)). The first design property (Section 3) states that the L1 distance between encodings should increase with the difference between corresponding label values. The difference between label values for pairs of encodings increases with the distance from the diagonal of this plot. Thus, the value of data points (i.e., the L1 distance between encodings) should increase with the distance from the diagonal of this plot. As shown in Figure 4a, without regularization, the distance between encodings is less for faraway label values (blue-colored data points away from diagonal), which shows that learned encodings do not follow the proposed design property. As shown in Figure 4b, the introduction of regularization function R2 remedies this and increases the L1 distance between encodings for faraway labels. Similar observations are made for FLD2_s benchmarks, as shown in Figure 5a and Figure 5b. (a) FLD_1 benchmark (b) FLD_1 benchmark Figure 6: (a) and (b) plot the L1 distance between pairs of encodings versus distance between corresponding label values for FLD1_s and FLD2_s benchmarks. Figure 6 plots the L1 distance between encodings versus the difference between corresponding label values for benchmarks FLD1_s and FLD2_s. For both the benchmarks, the proposed regularizer R1 helps enforce the first design property for real-valued label encodings and results in better label encodings with lower error (Table 3). Effect of the scaling parameter in equation 3 We use the scaling parameter 2 in equation 3. Our intuition behind using the scaling parameter 2 is based on binary-encoded labels. For two adjacent labels (i.e., \|y i − y j \| = 1), the loss function encourages \|\|Ẑ i −Ẑ j \|\| 1 to be greater than 2. Here,Ẑ is the output encodings. In the case of binarized label encoding (−1 if Z < 0 and +1 if Z > 0), \|\|Z i − Z j \|\| 1 = 2 signifies that two encodings differ in at least one bit. We also analyzed the effect of changing this parameter for two benchmarks. Table 6 shows the impact of changing this scaling parameter for two benchmarks. We observe that the error is higher if the scaling parameter is too low, as encodings for two adjacent labels can not be discriminated against. If this parameter is set too high, the encoding space is more constrained and consequently the performance is degraded. Based on this intuition and empirical verification on two benchmarks, we use the value 2 for all benchmarks. A.1.2 IMPACT OF REGULARIZER R2 The regularization function R2 is proposed to reduce the number of bit transitions in the learned label encoding. Figure 7 compares the label encodings learned for LFH1 benchmark for different values of α, where α is the weight of regularization function R2 (Equation 5). Each row k is the encoding for label value k. Each column k represents the output of the encoding position k for different label values. The regularization function is proposed to decrease the transitions in an encoding bit (blue→red and red→blue) over the range of label values. Section 4.4 provided quantitative results to demonstrate that increasing the value of α reduces the number of bit transitions. We observe similar trends in the plots of learned label encodings shown in Figure 7; increasing the value of α decreases bit transitions in the learned label encodings and improves MAE. A.1.3 EFFECT OF HYPERPARAMETERS IN RLEL The RLEL approach introduces two hyperparameters. We first evaluate the sensitivity to these hyperparameters to determine the complexity of hyperparameter tuning. Figure 8 shows the NME for FLD1_s benchmark for different values of α and β values in Equation 5. As shown in the figure, the error is not sensitive to small changes in these hyperparameters' values, suggesting that a sparse search in the hyperparameter space suffices. Furthermore, several approaches have been proposed for efficient hyperparameter search (Li et al., 2017;Falkner et al., 2018), and any off-the-shelf hypermeter tuners/libraries can be used to automatically find these values without manual efforts. In contrast, hand-designed codes need human intervention to design codes. Also, multiple training runs are still required to find suitable codes for a given benchmark from a set of hand-designed codes. On the other hand, RLEL provides an end-to-end automated approach for label encoding learning. A.1.4 IMPACT OF THE NUMBER OF FULLY-CONNECTED LAYERS: For RLEL , we use an extra fully connected bottleneck layer in the regressor as proposed by the prior work on regression by binary classification (Shah et al., 2022). We provide an ablation study (reproduced from (Shah et al., 2022)) to show the impact of additional fully connected layers in direct regression and multiclass classification. Table 7 provides the error (MAE or NME) for direct regression and multiclass classification with one or two fully connected layers after the feature extractor. As shown in the table, increasing the number of fully connected layers in direct regression and multiclass classification does not reduce the error for most benchmarks (possibly due to overparameterization). A.1.5 COMPARISON WITH DEEP HASHING APPROACHES Deep supervised hashing approaches use neural networks as a hash function and learn hash codes in an end-to-end manner. The loss function for deep supervised hashing is designed to preserve the similarity between inputs in the hashing space. Often, these approaches use the label information to determine the similarity between images (i.e., same label) (Liu et al., 2016;Xia et al., 2014). Some deep hashing approaches have proposed to augment the loss function with classification loss to improve the performance. We adapt two widely used deep-hashing approaches to regression and compare RLEL with deep hashing approaches. Liu et al. (2016) proposed a deep supervised hashing (DSH) approach with a loss function based on the pairwise similarity between images. The proposed approach introduces a loss function to preserve the similarity between output codes for similar training images (e.g., same class) and maximize discriminability between output codes for different training images (e.g., different class). Further, they propose using relaxation on the binary output and a regularizer to encourage the output code to be close to discrete values +1/ − 1. The hamming distance between output codes can be computed for binary-like outputs using the L2 norm. We use DSH for regression with some modifications (DSH-reg). We used the quantized label to determine the class of a training sample. Lai et al. (2015) proposed a triplet ranking loss to learn a hash function that preserves relative similarities between images (SFLH). For images (I, I+, I−), where I is closer to I+ than I−, the loss function is designed to encourage higher hamming distance between codes for (I, I−) than (I, I+). For classification datasets, triplets are typically formed using two images from the same class and one from a different class (Norouzi et al., 2012). They proposed to use a piece-wise threshold function to encourage binary-like outputs. We use the above approach (SFLH) for regression with a few modifications (SFLH-reg). To generate triplets, we pick sets of three images from a given batch and determine the similarity between images using differences between the label values. We use K 2 triplets for a minibatch of K training samples. Further, for both DSH-reg and SFLH-reg, we augment the loss function with regression loss. We add a fully-connected layer between the output code and prediction. The MSE loss between the final outputs and target labels is added to the loss function (DSH-reg-L2, SFLH-reg-L2). Table 8 compares the modified deep hashing approaches with RLEL. The gap between loss functions with and without regression loss is significant, which shows that a loss function that only aims to preserve the similarity between output codes is not sufficient and needs to account for the error between decoded output and target (i.e., regression loss). RLEL results in a lower error as it is designed for regression problems that account for classifiers' nonuniform error probability distribution. Regularizer R1 encourages the distance between output codes for images to be proportional to the difference between label values, similar to pairwise or ranking-based loss functions proposed by deep hashing. However, deep hashing approaches use the hamming distance between binary outputs. As we show in Section 3.2, the hamming distance between codes does not account for the error probability of classifiers. Thus we use the L1 distance between the real-valued outputs to account for the confidence of the classifiers. R1 does not use regularizer or nonlinear activation on the output codes to encourage binary-like outputs, as typically done in deep hashing approaches. In contrast, we show that suitable regression codes can be learned by not using this constraint. Thus RLEL with only R1 regularizer results in lower error than deep hashing approaches. A.1.6 EVALUATION Table 9 compares RLEL with direct regression and multiclass classification using geometric mean and Pearson coefficient as evaluation metrics. The geometric mean represents the geometric mean of absolute error for the test dataset. The Pearson coefficient represents the correlation between the target and predicted labels for the test dataset. As shown in the table, RLEL results in significant reduction in the error compared to other generic regression approaches. Regularization function R2: We used matrix D instead of label encoding E to apply regularizer R2 in equation 4. We insight into this decision as follows. First, note the output encodings are multiplied with D to generate the correlation vectorĈ i (Figure 3). We use the multiclass classification loss betweenĈ i and the target labels for training. Due to this, label encoding E and decoding matrix D are related, and use of matrix D proves to be effective for regularizer R2. We further explain this in detail below. Let E represent an encoding matrix of size N × M . Each row E k: represents the encoding output when the label is k. D is the decoding matrix of size M × N . LetĈ k represent the output correlation row vector of size 1 × N when the target label is k. Here,Ĉ k is obtained by multiplying E k,: with D ( Figure 3).Ĉ k = E k,: D Since we apply softmax on the output vector to find the predicted label (Figure 3), ideally,Ĉ k k should have the highest value as the target label value is k. ∴Ĉ k k >Ĉ x k , where, x = k, x ∈ {1, 2, ..., N } ∴ E k, Shah et al. (2022) used a hand-crafted decoding matrix D with an equal number of 1s and 0s in each column for binary-encoded labels. Hence the L2 norm of each column is the same. In label encoding learning, parameters of matrix D are learned during training and are not constrained to have the same L2 norm for each column. However, we observe a similar trend empirically. Figure 9c plots the distribution of \|\|D :,x \|\| for different benchmarks. As shown in the figure, for most benchmarks, we observe a small variance in the distribution of \|\|D :,x \|\|. Based on this intuition and empirical validation, we assume that \|\|D :,x \|\| ≈ \|\|D :,y \|\| for x ∈ [1, N ] and y ∈ [1, N ] to simplify the analysis. Using this assumption in equation 7 leads to the following inequality: cos(θ k,k ) > cos(θ k,x ), where, x = k, x ∈ {1, 2, ..., N } Thus the cosine similarity between E k,: and D :,k should be the highest to predict the label k. The optimization process to reduce the loss between the target and prediction will try to maximize this cosine similarity. In the best case, the angle between E k,: and D :,k will be zero, and both vectors are parallel. This simplification leads to the following relation between E and D. E k,: = tD :,k , where t > 0 Similarly, E k+1,: = t D :,k+1 , where t > 0 Since t and t both are positive values, reducing D i,k − D i,k+1 also reduces E k,i − E k+1,i . Regularizer rule R2 proposes to regularize the number of decision boundaries by regularizing M i=1 N −1 j=1 \|E j,i − E j+1,i \| as per equation 2. Based on the analysis above, regularizing M i=1 N −1 j=1 \|D i,j − D i,j+1 \| helps with the above goal as E j,i − E j+1,i reduces with D i,j − D i,j+1 . Regularization function R1: The first property suggests \|\|E i,: − E j,: \|\| 1 ∝ \|i − j\|. So ideally, \|\|E i,: − E j,: \|\| 1 = λ\|i − j\| Since E x,: is average ofẐ i for samples with label value x (equation 1), the above condition leads to: \|\|Ẑ i −Ẑ j \|\| 1 = λ\|y i − y j \|(8) Based on this requirement, we add a regularization function max(0, λ\|y i − y j \| − \|\|Ẑ i −Ẑ j \|\| 1 ), which penalizes the encodings if \|\|Ẑ i −Ẑ j \|\| 1 < λ\|y i − y j \|. It does not strictly impose equation 8. However, it approximately imposes the constraint as per shown in empirical verification in Section A.1.1. Our intuition behind using the scaling parameter 2 is based on binary-encoded labels. For two adjacent labels (i.e., \|y i − y j \| = 1), the loss function encourages \|\|Ẑ i −Ẑ j \|\| 1 to be greater than 2. Here,Ẑ is the output encodings. In the case of binarized label encoding (−1 if Z < 0 and +1 if Z > 0), \|\|Z i − Z j \|\| 1 = 2 signifies that two encodings differ in at least one bit. A.1.8 IMPACT OF THE NUMBER OF QUANTIZATION LEVELS (N ) The number of quantization buckets is treated as a design parameter for binary-encoded labels. Shah et al. (2022) showed that the error changes with the number of quantization levels. Fewer levels introduce quantization error, and more levels increase the number of bits in the encoding. They showed a trade-off between these two factors to decide the number of quantization levels. Our work focuses on the design space of encoding and decoding functions. Hence we use the same values for the quantization levels (N ) as BEL Shah et al. (2022). Parameter N tuning can be integrated into hyperparameter tuning or included in the optimization process. We further analyze the effect of the number of quantization levels for RLEL. Table 10 shows the NME (Normalized Mean Error) for different values of N for FLD1 benchmark. This suggests that the proposed method RLEL is less sensitive to the number of quantization levels for higher values. For RLEL, the decoding matrix that converts the encodings to the predicted label is also learned during the training (Figure 3). This matrix is of size M × N , where each column represents the weight parameters for one quantization level. One possible reason for the above results is that matrix D adaptively learns the number of quantization levels suitable for this problem. There is a potential to adaptively learn the number of quantization levels and non-uniform quantization using the proposed RLEL framework. For example, in Figure 3-step (4), fixed parameters j are used to scale the correlation vectorĈ j i and find the expected predictionŷ i . These parameters represent quantization levels. One possible approach to learning the quantization levels is to make these parameters trainable. In this case, L1/L2 loss between the expected predictionŷ i and target labels y i can be used to train the network. A.1.9 IMPACT OF DATASET SIZE ON ERROR FOR RLEL AND BEL In order to compare the effect of dataset size on encoding design, we run BEL and RLEL approaches with the same training loss function (cross entropy loss in equation 5). We take the dataset FLD1 and use a fraction of the dataset for training. The entire test dataset is used for testing here. Table 11 summarizes the error achieved by RLEL and BEL for different fractions of the training dataset. The evaluation shows that the gap between the performance of RLEL and BEL decreases with the increase in dataset size, which suggests that RLEL might be able to achieve lower error for larger datasets. A.1.10 COMPARISON OF LEARNED AND MANUALLY DESIGNED ENCODINGS We visually compare the encoding learned by RLEL with BEL manually designed code for one benchmark. Figure 10 shows the learned and manually designed encodings. Here, row k represents an encoding for label k. Column j represents the bit values for classifier-k over the numeric range of labels. We notice some common characteristics between both encodings. For example, the codes for nearby labels differ by fewer bits than faraway labels. Both the codes also have fewer bit transitions (0 → 1 and 1 → 0 transitions in a column). These characteristics in the learned encodings are encouraged by the proposed regularizers R1 and R2. There are a few differences between learned and hand-crafted encodings. In contrast to hand-crafted labels, encodings for adjacent labels do not differ in some cases, where hand-crafted encoding assures at least one or two bits of difference between adjacent labels. A.2 LABEL ENCODING DESIGN We evaluate different label encoding design approaches, including simulated annealing and autoencoder. These approaches have been used to design encodings for multiclass classification by prior works (Song et al., 2021;Cissé et al., 2012). We adapt these approaches to design encodings for regression tasks and compare RLEL with these code design techniques. This section provides the methodology for simulated annealing and autoencoder-based label encoding design. A.2.1 SIMULATED ANNEALING Simulated annealing is a probabilistic approach to find a global optimum of a given function. It is often used for combinatorial optimization, where the search space is discrete. Algorithm 1 represents the pseudo-code for label encoding design using simulated annealing. This algorithm takes two hyperparameters, K max (number of iterations) and T (initial temperature). It designs a code matrix C of size N × M , where N is the number of values and M is the number of bits. Each row k in this code matrix represents encoding for value k. Code matrix C is initialized with a random matrix of 0 and 1 (Line 1). For each iteration, a new code matrix C new is sampled from the current code matrix C using a Move function (Line 4). For example, a move function can be designed to randomly flip a few bits in C. The difference between the errors of the current and new code matrix is measured (Line 5). The error of a code matrix, i.e., expected regression error for this problem, is measured using function E. For example, E can be replaced by training a regression network for a given code matrix to measure the regression error. Finally, the current code matrix C is updated with the new matrix C new probabilistically. The probability is determined using the decrease in regression error and current temperature t (Line 6-8). The current temperature is updated for each iteration (Line 9). There are mainly two design parameters in the above algorithm: the error measurement function E and the move function Move. We further explain the design of these functions. Error measurement: We used the expected absolute error between targets and decoded predictions for a given code matrix as its error, as the goal is to design a code matrix that results in lowest regression error. However, training a regression network for each sample code matrix to measure its regression error is computationally expensive and time-consuming (∼ 200 training runs). Hence we use an analytical model to estimate the regression error for a given code matrix. Regression error is the absolute error between targets Q i and decoded predictionsQ i . For a given target Q i and target code B i = C Qi,: , the predicted code (B i ) will be erroneous due to classification errors. This erroneous predicted code is decoded to a predicted value (Q i ). The following equation is used to predictQ i in expected-correlation-based decoding (Shah et al., 2022). D GEN-EX (B i , C) = N k=1 kσ k , where σ k = eB i·Ck,: N j=1 eB i·Cj,:(9) The regression error can be estimated given sufficient samples of B i andB i . Shah et al. (2022) provided an approximate model of classification errors. They showed that for each classifier, its error probability distribution can be approximated using a combination of p Gaussian distributions, where p is the number of bit transitions. Each Gaussian distribution is centered around a bit transition. For example, for bit-k in unary code with bit transition between Q = k and Q = k + 1, the error probability of the classifier-k for different target labels Q i can be approximated as: e k (Q i ) = rf N (µ k ,σ 2 ) (Q i ), where, µ k = k + 0.5(10) B i can be sampled for the given Q i and C using the above error-probability model. Equation 9 is then used to find the decoded predictionQ i . We measure the expected absolute error betweenQ i and Q i using 100 × N samples. We further verify the validity of this analytical model by finding the correlation between regression error measured by this model and trained networks. Figure 11 plots the analytical regression error versus actual regression error for FLD_1 benchmarks. Here, each point is for a different code matrix. The Y -axis represents the absolute error approximated by the proposed analytical model. The X-axis represents the absolute test error of a trained network for a given code matrix. The figure shows that the proposed analytical model for error measurement approximates error with significant speedup. Move function: The move function flips some bits in the current code matrix to sample a new one. A naive approach would be to randomly flip b bits. We further optimize the move function to consider the regression task objective. For a given code matrix, using the proposed analytical model, we find a matrix F of size N × N , where F i,j = \|i − j\| × P r(Round(D(B, C)) = j\|Q = i, B = C i,: ). Thus, each element represents a pair (C i,: , C j,: ) of encodings' contribution to expected error. We select top-b pairs from this matrix. For each pair of encodings, we find bit-positions that have equal bit-values between two encodings, and a randomly selected bit-position from this list is flipped in encoding C i,: . This procedure increases the hamming distance between pairs of encodings that contribute the highest to the regression error. Figure 12 compares the convergence of the proposed move function and a random-flip-based move function. Here the Y -axis represents the approximated error for the current code matrix, and X-axis represents the iteration identifier. The figure shows that the proposed error-based move function results in faster convergence and lower error. We use the proposed move function with the analytical model to approximate regression error in Algorithm 1 to design label encoding for regression using simulated annealing. A.2.2 AUTOENCODER Cissé et al. (Cissé et al., 2012) proposed an autoencoder-based approach to design encodings for a multiclass classification problem. Figure 13 represents the network architecture used for encodings design. Input S i is an N -dimensional vector for class i. Here, each element S i [j] represents the similarity between class i and j. The output of the bottleneck layer C i is the designed encodings for class i. For regression problems, we set S i [j] = \|i − j\|. Let W represent the weight parameters of the network. The network is trained using SGD optimization, where each batch consists of randomly sampled i and j. The following loss function is used for training: L = \|\|Ŝ i − S i \|\| 2 + \|\|Ŝ j − S j \|\| 2 + βmax(0, b − \|\|C i − C j \|\| 1 ) + γ\|\|W \|\| 2(11) Here, the first and second terms represent reconstruction losses for inputs S i and S j . The third term encourages a minimum distance of b between any pair of encodings to yield unique encodings for different classes. The fourth term is an L2-regularizer. Once the network is trained, the real-valued encodings C i are converted to binary encodings such that it has equal numbers of 0s and 1s. This formulation introduces three hyperparameters. We determine the number of bit transitions in the designed label encodings and select hyperparameters that result in the lowest number of bit transitions. Note that this autoencoder network is decoupled from the regression network and design codes agnostic to classifiers' characteristics for a given regression problem. A.3 EXPERIMENTAL METHODOLOGY We use 11 benchmarks covering four different regression tasks for evaluation. This section summarizes the experimental setup, including datasets, evaluation metrics, hyperparameters, and related work for each of these tasks. We also report the training time using an Nvidia RTX 2080 Ti GPU with 11GB of memory for each benchmark. A.3.1 HEAD POSE ESTIMATION In landmark-free 2D head pose estimation, for a given 2D image, the head pose of a human is directly estimated in terms of three angles: yaw, pitch, and roll. We use loose cropping around the center with random flipping for data augmentation. We use the ResNet50 network as the feature extractor. This network is initialized using pre-trained parameters for ImageNet (Russakovsky et al., 2015) dataset. During the training for RLEL the entire network, including the feature extractor, is trained. Datasets: We use the evaluation methodology followed by prior works (Ruiz et al., 2018;Yang et al., 2019). Two protocols are used for evaluation. Evaluation metrics: We report the Mean Absolute Error (MAE) between the targets (y i ) and predictions (ŷ i ). Let N represent the number of samples, and P represent the number of labels (three in head pose estimation). The MAE is defined as: MAE = 1 N N i=1 1 P P j=1 \|y i,j −ŷ i,j \|(12) Network architecture and training parameters: Table 12 summarizes the hyperparameters used for RLEL . The learning rate of the decoding matrix D is kept 10× higher than the learning rate of the feature extractor. L2 regularization with weight of 0.0001 is used for direct regression. Related work Head pose estimation is a widely studied problem. Existing task-specialized approaches propose different loss formulations or feature extractors to improve the error. HopeNet Table 13 and Table 14 compare the performance of RLEL with related work. Evaluation metrics: We report the Normalized Mean Error (NME) between the targets y i and predictionsŷ i . Inter-ocular distance normalization is used for all datasets. For N test samples, P facial landmarks, and L normalization factor, the NME is defined as: NME = 1 N N i=1 1 P · 1 L P j=1 \|y i,j −ŷ i,j \| 2(13) Datasets: We use three datasets widely used for facial landmark detection: COFW (Burgos-Artizzu et al., 2013), 300W (Sagonas et al., 2013), and WFLW (Wu et al., 2018). HRNetV2-W18 network architecture for feature extraction (Wang et al., 2020) and the modified regressor architecture for label encoding proposed by BEL (Shah et al., 2022) are used in this work. Random flipping, scaling (0.75 − 1.25), and rotation (±30) are used for data augmentation. The COFW dataset consists of 1, 345 training and 507 testing images annotated with 29 landmarks. The training set of the 300W dataset has 3, 148 images annotated with 68 facial landmarks. This dataset provides four test sets: full test set, common subset, challenging subset, and the official test set with 300 indoor and 300 outdoor images. WFLW dataset is a comparatively larger dataset with 7, 500 training and 2, 500 testing images. Each image is annotated with 98 facial landmarks. The test set is divided into six subsets: large pose, expression, illumination, make-up, occlusion, and blur. Training parameters: Table 15 provides a summary of all the training parameters. The learning rate of the decoding matrix D is kept 20× higher than the learning rate of the feature extractor. The HRNetV2-W18 network is initialized with pretrained weight parameters for the ImageNet dataset. We refer to HRNetV2-W18 evaluated on COFW as FLD1/FLD1_s, on 300W as FLD2/FLD2_s, and on WFLW as FLD3/FLD3_s. Related work Facial landmark detection is a widely studied problem. A common approach is to use heatmap regression, where the target heatmaps are generated by assuming a Gaussian distribution around the ground truth landmark location. Prior works proposed the use of binary heatmaps with pixel-wise binary cross-entropy loss (Bulat & Tzimiropoulos, 2016 Training parameters: Table 19 summarizes the training parameters for AE1 (MORPH-II) and AE2 (AFAD) benchmarks. The learning rate of the decoding matrix D is kept 10× higher than the learning rate of the feature extractor. L2 regularization with weight of 0.001 is used for direct regression. Training for AE1 and AE2 consumes ∼ 2 and ∼ 7 hours, respectively. et al., 2018) proposed to penalize the prediction based on the variance of the age distribution. We compare CLL with related work in Table 20 and Table 21. A.3.4 END-TO-END SELF DRIVING For the regression task of end-to-end autonomous driving, we use the NVIDIA PilotNet dataset, and PilotNet model (Bojarski et al., 2016). In this task, for a given image of the road, the angle of the steering wheel that should be taken next is predicted. MAE (Equation 12) is used as the evaluation metric. Dataset The PilotNet driving dataset consists of 45, 500 images taken around Rancho Palos Verdes and San Pedro, California (Chen). We use the data augmentation technique used by prior works (Bojarski et al., 2016). Training parameters Table 22 summarizes the training parameters. The learning rate of the decoding matrix D is kept 10× higher than the learning rate of the feature extractor. Related work End-to-end autonomous driving is an interesting task with increasing attention. PilotNet (Bojarski et al., 2017) used a small, application-specific network. We compare RLEL with the baseline PilotNet architecture in Table 23. Figure 1 : 1(a) and (b) demonstrate the effect of proposed regularizers on learned label encodings and the regression error (NME/MAE) for FLD1_s and LFH1 benchmarks Figure 3 : 3The flow for combined training of feature extractor and label encoding. Figure 4 : 4(a) and(b) show the L1 distance between pairs of encodings for FLD1_s benchmark for β = 0 and β = 5.0, respectively. Each cell (i,j) in this matrix represents the L1 distance between learned encodings for label i and j, i.e., \|\|Ei,: − Ej,:\|\|1. Figure 5 : 5(a) and (b) show the L1 distance between pairs of encodings for FLD2_s benchmark for β = 0 and β = 5.0, respectively. Each cell (i,j) in this matrix represents the L1 distance between learned encodings for label i and j, i.e., \|\|Ei,: − Ej,:\|\|1. Figure 7 : 7(a)-(d) represent the label encodings learned by RLEL for different values of weight α for regularizer R2 (Equation 5). Figure 8 : 8Impact of hyperparameters α and β from Equation 5 on NME for FLD1_s benchmark. Figure 9 : 9(a) and (b) plot the distribution of \|\|D:,x\|\| for LFH2 benchmark. (c) plots the distribution of \|\|D:,x\|\| for LFH1 benchmark. Here the variance is very low, which suggests that the assumption \|\|D:,x\|\| ≈ \|\|D:y\|\|, x ∈ [1, N ], y ∈ [1, N ] is valid. For the LFH1 benchmark, the variance is higher than LFH2. However, all outliers are for label values with very few (or even zero) training examples. : .D :,k > E k,: .D :,x , where, x = k, x ∈ {1, 2, ..., N } (Using equation 6). Let θ k,x represent the angle between row vector E k,: and column vector D :,x . This leads to the below equation: \|\|E k,: \|\|\|\|D :,k \|\|cos(θ k,k ) > \|\|E k,: \|\|\|\|D :,x \|\|cos(θ k,x ), where, x = k, x ∈ {1, 2, ..., N } Hand-crafted encoding (BEL)Figure 10: (a) and(b) give examples of learned and hand-crafted encodings. Here, row k represents an encoding for label k. Column j represents the bit values for classifier-k over the numeric range of labels. Algorithm 1 1Simulated annealing for encodings design Input: K max , T, M , N ; Output: C ∈ {0, 1} M ×N ; 1: C = C0 ∈ {0, 1} N ×M , where Pr(C0 i,j = 0) = Pr(C0 i,j Figure 11 :Figure 12 : 1112Comparison of Mean Absolute Error (MAE) approximated by the proposed analytical model and trained network for different code matrices.Each point in this plot is a different code matrix. Iteration Comparison of convergence of random-flip and proposed error-based flip move functions. Figure 13 : 13Network architecture for autoencoder-based encodings design. (Ruiz et al., 2018) proposed a combination of regression and classification loss. SSR-Net (Yang et al., 2018) and FSA-Net (Yang et al., 2019) proposed stage-wise soft regression. QuatNet (Hsu et al., 2019) proposed to use MSE loss with custom ordinal regression loss. RAFA-Net (Behera et al., 2021) proposed an attention-based feature extractor architecture. Related work Different approaches including ordinal regression(Niu et al., 2016; Cao et al., 2020; Pan et al., 2018; Gao et al., 2018), soft stage-wise regression (Yang et al., 2018; 2019), soft labels (Díaz & Marathe, 2019) have been proposed for age estimation. OR-CNN (Niu et al., 2016) and CORAL-CNN (Cao et al., 2020) proposed ordinal regression by binary classification with threshold-based encodings (i.e., unary codes). DLDL (Gao et al., 2018) augmented the loss function with KL-divergence between softmax output and soft target probability distributions. MV-Loss (Pan Table 1 : 1Summary of notations used in this workNotation Description xi, yi, Qi Input, real-valued target label, and quantized target label for training example i. yi ∈ [1, N ] and Qi ∈ {1, 2, ..., N } N The range of target labels yi; Number of quantization levels for Qi M Number of bits/values for label encoding Bi,Bi Target and predicted binary-encoded labels (used for hand-crafted label encodinĝ Zi Predicted real-valued encodings; activation values of the output code layer in Figure 3 E Learned label encoding through RLEL; calculated fromẐi for all training examples using equation 1 Ci Table 2 : 2Benchmarks used for evaluationTask Feature Extractor Dataset Benchmark Label range/ Quantization levels θ Landmark-free 2D head pose estimation ResNet50 (He et al., 2016) 300LP (Zhu et al., 2016)/AFLW2000 (Zhu et al., 2016) LFH1 0-200/200 10 BIWI (Fanelli et al., 2013) LFH2 0-150/150 10 Facial Landmark Detection HRNetV2- W18 (Wang et al., 2020) COFW (Burgos-Artizzu et al., 2013) FLD1/FLD1_s (100%/10% training dataset) 0-256/256 10 300W (Sagonas et al., 2013) FLD2/FLD2_s (100%/10% training dataset) 0-256/256 10 WFLW (Wu et al., 2018) FLD3/FLD3_s (100%/10% training dataset) 0-256/256 10 Age estimation ResNet50/ ResNet34 MORPH-II (Ricanek & Tesafaye, 2006) AE1 0-64/64 10 AFAD (Niu et al., 2016) AE2 0-32/32 10 End-to-end autonomous driving Pilot- Net(Bojarski et al., 2017) PilotNet PN 0-670/670 10 this loss function for RLEL and multiclass classification. Complete loss function with regularizers (Equation 3 Table 3 : 3Comparison of RLELwith different label encoding design approaches. The bold and underlined numbers represent the first and second best errors, respectively.Error (MAE or NME) Table 4 : 4Comparison of RLELwith different regression approaches and state-of-the-art task-specialized approaches (more details in Appendix A.3). "/xM" represents the model size. This uses different network architecture, data augmentation, and training process.Direct regression Multiclass classification RLEL Task-specialized approach* LFH1 4.22±0.13/23.5M 4.49±0.24/24.2M 3.55±0.10/23.6M 3.30±0.04/69.8M LFH2 5.32±0.12/23.5M 5.31±0.05/24.8M 4.77±0.05/23.6M 3.90±0.03/69.8M FLD1 3.60±0.02/10.2M 3.48±0.03/25.6M 3.36±0.01/10.6M 3.34±0.02/10.6M FLD1_s 32.70±1.37/10.2M 5.36±0.03/25.6M 4.71±0.04/10.6M - FLD2 3.54±0.03/10.2M 3.46±0.02/45.2M 3.37±0.02/11.2M 3.07/25.1M FLD2_s 5.04±0.02/10.2M 4.50±0.04/45.2M 4.15±0.05/11.2M - FLD3 4.64±0.03/10.2M 4.46±0.01/61.3M 4.35±0.01/11.7M 4.32/- FLD3_s 6.35±0.07/10.2M 6.05±0.01/61.3M 5.58±0.01/11.7M - AE1 2.37±0.01/23.5M 2.75±0.03/24.2M 2.28±0.01/23.6M 1.96/3.7M AE2 3.16±0.02/23.5M 3.38±0.05/24.8M 3.14±0.01/23.6M 3.47/21.3M PN 4.24±0.45/10.2M 5.54±0.03/25.6M 3.01±0.03/10.6M 4.24/10.2M * Table 5 : 5Effect of regularization R2 on bit-transitions in binarized and real-valued label encodings.α value Proposed regularizer using Decoding matrix (Equation 4) #Bit transitions in binarized label encoding Approximated bit transitions in label encoding (Equation 2) 0 6816.1 5097 391.88 0.1 215.3 3596 168.52 0.5 130.8 3180 104.19 Mohammad Norouzi, David J. 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Face alignment across large poses: A 3d solution. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 146-155, 06 2016. A APPENDIX A.1 ABLATION STUDY Section A.1.1, Section A.1.2, and Section A.1.3 provide an ablation study and supporting data on impact of proposed regularization functions and hyperparameters on label encoding learning. Section A.1.4 covers an ablation study on the impact of the number of fully connected layers in direct regression and multiclass classification. Section A.1.5 explains and compares deep hashing approaches (adapted for regression) with RLEL. Section A.1.6 provides results for geometric mean and Pearson coefficient as evaluation metrics. A.1.1 IMPACT OF REGULARIZER R1 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 0 50 100 150 200 250 300 (a) Without regularization: β = 0 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 0 Table 6 : 6Effect of the scaling parameter on error for FLD1s and FLD2s benchmarks.Value of the scaling parameter NME (FLD1 s ) NME (FLD2 s )1 4.89 4.15 2 4.71 4.15 3 4.83 4.20 4 4.97 4.27 5 4.95 4.28 6 5.06 4.41 Table 7 : 7Impact of the number of fully-connected layers in direct regression and multiclass classification on the error (MAE or NME). This table is reproduced from(Shah et al., 2022).Benchmark Direct regression Multiclass classification 1 FC layer 2 FC layers 1 FC layer 2 FC layers LFH1 4.76 5.19 4.49 4.82 LFH2 5.65 5.59 5.31 5.42 FLD1 3.60 3.63 3.58 3.56 FLD2 3.54 3.58 3.51 3.62 FLD3 4.64 4.63 4.50 4.64 FLD4 1.51 1.51 1.56 1.53 AE1 2.44 2.35 2.75 2.81 AE2 3.21 3.14 3.38 3.40 PN 4.24 4.33 4.56 5.74 Table 8 : 8Comparison of RLELwith different deep hashing approaches adapted for regression.Method MAE DSH-reg 71.3 DSH-reg-L2 4.11 SFLH-reg 69.8 SFLH-reg-L2 4.73 RLEL ( only R1 ) 3.93 RLEL ( R1 + R2 ) 3.55 Table 9 : 9Comparison of RLELwith different regression approaches using Geometric mean and Pearson coefficient as evaluation metrics.RLEL Direct Regression Multiclass Classification GeoMean Pearson Coeff. GeoMean Pearson Coeff. GeoMean Pearson Coeff. LFH1 1.95 97.68 2.91 97.10 2.30 94.60 LFH2 2.09 92.22 2.49 91.06 2.40 88.76 FLD1 0.96 99.94 1.07 99.93 1.04 99.93 FLD1_s 1.31 99.87 6.38 99.81 1.81 99.80 FLD2 1.92 99.97 2.12 99.97 2.07 99.97 FLD2_s 2.44 99.96 3.03 99.98 3.22 99.94 FLD3 0.96 99.99 1.05 99.99 1.01 99.97 FLD3_s 1.21 99.98 1.56 99.97 1.37 99.97 Table 10 : 10Impact of the number of quantization levels on error for FLD1 benchmarkQuantization levels (N) NME 32 3.49 64 3.36 128 3.36 256 3.36 384 3.37 512 3.37 Table 11 : 11Effect of dataset size on the error for FLD1 benchmark.%Dataset used RLEL BEL Difference (RLEL-BEL) 100 3.36 3.35 0.01 80 3.43 3.42 0.01 60 3.53 3.47 0.06 40 3.77 3.72 0.05 20 4.08 4.04 0.04 10 4.71 4.63 0.08 Protocol 1 (LFH1): This protocol uses the BIWI (Fanelli et al., 2013) dataset for training and evaluation. This dataset consists of 15, 128 frames of 20 subjects. Random 70% − 30% splits are used for training and evaluation. The ranges of yaw, pitch, and roll angles are [−75 • , 75 • ], [−65 • , 85 • ], and [−55 • , 45 • ], respectively. Protocol 2 (LFH2): In this protocol, the network is trained using the 300W-LP (Zhu et al., 2016) dataset consisting of 122, 450 samples. AFLW2000 (Zhu et al., 2016) dataset is used for evaluation. The range of all labels is [−99 • , 99 • ] in this setting. Table 12 : 12Training parameters for LFH1.Approach Label range/ Quantization levels Optimizer Epochs Batch size Learning rate Learning rate schedule β α Training time (GPU hours) LFH1 Yaw: [−75 • , 75 • ]/150, Pitch:[−65 • , 85 • ]/150 , Roll: [−55 • , 45 • ]/100 Adam, weight decay=0, momentum = 0 50 8 0.0001 1/10 after 30 Epochs 0.5 1.0 2 LFH2 [−99 • , 99 • ]/200 Adam, weight decay=0, momentum = 0 20 16 0.00001 1/10 after 10 Epochs 2.0 0.0 4 Table 13 : 13Landmark-free 2D Head poses estimation evaluation for protocol 1 (HPE1 and HPE3).Facial landmark detection focuses on finding the (x, y) coordinates of facial keypoints for a given 2D image.Approach Feature Extractor #Params (M) Yaw Pitch Roll MAE SSR-Net-MD (Yang et al., 2018) (Soft regression) SSR-Net 1.1 4.24 4.35 4.19 4.26 FSA-Caps-Fusion (Yang et al., 2019) (Soft regression) FSA-Net 5.1 2.89 4.29 3.60 3.60 RAFA-Net (Behera et al., 2021) (Direct Regression) RAFA-Net 69.8 3.07 4.30 2.82 3.40 Direct regression (L2 loss) ResNet50 23.5 3.80 4.63 4.28q 4.22 ± 0.35 BEL (Shah et al., 2022) ResNet50 23.6 3.32 3.80 3.53 3.56 ± 0.11 RLEL ResNet50 23.6 3.41 3.20 3.97 3.55 ± 0.10 A.3.2 FACIAL LANDMARK DETECTION Table 14 : 14Landmark-free 2D Head poses estimation evaluation for protocol 2 (HPE2 and HPE4).Approach Feature Extractor #Params (M) Yaw Pitch Roll MAE SSR-Net-MD Yang et al. (2018) (Soft regression) SSR-Net 1.1 5.14 7.09 5.89 6.01 FSA-Caps-Fusion Yang et al. (2019) (Soft regression) FSA-Net 5.1 4.50 6.08 4.64 5.07 RAFA-Net Behera et al. (2021) (Direct Regression) RAFA-Net (HPE4) 69.8 3.60 4.92 3.88 4.13 HopeNet* (α = 2) Ruiz et al. (2018) (classification + regression loss) ResNet50 23.9 6.47 6.56 5.44 6.16 Direct regression (L2 loss) ResNet50 23.5 5.61 6.13 4.18 5.32 ± 0.12 BEL Shah et al. (2022) ResNet50 23.6 4.54 5.76 3.96 4.77 ± 0.05 RLEL ResNet50 23.6 4.69 5.79 3.86 4.77 ± 0.05 Table 15 : 15Training parameters for facial landmark detection for HRNetV2-W18 feature extractor.Dataset/ Benchmark Optimizer Epochs Batch size Learning rate (BEL/Direct regres- sion/Multiclass classification) Learning rate schedule β α Training time (GPU hours) COFW/ FLD1 Adam, weight decay=0, momentum = 0 60 8 0.0005/0.0003/ 0.0003 1/10 after 30 and 50 Epochs 3.0 0.0 1 2 COFW/ FLD1_s Adam, weight decay=0, momentum = 0 60 8 0.0005/0.0003/ 0.0003 1/10 after 30 and 50 Epochs 4.0 0.0 1 2 300W/ FLD2 Adam, weight decay=0, momentum = 0 60 8 0.0007/0.0003/ 0.0003 1/10 after 30 and 50 Epochs 5.0 1.0 3 300W/ FLD2_s Adam, weight decay=0, momentum = 0 60 8 0.0007/0.0003/ 0.0003 1/10 after 30 and 50 Epochs 5.0 0.05 3 WFLW/ FLD3 Adam, weight decay=0, momentum = 0 60 8 0.0003/0.0003/ 0.0003 1/10 after 30 and 50 Epochs 0.0 0.1 5 WFLW/ FLD3_s Adam, weight decay=0, momentum = 0 60 8 0.0003/0.0003/ 0.0003 1/10 after 30 and 50 Epochs 5.0 0.1 5 ). HRNet(Wang et al., 2020) proposed a feature extractor that maintains high-resolution representations and uses heatmap regression. AWing (Wang et al.,2019) proposed a modified heatmap regression loss function with adapted wing loss. AnchorFace (Xu et al., 2020) used anchoring of facial landmarks on templates. LUVLi(Kumar et al., 2020) proposed a landmark's location, uncertainty, and visibility likelihood-based loss.Table 16-18 compare RLEL with related work. Table 16 : 16Facial landmark detection results on COFW dataset (FLD1).Approach Feature Extractor #Params/ GFlops Test NME FR0.1 LAB (w B) (Wu et al., 2018) Hourglass 25.1/19.1 3.92 0.39 AWing (Wang et al., 2019)* Hourglass 25.1/19.1 4.94 - HRNetV2-W18 (Wang et al., 2020) (Heatmap regression) HRNetV2-W18 9.6/4.6 3.45 0.19 Direct regression (L2 loss) HRNetV2-W18 10.2/4.7 3.96 ± 0.02 0.29 Direct regression (L1 loss) HRNetV2-W18 10.2/4.7 3.60 ± 0.02 0.29 BEL (Shah et al., 2022) HRNetV2-W18 10.6/4.6 3.34 ± 0.02 0.40 RLEL HRNetV2-W18 10.6/4.6 3.36 ± 0.01 0.20 * Uses different data augmentation for the training Table 17 : 17Facial landmark detection results on 300W dataset (FLD2). 37 ± 0.02 * Uses different data augmentation for the trainingApproach Feature Extractor #Params/ GFlops Test Common Challenging Full DAN (Kowalski et al., 2017) - - - 3.19 5.24 3.59 LAB (w B) (Wu et al., 2018) Hourglass 25.1/19.1 - 2.98 5.19 3.49 AnchorFace (Xu et al., 2020) ShuffleNet-V2 - - 3.12 6.19 3.72 AWing (Wang et al., 2019)* Hourglass 25.1/19.1 - 2.72 4.52 3.07 LUVLi (Kumar et al., 2020) CU-Net - - 2.76 5.16 3.23 HRNetV2-W18 (Wang et al., 2020) (Heatmap regression) HRNetV2-W18 9.6/4.6 - 2.87 5.15 3.32 Direct regression (L2 loss) HRNetV2-W18 10.2/4.7 4.40 3.25 5.65 3.71 ± 0.05 Direct regression (L1 loss) HRNetV2-W18 10.2/4.7 4.26 3.10 5.42 3.54 ± 0.03 BEL (Shah et al., 2022) HRNetV2-W18 11.2/4.6 4.09 2.91 5.50 3.40 ± 0.02 RLEL HRNetV2-W18 11.2/4.6 4.03 2.90 5.39 3. Table 18 : 18Facial landmark detection results (NME) on WFLW test (FLD3) and 6 subsets: pose, expression (expr.), illumination (illu.), make-up (mu.), occlusion (occu.) and blur. θ = 10 is used for BEL. Expr. Illu. MU Occu. BlurDatasets MORPH-II(Ricanek & Tesafaye, 2006) andAFAD (Niu et al., 2016) datasets are used for evaluation. We follow the protocols for preprocessing and data augmentation of datasets as per prior works(Shah et al., 2022; Raschka, 2018). MORPH-II dataset consists of 55, 608 images with random split of 39, 617 training, 4, 398 validation, and 11, 001 test images. The AFAD dataset consists of 164, 432 images with random split of 118, 492 training, 13, 166 validation, and 32, 763 test images.Approach Feature Extractor #Params/ GFlops Test Pose Table 19 : 19Training parameters for age estimation using MORPH-II and AFAD datasetBench- mark Optimizer Epochs Batch size Learning rate Learning rate schedule β α AE1 Adam, weight decay=0, momentum=0 50 64 0.0001 - 0.0 2.0 AE2 Adam, weight decay=0, momentum=0 50 64 0.0001 - 0.0 5.0 Table 20 : 20Age estimation results on MORPH-II dataset (AE1).Approach Feature extractor #Parameters (M) MORPH-II (MAE) MORPH-II (CSθ = 5) OR-CNN (Niu et al., 2016) (Ordinal regression by binary classification ) - 1.0 2.58 0.71 MV Loss (Pan et al., 2018) (Direct regression) VGG-16 138.4 2.41 0.889 DLDL-v2 (Gao et al., 2018) (Ordinal regression with multi-class classification) ThinAgeNet 3.7 1.96* - CORAL-CNN (Cao et al., 2020) (Ordinal regression by binary classification) ResNet34 21.3 2.49 - Direct Regression (L2 Loss) ResNet50 23.1 2.37 ± 0.01 0.903 ± 0.002 BEL (Shah et al., 2022) ResNet50 23.1 2.27 ± 0.01 0.928 ± 0.001 RLEL ResNet50 23.1 2.28 ± 0.01 0.901 ± 0.002 Table 21 : 21Age estimation results on AFAD dataset (AE2).Approach Feature extractor #Parameters (M) AFAD (MAE) AFAD (CSθ = 5) OR-CNN (Niu et al., 2016) (Ordinal regression by binary classification ) - 1.0 3.51 0.74 CORAL-CNN (Cao et al., 2020) (Ordinal regression by binary classification) ResNet34 21.3 3.47 - Direct Regression (L2 Loss) ResNet50 23.1 3.16 ± 0.02 0.810 ± 0.02 BEL (Shah et al., 2022) ResNet50 23.1 3.11 ± 0.01 0.823 ± 0.001 RLEL ResNet50 23.1 3.14 ± 0.01 80.78 ± 0.002 Table 22 : 22Training parameters for end-to-end autonomous driving using PilotNet.Optimizer Epochs Batch size Learning rate Learning rate schedule β α SGD with weight decay=1e-5, momentum=0 50 64 0.1 1/10 at 10, 30 epochs 0.0 2.0 Table 23 : 23End-to-end autonomous driving results on PilotNet dataset (PN) and architecture(Bojarski et al., 2017; 2016).Approach Feature extractor #Parameters (M) MAE PilotNet (Bojarski et al., 2017) PilotNet 1.8 4.24 ± 0.45 BEL (Shah et al., 2022) PilotNet 1.8 3.11 ± 0.01 RLEL PilotNet 1.8 2.94 ± 0.01 ACKNOWLEDGEMENTSThis research has been funded in part by the National Sciences and Engineering Research Council of Canada (NSERC) through the NSERC strategic network on Computing Hardware for Emerging Intelligent Sensory Applications (COHESA) and through an NSERC Strategic Project Grant. Tor M. Aamodt serves as a consultant for Huawei Technologies Canada Co. Ltd and recently served as a consultant for Intel Corp. . Wu, LAB (w B. LAB (w B) (Wu et al., 2018) . ( Anchorface, Xu, AnchorFace (Xu et al., 2020)* HRNetV2-W18 . ( Awing, Wang, AWing (Wang et al., CU-Net -4. ( Luvli, Kumar, 37LUVLi (Kumar et al., 2020) CU-Net - 4.37 - - - - - - . -W18 ( Hrnetv2, Wang, HeatmapHRNetV2-W18 (Wang et al., 2020) (Heatmap . Bel (shah, BEL (Shah et al., 2022) HRNetV2-W18 This task focuses on predicting a person's age from a given 2D image. MAE (Equation 12) and Cumulative Score (CS) are used as the evaluation metrics, and ResNet50 (He et al., 2016) is used as the feature extractor. CSθ is the percentage of test samples with absolute error less than θ years. This task focuses on predicting a person's age from a given 2D image. MAE (Equation 12) and Cumulative Score (CS) are used as the evaluation metrics, and ResNet50 (He et al., 2016) is used as the feature extractor. CSθ is the percentage of test samples with absolute error less than θ years.	[ "https://github.com/ubc-aamodt-group/RLEL_regression.Reproducibility:We" ]
[ "An investigation of the reconstruction capacity of stacked convolutional autoencoders for log-mel-spectrograms", "An investigation of the reconstruction capacity of stacked convolutional autoencoders for log-mel-spectrograms" ]	[ "Anastasia Natsiou [email protected] \nTechnological University of Dublin Dublin\nIreland\n", "Luca Longo [email protected] \nTechnological University of Dublin Dublin\nIreland\n", "Seán O'leary [email protected] \nTechnological University of Dublin Dublin\nIreland\n" ]	[ "Technological University of Dublin Dublin\nIreland", "Technological University of Dublin Dublin\nIreland", "Technological University of Dublin Dublin\nIreland" ]	[]	In audio processing applications, the generation of expressive sounds based on high-level representations demonstrates a high demand. These representations can be used to manipulate the timbre and influence the synthesis of creative instrumental notes. Modern algorithms, such as neural networks, have inspired the development of expressive synthesizers based on musical instrument timbre compression. Unsupervised deep learning methods can achieve audio compression by training the network to learn a mapping from waveforms or spectrograms to low-dimensional representations. This study investigates the use of stacked convolutional autoencoders for the compression of time-frequency audio representations for a variety of instruments for a single pitch. Further exploration of hyper-parameters and regularization techniques is demonstrated to enhance the performance of the initial design. In an unsupervised manner, the network is able to reconstruct a monophonic and harmonic sound based on latent representations. In addition, we introduce an evaluation metric to measure the similarity between the original and reconstructed samples. Evaluating a deep generative model for the synthesis of sound is a challenging task. Our approach is based on the accuracy of the generated frequencies as it presents a significant metric for the perception of harmonic sounds. This work is expected to accelerate future experiments on audio compression using neural autoencoders.	10.1109/sitis57111.2022.00038	[ "https://export.arxiv.org/pdf/2301.07665v1.pdf" ]	255,998,348	2301.07665	1193e263cd88bb0692c226918a0516872bcd70bd	An investigation of the reconstruction capacity of stacked convolutional autoencoders for log-mel-spectrograms Anastasia Natsiou [email protected] Technological University of Dublin Dublin Ireland Luca Longo [email protected] Technological University of Dublin Dublin Ireland Seán O'leary [email protected] Technological University of Dublin Dublin Ireland An investigation of the reconstruction capacity of stacked convolutional autoencoders for log-mel-spectrograms Index Terms-Log-mel-spectrogram reconstructionautoen- codersmachine learning In audio processing applications, the generation of expressive sounds based on high-level representations demonstrates a high demand. These representations can be used to manipulate the timbre and influence the synthesis of creative instrumental notes. Modern algorithms, such as neural networks, have inspired the development of expressive synthesizers based on musical instrument timbre compression. Unsupervised deep learning methods can achieve audio compression by training the network to learn a mapping from waveforms or spectrograms to low-dimensional representations. This study investigates the use of stacked convolutional autoencoders for the compression of time-frequency audio representations for a variety of instruments for a single pitch. Further exploration of hyper-parameters and regularization techniques is demonstrated to enhance the performance of the initial design. In an unsupervised manner, the network is able to reconstruct a monophonic and harmonic sound based on latent representations. In addition, we introduce an evaluation metric to measure the similarity between the original and reconstructed samples. Evaluating a deep generative model for the synthesis of sound is a challenging task. Our approach is based on the accuracy of the generated frequencies as it presents a significant metric for the perception of harmonic sounds. This work is expected to accelerate future experiments on audio compression using neural autoencoders. I. INTRODUCTION Limited memory and the requirements for fast transmission of data influenced researchers and engineers to investigate methods that reduce the dimensionality of the data. Computational methods have previously been proposed to tackle the issue. Principle Component Analysis (PCA) [1] constitutes an algorithm that linearly projects every sample to a lower dimensional space, by extracting the most significant information as a set of new orthogonal variables. More recently, the rise of deep learning methods presented the ability of the non-linear projection of data to a lower dimension. Unsupervised networks such as autoencoders (AE) [2] or Generative Adversarial Networks (GANs) [3] have demonstrated promising results in extracting information from relatively big and complex datasets into a latent space. Architectures like these were originally used to extract meaningful patterns from images [4] or videos [5]. However, a few studies have also been proposed on speech [6] or even music compression tasks [7]. In this work, we aim to investigate stacked convolutional neural autoencoders for reducing the dimensionality of timefrequency representations of harmonic sounds. An autoencoder (AE) is a type of neural architecture composed of an encoder and a decoder. The encoder projects the input representation to a lower dimensional space while the decoder attempts to restore the representation back to its original dimensions. Convolutional neural networks belong to a sub-category of artificial neural networks (ANN) that aim to extract features from the input samples based on local pattern detection. The network is characterized by spatially local connections between nodes using a feature map that sweeps the input matrix. Additional techniques such as pooling layers can further decrease the dimensionality of the data. This paper presents a design of convolutional autoencoders and their improvements using regularization methods. Using multiple instruments in a specific pitch, this work demonstrates an exploration of timbre compression. The network attempts to project the log-mel-spectrogram of monophonic and harmonic sounds to a lower dimensional space. This compressed highlevel representation is used for musical instrument timbre synthesis. The paper is organised as follows. In Section II we review previous works of autoencoders for the compression of audio representations. Section III includes details on autoencoders, convolutional networks, regularization techniques, and the mel-spectrogram while in Section IV we demonstrate the conducted experiments. In Section V we present the evaluation methods used and we introduce a novel evaluation metric. Finally, Section VI demonstrates the results and a discussion while Section VII represents a brief conclusion. II. RELATED WORK Neural autoencoders have been used in previous studies aiming to extract meaningful information from audio representations. In the majority of this research, neural networks were developed to compress magnitude short-time Fourier transform frames [7]- [11]. One of the first attempts was in [8] where shallow feedforward neural autoencoders managed to decrease the original representation by 25%. Denoising techniques were also applied for additional regularization of the autoencoders. On the improvement of the previous work, Colonel et al [7] developed a four-layer multilayer perceptron with Adam optimizer and additive bias using L1 and L2 for weight regularization. In both of the previous studies, they evaluated their methods of measuring the mean square error (MSE) between the original and generated frame of the magnitude spectrogram frame. Feedforward autoencoders were also applied in [9] for transforming a 2049 Fast Fourier Transform (FFT) frame to a latent space of 8 values. In order to improve the performance of the network, they used augmentation techniques with first and second-order differences of the magnitude spectrum as well as additional MFCCs. The architecture was evaluated by applying Inverse-STFT (ISTFT) on the generated frames developing also a deterministic method for reconstructing the phase. Furthermore, Colonel et al [11] also conducted an investigation on activation functions for timbre synthesis using autoencoders. More thorough research for the generation of magnitude STFT frames was conducted in [10]. Shallow and deep feedforward networks as well as recurrent and variational autoencoders were compared to PCA. The study demonstrated that only deep autoencoders and LSTMs were able to adequately reconstruct the original samples. The model was evaluated subjectively using the MSE but also objectively by statistically analysing quality ratings. The waveform was synthesized by ISTFT with the original phase when there were no modifications in the latent space or using the Griffin Lim algorithm [12] when the phase needed to be estimated. Our method is heavily based on the baseline autoencoder developed in [13]. In this work, neural autoencoders were developed to extract meaningful information from a sound clip. They tried different input representations such as raw audio, the real and imaginary part of the STFT, or the log-magnitude part of the power spectra. The model was objectively evaluated using MSE between the original and generated spectrogram but also visually compared using rainbowgrams, which are instantaneous frequency colored spectrograms. III. METHODOLOGY This research work is devoted to understanding the capacity of stacked convolutional autoencoders for the reconstruction of the log-mel-spectrogram. The design of the network is illustrated in Fig. 1, where an encoder based on convolutional networks aims to compress instrumental sounds to a lower dimensional space and a mirrored decoder attempts to reconstruct the samples from this high-level representation. We encourage the reader to listen to the synthesized sounds of the conducted experiments while moving forward with the paper. The waveform of the audio files 1 is synthesized using the Griffith Lim algorithm [14] as well as a sinusoidal signal reconstruction method [15]. Towards the improvement of the design of convolutional autoencoders, regularization techniques are adopted to prevent overfitting and increase the performance of the network. In this research, we evaluate the effectiveness of autoencoders for sound synthesis purposes and we measure possible improvement using additional techniques that are expected to calibrate the model. This section presents a detailed explaination of autoencoders, convolutional neural networks and their components, a review of regularization methods, and the melspectrogram. A. Autoencoders An autoencoder is a neural architecture composed of an encoder and a decoder [2] trained together in an unsupervised manner. The encoder attempts to reduce the dimensionality of the original samples by extracting meaningful information in a non-linear procedure. Then, the decoder aims to reconstruct the original sample from the intermediate representation. This intermediate generated representation is called embedding or latent representation. For a feedforward single layer model, the encoder maps an input vector x ∈ R d to an encoding z ∈ R e where d > e using a non-linear activation function f (.) y = f (W x + b)(1) where W ∈ R (e×d) represents the weights of the connections between the neurons and b ∈ R e accounts for the bias term. The decoder maps z back to the reconstructed x ∈ R d using a similar approach x = f (W out y + b out )(2) where W out ∈ R (d×e) and b out ∈ R d . To expand the initial architecture to a deep neural autoencoder, a similar approach is adopted. The encoder maps the input of a neuron x i to an output representation x j for i, j < n as x → x 1 → x 2 → ... → x n(3) while the decoder reverses the previous procedure in a general way as x n → x n+1 → x n+2 → ... → x(4) The autoencoder is trained using a cost function that tries to minimize the error between the original and generated sample and updates the values of W and b using backpropagation. The activation and cost function used in autoencoders varies depending on the application and the training data. B. Convolutional Networks Convolutional neural networks (CNNs) consitute a regularized version of multilayer perceptrons. Their regularization depends on the fact that while feedforward neural networks investigate relationships within the whole sample, convolutional networks search for patterns locally. Their architecture is composed by three main components named convolutional layers, pooling, and fully connected layers. 1) Convolutional Layers: The convolutional layer is the most significant block of the CNNs. It is composed of a set of filters with smaller dimensions than the training samples and a set of learnable parameters. The filters are applied across the input data and the result creates the activation map. A few parameters define the convolutional layer, such as the number of filters, the number of moves over the input data (stride) and zero-padding which is the process of adding zeros outside the input matrix in order for the filters to fit the input size. 2) Pooling: Optionally, after every convolutional layer, a pooling layer is following the convolved output. Pooling is a dimensionality reduction process where a max or average filter matrix is applied across the convolved output. Although after pooling some information is lost, this technique can improve efficiency, reduce complexity, and limit the risk of overfitting. 3) Fully Connected Layer: In opposition to convolutional layers, fully connected layers directly connect each node of the output to the previous layer. This layer can be used to perform classification based on the features extracted from the convolutional layers and to manipulate the dimensionality of the outcome. C. Regularization Techniques One of the most considerable issues with deep neural networks is that they tend to overfit a training dataset. An overfitted model is a deep learning model that explicitly mimics the training samples by taking noise into consideration. These models are formulated with more parameters than necessary and become less accurate in predicting new data. To prevent overfitting, many techniques have been proposed including dropout [16], L1 and L2 regularization [17], data augmentation [18], or early stopping [19]. 1) Dropout: A technique where in every iteration, the network selects some random nodes and excludes them from the learning process [16]. Therefore, in each iteration a different set of nodes predicts the output. Dropout can also be considered as an ensemble technique since numerous subnetworks are created for each learning step. 2) L1 and L2 Regularization: They constitute methods where a regularization term is added to the cost function of the network. This additional term produces an additional residual to the loss function and therefore overfitting is reduced [17]. L1 regularization (also known as Lasso) updates the cost function using the Least Absolute Deviations (LAD) of the weight parameters as: Costf unction = Loss + λ 2m W (5) where λ corresponds to the regularization parameter and m to the number of parameters. In L2 regularization the cost function is updated with the Least Square Errors (LSE) of the weight matrix: Costf unction = Loss + λ 2m W 2(6) Regularization can be applied only on the weights W (kernel regularization), or only on the bias term b (bias regularization), or on the output layer y = W x + b (activity regularization). 3) Data Augmentation: An attempt to increase the training dataset by alternating the original samples. These additional data create extra noise to the network which can be used to prevent overfitting. In image processing, data augmentation can be achieved by flipping, scaling, or shifting the original images [18] while in audio processing, common techniques include noise injection, time shifting, or speed alternation [20]. 4) Early Stopping: A validation dataset is used to calculate the loss function after each epoch. If the performance of the network in the training set increases while the performance in the validation set does not improve, then the model starts overfitting. Early stopping is used to prevent the network from mimicking the pattern of the training data [19]. Patience denotes the number of epochs with no further improvement after which the training will be stopped. D. Mel-Spectrogram The spectrogram constitutes the time-frequency representation obtained by the application of the Fourier transform in overlapping fragments of time in an audio sample. The melspectrogram demonstrates a compressed form of the spectrogram based on human perception. The development of this mel-scale is based on the observation that the human ear demonstrates greater resolution at lower frequencies [21]. The association between the frequencies in hertz f and the melscale mel (mel from 'melody') is shown in Eq.7. mel = 2595log 10 (1 + f 700 )(7) The mel-spectrogram captures the most significant properties of the sound in a compressed form. Therefore, this timefrequency representation has been proven beneficial for deep learning applications since the memory and power requirements are reduced. IV. EXPERIMENTS A. Dataset For the conducted experiments, we used a subsample of the NSynth dataset 2 , which is a dataset of four-second monophonic notes. The subsample is composed of 3750 samples from a variety of instruments: guitar, bass, brass, synth, keyboard, flute, organ, mallet, vocal, reed, and string for a single pitch. The sounds were acoustic, electronic, or synthetic and could belong in different categories as per their velocity or acoustic quality. The data preprocessing included the computation of the logmel-spectrogram using a Blackman window of 690 samples, an FFT window of 1024 and 128 mel filter bands. The logmel-spectrograms were later normalized to be transformed into the range [0, 1]. The dataset was split into training, validation, and testing as 80/10/10. B. Models The proposed methodology demonstrates a stacked convolutional autoencoder with a mirrored encoder and decoder, like the one presented in Fig. 1. The two components are composed of three 2D convolutional layers with a kernel size of 4, stride of 2 and same padding. After experimental research, the hyperbolic tangent was used as an activation function for the convolutional layers while the softmax function was applied to the output layer to form the generated log-mel-spectrogram. Further investigation was conducted around kernel dimensions, filters, and pooling techniques. Experiments with additional regularization methods were also conducted, adopting techniques such as early stopping, dropout, and L1 and L2 kernel regularization and activity regularization. The network was trained using the ADAM optimizer [22] with an initial learning rate of 0.001, and a mean square error loss function in batches of size 64. An early stopping patience limit was set equal to 10 to avoid wasting resources during training. The experiments were conducted on a Tesla P100 GPU using the TensorFlow library 3 . V. EVALUATION To evaluate the effectiveness of a generative network, many methods have been proposed. In these experiments, the root mean squared error (RMSE) between the original and generated spectrogram has been used. Another metric applied to measure the accuracy of the generated spectrogram was the structural similarity index (SSIM) which also demonstrated as a sufficient initial indicator for the comparison between the two spectrograms. However, these two objective metrics cannot accurately capture the quality of the spectrogram. A difference in any value of the spectrogram will produce the same mean square error regardless of the position of the value but the synthesized sound could be nonidentical. Furthermore, as it is depicted in Fig. 2, the majority of the spectrogram values are zero and therefore a small error can demonstrate dissimilarities between the two spectrograms. In order to evaluate the generated spectrograms, we developed a metric that highlights the significance of harmonics in timbre synthesis. Our approach is based on a peak detection algorithm to estimate the frequencies of the original spectrogram (OrigFreq) and the frequencies of the generated spectrogram (GenFreq) for every frame. Following, the identical frequencies (IdFreq) constitute the retrieved frequencies that matched the relevant ones within a threshold of ±3%. This threshold can ensure that the detected peaks point to the same harmonic. In addition, to eliminate potential detected noise, harmonics with 30db from the peak amplitude are not taken into consideration. The calculation of the original, generated and identical frequencies can then be used to evaluate the predicted samples using a precision-recall schema [23]. Precision is computed as the fraction of the true positive values which are represented by the identical frequencies among the predicted frequencies as it is depicted in Eq. 8. P recision = IdF req GenF req (8) Recall is the number of identical frequencies among the number of frequencies estimated in the original spectrogram. Recall, which can also be referred to as sensitivity, is demonstrated in Eq. 9. Recall = IdF req OrigF req (9) Precision and recall are metrics to compute the relevance between the retrieved and relevant values. These two errors can later be combined into a single measurement called F1 score according to Eq. 10. F1 score demonstrates the harmonic mean of precision and recall indicating a more accurate metric for the overall system. F 1 score = 2 · P recision · Recall P recision + Recall(10) Precision, recall and F1 score can score between 0 and 1, with higher value presenting a better performance. A high precision and low recall indicates a generated sample with VI. RESULTS AND DISCUSSION This research work's main objective is to provide insight into the synthesis of timbre using stacked convolutional autoencoders. Additional parametrization and regularizations methods aim to minimize the error between the original and predicted audio samples. Each section of this paragraph provides results for an experimental set of parameters. Initially, we present the baseline autoencoder and then we discuss about experiments using regularization methods, pooling, dimensionality of the latent space and convolutional networks without a fully connected layer. A. Baseline The baseline design constitutes a neural autoencoder with three convolutional layers followed by a max pooling layer, and a fully connected layer. The encoder produces a latent representation of dimension 8192 and the network is trained with no other regularization methods than early stopping. The original and generated log-mel-spectrogram are illustrated in Fig. 2. As one can see, the reconstructed spectrogram is able to capture the general form of the original samples 4 . B. Regularization Additional enhancements were investigated in order to improve the performance of the baseline network. Table I demonstrates an experimentation with different regularization techniques. More specifically, regularization methods are applied in order to improve the effectiveness of the autoencoder for the reconstruction of the log-mel-spectrogram. Dropout with a probability of 0.3 was applied only to the encoder (E), only on the decoder (D), or on both encoder and decoder (ED). Furthermore, experiments on kernel regularizers (KR) and activity regularizers (AR) using L1 or L2 regularization on encoder, decoder or both networks were conducted. The results showed that dropout did not improve the performance of the network. Specifically, dropout only on the decoder presents the same RMSE as the baseline but a lower F1 score. However, an increased recall and decreased precision indicates that the model produces additional noise in the generated samples. Furthermore, dropout on the encoder or both encoder and decoder significantly deteriorated the performance of the network. Using kernel regularizers as regularization methods, an improvement can be identified. Either L1 or L2 regularization function on the decoder produces the most accurate results. In particular, an L2 kernel regularizer on the decoder demonstrated the highest F1 score and SSIM, and the lowest RMSE. The results are characterized by high precision and recall, and therefore generate the most promising spectrograms. Kernel regularizers on the encoder, or on both encoder and decoder also provide a slight improvement. Finally, activity regularizers spoiled the power of the neural networks yielding to a non-harmonic outcome. In most cases, the generated sounds produced are characterized by simple noise. Overall, stacked convolutional autoencoders can synthesize musical instrument timbre from a low dimensional representation. In some cases, regularization techniques such as L1 or L2 kernel regularization are able to enhance the performance of the network producing more accurate spectrograms. However, other regularization techniques, such as dropout, do not present any improvement and in even more extreme cases, like activity regularizers, the decoder generates random noise. C. Pooling The next set of experiments is based on the parametrization of the convolutional networks. More specifically, an inves- tigation on pooling layers was conducted. Initially, max or average pooling layers were adopted after each convolutional layer of both the encoder and decoder. Lastly, a last experiment without pooling layers was also studied. In order for the network with no pooling layers to match the latent dimensions of the network with pooling, three additional convolutional layers were added. Table II demonstrates the results from experiments based on pooling techniques. Based on the root mean square error, the structural similarity index and the F1 score, networks with max or average pooling present similar performance. Although the F1 score is almost identical, average pooling demonstrates a higher recall and a lower precision, indicating samples with additional noise. Finally, results from networks without pooling layers presented synthetic sounds of lower quality. Based on the results above, we can conclude that stacked convolutional autoencoders with a rough pooling approach, such as max pooling, can generate a more accurate audio time-frequency representation from a compressed low dimensional space of instrumental pitched sound. D. Dimensionality of the latent space To evaluate the effectiveness of neural autoencoders regarding their compression ability, an experimentation on the latent space dimension was conducted. Additional downsampling was studied by creating a latent representation of 4096 and 2048. The results are presented in Table III. According to the metrics, a more compressed representation is able to generate more accurate log-mel-spectrograms. Although the lower latent dimension was expected to reduce the performance of the network, the experiments revealed that a smaller latent space is able to synthesize spectrograms with a more smooth distribution of the higher partials leading to more natural sounds. E. Without a fully connected layer In the final set of experiments, we attempted to remove the dense layer of the autoencoder. A dense layer is necessary to manipulate the dimensions of the latent space and apply TABLE IV NETWORK WITHOUT A FULLY CONNECTED LAYER AT THE END OF THE CONVOLUTIONAL BLOCKS further compression. However, Table IV demonstrate that convolutional autoencoders without a dense layer generates spectrograms of better quality. That could be explained because the network preserves spatial information of the original samples. F. Discussion From the above experiments, one can identify that stacked convolutional autoencoders are able to compress and reconstruct spectral representations of sound with adequate accuracy. As it is illustrated in Fig. 2, most of the frequency bands in the log-mel-spectrogram have zero values and it can be debatable whether the whole representation can be considered as the actual encoded information. However, hearing is a sense that demonstrates sensitivity and a slight change in the spectrogram can synthesize a significantly different sound. Therefore, the network needs to be exceptionally precise in the position and amplitude of the non-zero values. Comparing the results with the state-of-the-art [13], our sounds demonstrate phase coherence, making the sounds more pleasant. However, training the autoencoder in the whole NSynth dataset conditioned by the pitch can expand the variety of the produced sounds but also creates an additional perplexity which can reduce the performance of the network. Furthermore, using Wavenet [24] as a vocoder may not preserve the phase continuity in the waveform and therefore generate more noisy instrumental sounds. VII. CONCLUSION Deep generative models have been previously studied to synthesize timbre from a compressed representation. Most of these studies generate sound from frames of magnitude shorttime Fourier transform. For this purpose, recurrent or fullyconnected autoencoders have been proposed. In this work, we demonstrated an investigation on stacked convolutional autoencoders for the reconstruction of a spectrogram based on the perception of hearing and experimented with parametrization techniques and regularizations of the autoencoders. Moreover, we presented an evaluation method for calculating the accuracy of predicted frequencies in monophonic and harmonic musical sounds. The conducted experiments revealed that a latent representation compressed to 3% of the size of the original data can synthesize timbre with adequate accuracy. Also, from a variety of regularization techniques applied to the network, some of them demonstrated improvement compared to the baseline autoencoder while others limited the abilities of the neural networks. The conducted research is expected to accelerate future studies on the compression of timbre and the evaluation of generated harmonic sounds. Future research work will expand the capabilities of the autoencoders including more diverse sounds in terms of pitch and instrument. An attempt for further compression will be studied by conditioning the network on additional temporal or spectral information. Moreover, incorporating temporal layers such as GRUs or LSTMs at the end of the encoder section is expected to influence the performance of the network. Finally, future work will outline an experimentation of the latent space using more perceptual properties. Following the previous work of Esling et al [25] and Roche et al [26], an investigation around timbre descriptors will be conducted to enhance the structure of the latent representation generated by autoencoders and increase the accuracy of the synthesized audio. Numerous techniques of numeric or visual explanations can be adopted for the interpretation and evaluation of the generated latent space [27], [28]. Fig. 1 . 1Illustration of a stacked convolutional autoencoders architecture for the reconstruction of the log-mel-spectrogram. (A) Original log-mel-spectrogram. (B) Stacked convolutional autoencoder. (C) Encoder based on three convolutional layers. (D) Projection of the latent representation of multiple samples. (E) Decoder based on symmetric encoder. (F) Reconstructed log-mel-spectrogram. Fig. 2 . 2Original and generated log-mel-spectrogram less detected harmonics while a low precision and high recall indicates additional noise in the synthesized spectrogram. Fig. 3 . 3Peak detection between the original and generated spectrogram I REGULARIZATION TECHNIQUES: A COMPARISON BETWEEN A NETWORK WITHOUT REGULARIZATION METHODS TO A NETWORK WITH DROPOUT ON ENCODER (E), DECODER (D), OR ENCODER AND DECODER (ED), AND TO KERNEL REGULARIZERS (KR) AND ACTIVITY REGULARIZERS (AR) TABLE https://anastasianat57.github.io/StackedConvolutionalAutoencoders https://magenta.tensorflow.org/datasets/nsynth 3 https://www.tensorflow.org/ https://anastasianat57.github.io/StackedConvolutionalAutoencoders ACKNOWLEDGMENT This work was funded by Science Foundation Ireland and its Centre for Research Training in Machine Learning (18/CRT/6183). Principal component analysis: Principal component analysis. 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[ "EMBEDDED TREFFTZ DISCONTINUOUS GALERKIN METHODS", "EMBEDDED TREFFTZ DISCONTINUOUS GALERKIN METHODS" ]	[ "Christoph Lehrenfeld ", "Paul Stocker " ]	[]	[]	In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves.	10.1002/nme.7258	[ "https://export.arxiv.org/pdf/2201.07041v2.pdf" ]	246,035,733	2201.07041	b6ce00dd34437db29de475ed91ba4f241d0c1b76	EMBEDDED TREFFTZ DISCONTINUOUS GALERKIN METHODS Christoph Lehrenfeld Paul Stocker EMBEDDED TREFFTZ DISCONTINUOUS GALERKIN METHODS In Trefftz discontinuous Galerkin methods a partial differential equation is discretized using discontinuous shape functions that are chosen to be elementwise in the kernel of the corresponding differential operator. We propose a new variant, the embedded Trefftz discontinuous Galerkin method, which is the Galerkin projection of an underlying discontinuous Galerkin method onto a subspace of Trefftz-type. The subspace can be described in a very general way and to obtain it no Trefftz functions have to be calculated explicitly, instead the corresponding embedding operator is constructed. In the simplest cases the method recovers established Trefftz discontinuous Galerkin methods. But the approach allows to conveniently extend to general cases, including inhomogeneous sources and non-constant coefficient differential operators. We introduce the method, discuss implementational aspects and explore its potential on a set of standard PDE problems. Compared to standard discontinuous Galerkin methods we observe a severe reduction of the globally coupled unknowns in all considered cases, reducing the corresponding computing time significantly. Moreover, for the Helmholtz problem we even observe an improved accuracy similar to Trefftz discontinuous Galerkin methods based on plane waves. Introduction In this manuscript we propose a novel numerical method closely related to Trefftz discontinuous Galerkin methods. The main idea of Trefftz methods, originating from [55], is to choose optimal discretization spaces that provide the same approximation quality as comparable discrete spaces with a significant reduction of the number of degrees of freedom (ndofs). For an overview on Trefftz methods see [26,34,52]. Polynomial Trefftz functions have been obtained for several linear partial differential operators with constant coefficients such as Laplace equation [42,43], acoustic wave equation [37,38,46], heat equation [53], plate vibration and beam vibration equation [1,35], and time-dependent Maxwell's equation [27]. Efforts to generate Trefftz polynomials in a general case have been undertaken, see [21,25]. The idea of finding Trefftz polynomials via Taylor series, used in [36], was extended in [57] to construct Trefftz-like polynomials in the case of non-linear elliptic PDEs possibly with smooth coefficients. The method requires a Taylor expansion of the coefficients of the PDE to construct these polynomials via a recursive procedure for the coefficients of a Taylor polynomial. Discontinuous Galerkin (DG) schemes can be easily combined with Trefftz functions as the basis construction of elements decouples. Trefftz-DG schemes for Laplace equations have been analyzed in [20,32,33]. Different time-dependent problems have been discussed recently, such as wave problems in one space dimension [28,29,51], the acoustic wave equation [3,46,50], elastoacoustics [5], a class of Friedrichs systems coming from linear transport [6,47], for time-dependent Maxwell's equation see [12,27], and the linear Schrödinger equation [16]. Very popular applications of Trefftz methods are wave propagation problems in frequency domain. There, no polynomial Trefftz space exists and plane wave functions are used instead. A DG scheme for Helmholtz equation has been presented and analyzed in [7,15,17,41,44], for more information on Trefftz methods for Helmholtz equation see the recent survey [19] and the references therein. Plane waves have also been used for linear elasticity [45] and time-harmonic Maxwell's equation [14,18,44]. Apart from combining Trefftz spaces with DG schemes, harmonic polynomials and plane waves have also been combined with virtual element methods, see [39,40,49]. In the case of a differential operator with smooth coefficients the local solutions in a Trefftz space are not sufficient to provide high-order approximation. This can be circumvented by weakening the requirements of the Trefftz space, providing again a sufficiently large basis. In [24] a quasi-Trefftz DG method is introduced and analyzed for the acoustic wave equation with smooth coefficients. Plane waves have been generalized to work with a DG method for the Helmholtz equation with smooth coefficient in [23]. Trefftz methods usually are applied to homogeneous problems (problems with no volume source term). To treat inhomogeneous problems with a Trefftz approach, one needs to construct a particular solution. Then it is possible to apply Trefftz methods to solve for the difference. In [56] the Poisson problem is treated in two separate ways, once a fundamental solutions are used as trial functions, while for the other approach radial basis functions are used to express the inhomogeneous part. Another approach for Helmholtz and time-harmonic Maxwell's equation is presented in [22], where a series of inhomogeneous local problems are solved using spectral elements to obtain local particular solutions. 1.1. Main contributions and outline of the paper. The embedded Trefftz method proposed here side-steps the explicit construction of a Trefftz space, instead we present a simple method of constructing a corresponding embedding. In its simplest form, the method proposed here can be seen as a convenient way to set up the linear system of a Trefftz DG discretization by means of a Galerkin projection of a standard DG method onto its Trefftz subspace. Instead of implementing Trefftz functions explicitly, the characterization of the Trefftz function space as the kernel of an associated differential operator is exploited to construct an embedding of a Trefftz subspace into the DG space in a very generic way. We denote this embedding as the Trefftz embedding. The construction requires only element-local operations of small matrices and is embarrassingly parallel. To compute the discrete kernel numerically several established methods exists, e.g. one can use QR factorizations, singular, or eigenvalue decompositions. We denote this approach as embedded Trefftz DG method. When the embedded Trefftz DG approach is used to implement an existing Trefftz DG method one trades the convenient setup for slightly larger costs in the setup of the linear system. However, the strongest feature of the embedded Trefftz DG approach is the following: The generic way the linear systems are set up allows to apply the concept of Trefftz methods beyond their previous limitations. Two of these limitations that can be conveniently exceeded are: • In many interesting applications a standard Trefftz space is either not polynomial or unfeasible to set up at all limiting the application of Trefftz DG methods for these cases. • If a standard Trefftz DG subspace exists, still only homogeneous equations (equations with no volume source terms) are easily dealt with. Inhomogeneous equations are difficult to deal with for standard Trefftz DG methods. In cases where a polynomial Trefftz space is not embedded in the discretization space or when a Trefftz basis is unknown, the proposed embedded Trefftz DG method can still be applied. This is done by slightly weakening the properties of the Trefftz space we are looking to embed. With this we are able to treat for example the Helmholtz equation and PDEs with piecewise smooth coefficients, e.g. the acoustic wave equation. The embedded Trefftz DG method offers a more generic way of obtaining local particular solutions exploiting the underlying DG discretization. Computing element-wise particular solutions in the DG space can be done with little computational effort and then be used to homogenize the DG system. The paper continuous as follows: In Section 2 we present the method, recovering traditional Trefftz DG methods and its extensions to the case of weaker Trefftz spaces and inhomogeneous equations. In Section 3 we address implementational aspects and turn to applications of the approach to a set of PDE problems in Section 4. We present numerical examples for Laplace equation, Poisson equation, acoustic wave equation with piecewise constant and also with smooth coefficient, Helmholtz equation, and a linear transport equation. In Section 5 we compare the Trefftz DG method to another popular acceleration technique for DG methods, the Hybrid DG method. The method was implemented using NGSolve and NGSTrefftz 1 . Conclusion and outlook are presented in Section 6. The method We introduce the method in several steps. After some preliminaries, we first introduce the basic concepts of the method in Section 2.2. Here, we will assume that a suitable Trefftz space exists, and is embedded in the discrete DG space V p (T h ). Here, we understand as the Trefftz space the subspace of functions that locally fulfill the PDE pointwise. This case only applies to a small set of problems where (polynomial) Trefftz spaces exist, like the Laplace equation and the acoustic wave equation with piece-wise constant wavespeed. We then turn to cases where a Trefftz space is not embedded in the discrete space. In this case we consider a weak Trefftz space instead. This is presented in Section 2.3. How inhomogeneous equations can easily be dealt with is explained in Section 2.4 resulting in a method that allows to treat a large class of PDE problems in a uniform way. 2.1. Preliminaries. We consider a linear partial differential operator L and a corresponding boundary (and possibly initial) value problem Lu = f in Ω ⊂ R d(1) supplemented by suitable boundary conditions and paired with a suitable discontinuous Galerkin formulation Find u hp ∈ V p (T h ), s.t. a h (u hp , v hp ) = (v hp ) ∀v hp ∈ V p (T h ).(2) The DG formulation is set on a mesh T h of the domain Ω, and we assume that the space V p (T h ) is the product of local spaces V p (K) for K ∈ T h where the index p indicates a polynomial degree, e.g. V p (K) = P p (K) the space of polynomials up to degree p. We note that space-time DG formulations are allowed in this setting, so that d ∈ N is either the dimension of the space or space-time domain. 2.2. Embedded Trefftz method. At first, in this subsection, we make the following simplifying assumptions: Firstly, we assume f = 0. Second, we assume that the differential operator has the form L = d l=1 α l D β l l for α l ∈ R and β l ∈ N and that all mesh elements are straight. Valid examples are L = −∆, L = b · ∇ for b ∈ R d , L = ∂ t + b · ∇, but not L = −∆ ± id, L = −∆ + b · ∇ or L = − div(α∇·) for a non-constant field α. This assumption ensures that the subspace of V p (K) that we define next is sufficiently large to allow for reasonable approximations of the solution. We define the Trefftz space T p (T h ) = {v ∈ V p (T h ), Lv = 0 on each K ∈ T h } ⊂ V p (T h ).(3) The Trefftz version of (2) is its Galerkin projection to T p (T h ): Find u T ∈ T p (T h ), s.t. a h (u T , v T ) = (v T ) ∀v T ∈ T p (T h ).(4) In contrast to previous works on Trefftz methods, we do not construct the space T p from scratch based on accordingly defined basis function, but rather aim to build an embedding operator for the Trefftz space into the underlying piecewise polynomial space V p (K). We call this the embedded Trefftz DG method. Construction of the embedding operator. Let {φ i } be the set of basis functions of V p (T h ), N = dim(V p (T h )) and G : R n → V p (T h ) be the Galerkin isomorphism, G(x) = N i=1 x i φ i . With e i , i = 1, . ., N the canonical unit vectors in R n , so that G(e i ) = φ i , we define the following matrices and vector for i, j = 1, . . . , N (A) ij = a h (G(e j ), G(e i )) = a h (φ j , φ i ), (l) i = (G(e i )) = (φ i ) (5a) (W) ij = Lφ j , Lφ i 0,h ,(5b) where ·, · 0,h = K∈T h ·, · K is the element-wise L 2 -inner product. We are interested in the kernel of L (in an element-wise and pointwise sense) in V p (T h ) as this is the part where the Trefftz DG method operates. We note that ker(L) = G(ker(W))(6) and hence are looking for a basis of ker(W). As W is block diagonal, with blocks corresponding to the elements K ∈ T h we will construct the basis element by element which implies that the following calculations can be done in parallel. For a fixed K ∈ T h let W K ∈ R N K ×N K be the corresponding block with N K = dim(V p (K)). For the dimension of the kernel we have M K := dim(ker(L)) = N K − L K with L K = dim(range(L)) where (for the case of differential operator L in the assumed form) L K is typically known. Now, we can determine the kernel of W K and collect a set of orthogonal basis vectors in a matrix T K ∈ R N K ×M K so that ker(W K ) = T K · R M K . The kernel matrix T K can be determined numerically, e.g. by a QR decomposition or a singular value decomposition, we discuss this in more depth in Section 3.4. Composing the individual element matrices T K into a block matrix T ∈ R N ×M with M = dim(ker(W)) = T ∈T h M T we easily obtain the characterization of the kernel as ker(W) = T·R M . Further note that there holds T T T = I M ×M . We define the Galerkin isomorphism between R M and T p (T h ) as G T : R M → T p (T h ), x → G(Tx) and denote a discrete operator corresponding to T as the Trefftz embedding T : T p (T h ) → V p (T h ), v h → G(TG −1 T (v h )) . To setup the linear systems corresponding to (4) we first assemble A, l and T and arrive at: Find u T (= G −1 T (u T )) so that T T AT u T = T T l.(7) The resulting linear system is well controlled in terms of its conditioning. Lemma 1 (Conditioning of the embedded Trefftz method). The spectral relative condition number of the embedded Trefftz DG method is bounded by the corresponding condition of the DG method, κ 2 (T T AT) ≤ κ 2 (A).(8) Proof. By construction of T all its column vectors are orthogonal. Remark 2 (Matrix representing the kernel of L). There are other possibilities to characterize the kernel of L in V p (K). Here, we chose the specific form as it most obviously displays that u ∈ ker W is equivalent to LG(u) = 0 pointwise as there has to hold LG(u), LG(u) 0,h = 0. In Section 2.3 we will replace W with a proper generalization. Remark 3 (Degrees of freedom). One major advantage of the Trefftz method is the reduction of the number of (globally coupled) degrees of freedom (ndofs) without harming the approximation too much. Let us first discuss the reduction on an example. Assume L is a second order operator, say −∆, the unknown field is scalar and the mesh is triangular. Then N K = (p + 1)(p + 2)/2 and −∆V p (K) = V p−2 (K) and hence L K = (p − 1)p/2 yields M K = 2p + 1. This holds for every element and similar calculations can be made for different differential operators and vectorial problems. In general, the Trefftz method decreases ndofs from O(p d ) to O(p d−1 ) and is hence especially attractive for higher order methods. In the lowest order case p = 0 or also p = 1 (depending on the differential operator) one often has T p (T h ) = V p (T h ). A similar reduction of ndofs from O(p d ) to O(p d−1 ) in a discontinuous Galerkin setting is also achieved in methods that associate dofs primarily to facets and rely in some way on static condensation such as Hybrid DG methods [8,9] or Hybrid High order (HHO) methods [11]. We discuss the comparison between these approaches in more detail in Section 5. Remark 4 (Uniqueness of the Trefftz embedding). We notice that while the embedding T : T p (T h ) → V p (T h ) is unique the matrix T and hence, the basis for T p (T h ) resulting from the previous procedure is not. Remark 5 (Volume integrals). One advantage of Trefftz DG methods that is sometimes advertised is that volume integrals that involve Lu (possibly after partial integration) can be removed during assembly as Lu = 0 on the Trefftz space. This is also possible when using the embedded Trefftz method, see also Remark 10. Weak Trefftz embedding. In the previous section we restricted the differential operator to have a special form, here we aim to lift some of the restrictions, specifically we want to treat non-constant coefficients and mixed differential orders also including id. In these situations the Trefftz space as defined in (3) will typically much too small, i.e. the requirement Lv = 0 on the polynomial spaces may restrict the discrete solution too much leading to a version of locking. To circumvent this problem, we weaken our condition in the Trefftz space. We introduce a projection Π that is yet to be defined and define the weak Trefftz space and the embedded weak Trefftz DG method : Find u T ∈ T p (T h ), s.t. a h (u T , v T ) = (v T ) ∀v T ∈ T p (T h ) with (9a) T p (T h ) = {v ∈ V p (T h ), ΠLv = 0 on each K ∈ T h }.(9b) This replaces the pointwise condition Lv = 0 with something that is potentially weaker. The projection Π will be designed to ensure that the resulting weak Trefftz space is reasonably large to allow for proper approximations of the PDE solution. A projection that has proven fruitful is given by Π : L 2 (Ω) → V q (T h ), s.t. Πw, v 0,h = w, v 0,h ∀ v ∈ V q (T h ).(10) The multi-index q ∈ N d on V q denotes the polynomial space of maximal order q = (q 1 , . . . , q d ) in each variable (x 1 , . . . , x d ). (If all entries of the multi-index are equal, we will continue to write V q .) Let us now consider the differential operator L = j∈N d α j D j with α j ∈ C(T h ). Then we choose q i equal to the largest appearing differential order, i.e. q i = p − max α j ≡0 j i , i = 1, . . . , d. With these definitions, the analogue to W from Section 2.2, i.e. the matrix that can be used to numerically extract the weak Trefftz space, becomes for j = 1, ..., N and i = 1, ...,Ñ = dim V q (T h ) and {ψ i } a basis for V q (T h ) (W) ij = Lφ j , ψ i . (11) In Section 3.1 we present a slightly more convenient version (with the same kernel) that we also used in the numerical examples in this manuscript. Remark 6 (Consistency with Section 2.2). We recover the Trefftz method from Section 2.2 if we choose V q = LV p . For instance for L = −∆ we can choose V q = ∆V p = V p−2 . From the constant coefficient case treated in Section 2.2, it becomes clear that smaller values in q would make the space larger than the Trefftz space, resulting in a sub-optimal reduction of the space. Remark 7 (Quasi-Trefftz). In [24] weak Trefftz spaces have been used to treat the acoustic wave equation with smooth coefficients, we recall the quasi-Trefftz spaces used there in (30). We recover this quasi-Trefftz method if we replace Π in (9b) with a Taylor polynomial expansion of order p − 1 in the element center. 2.4. Inhomogeneous PDEs. The Trefftz methods discussed in Section 2.3 and Section 2.2 as well as those known in the literature are designed for the homogeneous problem Lu = 0 only and hence are not able to solve for problems with non-zero source term. A simple adjustment however allows to deal also with the inhomogeneous case for the embedded Trefftz DG method. In this section we treat equations of the form Lu = f in Ω for f ∈ L 2 (Ω), supplemented by suitable boundary conditions for which a suitable DG discretizations is given in the form Find u hp ∈ V p (T h ), s.t. a h (u hp , v hp ) = (v hp ) = g(v hp ) + f, v hp 0,h ∀v hp ∈ V p (T h ). (12) where the linear form g(·) corresponds to the weak imposition of boundary conditions. On the continuous level of the PDE we can introduce an affine shift and decompose the solution to Lu = f with u = u 0 + u f . Here u f is a particular solution to Lu f = f and u 0 (uniquely) solves Lu 0 = 0 (with boundary data depending on u f ). As the embedded Trefftz (or weak Trefftz) method is based on an underlying DG discretization we are able to construct a reasonable particular solution in the DG space V p (T h ) and apply the same homogenization strategy. We then write u h = u T + u h,f for u T ∈ T p (T h ) and u h,f ∈ V p (T h ) . u h,f is a (non-unique) particular solution to ΠLu h,f = Πf that we can compute in an element-local fashion, see Section 3.5. For the existence of a particular solution we require that ΠL : V p (T h ) → V q (T h ) is surjective. This essentially depends on the choice of L, p and q. It always holds true in the setting of Section 2.2. In the remainder we assume that a particular solution exists. For u h,f a particular solution, we are looking for a solution u h ∈ T p (T h ) + u h,f so that a h (u h , v T ) = (v T ) ∀ v T ∈ T p (T h ).(13) After homogenization this means that we are looking for u T ∈ T p (T h ) that (uniquely) solves a h (u T , v T ) = (v T ) − a h (u h,f , v T ) ∀ v T ∈ T p (T h ).(14) This translates to the solution of the linear system T T ATu T = T T (l − Au f ).(15) Remark 8 (Degenerate cases). Let us briefly discuss two extreme cases of the Trefftz embedding. If L = id, then T p (T h ) = {0} and the discrete solution of Lu = u = f will effectively be computed when determining the -in this case unique -particular solution u h,f . The other extreme case is L = 0 for which we recover T p (T h ) = V p (T h ). Assuming a coercive problem and a DG formulation that provides good discretization properties, on the embedded (weak) Trefftz DG subspace we can still apply all essential analysis tools to obtain a Céa-type result: Lemma 9. Let u ∈ V (Ω) be a weak solution to the PDE problem under consideration and u h ∈ T p (T h ) + u h,f be the solution to (13) and · h be a suitable norm on V p (T h ) ∪ V (Ω). Assume that problem (2) is well-posed, specifically a h and are continuous with respect to · h on V p (T h )∪V (Ω) and a h is coercive (with respect to · h ) on V p (T h ). Then u − u h h inf v h ∈V p (T h ) ΠLv h =Πf u − v h h Proof. Let v h ∈ T p (T h ) + u h,f = V p (T h ) ∩ {v s.t. ΠLv = Πf }, and w h = u h − v h ∈ T p (T h ). Then there holds u h − v h 2 h a h (u − v h , w h ) + =0 a h (u h − u, w h ) u − v h h u h − v h h where we used coercivity and continuity and that a h (u − u h , v) = 0 ∀v ∈ T p (T h ). Hence u h − v h h u − v h h . Adding a triangle inequality and taking the infimum yields the result. The crucial question is of course on the approximation quality of the (affinely shifted) weak Trefftz space. We leave that open for future research. Implementational aspects Let us consider several implementational aspects in this separate section. First, in Section 3.1 we discuss how to set up a version of W -the matrix that characterizes the (weak) Trefftz subspace -in a more convenient way than what directly follows from Section 2.3. Next, we discuss how the dimension of the (weak) Trefftz subspace can be obtained in an implementation. Afterwards, we present the algorithmic structure of the embedded Trefftz DG method in comparison to a standard DG method. We discuss how to numerically compute the kernel of the matrix W and the particular solutions in Sections 3.4 and 3.5. In Section 3.6 we discuss the computational complexity of the embedded Trefftz DG method. 3.1. Characterization and implementation of the (weak) Trefftz DG space. Involving a new space V q (T h ) in the implementation and assembling of the matrix W K as used in (11) may occur somewhat unneccessarily cumbersome. In the following, we propose a different but equivalent formulation that is more convenient and is also applied in the numerical examples later. Instead of using V q (T h ) we again use the finite element space V p (T h ), and find a differential operatorL withL : V p (T h ) → V q (T h ) surjective. The operatorL is a modification of L which only keeps the highest order derivatives in each direction and has only constant coefficients. In the simplest cases, i.e. in the setting of Section 2.2 we simply chooseL = L. Otherwise, in the general setting of Section 2.3, for example for the Helmholtz equation with variable wavespeed, with the operator L = − id −ω(x)∆, we chooseL = ∆ (which would correspond to q i = p − 2). This way, we can re-define the matrix W from (11) for the general case as (W) ij = Lφ j ,Lφ i 0,h .(16) Note that this matrix W has dimension N × N as in the original definition (5b) in the setting of Section 2.2. 3.2. Dimension of the (weak) Trefftz subspace. On each element K ∈ T h the dimension of the ker ΠL, M K , can be computed as N K − dim(range ΠL) also for the (weak) Trefftz spaces. In most cases this expression could easily be precomputed and be given to the algorithm for the numerical kernel extraction. Alternative -and this is a bit more convenient as it requires less user input -we can also try to read that information from the matrix directly. Next, we explain how this can be achieved and notice that this approach has been used in all numerical examples below. When numerically computing the kernel of a matrix, we use a QR, singular value or eigenvalue decomposition. In all these factorizations the diagonal of one of these factors (e.g. the diagonal matrix of the SVD or the triangular matrix of the QR decomposition) would have M K zeros assuming exact arithmetics. Due to inexact computer arithmetics this will not be the case exactly and hence we use a truncation parameter ε > 0 to determine which values are considered as (numerical) zeros. A numerical study on the choice of the threshold for the Laplace problem is done in Section 4.2. 3.3. The algorithm. In Algorithm 1 we show how the embedded Trefftz DG method is implemented. We display the method in pseudo-code and a python code example in NGSolve, using the operatorL as described in Section 3.1. Let us briefly comment on the algorithm. 1. First of all, we notice that the computation and the application of the Trefftz embedding is separated from the setup of the linear system of the DG method. On the one hand this obviously leaves room for (efficiency) improvements as an integration into the DG assembly could allow to reduce the need for storing the DG matrices (and vectors in the homogeneous case) in the first place. On the other hand it also facilitates the realization of the embedded Trefftz DG method in a DG software framework without the need to interfere with optimized routines such as the assembly. We focus on the second approach in the presentation of the implementation in Algorithm 1. Both approaches are implemented in NGSolve+NGSTrefftz and can be found in the software documentation, see [54]. 2. Further, we observe that the setup of the matrix T is done element-by-element and can hence be carried out in an embarrassingly parallel manner. The resulting matrix is block diagonal such that the setup of T T AT and the r.h.s. vector can also be done efficiently. 3. Assuming a corresponding DG method already exists for a user, he only needs to additionally specify the operators L andL,the truncation parameter ε and the r.h.s. to obtain the corresponding embedded Trefftz DG method. All of these inputs are canonical in most cases rendering the approach and its implementation quite easily accessible. Require: Basis functions {φi}i, DG formulation (a h , l), operators L,L, truncation parameter ε, r.h.s. f 1: function dg matrix 2: Algorithm 1: Embedded Trefftz DG algorithm in the version discussed in Section 3. Pseudo-code (left) and running python code (using NGSolve and NGSTrefftz, right). The parts used by a standard DG scheme are colored black, the parts using the embedded Trefftz DG method, as in Section 2.3, are colored in teal. Code needed to treat the inhomogeneous problem, presented in Section 2.4, is colored purple. (A)ij = a h (φj, φi) 3: (l)i = (φi) 4: for K ∈ T h do 5: (WK )ij = Lφj,Lφi 0,h 6: TK = ker h (ε; WK ) 7: if f = 0 then 8: (wK )i = f,Lφi 0,h 9: (u f )K = W † K wK 10: Solve T T AT u T = T T (l−Au f ) 11: u h = Tu T +u f 12: output u h 1 def Solve ( mesh , 3.4. Computing the kernel matrix T K . In this section we briefly discuss how to numerically extract the kernel matrix T K from W T K so that ker(W T K ) = T K · R M K . The elementwise matrix W K is of sizeÑ K × N K , whereÑ K depends on the approach: If the matrix W K stems from either (5b) or (16), then W K is a square matrix andÑ K = N K . If (11) is used, theñ N K = dim V q (K). We discuss two possibilities: One based on a QR decomposition and one a singular value decomposition. QR decomposition. The most obvious option seems to be a QR decomposition of W T K , s.t. W T K = Q K · R K for an orthogonal matrix Q K , of size N K ×N K , and an upper triangular matrix R K , sized N K ×Ñ K , where we assume an ordering such that ( R K ) ij = 0 for i = L K + 1, ..,Ñ K . Then we have W K = R T K · Q T K and that the last M K columns of Q K span the kernel of W K . We denote this submatrix as T K , i.e. (T K ) i,j := (Q K ) i,L K +j for i = 1, .., N K , j = 1, .., M K . One easily checks that there holds ker(W K ) = T K · R M K . Singular value decomposition. Applying a singular value decomposition (SVD) to W K yields W K = U K Σ K V T K for two orthogonal matrices U K , V K and a diagonal matrix Σ K with (Σ K ) jj = 0 for j > L K . Then one easily sees that the last M K columns of V K span the kernel of W K and we choose this submatrix as T K , i.e. (T K ) i,j := (V K ) i,L K +j for i = 1, .., N K , j = 1, .., M K . 3.5. Computing particular DG solutions. To compute a (local) particular solution needed to solve inhomogeneous PDEs, see Section 2.4, on an element K ∈ T h . We assemble (w K ) i = (f, G(e i )) = (f,Lφ i ) and define (u f ) K = W † K w K . Here W † K denotes the pseudoinverse of the matrix W K , which can be obtained using the QR decomposition or SVD of the matrix W K which may have already been computed when numerically computing the kernel of W K , cf. Section 3.4. 3.6. Algorithmic complexity. The computation of the kernel of ΠL or a particular solution scales well in the mesh size and can easily be parallelized. However, the scaling in the local unknowns per element is severe. For a discretization of order p the unknowns on each element scale like O(p d ). Hence, the costs for element-wise QR or SV decompositions (including the setup of the adjusted matrices and vectors and neglecting parallelization effects) are O(h −d p 3d ). Let us compare these costs with the solver costs of standard DG and classical Trefftz DG methods. For a standard DG method we have O(h −d p d ) unknowns globally and assuming a solver complexity O(N α ), where N denotes the ndofs with α ∈ R, the costs are O((h −d p d ) α ). For a classical Trefftz DG method the costs are O((h −d p (d−1) ) α ). The proposed method has a bad asymptotic complexity if h = const and p → ∞. In this case the computational costs will soon be dominated by the QR or SVD operations in the Trefftz DG setup. However, in the opposite case, p = const and h → 0 asymptotically the costs of the embedded Trefftz setup are only relevant for an optimal solver complexity α = 1 and are negligible otherwise. In the applications below we observe that at least moderately high (fixed) polynomial degrees can be easily used with the embedded Trefftz DG method without dominating the computational costs by the computation of the embedding. Applications We consider a set of different PDE problems and aim to compare the embedded Trefftz DG method to the underlying DG method (with full polynomial basis) and to a standard Trefftz basis, whenever available. We provide the scripts used to obtain the numerical results presented in this section 2 and interactive code examples to try out the implementation online without any prerequisites 3 . In Preliminaries and Notation. Although there exist many DG schemes for each of the problems considered here, in order to focus on the comparison of the different bases, we will stick to one DG scheme per equation. Further, we concentrated on simplicial meshes here, but the approach carries directly over to hexahedral and even polygonal meshes immediately. If not explicitly mentioned we do not exploit the possibility to remove volume integrals, cf. Remark 5. In the following numerical experiments we will always use sparse direct solvers, i.e. umfpack [10] for non-symmetric and a sparsecholesky solver implemented in NGSolve for symmetric linear systems. In all examples, for the embedded Trefftz DG method we used the local SVD to compute the numerical kernel and particular soluations. For the purpose of brevity in the plots below we label the standard DG method with DGand the corresponding Trefftz method with T p . For the description of the DG schemes in the following we introduce some standard DG notation. We denote by F h the set of facets and distinguish F int h , the set of interior facets, from F bnd h , the set of boundary facets. Let K and K be two neighboring elements sharing a facet F ∈ F int h . On F the functions u K and u K denote the two limits of a discrete function from the different sides of the element interfaces. n K and n K are the unit outer normals to K and K . We define and consider a symmetric IP-DG discretization, cf. [2], given by [[v]] := v K · n K + v K · n K , { {v} } := 1 2 v K + 1 2 v K .a h (u, v) = Ω ∇u∇v dV − F int h { {∇u} }[[v]] + { {∇v} }[[u]] − αp 2 h [[u]][[v]] dS − F bnd h n x · ∇uv + n x · ∇vu − αp 2 h uv dS (v) = F bnd h αp 2 h gv − n x · ∇vg dS.(17) As interior penalty parameter we choose α = 4. We apply our method as described in Section 2.2 with L = −∆. The (embedded) Trefftz DG space, as in (3), is now the space of harmonic polynomials. For the numerical example we set the boundary condition g such that the exact solution is given by u = exp(x) sin(y) on Ω = (0, 1) 2 ,(18)u = exp(x + y) sin( √ 2z) on Ω = (0, 1) 3 ,(19) and consider nested simplicial unstructured meshes created by refining a coarse simplicial unstructured mesh of initial mesh size h ≈ 0.5. In Figure 1 we observe exponential convergence in terms of p and ndofs. In [20] exponential convergence for harmonic polynomials in terms of ndofs is seen to be superior to standard polynomials, this can be seen in the center of Figure 1. We measure the error in the DG-norm given by w DG = K∈T h ∇w L 2 (K) + E∈∂T h αp 2 h −1 [[w]] L 2 (E) . In Figure 1 we also compare the condition number of the different system matrices. As expected from Lemma 1, the conditioning of the embedded Trefftz method is bounded by the conditioning of the full system, and even outperforms it. The very basic implementation of the harmonic polynomials that we used here appears very ill-conditioned. We show plots of the runtime in Figures 2 and 3 for the two and three dimensional example, respectively. In Figure 2 it can be seen that the embedded Trefftz DG method benefits greatly from parallelization. The runtime is broken into parts of assembling the linear system and solving the linear system in Figures 2 and 3. For the embedded Trefftz DG method we plot also the time spent finding the local kernels of the operator using SVD, as described in Section 3.4. Note that in the case of the embedded Trefftz DG method the solving step includes the matrix multiplications needed in (7), therefore the time spent is not equal to that of the standard Trefftz method. In 2d performing the SVD sequentially is time consuming, as shown in Figure 2, a QR decomposition could improve the performance. However, in 3d, see Figure 3, the time spent on the SVD is completely negligible compared to the solver costs. It is possible to automatically find the dimension of the Trefftz space when computing the null space of the local operator. To account for numerical errors in the computations of the singular values we choose a threshold of ε = 10 −7 , identifying values below as zero singular values. In Figure 2 we plot the largest singular value that still needs to be identified as a zero singular value, as well as the smallest non-zero singular value. Note that prior knowledge of the dimension of the Trefftz space, which we do not exploit here in the implementation, would eliminate this problem completely, cf. Section 3.2. In [32,33] convergence rates in h for a mixed formulation over harmonic polynomials are shown. The results in Figure 3 show that we recover the expected convergence rate of u − u h L 2 (Ω) = O(h min(m,p)+1 ) for u ∈ H m (Ω). Poisson equation. Now, we consider the Poisson equation with Dirichlet boundary conditions ∆u = f in Ω, u = g on ∂Ω. We can use the symmetric IP-DG discretization given in (17) with the new right hand side (v) = Ω f v dV + F bnd h αp 2 h gv − n x · ∇vg dS.(20) For the numerical example we set the right hand side and the boundary conditions such that the exact solution is given by u = sin(x) sin(y) sin(z) on Ω = (0, 1) 3 .(21) To apply the embedded Trefftz method we use the approach described in Section 2.4 to find a particular solution and homogenize the system. Results in terms of accuracy and computing time are shown in Figure 4. The embedded Trefftz DG method matches the convergence rates of the DG scheme using the standard polynomial space. We consider the runtime for fixed mesh sizes and varying polynomial degree from p = 1, . . . , 7. As discussed in Section 3.6 we expect worse performance for sparse meshes and large polyomial degree, due to the increased cost of computing the local kernel of the operator. We can observe this on the mesh of the unit cube with mesh size h = 0.5, where the embedded Trefftz method does not show significant improvement of the runtime. The runtime on the fine mesh with h = 0.125 is considerably improved even for rather large values of p. Poisson equation with varying coefficient. We consider the Poisson equation with varying coefficients and Dirichlet boundary conditions, given by ∇ · (M∇u) = f in Ω, u = g on ∂Ω. We can use the symmetric IP-DG discretization given in (17) and right hand side (20), by inserting M at the appropriate locations. For the numerical example we fix the right hand side f and the boundary conditions such that the exact solution is given by u = sin(x) sin(y) with M = 1 + x 0 0 1 + y , and Ω = (0, 1) 2 . To apply the embedded Trefftz method we apply again use the approach detailed in Section 2.4 to homogenize the system. Due to the varying coefficient we cannot find a standard Trefftz space, thus we follow the approach details in Section 2.3, embedding a weak Trefftz space. To construct the Trefftz embedding we apply Eqs. (10) and (11), resulting in (W) ij = −∇ · (M∇φ j ), ψ i , ψ i ∈ V q (T h ).(23) We compare different choices for the polynomial degree of the space V q , with q = p−1, p−2, p−3. Note that the considerations in Section 2.3 imply that the optimal order is given by q = p − 2 since ∆ : V p → V p−2 . The results presented in Figure 5 show the expected behavior. Testing with an increased number of test functions, using q = p−1, leads to an embedding of a smaller subspace than the actual weak Trefftz space. This results in a loss of the good approximation properties, as we see in Figure 5 on the left. Using a smaller test space V q with q = p − 3, is less efficient since the approximation space holds more degrees of freedom, as shown in Figure 5 on the right. While the error for the choices q = p − 2 and q = p − 3 shows slight differences, most likely due to different conditioning, the behavior is the same, as predicted by Lemma 9. Specifically, the different computation of the particular solution does not affect the approximation. Recall that the particular solutio satisfies ΠLu h,f = Πf , i.e. Lu h,f , v = f, v ∀v ∈ V q , as described in Section 2.4. Hence, for the choice of q = p − 3, the approximation of the particular solution might be worse than for the choice q = p − 2, however, the loss in accuracy is made up for by the larger Trefftz space used after homogenization. 4.5. Acoustic wave equation with piecewise constant wave speed. Next, we consider the first order wave equation in a space-time setting, given by          ∇ · σ + c −2 ∂v ∂t = 0 in Ω × [0, T ], ∇v + ∂σ ∂t = 0 in Ω × [0, T ], v(·, 0) = v 0 , σ(·, 0) = σ 0 on Ω × {0}, v = g D on ∂Ω × [0, T ],(24) where (σ, v) are the unknowns -acoustic speed and pressure, (σ 0 , v 0 ) are the initial conditions, g D is Dirichlet boundary data, and T is the final time. In this section we will assume that the wavespeed c is piecewise constant. To apply our framework we take L = ∇· c −2 ∂ ∂t ∇ ∂ ∂t u = σ v The local space-time Trefftz space was introduced and analyzed in [46], and is given by T p (K) = (w, τ ) ∈ P p (K) n+1 ∇w + ∂ t τ = 0 ∇ · τ + c −2 ∂ t w = 0(25) We consider the space-time DG-scheme used in [4,24,46,50]: Find (v hp , σ hp ) ∈ (V p (T h )) d+1 s.t. a h (v hp , σ hp ; w, τ ) = (w, τ ) ∀(w, τ ) ∈ (V p (T h )) d+1 ,(26) with a h (v hp , σ hp ; w, τ ) = − K∈F h K v hp ∇ · τ + c −2 ∂ t w + σ hp · (∂ t τ + ∇w) dV (27) + F space h c −2 v − hp [[w]] t + σ − hp · [[τ ]] t + v − hp [[τ ]] N + σ − hp · [[w]] N dS + F time h ({ {v hp } }[[τ ]] N + { {σ hp } } · [[w]] N + α[[v hp ]] N · [[w]] N + β[[σ hp ]] N [[τ ]] N ) dS + F T h c −2 v hp w + σ hp · τ dS + F D h (σ · n x Ω + αv hp )w dS(28) and (w, τ ) = F 0 h c −2 v 0 w + σ · τ dS + F D h g D (αw − τ · n x Ω ) dS. For the numerical example we set boundary and initial conditions such that the exact solution with wavespeed c = 1 is given by v = √ 2 cos( √ 2t + x + y), σ = (− cos( √ 2t + x + y), − cos( √ 2t + x + y)), Ω = (0, 1) 2 .(29) The penalization parameters in (27) are chosen as α = β = 0.5, as in [50]. The considered spacetime meshes are made up from a simplicial unstructured mesh of the spatial domain and a tensor product mesh in time. The height of the time slabs is chosen approximately as the mesh size of the spatial domain. In Fig. 6 we observe that both Trefftz method maintain the accuracy of the underlying DG method while yielding almost an order of magnitude speed up in the computation time on the finest considered level. Also on the finest level the solver costs seem to dominate so that the embedded Trefftz DG method and the Trefftz method perform equally well. 4.6. Acoustic wave equation in inhomogeneous media. Instead of the piecewise constant case from the previous section, we now consider (24) with wavespeed smoothly varying in space c = c(x). The construction of a basis for (25) with smooth wavespeed is non-viable. A space with similar properties, for which a basis can be constructed, has been introduced in [24] -the quasi-Trefftz space for an element K ∈ T h is given by QT p (K):= (w, τ ) ∈ P p (K) n+1 D i (∇w + ∂ t τ )(x K , t K ) = 0 D i (∇ · τ + c(x) −2 ∂ t w)(x K , t K ) = 0 ∀i ∈ N n+1 0 , \|i\| ≤ p−1 . (30) where we use the notation i = (i x , i t ) = (i x1 , . . . , i xn , i t ) ∈ N n+1 0 for integer non-negative multiindices and D i f := ∂ ix 1 x1 · · · ∂ ix n xn ∂ it t f for the derivatives. The polynomials in the space are constructed such that the Taylor polynomial of their image under the wave operator vanishes at the element center (x K , t K ) up to order p − 2. The conditions of the quasi-Trefftz space are formulated in a way to allow for recursive construction of the basis functions. It is possible to write this space in the way of (9b), see Remark 7. However, this is not the most practical way to implement the embedding. Thus, we proceed with the construction proposed in Eqs. (10) and (11). While an explicit construction of the weak Trefftz space is unfeasible, the embedded Trefftz method is still able to project on such a space. We construct the weak Trefftz embedding as described in Sections 2.3 and 3.3, using the following condition for the weak Trefftz space (27) can be dropped when using the embedded Trefftz DG method, which we also did in the numerical examples in this section, cf. Remark 10. L(σ, v), (τ , w) = 0, ∀(τ , w) ∈ (V p−1 (T h )) d+1 . The volume term in We consider the exact solution given by v = − 2κ(κ − 1)(x + y + 1) κ e − √ 2κ(κ−1)t , σ = −κ(x + y + 1) κ−1 e − √ 2κ(κ−1)t −κ(x + y + 1) κ−1 e − √ 2κ(κ−1)t(31) with κ = 2.5 and wavespeed c(x, y) = x+y+1 on the space-time domain Ω×(0, 1) with Ω = (0, 1) 2 . In Fig. 7 we observe a similar performance as for the homogeneous case. When looking carefully we can now see a difference between Quasi-Trefftz and embedded Trefftz DG method. While in the homogeneous case the embedded Trefftz space coincides with the Trefftz space, in the inhomogeneous case the two spaces do not coincide, compare (9b) and (30). However, the asymptotics seem to be unaffected and all methods perform equally well in terms of accuracy on a given mesh. The timings are again in agreement with the experience from the homogeneous case. Remark 10. The volume term in (27) can be omitted for the case of homogeneous media when using Trefftz or embedded Trefftz DG methods. In the inhomogeneous case, in [24] the quasi-Trefftz methods also require the volume terms for stability, however, even in the case of smooth coefficients, these terms can still be omitted when using embedded Trefftz DG method as described in Section 4.6. Note that in [4,24] there appear additional Galerkin-least squares terms needed for the analysis, however, already in [24] it was observed that they do not seem to play a significant role in the numerical examples, which is why they are neglected here. We consider the DG-scheme used in [7,15,17,41,44] with (bi)linear forms a h (u, v) = K∈T h K ∇u∇v − ω 2 uv dV − F int h { {∇u} }[[v]] + [[u]]{ {∇v} } dS + F int h iαω[[u]][[v]] − β iω [[∇u]][[∇v]] dS − F bnd h δ n x · ∇uv + un x · ∇v dS (32a) + F bnd h i(1 − δ)ωuv − δ iω ∇u∇v dS (v) = F bnd h (1 − δ)gv − δ iω gn x · ∇v dS (32b) with the choices α = β = δ = 0.5 used in [7]. A Trefftz space for the Helmholtz equation in two dimensions is given by the (non-polynomial) space of plane wave functions T p = {e −iω(dj ·x) s.t. j = −p, . . . , p}.(33) The combination of DG (bi)linear forms and this trial and test space is known as plane wave DG method. In the numerical experiment we consider the exact solution given by u = H (1) 0 (ω\|x − x 0 \|), x 0 = (−0.25, 0), Ω = (0, 1) 2 .(34) where H (1) 0 is the zero-th order Hankel function of the first kind. Results are shown in Fig. 8 for ω = 1. Surprisingly, the embedded Trefftz DG solution is extremely close to the plane wave DG method although the one space is piecewise polynomial and the other one is not. While all methods exhibit the same convergence rate, both Trefftz methods yield the significantly smaller error compared to the DG method. This can certainly not be attributed to different approximation spaces as the embedded Trefftz DG space is a subspace of the DG space. Hence, the results suggest that the embedded Trefftz method has better stability properties than the standard DG method, probably comparable to that of the plane wave DG method, cf. also Remark 11. In Fig. 8 on the right, we consider p-convergence on two different meshes. On the finer mesh, with h = 2 −3 , the plane waves fail to converge for p = 5. We have implemented the simplest form of the plane wave basis functions, given by the functions in (33), which are notoriously haunted by ill-conditioning. While more stable constructions for the plane wave basis exist, see for example [44,Sec. 3.4.1], it is interesting to note that the embedded Trefftz space shows improved conditioning without any additional effort. Remark 11 (Approximation of plane wave functions with the embedded Trefftz DG space). The remarkable proximity of both Trefftz DG method in the previous experiment may suggest that the embedded Trefftz space approximates the plane wave DG space. We investigate this in more detail for a one-dimensional example. On [0, 1] we consider the embedded Trefftz space for p = 1, ..., 5. The embedded Trefftz space is only two-dimensional as in 1D dim V p −dim V q = 2 (independent of p). On this space we approximate the two plane wave functions sin(ωx) and cos(ωx) for ω = 2π. The result is shown in Fig. 9 (top row). For comparison we approximate sin(ωx) and cos(ωx) on the (much) larger space V p (T h ) (bottom row). We observe that starting from p = 2 the approximation quality of the embedded Trefftz space is close to the that of the DG space and for p = 5 the plane waves are extremely well resolved rendering the embedded Trefftz space in close proximity to the plane wave space. Linear transport equation. In this section we finally consider an example that is typically not related to Trefftz method: A scalar linear transport problem, the advection equation. It reads as b · ∇u = f in Ω, u = u D on ∂Ω in := {x ∈ ∂Ω \| b · n x < 0}. for a given velocity field b which we assume to be divergence-free. As underlying DG discretization we choose the standard Upwind DG formulation which reads as Here, we used the upwind notationû(x) = lim t→0 + u(x − bt). We do not assume b to be piecewise constant so that there will not be a suitable polynomial Trefftz space in general. To L = b · ∇ we hence choose the weak Trefftz space with V q with q = p − 1.The local Trefftz space on an element K will hence have the dimension M = #P p (K) − #P p−1 (K) which in 1D is M = 1, in 2D is M = p + 1 and in 3D is (p + 1)(p + 2)/2. In comparison to scalar second order problems we hence only have approximately half the degrees of freedoms in the resulting weak Trefftz space. In Fig. 10 we display a set of weak Trefftz basis functions that is obtained in 2D for a non-constant flow field and p = 4. The basis functions are approximately -but not exactly -constant along the flow trajectories. a h (u, v) = K∈T h K −u b · ∇v dV + ∂K\∂Ωin b · n xû v dS (35a) (v) = K∈T h K f v dV − ∂Ωin b · n x u D v dS (35b) For the numerical study we choose Ω = (0, 1) 3 and b = (− sin(x 2 ), cos(x 1 ), x 1 ) T and choose the r.h.s. data f and u D so that u = sin(x 1 ) sin(x 2 ) sin(x 3 ) is the exact solution. Starting from a coarse simplicial and unstructured mesh with mesh size h ≈ 0.5 we apply successive uniform refinements to compare the standard DG and the weak embedded Trefftz DG method. In the left half of Fig. 11 we observe the convergence behavior of the embedded Trefftz DG method compared to the standard Upwind DG method for p = 3, 4, 5. Both methods converge with optimal rate and we observe that there is only a marginal difference between the results. Moreover, a careful look at the numbers reveals that the solution of the embedded Trefftz DG method has a slightly smaller error on few occasions (see for instance the zoomed region in the left plot of Fig. 11). This may appear surprising at first glance as the approximation space is (by construction) smaller (T p (T h ) ⊂ V p (T h ) ), i.e. the approximation quality has not been improved. This suggests that the stability has been slightly improved in these cases. On the right half of Fig. 11 we compare the runtime of the two approaches for p = 3, 4, 5. We observe that on sufficiently fine meshes the costs associated to the two methods separate so that on the finest mesh level the computational costs differ already by more than an order of magnitude. Moreover, we observe that the embedded Trefftz DG method for p = 5 is still much cheaper than the standard DG method for p = 4 and only slightly more expensive than the standard DG method for p = 3, i.e. we can see the step from DG to embedded Trefftz DG as a way to obtain also two orders of accuracy more for the same computation time, at least in the given example. Although linear transport problems are not in the typical class of problems that are associated with Trefftz methods, the obtained results are very encouraging. Compared to the second order equations considered before the reduction of degrees of freedom is even higher and the gain in the computation time is remarkable. Comparison to Hybrid DG We saw in the previous examples that the embedded Trefftz DG approach has two applications: First, it can be seen as an acceleration technique to reduce the costs of DG methods when solving linear systems. Second, the restriction of the DG space to a suitable subset can also have an effect on the stability of the method, cf. the Helmholtz problem in Section 4.7. For the former aspect alternative approaches exists. One technique that became very popular in the last decade is the class of Hybrid DG methods, see [8,9,13]. In the remainder of this section we want to shed some light on the comparison between the two approaches. To this end, we will first recap the structure of HDG method and make a rough conceptual comparison especially in terms of asymptotic complexity in Section 5.1 and afterwards compare the methods on the scalar examples from Section 4.3 and Section 4.8 and close the section with a rough conclusion. 5.1. Introduction. In Hybrid DG methods additional unknowns are introduced on the element interfaces in order to allow for a decoupling of element unknowns. Element unknowns can then be removed from global linear systems by static condensation reducing the globally coupled ndofs (asymptotically for increasing p) from O(p d ) to O(p d−1 ) and thereby reducing the computational costs dramatically. Let us sketch the structure of an HDG method. To this end, we restrict to a scalar PDE here, but extensions to the vector case are obvious. Let F p (F h ) be the space of piecewise polynomials up to degree p on each facet of the mesh T h and F p D (F h ) and F p 0 (F h ) its subspaces with prescribed (inhomogeneous and homogeneous) values on Dirichlet-type boundaries. Then a typical discrete variational formulation of a Hybrid DG method is formulated in terms of the pair of volume and facet unknowns in the form: Find (u, u F ) ∈ V p (T h ) × F p D (F h ) so that a h ((u, u F ), (u, v F )) = ((v, v F )) for all (v, v F ) ∈ V p (T h ) × F p 0 (F h ).(36) Here the bilinear form a h (·, ·) can be very similar to a corresponding DG formulation with respect to the integral terms. However, direct couplings between volume unknowns of adjacent elements are avoided by involving the facet unknowns for the inter-element communication, so that the set of volume unknowns of one element is completely determined by the facet unknowns on the element boundary (and the r.h.s. data). This enables one of the main features of Hybrid DG schemes: the possibility to do static condensation. This allows to eliminate the interior unknowns in V p (T h ) completely based on the facet unknowns in F p (F h ). This can be done on the level of the variational formulation where with u = u(u F ) one can formulate a discrete formulation solely based on u F of the form: Find u F ∈ F p D (F h ) so that a h (u F , v F ) = (v F ) for all v F ∈ F p 0 (F h ). After solving for u F element-local problems can be solved to re-obtain u(u F ) ∈ V p (T h ). This elimination is often done merely on the linear algebra level based on a Schur complement strategy. We note that in many Hybrid DG formulations for second order PDEs an auxiliary (flux) variable is also introduced locally which however can be eliminated alongside u so that the structure of the global linear system for u F is not affected by this. The asymptotic complexity of the Hybrid DG approach is the same as the one of the Embedded Trefftz DG method. The total ndofs is determined by the element unknowns, i.e. O(h −d p d ), the globally coupled ndofs however only scales with O(h −d p d−1 ). Similarly the local operations, i.e. the SVD or QR decomposition for the Embedded Trefftz or the Schur complement strategy require O(h −d p 3d ) operations. Correspondingly, nnzesand arithmetic operations for general purpose linear algebra solvers have the same complexity, cf. Section 3.6. We summarize that the Hybrid DG and the embedded Trefftz methods achieve the same asymptotic complexity, however the global unknowns are associated with element interfaces in the Hybrid DG case and associated with volume elements in the case of embedded Trefftz methods. In Fig. 12 Figure 12. Sketch of global/local dofs of Hybrid DG, standard DG and Embedded Trefftz DG for 2D simplex elements and polynomials degree p = 3. Here q diff is the maximum differentiation index of L. In the first part of Section 3.3 we have discussed two possible implementations of the embedded Trefftz-DG method. For the timings in this section we use the faster method, which avoids the assembly of the full DG matrix. Poisson equation. To compare the Hybrid DG method with the Embedded Trefftz DG method on a concrete example, we reconsider the example from Section 4.3 and provide a simple hybrid DG discretization for it. A hybrid DG analogue to the interior penalty formulation takes the form of (36) with a h ((u, u F ), (u, v F )) = K∈T h K ∇u∇v dV − ∂K ∇u · n K (u − u F ) + ∇v · n K (v − v F ) dS + ∂K αp 2 h (u − u F )(v − v F ) dS ((v, v F )) = Ω f v dV(37) Note that there are no average and jump operators in (37) that directly involve neighboring element functions. Further, Dirichlet-type boundary conditions are directly imposed on the facet space which leads to the usual weak imposition of them on the element unknowns. As for the DG formulation chosen in Section 4.3 let us also mention that the Hybrid DG formulation is only one specific version (and a specifically simple one). For second order elliptic and diffusion dominated problems, based on the similarity to (hybrid) mixed methods, element-local postprocessing schemes can be devised to obtain an additional order of accuracy for a postprocessed solution field. We neglect this aspect here but refer to [8,30,31,48] for more details. In Fig. 13 we compare the computational results for the three methods, DG, Hybrid DG and Embedded Trefftz DG for fixed polynomial degrees and successively refined meshes. We observe that the differences in the error is negligible whereas the computation time of the Hybrid DG method is still smaller than the one of the Embedded Trefftz DG method. Comparing the overall computation time with the time spend only to solve the global linear system we observe that this makes up for most of the time. Linear transport equation. Let us now come to the first order example, the linear transport equation as in Section 4.8. A Hybrid DG version of (35) takes the form of (36) with a h ((u, u F ), (u, v F )) = K∈T h K −ub · ∇v dV + ∂K b · n xû v dS + ∂Kout b · n x (u F − u)v F dS ((v, v F )) = Ω f v dV(38) Here the upwind choiceû is as follows: On inflow boundaries (b · n x < 0) we setû = u F , i.e. the facet value is taken whereas on outflow boundaries (b · n x > 0) we setû = u. To ensure that the facet value taken on inflow boundaries is meaningful, it is glued together with the outflow trace of the corresponding adjacent element by the stabilization term that include v F . Note that testing with v = 0 yields that u F is exactly the upwind trace at a corresponding facet. Hence, the hybrid Upwind DG formulation is equivalent to the Upwind DG formulation used above, cf. [13]. Note that inflow boundary conditions are prescribed through the facet space here and do not appear in the linear form. In Fig. 14 we again compare DG, Hybrid DG and Embedded Trefftz DG methods. Again, the errors are comparable. In fact, DG and Hybrid DG are equivalent and hence yield exactly the same result. For the computation time we now observe that the Embedded Trefftz DG method is the fastest solution method as soon as the overhead in the system setup becomes negligible. 5.4. Some number crunching. In the previous two sections we considered first and second order scalar PDEs. Next, we would like to compare the sparsity patterns of DG, HDG and Trefftz DG for a generic setting: We consider an unstructured simplicial mesh with 54 elements in 2D and 729 elements in 3D and compare different methods w.r.t. the number of degrees of freedom (ndofs) and the number of non-zero entries (nnzes) for p = 0, .., 5. We note that although ndofs is the simpler measure nnzes has a more direct implication on the computational costs that are associated to solving linear systems. In Table 1 A standard DG scheme, a corresponding HDG scheme and two embedded Trefftz DG schemes are compared. Here the embedded Trefftz DG schemes denoted as TDG(1) and TDG(2) are distinguished depending on the leading order Table 1. Comparison of different measures (ndofs, nnzes) for the computational overhead for the solution of the largest global linear system related to the methods DG, Hybrid DG (after static condenstaion) and (embedded) Trefftz DG. of the differential operator involved. For first order problems a comparison of DG and HDG with TDG(1) is of interest, whereas TDG(2) is to be considered for comparison for second order formulations. For simplicity, we assume here that all element and facet unknowns of one element or facet couple with all unknowns of a corresponding facet or element (depending on the method) neighbor. We observe that for first order problems, the embedded Trefftz DG method even outperforms the HDG method. For second order problems HDG and embedded Trefftz DG methods are similar, in 2D HDG is cheaper, in 3D the embedded Trefftz DG method. However, optimized HDG formulations are able to reduce the polynomial degree on the element interfaces by one order in many cases without effecting the order of accuracy. In these cases the HDG method will still be cheaper. However, compared to the plain DG method the improvement of the embedded Trefftz DG method is already remarkable. Conclusion and possible extensions We have presented a method to reduce the matrix of a DG scheme post assembly, by projecting the polynomial basis onto Trefftz spaces. Several numerical examples have been presented, showing that the method matches the convergence rates of polynomial Trefftz spaces and showcasing its flexibility and potential in case of smooth coefficients and inhomogeneous equations. While the Trefftz spaces used in the polynomial case are well understood, analytical properties of the space used for the embedding in the case of smooth coefficients and plane wave Trefftz spaces will be addressed in a forthcoming paper. One remarkable finding are the convincing results that have been obtained for the linear transport problem, a problem that is typically not associated with Trefftz methods. Further, we saw that the method is more than an acceleration technique, as the restriction to a subspace can even improve stability as seen for the Helmholtz problem. The flexibility of the approach suggests to consider the approach in many more and possibly more complex cases. We focused on using the local operator and its kernel to emulate Trefftz spaces. However, the technique could also be applied to impose other constraints on the solution as an alternative to complicated constructions of special basis functions, weak impositions through the DG formulation or Lagrange multiplier techniques. Furthermore, the extraction done here for the kernel could similarly be applied for the range of certain differential operators or orthogonal (w.r.t. to a localizable inner product) complements. One obvious drawback is the confinement to DG schemes. It presents an alternative to other approaches that accelerate DG methods, such as HDG. However, using the method as a pure acceleration technique for DG methods may be less attractive, if sophisticated preconditioners and linear solvers for the DG discretization are available. Moreover, the scaling of the computing time of the embedding construction in p is not good, suggesting that really high polynomial degrees, e.g. beyond p = 10, may not be feasible. In these cases however, the method seems to at least offer a very convenient way to investigate Trefftz and Trefftz-type methods as a research tool. Another difficulty is, that combining the approach with (partially) conforming methods (such as those based on H(div) or H(curl) spaces) is not directly possible, due to the support of basis functions spanning multiple elements and overlapping partially the construction of a local embedding unclear. Additionally, the application of the PDE operator in strong form is only feasible element-wise. Sections 4.2 and 4.5 we consider the Laplace equation and the acoustic wave equation, where explicit polynomial Trefftz basis functions are available for comparison. For the Helmholtz equation, treated in Section 4.7, an explicit Trefftz basis is available, however it is not polynomial. In Section 4.3 we consider Laplace equation with a source term, applying the techniques described in Section 2.4. In Sections 4.6 and 4.8 we consider PDEs with non-constant coefficients. In Section 5 we present a comparison to HDG methods in terms of computational costs associated to the linear solvers. 2 for reproduction material see https://doi.org/10.25625/JIO1MP 3 find the documentation at https://github.com/PaulSt/NGSTrefftz On the boundary facets we set [[v]] = v K n x and { {v} } = v where n x denotes the (spatial) outer normal on the boundary. 4.2. Laplace equation. We start with the Laplace equation with Dirichlet boundary conditions −∆u = 0 in Ω, u = g on ∂Ω, Figure 1 . 1Results for the Laplace problem in 2 dimensions with exact solution (18) on a fixed mesh with h = 0.25 for different values of polynomial order p. Left: Convergence in terms of polynomial order p. Center: Convergence in terms of degrees of freedom of the linear system. Right: Condition number of the discrete system. Figure 2 . 2Results for the Laplace problem in 2 dimensions. Left: Comparison of the runtime on 4 threads (solid lines) and on 8 threads (dashed lines). Center: Comparison of timings for the different steps of each method, for p = 5 on a fixed mesh with h = 2 −4 . From left to right the bars correspond to 1,4,8, and 12 threads. Right: Comparison of the singular values obtained when determining the null space of the operator for p = 3, 6, 10 (full, dashed, dash-dotted line). Figure 3 . 3Results for the Laplace problem in 3 dimensions with exact solution(19). On the left: h-convergence for p = 2, 3, 4. On the right: Comparison of timings for the different steps of each method, for p = 5 on a fixed mesh with h = 2 −3 . The bars from left to right correspond to computations using 4, 8, 12 threads for each method. Figure 4 . 4Numerical results for Poisson equation in 3 dimensions for meshes h = 0.5, 0.25, 0.125 (solid, dashed, dotted dashed line, respectively). Left: pconvergence to the exact solution (21). Right: Timings on 12 threads. . Trefftz q = p − 1 emb. Trefftz q = p − 2 emb. Trefftz q = p − 3 Figure 5 . 5Numerical results for the Poisson equation with varying coefficient comparing different choices of q in (23). Results are obtained in 2 dimensions for meshes h = 1, 0.5, 0.25, 0.125. Left: h-convergence for p = 6 to the exact solution (22). Right: Comparison of the global number of degrees of freedom (ndofs) for different polynomial degrees on a fixed mesh with h = 0.125. Figure 6 . 6Numerical results for the wave equation in 2+1 dimensions. On the left: h-convergence in the space-time L 2 -error for p = 2, 3, 4 corresponding to full, dashed and dash-dotted line. Convergence is given with respect to exact solution(29). On the right: timings for p = 4 on 24 threads. Figure 7 . 7Numerical results for the wave equation with smooth coefficient in 2+1 dimensions, the exact solution given by(31). On the left: h-convergence for p = 2, 3, 4 corresponding to full, dashed and dash-dotted line. On the right: timings for p = 4 on 24 threads. 4. 7 . 7Helmholtz equation. We now switch to the time-harmonic case of wave equations and consider the Helmholtz equation with Robin boundary conditions −∆u − ω 2 u = 0 in Ω, ∂u ∂nx + iu = g on ∂Ω. Figure 8 . 8Numerical results for Helmholtz equation for the exact solution given in(34). On the left: h-convergence for p = 2, 3 corresponding to the full and dashed line. On the right: p-convergence for h = 2 −2 , 2 −3 corresponding to the full and dashed line, respectively. Figure 9 . 9Approximation of the real part of plane wave basis functions (dashed line) in 1 dimension by the embedded Trefftz basis in the top row and by full polynomial space in the bottom row (solid line) , for p = 1, . . . , 5. Figure 10 . 10(Non-constant) Flow field (left) and shape functions of the weak Trefftz space for p = 4 for a triangle and L = b · ∇ as considered in Section 4.8. Figure 11 . 11Comparison of L 2 -error (left) and timings (right) for standard DG method and embedded Trefftz DG method for p = 3, 4, 5 in 3D using 24 threads. Figure 13 . 13Comparison between DG, Hybrid DG and embedded Trefftz DG method for the Poisson problem from Section 4.3 in 3D. Left: L 2 -error for p ∈ {2, 3, 4} over successively refined meshes, center: computation time for p = 4 with 12 threads, right: computation time for sparse matrix factorization p = 4 with 12 threads. Figure 14 . 14Comparison between DG, Hybrid DG and embedded Trefftz DG method for the linear transport problem from Section 4.8 in 3D. Left: L 2 -error for p ∈ {3, 4, 5} over successively refined meshes, center: computation time for p = 5 with 12 threads, right: computation time for sparse matrix factorization p = 5 with 12 threads. we sketch the different dofs involved in the different methods for p = 3.: global dof : local dof : removed dof : removed dof if q diff = 1, global dof else. Hybrid DG standard DG Emb. Trefftz DG see https://ngsolve.org/ and https://github.com/PaulSt/NGSTrefftz. The method of solving polynomials in the beam vibration problem. M J Al-Khatib, K Grysa, A Maciąg, J. Theoret. Appl. Mech. 46M. J. Al-Khatib, K. Grysa, and A. Maciąg, The method of solving polynomials in the beam vibration problem, J. Theoret. Appl. Mech., 46 (2008), pp. 347-366. Unified analysis of discontinuous Galerkin methods for elliptic problems. 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J Yang, M Potier-Ferry, K Akpama, H Hu, Y Koutsawa, H Tian, D S Zézé, Arch. Comput. Methods Eng. 27J. Yang, M. Potier-Ferry, K. Akpama, H. Hu, Y. Koutsawa, H. Tian, and D. S. Zézé, Trefftz methods and Taylor series, Arch. Comput. Methods Eng., 27 (2020), pp. 673-690.	[ "https://github.com/PaulSt/NGSTrefftz", "https://github.com/PaulSt/NGSTrefftz." ]
[ "Neural-Symbolic Solver for Math Word Problems with Auxiliary Tasks", "Neural-Symbolic Solver for Math Word Problems with Auxiliary Tasks" ]	[ "Jinghui Qin [email protected] \nSun Yat-sen University\n\n", "Xiaodan Liang \nSun Yat-sen University\n\n\nDark Matter AI Inc\n\n", "Yining Hong [email protected] \nUniversity of California\nLos Angeles\n", "Jianheng Tang \nSun Yat-sen University\n\n", "Liang Lin [email protected] \nSun Yat-sen University\n\n\nDark Matter AI Inc\n\n" ]	[ "Sun Yat-sen University\n", "Sun Yat-sen University\n", "Dark Matter AI Inc\n", "University of California\nLos Angeles", "Sun Yat-sen University\n", "Sun Yat-sen University\n", "Dark Matter AI Inc\n" ]	[ "Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing" ]	Previous math word problem solvers following the encoder-decoder paradigm fail to explicitly incorporate essential math symbolic constraints, leading to unexplainable and unreasonable predictions. Herein, we propose Neural-Symbolic Solver (NS-Solver) to explicitly and seamlessly incorporate different levels of symbolic constraints by auxiliary tasks. Our NS-Solver consists of a problem reader to encode problems, a programmer to generate symbolic equations, and a symbolic executor to obtain answers. Along with target expression supervision, our solver is also optimized via 4 new auxiliary objectives to enforce different symbolic reasoning: a) self-supervised number prediction task predicting both number quantity and number locations; b) commonsense constant prediction task predicting what prior knowledge (e.g. how many legs a chicken has) is required; c) program consistency checker computing the semantic loss between predicted equation and target equation to ensure reasonable equation mapping; d) duality exploiting task exploiting the quasi duality between symbolic equation generation and problem's part-of-speech generation to enhance the understanding ability of a solver. Besides, to provide a more realistic and challenging benchmark for developing a universal and scalable solver, we also construct a new largescale MWP benchmark CM17K consisting of 4 kinds of MWPs (arithmetic, one-unknown linear, one-unknown non-linear, equation set) with more than 17K samples. Extensive experiments on Math23K and our CM17k demonstrate the superiority of our NS-Solver compared to state-of-the-art methods 1 . * Corresponding Author 1 The code and the new CM17k dataset are available at https://github.com/QinJinghui/NS-Solver.	10.18653/v1/2021.acl-long.456	[ "https://www.aclanthology.org/2021.acl-long.456.pdf" ]	235,731,777	2107.01431	50194e59ca170938c7022388cbf0319322763dd2	Neural-Symbolic Solver for Math Word Problems with Auxiliary Tasks August 1-6, 2021 Jinghui Qin [email protected] Sun Yat-sen University Xiaodan Liang Sun Yat-sen University Dark Matter AI Inc Yining Hong [email protected] University of California Los Angeles Jianheng Tang Sun Yat-sen University Liang Lin [email protected] Sun Yat-sen University Dark Matter AI Inc Neural-Symbolic Solver for Math Word Problems with Auxiliary Tasks Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language ProcessingAugust 1-6, 20215870 Previous math word problem solvers following the encoder-decoder paradigm fail to explicitly incorporate essential math symbolic constraints, leading to unexplainable and unreasonable predictions. Herein, we propose Neural-Symbolic Solver (NS-Solver) to explicitly and seamlessly incorporate different levels of symbolic constraints by auxiliary tasks. Our NS-Solver consists of a problem reader to encode problems, a programmer to generate symbolic equations, and a symbolic executor to obtain answers. Along with target expression supervision, our solver is also optimized via 4 new auxiliary objectives to enforce different symbolic reasoning: a) self-supervised number prediction task predicting both number quantity and number locations; b) commonsense constant prediction task predicting what prior knowledge (e.g. how many legs a chicken has) is required; c) program consistency checker computing the semantic loss between predicted equation and target equation to ensure reasonable equation mapping; d) duality exploiting task exploiting the quasi duality between symbolic equation generation and problem's part-of-speech generation to enhance the understanding ability of a solver. Besides, to provide a more realistic and challenging benchmark for developing a universal and scalable solver, we also construct a new largescale MWP benchmark CM17K consisting of 4 kinds of MWPs (arithmetic, one-unknown linear, one-unknown non-linear, equation set) with more than 17K samples. Extensive experiments on Math23K and our CM17k demonstrate the superiority of our NS-Solver compared to state-of-the-art methods 1 . * Corresponding Author 1 The code and the new CM17k dataset are available at https://github.com/QinJinghui/NS-Solver. Introduction Deep neural networks have achieved remarkable successes in natural language processing recently. Although neural models have demonstrated performance superior to humans on some tasks, e.g. reading comprehension (Rajpurkar et al., 2016;Devlin et al., 2019;Lan et al.), it still lacks the ability of discrete reasoning, resulting in low accuracy on math reasoning. Thus, it is hard for pure neural network approaches to tackle the task of solving math word problems (MWPs), which requires a model to be capable of natural language understanding and discrete reasoning. MWP solving aims to automatically answer a math word problem by understanding the textual description of the problem and reasoning out the underlying answer. A typical MWP is a short story that describes a partial state of the world and poses a question about an unknown quantity or multiple unknown quantities. To solve an MWP, the relevant quantities need to be identified from the text. Furthermore, the correct operators along with their computation order among these quantities need to be determined. Therefore, integrating neural networks with symbolic reasoning is crucial for solving MWPs. Inspired by the recent amazing progress on neural semantic parsing (Liang et al., 2017a) and reading comprehension , we address this problem by neural-symbolic computing. Recently, many researchers Huang et al., 2018;Wang et al., 2018bWang et al., , 2019Xie and Sun, 2019;Chiang and Chen, 2019), inspired by an encoder-decoder framework (Cho et al., 2014), apply neural networks to solve MWPs by learning the mapping function between problems and their corresponding equations, and achieve remarkable successes. The encoder uses a neural network to represent a problem as a real-valued vector, and the decoder uses another neural network to generate an equation or expression token by token. The main difference among previous methods is the way to decode expressions or equations. However, they only follow the encoder-decoder paradigm but lacking the ability to explicitly incorporate essential math symbolic constraints (e.g. commonsense constants, formulation regularization), leading to unexplainable and unreasonable predictions. Besides, most of them only focus on arithmetic MWPs without any unknown, preventing them from generalizing to various types of MWPs, such as equation set problems. To address the above issues, we propose a novel Neural-Symbolic Solver (NS-Solver), which explicitly and seamlessly incorporates different levels of symbolic constraints by auxiliary learning tasks. Our NS-Solver consists of three main components, a problem reader to encode the math word problems into vector representations, a programmer to generate the symbolic grounded equations, which are executed to produce answers, and a symbolic executor to obtain final results. In addition to the supervised training objective between generated symbolic grounded equations and groundtruth equations, our solver is also optimized by four novel auxiliary objectives that enforce four levels of problem understanding and symbolic reasoning. First, we apply number prediction task to predict both the number quantity and number location in the problem in a self-supervised manner. Second, we deploy commonsense constant prediction task to predict what prior commonsense knowledge (e.g. how many legs a chicken has) is required for our solver. Third, we propose program consistency checker to compute the semantic loss between the predicted program and ground-truth equation to ensure reasonable equation mapping. Finally, we also propose a novel duality exploiting task that exploits the quasi duality between symbolic grounded equation generation and the problem's part-of-speech generation to enhance the understanding ability of our solver. There are some key advantages of our solution. First of all, the above four auxiliary tasks can produce additional training signals, which improves the data efficiency in training and makes our solver more robust. Second, using the predicted constant to constrain the target symbolic table can reduce the search space greatly, which means that our solver can generate correct symbolic grounded equations easier and better. Third, the auxiliary tasks have been proven to help reduce the domain gap between seen and unseen MWPs (Sun et al., , 2020, thus improving the reasoning ability of our solver. Besides, beyond the current large-scale highquality MWP benchmark that only includes one type of problems, we also construct a large-scale challenging Chinese MWPs dataset CM17K, which contains 4 types of MWPs (arithmetic MWPs, oneunknown linear MWPs, one-unknown non-linear MWPs, equation set problems) with more than 17K samples, to provide a more realistic and challenging benchmark for developing a universal and scalable math solver. Extensive experiments on public Math23K and our proposed CM17k demonstrate the superiority of our NS-Solver compared to stateof-the-art methods in predicting final results while ensuring intermediate equation rationality. Related Work Deep learning-based MWP Solvers. Numerous methods have been proposed to tackle the MWP solving task, ranging from rule-based methods (Bakman, 2007;Yuhui et al., 2010), statistical machine learning methods (Kushman et al., 2014;Zhou et al., 2015;Roth, 2015, 2016;Mitra and Baral, 2016;Huang et al., 2016;Roy and Roth, 2018), semantic parsing methods (Shi et al., 2015;Koncelkedziorski et al., 2015;Huang et al., 2017;Liang et al., 2018a), to deep learning methods (Ling et al., 2017;Wang et al., 2017Wang et al., , 2018bHuang et al., 2018;Wang et al., 2018a;Xie and Sun, 2019;Wang et al., 2019;Zhang et al., 2020a,b;Shen and Jin, 2020;Wu et al., 2020;Chen et al., 2021;Hong et al., 2021a,b). However, most deep learning-based methods only follow the encoder-decoder framework without explicitly incorporating essential math symbolic constraints, resulting in some unexplainable and unreasonable predictions. Besides, most of them only focus on arithmetic MWPs, preventing them from generalizing to various types, such as equation set problems. Neural-Symbolic Computing. Neural-symbolic computing has greatly promoted the development of semantic parsing. Jia and Liang (2016); Dong and Lapata (2016); Zhong et al. (2017) applied neural sequence-to-sequence and sequence-to-tree models to semantic parsing with full supervision. Liang et al. (2017bLiang et al. ( , 2018b have advanced the stateof-the-art in weakly supervised semantic parsing on knowledge graphs and tabular databases. Al- Number Location Prediction Today there are chickens and rabbits … Figure 1: An overview of our NS-Solver. When a problem preprocessed by number mapping and replacement is entered, our problem reader encodes the problem text into context representation. Then our programmer generates a tree-structured symbolic grounded program explicitly. Finally, a symbolic grounded program will be executed to produce answers by the executor. In our NS-Solver, we apply four auxiliary tasks to enhance its problem understanding and symbol reasoning ability for generating better programs. though most of the successes of semantic parsing are limited to structured data sources, it is not expensive for MWPs since it is easy to crawl lots of problems with annotated equations and answers. Therefore, MWP solving can benefit from supervised neural-symbolic computing. Self-Supervised Learning. Self-supervised auxiliary tasks have been widely used in the fields of natural language understanding (Devlin et al., 2019;Lan et al.). Devlin et al. (2019) applied two selfsupervised auxiliary tasks, masked LM and next sentence prediction, to improve the understanding ability of BERT by pretraining. ALBERT (Lan et al.) introduces sentence-order prediction task to address the ineffectiveness of the next sentence prediction task in BERT. Hendrycks et al. (2019) show that self-supervised learning can improve model robustness and uncertainty. Dual Learning. Dual learning, first proposed by He et al. (2016), is a reinforcement training process that jointly trains a primal task and its dual task. Then Xia et al. (2017) considered it as a way of supervised learning and designed a probabilistic regularization term to exploit the duality. It has been widely applied in various fields, such as machine translation (He et al., 2016), sentiment classification (Xia et al., 2017), question answering (Tang et al., 2017), visual question answering (Li et al., 2018), machine reading comprehension , and code generation . To the best of our knowledge, we are the first to ex-ploit the duality in MWPs. Different from previous works, we design a quasi dual learning method between symbolic grounded equation generation and problem's part-of-speech generation to enhance the understanding ability by easing the difficulty of generating problems from symbolic equations. Neural-Symbolic Solver In this section, we present the design of the proposed NS-Solver. Its backbone mainly consists of a problem reader that encodes the math word problems into vector representations, a programmer to generate the symbolic grounded programs in prefix order, and a symbolic executor to obtain final results. The overview of our NS-Solver is visualized in Fig. 1. We first introduce the backbone of our NS-Solver in section 3.1, and then we introduce other auxiliary tasks in section 3.2. Backbone Problem Reader. Given a problem text P = {x i } n i=1 processed by number template replacement which maps numeric values in a problem to number templates (e.g., 26 and 82 to n 1 and n 2 in Fig. 1), the problem reader encodes each token x i in the problem text into an embedding e i . In this work, we deploy a two-layer bidirectional GRU to encode each token x i into an embedding e i = − → h i + ← − h i where − → h i and ← − h i are from forward and backward GRUs, respectively. Besides, our prob-lem encoder also outputs a problem representation g 0 = − → h n + ← − h 0 as the initial hidden state of our programmer, where − → h n and ← − h 0 are the last hidden state of forward and backward GRUs, respectively. Programmer. The programmer takes the output of the problem reader as input and the problem representation as the initial hidden state, and then decodes a problem as a sequence of tokens {y i } m i=1 which are organized as a prefix equation tree. In this work, we deploy a tree-structured decoder (Xie and Sun, 2019) with attention mechanism (Bahdanau et al., 2015) as the backbone of our programmer and modify them with UET representation to support more symbols for multiple types of MWPs. In our programmer, the symbolic table consists of four parts. For each problem, the problem-specific symbolic table contains math operators (+, −, * , /,, =, ;), unknown variable (x and y), a series of commonsense constants (1, 3.14, etc) predicted by the Commonsense Constant Prediction Task in 3.2, and the problemspecific number templates (n 1 , n 2 , n 3 , etc). It should be noticed that ; is a special operator with the lowest priority to integrate multiple equation trees as an ensemble equation tree, so that equation set problems can be handled as simple as arithmetic problems. Executor. We deploy sympy 2 , which is a python library for symbolic mathematics, as our symbolic executor for obtaining final results by solving generated equations. The Design of Auxiliary Tasks The MWP solving task remains challenging since previous methods did not take full advantage of the rich semantics contained in a problem and lacking the ability to explicitly incorporate essential math symbolic constraints. In this section, we introduce four auxiliary learning tasks to exploit additional training signals obtained from different tasks and exploit the result of the commonsense constant prediction task to explicitly constrain the constant symbolic table, which can reduce the search space for symbolic generation and ease the difficulty of generating correct constant. Self-supervised Number Prediction (SNP) Tasks. If a solver can fully understand the problem semantics, it should be able to identify the quantity of numbers in a problem (i.e., to count how many numeric values are in the problem) and 2 https://www.sympy.org/ their corresponding locations in the problem text accurately. For example, if the solver can understand the problem in Fig. 1, it should be able to predict there are two numbers(26 and 82) in the problem, and their positions are 15 and 18, respectively. Thus, number quantity prediction and number location prediction are two critical self-supervised tasks to help the problem reader fully understand the problem semantics and measure the ability of problem understanding of a solver. Both two number prediction tasks take the mean of the problem encoder's outputs {e i } n i=1 as their input and apply a single-layer feed-forward neural network to compute the distribution of number quantity and number locations. The training objectives of two tasks for each problem are formulated as: L N QP = − Q i=1 qt i log p (q i \|P ) , L N LP = − L i=1 lt i log p (l i \|P ) . (1) where L N QP and L N LP denote the loss for the Number Quantity Prediction (NQP) task and Number Location Prediction (NLP) task, respectively. Q and L are the maximum possible quantities of number and maximum possible number locations for a problem at the dataset level. qt i and lt i represent the ground-truth value on i-th index of the output probability distribution of NQP and NLP, respectively. Commonsense Constant Prediction (CCP) Task. Commonsense constants are important for solving some MWPs while most previous methods only consider the constants 1 and 3.14, which are not enough for a solver to solve problems that need other commonsense constants. However, attaching a lot of constants to the problem-specific symbolic table will enlarge the search space, increasing the difficulty of generating rational symbolic equations. Therefore, we propose a commonsense constant prediction task to predict what prior commonsense knowledge (e.g. a chicken has 2.0 legs and a rabbit has 4.0 legs for the problem in Fig. 1) is required for the solver to solve a problem according to the problem context. In this way, we can reduce the search space greatly, thus improving the performance of our solver. Similar to the number prediction tasks, the commonsense constant prediction task takes the mean of the problem encoder's output {e i } n i=1 as their input and apply a single-layer feed-forward neural network to compute the distribution of number quantity and number locations The training objective for each problem is formulated as: L CCP = − C i=1 ct j log p (c i \|P ) .(2) where C is the total number of constants in the symbolic table and ct i represents the true value on i-th index of the output probability distribution. Since it is impossible for the commonsense constant prediction task to achieve 100% accuracy, in addition to the predicted constants, we add three extra constants that are not predicted but with the highest probability into the symbolic table, making a better trade-off between the size of the search space and prediction accuracy. Program Consistency Checker (PCC). Although a problem can be solved by multiple equivalent but different equations, the predicted equations should be consistent with label equations as much as possible in the supervised learning setting. Therefore, we propose a program consistency checker to check the symbolic program consistency and regularize the model by computing semantic loss between the predicted symbolic program and ground-truth equation to ensure the reasonable symbolic equation mapping. Letŷ i and y i represent the predicted symbol and ground-truth symbol, p i represents the probability ofŷ i , the semantic loss is obtained by computing a distance between the predicted distribution and ground-truth distribution as: L P CC = −log i ŷ i =y i p i ŷ i =y i (1 − p i ) . (3) Duality Exploiting (DE) Task. Many previous works (He et al., 2016;Xia et al., 2017; have shown promising results by dual learning framework. Although intuitively, MWP solving and MWP generation are related to each other, i.e., the input of MWP solving is the output of MWP generation, and vice versa, it is very hard for the MWP generation task to generate good enough problems only by the equations without any topic information. Therefore, we propose a duality exploiting task to enhance the understanding ability of our solver by exploiting the quasi duality between symbolic grounded equation generation and the problem's part-of-speech generation. Given a pair of a problem and its corresponding equations (P ,T ), and P is the part-ofspeech of P 3 , the training objective of the duality exploiting task is formulated as: L dual = logp(P ) + log p (T \|P ) − logp(T ) − log p P \|T 2 .(4) wherep(P ) andp(T ) are marginal distributions, which can be modeled by their LSTM (Hochreiter and Schmidhuber, 1997)-based language models, respectively. Besides, we deploy a tree-structure encoder inspired by GTS (Xie and Sun, 2019) to encode equations in prefix for POS generation. Training Objective Given the training dataset D={(P i , T 1 ), (P 2 , T 2 ), · · · ,(P N , T N ) }, where T i is the universal expression tree of problem P i , we minimize the following loss function for our NS-Solver: L = (P,T )∈D [L ent1 + λ 1 * L dual + λ 2 * L P CC +λ 3 * (L N QP + L N LP ) + λ 4 * L CCP ] . (5) where L ent1 = − log m t=1 prob(y t \|P )(6) where m denotes the size of T, and y t denotes the t-th output. {λ i } 4 i=1 are empirical values that will be detailed in Section 4.2. For the duality exploiting task, there is another loss for training the branch of the problem's partof-speech generation: L P OS = (P ,T )∈D [L ent2 +λ 5 * L dual +λ 6 * L P CC ]. (7) where L ent2 = − log n t=1 prob(x t \|T )(8) where n denotes the size of P, and x t denotes the t-th output. L P CC is the semantic loss between predicted POS and the ground-truth POS. {λ i } 6 i=5 are empirical values that will also be detailed in Section 4.2. Experiments CM17K Dataset Most public MWPs datasets are quite small such as ALG514 or exist some incorrect labels such as Dolphin18K. An exception is the Math23K dataset, which contains 23161 problems labeled well with structured equations and answers. However, it only contains one-unknown linear math word problems, which is not sufficient to validate the ability of a math solver about solving multiple types of MWPs. Therefore, we introduce a new high-quality math word problems dataset, called CM17K, to validate the universality of a solver and provide a more realistic and challenging benchmark for developing a universal and scalable math solver. We collect CM17K from two education websites 4 . These problems are oriented grades 6-12, containing 4 types of MWPs with more than 17K samples, including 6215 arithmetic MWPs, 5193 one-unknown linear MWPs, 3129 one-unknown non-linear MWPs, and 2498 equation set problems. It should be noticed that our dataset is sufficient for validating the universality of math word problem solvers since these problems can cover most cases about MWPs. We label our data with structured equations and answers following Math23K . We split our CM17K into train/valid/test sets at a ratio of 8:1:1. The data statistics of Math23K and CM17K are shown in Table 1. From the statistics, we can see that all statistics of CM17K are larger than Math23K. This shows that our dataset is more challenging and difficult for math word problem solvers. Besides, since CM17K contains more types of MWPs than Math23K, CM17K is more suitable 4 http://www.zxxk.com/ and http://www.jyeoo.com/ for validating the reasoning ability of a solver than Math23K. Experimental Setup and Training Details Datasets, Baselines, and Metric We conduct experiments on Math23K and our CM17K. The main state-of-the-arts to be compared are as follows: DNS ) is a universal solver based on the seq2seq model with significant number identification (SNI). GTS (Xie and Sun, 2019) is a goal-driven tree-structured MWP solver. StackDecoder (Chiang and Chen, 2019) is an universal semantically-aligned math word problems solver. (Zhang et al., 2020a) is an enhanced GTS with teacher-student distillation and multi-decoder ensemble. Again, following prior works Chiang and Chen, 2019;Xie and Sun, 2019), we use answer accuracy as the evaluation metric: if the calculated value of the predicted equation tree equals to the true answer, it is thought as correct since the predicted expression is equivalent to the target expression. Implementation Details We use Pytorch 5 to implement our model on Linux with an NVIDIA RTX2080Ti GPU card. All those words with fewer than 5 occurrences are converted into a special token UNK. The size of word embeddings and all hidden states for other layers are set as 128 and 512, respectively. Our model is optimized by ADAM optimizor (Kingma and Ba, 2015) with β 1 = 0.9, β 2 =0.999, and = 1e −8 . The mini-batch size is set as 32. The initial learning rate is set as 1e −3 and then decreases to half every 40 epochs. To prevent overfitting, we set dropout rate as 0.5 and weight decay as 1e −5 . Finally, we conduct greedy search to generate symbolic equation trees. We set λ 1 , λ 2 , λ 3 , λ 5 , and λ 6 as 0.0005, 0.01, 1.0, 0.005, and 0.1 for both datasets, respectively. We set λ 4 as 0.000001 for Math23K while we set λ 4 as 1.0 for CM17K. All constants are extracted from the training set. In each epoch, all training data is shuffled randomly and then cut into mini-batches. Answer Accuracy Following prior works Chiang and Chen, 2019;Xie and Sun, 2019), we conduct 5fold cross-validation on Math23K. For CM17K, we evaluate the performance on the test set. The results are shown in Table 2. From Table 2, we can observe that benefiting from the four new auxiliary tasks and neural-symbolic paradigm, our NS-Solver outperforms the baselines on both datasets in terms of answer accuracy. Specifically, for Math23K and CM17K, the accuracy gains of NS-Solver over GTS are 1.37% and 5.93%, respectively. Comparing with TSN-MD, our solver outperforms it by about 0.6% on Math23K. It shows that our model is more feasible for solving multiple types of MWPs. It also shows that our NS-Solver is more effective than other state-of-the-art models on the real-world scenario that needs to solve various MWPs with a unified solver. Model Math23K CM17K DNS 58.1% 15.93% StackDecoder (Chiang and 66.0% 37.24% GTS (Xie and Sun, 2019) 74.3% 47.12% TSN-MD (Zhang et al., 2020a) 75.1% -NS-Solver (Ours) 75.67% 54.05% Comparisons on different subsets We drill down to analyze the generalization of DNS, GTS, and NS-Solver on different types of MWPs in the test subset of CM17K. Their answer accuracy on different types of MWPs is shown in Table 3. We can observe that our NS-Solver outperforms the other two models by a large margin on all subsets. Specifically, the accuracy gains of our NS-Solver over GTS on four subsets are 3.87%, 9.12%, 6.99%, and 9.44%. This shows that with the help of four auxiliary tasks, our NS-Solver obtains better generalization ability on multiple types of MWPs than baselines. Performance on Tree Length Intuitively, the size of the symbolic equation tree is proportional to the complexity of the mathematical relationship in the problem. The more complex the mathematical relationship is, the more difficult it is to solve the problem. Here, we compare our proposed NS-Solver with GTS on CM17K to show the superiority of our NS-Solver on different equation tree sizes. The answer accuracies for different sizes of expression trees on CM17K test subset are shown in Fig. 2. We can see that there is a tendency for answer accuracy to degrade with the growth of the problem complexity measured as the size of the equation tree, and our NS-Solver outperforms GTS on most cases of different equation tree sizes. This shows our NS-Solver can better model the mathematical relationships of the problem than GTS. It can also be noticed that the improvement of our NS-Solver over the GTS is increasing when the problems become more complex. However, although our model outperforms other methods, there still has room for improvement in semantic understanding and symbolic reasoning since longer equations often match with more complex MWPs which entail more complex math relationships. Ablation on different auxiliary tasks We study the contribution of different auxiliary tasks of our NS-Solver. For this purpose, we consider five different combinations: 1) only the backbone [NS-Solver -CCP -SNP -PCC -DE]; 2) backbone + duality exploiting task [NS-Solver -CCP -SNP -PCC]; 3) backbone + duality exploiting task + program consistent checker [NS-Solver -CCP -SNP]; 4) backbone + duality exploiting task + program consistent checker + number prediction tasks [NS-Solver -CCP]; and 5) the proposed NS-Solver [NS-solver]. For each of these combinations, each model was trained for 80 epochs on CM17K and validated on its test subset. The learning rate decreased to half every 20 epochs. The results are provided in Fig. 4. As one can see, all four auxiliary tasks can improve performance. Specifically, the accuracy gains of DE, PCC, SNP, and CCP are 1.00%, 1.41%, 1.11%, and 1.12%, respectively. Besides, the binary accuracies of the two SNP tasks are 97% (number quantity prediction) and 96.8% (number location prediction). Moreover, the accuracy of our CCP SNS-solver -CCP -NP -PCC -DE (Ours): x=n 2 /n 1 (error) SNS-solver -CCP -NP -PCC (Ours): x=n 2 /(n 0 n 1 ) (correct) SNS-solver -CCP -NP -PCC (Ours): x=n 1 /(n 0 1.0) (correct) SNS-solver -CCP -NP (Ours): x=n 1 /n 0 (correct) SNS-solver -CCP -NP (Ours): x=n 3 n 1 n 2 /10000 (error) SNS-solver -CCP (Ours): x=n 3 n 1 n 2 (correct) SNS-solver -CCP (Ours): x=n 0 n 1 /((n 2 /100)(n 2 /10)) (error) SNS-solver (Ours): x=n 0 n 1 /((n 2 /100)(n 2 /100)) (correct) GTS: n 2 x=n 1 +n 3 x (error) task is 97.8%. This shows that our auxiliary tasks can enhance our NS-Solver to enforce better problem understanding and symbol reasoning. Overall, our proposed NS-Solver achieves the best answer accuracy. SNS-solver (Ours Case Study We also present the results of our NS-Solver with different combinations of four auxiliary tasks in Fig. 3. Benefiting from explicitly exploiting the probabilistic correlation between two quasi dual tasks to regularize the training process in our duality exploiting ( To show that our auxiliary tasks can be adapted to other backbones, we replace GTS's encoder with BERT (BERT + Tree Decoder) and NS-Solver's encoder with BERT (NS-Solver + BERT), where we adopt a Chinese BERT-base pre-trained with whole word masking (Cui et al., 2020). We conduct experiments on CM17K. The results are shown in Table 4. We can observe that with auxiliary tasks, our NS-Solver + BERT still can outperform BERT + Tree Decoder, which shows that our auxiliary tasks' strong generalization. Conclusion In this work, we propose Neural-Symbolic Solver (NS-Solver) to explicitly and seamlessly incorporate different levels of symbolic constraints by four auxiliary tasks. Our NS-Solver consists of a problem reader to encode problems, a programmer to generate a symbolic grounded program, and a symbolic executor to obtain final results. In addition to supervised learning with target expression, our solver is also optimized via four new auxiliary objectives that enforce four levels of symbolic reasoning. Besides, we also construct a new dataset CM17K containing 4 types of MWPs with more than 17K samples, which provides a more realistic and challenging benchmark for developing a universal and scalable math solver. Extensive experiments on Math23K and CM17K demonstrate the superiority of our NS-Solver compared to state-ofthe-art methods in answer accuracy while ensuring intermediate equation rationality. Ethical Impact We collected CM17K from two online education websites, which is only used for academic research, and the copyright belongs to the original websites. This work may inspire research in the field of numerical reasoning. Figure 2 : 2Answer accuracies for different sizes of symbolic equation trees on CM17K. ) 元车费，他们买了几张票？ (Xiaojie went to Nanyueshan with his classmates. The ticket per person was NUM(n 0[22]) yuan. They spent a total of NUM(n 1 [154]) yuan. How many tickets were they bought?) Groundtruth: x=n 1 /n 0Case 3: 妈妈想给 NUM (n 0 [1]) 间长 NUM (n 1 [7]) 米，宽 NUM(n 2 [4]) 米的房间铺上地砖，每平方米的地砖价钱是 NUM(n 3 [60]) 元，那么铺好地砖至少要花多少钱？ (Mother wants to lay a floor tile in NUM(n 0 [1]) room with a length of NUM(n 1 [7]) meters and a width of NUM(n 2 [4]) meters. The price per square meter of floor tiles is NUM(n 3 [60]) yuan. So how much does it cost to lay the floor tiles?) Case 5: 甲、乙 NUM(n 0 [2]) 地相距 NUM(n 1 [200]) 千米，快车速度为 NUM(n 2 [120]) 千米每小时，慢车速度为 NUM(n 3 [80]) 千米每小时，慢车从甲地出发，快车从乙地出发。如果 NUM(n 4 [2]) 车同时出发，相向而行，出发后几时 NUM(n 5 [2]) 车相遇 ? (The distance between NUM(n 0 [2]) locations A and B is NUM(n 1 [200]) kilometers, the speed of express train is NUM(n 2 [120]) kilometers per hour, and the speed of slow train is NUM(n 3 [80]) kilometers per hour. If the NUM(n 4 [2]) cars depart at the same time \\ and travel towards each other, when will the NUM(n 5 [2]) cars meet after departure?) Case 4: 小胖家装修新房了，准备在客厅铺上地砖，客厅是长方形的地面，长 NUM (n 0 [5]) 米，宽 NUM(n 1 [6]) 米，他选中了边长为 NUM (n 2 [40]) 厘米的正方形地砖，他至少要购买多少块这样的地砖？(The chubby family has renovated a new house and is ready to lay floor tiles in the living room. The living room is a rectangular floor with a length of NUM(n 0 [5]) meters and a width of NUM (n 1 [6]) meters. He chose a square floor tile with a side length of NUM(n 2 [40]) cm. How many pieces of floor tiles should he buy at least?) Figure 3 : 3Typical cases. Note that the results are represented as infix order which is more readable than prefix order. The programs generated by NS-Solver are also translated into human-readable equations. Constants and number symbols are labelled in red and cyan, respectively. Figure 4 : 4Ablation Study on different auxiliary components. '-' represents we remove the component. Table 2 : 2Model comparison on answer accuracy Table 3 : 3Answer accuracy on CM17K's test subset. DE) task, our [NS-solver -CCP -SNP -PCC] can generate correct equations by understanding the problem better while [NS-solver -CCP -SNP -PCC -DE] generates error equations, as shown in Case 1. With the program consis-tency checker (PCC) that effectively regularizes the model's output by constraining the distance between predicted symbols and ground-truth symbols during training, [NS-solver -CCP -SNP] can generate more consistent equations with the ground-truth than [NS-solver -CCP -SNP -PCC], as shown in Case 2. With self-supervised number prediction (SNP), [NS-solver -CCP] can generate better results and avoid generating symbols that do not belong to the problem, as shown in Case 3. With commonsense constant prediction (CCP), our NS-Solver manages to choose correct constants by constraining the constant symbolic table using predicted results of CCP. As shown in Case 4, [NS-solver -CCP] chooses error constant 10 while NS-solver chooses two correct constants.Besides, although GTS and NS-Solver generate the same symbols sometimes, our NS-Solver generates correct equations with the help of our four auxiliary objectives, as shown in Case 5. Overall, all four auxiliary tasks can improve our NS-Solver's understanding and reasoning ability.Model BERT + Tree Decoder (Xie and Sun, 2019) NS-Solver + BERT CM17K 55.0% 60.68% Table 4 : 4Generalization to different backbone5878 deep bidirectional transformers for language understanding. 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[ "The planetary nebula NGC 6153 through the eyes of MUSE", "The planetary nebula NGC 6153 through the eyes of MUSE" ]	[ "Gómez-Llanos ", "V ", "García-Rojas ", "J ", "MorissetC ", "D Jones \nInstituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain\n\nInstituto de Astronomía\nUNAM\nApdo. postal 106C.P. 22800Ensenada, Baja CaliforniaMéxico\n", "H Monteiro \nInstituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain\n\nInstituto de Física e Química\nUniversidade Federal de Itajubá\nAv. BPS 1303-Pinheirinho37500-903ItajubáBrazil\n", "R Wesson \nInstituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain\n\nSchool of Physics and Astronomy\nCardiff University\nQueen's Buildings, The ParadeCF24 3AACardiffUK\n", "BoffinH M J ", "R L M Corradi \nDepartamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain\n\nEuropean Southern Observatory\nKarl-Schwarzschild-Str. 2D-85738Garching bei MünchenGermany\n\nGran Telescopio CANARIAS\n\n", "Pérez-Toledo ", "F ", "Rodríguez - Gil \nInstituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain\n\nDepartamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain\n\nGran Telescopio CANARIAS\n\n", "P ", "S A ", "/ Cuesta De San José ", "Breña Baja ", "Santa Cruz De Tenerife ", "Spain " ]	[ "Instituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain", "Instituto de Astronomía\nUNAM\nApdo. postal 106C.P. 22800Ensenada, Baja CaliforniaMéxico", "Instituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain", "Instituto de Física e Química\nUniversidade Federal de Itajubá\nAv. BPS 1303-Pinheirinho37500-903ItajubáBrazil", "Instituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain", "School of Physics and Astronomy\nCardiff University\nQueen's Buildings, The ParadeCF24 3AACardiffUK", "Departamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain", "European Southern Observatory\nKarl-Schwarzschild-Str. 2D-85738Garching bei MünchenGermany", "Gran Telescopio CANARIAS\n", "Instituto de Astrofísica de Canarias\nE-38205La Laguna, TenerifeSpain", "Departamento de Astrofísica\nUniversidad de La Laguna\nE-38206La LagunaTenerifeSpain", "Gran Telescopio CANARIAS\n" ]	[ "Proceedings of the XV Scientific Meeting of the Spanish Astronomical Society held on" ]	In this contribution, we present the results of a study on the high abundance discrepancy factor (ADF ∼ 10) planetary nebula (PN) NGC 6153 with MUSE. We have constructed flux maps for dozens of emission lines, that allowed us to build spatially resolved maps of extinction, electron temperature (T e ), electron density (n e ), and ionic abundances. We have simultaneously constructed ADF maps for O + and O 2+ and found that they centrally peak in this PN, with a remarkable spatial coincidence with the low T e found from recombination line diagnostics. This finding strongly supports the hypothesis that two distinct gas phases co-exist: one cold and metal-rich, and a second warm and with "normal" metal content. We show that to build T e ([N ii]) and ionic abundance maps of low-ionization species for these objects, recombination contribution to the auroral [N ii] and [O ii] lines must be properly evaluated and corrected.	null	[ "https://export.arxiv.org/pdf/2302.11525v2.pdf" ]	257,079,180	2302.11525	672ce48ff4164130efcfe9aa7215c2e98778677a	The planetary nebula NGC 6153 through the eyes of MUSE September 4 -9, 2022 Gómez-Llanos V García-Rojas J MorissetC D Jones Instituto de Astrofísica de Canarias E-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna E-38206La LagunaTenerifeSpain Instituto de Astronomía UNAM Apdo. postal 106C.P. 22800Ensenada, Baja CaliforniaMéxico H Monteiro Instituto de Astrofísica de Canarias E-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna E-38206La LagunaTenerifeSpain Instituto de Física e Química Universidade Federal de Itajubá Av. BPS 1303-Pinheirinho37500-903ItajubáBrazil R Wesson Instituto de Astrofísica de Canarias E-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna E-38206La LagunaTenerifeSpain School of Physics and Astronomy Cardiff University Queen's Buildings, The ParadeCF24 3AACardiffUK BoffinH M J R L M Corradi Departamento de Astrofísica Universidad de La Laguna E-38206La LagunaTenerifeSpain European Southern Observatory Karl-Schwarzschild-Str. 2D-85738Garching bei MünchenGermany Gran Telescopio CANARIAS Pérez-Toledo F Rodríguez - Gil Instituto de Astrofísica de Canarias E-38205La Laguna, TenerifeSpain Departamento de Astrofísica Universidad de La Laguna E-38206La LagunaTenerifeSpain Gran Telescopio CANARIAS P S A / Cuesta De San José Breña Baja Santa Cruz De Tenerife Spain The planetary nebula NGC 6153 through the eyes of MUSE Proceedings of the XV Scientific Meeting of the Spanish Astronomical Society held on the XV Scientific Meeting of the Spanish Astronomical Society held on2023September 4 -9, 20222 The planetary nebula NGC 6153 through the eyes of MUSE In this contribution, we present the results of a study on the high abundance discrepancy factor (ADF ∼ 10) planetary nebula (PN) NGC 6153 with MUSE. We have constructed flux maps for dozens of emission lines, that allowed us to build spatially resolved maps of extinction, electron temperature (T e ), electron density (n e ), and ionic abundances. We have simultaneously constructed ADF maps for O + and O 2+ and found that they centrally peak in this PN, with a remarkable spatial coincidence with the low T e found from recombination line diagnostics. This finding strongly supports the hypothesis that two distinct gas phases co-exist: one cold and metal-rich, and a second warm and with "normal" metal content. We show that to build T e ([N ii]) and ionic abundance maps of low-ionization species for these objects, recombination contribution to the auroral [N ii] and [O ii] lines must be properly evaluated and corrected. Introduction Since [16] first reported it, the abundance discrepancy problem, i. e., the long standing difference between the chemical abundances computed for a given metal ion from recombination lines (RLs) or collisionally excited lines (CELs) has cast doubt on the chemical abundances determinations in both planetary nebulae (PNe) and H ii regions. The RL/CEL ratio, the so-called abundance discrepancy factor (ADF) can show extreme values (up to 700) in some PNe. Some scenarios have been proposed to explain this problem (see [4] and references therein). However, for PNe there are several observational evidences pointing to the presence of two gas components with different chemical composition (and possibly kinematics) ( [2,6,15,11,3,9,12,13]). This bi-abundance scenario (first proposed by [14]) consists of a "normal" chemical composition gas with a relatively warm electron temperature (T e ∼10,000 K) that emits mainly the metal CELs and the H and He RLs , and an H-poor gas with a much lower temperature (∼1,000 K) and higher density whose emission is dominated by metals RLs. NGC 6153 is a southern PN with strong emission of metal RLs, thus making it a good object to address the abundance discrepancy problem ( [8,10,15,17,7,13]). The chemical composition of this PN has been extensively studied by different authors who hypothesised on the presence of two plasma components based on deep multi-wavelength spectroscopic data ( [8]) or on integral field spectroscopy ( [15]). Very recently, making use of a very highresolution spectra and position-velocity maps, [13] reached a similar conclusion with the addition of finding also differences in the kinematics of the gas between the two components. From the theoretical side, empirical, one-dimensional, and 3D photoionization models have been constructed for this object considering a chemically inhomogeneous gas, successfully fitting both the RLs and CELs ( [8,10,17,7]) and hence, strengthening the bi-abundance scenario for this object. The analysis of 2D spectroscopic data of PNe with large ADFs have revealed that extreme care should be taken when constructing physical conditions and ionic abundance maps, especially for low ionization species (see [5]). In this work we present some of the preliminary results we have obtained from the analysis of very deep MUSE data of NGC 6153. Observations NGC 6153 was observed with the Multi Unit Spectroscopic Explorer (MUSE) integral-field spectrograph ([1]) on the Very Large Telescope (VLT), in seeing-limited mode, on the night of 6 to 7 July 2016. We used the extended mode of MUSE (WFM-NOAO-E), which covers the wavelength range 460 − −930 nm with an effective spectral resolution that increases from R ∼ 1600 at the bluest wavelengths to R ∼ 3500 at the reddest wavelengths. The ontarget exposure time was 2320 s divided in several long and short exposures. The observing conditions, observation techniques and reduction process have been described by [5]. Electron temperature maps From the MUSE observations of NGC 6153, we have constructed flux maps and their uncertainties for more than 60 emission lines following the same methodology as described by [5]. We then built spatially resolved maps of extinction, electron temperature (T e ), electron density (n e ), and ionic abundances. The T e and n e maps were obtained using different line ratios as diagnostics (e.g., [N ii] λ5755/λ6548 1 and [S iii] λ6312/λ9069 for T e , and [S ii] λ6731/λ6716 and [Cl iii] λ5538/λ5518 for n e ). However, the diagnostics based on second-row elements such as O and N can have an important contribution from recombination to the low metastable levels, like [N ii] λ5755 and [O ii] λλ7320, 7730, that if not corrected, will lead to an overestimate of the temperature. This may be especially important for spatially resolved observations of PNe with high ADF, where in extreme cases, the [N ii] λ5755 emission can be dominated by recombination (see Fig. 7 in [5]). In this work, we present the recombination contribution to [N ii] λ5755 in the PN NGC 6153 (ADF∼10 [8,15]). In the left panel of Fig. 1 we show the spatially resolved emission of [N ii] λ5755. To estimate its recombination contribution we use Eq. 1 presented by [5], which is based on the emission of N ii λ5679 (middle panel of Fig. 1) and the recombination emissivities of j 5755 (T e , n e ) and j 5679 (T e , n e ). [13] presents an estimate of the electron temperature and density of the recombination emitting region in NGC 6153, giving an average value of n e = 10 4 cm −3 and T e = 2, 000 K. We use these values for the recombination emissivities. The corrected [N ii] λ5755 emission map is presented in the right panel of Fig. 1, which shows the main emission in two bright knots and on the edges of the nebula's main shell, while the uncorrected flux also shows a bright emission at inner regions of the nebula. To emphasize the importance of this correction, in Fig. 2 and T e = 2, 000 K, 4, 000 K, and 6, 000 K, respectively, for the recombination emission. nebula that are predicted without the correction. We also explore the effect of increasing the temperature of the recombination emitting region to 4,000 K and 6,000 K (bottom left and bottom right panels of Fig. 2), which results in a decrease on the temperature in the inner parts of the nebula. We tried to compute T e and n e from metal RL diagnostics in order to break the degeneracies found by [5] when trying to fully characterize the H-poor component. However, the most sensitive O ii and N ii RLs are either out of the wavelength range covered by MUSE or the maps constructed were too noisy to reach any conclusion. As already pointed out, [13] The abundance discrepancy maps Once we computed the ionic abundances from CELs and RLs, we constructed the abundance discrepancy factor maps for O + and O 2+ following the methodology described by [5]. [15] constructed the ADF(O 2+ ) map for NGC 6153, but only sampled a small area of the nebula. [13] made a study on the variation of the ADF over a position-velocity map, finding the highest values at positions close to the central star. These authors also find that the ADF was close to the unity in the diffuse emission beyond the receding side of the main shell of the nebula. As far as we know, this is the first time that the ADF is mapped for the whole nebula. On the other hand, we have constructed the H i RL temperature diagnostic from the ratio of the Paschen jump to the H i P9 λ9229 line, following the same methodology described by [5]. This temperature diagnostic can provide hints on the influence of the cold gas on the computation of the global physical conditions in the nebula (Gómez-Llanos et al. in prep.). In Fig. 3 we present the spatial distribution of log[ADF(O + )] (left panel) and log[ADF(O 2+ )] (central panel). In both maps, the ADF peak is clear in the central parts of the nebula, although the ADF variation in these central zones is not as extreme as that in the three high-ADF PNe presented in [5]. This is an expected behaviour if we take into account the lower ADF value in the integrated spectrum of NGC 6153. In the right panel of Fig. 3, we illustrate the spatial distribution of the T e obtained from the H i Paschen jump. The spatial coincidence between the high values of the ADF(O 2+ ) and the low T e 's in this map is remarkable. This behavior strongly supports the hypothesis of the presence of a cold, metal-rich gas phase embedded in a warm gas phase with"normal" metal content. The full analysis of the NGC 6153 MUSE data set will be presented in a forthcoming paper (Gómez-Llanos et al., in prep.). Figure 1 : 1Left panel: spatial distribution of the auroral [N ii] λ5755 emission line in the PN NGC 6153 prior to applying the recombination contribution. Middle panel: spatial distribution of the N ii λ5679 RL. Right panel: same as left panel but after applying the recombination contribution correction, considering a constant n e = 10 4 cm −3 and T e = 2, 000 K for the recombination emission. Figure 2 : 2we show the [N ii] λ5755/λ6548 temperature distribution map with and without applying the correction on the top right and left panels of Fig. 2, respectively. Notice the considerably higher temperatures in the inner parts Electron temperature map computed from the T e -sensitive [N ii] λλ5755/6548 line ratio. In the top left panel we show the map with no recombination correction to the [N ii] λ5755 auroral line; in the top right, bottom left, and bottom right panels we show the maps constructed with the recombination correction, considering a constant n e = 10 4 cm −3 Figure 3 : 3could estimate the physical conditions of the H-poor component of the gas where most of the metals recombination emission comes. These values can be used to qualitatively estimate the influence of the cold region on abundance determinations and to determine the oxygen mass ratio between the cold and the warm regions (Gómez-Llanos et al. in prep.) Spatial distributions of log[ADF(O + )] (left panel) and log[ADF(O 2+ )] (central panel). In the right panel we show the T e map obtained from the Paschen jump relative to H i P9 λ9229 line. Hereafter all wavelengths will be inÅ. AcknowledgementsThis paper is based on observations made with ESO Telescopes at the Paranal Observatory under program ID 097.D-0241. VG-LL and JG-R acknowledge financial support from the Canarian Agency for Research, Innovation and Information Society (ACIISI), of the Canary Islands Government, and the European Regional Development Fund (ERDF), under grant with reference ProID2021010074. JG-R acknowledges support from an Advanced Fellowship under the Severo Ochoa excellence program CEX2019-000920-S. DJ acknowledges support from the Erasmus+ program of the European Union under grant number 2020-1-CZ01-KA203-078200. VG-LL, JG-R, DJ, and RC acknowledge support under grant P/308614 financed by funds transferred from the Spanish Ministerio de Ciencia, Innovación y Universidades, charged to the General State Budgets and with funds transferred from the General Budgets of the Autonomous Community of the Canary Islands by the MCIU. 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[ "3DAvatarGAN: Bridging Domains for Personalized Editable Avatars", "3DAvatarGAN: Bridging Domains for Personalized Editable Avatars" ]	[ "Rameen Abdal \nKAUST\n\n", "Hsin-Ying Lee \nSnap Inc\n\n", "Peihao Zhu \nKAUST\n\n", "Menglei Chai \nSnap Inc\n\n", "Aliaksandr Siarohin \nSnap Inc\n\n", "Peter Wonka \nKAUST\n\n", "Sergey Tulyakov \nSnap Inc\n\n" ]	[ "KAUST\n", "Snap Inc\n", "KAUST\n", "Snap Inc\n", "Snap Inc\n", "KAUST\n", "Snap Inc\n" ]	[]	Figure 1. Editable 3D avatars. We present 3DAvatarGAN, a 3D GAN able to produce and edit personalized 3D avatars from a single photograph (real or generated). Our method distills information from a 2D-GAN trained on 2D artistic datasets like Caricatures, Pixar toons, Cartoons, Comics etc. and requires no camera annotations.AbstractModern 3D-GANs synthesize geometry and texture by training on large-scale datasets with a consistent structure. Training such models on stylized, artistic data, with often unknown, highly variable geometry, and camera information has not yet been shown possible. Can we train a 3D GAN on such artistic data, while maintaining multi-view consistency and texture quality? To this end, we propose an adaptation framework, where the source domain is a pre-trained 3D-GAN, while the target domain is a 2D-GAN trained on artistic datasets. We, then, distill the knowledge from a 2D generator to the source 3D generator. To do that, we first propose an optimization-based method to align the distributions of camera parameters across domains. Second, we propose regularizations necessary to learn high-quality texture, while avoiding degenerate geometric solutions, such as flat shapes. Third, we show a deformation-based technique for modeling exaggerated geometry of artistic domains, enabling-as a byproductpersonalized geometric editing. Finally, we propose a novel inversion method for 3D-GANs linking the latent spaces of the source and the target domains. Our contributions-for the first time-allow for the generation, editing, and animation of personalized artistic 3D avatars on artistic datasets. Project Page: https:/rameenabdal.github.io/3DAvatarGAN	10.48550/arxiv.2301.02700	[ "https://export.arxiv.org/pdf/2301.02700v2.pdf" ]	255,546,292	2301.02700	12f99b597fd65c9eb730cfef498b47f3fb3a5ec8	3DAvatarGAN: Bridging Domains for Personalized Editable Avatars Rameen Abdal KAUST Hsin-Ying Lee Snap Inc Peihao Zhu KAUST Menglei Chai Snap Inc Aliaksandr Siarohin Snap Inc Peter Wonka KAUST Sergey Tulyakov Snap Inc 3DAvatarGAN: Bridging Domains for Personalized Editable Avatars Project Page: https:/rameenabdal.github.io/3DAvatarGAN Figure 1. Editable 3D avatars. We present 3DAvatarGAN, a 3D GAN able to produce and edit personalized 3D avatars from a single photograph (real or generated). Our method distills information from a 2D-GAN trained on 2D artistic datasets like Caricatures, Pixar toons, Cartoons, Comics etc. and requires no camera annotations.AbstractModern 3D-GANs synthesize geometry and texture by training on large-scale datasets with a consistent structure. Training such models on stylized, artistic data, with often unknown, highly variable geometry, and camera information has not yet been shown possible. Can we train a 3D GAN on such artistic data, while maintaining multi-view consistency and texture quality? To this end, we propose an adaptation framework, where the source domain is a pre-trained 3D-GAN, while the target domain is a 2D-GAN trained on artistic datasets. We, then, distill the knowledge from a 2D generator to the source 3D generator. To do that, we first propose an optimization-based method to align the distributions of camera parameters across domains. Second, we propose regularizations necessary to learn high-quality texture, while avoiding degenerate geometric solutions, such as flat shapes. Third, we show a deformation-based technique for modeling exaggerated geometry of artistic domains, enabling-as a byproductpersonalized geometric editing. Finally, we propose a novel inversion method for 3D-GANs linking the latent spaces of the source and the target domains. Our contributions-for the first time-allow for the generation, editing, and animation of personalized artistic 3D avatars on artistic datasets. Project Page: https:/rameenabdal.github.io/3DAvatarGAN Introduction Photo-realistic portrait face generation is an iconic application demonstrating the capability of generative models especially GANs [29,31,32]. A recent development has witnessed an advancement from straightforwardly synthesizing 2D images to learning 3D structures without 3D supervision, referred to as 3D-GANs [10,43,58,67]. Such training † Part of the work was done during an internship at Snap Inc. arXiv:2301.02700v2 [cs.CV] 26 Mar 2023 is feasible with the datasets containing objects with highly consistent geometry, enabling a 3D-GAN to learn a distribution of shapes and textures. In contrast, artistically stylized datasets [26,68] have arbitrary exaggerations of both geometry and texture, for example, the nose, cheeks, and eyes can be arbitrarily drawn, depending on the style of the artist as well as on the features of the subject, see Fig. 1. Training a 3D-GAN on such data becomes problematic due to the challenge of learning such an arbitrary distribution of geometry and texture. In our experiments (Sec. 5.1), 3D-GANs [10] generate flat geometry and become 2D-GANs essentially. A natural question arises, whether a 3D-GAN can synthesize consistent novel views of images belonging to artistically stylized domains, such as the ones in Fig. 1. In this work, we propose a domain-adaption framework that allows us to answer the question positively. Specifically, we fine-tune a pre-trained 3D-GAN using a 2D-GAN trained on a target domain. Despite being well explored for 2D-GANs [26,68], existing domain adaptation techniques are not directly applicable to 3D-GANs, due to the nature of 3D data and characteristics of 3D generators. The geometry and texture of stylized 2D datasets can be arbitrarily exaggerated depending on the context, artist, and production requirements. Due to this, no reliable way to estimate camera parameters for each image exists, whether using an off-the-shelf pose detector [75] or a manual labeling effort. To enable the training of 3D-GANs on such challenging datasets, we propose three contributions. 1 An optimization-based method to align distributions of camera parameters between domains. 2 Texture, depth, and geometry regularizations to avoid degenerate, flat solutions and ensure high visual quality. Furthermore, we redesign the discriminator training to make it compatible with our task. We then propose 3 a Thin Plate Spline (TPS) 3D deformation module operating on a tri-plane representation to allow for certain large and sometimes extreme geometric deformations, which are so typical in artistic domains. The proposed adaptation framework enables the training of 3D-GANs on complex and challenging artistic data. The previous success of domain adaptation in 2D-GANs unleashed a number of exciting applications in the content creation area [26,68]. Given a single image such methods first find a latent code corresponding to it using GAN inversion, followed by latent editing producing the desired effect in the image space. Compared to 2D-GANs, the latent space of 3D-GANs is more entangled, making it more challenging to link the latent spaces between domains, rendering the existing inversion and editing techniques not directly applicable. Hence, we take a step further and explore the use of our approach to 3D artistic avatar generation and editing. Our final contribution to enable such applications is 4 a new inversion method for coupled 3D-GANs. In summary, the proposed domain-adaption framework allows us to train 3D-GANs on challenging artistic datasets with exaggerated geometry and texture. We call our method 3DAvatarGAN as it-for the first time-offers generation, editing, and animation of personalized stylized, artistic avatars obtained from a single image. Our results (See Sec. 5.2) show the high-quality 3D avatars possible by our method compared to the naive fine-tuning. Related Work GANs and Semantic Image Editing. Generative adversarial Networks (GANs) [20,50] are one popular type of generative model, especially for smaller high-quality datasets such as FFHQ [33], AFHQ [14], and LSUN objects [70]. For these datasets, StyleGAN [29,31,33] can be considered as the current state-of-the-art GAN [28,29,31,33,34]. The disentangled latent space learned by StyleGAN has been shown to exhibit semantic properties conducive to semantic image editing [1,3,16,23,37,46,54,59,65]. CLIP [49] based image editing [2,17,46] and domain transfer [15,73] are another set of works enabled by StyleGAN. GAN Inversion. Algorithms to project existing images into a GAN latent space are a prerequisite for GAN-based image editing. There are mainly two types of methods to enable such a projection: optimization-based methods [1,13,60,74] and encoder-based methods [5,7,51,61,72]. On top of both streams of methods, the generator weights can be further modified after obtaining initial inversion results [52]. Learning 3D-GANs with 2D Data. Previously, some approaches attempt to extract 3D structure from pre-trained 2D-GANs [44,55]. Recently, inspired by Neural Radiance Field (NeRF) [9,38,45,71], novel GAN architectures have been proposed to combine implicit or explicit 3D representations with neural rendering techniques [11,12,21,40,41,43,53,56,58,66,67]. In our work, we build on EG3D [11] which has current state-of-the-art results for human faces trained on the FFHQ dataset. Avatars and GANs. To generate new results in an artistic domain (e.g. anime or cartoons), a promising technique is to fine-tune an existing GAN pre-trained on photographs, e.g. [47,57,63]. Data augmentation and freezing lower layers of the discriminator are useful tools when fine-tuning a 2D-GAN [29,39]. One branch of methods [18,46,73] investigates domain adaptation if only a few examples or only text descriptions are available. While others focus on matching the distribution of artistic datasets with diverse shapes and styles. Our work also falls in this domain. Among previous efforts, StyleCariGAN [26] proposes invertible modules in the generator to train and generate caricatures from real images. DualStyleGAN [68] learns two mapping networks in StyleGAN to control the style and structure of the new domain. Some works are trained on 3D data or require heavy labeling/engineering [22,27,69] and use 3D morphable models to map 2D images of carica- tures to 3D models. However, such models fail to model the hair, teeth, neck, and clothes and suffer in texture quality. In this work, we are the first to tackle the problem of domain adaption of 3D-GANs and to produce fully controllable 3D Avatars. We employ 2D to 3D domain adaptation and distillation and make use of synthetic 2D data from StyleCari-GAN [26] and DualStyleGAN [68]. Domain Adaptation for 3D-GANs The goal of domain adaptation for 3D-GANs is to adapt (both texture and geometry) to a particular style defined by a 2D dataset (Caricature, Anime, Pixar toons, Comic, and Cartoons [25,26,68] in our case). In contrast to 2D-StyleGAN-based fine-tuning methods that are conceptually simpler [30,47], fine-tuning a 3D-GAN on 2D data introduces challenges in addition to domain differences, especially on maintaining the texture quality while preserving the geometry. Moreover, for these datasets, there is no explicit shape and camera information. We define the domain adaptation task as follows: Given a prior 3D-GAN i.e. EG3D (G s ) of source domain (T s ), we aim to produce a 3D Avatar GAN (G t ) of the target domain (T t ) while maintaining the semantic, style, and geometric properties of G s , and at the same time preserving the identity of the subject between the domains (T s ↔ T t ). Refer to Fig. 5 for the pipeline figure. We represent G 2D as a teacher 2D-GAN used for knowledge distillation fine-tuned on the above datasets. Note that as T t is not assumed to contain camera parameter annotations, the training scheme must suppress artifacts such as low-quality texture under different views and flat geometry (See Fig. 2). In the following, we discuss the details of our method. How to align the cameras? Selecting appropriate ranges for camera parameters is of paramount importance for high-fidelity geometry and texture detail. Typically, such parameters are empirically estimated, directly computed from the dataset using an off-theshelf pose detector [10], or learned during training [8]. In domains we aim to bridge, such as caricatures for which a 3D model may not even exist, directly estimating the camera distribution is problematic and, hence, is not assumed by our method. Instead, we find it essential to ensure that the camera parameter distribution is consistent across the source and target domains. For the target domain, we use StyleGAN2 trained on FFHQ, fine-tuned on artistic datasets [26,68]. Assuming that the intrinsic parameters of all the cameras are the same, we aim to match the distribution of extrinsic camera parameters of G s and G 2D and train our final G t using it (see illustration in Fig. 3). To this end, we define an optimization-based method to match the sought distributions. The first step is to identify a canonical pose image in G 2D , where the yaw, pitch, and roll parameters are zero. According to Karras et al., [32], the image corresponding to the mean latent code satisfies this property. Let θ, φ be the camera Euler angles in a spherical coordinate system, r, c be the radius of the sphere and camera lookat point, and, M be a function that converts these parameters into the camera-to-world matrix. Let I s (w, θ, φ, c, r) = G s (w, M(θ, φ, c, r)) and I 2D (w) = G 2D (w) represent an arbitrary image generated by G s and G 2D , respectively, given the w code variable. Let k d be the face key-points detected by the detector K d L kd (I s (w avg , 0, 0, c, r), I 2D (w avg )), (1) where L kd (I 1 , I 2 ) = k d (I 1 ) − k d (I 2 ) 1 and w avg and w avg are the mean w latent codes of G 2D and G s , respectively. In our results, r is determined to be 2.7 and c is approximately [0.0, 0.05, 0.17]. The next step is to determine a safe range of the θ and φ parameters. Following prior works, StyleFlow [3] and FreeStyleGAN [36] . Fine tuned EG3D (Gt) pipeline with TPS module. We design a framework to fine-tune a 3D-GAN i.e. EG3D (dotted yellow box) and adapt to an artistic domain that does not stem from a consistent 3D model. In addition, our framework modifies EG3D architecture to include a novel TPS module for geometric editing. What loss functions and regularizers to use? Next, although the camera systems are aligned, the given dataset may not stem from a consistent 3D model, e.g., in the case of caricatures or cartoons. This entices the generator G t to converge to an easier degenerate solution with a flat geometry. Hence, to benefit from the geometric prior of G s , another important step is to design the loss functions and regularizers for a selected set of parameters to update in G t . Next, we discuss these design choices: Loss Functions. To ensure texture quality and diversity, we resort to the adversarial loss used to fine-tune GANs as our main loss function. We use the standard non-saturating loss to train the generator and discriminator networks used in EG3D [11]. We also perform lazy density regularization to ensure consistency of the density values in the final finetuned model G t . Texture Regularization. Since the texture can be entangled with the geometry information, determining which layers to update is important. To make use of the fine-style information encoded in later layers, it is essential to update the tRGB layer parameters (outputting tri-plane features) before the neural rendering stage. tRGB are convolutional layers that transform feature maps to 3 channels at each resolution (96 channels in triplanes). Moreover, since the net- work has to adapt to a color distribution of T t , it is essential to update the decoder (MLP layers) of the neural rendering pipeline as well. Given the EG3D architecture, we also update the super-resolution layer parameters to ensure the coherency between the low-resolution and high-resolution outputs seen by the discriminator D. Geometry Regularization. In order to allow the network to learn the structure distribution of T t and at the same time ensure properties of W and S latent spaces are preserved, we update the earlier layers with regularization. This also encourages the latent spaces of T s and T t to be easily linked. Essentially, we update the deviation parameter ∆s from the s activations of the S space [65]. The s activations are predicted by A(w), where A is the learned affine function in EG3D. The s activations scale the kernels of a particular layer. In order to preserve the identity as well as geometry such that the optimization of ∆s does not deviate too far away from the original domain T s , we introduce a regularizer given by R(∆s) := ∆s 1 . Note that we apply R(∆s) regularization in a lazy manner, i.e., with density regularization. Interestingly, after training, we can interpolate between s and s + ∆s parameters to interpolate between the geometries of samples in T s and T t (See Fig. 9). Depth Regularization. Next, we observe that even though the above design choice produces better geometry for T t , some samples from G t can still lead to flatter geometry, and it is hard to detect these cases. We found that the problem is related to the relative depth of the background to the foreground. To circumvent this problem, we use an additional regularization where we encourage the average background depth of G t to be similar to G s . Let S b be a face background segmentation network [35]. We first compute the average background depth of the samples given by G s . This average depth is given by a d := 1 M M n=1 ( 1 N n D n S b (I n ) 2 F ).(3) Here, D n is the depth map of the image I n sampled from G s , represents the Hadamard product, M is the number of the sampled images, and N n is the number of background pixels in I n . Finally, regularization is defined as: R(D) := a d · J − (D t S b (I t )) F ,(4) where D t is the depth map of the image I t sampled from G t and J is the matrix of ones having the same spatial dimensions as D t . What discriminator to use? Given that the data in T s and T t is not paired and T t is not assumed to contain camera parameter annotations, the choice of the discriminator (D) used for this task is also a critical design choice. Essentially, we use the unconditional version of the dual discriminator proposed in EG3D, and hence, we do not condition the discriminator on the camera information. As a result, during the training, G t generates arbitrary images with pose using M(θ , φ , c , r ), and the discriminator discriminates these images using arbitrary images from T t . We train the discriminator from scratch and in order to adapt T s → T t , we use the StyleGAN-ADA [29] training scheme and use R1 regularization. How to incorporate larger geometric deformations between domains? While the regularizers are used to limit the geometric changes when adapting from T s to T t , modeling large geometric deformations, e.g., in the caricature dataset is another challenge. One choice to edit the geometry is to use the properties of tri-plane features learned by EG3D. We start out by analyzing these three planes in G s . We observe that the frontal plane encodes most of the information required to render the final image. To quantify this, we sample images and depth maps from G s and swap the front and the other planes from two random images. Then we compare the difference in RGB values of the images and the Chamfer distance of the depth maps. While swapping the frontal tri-planes, the final images are completely swapped, and the Chamfer distance changes by 80 ∼ 90% matching the swapped image depth map. In the case of the other two planes, the RGB image is not much affected and the Chamfer distance of the depth maps is reduced by only 20 ∼ 30% in most cases. Given the analysis, we focus to manipulate the 2D front plane features to learn additional deformation or exaggerations. We learn a TPS (Thin Plate Spline) [64] network on top of the front plane. Our TPS network is conditioned both on the front plane features as well as the W space to enable multiple transformations. The architecture of the module is similar to the standard StyleGAN2 layer with an MLP appended at the end to predict the control points that transform the features. Hence, as a byproduct, we also enable 3D-geometry editing guided by the learned latent space. We train this module separately after G t has been trained. We find that joint training is unstable due to exploding gradients arising from the large domain gap between T s and T t in the initial stages. Formally, we define this transformation as: T(w, f ) := ∆c,(5) where, w is the latent code, f is the front plane, and c are the control points. Let c I be the initial control points producing an identity transformation, (c 1 , c 2 ) be the control points corresponding to front planes (f 1 , f 2 ) sampled using W codes (w 1 , w 2 ), respectively, and (c 1 , c 2 ) be points with (w 1 , w 2 ) swapped in the TPS module. To regularize and encourage the module Figure 9. Interpolation of ∆s. Geometric deformation using the interpolation of learned ∆s parameters. to learn different deformations, we have R(T 1 ) := α 2 n=1 c I − c n 1 − β c 1 − c 2 1 − σ c 1 − c 2 1 .(6) We use initial control point regularization to regularize large deviations in the control points which would otherwise explode. Additionally, to learn extreme exaggerations in T t and 'in expectation', conform to the target distribution in the dataset, we add an additional loss term. Let S(I) be the softargmax output of the face segmentation network [35] given an image I and assuming that S generalizes to caricatures, then R(T 2 ) := S(G t (w)), S(I t ) 1 Eq. 6, Eq. 7, and adversarial training loss are used to train the TPS module. We adopt gradient clipping to make sure that the training does not diverge. See the illustrations in Fig. 4 and Fig. 5. Personalized Avatar Generation and Editing Although 3D domain adaptation adapts T s ↔ T t , it is still a challenge to effectively link the latent spaces of G s and G t to generate personalized 3D avatars using a single photograph as the reference image. Particularly, the challenge arises due to the discrepancy in the coupled latent spaces when dealing with the projection of real photographs on 3D generators. Moreover, one would like to edit and animate these 3D avatars. Projection. The task is to project a real image into the latent space of G s , transfer the latent to G t , and further optimize it to construct a 3D avatar. First, we use an optimizationbased method to find the w code that minimizes the similarity between the generated and the real image in G s . To achieve this, the first step is to align the cameras. We follow the steps mentioned in Sec. 3.1 for this step. Next, we use pixel-wise MSE loss and LPIPS loss to project the image into G s [1]. Additionally, to preserve the identity of the subject, we use attribute classifiers e.g. caricature dataset [25] provides the coupled attribute information of real images and caricatures. We use such attribute classifier [25,26] in a post-hoc manner as we notice that such networks can affect the texture in the target domain and could degenerate to narrow style outputs if applied during training. Moreover, such networks may not be available for all target domains. To avoid overfitting into G s and encourage the easier transfer of the optimized latent code to G t , we use W space optimization for this step. Finally, we initialize this w code for G t and use additional attribute classifier loss [26] for T t domain along with depth regularization R(D) (Eq. 4). As an approximation, we assume that attribute classifier [25,26] generalizes across all domains. We use W/W+ space optimization to control the quality and diversity of the outputs. Let x be the source image, let pixel-wise M SE loss be represented as L mse (x, w, G) = M SE(x, G(w, M(θ , φ , c , r ))), and let LP IP S loss be represented as L lpips (x, w, G) = LP IP S(x, G(w, M(θ , φ , c , r ))) where camera parameters are determined by Sec. 3.1. Let L d (w) be the depth regularizer (Eq. 4) and A t (x, w, G) be the attribute classifier loss. We define the algorithm of projection of a single source image into 3D avatars in Algorithm 1. Algorithm 1: Projection of single image into 3D Avatar. Input: source image x ∈ R n×n×3 ; G s , G t , gradient-based optimizer F and F . Output: the embedded code w for G t 1 Initialize() the code w = w avg ; 2 while not converged do 3 L ← L mse (x, w, G s ) + L lpips (x, w, G s ) + L d (w) + A t (x, w, G s ); 4 w ← w − ηF (∇ w L, w); 5 end 6 Initialize() the code w = w; 7 while not converged do 8 L ← L mse (x, w , G t ) + L lpips (x, w , G t ) + L d (w ) + A t (x, w , G t ); 9 w ← w − ζF (∇ w L , w ); end Editing and Animation. Since our 3D domain adaptation is designed to preserve the properties of W and S spaces, we can perform semantic edits via InterFaceGAN [54], GANSpace [23], StyleSpace [65] etc., and geometric edits using TPS (Sec. 3.4) and ∆s interpolation (Sec. 3.2). To perform video editing, we design an encoder for EG3D based on e4e [61] to encode videos and transfer the edits from G s to G t based on the w codes [4,6,62]. We leave a more fine-grained approach for video processing as future work. Results Quantitative Results In this section, we consider three important evaluations to verify the quality of the texture, geometry, and identity preservation in the new domain using the Caricature, Cartoons, and Pixar toons datasets. We also evaluate the ablation of our design choices and conduct a user study to assess the quality of generated avatars. In the evaluation, let G base be the baseline naïve fine-tuning method which is trained with all the parameters using the losses in EG3D fine-tuned from FFHQ trained prior G s . Note here we still align the cameras in G base using the method defined in Sec. 3.1 and use adaptive discriminator [29] with R1 regularization for a fair comparison. Texture Quality. To verify the quality of the texture, diversity of samples as well as to some extent, the geometry in the target domain T t , we compare the FID [24] scores using G base and G t in Table 1. Note that in the case of Caricatures, we report two scores i.e. with and without using the attribute classifier loss in the training as discussed in Sec. 4. Notice that our method outperforms the naïve baseline method by a huge margin in some cases, especially in Caricatures and Cartoons. We attribute these differences to the mode collapse prone training of G base which is correlated with flat geometry degenerate solution. We show visual results of the flat geometries learned by G base and comparison in Fig. 2. Geometric Quality. To quantify the flat geometries, in Table 2, we show three scores that help us understand such degenerate solutions. Here we consider coupled depth maps generated from sampling in the domains T s (G s ) and T t (G t and G base ). First, we compute the expectation of the absolute mean differences (M d ) of the corresponding foreground depth maps sampled from T s and T t . We also compute the expectation of the absolute standard deviation differences (S d ) for the same setting. Here, we assume that the flatter geometries have a large difference in the depth maps as compared to the prior as indicated by M d . Moreover, S d computes the distance in the distribution of the depth values, where a larger difference indicates a narrow distribution, and hence a flatter geometry. We also notice Table 1. FID Computation. FID (Fréchet Inception Distance) between the 2D dataset and the samples generated by the fine-tuned 3D GAN using baseline (G base ) and Ours (Gt). '' represents the score with the inclusion of the attribute classifier loss discussed in Sec. 3 M(0, 0, c, r). We hypothesize in the case of the flatter geometries, the model learns to pose information in the earlier layers instead of being camera view-dependent. To quantify this, since pose information may not be available for some domains (e.g. cartoons), we compute the R(T 2 ) scores between corresponding images in the domain T s (G s ) and T t (G t and G base ). Note that these scores are computed without the TPS module. Our scores are lower in all three metrics, hence, validating that our method avoids the degenerate solution and preserves the geometric distribution of the prior. Identity Preservation. Identity preservation score is another important evaluation to check the quality of latent space linking between G s and G t . In Table 3, we compute the attribute loss (BCE loss) between the domains T s and T t Table 3. Identity Preservation. Identity preservation using baseline (G base ) and Ours (Gt). Method Caricatures Cartoons Pixar Toons G base 1.28 0.92 0.85 G t (Ours) 0.87 0.81 0.73 Figure 11. Local edits. Local edits performed on the 3D avatars using the S space. using the attribute classifiers [25,26]. Note that our method is able to preserve the identity better across the domains. Ablation Study. In order to validate the importance of the losses, and components of our design choices, in Table 4, we show an ablation study of these components with regularizers. Note that we evaluate these design choices on the caricature dataset. Notice adding each component improves the corresponding scores in FID, M d , S d , and ID as discussed. Notice that by adding the T P S module, the FID is still comparable to G t . The slight drop is attributed to the stretching and squeezing of some parts of the texture (See Fig. 10). Nevertheless, by adding this module, we achieve better control over geometry and produce exaggerated features for a small drop in texture quality. In order to show that the T P S transformations are not random, we compute the FID scores with randomly perturbed front plane features which are derived from the perturbations of control points near the face e.g. taken at the early stages of training after it has stabilized a bit. This setup has less perturbation in the background. We found that the FID score is worse i.e. 25.5 hence validating non-random transformations. The purpose of adding the TPS module is to model the exaggerated geometries in the data and at the same time achieve geometric editing and animation capabilities (see Project Page). We ablate (see Table 5) the average face Keypoint Distance between the paired FFHQ-generated images and corresponding avatars and the face Keypoint Variation (standard deviation of keypoints) with and without the TPS module. The results show that with TPS module, the deviations are large and match the exaggerations. User Study. For results on real images refer to Fig. 7. We conducted a user study using 50 real images (25 caricatures, and 25 Pixar) on identity preservation and 3D consistency versus the baseline method. We asked 21 unique workers where each triplet was reviewed by 10 workers and our avatars were chosen 92% of the time. Qualitative Results For qualitative results, we show the results of the domain adaptation, as well as the personalized edits (geometric and semantic), performed on the resultant 3D avatars. First, in order to show the quality of domain adaptation, identity preservation, and geometric consistency, in Fig. 6, we show results from G s and corresponding results from 3D avatar generator G t trained on Caricatures, Pixar toons, Cartoons, and Comic domains. Next, in order to show that the method generalizes to real images, we use the method described in Sec. 4 to project and transfer the latent code from G s to G t to produce the 3D avatars. In Fig. 8, we show our results of real to 3D avatar transfer. Notice the quality both in terms of texture as well as geometry for both these results achieved by our method. Next, we show geometric and semantic edits possible to produce personalized 3D avatars: Geometry Edits. We show two type of geometric edits i.e. ∆s interpolation (Sec. 3.2) and deformation using TPS (Sec. 3.4). First, in Fig. 9, we show the geometry interpolation by interpolating between original s activations of G s and learned ∆s parameters. In Fig. 10, we show some additional exaggerations in caricatures using the learned 3D deformation latent space of TPS module. Semantic Edits and Animation. Since in our method, we encourage the latent regularization to preserve the properties of the latent space learned by the G s generator, in Fig. 11 we show S space edits performed on the 3D avatars. Notice the quality of edits in terms of locality and adaptability. Additionally, we can edit semantics like hair as opposed to 3D morphable model based methods. In Fig. 12, thanks to the latent space semantics preservation ensured by our method, we can perform some video edits to create a coherent animation based on the difference of w codes of video encoded in G s (Sec. 4) and applied to layers 7 − 10 in G t . Notice the quality of expressions, identity preservation, and 3D consistency across each identity in each row. Conclusion We tackled two open research problems in this paper. In the first part, we proposed the first domain adaptation method for 3D-GANs to the best of our knowledge. This part yields two linked EG3D generators, one in the photorealistic source domain of faces, and another EG3D generator in an artistic target domain. As possible target domains, we show results for cartoons, caricatures, and comics. In the second part, we built on domain adaptation to create 3D avatars in an artistic domain that can be edited and animated. Our framework consists of multiple technical components introduced in this paper. First, we propose a technique for camera space estimation for artistic domains. Second, we introduce a set of regularizers and loss functions that can regularize the fine-tuning of EG3D in such a way that enough of the 3D structure and geometry of the original model is kept, while the distinguishing attributes of the artistic domain, such as textures and colors and local geometric deformations can still be learned. Third, we introduce a geometric deformation module that can reintroduce larger geometric deformations in a controlled manner. These larger geometric deformations can interact and cooperate with EG3D so that semantic edits are still possible. Finally, we propose an embedding algorithm that is especially suitable for two linked EG3D generator networks. Limitations Our method also has some limitations. Overall, the visual quality is limited by the quality of StyleGAN2 pretraining. While we found the quality to be very high for the datasets shown in the paper, it relies on hundreds of images in the target domain to be available. Would be interesting to do few-shot domain adaptation in the future. Further, edits are largely limited to semantic edits of EG3D and global space deformations by TPS. Our method does not enable fine-grained geometric edits. Finally, a large part of our method is face specific. We justify this specialization by the importance of human models and the specific target domain of editable 3D avatars. We nevertheless believe that domain adaptation of general 3D-GANs will be an interesting avenue of future work. Ethical Concerns Deep learning-based image and video processing is a tool for image/video understanding, animation, and artistic expression. Similar to most software in this domain, our work could be used to produce offensive results. An application of concern would be if a user would generate offensive caricatures and cartoons of other people without consent. This can be used to insinuate biases against people or can have a detrimental effect on a person's autonomy, dignity, and/or privacy. The images used in this work are taken or derived from the FFHQ [42] dataset which has appropriate licenses for non-commercial research use and to the best of our knowledge, is committed to protecting the privacy of the individuals who do not wish to be included. Training Details We train our models on 4 V100 GPUs with a batch size of 8. Similar to EG3D, we start training from the neural rendering resolution of 64 2 which is increased during the training. Then we do fine-tuning on 128 2 resolution to produce the final 512 2 outputs. We sample 100k samples from each dataset. We train Caricatures on ∼ 880 kimgs, Pixar, and Cartoons on ∼ 500 kimgs. We fine-tune these models on 128 2 neural rendering resolution for an additional 80 − 160 kimgs. Other hyperparameters of learning rates are the same as the EG3D. We train the TPS module on ∼2000 kimgs. We set the weight for the regularization term R(∆s) as 0.001, and R(D) as 0.005. For the TPS training we use the weights for α, β, σ and R(T 2 ) as 150, 1, 3 and 1 respectively. For inversion, we perform 200 steps for the source domain inversion and 400 steps for the target domain to generate the final avatar. Comparison to 2D-GANs The quality drop is expected when the final model is compared with 2D-GANs as we derive our datasets from these 2D-GANs fine-tuned on avatar datasets. In Table 6, we compute the FID scores between the datasets (size ∼200 images for DualStyleGAN dataset) used to train these 2D-GANs and our corresponding avatar generators (size ∼10k images). We used the 3DCaricShop [48] dataset for Caricatures as the authors of WebCaricature did not reply with the download link. The scores are comparable, even better in the case of Caricatures and Pixar, probably due to our regularizers including 3D view consistency. Importance of Front Tri-plane Features As discussed in Sec. 3.4 of the main paper, the front triplane of the EG3D architecture encodes most of the texture and depth information in the output. In Fig. 13, we show two images with their front and other side plane swapped. Then we show the corresponding effect on the output image. Notice that the results are consistent with the analysis in Sec. 3.4 where the front tri-plane dominates the information for output texture and depth. Stylization To validate that our chosen layers in Sec. 3.2 are responsible for geometrical and texture changes, we resort to a stylization technique. For stylization given an arbitrary reference image e.g. painting, we use the Style Loss [19] to update the layers of G s or G t . Essentially, we use the same parameters used in Sec. 3.1. A critical technique to achieve multi-view consistency and circumvent the ghosting face artifact due to single image overfitting is to rotate the camera to cover the θ and φ ranges in Sec. 3.1 in the main paper uniformly during the optimization. In Fig. 14, we show some results using only the layers used in Texture Regularization (Sec. 3.2). Note the high-quality texture change in the images. In Fig. 15, we show results by adding layers of Geometry Regularization (Sec. 3.2). Note that the geometry is changed in the examples when we use this module. Note that in this example, the geometry is not expected to match as there is no such loss in the optimization. This example results in some arbitrary geometry change that is not flat. This validates our choice of geometry and texture layers used in this paper. Figure 13. Swaping tri-planes. Validation of the information stored in the front tri-plane. Given two images and their tri-plane representations, there is almost no change in the final output if the side planes are swapped. While the output changes completely when the front tri-plane is swapped. Figure 14. Validation of texture regularization. Style Transfer using the texture layers discussed in the main paper. Here the style of the image is changed using the texture layers. Failure cases Our method has some failure cases that stem from the samples of the 2D-GANs i.e. DualStyleGAN [68] and StyleCariGAN [26]. In the caricature domain, the random samples generated in the case of G t trained on StyleCari-GAN samples can have some severe artifacts (See Fig. 16). Although these do not appear often, such a sample can be improved by attribute classifier and depth regularization losses used on a single image as discussed in Sec. 4 of the main paper. Depth Map visualization In order to show some more samples and the corresponding depth maps for G base and G t , in Fig. 17, we show some Figure 15. Validation of geometry regularization. Geometry change (third row) using the geometry layers discussed in the main paper. Here the style (second row) is changed using the texture layers and the geometry (third row) is changed using the geometry layers. Note that the geometry is not correct as we apply Style Loss using a single image and is only used to demonstrate the usage of different layers. Figure 16. Failure cases. Failure cases in caricature generation that stems from the artifacts in the 2D-GAN from which the dataset is generated. Here the artifacts are the result of the generated results of StyleCariGAN [26] samples from both the generators and the corresponding depth maps with pose changes. Notice the flat geometry in the case of G base results. Next, in Fig. 18 and Fig. 19, we show some grid samples of our method with depth maps on the Caricature, Pixar toon, Cartoon, and Comic datasets. Video Results We also show our 3D avatar editing results in videos. We design a simple UI to show that the avatars can be edited in an interactive manner. Please refer to the webpage for editing videos and the interactive editing sessions. Figure 2 . 2Comparison with naive fine-tuning. Comparison of generated 3D avatars with a naïvly fine-tuned generator G base (left sub-figures) versus our generator Gt (right sub-figures). The corresponding sub-figures show comparisons in terms of texture quality (top two rows) and geometry (bottom two rows). See Sec. 5.1 for details. Figure 3 . 3Illustration of camera alignment. We define an optimization algorithm to determine the camera parameters of the new domain i.e. avatars. (see Fig.5 of the paper), we set these parameters as θ ∈ [−0.45, 0.45] and φ ∈ [−0.35, 0.35] in radians. Figure 4 . 4Illustration of TPS module. Our TPS module takes W latents and the front triplane as inputs and outputs deformed control points that deform the front triplane. Figure 5 5Figure 5. Fine tuned EG3D (Gt) pipeline with TPS module. We design a framework to fine-tune a 3D-GAN i.e. EG3D (dotted yellow box) and adapt to an artistic domain that does not stem from a consistent 3D model. In addition, our framework modifies EG3D architecture to include a novel TPS module for geometric editing. Figure 6 . 6Domain adaptation. Domain adaptation results of images from source domain Ts (top row in each sub-figure) to target domain Tt. Rows two to five show corresponding 3D avatar results from different viewpoints. Figure 7 . 7Real to avatar results. Our framework takes a single image as input and generates editable 3D Avatars. Also seeFig. 8 Figure 8 . 83D avatars from real images. Projection of real images on the 3D avatar generators. Figure 10 . 10Deformations using TPS. Geometric edits using our proposed TPS (Thin Plate Spline) module learned on the frontal tri-plane features. Each sub-figure shows a 3D avatar and three examples of TPS deformations sampled from the learned 3D deformation space. Figure 12 . 123D avatar animation. Animation of 3D avatars generated using a driving video encoded in source domain Ts and applied to samples in target domain Tt. The top row shows the driving video and the subsequent rows show generated animations using a random Caricature or Pixar toon. The head pose is changed in each frame of the generated animation to show 3D consistency. Figure 17 . 17Comparison with Naive method. Results of the Caricatures and Pixar toons were viewed from different viewpoints and compared with the baseline method. Note that the depth maps are also visualized highlighting flat geometry. For more results refer to the videos on the Project Page. Figure 18 . 18Grid samples. Samples from the source domain and corresponding results in the target domain. Corresponding images and depth outputs of the Caricatures and Pixar Toons are shown. Figure 19 . 19Grid samples. Extension of Fig. 18. Corresponding images and depth outputs of the Comics and Cartoons are shown. . 2 . 2Method Caricatures Cartoons Pixar ToonsTable 2. Geometry Evaluation. Comparing the geometry using baseline method (G base ) and Ours (Gt). For the definition of M d , S d and R(T2), refer to Sec. 5.1.G base 67.8 79.0 15.1 G t (Ours) 19.4/20.2 12.8 12.4 Metric Method Caricatures Cartoons Pixar M d ↓ G base 0.47 0.21 0.29 G t (Ours) 0.21 0.13 0.13 S d ↓ G base 0.22 0.14 0.15 G t (Ours) 0.15 0.10 0.09 R(T 2 ) ↓ G base 2.99 3.39 4.01 G t (Ours) 2.27 1.62 1.56 that the flat geometry is correlated with the generator learn- ing diverse poses when images are rendered under standard canonical camera parameters i.e. Table 4 .Table 5 . 45Ablation. Ablation of the design choices made in Sec. 3. Ablation of TPS on metrics based on facial keypoints.cam stands for the analysis in Sec. 3.1, Reg stands for the model after applying Eq. 2, DReg stands for the model after applying Eq. 4, and TPS stands for the model after applying Eq. 5 -7. This is the Gt used in the comparison. Note that by adding TPS the scores are affected as the geometry is exaggerated e.g. the identity is affected. This module can be added to do geometry editing. Method FID M d S d ID G base -cam 90.8 0.47 0.33 1.348 G base 67.8 0.47 0.22 1.272 + Reg 19.0 0.22 0.22 0.889 + DReg 19.4 0.21 0.15 0.879 + TPS 20.6 0.25 0.20 0.924 Metric with TPS without TPS Avg. Keypoint Distance 5.7 4.3 Avg. Keypoint Variation 0.06 0.04 Table 6 . 6FID comparison with SCG: StyleCariGAN, DSG: Du-alStyleGAN.Method Cari. (SCG) Cari. (DSG) Pixar (DSG) Cartoons (DSG) 2D-GANs 51.68 96.49 166.78 105.57 Ours 62.90 89.73 162.06 111.35 Im-age2stylegan: How to embed images into the stylegan latent space?. Rameen Abdal, Yipeng Qin, Peter Wonka, Proceedings of the IEEE/CVF International Conference on Computer Vision. the IEEE/CVF International Conference on Computer VisionSeoul, KoreaIEEE27Rameen Abdal, Yipeng Qin, and Peter Wonka. Im- age2stylegan: How to embed images into the stylegan la- tent space? In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4432-4441, Seoul, Korea, 2019. 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[ "Reverse Compute and Forward: A Low-Complexity Architecture for Downlink Distributed Antenna Systems", "Reverse Compute and Forward: A Low-Complexity Architecture for Downlink Distributed Antenna Systems" ]	[ "Song-Nam Hong \nUniversity of Southern California\n90089Los AngelesCA\n", "Giuseppe Caire \nUniversity of Southern California\n90089Los AngelesCA\n" ]	[ "University of Southern California\n90089Los AngelesCA", "University of Southern California\n90089Los AngelesCA" ]	[]	We consider a distributed antenna system where L antenna terminals (ATs) are connected to a Central Processor (CP) via digital error-free links of finite capacity R0, and serve L user terminals (UTs). This system model has been widely investigated both for the uplink and the downlink, which are instances of the general multiple-access relay and broadcast relay networks. In this work we focus on the downlink, and propose a novel downlink precoding scheme nicknamed "Reverse Quantized Compute and Forward" (RQCoF). For this scheme we obtain achievable rates and compare with the state of the art available in the literature. We also provide simulation results for a realistic network with fading and pathloss with K > L UTs, and show that channel-based user selection produces large benefits and essentially removes the problem of rank deficiency in the system matrix. 1I. SYSTEM AND PROBLEM DEFINITIONWe consider a distributed antenna system (DAS) with K user terminals (UTs) and L "antenna terminals" (ATs). All UTs and ATs have a single antenna each. The ATs are connected with a central processor (CP) via wired links of fixed rate R 0 . We study the downlink scenario, where the CP wishes to deliver independent messages to the UTs. This is a simple instance of a broadcast relay network, where the ATs operate as relays. In this work we focus on the symmetric rate, i.e., all messages have the same rate and assume that the CP and all UTs have perfect channel state information (more general results are provided in [1]). If R 0 → ∞, the problem reduces to the well-known vector Gaussian broadcast channel, the capacity region of which is achieved by Dirty Paper Coding (DPC). However, for fixed finite R 0 , DPC and other widely considered linear precoding schemes cannot be applied in a straightforward manner. A simple DAS system, the so-called Soft-Handoff model, was investigated in [2], by introducing a "compressed" version of DPC (CDPC), where the CP performs joint DPC under per-antenna power constraint and then sends the compressed (or quantized) codewords to the corresponding ATs via the wired links. While this scheme is expected to be near-optimal for very large R 0 , it is generally suboptimal at finite (possibly small) R 0 . Also, DPC is notoriously difficult to be implemented in practice, due to the nested lattice coding 1 This research was supported in part by the KCC (Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2011-11921-04001). construction and lattice quantization steps involved (See for example [3], [4]).Motivated by Compute-and-Forward (CoF) [5] (or quantized compute-and-forward (QCoF) [6]), we propose a novel coding strategy named Reverse QCoF (RQCoF) for the DAS downlink with finite backhaul link capacity R 0 . In QCoF and RQCoF the coding block length n can be arbitrarily large but the shaping block length is restricted to 1 (scalar quantization [6]). However, we would like to point out that the same approach can be straightforwardly applied to CoF based schemes, where also the shaping dimension becomes large (in this case, we would refer to the scheme as Reverse CoF (RCoF)).A. Overview of QCoFLet Z p = Z mod pZ denote the finite field of size p, with p a prime number, ⊕ denote addition over Z p , and g : Z p → R be the natural mapping of the elements of Z p onto {0, 1, ..., p− 1} ⊂ R. For a lattice Λ, let Q Λ (x) = argmin λ∈Λ { x − λ } denote the associated lattice quantizer, V = {x ∈ R n : Q Λ (x) = 0} the Voronoi region and define [x] mod Λ = x− Q Λ (x). For κ ∈ R, consider the two nested one-dimensional lattices Λ s = {x = κpz : z ∈ Z} and Λ c = {x = κz : z ∈ Z}, and define the constellation set S Λ c ∩ V s , where V s is the Voronoi region of Λ s , i.e., the interval [−κp/2, κp/2).The modulation mapping m : Z p → S is defined by v = m(u)[κg(u)] mod Λ s . The inverse function m −1 (·) is referred to as the demodulation mapping, and it is given byConsider the (real-valued)	10.1109/isit.2012.6283033	[ "https://arxiv.org/pdf/1202.0854v2.pdf" ]	15,761,756	1202.0854	b5c03018d432e189c50c36b7d7b6d91b52c5ba55	Reverse Compute and Forward: A Low-Complexity Architecture for Downlink Distributed Antenna Systems Song-Nam Hong University of Southern California 90089Los AngelesCA Giuseppe Caire University of Southern California 90089Los AngelesCA Reverse Compute and Forward: A Low-Complexity Architecture for Downlink Distributed Antenna Systems We consider a distributed antenna system where L antenna terminals (ATs) are connected to a Central Processor (CP) via digital error-free links of finite capacity R0, and serve L user terminals (UTs). This system model has been widely investigated both for the uplink and the downlink, which are instances of the general multiple-access relay and broadcast relay networks. In this work we focus on the downlink, and propose a novel downlink precoding scheme nicknamed "Reverse Quantized Compute and Forward" (RQCoF). For this scheme we obtain achievable rates and compare with the state of the art available in the literature. We also provide simulation results for a realistic network with fading and pathloss with K > L UTs, and show that channel-based user selection produces large benefits and essentially removes the problem of rank deficiency in the system matrix. 1I. SYSTEM AND PROBLEM DEFINITIONWe consider a distributed antenna system (DAS) with K user terminals (UTs) and L "antenna terminals" (ATs). All UTs and ATs have a single antenna each. The ATs are connected with a central processor (CP) via wired links of fixed rate R 0 . We study the downlink scenario, where the CP wishes to deliver independent messages to the UTs. This is a simple instance of a broadcast relay network, where the ATs operate as relays. In this work we focus on the symmetric rate, i.e., all messages have the same rate and assume that the CP and all UTs have perfect channel state information (more general results are provided in [1]). If R 0 → ∞, the problem reduces to the well-known vector Gaussian broadcast channel, the capacity region of which is achieved by Dirty Paper Coding (DPC). However, for fixed finite R 0 , DPC and other widely considered linear precoding schemes cannot be applied in a straightforward manner. A simple DAS system, the so-called Soft-Handoff model, was investigated in [2], by introducing a "compressed" version of DPC (CDPC), where the CP performs joint DPC under per-antenna power constraint and then sends the compressed (or quantized) codewords to the corresponding ATs via the wired links. While this scheme is expected to be near-optimal for very large R 0 , it is generally suboptimal at finite (possibly small) R 0 . Also, DPC is notoriously difficult to be implemented in practice, due to the nested lattice coding 1 This research was supported in part by the KCC (Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2011-11921-04001). construction and lattice quantization steps involved (See for example [3], [4]).Motivated by Compute-and-Forward (CoF) [5] (or quantized compute-and-forward (QCoF) [6]), we propose a novel coding strategy named Reverse QCoF (RQCoF) for the DAS downlink with finite backhaul link capacity R 0 . In QCoF and RQCoF the coding block length n can be arbitrarily large but the shaping block length is restricted to 1 (scalar quantization [6]). However, we would like to point out that the same approach can be straightforwardly applied to CoF based schemes, where also the shaping dimension becomes large (in this case, we would refer to the scheme as Reverse CoF (RCoF)).A. Overview of QCoFLet Z p = Z mod pZ denote the finite field of size p, with p a prime number, ⊕ denote addition over Z p , and g : Z p → R be the natural mapping of the elements of Z p onto {0, 1, ..., p− 1} ⊂ R. For a lattice Λ, let Q Λ (x) = argmin λ∈Λ { x − λ } denote the associated lattice quantizer, V = {x ∈ R n : Q Λ (x) = 0} the Voronoi region and define [x] mod Λ = x− Q Λ (x). For κ ∈ R, consider the two nested one-dimensional lattices Λ s = {x = κpz : z ∈ Z} and Λ c = {x = κz : z ∈ Z}, and define the constellation set S Λ c ∩ V s , where V s is the Voronoi region of Λ s , i.e., the interval [−κp/2, κp/2).The modulation mapping m : Z p → S is defined by v = m(u)[κg(u)] mod Λ s . The inverse function m −1 (·) is referred to as the demodulation mapping, and it is given byConsider the (real-valued) Abstract-We consider a distributed antenna system where L antenna terminals (ATs) are connected to a Central Processor (CP) via digital error-free links of finite capacity R0, and serve L user terminals (UTs). This system model has been widely investigated both for the uplink and the downlink, which are instances of the general multiple-access relay and broadcast relay networks. In this work we focus on the downlink, and propose a novel downlink precoding scheme nicknamed "Reverse Quantized Compute and Forward" (RQCoF). For this scheme we obtain achievable rates and compare with the state of the art available in the literature. We also provide simulation results for a realistic network with fading and pathloss with K > L UTs, and show that channel-based user selection produces large benefits and essentially removes the problem of rank deficiency in the system matrix. 1 I. SYSTEM AND PROBLEM DEFINITION We consider a distributed antenna system (DAS) with K user terminals (UTs) and L "antenna terminals" (ATs). All UTs and ATs have a single antenna each. The ATs are connected with a central processor (CP) via wired links of fixed rate R 0 . We study the downlink scenario, where the CP wishes to deliver independent messages to the UTs. This is a simple instance of a broadcast relay network, where the ATs operate as relays. In this work we focus on the symmetric rate, i.e., all messages have the same rate and assume that the CP and all UTs have perfect channel state information (more general results are provided in [1]). If R 0 → ∞, the problem reduces to the well-known vector Gaussian broadcast channel, the capacity region of which is achieved by Dirty Paper Coding (DPC). However, for fixed finite R 0 , DPC and other widely considered linear precoding schemes cannot be applied in a straightforward manner. A simple DAS system, the so-called Soft-Handoff model, was investigated in [2], by introducing a "compressed" version of DPC (CDPC), where the CP performs joint DPC under per-antenna power constraint and then sends the compressed (or quantized) codewords to the corresponding ATs via the wired links. While this scheme is expected to be near-optimal for very large R 0 , it is generally suboptimal at finite (possibly small) R 0 . Also, DPC is notoriously difficult to be implemented in practice, due to the nested lattice coding construction and lattice quantization steps involved (See for example [3], [4]). Motivated by Compute-and-Forward (CoF) [5] (or quantized compute-and-forward (QCoF) [6]), we propose a novel coding strategy named Reverse QCoF (RQCoF) for the DAS downlink with finite backhaul link capacity R 0 . In QCoF and RQCoF the coding block length n can be arbitrarily large but the shaping block length is restricted to 1 (scalar quantization [6]). However, we would like to point out that the same approach can be straightforwardly applied to CoF based schemes, where also the shaping dimension becomes large (in this case, we would refer to the scheme as Reverse CoF (RCoF)). A. Overview of QCoF Let Z p = Z mod pZ denote the finite field of size p, with p a prime number, ⊕ denote addition over Z p , and g : Z p → R be the natural mapping of the elements of Z p onto {0, 1, ..., p− 1} ⊂ R. For a lattice Λ, let Q Λ (x) = argmin λ∈Λ { x − λ } denote the associated lattice quantizer, V = {x ∈ R n : Q Λ (x) = 0} the Voronoi region and define [x] mod Λ = x− Q Λ (x). For κ ∈ R, consider the two nested one-dimensional lattices Λ s = {x = κpz : z ∈ Z} and Λ c = {x = κz : z ∈ Z}, and define the constellation set S Λ c ∩ V s , where V s is the Voronoi region of Λ s , i.e., the interval [−κp/2, κp/2). The modulation mapping m : Z p → S is defined by v = m(u) [κg(u)] mod Λ s . The inverse function m −1 (·) is referred to as the demodulation mapping, and it is given by u = m −1 (v) g −1 ([v/κ] mod pZ) with v ∈ S. Consider the (real-valued) L-user Gaussian multiple access channel with inputs {x ,i : i = 1, ..., n} for = 1, ..., L, output y i = L i=1 h x ,i + z i , for i = 1, . . . , n,(1) where the z i 's are i.i.d. ∼ N (0, 1). All users encode their information messages {w ∈ Z k p : = 1, . . . , L} using the same linear code C over Z p (i.e., denoting information sequences and codewords by row vectors, we have c = w G where G is a generator matrix for C), and produce their channel inputs according to x ,i = [m(c ,i ) + d ,i ] mod Λ s , i = 1, . . . , n,(2) where c ,i is the i-th symbol of c and d ,i 's are i.i.d. dithering symbols ∼ Uniform(V s ), known at the receiver. The channel inputs x ,i are uniformly distributed over V s and have second moment SNR E[\|x ,i \| 2 ] = κ 2 p 2 /12. The receiver's goal is to recover a linear combination c = q c of the transmitted users' codewords, for some coefficients q ∈ Z p . For this purpose, the receiver selects the integer coefficients vector a = (a 1 , ..., a L ) T ∈ Z L and produces the sequence of quantized observations u i = m −1 Q Λc αy i − a T d i mod Λ s ,(3) for i = 1, . . . , n. It easy to show [6] that (3) is equivalent to u i = L =1 q c ,i ⊕z i ,(4)with q = g −1 ([a ] mod pZ). Here,z i = m −1 ([Q Λc (ε)] mod Λ s ) where ε denotes the effective noise, capturing a Gaussian additive noise and non-integer penalty, and its variance [6] is σ 2 ε = a T (SNR −1 I + hh T ) −1 a.(5) By [6, Th. 1], the achievable computation rate of QCoF is given by R QCoF = log p − H(z).(6) Also, by [5,Th. 4], the achievable computation rate of CoF is given by R CoF (σ 2 ε ) = 1 2 log(SNR/σ 2 ε ).(7) We showed in [6] that, for fixed large SNR 1 and sufficiently large p (e.g., p ≥ 251), the (6) and (7) differ approximately by the shaping gain, i.e., ≈ 0.25 bits per real dimension. Remark 1: In order to achieve the CoF rate, p must grow to infinity in the lattice construction and the rank of the system matrix Q is the same as the rank of A over R, by [5,Th. 11]. II. REVERSE QUANTIZED COMPUTE-AND-FORWARD The main idea is that each UT decodes a linear combination (over the finite field) of the messages sent by the ATs using QCoF. In short, we exchange the role of the ATs and UTs and use QCoF in the reverse direction. However, decoding linear combination of the information messages is useful only when these combinations can be shared such that the individual messages can be recovered, provided that the resulting system of linear equations is invertible over Z p . Since the UTs do not cooperate, sharing the decoded linear combinations is impossible in the downlink. Nevertheless, thanks to algebraic structure of QCoF (or CoF), the messages from the ATs can be the precoded versions of the original information messages and hence, using an appropriate invertible precoding over Z p at the CP, the effect of the linear combination can be undone at the transmitter, so that every UT obtains just its own desired message. We present coding strategies considered in this work, assuming the K = L and (real-valued) channel matrix H ∈ R L×L . Let Q denote the system matrix whose elements in the -th row, denoted by q T = (q ,1 , ..., q ,L ), indicate the coefficients of the linear combination decoded at the -th UT as given in (4). For the time being we assume that these matrices are full rank over Z p , although they may be rank deficient since each UT chooses its own linear combination coefficients independently of the other nodes. The case of rank deficiency will be handled later. Letz be the discrete additive noise (over Z p ) at the -th UT. The detailed description of "reverse" QCoF (RQCoF) is as follows. • For the given Q, the CP precodes the user information messages {w ∈ Z k p : = 1, ..., L} using the inverse system matrix Q −1 . The precoded L-dimensional vectors of information symbols to be transmitted by the ATs are given by (µ 1,i , ..., µ L,i ) T = Q −1 (w 1,i , ..., w L,i ) T , for i = 1, . . . , k. (8) • The CP forwards each block µ = (µ ,1 , ..., µ ,k ) to the -th AT, during n time slots, corresponding to the duration of a codeword sent on the wireless channel. Therefore, we have the rate constraint (k/n) log p ≤ R 0 . • After receiving k symbols, the -th AT locally encodes its information symbols µ using the same linear code C over Z p (i.e., c = µ G), and produces its channel input according to x ,i = [m(c ,i ) + d ,i ] mod Λ s , for i = 1, . . . , n. (9) • By [6, Th. 1], the -th UT can recover a noiseless linear combination of ATs' information symbols if R ≤ log p − max {H(z )}. This is given by q T (µ 1,i , ..., µ L,i ) T = q T Q −1 (w 1,i , ..., w L,i ) T = w ,i , for i = 1, . . . , k.(10) Hence, the -th UT can successfully recover its desired message. The following rate is achievable by RQCoF: R RQCoF = min{R 0 , log p − max {H(z )}.(11) Similarly, from (7), we can get an achievable rate per user of RCoF R RCoF = min{R 0 , min {R CoF (σ 2 ε )}}.(12) Finally, the achievable rate of RQCoF (or RCoF) is maximized by minimizing the variance of effective noise in (5) with respect to A subject to the system matrix Q is full rank over Z p . This problem was solved in [6] using the LLL algorithm [7], possibly followed by Phost or Schnorr-Euchner enumeration (See [8]) of the non-zero lattice points in a sphere centered at the origin, with radius equal to the shortest vector found by LLL. III. COMPRESSED INTEGER-FORCING BEAMFORMING In short, the idea underlying RQCoF is that each UT converts its own downlink channel into a discrete additive-noise multiple access channel over Z p . Since each UT is interested only in its own message, the CP can precode the messages using zero-forcing linear precoding over Z p , at no transmit power additional cost (unlike linear zero-forcing over R). It is known that the performance of CoF (and therefore QCoF) is quite sensitive to the channel coefficients, due to the noninteger penalty, since the channel coefficients are not exactly matched to the integer coefficients of linear combinations [5], [6]. The same problem arises in RQCoF (or RCoF), due to their formal equivalence. In [9], it was shown that integerforcing linear receiver (IFLR) can eliminate this penalty by forcing the effective channel matrix to be integer. Here, we propose a new beamforming strategy named Integer-Forcing Beamforming (IFBF), that produces a similar effect for the downlink. We present the IFBF idea assuming R 0 = ∞, as the dual scheme of IFLR, and consider finite R 0 later. In IFBF, the beamforming vectors W = [w 1 , ..., w L ] are chosen such that the effective channel matrixH = HW is integer-valued. Then, the channel matrix is inverted over Z q by using RQCoF, as previously presented. In this case, sinceH ∈ Z L×L , RQCoF does not suffer from the non-integer penalty. Further, we extend IFBF to the case of finite R 0 by using quantization, as in done in [2], where CP forwards the quantized sequences to the ATs for which the quantization noise is determined from standard rate-distortion theory bounds. It is assumed that H ∈ R L×L is full rank and the detailed description of IFBF is as follows. For a given A ∈ Z L×L (optimized later), the CP uses the beamforming matrix W = H −1 A and the system matrix Q = [A] mod pZ as in Section I-A. Assuming that Q is full rank over Z p , the CP produces the downlink streams x = {x ,i : i = 1, . . . , n}, for = 1, . . . , L as follows. • The CP precodes the user information messages {w ∈ Z k p : = 1, ..., L} using the inverse system matrix Q −1 : (µ 1,i , ..., µ L,i ) T = Q −1 (w 1,i , ..., w L,i ) T ,(13) for i = 1, . . . , k. • The CP encodes the precoded information messages using the same linear code C over Z p (i.e., c = µ G) and produces the downlink stream according to x ,i = [m(c ,i ) + d ,i ] mod Λ s , for i = 1, . . . , n.(14) Using the predefined W, the CP produces the precoded channel inputs {v ,i : i = 1, . . . , n} using and forwards them to the ATs via the wired links. Consistently with our system definition, we impose a per-antenna power constraint equal to SNR (with suitable normalization). Hence, the second moment of x ,i is determined as E[\|x ,i \| 2 ] = SNR/ max { H −1 a 2 },(15) which guarantees that the power of the signal transmitted from the -th AT has the required power E[\|v ,i \| 2 ] = SNR. The received signal at the -th UT is given by y ,i = a T x ,i + z ,i , for i = 1, . . . , n.(16) Notice that thanks to the IFBF the non-integer penalty is equal to zero. So, every UT can recover its desired messages by decoding the linear combination of ATs' messages with integer coefficients a as shown in (10). Finally, the achievable rate of IFBF with RQCoF can be obtained by numerically computing the entropy of discrete additive noise over Z p corresponding to effective noise ε ∼ N (0, max {\|\|H −1 a \|\| 2 }) where the impact of power constraint is included in the effective noise. The following rate is achievable by IFBF with RQCoF: R IFBF = log p − max {H(z )}(17) for any full-rank matrix Q, wherez = m −1 ([Q Λc (ε )] mod Λ s ). From (7), the following rate is achievable by IFBF with RCoF: R IFBF = 1 2 log(SNR/ max {\|\|H −1 a \|\| 2 })(18) for any full-rank integer matrix A. For the case of finite R 0 , we propose a "compressed" IFBF (CIFBF) where the CP forwards the quantized channel inputŝ v ,i = v ,i +ẑ ,i for i = 1, . . . , n, to the -th AT, where {ẑ ,i : i = 1, . . . , n} denotes the quantization noise sequence, with variance (quantization mean-square error) equal to σ 2 z . From the standard rate-distortion theory, the CP can forward the {v ,i : i = 1, ..., n} to the -th AT if R 0 ≥ I(v ;v ),(19) where the index i is omitted for brevity. Using the well-known maximum entropy argument on (19) we have the bound I(v ;v ) ≤ 1 2 log(SNR/σ 2 z ).(20) From (19) and (20), we obtain σ 2 z = SNR/2 2R0 and E[\|v ,i \| 2 ] = SNR/(1 + 1/(2 2R0 − 1)), due to the power constraint. Accordingly, we have E[\|x ,i \| 2 ] = SNR/ max {\|\|H −1 a \|\| 2 }(1+1/(2 2R0 −1)). (21) Also, the effective noise at the -th UT is given by ε ,i = z ,i + L k=1 h ,kẑk,i ,(22) where the second term captures the impact of quantization noise and its variance is σ 2 ε = 1 + \|\|h \|\| 2 SNR/2 2R0 .(23) Finally, the achievable rate of CIFBF with RQCoF can be obtained numerically computing the entropy of discrete additive noise over Z p corresponding to the effective noise ε ∼ N (0, σ 2 ε ) where the impact of power constraint and quantization noise are included in the effective noise: σ 2 ε = max {\|\|H −1 a \|\| 2 }(1 + (1 + \|\|h \|\| 2 SNR)/(2 2R0 − 1)). (24) The following rate is achievable by CIFBF with RQCoF: R CIFBF = log p − max {H(z )}(25) for any full-rank matrix Q, wherez = m −1 ([Q Λc (ε )] mod Λ s ). From the (7), the following rate is achievable by CIFBF with RCoF: R CIFBF = R IFBF − 1 2 max {log(1 + (1 + \|\|h \|\| 2 SNR)/(2 2R0 − 1))} for any full-rank integer matrix A. Finally, the achievable rate is maximized by minimizing the max {\|\|H −1 a \|\| 2 } subject to full-rank constraint. This problem can be thought of as finding the L linearly independent "shortest lattice points" of the Ldimensional lattice generated by H −1 . This can be efficiently obtained using the LLL algorithm [7]. Specifically, for a given lattice Λ defined by Λ = {x = H −1 z : z ∈ Z L }, a reduced basis of lattice is obtained through a unimodular matrix U such that Λ = {x = H −1 Uz : z ∈ Z L }. Let F = H −1 U generates the same lattice but has "reduced" columns, i.e., the columns of F have small 2-norm. The solution of the original problem can be chosen as a = u where u denotes the -th column of U. While finding the optimal reduced basis for a lattice (e.g., finding the optimal U) is an NP-hard problem, the LLL algorithm finds a good reduced basis with low-complexity [7]. Remark 2: In CIFBF, relays (i.e., the distributed antenna elements) have very low-complexity and are oblivious to codebooks since they just forward the received signals from CP, not requiring modulation and encoding. Remark 3: In terms of performance, it is worthwhile understand the impact of non-integer penalty and quantization noise depending on parameters R 0 , SNR, and so on. As R 0 → ∞, the effect of quantization noise vanishes and thus, CIFBF would be better than RCoF. However, when R 0 is small, RCoF without beamforming may perform better than CIFBF since quantization noise would be severe in this case. A numerical result in a particular case is provided in Fig. 1. IV. SCHEDULING AND NUMERICAL RESULTS For the sake of comparison with CDPC we consider the same Soft-Handoff model of [2], with L ATs and L UTs for which the received signal at the -th UT is given by y ,i = x ,i + γx −1,i + z ,i ,(26) where γ ∈ [0, 1] represents the inter-cell interference level and z ,i ∼ CN (0, 1). The extension of results in previous sections to the (complex-valued) Soft-Handoff model is easy and done in the usual way [1]. In this example, thanks to the dualdiagonal structure of the channel matrix, the system matrix is guaranteed to have rank L. In Fig. 1, we compare various coding strategies where the upper bound and achievable rates for CDPC are provided by [2]. It is remarkable that RCoF can achieve the upper bound when R 0 ≤ 4 bits and outperforms the other schemes up to R 0 ≈ 6 bits. Notice that when γ = 1 (e.g., integer channel matrix), RCoF almost achieves the upper bound, showing better performance than other schemes. Also, from the Fig. 2, we can see that RCoF is a good scheme when R 0 is small and SNR is high, i.e., small cell networks with finite-backhaul capacity. Not surprisingly, RQCoF approaches the performance of RCoF within the shaping loss of ≈ 0.25 bits/symbol, as already noticed in the uplink case [6]. For the RQCoF, there would be a concern on rank-deficiency of system matrices Q in particular when p is small, since every UT selects its own linear combination coefficients independently of the other nodes. This problem can be avoid by scheduling since it can select a group of UTs (or ATs) for which the system matrix is invertible. In fact, this is a complex combinatorial optimization problem, which in some cases, can be formulated as the maximization of linear function over matroid constraint [10] and thus, greedy algorithm yields provably good performance. The following is the example that greedy algorithm is optimal. Consider a DAS system with K UTs and L ATs where K ≥ L and we consider the user selection that finds the subset of UTs to maximize the symmetric rate subject to full-rank constraint of system matrix. Independently of the user selection algorithm, we can obtain the coefficients of the linear combination of the k-th UT (e.g., k-th row of Q) and the variance of effective noise (i.e., σ 2 ε k for k = 1, . . . , K) that determines the achievable rate, for the given H ∈ R K×L . Let K be the subset of row indices [1 : K]. Also, let Q(K) denote the submatrix of Q consisting of k-th rows for k ∈ K. Assuming that Q has rank L, the user selection problem of finding L UTs can be formulated as arg min K⊂[1:K] max{σ 2 ε k : k ∈ K}(27) subject to Rank p (Q(K)) = L We first give the definition of matroid and subsequently, show that the above problem is equivalent to the maximization of linear function over matroid constraint. Matroids are structures that generalize the concept of linear independence for general sets. Formally, we have [10]: Definition 1: A matroid M is a tuple M = (Ω, I), where Ω is a finite ground set and I ⊆ 2 Ω (the power set of Ω) is a collection of independent sets, such that: 1) I is nonempty, in particular, φ ∈ I 2) I is downward closed; i.e., if Y ∈ I and X ⊆ Y, then X ∈ I 3) if X , Y ∈ I, and \|X \| < \|Y\|, then ∃y ∈ Y \ X such that X ∪ {y} ∈ I. This can be easily proved by the fact that Q has rank L and constraint is matroid. Rado and Edmonds proved that the Best-In-Greedy algorithm (See Algorithm 1) finds an optimal solution [10]. Detailed scheduling algorithms for various scenarios are omitted because of space limitation (See [1]). Algorithm 1 Best-In-Greedy Algorithm Input: M = (Ω, I) and w k = 1/σ 2 ε k for k ∈ [1 : K] step 0. Sort [1 : K] such that w 1 ≥ w 2 ≥ · · · ≥ w K Initially k = 1 and K = φ step 1. If Rank p (Q(K ∪ {k})) > Rank p (Q(K)), then K ← K ∪ {k} step 2. Set k = k + 1 step 3. Repeat until Rank p (Q(K)) = L In Fig. 3, we consider a DAS with channel matrix H ∈ R 20×5 , with i.i.d. Gaussian distributed elements ∼ N (0, 1). In our simulation we assumed that if the resulting system Fig. 3. Achievable rates per user as a function of SNRs, for finite capacity R 0 = 3 bits and p = 17 for RQCoF. matrix after greedy selection is rank deficient then the achieved symmetric rate of all users is zero, for that specific channel realization. Then, we computed the average achievable rate with user selection, by Monte Carlo averaging with respect to the random channel matrix. Random selection indicates that 5 UTs are randomly and uniformly chosen out of the 20 UTs. As shown in Fig. 3, RCoF has the rank-deficiency when using random selection, although the rank of the resulting 5 × 5 matrix over R is equal to 5 with probability 1. However, it is remarkable that RQCoF with greedy user selection does not suffer from the rank-deficiency problem, even for relatively small values of p (e.g., p = 17). This is indicated by the fact that the gap from the RCoF is essentially equal to the the shaping loss, as in the case where the full-rank system matrix is guaranteed by assumption. {y i : i = 1, ..., n} and coefficients h = (h 1 , ..., h L ) T ∈ R L , defined by (v 1 1,i , . . . , v L,i ) T = W(x 1,i , . . . , x L,i ) T , for i = 1, . . . , n, Fig. 1 .Fig. 2 . 12SNR = 20dB. Achievable rates per user as a function of finite capacity R 0 , for inter-cell interference γ ∼ Uniform(0.5, Achievable rates per user as a function of SNRs, for finite capacity R 0 = 2 or 4 bits, inter-cell interference level γ = 0.7, and p = 251 for RQCoF. Let Ω = [1 : K] and I = {K ⊂ [1 : K] : Q(K) has linearly independent rows}. From Definition 1, M = (Ω, I) forms a so-called linear matroid. Then, the optimization problem (27)-(28) 1 . 4 .SNR [dB]Rate per User RCoF, Greedy RCoF, Random RQCoF, Greedy RQCoF, Random This research was supported in part by the KCC (Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA-2011-11921-04001). A low-complexity architecture for distributed antenna system: Coding and Scheduling. S Hong, G Caire, In preparationS. Hong and G. Caire, "A low-complexity architecture for distributed antenna system: Coding and Scheduling," In preparation. Downlink Multicell Processing with Limited-Backhaul Capacity. O Simeone, O Somekh, H V Poor, S Shamai, EURASIP J. on Adv. in Signal Process. O. Simeone, O. Somekh, H. V. Poor, and S. Shamai (Shitz), "Downlink Multicell Processing with Limited-Backhaul Capacity," EURASIP J. on Adv. in Signal Process., Jan. 27-Feb. 1, 2009. A close-to-capacity dirty paper coding scheme. U Erez, S , Ten Brink, IEEE Trans. on Inform. Theory. 5110U. Erez and S. Ten Brink, "A close-to-capacity dirty paper coding scheme," IEEE Trans. on Inform. Theory, vol. 51, no. 10, pp. 3417- 3432, October 2005. Superposition coding for side-information channels. A Bennatan, D Burshtein, G Caire, S Shamai, IEEE Trans. on Inform. Theory. 525Shitz)A. Bennatan, D. Burshtein, G. Caire, and S. Shamai (Shitz), "Super- position coding for side-information channels," IEEE Trans. on Inform. Theory, vol. 52, no. 5, pp. 1872-1889, October 2006. Compute-and-forward: Harnessing interference through structured codes. B Nazer, M Gastpar, IEEE Trans. on Inform. Theory. 5710B. Nazer and M. Gastpar, "Compute-and-forward: Harnessing interfer- ence through structured codes," IEEE Trans. on Inform. Theory, vol. 57, no. 10, pp. 6463-6486, October 2011. Quantized compute and forward: A lowcomplexity architecture for distributed antenna systems. S Hong, G Caire, proceedings of IEEE Inform. Theory Workshop (ITW). IEEE Inform. Theory Workshop (ITW)S. Hong and G. Caire, "Quantized compute and forward: A low- complexity architecture for distributed antenna systems," in proceedings of IEEE Inform. Theory Workshop (ITW), October 2011. Factoring polynomials with rational coefficients. A K Lenstra, H W Lenstra, L Lovasz, Math. Ann. 261A. K. Lenstra, H.W. Lenstra, and L. Lovasz, "Factoring polynomials with rational coefficients," Math. Ann., Vol. 261, pp. 515.534, 1982. Lattice basis reduction: Improved practical algorithms and solving subset sum problems. C Schnorr, M Euchner, Mathematical programming. 661C. Schnorr and M. Euchner, "Lattice basis reduction: Improved practical algorithms and solving subset sum problems," Mathematical program- ming, vol. 66, no. 1, pp. 181-199, 1994. Integer-Forcing Linear Receivers. J Zhan, B Nazer, U Erez, M Gastpar, IEEE Trans. Inform. Theory. J. Zhan, B. Nazer, U. Erez, and M. Gastpar, "Integer-Forcing Linear Receivers," submitted to IEEE Trans. Inform. Theory, Jan. 2012. 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[ "Experimental Fock-State Bunching Capability of Non-Ideal Single-Photon States", "Experimental Fock-State Bunching Capability of Non-Ideal Single-Photon States" ]	[ "Petr Zapletal \nDepartment of Optics\nPalacký University\n17. listopadu 1192/1277146OlomoucCzech Republic\n", "Tom Darras \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance\n", "Hanna Le Jeannic \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance\n", "Adrien Cavaillès \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance\n", "Giovanni Guccione \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance\n", "Julien Laurat \nLaboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance\n", "Radim Filip \nDepartment of Optics\nPalacký University\n17. listopadu 1192/1277146OlomoucCzech Republic\n" ]	[ "Department of Optics\nPalacký University\n17. listopadu 1192/1277146OlomoucCzech Republic", "Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance", "Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance", "Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance", "Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance", "Laboratoire Kastler Brossel\nSorbonne Université\nCNRS\nENS-Université PSL, Collège de France\n4 Place Jussieu75005ParisFrance", "Department of Optics\nPalacký University\n17. listopadu 1192/1277146OlomoucCzech Republic" ]	[]	Advanced quantum technologies, as well as fundamental tests of quantum physics, crucially require the interference of multiple single photons in linear-optics circuits. This interference can result in the bunching of photons into higher Fock states, leading to a complex bosonic behaviour. These challenging tasks timely require to develop collective criteria to benchmark many independent initial resources. Here we determine whether n independent imperfect single photons can ultimately bunch into the Fock state \|n . We thereby introduce an experimental Fock-state bunching capability for single-photon sources, which uses phase-space interference for extreme bunching events as a quantifier. In contrast to autocorrelation functions, this operational approach takes into account not only residual multi-photon components but also vacuum admixture and the dispersion of the individual photon statistics. We apply this approach to high-purity single photons generated from an optical parametric oscillator and show that they can lead to a Fock-state capability of at least 14. Our work demonstrates a novel collective benchmark for single-photon sources and their use in subsequent stringent applications. arXiv:2012.08544v2 [quant-ph]	10.1364/optica.419230	[ "https://export.arxiv.org/pdf/2012.08544v2.pdf" ]	229,188,129	2012.08544	45a2b09ab88ba3e2f3196253c365ed3fb9815b6a	Experimental Fock-State Bunching Capability of Non-Ideal Single-Photon States Petr Zapletal Department of Optics Palacký University 17. listopadu 1192/1277146OlomoucCzech Republic Tom Darras Laboratoire Kastler Brossel Sorbonne Université CNRS ENS-Université PSL, Collège de France 4 Place Jussieu75005ParisFrance Hanna Le Jeannic Laboratoire Kastler Brossel Sorbonne Université CNRS ENS-Université PSL, Collège de France 4 Place Jussieu75005ParisFrance Adrien Cavaillès Laboratoire Kastler Brossel Sorbonne Université CNRS ENS-Université PSL, Collège de France 4 Place Jussieu75005ParisFrance Giovanni Guccione Laboratoire Kastler Brossel Sorbonne Université CNRS ENS-Université PSL, Collège de France 4 Place Jussieu75005ParisFrance Julien Laurat Laboratoire Kastler Brossel Sorbonne Université CNRS ENS-Université PSL, Collège de France 4 Place Jussieu75005ParisFrance Radim Filip Department of Optics Palacký University 17. listopadu 1192/1277146OlomoucCzech Republic Experimental Fock-State Bunching Capability of Non-Ideal Single-Photon States Advanced quantum technologies, as well as fundamental tests of quantum physics, crucially require the interference of multiple single photons in linear-optics circuits. This interference can result in the bunching of photons into higher Fock states, leading to a complex bosonic behaviour. These challenging tasks timely require to develop collective criteria to benchmark many independent initial resources. Here we determine whether n independent imperfect single photons can ultimately bunch into the Fock state \|n . We thereby introduce an experimental Fock-state bunching capability for single-photon sources, which uses phase-space interference for extreme bunching events as a quantifier. In contrast to autocorrelation functions, this operational approach takes into account not only residual multi-photon components but also vacuum admixture and the dispersion of the individual photon statistics. We apply this approach to high-purity single photons generated from an optical parametric oscillator and show that they can lead to a Fock-state capability of at least 14. Our work demonstrates a novel collective benchmark for single-photon sources and their use in subsequent stringent applications. arXiv:2012.08544v2 [quant-ph] INTRODUCTION Beyond its fundamental significance, the Hong-Ou-Mandel (HOM) effect [1], where two single photons interfere on a beamsplitter, has been central to the development of quantum technologies. With the advance of complex quantum information protocols and networking architectures [2,3], the availability of multiple indistinguishable photons is becoming a cornerstone. Multiphoton interference is required in quantum processing with optical [4][5][6][7][8][9][10][11][12][13][14] or microwave photons [15][16][17], ranging from boson sampling studies to quantum state engineering. It also plays a key role in quantum sensing [18][19][20], noiseless amplification [21], quantum key distribution [22,23] or error correction [24][25][26]. Multi-photon interference leads to a non-trivial redistribution of photons between optical modes. To achieve such interferences, all photons have to be indistinguishable. Several methods have been recently developed to investigate this indistinguishability using different benchmarks, e.g., fidelity [27] or specific photon correlation measures [28][29][30][31][32]. However, the joint impact of photon statistics from many imperfect single-photon states, i.e., exhibiting unwanted vacuum and residual multi-photon components, on multi-photon interference has remained elusive. The joint statistical influence of these parameters cannot be described by evaluating properties of single- ‡ Present address: Friedrich-Alexander University Erlangen-Nürnberg (FAU), Department of Physics, 91058 Erlangen, Germany. § photon states that are averaged over many experimental runs. Hence, we need criteria, experimental data and subsequent analysis to determine whether independently generated single photons can, in principle, produce the targeted multi-photon interference effects. Multi-photon interference effects come in a variety of flavors. An extreme event corresponds to the bunching of n single photons into the Fock state \|n [33][34][35]. Such bunching can appear in a linear-optical network with inputs fed by indistinguishable single photons, as shown in Fig. 1. The elementary example is the appearance of Fock state \|2 based on the HOM effect, as demonstrated in experiments with optical photons but also with microwave photons [16,36], phonons in trapped ions [37] or surface plasmons [38,39]. This extreme bunching event, i.e., the result of a clear operational procedure, enables to introduce a strong benchmark for single-photon states that evaluates their ability to undergo multi-photon interference [4]. This Fock-state bunching capability relies on negativities of the resulting Wigner function, which provide a very sensitive signature of the non-classicality of the generated higher Fock states [3,42]. In contrast to the well-known second-order autocorrelation function at zero time delay g (2) (0), which measures the suppression of the multi-photon contribution and affects the interference visibility [43], the capability is also strongly dependent on the vacuum admixture. Another crucial difference is that it collectively tests multiple photon statistics and determines the joint statistical impact of small discrepancies between them. This provides more stringent and accurate evaluation than other available characteristics. The previous theoretical study based on Monte-Carlo simulations has only predicted that the bunching of single photons is affected by vacuum and multi-photon contri-FIG. 1: Fock-state bunching capability of non-ideal single-photon states. A single-photon source provides photons with different vacuum admixture and residual multi-photon components, as depicted by the photon-number distributions (left). These states are used as inputs of a balanced linear optics networkÛ . In an extreme case, all photons can bunch into just one output mode whereas all other modes are in the vacuum state. This stage is done computationally and provides the expected photonnumber distribution Pn for the output mode (right). The negativities of the associated Wigner function are used to determine the Fock-state capability. In contrast to other measures, this collective bechmark depends not only on the vacuum admixture and multiple-photon statistics of the imperfect input photons but also on the small discrepancies between them. butions [4]. However, these contributions and their dispersion in non-ideal photon statistics of many independent copies are too complex to be described, specifically when the number of photons increases. Experimental data are necessary to confirm this prediction. Here, we employ the bunching capability to collectively benchmark experimental single-photon states using heralded single photons generated by parametric down-conversion from an optical parametric oscillator (OPO). By tuning the photon source properties, we address the scaling of the capability with the statistics of non-ideal single photons. We hereby provide a crucial insight into the combined effects of non-ideal photon statistics of independently generated single photons. We demonstrate that experimentally generated single photons can bunch into the Fock state \|14 with high fidelity and suppressed higher Fock states contributions. We show that the Fock state capability non-linearly decreases with photon loss, providing a more stringent characterization than g (2) (0), which is independent of photon loss, and also than negative Wigner function that decreases only linearly. Our results indicate that despite the negative impact of multi-photon contributions typically reported using g (2) (0), they prevent the bunching of single-photon states into a respective Fock state less severely than optical loss. QUANTIFIER PRINCIPLE We first describe the quantifier principle. To collectively test the ability of the generated single photons to undergo multi-photon interference, we computationally determine the Wigner function of the higher Fock state, which can, in principle, appear from multiple copies of the single-photon state, as depicted in Fig. 1. The area in phase space, where the Wigner function of the ideal Fock state \|n is negative, is composed of n/2 or (n−1)/2 concentric annuli if n is an even number or an odd number, respectively. By definition, a single-photon state has the capability of the Fock state \|n if the Wigner function of the state, which can be generated from n independent copies of the single-photon state, has the same number of negative annuli as the ideal Fock state \|n [4]. The negative annuli in the Wigner function witness the nonclassical nature of the multi-photon interference in phase space. The Fock-state capability, which is determined computationally, collectively tests the copies of a singlephoton state, even though any multi-copy procedure is not implemented in the laboratory. In theory, copies of the ideal single-photon state \|1 have the capability of an arbitrary Fock state \|n . For states generated by single-photon sources, the negative annuli in the Wigner function are sensitive to the presence of vacuum and multi-photon contributions. Also, the exact distribution of residual multi-photon statistics in many non-ideal single-photon states is not known. As a consequence, the joint effect of small discrepancies between individual single-photon copies on multi-photon interference has to be investigated by applying the quantifier on photon statistics measured in an experiment. In this way, we can determine whether the single-photon sources have a sufficient quality for applications in quantum technology that require multi-photon interference. I: Photon-number statistics of heralded single photons. Each set is obtained by successive measurements under the same conditions (in particular pump power). The table displays the single-photon component P1, the multi-photon probability P2+, the second-order correlation function g (2) (0), and the negativity at the origin of the Wigner function. To study this benchmark, we used heralded singlephoton states generated using a two-mode squeezer, i.e., a type-II phase-matched optical parametric oscillator operated well below threshold (see Appendix). The signal and idler photons at 1064 nm are separated on a polarizing beam-splitter and the idler photon is detected via a high-efficiency superconducting nanowire single-photon detector. This detection event heralds the generation of a single photon in the signal mode. The generated state is emitted into a well-defined spatio-temporal mode [44], with a bandwidth of about 65 MHz. The state is measured via high-efficiency homodyne detection, with a visibility of the interference with the local oscillator above 99%, and reconstructed via maximum-likelihood algorithms [2]. The experimental setup has been described elsewhere [1,46]. Importantly, the OPO used in this work exhibits a close-to-unity escape efficiency, i.e., the transmission of the output coupler is much larger than the intracavity losses [48]. As a result, a large heralding efficiency can be obtained, i.e., a very low admixture of vacuum. A single-photon component up to 91% is achieved. Also, by changing the pump power the multi-photon component can be increased at will. These features enable us to explore different combinations of state imperfections. Seven sets of data were recorded, each of them being obtained by a repetitive measurement of the single-photon states generated under the same conditions. Parameters of the sets are given in Table I. They include the singlephoton component P 1 and the probability P 2+ of finding two or more photons. These measured quantities give also access to the conditional second-order autocorrelation function at zero-time delay g (2) (0) [1]. P1 P2+ g (2) (0) 2π × W (0, EXPERIMENTAL FOCK-STATE CAPABILITY To test a particular data set for the Fock-state capability n, the data are randomly partitioned in n subsets from which n photon-number statistics are obtained and used as the quantifier inputs. The output-state Wigner function of the computational quantifier is averaged over 30 such random choices. From the averaged output-state Wigner function, it is determined whether the data set has the Fock-state capability n (see Appendix). The capability for all data sets is depicted in Fig. 2 as a function of P 1 and P 2+ . The quantifier is presently computationally limited by the Fock-state capability 14 (see Appendix), which is already a very large number in this operational context. All data sets for which this capability 14 is obtained may also have the capability of a higher Fock state. In the following, we describe the different measured points and typical trends. First, single-photon states with a low purity due to a vacuum component close to 50% (brown bars in Fig. 2, sets 1 and 2 in Table I) have only the trivial capability of the Fock state \|1 , despite their very low g (2) (0). This shows that the broadly used autocorrelation function does not fully characterize the ability to bunch into higher Fock states exhibiting non-classical signatures. In particular this example demonstrates that the capability is more sensitive to vacuum mixture, as a state obtained from two copies of these single photons would have a positive Wigner function. Due to their trivial capability, such states are not a useful resource for the preparation Table I. These parameters are averaged over photon-number statistics from a given data set obtained by successive measurements under the same experimental conditions. Colors denote the Fock-state capability. The gray-shaded area excludes the unphysical probabilities P1 + P2+ > 1. The standard deviation of the probabilities are given by the thicknesses of the color bars. of large Fock states that could be used e.g. for quantum metrology [18][19][20] or error correction [24][25][26]. The necessary condition for a non-trivial capability n > 1 is to reach a single-photon component P 1 > 2/3 [4]. Above this threshold, the capability moderately grows with P 1 . As can be seen in Fig. 2, the state corresponding to the green bar (set 3 in Table I) has a multi-photon component P 2+ = 0.02 and the capability of the Fock state \|3 . The state associated to the red bar (set 4) has the capability of the Fock state \|4 despite having a similar single-photon component as the previous state but a larger, still low, probability P 2+ = 0.05. For a given P 1 , an increase in P 2+ may thereby lead to a larger capability. Actually, this increase in P 2+ comes in that case with a decrease in the vacuum component, indicating that the bunching is less affected by multi-photon contributions than vacuum admixture. We have shown in additional simulations that at fixed vacuum the capability decreases with the multi-photon component. Finally, for P 1 > 0.8, the capability is expected to rapidly increase and to diverge at P (∞) 1 = 0.885, where an arbitrary capability can be reached [4]. The experimental results agree well with this prediction and highlight the nonlinearity of the quantifier. The verification of this trend is an important benchmark for the development of single-photon sources. The data sets indicated with blue bars have at least the capability 14. For the set 7, note that its g (2) (0) = 0.05 does not significantly differ from that of the states with the trivial Fock-state capability. The capability 14 is also achieved for lower single-photon fidelities P 1 and higher multi-photon contributions P 2+ , even for a state with four times larger g (2) (0) = 0.2. However, these states might have a lower capability than the set 7 due to the saturation to 14 for reason of computational power. Figure 3 presents the output of the computational quantifier with fourteen input states randomly chosen from the data set 7, i.e., the set with the highest heralding efficiency and lowest multi-photon component. Figure 3a first provides the cut through the Wigner function. The output Wigner function is fitted by the one of a lossy Fock state \|14 , with a fitted attenuation parameter η = 0.9205±0.0005. The fit shows that the oscillations of the output Wigner function in phase space coincide with the ones of the attenuated Fock state \|14 . The photonnumber statistics of the output state and attenuated Fock state are compared in Fig. 3b. The good cut-off of the multi-photon contributions with more than fourteen photons in the statistics of the output state is another feature that further demonstrates the high quality of the initial single-photon states. Such result was made possible only by considering single-photon states with limited multiphoton contributions and very low vacuum admixture, as provided by the OPO-based source used in this work. DISCUSSION: EFFECT OF LOSS AND TRUNCATION We now come to an additional characterization of the quantifier, i.e., its evolution with optical losses. This quantifier depth, in analogy to non-classicality depth [49], is tested by considering attenuation for two states randomly chosen from different data sets. Figure 4 shows the Fock-state capability as a function of the attenuation parameter η, for the state with P 1 = 0.91 and P 2 = 0.02 (blue in Fig. 2) and the state with P 1 = 0.74 and P 2 = 0.05 (red in Fig. 2). Both states exhibit a similar g (2) (0) parameter (which is preserved with attenuation), but different initial capabilities 14 and 4, respectively. As it can be seen, the capability depends nonlinearly on the attenuation η. This is in contrast to the negativity of the single-photon Wigner function which decreases linearly with the attenuation. As a result, the capability allows more sensitive benchmarking of singlephoton states than the negativity of the Wigner function. The results in Fig. 4 are also superimposed with two plots that give the evolution of the capability with optical losses for states whose photon-number statistics are truncated, i.e., neglecting the multi-photon contribution. The discrepancy in the Fock-state capability between the experimental states and the truncated ones demonstrates that the multi-photon contributions play a significant role in such bunching experiments. The truncation of multi-photon contributions can be a limiting approximation when multi-photon interference is involved. CONCLUSION In conclusion, with the advance of quantum technologies, novel procedures and applications put challenging demands on resources and required benchmarking [50]. In this broad context of utmost importance, we have employed the Fock-state bunching capability to collectively benchmark experimental single-photon states for the first time. We have investigated the behavior of this test with photon statistics and loss. This quantifier, which is highly non-linear, has a clear operational meaning in terms of photon merging and moreover takes into account the unavoidable dispersion of individual copies of singlephoton states. Thanks to high-purity states based on a state-ofthe-art OPO, this work has experimentally verified the numerically-predicted threshold, P 1 > 0.885, to observe a large Fock-state capability. Capability of at least 14 has been demonstrated thanks to the very low two-photon component and the large heralding efficiency. Importantly, we have shown that the capability is more sensitive to optical losses than the single-photon negativity of the Wigner function and fidelity. Based on our numerical data, we also deduced that a moderate increase in the ratio of the multi-photon contributions to the vacuum does not decrease the capability. This shows that despite the negative impact of multi-photon contributions, they prevent the bunching of single-photon states into a single Fock state less severely than optical losses. In the present implementation, we have estimated photon-number distributions from homodyne detection. Multiplexed single-photon detectors [51,52] or photon-number resolving superconducting detectors [53][54][55] should enable a direct measurement of the Fock-state capability. Also, this benchmark does not depend on the nature of the source and can thereby be used to characterize microwave photons in superconducting circuits [15], plasmons at metal-dielectric surfaces [38,39], phonons in trapped-ion [56] or optomechanics experiments [57], and collective excitations in atomic ensembles [58][59][60]. Finally, the multi-photon interference quantifier can be modified to investigate the capability of other resource states, e.g. squeezed states or Schrödinger cat states [13,61], to produce different target states such as NOON states [18][19][20] or superpositions of squeezed states (GKP states) [62], opening a new avenue for testing the potential of light emitters for advanced quantum state engineering. The computational demand is substantially reduced if one chooses a single photon-number distribution from the experimental data set and use it as an identical input, which is fed into all channels of the linear optics network. In this way, we calculate the output Wigner function for several random choices of the photon-number distribution. Then the Wigner function is averaged over these random choices. Hence the differences between the individual copies of single-photon states are not taken into account. Using this simplified method, we determine the Fock-state capabilities of the experimental data sets, which agree with the capabilities depicted in Fig. 2 obtained by the full, unsimplified multi-photon interference quantifier. However, note that the full quantifier should always be used to confirm the results of the simplified quantifier, which is not able to correctly estimate the propagation of the input state's discrepancies through the quantifier. In order to estimate the capability of a very high Fock state, the quantifier can be even further simplified by neglecting the discrepancies between photon-number distributions in the experimental data set. Working only with the average photon-number distribution, we estimate that the experimental data set 7 with P 1 = 0.91 and P 2 = 0.02 has the capability of at least the Fock state \|50 . This agrees with the theoretical prediction [4] that P 1 > P (∞) 1 = 0.885 is sufficient to reach the capability of an arbitrary Fock state, if multi-photon contributions and discrepancies between photon-number distributions are neglected. FIG. 2 : 2Fock-state bunching capability of the experimentallygenerated single-photon states. The capability is given as a function of the single-photon component P1 and of the multiphoton probability P2+ for the different sets given in FIG. 3 : 3Quantifier output. (a) Cut through the output Wigner function of the computational quantifier with fourteen experimental photon-number statistics as inputs. The line thickness provides the 3 σ interval for the values of the Wigner function. The black fit corresponds to the attenuated Fock state \|14 with an attenuation η = 0.9205. (b) Associated photon-number distribution (blue points) compared to the one of the attenuated Fock state \|14 (black crosses). The output Wigner function and the photon-number distribution are averaged over thirty random choices of photon-number statistics from the data set 7, with P1 = 0.91 and P2+ = 0.02. FIG. 4 : 4Fock-state bunching capability and optical loss. The blue and red lines provide the Fock-state capability for a single random choice of n attenuated photon-number statistics obtained from the two data sets 7 and 4, with a capability of 14 and 4 respectively. These capabilities are compared to the capabilities of truncated states, i.e., with neglected multiphoton contributions (light blue and red lines). This work was supported by the European Union's Horizon 2020 research and innovation framework programme under grant agreement No 731473 (QuantERA ERA-NET Cofund in Quantum Technologies, project ShoQC), No 820445 (FETFLAG Quantum Internet Alliance), and No 951737 (Twinning project NonGauss). This work was also funded by the French National Research Agency (HyLight project ANR-17-CE30-0006). R.F. acknowledges grant 20-16577S of the Czech Science Foundation and national funding from the MEYS (project 8C20002). G.G. acknowledges the support by the European Union (Marie Curie Fellowship HELIOS IF-749213), T.D. by the Région Ile-de-France in the framework of DIM SIRTEQ, and P.Z. by the European Union's Horizon 2020 research and innovation framework programme under grant agreement No 732894 (FET Proactive, project HOT). We also thank K. Huang and O. Morin for their contributions in the early stage of the experiment. FIG. 5 : 5Determining the Fock-state capability of n independent non-ideal single photons. The Wigner function W (x, 0) of the output state of the computational quantifier for n = 5 is averaged over 30 randomly chosen batches of n = 5 photon number statistics that are randomly chosen from data set 4 (red line) and data set 7 (blue line). The line thickness provides the 3σ interval for the values of the Wigner function. The black line corresponds to the attenuated Fock state \|5 with the attenuation η = 0.72. TABLE Appendix A: Single-photon state generation and detectionThe triply-resonant optical parametric oscillator is based on a 1-cm long type-II phase-matched KTP crystal (Raicol), pumped at 532 nm with a frequency-doubled Nd:YAG laser (InnoLight GmbH). The input face of the crystal is coated for high-reflection at 1064 nm and R = 95% for the pump, while the output curved mirror (38-mm radius of curvature) is highly reflective for the pump and has R = 90% for the infrared light. The OPO is operated well below threshold. Photon pairs emitted at 1064 nm are orthogonally polarized and separated on a polarizing beam-splitter. After frequency filtering via an interferential filter and a home-made narrow-band cavity, the idler photons are detected with a high-efficiency WSi superconducting nanowire single-photon detector[1]. The heralded single photons are characterized using homodyne detection[2]. The overall detection loss, which includes propagation losses, electronic noise, interference visibility and detector efficiency, amounts to about 15%.Appendix B: Evaluation of photon statisticsThe Fock-state capability is determined for n independent photon-number statistics p mj , where m j is the photon number and j = 1, ..., n labels the individual statistics. We reconstruct the photon-number statistics from multiple runs of the single-photon source under the same experimental conditions. In the computer, the statistics p mj conditionally merge into a single output statistics following the computational procedure described in the main text. In the first step of the procedure, the input statistics p mj are mixed in a linear optics network represented by the unitary operator U . To keep the quantifier unbiased, the linear optics network symmetrically mixes the inputs. In the second step, all photons are conditionally merged into a single output mode considering all the other modes in vacuum. In this way, we calculate the photon-number distribution for the output mode and the associated Wigner function. The single-photon source producing n photon-number statistics p mj has the capability of the Fock state \|n if the computed Wigner function has the same number of negative annuli as that for the ideal Fock state \|n[3].Appendix C: Capability determination: an exampleWe illustrate how the Fock-state capability is determined using the computational quantifier via a specific example. We start by taking a batch of photon number statistics randomly selected from a given data set. This batch contains a fix number n of photon statistics. For concreteness we set for instance n = 5 and consider data sets 4 and 7 (Table 1of the main text). We apply the computational quantifier(Fig. 1of the main text) on this batch and determine the Wigner function of the quantifier output state. The output Wigner function is averaged over 30 randomly chosen batches from a given data set and plotted inFig. 5. The area in phase space, where the rotationally symmetric Wigner function for data set 7 is negative, forms two negative annuli and one negative circle at the origin. This can be seen in the cut through the Wigner function (blue line) plotted inFig. 5. Following the discussion in Sec. 2 of the main text, we conclude that single photons in the data set 7 have the capability 5. On the other hand, the output Wigner function for data set 4 has only one negative annulus and one negative circle (inset ofFig. 5). We conclude that data set 4 does not have capability 5. By applying the quantifier to batches with different number n of single photons we determine the maximal capability that each data set exhibits and plot this capability inFig. 2of the main text.Appendix D: Computational limitationsThe data sets labeled by the blue color inFig. 2of the main text are expected to have a Fock-state capability higher than 14. However, it cannot be tested due to computational limitations. The quantifier is based on the processing of multiple distinct photon-number statistics estimated from the experiment. The computational time of this processing grows exponentially with the number of input photon-number statistics. To test the Fock-state capability n, m n distinct terms are evaluated if one considers m lowest photon components in the photon-number statistics. For practical reasons, we restrict our analysis to test the capability 14, which is sufficient for our purpose due to the nonlinearity of the capability. 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[ "ON THE AUTOMORPHISM GROUP OF NON-NECESSARILY NORMAL AFFINE TORIC VARIETIES", "ON THE AUTOMORPHISM GROUP OF NON-NECESSARILY NORMAL AFFINE TORIC VARIETIES" ]	[ "Roberto Díaz ", "Alvaro Liendo " ]	[]	[]	Our main result is the following: let X be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface X ′ with automorphism group isomorphic to the automorphism group of X if and only if X is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety X of any dimension. Recall that normalized additive group actions on X are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of X. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.	10.1093/imrn/rnad050	[ "https://export.arxiv.org/pdf/2208.11755v2.pdf" ]	251,800,158	2208.11755	ab228486b94f84a68bdbe5641a1bf6825d010708	ON THE AUTOMORPHISM GROUP OF NON-NECESSARILY NORMAL AFFINE TORIC VARIETIES 10 Jan 2023 Roberto Díaz Alvaro Liendo ON THE AUTOMORPHISM GROUP OF NON-NECESSARILY NORMAL AFFINE TORIC VARIETIES 10 Jan 2023 Our main result is the following: let X be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface X ′ with automorphism group isomorphic to the automorphism group of X if and only if X is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety X of any dimension. Recall that normalized additive group actions on X are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of X. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid. INTRODUCTION Let k be an algebraically closed field of characteristic 0. We denote by G a and G m the additive group and the multiplicative group over k, respectively. Furthermore, we let T be the algebraic torus T ≃ G n m , for some integer n > 0. A toric variety is an irreducible variety X containing a torus T as a Zariski open subset and such that the action of T on itself extends to an algebraic action of T on X. There is a well known correspondence between normal affine toric varieties of dimension n and strongly convex polyhedral cones in Q n . This correspondence extends to a duality of categories once morphisms are properly defined and, moreover, extends to non-affine toric varieties by taking certain collections on cones called fans [Oda83,Ful93,CLS11]. On the toric side, toric morphisms are regular maps that restrict to group homomorphisms in the corresponding acting tori [CLS11, Proposition 1.3.13]. If we drop the normality assumption, then there is still a duality of categories between affine toric varieties with toric morphisms and affine monoids with semigroup homomorphisms [CLS11, Theorem 1.1.17]. Recall that an affine monoid is a finitely generated semigroup with identity that admits an embedding into Z n for some positive integer n. These correspondences between toric varieties and combinatorial objects have allowed for toric varieties to turn into an important area of algebraic geometry, mostly as a source of new examples and testing ground for new theories. Motivated by the study of the Cremona Groups, Demazure first defined toric varieties in its seminal paper [Dem70]. In this paper, Demazure also introduced a combinatorial gadget that classifies regular G a -actions on toric varieties that are normalized by the acting torus, see also [LLA22, Section 1] and [AL12, Section 1.3] for the affine case. Nowadays, this gadget is known as the Demazure roots. The Demazure roots are in one-to-one correspondence with normalized G a -actions on a toric variety X via the derivation obtained from the G a -action G a × X → X regarded as the flow of a vector field on X. In the affine case, such derivations correspond to homogeneous locally nilpotent derivations on the ring of regular functions k[X] and the Demazure roots correspond to the degree of such derivations. Recall that a derivation ∂ : k[X] → k[X] is called locally nilpotent if for every f ∈ k[X] there exists an integer n such that ∂ n (f ) = 0. The second named author of this article presented in [Lie10] a modern account of this correspondence that has proved being a useful tool in the study of the automorphism groups of affine toric varieties, see for instance [AZ13,LRU18,AKZ19,RvS21a,KRvS21,BG21]. The main result in [LRU18, Theorem 1.3] is the following. Let X be an affine toric surface and let X ′ be a normal affine surface. If the automorphism groups of X and X ′ are isomorphic as abstract groups, then X and X ′ are isomorphic as algebraic varieties. Also in [LRU18, Proposition 6.2], it is proven that the assumption of normality of X ′ can be dropped if X is the affine plane. On the other hand, in [RvS21b,Proposition 9.1] a counter-example is given if we drop the normality assumption with a particular choice of a normal affine toric surface taking the role of X. Our main theorem in this paper states that for any normal affine toric surface X without torus factor different from the affine space, there exists a non-normal affine toric surface X ′ with automorphism group isomorphic to the automorphism group of X, see Theorem 4.5. Our main result is obtained after generalizing the notion of Demazure root described above to the case of cancellative monoids that are non-necessarily affine nor saturated, see Definition 3.1. This allows us to provide a full classification of homogeneous locally nilpotent derivations on the semigroup algebras of cancellative monoids generalizing the one given in [Lie10], see Theorem 3.11. This generalization is not straightforward and new techniques are introduced along the way. The paper is organized as follows. In Section 1 we collect and review the essential definitions and results on cancellative monoids, locally nilpotent derivation and affine toric varieties. Section 2 is devoted to the study of homogeneous locally nilpotent derivations on semigroup algebras of cancellative monoids. In Section 3, we generalize the notion of Demazure root to the case of cancellative monoids. Finally, in Section 4, we prove our main result concerning affine toric surfaces. PRELIMINARIES In this section we recall the necessary notions of commutative cancellative monoid, homogeneous locally nilpotent derivations, affine toric geometry and related concepts needed for this paper. 1.1. Cancellative monoids. A semigroup is a non-empty set S with an associative binary operation + : S × S → S where +(m, m ′ ) will be denoted by m + m ′ . An element 0 ∈ S is an identity if for every m ∈ S, the equations 0 + m = m and m + 0 = m hold. If such identity exists it is unique. Semigroups with identity are called monoids. A monoid is called commutative if m + m ′ = m ′ + m for all m, m ′ ∈ S. All monoids in this paper will be assumed to be commutative without explicitly mention it. A monoid is cancellative if for every m, m ′ , m ′′ ∈ S with m + m ′ = m + m ′′ we have m ′ = m ′′ . An affine monoid is a finitely generated commutative monoid S that admits an embedding in Z n for some integer n ≥ 0. By [Nag01, Theorem 3.10], a commutative monoid S is cancellative if and only if S can be embedded in a group. Furthermore, a minimal such group M S can be obtained as follows. Let ∼ be the equivalence relation on S × S given by (m 1 , m 2 ) ∼ (m ′ 1 , m ′ 2 ) if and only if m 1 + m ′ 2 = m ′ 1 + m 2 in S. We denote the class of (m 1 , m 2 ) in (S × S)/ ∼ by m 1 − m 2 . Then M S = {m 1 − m 2 \| m 1 , m 2 ∈ S}. The identity in M S is m − m for any m ∈ S and m 2 − m 1 is the inverse of m 1 − m 2 . An embedding of S in M S is given by m → m − 0. We will always regard S as a subset of M S via this embedding and, if no confusion arises, we will denote M S simply by M . Let now N = Hom(M, Z) be the group dual to M . We define the dual monoid S * of S as S * = {u ∈ N \| u(m) ≥ 0, for all m ∈ S} . A face of a monoid S is a submonoid F satisfying that m 1 + m 2 ∈ F implies m 1 , m 2 ∈ F . The trivial monoid 0 is always a face of the dual monoid S * . Let now S be a monoid having 0 as a face. A face of S isomorphic to Z ≥0 is called a ray of S. Let R be a ray of S, we define the primitive element of R as the unique generator ρ of R as monoid. We denote by S(1) the set of primitive elements of all the rays of S. The saturation of a monoid S is the submonoid S sat of M of the elements a verifying k · a is an element in S for some positive integer k. 1.2. Algebras graded by monoids. Let S be a monoid. An S-graded algebra is an algebra B with a direct sum decomposition B = m∈S B m where B m are k-submodules and B m B m ′ ⊂ B m+m ′ . The submodules B m are called the homogeneous components of B. We say that f ∈ B m is a homogeneous element of degree m. Moreover an S-graded algebra can also be regarded as an M S -graded algebra by setting B m = {0} whenever m ∈ M S \ S. The most natural S-graded algebra that we can associate to a monoid is the semigroup algebra defined as follows. Given a monoid S, we let χ m be new symbols for every m ∈ S and we define the semigroup algebra k[S] as k[S] = m∈S k · χ m with multiplication rule given by χ m · χ m ′ = χ m+m ′ and χ 0 = 1 . 1.3. Homogeneous locally nilpotent derivation. Let B be a commutative k-algebra. A k-derivation on B is a linear map ∂ : B → B such that ∂(k) = 0 and for every f, g ∈ B we have ∂(f g) = f ∂(g) + g∂(f ). This last condition is known as the Leibniz rule. All our algebras will be k-algebras and all our derivations will be k-derivations, hence we will drop k from the notation. A derivation on B is said to be locally nilpotent if for every f ∈ B there exists a non-negative integer n depending on f such that ∂ n (f ) = 0, where ∂ n denotes the nth iteration of ∂. Locally nilpotent derivations have a geometric counterpart. If X is an affine variety and k[X] is the ring of regular functions, there exists a correspondence between locally nilpotent derivations on k[X] and regular additive group actions on X. The correspondence is defined as follows, see [Fre06, Section 1.5] for details. Let µ : G a × X → X be a regular G a -action on X. We associate a locally nilpotent derivation ∂ to µ via: ∂ µ : k[X] → k[X] defined by f → d dt • µ * (f ) t=0 . Furthermore, every regular G a -action on X arises from such a locally nilpotent derivation ∂. The regular G a -action µ corresponding to ∂ is given as the comorphism of the morphism µ * : k[X] → k[t] ⊗ k[X] ≃ k[X][t] defined by f → exp(t∂)(f ) = ∞ i=0 t i ∂ i (f ) i! . Let now S be a monoid and let B be an S-graded algebra. A derivation ∂ : B → B is said to be homogeneous if it sends homogeneous elements into homogeneous elements. For every homogeneous f ∈ B m such that ∂(f ) = 0 we define the element α(f ) ∈ M S such that ∂(f ) ⊂ B m+α(f ) . In the following lemma we show that α(f ) does not depend on the choice of f and so will be denoted simply by α. The element α ∈ M S is called the degree of ∂. Lemma 1.1. Let B be an S-graded algebra, where S is a cancellative monoid. If ∂ is a homogeneous derivation ∂ : B → B, then α(f ) ∈ M S defined above does not depend on f ∈ B \ ker ∂. Proof. Let f ∈ B m and g ∈ B m ′ such that ∂(f ) = 0 and ∂(g) = 0. By Leibniz rule we have ∂(f g) = f ∂(g) + g∂(f ) . The left-hand side is homogeneous of degree m+m ′ +α(f g) and so the degree of both summands on the right must agree, i.e., m + m ′ + α(g) = m + m ′ + α(f ) and by the cancellative property we have α(f ) = α(g). 1.4. Affine toric variety. A (split) algebraic torus T is an algebraic group isomorphic to G n m where n is a non-negative integer. There are two mutually dual free abelian groups of rank n canonically associated to T , namely, the character lattice M = Hom(T, G m ) and the 1-parameter subgroup lattice N = Hom(G m , T ) with the duality pairing given by (m, u) → k where k is the unique integer such that m • u(t) = t k . It is customary to regard M and N as abstract groups with additive notation and to denote the duality pairing by m, u = u(m). This yields that the algebra of regular functions of T corresponds to k[M ] and the character associated to m ∈ M corresponds to χ m . An affine toric variety is an irreducible affine algebraic variety X containing a torus T as a Zariski open subset such that the action of T on itself by translation extends to an algebraic action of T on X. Remark that, following [CLS11], we do not assume normality unlike other authors, see, e.g., [Oda83,Ful93]. The category of affine toric varieties is dual with the category of affine monoids S. To obtain the correspondence between objects, if S is an affine monoid, the affine algebraic variety X S = Spec S X associated to X is the monoid {m ∈ M \| χ m ∈ k[X]}. In Section 4 our main interest will be non-normal affine toric varieties. Nevertheless, normal affine toric varieties will also be used as a tool throughout this paper. Letting S be an affine monoid, we let M = M S and N = Hom(M, Z). We also define the corresponding rational vector spaces M Q = M ⊗ Q ≃ Q n and N Q = N ⊗ Q ≃ Q n . The duality pairing extends naturally to a pairing M Q × N Q → Q. The dual cones σ = {u ∈ N Q \| u(m) ≥ 0 for all m ∈ S} and σ ∨ = {m ∈ M Q \| u(m) ≥ 0 for all u ∈ σ} are the convex polyhedral cones associated to X S . By this construction, the monoid σ ∨ ∩ M coincides with the saturation of S. Furthermore, S is saturated if and only if k[S] is integrally closed if and only if X S is normal. Moreover, if X S is a non-normal affine toric variety, X σ ∨ ∩M is the normalization of X S with normalization morphism η : X σ ∨ ∩M → X S obtained from the inclusion S ֒→ σ ∨ ∩ M . It is customary to denote X σ ∨ ∩M simply by X σ . In this paper we follow the notational conventions in [CLS11]. In particular, the orthogonal space of A ⊂ M Q is A ⊥ = {u ∈ N Q \| u(m) = 0 for all m ∈ A} We let H m = {m} ⊥ be the hyperplane orthogonal to m. Let σ ⊂ N Q be a polyhedral cone. A face of σ is the intersection τ = σ ∩ H m for some m ∈ σ ∨ . By σ(1) we denote the set of one-dimensional faces of σ that we call rays. A ray will be identified and denoted by its primitive element i.e., the first non-trivial vector in ρ∩N . The analogous definitions hold when we exchange the roles of M and N . The submonoid {0} is a face of the affine monoid S if and only if X S is not isomorphic to Y ×G m . In the case of a normal affine toric variety X σ this condition is also equivalent to σ ⊂ N Q having dimension rank N . In this case, we say that σ is a full-dimensional cone meaning that the dimension of σ equals the rank of N . On the other hand, the cone σ ∨ is always full-dimensional by construction. This translate into σ being a strongly convex cone meaning that {0} is a face of σ. 1.5. Locally nilpotent derivations on normal affine toric varieties. Let X be the normal affine toric variety associated to the polyhedral cone σ ∈ N Q . We let S = σ ∨ ∩ M so that the ring of regular functions of X is k[S]. There exists a combinatorial description of homogeneous locally nilpotent derivations on k[S] given in [Lie10, Theorem 2.7] by the second author that we recall here. Definition 1.2. A vector α ∈ M is a Demazure root of S if there exists ρ ∈ σ(1) such that ρ(α) = −1 and ρ ′ (α) ≥ 0, for every ρ ′ ∈ σ(1) \ {ρ}. The ray ρ will be called the distinguished ray of the root α. By R(σ) we will refer to the set of all roots of σ and by R ρ (σ) the set of all roots of σ with distinguished ray ρ, i.e., R ρ (σ) = α ∈ M \| ρ(α) = −1 and ρ ′ (α) ≥ 0 for all ρ ′ ∈ σ(1) different from ρ With the previous notation to every Demazure root α of σ with distinguished ray ρ we associate a homogeneous locally nilpotent derivation ∂ α : k[σ ∨ ∩ M ] → k[σ ∨ ∩ M ] defined via χ m → ρ(m)χ m+α . Furthermore, for every non-trivial homogeneous locally nilpotent derivation ∂ on k[S] there exists a root α of S and r ∈ k * such that ∂ = r∂ α [Lie10, Theorem 2.7]. HOMOGENEOUS DECOMPOSITION OF LOCALLY NILPOTENT DERIVATIONS Let B be a finitely generated S-graded algebra with unit element where S is a finitely generated cancellative monoid and let ∂ : B → B be a derivation. Since ∂ is a linear map, it can be decomposed into homogeneous pieces. In the following proposition, we show that only finitely many such homogeneous pieces are non-trivial and that each such homogeneous piece is again a derivation. k [n] α = {g ∈ k[x 1 , . . . , x n ] \| ϕ(g) is homogeneous of degree α}. Moreover, choosing f i ∈ ϕ −1 (∂(f i )), we can lift ∂ to a derivation ∂ satisfying ∂ • ϕ = ϕ • ∂ by setting ∂ : k [n] → k [n] given by x i → f i We can now decompose f i as a finite sum of homogeneous elements in f i = j∈M f i,j . For every α ∈ M S we define the homogeneous derivation ∂ α : k [n] → k [n] given by x i → f i,a i +α , so that α ∂ α = ∂ and all but finitely many ∂ α = 0. Moreover, since ∂ • ϕ = ϕ • ∂ we have ∂(I) ⊂ I and since I is homogeneous we have ∂ α (I) ⊂ I. Hence, ∂ α passes to the quotient k [n] /I and so defines a homogeneous derivation ∂ α : B → B of degree α ∈ M S satisfying We now provide examples showing that the hypothesis cancellative in Lemma 1.1 and the hypotheses cancellative and finitely generated in Proposition 2.1 are essential. Example 2.2 (Cancellative in Proposition 2.1). Let S = {0, a, b} be the monoid with sum operation given by the table below. + 0 a b 0 0 a b a a b b b b b b It is not cancellative since for instance a + a = a + b. Moreover, taking x = χ a we have χ b = χ a+a = x 2 . It is easily seen that k[S] = k ⊕ k · χ a ⊕ k · χ b ≃ k[x]/(x 2 − x 3 ) . A straightforward verification shows that the map ∂ : k[S] → k[S] given by χ a → χ a − χ b and χ b → 0 is a derivation. Nevertheless, it is impossible to exhibit ∂ as sum of homogeneous derivations since the map ∂ 0 : k[S] → k[S] given by χ a → χ a and χ b → 0 is not a derivation. Indeed, the map ∂ 0 does not satisfy Leibniz rule since 0 = ∂ 0 (χ b ) = ∂ 0 (χ a · χ a ) = 2χ a ∂(χ a ) = 2χ a χ a = 2χ b = 0 . Example 2.3 (Finitely generated in Proposition 2.1). Let S be the monoid of sequences of nonnegative integers with a finite number of non-zero entries, i.e., S = {(a 1 , a 2 , . . . ) \| a i ∈ Z ≥0 and all but finitely many a i = 0} . The semigroup algebra k[S] is the polynomial ring in infinitely many variables k[S] ≃ k[x 1 , x 2 , . . . ], where x i = χ e i , and e i is the sequence with the ith entry equal to one and all other entries equal to 0. We define the derivation ∂ : k[S] → k[S] given by χ e i → χ 2e i for all i ∈ Z >0 . The homogeneous components of ∂ are ∂ e j : k[S] → k[S] given by χ e j → χ 2e j , and χ e i → 0 for all i = j . These homogeneous components are indeed derivations, but there are infinitely many non-zero such homogeneous components. Example 2.4 (Cancellative in Lemma 1.1). Let S = Z 4 ≥0 / ∼ where ∼ is the equivalence relation given by (m 1 , m 2 , m 3 , m 4 ) ∼ (m ′ 1 , m ′ 2 , m ′ 3 , m ′ 4 ) if and only if m 1 = m ′ 1 ≥ 1, m 2 = m ′ 2 ≥ 1, and m 3 + m 4 = m ′ 3 + m ′ 4 By [How95, Theorem 1.5.2], the sum in the monoid Z 4 ≥0 induces a binary operation on S making it into a monoid if and only if for every m, m ′ , m ′′ ∈ Z 4 ≥0 such that m ∼ m ′ we have m + m ′′ ∼ m ′ + m ′′ . A straightforward verification shows that this is the case in our example, so S is a commutative monoid. Let now e i be the image of the ith standard basis vector on S. We define a derivation on ∂ : k[S] → k[S] via ∂(χ e 1 ) = χ e 1 +e 3 , ∂(χ e 2 ) = χ e 2 +e 4 , and ∂(χ e 3 ) = ∂(χ e 4 ) = 0 . Remark that ∂ is indeed homogeneous since by the Leibniz rule we have: ∂(χ (m 1 ,m 2 ,m 3 ,m 4 ) ) = χ m 3 e 3 · χ m 4 e 4 m 1 χ (m 1 −1)e 1 χ e 1 +e 3 χ m 2 e 2 + m 2 χ m 1 e 1 χ (m 2 −1)e 2 χ e 2 +e 2 = m 1 χ (m 1 ,m 2 ,m 3 +1,m 4 ) + m 2 χ (m 1 ,m 2 ,m 3 ,m 4 +1) =      m 2 χ (m 1 ,m 2 ,m 3 ,m 4 +1) if m 1 = 0 m 1 χ (m 1 ,m 2 ,m 3 +1,m 4 ) if m 2 = 0 (m 1 + m 2 )χ (m 1 ,m 2 ,m 3 +1,m 4 ) if m 1 , m 2 ≥ 1 In the last case of the above equation we used the fact that in S we have (m 1 , m 2 , m 3 + 1, m 4 ) = (m 1 , m 2 , m 3 , m 4 + 1) whenever m 1 , m 2 ≥ 1 . In Lemma 1.1 we defined the degree of an S-graded derivation when S is cancellative. In this example, S is not cancellative since e 1 + e 2 + e 3 = e 1 + e 2 + e 4 but e 3 = e 4 . Furthermore, it is impossible to define the degree since ∂(χ e 1 ) = χ e 1 +e 3 would give that the degree is e 3 while ∂(χ e 2 ) = χ e 2 +e 4 would give that the degree is e 4 which is a contradiction. In terms of algebras we have k[S] ≃ k[x, y, z, t]/I where I = (xyz − xyt) , and the derivation ∂ is given by ∂ = xz ∂ ∂x + yt ∂ ∂y . The Lemma 2.5. Let S be a finitely generated cancellative monoid. If S is saturated, then S = S ′ ⊕ T where S ′ is a saturated affine monoid and T is a finite group. Proof. Let M = M S be the smallest group containing S. By the structure theorem for finitely generated abelian groups, we have M = L ⊕ T where L is free and T is finite. Assume now that m = l + t ∈ S and let m ′ = l + t ′ ∈ M with l ∈ L and t, t ′ ∈ T . We have \|T \| · m ′ = \|T \| · (l + t ′ ) = \|T \| · l = \|T \| · (l + t) = \|T \| · m ∈ S . By saturation we conclude that m ′ ∈ S. This yields S = S ′ ⊕ T , where S ′ is the image of S in L under the first projection. Since L is free and finitely generated, we have S ′ is an affine monoid. Finally, since S is saturated, S ′ is also saturated, proving the lemma. Let S be a finitely generated cancellative monoid with associated group M S = M . Recall that we defined N S = N to be the group dual to M and S * = {u ∈ N \| u(m) ≥ 0, for all m ∈ S} to be the dual monoid in N . We now prove the following lemma. Lemma 2.6. Let S be a finitely generated cancellative monoid, then S = {m ∈ M \| u(m) ≥ 0, for all u ∈ S * } equals the saturation of S. Proof. Let S be a finitely generated cancellative monoid. Then the inclusion S sat ⊆ S is trivial. To prove the other inclusion, by the structure theorem for finitely generated abelian groups we have that M = L ⊕ T , where L is free and T is finite. We also let S ′ be the image of S in L under the first projection. Let now m ∈ S. Then m = l + t with l ∈ L and t ∈ T . We have \|T \| · m = \|T \| · l ∈ L. Since S ′ is an affine monoid we have that l belongs to the saturation of S ′ by [CLS11, Proposition 1.2.4]. Hence, l + t ′ ∈ S sat for some t ′ . Now by Lemma 2.5 we have that m = l + t is also in S sat . Corollary 2.7. Let S be a cancellative finitely generated monoid. If α ∈ M \ S sat then for every m ∈ S there exists a positive integer ℓ such that m + ℓα / ∈ S. Proof. Let α ∈ M \ S sat . By Lemma 2.6, we have α / ∈ S = S sat . Hence, there exists u ∈ S * such that u(α) < 0. Let now m ∈ S. Then u(m + ℓα) = u(m) + ℓu(α). Since u(α) < 0, taking ℓ big enough we can assume u(m + ℓα) < 0 so that m + ℓα / ∈ S sat ⊃ S. The following lemma is well known. In lack of a good reference, we provide a proof. ] ≃ k ⊕ · · · ⊕ k. We conclude that if f ∈ k[T ] is such that f n = 0 for some n ∈ Z >0 then f = 0. Example 2.9. The above proof may seem counter-intuitive to those unfamiliar with Wedderburn's Theorem so we provide as an example an isomorphism between k[Z/3Z] and k 3 = k × k × k with component-wise addition and multiplication. Let k[Z/3Z] = k ⊕ kχ 1 ⊕ kχ 2 . Since k is algebraically closed of characteristic zero, all three cubic roots of unity ω 1 , ω 2 and ω 3 are in k. Taking f i = 1 3 (1 ⊕ w i χ 1 ⊕ w 2 i χ 2 ) for i = 1, 2, 3 gives the desired isomorphism since the set {f 1 , f 2 , f 3 } is a basis of k[Z/3Z] as a vector space and f i · f j = 1 if i = j 0 if i = j For the proof of our main theorem in this section, we need the following remark showing that in k[S] a homogeneous element is never a zero divisor. Remark 2.10. By the cancellative property on S, the homogeneous elements χ m is not a zero divisor, for all m ∈ S. Indeed, assume χ m · f = 0 and take the homogeneous decomposition of f = i λ i χ m i . Then χ m · f = χ m · i λ i χ m i = i λ i χ m i +m = 0. By the cancellative property, we have that all the m i + m are different and so λ i = 0 for all i. Let S be a finitely generated cancellative monoid with M S = L ⊕ T , where L is free and T is finite. The following lemma shows that each M S -graded piece of an L-homogeneous locally nilpotent derivation in k[S] is also locally nilpotent. Lemma 2.11. Let S be a finitely generated cancellative monoid with M S = L ⊕ T , where L is free and T is finite. Let also ∂ be a locally nilpotent derivation on k[S] with homogeneous decomposition ∂ = r i=1 ∂ i , where each ∂ i is a homogeneous derivation of degree α i ∈ M S . If α i = α + α ′ i with α ∈ L fixed and α i ∈ T for all i ∈ {1, . . . , r}, then ∂ i is a homogeneous locally nilpotent derivation on k[S] for all i ∈ {1, . . . , r}. Proof. Before proceeding with the proof, remark that a straightforward computation shows that we can extend every ∂ i to a derivation ∂ i : k[M S ] → k[M S ] by the Leibniz rule via ∂ i (χ m ) = ∂ i (χ m ) and ∂ i (χ −m ) = −∂ i (χ m )χ −2m for all m ∈ S. Similarly, ∂ is extended to ∂ : k[M S ] → k[M S ] verifying that ∂ = r i=1 ∂ i , see also [AA20, Theorem 2.3] . Now, letting m ∈ T , we let r be the smallest integer such that rm = 0. Then ∂(χ rm ) = r · χ (r−1)m · ∂(χ m ) = 0. Now, by Remark 2.10 we conclude that ∂(χ m ) = 0, for all m ∈ T . We conclude that χ m ∈ ker ∂ whenever m is a torsion element. If α i / ∈ S sat for all i ∈ {1, . . . , r}, then by Corollary 2.7 for every i ∈ {1, . . . , r} and every m ∈ S there exist ℓ such that m + ℓα i / ∈ S thus ∂ ℓ i (χ m ) = 0 and so ∂ i is locally nilpotent for all i ∈ {1, . . . , r}. Assume now that α i ∈ S sat for some i. We will show that in this case ∂ is the trivial derivation. By Lemma 2.5, we have that α is also in S sat . On the other hand, since each ∂ i is homogeneous of degree α i = α + α ′ i , we have that ∂ i (χ m ) = λ i (m)χ m+α+α ′ i with λ i : M S → k. The Leibniz rule implies that λ is a group homomorphism. A straightforward verification by induction show ∂ n (χ kα ) = n−1 i=0 (k + i) χ kα+nα (λ 1 (α)χ α ′ 1 + · · · + λ r (α)χ α ′ r ) n , for all n, k ≥ 0 . Now choosing k such that kα ∈ S and n such that ∂ n (χ kα ) = ∂ n (χ kα ) = 0, we have n−1 i=0 (k + i) χ kα+nα (λ 1 (α)χ α ′ 1 + · · · + λ r (α)χ α ′ r ) n = 0 . By Remark 2.10 we have (λ 1 (α)χ α ′ 1 + · · · + λ r (α)χ α ′ r ) n = 0 and by Lemma 2.8, we have λ 1 (α)χ α ′ 1 + · · · + λ r (α)χ α ′ r = 0. This yields ∂(χ α ) = 0 and moreover, λ i (α) = 0 for all i ∈ {1, . . . , r}. Now, taking into account that χ α and χ α ′ i are in the kernel of ∂, again by induction we obtain ∂ n (χ m ) = ∂ n (χ m ) = χ m+nα (λ 1 (m)χ α ′ 1 + · · · + λ r (m)χ α ′ r ) n , for all m ∈ S and all n ≥ 0 . For every m ∈ S we can choose n big enough so that ∂ n (χ m ) = 0. With the same argument as before, we conclude λ 1 (m)χ α ′ 1 + · · · + λ r (m)χ α ′ r = 0 and so ∂(χ m ) = 0 ending the proof. Remark 2.12. It is a byproduct of the above proof that the degree α ∈ M S of a non-trivial homogeneous locally nilpotent derivation ∂ : k[S] → k[S] satisfies α / ∈ S sat . This generalizes [Lie10, Corollary 2.9]. We can now state and prove our main theorem in this section. Theorem 2.13. Let S be a finitely generated cancellative monoid and let ∂ : k[S] → k[S] be a derivation. Then ∂ admits a decomposition into homogeneous pieces ∂ = ∂ 1 + ∂ 2 + . . . + ∂ k . Furthermore, if ∂ is locally nilpotent, then ∂ i is locally nilpotent as well for some i ∈ {1, . . . , k}. Proof. Let S be a finitely generated cancellative monoid and consider a derivation ∂ : k[S] → k[S]. The existence of a decomposition into homogeneous pieces was proven in Proposition 2.1. Assume first that S is finite. For every m ∈ S there exists a minimal ℓ := ℓ(m) ∈ Z ≥0 such that ℓ · m = 0. Hence, applying the derivation to χ ℓ·m = (χ m ) ℓ we obtain ℓχ (ℓ−1)m ∂(χ m ) = 0. Since the homogeneous element χ (ℓ−1)m is not a zero divisor (see Remark 2.10) we obtain ∂(χ m ) = 0 and so ∂ is the trivial derivation. Let now S be any finitely generated cancellative monoid. Let M = M S be the smallest group containing S. By the structure theorem for finitely generated abelian groups we have M = L ⊕ T , where L is free and T is composed by the torsion elements of M . Assume now that ∂ is locally nilpotent and let α i be the degree of ∂ i , for i ∈ {1, . . . , k}. We decompose α i = α ′ i + α ′′ i with α ′ i ∈ L and α ′′ i ∈ T . In the vector space L ⊗ Z R we take the polytope ∆ obtained as the convex hull of the set {α ′ i \| i = 1, . . . k}. Let α ′ be a vertex of ∆ and let ∂ ′ be the sum of all the pieces in the homogeneous decomposition of ∂ having degree with free part equal to α ′ , i.e., ∂ ′ = α∈I ∂ i with I = {i ∈ {1, . . . , k} \| α ′ i = α ′ } . Remark now that k[S] is also an L-graded algebra and that ∂ ′ is an L-homogeneous piece of ∂. By [Lie10, Lemma 1.10], we have that ∂ ′ is locally nilpotent. Finally, Lemma 2.11 yields that each ∂ i with i ∈ I is locally nilpotent, proving the theorem. DEMAZURE ROOTS OF A CANCELLATIVE MONOID AND LOCALLY NILPOTENT DERIVATIONS In this section we provide a natural generalization of the notion of Demazure roots to the more general setting of cancellative monoids and show how this notion classifies homogeneous locally nilpotent derivations on the corresponding semigroup algebra. 3.1. Demazure roots of a cancellative monoid. The following is our definition of Demazure roots of a non-cancellative monoid. Definition 3.1. Let S be a cancellative monoid and let M S = M be its associated group. We also let N = Hom(M, Z) and S * be the dual monoid. An element α ∈ M is called a Demazure root of S if (i) There exists ρ ∈ S * (1) such that ρ(α) = −1, and (ii) The element m + α belongs to S for all m ∈ S such that ρ(m) > 0. We say that ρ is the distinguished ray of α. We denote the set of roots of S by R(S) and the set of roots of S with distinguished ray ρ by R ρ (S). As proven in the following lemma, it is enough to check condition (ii) in the above definition for a generating set of S. Lemma 3.2. Let S be a cancellative monoid. Let α ∈ M and ρ ∈ S * (1) be such that ρ(α) = −1. If A is a generating set of S, then m ′ + α belongs to S for all m ′ ∈ S such that ρ(m ′ ) > 0 if and only if m + α belongs to S for all m ∈ A such that ρ(m) > 0. Proof. The only if part is trivial. As for the other direction, let m ′ ∈ S be such that ρ(m ′ ) > 0. Since A is a generating set, let m ′ = n 1 m 1 + · · · + n l m l with n i positive integer and m i ∈ A for all i ∈ {1, . . . , l}. Up to reordering, we can assume that ρ(m 1 ) > 0. Hence, m 1 + α ∈ S and so m ′ + α = (m 1 + α) + (n 1 − 1)m 1 + n 2 m 2 + · · · + n l m l ∈ S. ✲ r r r r r r r r r ✲ r r r r r r r r r 0 1 Example 3.5. Let S be the semigroup S = Z 2 ≥0 × Z/2Z \ {(0, 1, 0), (0, 0, 1), (1, 0, 1)}. The smallest group containing S is M = Z 2 × Z/2Z. This yields N = Z 2 . The rays are ρ 1 = (1, 0) and ρ 2 = (0, 1) in N . A direct computation shows that the roots of S are R ρ 1 = (−1, 2 + i, j) \| i ∈ Z ≥0 and j ∈ Z/2Z ∪ {(−1, 1, 1)} R ρ 2 = (2 + i, −1, j) \| i ∈ Z ≥0 and j ∈ Z/2Z ∪ {(1, −1, 1)} ✲ ✻ r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r ❜ r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r ✲ ✻ r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r ❜ ❜ r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r 0 1 Proposition 3.6. Let S be a cancellative monoid and let S sat be its saturation. Then every Demazure root of S is also a Demazure root of S sat . Proof. Letting M be the group associated to S, we let α ∈ M be a Demazure root of S with distinguished ray ρ. Pick any m ∈ S sat such that ρ(m) > 0 and let k be the smallest positive integer such that km ∈ S. Since ρ(km) > 0 we have α + km ∈ S. If k = 1 then α + m ∈ S sat and so the proof is complete. Assume now k > 1. We have ρ(km) ≥ k and so ρ(α+km) ≥ k −1. Hence α + (α + km) = 2α + km ∈ S and ρ(2α + km) ≥ k − 2 since α is a Demazure root of S. Proceeding inductively, we obtain that kα + km = k(α + m) ∈ S and by the definition of saturation we conclude α + m ∈ S sat ending the proof. 3.2. Homogeneous locally nilpotent derivation on semigroup algebras. In the previous section we define a generalization of Demazure roots to the case of cancellative monoids. In this section we prove that, as in the affine toric case, Demazure roots classify locally nilpotent derivations on the semigroup algebra of the corresponding cancellative monoid. Our proof is not a straightforward generalization of the affine toric case as one can imagine since many of the usual properties of locally nilpotent derivations are lost in this more general setting where the algebra B is not necessarily a domain. For instance, the kernel of a locally nilpotent derivation ∂ on a domain B is factorially closed, i.e., if a product ab belongs to ker ∂ with a, b = 0, then a and b belong to ker ∂. This is no longer the case if B is not a domain as the following example shows. Here, the function f = x(y − 1) does not belong to the kernel, since ∂(f ) = y − 1. On the other hand, f · (y + 1) = 0 and so f · (y + 1) ∈ ker ∂ . Nevertheless, the above phenomenon does not occur for homogeneous elements as we show in the following Lemma 3.8. For the proof, recall that a homogeneous element is never a zero divisor in k[S] as we showed in Remark 2.10 above. Proof. We follow [Fre06, Proposition 1.10 (c)] adapted to our context. Let m ∈ S and f ∈ k[S] with f = 0. We let a and b be the smallest non-negative integers such that ∂ a+1 (f ) = 0 and ∂ b+1 (χ m ) = 0, respectively. We have ∂ a+b+1 (f χ m ) = i+j=a+b+1 a + b + 1 i · ∂ i (f ) · ∂ j (χ m ) . Since i + j = a + b + 1 then we have either i > a or j > b so ∂ i (f )∂ j (χ m ) = 0 and so ∂ a+b+1 (f χ m ) = 0. On the other hand, Similarly to the affine toric case, for every Demazure root α of S with distinguished ray ρ ∈ S * (1), we define a homogeneous derivation ∂ a+b (f χ m ) = i+j=a+b a + b i · ∂ i (f ) · ∂ j (χ m ) = a + b a · ∂ a (f ) · ∂ b (χ m ) .∂ α : k[S] → k[S] via χ m → ρ(m)χ m+α . The degree of ∂ α is α. The Leibniz rule follows by a straightforward computation. The fact that α is a Demazure root ensures that ∂ α is well-defined on k[S] since χ m+α ∈ k[S] for every m with ρ(m) = 0. Lemma 3.9. Let S be a cancellative monoid and let α be a Demazure root of S. Then, the homogeneous derivation ∂ α is locally nilpotent. Proof. Let k be a positive integer and let m ∈ S. Let also ρ be the distinguished ray of α so that ρ(α) = −1. A straightforward computation shows that Hence ∂ k+1 (χ m ) = 0 and so ∂ α is locally nilpotent. ∂ k+1 (χ m ) = ρ(m + kα) · ∂ k (χ m ) · χ α . The following lemma characterizes the kernel of a homogeneous locally nilpotent derivation on a semigroup algebra. Proof. Let α be the degree of ∂ and let f = i∈I λ i χ a i ∈ k[S] with I finite and λ i = 0. Then we have ∂(f ) = i∈I λ i ∂(χ a i ) . Since the degree of ∂(χ a i ) = a i + α and these elements are distinct by the cancellative property, we conclude that f ∈ ker ∂ if and only if χ a i ∈ ker ∂ for all i ∈ I. Let now F = {a ∈ S \| χ a ∈ ker ∂}. The Leibniz rule ensures that F is a subsemigroup of S and the above consideration proves that ker ∂ = k[F ]. Let us now prove that F is a face of S. Let a, b ∈ S and assume that a + b ∈ F . Then ∂(χ a χ b ) = ∂(χ a+b ) = 0. By Lemma 3.8 we have ∂(χ a ) = ∂(χ b ) = 0 which yields a, b ∈ F proving the lemma. Theorem 3.11. Let ∂ : k[S] → k[S] be a homogeneous locally nilpotent derivation of degree α and ∂ = 0. Then α is a Demazure root of S and ∂ = λ∂ α for some λ ∈ k * . Proof. Since ∂ is homogeneous of degree α, we have that ∂(χ m ) = γ(m)χ m+α , where γ : S → k . (1) By the Leibniz rule, we have that γ is a semigroup homomorphism. Indeed, γ(m + m ′ ) · χ m+m ′ +α = ∂(χ m+m ′ ) = ∂(χ m · χ m ′ ) = (γ(m) + γ(m ′ )) · χ m+m ′ +α . Let M be the group associated to S. Since (k, +) is a group we can extend γ to a group homomorphism γ : M → k and in turn we can use γ to extend ∂ to a derivation in ∂ : k[M ] → k[M ] given by ∂ : k[M ] → k[M ] where χ m → γ(m) · χ m+α . Remark now that ∂(χ −α ) = −γ(α)χ 0 ∈ k and so ∂ 2 (χ −α ) = 0 since ∂ is a k-derivation. We let now S ′ be the smallest semigroup in M containing S and −α. By Lemma 3.10, we have that ker ∂ = k[F ], where F is a face of S. Let m ∈ S and let n be the smallest integer such that ∂ n+1 (χ m ) = 0. Then ∂ n (χ m ) belongs to ker ∂ and it is homogeneous of degree m 0 ∈ F . On the other hand, m = m 0 − nα. Since ∂ is non-trivial, we have γ(α) = 0 and so −α / ∈ F . We conclude that S ⊂ S ′ = F ⊕ (−α)Z ≥0 , and so M = M F ⊕ αZ ,(2) where M F is the minimal group where F is embedded. Letting now λ = γ(−α) we let ρ = γ/λ so that ρ(M F ) = 0 and ρ(α) = −1. We have that ρ : M → Z is an element in N = Hom(M, Z). Furthermore, by (2) we have ρZ ≥0 = S * ∩ F ⊥ and so ρ is a ray of S * with ρ(α) = −1. This provides condition (i) in Definition 3.1 of a Demazure root. To prove condition (ii), remark that m ∈ S satisfies ρ(m) > 0 if and only if χ m / ∈ ker ∂. Hence, m + α ∈ S since otherwise ∂ is not a well defined derivation. This proves that α is a Demazure root of S. To concludes the proof remark that ∂ = λ∂ α by (1). Remark 3.12. Locally nilpotent derivations on a k-algebra B are in one-to-one correspondence with additive group actions on X = Spec B. This is most often proved for B a domain as in [Fre06] but it also holds in our context as proven in [Dai03,Theorem 4.12]. Remark 3.13. The previous theorem could be combined with the results in [BG21] to strengthen the results therein. In particular, it looks like the proof of [BG21, Lemma 4] could be significantly simplified applying our Theorem 3.11. It is well known that the ring of invariants of a locally nilpotent derivation on a finitely generated k-algebra is not necessarily finitely generated itself, see [Rob90,DF99]. In contrast, we have the following corollary. Corollary 3.14. Let S be a finitely generated cancellative monoid and let ∂ : k[S] → k[S] be a homogeneous locally nilpotent derivation. Then ker ∂ is a finitely generated k-algebra. Proof. Recall that a semigroup algebra k[S] is finitely generated if and only if S is finitely generated as a semigroup. The Corollary now follows directly from Lemma 3.10 since every face of a finitely generated semigroup is finitely generated itself. AUTOMORPHISM GROUP OF NON-NECESSARILY NORMAL AFFINE TORIC SURFACE Our original motivation to study cancellative monoids is to study automorphism groups of nonnecessarily normal affine toric varieties. Our main result in this paper is that for every normal affine toric surface X σ different from G 2 m , G m × A 1 and A 2 , there exists a non-normal affine toric surface X S whose normalization is X σ such that Aut(X σ ) ≃ Aut(X S ). We begin by showing that both our definitions of Demazure roots coincide when S is a saturated affine monoid, i.e., the monoid associated to a normal affine toric variety. This fact also follows indirectly from [Lie10, Theorem 2.7] and Theorem 3.11 since these theorems show, in particular, that with both definitions, the Demazure roots corresponds to the weights of homogeneous locally nilpotent derivations on k[S]. Nevertheless, we provide a direct proof of this fact. Let S be an affine monoid. Recall that M = M S is the smallest group where S can be embedded and N = N S = Hom(M, Z). The cone associated to S is σ = {u ∈ N Q \| u(m) ≥ 0 for all m ∈ S}. Lemma 4.1. Let S an affine monoid. If S is saturated then R(S) = R(σ). Proof. It is enough to verify that R ρ (S) = R ρ (σ) for any fixed ray ρ ∈ σ(1). Letting α ∈ R ρ (S), we let ρ ′ ∈ σ(1) \ {ρ} and we fix m ∈ ((ρ ′ ) ⊥ ∩ S) \ ρ ⊥ . By the definition of Demazure root, we have α+m ∈ S so that ρ ′ (α) = ρ ′ (α+m) ≥ 0. Hence, α ∈ R ρ (σ) and we have R ρ (S) ⊂ R ρ (σ). As for the other inclusion, let α ∈ R ρ (σ) and let ρ ′ ∈ σ(1) \ {ρ}. Letting A be a generating set of S, we let m ∈ A with ρ(m) > 0. Then ρ ′ (m+α) ≥ 0 for all ρ ′ ∈ σ(1) so m+α ∈ σ ∨ ∩M = S by saturation. This yields α ∈ R ρ (S). If S is not saturated, we only have the inclusion R(S) ⊆ R(σ) and typically this inclusion is proper. For instance, if X σ is the affine space, then the equality never holds by [CRX19]. Nevertheless, in Proposition 4.3, we will show that for every normal affine toric variety X σ without torus factor that is not a product X σ ′ × A 1 , we can always find a non-normal affine toric variety X S such that X σ is the normalization of X S and R(S) = R(σ). Before we do so, we need to prove the following lemma. Lemma 4.2. Let σ be a strongly convex cone. Let α ∈ R be a Demazure root. If −α ∈ (σ ∨ ∩ M ) then X σ ≃ X σ ′ × A 1 with X σ ′ an affine toric variety Proof. Let ρ be the distinguished ray of α. Since −α ∈ (σ ∨ ∩ M ), we have ρ ′ (−α) = 1 if ρ ′ = ρ 0 if ρ ′ = ρ(3) Let N ′ = α ⊥ be the orthogonal to the Demazure root α. Since α is primitive, we can split N = N ′ ⊕ ρZ. Let now σ ′ = cone(ρ ′ \| ρ ′ = ρ). By (3), we have σ ′ ⊂ N ′ Q . Now, the cone generated by σ ′ and ρ equals σ and moreover, the splitting N = N ′ ⊕ ρZ yields X σ = X σ ′ × A 1 . Let σ be a full-dimensional strongly convex polyhedral cone σ ∈ N Q . Recall that σ is fulldimensional if and only if X σ is non-degenerate, i.e., if X σ does not have a torus factor. Let also H be the Hilbert basis of S = σ ∨ ∩ M , i.e., the set of element in S that are irreducible in the sense that they cannot be expressed as sum of two non-trivial elements in S. Recall that H is a generating set of the semigroup σ ∨ ∩ M . Proposition 4.3. Let X σ be a non-degenerated affine toric variety. If S = (σ ∨ ∩ M ) \ H, then R(σ) = R(S) if and only if X σ cannot be decomposed as a product X σ ′ × A 1 with X σ ′ an affine toric variety. Proof. If X σ = X σ ′ × A 1 , then M = M ′ × Z, N = N ′ × Z and σ is the cone generated by (0, 1) and (σ ′ , 0), where σ ′ is a cone in N ′ Q corresponding to the affine toric variety X σ ′ . In this case, a straightforward computation shows that α = (0, −1) ∈ M is a Demazure root of σ with respect to the ray ρ = (0, 1) ∈ N . Furthermore, (0, 1) ∈ M belongs to H. By Definition 3.1, we have that α is not a Demazure root of S = (σ ∨ ∩ M ) \ H since m = (0, 2) ∈ S, ρ(m) = 2 > 0 and m + α / ∈ S. This yields R(σ) = R(S), proving one direction of the proposition. To prove the other direction, assume that X σ cannot be decomposed as a product X σ ′ × A 1 . Let A be a generating set of S. Let ρ ∈ σ(1). By Proposition 3.6, we have R ρ (S) ⊂ R ρ (σ), so we only need to verify the converse inclusion R ρ (σ) ⊂ R ρ (S). A Demazure root α ∈ R ρ (σ) is contained in R ρ (S) if and only if α + a ∈ S for all a ∈ A such that ρ(a) > 0. Nevertheless, by Definition 3.1 and Lemma 4.1 we have α + a ∈ σ ∨ ∩ M . Hence, α ∈ R ρ (S) if and only if α + a / ∈ H. Assume α + a ∈ H. We have that a / ∈ H, hence a = h 1 + . . . + h ℓ with h i ∈ H and ℓ ≥ 2. Since ρ(a) > 0 we have ρ(h i ) > 0 for some i. Without loss of generality, we may and will assume ρ(h 1 ) > 0. Hence a = h 1 + a ′ with a ′ ∈ (σ ∨ ∩ M ) \ {0}. But now α + a = α + h 1 + a ′ . Again by definition of Demazure root we have α + h 1 ∈ σ ∨ ∩ M . Moreover, if α + h 1 = 0 the −α ∈ σ ∨ ∩ M and by Lemma 4.2, we have that X σ can be decomposed as a product X σ ′ × A 1 , which is a contradiction. Hence, we may and will assume α + h 1 = 0. This yields α + a = (α + h 1 ) + a ′ with (α + h 1 ), a ′ ∈ (σ ∨ ∩ M ) \ {0} and so α + a / ∈ H since it is not irreducible proving the proposition. Before stating our main theorem in this section, we need the following lemma. Lemma 4.4. Let X and Y be algebraic varieties and assume that X is the normalization of Y . Then every automorphism of Y lifts to an automorphism of X. Moreover, we have Aut(Y ) ⊆ Aut(X) with equality if and only if every automorphism of X is the lifting of an automorphism of Y . Proof. The first statement follows directly from the Universal Property of Normalization. Remark now that the normalization map X → Y is a birational isomorphism and under this isomorphism, both groups Aut(X) and Aut(Y ) are naturally subgroups of the group of birational maps of X. Now the second statement follows from the first one. Our main interest in this section is affine toric surfaces. For these surfaces, we will prove the following theorem. Theorem 4.5. Let X σ be a normal affine toric surface. Then, there exists a non-normal affine toric surface X S whose normalization is X σ with Aut(X σ ) = Aut(X S ) if and only if X σ is different from G 2 m , G m × A 1 and A 2 . Hence, by Lemma 4.4, we obtain that Aut(X S ) ⊆ (Z/2Z ⋉ G m ) ⋉ k[t, t −1 ] * and so Aut(X σ ) is not isomorphic Aut(X S ) ending the proof. (k[S]) is toric with acting torus T = Spec(k[M ]), where M = M S . As for the other direction, let X be an affine toric variety with acting torus T . The dominant inclusion T ⊂ X provides us with an injective homomorphism k[X] ֒→ k[M ] and the affine monoid Proposition 2 . 1 . 21Let B be a finitely generated S-graded algebra where S is a finitely generated cancellative monoid. Then every derivation ∂ : B → B can be decomposed as a finite sum ∂ = ∂ α of M S -homogeneous derivations of degree α ∈ M S . Proof. Let {f 1 , . . . , f n } be a set of homogeneous generators of B as algebra. Letting k [n] = k[x 1 , . . . , x n ], we have an isomorphism between B and k [n] /I where I is the kernel of the surjective homomorphism ϕ : k [n] → B given by ϕ(x i ) → f i . The M S -grading on B induces an M S -grading on k [n] /I. We can lift this M S -grading to k [n] by setting k [n] = α∈M k derivation ∂ defines indeed a derivation in the quotient k[S] = k[x, y, z, t]/I since ∂(xyz − xyt) = z∂(xy) − t∂(xy) = (z − t)∂(xy) = (z − t)(xyz − xyt) ∈ I . Let S be a finitely generated cancellative monoid. Let ∂ : k[S] → k[S] be a locally nilpotent derivation on the semigroup algebra k[S]. By Proposition 2.1, the derivation ∂ can be decomposed as a finite sum ∂ = ∂ α of M S -homogeneous derivations of degree α ∈ M S . Moreover, if S is affine and saturated, by [Lie10, Lemma 1.10] some ∂ α is again locally nilpotent. In the remaining of this section we prove the same result in the more general context of finitely generated cancellative monoids, see Theorem 2.13. The techniques required for the proof are not straightforward generalizations of the affine and saturated case. Lemma 2. 8 . 8If T is a finite commutative group and f ∈ k[T ] is nilpotent, then f = 0. Proof. By [AB12, Corollary 12.8], k[T ] is semisimple. Moreover, since k is algebraically closed, by Wedderburn's Theorem [AB12, Theorem 13.18] (see also[DF04, Section 18.2]) the semisimple k-algebra k[T ] is isomorphic to a direct sum of matrix algebras over k. Hence, k[T ] ≃ M n 1 (k)⊕ · · · ⊕ M nr (k) with r, n 1 , . . . , n r ∈ Z >0 where addition and multiplication are taken componentwise. By the commutativity of k[T ] we have n 1 = · · · = n r = 1 i.e., k[T Example 3.3. A numerical monoid is an affine monoid S ⊂ Z ≥0 such that Z ≥0 \ S is a finite set. Let S be a numerical monoid different from Z ≥0 . We claim that R(S) = ∅. Indeed, M = Z and so N = Zρ, where ρ is the linear map ρ : M → Z given by ρ(1) = 1. Hence, S * (1) = {ρ} and the only element in M satisfying ρ(α) = −1 is α = −1. Let now m be the smallest element in S\{0}. We have m > 1 since otherwise S = Z ≥0 . We have ρ(m) = m > 0 and m + α = m − 1 / ∈ S. Hence, α is not a Demazure root proving the claim.Example 3.4. Let S = Z ≥0 × Z/2Z. The group associated to S is M = Z × Z/2Z. The dual group Hom(M, Z) is isomorphic to Z and under this identification we have S * ≃ Z ≥0 . A straightforward computation shows that R(S) = {(−1, 0), (−1, 1) ∈ Z × Z/2Z ≃ M }. Example 3 . 7 . 37Let S = Z ≥0 × Z/2Z as in Example 3.4. The corresponding semigroup algebra B = k[S] is equal to k[x, y]/(y 2 − 1) by setting x = χ (1,0) and y = χ (0,1) . On B we have the homogeneous locally nilpotent derivation ∂ : B → B given by ∂ = ∂ ∂x . Lemma 3 . 8 . 38Letting S be a cancellative monoid, we let ∂ : k[S] → k[S] be a homogeneous locally nilpotent derivation. If ∂(f χ m ) = 0 for some non-zero f ∈ k[S] and some m ∈ S, then ∂(χ m ) = 0 and ∂(f ) = 0. The last equality follows with the same argument as above since all the summands are zero with the only exception in the case where i = a and j = b. This yields ∂ a+b (f χ m ) = 0. Indeed, ∂ a (f ) = 0, ∂ b (χ m ) = 0 and ∂ b (χ m ) is not a zero divisor by Remark 2.10 since it is a homogeneous element. Since f χ m = 0 and ∂(f χ m ) = 0 by hypothesis, we conclude a + b + 1 = 1 and so a = b = 0 which in turn implies ∂(χ m ) = ∂(f ) = 0 proving the lemma. Now, taking k = ρ(m) we have ρ(m + kα) = ρ(m) + kρ(α) = ρ(m) − ρ(m) = 0 . Lemma 3 . 10 . 310Let S be a cancellative monoid and let ∂ : k[S] → k[S] be a homogeneous locally nilpotent derivation. Then ker ∂ = k[F ] where F is a face of S. α ∂ α = ∂. Acknowledgments. We would like to thank Andriy Regeta for valuable suggestions on applying Lemma 4.9 to prove Theorem 4.5. We would also like to thanks both referees. Their reports helped us to improve the quality of this paper and correct inaccuracies.Proof. The affine monoid S ′ corresponding to the normalization G m × A 1 of X S is S ′ = Z × Z ≥0 . The set of Demazure roots of S ′ is R(S ′ ) = {(i, −1) ∈ M \| i ∈ Z}. Since S is not saturated, by[MS05,Exercice 7.15] (see also[BG21,Lemma 2]), we can choose (m 1 , m 2 ) ∈ S ′ \ S such that (j, m 2 + 1) ∈ S for all j ∈ Z. By Proposition 3.6, we have R(S) ⊂ R(S ′ ). Take now α = (i, −1) ∈ R(S ′ ) be any root of S ′ . Now the element m = (m 1 − i, m 2 + 1) is in S but m + α is not in S. This yields R(S) = ∅.Remark 4.7. If X σ is the normalization of X S and X σ is isomorphic to the algebraic torus, then X S = X σ . Indeed, in this case X σ = Spec k[M ] with M ≃ Z 2 which is a group. Assume that S is a monoid whose saturation is M . By[MS05,Exercise 7.15], we have S = M .Lemma 4.8. Let X σ be a normal affine toric surface. Then, there exists a non-normal affine toric surface X S whose normalization ism , then by Remark 4.7, there are no non-normal affine toric varieties with normalization X σ so the lemma holds trivially. If X σ = G m × A 1 , then X σ admits a G a -action by translation in the A 1 factor, so R(σ) = ∅ while R(S) = ∅ by Lemma 4.6. Finally, if X σ = A 2 the lemma follows from [LRU18, Proposition 6.2].To achieve the proof of Theorem 4.5, we need the following lemma borrowed from[AZ13].Lemma 4.9. The automorphism group of a non-degenerated normal affine toric surface X σ is generated by the torus and the images in Aut(X σ ) of all the G a -actions corresponding to roots α ∈ R(σ). In particular, if X S is a non-normal affine toric surface X S whose normalization is X σ with X σ non-degenerated and R(σ) = R(S), then Aut(X σ ) = Aut(X S ).Proof. The second assertion follows directly from the first by Lemma 4.4. The first assertion follows from[AZ13,Lemma 4.3]. Indeed, in their notation, Aut(X σ ) is a quotient of N d,e and by [AZ13, Proposition 4.4 and Lemma 4.5] N d,e consists of de Jonquières transformations and de Jonquières transformations consist precisely of compositions of the images in Aut(X σ ) of the acting torus and of all the G a -actions corresponding to roots α ∈ R(σ).We can now prove our main result in Theorem 4.5.Proof of Theorem 4.5. If X σ is different from G 2 m , G m × A 1 and A 2 , the theorem follows from Lemma 4.8 and Lemma 4.9 taking for X S the affine toric variety given by the monoid S = (σ ∨ ∩ M ) \ H. If X σ = A 2 the theorem follows from [LRU18, Proposition 6.2].If X σ = G 2 m , then by Remark 4.7, there are no non-normal affine toric varieties with normalization X σ so the theorem holds trivially.If, where the factor Z/2Z ⋉ G m corresponds to the automorphism of G m and the factor k[t, t −1 ] * ⋉ k[t, t −1 ] corresponds to automorphisms of G m × A 1 that preserve the A 1 -fibration fiberwise. Indeed, let ϕ ∈ Aut(X σ ). The semidirect product structure comes from the fact that the composition pr 1 •ϕ : G m × A 1 → G m is constant on the fibers {t} × A 1 with t ∈ G m since there are no non-constant maps A 1 → G m . On the other hand, k[S] does not have any homogeneous locally nilpotent derivation by Lemma 4.6 and so X S it does not admit any G a -action by Theorem 2.13. 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Acyclic curves and group actions on affine toric surfaces. Ivan Arzhantsev, Mikhail Zaidenberg, Affine algebraic geometry. Hackensack, NJIvan Arzhantsev and Mikhail Zaidenberg. Acyclic curves and group actions on affine toric surfaces. In Affine algebraic geometry, pages 1-41. World Sci. Publ., Hackensack, NJ, 2013. Automorphisms of nonnormal toric varieties. I A Boldyrev, S A Gaifullin, Mat. Zametki. 1106I. A. Boldyrev and S. A. Gaifullin. Automorphisms of nonnormal toric varieties. Mat. Zametki, 110(6):837- 855, 2021. Toric varieties. David A Cox, John B Little, Henry K Schenck, American Mathematical Soc124David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties, volume 124. American Mathematical Soc., 2011. Families of commuting automorphisms, and a characterization of the affine space. Serge Cantat, Andriy Regeta, Junyi Xie, arXiv:1912.01567arXiv preprintSerge Cantat, Andriy Regeta, and Junyi Xie. Families of commuting automorphisms, and a characterization of the affine space. arXiv preprint arXiv:1912.01567, 2019. Locally nilpotent derivations, lecture notes for the september school of algebraic geometry. Daniel Daigle, Daniel Daigle. Locally nilpotent derivations, lecture notes for the september school of algebraic geometry, 2003. Michel Demazure, Sous-groupes algébriques de rang maximum du groupe de cremona. Annales scientifiques de l'École normale supérieure. 3Michel Demazure. Sous-groupes algébriques de rang maximum du groupe de cremona. Annales scien- tifiques de l'École normale supérieure, 3(4):507-588, 1970. A counterexample to hilbert's fourteenth problem in dimension 5. Daniel Daigle, Gene Freudenburg, Journal of Algebra. 2212Daniel Daigle and Gene Freudenburg. A counterexample to hilbert's fourteenth problem in dimension 5. Journal of Algebra, 221(2):528-535, 1999. Abstract algebra. David Steven Dummit, Richard M Foote, Wiley3HobokenDavid Steven Dummit and Richard M Foote. Abstract algebra, volume 3. Wiley Hoboken, 2004. Algebraic theory of locally nilpotent derivations. Gene Freudenburg, Springer136Gene Freudenburg. Algebraic theory of locally nilpotent derivations, volume 136. Springer, 2006. Introduction to toric varieties. William Fulton, Number 131 in Annals of mathematics studies. Princeton university pressWilliam Fulton. Introduction to toric varieties. Number 131 in Annals of mathematics studies. Princeton university press, 1993. Number 12 in Monographs new series. John Mackintosh Howie, Oxford University PressFundamentals of semigroup theoryJohn Mackintosh Howie. Fundamentals of semigroup theory. Number 12 in Monographs new series. Oxford University Press, 1995. Hanspeter Kraft, Andriy Regeta, Immanuel Van Santen, Is the affine space determined by its automorphism group? International Mathematics Research Notices. 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Paul Roberts, Journal of Algebra. 1322Paul Roberts. An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem. Journal of Algebra, 132(2):461-473, 1990. Andriy Regeta, Immanuel Van Santen, arXiv:2112.04784Characterizing affine toric varieties via the automorphism group. arXiv preprintAndriy Regeta and Immanuel van Santen. Characterizing affine toric varieties via the automorphism group. arXiv preprint arXiv:2112.04784, 2021. Characterizing smooth affine spherical varieties via the automorphism group. Andriy Regeta, Immanuel Van Santen, Journal de l'École polytechnique-Mathématiques. 8Andriy Regeta and Immanuel van Santen. Characterizing smooth affine spherical varieties via the automor- phism group. Journal de l'École polytechnique-Mathématiques, 8:379-414, 2021. . Instituto De Investigación, Universidad Interdisciplinaria, De Talca, Chile Talca, address: [email protected] DE INVESTIGACIÓN INTERDISCIPLINARIA, UNIVERSIDAD DE TALCA, CASILLA 721, TALCA, CHILE. Email address: [email protected] Email address: [email protected]. Universidad Instituto De Matemática Y Física, De Talca, TALCA, CHILEclINSTITUTO DE MATEMÁTICA Y FÍSICA, UNIVERSIDAD DE TALCA, CASILLA 721, TALCA, CHILE. Email address: [email protected]	[]
[ "The production of ionizing photons in UV-faint z ∼ 3 − 7 galaxies", "The production of ionizing photons in UV-faint z ∼ 3 − 7 galaxies" ]	[ "Gonzalo Prieto-Lyon \nCosmic Dawn Center (DAWN)\n\n\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark\n", "Victoria Strait \nCosmic Dawn Center (DAWN)\n\n\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark\n", "Charlotte A Mason \nCosmic Dawn Center (DAWN)\n\n\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark\n", "Gabriel Brammer \nCosmic Dawn Center (DAWN)\n\n\nNiels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark\n", "Gabriel B Caminha \nPhysik-Department\nTechnische Universität München\nJames-Franck Str. 185748GarchingGermany\n\nMax-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Str. 185748GarchingGermany\n\nDimensions (ASTRO 3D)\nAustralia\n", "Amata Mercurio \nDipartimento di Fisica \"E.R. Caianiello\"\nUniversità Degli Studi di Salerno\nVia Giovanni Paolo III-84084Fisciano (SA)Italy\n\nINAF -Osservatorio Astronomico di Capodimonte\nVia Moiariello 16I-80131NapoliItaly\n", "Ana Acebron \nDipartimento di Fisica\nUniversità degli Studi di Milano\nvia Celoria 16I-20133MilanoItaly\n\nINAF -IASF Milano\nvia A. Corti 12I-20133MilanoItaly\n", "Pietro Bergamini \nDipartimento di Fisica\nUniversità degli Studi di Milano\nvia Celoria 16I-20133MilanoItaly\n\nOsservatorio di Astrofisica e Scienza dello Spazio di Bologna\nINAF -OAS\nvia Gobetti 93/3I-40129BolognaItaly\n", "Claudio Grillo \nDipartimento di Fisica\nUniversità degli Studi di Milano\nvia Celoria 16I-20133MilanoItaly\n\nINAF -IASF Milano\nvia A. Corti 12I-20133MilanoItaly\n", "Piero Rosati \nOsservatorio di Astrofisica e Scienza dello Spazio di Bologna\nINAF -OAS\nvia Gobetti 93/3I-40129BolognaItaly\n\nDipartimento di Fisica e Scienze della Terra\nUniversità di Ferrara\nVia Saragat 1I-44122FerraraItaly\n", "Eros Vanzella \nDipartimento di Fisica e Scienze della Terra\nUniversità di Ferrara\nVia Saragat 1I-44122FerraraItaly\n", "Marco Castellano \nINAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly\n", "Emiliano Merlin \nINAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly\n", "Diego Paris \nINAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly\n", "Kristan Boyett \nSchool of Physics\nUniversity of Melbourne\n3010Parkville\n\nVIC\nAustralia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n", "Antonello Calabrò \nINAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly\n", "Takahiro Morishita \nIPAC\nCalifornia Institute of Technology\n1200 E. California Boulevard314-6, 91125PasadenaMC, CAUSA\n", "Sara Mascia \nINAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly\n", "Laura Pentericci \nINAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly\n", "Guido Roberts-Borsani \nDepartment of Physics and Astronomy\nUniversity of California\n430 Portola Plaza90095Los Angeles, Los AngelesCAUSA\n", "Namrata Roy \nCenter for Astrophysical Sciences\nDepartment of Physics & Astronomy\nJohns Hopkins University\n21218BaltimoreMDUSA\n", "Tommaso Treu \nDepartment of Physics and Astronomy\nUniversity of California\n430 Portola Plaza90095Los Angeles, Los AngelesCAUSA\n", "Benedetta Vulcani \nINAF Osservatorio Astronomico di Padova\nvicolo dell'Osservatorio 535122PadovaItaly\n" ]	[ "Cosmic Dawn Center (DAWN)\n", "Niels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark", "Cosmic Dawn Center (DAWN)\n", "Niels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark", "Cosmic Dawn Center (DAWN)\n", "Niels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark", "Cosmic Dawn Center (DAWN)\n", "Niels Bohr Institute\nUniversity of Copenhagen\nJagtvej 128DK-2200Copenhagen NDenmark", "Physik-Department\nTechnische Universität München\nJames-Franck Str. 185748GarchingGermany", "Max-Planck-Institut für Astrophysik\nKarl-Schwarzschild-Str. 185748GarchingGermany", "Dimensions (ASTRO 3D)\nAustralia", "Dipartimento di Fisica \"E.R. Caianiello\"\nUniversità Degli Studi di Salerno\nVia Giovanni Paolo III-84084Fisciano (SA)Italy", "INAF -Osservatorio Astronomico di Capodimonte\nVia Moiariello 16I-80131NapoliItaly", "Dipartimento di Fisica\nUniversità degli Studi di Milano\nvia Celoria 16I-20133MilanoItaly", "INAF -IASF Milano\nvia A. Corti 12I-20133MilanoItaly", "Dipartimento di Fisica\nUniversità degli Studi di Milano\nvia Celoria 16I-20133MilanoItaly", "Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nINAF -OAS\nvia Gobetti 93/3I-40129BolognaItaly", "Dipartimento di Fisica\nUniversità degli Studi di Milano\nvia Celoria 16I-20133MilanoItaly", "INAF -IASF Milano\nvia A. Corti 12I-20133MilanoItaly", "Osservatorio di Astrofisica e Scienza dello Spazio di Bologna\nINAF -OAS\nvia Gobetti 93/3I-40129BolognaItaly", "Dipartimento di Fisica e Scienze della Terra\nUniversità di Ferrara\nVia Saragat 1I-44122FerraraItaly", "Dipartimento di Fisica e Scienze della Terra\nUniversità di Ferrara\nVia Saragat 1I-44122FerraraItaly", "INAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly", "INAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly", "INAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly", "School of Physics\nUniversity of Melbourne\n3010Parkville", "VIC\nAustralia", "ARC Centre of Excellence for All Sky Astrophysics in\n", "INAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly", "IPAC\nCalifornia Institute of Technology\n1200 E. California Boulevard314-6, 91125PasadenaMC, CAUSA", "INAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly", "INAF Osservatorio Astronomico di Roma\nVia Frascati 33, 00078 Monteporzio CatoneRomeItaly", "Department of Physics and Astronomy\nUniversity of California\n430 Portola Plaza90095Los Angeles, Los AngelesCAUSA", "Center for Astrophysical Sciences\nDepartment of Physics & Astronomy\nJohns Hopkins University\n21218BaltimoreMDUSA", "Department of Physics and Astronomy\nUniversity of California\n430 Portola Plaza90095Los Angeles, Los AngelesCAUSA", "INAF Osservatorio Astronomico di Padova\nvicolo dell'Osservatorio 535122PadovaItaly" ]	[]	Aims. The demographics of the production and escape of ionizing photons from UV-faint early galaxies is a key unknown in discovering the primary drivers of reionization. With the advent of JWST it is finally possible to observe the rest-frame optical nebular emission from individual sub-L * z > 3 galaxies to measure the production rate of ionizing photons, ξ ion . Methods. Here we study a sample of 370 z ∼ 3 − 7 galaxies spanning −23 < M UV < −15.5 (median M UV ≈ −18) with deep multiband HST and JWST/NIRCam photometry covering the rest-UV to optical from the GLASS and UNCOVER JWST surveys. Our sample includes 102 galaxies with Lyman-alpha emission detected in MUSE spectroscopy. We use Hα fluxes inferred from NIRCam photometry to estimate the production rate of ionizing photons which do not escape these galaxies ξ ion (1 − f esc ). Results. We find median log 10 ξ ion (1 − f esc ) = 25.33 ± 0.47, with a broad intrinsic scatter 0.42 dex, implying a broad range of galaxy properties and ages in our UV-faint sample. Galaxies detected with Lyman-alpha have ∼ 0.1 dex higher ξ ion (1 − f esc ), which is explained by their higher Hα EW distribution, implying younger ages, higher sSFR and thus more O/B stars. We find significant trends of increasing ξ ion (1 − f esc ) with increasing Hα EW, decreasing UV luminosity, and decreasing UV slope, implying the production of ionizing photons is enhanced in young galaxies with assumed low metallicities. We find no significant evidence for sources with very high ionizing escape fraction ( f esc > 0.5) in our sample, based on their photometric properties, even amongst the Lyman-alpha selected galaxies.Conclusions. This work demonstrates that considering the full distribution of ξ ion across galaxy properties is important for assessing the primary drivers of reionization.	10.1051/0004-6361/202245532	[ "https://export.arxiv.org/pdf/2211.12548v2.pdf" ]	253,801,884	2211.12548	a54459436073fa9a45317aaa3394307bc90520b8	The production of ionizing photons in UV-faint z ∼ 3 − 7 galaxies January 26, 2023 Gonzalo Prieto-Lyon Cosmic Dawn Center (DAWN) Niels Bohr Institute University of Copenhagen Jagtvej 128DK-2200Copenhagen NDenmark Victoria Strait Cosmic Dawn Center (DAWN) Niels Bohr Institute University of Copenhagen Jagtvej 128DK-2200Copenhagen NDenmark Charlotte A Mason Cosmic Dawn Center (DAWN) Niels Bohr Institute University of Copenhagen Jagtvej 128DK-2200Copenhagen NDenmark Gabriel Brammer Cosmic Dawn Center (DAWN) Niels Bohr Institute University of Copenhagen Jagtvej 128DK-2200Copenhagen NDenmark Gabriel B Caminha Physik-Department Technische Universität München James-Franck Str. 185748GarchingGermany Max-Planck-Institut für Astrophysik Karl-Schwarzschild-Str. 185748GarchingGermany Dimensions (ASTRO 3D) Australia Amata Mercurio Dipartimento di Fisica "E.R. Caianiello" Università Degli Studi di Salerno Via Giovanni Paolo III-84084Fisciano (SA)Italy INAF -Osservatorio Astronomico di Capodimonte Via Moiariello 16I-80131NapoliItaly Ana Acebron Dipartimento di Fisica Università degli Studi di Milano via Celoria 16I-20133MilanoItaly INAF -IASF Milano via A. Corti 12I-20133MilanoItaly Pietro Bergamini Dipartimento di Fisica Università degli Studi di Milano via Celoria 16I-20133MilanoItaly Osservatorio di Astrofisica e Scienza dello Spazio di Bologna INAF -OAS via Gobetti 93/3I-40129BolognaItaly Claudio Grillo Dipartimento di Fisica Università degli Studi di Milano via Celoria 16I-20133MilanoItaly INAF -IASF Milano via A. Corti 12I-20133MilanoItaly Piero Rosati Osservatorio di Astrofisica e Scienza dello Spazio di Bologna INAF -OAS via Gobetti 93/3I-40129BolognaItaly Dipartimento di Fisica e Scienze della Terra Università di Ferrara Via Saragat 1I-44122FerraraItaly Eros Vanzella Dipartimento di Fisica e Scienze della Terra Università di Ferrara Via Saragat 1I-44122FerraraItaly Marco Castellano INAF Osservatorio Astronomico di Roma Via Frascati 33, 00078 Monteporzio CatoneRomeItaly Emiliano Merlin INAF Osservatorio Astronomico di Roma Via Frascati 33, 00078 Monteporzio CatoneRomeItaly Diego Paris INAF Osservatorio Astronomico di Roma Via Frascati 33, 00078 Monteporzio CatoneRomeItaly Kristan Boyett School of Physics University of Melbourne 3010Parkville VIC Australia ARC Centre of Excellence for All Sky Astrophysics in Antonello Calabrò INAF Osservatorio Astronomico di Roma Via Frascati 33, 00078 Monteporzio CatoneRomeItaly Takahiro Morishita IPAC California Institute of Technology 1200 E. California Boulevard314-6, 91125PasadenaMC, CAUSA Sara Mascia INAF Osservatorio Astronomico di Roma Via Frascati 33, 00078 Monteporzio CatoneRomeItaly Laura Pentericci INAF Osservatorio Astronomico di Roma Via Frascati 33, 00078 Monteporzio CatoneRomeItaly Guido Roberts-Borsani Department of Physics and Astronomy University of California 430 Portola Plaza90095Los Angeles, Los AngelesCAUSA Namrata Roy Center for Astrophysical Sciences Department of Physics & Astronomy Johns Hopkins University 21218BaltimoreMDUSA Tommaso Treu Department of Physics and Astronomy University of California 430 Portola Plaza90095Los Angeles, Los AngelesCAUSA Benedetta Vulcani INAF Osservatorio Astronomico di Padova vicolo dell'Osservatorio 535122PadovaItaly The production of ionizing photons in UV-faint z ∼ 3 − 7 galaxies January 26, 2023Submitted November 22, 2022Astronomy & Astrophysics manuscript no. ms_MUSE_xi_ionGalaxies: emission lines -Galaxies: high-redshift -Galaxies: evolution Aims. The demographics of the production and escape of ionizing photons from UV-faint early galaxies is a key unknown in discovering the primary drivers of reionization. With the advent of JWST it is finally possible to observe the rest-frame optical nebular emission from individual sub-L * z > 3 galaxies to measure the production rate of ionizing photons, ξ ion . Methods. Here we study a sample of 370 z ∼ 3 − 7 galaxies spanning −23 < M UV < −15.5 (median M UV ≈ −18) with deep multiband HST and JWST/NIRCam photometry covering the rest-UV to optical from the GLASS and UNCOVER JWST surveys. Our sample includes 102 galaxies with Lyman-alpha emission detected in MUSE spectroscopy. We use Hα fluxes inferred from NIRCam photometry to estimate the production rate of ionizing photons which do not escape these galaxies ξ ion (1 − f esc ). Results. We find median log 10 ξ ion (1 − f esc ) = 25.33 ± 0.47, with a broad intrinsic scatter 0.42 dex, implying a broad range of galaxy properties and ages in our UV-faint sample. Galaxies detected with Lyman-alpha have ∼ 0.1 dex higher ξ ion (1 − f esc ), which is explained by their higher Hα EW distribution, implying younger ages, higher sSFR and thus more O/B stars. We find significant trends of increasing ξ ion (1 − f esc ) with increasing Hα EW, decreasing UV luminosity, and decreasing UV slope, implying the production of ionizing photons is enhanced in young galaxies with assumed low metallicities. We find no significant evidence for sources with very high ionizing escape fraction ( f esc > 0.5) in our sample, based on their photometric properties, even amongst the Lyman-alpha selected galaxies.Conclusions. This work demonstrates that considering the full distribution of ξ ion across galaxy properties is important for assessing the primary drivers of reionization. Introduction In recent years, we have obtained increasing evidence that the reionization of hydrogen happened fairly late, approximately one billion years after the Big Bang (z ∼ 5.5 − 10), with a midpoint around z ∼ 7 − 8 (e.g., Fan et al. 2006;Stark et al. 2010;McGreer et al. 2015;Mason et al. 2018;Davies et al. 2018;Qin et al. 2021;Planck Collaboration et al. 2020;Bolan et al. 2022). However, there is evidence for significant star formation before E-mail: [email protected] this time (e.g., Oesch et al. 2018;Hashimoto et al. 2018;McLeod et al. 2021) thus it appears that reionization lags behind galaxy formation. The reason for this lag is unknown: we are still lacking a full physical understanding of the reionization process. In particular, we still do not know which types of galaxies drive the process, i.e. which physical mechanisms mediate the production and escape of ionizing photons from galaxies. In order to produce such a late and fairly rapid reionization, the ionizing population could have been dominated by low mass, UV-faint galaxies with a low (∼ 5%) average escape fraction (e.g., Mason et al. Article number, page 1 of 10 arXiv:2211.12548v2 [astro-ph.GA] 25 Jan 2023 A&A proofs: manuscript no. ms_MUSE_xi_ion 2019; Qin et al. 2021). Alternatively, rarer more massive galaxies with higher escape fractions could have been responsible (e.g., Sharma et al. 2017;Naidu et al. 2020). With only measurements of the timing of reionization, these scenarios are degenerate, thus physical priors on the ionizing properties of galaxies across cosmic time are necessary to pinpoint the sources of reionization. The total ionizing output of galaxies can be simply parameterized (e.g., Madau et al. 1999;Robertson et al. 2010) as the product of the production rate of ionizing photons relative to non-ionizing UV photons, ξ ion (determined by the stellar populations, e.g., Stanway et al. 2016), and the fraction of ionizing photons which escape the interstellar medium (ISM) into the intergalactic medium (IGM), f esc (determined by the structure and ionization state of the ISM, likely shaped by star formation and feedback, e.g., Trebitsch et al. 2017;Ma et al. 2020). Both of these quantities are also expected to vary with time in an individual galaxy, for example due to the lifetime and properties of young stellar populations, and the effects of feedback and bursty star formation on the ISM. While we can easily observe the non-ionizing UV photons from galaxies, the high optical depth of the IGM to ionizing photons makes direct measurements of the escaping ionizing spectrum statistically unlikely at z ∼ > 3 (Inoue et al. 2014;Becker et al. 2021;Vanzella et al. 2018). Alternatively, fluxes of nonresonant recombination lines, emitted by gas which was ionized in HII regions around massive stars, can crucially measure the flux of ionizing photons which do not escape galaxies. In particular, Hα emission can be used to directly estimate (1 − f esc )ξ ion (e.g., Leitherer & Heckman 1995;Bouwens et al. 2016;Shivaei et al. 2018;Emami et al. 2020). As f esc is inferred to be low ( ∼ < 10%) on average for Lyman-break galaxies at z ∼ 2 − 4 (Steidel et al. 2018;Begley et al. 2022;Pahl et al. 2022) measurements of Hα should trace the intrinsic production of ionizing photons reasonably well. ξ ion can also be inferred from the strength of [OIII]+Hβ emission, though due to the dependence of [OIII] emission on metallicity and ionization parameter, the correlation is not as tight as with Hα (e.g., Chevallard et al. 2018). Previous work at z ∼ < 2.5 , where direct Hα spectroscopy has been possible from the ground, has found a mean log 10 ξ ion [erg Hz −1 ] ≈ 25.3, with a scatter of ∼ 0.3 dex, likely dominated by variations in stellar populations between galaxies (e.g., Shivaei et al. 2018;Tang et al. 2019). At higher redshifts where Hα redshifts into the infra-red, broad-band photometry with Spitzer has been used extensively to estimate Hα line fluxes (e.g., Schaerer & de Barros 2009;Shim et al. 2011;Stark et al. 2013;Smit et al. 2015;Bouwens et al. 2016;Lam et al. 2019;Maseda et al. 2020;Stefanon et al. 2022). However, due to the limited spatial resolution and sensitivity of Spitzer, previous works had been limited to studying ξ ion in isolated, bright (> L * ) galaxies where deblending IRAC photometry was possible (e.g,. Bouwens et al. 2016) and stacks for fainter galaxies (e.g., Lam et al. 2019;Maseda et al. 2020). With JWST it is finally possible to extend these studies to individual UV-faint galaxies , and obtain rest-frame optical spectroscopy at z > 3 (e.g., Sun et al. 2022;Williams et al. 2022). Results from previous analyses have been intriguing but require further investigation. Using stacked IRAC photometry Lam et al. (2019) found no significant evidence for a strong correlation of ξ ion with M uv . However, Maseda et al. (2020) found a population of extremely UV-faint (M uv > −16) galaxies selected as Lyα emitters in deep MUSE observations, which have very elevated ξ ion compared to higher luminosity galaxies and at fixed Hα EW, implying these efficient ionizing galaxies are particu-larly young and low metallicity. It is thus important to examine the distribution of ξ ion at low UV luminosities, and to compare galaxies with and without Lyα emission to better understand the demographics of the ionizing population. Furthermore, using early JWST NIRCam data Endsley et al. (2022) discovered a population of UV-faint (M uv ∼ −19) galaxies at z ∼ 6.5 − 8 with high sSFR but low EW [OIII]+Hβ inferred from photometry. The high sSFR would imply high ξ ion due to the increased abundance of O and B stars. To explain the low [OIII]+Hβ EW, Endsley et al. (2022) suggest either these galaxies have extremely low metallicities (reducing oxygen abundance), or alternatively, all nebular lines are reduced. A reduction in all nebular lines could be due to either them being produced in density-bounded HII regions with very high ionizing escape fraction (e.g. Zackrisson et al. 2013;Marques-Chaves et al. 2022), or recent cessation of star formation. At z ∼ 3 − 7 both [OIII]+Hβ and Hα are visible in NIRCam photometry, enabling us to test these scenarios. In this paper we make use of deep multi-band HST/ACS, WFC3 and JWST/NIRCam imaging with overlapping MUSE observations, enabling us to blindly detect a spectroscopic sample with precision rest-frame ultra-violet to optical photometry. We measure the distribution of ξ ion over a broader luminosity range (−23 ∼ < M uv ∼ < − 15.5) than previously possible in individual galaxies, due to the excellent resolution and sensitivity of NIRCam at rest-optical wavelength compared to Spitzer/IRAC, and the power of gravitational lensing. We explore correlations of ξ ion with empirical galaxy properties. We find significant trends of increasing ξ ion with decreasing UV luminosity, UV β slope and with increasing Hα EW, all implying the strongest ionizers are young sources with expected low metallicities. We also explore whether our sample shows evidence for very low metallicities or extremely high escape fraction. The paper is structured as follows. In Section 2 we describe the photometric and spectroscopic data for our study. In Section 3 we describe how we infer the ionizing production rate ξ ion and we describe our results of correlations between ξ ion and other galaxy properties, and comparison to the literature in Section 4. We discuss our results and state our conclusions in Section 6. We assume a flat ΛCDM cosmology with Ω m = 0.3, Ω Λ = 0.7, h = 0.7, all magnitudes are in the AB system. Data For this work we select fields with multi-band HST/ACS and JWST/NIRCam imaging and overlapping MUSE spectroscopy. We select sources detected with Lyα emission (z ∼ 2.9 − 6.7 in MUSE) and sources with high probability of being in the same redshift range based on photometric redshift, and use the HST + JWST photometry to extract optical emission line fluxes. Below we describe the datasets and the selection of our sample. Imaging We use JWST NIRCam imaging in parallel to and of the cluster Abell 2744 from the GLASS-JWST program ERS-1324 (PI Treu Treu et al. 2022) and UNCOVER 1 program GO-2561 (co-PIs Labbé, Bezanson). The GLASS-JWST NIRCam observations discussed in this paper were taken in parallel to NIRISS observations of the cluster Abell 2744 on June 28-29 2022. They are centered at In our analysis, we also include new and archival HST imaging, from which ACS imaging is particularly important for constraining photometric redshifts. This includes new HST/ACS data in F606W (59530 s.), F775W (23550 s.), and F814W (123920 s) from HST-GO/DD program 17231 2 (PI Treu), as well as archival data acquired under the Hubble Frontier Fields program (HST-GO/DD-13495, PI Lotz, Lotz et al. 2017), BUF-FALO (HST-GO-15117, PI Steinhardt, Steinhardt et al. 2020) and programs HST-GO-11689 (PI Dupke), HST-GO-11386 (PI Rodney), HST-GO-13389 (PI Siana), HST-GO-15940 (PI Ribeiro) and HST-SNAP-16729 (PI Kelly). Not all HST bands cover every object in our sample, but we only keep objects in our sample that have a well-constrained photometric redshift, usually meaning that there is ACS coverage (see Section 2.4). We also include HST/WFC3 imaging for completeness, but these are generally not as constraining as the NIRCam fluxes. The image reduction and calibration, and the methods used to detect sources and measure multi-band photometry in both fields closely follow that of Brammer et al., (in prep). Briefly, we pull calibrated images from the MAST archive 3 and process them with the grizli pipeline (Brammer et al. 2022). The pipeline first aligns the exposures to external catalogs and to each other and corrects for any distortion within the image. Following this, we subtract a sky-level background, divide out flatfield structure using custom flat-field images, and correct for 1/ f noise. We also correct for NIRCam image anomalies, which include persistence, any remaining cosmic rays, and 'snowballs' (see Rigby et al. 2022). Finally, we apply zeropoint corrections calculated by G. Brammer 4 and drizzle all exposures to a common pixel grid. For source detection, we use SEP (Barbary 2018) to perform aperture photometry on the F444W detection image in each field. VLT/MUSE spectroscopy MUSE spectroscopy in the Abell 2744 cluster were obtained through ESO program 094.A-0115 (PI Richard) and are described by Mahler et al. (2018) and Richard et al. (2021). We use their publicly available catalog to select Lyα emitting galaxies. The data comprise a 4 sq. arcmin mosaic centered on the cluster core. Four 1 sq. arcmin quadrants were observed for a total of 3.5, 4, 4 and 5 hours respectively, and the center of the cluster was observed for an additional 2 hours. The median line flux 1σ uncertainty in the MUSE data is 3.6 × 10 −19 erg s −1 cm −2 . This corresponds to an 5σ EW limit of ∼ 4 − 30 Å over z ∼ 3 − 7 for a galaxy with M uv = −19 (the median for our sample, before accounting for magnification, as EW is invariant under magnification). VLT/MUSE spectroscopy in the GLASS-JWST NIRCam fields were obtained through a new ESO DDT program 109.24EZ.001 (co-PIs Mason, Vanzella) on the nights of July 28 and August 20 2022. The data comprise 5 pointings (4 pointings -4 sq. arcmin -overlapping with NIRCam imaging) each with 1 hour exposure time. The raw data are publicly available on the ESO archive 5 . The reduction, calibration and source detection methods used for this work are identical to techniques described in previous works (Caminha et al. 2017;Caminha et al. 2019). A full assessment of the depth is ongoing but based on the ∼ 4 hour depth of the Mahler et al. (2018) observations described above, we estimate a 5σ EW limit of ∼ 8 − 60 Å in these shallower data. In this work we use 102 spectroscopic confirmations at z ∼ 2.9−6.7: 42 from the GLASS-JWST NIRCam fields and 60 from the Abell 2744 cluster field. Gravitational lensing magnification For the galaxies detected in the core of the Abell 2744 cluster we correct for gravitational lensing magnification using the model by Bergamini et al. (2022). The median magnification of the sample is µ = 3.54 with 90% of the galaxies having µ = 2 − 20. We remove sources with a magnification with µ > 50 (12 sources) due to high uncertainties in the model near the critical curves. The galaxies in the parallel fields are ∼ 3 − 10 away from the cluster core where the magnification is expected to be modest (µ ≈ 1), we do not account for magnification of those sources. Sample selection For this work, we focus on selecting a sample of galaxies at z ∼ 3 − 7 with high purity. We select 102 MUSE Lyα-detected galaxies with overlapping HST/ACS and JWST/NIRCam data as described above. We also select a comparison sample of galaxies based on peak photometric redshift, within the same footprint as the MUSE observations, which we expect to have slightly lower Hα EW than the Lyα selected sample. We find the photometric redshift distribution of all sources detected as described in Section 2.1 using EAZY (Brammer et al. 2008), given all available photometric bands. To build a photometric sample with high purity, following Bouwens et al. (2016), we select sources with the peak of their photometric redshift between 2.9 < z < 6.7, and keep only sources which have 90% of the redshift probability density between ∆z ∼ 1 of the peak of their distribution. The resulting high purity photometric sample consists of 268 galaxies. The redshift and UV magnitude distribution of our sample is shown in Figure 1. The median redshift of the full sample is 4.02, with the Lyα-selected sample having median redshift 3.95. The median M uv is -18.1, with a Kolmogorov-Smirnov (KS) test showing no significant difference between the Lyα and photometrically selected samples. Inferring the ionizing photon production rate Inferring nebular emission line strengths from photometry To estimate nebular emission line fluxes from broad-band photometry, we follow approaches in the literature and fit the spec- Galaxies studied in this work; in purple Lyα detected galaxies, and in gray galaxies photometrically selected (with no Lyα detected). Top: Distribution of redshifts for the spectroscopic and photometric samples. We show the spectroscopic redshift where available, or the peak photometric redshift. Bottom: UV magnitude distribution for our sample, we find a median value of −18.14 ± 1.58, with no statistically significant difference between both samples. tral energy distribution (SED) to the full photometry, excluding bands we expect to contain strong nebular emission lines (e.g., Shim et al. 2011;Stark et al. 2013;Mármol-Queraltó et al. 2016;Bouwens et al. 2016). This provides us with a model for the continuum flux in those bands which we can subtract from the observed photometry to infer the line flux. We use BAGPIPES to fit SEDs (Carnall et al. 2018). We adopt BC03 (Bruzual & Charlot 2003) templates, and exclude any nebular emission contribution. We do not consider any broadbands where Hα or [OIII]+Hβ are observed according to each galaxy's redshift. For ease of comparison to the literature (e.g., Maseda et al. 2020;Lam et al. 2019), we assume a Chabrier (2003) Initial Mass Function and an Small Magellenic Cloud (SMC, Prevot et al. 1984) dust attenuation law, allowing A V to vary from 0−3 mag. Because metallicity is not well known at the range of redshifts we explore, we allow metallicity to vary from 0 − 2 Z . Because star formation histories are notoriously diffi-cult to constrain at high-redshift (Strait et al. 2021), we assume an exponentially rising delayed τ star formation history, allowing τ to vary freely. For the spectroscopically confirmed Lyman α emitters, we fix the redshift at the Lyα redshift. For our photometric sample, we use the photometric redshift obtained from EAZY with uniform prior with ∆z = 1. (see Section 2.4). We then compare the SED model of the galaxy's continuum to the broadbands where Hα or [OIII]+Hβ fall. We multiply the non-nebular SED posteriors with the transmission of the aforementioned broadbands to obtain the contribution of the galaxy's continuum to the observed flux. By subtracting this continuum flux contribution to the observed photometry, we then are able to recover the flux distributions of Hα and [OIII]+Hβ emission lines for each galaxy. We compared our measurements with a sample of 6 galaxies with [OIII]+Hβ EW measurements from the GLASS-ERS program using JWST/NIRISS (Boyett et al. 2022a), finding that our method recovers the EW of these sources within ∼ 20 − 40%. A full comparison of these photometric inference methods is left to future work. There are some limitations to our method for obtaining line fluxes, such as contamination from the 4000-Å break in the broadband containing [OIII]+Hβ, the chance that the line falls outside the effective width of any of our broadband filters, or Hα and [OIII]+Hβ falling on the same band. We consider a contribution of 6.8% from [NII] to the calculated Hα flux, and 9.5% from [SII] according to Anders & Fritze-v. Alvensleben (2003). We remove galaxies with a poor χ 2 (>50) score on their SED fit, we choose these value by ignoring all galaxies on the high-end of the χ 2 distribution. The advantage of this approach, unlike estimating line fluxes directly from the SED fitting, is that it does not depend strongly on star formation history assumptions and allows us to make a mostly empirical measurement of the line fluxes. We obtain comparable results using the flux in the band red-ward of Hα as the continuum flux, assuming a flat optical continuum (see also e.g., Maseda et al. 2020). Estimating other physical parameters such as star-formation rate and stellar mass from the SED fitting did not give reliable results, since it was too dependent on the initial assumptions, and needed extremely young ages (<10Myrs) and an instantaneous burst of star-formation to recreate the observed nebular emissions.. The following results will consist of: 83 and 64 Lyα-emitting galaxies with Hα and [OIII]+Hβ emission line measurements respectively, a photometric sample of 220 and 203 galaxies with Hα, and [OIII]+Hβ emission line measurements respectively. We see both lines in 62 Lyα galaxies and 177 photometrically selected galaxies. We see no apparent biases in our M uv distribution after narrowing down the sample. Nebular emission flux errors are derived from the 68% confidence interval of the resulting distributions. Measuring UV absolute magnitude and slope To infer the UV absolute magnitude, M uv (magnitude at 1500Å) and β slope we fit a power law (e.g., Rogers et al. 2013) f λ ∝ λ β to the fluxes from the HST and JWST bands. We perform the fit using a Markov Chain Monte Carlo sampling using the python module emcee (Foreman-Mackey et al. 2013). We assume flat priors for β and M uv , with bounds −4 < β < 1 and −25 < M UV < −12, sufficient to explore the common value ranges for galaxies (e.g., Bouwens et al. 2014). To obtain the photometric bands that are observing the UVrestframe of our galaxies, we exclude any bands that fall blue-wards from Lyman-α that might be affected by the Lyman-Break. For the same reason, we exclude bands redwards from the 4000Å break in the restframe. After these requirements, we are left with between 3 and 4 bands for each source. In the case of galaxies with Lyman-α detected in MUSE, we use the line's redshift. For photometrically-selected galaxies, in each call of the likelihood, we randomly draw a redshift from a Normal distribution N(µ = z phot , σ = 0.5), and select the appropriate photometric bands. For lensed sources, we consider magnifications and apply them following the same random draw method as redshift. We use the corresponding magnification and error obtained from the Bergamini et al. (2022) lensing model. Determination of ξ ion We define the production rate of ionizing photons, ξ ion . This is given by the ratio between the luminosity of observed ionizing photons and the intrinsic luminosity of the ionizing UV photons (e.g., Leitherer & Heckman 1995): ξ ion = L Hα (1 − f esc )L intr UV,ν × 7.37 × 10 11 Hz erg −1(1) where L Hα is the unattenuated Hα luminosity in erg s −1 and L ν,UV,intr is the intrinsic UV luminosity density at 1500Å. The models from where the conversion factor is derived, assume a young population of massive stars equivalent to a massive HII region. We assume this type of environment to be similar to what we would find in young galaxies. Because Hα is produced by the excitation of hydrogen gas from ionizing radiation that does not escape the galaxy, and we cannot measure f esc directly in our sample, we note that we are obtaining the production rate of ionizing photons which did not escape the galaxy, ξ ion (1 − f esc ). We first calculate L Hα directly from the SED obtained in Section 3.1, after accounting for dust attenuation (Prevot et al. 1984). To obtain the intrinsic value of the UV luminosity, we take into account the dust attenuation following Lam et al. (2019), where the intrinsic UV luminosity is defined as L intr UV,ν = L UV,ν / f esc,UV . f esc,UV is the fraction of escaping UV photons not absorbed by the dust. For this, we use the Small Magellanic Cloud dust law defined by Prevot et al. (1984): f esc,UV = 10 −1.1(β+2.23)/2.5 , β > −2.23 (2) Where β is the UV slope obtained in Section 3.2. Galaxies with slopes bluer than β < -2.23 are assumed dust-free and we do not correct for dust. In the following, uncertainties on ξ ion are the 68% confidence intervals, obtained from propagating the uncertainty in the Hα flux from its resulting distribution as described in Section 3.1, and the posterior distributions for β and M uv as described in Section 3.2. Correlation Analysis For the purpose of studying the correlations between galaxy properties we use the python package linmix 6 to do Bayesian linear regression including intrinsic scatter and accounting for two-dimensional errors (Kelly 2007). We fit for log 10 [(1 − f esc )ξ ion ] = α + βX + , where is the intrinsic scatter which is assumed to be normally distributed with variance σ 2 . We recover the best fit trend line from the posteriors, as well the 68% Distribution of (1f esc )ξ ion . The purple histogram includes galaxies with; Lyα emission detection, and in grey galaxies without Lyα detected. Overall, Lyα emitting galaxies show stronger ionizing photon production than galaxies with no Lyα emission selected galaxies, with median values 25.39±0.64 and 25.31±0.43 respectively. We show the median relation from the literature at z ∼ 2 − 5 as a dashed black line (e.g., Shivaei et al. 2018;Bouwens et al. 2016;Lam et al. 2019) confidence interval on the parameters. We report the results in Table 1 and show the best-fit line on plots. Results In the following section we present our results. In Section 4.1 we study the trends between ξ ion , Hα EW, M uv , and β slope, and in Section 4.2 we investigate whether our sample shows evidence for very high ionizing photon escape fraction and/or very low metallicity galaxies. Figure 2 shows the distribution of (1 − f esc )ξ ion for our Lyαselected and photometric samples. We find median values of log 10 ξ ion 25.39±0.64 and 25.31±0.43 respectively; with 25.33±0.47 [Hz erg −1 ] for the complete data set. We find an intrinsic scatter of 0.42 dex, obtained by subtracting in quadrature the average uncertainty in log 10 ξ ion (= 0.21 dex) from the standard deviation of the observed distribution. The recovered intrinsic scatter is broader by ∼ 0.1 dex than that found by Bouwens et al. (2016) and Shivaei et al. (2018) in M uv ∼ < − 20 galaxies. The broad distribution of ξ ion is likely an outcome of the broad range of stellar populations in these galaxies, i.e. due to a range of star formation histories and thus ages, and stellar metallicity (see e.g., Shivaei et al. 2018). Behaviour of ξ ion We perform a two sample Kolmogorov-Smirnov (KS) test to test whether the Lyα-selected and photometric samples are drawn from the same distribution. We recover p-value of 0.03, meaning it is likely that the underlying distributions are different, consistent with the results from (Saldana-Lopez et al. 2022), where a statistically significant difference is found between the ξ ion distributions of LAEs and non-LAEs at z ∼ 3 − 5. Given that galaxies with strong Lyα emission also likely have high ionizing photon escape fractions (e.g., Verhamme et al. 2015;Dijkstra et al. 2016) it is likely that the intrinsic ionizing photon produc-Article number, page 5 of 10 (2019) and Bouwens et al. (2016) as colored boxes for comparison. We find evidence for an increase in log 10 (1 − f esc )ξ ion towards fainter UV magnitude, with a slope of 0.03 ± 0.02, only considering the range where our sample is M uv complete (M uv < −18.1). We show literature constraints at similar redshifts as colored shapes (Bouwens et al. 2016;Harikane et al. 2018;Lam et al. 2019;Maseda et al. 2020), noting that all constraints fainter than M uv ∼ > − 20 were obtained by stacking IRAC photometry. tion efficiency of these galaxies is even higher than what we can infer based on Hα emission. Figure 3 shows (1 − f esc )ξ ion versus UV magnitude and demonstrates the revolutionary capabilities of MUSE and JWST/NIRCam: we are able to spectroscopically confirm extremely UV-faint galaxies via their high Lyα EW and we are able to infer Hα, and therefore ξ ion , from much fainter individual galaxies than was previously possible with Spitzer, where stacking was necessary at M uv ∼ > − 20 (e.g., Lam et al. 2019;Maseda et al. 2020). We reach ∼ 1 dex lower than any previous studies at similar redshifts without the need of stacking methods. We can reach individual detections of very faint galaxies, M uv < −17. We also find results consistent with those at z ∼ 2 (Shivaei et al. 2018) and at z ∼ 4 − 5 for > L * galaxies (Bouwens et al. 2016) and < L * galaxies (Lam et al. 2019, where a stacking analysis was used), as shown in Figure 2. We note our observations demonstrate the large scatter in (1 − f esc )ξ ion at fixed M uv which was not possible to observe in previous analyses which used stacking of Spitzer photometry for UV-faint galaxies. As described in Section 3.4 we perform a linear regression to assess correlations in our data. In contrast to Lam et al. (2019) we find significant evidence for a weak trend between ξ ion and M uv , where the highest ξ ion tends to come from the faintest galaxies. Since our sample is not M uv complete, we only study the correlation up to the peak of our M uv distribution (=-18.14) in Figure 1. We find log 10 [(1 − f esc )ξ ion ] = (0.03 ± 0.02)(M uv + 20) + 25.36 ± 0.03, but with large scatter (see Table 1). Figure 4 shows that ξ ion follows a strong trend with Hα EW, as found in previous work (Harikane et al. 2018;Lam et al. 2019;Tang et al. 2019). Previous works were limited to only the highest Hα EW values, while we reach ∼ 0.75 dex lower due to the sensitivity of NIRCam. This trend is consistent with a picture where ξ ion is elevated in the youngest, most highly star-forming galaxies (e.g., Tang et al. 2019). We find log 10 [(1 − f esc )ξ ion ] = (0.73 ± 0.04)(log 10 EW Hα − 2.5) + 25.15 ± 0.02. The measurement by Maseda et al. (2020), obtained from a stack of extremely UV-faint galaxies with high Lyα EW, lies significantly above our sample and the rest of the literature, with higher (1 − f esc )ξ ion at fixed Hα EW. As discussed by Maseda et al. (2020) this likely implies their sources have a much lower gas-phase metallicity than other samples. We also find that Lyα-selected galaxies have a higher Hα EW than the photometrically-selected sample, (median EW=732 ± 187 Å compared to 457 ± 161 Å for the photometric sample). A two-sample Kolmogorov-Smirnov test establishes that the EW distributions of the two samples are different (p-value 0.01). This is likely to be the primary driver of the increased ξ ion distribution for the Lyα-selected sample (Figure 2). At fixed Hα EW we see a clear tendency for galaxies having very blue β UV slopes to have elevated ξ ion (Figure 4). This trend is also seen in the full sample - Figure 5, where we find high ξ ion is weakly correlated to blue β slope, but with large scatter. We find log 10 (1 − f esc )ξ ion = (−0.20 ± 0.04)(β + 2) + 25.41 ± 0.01 (see Table 1). Similar correlations have been seen at z∼6 (i.e. Ning et al. 2022). Using KS test we find no significant difference in the β distributions for the Lyα and photometric samples. Our sample has a median β = −2.1. A search for high escape fraction and extremely low metallicity galaxies As well as being a tracer for the ionizing photon production of galaxies, nebular emission lines are also sensitive to the escape fraction. Zackrisson et al. (2013) proposed that in galaxies with very high ionizing escape fraction, one would expect a reduction in nebular emission line strength (Hβ EW ∼ < 30 Å) and extremely blue UV slopes (β < −2.5) due to the lack of nebular continuum. Early JWST observations have discovered potentially very blue galaxies (Topping et al. 2022, though c.f. Cullen et al. 2022) and galaxies with weak nebular line emission yet high sSFR (via [OIII]+Hβ, Endsley et al. 2022), potentially indicating a population with high ionizing escape fraction. However, the observation of low [OIII]+Hβ line strengths could also be caused by very low gas-phase metallicity (decreasing the strength of [OIII] emission) or a recent turnoff in star formation (which would also decrease all nebular emission lines). Given the redshift range of our sample we can infer both Hα and [OIII]+Hβ line strengths for 241 galaxies, allowing us to test these scenarios and to search for galaxies with high escape fraction. We obtained the [OIII]+Hβ nebular line fluxes as described in Section 3.1). In Figure 6 we show UV β slopes as a function of intrinsic Hα EW for our sample (where we correct for dust attenuation as described in Section 3.1. We compare our sample to the region proposed by Zackrisson et al. (2017) to have f esc > 0.5. While several sources fall into this region, and also have low [OIII]+Hβ EW ( ∼ < 100 Å) the uncertainties are too large to make these robust candidates. We discuss this further in Section 5.2. Figure 7 shows Hα EW as a function of [OIII]+Hβ for our sample. We see the expected positive correlation between both nebular emission lines, as these lines are all generated by the effects of stellar ionizing radiation. We see a very large scatter (with a range ∼ 1.5 dex) as expected due to variations in metallicity, temperature, and ionization parameter which will affect the strength of [OIII] individual galaxies (e.g., Maiolino . We compare Hα equivalent width with the ionizing photon production that does not escape the galaxy. As stars we show Lyα detected galaxies, as circles are photometrically selected galaxies with no Lyα. As above, error bars are only shown for 30% of the sources for clarity. We color code these two samples by UV β slope. On top we show the distribution of Hα EW for the same two samples, compared to the values found by Tang et al. (2019). We add data from Harikane et al. (2018) and Lam et al. (2019) for comparison, which is at the high end of our observed Hα EW distribution. We see that stronger ξ ion very strongly correlates with Hα EW. Galaxies with detected Lyα emission have an Hα EW distribution with higher values, median 732 ± 187 Å compared to 457 ± 161 with a Kolmogorov-Smirnov test p-value 0.01. The sources with the reddest UV slopes lie systematically below the best-fit relation at fixed Hα EW. Steidel et al. 2014;Sanders et al. 2021). We find log 10 EW(Hα) = 0.97 ± 0.06(log 10 EW([OIII] + Hβ) − 2.5) + 2.52 ± 0.03. Galaxies with detected Lyα tend to occupy the top right of the plot, with strong nebular emission lines, suggesting these are young star-forming galaxies with low metallicity and large ionization parameters, producing copious ionizing photons needed to power these emission lines (see e.g., Yang et al. 2017;Du et al. 2020;Tang et al. 2021, for more detailed studies). We find the Lyα selected galaxies have stronger [OIII]+Hβ EW compared to the photometric population, following the trend with Hα EW in Figure 4. Though, as discussed by Tang et al. (2021), not all galaxies with strong nebular emission are detected in Lyα, indicating that Lyα transmission is reduced due to a high column density of neutral gas in these systems and/or inclination effects. We compare our data to a z ∼ 2 sample by Tang et al. (2019), which was selected based on strong [OIII] emission. We find a similar correlation, but overall our ratio of Hα EW/[OIII]+Hβ EW is higher by ∼0.1 dex. Given that the Tang et al. (2019) sample has significantly sub-solar gas-phase metallicity Z < 0.3Z (Tang et al. 2021), the decrease we observe in [OIII] at fixed Hα EW would likely imply an overall lower metallicity due to a lower number of metal atoms in our sample. Discussion The profile of a strong ionizer Thanks to the depth of JWST/NIRCam we have been able to assess trends of ξ ion at z > 3 across the broadest range of galaxy properties to-date. From these results, we corroborate previous work at lower redshift and high luminosities, but push the measurement of ξ ion to a large sample of individual UV-faint galaxies for the first time. We find that galaxies with strong ionizing photon emission tend to have high Hα EW, low UV luminosity, blue UV β slope and Lyα emission -all implying that these galaxies are young, with likely low dust content and metallicity, and have a high O/B star population, capable of producing hard ionizing photons (e.g., Tang et al. 2019;Boyett et al. 2022b). This picture of the integrated emission from galaxies is complemented by high spatial resolution observations of highly magnified arcs with JWST. These have revealed extremely young star clusters ( ∼ < 10 Myr) with [OIII]+Hβ EW > 1000 Å, which dominate the ionizing photon production in their galaxy (Vanzella et al. 2022a,b), indicating that there can be large variations in ξ ion in individual galaxies, if they contain multiple stellar populations, but that the variation is primarily driven by the age of the stellar populations. (1-f esc ). In purple we show Lyα detected galaxies. In gray is the photometrically selected sample with no Lyα detected. As above, error bars are only shown for 30% of the sources for clarity. We add the stacked measurements from Lam et al. (2019) for comparison. We find a very weak trend of increasing ξ ion with decreasing β, with a linear slope of −0.10 ± 0.06. Comparison between intrinsic (unattenuated) EW of Hα and UV β slope, color-coded by [OIII]+Hβ EW. Lyα galaxies are shown with star shaped markers, and the photometric sample as circles. Galaxies shown in gray, do not have [OIII]+Hβ EW measurements. We show the region predicted by Zackrisson et al. (2017) to show f esc > 0.5. We rescale from Hβ EW to intrinsic Hα with a case B recombination scenario of factor 2.89, assuming a flat optical continuum in f λ , which we confirm from our SED fitting done in 3.1. We also find that overall, our Lyα galaxy sample has higher ξ ion than the photometrically selected one; the primary reason for this difference is that the former has higher Hα EW (Figure 4). The enhanced prevalence of Lyα emission in strong Hα emitters is likely a combination of increased production of Lyα photons due to the young stellar population implied by strong Hα, and potentially also an increase in the Lyα escape fraction in the interstellar medium (Tang et al. 2021;Naidu et al. 2022). In these rapidly star forming galaxies, the hard ionizing radiation may be ionizing the ISM and/or feedback may disrupt the ISM gas leading to a reduced HI column density and dust cover. We note that the galaxies with the highest (1 − f esc )ξ ion are not necessarily all Lyα-emitters, likely due to variance in the geometry and column density of neutral gas and dust in these sources. (Ning et al. 2022) has shown this same correlation between ξ ion and Lyα for a broad range of luminosities and equivalent widths. The ionizing photon escape fraction In Section 4.2 we explored whether our sample shows signs of high ionizing photon escape fraction, f esc , using the low Hβ EW -blue UV β slope region defined by Zackrisson et al. (2017) for f esc > 0.5. While several sources fell into this region, with both low Hα EW and [OIII]+Hβ EW ( ∼ < 100 Å), the uncertainties on the line flux measurements are too large for these to be robust candidates. More precise emission line measurements with JWST spectroscopy will be vital for identifying such candidates and their relative abundance in the galaxy population. The lack of high f esc candidates amongst the Lyα-selected galaxies is also surprising. As the same conditions (low neutral gas covering fraction) facilitate both Lyα escape and Lyman continuum escape, a correlation between Lyα and Lyman continuum escape fraction is expected (e.g., Verhamme et al. 2015;Dijkstra et al. 2016;Reddy et al. 2016). As discussed by Topping et al. (2022), however, it is possible for galaxies with high f esc but with very young ages to still have high nebular emission due to high ionizing photon production. It is likely that the criteria proposed by Zackrisson et al. (2017) can only find high f esc systems within the bounds of the assumptions made for their model, such as galaxy SFH, ages, metallicities, dust, but also the stellar models used. Our results suggest the low luminosity galaxies with high sSFR but low [OIII]+Hβ EW observed by Endsley et al. (2022) may be more likely due to variation in metallicity than high f esc . Conclusions We have inferred the hydrogen ionizing photon production rate, modulo the escape fraction, in the largest sample of individual sub-L * z > 3 galaxies to-date, spanning −23 ∼ < M uv ∼ < − 15.5, with a median M uv = −18.1, thanks to deep JWST/NIRCam imaging, enabling us to track the demographics of the ionizing population. Our conclusions are as follows: 1. The median log 10 (1 − f esc )ξ ion of our sample is 25.33 ± 0.47 with an intrinsic scatter of 0.42 dex. The inferred ξ ion distribution of our sample has values in a range of ∼ 1.5 dex, implying a wide range of galaxy properties and ages. 2. We find significant trends of increasing (1 − f esc )ξ ion with increasing Hα EW, decreasing UV luminosity, and with decreasing UV slope, all suggesting galaxies which are most efficient at producing ionizing photons are young, highly star forming, which are normally expected to be low metallicity and dust-poor. 3. We find galaxies selected with strong Lyα emission to have higher ξ ion than photometrically-selected galaxies, with median log 10 (1− f esc )ξ ion values of 25.39±0.64 and 25.31±0.43 respectively. We find the Lyα-detected galaxies have an elevated Hα EW distribution, thus the increased ξ ion is likely driven by the selection based on Lyα selecting a younger population. As strong Lyα emitters also likely have high ionizing photon escape fractions, this implies the intrinsic production rate of ionizing photons in these galaxies could be significantly higher than what we can infer from Hα luminosities. 4. We examine our sample for signs of very high f esc by comparing the inferred strengths of nebular emission lines ([OIII]+Hβ and Hα) and the strength of the nebular continuum via the UV β slope. We find no significant evidence for sources with high escape fraction galaxies with low nebular emission line strength and very blue UV β slopes. The reduced strength of [OIII]+Hβ EW in our z > 3 sample compared to a sample at z ∼ 2 from Tang et al. (2019) implies our sample has likely lower gas-phase metallicity and/or ionization parameter. We have demonstrated the power of JWST/NIRCam photometry to more precisely constrain the rest-frame optical emission of UV-faint high redshift galaxies than previously possible with Spitzer/IRAC. These observations allow us to constrain the production rate of ionizing photons from early galaxies, corroborating the picture obtained from previous stacking analyses that ξ ion is elevated in young, highly star forming galaxies, but that there is a broad distribution of ξ ion , likely driven by variation in galaxy properties and ages. With JWST spectroscopy it is becoming possible to obtain direct measurements of optical emission lines in large samples (e.g., Sun et al. 2022;Williams et al. 2022;Matthee et al. 2022). Deriving a census of the ionizing photon production rate across the full galaxy population will be necessary to fully understand reionization. Here we have shown that ξ ion is elevated in UV-faint galaxies with strong nebular emission lines, likely due to young ages. While a thorough analysis of the implications of our results for reionization are beyond the scope of this work, this becomes more prominent at high redshift (e.g., Boyett et al. 2022b;Endsley et al. 2022), implying that it would be possible to complete reionization with modest f esc . Considering the full distributions of ξ ion and f esc across galaxy properties will be required to assess the primary drivers of reionization. .5760475 deg and Dec= −30.37946 deg and consist of imaging in seven bands: F115W (10823 s.), F150W (10823 s.), F200W (6700 s.), F277W (6700 s.), F356W (6700 s.), F410M (6700 s.) and F444W (8246 s.). Fig. 1. Galaxies studied in this work; in purple Lyα detected galaxies, and in gray galaxies photometrically selected (with no Lyα detected). Top: Distribution of redshifts for the spectroscopic and photometric samples. We show the spectroscopic redshift where available, or the peak photometric redshift. Bottom: UV magnitude distribution for our sample, we find a median value of −18.14 ± 1.58, with no statistically significant difference between both samples. Fig. 2. Distribution of (1f esc )ξ ion . The purple histogram includes galaxies with; Lyα emission detection, and in grey galaxies without Lyα detected. Overall, Lyα emitting galaxies show stronger ionizing photon production than galaxies with no Lyα emission selected galaxies, with median values 25.39±0.64 and 25.31±0.43 respectively. We show the median relation from the literature at z ∼ 2 − 5 as a dashed black line (e.g., Shivaei et al. 2018; Bouwens et al. 2016; Lam et al. 2019) Fig. 4. We compare Hα equivalent width with the ionizing photon production that does not escape the galaxy. As stars we show Lyα detected galaxies, as circles are photometrically selected galaxies with no Lyα. As above, error bars are only shown for 30% of the sources for clarity. We color code these two samples by UV β slope. On top we show the distribution of Hα EW for the same two samples, compared to the values found by Tang et al. (2019). We add data from Harikane et al. (2018) and Lam et al. (2019) for comparison, which is at the high end of our observed Hα EW distribution. We see that stronger ξ ion very strongly correlates with Hα EW. Galaxies with detected Lyα emission have an Hα EW distribution with higher values, median 732 ± 187 Å compared to 457 ± 161 with a Kolmogorov-Smirnov test p-value 0.01. The sources with the reddest UV slopes lie systematically below the best-fit relation at fixed Hα EW. Fig. 5 . 5UV β slope vs ξ ion Fig. 6 . 6Fig. 6. Comparison between intrinsic (unattenuated) EW of Hα and UV β slope, color-coded by [OIII]+Hβ EW. Lyα galaxies are shown with star shaped markers, and the photometric sample as circles. Galaxies shown in gray, do not have [OIII]+Hβ EW measurements. We show the region predicted by Zackrisson et al. (2017) to show f esc > 0.5. We rescale from Hβ EW to intrinsic Hα with a case B recombination scenario of factor 2.89, assuming a flat optical continuum in f λ , which we confirm from our SED fitting done in 3.1. A&A proofs: manuscript no. ms_MUSE_xi_ion22 20 18 16 14 M UV, 1500Å 23.5 24.0 24.5 25.0 25.5 26.0 26.5 log 10 ion (1 f esc ) Ly-Detected (This work) no Ly-Detected (This work) Maseda+20 Harikane+18 Lam+19 Bouwens+16,z~4.5 Bouwens+16,z~5.2 Fig. 3. M uv vs (1-f esc )ξ ion . In purple circles we show Lyα detected galax- ies. In gray circles are the photometrically selected sample with no Lyα detected. We show data from Maseda et al. (2020); Harikane et al. (2018); Lam et al. Table 1 . 1Linear fitting parameters for trends with log 10 (1 − f esc )ξ ionParameter Slope, α Intercept, β Scatter variance, σ 2 M uv + 20 0.03 ± 0.02 25.33 ± 0.03 0.027 ± 0.006 log 10 EW Hα − 2.5 0.73 ± 0.04 25.14 ± 0.02 0.003 ± 0.001 β + 2 −0.20 ± 0.04 25.38 ± 0.01 0.032 ± 0.005 Notes. We fit for log 10 [(1 − f esc )ξ ion ] = α + βX + , where is the intrinsic scatter which is assumed to be normally distributed with variance σ 2 . Fig. 7. Comparison between EW of Hα and [OIII]+Hβ. We show Lyα detected galaxies as stars and the photometrically selected sample with no Lyα detected as circles. On top we show the distribution of [OIII]+Hβ EW for both of our samples. We find a very strong correlation between Hα EW and [OIII]+Hβ EW, though with large scatter. The dashed lines are the correlation trends found for this work (red) andTang et al. (2019) (green). We find higher Hα EW/[OIII]+Hβ EW than the z ∼ 2 sample fromTang et al. (2019), which was selected to have strong [OIII] EW, implying that we might be observing lower metallicity galaxies.0 25 Tang+19 Ly-Detected no Ly-Detected 1.0 1.5 2.0 2.5 3.0 3.5 4.0 log 10 EW 0 ([OIII] + H )[Å] 1 2 3 4 log 10 EW 0 (H )[Å] Tang+19 This Work Ly-Detected no Ly-Detected https://www.stsci.edu/jwst/science-execution/ program-information.html?id=2561Article number, page 2 of 10 Gonzalo Prieto-Lyon et al.: The production of ionizing photons in UV-faint z ∼ 3 − 7 galaxies https://www.stsci.edu/cgi-bin/get-proposal-info?id= 17231&observatory=HST 3 https://archive.stsci.edu 4 https://github.com/gbrammer/grizli/pull/107 http://archive.eso.org/wdb/wdb/eso/sched_rep_arc/ query?progid=109.24EZ.001Article number, page 3 of 10 A&A proofs: manuscript no. ms_MUSE_xi_ion Article number, page 7 of 10 A&A proofs: manuscript no. ms_MUSE_xi_ion Acknowledgements. We thank the co-PIs Ivo Labbé and Rachel Bezanson for the conception and public availability of the UNCOVER JWST Program (GO-2561), which made much of this work possible. We thank Mengtao Tang for sharing data the emission line catalog fromTang et al. (2019). This work is based on observations collected at the European Southern Observatory under ESO programmes 109.24EZ.001 and 094.A-0115. This work is based on NASA/ESA HST and JWST data which were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA con- . P Anders, U Fritze-V. Alvensleben, A&A. 4011063Anders, P. & Fritze-v. Alvensleben, U. 2003, A&A, 401, 1063 SEP: Source Extraction and Photometry. K Barbary, 1811.004Astrophysics Source Code Library. Barbary, K. 2018, SEP: Source Extraction and Photometry, Astrophysics Source Code Library, record ascl:1811.004 . 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[ "LaMini-LM: A Diverse Herd of Distilled Models from Large-Scale Instructions", "LaMini-LM: A Diverse Herd of Distilled Models from Large-Scale Instructions" ]	[ "Minghao Wu [email protected] \nMohamed bin Zayed University of Artificial Intelligence\n\n\nMonash University\n\n", "Abdul Waheed [email protected] \nMohamed bin Zayed University of Artificial Intelligence\n\n", "Chiyu Zhang [email protected] \nMohamed bin Zayed University of Artificial Intelligence\n\n\nThe University of British Columbia\n\n", "Muhammad Abdul-Mageed [email protected] \nMohamed bin Zayed University of Artificial Intelligence\n\n\nThe University of British Columbia\n\n", "Alham Fikri [email protected] \nMohamed bin Zayed University of Artificial Intelligence\n\n", "Aji " ]	[ "Mohamed bin Zayed University of Artificial Intelligence\n", "Monash University\n", "Mohamed bin Zayed University of Artificial Intelligence\n", "Mohamed bin Zayed University of Artificial Intelligence\n", "The University of British Columbia\n", "Mohamed bin Zayed University of Artificial Intelligence\n", "The University of British Columbia\n", "Mohamed bin Zayed University of Artificial Intelligence\n" ]	[]	Large language models (LLMs) with instruction fine-tuning demonstrate superior generative capabilities. However, these models are resource-intensive. To alleviate this issue, we explore distilling knowledge from instructiontuned LLMs into much smaller ones. To this end, we carefully develop a large set of 2.58M instructions based on both existing and newlygenerated instructions. In addition to being sizable, we design our instructions to cover a broad set of topics to ensure diversity. Extensive analysis of our instruction dataset confirms its diversity, and we generate responses for these instructions using gpt-3.5-turbo. Leveraging these instructions, we fine-tune a diverse herd of models, collectively referred to as LaMini-LM, which includes models from both the encoder-decoder and decoder-only families, with varying sizes. We evaluate the performance of our models using automatic metrics on 15 different natural language processing (NLP) benchmarks, as well as through human assessment. The results demonstrate that our proposed LaMini-LM models are comparable to competitive baselines, while being nearly ×10 smaller in size. 1 * work done while visiting MBZUAI 1 Our code, model checkpoints, and dataset are available at	10.48550/arxiv.2304.14402	[ "https://export.arxiv.org/pdf/2304.14402v2.pdf" ]	258,352,678	2304.14402	389ec3e8902a5dcfcde1adec735854e93f845937	LaMini-LM: A Diverse Herd of Distilled Models from Large-Scale Instructions Minghao Wu [email protected] Mohamed bin Zayed University of Artificial Intelligence Monash University Abdul Waheed [email protected] Mohamed bin Zayed University of Artificial Intelligence Chiyu Zhang [email protected] Mohamed bin Zayed University of Artificial Intelligence The University of British Columbia Muhammad Abdul-Mageed [email protected] Mohamed bin Zayed University of Artificial Intelligence The University of British Columbia Alham Fikri [email protected] Mohamed bin Zayed University of Artificial Intelligence Aji LaMini-LM: A Diverse Herd of Distilled Models from Large-Scale Instructions Large language models (LLMs) with instruction fine-tuning demonstrate superior generative capabilities. However, these models are resource-intensive. To alleviate this issue, we explore distilling knowledge from instructiontuned LLMs into much smaller ones. To this end, we carefully develop a large set of 2.58M instructions based on both existing and newlygenerated instructions. In addition to being sizable, we design our instructions to cover a broad set of topics to ensure diversity. Extensive analysis of our instruction dataset confirms its diversity, and we generate responses for these instructions using gpt-3.5-turbo. Leveraging these instructions, we fine-tune a diverse herd of models, collectively referred to as LaMini-LM, which includes models from both the encoder-decoder and decoder-only families, with varying sizes. We evaluate the performance of our models using automatic metrics on 15 different natural language processing (NLP) benchmarks, as well as through human assessment. The results demonstrate that our proposed LaMini-LM models are comparable to competitive baselines, while being nearly ×10 smaller in size. 1 * work done while visiting MBZUAI 1 Our code, model checkpoints, and dataset are available at Introduction Large language models (LLMs) with instruction tuning have demonstrated impressive capabilities in generating high-quality outputs across a wide range of use cases (Ouyang et al., 2022;Wei et al., 2022;Sanh et al., 2022;Chung et al., 2022;Ope-nAI, 2023).However, these models usually have billions of parameters, which require massive computational resources for both training and inference (Brown et al., 2020;Thoppilan et al., 2022;Hoffmann et al., 2022;Chowdhery et al., 2022). Kaplan et al. (2020) suggest that the performance of LLMs scales proportionally with model and dataset size. Consequently, scaling the models raises many issues such as those related to the energy footprint (Strubell et al., 2019). Moreover, the accessibility of large models is a real concern for many NLP practitioners due to limited access to computing resources (Nityasya et al., 2020). In this work, we introduce LaMini-LM, a collection of language models that stand out due to their smaller size compared to most instruction-tuned models currently available. We develop LaMini-LM models by employing sequence distillation (also known as offline distillation) (Kim and Rush, 2016) from LLMs. While previous studies (e.g., (Taori et al., 2023;Chiang et al., 2023;Anand et al., 2023)) have attempted similar approaches, there exist several gaps in the current literature that we aim to address. These gaps include: (i) the provision of a small-scale distilled dataset, (ii) a lack of diversity in the dataset, (iii) a limited number of models (typically only one), and (iv) an absence of comprehensive evaluation and analysis of the models' performance. Additionally, it is worth noting that many of the distilled models resulting from prior work remain computationally intensive. These recent models typically range from 7B to 13B parameters, which poses challenges for deploy-ment in resource-constrained settings, particularly for under-resourced institutions. Therefore, our goal is to develop a solution that overcomes these limitations and enables easier deployment in such settings. To alleviate these issues, we firstly generate a large-scale offline distillation dataset comprising 2.58M instructions, and then fine-tune a collection of language models to obtain the LaMini-LM models, as shown in Figure 1. We collect instructions from various existing datasets such as self-instruct (Wang et al., 2022a), P3 (Sanh et al., 2022), FLAN (Longpre et al., 2023), and Alpaca (Taori et al., 2023). Additionally, we leverage the power of ChatGPT (gpt-3.5-turbo) to generate additional diverse instructions that align with the quality and style of the human-written prompts. This approach is known as Example-Guided Instruction Generation. To further enrich the variety of generated text, we introduce the Topic-Guided Instruction Generation technique. This method aims to expand the scope of generated instructions by utilizing specific topics of interest collected from Wikipedia. We then utilize gpt-3.5-turbo to produce responses for each instruction, leveraging its advanced language modeling capabilities. We refer to this large-scale instruction dataset as LaMini instruction dataset. After creating the dataset, we proceed to finetune multiple smaller language models with different sizes (ranging from 61M to 1.5B) and architectures, including encoder-decoder and decoder-only models. Additionally, we conduct a comparative analysis of various model variations within each architecture. What sets our work apart from previous research is our comprehensive evaluation of the resulting models. We assess their performance on diverse NLP downstream tasks and also incorporate human evaluation to gauge the quality of the model outputs. This in-depth analysis allows us to gain a profound insight into the strengths and weaknesses of the models. Our contributions can be summarized as follows: 1. We introduce the LaMini instruction dataset, consisting of over 2.58 million examples. To the best of our knowledge, this dataset is currently the largest instruction dataset available. Notably, it is ×50 larger than the dataset released by Taori et al. (2023). 2. We investigate the process of distilling knowledge from large language models (LLMs) into smaller architectures, resulting in a family of distilled language models. Our research explores models of varying sizes, with our largest model being ×110 smaller and our smallest model being ×2800 smaller than GPT-3 (Brown et al., 2020). 3. We conduct extensive experiments on both our proposed models and several publicly available LLMs. These experiments involve automatic evaluation on 15 NLP tasks as well as human evaluation. The results of both our automatic and human evaluations demonstrate that our proposed models achieve comparable performance to Alpaca (Taori et al., 2023), despite their significantly smaller size. Specifically, our models are nearly ×10 smaller in size while maintaining comparable performance. 2 Related Work Instruction Tuning Instruction tuning is an emerging paradigm in the field of Natural Language Processing (NLP). This approach combines natural language instructions with language models to achieve zero-shot performance on tasks that have not been encountered before. Several studies have demonstrated that vanilla language models can effectively follow general language instructions when fine-tuned using human-written instructions (Weller et al., 2020;Wang et al., 2022b;Wei et al., 2022;Sanh et al., 2022;Ouyang et al., 2022;Parmar et al., 2022;Scialom et al., 2022;Chung et al., 2022;Yin et al., 2022;Gupta et al., 2022;Muennighoff et al., 2022). Moreover, a recent study by Wang et al. (2022a) showed that model-generated instructions can be utilized for instruction tuning, leading to significant improvements in the capabilities of vanilla language models in responding to instructions. Building upon this research, several other works have focused on instruction tuning vanilla language models using model-generated instructions (Taori et al., 2023;Chiang et al., 2023;Anand et al., 2023;Chen et al., 2023). In this study, we present the largest instruction dataset generated by gpt-3.5-turbo to date. We then fine-tune a collection of language models to create our LaMini-LM models. <example>What are some things you can do to de-stress?</example> <example>How can individuals and organizations reduce unconscious bias?</example> <example>Write a program to compute the sum of integers from k to n.</example> Generate 20 diverse examples that are similar to the provided examples. You do not need to provide a response to the generated examples. Each example must include an instruction. Each generated instruction can be either an imperative sentence or a question. Each example must start with the label "<example>" and end with the label "</example>". Knowledge Distillation Knowledge distillation is a technique used to train a smaller model, known as the student, by leveraging knowledge from a larger model, referred to as the teacher (Hinton et al., 2015). One popular method of knowledge distillation involves training the student with an additional objective of matching the teacher's representation, such as logits, output probability, or intermediate activation (Sanh et al., 2019;Jiao et al., 2020;Mirzadeh et al., 2020;. For sequence-to-sequence or generative models, the concept of sequence-level distillation was introduced by Kim and Rush (2016). This approach involves generating a synthetic output by performing inference with the teacher model, which is then used to train the student model. Sequencelevel distillation is efficient as it only requires running the typically large teacher model once. Previous research has demonstrated the effectiveness of sequence-level distillation. For instance, Costajussà et al. (2022) used sequence-level distillation to reduce the size of an NLLB machine translation system to 600M parameters. Similarly, by combining sequence-level distillation with model pruning and quantization, Behnke et al. (2021); Bogoychev et al. (2020) managed to train a translation system that was approximately ×25 smaller than the teacher model without a significant decrease in BLEU score. In our work, we adopt a sequence-level distillation approach by training our model using the output of gpt-3.5-turbo. While other researchers have also trained language models based on the output of GPT models, our approach stands out as we train our model on a substantially larger dataset and distill it into much smaller models. Moreover, we provide various student models as part of our contributions. Dataset Generation Our approach involves distilling knowledge from large language models through sequence/offline distillation (Kim and Rush, 2016). The student model learns from the outputs of a teacher model in this process. To create our dataset, we leverage various existing resources of prompts, which include self-instruct (Wang et al., 2022a) and Alpaca (Taori et al., 2023) as well as random subsets of P3 (Sanh et al., 2022) and FLAN (Longpre et al., 2023). By utilizing these resources, we generate a total of 2.58M pairs of instructions and responses using ChatGPT (gpt-3.5-turbo). Additionally, we conduct an exploratory analysis of the resulting text to gain further insights. Instruction Generation In this section, we present two strategies for generating instructions: the example-guided strategy and the topic-guided strategy. Additionally, we provide an overview of our approach to generating responses. Example-Guided Instruction Generation Inspired by the works of Wang et al. (2022a) and Taori et al. (2023), we develop a prompt for generating instructions. Our approach involves presenting a prompt with a few examples and constraints, as demonstrated in Figure 2. we include only three random examples and a limited number of constraints within each prompt. Instead of explicitly specifying language restrictions, output length limitations, or instruction types, our instruction to gpt-3.5-turbo is to generate a variety of examples that align with the provided examples and adhere to the desired output format. To optimize the generation process, we randomly sample three seed tasks from self-instruct and generate 20 instructions at once. These instructions are referred to as X X X SI . When the selected instructions are associated with specific inputs, we con-catenate them using a colon ":" symbol in the format "$instruction:$input". For datasets P3 and FLAN, we randomly select three examples from the same subset. Our preliminary study indicates that gpt-3.5-turbo requires a minimum of two examples to generate desirable instructions. To ensure more consistent output formatting, we include an additional example. Examples from P3 and FLAN tend to be longer compared to those from self-instruct. To ensure that we stay within the output length limit, we generate only 10 instructions at a time for P3 and FLAN.We refer to the original set of prompts from P3 and FLAN as X X X P3 and X X X FLAN , respectively. The instructions generated from these prompts are denoted as X X X P3 and X X X FLAN , respectively. Additionally, we denote the prompts from Alpaca as X X X A , although they are not utilized in this stage. Topic-Guided Instruction Generation It is of concern that gpt-3.5-turbo may not possess the ability to generate diverse text without explicit guidance. To address this concern, we collect several common topics from Wikipedia to guide the generation process. We first collect a total of 2.2M categories from Wikipedia. These categories are filtered based on two requirements. Firstly, the category must consist of less than three words. Secondly, the category must comprise more than 10 sub-categories and 50 pages. Upon manual inspection, we note that lengthy category titles are more likely to be associated with specific and niche information, while a common category can be divided into several sub-categories and discussed across multiple pages. For instance, the category "machine learning" contains 35 sub-categories and 200 pages. 2 After filtering, we obtain a list of 3.5K categories that serve as common topics. An example of the prompt with topics is presented in Appendix A. In this study, we generate topicguided instructions solely from self-instruct seed tasks, represented as X X X t,SI . This decision is based on our observation that gpt-3.5-turbo frequently struggles to produce the appropriate context for instructions. Conversely, examples from P3 and FLAN typically contain extensive contextual information. Therefore, to maintain generation quality, we limit our topic-guided instruction generation to X X X t,SI . (a) The t-SNE visualization of the sentence embeddings of X X XSI(ours) and X X XA. (b) The t-SNE visualization of the sentence embeddings of X X XP3(ours) and X X XP3. Response Generation To perform sequence-level distillation, we generate responses from the instructions described in the previous section. We generate the responses for all the generated instructions, including X X X SI , X X X t,SI , X X X P3 , X X X FLAN . As we observe that gpt-3.5-turbo is less capable of providing the necessary context for the instructions, we also directly generate responses for the collected instructions, including X X X A , X X X P3 and X X X FLAN . Hence, we denote the resulting pairs as D D D SI = { X X X SI , Y Y Y SI }, D D D t,SI = { X X X t,SI , Y Y Y t,SI }, D D D P3 = { X X X P3 , Y Y Y P3 }, D D D FLAN = { X X X FLAN , Y Y Y FLAN }, D D D A = { X X X A , Y Y Y A }, D D D P3 = {X X X P3 , Y Y Y P3 } and D D D FLAN = {X X X FLAN , Y Y Y FLAN }. 3 The complete dataset D D D ALL is the union of all the aforementioned instructionresponse pairs. Exploratory Data Analysis In this section, we conduct an exploratory analysis of the generated text. Our analysis focuses on several aspects of the dataset, including basic statistics, diversity, and human evaluation. Statistics We present the dataset statistics in Table 1. As we claimed before, gpt-3.5-turbo often fails to provide the necessary context for the generated instruction, given that the average length of X X X P3 and X X X FLAN is significantly short than that of X X X P3 and X X X FLAN . Another observation is that if the instructions are generated from the same source, such as self-instruct, the corresponding responses have a similar length. Table 1: Data statistics of the generated dataset. The average instruction length and average response length are measured in tokens. Dataset X X X {·} or X X X {·} Y Y Y {·} or Y Y Y {·} D D Diversity Semantic Diversity To explore the semantic diversity of the generated instructions, we sample 50K instructions from X X X SI , X X X A , X X X P3 and X X X P3 , compute their sentence embeddings using Sentence Transformer (Reimers and Gurevych, 2019), 4 and visualize the t-SNE of instruction sentence embeddings in Figure 3. We omit the comparison between X X X FLAN and X X X FLAN as it yields the same results as the comparison between X X X P3 and X X X P3 . We observe that X X X SI exhibits greater diversity than X X X A as shown in Figure 3a and X X X P3 is slightly more diverse than X X X P3 as shown in Figure 3b. It appears that this observation can be attributed to the enhanced generative capabilities of gpt-3.5-turbo. Lexical Diversity We use Moving-Average Type-Token Ratio (MATTR) (Covington and Mc-Fall, 2010) to measure the lexical diversity with the window size of 50, because each subset of D D D ALL varies in size and MATTR is free from the impact of text length. As shown in Table 2, the model-generated instructions X X X {·} given by gpt-3.5-turbo are not as diverse as the humanwritten instructions X X X {·} and X X X A generated by text-davinci-003. It is noteworthy that X X X t,SI is more diverse than X X X SI and Y Y Y t,SI is the most diverse subset of responses, which demonstrates the effectiveness of the topic-guidance. Furthermore, D D D ALL illustrates the greatest lexical diversity, compared with all the subsets. X X X S I X X X t , S I X X X P 3 X X X F L A N X X X A X X X P 3 X X X F L A N # of samples Rate-A Rate-B Rate-C Rate-D (a) Human evaluation for the instruction (X X X {·} or X X X {·} ). Y Y Y S I Y Y Y t , S I Y Y Y P 3 Y Y Y F L A N Y Y Y A Y Y Y P 3 Y Y Y F L A N # of samples Rate-A Rate-B Rate-C Rate-D (b) Human evaluation for the responses (Y Y Y {·} or Y Y Y {·} ). Human Evaluation We follow the human evaluation protocol given by Wang et al. (2022a), which categorizes the quality of the generated text into four levels: • Rate-A: The generated text is of high quality; • Rate-B: The generated text is acceptable but has minor errors; • Rate-C: The generated text has significant errors in content. • Rate-D: The generated text is completely unacceptable. More details about the human evaluation protocol are presented in Appendix C. We randomly sample 20 examples from each subset of D D D ALL and one of the co-authors scores the generated text. In general, both the generated instructions and the generated responses are of high quality as shown in Figure 4. During the annotation process, we observe that examples from X X X P3 and X X X FLAN are much shorter than those from X X X P3 and X X X FLAN and their associated context are significantly shorter and easier, which confirms our observation in Table 1. Another noteworthy observation is that gpt-3.5-turbo is even more prone to generated the responses with factual errors when we provide the topics. Experiment Training LaMini-LM We present LaMini-LM, a family of language models instruction-tuned on our 2.58M instructions dataset D D D ALL . We train two types of models, encoder-decoder and decoder-only, for architectural comparison. The size for both categories of models ranges from 61M to 1.5B to facilitate size comparison. The underlying models for initialization are from five sources, including T5 (Raffel et al., 2020), Flan-T5 (Chung et al., 2022), Cereberas-GPT (Dey et al., 2023), GPT-2 (Radford et al., 2019), and GPT-Neo (Gao et al., 2021a). The details of our LaMini-LM series are summarized in Table 3. Optimization We finetune all models over 5 epochs and a batch size of 1024. For our encoderdecoder models, we use a learning rate of 5 × 10 −4 following Chung et al. (2022). For our decoderonly models, we follow the same configuration as Alpaca (Taori et al., 2023) including the learning rate of 2 × 10 −5 . We use HuggingFace's transformers for training. Moreover, we use the same prompt wrapper as Alpaca (Taori et al., 2023), hence we also wrap our instruction similarly during inference. We perform all of our experiments on 8×V100 (32G) and 8×A100 (40G) GPUs. Our models are publicly available. Name Architecture Initialization Table 3: LaMini-LM collection. Models with † are those with the best overall performance given their size/architecture, hence we recommend using them. C-GPT indicates Cerebras-GPT. LaMini-T5-61M enc-dec T5-small LaMini-T5-223M enc-dec T5-base LaMini-T5-738M enc-dec T5-large LaMini-Flan-T5-77M † enc-dec Flan-T5-small LaMini-Flan-T5-248M † enc-dec Flan-T5-base LaMini-Flan-T5-783M † enc-dec Flan-T5-large LaMini-Neo-125M dec-only GPT-Neo-125M LaMini-Neo-1.3B dec-only GPT-Neo-1.3B LaMini-Cerebras-111M dec-only C-GPT-111M LaMini-Cerebras-256M dec-only C-GPT-256M LaMini-Cerebras-590M dec-only C-GPT-590M LaMini-Cerebras-1.3B dec-only C-GPT-1.3B LaMini-GPT-124M † dec-only GPT-2 LaMini-GPT-774M † dec-only GPT-2 large LaMini-GPT-1.5B † dec-only GPT-2 xl Model Evaluation We then evaluate the performance based on several downstream NLP tasks as well as human evaluation on user-oriented instruction. Automatic Evaluation on Downstream NLP Tasks We conduct a zero-shot evaluation on the downstream NLP tasks for our LaMini-LM. We use language model evaluation harness (Gao et al., 2021b) to evaluate our instruction-tuned models. 5 We select 15 diverse NLP tasks, covering QA, sentiment analysis, paraphrase identification, natural language inference, coreference resolution, word sense disambiguation, and sentence completion. The details for these NLP tasks can be found in Appendix D. Human Evaluation on User-Oriented Instructions The NLP tasks in Appendix D are designed for academic-oriented tasks, and are focused on classification. To complete the evaluation, we additionally evaluate the practicality of both our LaMini-LM and our baseline models by utilizing the user-oriented instructions from Wang et al. (2022a), which consists of 252 instructions covering 71 commonly used apps use-cases. In contrast with downstream NLP tasks, there is no single gold answer for many of these questions, therefore manual human evaluation is needed to benchmark T5 LaMini-T5 F-T5 LaMini-F-T5 C-GPT LaMini-C GPT-2 LaMini-GPT LLaMA Alpaca the performance. We follow the guideline as in Appendix C for measuring the model's response quality. To reduce the annotation cost yet ensure the instruction diversity, we keep no more than 2 instructions for each app and manually filter out those instructions that are already covered in downstream NLP tasks, such as natural language inference, sen-timent analysis, and summarization. Finally, we obtain a test set for human evaluation with 114 instructions. we organize a team of 8 human experts for human evaluation, with each expert responsible for evaluating the responses to 15 instructions across all chosen models. Arguably, human annotation is subjective. Thus, to ensure consistency, all model responses from the same instruction are scored by the same annotator, as the scores for that particular instruction is based on the same standard. Result and Discussions In this section, we provide evaluation result and discussion of LaMini-LM for both the downstream NLP tasks and human evaluation on user-oriented instruction. For NLP downstream task, larger models yield better average performance, as seen in Figure 5. Therefore to save space, we present the broken-down results given by the largest models in each group (Table 4). We also compare their performance with LLaMA-7B (Touvron et al., 2023) and Alpaca-7B (Taori et al., 2023). Surprisingly, we also observe that the instruction-tuned models, including ours and Alpaca, always underperform their baselines on the ReCoRD benchmark. We leave the further investigation of this observation to future work. Breakdown results of other models can be found in Appendix E. We ure 6. Similar to downstream NLP performance, larger models generally perform better. Interestingly, encoder-decoder models from T5 are performing exceptionally well, given their rather small size. Encoder-Decoder vs. Decoder-Only The encoder-decoder LaMini language models (LaMini-T5 series and LaMini-Flan-T5 series) outperform the decoder-only LaMini language models (the LaMini-GPT series) when the number of parameters is limited (less than 500M parameters). LaMini-Flan-T5-248M even outperforms LLaMA-7B on downstream NLP tasks. When the model size is higher, LaMini-Flan-T5 is comparable to LaMini-GPT. Yet, both LaMini-Flan-T5 and LaMini-T5 demonstrate strong human evaluation results for user-oriented instructions, despite their relatively small size. Especially, T5-based models of 200M parameters is competitive against LaMini-GPT-1.5B for human evaluation result. We recommend further exploration of the encoder-decoder architecture for language models, given their potential, as demonstrated in our experiments. GPT-2 vs. Cerebras-GPT Among all the decoder-only models that we fine-tune, we observe a performance discrepancy among models that are of comparable size. Based on the results in Table 4 and Figure 5, LaMini-GPT series is significantly superior on downstream NLP tasks compared to LaMini-Cerebras, despite both having similar architecture and size. Even more, LaMini-GPT of 774M is outperforms LaMini-Cerebras-1.3B despite being half in size. We also observe similar results on human evaluation. Generally, vanilla GPT-2 also outperforms Cerebras-GPT models of comparable size on downstream tasks, as shown in Appendix E. Therefore, there may be a correlation between the initial model's performance and the performance achieved after instruction tuning. T5 vs. Flan-T5 LaMini-Flan-T5 series exhibit better performance than LaMini-T5 in the downstream NLP tasks. Considering the original Flan-T5 is already instruct-tuned, this result is not surprising. But, the original Flan-T5 perform poorly under general user-oriented instruction, hence improvement on that aspect is possible by further finetuning the model with a diverse set of instruction. After fine-tuning with our dataset, both LaMini-Flan-T5 and LaMini-T5 achieve comparable performance according to human evaluation. Qualitative Analysis We present a comparison of model responses based on user-oriented human evaluation in Table 5 and Table 6. Our analysis reveals that the responses generated by LaMini-LM tend to be shorter in length when compared Model Response Rate Alpaca-7B Key Points to Remember About History Of The United States 1607-present D LaMini-Flan-T5-783M History of the USA: -The United States as it was founded on July 4, 1776. It is known for its unique culture, economy and lifestyle, technological advancements, political system such as democracy, nationalism and civil rights movements leading up to these events in particular. -This early American history begins with European colonization by Native Americans who established their own independent nations around America after crossing this waterway over thousands of years ago. LaMini-Flan-T5-783M "The Witcher" is a German crime drama film directed by Christopher Nolan, first released in 2019. The movie follows the story of Christian Fenker, who becomes obsessed with killing Frank Zappa and becoming the next prime suspect after being tricked into selling voodoo souls on black market exchanges. As he races against time to bring down these criminal masterminds and make peace with humanity, he begins to have personal struggles that threaten both his own existence as well as those around him. C LaMini-T5-738M "The Witcher" is a 2019 film that follows the story of a former witch who is now a powerful witch and embarks on a perilous adventure through a magical world filled with dangerous creatures. C Table 6: Model responses to the instruction "Write a short description about the given movie or series: "The Witcher (2019)"", where LaMini language models fails but Alpaca-7B manages to respond. The high-quality contents are highlighted in blue. The errors are highlighted in red. to those generated by the Alpaca-7B model. This phenomenon can be attributed to the fact that we have imposed a constraint on the gpt-3.5-turbo model to ensure that its responses are as concise as possible during the generation process described in Section 3.2. As shown in Table 5, LaMini-LM correctly respond to the instruction and generate coherent responses with minor errors, while Alpaca fails to respond the instruction. However, LaMini-LM hallucinate when responding the instruction, while Alpaca generates the response with accurate information. From both examples, we conclude that current language models are still prone to generate hallucinated and nonfactual information. We present more discussions on the limitations of LaMini-LM in Section 8. Total DNH FF NS Ob. Hallucination and Toxicity Hallucination LLMs often face the issue of generating hallucinations, resulting in textual outputs that either contain factual inaccuracies or lack co-herence. To thoroughly investigate the extent of this problem, we simplify it as a "question rejection" challenge, which can be treated as a binary classification task. The objective is to determine whether an LLM can correctly identify and reject questions that cannot or should not be answered. To accomplish this objective, we have manually curated the LaMini-Hallucination test set, which encompasses four distinct categories: "did not happen (DNH)", "far future (FF)", "nonsense (NS)", and "obscure (Ob.)". Each category contains 10 questions. We utilize the recommended models listed in Table 3 to address these questions and conduct human evaluation to assess the quality of generated responses. In contrast to the human evaluation described in Section 4.2, an ideal model should be capable of properly rejecting a question with appropriate justification (if generated). If a model rejects a question with a hallucinated justification, the response is considered incorrect. The evaluation results regarding hallucination are presented in Table 7. After fine-tuning on our LaMini instruction dataset, our LaMini language models outperform Alpaca in handling "far future" and "obscure" questions. However, it is evident that current lightweight instruction-tuned models, including Alpaca and our LaMini language models, struggle particularly with answering "did not happen" and "nonsense" questions. These models are highly prone to generating hallucinations when attempting to respond to such question types. In contrast, gpt-3.5-turbo successfully identifies and responds to these questions. Furthermore, it is important to acknowledge that our LaMini-Hallucination test set may not provide a sufficient level of challenge for gpt-3.5-turbo. It is also essential to emphasize that although our LaMini-LM, as well as Alpaca, perform well on various downstream NLP tasks, they still suffer significantly from the hallucination problem. Toxicity LLMs have been observed to demonstrate a tendency to generate toxic language, which poses challenges for their safe deployment. To evaluate the extent to which LLMs can generate toxic language when fine-tuned on our LaMini instruction dataset, we utilize the RealToxicityPrompts dataset (Gehman et al., 2020). We randomly select 1K prompts with a toxicity score below 0.1 as non-toxic prompts, and another 1K prompts with a toxicity score above 0.9 as toxic prompts. Using the prefixed instruction "Complete the sentence:", Non-Toxic Toxic Flan-T5-small 1 25 LaMini-Flan-T5-77M 1 46 Flan-T5-base 1 30 LaMini-Flan-T5-248M 0 we generate outputs using both the recommended LaMini models and their corresponding baselines. We then employ the OpenAI Moderation API to detect the toxicity of the generated outputs. 6 The toxicity results are presented in Table 8. Interestingly, after fine-tuning on our LaMini instruction dataset, we observe that the encoder-decoder models (the LaMini-Flan-T5 series) are more prone to generating toxic text, while the decoder-only models (the LaMini-GPT series) are less likely to produce toxic text. We hypothesize that the LaMini-Flan-T5 models possess stronger instruction-following capabilities, which may lead to the generation of more toxic outputs when the given prompt itself is toxic, potentially to maintain sentence coherence. We leave the in-depth investigation of this phenomenon to future work. Conclusion In this work, we release a large-scale instruction dataset distilled from ChatGPT with more than 2.58M examples. To the best of our knowledge, this dataset is currently the largest dataset of its kind. We explore distilling knowledge from LLMs to various smaller and more efficient model architectures. We refer to the resulting family of language models as LaMini, which includes 6 encoder-decoder models and 9 decoder-only models with varying model sizes. We also conduct a comprehensive evaluation in this work, including the automatic evaluation of the downstream NLP tasks and human evaluation. Both evaluation strategies highlight that our proposed models achieve comparable performance with Alpaca (Taori et al., 2023) while is nearly ×10 smaller in size. This work sheds light on distilling knowledge from LLMs to much smaller model architectures and demonstrates the potential of training efficient yet effective language models. Limitations In this paper, we explore instruction tuning on various small-size language models and performe evaluation across multiple benchmarks. However, our work still has some limitations: • Model Variations: Compared to previous studies that often only offer a single model without comprehensive evaluation, our work stands out by providing thorough analysis across multiple models with varying configurations. However, our current model selection is somewhat limited, consisting of T5, GPT-2, Cerebras GPT, and GPT-Neo as our base models. Furthermore, we have only explored models with a size of up to approximately 1B parameters. To enhance our understanding of performance trends and enable more meaningful comparisons with prior research, it would be advantageous to expand our exploration to include larger models. • Single Turn Dialog: Although our training data and user-oriented evaluation primarily focus on "dialog-like" instructions, it is essential to acknowledge that our models are not currently optimized for handling multi-turn dialogues. • Error Propagation: Our models have undergone training utilizing condensed knowledge obtained from gpt-3.5-turbo, thereby inheriting the potential risks associated with it. The presence of hallucination and toxicity in LaMini-LM models is evident from the findings presented in Section 6. Furthermore, our evaluation involving human feedback revealed unsatisfactory performance of LaMini-LM models in coding, mathematical problem-solving, and tasks demanding logical reasoning skills. We leave these limitations to be addressed in the future work. Ethical Consideration We demonstrate that training small language models on large-scale instruction can significantly en-hance their performance on downstream NLP tasks, as well as in human evaluation. These instructiontuned models exhibit superior performance compared to significantly larger models and are particularly adept at engaging in open-ended conversation. Despite these advantages, it is important to acknowledge that these instruction-tuned models are not fully aligned with human objectives. They may frequently generate discriminatory responses and propagate biases or other forms of discrimination originating from the teacher model. Moreover, as we detail in Section 6, these models often generate false information, which may have unintended consequences. To mitigate any potential harm arising from the use of these models, we intend to minimize the risks associated with their use in future research. We advocate for the responsible use of our models to prevent any harm. For the prompts with topics, besides three random examples, we sample three random topics from the common topic list and present an example in Figure 7. References B Response Generation The Python code used to generate the response can be found in Figure Figure 8. Before asking gpt-3.5-turbo to generate responses, we firstly send a message as the "system" that requires gpt-3.5-turbo to respond the instructions as concise as possible to avoid the overly lengthy responses. C Human Evaluation Protocol We present the human evaluation protocol as well as the corresponding example for each rating level in Table 9. D Automatic Evaluation Datasets We present the details of 15 downstream NLP tasks, including the number of test examples and the corresponding evaluation metrics, in Table 10. E Automatic Evaluation Results The breakdown results given by LaMini-T5, LaMini-Flan-T5, LaMini-Neo, LaMini-Cerebras and LaMini-GPT are presented in Table 11,Table 12,Table 13,Table 14 and Table 15 respectively. Each example must include an instruction. Each generated instruction can be either an imperative sentence or a question. Each example must start with the label "<example>" and end with the label "</example>".". import openai def send_request ( instruction ): response = openai . ChatCompletion . create ( model =" gpt -3.5 -turbo " , messages =[ {" role ": " system " , " content " : " You are a helpful assistant , but you must respond the provided instructions as concise as possible . " } , {" role ": " user " , " content ": instruction } ] ) return response Figure 8: The Python code of sending request via Ope-nAI API to generate the response for an instruction. F Qualitative Analysis Level Criteria Example Rate-A • Valid, acceptable and satisfying (subject to the annotator) response; • Accurate in terms of facts, yet comparable to human standards; • The response meets the required criteria, but it may not be in the expected format. • Instruction: Make a list of the pros and cons of the given decision: "Implementing a Remote Working Policy" • Response: Pros of implementing a remote working policy include improved productivity, reduced commuting time to work remotely, access to healthcare services for patients with chronic illnesses or injuries from their employment activities, potential cost savings by avoiding long commutes between workers instead of relying solely on trains, and better employee job security since individuals are constantly recharging into each other's networks. However, it may still face challenges in terms of accessibility due to restrictions imposed upon those living within certain walking distance limits and areas without reliable transportation infrastructure such as high density housing units or public transit systems. Rate-B • The response is acceptable but has minor errors that can be improved; • Mirror errors include out-of-context content, minimal factual errors, partially responding to the instruction, etc. • Instruction: Make a list of the pros and cons of the given decision: "Implementing a Remote Working Policy" • Response: Pros -1) Increased flexibility in work schedule, allowing for better job satisfaction with fewer commuting obligations 2) Improved workplace culture by offering opportunities to connect from another location or country leading off site; this can lead to more diverse employees 3) Reduced turnover rate as companies save costs associated with office space expenses and maintain separate offices which may attract new talent looking for quality Workforces due to remote working Rate-C • The response is relevant and responds to the instruction, but it has significant errors in the content. • Instruction: Make a list of the pros and cons of the given decision: "Implementing a Remote Working Policy" • Response: Effective Remote Working Policy A Remote Working Policy could significantly reduce the level of work done as a result of a saht process, including remotely working, or a project being carried out on the client computer without the physical workplace equipment being available. Rate-D • Invalid and unacceptable response. • Instruction: Make a list of the pros and cons of the given decision: "Implementing a Remote Working Policy" • Response: Implementing a Remote Working Policy Figure 1 : 1Overview of LaMini-LM Figure 2 : 2An example of instruction generation prompt based on three random examples from self-instruct. Figure 3 : 3The t-SNE visualizations of instruction sentence embeddings. Figure 4 : 4Human evaluation results for the generated instruction dataset. Figure 5 : 5The performance comparison between encoder-decoder models and decoder-only models of LaMini-LM on the downstream NLP tasks. The horizontal dash lines indicate the average performance given by Alpaca-7B and LLaMa-7B. Figure 6 : 6present the human evaluation results in Fig-Human evaluation results of the selected models on our 114 user-oriented instructions. study notes for the history of the USA include: -The US was created in 1776, three decades after the invention of the printing press. -The United States became an independent nation on April 15, 1787. -In the early days of the Republic, the US played a role in colonization and commerce. -The American Revolution took place during the Revolutionary War, from 1775 to 1803, and it marked the beginning of the end of slavery and the end of democracy. -The Civil War is characterized by the draft, which led to the expansion of the military and the a B Figure 7 : 7An example of instruction generation prompt based on three random examples from self-instruct and three random topics. Dataset # of samples # of ins. tokens avg. ins. len. # of res. tokens avg. res. len.D D DSI 0.27M 3.82M 14.27 17.64M 65.90 D D Dt,SI 0.28M 3.75M 13.26 17.61M 62.38 D D DP3 0.30M 14.63M 49.22 6.35M 21.34 D D DFLAN 0.29M 10.69M 36.37 8.62M 29.33 D D DA 0.05M 0.89M 17.11 2.84M 54.72 D D DP3 0.46M 39.37M 84.78 9.84M 21.19 D D DFLAN 0.93M 57.45M 61.91 21.88M 23.58 D D DALL 2.58M 130.60M 50.62 84.78M 32.86 Table 2 : 2MATTR (up-scaled by ×100) of the generated dataset. Table 4 : 4Automatic evaluation results of selected language models on 15 NLP tasks. "Average" indicates the micro-average of the individual task results. The best average results are highlighted in bold. F-T5 and LaMini-F-T5 indicate Flan-T5 and LaMini-Flan-T5 respectively. C-GPT and LaMini-C indicate Cerebras-GPT and LaMini-Cerebras respectively. Note: We are using lm-eval-harness to evaluate our performance. Therefore, LLaMA numbers are not supposed to be compared from the original paper since we are using different method of measurement. Table 5 : 5Model responses to the instruction "Include important study notes and key points that someone should know about the given subject: 'history of the USA'", where Alpaca-7B fails but LaMini language models manage to respond. The high-quality contents are highlighted in blue. The errors are highlighted in red.Model Response Table 7 : 7The number of hallucinations (lower is better) on our LaMini-Hallucination test set. The worst score for each category is 10. Table 8 : 8The number of toxic outputs given the non- toxic and toxic prompts, out of 1K prompts each. The lower, the better. <example>Try coming up with a creative way to stay motivated during a workout.</example> <example>In your opinion, what are the qualities of an effective sports coach?</example> <example>Return the SSN number for the person: "Yann LeCun"</example> Generate 20 diverse examples that are similar to the provided examples with the topics "Design bureaus, Conidae, Infantry". You do not need to provide a response to the generated examples.→ Table 9 : 9Human evaluation protocol with examples. Table 10 : 10Details of 15 downstream NLP tasks. Acc norm indicates the output probability used for computing the accuracy is normalized by the target sequence length.T5 LaMini-T5 T5 LaMini-T5 T5 LaMini-T5 # of params. 61M 223M 738M OpenBookQA 30.2 31.8 34.8 32.0 32.8 36.0 SciQ 58.0 69.7 71.7 82.9 82.4 84.5 RACE 26.4 29.0 31.1 32.6 31.5 32.6 ARC 22.7 23.0 24.4 26.5 25.4 29.0 PIQA 55.3 59.0 55.7 64.0 55.9 67.2 ReCoRD 53.4 51.7 64.6 59.1 73.1 68.7 SST 71.0 76.8 57.3 91.2 50.2 90.3 MRPC 48.0 68.4 31.6 73.5 34.3 71.1 RTE 53.4 52.7 61.4 71.5 79.8 57.0 MultiNLI 35.4 36.3 56.7 54.7 61.3 54.7 MultiNLI (mis) 35.2 36.2 57.1 55.5 63.1 55.8 WSC273 50.9 52.7 53.8 54.2 60.4 59.0 WinoGrande 48.9 49.3 50.4 51.9 55.2 54.9 WiC 50.0 50.0 52.0 56.0 49.4 50.5 HellaSwag 26.8 27.9 31.0 32.0 38.9 40.6 Average 44.4 47.6 48.9 55.8 52.9 56.8 Table 11 : 11Automatic evaluation results of LaMini-T5 language models and their baselines on 15 NLP tasks. "Average" indicates the micro-average of the individual task results.Flan-T5 LaMini-Flan-T5 Flan-T5 LaMini-Flan-T5 Flan-T5 LaMini-Flan-T5# of params. 77M 248M 783M OpenBookQA 27.0 30.0 28.8 33.0 31.2 34.0 SciQ 89.0 79.4 93.0 86.2 93.8 86.7 RACE 29.7 28.9 35.9 34.4 40.9 32.8 ARC 22.3 24.0 25.1 27.3 30.7 31.8 PIQA 61.9 61.9 67.0 65.7 72.2 70.6 ReCoRD 57.7 53.8 68.2 61.3 76.7 70.4 SST 87.3 85.7 92.3 92.2 94.0 93.1 MRPC 63.2 58.6 71.3 74.8 82.6 77.9 RTE 60.3 56.3 78.7 66.1 87.4 65.0 MultiNLI 42.4 53.2 66.7 66.6 72.4 61.4 MultiNLI (mis) 42.5 53.2 66.9 66.8 72.0 61.0 WSC273 53.1 54.6 57.5 60.4 66.7 64.1 WinoGrande 50.0 50.1 54.2 53.0 59.9 56.0 WiC 51.3 50.8 52.7 60.8 64.7 63.8 HellaSwag 29.1 28.6 36.4 34.6 48.7 43.7 Average 51.1 51.3 59.7 58.9 66.3 60.8 Table 12 : 12Automatic evaluation results of LaMini-Flan-T5 language models and their baselines on 15 NLP tasks. "Average" indicates the micro-average of the individual task results.GPT-Neo LaMini-Neo GPT-Neo LaMini-Neo# of params. 135M 1.3B OpenBookQA 26.2 31.6 33.6 36.4 SciQ 68.8 66.8 77.1 84.2 RACE 27.6 28.7 34.1 34.3 ARC 23.1 24.2 25.9 32.9 PIQA 62.5 63.5 71.1 71.7 ReCoRD 65.6 62.1 81.4 75.2 SST 53.9 52.2 65.7 91.2 MRPC 68.4 64.2 68.4 70.3 RTE 54.9 53.1 60.3 71.1 MultiNLI 35.5 31.9 35.8 49.3 MultiNLI (mis) 35.4 32.0 36.2 49.7 WSC273 55.3 52.7 75.1 66.7 WinoGrande 50.4 50.6 54.9 54.8 WiC 50.0 50.0 50.0 50.2 HellaSwag 30.4 29.9 48.9 47.5 Average 47.2 46.2 54.6 59.0 Table 13 : 13Automatic evaluation results of LaMini-Neo language models and their baselines on 15 NLP tasks. "Average" indicates the micro-average of the individual task results. C-GPT LaMini-C C-GPT C-GPT C-GPT LaMini-C C-GPT LaMini-C# of params. 111M 256M 590M 1.3B OpenBookQA 29.6 30.8 25.4 30.6 28.0 33.0 29.0 34.0 SciQ 52.8 60.0 65.7 68.8 68.2 71.7 73.0 79.4 RACE 25.6 27.1 27.5 27.1 28.4 29.0 30.3 32.9 ARC 22.9 23.3 21.9 26.1 23.5 26.9 25.3 30.3 PIQA 58.4 60.3 61.4 61.4 62.8 63.2 66.8 66.9 ReCoRD 52.4 51.6 61.2 58.6 67.2 63.6 75.0 66.3 SST 60.1 61.2 49.8 76.9 56.0 85.8 51.3 90.3 MRPC 68.4 68.4 68.4 68.4 68.4 68.4 68.4 71.3 RTE 53.1 49.8 52.3 55.6 52.3 60.6 53.1 65.7 MultiNLI 35.1 34.4 35.2 39.0 35.0 49.0 35.2 47.4 MultiNLI (mis) 35.0 35.2 35.1 40.3 35.1 50.8 35.4 49.2 WSC273 51.3 54.2 54.6 49.5 61.9 54.2 62.3 57.1 WinoGrande 50.2 49.3 51.3 52.0 49.8 50.9 51.9 51.8 WiC 50.0 50.0 50.0 50.0 50.0 50.0 50.2 50.2 HellaSwag 26.4 27.2 28.6 29.3 32.3 32.3 38.4 38.7 Average 44.8 45.5 45.9 48.9 47.9 52.6 49.7 55.4 Table 14 : 14Automatic evaluation results of LaMini-Cerebras language models and their baselines on 15 NLP tasks. "Average" indicates the micro-average of the individual task results. C-GPT and LaMini-C indicate Cerebras-GPT and LaMini-Cerebras respectively.GPT-2 LaMini-GPT GPT-2 LaMini-GPT GPT-2 LaMini-GPT# of params. 124M 774M 1.5B OpenBookQA 28.2 30.4 31.2 37.0 32.0 39.8 SciQ 66.1 64.4 69.4 78.3 76.1 80.4 RACE 28.7 31.8 31.6 37.6 33.1 39.1 ARC 23.3 26.4 25.1 30.6 28.5 35.8 PIQA 61.2 62.4 69.2 69.9 70.5 71.3 ReCoRD 70.7 66.8 81.9 77.5 84.4 78.5 SST 52.8 84.5 49.4 91.5 49.1 93.5 MRPC 67.6 68.4 65.2 70.6 63.2 76.0 RTE 54.2 55.2 52.7 74.4 52.3 67.9 MultiNLI 35.6 38.9 35.9 62.5 36.5 67.5 MultiNLI (mis) 35.1 40.2 36.0 65.6 37.0 69.3 WSC273 55.7 57.1 72.5 68.1 73.3 69.6 WinoGrande 51.5 51.9 55.3 54.7 58.3 56.0 WiC 50.0 50.0 49.7 50.0 49.8 52.4 HellaSwag 30.8 30.7 45.3 43.5 50.9 48.3 Average 47.4 50.6 51.4 60.8 53.0 63.0 Table 15 : 15Automatic evaluation results of LaMini-GPT language models and their baselines on 15 NLP tasks. "Average" indicates the micro-average of the individual task results. https://en.wikipedia.org/wiki/Category: Machine_learning We denote the model-generated text as X X X {·} or Y Y Y {·} and the human-written text as X X X {·} or Y Y Y {·} , except for Y Y Y P3 and Y Y Y FLAN that are also generated by gpt-3.5-turbo.4 Model signature: all-mpnet-base-v2. https://github.com/EleutherAI/ lm-evaluation-harness https://platform.openai.com/docs/guides/ moderation/overview . Mostafa Dehghani, Siddhartha Brahma, Albert Webson, Shane Shixiang, Zhuyun Gu, Mirac Dai, Xinyun Suzgun, Aakanksha Chen, Sharan Chowdhery, Gaurav Narang, Adams Mishra, Vincent Y Yu, Yanping Zhao, Andrew M Huang, Hongkun Dai, Slav Yu, Ed H Petrov, Jeff Chi, Jacob Dean, Adam Devlin, Denny Roberts, Quoc V Zhou, Jason Le, Wei, 10.48550/arXiv.2210.11416Scaling instruction-finetuned language models. 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CoRR, abs/1810.12885. Task Category Dataset Size Metric Multiple-Choice QA OpenBookQA. Borui Zhao, Quan Cui, Renjie Song, Yiyu Qiu, Jiajun Liang, ; Mihaylov, 10.1109/CVPR52688.2022.01165CVPR 2022172 Accnorm PIQA (Bisk et al., 2020) 1,838 Accnorm Extractive QA ReCoRD. Accnorm SciQ (Welbl et al., 2017) 1,000 Accnorm RACE (Lai et al.,New Orleans, LA, USAIEEE500872IEEE/CVF Conference on Computer Vision and Pattern Recognition. 000 F1 Sentiment Analysis SSTBorui Zhao, Quan Cui, Renjie Song, Yiyu Qiu, and Jia- jun Liang. 2022. Decoupled knowledge distillation. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, CVPR 2022, New Orleans, LA, USA, June 18-24, 2022, pages 11943-11952. IEEE. Task Category Dataset Size Metric Multiple-Choice QA OpenBookQA (Mihaylov et al., 2018) 500 Accnorm SciQ (Welbl et al., 2017) 1,000 Accnorm RACE (Lai et al., 2017) 1,045 Acc ARC (Clark et al., 2018) 1,172 Accnorm PIQA (Bisk et al., 2020) 1,838 Accnorm Extractive QA ReCoRD (Zhang et al., 2018) 10,000 F1 Sentiment Analysis SST (Socher et al., 2013) 872 Acc Paraphrase Identification MRPC. Dolan and Brockett408Acc Paraphrase Identification MRPC (Dolan and Brockett, 2005) 408 Wang, Acc Natural Language Inference RTE. 277Acc Natural Language Inference RTE (Wang et al., 2019) 277 . Acc Multinli, ( Williams, 9815Acc MultiNLI (Williams et al., 2018) 9,815 . Acc Multinli, ; Williams, 9832misAcc MultiNLI (mis) (Williams et al., 2018) 9,832 . Levesque, Acc Coreference Resolution. 273273Acc Coreference Resolution WSC273 (Levesque et al., 2012) 273 . Acc Winogrande, ( Sakaguchi, 1267Acc WinoGrande (Sakaguchi et al., 2020) 1,267 Acc Word Sense disambiguation WiC (Pilehvar and Camacho-Collados. 638Acc Word Sense disambiguation WiC (Pilehvar and Camacho-Collados, 2019) 638 Zellers, Acc Sentence Completion HellaSwag. 42Acc Sentence Completion HellaSwag (Zellers et al., 2019) 10,042 Accnorm	[ "https://github.com/EleutherAI/" ]
[ "Semi-classical BMS-blocks from the Oscillator Construction", "Semi-classical BMS-blocks from the Oscillator Construction" ]	[ "Martin Ammon [email protected] \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Seán Gray [email protected] \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Claire Moran [email protected] \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Michel Pannier [email protected] \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n", "Katharina Wölfl [email protected] \nTheoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany\n" ]	[ "Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany", "Theoretisch-Physikalisches Institut\nFriedrich-Schiller-Universität Jena\nMax-Wien-Platz 1D-07743JenaGermany" ]	[]	Flat-space holography requires a thorough understanding of BMS symmetry. We introduce an oscillator construction of the highest-weight representation of the bms 3 algebra and show that it is consistent with known results concerning the bms 3 module. We take advantage of this framework to prove that bms 3 -blocks exponentiate in the semi-classical limit, where one of the central charges is large. Within this context, we compute perturbatively heavy, and heavy-light vacuum bms 3 -blocks.	10.1007/jhep04(2021)155	[ "https://arxiv.org/pdf/2012.09173v1.pdf" ]	229,297,881	2012.09173	2e0f6136701b825c154a943932fe1358e289514f	Semi-classical BMS-blocks from the Oscillator Construction 16 Dec 2020 Martin Ammon [email protected] Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Seán Gray [email protected] Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Claire Moran [email protected] Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Michel Pannier [email protected] Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Katharina Wölfl [email protected] Theoretisch-Physikalisches Institut Friedrich-Schiller-Universität Jena Max-Wien-Platz 1D-07743JenaGermany Semi-classical BMS-blocks from the Oscillator Construction 16 Dec 2020 Flat-space holography requires a thorough understanding of BMS symmetry. We introduce an oscillator construction of the highest-weight representation of the bms 3 algebra and show that it is consistent with known results concerning the bms 3 module. We take advantage of this framework to prove that bms 3 -blocks exponentiate in the semi-classical limit, where one of the central charges is large. Within this context, we compute perturbatively heavy, and heavy-light vacuum bms 3 -blocks. Contents Introduction The Holographic Principle promises a great amount of insight into a possible formulation of quantum gravity [1][2][3]. Over the past two decades, a number of statements concerning quantum gravity could be made rigorous within the framework of the AdS/CFT correspondence [4][5][6]. Low-dimensional examples of the duality, such as the AdS 3 /CFT 2 correspondence, have proven to be valuable toy models; explicit computations are possible on both sides of the duality and can be matched, see [7] for a recent example. In AdS 3 /CFT 2 the central charge of the two-dimensional boundary CFT is related to the inverse of Newton's constant in the bulk theory [8]. Remarkably, semi-classical quantumgravitational phenomena generically emerge from universal properties of two-dimensional conformal field theories at large central charge; in addition to the Virasoro symmetry of the field theory, this statement requires a few reasonable assumptions, including modular invariance, locality, and a sparse spectrum of operators whose conformal dimension remain bounded for large central charge. At finite temperature such CFTs display universal thermodynamics [9] which is in agreement with computations in thermal AdS or hairless BTZ black holes [10]. Similar universal behaviour holds for one-and two-point correlation functions of the bounded operators operators, up to exponentially suppressed correction terms [11]. So-called heavy-light conformal blocks show the approximate thermal nature of heavy operators, which may be viewed as black-hole microstates of the dual gravity side [12]; these blocks may be computed with Wilson-line networks on the dual gravity side [13,14]. However, any conclusion made in AdS 3 /CFT 2 is naturally followed by the question to which extent the insight can be carried over to more realistic cases, such as higher-dimensional models or dualities involving gravity in asymptotically flat spacetimes -the scope of the present work is in connection to the latter aspect. Flat-space holography is the moniker given to the expected duality between quantum gravity in asymptotically flat cosmologies and BMS-invariant field theories [15,16]. It remains a challenging task to identify an explicit example of such a correspondence; for recent results regarding flat-space holography see for example . A key ingredient in flat-space holography is the infinite-dimensional Lie algebra bms 3 , which is the asymptotic symmetry algebra of three-dimensional gravity with vanishing cosmological constant, at null infinity [41,42]. The bms 3 algebra is generated by super-rotations L m and super-translations M m , with m ∈ Z, and contains two central charges c L , c M ∈ R [43]. Classical Einstein gravity demands c L = 0; the case c L = 0 emerges in the consideration of so-called chiral gravity [44,45]. Assuming the existence of a flat-space holographic duality, the large central charge limit c M → ∞ of a BMS-invariant field theory should correspond to a semi-classical theory of gravity in asymptotically flat spacetime. Some results regarding BMS-invariant theories concern the bootstrap program [46,47], bms 3 characters and modular invariance [48], torus bms 3 -blocks [49], as well as the computation of heavy-light bms 3 -blocks using the monodromy method [50]. In the context of flat-space holography, heavy-light bms 3 -blocks contain information about probe fields in non-trivial asymptotically flat spacetimes in a dual theory of Einstein gravity. The cosmological solutions generated by heavy operators are then quotients of Minkowski spacetimes and are labelled by their mass M = −1 + 24ξ/c M and angular momentum J/ √ M = 12∆/c M , where ∆ and ξ denote the scaling dimension and rapidity of the heavy operator, respectively [50]. The bms 3 algebra is related to two copies of the Virasoro algebra by suitableİnönü-Wigner contractions: there exist two such contractions, known as the ultra-relativistic and non-relativistic limits; both contractions lead to isomorphic symmetry algebras but with different representation theories [16]. One typically finds unitary (induced) representations by ultra-relativistic contraction of the conformal symmetry algebra [29]. The oscillator construction may be employed in order to express highest-weight representations of the Virasoro algebra in terms of oscillator variables [51]; we review this construction in section 2. This formulation provides an alternative approach to the calculation of various quantities of two-dimensional CFTs, such as correlation functions and conformal blocks, and has recently been employed in the context of the eigenstate thermalisation hypothesis [52] and the proof of Virasoro-block exponentiation [53]. In the present work, we introduce an oscillator construction for highest-weight representations of bms 3 . 1 In section 3 we express the bms 3 generators in terms of oscillator variables and define a suitable measure on the function space. The origin of our expressions is a nonrelativistic limit of a linear-dilaton like theory. The oscillator formalism allows us to prove the exponentiation of bms 3 blocks in the semi-classical limit, which we do in section 4. Moreover, in section 4 we also demonstrate the applicability of the oscillator construction by calculating a perturbatively heavy vacuum bms 3 -block as well as a heavy-light vacuum bms 3 -block. Oscillator Construction of the Virasoro Algebra Before approaching the bms 3 oscillator construction it is advantageous to first introduce the oscillator construction and its meaning for two-dimensional conformal field theories. The conformal symmetry algebra decomposes into a direct sum of two copies of the Virasoro algebra: one holomorphic and one anti-holomorphic copy. In the following review we will focus on the holomorphic sector but the analysis straightforwardly extends to the anti-holomorphic sector. Highest-Weight Representation of the Virasoro Algebra The generators of the Virasoro algebra, which we denote L vir m , obey the commutation relations [L vir m , L vir n ] = (m − n)L vir m+n + c 12 m m 2 − 1 δ m,−n , (2.1.1) where c denotes the central charge and m, n ∈ Z. We consider highest-weight representations of the above Virasoro algebra; such representations are built from a primary state \|h with conformal weight h, satisfying L vir 0 \|h = h \|h and L vir m \|h = 0 for m ≥ 1. Descendant states are generated by basis vectors of the form \| (m 1 , . . . , m k ) ; h =   k i=1 L vir −m i   \|h (2.1.2) with m 1 ≥ . . . ≥ m k ≥ 1. A primary state and its descendants span a vector space called the Verma module V c,h . Imposing that the central charge c is a real number, as well as the adjoint relation L vir m † = L vir −m , there exists a unique Hermitian product q\|p on the Verma module V c,h for which a primary state has unit norm. For c, h > 0 and in the absence of singular states the Hermitian product is positive definite and the corresponding representation is irreducible and unitary. Oscillator Construction We now employ the oscillator construction to study the highest-weight representation of the Virasoro algebra. In this context we write the Virasoro generators in cursive lowercase and they take the form 2 ℓ 0 = h + ∞ n=1 nu n ∂ un , (2.2.1a) ℓ k = ∞ n=1 nu n ∂ u n+k − 1 4 k−1 n=1 ∂ un ∂ u k−n + (µk + iλ) ∂ u k , (2.2.1b) ℓ −k = ∞ n=1 (n + k)u n+k ∂ un − k−1 n=1 n(k − n)u n u k−n + 2k (µk − iλ) u k , (2.2.1c) where k ≥ 1 and the constants λ, µ ∈ R are related to the central charge and conformal weight by 3 c = 1 + 24µ 2 and h = λ 2 + µ 2 . (2.2.2) In this language, the states of a Verma module V c,h are mapped to functions which depend on a full set of infinitely many representation space variables u n ∈ C with n ∈ N; in accordance with previous literature we will refer to these variables as oscillator variables. Equivalence between the oscillator construction and the formalism presented in the previous subsection follows from the definition f p (u) ≡ u\|p , (2.2.3) where u\| ≡ \|ū † is a generalised coherent state of the Verma module V c,h , \|p is a generic state of the Verma module, and we omit the index of the oscillator variables to indicate a full 2 These generators are obtained by considering a linear dilaton conformal field theory. See appendix A of [52] for further details. 3 To gain access to Verma modules with h < (c − 1)/24 the constant λ must be analytically continued to imaginary values. Similarly, minimal models with c < 1 require an analytic continuation in µ. set. The presence of a derivative in ℓ 0 makes it natural that the primary state of V c,h maps to a constant, which we choose to be unity, i.e. f h (u) = u\|h ≡ 1 . (2.2.4) The basis vectors \|(m 1 , . . . , m k ) ; h , which were defined in equation (2.1.2), are mapped to u\|   k i=1 L vir −m i   \|h =   k i=1 ℓ −m i   · 1 . (2.2.5) Given the form of ℓ −k in (2.2.1c), the basis vector on the right-hand side of the above equation is a polynomial in the oscillator variables. The dual function is defined by f q (u) = u\|q ≡ q\|ū , where \|q is a state in V c,h and the overline operation acts as u n →ū n as well as complex conjugation. Such dual functions are acted on byl −m , which results in the action q\|L vir n \|ū . Note that the barred quantities belong to the same holomorphic sector of the two-dimensional conformal algebra as the non-barred quantities. Unitarity requires the adjoint property ℓ † m = ℓ −m , which is defined by the Hermitian product; the appropriate expression in the oscillator construction may be found by inserting a completeness relation C ∞ [d 2 u] h \|ū u\| = 1 (2.2.6) into the product q\|p defined in the previous subsection. The Hermitian product of the Virasoro oscillator construction thus reads f q , f p = C ∞ [d 2 u] h f q (u)f p (u) , (2.2.7) with the measure given by [d 2 u] h = ∞ n=1 d 2 u n 2n π e −2nunūn , (2.2.8) where d 2 u n = du n dū n . Using the Hermitian product given in (2.2.7), monomials of oscillator variables form an orthogonal basis which satisfies u m 1 1 u m 2 2 · · · , um 1 1 um 2 2 · · · = ∞ n=1 m n ! (2n) mn δ mn,mn . (2.2.9) The above orthogonality relation proves to be useful when evaluating certain quantities, for instance the Gram matrix [52]. Correlators and Wave Functions Building upon the Virasoro symmetry and its representations, a two-dimensional conformal field theory is defined in terms of its operators and correlation functions. A holomorphic primary operator O h (z) is defined via the operator-state correspondence as \|h ≡ lim z→0 O h (z) \|0 , (2.3.1) where z is a holomorphic coordinate on the complex plane and the vacuum state \|0 is characterised by h = 0 and L vir m \|0 = 0 for m ≥ −1. Virasoro generators L vir m act on O h (z) via [L vir m , O h (z)] = −L vir m O h (z) , (2.3.2) where L vir m is a differential operator given by L vir m = −z m+1 ∂ z − (m + 1)hz m . (2.3.3) Note that L vir m satisfies the Virasoro algebra without a central extension. Two-point Correlation Functions The two-point correlation function is given by 0\|O h 1 (z 1 )O h 2 (z 2 )\|0 ; inserting a complete set of states as defined in equation (2.2.6) results in the expression 0\|O h 1 (z 1 )O h 2 (z 2 )\|0 = C ∞ [d 2 u] h 0\|O h 1 (z 1 )\|ū u\|O h 2 (z 2 )\|0 . (2.3.4) The above result defines the level-one wave functions ψ h 2 ;h (z 2 ; u) = u\|O h 2 (z 2 )\|0 , (2.3.5a) χ h 1 ;h (z 1 ;ū) = 0\|O h 1 (z 1 )\|ū , (2.3.5b) where the subscripts h i denote the conformal dimensions of the external operators O h i while h labels the Verma module V c,h . When treated individually we refer to ψ h 2 ;h (z 2 ; u) as the wave function and χ h 1 ;h (z 1 ;ū) as the dual wave function; the relationship between them is 4 χ h 1 ;h (z 1 ;ū) = z −2h 1 1 ψ h 1 ;h (z −1 1 ; u) , (2.3.6) which follows from the property that bra-and ket-states of a conformal field theory are related by a coordinate inversion on the complex plane. As a consequence of L vir m \|0 = 0 for m ≥ −1, together with (2.3.2), the level-one wave functions satisfy the differential equations ℓ (h) m + L (h 2 ) m ψ h 2 ;h (z 2 ; u) = 0 , (2.3.7a) l (h) m − L (h 1 ) −m χ h 1 ;h (z 1 ;ū) = 0 . (2.3.7b) Since the level-one wave functions connect oscillator variables and coordinates of the complex plane, we here introduce the superscripts h i and h to emphasise the generators' respective domains of action. Note that the above differential equations allow for non-trivial solutions only if h 1 = h or h 2 = h. Three-point Correlation Functions and Conformal Blocks Inserting a completeness relation into the definition of a three-point correlation function, in two different ways, gives 0\|O h 1 (z 1 )O h 2 (z 2 )O h 3 (z 3 )\|0 = C ∞ [d 2 u] h 0\|O h 1 (z 1 )\|ū u\|O h 2 (z 2 )O h 3 (z 3 )\|0 (2.3.8a) = C ∞ [d 2 u] h 0\|O h 1 (z 1 )O h 2 (z 2 )\|ū u\|O h 3 (z 3 )\|0 . (2.3.8b) The structure of the above expressions may be used to define the level-two wave functions ψ h 1 ,h 2 ;h (z 1 , z 2 ; u) = u\|O h 1 (z 1 )O h 2 (z 2 )\|0 , (2.3.9a) χ h 3 ,h 4 ;h (z 3 , z 4 ;ū) = 0\|O h 4 (z 4 )O h 3 (z 3 )\|ū . (2.3.9b) Following the same reasoning as in the previous paragraph the level-two wave functions are related by χ h 3 ,h 4 ;h (z 3 , z 4 ;ū) = z −2h 3 3 z −2h 4 4 ψ h 3 ,h 4 ;h (z −1 3 , z −1 4 ; u) , (2.3.10) and they satisfy the differential equations ℓ (h) m + L (h 1 ) m + L (h 2 ) m ψ h 1 ,h 2 ;h (z 1 , z 2 ; u) = 0 , (2.3.11a) l (h) m − L (h 3 ) −m − L (h 4 ) −m χ h 3 ,h 4 ;h (z 3 , z 4 ;ū) = 0 . (2.3.11b) We may now appropriately express conformal blocks in terms of wave functions. Virasoro blocks are defined by the four-point function 0\|O h 4 (z 4 )O h 3 (z 3 )P h O h 1 (z 1 )O h 2 (z 2 )\|0 , where P h denotes the projector onto the Verma module V c,h . It follows directly from the argumentation and definitions presented in this subsection that a conformal block reads 0\|O h 4 (z 4 )O h 3 (z 3 )P h O h 1 (z 1 )O h 2 (z 2 )\|0 = C ∞ [d 2 u] h χ h 3 ,h 4 ;h (z 3 , z 4 ;ū) ψ h 1 ,h 2 ;h (z 1 , z 2 ; u) . (2.3.12) The operator-state correspondence allows for the primary and descendant states of the Verma module V c,h to be interpreted as internal operators which arise due to the operator product expansion of the four-point correlation function. Although the differential equations (2.3.7) allow for solutions for general values of h and c, the level-two equations (2.3.11) have known closed-form solutions for the case h i = 1/16 and c = 1, only [51]. Nevertheless, in the semi-classical limit, i.e. c → ∞, it is possible to find approximate solutions for the level-two wave functions, and by extension the Virasoro conformal blocks [53]. Oscillator Construction of the Highest-Weight Representation of bms 3 The bms 3 algebra is a semi-direct sum of one Virasoro algebra and an infinite-dimensional Abelian algebra of super-translations. Hence, in contrast to the Virasoro case, the bms 3 algebra cannot be decomposed into commuting sectors; thus it needs to be treated en bloc. In this section we first review highest-weight bms 3 modules, after which we present our oscillator construction of such modules. We also discuss the computation of correlation functions and bms 3 -blocks in the language of oscillator variables and wave functions. Modules of the bms 3 Algebra The bms 3 algebra is generated by L n and M n , and defined by the Lie bracket [L m , L n ] = (m − n)L m+n + c L 12 m(m 2 − 1)δ m,−n , (3.1.1a) [L m , M n ] = (m − n)M m+n + c M 12 m(m 2 − 1)δ m,−n , (3.1.1b) [M m , M n ] = 0 , (3.1.1c) where c L and c M are central charges, which we take to be non-negative real numbers, and m, n ∈ Z. In the context of asymptotically flat gravity in three spacetime dimensions L n are the generators of super-rotations and the M n generate super-translations. We define a primary state \|∆, ξ to be a state that satisfies the eigenvalue equations L 0 \|∆, ξ = ∆\|∆, ξ and M 0 \|∆, ξ = ξ\|∆, ξ , (3.1.2) where ∆ is the scaling dimension and ξ is the rapidity, as well as L n \|∆, ξ = 0 and M n \|∆, ξ = 0 , is built by acting with an ordered string, of arbitrary length, of operators L −n and M −n with n > 0 on a state \|∆, ξ . Hence, the vector space associated to the highest-weight representation of bms 3 is spanned by the basis vectors \|(m 1 , . . . , m s ), (n 1 , . . . , n l ); ∆, ξ = L −m 1 · · · L −ms M −n 1 · · · M −n l \|∆, ξ , (m ′ i ) s ′ i=1 , (n ′ j ) l ′ j=1 ; ∆, ξ (m i ) s i=1 , (n j ) l j=1 ; ∆, ξ = ∆, ξ 1 j=l ′ M n ′ j 1 i=s ′ L m ′ i s i=1 L −m i l j=1 M −n j ∆, ξ . (3.1.5) 5 In the following we will use the short-hand notation (mi) s i=1 ≡ (m1, . . . , ms). For the highest-weight representation of bms 3 the Hermitian product is not generically positive semi-definite and hence the corresponding representation will not necessarily be unitary. 6 We return to this point at the end of the next subsection. Oscillator Construction The expressions for the bms 3 generators in terms of the complex oscillator variables v (1) n and v (2) n , with n ∈ N, may be found by taking a non-relativistic limit of a two-dimensional linear-dilaton like conformal field theory; we refer the reader to appendix A for details. The resulting generators, which we denote in lowercase, read l 0 = ∆ + ∞ n=1 n v (1) n ∂ v (1) n + v (2) n ∂ v (2) n , (3.2.1a) l k = ∞ n=1 n v (1) n ∂ v (1) k+n + v (2) n ∂ v (2) k+n − 1 4 k−1 n=1 ∂ v (1) n ∂ v (2) k−n + A k ∂ v (1) k + B k ∂ v (2) k , (3.2.1b) l −k = ∞ n=1 (k + n) v (1) k+n ∂ v (1) n + v (2) k+n ∂ v (2) n − 4 k−1 n=1 n(k − n)v (1) n v (2) k−n + 4kB k v (1) k + 4kÂ k v (2) k , (3.2.1c) and, m 0 = ξ + ∞ n=1 nv (1) n ∂ v (2) n , (3.2.2a) m k = ∞ n=1 nv (1) n ∂ v (2) k+n − 1 8 k−1 n=1 ∂ v (2) k−n ∂ v (2) n + A k ∂ v (2) k , (3.2.2b) m −k = ∞ n=1 (k + n)v (1) k+n ∂ v (2) n − 2 k−1 n=1 n(k − n)v (1) k−n v (1) n + 4kÂ k v (1) k , (3.2.2c) for k > 0. The above generators satisfy the bms 3 commutation relations (3.1.1). In the above expressions we have made the identifications 7 A k = − i 2 2ξ − c M 12 − k c M 48 , B k = i c L − 2 − 24∆ 48 2ξ − c M 12 − k c L − 2 48 c M 12 , (3.2.3a) A k = i 2 2ξ − c M 12 − k c M 48 ,B k = −i c L − 2 − 24∆ 48 2ξ − c M 12 − k c L − 2 48 c M 12 . (3.2.3b) 6 There exists an exceptional case for cM = 0 and ξ = 0 where the bms3 representation constructed above reduces to a Virasoro highest-weight representation with central charge cL and conformal dimension h = ∆, provided that we take a quotient with respect to the null states M−n\|∆, 0 with n ∈ N [54]. 7 If ξ ≥ cM/24 these coefficients are related by complex conjugation,Â k = A * k andB k = B * k ; in this case the adjoint property l † n = l−n holds. However, for ξ < cM/24 all coefficients are real and hence independent of each other; preserving the adjoint property requires analytic continuation, as discussed in section 4.3.1. Unless otherwise stated we will assume ξ ≥ cM/24. Analogously to the Verma module, a state \|p ∈ B c L ,c M ∆,ξ is mapped to a function via f p (v) ≡ v\|p , (3.2.4) where v\| ≡ \|v † is a generalised coherent state of the bms 3 module, and v denotes the collective set of all oscillator variables v (1) n and v (2) n . 8 It follows from the form of the generators (3.2.1) and (3.2.2) that the requirements for a primary state are fulfilled by a constant function, hence we choose f ∆,ξ (v) = v\|∆, ξ ≡ 1 . (3.2.5) The properties (3.1.2) and (3.1.3) translate to l 0 · 1 = ∆ , m 0 · 1 = ξ , (3.2.6a) l k · 1 = 0 , m k · 1 = 0 , (3.2.6b) for k > 0. The basis vectors of the form (3.1.4) are thus given in terms of the polynomials s i=1 l −m i l j=1 m −n j · 1. Additionally, we define f q (v) = v\|q ≡ q\|v for the state \|q ∈ B c L ,c M ∆,ξ , where the overline acts as v (i) n →v (i) n and complex conjugation. The barred functions are acted on with the generatorsl −n andm −n , which in terms of states is expressed as q\|L n \|v and q\|M n \|v , respectively. Finally, the adjoints l † n = l −n and m † n = m −n specify the unique Hermitian product of the oscillator construction (f q , g p ) = C ∞ [d 2 v] ∆,ξ f q (v)g p (v) , (3.2.7) where the measure is given by [d 2 v] ∆,ξ = ∞ n=1 16n 2 exp −4n v (1) nv (2) n + v (2) nv (1) n d 2 v (1) n d 2 v (2) n , (3.2.8) with d 2 v (i) n = dv (i) n dv (i) n . The motivation for the form of the measure [d 2 v] ∆,ξ is deferred to the appendix A.2. We arrived at the expression (3.2.7) in the same way as one does for Virasoro, i.e. by inserting a complete set of states of the form C ∞ [d 2 v] ∆,ξ \|v v\| = 1 (3.2.9) into the Hermitian product q\|p . The Hermitian product of oscillator monomials is given by the formula v (1) m a v (2) m b , v (1) m c v (2) m d = a! b! (4m) a+b δ a,d δ b,c , (3.2.10) which is a result of a calculation that is sketched in appendix A.2. By calculating the Hermitian product of basis vectors we find the Gram matrix; in the oscillator construction a general element of the form (3.1.5) is given by l −m ′ 1 · · · l −m ′ s ′ m −n ′ 1 · · · m −n ′ l ′ · 1 , l −m 1 · · · l −ms m −n 1 · · · m −n l · 1 . (3.2.11) The basis vectors are polynomials in v (1) n and v (2) n , hence to calculate a specific element of the Gram matrix one may advantageously apply the orthogonality relation (3.2.10) to their constituent monomials. For the lowest-level bms 3 Gram matrix we obtain the entries (l −1 · 1, l −1 · 1) = 2∆ , (l −1 · 1, m −1 · 1) = 2ξ , (m −1 · 1, l −1 · 1) = 2ξ , (m −1 · 1, m −1 · 1) = 0 . (3.2.12) The above matrix components, as well as the second-level Gram matrix entries (which are not displayed here) match the results of [48]. This serves as a check of our oscillator construction of bms 3 . The bms 3 module is in general not unitary. This can be seen from the lowest-level Gram matrix since at least one of its eigenvalues is negative if ξ = 0; thus the Hermitian product is negative definite, or indefinite, and hence the highest-weight representation is non-unitary. Correlators and Wave Functions We define a bms 3 primary operator O ∆,ξ (t, x) by means of the operator-state correspondence \|∆, ξ ≡ lim t,x→0 O ∆,ξ (t, x) \|0 , (3.3.1) where t and x are coordinates on the plane. The vacuum state \|0 is defined as the primary state with ∆ = ξ = 0 and L n \|0 = M n \|0 = 0 for n ≥ −1. The generators of bms 3 act on O ∆,ξ (t, x) as [L n , O ∆,ξ (t, x)] = −L n O ∆,ξ (t, x) , (3.3.2a) [M n , O ∆,ξ (t, x)] = −M n O ∆,ξ (t, x) , (3.3.2b) and the differential operators take the forms [55] L n = −t n+1 ∂ t − (n + 1)t n x∂ x − (n + 1)(t n ∆ + nt n−1 xξ) , (3.3.3a) M n = −t n+1 ∂ x − (n + 1)ξt n . (3.3.3b) Note that L n and M n satisfy the commutation relations (3.1.1) with c L = c M = 0. Two-point Correlation Functions Inserting the completeness relation (3.2.9) into the definition of a two-point correlation function of primary operators gives 0\|O ∆ 1 ,ξ 1 (t 1 , x 1 )O ∆ 2 ,ξ 2 (t 2 , x 2 )\|0 = C ∞ [d 2 v] ∆,ξ χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v)ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) ,(3. 3.4) where we have defined the level-one wave functions ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) = v\|O ∆ 2 ,ξ 2 (t 2 , x 2 )\|0 , (3.3.5a) χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) = 0\|O ∆ 1 ,ξ 1 (t 1 , x 1 )\|v . (3.3.5b) We will refer to ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) as the wave function and χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) as the dual wave function. The subscripts ∆ i and ξ i label the scaling dimension and rapidity of the external operators, respectively, while ∆ and ξ label the bms 3 module B c L ,c M ∆,ξ . Using that L n \|0 = M n \|0 = 0 for n ≥ −1, as well as the definitions of the differential operators (3.3.2), we find two sets of differential equations for the wave function ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v), l (∆,ξ) n + L (∆ 2 ,ξ 2 ) n ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) = 0 , (3.3.6a) m (∆,ξ) n + M (∆ 2 ,ξ 2 ) n ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) = 0 , (3.3.6b) for n ≥ −1. By similar arguments we find that the dual wave function χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) is constrained by the set of differential equations l (∆,ξ) n − L (∆ 1 ,ξ 1 ) −n χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) = 0 , (3.3.7a) m (∆,ξ) n − M (∆ 1 ,ξ 1 ) −n χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v) = 0 , (3.3.7b) for n ≥ −1. Above we use superscripts for the generators l n , m n as well as L n and M n for the same reason as discussed below equations (2.3.7). The wave functions may be determined by solving both sets of differential equations for n ∈ {−1, 0, 1, 2}, which also ensures the validity of the solutions for all n > 2. 9 The differential equations (3.3.6) and (3.3.7) have non-trivial solutions only if ∆ 2 = ∆, ξ 2 = ξ, and ∆ 1 = ∆, ξ 1 = ξ, respectively. 10 We find the following expressions for the level-one bms 3 wave functions, where the coefficients A 1 , B 1 andÂ 1 ,B 1 are given by equations (3.2.3a) and (3.2.3b) for k = 1, respectively. 11 The relationship between the above level-one wave functions is thus ψ ∆,ξ (t 2 , x 2 ; v) = exp   4Â 1 ∞ n=1 t n 2 v (2) n + n x 2 t n−1 2 v (1) n + 4B 1 ∞ n=1 t n 2 v (1) n   , (3.3.8a) χ ∆,ξ (t 1 , x 1 ;v) = t −2∆ 1 e −2ξ x 1 t 1 exp   4A 1 ∞ n=1 t −n 1v (2) n − n x 1 t −n−1 1v (1) n + 4B 1 ∞ n=1 t −n 1v (1) n   ,(3.χ ∆,ξ (t 1 , x 1 ;v) = t −2∆ 1 e −2ξ x 1 t 1 ψ ∆,ξ t −1 1 , −x 1 t −2 1 ; v , (3.3.9) where the overline acts in accordance with footnote 7. The above relation can be determined from the expressions (3.3.8) or motivated by the operator-state correspondence for bra-states ∆, ξ\| = lim t→∞ t 2∆ e 2ξ x t 0\| O ∆,ξ (t, x) . (3.3.10) Inserting the wave functions (3.3.8) in (3.3.4), power expanding, and using the orthogonality relation (3.2.10), we arrive at the expression for the bms 3 two-point correlation function 0\| O ∆ 1 ,ξ 1 (t 1 , x 1 )O ∆ 2 ,ξ 2 (t 2 , x 2 ) \|0 = (t 1 − t 2 ) −2∆ e − 2ξ(x 1 −x 2 ) (t 1 −t 2 ) , (3.3.11) if ∆ 1 = ∆ 2 = ∆ and ξ 1 = ξ 2 = ξ, and zero otherwise. The above expression is in agreement with previously known results [55]. Three-point Correlation Functions and bms 3 -blocks In terms of our oscillator construction for bms 3 modules, the three-point correlation function reads 0\|O ∆ 1 ,ξ 1 (t 1 , x 1 )O ∆ 2 ,ξ 2 (t 2 , x 2 )O ∆ 3 ,ξ 3 (t 3 , x 3 )\|0 = C ∞ [d 2 v] ∆,ξ χ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t 1,2 , x 1,2 ;v)ψ ∆ 3 ,ξ 3 ;∆,ξ (t 3 , x 3 ; v) , (3.3.12a) = C ∞ [d 2 v] ∆,ξ χ ∆ 1 ,ξ 1 ;∆,ξ (t 1 , x 1 ;v)ψ ∆ 2,3 ,ξ 2,3 ;∆,ξ (t 2,3 , x 2,3 ; v) , (3.3.12b) where we have defined the level-two wave functions by ψ ∆ 2,3 ,ξ 2,3 ;∆,ξ (t 2 , x 2 , t 3 , x 3 ; v) = v\|O ∆ 2 ,ξ 2 (t 2 , x 2 )O ∆ 3 ,ξ 3 (t 3 , x 3 )\|0 , (3.3.13a) χ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t 1 , x 1 , t 2 , x 2 ;v) = 0\|O ∆ 2 ,ξ 2 (t 2 , x 2 )O ∆ 1 ,ξ 1 (t 1 , x 1 )\|v , (3.3.13b) where the shorthand notation ∆ i,j and ξ i,j indicates the dependence on both ∆ i , ∆ j and ξ i , ξ j . The left-and right-hand sides of (3.3.12) are related by an insertion of a complete set of states (3.2.9). The transformation between bra-and ket-states (3.3.10) means that the level-two wave functions are related by χ ∆ 3,4 ,ξ 3,4 ;∆,ξ (t 3 , x 3 , t 4 , x 4 ;v) = t −2∆ 3 3 e −2ξ 3 x 3 t 3 t −2∆ 4 4 e −2ξ 4 x 4 t 4 ψ ∆ 3,4 ,ξ 3,4 ;∆,ξ (t −1 3 , −x 3 t −2 3 , t −1 4 , −x 4 t −2 4 ; v) . (3.3.14) 11 We detail the solution procedure for ψ ∆,ξ (t2, x2; v) in appendix C. Note that the level-one wave functions are completely fixed by the differential equations (3.3.6) and (3.3.7) for n ∈ {−1, 0, 1}. This is connected to the fact that two-point functions are completely determined by the globally well-defined generators Ln and Mn with n ∈ {−1, 0, 1}. Nevertheless, the set of n = 2 differential equations must still be satisfied in order to guarantee a general solution for all n. Following the same reasoning as in the previous subsection we find that the wave function must satisfy the set of differential equations l (∆,ξ) n + L (∆ 1 ,ξ 1 ) n + L (∆ 2 ,ξ 2 ) n ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t 1 , x 1 , t 2 , x 2 ; v) = 0 , (3.3.15a) m (∆,ξ) n + M (∆ 1 ,ξ 1 ) n + M (∆ 2 ,ξ 2 ) n ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t 1 , x 1 , t 2 , x 2 ; v) = 0 , (3.3.15b) and similarly the dual wave function must obey l (∆,ξ) n − L (∆ 3 ,ξ 3 ) −n − L (∆ 4 ,ξ 4 ) −n χ ∆ 3,4 ,ξ 3,4 ;∆,ξ (t 3 , x 3 , t 4 , x 4 ;v) = 0 , (3.3.16a) m (∆,ξ) n − M (∆ 3 ,ξ 3 ) −n − M (∆ 4 ,ξ 4 ) −n χ ∆ 3,4 ,ξ 3,4 ;∆,ξ (t 3 , x 3 , t 4 , x 4 ;v) = 0 , (3.3.16b) for n ≥ −1. Like for the Virasoro case, we have not found any closed-form solutions for the level-two wave functions. Nevertheless, in section 4 we make use of the semi-classical limit in order to approximate solutions for the above set of equations. Finally, bms 3 -blocks are given by 0\|O ∆ 4 ,ξ 4 (t 4 , x 4 )O ∆ 3 ,ξ 3 (t 3 , x 3 )P ∆,ξ O ∆ 1 ,ξ 1 (t 1 , x 1 )O ∆ 2 ,ξ 2 (t 2 , x 2 )\|0 = C ∞ [d 2 v] ∆,ξ χ ∆ 3,4 ,ξ 3,4 ;∆,ξ (t 3 , x 3 , t 4 , x 4 ;v) ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t 1 , x 1 , t 2 , x 2 ; v) , (3.3.17) where P ∆,ξ is the projector onto the bms 3 module B c L ,c M ∆,ξ , which when restricted to the module acts as the unit operator, and we have inserted a completeness relation. Without loss of generality we may consider the point configuration {(t i , x i )} = {(t, x), (0, 0), (1, 0), (∞, 0)} , (3.3.18) such that the bms 3 -block B ∆tot,ξtot;∆,ξ (t, x) is given by B ∆tot,ξtot;∆,ξ (t, x) = lim t 4 →∞ x 4 →0 t 2∆ 4 4 e 2ξ 4 x 4 t 4 C ∞ [d 2 v] ∆,ξ χ ∆ 3,4 ,ξ 3,4 ;∆,ξ (1, 0, t 4 , x 4 ;v)ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0; v) , (3.3.19) where ∆ tot = {∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 } and ξ tot = {ξ 1 , ξ 2 , ξ 3 , ξ 4 }. Furthermore, in the point configuration (3.3.18) the relationship between the wave function and its dual simplifies to χ ∆ 3,4 ,ξ 3,4 ;∆,ξ (1, 0, t 4 , x 4 ;v) = t −2∆ 4 4 e −2ξ 4 x 4 t 4 ψ ∆ 3,4 ,ξ 3,4 ;∆,ξ (1, 0, t −1 4 , −x 4 t −2 4 ; v) . (3.3.20) Plugging in the above relation into (3.3.19) and implementing the limits, we reach the compact formula B ∆tot,ξtot;∆,ξ (t, x) = C ∞ [d 2 v] ∆,ξ ψ ∆ 3,4 ,ξ 3,4 ;∆,ξ (1, 0, 0, 0; v) ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0; v) . (3.3.21) The set of equations (3.3.15) with n = 0 fixes the wave function to be of the form ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0; v) = t ∆−∆ 1 −∆ 2 e x t (ξ−ξ 1 −ξ 2 ) F (η, ν) , (3.3.22) where F (η, ν) is an unknown function, and we have introduced the combinations of oscillatorand spacetime variables η n = t n v (1) n , ν n = nt n−1 xv (1) n + t n v (2) n , (3.3.23) where n ∈ N. The dual wave function follows from the transformation (3.3.20). In appendix B.1 we prove that F (η, ν) is uniquely determined by the remaining differential equations. Semi-classical bms 3 -blocks In this section we will apply the machinery of our oscillator construction of the highest-weight representation of bms 3 in order to calculate bms 3 -blocks in the semi-classical limit c M → ∞, with ∆ c M , ∆ i c M ; ξ c M , ξ i c M , and c L c M kept fixed. Here ∆, ξ ∈ B c L ,c M ∆,ξ are the scaling dimension and rapidity of the internal primary operator of each block, while ∆ i , ξ i with i = 1, .., 4 denote the scaling dimensions and rapidities of the external operators. In order to implement the semi-classical limit we express the affected quantities in terms of an auxiliary parameter µ as follows, c M = µ 2c M , c L = µ 2c L , ∆ = µ 2∆ , ∆ i = µ 2∆ i , ξ = µ 2ξ , ξ i = µ 2ξ i ; (4.0.1) hence the semi-classical limit corresponds to µ → ∞ while keeping the tilde-quantities fixed. Semi-classical Differential Equations The semi-classical limit allows us to evaluate the bms 3 -block, using the integral expression (3.3.21), by means of the saddle-point approximation. In order to express the integrand of (3.3.21) in a form which is suitable for the saddle-point approximation we must consider the contribution due to the exponential function in the measure (3.2.8); we rescale the oscillator variables v In accordance with our intention to employ the saddle-point approximation, we introduce an exponential ansatz for the wave function (3.3.22) F (σ, κ) = exp µ 2 S(σ, κ) . (i) n andv (i) n as follows v (i) n → µv (i) n ,v (i) n → µv (i) n ,(4.0 = ∞ n=1 n σ n ∂ σ k+n S − ∂ σn S + κ n ∂ κ k+n S − ∂ κn S − 1 4 k−1 n=1 ∂ σn S ∂ κ k−n S +Ã k ∂ σ k S +B k ∂ κ k S − ∆ + k∆ 1 −∆ 2 , (4.1.4a) 0 = ∞ n=1 nσ n ∂ κ k+n S − ∂ κn S − 1 8 k−1 n=1 ∂ κ k−n S ∂ κn S +Ã k ∂ κ k S − ξ + kξ 1 −ξ 2 , (4.1.4b) were we have defined A k = µ − i 2 2ξ −c M 12 − k c M 48 ≡ µ ·Ã k , (4.1.5a) B k = µ   ic L − 24∆ 48 2ξ −c M 12 − kc L 48 c M 12    ≡ µ ·B k . (4.1.5b) We arrived at the equations (4.1.4) after dividing by a leading factor of µ 2 and getting rid of the overall exponential function; the resulting terms proportional to 1 µ are sub-leading in the limit µ → ∞ and have hence been dropped, leaving two first-order differential equations for S(σ, κ). Proof of Exponentiation Our ansatz (4.1.3) has consequences for the general form of the bms 3 -block. Using the exponential ansatz (4. B ∆tot,ξtot;∆,ξ (t, x) ∼ t ∆−∆ 1 −∆ 2 e x t (ξ−ξ 1 −ξ 2 ) C ∞   ∞ n=1 16n 2 µ 4 d 2 v (1) n d 2 v (2) n   exp µ 2 I(t, x; v,v) ,(4.I(t, x; v,v) = −4 ∞ n=1 n v (1) nv (2) n + v (2) nv (1) n + S(t, x; v) + S(v) .(4 .2.2) Note that the oscillator variables in the sum are those which have been rescaled, providing the overall factor of µ 2 in the exponential function. In the semi-classical limit, i.e. when µ → ∞, the exponent of the exponential function in (4.2.1) is large and hence the integral is dominated by the stationary points of I(t, x; v,v); we may then use the saddle-point approximation in order to evaluate the integral. The stationary points (w,w) are found by extremising I(t, x; v,v) and they satisfy w (1) m = 1 4m ∂S ∂v (2) m v (i) =w (i) , w (2) m = 1 4m ∂S ∂v (1) m v (i) =w (i) , (4.2.3a) w (1) m = 1 4m ∂S ∂v (2) m v (i) =w (i) ,w (2) m = 1 4m ∂S ∂v (1) m v (i) =w (i) . (4.2.3b) To approximate the bms 3 -block we plug the stationary points into the integrand of (4.2.1), which gives 12 B ∆tot,ξtot;∆,ξ (t, x) ≈ t ∆−∆ 1 −∆ 2 exp x t (ξ − ξ 1 − ξ 2 ) + µ 2 I(t, x; w,w) , (4.2.4) where the overall factor 16n 2 µ 4 in (4.2.1) cancels with the determinant of the Hessian evaluated at the stationary point. The factor t ∆−∆ 1 −∆ 2 may be expressed as an exponential of a logarithm. We have thus shown that the bms 3 -block takes a unique exponential form in the semi-classical limit. In the following subsections we present two illustrative examples for how to compute bms 3 -blocks using the oscillator construction. We will focus on bms 3 -blocks of the vacuum module B c L ,c M 0,0 , namely the perturbatively heavy vacuum block and the heavy-light vacuum block. Our computations will take advantage of the saddle-point approximation presented in this subsection. Perturbatively Heavy Vacuum bms 3 -block The perturbatively heavy vacuum bms 3 -block is defined by ∆ = ξ = 0 as well as the external parameters ∆ i and ξ i being infinitesimal and of the same order ǫ. 13 As a simplifying assumption we pairwise identify the external operators such that ∆ 1 = ∆ 2 , ξ 1 = ξ 2 and ∆ 3 = ∆ 4 , ξ 3 = ξ 4 . Analytic Behaviour As a consequence of setting ξ = ∆ = 0 the quantities defined in equations (3.2.3) will have negative expressions under a square root; we remedy this issue by analytically continuing 12 Generically there exits more than one stationary point. However, to keep the formula simple, we assume that there is only a single stationary point (w,w). 13 We set ∆ = ξ = 0 by choosing∆ =ξ = 0 before taking µ → ∞. these quantities, resulting in the coefficients A k = − c M 48 (k − 1) ≡ A + k , B k = − c L 2 √ 48c M (k − 1) ≡ B + k , (4.3.1a) A k = − c M 48 (k + 1) ≡ A − k ,B k = − c L 2 √ 48c M (k + 1) ≡ B − k , (4.3.1b) where we have chosen the branch √ −1 = +i. The absence of factors of i in the above expressions means that the bms 3 generators l n and m n , and by extension the differential equations (4.1.4), are unaffected by the complex conjugation defined with the overline operation; hence the relation between the level-two wave functions (3.3.14) is invalidated. This situation was alluded to in footnote 7. Under the current circumstances we have two independent wave functions which we introduce in a way that is consistent with equation (3.3.21), i.e. ψ ∆ 1 ,ξ 1 ;0,0 (t, x, 0, 0; v) ≡ ψ + ∆ 1 ,ξ 1 ;0,0 (t, x, 0, 0; v) , (4.3.2a) ψ ∆ 3 ,ξ 3 ;0,0 (1, 0, 0, 0; v) ≡ ψ − ∆ 3 ,ξ 3 ;0,0 (1, 0, 0, 0;v) , (4.3.2b) where the superscripts of the wave functions correlate with the superscripts of the coefficients (4.3.1) in their equations. 14 Subsequently, since the n = 0 equations are independent of the coefficients (4.3.1), we use the partial solution (3.3.22) to immediately deduce The two sets of differential equations for S + and S − must be considered separately. ψ + ∆ 1 ,ξ 1 ;0,0 (t, x, 0, 0; v) = t −2∆ 1 e −2ξ 1 x t F + (η, ν) , (4.3.3a) ψ − ∆ 3 ,ξ 3 ;0,0 (1, 0, 0, 0;v) = F − (η,ν) , Solving the Differential Equations The analysis leading up to the set of differential equations (4.1.4) is general enough that the expressions may serve as equations for the wave functions (4.3.2) if we make the appropriate substitutions. For ψ + ∆ 1 ,ξ 1 ;0,0 (t, x, 0, 0; v) we get 0 = ∞ n=1 n σ n ∂ σ k+n S + − ∂ σn S + + κ n ∂ κ k+n S + − ∂ κn S + − 1 4 k−1 n=1 ∂ σn S + ∂ κ k−n S + +Ã + k ∂ σ k S + +B + k ∂ κ k S + −∆ 1 (k − 1) , (4.3.5a) 0 = ∞ n=1 nσ n ∂ κ k+n S + − ∂ κn S + − 1 8 k−1 n=1 ∂ κ k−n S + ∂ κn S + +Ã + k ∂ κ k S + −ξ 1 (k − 1) ; (4.3.5b) 14 Although we will not use it here, the analytic continuation motivates an analogous notation for the bms3 generators, i.e. ln ≡ l + n , mn ≡ m + n andln ≡ l − n ,mn ≡ m − n . and for ψ − ∆ 3 ,ξ 3 ;∆,ξ (1, 0, 0, 0;v) we get 0 = ∞ n=1 n σ n ∂σ k+n S − − ∂σ n S − +κ n ∂κ k+n S − − ∂κ n S − − 1 4 k−1 n=1 ∂σ n S − ∂κ k−n S − +Ã − k ∂σ k S − +B − k ∂κ k S − −∆ 3 (k − 1) , (4.3.6a) 0 = ∞ n=1 nσ n ∂κ k+n S − − ∂κ n S − − 1 8 k−1 n=1 ∂κ k−n S − ∂κ n S − +Ã − k ∂κ k S − −ξ 3 (k − 1) , (4.3.6b) where the tilde references the dependence on fixed quantities, as in equations (4.1.5). S + and S − may be treated as expansions in the infinitesimal parameters∆ 1 ,ξ 1 and ∆ 3 ,ξ 3 , respectively. We choose S + and S − to be of leading order ǫ. 15 The above set of differential equations has the leading order ǫ; hence the terms quadratic in derivatives of S ± are sub-leading of order ǫ 2 and may be dropped. For the expansions of S ± which we are assuming, the stationary point coordinates (4.2.3) have the leading order ǫ. This means that products of oscillator variables with S ± give rise to terms which are sub-leading when evaluated at the stationary point -such terms will be dropped, too. This reasoning leaves us with the sets of linear differential equations 0 =Ã + k ∂ σ k S + +B + k ∂ κ k S + −∆ 1 (k − 1) + O ǫ 2 , (4.3.7a) 0 =Ã + k ∂ κ k S + −ξ 1 (k − 1) + O ǫ 2 ; (4.3.7b) as well as 0 =Ã − k ∂σ k S − +B − k ∂κ k S − −∆ 3 (k − 1) + O ǫ 2 , (4.3.8a) 0 =Ã − k ∂κ k S − −ξ 3 (k − 1) + O ǫ 2 , (4.3.8b) where the omitted terms are indicated by their order at the stationary point. Plugging the relevant expressions into the differential equations (4.3.7b) and (4.3.8b), and integrating, we get S + (σ, κ) = − 48 c Mξ 1 ∞ n=1 κ n + f (σ) + O ǫ 3 , (4.3.9a) S − (σ,κ) = − 48 c Mξ 3 ∞ n=1 n − 1 n + 1κ n + g(σ) + O ǫ 3 , (4.3.9b) where f (σ) and g(σ) arise as constants of integration. The functions f (σ) and g(σ) are determined by integrating the equations (4.3.7a) and (4.3.8a) after the insertion of the above 15 In principle the leading terms of S + (σ, κ) and S − (σ,κ) may be constant functions of order one. Such constant functions may be dropped since they only contribute to the bms3-block as inconsequential multiplicative factors. results; we get f (σ) = − 48 c M ∆ 1 −ξ 1c L 2c M ∞ n=1 σ n + O ǫ 3 , (4.3.10a) g(σ) = − 48 c M ∆ 3 −ξ 3c L 2c M ∞ n=1 n − 1 n + 1σ n + O ǫ 3 . (4.3.10b) Using the variable transformations (3.3.23) and the rescalings (4.1.1), the leading order solutions for S ± read S + (t, x, 0, 0; v) ≈ − 48 c M   ∆ 1 −ξ 1c L 2c M ∞ n=1 t n v (1) n +ξ 1 ∞ n=1 nt n−1 xv (1) n + t n v (2) n   , (4.3.11a) S − (1, 0, 0, 0;v) ≈ − 48 c M   ∆ 3 −ξ 3c L 2c M ∞ n=1 n − 1 n + 1v (1) n +ξ 3 ∞ n=1 n − 1 n + 1v (2) n   . (4.3.11b) Note that our solutions for S + and S − are true up to order ǫ 2 when evaluated at the saddle point. Implementing the Saddle-point Approximation Evaluating the bms 3 -block according to the formula (4.2.4) requires us to determine I(t, x; w,w) at the stationary point (w,w). Appropriately substituting the solutions (4.3.11) into the formulae (4.2.3) we find that the coordinates of the stationary point are w (1) m = 1 4m ∂S − ∂v (2) m = − 1 4m 48 c M m − 1 m + 1ξ 3 , (4.3.12a) w (2) m = 1 4m ∂S − ∂v (1) m = − 1 4m 48 c M m − 1 m + 1 ∆ 3 −ξ 3c L 2c M , (4.3.12b) w (1) m = 1 4m ∂S + ∂v (2) m = − 1 4m 48 c Mξ 1 t m , (4.3.12c) w (2) m = 1 4m ∂S + ∂v (1) m = − 1 4m 48 c M t m ∆ 1 −ξ 1c L 2c M +ξ 1 m x t . (4.3.12d) It is practical to consider the terms of I(t, x; w,w), as defined in equation (4.2.2), separately. Plugging the values (4.3.12) into the sum of oscillator variables yields n − 1 n(n + 1) t n = t 2 6 F 2 1 (2, 2; 4; t) ≡ 1 6 F(t) , (4.3.14) −4 ∞ n=1 n w (1) nw (2) n + w (2) nw (1) n = − 2 c M ∆ 3ξ1 +∆ 1ξ3 −ξ 3ξ1c L c M F(t) +ξ 3ξ1 x∂ t F(t) , where F 2 1 (2, 2; 4; t) is a hypergeometric function. The functions S + (t, x; v) and S − (v) coincide at the stationary point, i.e. evaluating them at the coordinates (4.3.12) gives the same result for both, S + (t, x; w) = S − (w) = 2 c M ∆ 3ξ1 +∆ 1ξ3 −ξ 3ξ1c L c M F(t) +ξ 3ξ1 x∂ t F(t) ,(4. 3.15) which follows from the same identifications as for the expression (4.3.13). Using the above results in the definition I(t, x; w,w) = −4 ∞ n=1 n w (1) nw (2) n + w (2) nw (1) n + S + (t, x; w) + S − (w) ,(4.B ∆ 1,3 ,ξ 1,3 ;0,0 (t, x) ≈ t −2∆ 1 exp   −2ξ 1 x t + 2 c M ∆ 3 ξ 1 + ∆ 1 ξ 3 − ξ 3 ξ 1 c L c M F(t) + ξ 3 ξ 1 x∂ t F(t)   , (4.3.17) where factors of µ have been absorbed into the non-tilde quantities. Heavy-light Vacuum bms 3 -block The heavy-light vacuum bms 3 -block has two heavy and two light external operators; the light operators have infinitesimal scaling dimensions and rapidities of order ǫ, while those of the heavy operators are of order one. 16 Continuing with the pairings used in the previous subsection, we assign ∆ 1 , ξ 1 to the light operators and ∆ 3 , ξ 3 to the heavy operators; hence, in the semi-classical limit∆ 1 ,ξ 1 are of infinitesimal order ǫ while∆ 3 ,ξ 3 are of order one. The vacuum block dictates that ∆ = ξ = 0. Analysing the Differential Equations The prescription for determining the exponents of the semi-classical wave-function ansätze (4.3.4) is analogous to the procedure in subsection 4.3.2. In fact, the properties of the equations containing∆ 1 andξ 1 are unchanged and hence the solution for S + is identical to the expression (4.3.11a); thus also the coordinates of the stationary point (4.3.12c) and (4.3.12d) 16 Other authors define light operators to scale as order (cM) 0 in the semi-classical limit cM → ∞. Hence a perhaps more correct denomination would be to call the operators under consideration 'perturbatively heavy'. are the same and of infinitesimal order. However, since∆ 3 andξ 3 are of finite order the discussion below equations (4.3.6) no longer applies to S − ; we must thus re-consider its differential equations and solution. We motivate an appropriate ansatz for S − (σ,κ) by analysing the lowest non-trivial infinitesimal order of I(t, x; w,w), as given in (4.3.16). As mentioned above, the stationarypoint coordinatesw (i) m are of infinitesimal order ǫ; the coordinates w (i) m are of order one, hence the contribution of the measure to I(t, x; w,w) is of order ǫ. Similar argumentation leads to the conclusion that S + (t, x; w) is of order ǫ, too. Hence, any terms of higher order than ǫ which arise from S − (w) are sub-leading. Since the variablesκ n andσ n are of order ǫ when evaluated at the stationary point, we conclude that S − (σ,κ) should be at most linear in the variables and thus a suitable ansatz is 17 S − (σ,κ) = ∞ n=1 C nσn + ∞ n=1 D nκn , (4.4.1) where the coefficients C n and D n depend onξ 3 ,∆ 3 . Plugging the above ansatz into the differential equations (4.3.5) yields 0 = ∞ n=1 n σ n (C k+n − C n ) +κ n (D k+n − D n ) − 1 4 k−1 n=1 C n D k−n +Ã − k C k +B − k D k −∆ 3 (k − 1) , (4.4.2a) 0 = ∞ n=1 nσ n (D k+n − D n ) − 1 8 k−1 n=1 D k−n D n +Ã − k D k −ξ 3 (k − 1) . (4.4.2b) The leading terms of the above equations are of order one. Invoking the perspective of the saddle-point approximation, the variablesσ n ,κ n give rise to sub-leading terms of order ǫ; thus terms containing these variables may be dropped. The remaining set of equations takes the form of recurrence relations, i.e. 0 = − 1 4 k−1 n=1 C n D k−n − (k + 1) c M 48 C k − (k + 1)c L 2 √ 48c M D k −∆ 3 (k − 1) + O ǫ , (4.4.3a) 0 = − 1 8 k−1 n=1 D k−n D n − (k + 1) c M 48 D k −ξ 3 (k − 1) + O ǫ , (4.4.3b) where the omitted terms are indicated by their order at the stationary point (w,w), and we used the definitions (4.3.1b) forÃ − k andB − k . Note that for k = 1 the above recurrence relations fix the initial values C 1 = D 1 = 0. The recurrence relations (4.4.3) may be turned into differential equations by using the method of generating functions. To this end, we make the ansätze C(τ ) = ∞ n=1 C n τ n and D(τ ) = ∞ n=1 D n τ n . (4.4.4) As a consequence of the above ansätze we have C(0) = D(0) = 0 and the constraints C 1 = D 1 = 0 become the boundary conditions C ′ (0) = 0 = D ′ (0) = 0. Using standard techniques, the recurrence relations (4.4.3) are transformed into two coupled differential equations for C(τ ) and D(τ ), ∂ τ τ · C(τ ) = − 3 c M C(τ )D(τ ) − 48 c M∆ 3 τ 2 (1 − τ ) 2 −c L 2c M ∂ τ τ · D(τ ) + O ǫ , (4.4.5a) ∂ τ τ · D(τ ) = − 1 8 48 c M D(τ ) 2 − 48 c Mξ 3 τ 2 (1 − τ ) 2 + O ǫ . (4.4.5b) We present the details of the solution procedure of the above differential equations in appendix D; the solutions, to leading order in infinitesimal quantities, read C(τ ) ≈ − τ (1 − τ ) β 3 −1 24∆ 3 +c L (β 2 3 − 1) √ 3c M (1 − τ ) β 3 − 1 2 ln(1 − τ ) − 12∆ 3 τ √ 3c M β 3 (1 − τ ) β 3 + 1 (τ − 1) (1 − τ ) β 3 − 1 −c L 2 √ 3c M (τ − 1) τ − 2 − τ β 3 (1 − τ ) β 3 + 1 (1 − τ ) β 3 − 1 , (4.4.6a) D(τ ) ≈ − c M 3 1 1 − τ 2 − τ + β 3 τ 1 − 2 1 − (1 − τ ) β 3 , (4.4.6b) with β 3 = 1 −ξ 3 24 c M . (4.4.7) Although the ansatz (4.4.1) is expressed in terms of the coefficients C m and D m , which can be extracted from the functions C(τ ) and D(τ ), we may also continue using the solutions as they are given in (4.4.6). Implementing the Saddle-point Approximation Returning to the rescaled oscillator variables (4.1.1) via (3.3.23), and remembering the solution for S + given in equation (4.3.11a), we have S + (t, x, 0, 0; v) ≈ − 48 c M   ∆ 1 −ξ 1c L 2c M ∞ n=1 t n v (1) n +ξ 1 ∞ n=1 nt n−1 xv (1) n + t n v (2) n   , (4.4.8a) S − (1, 0, 0, 0;v) = ∞ n=2 C nv (1) n + D nv (2) n . (4.4.8b) Note that the sum in S − starts at n = 2. Being cognisant of the saddle-point approximation of the bms 3 -block, given in equation (4.2.4), the next step is to evaluate I(t, x; w,w). Making use of the expressions (4.4.8), the coordinates of the stationary point read w (1) m = 1 4m D m , w (2) m = 1 4m C m , (4.4.9a) w (1) m = − 1 4m 48 c Mξ 1 t m ,w (2) m = − 1 4m 48 c M t m ∆ 1 −ξ 1c L 2c M + mξ 1 x t . (4.4.9b) Plugging the above values into equation (4.3.16) yields hence where factors of µ are absorbed by the non-tilde quantities. Choosing ∆ 3 , ξ 3 to be infinitesimal; expanding β 3 to first order in ξ 3 ; keeping terms up to second order in infinitesimal quantities, transforms the above result into the perturbatively heavy vacuum bms 3 -block (4.3.17), modulo constant prefactors. 18 Furthermore, (4.4.13) generalises the perturbatively heavy vacuum bms 3 -block presented in [50]. I(t, x; w,w) = − 48 c M ∞ m=2 1 4m t m ξ 1 C m +∆ 1 D m −ξ 1c L 2c M D m + mξ 1 x t D m .I(t, x; w,w) = − 1 4 48 c M ξ 1 t 0 dτ C(τ ) τ +∆ 1 t 0 dτ D(τ ) τ −ξ 1c L 2c M t 0 dτ D(τ ) τ +ξ 1 x t D(t) ,B HHLL ∆ 1,3 ,ξ 1,3 ;0,0 (t, x) ≈ (1 − t) β 3 −1 (1 − (1 − t) β 3 ) 2 ∆ 1 exp −ξ 1 x (1 − t) β 3 (1 + β 3 ) + β 3 − 1 (1 − t)(1 − (1 − t) β 3 ) × exp   12ξ 1 c M β 3 ξ 3 c L c M − ∆ 3 1 + (1 − t) β 3 1 − (1 − t) β 3 ln(1 − t)   , Summary and Outlook In this paper we presented an oscillator construction of highest-weight bms 3 modules. We used this construction to obtain general expressions for two-and three-point correlation functions and the bms 3 -block in terms of wave functions, where the wave functions must satisfy an infinite set of partial differential equations. We found an expression for the level-one wave function and its dual -we could not, however, obtain closed-form solutions for the level-two wave functions; nevertheless, we proved that they are uniquely determined in generic cases. We showed the strengths of the oscillator construction by considering the semi-classical limit defined by c M → ∞ while keeping all fractions of the form ∆/c M , ∆ i /c M ; ξ/c M , ξ i /c M and c L /c M fixed. Our oscillator construction allowed us to prove the exponentiation of bms 3 -blocks in this limit. Finally, we explicitly determined the vacuum bms 3 -block in the perturbatively heavy and the heavy-light regimes; the heavy-light vacuum bms 3 -block is in agreement with previous results stemming from holographic computations involving probe particles propagating in flat-space cosmologies within Einstein gravity [50]. It would be interesting to check our predictions against holographic calculations in the semi-classical limit of chiral gravity theories with c L = 0. The work presented in this paper allows for a number of possible directions which may be explored in the future. In our proof of uniqueness of the level-two wave functions we could identify a discrete set of values for c L , c M , ∆ and ξ which must be excluded. Analysing the analogous proof for the Virasoro algebra [53], we notice that the set of excluded parameters in this case include the known values of Virasoro minimal models. Following this reasoning, we speculate that this method may be of use for identifying new bms 3 minimal models. It would be interesting to study bms 3 -blocks beyond the vacuum module, as well as blocks in which all external operators are heavy. In addition, 1/c M corrections to the semi-classical limit are of interest since they give rise to quantum contributions to the bulk gravitational theory. Another avenue worth exploring is the computation of torus bms 3 -blocks in the semi-classical limit; we suspect that the oscillator construction would be beneficial for this task. The study of one-point functions on the torus was recently initiated in [48,49]. We expect that modularity gives similar stringent constraints on the thermodynamics of the dual gravitational theory as in the case of asymptotically-AdS spacetimes. Furthermore, it would be valuable to have access to an oscillator construction for induced representations of bms 3 , since these representations are unitary [29,56]. As a first step in this direction one could approach an oscillator construction of induced isl 2 modules. In view of the recent results of [57][58][59], it would also be interesting to obtain an oscillator construction of bms 4 algebras and investigate its relation to conformal field theories on the celestial sphere. In conclusion, all of the above tasks are of importance in view of a putative holographic duality involving asymptotically flat spacetimes and BMS-invariant field theories. We hope to return to some of these aspects in future work. A Realising the Oscillatory Construction of bms 3 In section 3 of the main text we state that the oscillator construction of the highest-weight representation of bms 3 may be reached by taking a non-relativistic limit of a two-dimensional linear-dilaton like theory; in this section we detail the procedure. 19 Generalising the linear dilaton theory, the components of the stress-tensor T ≡ T zz and T ≡ Tzz, where the bar denotes quantities belonging to the anti-holomorphic sector of the two-dimensional conformal algebra, may be expressed as We will take the non-relativistic limit at the level of the generators L vir m andL vir m . A.1 Non-relativistic Limit In order to implement the non-relativistic limit we introduce a pair of new oscillators Furthermore, we introduce the new constants β m = 1 √ ǫ (α m − iᾱ m ) , γ m = √ ǫ (α m + iᾱ m ) .W L = 1 2 √ ǫ V − iV , W M = √ ǫ 2 V + iV .a † 0 = a 0 + 2i(W M + W L ) ,â † 0 =â 0 − 2i(W M − W L ) . (A.1.9) The eigenvalues of the oscillators a m andâ m are in general complex: the real parts of the eigenvalues are arbitrary and parametrised by λ 1 and λ 2 , respectively; the imaginary parts are fixed by the above expressions. From the transformations (A.1.9) we read off the eigenvalues a 0 ≡ √ 2λ 1 + i √ 2µ 1 ,â 0 ≡ √ 2λ 2 + i √ 2µ 2 , (A.1.10) where the factors of √ 2 are due to a choice of normalisation, and we defined µ 1 = −(W M + W L )/ √ 2 and µ 2 = (W M − W L )/ √ 2. A.2 Differential Operators and Measure We set up the oscillator construction of the bms 3 generators (A.1.5) by assigning differential operators to the modes a m andâ m ; we do so in the following way: a m = i 2 √ 2 ∂ v (1) m + ∂ v (2) m , a −m = −im √ 2 v (1) m + v (2) m , (A.2.1a) a m = i 2 √ 2 ∂ v (1) m − ∂ v (2) m ,â −m = im √ 2 v (1) m − v (2) m . (A.2.1b) The adjoint transformation properties a † m = a −m andâ † m =â −m are implied by the Hermitian product (f, g) = C ∞ [d 2 v] f (v)g(v) , (A.2.2) with the measure [d 2 v] = ∞ n=1 16n 2 exp −4n v (1) nv (2) n + v (2) nv (1) n d 2 v (1) n d 2 v (2) n , (A.2.3) where the normalisation is such that (1, 1) = 1. The exponential factor of the measure is made plausible by introducing the linear combinations q m = v (1) m + v (2) m and j m = v (1) m − v (2) m . Then a −m only contains q m and a −m only contains ∂ qm ; whileâ −m andâ m contain j m and its derivative, respectively. The measure of the oscillator construction of the Virasoro algebra, given by (2.2.8), corresponds to an assignment akin to (A.2.1a) [52]; hence, replacing u n with q n , the transformation a † m = a −m requires a measure with the exponential factor exp −2n v (1) n + v (2) n v (1) n +v (2) n . (A.2.4) Taking the overall sign difference between a −m andâ −m into consideration, similar reasoning motivates the exponential factor that allows forâ † m =â −m to be of the form 2) for a general function f (v,v) is made possible by the analytic continuation associated with the mapping n → in. Such an analytic continuation allows us to exploit the behaviour of the complex delta distribution and its derivatives, defined by exp 2n v (1) n − v (2) n v (1) n −v (2)C dv dv f (v,v) ∂ a v ∂ b v δ(v,v) = (−1) a+b ∂ a v ∂ b v f (v,v) v = 0,v = 0 , (A.2.6) for any pair of non-negative integers a, b ∈ N 0 . Monomials of complex variables integrate to derivatives of the complex delta distribution according to C dw dww a w b e iκ(vw+vw) = − i κ a+b+2 ∂ a v ∂ b v δ(v,v) , (A.2.7) for any κ ∈ R and v ∈ C; it is then straightforward to determine the normalisation of the Hermitian product, as well as the orthogonality relation for monomials of the oscillator variables v (1) m a v (2) m b , v (1) m c v (2) m d = a!b! (4m) a+b δ a,d δ b,c . (A.2.8) A.3 Generators To find the oscillator construction of bms 3 we take care of the normal ordering in the bms 3 generators (A.1.8); plug in the differential representations of the oscillators, as given in equations (A.2.1a) and (A.2.1b); use the definitions (A.1.10), and then simplify. Denoting the generators in the oscillator construction by lowercase, we arrive at l 0 = ∆ + ∞ n=1 n v (1) n ∂ v (1) n + v (2) n ∂ v (2) n , (A.3.1a) l k = ∞ n=1 n v (1) n ∂ v (1) k+n + v (2) n ∂ v (2) k+n − 1 8 k−1 n=1 ∂ v (1) n ∂ v (2) k−n + ∂ v (2) n ∂ v (1) k−n + i 2 (λ 1 − λ 2 ) − ik(µ 1 − µ 2 ) ∂ v (1) k + i 2 (λ 1 + λ 2 ) − ik(µ 1 + µ 2 ) ∂ v (2) k , (A.3.1b) l −k = ∞ n=1 (k + n) v (1) k+n ∂ v (1) n + v (2) k+n ∂ v (2) n − 2 k−1 n=1 n(k − n) v (1) n v (2) k−n + v (2) n v (1) k−n − i2k (λ 1 + λ 2 ) + ik(µ 1 + µ 2 ) v (1) k − i2k (λ 1 − λ 2 ) + ik(µ 1 − µ 2 ) v (2) k , (A.3.1c) as well as m 0 = ξ + ∞ n=1 nv (1) n ∂ v (2) n , (A.3.2a) m k = ∞ n=1 nv (1) n ∂ v (2) k+n − 1 8 k−1 n=1 ∂ v (2) k−n ∂ v (2) n + i 2 (λ 1 − λ 2 ) − ik(µ 1 − µ 2 ) ∂ v (2) k , (A.3.2b) m −k = ∞ n=1 (k + n)v (1) k+n ∂ v (2) n − 2 k−1 n=1 n(k − n)v (1) k−n v (1) n − i2k (λ 1 − λ 2 ) + ik(µ 1 − µ 2 ) v (1) k . (A.3.2c) In (A.3.1a) and (A.3.2a) we have identified the scaling dimension ∆ and rapidity ξ as ∆ ≡ λ 2 1 − λ 2 2 + µ 2 1 − µ 2 2 , (A.3.3a) ξ ≡ 1 2 (λ 1 − λ 2 ) 2 + (µ 1 − µ 2 ) 2 . (A.3.3b) Furthermore, the above generators satisfy the bms 3 algebra with central charges c L = 2 + 24 µ 2 1 − µ 2 2 and c M = 12 (µ 1 − µ 2 ) 2 . Finally, we will make use of the abbreviations A k = i 2 λ 1 − λ 2 − ik(µ 1 − µ 2 ) , B k = i 2 λ 1 + λ 2 − ik(µ 1 + µ 2 ) , (A.3.4a) A k = − i 2 λ 1 − λ 2 + ik(µ 1 − µ 2 ) ,B k = − i 2 λ 1 + λ 2 + ik(µ 1 + µ 2 ) . (A.3.4b) Using the definitions (A.3.3) together with the values of the central charges, the coefficients defined above may be expressed in terms of familiar quantities as A k = ∓ i 2 2ξ − c M 12 ± k c M 48 , B k = ±i c L − 2 − 24∆ 48 2ξ − c M 12 ± k c L − 2 48 c M 12 , (A.3.5a) A k = ± i 2 2ξ − c M 12 ± k c M 48 ,B k = ∓i c L − 2 − 24∆ 48 2ξ − c M 12 ± k c L − 2 48 c M 12 , (A.3.5b) where the sign choices for the k-dependent terms are correlated with each other, and the sign choices for the k-independent terms are correlated with each other. B Proof of Unique Solution In this section we prove that the level-two bms 3 wave functions admit unique solutions; we show it for finite central charge c M and comment on the semi-classical limit c M → ∞. The discussion for the semi-classical limit may be considered to be part of the proof of exponentiation presented in subsection 4.2. Parts of the analysis below follow methods presented in [53]. B.1 Proof of Unique Solution for F (η, ν) We focus on ψ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0; v), similar arguments apply to the dual wave function χ ∆ 1,2 ,ξ 1,2 ;∆,ξ (t, x, 0, 0;v). To get a valid wave function for the point configuration under consideration, F (η, ν) in the ansatz (3.3.22) must satisfy the differential equations (3.3.15) for n ≥ 1. 21 In order to express the differential equations in terms of F (η, ν) it is beneficial to write the bms 3 generators l 0 , m 0 and l k , m k , with k ≥ 1, using the variables η m and ν m ; setting t = 1 and x = 0 for simplicity, we get l 0 = ∆ + ∞ n=1 n η n ∂ ηn + ν n ∂ νn , (B.1.1a) l k = ∞ n=1 n η n ∂ η k+n + ν n ∂ ν k+n − 1 4 k−1 n=1 ∂ ηn ∂ ν k−n + A k ∂ η k + B k ∂ ν k , (B.1.1b) m 0 = ξ + ∞ n=1 n η n ∂ νn , (B.1.1c) m k = ∞ n=1 nη n ∂ ν k+n − 1 8 k−1 n=1 ∂ ν k−n ∂ νn + A k ∂ ν k . (B.1.1d) Moreover, to implement the action of L (∆ 1 ,ξ 1 ) n and M (∆ 1 ,ξ 1 ) n we make use of the identities ∂ x F (η, ν) t=1,x=0 = ∞ m=1 m η m ∂ νm F (η, ν) , (B.1.2a) ∂ t F (η, ν) t=1,x=0 = ∞ m=1 m η m ∂ ηm + ν m ∂ νm F (η, ν) , (B.1.2b) which result from applying the chain rule to F (η, ν). Taking all of the above quantities together, the differential equations (3.3.15) with n ≥ 1 give rise to the equations l k F (η, ν) = (l 0 + k∆ 1 − ∆ 2 )F (η, ν) , (B.1.3a) m k F (η, ν) = (m 0 + kξ 1 − ξ 2 )F (η, ν) . (B.1.3b) We now assume that F (η, ν) takes the form of a power series expansion in the variables η m and ν m , F (η, ν) = ∞ n=0 F n , (B.1.4) where each F n is a linear combination of every possible linearly independent monomial of the form η j 1 η j 2 · · · η jp ν k 1 ν k 2 · · · ν kq , with the ordering of variables such that j 1 ≤ j 2 ≤ · · · ≤ j p and k 1 ≤ k 2 ≤ · · · ≤ k q , and the sum of the indices satisfies p i=1 j i + q i=1 k i = n. Hence, the number of terms in F n is given by p 2 (n) = n k=0 p(k)p(n − k) , (B.1.5) where p(k) is the number of partitions of the number k. We choose the normalisation F 0 = 1. We may view F n as an element of a vector space V n which is spanned by the monomials η j 1 η j 2 · · · η jp ν k 1 ν k 2 · · · ν kq . The dimension of the vector space V n is p 2 (n), i.e. equal to the number of basis monomials. Due to the action of the derivative in the differential operators l k and m k we note that l k (V n ) ⊆ V n−k as well as m k (V n ) ⊆ V n−k for k ≤ n. B.1.1 Building a Linear Set of Equations To specify F n , and in turn F (η, ν), we have to know the coefficients in front of the p 2 (n) basis vectors of V n . Our strategy is to show that there are as many coefficients as equations for these coefficients, and that the solution to this set of linear equations is unique. In terms of F n the equations (B.1.3) read, for k ≤ n, l k F n = l 0 F n−k + (k∆ 1 − ∆ 2 )F n−k , (B.1.6a) m k F n = m 0 F n−k + (kξ 1 − ξ 2 )F n−k , (B.1.6b) where we have used that l 0 (V m ) ⊆ V m and m 0 (V m ) ⊆ V m for m ∈ N 0 , and the right-hand side contains F n−k because the two sides must belong to the same vector space V n−k . The form of l 0 gives rise to the eigenvalue equation l 0 v = (∆ + m) v for any element v ∈ V m ; this observation allows us to simplify (B.1.6a) further, arriving at l k F n = β k,n F n−k , (B.1.7a) m k F n = m 0 F n−k + (kξ 1 − ξ 2 )F n−k , (B.1.7b) with β k,n ≡ ∆ + n − k + k∆ 1 − ∆ 2 . Unfortunately, we cannot constrain m 0 F n−k further since m 0 does not give rise to a simple action on the vector space V m . The set of equations (B.1.7) displays some recursive traits which we will use to our advantage; remembering that F 0 = 1, we may reduce the index of F n until the right-hand side only contains numbers. To this end, we evaluate a string of operators l kq · · · l k 2 l k 1 acting on F n by applying equation (B.1.6a) recursively; we obtain l kq · · · l k 1 F n = β kq,n−K+kq · · · β k 2 ,n−k 1 β k 1 ,n F n−K , (B.1.8) provided that K ≡ q i=1 k i is less than n. Since m 0 does not have a simple eigenvalue equation, we cannot evaluate a string of operators m jp · · · m j 2 m j 1 acting on F n−K unless J ≡ p i=1 j i is equal to n − K; in this case we find m jp · · · m j 2 m j 1 F n−K = (m 0 + j p ξ 1 − ξ 1 ) · · · (m 0 + j 2 ξ 1 − ξ 1 )(m 0 + j 1 ξ 1 − ξ 1 )F 0 , (B.1.9) where we used that the generators m j i mutually commute so that each product of m j i acts directly on F κ , for the suitable level κ. We may apply m 0 F 0 = ξ recursively to equation (B.1.9), which yields m jp · · · m j 2 m j 1 F n−K = γ jp · · · γ j 2 γ j 1 , (B.1.10) where γ j = ξ + jξ 1 − ξ 1 , for the case J = n − K. The two equations (B.1.8) and (B.1.10) bring us to the general form of repeated action of generators to be 22 m jp · · · m j 2 m j 1 l kq · · · l k 1 F n = γ jp · · · γ j 2 γ j 1 β kq,n−K+kq · · · β k 2 ,n−k 1 β k 1 ,n (B. 1.11) in the case of p i=1 j i + q i=1 k i = n. B.1.2 Existence and Uniqueness of the Solution The left-hand side of equation (B.1.11) is a linear combination of the p 2 (n) unknown coefficients of the basis monomials. There are p 2 (n) possible strings of ordered operators m jp · · · m j 2 m j 1 l kq · · · l k 1 with J + K = n. Hence, by considering all actions of the form m jp · · · m j 2 m j 1 l kq · · · l k 1 F n we form a linear system of p 2 (n) equations with p 2 (n) unknowns. To prove that this system is solvable it is sufficient to show that the matrix M n , whose entries are given by m j 1 · · · m jp l k 1 · · · l kq (η j ′ 1 · · · η j ′ r ν k ′ 1 · · · ν k ′ s ) and which characterises the system of equations, is invertible except for at most a discrete set of exceptional cases; we will do this by arguing that the determinant of M n is generically non-vanishing. We will continue in a two-step procedure. In the first step we simplify the task by replacing the operators l k by l • k = A k ∂ η k + B k ∂ ν k and m k by m • k = A k ∂ ν k . It is always possible to find a basis such that the simplified matrix M • n has an upper triangular form, with products of A k on its diagonal; we postpone the proof of this statement to the next subsection. The determinant of M • n is the product of its diagonal elements, i.e. products of A k . Using that A k is given by with k ∈ N; this means that there is only a discrete set of values of ξ for which the determinant of M • n vanishes. As the second step we need to consider the matrix M n corresponding to the equations using the full operators l k and m k . A product of the operators l k , m k can be re-expressed in terms of sums of l • k , m • k and l k − l • k , m k − m • k ; where the differences between the full and simplified operators are independent of ξ and c M . Hence, the determinant of M n and the determinant of M • n have features in common: both are polynomials of the same degree in y ≡ 2ξ − c M /12, and they have the same coefficient of the largest power of y. Consequently, the determinant of M n may vanish only for a discrete set of values y, generically without relation to (B.1.12). We have thus shown that the system of equations for the coefficients of F n admits a unique solution; this means that a solution for F (η, ν) is unique, too. B.1.3 Proof of Unique Solution for S(σ, κ) In the semi-classical limit we make the ansatz F (σ, κ) = exp µ 2 S(σ, κ) . The proof of a unique solution for S(σ, κ) goes along the same lines as above. However, there are two differences which we would like to highlight. First, we treat the equations in terms of S(σ, κ), not in terms of F (σ, κ). Hence, the expansion of S(σ, κ) in terms of its variables is reorganised compared to the expansion of F (σ, κ). Second, due to the semi-classical limit we may drop the second derivative terms in l k and m k . Neither of the changes mentioned in this subsection invalidate the arguments above concerning the uniqueness of the wave function. partition just containing n + 1 times the number one, may be obtained from a partition of the number n by raising one of the numbers by one without violating the (decreasing) ordering of the numbers in the partition. For example, the partition (4, 2) of the number 6 can be obtained from the partition (4, 1) of the number 5. By the induction assumption, the matrix M • n is upper triangular and hence also the block corresponding to the partition n. Thus, keeping the ordering of the operators and monomials (but raising one of the indices to match the partition of n + 1) the block of the matrix M • n+1 corresponding to that partition of n + 1 is also upper triangular. By this construction, the diagonal of M • n+1 contains products of A k only. We are left with the trivial partition 1 + · · · + 1 of n + 1. In this case we have to find a suitable ordering of the operators (m • 1 ) p (l • 1 ) q with p + q = n + 1 and the monomials (η 1 ) r (ν 1 ) s with r + s = n + 1. Since m • 1 is a derivative with respect to ν 1 we have (m • 1 ) p (l • 1 ) q (η 1 ) r (ν 1 ) s = 0 for p > s . (B.2.1) Hence ordering the operators (m • 1 ) p (l • 1 ) n+1−p with increasing p = 0, . . . , n + 1 and the monomials (η 1 ) n+1−r (ν 1 ) r with increasing r = 0, . . . , n + 1 we obtain an upper triangular matrix with powers of A 1 on its diagonal. This completes the proof by induction. Hence we conclude that the matrix M • n has an upper-triangular form for a specific ordering of the operators and monomials, with products of A k on its diagonal. C Determining the Level-one Wave Function In this appendix we solve the differential equations (3.3.6) for the level-one wave function. The result for the dual wave function follows from applying the same procedure detailed below to the differential equations (3.3.7). We first turn our attention to equation ( − t 2 ∂ t 2 Ψ − x 2 ∂ x 2 Ψ + ∞ n=1 n v (1) n ∂ v (1) n + v (2) n ∂ v (2) n Ψ + (∆ − ∆ 2 ) Ψ = 0 , (C.0.2) where we have abbreviated Ψ ≡ ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) for simplicity. The above differential where h ( · · · ) is again an unknown function arising as a constant of integration. With the above expression the wave function takes the form Imposing the boundary condition C ′ (0) = 0 fixes c 2 = −c L √ 3c M . The solution for C(τ ) is thus C(τ ) = − τ (1 − τ ) β 3 −1 24∆ 3 +c L (β 2 3 − 1) √ 3c M (1 − τ ) β 3 − 1 2 ln(1 − τ ) − 12∆ 3 τ √ 3c M β 3 (1 − τ ) β 3 + 1 (τ − 1) (1 − τ ) β 3 − 1 −c L 2 √ 3c M (τ − 1) τ − 2 − τ β 3 (1 − τ ) β 3 + 1 (1 − τ ) β 3 − 1 . (D.2.5) In the same way as for D(τ ), evaluating the heavy-light vacuum bms 3 -block requires us to determine the integral t 0 dτ C(τ ) τ . This integral can be solved using partial integration and the partial fraction expansion 1 + (1 + τ ) β 3 (1 − τ )(1 − (1 − τ ) β 3 = 1 (1 − τ ) + 2(1 + τ ) β 3 −1 1 − (1 − τ ) β 3 , (D.2.6) which simplifies the integrand. The integral is then given by t 0 dτ C(τ ) τ =c L √ 3c M ln 1 − (1 − t) β 3 − ln(t) + 24 √ 3c M β 2 3 ∆ 3 −c L c Mξ 3 + 1 2 √ 3c M β 3    24∆ 3 +c L (β 3 − 1) + (1 − t) β 3 24∆ 3 +c L (β 3 − 1)(1 + 2β 3 ) 1 − (1 − t) β 3    ln(1 − t) . > 0 . 0Analogously to the construction of a Verma module, a bms 3 module B c L ,c M ∆,ξ 1 ≥ . . . ≥ m s ≥ 1 and n 1 ≥ . . . ≥ n l ≥ 1. 5 The Hermitian product q\|p for \|p , \|q ∈ B c L ,c M ∆,ξ is uniquely defined by imposing the adjoint relations L † n = L −n and M † n = M −n . The Hermitian product of two basis vectors is thus given by turn motivates the introduction of the variables σ n and κ n defined by η n = µ σ n , ν n = µ κ n . comment on the proof of uniqueness for the function S(σ, κ) in the exponential ansatz.Using (4.1.3) together with the expressions for l k given in (3.2.1b), m k given in (3.2.2b), and L k and M k given in(3.3.3), the differential equations (3.3.15a) and (3.3.15b) for n ≥ 1 read, respectively, 1.3) in the n = 0 solution for the wave function (3.3.22); transforming the result using the wave function relation (3.3.14); returning back to the oscillator variables v remembering the rescaling (4.1.1), the expression for the bms 3 -block given in (3.3.21) reads F + (σ, κ) = exp µ 2 S + (σ, κ) , F − (σ,κ) = exp µ 2 S − (σ,κ) . definitions (4.4.4) we may express the coefficients C m and D m as the integrals last term follows directly from (4.4.4). After evaluating the integrals, for which the expressions are presented in equations (D.2.7) and (D.1.8), and using (4.2.4), we arrive at vir m ,L vir m are generators of the Virasoro algebra, and V ,V are complex constants which are not related by complex conjugation. 20 We impose L vir m † = L vir −m and L vir m † =L vir −m . The oscillators α m ,ᾱ m satisfy the commutation relations [α m , α n ] = [ᾱ m ,ᾱ n ] = m δ m+n,0 , [α m ,ᾱ n ] = 0 . (A.0.3) from the commutation relations (A.0.3) that the oscillators defined above satisfy [β m , γ n ] = 2mδ m+n,0 , [β m , β n ] = [γ m , γ n ] = 0 . (A.1.2) :::: adjoint properties β † m = β −m and γ † m = γ −m means that the Virasoro oscillators transform like α † m =α −m andᾱ † m = −ᾱ −m for m = 0. Preserving the adjoint property of the Virasoro generators in turn requiresV * = V andV * = −V , together with α † 0 = α 0 + 2iV andᾱ † 0 = −ᾱ 0 − 2iV .The bms 3 generators are reached by taking a non-relativistic limit of the relativistic Virasoro generators. In practice this is achieved by taking the linear combinations[54,60] vir m andL vir m in terms of the oscillators (A.1.1), applying the linear combinations above, and taking ǫ → 0β m−n γ n + γ m−n β n : +i(m + 1) (W L γ m + W M β m ) , γ m−n γ n : +i(m + 1)W M γ m . (A.1.5b) The generators (A.1.5) satisfy the bms 3 algebra (3.1.1) with central charges c L = 2+48W L W M and c M = 24W 2 M . Enforcing that L † m = L −m and M † m = M −m requires, in addition to the adjoint properties of β m and γ m for m = 0, that β † 0 = β 0 + 4iW L and γ † 0 = γ 0 + 4iW M . It is advantageous to work with more familiar commutation relations than those in (A.1.2). We introduce a third set of oscillators as m , a n ] = mδ m+n,0 , [â m ,â n ] = −mδ m+n,0 , [a m ,â n ] = 0 . (A.1.7)In terms of the above definitions the bms 3 generators (a m−n a n −â m−nân : +i(m + 1) (W M + W L ) a m + (W M − W L )â m , a m−n a n − (a m−nân +â m−n a n ) +â m−nân : +i(m + 1)W M (a m −â m ) .(A.1.8b) Preserving the adjoint properties L † m = L −m and M † m = M −m requires a † m = a −m , a † m =â −m for m = 0, as well as products of the factors (A.2.4) and (A.2.5) we arrive at the exponential of the measure (A.2.3). Evaluating the integral in equation (A.2. ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , x 2 ; v) = 0 . (C.0.1)Plugging in the generators (3.2.1a) and (3.3.3a) the above equation reads equation may be solved by applying the method of characteristics. The Lagrange-Charpit equations corresponding to (C. term on the first line of the above solution contributes to the heavy-light vacuum bms 3 -block (4.4.13) are of no consequence and may be neglected. 3.8b) This can be proven by induction, remembering that for k > 1 we may express the generators l k+1 and L k+1 in terms of commutators [l1, l k ] and [L1, L k ].10 To keep the notation compact we drop degenerate subscripts.9 Highest-weight representations of bms3 are also known as gca 2 . The overline operation does not affect the holomorphic properties of the coordinates zi, even though they are complex. To flesh out the notation, we could equivalently write \|v = \|v(1) , v(2) . We drop the index to indicate the infinite collection. As before, we may ignore any constant terms in the ansatz for S − since those only contribute as multiplicative factors to the bms3-block. Seeing this requires the use of the identity F(t) = 6 t−2 t ln(1 − t) − 2 . The analogous procedure for the Virasoro case is reviewed in appendix A.2.1 of[52].20 Note that there is no action principle associated to this kind of theory, which is why we use the wording 'dilaton-like'. If we consider a general point configuration with t2 and x2 being non-zero, the n = −1 equation may be used to reinstate the dependence on these coordinates. Note that the ordering of mj p · · · mj 2 mj 1 l kq · · · l k 1 is crucial in equation (B.1.11). In particular, we are not able to easily evaluate the reversed ordering of l and m operators, i.e. l kq · · · l k 1 mj p · · · mj 2 mj 1 . B.2 Upper-triangularity of M • nWe now analyse the evaluation of the string of operators m • j 1 · · · m • jp l • k 1 · · · l • kq on monomials of the form η j ′ 1 · · · η j ′ r ν k ′ 1 · · · ν k ′ s . Both sets of indices are ordered and add up to n, i.e. both sets are partitions of n. We will prove that there exists an ordering of the partitions of the index variables of the operators and monomials such that the matrix M • n , with entries m • j 1 · · · m • jp l • k 1 · · · l • kq (η j ′ 1 · · · η j ′ r ν k ′ 1 · · · ν k ′ s ), is upper triangular. We will prove our claim by induction; we start with the case n = 1. In this case the matrix readswhich illustrates our choice of basis (l • 1 , m • 1 ) and (η 1 , ν 1 ), where partitions of generator indices are associated to rows, whereas partitions of oscillator indices are associated to columns of the respective matrix M • n . Since m • 1 η 1 = 0 we find that M • 1 is upper triangular with A 1 on the diagonal.Assuming the matrix M • n to be upper triangular and having products of A k as entries on its diagonal, our task is to show that M • n+1 has the same properties. The ma-can be non-zero only if the sets {j 1 , . . . j p , k 1 , . . . k q } and {j ′ 1 , . . . j ′ r , k ′ 1 , . . . k ′ s } belong to the same partition of n + 1; this is evident from the action of the differential operators l • k and m • k on the basis monomials. Hence, the matrix M • n+1 is block diagonal: each block of the matrix M • n+1 corresponds to a particular partition of n + 1.The next step is to prove that for a given partition of n + 1 its corresponding block of M • n+1 is upper-triangular. Note that any partition of the number n + 1, apart from the trivial The characteristic curves c i are given byand using the method of characteristics we find the solution of the differential equation (C.0.2) to beis an unknown function to be determined, and we have returned to the original notation of the wave function. It follows from the definition of the primary function given in (3.2.5) that we may impose the boundary condition lim t 2 →0 ψ ∆ 2 ,ξ 2 ;∆,ξ (t 2 , 0; v) = 1. This boundary condition implies that the unknown function in (C.0.5) is an exponential function, and the prefactor t ∆−∆ 2 enforces that ∆ = ∆ 2 ; hence the wave function is of the formInserting the above partial solution into equation (3.3.6b) for n = 0 and n = 1, leads to two differential equations, which when combined becomewhere we have relabeled the arguments as a = x 2n t n 2 , as well as c = ∞ n=1 v(2) n t n 2 , and A 1 is defined by (3.2.3). Integrating the above equation and reinserting the solution into the wave function, we arrive atwhere g ( · · · ) is an unknown function which arises as a constant of integration. Inserting the partial solution (C.0.8) back into equation (3.3.6b) with n = 0 results in the differential equationn t n 2 . 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[ "Prequantization to Phase Space with Boundaries * Symplectic geometrical description of classical mechanics and its geometric quantiza- tion are essentially globalization of, respectively, Hamiltonian mechanics and canonical", "Prequantization to Phase Space with Boundaries * Symplectic geometrical description of classical mechanics and its geometric quantiza- tion are essentially globalization of, respectively, Hamiltonian mechanics and canonical" ]	[ "Ming-Xue Shao [email protected]‡e-mail:[email protected] \nInstitute of Theoretical Physics\nCCAST(World Laboratory)\nAcademia Sinica\nP.O.Box 8730, P.O.Box 2735100080, 100080Beijing, BeijingP.R.China, P.R.China\n", "Zhong-Yuan Zhu \nInstitute of Theoretical Physics\nCCAST(World Laboratory)\nAcademia Sinica\nP.O.Box 8730, P.O.Box 2735100080, 100080Beijing, BeijingP.R.China, P.R.China\n" ]	[ "Institute of Theoretical Physics\nCCAST(World Laboratory)\nAcademia Sinica\nP.O.Box 8730, P.O.Box 2735100080, 100080Beijing, BeijingP.R.China, P.R.China", "Institute of Theoretical Physics\nCCAST(World Laboratory)\nAcademia Sinica\nP.O.Box 8730, P.O.Box 2735100080, 100080Beijing, BeijingP.R.China, P.R.China" ]	[]	The Weil's integrality condition of prequantization line bundle is generalized to phase space with boundaries. The proofs of both necessity and sufficiency are given.It is pointed out via the method of topological current that Weil's integrality condition is closely connected with the summation of index of isolated singular points of sections of prequantization line bundle.	null	[ "https://export.arxiv.org/pdf/math-ph/9808009v1.pdf" ]	16,143,471	math-ph/9808009	d83a6b1839c4248331464c28feccde96f7fb7121	Prequantization to Phase Space with Boundaries * Symplectic geometrical description of classical mechanics and its geometric quantiza- tion are essentially globalization of, respectively, Hamiltonian mechanics and canonical Ming-Xue Shao [email protected]‡e-mail:[email protected] Institute of Theoretical Physics CCAST(World Laboratory) Academia Sinica P.O.Box 8730, P.O.Box 2735100080, 100080Beijing, BeijingP.R.China, P.R.China Zhong-Yuan Zhu Institute of Theoretical Physics CCAST(World Laboratory) Academia Sinica P.O.Box 8730, P.O.Box 2735100080, 100080Beijing, BeijingP.R.China, P.R.China Prequantization to Phase Space with Boundaries * Symplectic geometrical description of classical mechanics and its geometric quantiza- tion are essentially globalization of, respectively, Hamiltonian mechanics and canonical arXiv:math-ph/9808009v1 20 Aug 1998 Generalization of Integrality Condition of0240Ma Keywords: integrality conditionboundaryprequantization The Weil's integrality condition of prequantization line bundle is generalized to phase space with boundaries. The proofs of both necessity and sufficiency are given.It is pointed out via the method of topological current that Weil's integrality condition is closely connected with the summation of index of isolated singular points of sections of prequantization line bundle. quantization [1]. Geometric quantization has been considered as a so far most mathematically thorough approach to quantization. The first step in geometric quantization is prequantization which needs a Hermitian line bundle to realize the representation of the Poission brackets of classical observables. Furthermore in order to satisfy the Dirac condition [2] in prequantization, the pull back of curvature form Ω of the bundle should be the same as the symplectic formh −1 ω of the symplectic manifold M. The question is whether or not any ω can be used to construct the bundle . The answer is as follows. Such a bundle and connection exist if and only if ω satisfies Weil ′ s integrality condition [3] [4]: 'The integral of ω over any closed oriented 2-surface in M is an integral multiple of 2πh.' This condition is closely related to the quantization rule in the old quantum theory [5] [6]. In this paper we use the technique of decomposition of connection [7] [8] and topological current [9][10] to prove its necessity. We also generalize it to the case of phase spaces with boundary. Besides a proof of sufficiency via geometric construction is given when M is simply connected. Let B → M be a Hermitian line bundle with M the phase space of classical system and Ψ = ψs be the section of B with s the unit section. The Dirac condition in prequantization requires[1] Ω = 1 h ω,(1) in which Ω is the curvature form of the bundle B and ω the symplectic form of symplectic manifold M. The connection on B is defined as Ds = −iΘs,(2) where Θ denotes the connection 1-form. The covariant derivative of Ψ is Dψs = (dψ − iΘψ)s(3) which can be rewritten as Dψ = dψ − iΘψ,(4) from which we can obtain Θ = −i 1 ψ dψ + i 1 ψ Dψ.(5) Now we write ψ by its real part ψ 1 and imaginary part ψ 2 ψ = \|\|ψ\|\|(n 1 + in 2 ), where \|\|ψ\|\| = ψ 1 2 + ψ 2 2 ; (7) n 1 = ψ 1 ψ 1 2 + ψ 2 2 , n 2 = ψ 2 ψ 1 2 + ψ 2 2 .(8) Denote ψ = (ψ 1 , ψ 2 ) which can be considered as a 2-component vector field and n = (n 1 , n 2 ) a unit vector field n 1 n 1 + n 2 n 2 = 1. Putting (6) into (4), we obtain Dψ = d\|\|ψ\|\|(n 1 + in 2 ) + \|\|ψ\|\|(Dn 1 + iDn 2 ),(10) in which Dn 1 = dn 1 + Θn 2 ; Dn 2 = dn 2 − Θn 1 .(11) Recall the so(2) covariant derivative Dn a = dn a + ω ab n b .(12) The comparison of (11) and (12) implies Θ = ω 12 , for which the underlined reason is the homomorphism of Lie algebra of su(1) and so (2). Putting Eqs. (10) and (6) into (5), we get Θ = ǫ ab n a dn b − ǫ ab n a Dn b ,(13) where we have used Eq. (9). From Eq. (13), the curvature of the line bundle is Ω = dΘ = ǫ ab dn a ∧ dn b − d(ǫ ab n a Dn b ).(14) If there is no zero points of ψ, or no singular points of n, the integral of (14) over any closed 2-surface will vanish by Stocks theorem.. However in general (14) can not be globally correct if ψ as a global section has zero points which depend on topological property of the bundle. That is to say, (14) has singular points. In this case a convenient method is to use the so called topological current technique [9] [7] [10]as follows. The first term of rhs of (14) is ǫ ab dn a ∧ dn b = 2πT dx 1 ∧ dx 2(15) where x µ , µ = 1, 2 is two dimensional local coordinates of oriented 2-surface Σ in M, and T is defined by T = 1 2π ǫ µν ǫ ab ∂ µ n a ∂ ν n b .(16) Substituting (8) into (16) and using ∂ µ n a = δ al \|\|ψ\|\| 2 − ψ a ψ l \|\|ψ\|\| 3 ∂ µ ψ l(17) and ∂ ∂ψ l (ln 1 \|\|ψ\|\| ) = − ψ l \|\|ψ\|\| 2 (18) results in T = − 1 2π ǫ µν ǫ ab ∂ µ ψ l ∂ ν ψ b ∂ ∂ψ l ∂ ∂ψ a (ln 1 \|\|ψ\|\| ).(19) Define Jacobian determinant as ǫ ab J([ ∂ψ ∂x ]) = ǫ µν ∂ µ ψ a ∂ ν ψ b ,(20) by virtue of the 2-dimensional Laplacian relation [11] ∂ ∂ψ l ∂ ∂ψ l (ln 1 \|\|ψ\|\| ) = −2πδ( ψ),(21) in which ( ∂ 2 ∂ψ l ∂ψ l ) is 2-dimensional Laplacian operator in ψ space, the δ-function-like density T = δ( ψ)J([ ∂ψ ∂x ])(22) is obtained. Suppose that the function ψ a (a = 1, 2) possess n isolated zeroes. Let the i-th zero be x = z i , one has ψ a ( z i ) = 0, with i = 1, 2, ..., n. Then we have δ( ψ) = n i=1 β i \|J([ ∂ψ ∂x ])\| x= z i ,(24) where β i is a positive integer called Hopf index [12] of the i-th singular point, which denote the times the function ψ covers the corresponding region while the point x covers the neighborhood of x = z i once. Substituting (24) into (22), the charge density T can be written in the form T = n i=1 β i η i δ 2 ( x − z i ),(25) where η i = J([ ∂ψ ∂x ]) \|J([ ∂ψ ∂x ])\| \| x= z i = ∓1(26) are called the Brouwer degrees [13], which reflects whether or not the covering of ψ has the same direction on bundle with that of x on 2-dimensional base manifold. Now consider the integral of (14) over a closed 2-surface Σ. . Using (25) and the fact Σ is closed, we get Σ Ω = 2π n i=1 β i η i = 2πg,(27)where g = n i=1 β i η i is topological charge. Considering (1), Σ ω = hg.(28) is obtained. This demonstrates the integral of ω over any closed 2-surface of phase space is an integer multiplied by plank constant h. Furthermore from this derivation we know this integer can be determined by the summation of index of zero-points of ψ. Now we consider the case that 2-surface Σ with boundary ∂Σ to be piece-wise smooth. Making use of Stocks theorem, we have Σ Ω = 2π n i=1 β i η i + ∂(Σ−Σ ′ ) ǫ ab n a dn b − ∂Σ ǫ ab n a Dn b ,(29) where Σ ′ is another 2-surface with boundary ∂Σ ′ which is chosen to be smooth and → ∂Σ in limit. Defining α to be the angle from e 1 = (1, 0) to n = (n 1 , n 2 ), we get ǫ ab n a dn b = dα. Obviously the change of α along ∂Σ ′ is 2πl with l an integer. So, considering this and (1)(29), we obtain Σ ω = 2πh(−l + n i=1 β i η i ) +h ∂Σ Θ,(31) in which we have used the Stocks theorem and (1)(29). Notice the second term in (31) is invariant under the transformation of connection. Condition (31)is necessary. A slight difference is that here we generalize to 2-surface with boundary in M. When M has boundary this generalization is needed because a closed path γ in M will deduce two types of 2-surfaces in M : (1) with γ as boundary (2) with γ and a closed path c ⊂ ∂M as boundary. The particular case is when M is closed both the second term in (31) and l in the first term vanish, we return to Σ ω = 2πh( n i=1 β i η i ).(32) For a general case that the phase space M has boundary ∂M, assuming that M is simply connected, (31) is also a sufficient condition. We will use geometric construction to prove it. Suppose that Ω = ω/h = dΘ is a closed 2-form with ω satisfy (31). Choose a base point m 0 in M and let K denote the set of all triples (m, z, γ) where m ∈ M, z ∈ C, and γ is a piecewise smooth path from m 0 to m. On K, define an equivalence relations ∼ by (m, z, γ) ∼ (m ′ , z ′ , γ ′ ),(33) whenever m = m ′ and z ′ = { zexp(i Σ Ω), zexp(−i Σ ′ Ω + i c Θ),(34) where γ is some fixed curve joining m 0 to m 1 . It follows from Stokes theorem and (33) that ψ 1 (m, w) is independent of the choice made for ξ. Therefore U 1 and ψ 1 from a local trivialization. If (U 2 , ψ 2 ) is another such local trivialization by replacing m 1 , U 1 , and Θ 1 by m 2 , U 2 , and Θ 2 , and if U 1 ∩ U 2 is simply connected, then ψ 2 (m, w) = c 12 (m)ψ 1 (m, w); m ∈ U 1 ∩ U 2 , w ∈ C,(38) where the transition function c 12 ∈ C ∞ C (U 1 ∩ U 2 ) satisfies c 12 (m ′ ) = c 12 (m)exp(i m ′ m (Θ 1 − Θ 2 )); m, m ′ ∈ U 1 ∩ U 2 ,(39) in which the integral is taken along any path from m to m ′ . It follows that dc 12 c 12 = i(Θ 1 − Θ 2 ),(40) Since a different base point will give an equivalent Hermitian line bundle-with connection, we complete the proof of sufficient condition. At last of this paper, we point out that the second term in (31) can be simplified by choosing unit vector n to be tangent to the boundary. Then n b , k b = −ǫ ab n a has the same orientation with the base manifold, then define − ǫ ab Dn a n b = k g ds (42) with k g the geodesic curvature along the boundary and s the parameter of the boundary ∂Σ. Consider the boundary is piece-wise with m angle-change points. Let the inner angle of i-th angle-change points be α i . From Eqs.(31) and (42), we get Σ ω = 2πh n i=1 β i η i −h m i=1 (π − α i ) −h ∂Σ k g ds.(43) where Σ is any surface with boundary made up of γ(γ ′ ) −1 and Σ ′ surface with boundary made up of γ ′ (γ) −1 c with c ⊂ ∂M a closed path. Because (31) hold, it does not matter which surface is chosen. We shall take as the total space of our bundle the manifold B = K/ ∼ with the obvious projection onto M: addition and scalar multiplication within the fibers are defined by [(m, z, γ) + (m, z ′ , γ)] = [(m, z + z ′ , γ)], square brackets denote equivalence classes). 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Soc. London Ser.A, 133(1925)642-653 . A Weil Variets, Kahleriennes, HermannParisA. Weil Variets Kahleriennes. Hermann, Paris(1958) Quantization and unitary representations. B Kostant, Lectures in modern analysis 3(e. C.T. Taam). Lecture notes in mathmatics. BerlinSpringer170B. Kostant, Quantization and unitary representations. in Lectures in modern analysis 3(e. C.T. Taam). Lecture notes in mathmatics, Vol.170 Springer, Berlin. . A. Messiah quantum Mechanics. 1North-Hoolland. AmsterdamA. Messiah quantum Mechanics, Vol. 1 (1961)North-Hoolland. Amsterdam. Geometric Quantization of the Harmonic Oscillator with diagonalised Hamiltonian. D J Simms, Colloquium on Group theoretical Methods in Physics. A. Janner et.al.HollandUniversity of NijmegenD. J. Simms, Geometric Quantization of the Harmonic Oscillator with diagonalised Hamiltonian. In Colloquium on Group theoretical Methods in Physics(ed. A. Janner et.al.,)University of Nijmegen, Holland. . Y S Duan, Z P Duan, Int. J. Engng. Sci. 24531Y. S. Duan and Z. P. Duan, Int. J. Engng. Sci. 24(1986)531 . Y S Duan, S L Zhang, Int. J. Engng. Sci. 281593Y. S. Duan and S. L. Zhang, Int. J. Engng. Sci. 28(1990)689; 29(1991)1593; . M X Shao, Lanzhou UniversityMaster ThesisM. X. Shao, Master Thesis, Lanzhou University (1995) . Y S Duan, J. Math. Phys. 359Y. S. Duan, J. Math. Phys.35(1994)9 . Y S Duan, X H Meng, J. Math. Phys. 341149Y.S. Duan and X.H. Meng, J. Math. Phys. 34(1993)1149 Generalized function. I M Gelfand, G E Shilov, National Press of Mathmatics LiteratureMoscoroI. M. Gelfand and G.E. Shilov, Generalized function, National Press of Mathmatics Literature, Moscoro(1958) . J Arafune, G O Freund, C J Geobel, J. Math. Phys. 16433J.Arafune, G. O. Freund and C. J. Geobel. J. Math. Phys. 16(1975)433 B A Dubrovin, A T Fomenko, S P Novikov, Modern Geometry methods and Applications. New YorkSpringerB.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry methods and Applications, Springer, New York(1984).	[]
[]	[ "Qian Zhang " ]	[]	[]	In this paper we study global nonlinear stability for the Dirac-Klein-Gordon system in two and three space dimensions for small and regular initial data. In the case of two space dimensions, we consider the Dirac-Klein-Gordon system with a massless Dirac field and a massive scalar field, and prove global existence, sharp time decay estimates and linear scattering for the solutions. In the case of three space dimensions, we consider the Dirac-Klein-Gordon system with a mass parameter in the Dirac equation, and prove uniform (in the mass parameter) global existence, unified time decay estimates and linear scattering in the top order energy space.Keywords Dirac-Klein-Gordon equations · global existence · sharp pointwise decay · linear scattering Mathematics Subject Classifications (2010) 35L70 · 35L52 · 35Q40[15]. Dong and Wyatt [13] first proved sharp asymptotic decay results for (1.1) with ( , ) = (2, 0) and small, regular initial data with compact support, using the method of hyperboloidal foliation of space-time. Recently Dong, Li, Ma and Yuan[12]removed the compactness condition on the support of the initial data in[13]. Precisely, by applying the ghost weight method introduced by Alinhac [1], they proved global existence, sharp time decay and linear scattering for (1.1) with ( , ) = (2, 0) and small, regular initial data. The hyperboloidal method used in[13]was introduced by Klainerman in [18] to prove global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. Later this method was developed by Klainerman, Wang and Yang[19,24]and by LeFloch and Ma [20,21]; see also[10,11,14].In this paper, we study the Cauchy problem (1.1)-(1.2) in two or three space dimensions. We consider the following two cases depending on the spatial dimension and the Dirac mass parameter : ) ( , ) = (2, 0); ) = 3 and ∈ [0, 1]. In both cases we consider small and regular initial data which decay at suitable rates at infinity. In each case we prove global existence with sharp time decay estimates and linear scattering in the high order energy space. In particular, in the case ), we give an alternative proof of the results in[12]. In the case ), our global existence and pointwise decay results are uniform in the Dirac mass parameter ∈ [0, 1] in the sense that, the smallness condition on the initial data is independent of , and we give explicit dependence of the pointwise decay estimates on the parameter . We use unified method to prove the global stability results in the cases )-). This method is inspired by a recent work by Huneau and Stingo [16], where the authors studied global existence of solutions to a certain class of quasilinear systems of wave equations on a product space with null nonlinearities.Major difficulties and key ideas.We set the initial data at time = 2 to simplify the computations and this is not essential by the invariance of (1.1) under time translations. We first consider (1.1)-(1.2) with ( , ) = (2, 0). The main difficulties in proving global existence of solutions in this case include: ) the free Dirac field and Klein-Gordon field decay at rates − 1 2 and −1 respectively, which means that neither of the nonlinearities in (1.1) is integrable in time; ) there is no compactness assumption on the support of the initial data, and we cannot use the hyperboloidal method as in[13]directly. For this, we adopt an idea by Huneau and Stingo [16], i.e., we divide the space-time into the exterior region ex = {( , ) : ≥ 2, \| \| ≥ − 1} and the interior region in = {( , ) : ≥ 2, \| \| < − 1}, and use weighted energy and Sobolev estimates on flat time slices in the exterior region, while using estimates on truncated hyperboloids in the interior region. However, there are several difficulties to overcome in using this method. Firstly, to close the bootstrap in the exterior region, we need to make use of the weighted space-time 2 estimates of the solution ( , ). Considering the growth of the top order energy, it is essential to treat the energy of both and carefully. Secondly, to bound the lower order energy of on truncated exterior hyperboloids, we need the improved pointwise estimate of [ ] − := − ( / ) 0 . Usually this is obtained from the conformal energy estimate of Ψ which is the solution to Ψ = . We can bypass the conformal energy and obtain pointwise estimates of \| 0 Ψ\| and \|ΓΨ\| from weighted energy estimates of Ψ, where 0 is the scaling vector field and Γ ∈ { , , Ω} (see Section 2.1). Thirdly, due to the growth of the top order energy of the solution ( , ) in the interior region, we need to bound the lower order energy of the solution by performing nonlinear transformations and give control of boundary integrals appearing in the energy estimates of the functions in these transformations. Furthermore, some new ingredients are required to prove the scattering results, since the linear scattering relies on flat 2 -type estimates. However, in the interior region, we only have 2 -type estimates on truncated hyperboloids. The main ideas are described below. For any fixed ≥ 2, let Σ ex := { : \| \| ≥ − 1} denote the truncated flat time slice in the exterior region and [2, ] := {( , ) : \| \| = − 1, 2 ≤ ≤ } denote the portion of the boundary of the light cone in the time strip [2, ]. We introduce two weight functions − and ( ) := (2 + ) 1+ ( ≥ 0, > 0 are constants), and define the weighted energy functionals for the Klein-Gordon field and the Dirac field as ex,	null	[ "https://export.arxiv.org/pdf/2303.08278v1.pdf" ]	257,532,557	2303.08278	e4f77afbd8637b15ebf4555677e767e4e5fa7203	14 Mar 2023 Qian Zhang 14 Mar 2023Global stability of the Dirac-Klein-Gordon system in two and three space dimensions In this paper we study global nonlinear stability for the Dirac-Klein-Gordon system in two and three space dimensions for small and regular initial data. In the case of two space dimensions, we consider the Dirac-Klein-Gordon system with a massless Dirac field and a massive scalar field, and prove global existence, sharp time decay estimates and linear scattering for the solutions. In the case of three space dimensions, we consider the Dirac-Klein-Gordon system with a mass parameter in the Dirac equation, and prove uniform (in the mass parameter) global existence, unified time decay estimates and linear scattering in the top order energy space.Keywords Dirac-Klein-Gordon equations · global existence · sharp pointwise decay · linear scattering Mathematics Subject Classifications (2010) 35L70 · 35L52 · 35Q40[15]. Dong and Wyatt [13] first proved sharp asymptotic decay results for (1.1) with ( , ) = (2, 0) and small, regular initial data with compact support, using the method of hyperboloidal foliation of space-time. Recently Dong, Li, Ma and Yuan[12]removed the compactness condition on the support of the initial data in[13]. Precisely, by applying the ghost weight method introduced by Alinhac [1], they proved global existence, sharp time decay and linear scattering for (1.1) with ( , ) = (2, 0) and small, regular initial data. The hyperboloidal method used in[13]was introduced by Klainerman in [18] to prove global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. Later this method was developed by Klainerman, Wang and Yang[19,24]and by LeFloch and Ma [20,21]; see also[10,11,14].In this paper, we study the Cauchy problem (1.1)-(1.2) in two or three space dimensions. We consider the following two cases depending on the spatial dimension and the Dirac mass parameter : ) ( , ) = (2, 0); ) = 3 and ∈ [0, 1]. In both cases we consider small and regular initial data which decay at suitable rates at infinity. In each case we prove global existence with sharp time decay estimates and linear scattering in the high order energy space. In particular, in the case ), we give an alternative proof of the results in[12]. In the case ), our global existence and pointwise decay results are uniform in the Dirac mass parameter ∈ [0, 1] in the sense that, the smallness condition on the initial data is independent of , and we give explicit dependence of the pointwise decay estimates on the parameter . We use unified method to prove the global stability results in the cases )-). This method is inspired by a recent work by Huneau and Stingo [16], where the authors studied global existence of solutions to a certain class of quasilinear systems of wave equations on a product space with null nonlinearities.Major difficulties and key ideas.We set the initial data at time = 2 to simplify the computations and this is not essential by the invariance of (1.1) under time translations. We first consider (1.1)-(1.2) with ( , ) = (2, 0). The main difficulties in proving global existence of solutions in this case include: ) the free Dirac field and Klein-Gordon field decay at rates − 1 2 and −1 respectively, which means that neither of the nonlinearities in (1.1) is integrable in time; ) there is no compactness assumption on the support of the initial data, and we cannot use the hyperboloidal method as in[13]directly. For this, we adopt an idea by Huneau and Stingo [16], i.e., we divide the space-time into the exterior region ex = {( , ) : ≥ 2, \| \| ≥ − 1} and the interior region in = {( , ) : ≥ 2, \| \| < − 1}, and use weighted energy and Sobolev estimates on flat time slices in the exterior region, while using estimates on truncated hyperboloids in the interior region. However, there are several difficulties to overcome in using this method. Firstly, to close the bootstrap in the exterior region, we need to make use of the weighted space-time 2 estimates of the solution ( , ). Considering the growth of the top order energy, it is essential to treat the energy of both and carefully. Secondly, to bound the lower order energy of on truncated exterior hyperboloids, we need the improved pointwise estimate of [ ] − := − ( / ) 0 . Usually this is obtained from the conformal energy estimate of Ψ which is the solution to Ψ = . We can bypass the conformal energy and obtain pointwise estimates of \| 0 Ψ\| and \|ΓΨ\| from weighted energy estimates of Ψ, where 0 is the scaling vector field and Γ ∈ { , , Ω} (see Section 2.1). Thirdly, due to the growth of the top order energy of the solution ( , ) in the interior region, we need to bound the lower order energy of the solution by performing nonlinear transformations and give control of boundary integrals appearing in the energy estimates of the functions in these transformations. Furthermore, some new ingredients are required to prove the scattering results, since the linear scattering relies on flat 2 -type estimates. However, in the interior region, we only have 2 -type estimates on truncated hyperboloids. The main ideas are described below. For any fixed ≥ 2, let Σ ex := { : \| \| ≥ − 1} denote the truncated flat time slice in the exterior region and [2, ] := {( , ) : \| \| = − 1, 2 ≤ ≤ } denote the portion of the boundary of the light cone in the time strip [2, ]. We introduce two weight functions − and ( ) := (2 + ) 1+ ( ≥ 0, > 0 are constants), and define the weighted energy functionals for the Klein-Gordon field and the Dirac field as ex, Introduction Let = 2 or 3. We consider the Dirac-Klein-Gordon system in space dimensions: In the above, = 0 + 1 1 + · · · + is the Dirac operator, = for ∈ {1, · · · , }, = ( , ) : R 1+ → C 0 is a spinor field with mass ≥ 0, = ( , ) : R 1+ → R is a scalar field, * denotes the complex conjugate transpose of the vector , and are the Dirac matrices, where 0 := 2 [ ( +1)/2] ([ ] is the largest integer which is smaller than or equal to , for ∈ R). Dirac matrices are defined by the identities + = := ,( , )+ = −2 , ( ) * = − ,(1.3) where , ∈ {0, 1, · · · , }, = ( ) = diag(−1, 1, · · · , 1) denotes the Minkowski metric in R 1+ , ( ) is the inverse matrix of ( ), := 0 is the 0 × 0 identity matrix and * is the conjugate transpose of the matrix . Throughout, Einstein summation convention is adopted. As usual, = = − 2 + Δ denotes the wave operator. The Dirac-Klein-Gordon system is a basic model of proton-proton interaction or neutron-neutron interaction in physics. This system and the Maxwell-Dirac equations form the foundations of Relativistic Electrodynamics (see Bjorken and Drell [4]). For = 3 and fixed Dirac mass parameter , the global well-posedness and global behavior of the solutions to (1.1)-(1.2) have been widely studied both in the low regularity setting and in the high regularity setting; see, for example, Bournaveas [5], D'Ancona, Foschi and Selberg [8], Bejenaru and Herr [3], Bachelot [2] and Choquet-Bruhat [7]. For = 2, Tsutsumi [23] and Delort, Fang and Xue [9] proved small data global existence with optimal pointwise decay and scattering for the massive Dirac-Klein-Gordon system. On the other hand, for ( , ) = (2, 0), Bournaveas [6] proved local existence for (1.1) with low regular initial data, and the global existence result was first proved by Grünrock + \| \| ≤ −1 Γ ex, 1,0, ,(1.7) where ≥ 6 is an integer, > 0, 0 < ≪ 1, denotes a multi-index and Γ denotes the set of compatible vector fields. To close the estimate of the top order energy in (1.7), we use a special structure generated in the energy estimate of (see Proposition 2.5) and the estimates of the space-time 2 norms ∫ 2 (2 + − ) 2 − 2 \|Γ \| + \|[ Γ ] − \| 2 2 (Σ ex ) d 1/2 , \| \| ≤ (see (1.4) and (1.5)) together with weighted Sobolev inequality on truncated flat time slices in the exterior region. For the lower order case, we conduct a nonlinear transformation˜ = − * 0 (which was also used in [13,12]). The refined exterior energy estimates of the solution obtained above imply corresponding estimates on truncated exterior hyperboloids. To derive (unweighted) lower order energy estimates of on exterior hyperboloids, we need the improved pointwise estimate of [ ] − , and this is obtained by the weighted energy estimates of the solution Ψ to the equation Ψ = , and weighted Sobolev and Hardy inequalities on flat time slices in the exterior region. The (unweighted) lower order energy estimates of the solution ( , ) on truncated exterior hyperboloids, combined with the bootstrap estimates in the interior region, give pointwise decay estimates for the solution on truncated interior hyperboloids, via Sobolev inequality on hyperboloids. We also point out that in deriving exterior energy estimates of the solution ( , ) and the functions in the nonlinear transformations, we also obtain refined estimates of integrals on the boundary of the light cone (see (1.6)). These are used to close the bootstrap estimates in the interior region. Next we consider (1.1)-(1.2) with = 3 and ∈ [0, 1]. In this case, the uniform (in the mass parameter) weighted space-time 2 estimates, and the faster pointwise decay of the solution in terms of these 2 estimates, give the boundedness of the top order energy of the solution ( , ) on flat time slices in the exterior region, and therefore on truncated exterior hyperboloids. Hence we obtain global existence which is uniform in the Dirac mass parameter . By exploiting the hidden structure of the Dirac equation, we obtain pointwise decay with explicit dependence on the mass parameter . Finally, we also prove linear scattering of the solutions in all the cases under consideration. These scattering results cannot be obtained from the pointwise decay estimates directly since the Sobolev norms of the nonlinearities are not integrable in time. We prove a technical lemma which gives a sufficient condition for the linear scattering of the solutions; see Lemma 2.12. In the sequel, we use to denote a universal constant whose value may change from line to line. As usual, means that ≤ for some constant . Given a vector or a scalar , we use Japanese bracket to denote := (1 + \| \| 2 ) 1/2 . Next we state our main results. We first present the results in two space dimensions. Consider = := , ( , ) ∈ [2, ∞) × R 2 , − + = := * 0 , ( , ) ∈ [2, ∞) × R 2 (1.8) with prescribed initial data at = 2 ( , , )\| =2 = ( 0 , 0 , 1 ). (1.9) Theorem 1.1. Let ∈ N with ≥ 6. Then there exists a constant 0 > 0 such that for any 0 < < 0 and all initial data ( 0 , 0 , 1 ) satisfying the smallness condition +1 =0 \| \| +1 ∇ 0 2 (R 2 ) + =0 \| \| +1 \|∇ 0 \| + \| \| +1 \|∇ 1 \| 2 (R 2 ) ≤ ,(1.\| ( , )\| + \| \| − 1 2 − \| \| − 1 2 , \| ( , )\| + \| \| −1 . In addition, the solution ( , ) scatters to a free solution in X −1 (R 2 ) := −1 (R 2 ) × (R 2 ) × −1 (R 2 ), i.e., there exists ( * 0 , * 0 , * 1 ) ∈ X −1 (R 2 ) such that lim →+∞ ( , , ) − ( * , * , * ) X −1 (R 2 ) = 0, where ( * , * ) is the solution to * = 0, ( , ) ∈ [2, ∞) × R 2 , − * + * = 0, ( , ) ∈ [2, ∞) × R 2 with the initial data ( * , * , * )\| =2 = ( * 0 , * 0 , * 1 ). We next state the results in three space dimensions. Consider + = := , ( , ) ∈ [2, ∞) × R 3 , − + = := * 0 , ( , ) ∈ [2, ∞) × R 3 (1.11) with prescribed initial data at = 2 1] and ∈ N with ≥ 5. Then there exists a constant 0 > 0, which is independent of the mass parameter , such that for any 0 < < 0 and all initial data ( 0 , 0 , 1 ) satisfying the smallness condition ( , , )\| =2 = ( 0 , 0 , 1 ). (1.12) Theorem 1.2. Let = 3, ∈ [0,+1 =0 \| \| +1 ∇ 0 2 (R 3 ) + =0 \| \| +1 \|∇ 0 \| + \|∇ 1 \| 2 (R 3 ) ≤ ,(1.\| ( , )\| + \| \| − \| \| 1 2 + 2 + \| \| 3 2 , \| ( , )\| + \| \| − 3 2 . In addition, the solution ( , ) scatters to a free solution in X (R 3 ) := (R 3 ) × +1 (R 3 ) × (R 3 ), i.e., there exists ( * 0 , * 0 , * 1 ) ∈ X (R 3 ) such that lim →+∞ ( , , ) − ( * , * , * ) X (R 3 ) = 0, where ( * , * ) is the solution to * + * = 0, ( , ) ∈ [2, ∞) × R 3 , − * + * = 0, ( , ) ∈ [2, ∞) × R 3 with the initial data ( * , * , * )\| =2 = ( * 0 , * 0 , * 1 ). The organization of this paper is as follows. In Section 2, we introduce some notations, present energy and Sobolev estimates both on flat time slices in the exterior region and on truncated hyperboloids. In Sections 3 -4, we prove the global existence result in Theorem 1.1. In Section 5, we show the uniform global existence result in Theorem 1.2. Section 6 is devoted to proving the scattering results in Theorems 1.1 and 1.2. Preliminaries In this section, we assume that the spatial dimension = 2 or 3. Unless otherwise specified, the definitions and conclusions hold for both = 2 and = 3. We denote 0 := 2 [ ( +1)/2] ,(2.1) i.e. 0 = 2 for = 2 and 0 = 4 for = 3, where [ ] denotes the largest integer which is smaller than or equal to , for ∈ R. Notations We work in the (1 + ) dimensional space-time R 1+ with Minkowski metric = (−1, 1, · · · , 1), which is used to raise or lower indices. The space indices are denoted by Roman letters , , · · · ∈ {1, 2, · · · , }, and the space-time indices are denoted by Greek letters , , , , · · · ∈ {0, 1, 2, · · · , }. Einstein summation convention for repeated upper and lower indices is adopted throughout the paper. We denote a point in R 1+ by ( , ) = ( 0 , 1 , 2 , · · · , ) with = 0 , = ( 1 , 2 , · · · , ), = , = 1, 2, · · · , , and its spatial radius is denoted by := \| \| = 2 1 + 2 2 + · · · + 2 . The following vector fields will be used frequently in the analysis (see Klainerman [17]): (i) Translations: := , for ∈ {0, 1, 2, · · · , }. (ii) Lorentz boosts: := + , for ∈ {1, 2, · · · , }. (iii) Rotations: Ω := − , for 1 ≤ < ≤ . (iv) Scaling: 0 = + . We also use the modified Lorentz boosts and rotations introduced by Bachelot [2], := − 1 2 0 , Ω := Ω − 1 2 , (2.2) which enjoy the following commutative property, i.e. [ , ] = [ Ω , ] = 0, where the commutator [ , ] is defined as We denote the constant time slices which foliate ex as [ , ] := − .Σ ex := { ∈ R : \| \| ≥ − 1}. The portion of exterior region in the time strip [2, ] for any fixed time is denoted by ex := {( , ) ∈ ex : 2 ≤ ≤ }. For 2 ≤ 1 < 2 , we denote the portion of the boundary of the light cone in the time interval [ 1 , 2 ] by [ 1 , 2 ] := {( , ) : = − 1, 1 ≤ ≤ 2 }. (2.3) For any ≥ 2, we use ℋ := {( , ) : 2 = 2 + \| \| 2 } (2.4) to denote the hyperboloid at hyperbolic time . We denote by ℋ in (resp. ℋ ex ) the portion of ℋ contained in the interior region in (resp. in the exterior region ex ), i.e., ℋ in := {( , ) ∈ ℋ : \| \| < ( 2 − 1)/2}, (2.5) ℋ ex := {( , ) ∈ ℋ : \| \| ≥ ( 2 − 1)/2}. (2.6) We set ( ) := 2 + 1 2 , ( ) := 2 − 1 2 . (2.7) In addition, we denote by ℋ in [2, ] the hyperbolic interior region limited by the hyperboloids ℋ 2 and ℋ , and by ℋ ex i.e., 0 = 6 for = 2 and 0 = 10 for = 3. We define the ordered sets {Γ } 0 =1 := ( ) 0≤ ≤ , ( ) 1≤ ≤ , (Ω ) 1≤ < ≤ { Γ } 0 =1 := ( ) 0≤ ≤ , ( ) 1≤ ≤ , ( Ω ) 1≤ < ≤ . (2.14) The following relations on commutators between the vector fields in {Γ } 0 =1 are well-known (see for example [22]): For any multi-index = ( 1 , · · · , 0 ) ∈ N 0 of length \| \| = 0 =1 , we denote Γ = 0 =1 Γ , where Γ = (Γ 1 , . . . , Γ 0 ), Γ = 0 =1 Γ , where Γ = ( Γ 1 , . . . , Γ 0 ). (2.15) For any = ( 0 , 1 , · · · , ) ∈ N 1+ , = ( 1 , · · · , ) ∈ N , = ( ) 1≤ < ≤ ∈ N 1 ,[ , ] = ′ 0≤ ≤ , [ , Ω ] = ′ 0≤ ≤ , [ , 0 ] = , 0 ≤ ≤ , 1 ≤ < ≤ , [ , ] = ′ ∈ , , ∈ := {( ) 1≤ ≤ , (Ω ) 1≤ < ≤ }, [ 0 , Ω ] = [ 0 , ] = 0, 1 ≤ < ≤ , 1 ≤ ≤ . (2.17) The following proposition is a direct result from the definitions (2.2) and (2.15). Proposition 2.1. Let 0 , 0 be as in (2.1) and (2.13) respectively. For any sufficiently smooth vector field = ( , ) : R 1+ → C 0 , and any , ∈ N 0 , we have Γ = Γ + \| ′ \|< \| \| ′ Γ ′ , Γ = Γ + \| ′ \|< \| \| ′ Γ ′ for some 0 × 0 constant matrices ′ , ′ . Energy estimates for wave and Klein-Gordon equations For any fixed constants 0 ≤ ≤ 1, > 0 and ≥ 0, we define ex, , ( , ) := ∫ Σ ex − (2 + − ) 1+ (\| \| 2 + 2 2 ) ( , )d . (2.18) When = 0, we denote for simplicity ex, ( , ) := ∫ Σ ex (2 + − ) 1+ (\| \| 2 + 2 2 ) ( , )d . (2.19) We also denote ex , ( , ) := ∫ Σ ex − (\| \| 2 + 2 2 ) ( , )d . (2.20) In particular, we define the unweighted energy ex ( , ) := ex ,0 ( , ) = ∫ Σ ex (\| \| 2 + 2 2 ) ( , )d . (2.21) We have the following weighted energy estimate in the exterior region. + ( , ) ex, , ,2 + ∫ 2 − 2 (2 + − ) 1+ 2 (− + 2 ) 2 (Σ ex ) d , where ex, , , : = [ ex, , ( , )] 1 2 + ∫ 2 ∫ Σ ex − (2 + − ) \| \| 2 + 2 2 d d 1 2 , ( , ) : = ∫ [2, ] − \| \| 2 + 2 2 d 1 2 ,(2. 22) and we recall the definitions (2.11) and (2.3). Proof. Let be a positive weight. Denote := − + 2 . Multiplying on both sides of this equality by − ( − ) and by direct calculation, we obtain − (\| \| 2 + 2 2 ) − 2 − + − ′ \| \| 2 + 2 2 + − −1 (\| \| 2 + 2 2 ) = 2 − . (2.23) We note that the upward unit normal to [2, ] is 2 − 1 2 (1, − / ). Taking ( ) = (2 + ) 1+ , and integrating (2.23) over the region ex , we have ex, , ( , ) + (1 + ) ∫ 2 ∫ Σ ex − (2 + − ) \| \| 2 + 2 2 d d + ∫ [2, ] − \| \| 2 + 2 2 2 − 1 2 d + ∫ 2 ∫ Σ ex − −1 (2 + − ) 1+ (\| \| 2 + 2 2 )d d = ex, , (2, ) + 2 ∫ 2 ∫ Σ ex − (2 + − ) 1+ d d . (2.24) Differentiating (2.24) with respect to and using Hölder inequality, we derive ex, , ( , ) + (1 + ) ∫ 2 ∫ Σ ex − (2 + − ) \| \| 2 + 2 2 d d + ∫ [2, ] − \| \| 2 + 2 2 2 − 1 2 d ≤ 2[ ex, , ( , )] 1 2 · − 2 (2 + − ) 1+ 2 2 (Σ ex ) , which implies the conclusion of the proposition. \| \| + \| \| ( ) 2 (R ) \| \| + \| \| (2) 2 (R ) + ∫ 2 (− + 2 ) ( ) 2 (R ) d , for any sufficiently smooth function which decays sufficiently fast at space infinity. Let ≥ 0 and ≥ 2. The energy functionals on the truncated hyperboloids ℋ in and ℋ ex are defined as E in,ℎ , ( , ) := ∫ ℋ in − ℎ [ ]d , (2.25) E ex,ℎ , ( , ) := ∫ ℋ ex − ℎ [ ]d (2.26) respectively, where ℎ [ ] : = \| \| 2 + 2 2 + 2 = 2 2 \| \| 2 + −2 \| \| 2 + 2 2 = 2 2 \|∇ \| 2 + −2 \| 0 \| 2 + −2 \|Ω \| 2 + 2 2 (2.27) and we recall the definitions (2.9) and (2.10). We denote for simplicity E in,ℎ ( , ) := E in,ℎ ,0 ( , ), E ex,ℎ ( , ) := E ex,ℎ ,0 ( , ). (2.28) The energy estimate on the truncated exterior hyperboloids ℋ ex is stated as follows. Proposition 2.3. Let ≥ 0. For any ∈ [2, ∞) and any sufficiently smooth function which is defined in ℋ ex [2, ] and decays sufficiently fast at space infinity, we have E ex,ℎ , ( , ) + ∫ [2, ( ) ] − \| \| 2 + 2 2 d ex , (2, ) + sup ∈ [2,∞) [ ex , ( , )] 1 2 · ∫ ∞ 2 − 2 (− + 2 ) 2 (Σ ) d , where ( ) = 2 +1 2 is as in (2.7), we recall (2.3) and (2.20) for the definitions of [2, ( ) ] and ex , ( , ), and Σ := ∈ R : \| \| ≥ − 1 , ≤ ( ); ∈ R : \| \| ≥ 2 − 2 , > ( ). (2.29) Proof. Recall the definition (2.4). We note that on ℋ we have nd = (1, − / )d , where n and d denote the upward unit normal and the volume element of ℋ respectively. Taking ≡ 1 in (2.23) and integrating over the region ℋ ex [2, ] , we obtain E ex,ℎ , ( , ) + ∫ [2, ( ) ] − \| \| 2 + 2 2 2 − 1 2 d + ∫ ℋ ex [2, ] − −1 (\| \| 2 + 2 2 )d d = ex , (2, ) + 2 ∫ ∞ 2 ∫ Σ − (− + 2 )d d . We note that Σ ⊂ Σ ex , hence the conclusion follows. On the truncated interior hyperboloids ℋ in , we have the following energy estimate. Proposition 2.4. Let ≥ 0. For any ≥ 2 and any sufficiently smooth function defined in ℋ in [2, ] , we have E in,ℎ , ( , ) E in,ℎ , (2, ) + ∫ [ (2) , ( ) ] − \| \| 2 + 2 2 d + ∫ 2 [E in,ℎ , ( , )] 1 2 · − 2 (− + 2 ) 2 (ℋ in ) d , where ( ) = 2 +1 2 is as in (2.7) and (2) = 5 2 , and we recall (2.3) for the definition of [ (2), ( ) ] . Proof. Taking ≡ 1 in (2.23), integrating over the region ℋ in [2, ] , and using the transformation ( , ) → ( , ), where = 2 − \| \| 2 , we obtain E in,ℎ , ( , ) + ∫ ℋ in [2, ] − −1 (\| \| 2 + 2 2 )d d = E in,ℎ , (2, ) + ∫ [ (2) , ( ) ] − \| \| 2 + 2 2 2 − 1 2 d + 2 ∫ 2 ∫ ℋ in − (− + 2 ) d d . Using Hölder inequality, the conclusion follows. Energy estimates for the Dirac equation Let be any C 0 -valued function defined on Σ ex , where 0 is as in (2.1). For any fixed constant > 0, we define ex, ( , ) := ∫ Σ ex (2 + − ) 1+ \| \| 2 ( , )d . (2.30) For the unweighted case we denote ex ( , ) := ∫ Σ ex \| \| 2 ( , )d .+ ( ( , )) 2 2 ex, ,2 + ∫ 2 (2 + − ) 1+ * 0 ( + ) 1 (Σ ex ) d , ex, , + ( , ) ex, ,2 + ∫ 2 (2 + − ) 1+ 2 ( + ) 2 (Σ ex ) d , where ex, , : = [ ex, ( , )] 1 2 + ∫ 2 ∫ Σ ex (2 + − ) \|[ ] − \| 2 d d 1 2 , ( , ) : = ∫ [2, ] \|[ ] − \| 2 d 1 2 . (2.33) Proof. Let be a positive weight. Set := + . Multiplying on both sides of this identity by − ( − ) * 0 , we obtain * + * 0 − * 0 = − * 0 . Taking the complex conjugate of the above equality, we have * + * 0 + * 0 = * 0 . Adding the last two equalities, we derive * + * 0 + 1 2 ′ \|[ ] − \| 2 = − ( * 0 − * 0 ), (2.34) where we use the relation * − * 0 = 1 2 − 0 2 . Let ( ) := (2 + ) 1+ and integrate (2.34) over the region ex , then we obtain ex, ( , ) + 1 + 2 ∫ 2 ∫ Σ ex (2 + − ) \|[ ] − \| 2 d d + 1 2 ∫ [2, ] \|[ ] − \| 2 2 − 1 2 d = ex, (2, ) − ∫ 2 ∫ Σ ex (2 + − ) 1+ ( * 0 − * 0 )d d . Differentiating this equality with respect to and using Hölder inequality, we obtain ex, ( , ) + 1 + 2 ∫ 2 ∫ Σ ex (2 + − ) \|[ ] − \| 2 d d + 1 2 ∫ [2, ] \|[ ] − \| 2 2 − 1 2 d [ ex, ( , )] 1 2 · (2 + − ) 1+ 2 2 (Σ ex ) . The conclusions of the proposition follow. The energy functionals for the Dirac field on the truncated hyperboloids ℋ in and ℋ ex are defined as ( ) 2 (R ) (2) 2 (R ) + ∫ 2 ( + ) ( ) 2 (R ) d ,E in,ℎ ( , ) := ∫ ℋ in \|( ) − \| 2 + \|( / ) \| 2 d , (2.36) E ex,ℎ ( , ) := ∫ ℋ ex \|( ) − \| 2 + \|( / ) \| 2 d . (2.37) We have the following energy estimate for the Dirac equation on the exterior hyperboloids ℋ ex . Proposition 2.6. For any ∈ R, any ∈ [2, ∞) and any sufficiently smooth C 0 -valued function which is defined in ℋ ex [2, ] and decays sufficiently fast at space infinity, we have E ex,ℎ ( , ) + ∫ [2, ( ) ] \|[ ] − \| 2 d ex (2, ) + ∫ ∞ 2 * 0 ( + ) 1 (Σ ) d ex (2, ) + sup ∈ [2,∞) [ ex ( , )] 1 2 · ∫ ∞ 2 + 2 (Σ ) d , where ( ) = 2 +1 2 is as in (2.7), Σ1 2 E ex,ℎ ( , ) + 1 2 ∫ [2, ( ) ] \|[ ] − \| 2 2 − 1 2 d ≤ ex (2, ) + 2 ∫ ∞ 2 ∫ Σ \| * 0 \|d d , where := + and we use the following identity on ℋ : * − * 0 = 1 2 − 0 2 + 2 2 \| \| 2 . The proof is done. The energy estimate for the Dirac equation on the interior hyperboloids is stated as follows. Proposition 2.7. For any ∈ R, any ≥ 2 and any sufficiently smooth C 0 -valued function defined in ℋ in [2, ] , we have E in,ℎ ( , ) E in,ℎ (2, ) + ∫ [ (2) , ( ) ] \|[ ] − \| 2 d + ∫ 2 [E in,ℎ ( , )] 1 2 · + 2 (ℋ in ) d , where ( ) = 2 +1 2 is as in (2.7) and (2) Proof. Taking ≡ 1 in (2.34), integrating over the region ℋ in [2, ] and using the transformation ( , ) → ( , ), where = 2 − \| \| 2 , we obtain 1 2 E in,ℎ ( , ) = 1 2 E in,ℎ (2, ) + 1 2 ∫ [ (2) , ( ) ] \|[ ] − \| 2 2 − 1 2 d − ∫ 2 ∫ ℋ in ( * 0 − * 0 ) d d , where := + . Using Hölder inequality, the conclusion of the proposition follows. Sobolev and Hardy inequalities We give the following weighted Sobolev inequality in the exterior region. Lemma 2.1. Let Λ ∈ R. For any ∈ [2, ∞) and any sufficiently smooth function defined on Σ ex , we have sup Σ ex (2 + − ) Λ −1 \| ( , )\| 2 \| \| ≤ −1 ∫ Σ ex (2 + − ) Λ+1 \| Ω \| 2 + (2 + − ) Λ−1 \|Ω \| 2 d , sup Σ ex (2 + − ) Λ −1 \| ( , )\| 2 \| \| ≤ −1 ∫ Σ ex (2 + − ) Λ \| Ω \| 2 + \|Ω \| 2 d ,(2. 38) where we recall (2.16). Proof. We only give the proof for = 2. The proof for = 3 is similar (using spherical coordinates instead) and was given in Hence for ∈ Σ ex we have ( ( )) Λ 2 ( , ) ∫ ∞ ( ( )) Λ−1 2 + ( ( )) Λ+1 \| \| 2 d . (2.40) If is compactly supported in , we take = ( , , ) ( = 0, 1) in (2.40), integrate over S 1 and use (2.39), and then obtain sup Σ ex ( ( )) Λ \| ( , )\| 2 sup ≥ −1 0≤ ≤1 ∫ S 1 ( ( )) Λ \| ( , , )\| 2 d 0≤ ≤1 ∫ S 1 ∫ ∞ −1 ( ( )) Λ−1 \| ( , , )\| 2 + ( ( )) Λ+1 \| ( , , )\| 2 d d 0≤ ≤1 ∫ Σ ex ( ( )) Λ−1 \| ( , )\| 2 + ( ( )) Λ+1 \| ( , )\| 2 d . (2.41) In the general case where is not compactly supported in , we choose a cut-off function ∈ ∞ 0 (R) and apply (2.41) to ( ) for any > 0, then we derive sup Σ ex ( ( )) Λ \| ( ) ( , )\| 2 0≤ ≤1 ∫ Σ ex ( ( )) Λ−1 \| ( ) \| 2 + ( ( )) Λ+1 \| ( ) \| 2 + \| ′ ( ) \| 2 d . Note that on Σ ex we have \| ′ ( )\| 2 ≤ ( ( )) 2 \| ′ ( )\| 2 ≤ \| ′ ( )\| 2 1. Let → 0, and we obtain sup Σ ex ( ( )) Λ \| ( , )\| 2 0≤ ≤1 ∫ Σ ex ( ( )) Λ−1 \| \| 2 + ( ( )) Λ+1 \| \| 2 d . Taking the same weight ( ( )) Λ on the right hand side of (2.40) and arguing as above, we also obtain sup Σ ex ( ( )) Λ \| ( , )\| 2 0≤ ≤1 ∫ Σ ex ( ( )) Λ \| \| 2 + \| \| 2 d . We note that = 1 2 − 2 1 = Ω 12 , hence the conclusions of the lemma follow. (2 + − ) Λ 2 d ∫ Σ ex (2 + − ) Λ+2 \| \| 2 d . Proof. The proof is similar to [16,Lemma 4.5]. Using that (2 + − ) Λ+1 −1 ≥ (Λ + 1) (2 + − ) Λ −1 , we obtain ∫ Σ ex (2 + − ) Λ 2 ( , )d ≤ 1 Λ + 1 ∫ Σ ex (2 + − ) Λ+1 −1 2 ( , )d = − 1 Λ + 1 ∫ \| \|= −1 2 ( , )d ( ) − 2 Λ + 1 ∫ Σ ex (2 + − ) Λ+1 d ≤ 2 Λ + 1 ∫ Σ ex (2 + − ) Λ 2 d 1 2 · ∫ Σ ex (2 + − ) Λ+2 \| \| 2 d 1 2 . The conclusion of the lemma follows. We . ( (2 + − ) Λ \| \| 2 d \| \| ≤1 ∫ Σ ex (2 + − ) Λ+2 \| \| 2 d , ∈ {1, · · · , 2 }, ∫ Σ ex (2 + − ) Λ \| 0 \| 2 d \| \| ≤1 ∫ Σ ex (2 + − ) Λ+2 \| \| 2 d . Proof. The second inequality follows from the first one, using the relation 0 = ( − ) + ( − ) + ( / ) (2.43) and the fact that \| − \| 2 + − in ex . (2.44) The proof of the first inequality is as in [16,Corollary 4.6]. We choose a cut-off function ∈ ∞ 0 (R), apply Lemma 2.2 to ( ) for any > 0, and obtain ∫ Σ ex (2 + − ) Λ \| ( ) \| 2 d ∫ Σ ex (2 + − ) Λ+2 \| ( ) \| 2 + \| ′ ( ) \| 2 d ∫ Σ ex (2 + − ) Λ+2 \| \| 2 + \| \| 2 d , where we use that \| \| \| \| on Σ ex and that \| ′ ( )\| 2 1. Let → 0, and the proof is done. Lemma 2.4. Let > 0. For any ∈ [2, ∞) and any sufficiently smooth function defined on Σ ex , we have sup Σ ex (2 + − ) −1 \| \| 2 \| \| ≤ ∫ Σ ex (2 + − ) +1 \| \| 2 d , ∈ {1, · · · , 2 }, sup Σ ex (2 + − ) −1 −1 \| 0 \| 2 \| \| ≤ +1 ∫ Σ ex (2 + − ) +1 \| Γ \| 2 d , where we recall (2.15) and (2.42). Proof. For any fixed we denote ( ) := 2 + − . By Lemmas 2.1 and 2.3, we have sup Σ ex ( ( )) −1 \| \| 2 \| \| ≤ −1 ∫ Σ ex ( ( )) +1 \| Ω \| 2 + ( ( )) −1 \|Ω \| 2 d \| \| ≤ ∫ Σ ex ( ( )) +1 \| \| 2 d , sup Σ ex ( ( )) −1 −1 \| 0 \| 2 \| \| ≤ −1 ∫ Σ ex ( ( )) −1 \| Ω 0 \| 2 + \|Ω 0 \| 2 d \| \| ≤ −1 ∫ Σ ex ( ( )) −1 \| 0 Ω \| 2 + \| Ω \| 2 + \| 0 Ω \| 2 d \| \| ≤ +1 ∫ Σ ex ( ( )) +1 \| Γ \| 2 d , where we use the estimates on commutators in (2.17). The proof is done. We give the following Sobolev inequality on hyperboloids; see [21]. Proof. The first inequality follows from the identities below (recall (2.12)): = ¯ = − , = − ¯ = 2 2 − . The proof of the second inequality can be found in [13,Proposition 4.2]. We sketch it below. Let := 0 denote the 0 × 0 identity matrix and = ( , ) := ( / ) for 1 ≤ ≤ . By direct computation, [ , − 0 ] = −( 0 + ) ( − 0 ), [ , 0 + ] = − − . It follows that [ , − 0 ] = − ( 0 + ) ( − 0 ) = −( 0 + ) ( − 0 ) + ( + ) ( − 0 ) = −( 0 + ) ( − 0 ) − ( 0 + ) ( − 0 ) + ( + ) ( − 0 ) = −( 0 + ) ( − 0 ) + ( 0 + 0 + 2 ) ( − 0 ), [ , − 0 ] = [ , − 0 ] + [ , − 0 ] = −( 0 + ) ( − 0 ) − ( 0 + ) ( − 0 ) + ( 0 + 0 + 2 ) ( − 0 ). The proof is done. Proof. We first recall the estimate of \| \| below (see for example [22]) Estimates on good derivatives and the \|( − ) \| \| 0 \| + \| \|=1 \| \|. The estimate of \| \| follows from this and the identities below: = 1 ( + ( − ) ) = 1 − ( − ) . The proof is done. where we recall (2.11). Γ ( ) = 1 + 2 = (Γ 1 ) ( Γ 2 ), (2.45) Γ ( * 0˜ ) = 1 + 2 = ( Γ 1 ) * 0 ( Γ 2˜ ), (2.46) \| * 0˜ \| \|[ ] − \| · \|˜ \| + \| \| · \|[˜ ] − \|, (2.47) \| * 0˜ \| \|( ) − \| · \|˜ \| + \| \| · \|(˜ ) − \| + \| 2 − 2 \| 2 \| \| · \|˜ \| in { < },(2. Proof. ) Recall the definition (2.14). We have ( ) = ( ) + ( ), for 1 ≤ ≤ and similarly for Ω , 1 ≤ < ≤ . We also have Using (2.50), we obtain ( * 0˜ ) = ( ) * 0˜ + * 0 ˜ = ( ) * 0˜ + 1 2 * 0 0˜ + * 0 ˜ + 1 2 * 0 0 ˜ = ( ) * 0˜ + * 0 ˜ , for 1 ≤ ≤ , Ω ( * 0˜ ) = (Ω ) * 0˜ + * 0 Ω ˜ = ( Ω ) * 0˜ + 1 2 * 0˜ + * 0 Ω ˜ + 1 2 * 0 ˜ = ( Ω ) * 0˜ + * 0 Ω ˜ , for 1 ≤ < ≤ .[˜ ] − = − 0 = − 0 . The proof is done. Nonlinear transforms The following lemma is obtained by straightforward computation; see also [13,12]. (2.54) We next give estimates of the null forms in the above lemma. Lemma 2.10. For any sufficiently smooth functions and , we have \| \| −1 =1 \| \| · \| \| + \| \| · \| \| + \| − \| · \| \| · \| \| , \|Γ ( ) \| \|( Γ ) ( )\| + \|( ) ( Γ )\|, = 1, · · · , 0 , where we recall (2.14). Proof. The second inequality is well-known; see for example [22]. For the first one, using the relation =¯ − ( / ) where¯ = −1 , we have − = 1 ( + − ) − ¯ − = −1 ( + ) −¯ = −1 ( − ) + ( − ) + ( / ) ) −¯ = − + − + ¯ −¯ . The proof is done. Lemma 2.11. For any sufficiently smooth scalar function and C 0 -valued vector field , we have \| \| 2 + − \| \| ≤1 \| Γ \| + \| − + \| in ex ∩ { ≤ 3 }, \| \| 1 − \| \|=1 \|Γ \| + − \| \| in in . (2.55) Proof. We write the d'Alembert operator − as − = ( − ) ( + ) 2 + 2 − 1 + − 2 . Then the estimate of follows from (2.44). The estimate of \| \| was given in [13,Lemma 4.3]. Using the relation = −1 − ( / ) , we rewrite the identity := as 0 − = − 1 + . Multiplying on both sides of this equation by − 0 − yields 2 − 2 2 \| \| 1 =1 \| \| + \| \| in in , which implies \| \| 1 − =1 \| \| + − \| \| in in . Using the relation = −1 − ( / ) again, we obtain the same estimate for . A scattering lemma In this subsection, we prove a technical lemma which will be used in Section 6 to prove the scattering results in Theorems 1.1 and 1.2. Let ( , ) be the global solution to (1.1)-(1.2). We denote ì = ( , ) ′ = , where ( 1 , 2 ) ′ denotes the transpose of a vector ì = ( 1 , 2 ) ∈ R 2 . Let ì = (0, ) ′ , ì 0 = ( 0 , 1 ) ′ . (2.56) By the linear theory of Dirac and Klein-Gordon equations, we have = S ( − 2) 0 − ∫ 2 S ( − ) 0 ( )d , (2.57) ì =S ( − 2)ì 0 + ∫ 2S ( − ) ì ( )d ,(2.≤ 1 < 2 < ∞, we have ∫ 2 1 S ( − ) ( ( ))d (R ) − 2 1 =0 ∫ 2 1 ∇ 2 2 (Σ ex ) · 1+ d 1 2 + ∫ 2 1 2 1 ∇ 2 2 (ℋ in ) · 1+2 d 1 2 , ∫ 2 1S ( − ) ( ì ( ))d H (R ) − 2 1 =0 ∫ 2 1 \| ì ∇ ì \| + \| 1 \| 2 2 (Σ ex ) · 1+ d 1 2 + ∫ 2 1 2 1 \| ì ∇ ì \| + \| 1 \| 2 2 (ℋ in ) · 1+2 d 1 2 , where \| ì ∇ ì \| := \|∇ +1 1 \| + \|∇ 2 \| and ∇ = ( 1 , · · · , ). ii) Let ∈ N and , ì , , ì , 0 , ì 0 be as in (2.56), (2.57) and (2.58) with 0 ∈ (R ), ì 0 ∈ H (R ) and ( ), ( ) ∈ (R ) for any fixed ∈ [2, ∞). If for some > 0, it holds that := ∫ 4 2 ( , ·) 2 (R ) d + ∫ +∞ 4 ( , ·) 2 2 (Σ ex ) · 1+ d + ∫ +∞ 2 2 2 (ℋ in ) · 1+2 d 1 2 < ∞, where := =0 \|∇ \| + \|∇ \| , then the solution ( , ) scatters to a free solution in (R ) × H (R ), i.e., there exist * 0 ∈ (R ) and ì * 0 = ( * 0 , * 1 ) ′ ∈ H (R ) such that lim →+∞ − * (R ) = 0, lim →+∞ ì − ì * H (R ) = 0, where ì * = ( * , * ) ′ , and ( * , * ) is the solution to * + * = 0, ( , ) ∈ [2, ∞) × R , − * + * = 0, ( , ) ∈ [2, ∞) × R , ( * , * , * )\| =2 = ( * 0 , * 0 , * 1 ). Proof. ) We only need to consider the case = 0. For any fixed ≥ 2, let ì ( ) :=S ( − ) ( ì ( )) = ( 1 , 2 ) ′ . For any 4 ≤ 1 < 2 < ∞, by standard energy inequalities, we have 1 2 . ∫ 2 1 ì ( )d H 0 (R ) ∫ R ∫ 2 1 ∇ 1 ( )d 2 + ∫ 2 1 2 ( )d 2 + ∫ 2 1 1 ( )d 2 d 1 2 ∫ R ∫ 2 1 \|∇ 1 \| + \| 2 \| + \| 1 \| 2 ( , ) · 1+ d · ∫ 2 1 −(1+ ) d d 1 2 − 2 1 ∫ 2 1 ∫ R \|∇ 1 \| + \| 2 \| + \| 1 \| 2 ( , ) · 1+ d d 1 2 := − 2 1 ∫ 2 1 ∫ ≥ −1 + ∫ < −1 \|∇ 1 \| + \| 2 \| + \| 1 \| 2 ( , ) · 1+ d d Denote˜ := \|∇ 1 \| + \| 2 \| + \| 1 \|. By a change of variables ( , ) → ( , ) with = 2 − \| \| 2 , we obtain − 2 1 ∫ 2 1 ∫ Σ ex \|˜ \| 2 ( , ) · 1+ d d 1 2 + − 2 1 ∫ 2 1 2 1 ∫ < 2 −1 2 \|˜ \| 2 ( 2 + \| \| 2 , ) · 1+ d d 1 2 − 2 1 ∫ 2 1 ˜ 2 2 (Σ ex ) · 1+ d 1 2 + ∫ 2 1 2 1 ˜ 2 2 (ℋ in ) · 1+2 d 1 2 . (2.59) Similarly, ∫ 2 1 S ( − ) ( ( ))d 2 (R ) − 2 1 ∫ 2 1 2 2 (Σ ex ) · 1+ d 1 2 + ∫ 2 1 2 1 2 2 (ℋ in ) · 1+2 d 1 2 . ) Let * 0 : = 0 − ∫ +∞ 2 S (2 − ) 0 ( )d , * = S ( − 2) * 0 , ì * 0 : = ì 0 + ∫ +∞ 2S (2 − ) ì ( )d , ì * =S ( − 2)ì * 0 . For any 4 ≤ 1 < 2 < ∞, by ), we have ∫ − * (R ) = ∫ +∞ S ( − ) 0 ( )d (R ) − 2 → 0 as → +∞, ì − ì * H (R ) = ∫ +∞ S ( − ) ì ( )d H (R ) − 2 → 0 as → +∞. The proof is done. = 2: Global existence and estimates in the exterior region In this section, we prove global existence of the solution to (1.8)-(1.9) in the exterior region. Bootstrap setting Fix + \| \| ≤ −1 Γ ex, 1,0, ≤ 1 ,(3.2) where (see ( In the above proposition is arbitrary, hence the solution ( , ) exists globally in time in ex (i.e., * = ∞ where * is as in (3.6)) and satisfies (3.2) for all ∈ [2, ∞). Below we give the proof of Proposition 3.1. In the sequel, the implied constants in do not depend on the constants 1 and appearing in the bootstrap assumption (3.2). Let ( ) : = ( 1 ) −2 \| \| ≤ ∫ Σ ex (2 + − ) \|[ Γ ] − \| 2 + − \|Γ \| 2 d + \| \| ≤ −1 ∫ Σ ex (2 + − ) \|Γ \| 2 d .                               \| \| ≤ (2 + − ) 1+ 2 \| Γ \| + − 2 \| Γ \| + \|Γ \| 2 (Σ ex ) 1 , \| \| ≤ −1 (2 + − ) 1+ 2 \| Γ \| + \|Γ \| 2 (Σ ex ) 1 , \| \| ≤ (2 + − ) 2 \|[ Γ ] − \| + − 2 \|Γ \| 2 (Σ ex ) 1 ( ), ∈ [2, ], \| \| ≤ −1 (2 + − ) 2 Γ 2 (Σ ex ) 1 ( ), ∈ [2, ].Ω 12 , 0 = 1 2 − 2 1 − 1 2 1 2 , 0 = 1 2 − 2 3 0 − 2 1 − 1 3 0 − 1 2 1 2 0 − 0 1 2 = 1 0 2 − 2 0 1 − 1 2 0 1 (−2 2 − 2 ) − 0 (−2 1 − 1 ) 2 = 0, hence the claim holds. Then by (3.9), we have We note that on Σ ex we have \| \| ≤ −2, \| \| ≤1 (2 + − ) 2 Ω [ Γ ] − 2 (Σ ex ) + (2 + − ) 2 Ω [ Γ ] − 2 (Σ ex ) \| \| ≤ −2, \| \| ≤1 (2 + − ) 2 [ Ω Γ ] − 2 (Σ ex ) + (2 + − ) 2 [ Ω Γ ] − 2 (Σ ex ) \| \| ≤ (2 + − ) 2 [ Γ ] − 2 (Σ ex ) 1 ( ), ∈ [2, ].(2 + − ) 1− 2 max{ 1− 2 , 1}. (3.12) By (3.11), we obtain              \| \| ≤ −3 sup Σ ex (2 + − ) 0 \| Γ \| + \| Γ \| + \|Γ \| 1 , \| \| ≤ −3 sup Σ ex (2 + − ) 1 2 0 \|[ Γ ] − \| + \|Γ \| 1 ( ), ∈ [2, ], (3.13) where 0 is as in (3.1). By Lemma 2.11, (3.11) and (3.13), on Σ ex ∩ { ≤ 3 }, we have \| \| ≤ −4 \|Γ \| 2 + − \| \| ≤ −3 \| Γ \| + \| \| ≤ −4 \|Γ ( * 0 )\| 2 + − \| \| ≤ −3 \| Γ \| + \| 1 \|+\| 2 \| ≤ −4 \| Γ 1 \| · \| Γ 2 \| 1 −1− 0 + ( 1 ) 2 (2 + − ) − −1 −1 1 −1 . (3.14) By ( \| \| ≤ −4 \|Γ \| 2 + − \| \| ≤ −3 \| Γ \| + \| \| ≤ −4 \|Γ ( * 0 )\| 2 + − \| \| ≤ −3 \| Γ \| + \| 1 \|+\| 2 \| ≤ −4 \|[ Γ 1 ] − \| · \| Γ 2 \| 1 −1− 0 + ( 1 ) 2 (2 + − ) − − 1 2 ( ) −1 1 −1− 0 + (2 + − ) − 1 2 ( ) −1 . (3.15) On the other hand, on Σ ex ∩ { ≥ 3 } we have 2 + − ∼ , hence by (3.11), we have \|Γ \| 1 −1− 2 , \| \| ≤ − 4. Improved estimates for the solution ( , ) in the exterior region Step 1. First, we refine the energy estimate of . Acting the vector fields Γ , \| \| ≤ on both sides of the first equation in (1.8) and applying Proposition 2.5, we obtain Γ 2 ex, , Γ 2 ex, ,2 + ∫ 2 (2 + − ) 1+ ( Γ ) * 0 Γ ( ) 1 (Σ ex ) d ,(2 + − ) 1+ ( Γ ) * 0 Γ ( ) 1 (Σ ex ) \| 1 \|+\| 2 \| ≤ (2 + − ) 1+ \|[ Γ ] − \| · \|Γ 1 \| · \| Γ 2 \| + \| Γ \| · \|Γ 1 \| · \|[ Γ 2 ] − \| 1 (Σ ex ) \| 1 \| ≤ −4 \| ′ \|, \| 2 \| ≤ (2 + − ) 2 [ Γ ′ ] − 2 (Σ ex ) · (2 + − ) 1 2 Γ 1 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 2 2 (Σ ex ) + \| 2 \| ≤ −3 \| ′ \|, \| 1 \| ≤ (2 + − ) 2 [ Γ ′ ] − 2 (Σ ex ) · (2 + − ) 2 Γ 1 2 (Σ ex ) · (2 + − ) Γ 2 ∞ (Σ ex ) + \| 2 \| ≤ −3 \| ′ \|, \| 1 \| ≤ (2 + − ) 1+ 2 Γ ′ 2 (Σ ex ) · (2 + − ) 2 Γ 1 2 (Σ ex ) · (2 + − ) 1 2 [ Γ 2 ] − ∞ (Σ ex ) ( 1 ) 3 ( ) − 1 2 − 0 + ( ) −1 + ( 1 ) 3 ( ) − 0 + 2 , (3.19) where we use (3.9), (3.18) and (3.13). Recall that 0 < ≪ 0 . Hence by (3.8), we have \| \| ≤ Γ ex, , + ( 1 ) 3 2 . (3.20) Step 2. We now refine the highest order energy estimate of . Acting the vector fields Γ , \| \| ≤ on both sides of the second equation in (1.8) and applying Proposition 2.2, we obtain Γ ex, 1, , Γ ex, 1, ,2 + ∫ 2 − 2 (2 + − ) 1+ 2 Γ ( * 0 ) 2 (Σ ex ) d , where we recall (3.4). For \| \| ≤ and ∈ [2, ], by (2.46) and (2.47) in Lemma 2.8, we have − 2 (2 + − ) 1+ 2 Γ ( * 0 ) 2 (Σ ex ) \| 1 \|+\| 2 \| ≤ − 2 (2 + − ) 1+ 2 \|[ Γ 1 ] − \| · \| Γ 2 \| 2 (Σ ex ) \| 1 \| ≤ −3 \| 2 \| ≤ − 2 [ Γ 1 ] − ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 2 2 (Σ ex ) + \| 2 \| ≤ −3 \| 1 \| ≤ [ Γ 1 ] − 2 (Σ ex ) · − 2 (2 + − ) 1+ 2 Γ 2 ∞ (Σ ex ) ( 1 ) 2 ( ) − 1 2 − 2 ,(3.21) where we use (3.9) and (3.11). Hence (3.8) gives \| \| ≤ Γ ex, 1, , + ( 1 ) 2 . (3.22) Step 3. Next we refine the lower order energy estimates of . Let˜ := − * 0 . For \| \| ≤ − 1, by ) in Lemma 2.9 and Proposition 2.2, we obtain Γ ˜ ex, 1,0, Γ ˜ ex, 1,0,2 + ∫ 2 (2 + − ) 1+ 2 Γ ˜ 2 (Σ ex ) d , where Γ ˜ ex, 1,0, = [ ex, 1 ( , Γ ˜ )] 1 2 + ∫ 2 ∫ Σ ex (2 + − ) \| Γ ˜ \| 2 + \|Γ ˜ \| 2 d d 1 2 (3.23) and (see (2.52))˜ := − ( * ) 0 + * 0 ( ) + 2 * 0 . (3.24) Here we recall (2.19) for the definition of ex, 1 ( , Γ ˜ ). For \| \| ≤ − 1 and ∈ [2, ], by (3.9) and (3.11), we have (2 + − ) 1+ 2 Γ ( * ) 0 2 (Σ ex ) \| 1 \|+\| 2 \|+\| 3 \| ≤ (2 + − ) 1+ 2 \|Γ 1 \| · \|Γ 2 \| · \|Γ 3 \| 2 (Σ ex ) \| 1 \|, \| 2 \| ≤ −3 \| 3 \| ≤ Γ 1 ∞ (Σ ex ) · Γ 2 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 3 2 (Σ ex ) + \| 2 \|, \| 3 \| ≤ −3 \| 1 \| ≤ Γ 1 2 (Σ ex ) · Γ 2 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 3 ∞ (Σ ex ) ( 1 ) 3 ( ) −1+ 2 . (3.25) For \| \| ≤ − 1 and ∈ [2, ], by Lemma 2.10, we obtain (2 + − ) 1+ 2 Γ * 0 2 (Σ ex ) \| 1 \|+\| 2 \| ≤ −1 (2 + − ) 1+ 2 ( Γ 1 ) * 0 Γ 2 2 (Σ ex ) \| 1 \|+\| 2 \| ≤ −1 2 =1 −1 (2 + − ) 1+ 2 \| Γ 1 \| · \| Γ 2 \| + \| − \| · \| Γ 1 \| · \| Γ 2 \| 2 (Σ ex ) \| 1 \|+\| 2 \| ≤ −1 \| \|= \| ′ \|=1 −1 (2 + − ) 1+ 1+ 2 \|Γ Γ 1 \| · \|Γ ′ Γ 2 \| 2 (Σ ex ) \| 1 \| ≤ −4 \| \|=1, \| 2 \| ≤ −1 (2 + − )Γ Γ 1 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 2 2 (Σ ex ) ( 1 ) 2 −1− 0 , (3.26) where we use (2.44), (3.9) and (3.13). The estimates (3.25) and (3.26) yield (2 + − ) 1+ 2 Γ ˜ 2 (Σ ex ) ( 1 ) 2 ( ) −1+ 2 + −1− 0 , \| \| ≤ − 1, ∈ [2, ],(3.+ ( 1 ) 2 . (3.28) We recall that˜ = − * 0 . For \| \| ≤ − 1, using (2.19), (3.9) and (3.11), we have [ ex, 1 ( , Γ ( * 0 ))] 1 2 \| ′ \| ≤ (2 + − ) 1+ 2 Γ ′ ( * 0 ) 2 (Σ ex ) \| 1 \| ≤ −3, \| 2 \| ≤ Γ 1 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 2 2 (Σ ex ) ( 1 ) 2 . By (2.46) and (2.47) in Lemma 2.8, we also have \| ′ \| ≤ ∫ 2 (2 + − ) 2 Γ ′ ( * 0 ) 2 2 (Σ ex ) d 1 2 \| 1 \|+\| 2 \| ≤ ∫ 2 (2 + − ) 2 \|[ Γ 1 ] − \| · \| Γ 2 \| 2 2 (Σ ex ) d 1 2 \| 1 \| ≤ −3, \| 2 \| ≤ ∫ 2 (2 + − ) 2 [ Γ 1 ] − 2 ∞ (Σ ex ) · Γ 2 2 2 (Σ ex ) d 1 2 + \| 2 \| ≤ −3, \| 1 \| ≤ ∫ 2 (2 + − ) 2 [ Γ 1 ] − 2 2 (Σ ex ) · Γ 2 2 ∞ (Σ ex ) d 1 2 ( 1 ) 2 . Recall ( (2 + − ) 1+ 2 Γ ( * 0 ) 2 (Σ ex ) \| 1 \|+\| 2 \|+\| 3 \| ≤ −1 (2 + − ) 1+ 2 \|[ Γ 1 ] − \| · \| Γ 2 \| · \| Γ 3 \| 2 (Σ ex ) \| 1 \|, \| 2 \| ≤ −3 \| 3 \| ≤ −1 [ Γ 1 ] − ∞ (Σ ex ) · Γ 2 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 3 2 (Σ ex ) + \| 2 \|, \| 3 \| ≤ −3 \| 1 \| ≤ −1 [ Γ 1 ] − 2 (Σ ex ) · (2 + − ) 1+ 2 Γ 2 ∞ (Σ ex ) · Γ 3 ∞ (Σ ex ) ( 1 ) 3 ( ) −1 (3.31) and (2 + − ) 1+ 2 Γ ( ) 2 (Σ ex ) \| 1 \|+\| 2 \|+\| 3 \| ≤ (2 + − ) 1+ 2 \|Γ 1 \| · \|Γ 2 \| · \|Γ 3 \| 2 (Σ ex ) \| 1 \|, \| 2 \| ≤ −3 \| 3 \| ≤ Γ 1 ∞ (Σ ex ) · Γ 2 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 3 2 (Σ ex ) + \| 2 \|, \| 3 \| ≤ −3 \| 1 \| ≤ Γ 1 2 (Σ ex ) · Γ 2 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 3 ∞ (Σ ex ) ( 1 ) 3 ( ) −1+ 2 ,(3.32) where we use (3.9) and (3.11). For \| \| ≤ − 1 and ∈ [2, ], by Lemma 2.10, we have (2 + − ) 1+ 2 Γ 2 (Σ ex ) \| 1 \|+\| 2 \| ≤ −1 \| \|= \| ′ \|=1 −1 (2 + − ) 1+ 2 \|Γ Γ 1 \| · \|Γ ′ Γ 2 \| + \| − \| · \| Γ 1 \| · \| Γ 2 \| 2 (Σ ex ) \| 1 \|+\| 2 \| ≤ −1 \| \|= \| ′ \|=1 −1 (2 + − ) 1+ 1+ 2 \|Γ Γ 1 \| · \|Γ ′ Γ 2 \| 2 (Σ ex ) \| 1 \| ≤ −4 \| \|=1, \| 2 \| ≤ −1 (2 + − )Γ Γ 1 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 2 2 (Σ ex ) + \| 2 \| ≤ −4 \| ′ \|=1, \| 1 \| ≤ (2 + − ) 1+ 2 Γ 1 2 (Σ ex ) · −1 (2 + − )Γ ′ Γ 2 ∞ (Σ ex ) ( 1 ) 2 −1− 0 + 2 ,(3.\| ′ \| ≤ (2 + − ) 1+ 2 Γ ′ ( ) 2 (Σ ex ) \| 1 \| ≤ −3 \| 2 \| ≤ Γ 1 ∞ (Σ ex ) · (2 + − ) 1+ 2 Γ 2 2 (Σ ex ) + \| 2 \| ≤ −3 \| 1 \| ≤ − 2 (2 + − ) 1+ 2 Γ 1 2 (Σ ex ) · 2 Γ 2 ∞ (Σ ex ) ( 1 ) 2 . This together with (3.35) gives the following estimate for all ∈ [2, ∞): [ ex, 0 ( , Γ Ψ)] 1 2 + ( 1 ) 2 , \| \| ≤ − 1. (3.36) By (3.1), (3.12) and Lemma 2.4, we have for all ∈ [2, ∞), where˜ is as in (3.30) and we use (3.34). \| \| ≤ −4 \| \|=1 sup Σ ex 0 \| 0 Γ Ψ\| + \| Γ Ψ\| \| \| ≤ −4 \| \|=1 sup Σ ex (2 + − ) −1 2 1 2 \| 0 Γ Ψ\| + \| Γ Ψ\| \| ′ \| ≤ −1 (2 + − ) 1+ 2 Γ ′ Ψ 2 (Σ ex ) + ( 1 ) 2 (3.37) for all ∈ [2, ∞),\|( Γ ) − \| = [ Γ ] − + − 0 Γ \|[ Γ ] − \| + \| − \| \| Γ Ψ\| \|[ Γ ] − \| + −1 \| \|=1 \| 0 Γ Ψ\| + \| Γ Ψ\| [ + ( 1 ) 2 ] − 0 −1 , \| \| ≤ − 4. Estimates of the solution ( , ) on truncated exterior hyperboloids Step 1. First, we derive the highest order energy estimates of ( , ) on exterior hyperboloids. By Propositions 2.6 and 2.3, for \| \| ≤ and any ∈ [2, ∞), we have In the above proposition the hyperbolic time˜ is arbitrary, hence the solution ( , ) exists globally in the interior region in (i.e.,˜ * = ∞ where˜ * is as in (4.3)) and satisfies the estimates (4.1)-(4.2) for all ∈ [2, ∞). Below we give the proof of Proposition 4.1. In the sequel, the implied constants in do not depend on the constants 2 and appearing in the bootstrap assumptions (4.1)-(4.2). We recall the definitions (2.36), (2.37), (2.28), (2.8) and that ℋ = ℋ in ∪ ℋ ex by (2.4)-(2.6). By where ( ) = 2 + 1 2 ,(2)Γ ( ) 2 (ℋ in ) \| 1 \|+\| 2 \| ≤ \|Γ 1 \| · \| Γ 2 \| 2 (ℋ in ) \| 1 \| ≤ −3 \| 2 \| ≤ ( / )Γ 1 ∞ (ℋ in ) · ( / ) Γ 2 2 (ℋ in ) + \| 2 \| ≤ −3 \| 1 \| ≤ Γ 1 2 (ℋ in ) · Γ 2 ∞ (ℋ in ) ( 2 ) 2 −1+ . (4.9) Note that the estimate of the second term on the right hand side of (4.7) was given in (3.42 ∈ [2, ∞) × R , − + = := * 0 , ( , ) ∈ [2, ∞) × R (1.1) with prescribed initial data at = 2 ( , , )\| =2 = ( 0 , 0 , 1 ). (1.2) + ( / ) , ∈ {1, 2} denote the good derivatives. The energy functionals on the truncated hyperboloids ℋ ex and ℋ in (see (2.6) and (2.5)) are controlled in terms of the energy on flat time slices in the exterior region above.The bootstrap assumption in the exterior region is the bound for the following quantity We decompose the space-time R 1+ into two regions: interior region in := {( , ) : ≥ 2, < − 1}, exterior region ex := {( , ) : ≥ 2, ≥ − 1}. [ 2 , 2] the portion of exterior region below ℋ ex , i.e.,ℋ in [2, ] := {( , ) ∈ in : 2 2 ≤ 2 − \| \| 2 ≤ 2 }, ℋ ex [2, ] := {( , ) ∈ ex : 2 − \| \| 2 ≤ 2 }.For any sufficiently smooth function = ( , ) defined on a hyperboloid ℋ , we denote ∫ index set Θ, a finite set { : ∈ Θ} of linear operators and a linear operator , we write = ′ ∈Θ if there exist some constants such that we have = ∈Θ . Proposition 2. 2 . 2Let > 0 and ≥ 0. For any ≥ 2 and any sufficiently smooth function which is defined in the region ex and decays sufficiently fast at space infinity, we have ex, , , state the energy estimates for the Dirac equation on flat time slices in the exterior region. Proposition 2.5. Let > 0. For any ∈ R, any ∈ [2, ∞) and any sufficiently smooth C 0 -valued function which is defined in ex and decays sufficiently fast at space infinity, we have 2 ex, , for any sufficiently smooth function = ( , ) : [2, ∞) × R → C 0 which decays sufficiently fast at space infinity.For any C 0 -valued function = ( , ), we denote = 5 2 2, and we recall (2.3) for the definition of [ (2), ( ) ] . [ 16 ,R 2 162Lemmas 4.1 and 4.2]. Let ( , ) be the polar coordinates in fixed we denote ( ) := 2 + − . Let be any smooth function which is compactly supported in , then in the region { ≥ − 1} we have ( ( )) Λ 2 ≥ Λ( ( )) Λ−1 2 + 2( ( )) Λ . Lemma 2. 2 . 2Let Λ > −1. For any ∈ [2, ∞) and any sufficiently smooth function which is defined on Σ ex and is compactly supported in , we have ∫ Σ ex denote { } 2 =1 := {( ) 1≤ ≤ , (Ω ) 1≤ < ≤ }, where 2 := ( 2 + )/2. For any multi-index = ( 1 , · · · , 2 ) ∈ N 2 of length \| \| = 2 =1, we denote Lemma 2. 5 . 5For any sufficiently smooth function defined in the cone := { < } and all ( , ) ∈ , we have \| ( , 2 , ( , /3) ⊂ R denotes the ball centered at with radius /3, and we recall (2.16). Lemma 2.6. For any sufficiently smooth scalar function and C 0 -valued function (recall (2.1)), the following estimates hold in the region := { < }: we recall (2.35), (2.2) and (2.16). 0 -structure Lemma 2. 7 . 7For any sufficiently smooth function defined in [2, ∞) × R , we have we recall (2.11) and (2.42). ) Let : [2, ∞) × R → R, ,˜ : [2, ∞) × R → C 0 be sufficiently smooth functions. Then 48)for any multi-index ∈ N 0 , where we recall (2.15), (2.32) and (2.35).) For any sufficiently smooth function : [2, ∞) × R → C 0 , let˜ := Lemma 2. 9 .)) 9Let ( , ) be the solution to (1.1)-(1.2) with = 2 and = 0. Then the following statements hold: ) Let˜ := − ( ). Then˜ solves the equation ˜ =˜ := ( * 0 Let˜ := − * 0 . Then˜ solves the equation − ˜ +˜ =˜ := − ( Let Ψ be the solution to Then we have = Ψ. LetΨ := Ψ − . ThenΨ solves the equation Ψ =˜ = ( * 0 ) − ( ) − 2 . is the propagator for the linear Dirac equation, and S ( ) = cos( ∇ ) sin( ∇ ) ∇ − ∇ sin( ∇ ) cos( ∇ ) . Let ∈ N. We denote H (R ) := +1 (R ) × (R ). Lemma 2.12. The following statements hold: i) Let ∈ N and > 0. For any scalar function and R 2 -valued function ì ( , ) = ( 1 , 2 ) ′ which are defined in [2, ∞) × R and satisfy ( , ·) ∈ (R ), ì ( , ·) ∈ H (R ) for any fixed ∈ [2, ∞), and any ≥ 2, 4 and H (R ) respectively. Similarly, using ) again, we have ≥ 6 , 6> chosen later. We introduce the following bootstrap setting for the solution ( , ) to ( recall (2.11) and (2.32) for the definitions of \| Γ \| and [ Γ ] − . We also recall (2.30), (2.18) and (2.19) for the definitions of the energy functionals above. Let * := sup{ > 2 : (3.2) holds for ∈ [2, ]}. (3.6) Proposition 3.1. There exist some constants 1 > 0 sufficiently large and 0 < 0 ≪ −1 1 sufficiently small such that, for any 0 < < 0 , if ( , ) is a solution to (1.8)-(1.9) in a time interval [2, ] and satisfies (3.2), then for ∈ [2, ] we have ∈ [2, ]. By (3.2), we obtain the following 2 -type estimates for the solution ( , ) on the time interval [2, ]: 0 . 0and (2.16). We claim that [ , ( / ) 0 ] = [ Ω 12 , ( / ) 0 ] = Indeed and (2.38) in Lemma 2.1, we obtain the following pointwise estimates for the solution ( , ) on [2, ]: Σ ex , for ∈ [2, ], \| \| ≤ − 4. (3.18) to close the bootstrap in the interior region in Section 4, we give the following estimate of the function˜ := − ( ). For \| \| ≤ − 1, by ) in Lemma 2.9 and Proposition 2. E. 1 . 1ex,ℎ ( , Γ ) + ∫ [2, ( ) ] \|[ Γ ] − \| 2 d ex (2, Γ ) + ∫ ∞ 2 ( Γ ) * 0 Γ ( ) 1 (Σ ex ) we use (3.19), (3.21) and (3.22) (recall (3.4) and (2.18)), and recall (2.37), (2.31), (2.26) and (2.20) for the definitions ofE ex,ℎ ( , Γ ), ex (2, Γ ), E ex,ℎ 1, ( , Γ ) and ex 1, ( , Γ ).Step 2. Next we show lower order energy estimates of on exterior hyperboloids. Let˜ = − * 0 be as in Step 3 in Section 3.2. By Proposition 2.3, for \| \| ≤ − 1 and all ∈ [2, ∞), we have There exist some constants 2 > 0 sufficiently large and 0 < 1 ≪ −1 2 sufficiently small such that for any 0 < < 1 , if ( , ) is a solution to (1.8)-(1.9) and satisfies the exterior estimate(3.2) globally in time as well as the interior bounds (4.1)-(4.2) for all ∈ [2,˜ ], then we have the following improved interior estimates for all ∈ [ (3.42),(3.45), (4.1) and (4.2), we obtain the following 2 -type estimates for the solution ( , ), Improved estimates for the solution ( , ) in the interior regionStep 1. First, we refine the highest order energy estimates of and . By Proposition 2.7, for \| \| ≤ , we haveE in,ℎ ( , Γ ) E in,ℎ (2, Γ ( ) 2 (ℋ in ) d ,(4.7) Remark 2.2. Taking ≡ 1 in (2.34) and integrating over [2, ] × R , we obtain the standard energy inequality for the Dirac equation, i.e., Proof. Taking ≡ 1 in (2.34) and integrating over the region ℋ ex[2, ] , we obtainis as in (2.29), and we recall (2.3), (2.31) and (2.32). 3.5) and(3.23). The last two estimates together with (3.28) give sufficiently large and 0 < ≪ −1 1 sufficiently small. Hence the proof of Proposition 3.1 is completed. We conclude that the solution ( , ) exists globally in the exterior region ex and satisfies(3.2) for all ∈ [2, ∞). We next give an improved pointwise estimate of [ Γ ] − for \| \| ≤ − 4. Let Ψ solve (2.53) and Ψ := Ψ − . For \| \| ≤ − 1, by ) in Lemma 2.9 and Proposition 2.2, we obtain For \| \| ≤ − 1 and ∈ [2, ], by (2.45), (2.46) and (2.47) in Lemma 2.8, we have\| \| ≤ −1 Γ ex, 1,0, + ( 1 ) 2 . (3.29) Combining (3.20), (3.22) and (3.29), we have strictly improved the bootstrap estimate (3.2) for 1 [ ex, 0 ( , Γ Ψ )] 1 2 [ ex, 0 (2, Γ Ψ )] 1 2 + ∫ 2 (2 + − ) 1+ 2 Γ ˜ 2 (Σ ex ) d for all ∈ [2, ∞), where we recall (2.19) for the definition of ex, 0 ( , Γ Ψ ) and (see (2.54)) := ( * 0 ) − ( ) − 2 (3.30) We recall thatΨ = Ψ − . For \| \| ≤ − 1, by (3.9), (3.11) and the definition (2.19), we have33) where we use (2.44), (3.9) and (3.13). Hence (3.31)-(3.33) yield (2 + − ) 1+ 2 Γ ˜ 2 (Σ ex ) ( 1 ) 2 ( ) −1+ 2 + −1− 0 + 2 , \| \| ≤ − 1, ∈ [2, ]. (3.34) Recall that 0 < ≪ 0 . Then the estimates (3.34) and (3.8) imply that for all ∈ [2, ∞) [ ex, 0 ( , Γ Ψ )] 1 2 + ( 1 ) 2 , \| \| ≤ − 1. (3.35) [ ex, 0 ( , Γ ( ))] 1 2 where we recall (2.42). By ) in Lemma 2.9, we have Then by (2.49) in Lemma 2.8, Lemma 2.7 and (3.37), we have the following estimate in the region ex :\|[ Γ ] − \| \| Γ Ψ\|Γ = Γ Ψ. (3.38) −1 \| \|=1 \| 0 Γ Ψ\| + \| Γ Ψ\| [ + ( 1 ) 2 ] −1− 0 , \| \| ≤ − 4. (3.39) We recall (2.32) and (2.35). By (3.38), Lemma 2.7, (3.39) and (3.37), in the region ex , it holds that By Proposition 2.4, for \| \| ≤ , we have). It follows that E in,ℎ ( , Γ ) 2 + ( 1 ) 3 + ( 2 ) 2 sup ∈ [2, ] [E in,ℎ ( , Γ )] 1 2 , \| \| ≤ , which implies \| \| ≤ [E in,ℎ ( , Γ )] 1 2 + ( 2 ) 3 2 . (4.10) E in,ℎ 1 ( , Γ ) E in,ℎ 1 (2, Γ ) + ∫ [ (2) , ( ) ] \| Γ \| 2 + \|Γ \| 2 d + ∫ 2 and Arguing as in (3.13), we have \| \| ≤ −3 sup Σ ex (2 + − ) 1 2 + 0 \| Γ \| + \| Γ \| + \|Γ \| 1 , (5.7) where 0 := 1 2 min{ , 1}. Hence similar to (3.17) (see (3.14) and (3.16)), by (5.6), (5.7) and Lemma 2.11, we have \|Γ \| 1 − 3 2 on Σ ex , for \| \| ≤ − 4. (5.8) By Propositions 2.5 and 2.2, for \| \| ≤ , we have Γ ex, , ,˜ is as in(3.24), and we use(3.27) and (3.28) (recall (3.23) and (2.19)). We also recall (2.28) and (2.21) for the definitions of E ex,ℎ 1 ( , Γ ˜ ) and ex 1 ( , Γ ˜ ). For \| \| ≤ − 1, by (2.28), and (2.46) and (2.48) in Lemma 2.8, we have (ℋ in ) ( 2 ) 2 , where we use (3.40),(3.42)and(3.11), and recall the definition(2.37). This together with(3.44)gives(3.45)= 2: Global existence and estimates in the interior regionIn this section, we prove global existence of the solution to (1.8)-(1.9) in the interior region.Bootstrap assumptionsLet ≥ 6, 0 < ≪ 0 and 1 ≫ 1 be as in Section 3.1. Let 2 ≫ 1 and 0 < ≪ −1 2 be chosen later. The bootstrap assumptions in the interior region are the following energy bounds for the solution ( , ) to (1.8)-(1.9):where we recall (2.36) and (2.28) for the definitions of the energy functionals above. Let * := sup{˜ > 2 :where ( ),(2)are as in(4.8We recall (2.3) and note that on [2, ( ) ] we have −2 − . Hence by (4.11), (3.43) and (4.12), we have(4.13)Step 2. Next, we refine the lower order energy estimates of . Let˜ := − ( ). For \| \| ≤ − 1, by ) in Lemma 2.9 and Proposition 2.7, we havewhere ( ),(2)are as in (4.8) and˜ is as in (3.30), i.e., We point out that the estimate of the second term on the right hand side of (4.14) was given by (3.41). It follows thatWe recall the definition (2.36) and that˜ = − ( ). We note that(4.20)Step 3. We turn to the lower order energy estimates of . Let˜ := − * 0 . For \| \| ≤ − 1, by ) in Lemma 2.9 and Proposition 2.4, we havewhere ( ),(2)are as in(4.8)and˜ is as in(3.24), i.e.,For \| \| ≤ − 1 and ∈ [2, ], we havewhere we use (4.4), (4.6) and(4.22). Combining the last two estimates, we obtainWe observe that the estimate of the second term on the right hand side of (4.21) was given by(3.44). It follows thatWe recall the definition (2.28) and that˜ = − * 0 . By (2.46) and (2.48) in Lemma 2.8, we haveand we recall (2.30) and(2.19). 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I Math. 2922Choquet-Bruhat, Y.: Solutions globales des équations de Maxwell-Dirac-Klein-Gordon (masses nulles). (French) C. R. Acad. Sci. Paris Sér. I Math. 292 (2) (1981), 153-158. Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system. P D'ancona, D Foschi, S Selberg, J. Eur. Math. Soc. 94D'Ancona, P., Foschi, D., Selberg, S.: Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system. J. Eur. Math. Soc. 9 (4) (2007), 877-899. Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions. J Delort, D Fang, R Xue, J. Funct. Anal. 2112Delort, J., Fang, D., Xue, R.: Global existence of small solutions for quadratic quasilinear Klein- Gordon systems in two space dimensions. J. Funct. Anal. 211 (2) (2004), 288-323. Asymptotic behavior of the solution to the Klein-Gordon-Zakharov model in dimension two. S Dong, Comm. Math. Phys. 3841Dong, S.: Asymptotic behavior of the solution to the Klein-Gordon-Zakharov model in dimension two. Comm. Math. Phys. 384 (1) (2021), 587-607. Global solution to the cubic Dirac equation in two space dimensions. S Dong, K Li, J. Differential Equations. 331Dong, S., Li, K.: Global solution to the cubic Dirac equation in two space dimensions. J. Differential Equations 331 (2022), 192-222. S Dong, K Li, Y Ma, X Yuan, arXiv:2205.12000Global behavior of small data solutions for the 2 Dirac-Klein-Gordon Equations. Dong, S. Li, K., Ma, Y., Yuan, X.: Global behavior of small data solutions for the 2 Dirac-Klein- Gordon Equations. arXiv:2205.12000 S Dong, Z Wyatt, arXiv:2105.13780Hidden structure and sharp asymptotics for the Dirac-Klein-Gordon system in two space dimensions. Dong, S., Wyatt, Z.: Hidden structure and sharp asymptotics for the Dirac-Klein-Gordon system in two space dimensions. arXiv:2105.13780 Global solutions of wave-Klein-Gordon systems in 2 + 1 dimensional space-time with strong couplings in divergence form. S Duan, Y Ma, SIAM J. Math. Anal. 543Duan, S., Ma, Y.: Global solutions of wave-Klein-Gordon systems in 2 + 1 dimensional space-time with strong couplings in divergence form. SIAM J. Math. Anal. 54 (3) (2022), 2691-2726. Global solutions for the Dirac-Klein-Gordon system in two space dimensions. A Grünrock, H Pecher, Comm. Partial Differential Equations. 351Grünrock, A., Pecher, H.: Global solutions for the Dirac-Klein-Gordon system in two space dimen- sions. Comm. Partial Differential Equations 35 (1) (2010), 89-112. C Huneau, A Stingo, arXiv:2110.13982Global well-posedness for a system of quasilinear wave equations on a product space. Huneau, C., Stingo, A.: Global well-posedness for a system of quasilinear wave equations on a product space. arXiv:2110.13982 The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics. S Klainerman, Lectures in Appl. Math. 231Amer. Math. SocKlainerman, S.: The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 293-326, Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986. Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. S Klainerman, Comm. Pure Appl. Math. 385Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. Comm. Pure Appl. Math 38 (5) (1985), 631-641. Global solution for massive Maxwell-Klein-Gordon equations. S Klainerman, Q Wang, S Yang, Comm. Pure Appl. Math. 731Klainerman, S., Wang, Q., Yang, S.: Global solution for massive Maxwell-Klein-Gordon equations. Comm. Pure Appl. Math. 73 (1) (2020), 63-109. The hyperboloidal foliation method. P G Lefloch, Y Ma, of Series in Applied and Computational Mathematics. 2World Scientific Publishing Co. Pte. LtdLeFloch, P.G., Ma, Y.: The hyperboloidal foliation method. volume 2 of Series in Applied and Computational Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2014). The global nonlinear stability of Minkowski space for self-gravitating massive fields. P G Lefloch, Y Ma, of Series in Applied and Computational Mathematics. 3World Scientific Publishing Co. Pte. LtdLeFloch, P.G., Ma, Y.: The global nonlinear stability of Minkowski space for self-gravitating massive fields. volume 3 of Series in Applied and Computational Mathematics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2018). C D Sogge, Lectures on non-linear wave equations. Boston, MAInternational PressSecond edition. x+205 ppSogge, C.D.: Lectures on non-linear wave equations. Second edition. International Press, Boston, MA, 2008. x+205 pp. Stability of constant equilibrium for the Maxwell-Higgs equations. Y Tsutsumi, Funkcial. Ekvac. 461Tsutsumi, Y.: Stability of constant equilibrium for the Maxwell-Higgs equations. Funkcial. Ekvac. 46 (1) (2003), 41-62. An intrinsic hyperboloid approach for Einstein Klein-Gordon equations. Q Wang, J. Differential Geom. 1151Wang, Q.: An intrinsic hyperboloid approach for Einstein Klein-Gordon equations. J. Differential Geom. 115 (1) (2020), 27-109. Ministry of Education Beĳing 100875. Qian Zhang School of Mathematical Sciences, Beĳing Normal University Laboratory of Mathematics and Complex SystemsChina Email: [email protected] Zhang School of Mathematical Sciences, Beĳing Normal University Laboratory of Mathematics and Complex Systems, Ministry of Education Beĳing 100875, China Email: [email protected]	[]
[ "LETTER A likely decade-long sustained tidal disruption event", "LETTER A likely decade-long sustained tidal disruption event" ]	[ "Dacheng Lin \nSpace Science Center\nUniversity of New Hampshire\n03824DurhamNHUSA\n", "James Guillochon \nThe Institute for Theory and Computation\nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA\n", "S Komossa \nQianNan Normal University for Nationalities\nLongshan Street\n\nDuyun City of Guizhou Province\nChina\n", "Enrico Ramirez-Ruiz \nDepartment of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA\n", "Jimmy A Irwin \nDepartment of Physics and As-tronomy\nUniversity of Alabama\nBox 87032435487TuscaloosaALUSA\n\nDepartment of Physics and Astronomy\nSeoul National University\n08826SeoulKorea\n", "W Peter Maksym \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA\n", "Dirk Grupe \nSpace Science Center\nMorehead State Uni-versity\n235 Martindale Drive40351MoreheadKYUSA\n", "Olivier Godet \nCNRS\nIRAP\n9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France\n\nUniversité de Toulouse\nUPS-OMP\nIRAP\nToulouseFrance\n", "Natalie A Webb \nCNRS\nIRAP\n9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France\n\nUniversité de Toulouse\nUPS-OMP\nIRAP\nToulouseFrance\n", "Didier Barret \nCNRS\nIRAP\n9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France\n\nUniversité de Toulouse\nUPS-OMP\nIRAP\nToulouseFrance\n", "B Ashley Zauderer \nCenter for Cosmology and Particle Physics\nNew York University\n4 Washington Place10003New YorkNYUSA\n", "Pierre-Alain Duc \nAIM Paris-Saclay Service d'astrophysique\nCEA-Saclay\n91191Gif sur YvetteFrance\n", "Eleazar R Carrasco \nSouth-ern Operations Center\nGemini Observatory/AURA\nCasilla 603, La SerenaChile\n", "Stephen D J Gwyn \nCanadian Astronomy Data Centre\nHerzberg Institute of Astrophysics\n5071 West Saanich RoadV9E 2E7VictoriaBritish ColumbiaCanada\n" ]	[ "Space Science Center\nUniversity of New Hampshire\n03824DurhamNHUSA", "The Institute for Theory and Computation\nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA", "QianNan Normal University for Nationalities\nLongshan Street", "Duyun City of Guizhou Province\nChina", "Department of Astronomy and Astrophysics\nUniversity of California\n95064Santa CruzCAUSA", "Department of Physics and As-tronomy\nUniversity of Alabama\nBox 87032435487TuscaloosaALUSA", "Department of Physics and Astronomy\nSeoul National University\n08826SeoulKorea", "Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA", "Space Science Center\nMorehead State Uni-versity\n235 Martindale Drive40351MoreheadKYUSA", "CNRS\nIRAP\n9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France", "Université de Toulouse\nUPS-OMP\nIRAP\nToulouseFrance", "CNRS\nIRAP\n9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France", "Université de Toulouse\nUPS-OMP\nIRAP\nToulouseFrance", "CNRS\nIRAP\n9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France", "Université de Toulouse\nUPS-OMP\nIRAP\nToulouseFrance", "Center for Cosmology and Particle Physics\nNew York University\n4 Washington Place10003New YorkNYUSA", "AIM Paris-Saclay Service d'astrophysique\nCEA-Saclay\n91191Gif sur YvetteFrance", "South-ern Operations Center\nGemini Observatory/AURA\nCasilla 603, La SerenaChile", "Canadian Astronomy Data Centre\nHerzberg Institute of Astrophysics\n5071 West Saanich RoadV9E 2E7VictoriaBritish ColumbiaCanada" ]	[]	Multiwavelength flares from tidal disruption and accretion of stars can be used to find and study otherwise dormant massive black holes in galactic nuclei 1 . Previous well-monitored candidate flares are short-lived, with most emission confined to within ∼1 year 2-5 . Here we report the discovery of a well observed super-long (>11 years) luminous soft X-ray flare from the nuclear region of a dwarf starburst galaxy. After an apparently fast rise within ∼4 months a decade ago, the X-ray luminosity, though showing a weak trend of decay, has been persistently high at around the Eddington limit (when the radiation pressure balances the gravitational force). The X-ray spectra are generally soft (steeply declining towards higher energies) and can be described with Comptonized emission from an optically thick low-temperature corona, a super-Eddington accretion signature often observed in accreting stellar-mass black holes 6 . Dramatic spectral softening was also caught in one recent observation, implying either a temporary transition from the super-Eddington accretion state to the standard thermal state or the presence of a transient highly blueshifted (∼ 0.36c) warm absorber. All these properties in concert suggest a tidal disruption event (TDE) of an unusually long super-Eddington accretion phase that has never been observed before.The X-ray source 3XMM J150052.0+015452 (XJ1500+0154 hereafter) was serendipitously detected in frequent observations 7 of the foreground galaxy group NGC 5813 by the X-ray observatories Chandra and XMM-Newton from 2005 to 2011. Our follow-up observation of the source with Chandra on February 23rd 2015 provided a well constrained X-ray position coincident with the center of the galaxy SDSS J150052.07+015453.8, to within 0. ′′ 18(Figure 1, see SI). The galaxy lies at a redshift of 0.145, or a luminosity distance of DL = 689 Mpc (for H0=70 km s −1 Mpc −1 , ΩM = 0.3, ΩΛ = 0.7), with strong emission lines indicative of intense star-forming activity. It has a total stellar mass of ∼6 × 10 9 M⊙ (see SI), comparable to that of the Large Magellanic Cloud. For such a small galaxy, we expect 8 the central supermassive black hole (SMBH) to have mass ∼10 6 M⊙.	10.1038/s41550-016-0033	[ "https://arxiv.org/pdf/1702.00792v1.pdf" ]	119,354,093	1702.00792	09c7ad036b6cd5a4d85cedbcbe50cb1ff7fc3a1d	LETTER A likely decade-long sustained tidal disruption event 2 Feb 2017 Dacheng Lin Space Science Center University of New Hampshire 03824DurhamNHUSA James Guillochon The Institute for Theory and Computation Harvard-Smithsonian Center for Astrophysics 60 Garden Street02138CambridgeMAUSA S Komossa QianNan Normal University for Nationalities Longshan Street Duyun City of Guizhou Province China Enrico Ramirez-Ruiz Department of Astronomy and Astrophysics University of California 95064Santa CruzCAUSA Jimmy A Irwin Department of Physics and As-tronomy University of Alabama Box 87032435487TuscaloosaALUSA Department of Physics and Astronomy Seoul National University 08826SeoulKorea W Peter Maksym Harvard-Smithsonian Center for Astrophysics 60 Garden St02138CambridgeMAUSA Dirk Grupe Space Science Center Morehead State Uni-versity 235 Martindale Drive40351MoreheadKYUSA Olivier Godet CNRS IRAP 9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France Université de Toulouse UPS-OMP IRAP ToulouseFrance Natalie A Webb CNRS IRAP 9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France Université de Toulouse UPS-OMP IRAP ToulouseFrance Didier Barret CNRS IRAP 9 avenue du Colonel Roche, BP 44346F-31028Toulouse Cedex 4France Université de Toulouse UPS-OMP IRAP ToulouseFrance B Ashley Zauderer Center for Cosmology and Particle Physics New York University 4 Washington Place10003New YorkNYUSA Pierre-Alain Duc AIM Paris-Saclay Service d'astrophysique CEA-Saclay 91191Gif sur YvetteFrance Eleazar R Carrasco South-ern Operations Center Gemini Observatory/AURA Casilla 603, La SerenaChile Stephen D J Gwyn Canadian Astronomy Data Centre Herzberg Institute of Astrophysics 5071 West Saanich RoadV9E 2E7VictoriaBritish ColumbiaCanada LETTER A likely decade-long sustained tidal disruption event 2 Feb 2017 Multiwavelength flares from tidal disruption and accretion of stars can be used to find and study otherwise dormant massive black holes in galactic nuclei 1 . Previous well-monitored candidate flares are short-lived, with most emission confined to within ∼1 year 2-5 . Here we report the discovery of a well observed super-long (>11 years) luminous soft X-ray flare from the nuclear region of a dwarf starburst galaxy. After an apparently fast rise within ∼4 months a decade ago, the X-ray luminosity, though showing a weak trend of decay, has been persistently high at around the Eddington limit (when the radiation pressure balances the gravitational force). The X-ray spectra are generally soft (steeply declining towards higher energies) and can be described with Comptonized emission from an optically thick low-temperature corona, a super-Eddington accretion signature often observed in accreting stellar-mass black holes 6 . Dramatic spectral softening was also caught in one recent observation, implying either a temporary transition from the super-Eddington accretion state to the standard thermal state or the presence of a transient highly blueshifted (∼ 0.36c) warm absorber. All these properties in concert suggest a tidal disruption event (TDE) of an unusually long super-Eddington accretion phase that has never been observed before.The X-ray source 3XMM J150052.0+015452 (XJ1500+0154 hereafter) was serendipitously detected in frequent observations 7 of the foreground galaxy group NGC 5813 by the X-ray observatories Chandra and XMM-Newton from 2005 to 2011. Our follow-up observation of the source with Chandra on February 23rd 2015 provided a well constrained X-ray position coincident with the center of the galaxy SDSS J150052.07+015453.8, to within 0. ′′ 18(Figure 1, see SI). The galaxy lies at a redshift of 0.145, or a luminosity distance of DL = 689 Mpc (for H0=70 km s −1 Mpc −1 , ΩM = 0.3, ΩΛ = 0.7), with strong emission lines indicative of intense star-forming activity. It has a total stellar mass of ∼6 × 10 9 M⊙ (see SI), comparable to that of the Large Magellanic Cloud. For such a small galaxy, we expect 8 the central supermassive black hole (SMBH) to have mass ∼10 6 M⊙. Multiwavelength flares from tidal disruption and accretion of stars can be used to find and study otherwise dormant massive black holes in galactic nuclei 1 . Previous well-monitored candidate flares are short-lived, with most emission confined to within ∼1 year 2-5 . Here we report the discovery of a well observed super-long (>11 years) luminous soft X-ray flare from the nuclear region of a dwarf starburst galaxy. After an apparently fast rise within ∼4 months a decade ago, the X-ray luminosity, though showing a weak trend of decay, has been persistently high at around the Eddington limit (when the radiation pressure balances the gravitational force). The X-ray spectra are generally soft (steeply declining towards higher energies) and can be described with Comptonized emission from an optically thick low-temperature corona, a super-Eddington accretion signature often observed in accreting stellar-mass black holes 6 . Dramatic spectral softening was also caught in one recent observation, implying either a temporary transition from the super-Eddington accretion state to the standard thermal state or the presence of a transient highly blueshifted (∼ 0.36c) warm absorber. All these properties in concert suggest a tidal disruption event (TDE) of an unusually long super-Eddington accretion phase that has never been observed before. The X-ray source 3XMM J150052.0+015452 (XJ1500+0154 hereafter) was serendipitously detected in frequent observations 7 of the foreground galaxy group NGC 5813 by the X-ray observatories Chandra and XMM-Newton from 2005 to 2011. Our follow-up observation of the source with Chandra on February 23rd 2015 provided a well constrained X-ray position coincident with the center of the galaxy SDSS J150052.07+015453.8, to within 0. ′′ 18 (Figure 1, see SI). The galaxy lies at a redshift of 0.145, or a luminosity distance of DL = 689 Mpc (for H0=70 km s −1 Mpc −1 , ΩM = 0.3, ΩΛ = 0.7), with strong emission lines indicative of intense star-forming activity. It has a total stellar mass of ∼6 × 10 9 M⊙ (see SI), comparable to that of the Large Magellanic Cloud. For such a small galaxy, we expect 8 the central supermassive black hole (SMBH) to have mass ∼10 6 M⊙. The upper panel in Figure 2 shows the long-term evolution of the X-ray luminosity LX. Our best fits to the spectra with sufficient counts are shown in the lower panel of the figure and are given in Table 1. One striking property of the source is the fast-rise-very-slow-decay outburst profile. It was not detected in the first Chandra observation on April 2nd 2005, with LX < 4.3 × 10 41 erg s −1 (3σ upper limit, assuming a powerlaw source spectrum of photon index 2.0). Less than 4 months later (July 23rd 2005), the source was detected in the first XMM-Newton observation, with LX ∼ 5.5 × 10 42 erg s −1 . It was detected at an even higher luminosity three years later, in one Chandra observation on June 5th 2008 and two XMM-Newton observations in February 2009, with LX ∼ 7.0 × 10 43 erg s −1 . The luminosity decreased only slightly to ∼ 3.0 × 10 43 erg s −1 in seven Chandra observations in March-April 2011. Similar luminosities were seen in our follow-up observations later, one by Swift on March 28th 2014, one by Chandra on February 23rd 2015, and seven by Swift in February 2016. Given the probably small central SMBH and correcting for emission outside the X-ray band, the source luminosity has been most likely at around the Eddington limit since it went into the outburst. Another special property of the source is the generally quasi-soft X-ray spectra, most evident in the XMM-Newton and Chandra observations in 2008-2011. These spectra can be roughly described with a dominant thermal disk of apparent maximum disk temperature kT diskbb ∼ 0.3 keV plus a weak powerlaw (see SI). However, such a model is physically unacceptable, because the standard thin accretion disk around a SMBH is expected to produce much cooler thermal emission (kT diskbb 0.1 keV) 9 . Instead, the spectra can be described well with the Comptonization model CompTT 10 , with an optically thick (τ ∼ 4-11) low-temperature (kTe ∼ 0.4-1.3 keV) corona (see SI). Such spectral parameters are commonly seen in ultraluminous X-ray sources (ULXs) 6, 11, 12 , most of which are believed to be super-Eddington accreting stellar-mass black holes, except that XJ1500+0154 had orders of magnitude higher luminosities. Therefore, we identified the observations in 2008-2011 as being in the super-Eddington accretion state. Surprisingly, we obtained a much softer X-ray spectrum in the Chandra observation in 2015, mostly due to a drop (by a factor of 7.0) in the count rate above ∼1 keV (Figure 2), compared with the observations in 2008-2011. This super-soft spectrum can be described with a dominant thermal disk of kT diskbb ∼ 0.13 ± 0.01 keV plus a very weak powerlaw (Figure 2). Such a cool thermal disk is ex-pected from accretion onto a SMBH below the Eddington limit. Then the source could be in the thermal state, which in turn supports the identification of the super-Eddington accretion state in previous observations. In this case the X-ray spectral evolution of XJ1500+0154 is very similar to a transient ULX in M31 6 , which changed from a super-Eddington accretion state to a thermal state within 20 days with the X-ray luminosity decreasing only slightly. However, we find that the spectrum can be described equally well with the CompTT model that fits the Chandra observations in 2011, except for additional absorption by a strong (NH ∼ 6 × 10 23 cm −2 ) ionized (log(ξ) = 2.8) absorber with a blueshifted velocity of 0.36c. Powerful sub-relativistic winds are expected in super-Eddington accreting black holes 13 , and highly blueshifted warm absorbers have been detected in ULXs 14 . Therefore, this interpretation for the Chandra observation in 2015 also supports the identification of the super-Eddington accretion state in previous observations. The recent Swift observations in 2016 have poor statistics, but some of them did not seem to show similar super-soft X-ray spectra. Therefore, the source has not completely settled to a new state of super-soft X-ray spectra, and the super-Eddington accretion seems to have lasted for 11 years (see SI). No sign of persistent nuclear activity is seen in the optical emission lines of the host galaxy, whose ratios are fully consistent with those expected from a starburst galaxy. There are many other properties of the source that argue against the possibility that it is a standard active galactic nucleus (AGN, see SI). In particular, no AGN is known to show X-ray spectra as soft as XJ1500+0154 within the 1-4.5 keV energy band or show dramatic quasi-soft to super-soft X-ray spectral change. The large X-ray variability (a factor of >97) is also extremely rare among AGNs. Therefore, although we cannot completely rule out that XJ1500+0154 is just a highly variable AGN at this point, its Xray outburst is best explained as tidal disruption of a star by the central black hole. This interpretation is strongly supported by our new discovery of two other sources that seemed to be in X-ray outbursts with similar quasi-soft X-ray spectra as XJ1500+0154 but have host galaxies showing no sign of nuclear activity in optical (see SI). The super-Eddington accretion phase from tidal disruption of a solar-type star by a 10 6 M⊙ black hole can last t Edd ≈ 2 years, with the peak mass accretion rate highly super-Eddington 1, 15 . One main property of a super-Eddington accretion disk is a lower radiative efficiency than a standard thermal thin disk, due to significant super-Eddington effects of photon trapping and outflows in the inner disk region 13, 16, 17 . These effects are more serious at higher accretion rates, with the disk luminosity sustained at around the Eddington limit. The Eddington-limited slow decay of our source thus agrees well with the super-Eddington accretion signatures suggested by the X-ray spectra. The long super-Eddington accretion phase of 11 years in our event would imply disruption of a very massive star (10 M⊙) based on the standard theory 15 . However, it has been realized that the evolution of TDEs heavily depends on how the streams of tidal debris intersect each other [18][19][20][21][22] . It should be common for circularization of the fallback mass onto the accretion disk to occur at a much larger distance, resulting in a much longer viscous time scale, than predicted from the standard theory 19 . Therefore, t Edd can be very long in a slow circularization process, unless the circularization is so slow that the peak accretion rate drops to be sub-Eddington. We plot in Figure 2 (solid line) the evolution of the luminosity from a full disruption of a 2 M⊙ star by a 10 6 M⊙ black hole, with the accretion of the mass slowed relative to the fallback time by 3 years. The super-Eddington effects were taken into account by introducing a logarithmic dependence of the radiative efficiency on the accretion rate above the Eddington limit 13 (see SI). We assumed that 25% of the radiation is in X-rays, as inferred from the spectral modeling. The model describes the data well. The total energy release and the total mass accreted onto the black hole until the last Swift observation would then be 6.4 × 10 52 ergs and 0.89 M⊙, respectively, which are orders of magnitude higher than seen in other known events 5, 23, 24 . Although disruption of a very massive star of 10 M⊙ with prompt circularization can also describe the data, such disruption is expected to be orders of magnitude rarer than disrupting a star of 2 M⊙ with slow circularization (see SI). Therefore our event provides the first convincing evidence of slow circularization effects in TDEs, which are expected to be very common when the black holes are small (∼10 6 M⊙) but were not clearly observed before probably due to observational bias 19 . We calculated the rate of events similar to XJ1500+0154 to be ∼4 × 10 −7 per galaxy per year (see SI), which is about two orders of magnitude lower than estimated for short TDEs 25 . One main reason for the low rate of events like XJ1500+0154 could be the relatively large mass (2M⊙) of the disrupted stars required, which is only possible in starburst galaxies 26 . Although events like XJ1500+0154 are rare, their extreme duration and radiative inefficiency mean that their contributions to the luminosity function of active galactic nuclei and to the growth of the black holes are comparable to or even higher than those of short events. TDEs with a shorter super-Eddington accretion phase than XJ1500+0154 could be more common. The discovery of our event opens up a new realm in which to search for super-Eddington accreting TDEs, that is, by investigating sources with quasi-soft X-ray spectra. Our discovery of the other two candidates is the result of applying this scheme. This is the first time that X-ray spectra resembling typical super-Eddington accreting stellar-mass black holes were observed in an accreting SMBH. If our interpretation of a super-Eddington accreting TDE for XJ1500+0154 is correct, it would have important implications for the growth of massive black holes. The detection of quasars at redshift z > 6 with black hole masses ∼ 10 9 M⊙ poses a problem to explain their growth with accretion via a standard thin disk at the Eddington rate 27 . However, their formation would be possible if the black holes can accrete at a super-Eddington rate during an early phase 28 . Our event shows that super-Eddington accretion onto massive black holes can occur, giving strong observational support to this model. The high absorption in these systems would mean that the search for them should be through radio and infrared 29 , because their X-ray spectra, if as soft as XJ1500+0154, would be completely absorbed. We expect the accretion rate to drop by an order of magnitude to be well sub-Eddington in the next ten years based on our model of full disruption of a 2 M⊙ star. By continued monitoring of the event, we will be able to test our TDE interpretation and to determine the duration of the super-Eddington accretion phase and the origin of spectral softening. We will also be given a rare opportunity to observe the spectral evolution of the event across different accretion regimes and to investigate its connection with short super-soft events that are mostly believed to accrete below the Eddington limit. The X1 and C10 spectra were rebinned to have at least one count per bin and were fitted by minimizing the C statistic, while the X2, X3, C2, and C3-C9 spectra were rebinned to have at least 20 counts per bin and were fitted by minimizing the χ 2 statistic. The fits used energy channels within 0.3-10 keV for XMM-Newton and energy channels within 0.3-7 keV for Chandra. All models include Galactic absorption of column density N H,Gal = 4.4 × 10 20 cm −2 . The absorption intrinsic to the X-ray source at redshift 0.14542 was also included and fixed at N H,i = 4.2 × 10 21 cm −2 , which was the best-fitting value from the simultaneous fit to the C2, X3, and C3-C9 spectra. L abs is the source rest-frame 0.34-11.5 keV luminosity, corrected for the Galactic absorption but not intrinsic absorption, and L unabs is the source rest-frame 0.34-11.5 keV luminosity, corrected for both Galactic and intrinsic absorption. Both L abs and L unabs are in units of 10 43 erg s −1 . All errors are at the 90%-confidence level. Parameters without errors were fixed in the fits. For C10, two models were tested: diskbb+PL and zxipcf(comptt). For the latter model, the CompTT component was fixed at the best fit of this model to C3-C9, and the luminosities L abs and L unabs was simply copied from those of C3-C9. The reduced χ 2 values are given for fits using the χ 2 statistic, but not for those using the C statistic. The long-term source rest-frame 0.34-11.5 keV unabsorbed luminosity curve. The Chandra, XMM-Newton and Swift observations are shown as blue squares, red triangles and green circles, respectively, with 90% error bars, but for the first Chandra observation C1 in 2005 the 3σ upper limit is shown with an arrow. We have merged the seven Chandra observations in 2011 to create a single coadded spectrum, given the lack of significant spectral/flux change in these observations. Similarly we also created a coadded spectrum from the combination of S2-S5 and another one from S6-S8. For clarity, we have offset S2-S5 to be one month earlier, because they are too close to S6-S8 in time. The solid line is a model of disrupting a 2 M ⊙ star by a black hole of mass 10 6 M ⊙ with slow circularization and super-Eddington effects (see SI). Such a model describes the data well. The dashed line plots t −5/3 , assuming a peak X-ray luminosity of 10 44 erg s −1 that is reached two months after disruption of the star; it represents the typical evolution trend for thermal TDEs 24, 30 , which obviously last much shorter than our event. Lower panels: The unfolded X-ray spectra. The X1 and C10 observations were fitted with a diskbb model (red dotted line) plus a PL (green dot-dashed line), and the C2, X2, X3, and C3-C9 observations were fitted with a CompTT model (the X2 spectrum is not shown but looks very similar to X3). Note that C10 can also be described with the CompTT fit to C3-C9 subject to a fast warm absorber. For clarity, we show only pn data for the XMM-Newton observations. Also for clarity, the spectra were rebinned to be above 2σ in each bin in the plot. Method XMM-Newton observations and data analysis XJ1500+0154 was serendipitously detected at off-axis angles of ∼13 ′ in three XMM-Newton observations (X1-X3 hereafter; see Supplementary Table 1) of NGC 5813. X1 was made in July 2005, while X2 and X3 were made in February 2009, only six days apart. The source was detected in all three European Photon Imaging Cameras (i.e., pn, MOS1/M1, and MOS2/M2) 31-33 in the imaging mode in X1, but only in pn and MOS2 in X2 and X3 because the source is outside the field of view (FOV) of MOS1 in these two observations. We used SAS 15.0.0 and the calibration files of 2016 February for reprocessing the X-ray event files and follow-up analysis. The data in strong background flare intervals, seen in all cameras in all observations, were excluded following the SAS thread for the filtering against high backgrounds. The final clean exposures used are given in Supplementary Table 1. We extracted the source spectra from all available cameras using a circular region of radius 20 ′′ . The background spectra were extracted from a large circular region near the source, using a radius of 100 ′′ for MOS1 and MOS2 and a radius of 50 ′′ for pn. The event selection criteria followed the default values in the pipeline 34 . For X2 and X3, in which the source was bright, we also extracted the pn light curves binned at 500 s using the SAS tool epiclccorr and performed variability tests using the ekstest tool. We used the 0.3-3 keV band, where the source counts dominated over those of the background. Chandra observations and data analysis XJ1500+0154 was serendipitously covered in nine Chandra observations (C1-C9 hereafter; see Supplementary Table 1) of NGC 5813, but all at large pointing offsets (11 ′ -15 ′ ). The dense observations in 2011 (C3-C9, ∼0.5 Ms) are from a Large Program (LP, PI: Dr. Scott Randall) on NGC 5813. All nine observations used the imaging array of the AXAF CCD Imaging Spectrometer (ACIS) 35 , and XJ1500+0154 fell in the back-illuminated chip S1 in all observations except C2, in which it fell in the front-illuminated chip I3. We had a Chandra followup observation of the source for 37 ks in February 2015 (C10 hereafter), with the aim point at the back-illuminated chip S3. We reprocessed all the data to apply the latest calibration (CALDB 4.6.7) using the script chandra repro in the Chandra Interactive Analysis of Observations (CIAO, version 4.7) package. No clear background flares were seen, and we used all data for all observations. The spectra of XJ1500+0154 were extracted for each observation. We used a circular source region enclosing 90% of the point spread function (PSF) at 1.0 keV and a circular background region of a radius of 50 ′′ near the source. However, there are three exceptions. For observations C1 and C2, we used the 70% PSF radius for the source region, considering that the source was not detected in C1 and was near the CCD edge in C2. For our follow-up observation C10, in which the source is near the aim point with minimum background contamination, we used the 95% PSF radius for the source region. We used the CIAO task mkacisrmf to create the response matrix files and the CIAO tasks mkarf and arfcorr to create the point-source aperture corrected auxiliary response files. Considering no significant difference between LP observations, which were taken within 13 days, and in order to improve the statistics for spectral modeling, we created a single spectrum combining LP observations. For observation C1 in which the source was not detected, we used the CIAO task aprates to determine confidence bounds of the flux. We measured the short-term variability within 0.4-3 keV adopting the Gregory-Loredo algorithm 36 implemented by the CIAO tool glvary 37 . It splits the events into multiple time bins and looks for significant deviations. The variation of the effective area with time was taken into account and was obtained by another CIAO tool dither region. The different degrees of confidence is indicated by the parameter of "variability index", which spans values within [0, 10] and is larger for variability of higher confidence 37 . Our C10 observation was intended to provide an accurate position of XJ1500+0154, utilizing the sub-arcsec resolution of Chandra near the aimpoint. We performed the source detection by applying the CIAO wavdetect wavelet-based source detection algorithm 38 on the 0.3-7 keV image binned at single sky pixel resolution. We then carried out the absolute astrometric correction for the X-ray sources by cross-correlating them with optical sources in the Canada-France-Hawaii Telescope (CFHT) MegaPrime/MegaCam 39 r ′ -band stacked images. We only used 19 matches that are outside the strong diffuse gas emission in NGC 5813 40 and have X-ray 95% statistical positional errors ≤1. ′′ 2 (based on Equation 12 in Kim et al. 41 ) and magnitude m r ′ < 24.0 AB mag for astrometric correction. These 19 matches do not include XJ1500+0154, in order to reduce the effect of the astrometric correction on the identification of its optical counterpart. The astrometric correction method in Lin et al. 42 was used, by searching for the translation and rotation of the X-ray frame that minimize the total χ 2 (χ is the ratio of the X-ray-optical separation to the total positional error) for 90% (i.e., 17, allowing 2 matches to be spurious or bad) of the matches that have the smallest χ values. The uncertainties of the translation and rotation and thus the systematic positional errors of the X-ray sources were estimated using 200 simulations. In order to calculate the statistical positional uncertainty for the source, we carried out 2000 ray-trace simulations with MARX 5.1.0 at positions near it and at the same off-axis angle. The spectrum from the thermal disk plus powerlaw fit to C10 was assumed. Swift observation and data analysis At our request, Swift 43 observed XJ1500+0154 in two epochs: one observation on March 28th 2014 (S1 hereafter), and seven observations between February 3rd-14th 2016 (S2-S8 hereafter, Supplementary Table 1). We analyzed the data with FTOOLS 6.18 and the calibration files released on July 31st 2015. The X-ray telescope (XRT) 44 was operated in the Photon Counting mode for all observations, and we reprocessed the data with the task xrtpipeline (version 0.13.2). The spectrum was extracted, using radii of 25 ′′ and 2 ′ for the circular source and background regions, respectively. The source was hardly detected by the XRT in S2-S5, and we created a co-added XRT spectrum from these observations. The source was clearly detected in S6-S8, and we also created a co-added XRT spectrum from these observations. The source net count rate was higher in S6-S8 than in S2-S5 at the 3σ confidence level. The UV-Optical Telescope (UVOT) 45 in S1 used three standard UV filters W1 (2.3 ks), M2 (2.3 ks), and W2 (2.3 ks). In S2-S8 the UVOT used the "Filter of the Day" mode, and we combined images for different filters, resulting in total exposures of 6.3 ks, 1.4 ks, 3.9 ks and 3.8 ks for the U, W1, M2, and W2 filters, respectively. To obtain the photometry, we used the task uvotsource with radii of 5 ′′ and 20 ′′ for the circular source and background regions, respectively. ROSAT Observations XJ1500+0154 was in the field of view of a ROSAT High-Resolution Imager pointed observation in 1998 for 1.9 ks (Supplementary Table 1). XJ1500+0154 was not detected in the observation, and we extracted the source and background spectra using a circular region of radius 10 ′′ and a circular region of radius 50 ′′ , respectively, in order to estimate the flux limit. CFHT MegaCam observations and data analysis There are seven CFHT/MegaPrime r ′ -band images, with exposure 345 second each and taken on May 5th 2014, and eight g ′ -band images, with exposure 345 second each and taken on April 26th 2014. We produced two stacked images, one for the r ′ band (the final seeing FWHM is 0. ′′ 65) and the other for the g ′ band (the final seeing FWHM is 0. ′′ 80), using MegaPipe 46 , and aligned their astrometry to the Sloan Digital Sky Survey (SDSS) 47 . The sources detected from these two stacked images were used to compare with the SDSS photometry to detect the source optical variability and to compare with X-ray sources to align the X-ray source astrometry. We also fitted the host galaxy profile in the CFHT/MegaPrime stacked images using GALFIT 48 . Ten stars within 2 ′ from XJ1500+0154 were used to construct the point spread function of the images. X-ray spectral fits The X-ray activity of XJ1500+0154 should be due to accretion onto a BH, and we fitted the X-ray spectra with several models typically used to study such an object. Given that XJ1500+0154 is most likely associated with a galaxy at z = 0.14542, we applied this redshift to all the spectral models that we tested with the convolution model zashift in XSPEC 49 . All models included the Galactic absorption 50 fixed at NH = 4.4 × 10 20 cm −2 using the TBABS model. Possible absorption intrinsic to the source was accounted for using the ztbabs model. The abundance tables from Wilms et al. 51 were used. Data availability statement The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Supplementary Information The X-ray source position and the host galaxy We obtained the position of XJ1500+0154 from C10 to be R.A.=15:0:52.068 and Decl.=+1:54:53.79, with the 95% positional error of 0. ′′ 18 (including both the statistical component and the systematic component from the astrometric correction procedure). This position is only 0. ′′ 07 away from the center of the galaxy SDSS J150052.07+015453.8, consistent within the uncertainty (Figure 1). The number density of the galaxies that are as bright as or brighter than SDSS J150052.07+015453.8 in the r ′ band within 10 ′ is 0.00016 per square arcsec, implying that the chance probability for our X-ray source to be within 0. ′′ 18 from the center of SDSS J150052.07+015453.8 is only 0.000017. Therefore we can securely identify this galaxy as the host of our X-ray source, with the X-ray emission most likely from the galaxy center (the projected offset ∼0. ′′ 18 or 0.5 kpc at the redshift of z = 0.14542 of the galaxy; see below). The host galaxy in the CFHT/MegaPrime r ′ and g ′ -band images can be fitted with a Sérsic profile, with effective radii of 0. ′′ 48±0. We fitted the spectrum with Penalized Pixel Fitting (pPXF) software 52 . As the stellar templates, we adopted the single-population models by Maraston and Strömbäck 53 that were created based on the ELODIE library of empirical stellar spectra of solar metallicity 54 and have UV extension (below around 3950Å) based on theoretical models 55 . We chose these models because of their extended age coverage (54 ages from 3 Myr to 12 Gyr), broad wavelength range (1000.2-6800.0Å), and high resolution (FWHM ∼ 0.55Å). We interpolated the models to make them distribute uniformly in logarithm of the age from 3 Myr to 12 Gyr with 54 grids. We fitted the SDSS spectrum over source rest-frame 3200-6800Å, considering that the spectrum at shorter wavelengths is very noisy. Fourteen Gaussian functions were also included in the fit to model the apparent gas emission lines. No additive or multiplicative polynomials were included in the fit. The spectrum was corrected for the Galactic dust reddening 56 of E(B − V) G = 0.05 mag before the fit. The intrinsic reddening was left as a free fitting parameter and was inferred to be E(B − V) i = 0.068 mag. The pPXF fit inferred intrinsic width of the emission lines to be about σ = 52 km s −1 . The stellar kinematics could not be constrained well, as expected considering the lack of strong absorption features. Therefore in the final fit we assumed the stellar velocity dispersion to be the same as that of the gas. The best-fit model is shown in Supplementary λ5007 line, indicating systematic blue excess. Fitting the star light subtracted spectrum around the [O III] λ5007 region with two Gaussian lines, one for the strong narrow component and the other for the broad wing, we infer the wing to have σ = 555 +283 −161 km s −1 and to be blueshifted relative to the narrow line center by 312 +271 −233 km s −1 . The flux of the blue wing is 1.3 ± 0.4 × 10 −17 erg s −1 cm −2 , corresponding to a 5.0σ detection. Compared with the fitting residuals in other places, the residuals here are about the strongest, and thus the blue wing could be real, but we cannot completely rule out that it is due to enhanced sky noise caused by a bright sky line near [O III] λ4959. A blue wing around [O III] λ5007 is commonly seen in AGNs and starburst galaxies and has been interpreted as due to outflowing ionized gas [60][61][62][63] . For XJ1500+0154, the blue wing around [O III] λ5007, if real, is more likely due to star-forming activity, instead of due to the X-ray outburst, as the nucleus is significantly obscured. The mass and light (source rest-frame 3200-6800Å) distributions of the stellar populations with respect to the age from the pPXF fit are shown in Supplementary Figure 3, indicating the presence of some very young (< 5 Myr) populations, although the mass is dominated by old populations. The mean age weighted by light is 1.6 Gyr, and the mean age weighted by mass is 5.0 Gyr. The total stellar mass is 6.4 × 10 9 M⊙, and the total stellar luminosity is 1.9 × 10 9 L⊙, suggesting a dwarf galaxy (comparable to the Large Magellanic Cloud). Using the relation between the BH mass and the total galaxy stellar mass 8 , we estimated the BH mass to be 1.5 × 10 6 M⊙ (the 1σ uncertainty is 0.55 dex). We do not see a clear broad component of the Hα line. Some TDEs, which would be the best interpretation for XJ1500+0154, did show broad emission lines. The well covered TDE ASASSN-14li had peak Hα luminosity of 1.1 × 10 41 erg s −1 with the corresponding FWHM of ∼3000 km s −1 , detected in early epochs 64 . Such an emission line, however, would be difficult to detect for XJ1500+0154 if we assume the Galactic reddening of E(B − V) G = 0.05 mag and intrinsic reddening of E(B − V) i = 0.71 mag, which is based on E(B − V) = 1.7 × 10 −22 NH and intrinsic column density NH,i = 0.42 × 10 22 cm −2 (inferred from our X-ray spectral fits; see below). Therefore, we cannot rule out that a broad Hα line might be present in the source but is hard to detect due to strong extinction. However, it is also possible that XJ1500+0154 has intrinsically very weak optical emission (either broad emission lines or the continuum) associated with the X-ray outburst, as observed in some other TDEs during the X-ray bright phase 65 The source evolution with normalized count rates In order to study the source behavior in a relatively modelindependent way, we calculated the count rates normalized to those expected based on the CompTT fit to the C3-C9 spectrum. Although the CompTT fit was used, other models that fit the C3-C9 spectra reasonably well (e.g., an absorbed PL) gave very similar results. Supplementary Figure 4 shows the long-term evolution of such normalized count rates in b = 0.4-3 keV, b1 = 0.4-1 keV and b2 = 1-3 keV. The source appears to show a prolonged outburst. It was not detected in C1 on April 2nd 2005, with the 0.4-3 keV normalized count rate R N,b < 2.3% (3σ upper limit, upper panel in Supplementary Figure 4). The source was detected less than 4 months later in X1 on July 23rd 2005, with R N,b = 7.0 ± 2.6% (90% error). It was detected much brighter three years later, with R N,b = 1.9 ± 0.1 for C2 on June 5th 2008 and R N,b = 2.3 ± 0.1 and 2.1 ± 0.1 for X2 and X3, respectively, in February 2009. The source flux decayed very slowly later, with R N,b = 1.01 ± 0.02 in C3-C9 in March-April 2011 and 0.35 ± 0.04 in C10 on February 23rd 2015. Similar normalized count rates were also observed in the Swift monitoring observations in 2014-2016, though they have poor statistics. Overall, the source count rate seemed to show a fast rise, by a factor of >97 and probably in just months, and experienced a very slow decay, by a factor of a few over a decade in time. Comparing the 0.4-1 keV and 1-3 keV count rates (middle and lower panels in Supplementary Figure 4), we detected significant softening of the source in C10, due to a significant drop in the 1-3 keV count rate. The normalized 0.4-1 keV and 1-3 keV count rates of C10 are R N,b1 = 0.68 ± 0.10 and R N,b2 = 0.15 ± 0.04, while C3-C9 has R N,b1 = 1.01 ± 0.03 and 1.02 ± 0.03 (close to one by definition). The R N,b2 to R N,b1 ratios are 0.21 ± 0.06 and 1.01 ± 0.04 (90% error) for C10 and C3-C9, respectively, indicating the softening of C10 with respect to C3-C9 at the 17.8σ confidence level. In comparison, C2, X2, and X3 have the R N,b2 to R N,b1 ratios of 0.90±0.10, 0.90±0.07, and 0.79 ± 0.06, respectively, close to that of C3-C9. Other observations have poor statistics. S1 and S6-S8 seems to be as bright and as hard as C3-C9. S2-S5 is fainter and could be consistent with C10. X1, in the rise phase of the outburst, could have a relatively hard spectrum too. Spectral state identification for X2, X3, C2, and C3-C9 Supplementary Table 2 gives the fit results using simple models: a PL, a MCD (diskbb in XSPEC), and their combination MCD+PL. The spectra of X2, X3, C2, and C3-C9 have similar best-fitting spectral parameters in these models, implying the same emission mechanism in these observations. The absorbed PL fits gave the photon index ΓPL ∼ 4-5 for X2, X3, and C2, and C3-C9, indicating very soft spectra and ruling out that the source was in the hard spectral state of an accreting BH in these observations. X2, X3, and C2 are relatively close in time (within ∼8 months), and their spectra are a little softer (ΓPL ∼ 5) than that of C3-C9 (ΓPL ∼ 4), taken two years later after X2 and X3. The X2, X3, and C2 spectra can be generally described with an absorbed MCD, but the fit with an absorbed MCD to C3-C9 shows systematic positive residuals at high energies >2 keV (Supplementary Figure 5). Adding a PL (i.e., total model MCD+PL) with ΓPL fixed at a value of 2.5, typically seen in the Galactic BHBs in the thermal state, improved the fit to C3-C9 most significantly, but it still shows some systematic positive residuals in 2-3 keV and negative ones above 3 keV (Supplementary Figure 5). In any case, the MCD+PL fits inferred a very hot disk of kTMCD ∼ 0.3 keV in all spectra (the MCD+PL fits were not sensitive to the ΓPL value assumed, as long as ΓPL was constrained to be 3.5; for higher values of ΓPL, the fits obtained would be dominated by a PL, instead of by a disk). We note that all fits to X2 tested above in fact showed some systematic residuals, whose origin will be discussed more later. The MCD+PL fits are roughly statistically acceptable, but there are problems to associate them with the thermal state, which is characterized by a dominant thermal disk at the sub-Eddington luminosity 66 . The inferred disk temperatures of kTMCD ∼ 0.3 keV are too high for such a state of an accreting SMBH. For a standard thermal disk, the maximum temperature kTMCD ∝ M −1/4 (LMCD/L Edd ). The Galactic BH X-ray binaries have maximum disk temperatures normally 66,67 1 keV. The MCD+PL fits inferred LX ∼ 1-2 × 10 43 erg s −1 . Therefore, for XJ1500+0154 to be at the sub-Eddington limit in the observations considered, the BH mass should be at least 10 5 M⊙. Then the maximum temperature kTMCD should be 0.1 keV, much lower than the values inferred. The high disk temperature might be possible if the SMBH considered is maximally spinning. To test this further, we tried to fit X3, C2, C3-C9 simultaneously with the more physical AGN model optxagnf 68 (in XSPEC). Because we are considering the thermal state, in the optxagnf model, we assumed that the gravitational energy released in the disk is emitted as a color-corrected blackbody down to a (coronal) radius rcor, while within this radius the available energy is released in a hard PL form Comptonization component in an optically thin hot corona of temperature 100 keV. The PL index was fixed at a value of 2.5. The BH mass, the BH spin and the intrinsic column density was tied to be the same but allowed to vary. The Eddington ratios were forced to be below 1.0 (thus sub-Eddington). With all these settings, the best fit required a maximally spinning BH with mass 2.8 × 10 5 M⊙, but still a relatively high reduced χ 2 value was obtained (1.35 for 280 degrees of freedom), and systematic fit residuals were seen clearly. Given the above problem of fitting the spectra with the standard thermal state model MCD+PL, we tried to test the Comptonization model CompTT 10 (in XSPEC). This model is commonly used to fit the X-ray spectra of ULXs 11 , most of which are believed to be the super-Eddington accreting stellar-mass BHs. We found that we cannot constrain all parameters well, and it seems that the fits can either assume cold seed photon solutions of kT0 0.1 keV or hot seed photon solutions of kT0 ∼ 0.1-0.2 keV, corresponding to two local/global minima in χ 2 , which differed by <6. Cold seed photon solutions were associated with higher column densities (NH ∼ 0.4-0.5 × 10 22 cm −2 ) than hot seed photon solutions (NH ∼ 0.1-0.2 × 10 22 cm −2 ). In the case of ULXs, the seed photon temperatures were often inferred 11 to be 0.3 keV. If our spectra have the same emission mechanism as most ULXs and the seed photons are from a thermal disk, we would expect the seed photons in our spectra to be very cool, < 0.1 keV. Therefore we will focus on the cold seed photon solutions. Still, we found that in order to obtain well constrained corona temperatures and optical depths for better comparison between observations, it is desirable to fix the intrinsic column density NH,i and the seed photon temperature kT0. From the simultaneous fits to X3, C2, and C3-C9 spectra with these parameters tied to be the same (X2 was not included, again due to presence of fit residuals), we inferred NH,i = 0.42 ± 0.06 × 10 22 cm −2 and kT0 = 0.04 ± 0.04 keV. Therefore in the final fits with CompTT, we fixed NH at 0.42 × 10 22 cm −2 and kT0 at 0.04 keV. The fit results are given in Table 1. The disk geometry was assumed (the fits of similar quality can also be obtained assuming a spherical geometry, which would infer similar corona temperatures but higher (by a factor of ∼2) values of the optical depth). From Table 1, we see that the CompTT fits suggest a cool (kTe ∼ 0.35-1.3 keV) optically thick (τ ∼ 4-11) corona for X2, X3, C2 and C3-C9. C3-C9 might have a little hotter corona and a lower optical depth than earlier observations X2, X3, and C2. To test the significance of the presence of a cool optically thick corona in these spectra, we still tried to fit X3, C2, and C3-C9 simultaneously, with the seed photon temperatures all fixed at 0.04 keV, the corona temperatures all fixed at 20.0 keV, and the column densities allowed to vary freely but all tied to be the same. Then we obtained a total χ 2 value of 257.6 for 282 degrees of freedom. In comparison, the total χ 2 value is 242.3 for 3 fewer degrees of freedom if the corona temperature parameters are all allowed to vary freely. Based on the F -test, this implies a 3.4σ improvement of the fits with a cool optically thick corona over those with a hot optically thin one, under the assumption of cool seed photons. Therefore there is evidence that XJ1500+0154 was in a super-Eddington accretion state in X2, X3, C2, and C3-C9. We note that all the spectral models tested above on X2 all showed systematic fit residuals, as shown in Supplementary Figure 5. X2 was taken only six days before X3, and they have very similar instrument configurations and source off-axis angles. The source spectral shape and count rates are also very similar in these two observations, which seems to suggest that the residuals in X2 could be due to some calibration uncertainty, or due to presence of a warm absorber in X2. Starting with the continuum model CompTT with NH,i fixed at 0.42 × 10 22 cm −2 and kT0 fixed at 0.04 keV, we found that the residuals can be significantly reduced by adding an edge of threshold energy E edge = 0.63 ± 0.03 keV and optical depth τ = 0.88 ± 0.30 (corresponding to a 5σ improvement) and another edge of E edge = 1.33 ± 0.07 keV and τ = 0.37 ± 0.21 (corresponding to a 3σ improvement). Replacing these edges with the more physical model zxipcf, we obtained two possible fits. One inferred an absorber of column density NH = 0.4 × 10 22 cm −2 , ionizing parameter log(ξ) = −0.55 (i.e., nearly neutral), and zero speed (i.e., the absorber is static relative to the source), and required the unabsorbed luminosity to be a factor of 6.7 higher than that of X3. The other fit inferred an absorber of NH = 3.5 × 10 22 cm −2 , log(ξ) = 2.7 and redshifted speed of 29% the speed of the light (thus an inflow), with the unabsorbed source luminosity similar to that of X3. Both fits accounted for most of the residuals around 0.6 keV and are clearly challenging to understand, with the former requiring a large change in luminosity in a short time and the latter requiring a fast warm inflow. Future observations would be helpful to determine whether the residuals are real. Spectral state identification for C10 We also have a relatively good spectrum from C10, which appears to be much softer than earlier observations, as shown above based on the normalized count rates in different energy bands. The absorbed PL fit inferred a very high photon index of ΓPL = 8.0 ± 1.2 for C10, compared with ΓPL ∼ 4-5 for X2, X3, C2 and C3-C9. The MCD+PL fit with ΓPL fixed at a value 2.5 also indicates that the spectrum in C10 is dominated by a disk with a much lower temperature in C10 than in X2, X3, C2 and C3-C9 (0.13 keV versus ∼0.3 keV). One explanation for the spectral softening in C10 is that the source showed a transition from the super-Eddington accretion state in earlier observations to the thermal state in C10. Then the MCD+PL fit would be a reasonable model. Table 1 lists such a fit with NH,i fixed at 4.2 × 10 21 cm −2 , as obtained from the simultaneous CompTT fits to X3, C2, and C3-C9. This model inferred LX in C10 only lower than that in C3-C9 by 15%. We note that in the second ULX in M31, similar state transition also occurred with luminosity changing 6 by <20%. Alternatively, the spectral softening in C10 could be due to a transient warm absorber in C10 that obscured the high-energy flux. To test this scenario, we added a warm absorber to the CompTT fit to C3-C9. The warm absorber was modelled with the zxipcf (in XSPEC) model. The CompTT parameters were fixed at the values obtained from the fit to C3-C9. The covering fraction was found to be consistent with 1.0 (the 90% lower confidence bound is 0.91), and we thus fixed it at the value of 1.0. Then we inferred the warm absorber to have NH = 64 × 10 22 cm −2 , log(ξ) = 2.8 and blueshifted velocity of 0.36c (Table 1). This fit has almost the same C statistic as the MCD+PL fit. If the normalization of the CompTT model is allowed to vary, considering that the source might become fainter, we found that the unabsorbed luminosity in C10 should be >88% (90% lower bound, the upper bound cannot be constrained, as there is degeneracy between the source flux and the column density of the absorber) of that of C3-C9. A highly ionized sub-relativistic outflowing absorber is expected in the super-Eddington accretion phase of TDEs 1 and had been inferred in a few cases 69,70 . Spectral state identification for Swift observations The Swift observations are all too faint to carry out detailed spectral fits. In S1, S2-S5, and S6-S8, we collected 18.3, 6.0 and 18.8 net counts within 0.3-10 keV in exposures of 8.7 ks, 9.0 ks, and 6.7 ks, respectively. Based on the normalized count rates shown in Supplementary Figure 4, the source spectra in S1 and S6-S8 are more consistent with that in C3-C9 than with that in C10. The normalized 1-3 keV count rates in S1 and S6-S8 are both consistent with that in C3-C9 to within 1σ error but both different from that in C10 at the 2.5σ confidence level. Therefore the source might be still in the super-Eddington accretion state in S1 and S6-S8. In contrast, the low 0.4-3 keV normalized count rate in S2-S5 makes it more consistent with C10 (within 1σ) than C3-C9, and the source could be in the thermal state or subject to transient absorption again in this observation. Given the low statistics of the Swift observations, however, we cannot completely rule out that the source had similar X-ray spectra in S2-S8 as in C10, which could suggest that the source had been completely settled to the thermal state since C10. Deeper observations of the source in the future are needed to firmly determine the duration of the super-Eddington accretion phase. Spectral state identification for X1 The spectrum of X1 has low quality too. Fitting it with a PL gave ΓPL = 2.7 +2.3 −0.7 . Thus it could be a little harder than other observations. We tried to fit it with the MCD+PL model with NH,i fixed at 4.2×10 21 cm −2 and the PL photon index fixed at the value of 2.5. We obtained a fit with a cool disk of kTMCD = 0.10 +0.06 −0.04 keV and a PL contributing about 25% of the rest-frame 0.34-11.5 keV luminosity. The identification of the spectral state for X1 is clearly non-trivial due to its low statistics, but it is consistent with a transitional state, as expected considering that it was at the beginning of the outburst. Fitting the long-term X-ray luminosity curve The evolution of the X-ray luminosity in TDEs depends on many factors and can involve physics that is still poorly understood, especially in the super-Eddington accretion regime. We explored the models that could describe the data assuming full disruption of a star by a 10 6 M⊙ black hole. The accretion rate of the mass was assumed to be slowed relative to the fallback rate by the viscous timescale τvisc, according to the equatioṅ M d (t) = 1 τvisc e −t/τ visc t 0 e t ′ /τ viscṀ fb (t ′ )dt ′ ,(1) whereṀ d is the mass accretion rate andṀ fb is the mass fallback rate 71 . The radiative efficiency was assumed to be 0.1 when the accretion rateṀ d is below the 0.5 isotropic Eddington limitṀ Edd . At higher accretion rates, the inner disk could begin to reach the local Eddington limit 72 , and we assumed the luminosity to scale logarithmically, in the form 13 of 1.0 + log(Ṁ d /0.5Ṁ Edd ). Based on the X-ray spectral fits, we assumed that 25% of the light was emitted in the X-ray band (the source rest-frame 0.34-11.5 keV). Supplementary Figure 6 shows the expected light curves for several models of different masses of the disrupted star and different viscous timescales. For comparison, models incorporating the super-Eddington accretion effects are shown as thick black lines, while those not are shown as thin gray lines (representing the evolution of the fallback rate in case of prompt circularization). We find that there is degeneracy between the stellar mass and τvisc. Our data can be described with the full disruption of a 2 M⊙ star with τvisc = 3 yr (Figure 2) or a 10 M⊙ star with τvisc = 0 (Supplementary Figure 6). The degeneracy can be broken with more future monitorings of the source, as the luminosity is expected to decay faster for disruption of a lower mass star. Assuming a higher efficiency in either the Eddington excess or in the mass-to-light conversion would adjust the favored stellar masses downward, with a maximally-spinning black hole driving the favored masses to values of 0.5 and 2.5 M⊙, respectively. While the case with a larger star describes our data just as well as the case with a star of nearly a solar mass, the steep stellar mass function strongly favors the disruptions of lower-mass stars, even in cases where a recent starburst has occurred 26 , with < 1% of all disruptions coming from stars with M ≥ 10 M⊙. Additionally, we expect that disruptions of stars around 10 6 M⊙ black holes will be slowed by viscous effects in the majority of disruptions, with two-thirds of disruptions having viscous timescales longer 19 than a year for M h < 10 6 M⊙. Because of these expectations, we argue that XJ1500+0154 most likely originated from the disruption of a nearly solar-mass star. A single strong disruption and prompt accretion of an evolved star, either ascending the red giant branch or on the horizontal branch, can produce a prolonged TDE with a fast rise and a prolonged super-Eddington accretion phase 73 , as seen in XJ1500+0154. However, TDEs of evolved stars are expected to be very rare for a black hole of mass 10 6 M⊙, accounting for ∼1% of the total TDEs 26 , and they should be dominated by repeated, partial disruption 74 . Single strong disruptions of evolved stars should be extremely rare, and we do not expect to detect any in the limited volumn of space searched (see below). ROSAT Observations XJ1500+0154 was not detected in the ROSAT All-Sky Survey in 1990. The survey had a detection limit 75 of the 0.1-2.4 keV flux of 5×10 −13 erg s −1 cm −2 , which is a factor of 3 higher than the peak absorbed 0.1-2.4 keV (observer-frame) flux of XJ1500+0154 (∼1.7 × 10 −13 erg s −1 cm −2 in X2 and X3). The source was not detected either in the ROSAT/HRI pointed observation in 1998. The 3σ upper limit of the source net count rate is 0.0038 counts s −1 , which is about 50% lower than that expected from X2 and X3. Therefore, although the source was not detected in ROSAT observations, they are not deep enough for us to rule out the emission level seen in the current outburst. Short-term X-ray variability Supplementary Figure 7 shows the light curves for all bright observations (X2-X3 and C2-C10). We obtained χ 2 probability of constancy of 0.13 and 2.9 × 10 −4 from the 0.3-3 keV pn light curves binned at 500 s in the X2 and X3 observations, respectively. Therefore, no short-term variability was detected from X2. X3 might show some variability, with a fast (within 2 ks) drop to almost zero count rate at around 20 ks into the observation. The average count rates of X2 and X3, which are six days apart, are, however, consistent with each other within 2σ (Supplementary Table 1). We detected no short-term variability in the Chandra observations C2-C10, all with a Gregory-Loredo variability index of 0. We note that the lower count rates in the Chandra observations (a factor of 2 lower than the XMM-Newton/pn in X2 and X3) could make it hard to detect the fast drop as is possibly present in X3. The observations C3-C9, taken within a 13 day period, have count rates consistent with each other within 2σ (Supplementary Table 1). UV and optical long-term variability The SDSS photometry of the host galaxy on May 23rd 2001 is u ′ = 21.71 ± 0.35 AB mag, g ′ = 20.96 ± 0.07 AB mag, r ′ = 20.34 ± 0.06 AB mag, i ′ = 19.97 ± 0.07 AB mag, and z ′ = 19.81 ± 0.27 AB mag (Petrosian magnitudes). The corresponding CFHT/MegaPrime r ′ and g ′ photometry can be obtained through the relation r ′ Mega = r ′ SDSS − 0.024(g ′ SDSS − r ′ SDSS) = 20.32 ± 0.06 AB mag, and g ′ Mega = g ′ SDSS − 0.153(g ′ SDSS − r ′ SDSS) = 20.87 ± 0.07 AB mag. In comparison, we measured r ′ Mega = 20.25±0.01 AB mag and g ′ Mega = 20.92 ± 0.01 AB mag from the CFHT/MegaPrime stacked images in 2014. Therefore the photometry in each filter was consistent with each other in these two epochs to within the 1σ uncertainty. The 3σ upper limit in the r ′ band in 2014 due to the flare is about λL λ < 1.3 × 10 42 erg s −1 or < 3.5 × 10 8 L⊙. If we adopt the reddening relation E(B − V) = 1.7 × 10 −22 NH and use the column density of NH,i = 4.2 × 10 21 cm −2 inferred from the X-ray spectral fits, the above limit after extinction correction would be an order of magnitude larger (i.e., < 3.5 × 10 9 L⊙). The three ASASSN TDEs have λL λ in the range between 10 9 and 10 10 L⊙ at λ = 5400Å (the source restframe central wavelength of the r ′ band for XJ1500+0154) at the very early epoch of the events 76 . Therefore we cannot rule out a similar level of optical emission as seen in the ASASSN TDEs in our event. There is a faint UV source from the Swift observation at the position of the SDSS galaxy. We obtained the W1, M2, and W2 magnitudes of 23.1 ± 0.5 AB mag, 23.3 ± 0.5 AB mag, and 23.8 ± 0.7 AB mag, respectively, from S1, and 23.5±1.4 AB mag, 22.8±0.3 AB mag, and 22.9 ± 0.3 AB mag, respectively, from S2-S8. The relatively large errors are due to the scattered light background from a nearby bright star. The UV source was also detected by GALEX on June 3rd, 2007, with the NUV and FUV magnitudes of 22.9 ± 0.3 AB mag and 23.8 ± 0.3 AB mag (the NUV filter has an effective wavelength similar to that of the Swift M2 filter, i.e., ∼2230Å), The blue UV emission is consistent with each other in these epochs and is consistent with that expected from emission from the young stellar populations based on the pPXF fit to the SDSS spectrum of the galaxy. The UV emission from the three ASASSN TDEs in the early epoch 76 would correspond to a W2 magnitude of >27 AB mag for XJ1500+0154 if we also applied the extinction based on E(B − V) = 1.7 × 10 −22 NH and used the column density of NH,i = 4.2 × 10 21 cm −2 . The U band magnitude in S2-S8 (22.1±0.3 AB mag) is also consistent with the stellar emission. Arguments against the AGN explanation The coincidence with an optical galactic center, with chance coincidence probability of only 0.000017, led us to conclude that XJ1500+0154 is due to nuclear activity in SDSS J150052.07+015453.8. The significant absorption column density other than the Galactic value required in the X-ray spectral fits reassures this association. Then we are left with two possible interpretations of the source: a persistent AGN or a TDE. As we have shown, we see no sign of persistent nuclear activity in the optical spectrum, which shows no clear broad emission lines but narrow ones, with the ratios fully consistent with those expected for starburst galaxies. For AGNs, there is a strong correlation between the persistent hard X-ray luminosity and the extinction-corrected [O III] λ5007 luminosity L c OIII . XJ1500+0154 has L c OIII = 2.9 ± 0.1 × 10 41 erg s −1 if we assume E(B − V) G = 0.05 mag and E(B − V) i = 0.71 mag, which is again based on E(B − V) = 1.7 × 10 −22 NH and intrinsic column density NH,i = 0.42 × 10 22 cm −2 inferred from our X-ray spectral fits. The 2-10 keV luminosity corresponding to this L c OIII is 77 3.6 × 10 42 erg s −1 (the dispersion is 0.63 dex). The corresponding 0.34-2.0 keV unabsorbed luminosity assuming a PL of photon index 2.0 is 3.9 × 10 42 erg s −1 , which is about an order of magnitude lower than the flux seen in the outburst. Given that the line ratios are fully consistent with the star-forming activity, the [O III] λ5007 flux should be dominated by the star-forming activity, instead of that of the nucleus. Then the persistent unabsorbed 0.34-2 keV luminosity implied by the optical spectrum would be much lower than observed in the ourburst, arguing against a persistent AGN explanation for XJ1500+0154. We can also compare XJ1500+0154 with 753 spectroscopically identified AGNs in Lin et al. 78 . These AGNs were included in that study because they had multiple observations, with at least one detection with S/N > 20, in the 2XMM-DR3 catalog. Supplementary Figure 8 shows an X-ray color-color diagram, adapted from Figure 2 in Lin et al., by using the 3XMM-DR5 catalog. The X-ray colors HR2 and HR3 are defined as (H − S)/(H + S), with S and H being the MOS1-medium-filter equivalent 0.5-1 keV and 1-2 keV counts rates for HR2 and 1-2 keV and 2-4.5 keV count rates for HR3, respectively. The MOS1-medium-filter equivalent count rates are those expected for an on-axis MOS1 observation using a "Medium" optical filter 78 . In terms of HR3, which characterizes the spectral shape within 1-4.5 keV, XJ1500+0154 is significantly softer (HR3 in the range between −0.78 and −0.86 for all bright observations C2, C3-C9, C10, X2 and X3) than AGNs (HR3 −0.70). The long-term variability of XJ1500+0154, a factor of >97 (3σ lower limit), is also extreme, compared with AGNs. Only one out of 753 AGNs in Lin et al. varied by a factor of >97 based on the 3XMM-DR5 catalog. The large variability in AGNs has been normally ascribed to all kinds of absorbers 79 . In order to explain the non-detection of XJ1500+0154 in C1 and the low count rate in X1, it would require a neutral absorber of NH > 6.8 × 10 22 cm −2 and NH = 2.5 × 10 22 cm −2 fully covering a X2 spectrum in C1 and X1, respectively. The problem with this explanation is that it would imply no detection of the source below 1 keV in X1, while we had detected the source in 0.4-1 keV at the 3.2σ confidence level in this observation, unless the absorber has a complex structure. Besides, this explanation has to require highly variable absorption of an AGN with unusually soft X-ray spectra. A very small number of high-amplitude very soft X-ray flares were detected from galactic nuclei that might have low persistent luminosities as inferred from the optical emission lines [80][81][82][83][84][85][86][87] . It is under debate on whether these flares are due to, e.g., disk instability in AGNs, or due to TDEs. They might even represent a mixed class, as they showed a variety of spectral shape and temporary evolution. XJ1500+0154 is different from them in several aspects (e.g., generally higher characteristic temperatures and dramatic spectral softening in XJ1500+0154). Therefore it is not clear whether XJ1500+0154 has the same origin as the other flares. There are several AGNs known to show significant spectral changes, between type 1 and type 2 in optical [88][89][90] and/or between Compton-thin and reflection-dominated states in X-rays 91,92 . There are various explanations for them, including variable absorption and changes in the accretion rate (due to, e.g., a TDE). Supplementary Figure 8 in fact includes some such changing-look AGNs, e.g., NGC 1365 and 1H 0707-495. These AGNs change between two AGN standard spectral states with hard X-ray spectra. XJ1500+0154 is different from them in that it changed between two states unseen in standard AGNs, with very soft X-ray spectra. Dwarf starburst galaxies hosting luminous AGNs are also extremely rare in the local Universe. Only a small number of dwarf galaxies are known to show active nuclei [93][94][95][96] . Among them, only Henize 2-10 is in a starburst galaxy, with extremely low luminosity in the nucleus 95 . The column density inferred from the X-ray spectral fits is optically thin to Thomson electron scattering 97 . Then the wind from the central source can sweep up the gas into a shell and push it outwards at a velocity 98 of vs = (fwL8π 2 G/fgasσ 2 ) 1/3 = 2700(fw/fgas) 1/3 km s −1 , where fgas is the mass fraction of the gas, σ = 52 km s −1 is the velocity dispersion, G is gravitational constant, L is the source bolometric radiation luminosity, ∼10 44 erg s −1 , and fwL is the wind energy. Given the super-Eddington accretion nature of the source, we expect fw/fgas to be > 1.0. Therefore vs > 2700 km s −1 , and gas within 1 kpc (the effective radius of the host galaxy) can be swept out in <0.4 Myr, which is very short. Therefore the persistent luminous AGN explanation is hard to be reconciled with the star-forming activity of the host galaxy. We note that we do not have deep radio observations of XJ1500+0154 yet and that it was not detected in the radio surveys NVSS or FIRST. The low sensitivity of these surveys did not allow us to rule out a persistent AGN based on the radio upper limit. Comparison with other candidate TDEs There are about 30 TDEs discovered thus far that are bright in Xrays 99, 100 . Most of them had peak X-ray luminosity <10 44 ergs s −1 , fast decay by about one order of magnitude in a year, and super-soft Xray spectra (kT 0.1 keV) 4, 30, 70, 101-108 . Figure 2 plots a dashed line showing a representative light curve for such super-soft TDEs. Our event, evolving on a timescale at least two orders of magnitude longer, having much harder X-ray spectra, and exhibiting dramatic spectral softening, is different from them. Recently, three TDEs with hard X-ray spectra (photon index around 2.0) and apparent peak luminosity highly super-Eddington were also found [109][110][111][112][113] . They are also short-lived, decreasing by more than two orders of magnitude in the first year. Our event is distinguished from them, with much softer X-ray spectra, long duration, and dramatic X-ray spectral change. Event rate We roughly estimated the rate of TDEs with a super-Eddington accretion phase, characterized by quasi-soft X-ray spectra. XJ1500+0154 was discovered through the search over the detections in the 3XMM-DR5 catalog that have S/N > 20. We selected out sources that showed large variability and/or soft spectra (e.g., HR3 < −0.7). Based on observations outside the Galactic plane (the galactic latitude \|b\| > 20 • ) and assuming that events like XJ1500+0154 have a mean luminosity three times lower than that in X2 but have similar quasi-soft X-ray spectra and last for 10 years, we estimated that we should be able to detect about 1.8 × 10 8 rn events, where r is the event rate per galaxy per year and n is the galaxy density per Mpc 3 . We assumed 25 n = 1.4 × 10 −2 Mpc −3 . We only discovered one with quasi-soft X-ray spectra, indicating r ∼ 4 × 10 −7 gal −1 yr −1 . This is a conservative estimate because our search was still preliminary and the absorption effect was not taken into account. The real rate could be a factor of a few larger. The above event rate calculation depends on the assumption of the duration of the super-Eddington accretion phase in the events. Assuming shorter durations would proportionally infer higher rates. Therefore we cannot rule out a much higher rate for all super-Eddington accreting TDEs as their super-Eddington accretion phase could be generally shorter than that of our event. Although super-Eddington accretion in TDEs is probably rarer than previously thought, due to slow circularization effects, a significant fraction of TDEs are still expected to show a super-Eddington accretion phase 19 . Guided by the discovery of XJ1500+0154, we tried to search for TDEs with quasi-soft X-ray spectra (thus suggesting a super-Eddington accretion phase) from newly public XMM-Newton observations and have discovered two candidates (Lin et al. 2017, in preparation). They have peak X-ray luminosity of ∼10 44 erg s −1 . One of them is clearly in an outburst, with a rising time less than 10 months, and the host galaxy showed no optical emission lines, ruling out the presence of a persistently bright nucleus. The other one is consistent to be in an outburst too, with two X-ray observations showing the flux decay by a factor of ∼6 in 1.5 years, and the host is a star-forming galaxy (or at the end phase of a starburst phase, given the strong Balmer absorption). The durations of these two events still need to be constrained with future observations. Discovery of additional quasi-soft X-ray objects in bright outbursts and in hosts showing no sign of persistent nuclear activity in the optical spectra strongly support that they are super-Eddington accreting TDEs and that such events might not be so rare. (2) the model; (3) intrinsic absorption column density; (4) the apparent inner disk temperature of the MCD; (5) the MCD normization N MCD ≡ ((R MCD /km)/(D/10kpc)) 2 cos θ, where R MCD is the apparent inner disk radius, D is the source distance, θ is the disk inclination; (6) the PL photon index; (7) the PL normalization; (8) the reduced χ 2 and degrees of freedom for fits using the χ 2 statistic (we rebinned the spectra to have at least 20 counts in each bin in order to use the χ 2 statistic; other fits without reduced χ 2 used the C statistic); (9) the 0.34-11.5 keV luminosity, corrected for the Galactic absorption but not instrinsic absorption (10) the 0.34-11.5 keV luminosity (the rest frame), corrected for the Galactic absorption and intrinsic absorption. The fits used spectra within 0.3-10 keV for XMM-Newton and Swift and used spectra within 0.3-7 keV for Chandra. Parameters without errors were fixed in the fits. All errors are at the 90%-confidence level. Supplementary N 02 Figure 1 \| 021H = 64 ± 10 × 10 22 cm −2 , log(ξ) = 2.78 ± 0.04, z = −0.36 ± 0.The CFHT/MegaPrime r ′ -band image around the field of XJ1500+0154 indicates its galactic nuclear origin. The origin of the image is at the center of the galaxy SDSS J150052.07+015453.8 (black cross). The blue circle of radius 0. ′′ 18 (0.5 kpc) represents the 95% positional uncertainty of XJ1500+0154. Figure 2 \| 2The long-term evolution of the X-ray luminosity and spectrum of XJ1500+0154. Upper panel: 31. Jansen, F. et al. XMM-Newton observatory. I. The spacecraft and operations. Astron. Astrophys. 365, L1-L6 (2001). 32. Strüder, L. et al. The European Photon Imaging Camera on XMM-Newton: The pn-CCD camera. Astron. Astrophys. 365, L18-L26 (2001). 33. Turner, M. J. L. et al. The European Photon Imaging Camera on XMM-Newton: The MOS cameras : The MOS cameras. Astron. Astrophys. 365, L27-L35 (2001). arXiv:astro-ph/0011498. 34. Watson, M. G. et al. The XMM-Newton serendipitous survey. V. The Second XMM-Newton serendipitous source catalogue. Astron. Astrophys. 493, 339-373 (2009 ′′ 01 (1.23±0.03 kpc) and 0. ′′ 41±0. ′′ 0.01 (1.04±0.01 kpc), axis ratios of 0.56±0.02 and 0.56±0.01, and indices of 2.2±0.2 and 3.0±0.2, respectively. Therefore it is a small galaxy. The SDSS took a spectrum of the galaxy on March 3rd 2011, which is shown in Supplementary Figure 1. It exhibits strong narrow emission lines at the redshift of z = 0.14542 ± 0.00001 (DL = 689 Mpc, assuming a flat universe with H0=70 km s −1 Mpc −1 and ΩM=0.3). Figure 1 . 1The gas emission lines are all generally consistent with a single Gaussian profile. The intrinsic reddening of the emission lines is E(B − V) i = 0.41 mag, assuming an intrinsic Hα/Hβ ratio of 2.85. Based on more careful Gaussian fits to the star light subtracted spectrum and adopting this amount of intrinsic reddening, we obtained the extinction-corrected line ratios of log([OIII]λ5007/Hβ) = 0.18 ± 0.02 (1σ error), log([NII]λ6583/Hα) = −0.63 ± 0.02, log([SII]/Hα) = −0.79 ± 0.04, and log([OI]λ6300/Hα) < −1.44 (3σ upper limit), which are consistent with a star-forming galaxy based on the BPT diagrams (Supplementary Figure 2) 57, 58 , as indicated by the SDSS pipeline. The extinction-corrected Hα luminosity is 2.13 ± 0.03 × 10 41 erg s −1 , which implies 59 a star formation rate of ∼1.7 M⊙/yr. However, there seem to be some small residuals around the [O III] Supplementary Figure 1 \| 114 Supplementary Figure 3 \|Supplementary Figure 4 \|Supplementary Figure 5 \|Supplementary Figure 6 \|Supplementary Figure 8 \| 111434568The SDSS optical spectrum of the candidate host galaxy of XJ1500+0154 taken on March 3rd 2011, showing only narrow emission lines consistent with a starburst galaxy. The upper two panels zoom into the Hβ-[O III] and Hα-[N II] regions, with the fit residuals. The pPXF fit is shown as a solid green line, while the star component is shown as a red line. The data points outside the emission line regions have been smoothed with a box function of width 5, for clarity. Supplementary Figure 2 \| XJ1500+0154 on the BPT diagrams, indicating it as a star-forming galaxy.The dashed and solid lines are used to separate galaxies into HII-region-like, AGN, and composite types The distribution of the mass and light with respect to the age from the pPXF fit to the SDSS optical spectrum of the candidate host galaxy of XJ1500+0154, indicating the presence of very young (< 5 Myr) populations. The light was integrated over source rest-frame 3200Å and 6800Å. The long-term evolution of the count rates in 0.4-3 keV, 0.4-1 keV and 1-3 keV (observer frame), normalized to those expected assuming the CompTT fit to C3-C9, with 90% errors or 3σ upper limit (for C1). For XMM-Newton observations, the normalized count rates are the mean of all available cameras weighted by the error. The fit residuals for the X2, X3, C2, and C3-C9 spectra, with the MCD, MCD+PL, and CompTT models. The modeling of the long-term X-ray luminosity curve of XJ1500+0154. The meanings of the symbols are the same as inFigure 2. The lines are for different models: solid lines for a 2 M ⊙ star with slow circularization, dashed lines for a 2 M ⊙ star with prompt circularization, and dotted lines for a 10 M ⊙ star with prompt circularization. The thick black lines incorporate the super-Eddington accretion effects, while the thin gray lines do not (see SI); they deviate from each other when the accretion rate is super-Eddington. All models assume a SMBH of mass 10 6 M ⊙ . The X-ray color-color diagram for identified AGNs in Lin et al. 201278 and XJ1500+0154 (big red squares) from the 3XMM-DR5 catalog. The colors HR2 and HR3 are defined using (H − S)/(H + S), with S and H being the MOS1-medium-filter equivalent 0.5-1 keV and 1-2 keV counts rates for HR2 and 1-2 keV and 2-4.5 keV count rates for HR3, respectively. The Chandra C2, C3-C9 and C10 observations are shown as big red diamonds. We overplot PL spectra (dotted lines) with Γ PL = 0.5 (top), 1, 2, 3, and 4 and N H varying from 0 (lower-left) to 10 23 cm −2 . The detections for each source are connected by solid lines in an increasing order of HR3. We only show AGN detections with S/N≥18 (based on the 0.2-4.5 keV flux), resulting in 2002 detections in total. XJ1500+0154 has lower values of HR3 (thus softer in 1-4.5 keV) than AGNs in all observations shown. Randall, S. W. et al. A Very Deep Chandra Observation of the Galaxy Group NGC 5813: AGN Shocks, Feedback, and Outburst History. Astrophys. J. 805, 112 (2015). 1503.08205. 8. Reines, A. E. & Volonteri, M. Relations between Central Black Hole Mass and Total Galaxy Stellar Mass in the Local Universe. Astrophys. J. Titarchuk, L. Generalized Comptonization models and application to the recent high-energy observations. Astrophys. J. 434, 570-586 (1994). 11. Gladstone, J. C., Roberts, T. P. & Done, C. The ultraluminous state. Krolik, J. H. & Piran, T. 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We thank the Swift PI Neil Gehrels for approving our ToO request to make several observations of XJ1500+0154.Supplementary Table 1 \| The X-ray Observation Log. Columns: (1) the observation ID with our designation given in parentheses, (2) the observation start date, (3) the detector, (4) the off-axis angle, (5) the exposures of data used in final analysis, (6) radius of the source extraction region, (7) the net count rate in the source extraction region (0.3-3 keV for XMM-Newton and 0.4-3 keV for Chandra and Swift observations), with 1σ error (but for observations rh601125n00, C1 and S2-S5, the 3σ confidence bounds are given).Obs. IDDate 3-3 keV for XMM-Newton observations and 0.4-3 keV for Chandra observations). For X2 and X3, we only show the pn data for clarity, and data in the strong background intervals have been excluded. The light curve bin size is 500 s for X2 and X3, 2 ks for C2-C9, and 6 ks for C10. There seems to be a. 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[ "A Quantum Extended Kalman Filter", "A Quantum Extended Kalman Filter" ]	[ "Muhammad F Emzir [email protected] \nSchool of Engineering and IT\nUniversity of New South Wales\nADFA\nACT 2600CanberraAustralia\n", "Matthew J Woolley \nSchool of Engineering and IT\nUniversity of New South Wales\nADFA\nACT 2600CanberraAustralia\n", "Ian R Petersen \nSchool of Engineering and IT\nUniversity of New South Wales\nADFA\nACT 2600CanberraAustralia\n" ]	[ "School of Engineering and IT\nUniversity of New South Wales\nADFA\nACT 2600CanberraAustralia", "School of Engineering and IT\nUniversity of New South Wales\nADFA\nACT 2600CanberraAustralia", "School of Engineering and IT\nUniversity of New South Wales\nADFA\nACT 2600CanberraAustralia" ]	[]	A stochastic filter uses a series of measurements over time to produce estimates of unknown variables based on a dynamic model[1]. For a quantum system, such an algorithm is provided by a quantum filter[2], which is also known as a stochastic master equation (SME)[3]. For a linear quantum system subject to linear measurements and Gaussian noise, the quantum filter reduces to a quantum Kalman filter[4,5]. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to non-commutative quantum stochastic differential equations (QSDEs). We will show that there are conditions under which a filter similar to the classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problems with 'state-dependent' covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems which involve multiple modes, nonlinear Hamiltonians and simultaneous jump-diffusive measurements.	10.1088/1751-8121/aa6e5e	[ "https://arxiv.org/pdf/1603.01890v1.pdf" ]	54,952,068	1603.01890	50639f060ec67f236366ae034e88692a1bfb2e9b	A Quantum Extended Kalman Filter March 2016 Muhammad F Emzir [email protected] School of Engineering and IT University of New South Wales ADFA ACT 2600CanberraAustralia Matthew J Woolley School of Engineering and IT University of New South Wales ADFA ACT 2600CanberraAustralia Ian R Petersen School of Engineering and IT University of New South Wales ADFA ACT 2600CanberraAustralia A Quantum Extended Kalman Filter March 2016 A stochastic filter uses a series of measurements over time to produce estimates of unknown variables based on a dynamic model[1]. For a quantum system, such an algorithm is provided by a quantum filter[2], which is also known as a stochastic master equation (SME)[3]. For a linear quantum system subject to linear measurements and Gaussian noise, the quantum filter reduces to a quantum Kalman filter[4,5]. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to non-commutative quantum stochastic differential equations (QSDEs). We will show that there are conditions under which a filter similar to the classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problems with 'state-dependent' covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems which involve multiple modes, nonlinear Hamiltonians and simultaneous jump-diffusive measurements. Introduction In light of recent advances in quantum engineering, the need to effectively measure and control complex quantum systems has become more crucial. One requirement is to infer unknown observables of a quantum system from noisy measurements based on a dynamic model, a process known as a filtering. The quantum filter was developed in the 1980's by Belavkin [2,6,7,8]. It has recently been used in experimental systems such as trapped ions [9], cavity QED systems [10], and optomechanical systems [11]. The real-time application of the quantum filter demands an efficient computational algorithm. The quantum filter in the Schrödinger picture, also known as the stochastic master equation, solves the stochastic evolution of the conditional density operator based on the measurement record. In practice, to numerically compute the filter, one has to truncate the Hilbert space basis. The computation time then scales exponentially with the size of the Hilbert space basis which makes the quantum filter difficult to implement in real-time. In the Heisenberg picture, the quantum filter dynamics reduce to the dynamics of the systems observables x t . However, for general nonlinear systems, the quantum filter arXiv:1603.01890v1 [quant-ph] 6 Mar 2016 in the Heisenberg picture cannot be interpreted as an explicit solution to the filtering problem [12]. These two facts lead to the need for new approximation techniques. Among these are a Gaussian approximation of the conditional density operator [13,14]. This work is rather heuristic and lacks an evaluation of the estimation error. Number phase Wigner particle filters have also been suggested [15], but like most particle filter based techniques, they are computationally expensive and not suitable for real-time filtering. Recently a Volterra series approximation [16] was introduced. Although this approach has a tractable error bound [17], estimating the Volterra kernel is complicated in general, and the filter structure is not recursive. Our approach is to use a commutative operator approximation of the non-commutative nonlinear QSDE. A first order Taylor expansion of the nonlinear quantum Markovian generator is then used to compute the filter gain. We refer to our approach as the quantum extended Kalman filter (quantum EKF), due to the similarity of its structure to the classical extended Kalman filter (EKF). Classical EKF estimation error convergence has been well studied [18,19,20,21,22], which has resulted in some criteria that guarantee local convergence behavior. The main difficulty in this approach is that the quantum system, in contrast to a classical system, is governed by a QSDE which involves non-commutative operators. This noncommutativity implies that there is no isomorphism that could map the dynamics into a standard stochastic differential equation (SDE). Due to the non-commutative nature of the QSDE, generally the ordinary partial differentiation with respect to a system observable may not be well defined [23,24]. Further, the sub-optimality condition of the estimation error has to be defined differently since the system's observables are operators in Hilbert spaces and not scalar random variables. In this article, we provide a mathematical description of a quantum EKF. We then establish a sufficient criterion under which the quantum EKF will satisfy a dissipativity condition, which ensures the boundedness of the quadratic estimation error. The cases where the quantum systems and their measurements have a state-dependent covariance, i.e. they are functions of the system observables, are treated as an extension of the quantum EKF. We show that it is sufficient to modify the behavior of the Riccati differential equation to guarantee an equivalent dissipativity condition as the original quantum EKF. Finally, we demonstrate the application of the quantum EKF to two estimation problems. The first problem is the estimation of quadratures of two cavity modes with Kerr nonlinearities [25] subject to homodyne detection. The second example is the estimation of quadratures of a cavity subject to simultaneous homodyne detection and photon counting measurements. The readers are referred to [26] for an introduction to quantum probability, quantum stochastic calculus, and quantum non-demolition measurements. We shall use the term 'classical' to refer to commutative dynamics and filters. We shall also use the term 'state-dependent' covariance to refer to covariance matrices that are functions of the system observables. The article is organized as follows. The second section will contain some preliminary facts that are used in this article. The main section will describe the main contribution of this work. The first is the mathematical description of the quantum EKF and its existence. The computational complexity of the quantum EKF is analyzed as compared to the SME. Next, we analyze the convergence of the quantum EKF. Lastly, we show an extension of the quantum EKF for state-dependent quantum systems. The third section will be examples of quantum EKF applications, and the last section is the conclusions. Notation Classical probability spaces are denoted by a triple (Ω, F, µ). Plain letters (e.g. P ) will be used to denote elements of an algebra. P will be used for a measure from a von Neumann algebra A to a complex number C, that is positive and normalized, i.e. P (A * A) ≥ 0 and P (1) = 1. We also use E P (·\|A) to denote a conditional expectation with measure P with respect to a commutative von Neumann algebra A. Script face (e.g. H for Hilbert space) is used to denote a spaces as well as any type of algebra. A class of operators will be denoted by calligraphic face, e.g., for bounded linear operators from a Hilbert space H, to a Hilbert space K, we denote B (H, K), and also we denote B (H) ≡ B (H, H). Bold letters (e.g. y) will be used to denote a matrix whose elements are operators on a Hilbert space. Hilbert space adjoints, are indicated by * , while the complex conjugate transpose will be denoted by †, i.e. (X * ) = X † . For single-element operators we will use * and † interchangeably. The commutator of x and y is given by [x, y] = xy − yx , while their anti-commutator is given by {x, y} = xy + yx . (1) α Preliminary In the classical stochastic case, the optimal nonlinear filter is given by the Kushner-Stratonovich equation [27]. This equation is based on the existence of a sample path, X t , whose dynamics is described by a stochastic differential equation, e.g. dX t = b(X t )dt + σ(X t )dW t . The nonlinear filter for a function of X t , f (X t ), is then given via the generator of the Markov process f (X t ), Lf(X t ) ≡ lim dt↓0 E [f (X t+dt )\|X t = x] − f (x) dt . In the nonlinear quantum filter, the sample path of the underlying process X t generally does not exist [26], but instead, we have an equivalent evolution of a unitary operator. By means of the evolution of the unitary operator, for open quantum system with Hamiltonian H t , and bath coupling operator L t , the equivalent quantum Markovian generator for a set of observables x t is given by [28] L( x t ) = − ī h [x t , H t ] + 1 2 L † t [x t , L t ] + 1 2 [L * t , x t ] L t .(2) We begin our formulation by considering Hilbert spaces for the system and the environment. First, let the system Hilbert space and the field Boson Fock space, be given by H s and Γ(h). The total Hilbert space is given by H T = H s ⊗ Γ(h) and H T[0,t] = H s ⊗ Γ(h) [0,t] . The unitary evolution of the system interaction with the field is described by U t ∈ U(H s ⊗ Γ(h) [0,t] ) , where U is the class of unitary operators on the associated Hilbert spaces. Let the evolution of a system operator X be given by X t = U * t (X ⊗ 1) U t where 1 is identity operator in Γ(h) . Furthermore, without loss of generality, we will assume that the field is initially on the vacuum state. Let the system and field's initial density operators be given by ρ ∈ S(H s ) , ω ∈ S(Γ(h)), where S is the class of unity trace operator on the associated Hilbert spaces. The fundamental quantum processes as in [29] are given by dA * t , dA t , dΛ t , the annihilation, creation and conservation processes. These processes are forward time differential, i.e. dA * t , dA t , dΛ t ∈ B(H T[t,t+δt] ) and hence commute with x t . The counting process dλ t is defined as the diagonal element of dΛ t . From now on we let B ≡ B (H T ) and B [t,t+δt] ≡ B H T[t,t+δt] , B t] ≡ B H T t] . Furthermore, for given system and field initial density operators (ρ ⊗ ω) there is a corresponding measure P, called a state, which is positive linear and normalized. For a bounded operator X ∈ B t] , P (X) = Tr [Xρ ⊗ ω], see [29,Proposition 9.19]. Let us now fix the von Neumann algebra N = B, and the set of bounded self adjoint operators in total Hilbert space, O = O (H T ) ⊂ N. We denote x t ∈ O t to be a set of system observables evolved up to time t. In the quantum probability setting, a quantum probability space is defined by specifying a von Neumann algebra N and a state P. Let A ⊂ N be a commutative von Neumann sub algebra. We call the set A = {B ∈ N : AB = BA, ∀A ∈ A} the commutant of A in N. The conditional expectation in the quantum probability setting is defined as follows. Definition 2.1. [26] For a given quantum probability space (N, P), let A ⊂ N be a commutative von Neumann sub-algebra. Then the map E P [·\| A] : A → A is called the conditional expectation from A onto A if P (E P [B\| A] A) = P (BA) , ∀A ∈ A, B ∈ A . The following theorem is fundamental to obtain the relation between quantum probability and the classical Kolmogorov probability axioms. In addition, we will use this theorem to show the implementation of the quantum EKF as a classical EKF in the following section. The proof of the theorem is given in [30, §1.1.8] [26]. Theorem 2.1. Let A be a commutative von Neumann algebra. Then A is * -isomorphic to L ∞ (Ω, F, µ), with * -isomorphism τ . Furthermore, a normal state P on A defines a probability measure µ P , which is absolutely continuous with respect to µ, such that, P (A) = E µ P (τ (A)), for all A ∈ A. We notice here that the commutative von Neumann algebra A corresponds to a classical field F, and every projection operator in A corresponds to a classical event. In the following discussion, we will use · the following semi-norm on B: if x ∈ B n×1 , then (3) x ≡ P x † x 1 2 . This quantity is not a norm since, P x † x can be zero for non zero x that is perpendicular to the density operator. Furthermore, a partial order of two operators, A > B is taken in the sense of P, where A > B in P denotes P (A − B) > 0. Now suppose we have two operator vectors x ∈ A ,x ∈ A, where A is a commutative von Neumann algebra. Under semi-norm(3), by Definition 2.1, we obtain (4) x −x = P (x −x) † (x −x) 1/2 = P E P (x −x) † (x −x) A 1/2 . The last equation implies that, for any > 0, {ω : x −x < } ∈ A. That is, the event that x −x < is A− measurable. Later on we will use this fact to define a Markov time when we are dealing with the stochastic stability of the quantum EKF, see Section 3.3. For a self adjoint element T ∈ N there is a * -isomorphism f → f (T ) from a continuous function in the spectrum of T , I T = sp(T ) ∈ R, namely C(I T ), onto the C * -subalgebra C * (T ) of N generated by T and the identity element 1. The following two propositions show how one can define a partial derivative for an operator differentiable mappings which will be used in the filter algorithm. Both have been proved in [31,32], but we mention the proof here again for the sake of completeness. For the two propositions, we will recall the following definitions, Definition 2.2. [33,31]If X and Y are Banach spaces with norm · X and · Y respectively, a mapping f : D → Y on a subset D of X is Fréchet differentiable at T ∈ D if there is a bounded linear operator D (f,T ) in B (X, Y), the class of linear bounded functions from X to Y, such that, (5) lim S →0 f (T + S) − f (T ) − D (f,T ) S Y S X = 0. If D (f,T ) is defined for every T ∈ X, then f is Fréchet differentiable on D. Proof. Without losing of generality, we may assume that I is bounded, i.e I = {x ∈ R : a ≤ i ≤ b}, and f : U → U. Set U = C b (I), the set of a bounded continuous functions on I. Since f ∈ C 1 op (I), the differentiability of f at a function g ∈ U implies that for every > 0, there exists δ > 0 such that for any function h, h < δ (6) f (g + h) − f (g) − D (f,g) h ≤ h , which shows that, D (f,g) h : h → (f • g) h. Suppose there exist two arbitrary points x 1 , x 0 ∈ I, satisfying \|x 1 − x 0 \| < δ. Now let g(x) equal the identity function, and h(x) = x 0 − x 1 constant. We get, h < δ, and at x = x 1 , f (x 1 + x 0 − x 1 ) − f (x 1 ) − f (x 1 ) (x 0 − x 1 ) = f (x 0 ) − f (x 1 ) − f (x 1 ) (x 0 − x 1 ). Next interchanging x 0 and x 1 , we get, f (x 1 ) − f (x 0 ) − f (x 0 ) (x 1 − x 0 ). Adding this and the previous equation, and dividing by x 1 − x 0 , we get f (x 0 ) − f (x 1 ) < which shows the continuity of f on I. Proposition 2.2. [31] If f ∈ C 1 op (I), for any two elements S, T ∈ U, a unital commutative C- * algebra, then D (f,T ) S = f (T )S. Proof. Let S, T ∈ U, a unital commutative C- * algebra on the space X. We write, f (T )(x) = f (T (x)), ∀x ∈ X, so that, as the previous proposition, f (T ) = f • T . Since f ∈ C 1 op (I), by definition, there exists a differential D T f ∈ B (U, U), such that, (7) f (T + S)(x) = f (T )(x) + D (f,T ) S(x) + R (x), where lim →0 R = 0. By Proposition 2.1, it follows that, D (f,T ) S(x) = f (T (x))S(x), as desired. By definition the last proposition is also true for a commutative von Neumann algebra, since any von Neumann algebra is a C- * subalgera of B (H) that is strongly closed and contains the identity, for H is any Hilbert space [34]. The condition that U is commutative in Proposition 2.2 is essential. In addition, it was proven that one can construct a C 1 function that is not operator differentiable [35]. Moreover, if f ∈ C 2 , then its extension is operator differentiable, see [36,37]. Hence C 2 ⊆ C 1 op ⊆ C 1 . Definition 2.4. An operator A in a Hilbert space H is single valued, if it can be written as a scalar times an identity, A = a1, a ∈ C.(8) A matrix A whose each elements are operators in H is single valued if each element of A is single valued. Main Statement Extended Kalman filter for a class of open quantum quantum systems with diffusive and Poissonian measurements Let G be an m channel open quantum system with parameters (S, L, H), S ∈ C m×m , L ∈ B m×1 t , H ∈ O m×1 t and SS † = S † S = I. The class of open quantum system that we consider here is the one which has covariance matrices independent of x t . A more general class of open quantum system with state-dependent covariances will be considered in the following section. The smallest von Neumann algebra generated from the measurement up to time t is given by Y t . As in the classical setting, we can always assume that the von Neumann algebra Y t is right continuous, by taking Y t = Y t + = s>t Y t [38]. We assume that the scattering matrix S is single valued. This implies that for any X t = U † t XU t , Tr ((S † X t S) − X t )dΛ t = 0. This ensures that the open quantum system evolution can be described by a diffusive QSDE as follows [39]: (9a) dx t = f (x t )dt + G(x t )dA * t + G(x t ) * dA t , with, f (x t ) = L(x t ), G(x t ) = [x t , L t ] S * t . We shall assume that the number of measurements subject to the open quantum system G also equal to m. The class of measurements of the quantum system above are assumed to be a collection of m functions of output field creation, annihilation, and conservation processes, as below (10a) dy t = E t dÃ * To satisfy the non-demolition and self-nondemolition properties, E t and N t have to satisfy the algebraic condition in Theorem 3.1 of [12]. We could then simplify (10a) to dy t =h(x t )dt + L(x t )dA * t + L(x t ) * dA t + N t dα t , (11a) h(x t ) =E * t L t + E t L * t + N t l t , (11b) L(x t ) = (E t + N tL ) S * t ,(11c) where N t ∈ O m×m , and L =    L 1,t · · · 0 0 L i,t 0 0 · · · L m,t    , l t =    L * 1,t L 1,t . . . L * m,t L m,t    , dα t = diag S * t dΛS t . Now, we define the variance and covariance of the system's observables and measurements as follows, (12a) Q t = 1 2dt E P [{dx t , dx t }\| Y t ] , (12b) R t = 1 2dt E P [{dy t , dy t }\| Y t ] , (12c) S t = 1 2dt E P [{dx t , dy t }\| Y t ] . The use of anti-commutator in (12) above ensures that each element of the variance matrices is a self adjoint operator. In the remainder of the article, we require that R t is a positive definite matrix of operators in Y t , i.e τ (R) t,ω > 0, ∀t ≥ 0, ω ∈ Ω, where τ is the * −-isomporhism from Y t to L ∞ (Ω, F, µ). We will use Proposition 2.2 to calculate the partial derivative of the nonlinear quantum Markovian process generator f (x). As we will show shortly, we could establish a mild condition on the quantum systems parameters S, L, H so that f i , h j ∈ C 1 op (R)∀i ≤ n, j ≤ m. The following proposition gives a sufficient condition under which f i , h j ∈ C 1 op (R)∀i ≤ n, j ≤ m belongs to C 1 op (R). Proposition 3.1. In order to have f i , h j ∈ C 1 op (R)∀i ≤ n, j ≤ m, it is sufficient to require that H, L ∈ C 1 op (R) and E t , N t ∈ C 1 op (R) Proof. To prove this, we will first claim that for any two function, g 1 , g 2 ∈ C 1 op (I), with I ⊆ R , then g 1 g 2 ∈ C 1 op (I). Without loss of the generality, we can fix any S ∈ U, where S U = 1. Letting ε → 0, the linear operator D g i ,T as in (5) is given by D g i ,T = lim ε→0 g i (T + εS) − g i (T ) ε . Therefore, we can write for g 1 g 2 , D g 1 g 2 ,T = lim ε→0 g 1 g 2 (T + εS) − g 1 g 2 (T ) ε = lim ε→0 (g 1 (T + εS) − g 1 (T )) g 2 (T + εS) ε + lim ε→0 g 1 (T ) (g 2 (T + εS) − g 2 (T )) ε = D g 1 ,T g 2 + g 1 D g 2 ,T , which shows that g 1 g 2 ∈ C 1 op (I). Since I is arbitrary, the desired result follows immediately by applying the claim to f and h. Since the system's observables belong to the commutant of the commutative von Neuman algebra Y t , Y t , it is reasonable to approximate each element of x t , x i,t , with a set of commutative estimatesx i,t ∈ Y t . The differencex i,t = x i,t −x i,t also belongs to the commutant algebra Y t . Then for eachx i,t , bothx i,t and Y t generate a larger commutative algebra Z i,t for which the Proposition 2.2 can be invoked to infer the partial derivative of f (x t ) with respect to x t atx t . We could not go further beyond the first order term of the Taylor series as in the classical nonlinear filter since the second order partial derivative of f (x t ) will generally involve multiplication of two elements of x i,txj,t ∈ Y t which generally do not commute with each other, and consequently a larger commutative algebra Z ij,t generally does not exist. Our approximation begins by conjecturing that it is possible to construct a filter algorithm such that if the estimatex t is initially in the measurement algebra, it will always be inside it in the future,x 0 ∈ Y 0 ⇒x t ∈ Y t ∀t ≥ 0. In contrast to the formulation of quantum Kalman filter in [40,5], since we will neglect the residual terms of Taylor series in our filter structure,x t is no longer optimal in the sense of the distance from x t to a projection on Y t , i.ex t = E P [x t \| Y t ]. Nonetheless, each element ofx t is commutative with respect to the other elements, as they are belong to the same commutative von Neumann algebra Y t . Since we require that f i ∈ C 1 op (R) andx t ∈ Y t ,x t ∈ Y t , the condition in Proposition 2.2 is satisfied. Consequently, we can write (13a) f (x t ) = f (x t ) + ∂f (x t ) ∂x t xt=xtx t + r f (x t ,x t ) , with (13b) F(x t ) ≡ ∂f (x t ) ∂x t xt=xt , and (13c) h(x t ) = h(x t ) + ∂h(x t ) ∂x t xt=xtx t + r h (x t ,x t ) , (13d) H(x t ) ≡ ∂h(x t ) ∂x t xt=xt , where r (·) is the residual term of the Taylor expansion. The measurement error operator is given by (14) dy t − dŷ t = (H(x t )x t + r h (x t ,x t )) dt + L(x t )dA * t + L(x t ) * dA t + N t dα t . To construct a quantum EKF, we define a matrix of operators P t ≡ 1 2 E P [{x t ,x t }\| Y t ] , that is the Hermitian variance of the estimation error. The quantum EKF problem is then given as follows. For a given open quantum system subjected to measurements with corresponding QSDEs given in (9a) and (11) respectively, the following conditions are assumed : (i) Variances and cross correlation matrices Q t , R t , S t are single valued. (ii) R t is invertible. (iii) Initiallyx 0 ∈ Y 0 . We find a matrix K t ∈ Y t corresponding to (15) below, such thatx t ∈ Y t , ∀t ≥ 0 and P t evolves according to the following Riccati differential equation upon neglecting the residual term of the Taylor series, (15) dx t = f (x t )dt + K t (dy t − dŷ t ) , (16) dP t dt = F(x t )P t + P t F(x t ) + Q t − P t H(x t ) + S t R −1 t P t H(x t ) + S t . We notice that since we neglect the non linear residual term in (16), P t is no longer interpreted as the variance of the estimation error, but rather a general positive definite matrix of operators which will be involved in the dynamics of the filter. Without loss of the generality, we can assume that P 0 ∈ Y 0 . The following theorem shows the existence of a quantum EKF satisfying the condition above. Consider an open quantum system described by the QSDEs given in (9a) subjects to the measurements given in (10a). Then, there exists a Kalman gain K t ∈ Y t , K t = P t H(x t ) + S t R −1 t ,(17) such that if the quantum extended Kalman filter is given by (15), thenx t ∈ Y t , ∀t ≥ 0 and P t evolves according to (16) upon neglecting the residual term of the Taylor series in (13). Proof. To establish the first part of the theorem, we need to show that there exists K t ∈ Y t , such thatx t ∈ Y t , ∀t ≥ 0. The condition thatx t ∈ Y t , ∀t ≥ 0 follows from the causality of the quantum EKF given in (15). But to make it clear, let K t ∈ Y t . We observe that the differential equation of the quantum EKF can be written in integral form as below t 0 dx s = t 0 [f (x s ) − K s h (x s )] ds + t 0 K s dy s , where the first term of the right hand side involves Reimann-Stieltjes integration. Since the integral above is defined then, there exists a partition of times 0 = t 0 ≤ t 1 ≤ . . . t N = t, ∆t i ≡ t i − t i−1 , such that we can define an infinite sum, by taking lim N →∞ sup i ∆t i = 0, regarding the second integration in dy s in the Itô sense, x t =x 0 + lim N →∞ N −1 i=0 [f (x t i ) − K t i h (x t i )] ∆t i+1 + lim N →∞ N −1 i=0 K t i y t i+1 − y t i . Hence, it is clear that ifx 0 , ∈ Y 0 , thenx t ∈ Y t , ∀t ≥ 0 , since it is in the span of y t i and x t i ∀t i < t. For the second part of the theorem, let us now expand dx t according to the Taylor expansion in (13a) and (13c). Doing that leads us to the following equation, dx t = f (x t )dt + K t ((H(x t )x t + r h (x t ,x t )) dt + L(x t )dA * t + L(x t ) * dA t + N t dα t ) . From the equation above, the estimation error can be given by the following equations, (18a) dx t = (F(x t ) − K t H(x t ))x t dt + rdt + (G(x t ) * − K t L(x t ) * ) dA t + (G(x t ) − K t L(x t )) dA * t − K t N t dα t , (18b) r t (x t ,x t ) = r f (x t ,x t ) − K t r h (x t ,x t ) . Using the quantum Itô multiplication rule [29], we can obtain the estimation variance dynamics as follow (19a) dP t dt = (F(x t ) − K t H(x t )) P t + P t (F(x t ) − K t H(x t )) + Θ(x t ,x t ) +h 2 E P [ {dx t , dx t }\| Y t ] , with, (19b) Θ(x t ,x t ) ≡ 1 2 E P [{r t (x t ,x t ) ,x t }\| Y t ] . Now, from (19a), by the definition of variances given in (12a),(12b) and (12c), and that K t ∈ Y t , we obtain (20) dP t dt = F(x t )P t + P t F(x t ) + Q t + K t R t K t − K t H(x t )P t + S t + H(x t )P t + S t K t + Θ(x t ,x t ). Since P t ,R t ,S t , andx t belong to Y t , then the Kalman gain given in (17) also belongs to Y t . Substituting (17) to (20), and ignoring the nonlinearity Θ we obtain the desired result as in (16). Before we go further, we would like to address the implementability of the quantum EKF in (15). In practical applications, we would often be given initial values ofx t and P t , rather than a complete description of a set of operators in an underlying Hilbert space. Furthermore, in many cases, the interest is only to estimate the mean value and covariance of x t . Given such conditions, we would like to see the relation between the quantum EKF and the classical EKF. From (15), we have the evolution ofx t ∈ Y t , which can be written as dx t = [f (x t ) − K t h (x t )] dt + K t dy t .(21) By definition dy t ∈ Y t and hence by Theorem 2.1 there exist a * − isomorphism τ from Y t to L ∞ (Ω, F, µ). From now on, we write τ (·) t,ω ∈ C as (·) t,ω . Then we write ∀ω ∈ Ω, ∀t ≥ 0, dx t,ω = [f (x t,ω ) − K t,ω h (x t,ω )] dt + K t,ω dy t,ω ,(22) which is an ordinary stochastic differential equation. Using the same * − isomorphism τ , the dynamics of the estimation error variance, (see the Riccati equations in (16)) can also be transformed into classical Riccati differential equations. This transformation in turn makes the quantum EKF implementable as a recursive filter in a digital signal processor. Remark 3.1. It is worth emphasizing that the essential difference between the quantum EKF and the classical EKF is the fact that the set of the system's observables x t do not belong to a commutative von Neumann algebra. In addition, the dynamics of x t generally consist of non-commutating operators and hence there is no * − isomorphism that can transform x t and its dynamics into a measurable function on a classical probability space (Ω, F, µ). Otherwise, if this is the case, then the quantum EKF problem is reduced to the classical EKF problem. For photon counting measurements, a Poisson processes can be written as a sum of two independent quantum Gaussian noises as in (10). This enables us to treat a filtering problem for both diffusive and jump measurements simultaneously. This is unique to quantum stochastic filtering, since in a classical settings, one can never have a transformation from a jump random process into a continuous processes, [26]. One Step Computational Complexity of the SME and the quantum EKF Here we present a comparison of the computational complexity of the SME and the quantum EKF. The SME for m output measurement channels is given by [12], (23) dρ t = −i [H, ρ t ] + L ρ t L * − 1 2 L † Lρ t − 1 2 ρ t L † L dt + ζ ρ Γ − dW. In this equation, ρ t is the system's conditional density operator and dW is the error vector between the expected value and the measurement. The weighting function ζ ρ Γ − relates the contribution of each measurement to the total increment of the conditional density operator. Now, suppose we have N m subsystems, and under truncation let each system's Hilbert space dimension be N s . The computational complexity of (23), will be of order O N m (N Nm s ) 3 = O N m N 3Nm s . Now, suppose we want to estimate n observables of the system. Then after propagating the SME, we need to calculate Tr [ρ t x i ] , i ≤ n, which is on the order of O n(N Nm s ) 3 . Consequently to propagate n observables from each subsystem from SME, we will need a calculation effort ∼ (ζ 1 n + ζ 2 N m ) N 3Nm s , for some ζ 1 , ζ 2 > 1. In contrast, after transforming the Riccati and quantum EKF equations to the standard SDE, the computational complexity of the quantum EKF is the same as that of the classical EKF. The EKF computational effort only depends on n, m and N m , and the complexity of evaluating f , and the Jacobian matrices F and H . In a single time step, one has to propagate the Riccati equation in (16), which has the complexity O n 3Nm , calculation of the Jacobian matrices F and H which could vary depend on the type of the function involved, plus solving the quantum EKF (15). Evaluating (15) involves the calculation of f which also can vary, and matrix-vector calculation in K t dy −dy , which is O(m 2 (n Nm )). Convergence analysis Here we will establish a stochastic convergence condition for the quantum EKF. The approach we pursue here is closely related to the stochastic convergence analysis of the classical extended Kalman filter and deterministic nonlinear observer design [22,41,42]. In essence, the main difference between the classical EKF stochastic convergence proof and what we present here is the use of the semi-norm in (3) in the place of Euclidean norm, and the quantum Markovian process generator. Moreover, due to the coupling nature of the measurement and process noise in every open quantum system, we need to assume the positive definiteness of Q t − S t R −1 t S t . The following assumptions, definitions, and lemmas will serve as a foundation for the local convergence condition of the quantum EKF estimation errors. (3), together with our previous assumption that R t is positive definite matrix of operators in Y t , we also need the following assumptions in our analysis. AI Let E f , E h ⊂ Y t . For E f = {x t : x t −x t ≤ f }, E h = {x t : x t −x t ≤ h }, f , h > 0, ∃r h , r f > 0, such that the residual term of r f and r h satisfy the following rate constraint r f < r f x t 2 , ∀x t ∈ E f , r h < r h x t 2 , ∀x t ∈ E h , AII The operator valued matrix H, and the cross correlation S are bounded from above, H (x) ≤h,(24) S ≤s. This assumption follows from the fact that we restrict our observable to the von Neumann algebra N = B. Moreover, since h ∈ C 1 op ⊂ C 1 , H is bounded. AIII P t is always greater than zero and bounded, i.e, 0 < pI ≤ P t ≤pI , ∀t ≥ 0. If we consider P t as an estimation error covariance, this condition will generally be valid in quantum mechanical system estimation since the Heisenberg inequality dictates that 0 < pI. Furthermore, it seems also natural to consider P t to be bounded from above. AIV Q t − S t R −1 t S t ≥ 0, ∀t ≥ 0.(27) As we will encounter later on in Lemma 3.3, to satisfy the dissipative inequality, we require that Q t − S t R −1 t S t > 0 is always satisfied. However, as we will show in the Proposition 3.2, we can only obtain a sufficient condition that mQ t −S t R −1 t S t ≥ 0. In the next section, we will show how to deal with this restriction. We will now show the inequality, mQ t − S t R −1 t S t ≥ 0, by using the following Lemma, Lemma 3.1. If a set of m measurement y t satisfying self-non demolition property, then [dy i,t , dy j,t ] = 0 ∀i, j ≤ m, t ≥ 0. Proof. To see this, we first examine that if y t satisfying self-non demolition property, we could write for i, j ≤ m d [y i,t , y j,t ] = 0 = [dy i,t , y j,t ] + [y i,t , dy j,t ] + [dy i,t , dy j,t ] , but [dy i,t , y j,t ] = [y i,t , dy j,t ] = 0. This can be seen from dy i,t = k U * t (z k ⊗ 1) U t db k , where db k ∈ {dA i , dA * i , dα i , dt} , i ≤ m, and z ∈ B (H s ), whilst y j,t = U * t (1 ⊗ y) U t , where y ∈ B (Γ(h) t }). Hence [dy i,t , dy j,t ] = 0. Proposition 3.2. For an open quantum system with QSDEs and measurement given in (9a) and (10a) respectively, the covariance matrices Q t , R t , S t satisfy the following inequality mQ t − S t R −1 t S t ≥ 0, ∀t ≥ 0.(28) Proof. Let x i,t and y j,t denote the i and j th elements of x t and y t respectively. From Lemma 3.1, we have [dy i,t , dy j,t ] = 0 ∀i, j ≤ m, t ≥ 0. LetS t = dx t dy t , S t = dy t dx t ,Q t = Q t = dx t dx t , andR t = R t = dy t dy t . We first claim that there exists a symmetric matrix of operators M t ∈ Y t such that {dx t , dx t } as below, (29) {dx t , dx t } =S t M t S t + S t M t S t . To see this, we can write {dx t , dx t } i,j = dx i,t m,m k,k =1 dy k,t M k,k dy k ,t dx j,t + dx j,t m,m k,k =1 dy k ,t M k,k dy k,t dx i,t , by requiring, (30) m,m i,j =1 dy i,t M i,j dy j,t = 1. From (30), since M t ∈ Y t , we have, m,m i,j =1 dy i,t M i,j dy j,t = 1 = m,m i,j=1 dy i,t dy j,t M i,j = m,m i,j=1 dy i,t dy j,t M j,i = Tr R t M t . Hence, selecting M t = 1 mR −1 t ∈ Y t ⊂ Y t the assertion is verified. Now since S t R −1 t S t is a convex function with respect to both S t and R t (see [43,Proposition 8.6.17 (xiv)]), by the Jensen inequality S t R −1 t S t = 1 dt E P 1 2 S t + S t Y t E P R t Y t −1 E P 1 2 S t + S t Y t ≤ 1 dt 1 4 E P S t + S t R −1 t S t + S t Y t ≤ 1 dt 1 2 E P S tR −1 t S t + S tR −1 t S t Y t = m dt 1 2 E P S t M t S t + S t M t S t Y t = m dt E P 1 2 Q t + Q t Y t = mQ, which completes the proof. The following Lemma gives a bound on the nonlinear residual rate based on Assumption 3.1, see also [22,Lemma 3.3] for an analogous of result for the classical EKF. Lemma 3.2. Consider a functional φ that is a function of the residual r t and the estimation errorx t given below, φ(r t ,x t ) ≡r t P −1 tx t +x t P −1 t r t .(31) Then under Assumptions 3.1, there exists positive and κ such that for every x t ≤ , then (32 ) φ(r t ,x t ) ≤ κ x t 3 . Proof. From (18b), we have φ(r t ,x t ) = r f P −1 tx t +x t P −1 t r f − r h K t P −1 tx t +x t P −1 t K t r h . Furthermore using AIII and AII, k ≡ K t ≤ ph +s r . From AI, taking = min( f , h ) φ(r t ,x t ) ≤2 r hk + r f p x t 3 ≡ κ x t 3 . To examine the convergence behavior of the coupling dynamics between the estimation error and the covariance, we consider a Lyapunov positive operator V ∈ O + (H T ) given by V (x t ) =x t P −1 tx t .(33) Definition 3.1 (Class K, and class KR function). [44] A function α(·) : R + → R + belongs to class K if it is continuous and strictly increasing, where α(0) = 0. A function α(·) is said to belong class KR if α is of class K and in addition, lim z→∞ α(z) = ∞. Notice that, under AIII, the Lyapunov candidate functional V is positive definite and decresent. That is we can find α( x ), β( x ) of class K, such that α( x t ) ≤ P (V (x t )) ≤ β( x t ). The following Lemma shows the dissipativity of the quantum EKF estimation error. L (V ) ≤ − γV (x t ) + δ.(34) Proof. We could write the time derivative of P −1 t as below. ∂P −1 t ∂t = − P −1 t ∂P t ∂t P −1 t . Then by Itô expansion, we have dV =x t P −1 t dx t + dx t P −1 tx t + dx t P −1 t dx t −x t P −1 t ∂P t ∂t P −1 tx t dt. The quantum Markov generator for the Lyapunov positive operator V is then given by L (V ) =x t P −1 t [(F(x t ) − K t H(x t ))x t + r] + [(F(x t ) − K t H(x t ))x t + r] P −1 tx t + Tr (G(x t ) * − K t L(x t ) * ) P −1 t (G(x t ) − K t L(x t )) −x t P −1 t ∂P t ∂t P −1 tx t . From Lemma 3.2, (18a), (17), and (16), there exists > 0 such that, L (V ) ≤ −x t P −1 t Q t + PH t R −1 t H t P − S t R −1 t S t + κ x t P 2 t P −1 t x t + δ ≤ −x t P −1 t Q t − S t R −1 t S t + P t H t R −1 t H t − κ x t P t P −1 t x t + δ. By assumption AIV, Q t − S t R −1 t S t > 0. Now select , such that, ∀ x t ≤ Q t − S t R −1 t S t + P t H t R −1 t H t − κ x t P t ≥ 0. This fulfilled by taking, = min q ē p 2 κ , , where q e = inf Q t − S t R −1 t S t . Taking x t ≤ , there exists γ = inf x x t P −1 t Q t − S t R −1 t S t P −1 t + H t R −1 t H t − κ x t x t P −1 txt . Moreover, since x t is bounded, then there exists δ which is given by, δ = sup xt Tr (G(x t ) * − K t L(x t ) * ) P −1 t (G(x t ) − K t L(x t )) . The result then follows immediately. In contrast to the proof of the stochastic convergence of a classical EKF given in [22,Theorem 3.2], having x 0 ≤ does not guarantee that in the future the estimation error will always remain in the region x t ≤ . Consequently, since the last dissipative inequality is only valid in the region x t ≤ then we can only state the quadratic bound of the estimation error before it leaves the region x t ≤ . From the definition of the semi-norm in (3), by (4) we obtain, (35) x t −x t = P E P (x t −x t ) † (x t −x t ) Y t 1/2 . Consequently, the set all events such that the estimation error is greater than is Y t measurable, i.e. { x t −x t > } ∈ Y t . Let E = {x t : x t −x t ≤ } ⊂ Y t . Now, define the Markov time τ (x 0 ) : {τ (x 0 ) ≤ t} ∈ Y t to be the first time that the trajectoryx t leaves E given that it begins inside it. The following theorem describes the quantum EKF's estimation error quadratic bound. Theorem 3.2. If the set of system observables x t has an evolution given by the QSDE in (9a) that satisfies Assumption 3.1 and if initially x 0 ≤ , where is given in Lemma 3.3, then for all t = min (τ (x 0 ) , t) ≥ 0 (36 ) E P x t x t Y t ≤p p E P x 0x 0 Y t e −γt + δ γp 1 − e −γt . Proof. The proof of this theorem makes use of the proof of the classical ultimate bound of quadratic moments given in [45], except that in this theorem, we replace the generator of the classical Markov process with the quantum Markov process generator (Lindblad generator) in (2). We also use the operator inequality with respect to the state P in Y t . First, we observe that t is a Y t measurable random variable [27,Lemma 1.5]. By our definition of a Lyapunov positive operator in (33), we may apply the Itô formula, and therefore we obtain, E P [ V (x t )\| Y t ] = V (x 0 ) + t 0 E P L (V (x s )) Y t ds.(37) The equation above indicates that the expectation E P [V (x t )\| Y t ] is absolutely continuous in t since L (V (x s )) is Lebesgue integrable, and hence for almost all s ≥ 0 dE P [V (x s )\| Y t ] ds ≤ −γE P [V (x s )\| Y t ] + δ. Multiplying by the factor e γs , we have d (e γs E P [V (x s )\| Y t ]) ds ≤ δe γs . By (37), e γs E P [V (x s )\| Y t ] is also absolutely continuous, and hence by integrating the last inequality, we obtain, E P [V (x t )\| Y t ] ≤ E P [V (x 0 )\| Y t ] e −γt + δ γ 1 − e −γt . From AIII, we obtain pE P x t x t Y t ≤ E P [V (x t )\| Y t ] ≤ E P [V (x 0 )\| Y t ] e −γt + δ γ 1 − e −γt ≤pE P x 0x 0 Y t e −γt + δ γ 1 − e −γt . Dividing the last result by p gives the desired result. Robust quantum nonlinear filter for a class of open quantum system with state-dependent noise In the previous subsection, the quantum EKF is developed for a class of an open quantum systems where the noise variances are known. We also notice that the dissipation inequality given in Lemma 3.3 is a very conservative condition and often rather difficult to validate. Moreover, for many open quantum systems, the variances of the system observables and the measurements are state-dependent. As an example, the covariance of photon counting is indeed a stochastic process, for which is a 'doubly stochastic Poisson' or Cox process [47,48]. In classical systems, doubly stochastic Poisson processes have been treated in a variety of ways, e.g. by solving a conditional probability density evolution, or a filtered martingale problem approach [47,49,50,51]. In this subsection, we will show how to modify the quantum EKF to include open quantum systems and measurements with state-dependent noise variance and cross correlation, while at the same obtain a stronger condition of convergence than Lemma 3.3. The treatment we present here uses Riccati differential equation shaping, which is to some extent similar to the treatment of the deterministic nonlinear observer in [41,42]. We will still use the same filter dynamics as in (15). We will also use the same notation P t to denote a positive definite matrix of operators, whose dynamics is shaped to achieve robustness of the estimation error dynamics. Since the variance and covariance are functions of x t , we will use the estimates of R t and S t in the Kalman filter, and therefore we have K t = P t H(x t ) + S(x t ) R(x t ) −1 .(38) Furthermore, we shape the Riccati differential equation, where for given µ, λ > 0, witĥ Q > µI, dP t dt =F(x t )P t + P t F(x t ) +Q + λP t 2 − K t R(x t )K t .(39) We state our result in the following theorem, Theorem 3.3. Consider an open quantum system QSDE given in (9a), the measurement given in (10a), and the quantum EKF given in (15). Also assume Assumptions AI, AII, and AIII, and x t ≤ , ∀t ≥ 0, where = min ( f , h ) in AI. Then there exist µ, λ > 0 in (39) and γ, δ > 0 such that the Lyapunov positive operator given in (33), satisfies the dissipation inequality below L (V ) ≤ − γV (x t ) + δ.(40) Proof. Using the same argument as in Lemma 3.3, L (V ) =x t P −1 t [(F(x t ) − K t H(x t ))x t + r] + [(F(x t ) − K t H(x t ))x t + r] P −1 tx t + Tr (G(x t ) * − K t L(x t ) * ) P −1 t (G(x t ) − K t L(x t )) −x t P −1 t ∂P t ∂t P −1 tx t . With the same δ as in the Lemma 3.3, then from Lemma 3.2, we can select sufficiently large = min ( f , h ), and use (18a), (38), and (39) to obtain L (V ) ≤ −x t P −1 t UP −1 tx t + δ t , with U = P t H (x) R(x t ) −1 H (x) + (λ − κ ) I P t + Q − S(x t )R(x t ) −1 S(x t ) , where I is the identity matrix. Furthermore, selectinĝ Q = µI + S(x t )R(x t ) −1 S(x t ) , and λ > κ , from the fact that R(x t ) −1 > 0, we infer that U > 0. Furthermore, with U = inf x U, γ is given by γ = U p , which completes the proof. The quadratic bound on the estimation error follows immediately by similar assertions in the Theorem 3.2, by replacing with . We notice that we could independently select sufficiently large due to extra freedom introduced by µ and λ. These two parameters remove the positive definiteness restriction of Q t − S t R −1 t S t , as in Assumption AIV. λ is selected to directly dominate the nonlinearity effect κ in the filter that can lead to the divergence of the estimation errors. On the other hand, µ increases the convergence, ensuring the inequality in P t AIII, while at the same time increases the noise level of the estimation. Application Examples In this section, we consider some examples of the application of the quantum EKF. We begin with the estimation of position and momentum quadratures of two optical cavity modes with Kerr nonlinearities and subject to homodyne detection. In this example the variance matrices Q, R and S are constant matrices. Next, we show the estimation of the position and momentum quadratures in an optical cavity subjected to simultaneous homodyne and photon counting detections, which corresponds to the case where the variance matrices R and S are a state-dependent matrices. The constanth is assumed to be one. As before, we will denote the isomorphically transformed operator in Y t , τ (·) t,ω ∈ C as (·) t,ω . Estimating the quadratures of multiple optical modes with a Kerr Hamiltonian In this example, we would like to estimate the quadratures of two cavity modes x i,t = [q i p i ] , i = 1, 2. Each mode has an identical Kerr Hamiltonian and a direct coupling Hamiltonian. For n cavity modes, the direct coupling Hamiltonian is given by [52], H int = n i,j=1 i =j i √ γ ij a j a † i − a i a † j = n i=1 j=i+1 x i,tS ij x j,t . The parameters for the optical cavity, are given by S = I, L i = √ γa i = C i x i,t , and H = n i=1 χ i a † i 2 a 2 i + n i=1 j=i+1 x i,tS ij x j,t . Let Σ = 0 1 −1 0 . The quantum Markovian generator for x t , is given by f i (x t ) = n j=0 A ij x j,t + f i,Kerr (x i,t ), while G i = iΣC i . Now A ij is given by the following matrices A ij =    −Σ C i C † i , i = j ΣS ij , i = j . f i,Kerr (x t ) is the nonlinearity from the Kerr Hamiltonian, we have f i,Kerr = χ 4 4p 3 i,t + p i,t q 2 i,t + q 2 i,t p i,t + 2q i,t p i,t q i,t − 8p i,t − 4q 3 i,t + q i,t p 2 i,t + p 2 i,t q i,t + 2p i,t q i,t p i,t − 8q i,t , F i (x) = χ q i,t p i,t + p i,t q i,t 3p 2 i,t + 0.5 q 2 i,t + p 2 i,t − 2 −(3q 2 i,t + 0.5 q 2 i,t + p 2 i,t − 2) −(q i,t p i,t + p i,t q i,t ) . We consider homodyne detection on each cavity mode as measurement such that E = I, N = 0. From Figure 1c, it can be seen that under a small error at time zero, the quantum EKF still maintains the estimation error to be bounded, which shows the Figure 1a shows the ratio of one time-step computational time between the SME and the quantum EKF for various number of modes, and order of the Hilbert space basis used for solving the SME. The ratio increases dramatically when the number of modes increases. Figure 1b shows a comparison of the mean integral of squared quadratic error, 1 T ( T 0x t,ωxt,ω dt) 1/2 , for the quantum EKF and SME estimation, with various different orders of the Hilbert space basis, against different initial coherent amplitudes α. The errorx t,ω is approximated by subtracting the estimate of the quantum EKF and the SME against the estimate of the SME with the largest Hilbert space basis (32). Figure 1c shows the estimation of the first mode quadrature with different initial errors. The initial value of the quantum EKF is given byx 0,ω =x SM E(0,ω) + ζ[1 − 1 − 1 1] . ζ corresponds to the magnitude of the initial estimation error. In this simulation, the trajectories of the SME and the quantum EKF estimation are generated from identical measurement records. Figure 1, this simulation shows a comparison of the quantum EKF, the SME, and the quantum Kalman Filter (KF) estimation results. The black line with a shaded area corresponds to the mean with one standard deviation range from 100 Monte Carlo trials of the SME's estimation (Hilbert space basis of order 20 for each mode). The red line with a shaded area is the quantum EKF and the blue line is the estimation of the quantum KF where we simply ignore the nonlinearity. Initial errors are set to zero for these simulations. convergence of the quantum EKF. We can observe from Figure 2 that the area of mean and one standard deviation of the quantum EKF on the q 1 ,p 1 ,q 2 and p 2 are quite narrow and contained in the estimation of SME. In contrast, the quantum KF estimation has a substantially larger deviation from the truncated SME mean although it has a narrower standard deviation range. Regarding the computational time, with a Hilbert space basis of order 20 on each mode, the SME requires nearly 200 times that of the quantum EKF. The SME will require 170,000 times that of the quantum EKF if the number of modes equals to three, see also Figure 1a. From Section 3.2, generally for every single mode increase in the quantum systems, the required time of the SME single step computation will be increased to by a factor of N 3 s . In terms of estimation error, the quantum EKF can mantain a fairly low slope of the mean integral of squared quadratic error, 1 T ( T 0x t,ωx t,ω dt) 1/2 regardless of the initial state of the estimated quantum system. This is another benefit, since for the SME, the MISE could make a huge difference if the initial state needs a higher order Hilbert space basis, see Figure 1b. 4.2. Estimating the quadratures of an optical cavity subject to simultaneous homodyne detection and photon counting Figure 3 shows an optical cavity with the simultaneous homodyne detection and photon counting setup as considered in [12]. The involvement of photon counting in this optical setup makes the covariance matrix of the measurement R state-dependent. Let G 1 be our system of interest, with parameters (1, √ γa, H). The vacuum noise is concatenated into our system by G 2 , whose parameters are (1, 0, 0). The beam splitter is given by G 3 , with parameters (S, 0, 0). By taking the series and the concatenation products [39], the parameters of the composite quantum system in Figures 4c-4d show a sample of Monte Carlo simulation of the SME (black), and the extension of the quantum EKF developed in Section 3.4 (rqEKF) (blue). In this trial, the system's density operator is a superposition between a coherent and a Fock state. It is clear from Figs. 4c-4d that rqEKF estimation gives a good approximation to the solution of the SME after a transient period. In the event of photon detection however, the rqEKF tends to have a slightly higher instantaneous jumps compared to the SME. Figures 4a-4b shows how the rqEKF performs against the truncated SME in terms of the average estimate of 100 Monte Carlo trials and their one standard average region. We first observe that the area of mean and one stdev of the rqEKF for q t and p t gives a qualitatively good approximation after a small transient time similar to those of the SME. Conclusion In this article we have developed a quantum EKF for a class of nonlinear QSDEs describing an open quantum system subject to measurement. We derived a sufficient condition for the quantum EKF to achieve local quadratic exponential convergence in the estimation error. We also extended the quantum EKF to the class of quantum systems and measurements with state-dependent covariance matrices. Finally, we have illustrated via two examples the effectiveness of the quantum EKF approximation. Definition 2. 3 . 3[31] Let U be any unitary C-* algebra, and S is the corresponding selfadjoint sub-algebra. For any continuous function f on the compact interval I, f ∈ C (I) is said to be operator differentiable if the operator function f : S I → U, is Fréchet differentiable on D = S I , symbolically f ∈ C 1 op (I). Proposition 2.1. [32, 31] If f ∈ C 1 op (I), then f ∈ C 1 (I) Assumption 3 . 1 . 31With the definition of semi-norm given in Lemma 3. 3 . 3Consider an open quantum system QSDE given in (9a), and measurement given in (10a). With the quantum EKF given in(15), and κ, ≥ 0 in Lemma 3.2. Under Assumptions 3.1, there exist > 0 such that if x t ≤ , ∀t ≥ 0, there exist γ, δ > 0 such that the Lyapunov positive operator (33) satisfies the dissipation inequality below Remark 3. 2 . 2Using a treatment similar to[22, Theorem 4.2], it can be shown that the condition in AIII can be replaced with the uniform detectability of the pair F (x) , H (x) , ∀x ∈ Y t to obtain the same result in the Theorem 3.2, see[46, Theorem 7]. Figure 1 . 1(color online) Application of the quantum EKF to estimate quadratures of two cavity modes with Kerr Hamiltonians. Here, we use a cavity with damping constant γ i = 32 and Kerr nonlinearity constant χ i = 0.3π. The Hilbert space basis for each modes is of order to 32. Figure 2 . 2(color online) As in Figure 3 . 3Simultaneous photon counting and homodyne detection at either output port of a beam splitter in a quantum optics experiment, and the corresponding quantum network depiction of the quantum optics setup,[12]. Figure 3 3are givenby G = G 1 G 2 G 3 , with S, S √ γa 0 , H .The beam splitter matrix S, bath coupling L and Hamiltonian H are given by the followingS = √ 1 − r 2 ir ir √ 1 − r 2 , r ≥ 0, L = SC x t , H = i η * a 2 − ηa † 2 , η ∈ C. Figure 4 . 4(color online) These figures show the estimation of the optical cavity quadratures subject to simultaneous homodyne detection and photon counting. The system's Hilbert space basis (N) is of order 40. 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[ "Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media", "Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media" ]	[ "Mamadou N'diaye \nEnergy Resources Engineering\nStanford University\nStanfordUnited States\n", "François P Hamon \nTotalEnergies E&P Research & Technology USA\nLLC\nUnited States\n", "Hamdi A Tchelepi \nEnergy Resources Engineering\nStanford University\nStanfordUnited States\n" ]	[ "Energy Resources Engineering\nStanford University\nStanfordUnited States", "TotalEnergies E&P Research & Technology USA\nLLC\nUnited States", "Energy Resources Engineering\nStanford University\nStanfordUnited States" ]	[]	This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approachand in particular, FSMSN-results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance-i.e., measured by wall-clock time-will be studied in a subsequent publication.	10.1007/s10596-023-10200-x	[ "https://export.arxiv.org/pdf/2205.05913v2.pdf" ]	248,721,978	2205.05913	d1e45a010af780207123fd90d1ed6653cd57d209	Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media Mamadou N'diaye Energy Resources Engineering Stanford University StanfordUnited States François P Hamon TotalEnergies E&P Research & Technology USA LLC United States Hamdi A Tchelepi Energy Resources Engineering Stanford University StanfordUnited States Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media Nonlinear solverfield-split preconditioning methodstwo-phase flowcoupled multi-physics problems This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, FSMSN is compared to an existing field-split preconditioner, Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN), and to standard solution strategies such as the Sequential Fully Implicit (SFI) method or the Fully Implicit Method (FIM). The comparison highlights the impact of the upwinding scheme in the algorithmic performance of the preconditioners and the importance of the dynamic adaptation of the subproblem tolerance in the preconditioning step. Our results demonstrate that the two-step nonlinear preconditioning approachand in particular, FSMSN-results in a faster outer-loop convergence than with the SFI and FIM methods. The impact of the preconditioners on computational performance-i.e., measured by wall-clock time-will be studied in a subsequent publication. Introduction The numerical simulation of multiphase flow and transport in geological porous media requires solving complex partial differential equations (PDEs) with highly nonlinear saturation-dependent coefficients [1,2]. In most subsurface applications, the heterogeneity of the porous medium generates large spatial variations in the flow regimes with high velocities near the wells and in high-permeability regions. This leads to severe stability constraints on the time step size in explicit discretization schemes. Therefore, unconditionally stable implicit schemes are often the temporal discretization method of choice for porous media flow problems. However, solving the nonlinear systems resulting from an implicit discretization of the PDEs is challenging and often represents most of the computational cost of the simulations. In the Fully Implicit Method (FIM), all the degrees of freedom of the system-typically, pressure, saturations, and compositions-are updated simultaneously using Newton's method with damping [3]. This requires solving large, ill-conditioned linear systems at each nonlinear iteration which is computationally expensive. In addition, for highly nonlinear problems and/or large time steps, Newton's method often fails to converge, in which case the time step is restarted from the previous converged state with a reduced size. To avoid these convergence failures, globalization methods have been developed to enlarge the convergence radius of Newton's method. Popular damping methods for the Newton updates include saturation chopping based on heuristics [4] or relying on the structure of the fractional flow function [5][6][7][8]. In the Sequential Fully Implicit (SFI) method [9][10][11][12][13], the system is decomposed into two subproblems-namely, flow and transport-to avoid solving large coupled linear systems involving all the degrees of freedom. In this approach, the subproblems are solved sequentially using specialized nonlinear solvers until convergence of the outer loop is attained. However, this decoupled approach often suffers from slow outer loop convergence for tightly coupled physics (e.g., in the presence of strong buoyancy and capillary effects) and may require the use of dedicated convergence accelerators for challenging problems [14][15][16][17]. In recent years, advanced nonlinear strategies have been proposed to overcome the limitations of Newton-based FIM and of SFI for multiphase flow in porous media. Although a comprehensive review is out of the scope of the present work, we mention some of the recently presented nonlinear solution strategies. They include homotopy methods [3,18,19], in which a sequence of prediction-correction steps is used to follow a suitably parameterized homotopy path leading to the solution. Homotopy methods are robust for large time steps and can achieve, in some cases, unconditional nonlinear convergence. Using a different approach, ordering-based methods [20][21][22][23][24] accelerate nonlinear convergence thanks to a reordering technique based on the flow direction. The reordered systems have block-triangular structure and can therefore be efficiently solved with backward substitution. To obtain highly scalable solvers able to take large time steps, multilevel solution algorithms relying on the application of the multigrid principles at the nonlinear level in a Full Approximation Scheme (FAS) have been proposed [25][26][27][28]. The efforts to design efficient nonlinear preconditioners to the Newton update are particularly relevant to this work. Two distinct directions have been explored. A class of nonlinear preconditioners leverages domain decomposition methods [29][30][31][32][33] to precondition the nonlinear system in a computationally inexpensive way and speed up convergence. In this work, we focus on nonlinear preconditioners obtained by splitting the system by physical field [34][35][36]. Instead of decomposing the domain in space, field-split preconditioners exploit the mathematical structure of the nonlinear system to split the problem into multiple subproblems solved nonlinearly at a loose tolerance before the computation of a global update for all the degrees of freedom. This approach provides an efficient framework to precondition the system field by field to compute more accurate nonlinear updates and in turn, accelerate nonlinear convergence [37,38]. In the context of mixed elliptic-hyperbolic multiphase flow and transport in porous media, the method is particularly attractive since it retains the best features of both SFI and FIM. Specifically, the preconditioning step resembles the SFI outer iteration based on a pressure update followed by a saturation update and therefore does not involve solving a large coupled linear system. The global step of field-split algorithms plays the role of the computation of the Newton update in FIM and maintains robust convergence properties for strongly coupled problems. In this work, we focus on immiscible two-phase flow in porous media and compare various field-split preconditioners based on the pressure-saturation decomposition. We consider the Multiplicative Schwarz Preconditioned for Inexact Newton (MSPIN) proposed in [39], in which the solution of the preconditioning step is used to compute an approximation of the preconditioned Jacobian system. The global step consists in solving this global preconditioned system to obtain the updated pressures and saturations. We also propose an alternative method, Field-Split Multiplicative Schwarz Newton (FSMSN) in which the preconditioning step is the same as in MSPIN, but the global step is simply the Newton update computed by linearizing the system at the preconditioned state. We compare MSPIN and FSMSN to SFI and Newton with damping for FIM on challenging two-phase flow test cases. The comparison takes into account the role of the upwinding scheme-Phase Potential Upwinding [40,41] or Implicit Hybrid Upwinding [11,[42][43][44][45][46][47][48]-as well as the role of the nonlinear tolerance used in the subproblems. To do so, we propose an adaptive method to compute the subproblem nonlinear tolerance at each preconditioning step, as typically done in inexact Newton methods [49][50][51][52][53][54]. We demonstrate that the two-step nonlinear preconditioners and, in particular, FSMSN, are successful at accelerating nonlinear convergence and are worth being explored as viable options to reduce the computational cost of reservoir simulation-which will be addressed in a future publication. The structure of the present article is as follows. We review the PDEs governing two-phase flow and transport in porous media in Section 2. The fully implicit finite-volume scheme is reviewed in Section 3. The two nonlinear preconditioning techniques considered here-namely, FSMSN and MSPIN-are reviewed in Sections 4 and 5, respectively. The method used to compute the adaptive subproblem tolerance is presented in Section 6. We compare the efficiency of the nonlinear preconditioners with numerical examples in Section 7. Governing equations Let Ω = Ω ∪ Γ be a closed set in R 2 , with Ω an open set and Γ its boundary. We note I = [0, t max ] a finite time interval with t max > 0 the maximal time. We denote the spatial coordinates by x ∈ Ω and the time coordinate by t ∈ I. We consider the immiscible flow of two fluid phases in an incompressible porous medium, with a wetting (w) and a non-wetting (nw) phase. For a phase ∈ {nw, w}, the mass conservation equation reads φ ∂(ρ s ) ∂t + ∇ · (ρ u (p, s)) = q , ∈ {nw, w}, on Ω × I,(1) where φ = φ(x) is the porosity of the medium, q = q (x) is the source/sink term with the convention that q > 0 for injection, and q < 0 for production. The saturation s = s (x, t) represents the fraction of the pore volume occupied by phase , with the following constraint: s = 1.(2) We choose the wetting-phase saturation as the primary saturation unknown, denoted by s := s w . Using the saturation constraint (2), we write all the saturation-dependent properties as a function of s only. The phase velocity u of a phase is given by Darcy's law u (p, s) = −kλ (∇p − ρ g∇d), ∈ {nw, w},(3) where we have neglected capillary pressure. In (3), p(x, t) is the pressure, k(x) is the scalar rock permeability, ρ is the phase density, and λ (s) = k r (s)/µ is the phase mobility, defined as the phase relative permeability, k r (s), divided by the phase viscosity µ . The gravitational acceleration is g and the depth is d (positive going downward). Inserting the expression of the phase velocity given by Darcy's law (3) into the mass conservation equation (1) gives rise to the following form of the two-phase flow and transport equation: φ ∂(ρ s ) ∂t − ∇ · (kρ λ (∇p − ρ g∇d)) = q , ∈ {nw, w}, on Ω × I.(4) In the sequential and field-split solution methods considered in this work, we employ a discretization scheme applied to a split form of the governing equations consisting of an elliptic pressure equation coupled with a hyperbolic transport problem. The pressure subproblem is obtained by summing the mass balance equations (4) over the two phases. Using the saturation constraint (2) and assuming that the two phases are incompressible, we obtain ∇ · u T (p, s) = q ,(5) where the total velocity u T is defined by u T (p, s) := u (p, s). The transport problem is obtained by eliminating the pressure variable in the flux term of (4) to obtain the following fractional flow formulation as explained in [44,55]. Precisely, we first write that u T (p, s) := u (p, s) = λ T (−k∇p) + (λ w ρ w + λ nw ρ nw )g∇d,(6) which gives an expression of −k∇p as a function u T and the gravity weights −k∇p = 1 λ T u T (p, s) − (λ w ρ w + λ nw ρ nw )g∇d .(7) Using this in Darcy's law, we obtain φ ∂(ρ s ) ∂t + ∇ · ρ λ λ T u T (p, s) + kρ λ m λ λ T (ρ − ρ m )g∇d = q , m, ∈ {nw, w}, m ,(8) where λ T (s) := λ (s) is the total mobility. Discretization scheme Given a mesh consisting of M cells, we use first-order finite-volume scheme. We consider 0 = t 0 ≤ t 1 ≤ · · · ≤ t N = t max , N ∈ N a finite discretization of the temporal axis, and ∆t n+1 := t n+1 −t n , 0 ≤ n ≤ N, n ∈ N the time step. We use a (backward-Euler) fully implicit scheme for the integration in time. We denote by p n+1 := [p n+1 1 , . . . , p n+1 M ] and s n+1 := [s n+1 1 , . . . , s n+1 M ] , the solution pair (p n+1 , s n+1 ) the vectors collecting respectively the pressure and saturation unknowns. To define the discrete problem, we first introduce the phase-based residual in cell K at time n + 1, r n+1 ,K , as r n+1 ,K (p n+1 , s n+1 ) := V K φ K ρ n+1 ,K s n+1 ,K − ρ n ,K s n ,K ∆t n+1 + L∈ad j(K) F n+1 ,KL (p n+1 , s n+1 ) − V K q ,K (p n+1 K , s n+1 K ).(9) In (9), F n+1 ,KL is the numerical flux for the interface (KL) between cells K and L, adj(K) is the set of neighbors of cell K and V K the volume of cell K. The computation of the numerical flux is based on a Two-Point Flux Approximation (TPFA). We consider both Phase-Potential Upwinding (PPU) [40,41] and Implicit Hybrid Upwinding (IHU) [11,[42][43][44][45][46][47]56] for the approximation of the transport coefficient of the flux term. In this work, we consider two equivalent formulations of the nonlinear problem. In the Newton-based FIM, we apply the standard fully coupled Newton's method with damping to find the solution pair (p n+1 , s n+1 ) that satisfies r n+1 (p n+1 , s n+1 ) := [r n+1 nw (p n+1 , s n+1 ), r n+1 w (p n+1 , s n+1 )] = 0.(10) In sequential and field-split nonlinear solution strategies, we split the problem into a discrete pressure problem and a discrete transport problem. The solution pair (p n+1 , s n+1 ) must in this case satisfy: f n+1 (p n+1 , s n+1 ) := [g n+1 (p n+1 , s n+1 ), h n+1 (p n+1 , s n+1 )] = 0,(11) where the discrete pressure problem is obtained by summing the phase-based residual equations, as in (5): g n+1 (p n+1 , s n+1 ) := r n+1 nw (p n+1 , s n+1 ) + r n+1 w (p n+1 , s n+1 ),(12) and the discrete saturation problem simply consists of one of the phase-based residual equations. Here, we choose the wetting-phase residual: h n+1 (p n+1 , s n+1 ) := r n+1 w (p n+1 , s n+1 ). From now on, we consider time step n ∈ N and describe two algorithms to compute the solution pair (p n+1 , s n+1 ) of nonlinear problems (10) and (11) at time n + 1. For simplicity, we drop the temporal superscript denoting the time step. Field-split multiplicative Schwarz Newton method The multiplicative Schwarz method has been used to split boundary value problems (BVP) into subproblems solver on smaller physical domains [29][30][31][32][33]. It has also been used to split a coupled BVP into subproblems based on the physics [34][35][36], each subproblem being solved on the full domain to update one of the fields (here, pressure and saturation). The motivation for splitting a coupled problem according to the physics is to solve the physical subproblems one at a time (ideally, with a specialized solver) and use the individually updated fields to precondition the nonlinear iteration, yielding a faster nonlinear convergence. In this section, the objective is to use the field-split approach to construct a predictor-corrector method that converges faster than commonly used algorithms based on the Newton iteration or the sequential fully implicit iteration. The preconditioning step of the FSMSN outer iteration consists in computing an intermediate value of the pressure and saturation, p k, and s k, , by solving individual pressure and transport problems sequentially. We first solve nonlinearly the pressure equation. Specifically, for a fixed s k , find the update δ 1 (p k , s k ) such that g(p k, , s k ) = 0,(14) where p k, := p k + δ 1 (p k , s k ). We solve (14) with Newton's method to obtain a pressure prediction, p k, . The preconditioning step continues by solving nonlinearly a transport problem. For a fixed p k, , find δ 2 (p k , s k ) such that h(p k, , s k, ) = 0,(15) where s k, := s k + δ 2 (p k , s k ). In (15), we explore two formulations of the discrete transport problem. In the first formulation, we use directly the intermediate pressure, p k, , and we consider a discrete transport equation approximating (4) for = w. In the second formulation, we compute an intermediate total velocity field with the intermediate pressure, and we consider a discrete transport equation approximation (8) for = w with a fixed total velocity. This technique is commonly used in the sequential fully implicit method in which the total velocity is also fixed during the resolution of the transport problem. In both options, we use Newton's method with damping to solve the discrete transport problem, although more efficient nonlinear solvers are available [20][21][22][23][24]. In the correction step, we re-evaluate the residual and Jacobian matrix with the intermediate pressure and saturation. Then, we compute the (k + 1)-th solution iterate as p k+1 ← p k, + δp k+1 , s k+1 ← s k, + τ k+1 δs k+1 ,(16) where τ k+1 is a diagonal matrix of damping parameters. In (16), the update is obtained by solving the linear system δx k+1 = −J(x k, ) −1 r(x k, ), x := p s .(17) Note that for the assembly of J in (17), the fluxes are computed using the same discretization as in the subproblems (14) and (15) to maintain a uniform discrete approach throughout the FSMSN nonlinear iteration. The implementation of FSMSN is described in Algorithm 1. Algorithm 1 FSMSN algorithm for two-phase flow and transport for k = 1, . . . , k itermax do Check convergence, and break nonlinear loop if convergence was achieved Pressure step: For a fixed s k , solve g(p k, , s k ) = 0 Transport step: For a fixed p k, or a fixed total velocity u k, T , solve h(p k, , s k, ) = 0 Coupled step: Recompute residual and Jacobian using (p k, , s k, ) Solve linear system: δx k+1 = −J(x k, ) −1 r(x k, ) Update solution: p k+1 ← p k, + δp k+1 ; s k+1 ← s k, + τ k+1 δs k+1 end for Preconditioned method based on field-split multiplicative Schwarz method In this section, we review the construction of a preconditioned method for the two-phase problem using a fieldsplit multiplicative Schwarz method (MSPIN). As in the previous section, the iteration consists of two steps, with first a preconditioning step following with a global step. The preconditioning step is identical to that of FSMSN. Using the same notations as in Section 4, the flow problem consists in finding a new pressure field, p k, := p k + δ 1 (p k , s k ), that satisfies (14) for a fixed saturation field, s k . Then, the transport problem consists in finding an updated saturation field, s k, := s k + δ 2 (p k , s k ), that satisfies (15) for a fixed pressure field, p k, . In this second step, we only consider the formulation in which the discrete transport equation approximates (4) for = w. The MSPIN algorithm differs from FSMSN in the global step. Once the pressure and transport problems are solved, we form the following preconditioned problem: F (p k , s k ) := [δ 1 (p k , s k ), δ 2 (p k , s k )] = 0.(18) The goal of the coupled step is to form a Jacobian system from (18) and solve it to obtain the pressure and saturation updates of outer iteration k. Forming the Jacobian system requires computing the partial derivatives of δ 1 and δ 2 with respect to the pressure and saturation variables. Using the chain rule, we obtain the derivatives of δ 1 by differentiating (14), which yields: ∂ p δ 1 (p k , s k ) = −I M ,(19)∂ s δ 1 (p k , s k ) = − ∂ p g(p k, , s k ) −1 ∂ s g(p k, , s k ).(20) I M denotes a M by M identity matrix. Differentiating (15) yields the derivatives of δ 2 : ∂ p δ 2 (p k , s k ) = 0,(21)∂ s δ 2 (p k , s k ) = −I M − ∂ s h(p k, , s k, ) −1 ∂ p h(p k, , s k, ) ∂ p g(p k, , s k ) −1 ∂ s g(p k, , s k ).(22) Using equations (19) to (22), the Jacobian matrix, J (p k , s k ) of (18) is then given by: J (p k , s k ) = − ∂ p g(p k, , s k ) 0 ∂ p h(p k, , s k, ) ∂ s h(p k, , s k, ) −1 ∂ p g(p k, , s k ) ∂ s g(p k, , s k ) ∂ p h(p k, , s k, ) ∂ s h(p k, , s k, ).(23) At this point, the second matrix on the right-hand side of (23) differs from the unpreconditioned Jacobian matrix of (10) because the first row is evaluated at (p k, , s k ), while the second row is evaluated at (p k, , s k, ). Following the methodology of [39], we approximate the preconditioned Jacobian matrix of the Multiplicative Schwarz Preconditioned Inexact Newton (MSPIN) method by setting δ 2 (p k , s k ) = 0 in (23), which results in: J (p k , s k ) ≈ − ∂ p g(p k, , s k ) 0 ∂ p h(p k, , s k ) ∂ s h(p k, , s k ) −1 J(p k, , s k ).(24) where J is the original, unpreconditioned Jacobian matrix of (10). We note that in (24), the two matrices on the righthand side are now fully evaluated using the state of the system after the pressure step. Then, we obtain the (k + 1)-th solution iterate as p k+1 ← p k + δp k+1 , s k+1 ← s k + τ k+1 δs k+1 ,(25) In (25), the update is obtained by solving the linear system: δx k+1 = −J (p k , s k ) −1 F (p k , s k ), x := p s .(26) The MSPIN iteration for two-phase flow and transport is summarized in Algorithm 2. The MSPIN strategy converges to the same solution as Newton based FIM, as proven in [39]. In [39], the authors provide numerical tests based on the Navier-Stokes equations showing that MSPIN is more robust than Newton's method and the Additive Schwarz-type preconditioning (ASPIN). In this work, we focus on the assessment of the nonlinear behavior of the MSPIN algorithm for the strongly coupled and highly nonlinear two-phase flow and transport problem. In our implementation, we form matrix (24) and solve (26) by calling a direct solver twice. In future work, we will exploit the structure of the blocktriangular matrix of (24) inside an iterative Krylov-type linear solver [57,58] to improve the efficiency of the approach on large-scale problems. Convergence check and adaptive nonlinear tolerance In Algorithms 1 and 2, the convergence checks involved in the outer and inner loops are performed using the 2 -norm of the normalized residual. Specifically, convergence of the full problem is achieved when: max diag r k+1 (p k+1 , s k+1 ) diag m (p n ) −1 2,2 < .(27) Algorithm 2 MSPIN algorithm for two-phase flow and transport for k = 1, . . . , k itermax do Check convergence, and break nonlinear loop if convergence was achieved Pressure step: For a fixed s k , solve g(p k, , s k ) = 0 Update and save ∂ p g(p k, , s k ), ∂ s g(p k, , s k ), ∂ p h(p k, , s k ), and ∂ s h(p k, , s k ) Transport step: For fixed p k, , solve h(p k, , s k, ) = 0 Coupled step: Form preconditioned residual F (p k , s k ) and Jacobian matrix J (p k , s k ) as in (18) and (23) Solve linear system: δx k+1 = −J (p k , s k ) −1 F (p k , s k ) Update solution: p k+1 ← p k + δp k+1 ; s k+1 ← p k + τ k+1 δs k end for where we have used the normalizer m (p n ) = [ρ n ,1 V 1 φ 1 , . . . , ρ n ,M V M φ M ] evaluated at the previous converged time step, and where diag(x) is the diagonal matrix obtained from the argument vector x. In the first inner loop of Algorithms 1 and 2, convergence of the pressure problem is reached when diag g(p k, , s k ) diag m nw (p n ) + m w (p n ) −1 2,2 < k p ,(28) while, in the second inner loop, the transport problem is converged when diag h(p k, , s k, ) diag m w (p n ) −1 2,2 < k s .(29) If these criteria are not satisfied, we keep iterating until we reach convergence or the maximum number of iterations. The choice of subproblem tolerances in (28) and (29) is key for the nonlinear behavior and the efficiency of the schemes. In the numerical examples, we explore the two approaches detailed below. In the first approach, we set the subproblem tolerances, k p and k s , to constant values, p and s , chosen to be stricter or equal to the full problem tolerance, , used in the outer loop. This is motivated by the fact that, in MSPIN, the computation of the preconditioned Jacobian in (23) assumes that the pressure and transport subproblems are fully converged and that (14) and (15) are satisfied. However, this approach requires a significant computational effort to solve the subproblems which is likely to undermine the efficiency of the scheme. In an alternative approach, we also explore the use of adaptive subproblem tolerances to minimize the number of subproblem iterations and reduce the computational cost of the schemes. We define the subproblem tolerances as: The parameter η k ∈ [0, 1[ depends on the outer iteration number and is used to control the subproblem tolerance. We choose a relatively large η k during the first outer iterations to use a relaxed tolerance, and we gradually reduce η k to obtain a tighter tolerance as the outer loop approaches convergence. A similar approach based on the Eisenstat-Walker algorithm is commonly used to reduce the cost of solving the linear systems in the inexact Newton method (see for instance [50,51]). Computing the parameter η k at each outer iteration is the critical part of the algorithm. We adapt the work of [50,51] to Algorithms 1 and 2 by setting: (A 1 ) η k = 0.1, k ≥ 0,(32)(A 2 ) η k = 2 −(k+1) , k ≥ 0,(33)(A 3 ) η 0 = 1, η k = r k (p k , s k ) 2 − r k−1 (p k−1 , s k−1 ) + J(x k−1 )δx k−1 2 r k−1 (p k−1 , s k−1 ) 2 , k ≥ 1.(34) In the next section, we refer to these approaches as A 1 , A 2 , and A 3 , respectively. The algorithm employed to compute the subproblem tolerance and use in the inner loop is illustrated in Algorithm 3 for the transport problem. Algorithm 3 Transport inner loop with adaptive tolerance Given k ∈ N * the outer-loop iteration number Given k−1 s ∈ [0, 1[ the tolerance used at outer-loop iteration k − 1 ( 0 = 1) Compute η k using (32), or (33), or (34) Compute the new tolerance k s = η k k−1 s for m = 1, · · · , m maxiter do Compute the residual, h(p k, , s k,m ), and the Jacobian matrix if \|\|h(p k, , s k, )/m w (p n )\|\| 2 < k s then Convergence is achieved, return the solution s k, end if Solve the Jacobian system to compute δs k,m+1 Update the saturation solution: s k,m+1 ← s k,m + τ m+1 δs k,m+1 end for Numerical results To compare the different algorithms discussed above, we consider various numerical examples with heterogeneous permeability fields. For a given time step ∆t, the local phase-based CFL number is defined as CFL ,K := ∆t L∈ad j(K) max 0, F n+1 ,KL (p n+1 , s n+1 ) V K φ K , = nw, w,(35) where V K is the volume of the grid cell K, φ K is the porosity and F ,KL is the numerical flux of the phase for the interface (KL) between cells K and L. The maximum CFL number is then computed as max CFL = arg max ,1≤K≤M CFL ,K ,(36) where the computational domain is discretized into M cells and K is the grid cell number. In the following sections, we compare four types of solution methods for the discrete two-phase problem (10): • FIM based on Newton's method with damping, in which the damping parameter is chosen to ensure that the largest saturation change between two Newton iterations is smaller than 0.2. The residual is computed by discretizing (4) and using PPU to approximate the mobilities. • The sequential fully implicit method [9][10][11][12][13] referred to as SFI-u T . The outer iteration consists in two steps. We first solve the flow problem nonlinearly and compute a new total velocity field using the updated pressure. In a second step, we solve the transport problem nonlinearly with a fixed total velocity. The residual is computed by discretizing (8) and using IHU to approximate the mobilities. We have observed very slow convergence rates for SFI-p-in which the pressure is fixed during the transport solve-and we therefore do not report these results in the next sections. • The FSMSN method of Algorithm 1. We consider two versions of the algorithm. In FSMSN-p, the transport problem is computed with a fixed pressure. The residual is computed by discretizing (4) and using PPU to approximate the mobilities. In FSMSN-u T , the transport problem is computed with a fixed total velocity. The residual is computed by discretizing (8) and using IHU to approximate the mobilities. • The MSPIN method of Algorithm 2 referred to as MSPIN-p. The transport problem is computed with a fixed pressure. The residual is computed by discretizing (4) and using PPU to approximate the mobilities. The algorithms have been implemented in Matlab (R2019b) and tested on a basic laptop (Intel Core i5-8250U QuadCore @ 1.60GHz; 8GB RAM; hard disk size 256GB; Windows 10). The linear solves required by the algorithms have been performed by the default direct solver in Matlab (the "backslash" operator). No external library has been used. SPE10 bottom layer We first consider a horizontal test case consisting of 13,200 cells in which the porosity and permeability fields are taken from the bottom layer of the SPE10 test case [59]. We inject the wetting phase (water) from the middle well and produce from the wells located in the four corners. Capillary pressure is neglected. The phase densities are set to ρ w = 1025.0 kg.m −3 and ρ nw = 849.0 kg.m −3 and the phase viscosities are set to µ w = 0.0003 Pa.s and µ nw = 0.003 Pa.s. We use quadratic Corey-type relative permeabilities. The domain is initially fully saturated with the non-wetting phase. We simulate 500 days of injection (0.1 total pore volume injected) with a constant time step size. The permeability and final saturation maps are shown in Fig. 1. The nonlinear behavior of the schemes is illustrated in Fig. 2 for two selected time steps and in Table 1 for the full simulation. We consider first a fixed tolerance of 10 −6 in the pressure and transport subproblems. We observe that for FSMSN-u T , FSMSN-p, and MSPIN-p, the residual norm decreases at a faster rate than with Newton-based FIM and with SFI. This faster rate results in a significant reduction in the number of outer iterations. We note, however, that this reduction in the number of outer iterations requires performing a large number of subproblem iterations in FSMSN and MSPIN, which is likely to make these algorithms unpractical. For instance, FSMSN-u T requires only 48 outer iterations-while 149 iterations are performed by Newton's method-but involves 81 pressure iterations and 205 transport iterations. Table 2 shows the results of the adaptive strategies introduced in Section 6 to relax the tolerance in the subproblems and reduce the computational cost of the pressure and transport steps in FSMSN and MSPIN. For both FSMSN and MSPIN, using adaptive tolerances in the subproblems results in a slight increase in the number of outer iterations, while the number of pressure iterations and-more significantly-the number of transport iterations are reduced. To conclude this section, we study the sensitivity of the results to the time step size-measured by the maximum CFL number observed during the simulation. The results with a fixed tolerance and an adaptive tolerance are shown in Fig. 3. We observe that when the time step is increased, the number of Newton iterations is slightly reduced for Solver Fixed total velocity Fixed pressure CFL numbers smaller than 138, but stagnates for CFL numbers larger than 138. This is not the case for FSMSN-u T , FSMSN-p, and MSPIN-p, as these schemes exhibit a significant reduction in the number of outer iterations as the time step is increased, even for large CFL numbers. With both fixed tolerance and adaptive tolerance in the pressure and transport subproblems, FSMSN-u T is the solution strategy that requires the smallest number of outer iterations. SFI-u T FSMSN-u T MSPIN-p FSMSN-p A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 SPE10 top layer In this example, we study the impact of the introduction of buoyancy forces on the nonlinear behavior of the schemes. To do that, we consider a tilted two-dimensional domain in which the porosity and permeability fields are taken from the top layer of the SPE10 test case. The fluid properties and the well locations are the same as in the previous section. We inject 0.08 total pore volume in two distinct configurations: • Case 1: tilting of 60 • in the y-direction and fast injection rate (9.352 m 3 /day) so that viscous forces dominate. • Case 2: same tilting and slower injection rate (0.9352 m 3 /day) so that buoyancy forces dominate. We note that the time step size and total simulation time are adapted to obtain approximately the same maximum CFL number and total PVI in the two cases. We start the simulation with a uniform initial saturation field equal to S 0 w = 0. The nonlinear behavior of the schemes with a maximum CFL number of approximately 69 is summarized in Tables 3-4 for Case 1, and in Tables 5-6 for Case 2. We observe that increasing the strength of buoyancy forces (relatively to viscous forces) makes the nonlinear convergence of SFI-u T very slow, and deteriorates slightly that of Table 3: SPE10 top layer: nonlinear behavior of the schemes with a fixed subproblem tolerance of 10 −6 (Case 1 with fast injection rate). Approximately 2% of the interfaces experience counter-current flow. The maximum CFL number is 69 and the total PVI is 0.08. Solver Fixed total velocity Fixed pressure Table 5: SPE10 top layer: nonlinear behavior of the schemes with a fixed subproblem tolerance of 10 −6 (Case 2 with a slower injection rate). Approximately 8% of the interfaces experience counter-current flow. The maximum CFL number is 69 and the total PVI is 0.08. SFI-u T FSMSN-u T MSPIN-p FSMSN-p A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 Solver Fixed total velocity Fixed pressure Table 6: SPE10 top layer: nonlinear behavior of the schemes with an adaptive subproblem tolerance computed using the strategies (A 1 ), (A 2 ), and (A 3 ) of Section 6 (Case 2 with slower injection rate). The maximum CFL number is 69 and the total PVI is 0.08. d.n.c. denotes lack of convergence. SFI-u T FSMSN-u T MSPIN-p FSMSN-p A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A MSPIN-p. However, for Newton's method, FSMSM-u T , and FSMSN-p, we observe a slight improvement in the nonlinear behavior when buoyancy forces are stronger. In this section, the nonlinear behavior of FSMSN-u T and FSMSN-p with adaptive tolerance in the subproblems remains excellent. For Cases 1 and 2, the reduction in the number of subproblem iterations obtained with the adaptive strategies is drastic but the increase in the number of outer iterations is small. However, for MSPIN-p, we observe that using adaptive tolerances deteriorates quite significantly the nonlinear behavior, with for instance an increase by 48% in the number of outer iterations for strategy (A 3 ). The sensitivity of the nonlinear behavior to the time step size is studied in Figs. 4 and 5. We note that these figures do not include the results obtained with SFI-u T as this approach does not converge for Case 2 when the CFL number is increased. As in Section 7.1, the number of Newton iterations stagnates for CFL numbers larger than 69, while the number of outer iterations required by FSMSN-u T , FSMSN-p, and MSPIN-p reduce for the range of time step sizes considered here. Gravity segregation test case We consider a two-dimensional The nonlinear behavior of the schemes is summarized in Fig. 7 and in Tables 7 and 9. In this challenging buoyancydriven test case in which flow and transport are strongly coupled, SFI requires significantly more outer iterations than FIM based on Newton's method for the case with small time steps (10 days) and fails to converge when the time step is increased. The nonlinear behavior of the solution strategies based on nonlinear preconditioning depends heavily on the formulation. Specifically, we note that nonlinear preconditioning approaches based on fixed pressure (MSPIN-p and FSMSN-p) do not perform as well as FSMSN-u T which used a fixed total velocity to couple the flow and transport problems. Figure 7 shows that both MSPIN-p and FSMSN-p fail to converge beyond a certain time step size (10 days for MSPIN-p and 50 days for FSMSN-u T ). Importantly, FSMSN-u T exhibits a steady reduction in the number of outer iterations as a function of time step size, which is not the case for FIM with Newton's method with which the number of iterations levels off for time step sizes larger than 50 days. Using an adaptive tolerance in the subproblems does not alter these conclusions as shown for two different time step sizes by Tables 8 and 10. In FSMSN-u T , the three adaptive strategies considered in this work to select the subproblem tolerance cause a slight increase in the total number of outer iterations (from 151 to 154) but achieve a large reduction in the number of subproblem iterations (from 200 to 155 for flow and from 430 to 185 for transport). Fractured heterogeneous two-dimensional model To conclude this study, we construct a test case in which the flow is driven by competing viscous and buoyancy forces. We consider a two-dimensional x-z domain with a channelized permeability field-the channels can be seen as fractures modeled by contiguous cells with a very large permeability. The fluid properties are the same as in Solver Fixed total velocity Fixed pressure the previous example. The domain is initially saturated with the non-wetting phase. The wetting phase is injected through a well perforating twenty cells in the top-right part of the domain, while a producer perforates twenty cells in the bottom-left part of the domain. We simulate 1,500 days of injection (0.56 total pore volume injected). The permeability maps as well as the saturation map at different times is shown in Fig. 8 The results with fixed subproblem tolerance and adaptive tolerance are in Tables 11 and 12, respectively. The sensitivity of the nonlinear behavior of the schemes to the time step (measured by CFL number) is shown in Fig. 9. Although all the solution strategies converge well in this case, the results confirm the observations made in Sections 7.1 and 7.2. In particular, comparing the slopes of the curves for Newton-based FIM and the nonlinear preconditioners in Fig. 9 SFI-u T FSMSN-u T MSPIN-p FSMSN-p A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 ASFI-u T FSMSN-u T MSPIN-p FSMSN-p A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 Conclusion Solving the nonlinear systems that result from a fully implicit discretization of the PDEs governing multiphase flow and transport in porous media is challenging. To address this issue, we propose a field-split preconditioner referred to as Field-Split Multiplicative Schwarz Newton (FSMSN). The FSMSN-preconditioned iteration relies on two steps: a preconditioning step in which we solve sequentially a flow problem followed by a transport problem Solver Fixed total velocity Fixed pressure SFI-u T FSMSN-u T MSPIN-p FSMSN-p A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 A 1 A 2 A 3 Nonlinear iterations 166 166 166 54 54 54 77 83 78 61 61 61 Pressure iterations 168 1356 168 56 56 56 82 87 83 64 64 64 Transport iterations 203 203 203 92 91 92 126 123 123 96 96 96 Table 12: Fractured heterogeneous model: nonlinear behavior of the schemes with an adaptive tolerance computed using the strategies (A 1 ), (A 2 ), and (A 3 ) of Section 6. The maximum CFL number is 209 and the total PVI is 0.56. using a loose nonlinear tolerance; a global step in which we compute a Newton update for pressure and saturations by linearizing the preconditioned system. We compare its nonlinear behavior to another preconditioner of the same class (Multiplicative Schwarz Preconditioned Inexact Newton) and to standard solution strategies like Newton's method with damping to the full system, and the Sequential Fully Implicit method. The numerical examples show that FSMSN can successfully reduce the number of outer iterations for challenging viscous-dominated and gravity-dominated problems, compared to the other solution strategies considered here. Our results also demonstrate that this robust nonlinear behavior is preserved for large CFL numbers (corresponding to large time steps). This is key to make sure that time step sizes can be chosen based on accuracy considerations and not constrained by the nonlinear behavior of the solution strategy. Two key steps have to be taken to show that the improved nonlinear behavior of FSMSN can result in a reduction of the computational cost of the simulation (i.e., reduction in wall-clock time). First, we plan to design a more adaptive FSMSN in which the preconditioning step would be used only when necessary, that is, for the first outer iterations of time steps with bad initial guesses and/or large sizes, when the large reduction in the number of outer iterations obtained with FSMSN is more likely to offset the overhead caused by the preconditioning step. As soon as the state of the system is close to the solution, the global step would be sufficient to enter the quadratic convergence regime. Second, we will substitute the direct solvers used in this work with inexact Krylov-type iterative solution strategies [57,58,60]. This will reduce the overhead caused by the preconditioning step by leveraging the efficiency of specialized solvers for the subproblems, like Algebraic MultiGrid (AMG) for the pressure problem and an optimized orderingbased solver for transport [21]. Switching to iterative solvers will also enable the use of an adaptive linear tolerance [51] in these subproblems. These improvements will enhance the efficiency of FSMSN without compromising its robust nonlinear behavior. Figure 1 : 1SPE10 bottom layer: permeability map and water saturation maps at various times. Figure 2 : 2SPE10 bottom layer: Residual norm as a function of the number of outer iterations for two time steps. The plot on the left (respectively, on the right) corresponds to the second time step (respectively, the last time step) in the simulation. The max CFL number for these two time steps is approximately 69. Figure 3 : 3SPE10 bottom layer: cumulative number of outer iterations for the full simulation as a function of the maximum CFL number observed during the simulation. Figure 4 : 4SPE10 top layer: cumulative number of outer iterations for the full simulation as a function of the maximum CFL number observed during the simulation (Case 1 with strong viscous forces relatively to buoyancy forces). Figure 5 : 5x-z domain of size 30.48 m × 30.48 m divided into 100 × 100 cells. The domain is initially saturated with the wetting phase on the left and with the non wetting phase on the right. The phase densities and viscosities are set to ρ w = 1025 kg.m −3 , ρ nw = 785 kg.m −3 , µ w = 0.0003 Pa.s, and µ nw = 0.003 Pa.s. The phase relative permeabilities are quadratic. The homogeneous permeability is equal to k = 200 mD. We simulate 500 days of gravity segregation with a constant time step size. The saturation maps at different times are showed inFig. SPE10top layer: cumulative number of outer iterations for the full simulation as a function of the maximum CFL number observed during the simulation (Case 2 with strong buoyancy forces relatively to viscous forces). Figure 6 :Figure 7 : 67Gravity segregation: water saturation maps at various times. Gravity segregation: cumulative number of outer iterations for the full simulation as a function of time step size. saturation after 900 days. (e) Water saturation after 1,200 days.(f) Water saturation after 1,500 days. Figure 8 :Figure 9 : 89Permeability map of the fractured heterogeneous 2D porous media and water saturation maps at various times. Fractured heterogeneous model: cumulative number of the nonlinear iterations as a function of the maximum CFL number observed during the simulation. Table 2 : 2SPE10 bottom layer: nonlinear behavior of the schemes with an adaptive subproblem tolerance computed using the strategies (A 1 ), (A 2 ), and (A 3 ) of Section 6. The maximum CFL number is 69 and the total PVI is 0.1. Table 4 : 4SPE10 top layer: nonlinear behavior of the schemes with an adaptive subproblem tolerance computed using the strategies (A 1 ), (A 2 ), and (A 3 ) of Section 6 (Case 1 with fast injection rate). The maximum CFL number is 69 and the total PVI is 0.08. d.n.c. denotes lack of convergence. Solver Fixed total velocity Fixed pressure Newton SFI-u T FSMSN-u T MSPIN-p FSMSN-p Nonlinear iterations 127 567 48 67 56 Iterations per time step 7.93 33.35 3 3.18 3.5 Pressure iterations - 1049 86 120 96 Transport iterations - 2166 188 244 214 Table 7 : 7Gravity segregation: nonlinear behavior of the schemes with a fixed subproblem tolerance of 10 −6 . We simulate 500 days with 50 time steps. Table 8 : 8Gravity segregation: nonlinear behavior of the schemes with an adaptive subproblem tolerance computed using the strategies (A 1 ), (A 2 ), and (A 3 ) of Section 6. We simulate 500 days with 50 time steps.Solver Fixed total velocity Fixed pressure Newton SFI-u T FSMSN-u T MSPIN-p FSMSN-p Nonlinear iterations 151 824 80 d.n.c. 110 Iterations per time step 7.55 41.2 4 d.n.c. 5.5 Pressure iterations - 940 115 d.n.c. 190 Transport iterations - 1378 265 d.n.c. 509 Table 9 : 9Gravity segregation: nonlinear behavior of the schemes with a fixed subproblem tolerance of 10 −6 . We simulate 500 days with 20 time steps. d.n.c. denotes lack of convergence.Solver Fixed total velocity Fixed pressure Table 10 : 10Gravity segregation: nonlinear behavior of the schemes with an adaptive subproblem tolerance computed using the strategies (A 1 ), (A 2 ), and (A 3 ) of Section 6. We simulate 500 days with 20 time steps. d.n.c. denotes lack of convergence. is very insightful, as it shows the improved robustness of FSMSN and MSPIN for large time step sizes.Solver Fixed total velocity Fixed pressure Newton SFI-u T FSMSN-u T MSPIN-p FSMSN-p Nonlinear iterations 104 158 54 68 62 Iterations per time step 6.93 10.53 3.6 4.53 4.13 Pressure iterations - 330 124 174 137 Transport iterations - 338 176 198 166 Table 11 : 11Fractured heterogeneous model: nonlinear behavior of the schemes with fixed subproblem tolerance of 10 −6 . Approximately 2% of the interfaces experience counter-current flow. The maximum CFL number is 209 and the total PVI is 0.56. AcknowledgmentsFunding was provided by TotalEnergies through the FC-MAELSTROM project. The authors thank the SUPRI-B affiliates program at Stanford University and well as Joshua A. White, Nicola Castelletto (Lawrence Livermore National Laboratory), and Hervé Gross (TotalEnergies) for their insight and guidance. 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[ "Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks", "Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks", "Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks", "Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks" ]	[ "Tiantian Chen ", "Member, IEEEJianxiong Guo ", "Senior Member, IEEEWeili Wu ", "Tiantian Chen ", "Member, IEEEJianxiong Guo ", "Senior Member, IEEEWeili Wu " ]	[]	[]	Online social network has been one of the most important platforms for viral marketing. Most of existing researches about diffusion of adoptions of new products on networks are about one diffusion. That is, only one piece of information about the product is spread on the network. However, in fact, one product may have multiple features and the information about different features may spread independently in social network. When a user would like to purchase the product, he would consider all of the features of the product comprehensively not just consider one. Based on this, we propose a novel problem, multi-feature budgeted profit maximization (MBPM) problem, which first considers budgeted profit maximization under multiple features propagation of one product.Given a social network with each node having an activation cost and a profit, MBPM problem seeks for a seed set with expected cost no more than the budget to make the total expected profit as large as possible. We mainly consider MBPM problem under the adaptive setting, where seeds are chosen iteratively and next seed is selected according to current diffusion results. We study adaptive MBPM problem under two models, oracle model and noise model. The oracle model assumes conditional expected marginal profit of any node could be obtained in O(1) time and a (1 − 1/e) expected approximation policy is proposed. Under the noise model, we estimate conditional expected marginal profit of a node by modifying the EPIC algorithm and propose an efficient policy, which could achieve a (1 − e −(1− ) ) expected approximation ratio. Several experiments are conducted on six realistic datasets to compare our proposed policies with their corresponding non-adaptive algorithms and some heuristic adaptive policies. Experimental results show efficiencies and superiorities of our policies.	10.1007/s13278-022-00989-3	[ "https://export.arxiv.org/pdf/2006.03222v3.pdf" ]	219,401,582	2006.03222	8bbe918fa9fa2bca88f6db00f8ecaee942ca04c0	Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks Tiantian Chen Member, IEEEJianxiong Guo Senior Member, IEEEWeili Wu Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks 1Index Terms-Multi-feature DiffusionAdaptive Budgeted Profit MaximizationApproximation AlgorithmSocial Network Online social network has been one of the most important platforms for viral marketing. Most of existing researches about diffusion of adoptions of new products on networks are about one diffusion. That is, only one piece of information about the product is spread on the network. However, in fact, one product may have multiple features and the information about different features may spread independently in social network. When a user would like to purchase the product, he would consider all of the features of the product comprehensively not just consider one. Based on this, we propose a novel problem, multi-feature budgeted profit maximization (MBPM) problem, which first considers budgeted profit maximization under multiple features propagation of one product.Given a social network with each node having an activation cost and a profit, MBPM problem seeks for a seed set with expected cost no more than the budget to make the total expected profit as large as possible. We mainly consider MBPM problem under the adaptive setting, where seeds are chosen iteratively and next seed is selected according to current diffusion results. We study adaptive MBPM problem under two models, oracle model and noise model. The oracle model assumes conditional expected marginal profit of any node could be obtained in O(1) time and a (1 − 1/e) expected approximation policy is proposed. Under the noise model, we estimate conditional expected marginal profit of a node by modifying the EPIC algorithm and propose an efficient policy, which could achieve a (1 − e −(1− ) ) expected approximation ratio. Several experiments are conducted on six realistic datasets to compare our proposed policies with their corresponding non-adaptive algorithms and some heuristic adaptive policies. Experimental results show efficiencies and superiorities of our policies. Adaptive Multi-Feature Budgeted Profit Maximization in Social Networks Tiantian Chen, Jianxiong Guo, Member, IEEE Weili Wu, Senior Member, IEEE Abstract-Online social network has been one of the most important platforms for viral marketing. Most of existing researches about diffusion of adoptions of new products on networks are about one diffusion. That is, only one piece of information about the product is spread on the network. However, in fact, one product may have multiple features and the information about different features may spread independently in social network. When a user would like to purchase the product, he would consider all of the features of the product comprehensively not just consider one. Based on this, we propose a novel problem, multi-feature budgeted profit maximization (MBPM) problem, which first considers budgeted profit maximization under multiple features propagation of one product. Given a social network with each node having an activation cost and a profit, MBPM problem seeks for a seed set with expected cost no more than the budget to make the total expected profit as large as possible. We mainly consider MBPM problem under the adaptive setting, where seeds are chosen iteratively and next seed is selected according to current diffusion results. We study adaptive MBPM problem under two models, oracle model and noise model. The oracle model assumes conditional expected marginal profit of any node could be obtained in O(1) time and a (1 − 1/e) expected approximation policy is proposed. Under the noise model, we estimate conditional expected marginal profit of a node by modifying the EPIC algorithm and propose an efficient policy, which could achieve a (1 − e −(1− ) ) expected approximation ratio. Several experiments are conducted on six realistic datasets to compare our proposed policies with their corresponding non-adaptive algorithms and some heuristic adaptive policies. Experimental results show efficiencies and superiorities of our policies. Index Terms-Multi-feature Diffusion, Adaptive Budgeted Profit Maximization, Approximation Algorithm, Social Network I. INTRODUCTION Online social network, like Facebook, Twitter, Linkedln, etc., has been one of the most important platforms for marketing and communication. Many companies have taken social network as main method to promote products by wordof-mouth effects. To maximize the product influence and obtained profit, companies may apply many methods, such as distributing coupons, free samples or offering some discounts when purchasing. Many researches have been focused on the diffusion phenomenon on social networks, including diffusion of ideas, news, adoptions of new products, etc. One topic extensively studied is the Influence Maximization (IM) problem [1]- [5], which asks for k seeds to maximize the expected number of influenced users under some diffusion model. There are two classical diffusion models: Independent Cascade (IC) model and Linear Threshold (LT) model. However, it has been proved IM problem is NP-hard and computing the expected spread from a node set is #P-hard in general under IC and LT model [3], [4]. Kempe et al. [5] presented a (1 − 1/e)-approximation scheme, classical greedy algorithm, for IM problem and they used Monte Carlo method to estimate expected spread of a seed set, but it is time-consuming. Many recent works, like [6]- [12], have been focused on solving this problem, which not only could obtain a (1 − 1/e − )-approximation solution with high probability but are efficient even for large-scale datasets. Most of existing papers related to IM problem only consider a single diffusion. That is, only one piece of information about the product is spread on social networks. Some papers like [13]- [16] indeed consider multiple diffusions of products. But the diffusions are for multiple products and each diffusion is for one product. However, in reality, one product may have multiple features and the information about all these features can spread on social network. For instance, when a customer wants to buy a phone, he may consider many features, such as price, brand, camera, display, speed, etc. He has his own preference for each feature, which can be regarded as weight for the feature, and has a threshold to purchase the phone. He heard the information about features of the phone on networks, and he will purchase the phone only when the sum of weights of features satisfying his requests is larger than or equal to the threshold. Guo et al. [17] first proposed a multi-feature diffusion model (MF-model) to describe multiple features about one product spreading on the social network, where different feature information spreads independently according to different successful probabilities and whether to accept a product is determined by overall evaluation on all these features. They considered the rumor blocking problem under this model, but they assume the weights of each feature for each node are equal when solving the problem. Based on the MF-model [17], we propose a novel budgeted profit maximization problem, MBPM problem. Given a social network, MBPM problem assumes that multiple information about multiple features of a product are spread on it. Each feature has its own propagation probability when spreading from one user to another, and each user has its own weights for each feature. A user will purchase the product only when the sum of weights of features he accepts is larger than or equal to his threshold. Each node has an activation cost and a profit. MBPM problem seeks for a seed set with expected cost no more than the budget to make the obtained profit as large as possible. We consider MBPM problem under the adaptive setting, in which next seed is selected based on the diffusion result of current seeds. That is, we first select a seed and then observe which nodes would be activated by the seed. According to diffusion result, we would select next seed to maximize the profit as much as possible. Main contributions of this work are as follows: • We propose a novel practical problem, MBPM problem, consider it under both the non-adaptive and adaptive settings and propose efficient strategies to solve them, respectively. • For the non-adaptive MBPM problem, we show its objective is monotone submodular and give a randomized algorithm which could achieve (1 − 1/e) expected approximation guarantee. • For the adaptive MBPM problem, we prove its objective function is adaptive monotone and adaptive submodular. We consider adaptive MBPM problem under two models, oracle model and noise model. An policy with (1 − 1/e) expected approximation ratio is given in oracle model. Under noise model, we estimate the conditional expected marginal profit of any node by modifying the EPIC algorithm and propose a sampled adaptive greedy policy which could achieve a (1 − e −(1− ) ) expected approximation guarantee, where 0 < < 1. • Experimental results on realistic datasets confirm effectiveness and superiority of our algorithm. Organization. In Section II, we introduce related works of MBPM problem. The multi-feature diffusion model (MFmodel) and IM problem under the MF-model are discussed in Section III. Section IV presents the MBPM problem under the non-adaptive setting and its solving algorithm. Section V gives definition of adaptive MBPM problem and property of its objective function. Section VI gives the algorithm to solve the adaptive MBPM problem and corresponding proofs for theoretical guarantee. Section VII is dedicated to experiments and Section VIII concludes the paper. II. RELATED WORKS Kempe et al. [5] first formulated IM problem as a combinatorial optimization problem, which aims to choose k seeds to make the expected influence as large as possible. They presented a (1 − 1/e)-approximation algorithm, classical greedy scheme, to solve IM. Later, many variants of IM problem appeared, such as coupon based profit maximization [18]- [20], multiple products profit maximization [13], [15], [16], etc. The one related to our work is cost-aware targeted viral marketing (CTVM) problem [21], which maximizes the expected total benefit by choosing a seed set under the budget. The difference between CVTM and our MBPM problem is CVTM only considered one information diffusion on networks under the classical IC and LT model. And they studied CVTM problem under non-adaptive setting and designed a (1 − 1/ √ e − )-approximation algorithm. Banerjee et al. [22] considered targeted CVTM problem where only nodes in target set have an activation profit, and they proposed a (1 − 1/ √ e)-approximation algorithm. However, we consider multiple diffusions of a product's features and studied the MBPM problem under the adaptive setting. [23] considered a diffusion model of multiple diffusions of a product's features, which is similar to ours, but the activation threshold for each node in their model is fixed. They measured the amount of information that a user received as the probability that the user is activated in an information cascade. However, in our MF-model, each node has a weight for each feature which measures how the user cares about the feature, and the activation threshold for each node is distributed uniformly in [0, 1]. For adaptive problem related research, Golovin et al. [24] proved the objective of adaptive IM problem is adaptive monotone and submodular under full-adoption model and IC model. They proposed an adaptive greedy scheme, which is a (1 − 1/e)-approximation scheme for adaptive IM problem. They also proved this algorithm can be used when a set function with adaptive monotonicity and submodularity subjects to the knapsack constraint. Han et al. [10] considered an variant of the adaptive IM problem, where k seeds are selected in batches of equal size b. They designed an AdaptGreedy framework instantiated by scalable IM algorithms, which could achieve a (1 − e −(1−1/e)+ ) approximation guarantee with high probability. Sun et al. [25] studied a Multi-Round Influence Maximization problem under the adaptive setting where information spreads in multiple rounds independently from probably different seed sets. They proposed an adaptive algorithm instantiated by the IMM [7], which could guarantee (1−e −(1−1/e) − ) approximation. Recently, Huang et al. [11] pointed out there are some gaps in analysis of approximation guarantee for adaptive policies in [10] and [25]. They fixed the previous AdaptGreedy framework in [10] by instantiating with their improved EPIC algorithm and showed it could provide a (1 − e ρ b ( −1) ) expected approximation guarantee. [26] considered the adaptive influence maximization with multiple activations problem, where a selected node in each iteration can be unwilling to be the seed and a node not being the seed in previous iteration can be activated again later but with higher activation cost. The goal is to find a randomized policy subject to expected knapsack constraint to maximize the expected influence spread. They designed an adaptive greedy policies by modifying EPIC algorithm in [11]. Peng et al. [27] showed that the adaptivity gap of the IM problem under the IC model with myopic feedback is at least e/(e−1) and at most 4, and that both the non-adaptive and adaptive greedy algorithms achieve a 1 4 (1 − 1/e)-approximation to the adaptive optimum. [28] showed the adaptivity gap of the IM problem under the IC model with the full-adoption feedback on several families of well-studied influence graphs. [29] proposed the concept of greedy adaptivity gap, comparing the performance of adaptive greedy algorithm to its non-adaptive counterpart. III. INFLUENCE MAXIMIZATION PROBLEM UNDER THE MF-MODEL A social network is generally denoted by a directed graph G = (V, E), where \|V \| = n and \|E\| = m. For each (v, w) ∈ E, v is named the in-neighbor of w and w is called the out-neighbor of v. Here, each node u ∈ V represents a user (customer) in this paper. A. Multi-Feature Diffusion Model Consider a product with multiple features and the information about each feature may spread from one customer to another. To characterize it, we consider the multi-feature diffusion model (MF-model) [17] in this paper. 1. Given a social network G = (V, E), q pieces of information about q features of a product are spread on it, respectively. For each (u, v) ∈ E, there is a q-dimensional propagation probability vectorp u,v = (p 1 u,v , . . . , p q u,v ) associated with it, where p i u,v ∈ (0, 1] is the successful probability when u tries to motivate v to accept the information about feature i of the product. 2. Each u ∈ V has a threshold θ u distributed in [0,1] uniformly and a weight vectorw u = (w 1 u , . . . , w q u ), where w i u denotes the weight of feature i for user u and q i=1 w i u = 1. 3. When user u accepts feature i at timestamp t (called i-accepted, otherwise called i-unaccepted), it will attempt to motivate its i-unaccepted out-neighbor v with successful probability p i u,v at timestamp t+1. The information about different features is diffused independently on the social network, and user v will purchase the product if and only if the sum of corresponding weights of features that have already been accepted by v is no less than θ v (called purchase condition). 4. Initially, a set of seeds is activated to spread all of the q features. At every step, each node that hasn't purchased the product would check whether the purchase condition is satisfied. The diffusion process will continue until there is no more node activated. To further illustrate the model by the phone example, say, the five features of a phone corresponding to price, brand, camera, display, and speed are 1, 2, 3, 4, and 5, respectively. Each node can either accept or not accept each feature. For example, a potential customer may either be convinced that the display is good or not. Each node v ∈ V also has a weight for each of the five features: w 1 v , w 2 v , w 3 v , w 4 v , and w 5 v , which measures how much he cares about each feature. Before the diffusion, each node also has to sample a threshold θ v ∈ [0, 1]. Initially, all features for each seed are accepted. Then, we have five different cascade processes corresponding to the five features. Each of them follows the IC model, and the five cascade processes are independent. Now, at the end of the cascade process, each node is infected by some of the five features. A node will eventually buy the product if the sum of the weights of the accepted features exceeds its threshold. For example, if v accepts features 2 and 4 but not 1, 3, 5, then v will be considered activated if the weighted sum w 2 v + w 4 v exceeds his threshold θ v . Even though Guo et al. [17] first proposed this MF-model, they assume the weight of each feature for each node is the same, namely w i u = w i v for any user u, v ∈ V , in their submodularity proof and algorithm analysis. This is only a special case and not that realistic. In this paper, we consider the general case where the weight vectorw u = (w 1 u , . . . , w q u ) for each node u is arbitrary. Denote by σ(S) the expected number of nodes (users) purchasing the product when S is the seed. Actually, we could prove that σ(S) is monotone non-decreasing and submodular with respect to S under the general MF-model. Our following analysis is based on the general MF-model, which is an important improvement and extension. Before showing properties of σ(S), let's first see the equivalent diffusion process of the MF-model. Remark 1. For convenience, we still use G = (V, E) to represent the social network G = (V, E) with propagation probability p : E → (0, 1] q , threshold θ : V → [0, 1], and weight w : V → [0, 1] q . B. Equivalent Diffusion Process Since information of different features is spread independently on the social network in MF-model, that is, the diffusion of one piece of information about one feature has no interference on information of other features, we can view the propagation process of MF-model as follows. Definition 1 (Multi-level Graph). Given a social network G = (V, E), define its multi-level graph as G = ( V , E) = G 1 ∪ G 2 ∪ · · · G q , where G i = (V i , E i ) and each node v i ∈ V i is a copy node of v ∈ V , called feature node of v. For each (u, v) ∈ E, there is a corresponding edge (u i , v i ) ∈ E i , i = 1, · · · , q, and the propagation probability on (u i , v i ) is p i u,v , that is, the successful probability when u attempts to motivate v to accept feature i. An example of the multi-level graph can be seen in Fig. 1. For each node set S ⊆ V , denote by S = S 1 ∪ . . . ∪ S q the corresponding feature node set in G of nodes in S, where S i = {u i ∈ V i : u i is the corresponding feature node of u ∈ S for feature i}. For each u ∈ V , denote the corresponding feature node set of u as u = {u 1 , . . . , u q }. Then we could give the equivalent diffusion process of the general MF-model. 1. Given a social network G = (V, E) and its multi-level graph G = G 1 ∪ . . . ∪ G q , q pieces of information about q different features are spread on G, but the information about feature i is only spread on G i . Each node v ∈ V samples a threshold θ v independently uniformly at random from [0,1]. 2. Initially, we choose the seed set S ⊆ V for the product. Then nodes in S are seeds of the corresponding features. 3. The information about different features is diffused independently from their own seeds according to the classic IC model. A node in G i can only have two states: i-accepted or iunaccepted. A node in G i accepting the information of feature i is called i-accepted. Otherwise, it is called i-unaccepted. 4. After the propagation process of all features terminates, we could determine whether each node v ∈ V would purchase the product. That is, we would check whether the sum of weights of i-accepted nodes v i , i = 1, · · · , q, is larger than or equal to θ v . If it satisfies the purchase condition, then node v would purchase the product and we call v active. Otherwise, v is called inactive. Remark 2. To avoid confusion, we will use "infect" when we say a feature node u i tries to activate its out-neighbor v i to accept feature i, and use "activate" for user node. C. Property of σ(S) Definition 2 (Realization). Given a social network G = (V, E) with probability p : E → (0, 1], a (full) realization φ of G is defined as φ : E → {0, 1}. For e ∈ E, φ(e) = 0 (resp. 1) means edge e is blocked (resp. live) under φ. Let Φ be a random variable denoting a random realization. Then we have Pr [φ] := Pr[Φ = φ] = e∈E: φ(e)=1 p e e∈E: φ(e)=0 (1 − p e ) Denote by Ω the set of all possible realizations of multilevel graph G. Let S be the seed set of the product and S = S 1 ∪ . . . ∪ S q be its corresponding feature node set. Then S i is the seed set of feature i in G i . Given a realization φ ∈ Ω, for each node u i in G, define x φ (S i , u i ) = 1, u i accepts feature i in φ under S i 0, otherwise Therefore, node u ∈ V will purchase the product under φ if and only if q i=1 x φ (S i , u i ) · w i u ≥ θ u . Denote I φ (S i ) as the node set in G i containing the i-accepted nodes in φ under S i .That is, I φ (S i ) = {u i ∈ V i \|x φ (S i , u i ) = 1}. Let I φ (S) be the set of active nodes in V when diffusion process of nodes in S on φ terminates. Then for any u ∈ V , we have Pr[u ∈ I φ (S)] = 1≤i≤q: ui∈I φ (S i ) w i u . Remark 3. I φ (S i ), i = 1, . . . , q are deterministic sets while I φ (S) is a random set. Theorem 1. σ(S) is monotone non-decreasing and submodular with respect to S. Proof: For any node set S ⊆ T ⊆ V , denote byŜ = S 1 ∪ . . . ∪ S q andT = T 1 ∪ . . . ∪ T q their corresponding copy node set inĜ, respectively. Then σ(S) under the MF-model can be represented as: σ(S) = φ∈Ω Pr[φ] · v∈V Pr[v ∈ I φ (S)] = φ∈Ω Pr[φ] · v∈V 1≤i≤q: vi∈I φ (S i ) w i v Since S ⊆ T , then S i ⊆ T i , i = 1, . . . , q. Clearly, I φ (S i ) ⊆ I φ (T i ) since any node in I φ (S i ) can also be i-accepted under T i in φ. Therefore, vi∈I φ (S i ) w i v ≤ vi∈I φ (T i ) w i v and σ(S) is monotone with respect to S. For any u ∈ V \ T , σ(S ∪{u})−σ(S) = φ∈Ω Pr[φ]· v∈V 1≤i≤q: vi∈(I φ (S i ∪{ui})\I φ (S i )) w i v I φ (S i ∪ {u i }) \ I φ (S i ) contains the nodes in G i that can only be infected by u i but cannot by S i under φ. Clearly, I φ (S i ∪ {u i }) \ I φ (S i ) ⊇ I φ (T i ∪ {u i }) \ I φ (T i ) , since u i could infect more feature nodes when adding to S i than T i under φ. Therefore, σ(S ∪ {u}) − σ(S) ≥ σ(T ∪ {u}) − σ(T ) and the proof of Theorem 1 is completed. IV. MULTI-FEATURE BUDGETED PROFIT MAXIMIZATION PROBLEM A company wants to promote a new product by distributing coupons on social networks to maximize its profit as much as possible. However, the advertisement budget is usually limited. Thus, it is important to wisely select customers to allocate coupons. The product has multiple features and the information about each feature spreads from one customer to another. Given the social network G = (V, E), for each u ∈ V , assume the cost of picking u as the seed of product and profit obtained when u purchases the product are c(u) and b(u), respectively. For any S ⊆ V , the activation cost and profit of S are defined as c(S) = u∈S c(u) and u∈S b(u), respectively. Since we will consider randomized algorithm in the adaptive case later, so in this section we will also consider the randomized algorithm for comparison. Given a budget B, we want to find a seed set S with expected cost at most B to maximize the obtained profit. Given the equivalent diffusion process of the MF-model, we could solve the MBPM problem by solving the profit maximization problem on the multi-level graph. A. Problem Definition Then the MBPM problem can be formulated as: max P (S) = φ∈Ω Pr[φ] · u∈V Pr[u ∈ I φ (S)] · b(u) s.t. E[c(S)] ≤ B(1) Based on the proof of Theorem 1, we have the following result. Theorem 2. The objective function of MBPM Problem is monotone submodular with respect to the seed set of the product. B. Algorithm Before presenting the algorithm of MBPM problem, we first introduce another problem, maximization of a monotone submodular function under the cardinality constraint. Let g : 2 V → R ≥0 be a monotone submodular function. For u ∈ V and S ⊆ V , the marginal gain by adding v to S is denoted as g v (S) = g(S ∪ {v}) − g(S). For the problem max S⊆V,\|S\|≤k g(S), classical greedy scheme could return (1 − 1/e)-approximation solutions [30]. The algorithm always selects the element with largest marginal gain to current selected set until k nodes are chosen. That is, for current selected set S 0 , the algorithm will select v * = arg max v∈V \S0 g v (S 0 ) and add it into S 0 . Under cardinality constraint, each node actually has a cost of 1 and greedy scheme always selects the element with the largest marginal gain per unit cost. Since the objective of MBPM problem is monotone submodular, inspired by ideas of classical greedy scheme, we could utilize Algorithm 1 to solve it. Assume current selected set is S. Alg. 1 always selects the node v * with largest ratio of marginal gain to S to cost among remaining nodes. If c(S) + c(v * ) ≤ B, v * will be added into S. Otherwise, add v * to S with B−c(S) c(v * ) probability or break with probability 1 − B−c(S) c(v * ) . It could guarantee that the output S satisfies E[c(S)] ≤ B. For the node found in the last iteration of our Algorithm, it will be selected with a very low probability if it is far more than the remaining budget. Let v 1 , . . . , v n be the result sorted by increasing order of activation cost of nodes in V . That is, c(v 1 ) ≤ c(v 2 ) ≤ . . . ≤ c(v n ). Denote p as the minimum number satisfying p i=1 c(v i ) ≥ B. Assume p < n. Otherwise, we could select all nodes as the seeds. Then we know Algorithm 1 would execute at most p iterations. Proof: Since the expected knapsack constraint is somewhat different from the classical knapsack constraint, we think it's necessary to provide the proof for this theorem here. Our proof is inspired by [30]. Assume S * = {u 1 , . . . , u t } is an optimal solution to MBPM problem. Let S r = {v 1 , . . . , v r } be the node set obtained by Algorithm 1 after r iterations and S 0 = ∅. Assume v r+1 = arg max v∈(V \Sr) { P (Sr∪{v})−P (Sr) c(v) }. Assume c(S r ) ≤ B and c(S r ) + c(v r+1 ) > B. Let S G be the node set returned by Algorithm 1. Denote P v (S i ) = P (S i ∪ {v}) − P (S i ). Since P (·) is monotone submodular, for 1 ≤ i ≤ r, we have P (S * ) ≤ P (S * ∪ S i ) = P (S i ) + P u1 (S i ) + P u2 (S i ∪ {u 1 }) + . . . + P ut (S i ∪ {u 1 , . . . , u t−1 }) ≤ P (S i ) + P u1 (S i ) + P u2 (S i ) + . . . + P ut (S i ) ≤ P (S i ) + c(u 1 ) · P vi+1 (S i ) c(v i+1 ) + . . . + c(u t ) · P vi+1 (S i ) c(v i+1 ) ≤ P (S i ) + B · P (S i+1 ) − P (S i ) c(v i+1 ) . Denote a i = P (S * )−P (S i ). Then we know a i ≤ B c(vi+1) (a i − a i+1 ). Thus, a i+1 ≤ (1 − c(v i+1 ) B )a i ≤ Π i+1 j=1 1 − c(v j ) B a 0 Therefore, P (S i+1 ) ≥ 1 − Π i+1 j=1 1 − c(v j ) B P (S * ). Then we have E[P (S G )] = B − c(S r ) c(v r+1 ) P (S r+1 ) + 1 − B − c(S r ) c(v r+1 ) P (S r ) ≥ P (S r ) + (B − c(S r )) · P (S * ) − P (S r ) B = 1 − c(S r ) B P (S * ) + c(S r ) B P (S r ) ≥ 1 − c(S r ) B P (S * ) + c(S r ) B   1 − r j=1 1 − c(v j ) B   P (S * ) =   1 − r j=1 1 − c(v j ) B · c(S r ) B   P (S * ) =   1 − r j=1 1 − c(v j ) B 1 − 1 − c(S r ) B   P (S * ) ≥ 1 − exp   − r j=1 c(v j ) B   · exp − 1 − c(S r ) B P (S * ) = (1 − 1/e)P (S * ) V. ADAPTIVE MULTI-FEATURE BUDGETED PROFIT MAXIMIZATION PROBLEM In practice, the decision maker may select one seed at a time and then observe the propagation result. He could make choice to select the next seed based on currently observed results. And this strategy is usually called adaptive seed selection strategy. This strategy may bring more advantages and profits since the decision maker could adaptively revise the strategy according to the current situation rather than select all seeds once before the actual propagation process starts. Therefore, it is worth considering whether adaptive selection strategy helps a lot or not. In this section, we will introduce the adaptive MBPM problem and some related definitions. A. Problem Definition In the adaptive MBPM problem, we also choose seeds S from G = (V, E) and observe the propagation process of corresponding seeds S in its multi-level graph G like in the non-adaptive MBPM problem. But under the adaptive setting, seeds are selected one by one and we need observe the diffusion result once a seed u is chosen. Specifically, thresholds for each node v ∈ V are sampled independently uniformly at random from [0, 1] at first. Then we select one seed at each step. When node u is selected as the next seed, equivalently we infect all of its feature nodes u 1 , . . . , u q . Then we need to observe states of edges in G: observe the propagation result of u i on G i (related edges are live or blocked), i = 1, . . . , q. After all q diffusions on G stop, we could determine whether nodes in G not buying the product before selecting u would purchase the product or not now, according to the current propagation results on G. Then we select next seed and repeat this process until there is no seed budget. In this adaptive seeding process, after selecting a node u ∈ V and all feature nodes u 1 , . . . , u q of u are infected, we could observe all edges exiting v i , i = 1, . . . , q, which can be reached from u i by currently live edges in G i . That is, the full-adoption feedback model [24] is considered in this paper. Our observation so far could be described by the partial realization ϕ, a function mapping from currently observed items to their states. For (u i , v i ) ∈ E, ϕ((u i , v i )) ∈ {0, 1, ?} and ϕ((u i , v i )) = 1 (resp. 0) if edge (u i , v i ) has been observed live (resp. blocked). ϕ((u i , v i )) =? if the status of (u i , v i ) is not known yet. For any partial realization ϕ, define the domain dom(ϕ) of ϕ as the seed set for the product that have already been picked from V . Denote dx(ϕ) as the set of edges in E whose states have been known under ϕ. Let φ : E → {0, 1} be a full realization of G. We say a partial realization ϕ is consistent with φ if they are equal everywhere in the domain of ϕ, denoted by φ ∼ ϕ. If ϕ and ϕ are both consistent with some full realization φ, satisfying dom(ϕ) ⊆ dom(ϕ ), we say ϕ is a subrealization of ϕ , denoted as ϕ ⊆ ϕ . Let π(τ ) be a randomized policy where τ represents all random source of the randomized policy. Specifically, π(τ ) is a function mapping from an already chosen seed set S ⊆ V and a set of partial realizations to V , specifying which node to select as the next seed of the product within the budget. Let S(π(τ ), φ) be the set of nodes in V chosen by π(τ ) under realization φ. Let I i φ (S(π(τ ), φ)) be the set of nodes in V i accepting feature i when diffusion process of feature nodes of S(π(τ ), φ) on φ terminates. The profit obtained by policy π(τ ) under realization φ is defined as: f (S(π(τ ), φ), φ) = u∈V b(u) · [ 1≤i≤q: ui∈I i φ (S(π(τ ),φ)) w i u ]. Thus, the expected profit obtained by policy π(τ ) can be formulated as: f avg (π(τ )) = E Φ [f (S(π(τ ), Φ), Φ)] . Definition 4 (Adaptive Multi-feature Budgeted Profit Maximization (AMBPM) Problem). Given G = (V, E), assume q pieces of information about q features of a product are spread on G according to the MF-model. The AMBPM problem seeks for a randomized policy to maximize the total expected profit obtained: max π E τ [f avg (π(τ ))] s.t. E τ [c(S(π(τ ), φ))] ≤ B, for any realization φ Definition 5 (Conditional Expected Marginal Profit). Given a partial realization ϕ and a node u, the conditional expected marginal profit of u conditioned on having observed ϕ is defined as: ∆(u\|ϕ) = E [f (dom(ϕ) ∪ {u}, Φ) − f (dom(ϕ), Φ)\|Φ ∼ ϕ] where the expectation is taken over p(φ\|ϕ) = P(Φ = φ\|Φ ∼ ϕ). Proof: We first show adaptive monotonicity of f . Consider a fixed partial realization ϕ. For a node u / ∈ dom(ϕ), when selecting u as the seed under ϕ, if all feature nodes u 1 , . . . , u q of u have been infected before u is selected under ϕ, then for any realization φ ∼ ϕ, we have f (dom(ϕ) ∪ {u}, φ) = f (dom(ϕ), φ). Otherwise, there exists at least one of u 1 , . . . , u q not infected before u is selected, and assume u 1 is one of the feature node satisfying the condition. Then for any realization φ ∼ ϕ, we have f (dom(ϕ) ∪ {u}, φ) − f (dom(ϕ), φ) ≥ b(u) · w 1 u ≥ 0. Thus, no matter which case happens, for any realization φ ∼ ϕ, f (dom(ϕ) ∪ {u}, φ) ≥ f (dom(ϕ), φ) always holds. Since ∆(u\|ϕ) is a linear combination of each realization φ ∼ ϕ, we know that ∆(u\|ϕ) ≥ 0. Next we prove the adaptive submodularity of f . For any pairs of partial realizations ϕ, ϕ satisfying ϕ ⊆ ϕ and any u / ∈ dom(ϕ ), we have to show ∆(u\|ϕ) ≥ ∆(u\|ϕ ). Our proof is inspired by the proof technique in [24] and [31]. Consider two fixed partial realizations ϕ, ϕ satisfying ϕ ⊆ ϕ . Assume there are two realizations φ and φ with We first show that M ⊆ M . Fix w i ∈ M . Then there must exist a path P i from some feature node v i of v ∈ dom(ϕ) to w i . Therefore, edges on path P i are observed to be live by ϕ. Since φ ∼ ϕ, φ ∼ ϕ and ϕ ⊆ ϕ , edges observed by ϕ have same states in φ and φ . That is, each edge on P i is also live under φ . Since ϕ ⊆ ϕ , it is clear that v ∈ dom(ϕ ). Therefore, w i will be i-accepted when feature nodes of dom(ϕ ) are seeds in G under realization φ , i.e., w i ∈ M . φ ∼ ϕ, φ ∼ ϕ , satisfying φ((u i , v i )) = φ ((u i , v i )) for all (u i , v i ) / ∈ dx(ϕ ). Thus, φ and φ have the same area β = ϕ ∪ (φ \ ϕ ). Let σ(dom(ϕ) ∪ {u}, φ) = ∪ q i=1 I i φ (dom(ϕ) ∪ {u}, φ)) We next show N ⊆ N . We prove this by contradiction. Fix v j ∈ N . Assume v j / ∈ N . Since v j ∈ N and M ∩ N = ∅, we have that v j / ∈ M . Since we have proven M ⊆ M , it is obvious that v j / ∈ M . As v j ∈ N , there must exist some path P j from u j to v j in φ but at least one edge on path P j is blocked in φ. Assume one such edge is (s j , t j ). Since the status of edge (s j , t j ) is different in realization φ and φ , and φ and φ have the same area β, thus (s j , t j ) must be observed by ϕ but not by ϕ. Since (s j , t j ) is observed by ϕ , s j must be infected after selecting dom(ϕ ) according to the full-adoption feedback model. That is, s j and the nodes that can be reachable from s j must be infected after we select dom(ϕ ). Therefore, s j and the nodes that can be reachable from s j , including v j , will belong to M , a contradiction. Define δ(u\|φ, φ ∼ ϕ) = f (dom(ϕ) ∪ {u}, φ) − f (dom(ϕ), φ) = v∈V b(v) 1≤i≤q: vi∈T w i v − v∈V b(v) 1≤i≤q: vi∈M w i v = v∈V b(v) 1≤i≤q: vi∈(T \M ) w i v = v∈V b(v) 1≤i≤q: vi∈N w i v Since we have shown that N ⊆ N , we could obtain that δ(u\|φ, φ ∼ ϕ) ≥ δ(u\|φ , φ ∼ ϕ ). Since φ∼β Pr[φ\|φ ∼ β] = 1, we know ∆(u\|ϕ) = φ∼ϕ Pr[φ\|φ ∼ ϕ] · δ(u\|φ, φ ∼ ϕ) = φ ∼ϕ Pr[φ \|φ ∼ ϕ ] φ∼β Pr[φ\|φ ∼ β] · δ(u\|φ, φ ∼ ϕ) ≥ φ ∼ϕ Pr[φ \|φ ∼ ϕ ] φ∼β Pr[φ\|φ ∼ β] · δ(u\|φ , φ ∼ ϕ ) = φ ∼ϕ Pr[φ \|φ ∼ ϕ ] · δ(u\|φ , φ ∼ ϕ ) = ∆(u\|ϕ ), which completes the proof. VI. ALGORITHM AND THEORETICAL ANALYSIS Since the objective f (·, φ) of AMBPM problem is adaptive monotone and adaptive submodular, we could utilize adaptive greedy policy proposed in [24] to solve it. The seed selection rule of adaptive greedy policy is straightforward, i.e., always selecting the node with largest ratio of conditional expected marginal profit to cost. However, given a partial realization ϕ and a node u / ∈ dom(ϕ), it is difficult to compute the conditional expected marginal profit ∆(u\|ϕ) since there are almost exponential possible realizations φ with φ ∼ ϕ. This section would consider algorithms of AMBPM problem under both the oracle model and noise model. A. Adaptive Greedy Algorithm under the Oracle Model Under the oracle model, assume conditional expected marginal profit of any node under any partial realization can be obtained in constant time. Define a randomized adaptive greedy policy π ag (τ ). The main idea of adaptive greedy policy to solve this problem can be seen in Algorithm 2, which is based on the adaptive greedy policy proposed in [24]. Under the current partial realization ϕ and seed set S, the π ag (τ ) would select a node v max satisfying v max : = arg max v∈V \S { ∆(v\|ϕ) c(v) }. If c(S) + c(v max ) ≤ B, then v max is the next seed. Otherwise, π ag (τ ) would select v max as the next seed with probability B−c(S) c(vmax) . After selecting v max , we observe the nodes infected by feature nodes of v max , denoted by A(v max ) and update the partial realization ϕ by changing states of edges related to nodes in A(v max )∪v max from ? to 0 or 1. The algorithm repeats the above process, and terminates until c(S) ≥ B, or terminates with a probability. In this way, we could guarantee E[c(S)] ≤ B. The random source τ in this adaptive greedy policy indicates whether to contain the node found in the last iteration. Algorithm 2 Adaptive-Greedy Input: G = (V, E), its multi-level graph G and B Output: A seed set S ⊆ V and f (S, ϕ). Observe the node set A(v max ) infected by feature nodes of v max , 1 ≤ i ≤ q; 9: Update ϕ by updating states of edges related to nodes in A(v max ) ∪v max ; 10: Return S, f (S, ϕ) Since the objective f (S, φ) of AMBPM problem is adaptive monotone and adaptive submodular, according to the result in [24], we have the following conclusion. B. Adaptive Greedy Algorithm under the Noise Model This section will present algorithms of AMBPM problem under the noise model. The basic seed selection strategy is similar to that in oracle model, but the difference is we will estimate the conditional expected marginal profit of any node under a fixed partial realization, ∆(u\|ϕ), by the reverse influence sampling technique. However, maximizing the estimation of ∆(u\|ϕ) by sampling technique is likely to obtain an extremely worse node with some probability, although the probability is very small. That is, the node u * maximizing the estimation may not be the optimal solution to max v∈V \S ∆(u\|ϕ)/c(u). In this case, the expected approximation ratio in Theorem 5 is not guaranteed. 1) Technique: Definition 8 (Reverse Reachable (RR) set [6]). For any graph realization φ ∈ Ω and v i ∈ V , the RR set for v i is denoted by R φ (v i ), which contains all nodes that could reach v i in φ. v i is called the target node of R φ (v i ). Intuitively, RR set R φ (v i ) of v i contains feature nodes that are likely to infect v i during the propagation. A random RR set is an RR set whose target node v i is selected randomly from V . Given a random RR set R φ (v i ) and S ⊆ V , we say S covers R φ (v i ) if S ∩ R φ (v i ) = ∅. A set with larger expected influence has a higher probability to cover a random RR set. Specifically, given a graph G = (V, E) and a random RR set R, the expected influence [6]. Therefore, if we could generate a large number of random RR sets, a set with large expected influence would cover a large amount of the generated random RR sets. We will use this idea in our estimation of the conditional expected marginal profit of a node. E[I(S)] of a set S in G is E[I(S)] = \|V \| · Pr[S ∩ R = ∅] Given a partial realization ϕ, let G ϕ = (V ϕ , E ϕ ) be the subgraph induced by the i-unaccepted nodes under ϕ, i = 1, . . . , q. That is, G ϕ is obtained by deleting all of the i-accepted feature nodes and their related edges in G, i = 1, . . . , q. Let Ω ϕ be the set containing all realizations of G ϕ . Denote W = vi∈Vϕ b(v) · w i v . Assume each node v i ∈ V ϕ is selected randomly from V ϕ with probability b(v)·w i v W as the target node of an RR set. Given a partial realization ϕ and u ∈ V , let R ϕ be a random RR set generated from a realization φ ∈ Ω ϕ . Define h(u, R ϕ ) = 1, if u ∩ R ϕ = ∅ 0, otherwise By the reverse Breadth First Search algorithm [32], we could produce a large number of random RR sets R(ϕ) = {R 1 , R 2 , . . . , R α } of G ϕ . Define F R(ϕ) (u) = 1 α α j=1 h(u, R j ). Denote ρ(u\|ϕ) = W · F R(ϕ) (u) = W · 1 α α j=1 h(u, R j ).(2) Then the following result holds. Theorem 6. Given a node u ∈ V and a partial realization ϕ, we have E [ρ(u\|ϕ)] = ∆(u\|ϕ). Proof: Given a realization φ ∈ Ω ϕ and a user node u ∈ V , let I i φ (u) be the feature nodes infected by feature node u i under φ. Then we have E [ρ(u\|ϕ)] = E[W · 1 α α j=1 h(u, R j )] = W · E[ 1 α α j=1 h(u, R j )] =W · φ∼Ωϕ Pr[φ] vi∈Vϕ Pr[v i ] · h(u, R ϕ (v i )) = φ∼Ωϕ Pr[φ] vi∈Vϕ b(v) · w i v · h(u, R ϕ (v i )) = φ∼Ωϕ Pr[φ] v∈V b(v) · 1≤i≤q: vi∈I i φ (u) w i v = φ∼Ω Pr[φ\|φ ∼ ϕ] · v∈V b(v) 1≤i≤q: vi∈(I i φ (dom(ϕ)∪{u})\I i φ (dom(ϕ)) w i v = φ∼Ω Pr[φ\|φ ∼ ϕ] · [f (dom(ϕ) ∪ {u}, φ) − f (dom(ϕ), φ)] =∆(v\|ϕ) Given a partial realization ϕ and a set of random RR sets R(ϕ) generated from subgraph G ϕ , define Q R(ϕ) (u) = F R(ϕ) (u)/c(u). According to Theorem 6, we know E[W · Q R(ϕ) (u)] = W · E[Q R(ϕ) (u)] = ∆(u\|ϕ)/c(u). Thus, W ·Q R(ϕ) (u) is an unbiased estimation of ∆(u\|ϕ)/c(u). When \|R(ϕ)\| is sufficiently large, W · Q R(ϕ) (u) could be convergent to ∆(u\|ϕ)/c(u). Thus, we could use W · Q R(ϕ) (u) as an estimation for ∆(u\|ϕ)/c(u). Algorithm 3 Sampled-AdapGreedy (SAG) Input: A graph G = (V, E), its multi-level graph G and a budget B ∈ R + , an error parameter . Output: A seed set S ⊆ V and f (S, ϕ). S ← S ∪ {v * }; 13: Observe the node set A(v * ) infected by the feature nodes of v * , 1 ≤ i ≤ q; 14: Update ϕ by updating states of edges related to nodes in A(v * ) ∪ v * ; 15: W = W − ui∈A(v * ) b(u)w i u − vi∈ v * ∩Gϕ b(v * )w i v; 16: Update G ϕ by removing nodes in A(v * )∪ v * and their corresponding edges; 17: return S and f (S, ϕ); Algorithm 3 show the adaptive greedy policy with the above sampling technique, named Sampled-AdapGreedy. It is denoted by π sag (τ, ω) where ω usually represents the random source of sampling. At each iteration, instead of finding a node maximizing ∆(u\|ϕ)/c(u) from currently unselected user nodes, we select a node v * which could maximize Q R(ϕ) (u), which is obtained by Algorithm 4 [11]. If c(v * ) is larger than the current remaining budget, then we add v * into the current seed set with (B − c(v * ))/c(v * ) probability. Otherwise, we add v * to the current seed set, observe the corresponding propagation result on G ϕ , and update partial realization ϕ and subgraph G ϕ . 2) Theoretical Analysis: At each iteration of Algorithm 3, it needs use Algorithm 4 (line 9 of Alg. 3) to obtain a node which could achieve the maximum of function Q R(ϕ) (u). Alg. 4 is obtained by modifying the EPIC algorithm proposed in [11]. However, there are some difference between EPIC and Modified-EPIC (MEPIC) algorithm: (1) The seed selected at each iteration is one in MEPIC. (2) The target estimation function in MEPIC is Q R(ϕ) (u) instead of F R(ϕ) (u). Algorithm 4 Modified-EPIC (MEPIC) [11] Input: A graph G ϕ = (V ϕ , E ϕ ), T, W, W * , n ϕ , Output: An approximately optimal node u ∈ V \ S. = ( − δ · W )/(1 − δ · W ); 3:¯ = /(1 − ); 4: i max = log 2 (2+2¯ /3)·W 2 + 1 and a = ln( 2·imax δ ); 5: θ 0 = 1 W * ln 2 δ + ln (n ϕ ) ; 6: Generate two sets of random RR sets R 1 (ϕ) and R 2 (ϕ) of G ϕ with \|R 1 (ϕ)\| = \|R 2 (ϕ)\| = θ 0 ; 7: for i = 1 to i max do 8: v * = arg max v∈T Q R1(ϕ) (v); 9: Q u (v max ) ← Q R1(ϕ) (v * ); 10: Q l (v * ) ← Q R2(ϕ) (v * ) + 2a 9\|R2(ϕ)\| − a 2\|R2(ϕ)\| 2 − a 18·\|R2(ϕ)\| ; 11: if Q l (v * ) Q u (vmax) ≥ 1 − or i = i max then 12: return v * ; 13: Double the sizes of R 1 (ϕ) and R 2 (ϕ) with new random RR sets; At each iteration of Algorithm 3, denote the current partial realization as ϕ. We could obtain its corresponding induced subgraph G ϕ . Alg. 4 first initializes some parameters and then generates two same size sets of random RR sets of G ϕ , R 1 (ϕ) and R 2 (ϕ). At each iteration, it chooses a node v * maximizing Q R1(ϕ) (·), which can be achieved in polynomial time. As- sume v max = arg max v∈T ∆(v\|ϕ)/c(v).Then Q u (v max ) = Q R1(ϕ) (v * ) ≥ Q R1(ϕ) (v max ). That is, Q u (v max ) is an upper bound of Q R1(ϕ) (v max ). And W · Q l (v * ) is an accurate lower bound of ∆(v * \|ϕ)/c(v * ) with high probability. Then MEPIC checks whether the stopping condition (line 11) is satisfied. If satisfied, it returns v * as output. Otherwise, it doubles the size of R 1 (ϕ) and R 2 (ϕ), and repeats the above process. Lemma 1. Given a partial realization ϕ and its corresponding induced subgraph G ϕ = (V ϕ , E ϕ ), denote by T the set of current unselected nodes in V . Then MEPIC algorithm could return a user node v * satisfying that E τ ∆(v * \|ϕ) c(v * ) ≥ (1 − ) · max v∈T ∆(v\|ϕ) c(v) , within O((\|V ϕ \| + \|E ϕ \|)(log(\|T \|) + log 1 )/ 2 ) expected time. Proof: Given a partial realization ϕ and its correspond- ing induced subgraph G ϕ = (V ϕ , E ϕ ), the target function Q R1(ϕ) (v) = F R1(ϕ) (u)/c(u) is a weighted coverage function on R 1 (ϕ), where the weight for each u ∈ (V \ dom(ϕ)) is 1/c(u). Since Q R1(ϕ) (v) is a monotone submodular function and maximizing a monotone submodular weighted coverage function can be solved in polynomial time, thus the node v * = arg max v∈T Q R1(ϕ) (v) can be obtained in polynomial time. Also, the expected approximation guarantee can be obtained accordingly from results of EPIC algorithm in [11]. Recall that p is the minimum number satisfying p i=1 c(v i ) ≥ B. Then we know Algorithm 3 would execute at most p iterations. Now, we could give the approximation guarantee of our AG algorithm. Theorem 7. Given ∈ (0, 1), the sampled adaptive greedy policy π sag (τ, ω) (Algorithm 3) could achieve a (1 − e −(1− ) ) expected approximation ratio within O(pq · (n + m)(log(n + log 1 )/ 2 ) expected time. That is, for any realization φ and any policy π(τ ) satisfying E τ [c(S(π(τ ), φ))] ≤ B, we have E τ [E ω [f avg (π sag (τ, ω))]] ≥ (1 − e −(1− ) ) · E τ [f avg (π(τ ))] Proof: According to Lemma 1, the node selected in each iteration of Algorithm 3 satisfies (1 − ) expected approximation. Since Algorithm 3 would execute no more than p iterations, thus the total expected error of all iterations is (1/p) · p i=1 = . According to the Theorem 3 and Lemma 1, Theorem 7 holds by inferring from Theorem 6 in [11]. VII. EXPERIMENTS We verify efficiencies of our proposed policy by comparing the running time and its obtained profit with other algorithms. We run experiments on a Linux machine with an Intel Xeon 3.5GHz CPU and 32GB RAM. For each dataset, 30 possible realizations are produced randomly and the average performance of each algorithm is reported. A. Experimental Setup Datasets. Five real-world social network datasets are used in this paper and detailed statistics are shown in table I. Epinions dataset could be found in [33] and all other datasets are from [34]. According to the structure of MF-model, the number of nodes in multi-level graph is different from these basic information, which is also determined by the number of features. For the undirected graph, we replace each edge with two reversed directed edges. Propagation Model and Parameters. We use the MFmodel as diffusion model in experiments and for each edge e = (u, v) ∈ E, set p i u,v = 1/\|N in (v)\|, i = 1, . . . , q, where N in (v) is the set of in-neighbors of v. This setting is widely used in prior works [6], [35], [36]. For u ∈ V , the weight vector of u is generated randomly from (0, 1] q such that the sum of weights of all features for u is 1. Also, we generate random numbers from (0, 1] as the cost and profit of each node. For each dataset, we vary budget B such that B ∈ {0, 10, 20, 30, 40, 50}. We conduct two groups of experiments to test the time efficiency and performance of our proposed policy, respectively. The first group of experiments is performed to verify the time efficiency of adaptive greedy policy (Algorithm 2) and sampled adaptive greedy policy (Algorithm 3). We compare running time and obtained profit of adaptive greedy policy and sampled adaptive greedy policy with their non-adaptive corresponding algorithms, with different implementations. (1) Modified greedy algorithm sampled by Monte Carlo (MGMC): Shown as Algorithm 1 and the profit P (S) of any node set S is estimated by Monte Carlo method. (2) Modified greedy algorithm sampled by reverse influence sampling (MGRIS): Shown as Algorithm 1 and the profit P (S) of any node set S is estimated by reverse influence sampling method. Let Q = u∈V b(u). Each feature node v i in multi-level graph G is selected as a target node of a RR set with b(v) · w i v /Q probability. Let R = {R 1 , . . . , R λ } be a set of random RR sets generated from G. Then it can be proved K R (S) = Q · F R (S) is an unbiased estimation of P (S). According to Chernoff Bounds [37], if λ ≥ (2+η)Q η 2 P (S) · ln 1 δ , then for any node set S with c(S) ≤ B, we have Pr[\|P (S) − K R (S)\| > η · P (S)] < δ . Let p * be the minimum number such that n j=p * c(v j ) ≤ B and Q * = n j=p * b(v j ). By setting λ = (2+η)Q η 2 Q * · ln 1 δ , we could guarantee λ ≥ (2+η)Q η 2 P (S) · ln 1 δ . Here we set η = δ = 0.1. The second group of experiments is to compare the performance of our SAG policy with three heuristic adaptive policies: Adaptive Random (AR), Adaptive Max-degree (AMD) and Adaptive Max-profit (AMP). (1) AR is the adaptive version of the simple random algorithm. It uniformly selects currently unselected nodes in V as the next seed. (2) AMD picks the node with maximum out-degree from currently unselected nodes in V as the next seed. (3) Given the partial realization ϕ and currently selected seed set S, AMP selects the node u * satisfying u * ∈ arg max u∈(V \S) ∆(u\|ϕ) and estimates ∆(u\|ϕ) by reverse influence sampling technique. According to Theorem 6, we know ρ(u\|ϕ) is an unbiased estimation of ∆(u\|ϕ). According to Chernoff Bounds [37], if α ≥ (2+ˆ )Ŵ 2 ∆(u\|ϕ) · ln 1 δ , then we have Pr[\|ρ(u\|ϕ) − ∆(u\|ϕ)\| >ˆ · ∆(u\|ϕ)] < δ . Let W * = min vi∈ V b(v) · w i v . By setting α = (2+ˆ )Ŵ 2 W * · ln 1 δ , we could guarantee α ≥ (2+ˆ )Ŵ 2 ∆(u\|ϕ) ·ln 1 δ . Here we setˆ = δ = 0.1. B. Experimental Results 1) Results of first group of experiment: Fig. 2 and Fig. 3 present results of the first group of experiments on Twitter and Wiki datasets. Fig. 2 and Fig. 3 present profits obtained by our proposed adaptive greedy policy (AG and SAG) with their non-adaptive versions (MGMC and MGRIS). We implement the experiments under two values of q, 3 and 5. Since AG policy and MGMC algorithm are implemented by Monte Carlo method and they are time-consuming, thus we only conduct the first group of experiments on two small datasets. Here the number of Monte Carlo simulations is set to 500. The results show that AG and SAG policy are always evidently superior than MGMC and MGRIS algorithm with respect to the obtained profit, which shows the benefits of adaptive policies. The profits obtained by AG and SAG policies are very close, which indicates effectiveness of our sampling technique. The profits obtained by MGRIS and MGMC algorithms are close at most time, but in some cases, profits of MGMC are smaller than those of MGRIS. This may be because the number of Monte Carlo simulations is not enough. 10,30 and 50, respectively. To compare running time of different strategies fairly, parallel computing is not used here. We can see that MGRIS is fastest among all of the four algorithms since it only need to generate a set of random RR sets once and choose seeds once. Our SAG policy is the second fastest strategy and faster than AG and MGMC. AG is much faster than MGMC and this may be because the induced subgraph of partial realization becomes smaller and smaller. 2) Results of second group of experiment: Fig. 4 to Fig. 6 present the performance of our SAG policy and other three heuristic adaptive policies on all of the six listed datasets. In this group of experiments, the value of q is set to 3. We can see that the profits obtained by any policies increase with the value of the budget. And profits obtained by our SAG policy are always higher than those obtained by other three heuristic adaptive policies no matter on which one of the six datasets. Among the three heuristic adaptive policies, adaptive max-profit (AMP) policy performs better than AR and AMD policies at most time. This is intuitive since AMP considers the profit not just the degree and a node with large degree may not bring many profits. And our SAG policy usually can obtain about 10% higher profits than AMP policy, which also indicates the effectiveness of our SAG policy. But the results are not so stable and this may be due to the different graph structures and other features of different datasets. VIII. CONCLUSION This work proposes a novel problem, multi-feature budgeted profit maximization problem (MBPM), which asks for a seed set with expected cost no more than the budget to make expected profit as large as possible. We mainly consider the adaptive MBPM problem, where the seeds are selected iteratively and next seed is chosen based on the current diffusion result. We study the adaptive MBPM problem under two models, oracle model and noise model. Specifically, a (1 − 1/e) expected approximation policy is proposed in the oracle model. Under the noise model, we compute conditional expected marginal profit of a node under a partial realization by reverse influence sampling technique and propose an efficient algorithm, which could achieve a (1 − e −(1− ) ) expected approximation ratio, where 0 < < 1. To evaluate the performance of our algorithms, extensive experiments are done on six realistic datasets with the comparison of our proposed policies to their corresponding non-adaptive algorithms and some heuristic adaptive policies. IX. ACKNOWLEDGEMENT This work is supported in part by NSF under grants 1747818 and 1907472. Fig. 1 . 1An example of G = (V, E) with its multi-level graph G. Definition 3 ( 3Multi-feature Budgeted Profit Maximization (MBPM) Problem). Given G = (V, E), q pieces of information about q features of a product are spread on the social network according to the MF-model. The MBPM problem seeks for a seed set S ⊆ V with expected activation cost at most B, i.e., E[c(S)] ≤ B, to maximize the total expected profit P (S). Algorithm 1 1Modified Greedy Algorithm Input: G = (V, E) and a positive number B Output: A seed set S ⊆ V . Theorem 3 . 3Algorithm 1 could achieve a (1 − 1/e) expected approximation guarantee of MBPM problem. The algorithm requires O(n 2 ) function value computation. Definition 6 ( 6Adaptive Monotonicity). A function f (·, φ) is adaptive monotone with respect to distribution p(φ), if for all partial realization ϕ with Pr[Φ ∼ ϕ] > 0 and all u / ∈ dom(ϕ), we have ∆(u\|ϕ) ≥ 0.Definition 7 (Adaptive Submodularity). A function f (·, φ) is adaptive submodular with respect to distribution p(φ), if for all partial realizations ϕ and ϕ satisfying ϕ ⊆ ϕ and for all u / ∈ dom(ϕ ), we have ∆(u\|ϕ) ≥ ∆(u\|ϕ ). Theorem 4 . 4The objective function f (·, φ) is adaptive monotone and adaptive submodular. be the set of infected feature nodes in G when feature nodes of dom(ϕ) ∪ {u} are seeds under the realization φ. Denote T = σ(dom(ϕ) ∪ {u}, φ) and M = σ(dom(ϕ), φ). Let N = T \ M . Similarly, denote T = σ(dom(ϕ ) ∪ {u}, φ ) and M = σ(dom(ϕ ), φ ), and let N = T \ M . = arg max v∈V \S ∆(v\|ϕ)/c(v); 5:if c(S) + c(v max ) Theorem 5 . 5The adaptive greedy policy shown in Algorithm 2 could obtain a (1 − 1/e) expected approximation solution of the AMBPM problem. It requires O(n 2 ) function value computations. 1 :v 1S = ∅; 2: ϕ = {?} E ; 3: G ϕ = G; 4: W = u∈V b(u); 5: W * = min vi∈ V b(v) * ← Modified-EPIC(G ϕ , T, W, W * , n ϕ , ) 10: if c(S) + c(v * ) ( 3 ) 3Adaptive greedy policy (AG): Shown as Algorithm 2 and the conditional expected marginal profit of a node u under any partial realization, ∆(u\|ϕ), is estimated by Monte Carlo method. (4) Sampled adaptive greedy policy (ASG): Shown as Algorithm 3 and set the error parameter = 0.5. Fig. 2 . 2Profit VS budget on Twitter. Fig. 3 . 3Profit VS budget on Wiki. Fig. 4 . 4Profit VS budget on Twitter and Wiki. Fig. 5 . 5Profit VS budget on Hamsterster and DBLP. Fig. 6 . 6Profit VS budget on HepPh and Epinions. TABLE I DATASET ICHARACTERISTICSDataset n m Type Average degree Twitter 0.8k 1k directed 2 Wiki 0.9k 3k directed 6 Hamsterster 2.4k 16.6k undirected 13 DBLP 12.6k 49.7k undirected 7.9 HepPh 12k 118k undirected 19 Epinions 75.9k 508.8k directed 13 Table II presents IIthe running time of our proposed AG and SAG policies with MGMC and MGRIS on Twitter and Wiki datasets under budget TABLE II RUNNING IITIME VS BUDGET ON TWITTER AND WIKITwitter 1 q = 3 q = 5 Algorithm 10 30 50 10 30 50 MGRIS(s) 5.36 20.64 39.57 9.1 31.54 61.3 MGMC(h) 0.7 3.01 5.46 0.97 4.35 7.57 AG(s) 175.23 419.59 667.5 271.48 677.69 1012.19 SAG(s) 42.64 137.58 231.2 59.84 186.75 313.62 Wiki 1 q = 3 q = 5 Algorithm 10 30 50 10 30 50 MGRIS(s) 7.31 25.82 46.75 10.37 39.03 68.52 MGMC(h) 3.56 11.41 19.01 6.67 22.2 36.92 AG(s) 186.81 445.79 690.42 268.85 639.21 970.45 SAG(s) 39.29 143.84 239.65 67.55 190.42 323.28 Threshold models for competitive influence in social networks. A Borodin, Y Filmus, J Oren, SpringerA. Borodin, Y. 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Cambridge university pressR. Motwani and P. Raghavan, Randomized algorithms. Cambridge university press, 1995. She received her B.S. degree in Mathematics and Applied Mathematics, and M.S. degree in Operational Research and Cybernetics from Ocean University of China in 2016 and 2019, respectively. Her research focuses on design and analysis of approximation algorithms and social networks. Department of Computer Science, The University of Texas at DallasTiantian Chen is a Ph.D. candidate in theTiantian Chen is a Ph.D. candidate in the Depart- ment of Computer Science, The University of Texas at Dallas. She received her B.S. degree in Mathemat- ics and Applied Mathematics, and M.S. degree in Operational Research and Cybernetics from Ocean University of China in 2016 and 2019, respectively. Her research focuses on design and analysis of approximation algorithms and social networks.	[]
[ "A GAUCHE PERSPECTIVE ON ROW REDUCED ECHELON FORM AND ITS UNIQUENESS", "A GAUCHE PERSPECTIVE ON ROW REDUCED ECHELON FORM AND ITS UNIQUENESS" ]	[ "Eric Grinberg " ]	[]	[]	Using a left-to-right "sweeping" algorithm, we define the Gauche basis for the column space of a matrix M . By means of the Gauche basis we interpret the row reduced echelon form of M , and give a direct proof of its uniqueness. We conclude with pedagogical reflections.The Remembrance of Row Reduced Echelon Form (RREF)Given a matrix M , viewed as the coefficient portion of a linear system M x = b, we can apply row operations to M , or to the augmented matrix (M \| b), and corresponding equation operations on the system M x = b, to yield a simpler system that is solution-equivalent to the original. These operations include scaling a row by a non-zero scalar, interchanging two rows, and subtracting a scalar multiple of 2010 Mathematics Subject Classification. 15A03, 97H60, 15A06 .	10.1080/00029890.2022.2027717	[ "https://arxiv.org/pdf/2005.06275v1.pdf" ]	218,614,022	2005.06275	31c5b6699ff7afd58f45eec25653828caa800568	A GAUCHE PERSPECTIVE ON ROW REDUCED ECHELON FORM AND ITS UNIQUENESS 12 May 2020 Eric Grinberg A GAUCHE PERSPECTIVE ON ROW REDUCED ECHELON FORM AND ITS UNIQUENESS 12 May 2020 Using a left-to-right "sweeping" algorithm, we define the Gauche basis for the column space of a matrix M . By means of the Gauche basis we interpret the row reduced echelon form of M , and give a direct proof of its uniqueness. We conclude with pedagogical reflections.The Remembrance of Row Reduced Echelon Form (RREF)Given a matrix M , viewed as the coefficient portion of a linear system M x = b, we can apply row operations to M , or to the augmented matrix (M \| b), and corresponding equation operations on the system M x = b, to yield a simpler system that is solution-equivalent to the original. These operations include scaling a row by a non-zero scalar, interchanging two rows, and subtracting a scalar multiple of 2010 Mathematics Subject Classification. 15A03, 97H60, 15A06 . Introduction The Row Reduced Echelon Form of a matrix M , or RREF (M ), is a useful tool when working with linear systems [2,6], and its uniqueness is an important property. A survey of papers and textbooks yields a variety of proofs. Some are simpler [8] and shorter than others. Generally they begin with two candidates for RREF (M ) and conclude that these are equal. It is deemed desirable to have a direct proof, one that simply identifies every atom and molecule of RREF (M ) in terms of properties of M and standard conventions. We use the Gauche basis of the column space of M to give such a proof, taking the opportunity to view RREF from a shifted perspective and offering some reflections on teaching. Conventions and Notations We will work in the vector space R p , consisting of p × 1 column vectors, and sometimes denote these as transposed row vectors, e.g., ( 01...0 ) t . We'll adhere to the ordering conventions of left to right and up to down. Thus the first column of a matrix is the leftmost, and first entry of a column is its top entry. Recall the notation for the "canonical" or "standard" basis of R p : { e j } , where e j stands for the p × 1 column vector ( 0 ...0 1 0...0 ) t with zeros throughout, except for a 1 in the j th entry. Recall also that the span of a set S of vectors in R p is the collection of all linear combinations of these vectors. Thus the span of the singelton set { v} consists of the set of all scalar multiples of v, i.e., a line in R p , unless v = 0, in which case the span of { v} is { 0}. We also have the degenerate case where S is the empty set; by convention, the span of the empty set is { 0}. one row from another row. This last operation is the most commonly used, and is sometimes called a workhorse row operation. Starting with a matrix M and applying carefully chosen row operations, one can obtain a matrix E with, arguably, the "best possible" form among all matrices row equivalent to M . This is the Row Reduced Echelon Form of M , or RREF (M ), or just RREF . We use the definite article the because this form turns out to be be unique, as we'll see. A matrix E is in RREF if it satisfies the follow conditions. • Pivots Sweeping each row of E from the left, the first nonzero scalar encountered, if any, is a 1. We call this entry, along with its column a pivot. • Pivot Column Insecurity In a pivot column, the scalar 1 encountered in the row sweep is the only non-zero entry in its column. • Downright Conventional If a pivot scalar 1 is to the right of another, it is also lower down. • Bottom Zeros Rows consisting entirely of zeros, if they appear, are at the bottom of the matrix. The label Pivot Insecurity requires explanation. We think of the pivot scalar 1s as insecure: they don't want competition from other nonzero entries along their column. Sorry pivots-row insecurity cannot be accommodated. A Gauche Basis for a Matrix with A Fifth Column For the purpose of introduction and illustration we'll begin with a specific matrix: T ≡   2 1 7 −7 2 −3 4 −5 −6 3 1 1 4 −5 2   . We will "sweep" the columns of T from left to right, and designate each column as a keeper or as subordinate. These are meant to be value-neutral, not value judgements, and we hope that no vectors will take offense. For each column we ask Can we present this column as a linear combination of keeper columns to its left? (LLQ) We will call this the Left-Leaning Question, or LLQ for short. Columns for whom the answer is no will be designated as keepers and the rest as subordinates. When focusing on the first column of T , we recall the convention that a linear combination of the empty set is, in the context of a vector space V , the zero vector of V . Thus the LLQ for the first column of T is tantamount to asking: Is this vector non-zero? For T the answer is yes. Therefore, we adorn column one with the adjective keeper. In the aim of responsible accounting, we "journal" our action with the vector J 1 ≡ e 1 . (Recall that in our context e 1 is the 3 × 1 column vector with a 1 in the first entry and zeros elsewhere.) Next, we focus on column two and the LLQ, which, in the current context, asks: Is this column a scalar multiple of column one? The answer is no, so column two is a keeper, and we journal it with J 2 ≡ e 2 . The LLQ for third column asks if this column is a linear combination of the first two (keeper) columns of T ; by inspection, column three is presentable as a linear combination of columns one and two, with scalings 3, 1 respectively. So column three is subordinate and we journal our action with the vector J 3 ≡ 3 e 1 + 1 e 2 , which encodes the manifestation of this vector as a linear combination of keeper columns to its left:   7 −5 4   = 3 ·   2 −3 1   + 1 ·   1 4 1   ; J 3 ≡   3 1 0   . Similarly, the fourth column of T is subordinate, and journaled with J 4 ≡ (−2) · e 1 + (−3) e 2 . The fifth and final column vector of T is not presentable as a linear combo of previous keepers. The reader is invited to prove this or, alternatively, perform a half-turn on the solution box below. (Take a times the first column of T and add it to b times the second column, and look at the top and bottom entries. To produce the fifth column of T , we need 2a + b = 2 and also a + b = 2. This implies that a = 0, and then we run into trouble with the middle entries of our vectors.) We declare the fifth column a keeper (at our peril), and journal it with J 5 ≡ e 3 . Now form a 3 × 5 matrix using the vectors we journaled, in the order we journaled them: J 1 J 2 · · · J 5 , or   1 0 3 −2 0 0 1 1 −3 0 0 0 0 0 1   .(1) This turns out to be the RREF of T , perhaps surprisingly. For an independent verification, using Gauss-Jordan elimination on the same matrix T , see Example SAE in [3]. Notice that our procedure does not show that (1) is row-equivalent to T , whereas the Gauss-Jordan algorithm, e.g., as in [3], does. It is possible, however, to show directly that the Gauche procedure yields a matrix that is row-equivalent to the original. In case anyone insists, we will prove this later on. RREF is unique Lemma. Let M be a matrix and let E a matrix in RREF which is row-equivalent to M . If the pivots of E are in columns j 1 , . . . , j ℓ , then This assertion requires some reflection and interpretation. It is inspired, in part, by a deep principle in the analysis of meromorphic functions [9]. For instance, the statement The k th column of M is in the span of columns j 1 , . . . j ℓ of M . is equivalent to the statement There are scalars α 1 , . . . α ℓ so that α 1 e j 1 + . . . + α 1 e j ℓ − e k is a solution to M x = 0. And ditto for E. Iterating the idea, we can express in this way the statement Columns j 1 , . . . , j ℓ form the Gauche basis of the column space of (·). and others like it. Indeed, in this way, all assertions in the statement of the lemma may be translated into assertions about solutions of the respective homogeneous systems. Hence these are shared values [10] for E and M . Theorem. Let M be a matrix. Then there is one and only one matrix E in RREF which is row equivalent to M . Proof. The lemma above describes every entry of E in terms of conventions and properties of M , without reference to any process for row reducing M to yield E, e.g., Gauss-Jordan elimination. This proves uniqueness. For existence, one can invoke the Gauss-Jordan algorithm, or prove directly that E is row equivalent to M , as is done elsewhere, and independently in this writing. Beyond the Fifth Column: a General Procedure Here we detail the procedure for generating the Gauche basis for an arbitrary matrix and use it to produce the corresponding RREF. For student readers, we suggest following the How To Read Mathematics ideas of John H. Hubbard and Bill Thurston [5]: jump to the illustrative concrete example above, when a point in the general procedure below appears nebulous. Let M be a p × q matrix. We outline a general algorithm that transforms M into row reduced echelon form without invoking row reduction. This manifests, among other things, the uniqueness of the row reduced echelon form. Sweeping the columns of M from left to right, we will adorn some of the columns with the title of keeper. Initially, the set of keepers is empty. Going from left to right, we take a column of M and ask: Is this column a linear combination of the keeper columns to its left? We will call this the Left Leaning Question, or LLQ. For the first column of M this is tantamount to asking Is this column nonzero? If so, we declare it a keeper and journal our action with the vector J 1 ≡ e 1 . If the first column is zero, we do not adorn it with the title of keeper; we call it subordinate and we journal our action with the vector J 1 ≡ 0. In general, we examine the n th column of M and ask the LLQ. If this column is not in the span of the current keeper set, we adorn this column with the keeper designation and journal our action with the vector J ℓ + 1 ≡ e ℓ+1 , where ℓ + 1 is the number of keepers adorned up to this step, current column included. If the current column is presentable as a linear combination of (already designated) keepers, say α 1 k 1 + . . . + α ℓ k ℓ , where the already designated keeper columns are { k i } ℓ i=1 , then we call the current column subordinate and journal our action with the vector J n ≡ α 1 e 1 + . . . + α n e ℓ , recalling that we are focusing on column n and we have ℓ keeper columns already designated. The careful (or fussy, or both) reader may object that the current column may be expressible as a linear combination of keepers in more than one way. However, induction readily shows that at each stage the keeper set is linearly independent. At the end of this procedure we obtain a matrix E of the same size as M . Lemma. The matrix E is in row reduced echelon form. Proof. In this discussion we will sometimes tacitly identify columns of E with corresponding columns of M . We take row i of E and "sweep" it from the left. We encounter a first non-zero entry in only one circumstance: where we meet a pivot of E, i.e., a journaled vector J ℓ corresponding to keeper column of M . In the Gauche algorithm, whenever we introduce a new journal vector J ℓ , corresponding to a pivot, the scalar 1 appears in a lower slot than those of any prior keepers, and prior subordinate columns are linear combinations of prior keepers, so their entries are zero at this level as well. What about the downright condition? If a pivot 1 is to the right of another, it is also lower down, as it gets adorned with the keeper designation at a later stage and is journaled as e k with a larger value of k. When the pivot journals stop, no further nonzero entries are journaled in rows lower than the row of the 1 entry in the last pivot column. Hence, in particular, all pure-zero rows are at the bottom of E. Thus we have verified that E is in RREF. An Existential Question The Gauche procedure takes a matrix M and associates with it a matrix E which is in RREF. But how do we know that there exists a sequence of row operations taking M to E, i.e., why is E row equivalent to M ? We can invoke the Gauss-Jordan elimination algorithm which yields a matrix in RREF that is rowequivalent to M and then cite uniqueness considerations to conclude that our Gauche E must be that matrix. But this is unsatisfying-we should be able to show directly that the Gauche-produced matrix E is row-equivalent to M and, if one insists, we can. Proposition. Given a matrix M , the Gauche-produced RREF matrix E ≡ E(M ) is row equivalent to M . Proof. If M is the zero matrix then E = M and we are done. Otherwise, E has a first pivot column, which corresponds to the first non-zero column of M , say column j 1 . Taking M and permuting rows we obtain a matrix whose first non-zero column is number j 1 , and which has a non-zero entry in the first slot; after scaling the first row we can assume that this entry is 1. Subtracting scalar multiples of the first row from each of the other rows, i.e., employing workhorse row operations, we obtain a matrix whose first non-zero column is the j th 1 , with entries equal to those of e 1 . If E has no other pivot columns, all later columns are scalar multiples of the j st 1 , and we are done. If E has a second pivot column, say in slot j 2 , then this column must have a non-zero entry below the first pivot. Permuting rows other than the first and then applying workhorse-type operations and rescaling the top non-zero entry in this column we obtain e 2 in the j nd 2 slot while retaining e 1 in the first slot. Continuing this way we produce row operations that place appropriate canonical vectors of the form e ℓ in each of the pivot slots. Each of the non-pivot columns is a linear combination of the pivot columns to its left, and requires no additional "processing" by row operations. Thus we have manifested E ≡ E(M ) as the result of a sequence of row operations applied to M . Reflections on Teaching The method of elimination via row reduction may be introduced at the very start of a course on linear algebra. Taking the Gauche approach to echelon form, we are led naturally, directly and concretely to the notions of linear combination, span, and linearindependence. Definition and application are allied and threaded. There is no need for a separate introduction with rationale for use. This brings to mind the introduction, into a course on mathematical proof, of H. Furstenberg's topological proof of the infinitude of primes [4,7]. Every such course covers Euclid's proof of the same. Adding Fursternberg's topological proof leads directly to the basic set operations of intersection, union and complement. Here too, definition and application are allied and threaded. They provide pedagogical advantages. • Columns j 1 , . . . , j ℓ of M form the Gauche basis for the column space of M . • Each non-pivot column c of E is a linear combination of the pivot columns to its left. This combination manifests the presentation of the corresponding column of M as a linear combination of Gauche basis vectors to its left. The "top" entries of c encode this (unique) linear combination, and the rest of the entries of c are "padded" zeros.Note that if E has no pivots at all then M = E = 0, which is consistent with the vacuous interpretation of the statement of the lemma.Proof. It is well known that if M and E are row equivalent then the associated homogeneous linear systems M x = 0 and E x = 0 have the same solutions. At the risk of slightly abusing language we state that Every linear property of the columns of M is also enjoyed by the columns of E, and conversely. We conclude with a question: Is there a book proof (see[1]) of the uniqueness of RREF ?AcknowledgementThe Gauche idea emerged from a conversation with Professor Gilbert Strang in the fall of 2019 at MIT's Endicott House. The author is grateful to Professor Strang for the conversation and for his inspiring writings through the years. He is also grateful to MIT for the invitation to the Endicott House event, and he would happily repeat the experience. . Martin Aigner, ; Günter Ziegler, The Book. 6th Ed., illust Karl H. Hofmann. Springerviii+326 ppMartin Aigner; Günter Ziegler, Proofs from The Book. 6th Ed., illust Karl H. Hofmann. Springer, 2018. viii+326 pp. Extended echelon form and four subspaces. Robert A Beezer, Amer. Math. Monthly. 1213229116Robert A. Beezer, Extended echelon form and four subspaces. Amer. Math. Monthly 121 (2014). MR3229116 A First Course in Linear Algebra. Robert A Beezer, edition 3.50, online opensource edition 3.50 linear.ups.eduRobert A. Beezer, A First Course in Linear Algebra, edition 3.50, online opensource edition 3.50 linear.ups.edu, accessed April 15, 2020. On the infinitude of primes. H Furstenberg, Amer. Math. Monthly. 62H. Furstenberg On the infinitude of primes, Amer. Math. Monthly 62 (1955). John Hubbard, Reading Mathematics In, J H Hubbard, Burke Hubbard, B , Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Prentice Hall2nd edn. chapter 0.1: Reading MathematicsJohn Hubbard, Reading Mathematics in Hubbard, J. H. and Burke Hubbard, B. (2002), Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 2nd edn, Prentice Hall, chapter 0.1: Reading Mathematics. Accessed April, 2020. http://pi.math.cornell.edu/~hubbard/readingmath.pdf . Gilbert Strang, 10.1090/noti1174Not. Am. Math. Soc. 61Gilbert Strang, Not. Am. Math. Soc. 61 (2014) doi:10.1090/noti1174. Multiplying and factoring matrices. Gilbert Strang, Amer. Math. Monthly. 12533768030Gilbert Strang, Multiplying and factoring matrices. Amer. Math. Monthly 125 (2018), no. 3, 223?230 MR3768030 Thomas Yuster The Reduced Row Echelon Form of a Matrix is Unique: A Simple Proof. Math. Mag. 57Thomas Yuster The Reduced Row Echelon Form of a Matrix is Unique: A Simple Proof, Math. Mag. 57 (1984) A Heuristic Principle in Complex Function Theory Amer. Lawrence A Zalcman, Math. Monthly. 828Lawrence A. Zalcman, A Heuristic Principle in Complex Function Theory Amer. Math. Monthly, Vol. 82, No. 8, 1975. Normal families and shared values. Xuecheng Pang, A¿ Lawrence, Zalcman, Bull. London Math. Soc. 321750485Xuecheng Pang and Lawrence A¿ Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 (2000) MR1750485 UMass Boston E-mail address: eric. [email protected] Boston E-mail address: [email protected]	[]
[ "All Vacuum Near-Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries", "All Vacuum Near-Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries" ]	[ "Stefan Hollands \nSchool of Mathematics\nCardiff University Cardiff\nUnited Kingdom\n\nKEK Theory Center\nInstitute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK) Tsukuba\nJapan\n", "Akihiro Ishibashi \nKEK Theory Center\nInstitute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK) Tsukuba\nJapan\n" ]	[ "School of Mathematics\nCardiff University Cardiff\nUnited Kingdom", "KEK Theory Center\nInstitute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK) Tsukuba\nJapan", "KEK Theory Center\nInstitute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK) Tsukuba\nJapan" ]	[]	We explicitly construct all stationary, non-static, extremal near horizon geometries in D dimensions that satisfy the vacuum Einstein equations, and that have D − 3 commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in D = 4, 5. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology S 2 × T D−4 , or S 3 × T D−5 , or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as (D−2)(D−3)/2 continuous parameters. Not all of our metrics in D ≥ 6 seem to arise as the near horizon limits of known black hole solutions. * [email protected]	10.1007/s00023-010-0022-y	[ "https://arxiv.org/pdf/0909.3462v3.pdf" ]	119,198,990	0909.3462	10a0655f286f22edd416787e4e9863bb455f6521	All Vacuum Near-Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries 27 Jan 2010 19 January 2010 Stefan Hollands School of Mathematics Cardiff University Cardiff United Kingdom KEK Theory Center Institute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK) Tsukuba Japan Akihiro Ishibashi KEK Theory Center Institute of Particle and Nuclear Studies High Energy Accelerator Research Organization (KEK) Tsukuba Japan All Vacuum Near-Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries 27 Jan 2010 19 January 2010 We explicitly construct all stationary, non-static, extremal near horizon geometries in D dimensions that satisfy the vacuum Einstein equations, and that have D − 3 commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in D = 4, 5. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology S 2 × T D−4 , or S 3 × T D−5 , or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as (D−2)(D−3)/2 continuous parameters. Not all of our metrics in D ≥ 6 seem to arise as the near horizon limits of known black hole solutions. * [email protected] Introduction Many known families of black hole solutions possess a limit wherein the black hole horizon becomes degenerate, i.e. where the surface gravity tends to zero; such black holes are called extremal. While extremal black holes are not believed to be physically realized as macroscopic objects in nature, they are nevertheless highly interesting from the theoretical viewpoint. Due to the limiting procedure, they are in some sense at the fringe of the space of all black holes, and therefore possess special properties which make them easier to study in various respects. For example, in string theory, the derivation of the Bekenstein-Hawking entropy of black holes from counting microstates (see e.g. [13] for a review) is best understood for extremal black holes. Furthermore, many black hole solutions that have been constructed in the context of supergravity theories (see e.g. [17,18]) have supersymmetries, and are thus automatically extremal. Many of the arguments related to the derivation of the black hole entropy-especially in the context of the "Kerr-CFT correspondence" [19,34,10,3,21,1]-actually only involve the spacetime geometry in the immediate (actually infinitesimal) neighborhood of the black hole horizon. More precisely, by applying a suitable scaling process to the spacetime metric which in effect blows up this neighborhood, one can obtain in the limit a new spacetime metric, called a "near horizon geometry." It is the near horizon geometry which enters many of the arguments pertaining to the derivation of the black hole entropy. The near horizon limit can be defined for any spacetime (M, g) with a degenerate Killing horizon, N-not necessarily a black hole horizon. The construction runs as follows 1 . First, recall that a spacetime with degenerate Killing horizon by definition has a smooth, codimension one, null hypersurface N, and a Killing vector field K whose orbits are tangent N, and which on N are tangent to affinely 2 parametrized null-geodesics. Furthermore, by assumption, there is a "cross section", H, of codimension one in N with the property that each generator of K on N is isomorphic to R and intersects H precisely once. In the vicinity of N, one can then introduce "Gaussian null coordinates" u, v, y a as follows, see e.g. [36]. First, we choose arbitrarily local 3 coordinates y a on H, and we Lie-transport them along the flow of K to other places on N, denoting by v the flow parameter. Then, at each point of N we shoot off affinely parametrized null-geodesics and take u to be the affine parameter along these null geodesics. The tangent vector ∂/∂u to these null geodesics is required to have unit inner product with K = ∂/∂v on H, and to be orthogonal to the Lie-transported cross-section H. It can be shown that the metric then takes the Gaussian null form g = 2dv(du + u 2 α dv + uβ a dy a ) + γ ab dy a dy b , (1.1) where the function α, the one-form β = β a dy a , and the tensor field γ = γ ab dy a dy b do not depend on v. The Killing horizon N is located at u = 0, and the cross section H at u = v = 0. The near horizon limit is now taken by applying to g the diffeomorphism v → v/ǫ, u → ǫu 1 The general definition of a near-horizon limit was first considered in the context of supergravity black holes in [39], and in the context of extremal but not supersymmetric black holes in [12] for the static case and in [30] for the general case. The concept of near-horizon geometry itself has appeared previously in the literature, e.g., [20] for 4-dimensional vacuum case (also see [33] for the isolated horizon case). 2 For a non-degenerate horizon, the orbits on N of K would not be affinely parametrized. 3 Of course, it will take more than one patch to cover H, but the fields γ, β, α on H below in eq. (1.1) are globally defined and independent of the choice of coordinate systems. (leaving the other coordinates y a unchanged), and then taking ǫ → 0. The so-obtained metric looks exactly like eq. (1.1), but with new metric functions obtained from the old ones by evaluating them at u = 0. Thus, the fields α, β, γ of the near horizon metric neither depend on v nor u, and can therefore be viewed as fields on H. If the original spacetime with degenerate Killing horizon satisfied the vacuum Einstein equation or the Einstein equation with a cosmological constant, then the near horizon limit does, too. The near horizon limit is simpler than the original metric in the sense that it has more symmetries. For example, if the limit procedure is applied to the extremal Kerr metric in D = 4 spacetime dimensions with symmetry group R × U(1), then-as observed 4 first by [4] (see also [5,7])-the near horizon metric has an enhanced symmetry group of O(2, 1) ×U(1). The first factor of this group is related to an AdS 2 -factor in the metric. A similar phenomenon occurs for stationary extremal black holes in higher dimensions with a comparable amount of symmetry: As proved in [30], if (M, g) is a D-dimensional stationary extremal black hole with isometry group 5 R × U(1) D−3 and compact horizon cross section H, then the near horizon limit has the enhanced symmetry group O(2, 1) × U(1) D−3 . In D ≥ 5 dimensions, it is not known at present what is the most general stationary extremal black hole solution with symmetry group R×U(1) D−3 , so one can neither perform explicitly their near horizon limits. Nevertheless, because the near horizon metric has an even higher degree of symmetry-the metric functions essentially only depend non-trivially on one coordinate-one can try to classify them directly. This was done for the vacuum Einstein equations in dimensions D = 4, 5 by [29], where a list of all near horizon geometries, i.e. metrics of the form (1.1) with metric functions α, β, γ independent of u, v, was obtained. It is a priori far from obvious that all these metrics are the near horizon limits of actual globally defined black holes. Remarkably though, [29] could prove that the metrics found are indeed the limits of the extremal black ring [14], boosted Kerr string, Myers-Perry [37], and the Kaluza-Klein black holes [38,32], respectively. In this paper, we give a classification of all possible vacuum near horizon geometries with symmetry group O(2, 1) × U(1) D−3 in arbitrary dimensions D. The method of analysis used in [29] seems restricted to D = 4, 5, so we here use a different method based on a matrix formulation of the vacuum Einstein equations that works in arbitrary dimensions. The metrics that we find come in three families depending on the topology of H, which can be either S 3 × T D−5 , S 2 × T D−4 or L(p, q) × T D−5 , where L(p, q) is a Lens space. The metrics in each of these families depend on (D − 2)(D − 3)/2 real parameters; they are given explicitly in Thm. 1 below. When specialized to D = 5, our first two families of metrics 4 By construction, the near horizon geometry has the Killing fields ∂/∂v and u∂/∂u − v∂/∂v, which generate a two-parameter symmetry group. The non-trivial observation by [4] is that this actually gets enhanced to the three-parameter group O(2, 1). 5 The "rigidity theorem" [23] guarantees that a stationary extremal black hole has a symmetry group that contains R × U (1), i.e. guarantees only one axial Killing field in addition to the assumed timelike Killing field. Therefore, in D ≥ 5, assuming a factor of U (1) D−3 is a non-trivial restriction, while it is actually a consequence of the rigidity theorem in D = 4. must coincide with those previously found in [29], whereas the last family is shown to arise from the first one by taking quotients (this last properties generalizes to arbitrary D). In all dimensions, examples for near horizon geometries with topology S 2 × T D−4 are provided by the near horizon limit of the "boosted Kerr-branes" see e.g. [30,15]. This family of metrics depends on (D − 2)(D − 3)/2 real parameters and it is conceivable that all near horizon geometries of this topology can be obtained in this way. The analogous construction is also possible when the horizon topology is S 3 ×T D−5 . However, in this case, the resulting metrics depend on fewer parameters. We should also point out that there are vacuum near-horizon geometries that possess fewer symmetries than R × U(1) D−3 . For example, the near-horizon geometry of the extremal Myers-Perry black holes, constructed explicitly in [15], has the smaller symmetry group, R × U(1) [(D−1)/2] . In this paper we are not going to classify such less symmetric vacuum near-horizon geometries. Also, we are not going to consider the case of a non-vanishing cosmological constant, since, as far as we are aware, there has appeared no successful reduction of the Einstein gravity with a cosmological constant to a suitable nonlinear sigma model, which is however required in our approach. The same remark would apply to other theories with different matter fields. On the other hand, we expect our approach to be applicable to theories that can be reduced to suitable sigma-models. For D = 5 minimal gauged and ungauged supergravity, the near horizon geometries were classified in [31,39] using a method different from ours. Also for D = 4 Einstein-Maxwell theory with a cosmological constant, see e.g. [28]. Geometrical coordinates The aim of this paper is to classify the near horizon geometries in D dimensions. As explained in the previous section, by this we mean the problem of finding all metrics g of the form (1.1) with vanishing Ricci tensor (i.e. vacuum metrics), where γ = γ ab dy a dy b is a smooth metric on the compact manifold H, β = β a dy a is a 1-form on H and α is a scalar function on H. These fields do not depend on u, v, and the near horizon geometries therefore have the Killing vectors K = ∂/∂v and X = u∂/∂u − v∂/∂v. We do not assume a priori that the near horizon metrics arise from a black hole spacetime by the limiting procedure described above. Unfortunately, this problem appears to be difficult to solve in this generality, so we will make a significant further symmetry assumption. Namely, we will assume that our metrics do not only have the Killing vectors K, X, but in addition admit the symmetry group U(1) D−3 , generated by (D − 3) commuting Killing fields ψ 1 , . . . , ψ D−3 that are tangent to H and also commute with K, X. Thus, the full isometry group of our metric is (at least) G 2 × U(1) D−3 , where G 2 denotes the Lie-group that is generated by K, X. This means roughly speaking that the metric functions can nontrivially depend only on a single variable, and our metrics may hence be called "cohomogeneity-one." As a consequence, Einstein's equations reduce to a coupled system of non-linear ordinary differential equations in this variable. Our aim is to solve this system in the most general way and thereby to classify all near horizon geometries with the assumed symmetry. It seems that this system becomes tractable only if certain special coordinates are introduced that are adapted in an optimal way to the geometric situation under consideration. These coordinates are the well-known Weyl-Papapetrou coordinates up to a simple coordinate transformation. However, to introduce these coordinates in a rigorous and careful manner is more subtle in the present case than for non-extremal horizons. These technical difficulties are closely related to the fact that the usual Weyl-Papapetrou coordinates are actually singular on H, the very place we are interested in most. To circumvent this problem, we follow the elegant alternative procedure introduced in [30,29]. That procedure applies in the form presented here to non-static geometries, and we will for the rest of this paper make this assumption. The static case has been treated previously in [12,27]. We first observe that the horizon H is a compact (D − 2)-dimensional manifold with an action of U(1) D−3 . By general and rather straightforward arguments (see e.g. [26,24]) it follows that, topologically, H can only be of the following four types: H ∼ =          S 3 × T D−5 , S 2 × T D−4 , L(p, q) × T D−5 , T D−2 . (2.2) Furthermore, in the first three cases, the quotient space H/U(1) D−3 is a closed intervalwhich we take to be [−1, 1] for definiteness-whereas in the last case, it is S 1 . We will not treat the last case in this paper 6 , but we note that the topological censorship theorem [11] implies that there cannot exist any extremal, asymptotically flat or Kaluza-Klein vacuum black holes with H ∼ = T D−2 . Thus, while there could still be near horizon geometries with H ∼ = T D−2 , they cannot arise as the limit of a globally defined black hole spacetime. In this paper, we will focus on the first three topology types. In these cases, the Gram matrix f ij = γ(ψ i , ψ j ) (2.3) is non-singular in the interior of the interval and it has a one-dimensional null-space at each of the two end points [24]. In fact, there are integers a i ± ∈ Z such that f ij (x)a i ± → 0 at boundary points ±1. (2.4) The integers a i ± determine the topology of H (i.e. which of the first three cases we are in), as we explain more in Thm. 1 below. The first geometric coordinate, x, parametrizes the interval [−1, +1], and is introduced as follows. Consider the 1-form on H defined by Σ = (det f ) ⋆ γ (ψ 1 ∧ · · · ∧ ψ D−3 ), where the Hodge dual is taken with respect to the metric γ on H. Using the fact that the ψ i are commuting Killing fields of γ, one can show that Σ is closed, and that it is Lie-derived by all ψ i . Hence Σ may be viewed as a closed 1-form on the orbit space H/U(1) D−3 , which, as we have said, is a closed interval. It can be seen furthermore that Σ does not vanish anywhere within this closed interval, so there exists a function x, such that dx = CΣ . (2.5) The constant C is chosen so that x runs from −1 to +1. We take x to be our first coordinate, and we take the remaining coordinates on H to be angles ϕ 1 , . . . , ϕ D−3 running between 0 and 2π, chosen in such a way that ψ i = ∂/∂ϕ i . In these coordinates, the metric γ on H takes the form γ = 1 C 2 det f dx 2 + f ij (x)dϕ i dϕ j . (2.6) To define our next coordinate, we consider the 1-form field β on H, see eq. (1.1). Standard results on the Laplace operator ∆ γ on a compact Riemannian manifold (H, γ) guarantee that there exists a smooth function λ on H such that ⋆ γ d⋆ γ β = ∆ γ λ , (2.7) where ⋆ γ is the Hodge star of γ. The function λ is unique up to a constant. Because β and γ are Lie-derived by all the rotational Killing fields ψ i , it follows that L ψ i λ = c i are harmonic functions on H, i.e. constants. Furthermore, these constants must vanish, because the ψ i have periodic orbits. Thus, λ is only a function of x. We also claim that the 1form β − dλ has no dx-part. To see this, we let h be the scalar function on H defined by h = ⋆ γ (ψ 1 ∧ · · · ∧ ψ D−3 ∧ [β − dλ] ). Using eq. (2.7) and the fact that the ψ i are commuting Killing fields of γ, it is easy to show that dh = 0, so h is constant. Furthermore, by eq. (2.4) there exist points in H where the linear combinations a i ± ψ i = 0, and it immediately follows from this that h = 0 on H. This shows that β − dλ has no dx-part, hence we can write β = dλ + Ce λ k i dϕ i , (2.8) where we have introduced the quantities k i := C −1 e −λ ψ i · β . (2.9) The next coordinate is defined by r := ue λ , (2.10) and we keep v as the last remaining coordinate. The coordinates ϕ i , r, x, v are the desired geometrical coordinates. In these, the metric takes the form g = e −λ [2dvdr+r 2 (2αe −λ −e λ k i k i ) dv 2 ]+ dx 2 C 2 det f +f ij (dϕ i +Cr k i dv)(dϕ j +Cr k j dv) . (2.11) We have also determined that the quantities k i , f ij , α, λ are functions of x only. The indices i, j, ... are raised with the inverse f ij of the Gram matrix, e.g. k i = f ij k j . So far, we have only used the symmetries of the metric, but not the fact that it is also required to be Ricci flat. This imposes significant further restrictions [29,30]. Namely, one finds that k i are simply constants, and that (2αe −λ − e λ k i k i ) is a negative 7 constant, which one may choose to be −C 2 after a suitable rescaling of the coordinates r, v and the constants k i , and by adding a constant to λ. Then the Einstein equations further imply that ∂ 2 x (e −λ det f ) = −2; hence e −λ = −(x − x − )(x − x + )(det f ) −1 for real numbers x ± . Furthermore, λ is smooth and det f vanishes only at x = ±1 by eq. (2.4), so x ± = ±1 and consequently e −λ = (1 − x 2 )(det f ) −1 . (2.12) Thus, in summary, we have determined that the near horizon metric is given by g = 1 − x 2 det f (2dvdr − C 2 r 2 dv 2 ) + dx 2 C 2 det f + f ij (dϕ i + rCk i dv)(dϕ j + rCk j dv) (2.13) where k i , C are constants, and where f ij depends only on x. In the remainder of the paper, we will work with above form of the metric (2.13). However, we will, for completeness, also give the relation to the more familiar Weyl-Papapetrou form: For r > 0 (i.e., strictly outside the horizon), we define new coordinates (t, ρ, z, φ i ) by the transformation [16] z := rx (2.14) ρ := r √ 1 − x 2 (2.15) t := Cv + (Cr) −1 (2.16) φ i := ϕ i + C −1 k i log r . (2.17) In the new coordinates (t, ρ, z, φ i ), the metric then takes the Weyl-Papapetrou form g = − ρ 2 dt 2 det f + e −λ C 2 r 2 (dρ 2 + dz 2 ) + f ij (dφ i + rk i dt)(dφ j + rk j dt) , (2.18) where it is understood that r 2 = ρ 2 + z 2 . Note that, by contrast with the coordinate system (v, r, x, ϕ i ), the Weyl-Papapetrou coordinate system does not cover the horizon itself, i.e., it is not defined for r = 0 but only for r > 0. This can be seen in several ways, for example by noting that the coordinate transformation is singular at r = 0, i.e. on the horizon, or alternatively, by noting that the horizon corresponds in the new coordinates to the single point ρ = z = 0. This behavior is characteristic for extremal horizons and does not happen in the non-extremal case. In obtaining our form (2.13) for the near horizon metric, we have used up all but the ij-components of the Einstein equations. The remaining Einstein equations determine the matrix of functions f ij (x). As is well-known [35], a beautifully simple form of these equations can be obtained by introducing the twist potentials of the rotational Killing fields as auxiliary variables. These potentials χ i are defined up to a constant by dχ i = ⋆(ψ 1 ∧ · · · ∧ ψ D−3 ∧ dψ i ) . (2.19) To see that this equation makes sense, one has to prove that the right side is an exact form. Indeed, taking d of the right side and using the vanishing of the Ricci tensor together with the fact that the Killing fields all commute, one gets zero. To see that the right side is even exact, it is best to pass to the orbit space M/(G 2 × U(1) D−3 ) first, which can be identified with the interval [−1, 1]. Then the χ i can be defined on this orbit space and lifted back to functions on M. It also follows from this construction that χ i only depends on the coordinate x parametrizing [−1, 1]. Setting Φ = (det f ) −1 −(det f ) −1 χ i −(det f ) −1 χ i f ij + (det f ) −1 χ i χ j ,(2.20) it is well-known that the vanishing of the Ricci-tensor implies that ∂ x [(1 − x 2 )Φ −1 ∂ x Φ] + ∂ r [r 2 Φ −1 ∂ r Φ] = 0 . (2.21) These equations are normally written in the Weyl-Papapetrou coordinates ρ, z (see e.g. [24]), and the above form is obtained simply by the change of variables eq. (2.14). Since Φ is a function of x only in our situation (but would not be e.g. for black holes without the near horizon limit taken) an essential further simplification occurs: The second term in the above set of matrix equations is simply zero! Hence, the content of the remaining Einstein equations is expressed in the matrix of ordinary differential equations ∂ x [(1 − x 2 )Φ −1 ∂ x Φ] = 0 . (2.22) In fact, this equation could be derived formally and much more directly by simply assuming the Weyl-Papapetrou form of the metric, introducing r, x as above, and then observing that, in the near horizon limit, the dependence on r is scaled away, so that the matrix partial differential equations (2.21) reduce to the ordinary differential equations (2.22). Classification To determine all near horizon metrics (2.13), we must solve the matrix equations (2.22), i.e. find f ij , χ i . Then the constants k i are given by 23) and this determines the full metric up to the choice of the remaining constant C. We must furthermore ensure that, among all such solutions, we pick only those that give rise to a smooth metric g. k i = 1 − x 2 det f f ij ∂ x χ j ,(3. The equations (2.22) for Φ are easily integrated to Φ(x) = Q exp [2 arcth(x) · L] = Q 1 + x 1 − x L . (3.24) Here, Q = Φ(0), L = 1 2 (1 − x 2 )Φ(x) −1 ∂ x Φ(x) are both constant real (D − 2) × (D − 2) matrices, and we mean the matrix exponential etc. It follows from the definition that Φ has the following general properties: It is symmetric, det Φ = 1, and it is positive definite. It is an easy consequence of these properties that det Q = 1, Tr L = 0 (taking the determinant of the equation), that Q = Q T is positive definite, and that L T Q = QL. These relations allow us to write Q = S T S for some real invertible matrix S = (s IJ ) of determinant ±1, and to conclude that SLS −1 is a real symmetric matrix. By changing S to V S, where V is a suitable orthogonal transformation, we can achieve that SLS −1 =      σ 0 0 . . . 0 0 σ 1 . . . 0 . . . . . . 0 0 . . . σ D−3      (3.25) is a real diagonal matrix, while leaving Q unchanged. It then follows that Φ(x) = S T exp [2 arcth(x) · SLS −1 ] S, that is Φ IJ (x) = D−3 K=0 1 + x 1 − x σ K s KI s KJ . (3.26) This is the most general solution to the field equation for Φ in the near horizon limit, and it depends on the real parameters s IJ , σ I , which are subject to the constraints det(s IJ ) = ±1 , D−3 I=0 σ I = 0 . (3.27) The near horizon metric is completely fixed in terms of Φ. It can be obtained combining eqs. (3.26) with eq. (2.20) to determine f ij , χ i , which in turn then fix the remaining constants k i , C in the near horizon metric. In the rest of this section, we explain how this can be done. It turns out that the smoothness of the near horizon metric also implies certain constraints on the parameters σ I , s IJ , and we will derive the form of these. Our analysis applies in principle to all dimensions D ≥ 4. The case D = 4, while being simplest, is somewhat different from the remaining cases D ≥ 5 and would require us to distinguish these cases in many of the formulae below. Therefore, to keep the discussion simple, we will stick to D ≥ 5 in the following. First, we consider the ij-component of Φ in eq. (3.26). By eq. (2.20) this is also equal to D−3 I=0 1 + x 1 − x σ I s Ii s Ij = Φ ij = f ij + (det f ) −1 χ i χ j . (3.28) Now, the coordinate x ∈ [−1, 1] parametrizes the orbit space H/U(1) D−3 of the horizon, which is topologically a finite interval. The boundary points x = ±1 correspond to points on the horizon where an integer linear combination a i ± ψ i of the rotational Killing fields vanishes. This is equivalently expressed by the condition f ij (x)a j ± → 0 as x → ±1. By contrast, for all values of x ∈ (−1, +1), no linear combination of the rotational fields vanishes. Therefore, det f = 0 for x ∈ (−1, +1), while det f → 0 as x → ±1. In fact, using eq. (2.12) one sees that (det f ) −1 = 2c 2 + (1 − x) −1 + 2c 2 − (1 + x) −1 + . . . as x → ±1,(3.29) where the dots represent contributions that go to a finite limit, and where c ± are non-zero constants related to λ by 4c 2 ± = e −λ(±1) = 0. The twist potentials χ i also go to a finite limit as x → ±1. By adding suitable constants to the twist potentials if necessary, we may achieve that χ i → 1 c ± µ i as x → ±1 ,(3.30) where µ i ∈ R are constants. The upshot of this discussion is that, as one approaches the boundary points, the components Φ ij are dominated by the rank-1 part (det f ) −1 χ i χ j , which diverges as 2(1 ∓ x) −1 µ i µ j as x → ±1. This behavior can be used to fix the possible values of the eigenvalues σ I as follows. First, it is clear that at least one of the eigenvalues must be non-zero, for otherwise the right side of eq. (3.28) would be smooth as x → ±1, which we have just argued is not the case. Let us assume without loss of generality then that σ D−3 ≥ · · · ≥ σ D−3−n > 0 are the n positive eigenvalues. Multiplying eq. (3.28) by 1 − x and taking x → +1, we see that σ D−3 = 1, that µ i = s (D−3)i , and that all other remaining positive eigenvalues must be strictly between 0 and 1. If we now subtract (1 − x 2 ) −1 µ i µ j from both sides of the equation, then the right side of eq. (3.28) goes to a finite limit as x → 1, and so the left side has to have that behavior, too. This is only possible if there are no other remaining positive eigenvalues besides σ D−3 . A similar argument then likewise shows that there is only one negative eigenvalue, which has to be equal to −1 (without loss of generality we may take σ D−4 = −1) and that µ i = s (D−4)i . In summary, we have shown that and we also see that σ I =      0 if I ≤ D − 5, −1 if I = D − 4, 1 if I = D − 3,µ i = s (D−3)i = s (D−4)i , c + = s (D−3)0 , c − = s (D−4)0 . (3.32) The condition that det S = ±1 then moreover gives ±1 = (c + − c − ) ǫ ijk...m s 0i s 1j s 2k · · · µ m . (3.33) We may now combine this information with the equations (3.26) and (2.20) and solve for f ij , χ i . The result can be expressed as: f ij ξ i ξ j = 2 1 + x 2 1 − x 2 (µ · ξ) 2 + D−5 I=0 (s I · ξ) 2 (3.34) − e λ(x) 1 − x 2 (1 − x 2 ) D−5 I=0 s I0 (s I · ξ) + [c + (1 + x) 2 + c − (1 − x) 2 ](µ · ξ) 2 χ i ξ i = e λ(x) (1 − x 2 ) D−5 I=0 s I0 (s I · ξ) + [c + (1 + x) 2 + c − (1 − x) 2 ](µ · ξ) . (3.35) Here, we are using shorthand notations such as µ · ξ = µ i ξ i or s I · ξ = s Ii ξ i , and (3.36) in order to have a reasonably compact notation. This function λ agrees with that previously defined in eq. (2.7) by eq. (2.12). From eq. (3.34), one now finds after a short calculation that the conditions (2.4) are equivalent to exp[−λ(x)] = c 2 + (1 + x) 2 + c 2 − (1 − x) 2 + (1 − x 2 ) D−5 I=0 s 2 I0 ,s I0 µ i a i + = c + s Ii a i + , s I0 µ i a i − = c − s Ii a i − , for I = 0, . . . , D − 5. (3.37) Either of these equations "±" can be used to solve for s I0 , because 8 µ i a i ± = 0 for both "±". We will do this in the following. As we have explained, the constants k i in the near horizon metric are given by (3.23). A longer calculation using eqs. k i = 2c + c − c + − c − a i + µ j a j + + a i − µ j a j − . (3.38) 8 Indeed, let us assume that, say µ i a i + = 0. Then, since c + = 0, we know that also s Ii a i + = 0. It then would follow that 0 = ǫ ijk...m s 0i s 1j s 2k · · · µ m , which however is in contradiction with eq. (3.33). To avoid conical singularities in the near horizon metric (2.13), we must furthermore have 9 (3.39) and this determines C. A longer calculation using eqs. (3.34), (3.37) shows that (1 − x 2 ) 2 det f · f ij a i ± a j ± → C 2 as x → ±1,C = 4c 2 + (c + − c − )µ i a i + = 4c 2 − (c + − c − )µ i a i − . (3.40) Thus, we have determined all quantities C, k i , f ij in the near horizon metric (2.13). We substitute these, and make the final coordinate change x = cos θ , 0 ≤ θ ≤ π . (3.41) Then, after performing some algebraic manipulations, we get the following result, which summarizes our entire analysis so far: g = e −λ (2dvdr − C 2 r 2 dv 2 + C −2 dθ 2 ) + e +λ (c + − c − ) 2 (sin 2 θ) Ω 2 +(1 + cos θ) 2 c 2 + I ω I − s I · a + µ · a + Ω 2 + (1 − cos θ) 2 c 2 − I ω I − s I · a − µ · a − Ω 2 + c 2 ± sin 2 θ (µ · a ± ) 2 I<J (s I · a ± )ω J − (s J · a ± )ω I 2 . (3.42) Here, the sums run over I, J from 0, . . . , D − 5, the function λ(θ) is given by exp[−λ(θ)] = c 2 + (1 + cos θ) 2 + c 2 − (1 − cos θ) 2 + c 2 ± sin 2 θ (µ · a ± ) 2 I (s I · a ± ) 2 , (3.43) C is given by C = 4c 2 ± [(c + − c − )(µ · a ± ) ] −1 , and we have defined the 1-forms Ω(r) = µ · dϕ + 4Cr c + c − c + − c − dv (3.44) ω I (r) = s I · dϕ + r 2 C 2 (s I · a + + s I · a − ) dv . (3.45) 9 Here the constants a i ± ∈ Z are normalized so that the greatest common divisor of a i + , i = 1, . . . , D − 3 is equal to 1, and similarly for a i − . We are also using the shorthand notations such as s Ii a i + = s I · a + , or µ · dϕ = µ i dϕ i , etc. The parameters are subject to the constraints µ · a ± = 0 and c 2 + µ · a + = c 2 − µ · a − , c + (s I · a + ) µ · a + = c − (s I · a − ) µ · a − , ±1 = (c + −c − ) ǫ ijk. ..m s 0i s 1j s 2k · · · µ m (3.46) but they are otherwise free. The coordinates ϕ i are 2π-periodic, 0 ≤ θ ≤ π, and v, r are arbitrary. When writing "±", we mean that the formulae hold for both signs. Remarks: (1) The function λ(θ) was invariantly defined in eq. (2.7), and therefore evidently has to be a smooth function. This is manifestly true, because both c ± = 0. Because also µ · a ± are both non-zero, we explicitly see that the above metrics are smooth (in fact analytic). (2) The part 2dvdr − C 2 r 2 dv 2 of the metric is that of AdS 2 with curvature C 2 . This is the cause for the enhanced symmetry group of O(2, 1) × U(1) D−3 . Let us finally discuss the meaning of the parameters on which the near horizon metrics depend. The parameters a i ± ∈ Z are related to the horizon topology. Up to a globally defined coordinate transformation of the form ϕ i → A i j ϕ j mod 2π , A ∈ SL(Z, D − 3) , we have a + = (1, 0, 0, . . . , 0) , a − = (q, p, 0, . . . , 0) , p, q ∈ Z , g.c.d.(p, q) = 1 . (3.47) A general analysis of compact manifolds with a cohomogeneity-1 torus action (see e.g. [24]) implies that the topology of H is H ∼ =      S 3 × T D−5 if p = ±1, q = 0, S 2 × T D−4 if p = 0, q = 1, L(p, q) × T D−5 otherwise. (3.48) The constants µ i , c ± , a i ± are directly related to the horizon area by A H = (2π) D−3 (c + − c − ) 2 (µ · a ± ) 2 8c 4 ± ,(3.49) and we also have J i := 1 2 H ⋆(dψ i ) = (2π) D−3 c + − c − 2c − c + µ i . (3.50) In an asymptotically flat or Kaluza-Klein black hole spacetime with a single horizon H, the above integral for J i could be converted to a convergent integral over a cross section at infinity using Stokes theorem and the vanishing of the Ricci tensor. Then the J i would be equal to the Komar expressions for the angular momentum. The near horizon limits that we consider do not of course satisfy any such asymptotic conditions, and hence this cannot be done. Nevertheless, if the near horizon metric under consideration arises from an asymptotically flat or asymptotically Kaluza-Klein spacetime, then the J i are the angular momenta of that spacetime. Hence, we see that the parameters c ± , µ i , a i ± are directly related to geometrical/topological properties of the metric. This seems to be less clear for the remaining parameters s Ii . The number of continuous parameters on which our metric depend can be counted as follows. First, the matrix s Ii has (D −3)(D −4) independent components, µ i has (D −3) and c ± has 2 components. These parameters are subject to the (D − 2) constrains, eqs. It is instructive to compare this number to the number of parameters of a boosted Kerrbrane. If we start from a direct product of a 4-dimensional extremal Kerr metric with a flat torus T D−4 and apply a boost in an arbitrary direction, then the resulting family of metrics has (D − 2)(D − 3)/2 parameters, and the horizon topology is S 2 × T D−4 . It is plausible that all our metrics in our Thm. 1 for this topology can be obtained by taking the near horizon limit of these boosted Kerr-branes. By contrast, if we start with a direct product of a 5-dimensional extremal Myers-Perry black hole with a flat torus T D−5 , then we similarly get a family of metrics which depends only on (D − 3)(D − 4)/2 + 1 parameters. Therefore in this case, we get metrics depending on fewer parameters than those in Thm. 1. Examples Let us first illustrate our classification in D = 5 spacetime dimensions. According to our general result, the metrics have the discrete parameters a 1 ± , a 2 ± as well as the 6 continuous parameters µ 1 , µ 2 , s 01 , s 02 , c + , c − which are subject to 3 constraints. Thus, the number of free parameters is 3, and we take C [given by eq. (3.40)] as one of them for convenience. We have the following cases to consider, depending on the possible values of the discrete parameters, see eq. (3.47): Topology H ∼ = S 1 × S 2 : This case corresponds to the choice a + = a − = (1, 0). The constraints (3.46) read explicitly c 2 + µ 1 = c 2 − µ 1 , c + s 01 µ 1 = c − s 01 µ 1 , (c + − c − ) µ 1 s 01 µ 2 s 02 = 1 (4.51) in this case. We know that µ 1 cannot vanish, so the first and third equation imply together that c ± = ±B for some non-zero constant B. As a consequence, the second equation then gives s 01 = 0, from which the third equation then gives s 02 = 1/(2c + µ 1 ). Putting all this into our formula (3.42) for the near horizon metric gives g = 2B 2 (1 + cos 2 θ)(2dvdr − C 2 r 2 dv 2 + C −2 dθ 2 ) + C 2 16B 4 (dϕ 2 ) 2 + 8B 2 sin 2 θ C 2 (1 + cos 2 θ) dϕ 1 + A dϕ 2 + C 2 r dv 2 ,(4.52) where we have put A = µ 2 /µ 1 .We can explicitly read off from the metric that the norm of ∂/∂ϕ 1 [i.e., the coefficient of (dϕ 1 ) 2 ] vanishes at θ = 0, π, whereas the norm of ∂/∂ϕ 2 [i.e., the coefficient of (dϕ 2 ) 2 ] never vanishes. This is the characteristic feature of the action of U(1) 2 on S 2 × S 1 . Topology H ∼ = S 3 : In this case, a + = (1, 0), a − = (0, 1). The constraints (3.46) are c 2 + µ 2 = c 2 − µ 1 , c + s 01 µ 2 = c − s 02 µ 1 , (c + − c − ) µ 1 s 01 µ 2 s 02 = 1 . (4.53) The constraints allow us e.g. to express µ 1 , µ 2 , s 01 , s 02 in terms of A := c + , B := c − and C given by eq. (3.40). The result must then be plugged back into the equation for the near horizon metric (3.42). After some calculation, one ends up with the result g = e −λ (2dvdr − C 2 r 2 dv 2 + C −2 dθ 2 ) (4.54) +e +λ 4 C 2 sin 2 θ A 2 dϕ 1 + B 2 dϕ 2 + rABC 2 dv 2 + C 4 2 (1 + cos θ) 2 A −1 dϕ 2 + r(2B) −1 C 2 dv 2 + C 4 2 (1 − cos θ) 2 B −1 dϕ 1 + r(2A) −1 C 2 dv 2 , where exp[−λ(θ)] = A 2 (1 + cos θ) 2 + B 2 (1 − cos θ) 2 + C 2 16AB 2 sin 2 θ . (4.55) The quantity A−B must be non-zero on account of the third constraint. Note that exp λ(θ) = 0 for 0 ≤ θ ≤ π, so we can explicitly read off from the metric that the norm of ∂/∂ϕ 2 [i.e., the coefficient of (dϕ 2 ) 2 ] vanishes at θ = π, whereas the norm of ∂/∂ϕ 1 [i.e., the coefficient of (dϕ 1 ) 2 ] vanishes at θ = 0. This is the characteristic feature of the action of U(1) 2 on the 3-sphere. Topology H ∼ = L(p, q): In this case, a + = (1, 0), a − = (q, p), where p, q ∈ Z and p = 0. The constraints (3.46) are explicitly c 2 + (qµ 1 + pµ 2 ) = c 2 − µ 1 , c + s 01 (qµ 1 + pµ 2 ) = c − (qs 01 + ps 02 ) µ 1 , (c + − c − ) µ 1 s 01 µ 2 s 02 = 1 . (4.56) We choose as the independent parameters A := c + /p, B := c − /p, and C given by eq. (3.40), and solve for the remaining ones using the constraints. The result is plugged back into the equation for the near horizon metric (3.42). After some calculation, one ends up with the result g = e −λ (2dvdr − C 2 r 2 dv 2 + C −2 dθ 2 ) (4.57) +p 2 e +λ 4p C 2 sin 2 θ A 2 (1/p)dϕ 1 + B 2 (dϕ 2 − (q/p)dϕ 1 ) + rABC 2 dv 2 + C 4p 2 (1 + cos θ) 2 A −1 (dϕ 2 − (q/p)dϕ 1 ) + r(2B) −1 C 2 dv 2 + C 4p 2 (1 − cos θ) 2 (pB) −1 dϕ 1 + r(2A) −1 C 2 dv 2 , where exp[−λ(θ)] = p 2 A 2 (1 + cos θ) 2 + B 2 (1 − cos θ) 2 + C 2 16p 2 AB 2 sin 2 θ . (4.58) We note that at θ = π, the Killing field ∂/∂ϕ 1 has vanishing norm, while at θ = 0, the Killing field q∂/∂ϕ 1 + p∂/∂ϕ 2 has vanishing norm. This is the characteristic feature of the action of U(1) 2 on the Lens space L(p, q). The metrics with H ∼ = L(p, q) just described are closely related to those in the case H ∼ = S 3 described in the previous example. Indeed, in the case H ∼ = S 3 , consider the map given by (ϕ 1 , ϕ 2 ) → (ϕ 1 + 2π/p, ϕ 2 + 2πq/p), leaving invariant the other coordinates, where ϕ 1 , ϕ 2 are 2π-periodic. This map is an isometry of the metric with H ∼ = S 3 , and by repeated application generates the subgroup Z p of the full isometry group. If we factor by this group, then we get a metric with H ∼ = L(p, q), and we claim that this metric is exactly the one just given. To see this more explicitly, we note that factoring by the above group Z p of isometries in effect imposes the further identifications (ϕ 1 , ϕ 2 ) ∼ = (ϕ 1 + 2π/p, ϕ 2 + 2πq/p) (4.59) on the angular coordinates in the metric (4.54), which were initially 2π-periodic. If we let f : (r, v, θ, ϕ 1 , ϕ 2 ) → (r, p 2 v, θ, (1/p)ϕ 1 , ϕ 2 − (q/p)ϕ 1 ) (4.60) then f provides an invertible mapping from the ordinary 2π-periodic coordinates to the coordinates with the identifications (4.59). If we now take the metric (4.54) in the case H ∼ = S 3 , factor it by Z p , pull it back by f , and furthermore put C → C/p, then we get precisely the H ∼ = L(p, q) metrics (4.57). Thus, all metrics in the case H ∼ = L(p, q) arise from the case H ∼ = S 3 by taking quotients. The same statement (with similar proof) is true in all dimensions D. Let us finally briefly discuss an example of our classification in D = 6 dimensions. In this case, the metrics are classified by the discrete parameters a ± [see eq. (3.47)] and 7 real continuous parameters. An example is Topology S 3 × S 1 : In this case, a + = (1, 0, 0), a − = (0, 1, 0). The constraints are explicitly (4.61) To simplify the formulae somewhat, we consider the special case that c + = −c − =: A/2. Then the constraints may be solved easily for the remaining parameters. To obtain a halfway simple expression, we also consider the special case s 11 = s 03 = 0, and we denote the remaining free parameters as B := s 01 , D = µ 3 , and C as usual. The resulting metric is still rather complicated and is given by c + s 01 µ 2 = c − s 02 µ 1 c + s 11 µ 2 = c − s 12 µ 1 c 2 + µ 2 = c 2 − µ 1 , (c + − c − )g = e −λ(θ) 2dvdr − C 2 r 2 dv 2 + C −2 dθ 2 +e +λ(θ) A 4 C −2 sin 2 θ dϕ 1 + dϕ 2 + A −1 CD dϕ 3 − rC 2 dv 2 + A 2 B 2 4 (1 + cos θ) 2 (2dϕ 2 + A −1 CD dϕ 3 − rC 2 dv) 2 This special family of metrics depends on only 4 parameters. It is easy to write down the general 7 parameter family of metrics. + A 2 B 2 4 (1 − cos θ) 2 (2dϕ 1 + A −1 CD dϕ 3 − rC 2 dv) 2 + C 2 4A 4 B 2 (dϕ 3 ) 2 . Conclusion We have determined explicitly what are the possible (non-static) stationary smooth, cohomogeneity-one near horizon geometries satisfying the vacuum Einstein equations. We excluded by hand 10 the case that the horizon topology is T D−2 . The solution, described in thm. 1, is given in closed form in terms of real and discrete parameters (corresponding to the possible topology types other than T D−2 ), which are subject to certain constraints that take the form of algebraic equations. After taking into account these constraints, the metrics depend on (D − 2)(D − 3)/2 independent real parameters, and two discrete ones. For example, in D = 5, we initially have 3 real continuous parameters. We have worked out explicitly this case as did [29], but our metrics are presented in different coordinates 11 for the case H ∼ = S 3 . In D ≥ 6, not all of our metrics can be obtained as the near horizon limit of a known black hole solution, so in this sense some of our metrics are new for D ≥ 6. By contrast to D ≤ 5, not all near horizon metrics that we have found can be obtained as the near horizon limits of known black hole solutions in dimensions D ≥ 6. It is conceivable that there are further extremal black hole solutions-to be found-which give our metrics in the near horizon limit, but it is also possible that some of our metrics in D ≥ 6 simply do not arise in this way. Our method as described only works for vacuum solutions. However, we expect that it can be generalized to any theory whose equations can be recast into equations of the sigma-model type that we encounter. Thus we expect our method to be applicable e.g. to 5-dimensional minimal supergravity, see e.g. [6,9,8,40]. By contrast, our method does not seem applicable straightforwardly to the case of a cosmological constant. In our proof, we also assumed that the metrics are not static. All static near horizon geometries were found in [27] in D = 5 and in [12] in arbitrary dimensions. It would be interesting to see whether our classification can be used to prove a black hole uniqueness theorem in arbitrary dimensions for extremal black holes along the lines of [2,16], thereby generalizing [24,25]. It would also be interesting to investigate whether our analysis can be used to obtain new structural insights into the origin of the Bekenstein-Hawking entropy, e.g. by considering a suitably quantized version of eq. (2.22). formula (4.62). This work is supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. Note added in proof: In our analysis, we excluded by hand the horizon topology T D−2 . There cannot exist any asymptotically flat or Kaluza-Klein black hole solutions with this topology by general arguments [11,24]. At any rate, these could not arise as the near horizon limits of a black hole. After we finished this work, it was confirmed by J. Holland that there cannot be any non-static cohomogeneity-one near horizon geometries with topology H ∼ = T D−2 [22]. Hence our main theorem 1 covers all possibilities with D − 3 commuting rotational symmetries. The static case is covered by the results of [12]. (3.34), (3.37), (3.33) and (3.36) reveals that Theorem 1 . 1All non-static near horizon metrics (except topology type H ∼ = T D−2 ) are parametrized by the real parameters c ± , µ i , s Ii , and the integers a i ± where I = 0, . . . , D − 5 and i = 1, . . . , D − 3, and g.c.d.(a i ± ) = 1. The explicit form of the near horizon metric in terms of these parameters is (3.46). However, changing s Ii to D−5 J=0 R J I s Ji , with R J I an orthogonal matrix in O(D −4), does not change the metric. Since such a matrix depends on (D −4)(D −5)/2 parameters, our metrics depend only on (D − 3)(D − 4) + (D − 3) + 2 − (D − 2) − (D − 4)(D − 5)/2 = (D − 2)(D − 3)/2 real continuous parameters. µ 1 s 101 s 11 µ 2 s 02 s 12 µ 3 s 03 s 13 = 1 . See, however, the note added in proof. Here one must use that the metric is not static, i.e. that not all k i vanish. See, however, the note added in proof.11 We also do not distinguish between the subcases "A" and "B" as in[29] but instead give a unified expression for the metric. Acknowledgements: S.H. would like to thank the Centro de Ciencias de Benasque Pedro Pascual for its hospitality during the inspiring programme on "Gravity -New perspectives from strings and higher dimensions", where a key part of this work was done. He would also like to thank P. Figueras, H. Kunduri and especially J. Lucietti for numerous useful discussions. We especially would like to thank the unknown referee for pointing out an error in the counting of parameters of our solutions and for suggesting a simplification of No Dynamics in the Extremal Kerr Throat. A J Amsel, G T Horowitz, D Marolf, M M Roberts, arXiv:0906.2376hep-thAmsel, A. J., Horowitz, G. T., Marolf, D., and Roberts, M. 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[ "Sparticle Spectroscopy from SO(10) GUT with a Unified Higgs Sector", "Sparticle Spectroscopy from SO(10) GUT with a Unified Higgs Sector" ]	[ "M Adeel Ajaib \nDepartment of Physics and Astronomy\nBartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA\n\nAndronikashvili Institute of Physics\n0177TbilisiGeorgia\n", "Ilia Gogoladze \nDepartment of Physics and Astronomy\nBartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA\n\nAndronikashvili Institute of Physics\n0177TbilisiGeorgia\n", "Qaisar Shafi \nDepartment of Physics and Astronomy\nBartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA\n\nAndronikashvili Institute of Physics\n0177TbilisiGeorgia\n" ]	[ "Department of Physics and Astronomy\nBartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA", "Andronikashvili Institute of Physics\n0177TbilisiGeorgia", "Department of Physics and Astronomy\nBartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA", "Andronikashvili Institute of Physics\n0177TbilisiGeorgia", "Department of Physics and Astronomy\nBartol Research Institute\nUniversity of Delaware\n19716NewarkDEUSA", "Andronikashvili Institute of Physics\n0177TbilisiGeorgia" ]	[]	We study the low energy implications, especially the particle spectroscopy, of SO(10) grand unification in which the SO(10) symmetry is broken to the Standard Model gauge group with a single pair of (144 + 144) dimensional Higgs multiplet (unified Higgs sector). In this class of models, the asymptotic relation Y b ≈ Y τ ≈ Y t /6 among the third generation quark and lepton Yukawa couplings can be derived. This relation leads to the prediction tan β ≈ 14, where tan β is the well known MSSM parameter. We find that this type of Yukawa coupling unification (YU) is realized only by employing non-universal soft supersymmetry breaking terms, dictated by SO(10) symmetry, for the gauginos. A 125 GeV Higgs boson mass is also found to be consistent with YU at the ∼ 5% level. Without imposing a constraint on the relic abundance of dark matter in these models, the squark and slepton masses, with the exception of the stop, exceed 2 TeV and the gluino is heavier than 1 TeV. We show that the neutralino in this model is an acceptable dark matter candidate through the neutralino-stop coannihilation scenario, with the stop quark being relatively light ( 500 GeV).	10.1103/physrevd.88.095019	[ "https://arxiv.org/pdf/1307.4882v1.pdf" ]	118,597,292	1307.4882	d848011b24b7dac6e8cc858393b4a20963af6ce7	Sparticle Spectroscopy from SO(10) GUT with a Unified Higgs Sector 18 Jul 2013 M Adeel Ajaib Department of Physics and Astronomy Bartol Research Institute University of Delaware 19716NewarkDEUSA Andronikashvili Institute of Physics 0177TbilisiGeorgia Ilia Gogoladze Department of Physics and Astronomy Bartol Research Institute University of Delaware 19716NewarkDEUSA Andronikashvili Institute of Physics 0177TbilisiGeorgia Qaisar Shafi Department of Physics and Astronomy Bartol Research Institute University of Delaware 19716NewarkDEUSA Andronikashvili Institute of Physics 0177TbilisiGeorgia Sparticle Spectroscopy from SO(10) GUT with a Unified Higgs Sector 18 Jul 2013On leave of absence from: 1 We study the low energy implications, especially the particle spectroscopy, of SO(10) grand unification in which the SO(10) symmetry is broken to the Standard Model gauge group with a single pair of (144 + 144) dimensional Higgs multiplet (unified Higgs sector). In this class of models, the asymptotic relation Y b ≈ Y τ ≈ Y t /6 among the third generation quark and lepton Yukawa couplings can be derived. This relation leads to the prediction tan β ≈ 14, where tan β is the well known MSSM parameter. We find that this type of Yukawa coupling unification (YU) is realized only by employing non-universal soft supersymmetry breaking terms, dictated by SO(10) symmetry, for the gauginos. A 125 GeV Higgs boson mass is also found to be consistent with YU at the ∼ 5% level. Without imposing a constraint on the relic abundance of dark matter in these models, the squark and slepton masses, with the exception of the stop, exceed 2 TeV and the gluino is heavier than 1 TeV. We show that the neutralino in this model is an acceptable dark matter candidate through the neutralino-stop coannihilation scenario, with the stop quark being relatively light ( 500 GeV). Introduction An SO (10) gauge symmetry provides an elegant framework for unifying the strong and electroweak interactions. A single generation of quarks and leptons including a right handed neutrino, nicely fits in an irreducible 16 dimensional representation [1]. The right handed neutrino (ν R ) helps generate the observed light neutrino masses via the see-saw mechanism [2]. It can also naturally account for the observed baryon asymmetry of the universe via leptogenesis [3]. Another virtue of SO (10) is that, in principle, the two MSSM Higgs doublets can be accommodated in a single ten dimensional representation (10 H ), which then yields the following Yukawa couplings Y ij 16 i 16 j 10 H .(1) Here i, j = 1, 2, 3 stand for family indices and the SO (10) indices have been omitted for simplicity. Considering only the third generation quarks and leptons, the interaction in Eq.(1) yields the following Yukawa coupling unification (YU) condition at M GU T : Y t = Y b = Y τ = Y ντ .(2) where Y ντ denotes the tau neutrino Dirac coupling. Consequently, large tan β ∼ 50 is predicted [4] in order to get compatibility with experimental observations. It is interesting to note that in the gravity mediation SUSY breaking scenario [5], t-b-τ YU condition leads to LHC testable sparticle spectrum [6] and it even 'predicts' a 125 GeV light CP-even Higgs boson mass [7]. One potential drawback of SO (10) grand unification is the lack of a unique minimal model due to the various possibilities available in the Higgs sector for breaking SO (10) to SU (3) C × U (1) em . Typically, one needs a 16 + 16 or a 126 + 126 Higgs representation to reduce the rank of the group from five to four together with either a 45 or 210-dimensional representation for breaking the symmetry down to SU (3) C × SU (2) L × U (1) Y . Furthermore, one needs a 10-dimensional Higgs multiplet to break the electroweak symmetry. These requirements imply that in principle, two distinct superheavy mass scales are involved in the breaking of SO(10), one associated with the reduction of the rank, and the other for breaking the symmetry all the way down to the Standard Model (SM). In order to maintain gauge and Yukawa coupling unification, one needs to assume that the various vacuum expectation values (VEVs) are of the same order of magnitude. This requires suitable relations among the parameters in the superpotential which may not appear very natural. Recently a new class of SO(10) models was presented [8,9] where the SO(10) symmetry breaking down to the SM gauge group involves just a single pair of (144 + 144)-dimensional vector-spinor Higgs multiplet. It was also shown that this pair of multiplets can contain a pair of light Higgs doublets, necessary for breaking the electroweak symmetry. In order to solve the doublet-triplet splitting problem in this class of models, a (560+560)-dimensional vector-spinor representation was introduced instead of 144 + 144 [10]. In this case the doublet-triplet splitting problem was solved via the missing partner mechanism. In this paper we study the low energy spectrum of supersymmetric SO(10) GUT with 144+144 dimensional Higgs multiplet. The Yukawa coupling for third generation quarks and leptons is given by Y i 16 i 16 i 144 H 144 H M * ,(3) where M * is a super heavy mass. It was shown in ref. [8] that there corresponds a parameter space in this class of model where the following relation among third generation quark and lepton Yukawa couplings is obtained: Y b ≈ Y τ ≈ Y t 6 .(4) In order to be compatible with observations, the theory predicts an intermediate value ∼ 10 for tan β. In this paper we seek the low scale sparticle spectrum which is consistent with the asymptotic relation presented in Eq. (4). We find that this requires non-universal gaugino masses at M GU T which, as previously discussed, can be incorporated in the SO(10) framework. The outline for the rest of the paper is as follows. In Section 2 we present the parameter space that we randomly scan and describe how the MSSM gaugino mass relations can be obtained at M GUT . In Section 3 we summarize the scanning procedure and the experimental constraints applied in our analysis. The results are presented in Section 4. The table in this section lists some benchmark points which can be tested at the LHC. Our conclusions are summarized in Section 5. Fundamental Parameter Space We first comment on our results when the asymptotic relation among the Yukawa couplings presented in Eq.(4) is applied assuming universal gaugino masses at M GU T . We find that in this case the Yukawa coupling unification is not better than the 30% level, regardless of whether universal or non-universal Higgs soft supersymmetry (SUSY) breaking mass terms are imposed. Based on the experience (see for instance ref. [7]) that non-universal gaugino masses help achieve conventional Yukawa unification (Y t = Y b = Y τ ), we employ the same non-universal gaugino mass condition in this analysis as well. We will show in section 4 that non-universal gaugino masses also leads to unification of Yukawa couplings according to Eq.(4). It has been pointed out [11] that non-universal MSSM gaugino masses at M GUT can arise from non-singlet SUSY breaking F-terms, compatible with the underlying GUT symmetry. The SSB gaugino masses in supergravity [5] can arise from the following dimension five operator: − F ab 2M P λ a λ b + c.c.(5) Here λ a is the two-component gaugino field, F ab denotes the F-component of the field which breaks SUSY, and the indices a, b run over the adjoint representation of the gauge group. The resulting gaugino mass matrix is F ab /M P , where the SUSY breaking parameter F ab transforms as a singlet under the MSSM gauge group SU (3) c × SU (2) L × U (1) Y . The F ab fields belong to an irreducible representation in the symmetric part of the direct product of the adjoint representation of the unified group. In SO (10), for example, (45 × 45) S = 1 + 54 + 210 + 770. If F transforms as a 54 or 210 dimensional representation of SO(10) [11], one obtains the following relation among the MSSM gaugino masses at M GUT : M 3 : M 2 : M 1 = 2 : −3 : −1,(7) where M 1 , M 2 , M 3 denote the gaugino masses of U (1), SU (2) L and SU (3) c respectively. Notice that in general, if F ab transforms non trivially under SO (10), the SSB terms such as the trilinear couplings and scalar mass terms are not necessarily universal at M GU T . However, we can assume, consistent with SO(10) gauge symmetry, that the coefficients associated with terms that violate the SO(10)-invariant form are suitably small, except for the gaugino terms in Eq. (7). We also assume that D-term contributions to the SSB terms are much smaller compared with contributions from fields with non-zero auxiliary F-terms. Employing the boundary condition from Eq. (7), we define the MSSM gaugino masses at M GUT in terms of the mass parameter M 1/2 : M 1 = M 1/2 , M 2 = 3M 1/2 and M 3 = −2M 1/2 .(8) We have performed random scans for the following parameter range: 0 ≤ m 16 ≤ 20 TeV, 0 ≤ m 144 ≤ 20 TeV, 0 ≤ M 1/2 ≤ 5 TeV, − 3 ≤ A 0 /m 16 ≤ 3, 2 ≤ tan β ≤ 60.(9) Here m 16 is the universal SSB mass for MSSM sfermions, m 144 is the universal SSB mass term for up and down MSSM Higgs masses, M 1/2 is the gaugino mass parameter, tan β is the ratio of the vacuum expectation values (VEVs) of the two MSSM Higgs doublets, and A 0 is the universal SSB trilinear scalar interaction (with corresponding Yukawa coupling factored out). We use the central value m t = 173.1 GeV and 1σ deviation (m t = 174.2 GeV) for top quark in our analysis [17]. A +1σ increase in m t slightly raises the Higgs mass which is desirable in our analysis. Our results however are not too sensitive to one or two sigma variation in the value of m t [18]. We use m b (m Z ) = 2.83 GeV which is hard-coded into Isajet. Phenomenological Constraints and Scanning Procedure We employ the ISAJET 7.84 package [12] to perform random scans over the fundamental parameter space. In this package, the weak scale values of gauge and third generation Yukawa couplings are evolved to M GUT via the MSSM renormalization group equations (RGEs) in the DR regularization scheme. We do not strictly enforce the unification condition g 3 = g 1 = g 2 at M GUT , since a few percent deviation from unification can be assigned to unknown GUT-scale threshold corrections [13]. The deviation between g 1 = g 2 and g 3 at M GUT is no worse than 3 − 4%. For simplicity we do not include the Dirac neutrino Yukawa coupling in the RGEs, whose contribution is expected to be small. The various boundary conditions are imposed at M GUT and all the SSB parameters, along with the gauge and Yukawa couplings, are evolved back to the weak scale M Z . In the evaluation of Yukawa couplings the SUSY threshold corrections [14] are taken into account at the common scale M SUSY = √ mt L mt R , where mt L and mt R are the third generation left and right handed stop quark masses. The entire parameter set is iteratively run between M Z and M GUT using the full 2-loop RGEs until a stable solution is obtained. To better account for leading-log corrections, one-loop stepbeta functions are adopted for gauge and Yukawa couplings, and the SSB parameters m i are extracted from RGEs at their appropriate scales m i = m i (m i ). The RGEimproved 1-loop effective potential is minimized at M SUSY , which effectively accounts for the leading 2-loop corrections. Full 1-loop radiative corrections are incorporated for all sparticle masses. An important constraint comes from limits on the cosmological abundance of stable charged particles [15]. This excludes regions in the parameter space where charged SUSY particles become the lightest supersymmetric particle (LSP). We accept only those solutions for which one of the neutralinos is the LSP and saturates the WMAP bound on relic dark matter abundance. An approximate error of around 2 GeV in the estimate of the Higgs mass in Isajet largely arises from theoretical uncertainties in the calculation of the minimum of the scalar potential, and to a lesser extent from experimental uncertainties in the values for m t and α s . Micromegas 2.4 [16] is interfaced with Isajet to calculate the relic density and branching ratios BR(B s → µ + µ − ) and BR(b → sγ). We implement the following random scanning procedure: A uniform and logarithmic distribution of random points is first generated in the parameter space given in Eq. (9). The function RNORMX [21] is then employed to generate a Gaussian distribution around each point in the parameter space. The data points collected all satisfy the requirement of radiative electroweak symmetry breaking (REWSB), with the neutralino in each case being the LSP. After collecting the data, we impose the mass bounds on all the particles [15] and use the IsaTools package [20] to implement the various phenomenological constraints. We successively apply the following experimental constraints on the data that we acquire from SuSpect and Isajet: 0.8 × 10 −9 ≤ BR(B s → µ + µ − ) ≤ 6.2 × 10 −9 (2σ) [22] 2.99 × 10 −4 ≤ BR(b → sγ) ≤ 3.87 × 10 −4 (2σ) [23] 0.15 ≤ BR(Bu→τ ντ ) MSSM BR(Bu→τ ντ ) SM ≤ 2.41 (3σ) [23] 0 ≤ ∆(g − 2) µ /2 ≤ 55.6 × 10 −10 [24] In order to quantify Yukawa coupling unification, we define the quantity R tbτ as, R tbτ = max(y t /6, y b , y τ ) min(y t /6, y b , y τ ) . 4 Sparticle Spectroscopy and the Higgs mass Green points form a subset of the gray and satisfy sparticle mass and B-physics constraints described in section 3. The green points also satisfy the Higgs mass bound 123 GeV ≤ m h ≤ 127 GeV. Brown points form a subset of the green points and satisfy Ωh 2 ≤ 10. We chose to concentrate on Ωh 2 ≤ 10 because in this model neutralino is mostly a bino like particle and it is heavier than a 100 GeV. In this case, without any additional contribution, Ωh 2 can be around O(10 2 ) or even O(10 3 ) [25]. So, Ωh 2 ≤ 10 already indicates that there is some additional mechanism which significantly reduces the relic abundance close to the desired value with some fine tuning in the SSB parameter space. Figure 1: Plots in the R tbτ −M 1/2 , R tbτ −m 144 , R tbτ −m 16 and R tbτ −tan β planes. The data points shown are collected using Isajet 7.84. Gray points are consistent with REWSB and LSP neutralino. Green points form a subset of the gray and satisfy sparticle mass and B-physics constraints. The green points also satisfy the Higgs mass bound 123 GeV ≤ m h ≤ 127 GeV. Moreover, we require that the green points do no worse than the SM in terms of (g − 2) µ . Brown points form a subset of the green points and satisfy Ωh 2 ≤ 10. Figure 1. In addition blue points are subset of the green and satisfy R tbτ < 1.2. Brown points form a subset of the blue points and satisfy Ωh 2 ≤ 10. Figure 1 shows that our analysis does not yield better than ∼ 5% YU consistent with the constraints described in section 3. The prediction for YU essentially remains the same if we require the relic density to be small, Ωh 2 ≤ 10. We also observe that requiring good YU leads to narrow ranges of the fundamental parameters in the model. The gaugino mass parameter (M 1/2 ) consistent with good YU is ∼ 200 GeV whereas the Higgs mass parameter (m 144 ) lies in the range 2 − 4 TeV. Similarly, good YU prefers the GUT scale scalar mass parameter m 16 ∼ 2 TeV and tan β ∼ 14. Note that this value for tan β is notably different from the prediction tan β ∼ 47 in refs. [7], which studied the same model but with the condition Y t = Y b = Y τ at M GU T . Note also that requiring the neutralino relic abundance of the neutralino to satisfy Ωh 2 ≤ 10 affects the above mentioned predictions. While the preferred value for tan β essentially remains the same, the smallest values of the parameters M 1/2 , m 144 and m 16 consistent with YU ∼ 5% are pushed to higher values, namely, M 1/2 ∼ 300 GeV, Figure 3: Plots in the R tbτ − m h and R tbτ − mχ± planes. The color coding is the same as in Figure 1. m 144 ∼ 1.6 TeV and m 16 ∼ 2.8 TeV. Figure 2 shows plots in the mq − mg, mτ 1 − M χ 0 1 , m A − M χ 0 1 and mt 1 − M χ 0 1 planes. The green points have the same definition as in Figure 1. The blue points are a subset of the green and satisfy R tbτ < 1.2. Brown points form a subset of the blue points and satisfy Ωh 2 ≤ 10. The mq − mg plane shows that 20% or better YU predicts the first and second generation squark masses to be 2 TeV. The mt 1 − M χ 0 1 shows that neutralino-stop coannihilation is consistent with good YU. We can see from the mq − mg plane that for Ωh 2 ≤ 10 (brown points) the first two generation squarks have masses 3 TeV and gluinos are heavier than 1.5 TeV or so. We can conclude, based on the location of blue points in the mτ 1 − M χ 0 1 plane, that neutralino-stau coannihilation is impossible to realize in this model. From the m A − M χ 0 1 plot, we observe that good YU is not consistent with the M A resonance solution for neutralino dark matter. In the mt 1 − M χ 0 1 plane, we observe that 20% or better YU can be achieved with the neutralino-stop coannihilation scenario. This is a prediction of this model if we require neutralino to be the sole dark matter candidate. A lower limit on the mass of the NLSP stop quark was obtained in refs. [26] in light of 7 TeV LHC data corresponding to 1 fb −1 integrated luminosity. It was shown that the NLSP stop mass below 140 GeV is essentially excluded. Note that the analysis in refs. [7] employ similar GUT scale boundary conditions for the SSB terms but with a different relation for the Yukawa couplings. The model in [7] predicts neutralino-stau coannihilation scenario to be consistent with good YU. It also predicts relatively low values of m A and mτ 1 , while the stop and the gluino masses are 5 TeV. Figure 3 shows the results in the R tbτ − m h and R tbτ − mχ± planes. coding is the same as in Figure 1. We observe that the model accommodates the Higgs mass range, 123 GeV ≤ m h ≤ 127 GeV, while exhibiting good (∼ 5%) YU. Requiring Ωh 2 ≤ 10, the Higgs mass can still be within the favorable range with YU still at the ∼ 5% level. The R tbτ − mχ± plane indicates that the chargino, which in this model is mostly a wino like particle, can be as light as 500 GeV. Imposing the relic abundance bound implies that the lightest chargino mass is ∼ 1 TeV. We therefore conclude that compatibility of YU with neutralino dark matter scenario in this model predicts that only the stop quark will be accessible at the LHC. If we assume that neutralino is not the dark matter candidate we can see from Figure 2 that the gluino can be around 1 TeV, while the squarks can lie around 2 TeV or so. In Table 1 we present three characteristic benchmark points which summarize the salient features of this model. The three points satisfy 123 GeV ≤ m h ≤ 127 GeV as well as the sparticle mass and B-physics constraints described in section 3. The mass of the gluino decreases from 4 TeV (point 1) to 1.5 TeV (point 3). For point 1, YU is at the level of 11% and the neutralino relic abundance satisfies the 5σ WMAP bound. For point 2, the neutralino relic abundance is relatively large but YU is at the few percent level. Point 3 shows acceptable YU with a lighter gluino and stop compared to point 2. The stop is the NLSP for the three points with the lightest being mt 1 = 488 GeV for the first point. Note that in some SO (10) GUT models with a unified Higgs sector, it is possible to have relations among Yukawa couplings [8,9] different from what we employed in Eq. (4). For instance, in ref. [8] it is discussed that there is parameter space available consistent with the relation: Y t /48 = Y b = Y τ ,(11) which predicts tan β ≈ 2. We find that in this case the solutions yield YU no better than the 70% level, which therefore indicates there is no YU at all. A different scenario [9] allows: Y t /8.35 = Y b = 0.7 Y τ .(12) The predicted value of the parameter tan β for this case is ≈ 10 and is therefore preferable. However, in this case also the best unification is only at the 12% level. Because of these unfavorable results we are not presenting a more detailed analysis for these relations (Eqs (11) and (12)). Conclusion We have explored a class of SO(10) GUT models with a unified Higgs sector which yield the asymptotic relation Y b ≈ Y τ ≈ Y t /6 among the third generation quark and lepton Yukawa couplings. This relation among the Yukawa couplings is compatible with the various phenomenological constraints only with non-universal SSB mass terms at the GUT scale. The best YU (∼ 5%) we found in our analysis is consistent with the lightest CP-even Higgs boson mass to be in the interval 123 GeV ≤ m h ≤ 127 GeV. By scanning the fundamental parameter space of this model we showed that tan β is constrained in a very narrow interval, namely, tan β ≈ 14. Without imposing the constraint on the relic abundance of dark matter in these models, the squark and slepton masses, except for the stop, exceed 2 TeV while the gluino can be more than 1 TeV. On the other hand, the LSP neutralino as a dark matter candidate in this model can only be realized through the neutralino-stop coannihilation scenario. We found that requiring good YU can lead to a light stop ( 500 GeV) with all other sfermions having masses possibly beyond the reach of the 14 TeV LHC. Figure 1 1shows our results in the R tbτ − M 1/2 , R tbτ − m 10 , R tbτ − m 16 and R tbτ − tan β planes. Gray points in the figure are consistent with REWSB and LSP neutralino. Figure 2 : 2Plots in the R tbτ − M χ 0 1 , mτ 1 − M χ 0 1 , m A − M χ 0 1 and mt 1 − M χ 0 1 planes.Color coding is the same as in The Higgs mass for the three points is within the favored range, 123 GeV ≤ m h ≤ 127 GeV. Stop is the NLSP for the three points.The color Point 1 Point 2 Point 3 m 10 5936.55 1479.57 1086.04 m 16 4958.8 2922.23 2741.97 M 1 928.11 406.53 315.61 M 2 2784.33 1219.58 946.83 M 3 −1856.22 −813.06 −631.22 A 0 /m 16 −2.88 −2.98 −2.99 tan β 14.06 13.42 13.44 µ 3652 3448 3315 m h 126 124 125 m H 6776 3558 3270 m A 6732 3535 3249 m H ± 6777 3559 3271 mχ0 1,2 456, 2478 196, 1083 153, 846 mχ0 3,4 3665, 3667 3434, 3434 3303, 3304 mχ± 1,2 2493, 3675 1089, 3442 853, 3310 mg 4096 1922 1534 mũ L,R 6172, 5945 3364, 3285 3031, 2980 mt 1,2 488, 4325 934, 2428 650, 2122 md L,R 6173, 5944 3365, 3284 3032, 2979 mb 1,2 4382, 5562 2427, 3077 2118, 2774 mν 1 5263 3026 2810 mν 3 5088 2936 2724 mẽ L,R 5254, 4965 3024, 2922 2808, 2741 mτ 1,2 5066, 4567 2926, 2727 2715, 2559 ∆(g − 2) µ 1.12 × 10 −11 2.52 × 10 −11 2.90 × 10 −11 σ SI (pb) 3.12 × 10 −13 5.38 × 10 −13 4.89 × 10 −13 σ SD (pb) 3.67 × 10 −10 7.93 × 10 −12 7.05 × 10 −13 Ω CDM h 2 0.11 22 371 R tbτ 1.12 1.06 1.05 Table 1: Benchmark points with good Yukawa unification. Point 1 has a small neutralino relic abundance with YU around 11%. For point 2, the relic abundance is relatively large and the stop is twice as heavy compared to point 1 while the gluino is lighter. For point 3, YU is around the best value we obtained in our analysis (∼ 5%). Point 3 also has good YU with a lighter gluino and stop compared to point 2. AcknowledgmentsWe are extremely grateful to Pran Nath for collaborating with us in the early stages of this work and for numerous encouraging discussions.This work is supported in part by the DOE Grants No. DE-FG02-12ER41808 and by Rustaveli National Science Foundation No. 03/79 (I.G.). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation grant number OCI-1053575.I.G. wishes to thank the Center for Theoretical Underground Physics and Related Areas (CETUP*) where some part of this project was done. H Georgi, Particles and Fields. C.E. 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[ "Adiabatic quantum pumping in graphene NIS junctions", "Adiabatic quantum pumping in graphene NIS junctions" ]	[ "M Alos-Palop \nKavli Institute of Nanoscience\nDelft University of Technology\nLorentzweg 12628 CJDelftThe Netherlands\n", "M Blaauboer \nKavli Institute of Nanoscience\nDelft University of Technology\nLorentzweg 12628 CJDelftThe Netherlands\n" ]	[ "Kavli Institute of Nanoscience\nDelft University of Technology\nLorentzweg 12628 CJDelftThe Netherlands", "Kavli Institute of Nanoscience\nDelft University of Technology\nLorentzweg 12628 CJDelftThe Netherlands" ]	[]	We investigate adiabatic quantum pumping through a normal metal-insulator-superconductor (NIS) junction in a monolayer of graphene. The pumped current is generated by periodic modulation of two gate voltages, applied to the insulating and superconducting regions respectively. In the bilinear response regime and in the limit of a thin high insulating barrier, we find that the presence of the superconductor enhances the pumped current per mode by a factor of 4 at resonance. Compared to the pumped current in an analogous semiconductor NIS junction, the resonances have a π/2 phase difference. We also predict experimentally distinguishable differences between the pumped current and the tunneling conductance in graphene NIS junctions.	10.1103/physrevb.84.073402	[ "https://arxiv.org/pdf/1102.0926v1.pdf" ]	119,275,224	1102.0926	52c102da12b835b9191e8fc1c3f8d7ffcdcb8f22	Adiabatic quantum pumping in graphene NIS junctions 4 Feb 2011 M Alos-Palop Kavli Institute of Nanoscience Delft University of Technology Lorentzweg 12628 CJDelftThe Netherlands M Blaauboer Kavli Institute of Nanoscience Delft University of Technology Lorentzweg 12628 CJDelftThe Netherlands Adiabatic quantum pumping in graphene NIS junctions 4 Feb 2011(Dated: January 11, 2013) We investigate adiabatic quantum pumping through a normal metal-insulator-superconductor (NIS) junction in a monolayer of graphene. The pumped current is generated by periodic modulation of two gate voltages, applied to the insulating and superconducting regions respectively. In the bilinear response regime and in the limit of a thin high insulating barrier, we find that the presence of the superconductor enhances the pumped current per mode by a factor of 4 at resonance. Compared to the pumped current in an analogous semiconductor NIS junction, the resonances have a π/2 phase difference. We also predict experimentally distinguishable differences between the pumped current and the tunneling conductance in graphene NIS junctions. PACS numbers: 72.80.Vp,74.45+c,73. 23.-b Adiabatic pumping is a transport mechanism in mesoand nanoscale devices by which a finite dc current is generated in the absence of an applied bias by low-frequency periodic modulations of at least two system parameters (typically gate voltages or magnetic fields) [1,2]. In order for electrical transport to be adiabatic, the period of the oscillatory driving signals has to be much longer than the dwell time τ dwell of the electrons in the system, T = 2πω −1 ≫ τ dwell . Adiabatic quantum pumping [3] refers to pumping in open systems in which quantummechanical interference of electron waves occurs. In the last decade, many different aspects of quantum pumping have been investigated in a diverse range of nanodevices, for example charge and spin pumping in quantum dots [4], the relation of quantum pumping to geometric (Berry) phases [5] and the role of electron-electron interactions [6]. Quantum pumped currents have also been studied in hybrid systems consisting of normal-metal (N) and superconducting (S) parts, such as NS and SNS junctions [7][8][9][10][11]. Recently, investigations of quantum pumping in graphene mono-and bilayers have appeared [12]. In this Letter we investigate adiabatic charge pumping through a normal metal-insulating-superconductor (NIS) junction in graphene. The pumped current is generated by adiabatic variations of two gate voltages U 0 (t) and V 0 (t) which change, respectively, the Fermi level in the superconducting region and the height of the insulating tunnel barrier. The central question we aim to answer is what the effect of electron-hole (Andreev) reflection is on pumped charge currents in graphene. Using the scattering matrix formalism, we calculate the adiabatically pumped current at zero temperature in the linear response regime, i.e., for small variations of the pumping parameters U 0 and V 0 , and we compare this with the pumped current in the absence of the superconducting lead. Our main result is that the presence of the superconducting lead enhances the pumped current per mode by a factor of 4 and the total pumped current by a factor of 3 √ 2/2 at the resonant tunneling condition. Off resonance, the pumped current is an order of magnitude smaller than the analogous current in a semiconductor NIS junction. We also find that whereas the conductance increases with U 0 for thin barriers, the pumped current decreases with U 0 . This difference might be used to discriminate between conductance and pumped currents in graphene NIS junctions. In the last part of the Letter, we briefly comment on the pumped current in the so-called specular reflection regime (where ∆ 0 ≥ E F with ∆ 0 the superconductor gap and E F the Fermi energy) [13]. (a) (b) (c) EF 0 −d U0 U0 V0 Consider the geometry depicted in Fig. 1. A ballistic sheet of graphene in the (x, y)-plane contains a potential barrier of height V 0 and length d and a superconducting contact in the region x ≥ 0. The barrier can be implemented by employing the electric field effect [14,15] via a gate voltage and superconductivity can be induced in the region x ≥ 0 via the proximity effect. We assume the potential step V 0 to be abrupt on both sides, which is justified close to the Dirac point where the Fermi wavelength λ F ≫ d and which can be realized experimentally [15]. The conductance G(eV ) through such a graphene NIS junction has recently been studied, both for potential barriers of finite length [16] and in the limit of a thin barrier [17]. Here we investigate the adiabatically pumped current through the latter junction. The pumped current is induced by periodic variations of U 0 (t) = U 0 +δU 0 cos(ωt) and V 0 (t) = V 0 +δV 0 cos(ωt+φ). The total pumped current I into the normal lead (the left contact in Fig. 1) can be expressed as an integral over the area A that is enclosed in (U 0 , V 0 ) parameter space during one period, and is given by the scattering matrix expression [8] I ≡ I N = ωe 2π 2 A dU 0 dV 0 α,β∈N Π αβ (U 0 , V 0 ) (1) ≈ ωe 2π δU 0 δV 0 sin φ α,β∈N Π αβ (U 0 , V 0 ) (2) with Π αβ (U 0 , V 0 ) ≡ Im ∂S ee⋆ αβ ∂U 0 ∂S ee αβ ∂V 0 − ∂S he⋆ αβ ∂U 0 ∂S he αβ ∂V 0 .(3) Eq. (2) is valid in the bilinear response regime where δU 0 ≪ U 0 and δV 0 ≪ V 0 and the integral in Eq. (1) becomes independent of the pumping contour. The indices α and β sum over all modes in the normal lead and S denotes the Landauer-Büttiker scattering matrix whose elements S he αβ,nm describe the scattering of an electron in mode m in lead β to a hole in mode n in lead α. The low-energy excitations in the NIS junction close to the Dirac point K(K ′ ) are described by the 4 × 4 Dirac-Bogoliubov-deGennes Hamiltonian [13] H = H a − E F + U (x) ∆(x) ∆ * (x) E F − U (x) − H a (4) with H a = −ihv F (σ x ∂ x + sgn(a)σ y ∂ y ), where sgn(a) is ± for a = K(K ′ ) , v F denotes the Fermi velocity of the quasi-particles, and the potential U (x) = −U 0 θ(x) + V 0 θ(−x)θ(x + d). The pair potential ∆(x) which couples the electron and hole excitations has the form ∆ 0 e iφ and we assume that (E F + U 0 ) ≫ ∆ 0 , the mean-field condition for superconductivity. Two regimes can be distinguished: E F ≫ ∆ 0 where the usual retro Andreev reflection dominates, and E F ≤ ∆ 0 so U 0 ≫ ∆ 0 , where specular Andreev reflection dominates [13]. Our analysis focuses on the retro reflection regime unless we explicitly mention that we study the specular reflection regime. The Hamiltonian (4) acts on the four-component wavefunction ψ a = (ψ Aa , ψ Ba , ψ * Aā , −ψ * Bā ), where A and B denote the two nonequivalent sides of the graphene unit lattice andā = K ′ (K) for a = K(K ′ ). Following Ref. [17], we also introduce the dimensionless barrier strength χ = dV 0 /(hv F ) which allows us to consider the limit of a thin barrier where V 0 → ∞ and d → 0 such that χ remains finite. After applying continuity of the wavefunctions at the boundaries x = −d and x = 0, one obtains the reflection and transmission coefficients of the NIS junction, see Eqns. (9) in Ref. [17]. At the Dirac point ǫ = 0, where ǫ denotes the energy of the electrons measured from the Fermi level [18], and defining δ = E F /(E F + U 0 ), the derivatives of the coefficients for normal and Andreev reflection, r and r A , with respect to the gate voltages U 0 and V 0 are given by ∂r Here sin γ ≡ δ sin α with α the angle of incidence of the electron. Substituting Eq. (5) into Eq. (2) and integrating over the angle of incidence yields the pumped current at ǫ = 0 I NIS g = I g,0 δ 2 π π/2 0 dα sin γ sin α cos 4 α (1 + sin γ sin α cos 2χ) 3 (6) U0=0 = I g,0 4 cos 4 χ − 12 cos 2 χ + 8 √ 2\| cos χ\| − 3 16 √ 2\| cos χ\|(2 cos 2 χ − 1) 3(7) where I g,0 ≡ ωe d hvF EF δU 0 δV 0 sin φ. Eq. (7) is valid for U 0 = 0. Note from Eq. (6) that there is no contribution to I NIS g from normally incident electrons with α = 0. This is a display of the Klein tunneling effect where the Andreev reflection coefficient \|r A (α = 0)\| = 1 independent of χ and U 0 . Figure 2(a) shows the pumped current, Eq. (7), as a function of χ. Notice that I NIS g , just as the conductance in this system [17], is periodic in χ with a period π. We see that I NIS g reaches maximum values when V 0 d = (n + 1/2)πhv F , where n is an integer. This is the condition for resonant transmission (i.e., r = 0) when the conductance of the system reaches a sharply peaked maximum [17]. Due to the sharp changes in the derivative of r the pumped current diverges at this point. Expanding Eq. (7) with respect to χ around χ = 0, we find that I NIS g scales as I NIS g /I g,0 ∼ a 0 + a 1 χ 2(8) where a 0 = 16−11 √ 2 32 ≈ 0.014 and a 1 = 3(1 − 45 32 √ 2 ) ≈ 0.017. The pumped current thus increases with the barrier height close to χ = 0, which is another manifestation of the Klein paradox. Next, we analyze the behavior of the current with respect to U 0 . We expand Eq. (6) with respect to U 0 around U 0 = 0 at χ = 0, which results in I NIS g /I g,0 ∼ b 0 − b 1 U 0(9) where b 0 = a 0 and b 1 = 3 √ 2 128 ≈ 0.033. Eq. (9) shows that the pumped current decreases with increasing U 0 . Figure 2(b) shows the corresponding pumped current I NIS sc in a semiconductor NIS junction. We model the insulating region again as a barrier of height V 0 and width d, and define the dimensionless barrier strength χ = √ 2mV 0 d/h. Solving the Bogoliubov-deGennes equation, matching the wavefunction and its derivative at the boundaries, obtaining r and r A and calculating the derivatives with respect to the gate voltages V 0 and U 0 yields the pumped current (at the Dirac point ǫ = 0 and for U 0 = 0); I NIS sc = I sc,0 π/2 0 dα cos α k 3 d 3 χ 3 (10) 2χ(k 2 d 2 − χ 2 ) − (k 2 d 2 + χ 2 ) sin 2χ (2k 2 d 2 χ 2 cos 2 χ + (k 4 d 4 + χ 4 ) sin 2 χ) 3 , where I sc,0 ≡ (ωe/π)(2m 2 d 4 /h 4 )δU 0 δV 0 sin φ and k ≡ ( √ 2mE F /h) cos α. Eqns. (7) and (10) are the main results of this Letter. From Eq. (10) we notice that, in contrast to graphene, the normally incident electrons in the semiconductor junction do contribute to the pumped current, illustrating the absence of Klein tunneling in a semiconductor NIS junction. Figure 2 (b) displays I NIS sc as a function of the barrier strength χ. We observe that I NIS sc also oscillates as a function of χ with a period of π. However, the maxima at resonant transmission occur when χ = nπ and are thus shifted by π/2 with respect to the maxima of I NIS g . Notice that I NIS sc and I NIS g mostly have opposite signs and that I NIS sc switches sign at several points. In addition, the pumped current in a semiconductor NIS junction is roughly one order of magnitude larger than in graphene [19]. An important question arising when thinking about experimental detection of pumped currents is how to distinguish them from the conductance G in the system. In order to answer this question we explicitly compare both quantities. The conductance of the NIS junction in graphene was considered in Ref. [17] and is given by, at ǫ = 0 and U 0 = 0, G NIS g = G 0 √ A(A + 3) + (A 2 − 2A − 3) arctan ( √ A) A 5/2 ,(11) where G 0 ≡ 4e 2 h EF w πhvF with w the width of the sample and A ≡ cos 2χ. Expanding G NIS g for small χ around χ = 0 at U 0 = 0 and for small U 0 around U 0 = 0 at χ = 0 yields, respectively, G NIS g /G 0 ∼ (4 − π) + (16 − 5π)χ 2 ,(12)G NIS g /G 0 ∼ (4 − π) + (2 − π/2)U 0 /E F .(13) We find that G NIS g = (4 − π)G 0 for U 0 = χ = 0, somewhat below the ballistic value G 0 due to mismatch in Fermi wavelength in the normal and superconducting leads, as mentioned earlier [13]. Comparing the scaling behavior of G NIS g and I NIS g as a function of χ, Eqns. (8) and (12), we find that both transport quantities increase with increasing χ. However when comparing Eqns. (9) and (13) we see that whereas the pumped current I NIS g decreases with increasing U 0 , the conductance G NIS g increases with increasing U 0 . Intuitively, switching on U 0 increases the Fermi level mismatch, which increases the conductance [20] however decreases the pumped current. This difference can be used to discriminate the pumped current from the conductance in an actual experiment. We now investigate the influence of the superconducting lead on the pumped current by comparing I NIS g with the pumped current I NIN g through an entirely normal NIN junction in graphene. At the Dirac point ǫ = 0 and for U 0 = 0 the latter is given by I NIN g = I g,0 (1 − \| cos χ\|) 2 16\| cos χ\| sin 4 χ .(14) Similarly, the conductance through a NIN junction is given by Figure 3 shows the ratios I NIS g /I NIN g and G NIS g /G NIN g versus χ. In both cases, the superconducting lead enhances the transport reaching a maximum at χ = π/2. For the conductance this maximum enhancement is 2 due to the contribution of the holes [21], while the enhancement of the pumped current rises from 2(16 − 11 √ 2) ≈ 0.89 at χ = 0 to a factor of 3 √ 2/2 ≈ 2.12 at χ = π/2. However, when comparing I NIS g (Eq. (6)) and I NIN g per mode at χ = π/2, i.e., before integration over α, we see that the superconducting lead enhances the pumped current of each mode by a factor of 4. This last result is due to both the holes which contribute a factor of 2 and the asymmetry of the NIS junction with respect to injection of charge carriers which contribute another factor of 2 [8]. G NIN g = G 0 sin χ − cos 2 χ arctanh(sin χ) sin 3 χ .(15) At this point we briefly mention the behavior of the pumped current as a function of an applied bias voltage eV both in the normal (retro) and in the specular Andreev reflection regime [22]. For bias voltages below the gap eV ≤ ∆ 0 , the pumped current I NIS g in the retro reflection regime decreases from a finite value at eV = 0 (see Eq. (7)) to zero at eV = ∆ 0 . At eV = ∆ 0 , the particles are fully Andreev reflected (\|r A \| = 1 independent of χ and U 0 ) and therefore I NIS g = 0. In the specular reflection regime [13] where E F ≤ ∆ 0 and U 0 ≫ E F , the pumped current exhibits different behavior. First, the pumped current, just as the conductance [17], is insensitive to χ for energies below the gap. The large mismatch in Fermi energies of the normal and the superconducting leads already acts as a barrier, and as a result the addition of another barrier is irrelevant, explaining this behavior. Furthermore, the pumped current I NIS g is zero for bias voltages equal to the Fermi level eV = E F , due to the absence of Andreev reflection [13], and also for eV = ∆ 0 , see the discussion above. As a final remark, the pumped current is several orders of magnitudes smaller than in the retro reflection regime. Finally, we briefly comment on possibilities for experimental observation of our predictions. Experiments with superconducting electrodes on top of graphene in which multiple Andreev reflections were observed have already been carried out [23,24]. From these experiments we can estimate the order of magnitude of the pumped current. Some typical parameters are ω/(2π) = 5 GHz, E F = 80 meV , v F = 10 6 m/s and barrier width d = 10 − 20 nm [15,25]. For gate voltages on the order of 10 meV , the pumped current is on the order of 10f A far from the resonant tunneling condition, going up to 0.1 − 1 pA or higher close to resonance. In conclusion, we have investigated adiabatic quantum pumping in a graphene NIS junction, which is generated by periodic modulation of the insulating barrier V 0 and the Fermi level on the superconductor side. We have demonstrated that the presence of the superconducting lead can enhance the pumped current per mode by a factor of 4 (at resonance) and suggested experimentally observable differences between the conductance and the pumped current in this system. This work has been supported by the Netherlands Organization for Scientific Research (NWO/FOM). FIG . 1. (a) Sketch of the graphene NIS junction. The variable gate voltage V0 creates the insulating barrier I of length d and the gate voltage U0 is applied to the superconducting electrode (yellow). (b) Schematic of the energy levels in the three regions. (c) The area enclosed in the (U0, V0) parameter space during one pumping cycle. the potential barrier strength χ in a graphene (g) NIS junction. (b) The analogous pumped current I NIS sc (Eq. (10)) in a semiconductor (sc) NIS junction, where √ 2mEF d/h = 10. (A) /∂U 0 = (∂r (A) /∂δ)(∂δ/∂U 0 ) and ∂r (A) /∂V 0 = (∂r (A) /∂χ)(∂χ/∂V 0 ) with ∂r ∂χ = 2e iα sin γ cos α(cos 2χ + sin γ sin α − i cos α sin 2χ) (1 + sin γ sin α cos 2χ) 2 , ∂r * ∂δ = −ie −iα sin α cos α(cos 2χ cos α + i sin 2χ) (1 + sin γ sin α cos 2χ) 2 , ∂r A ∂χ = −2ie −iφ sin γ cos γ sin α cos α sin 2χ (1 + sin γ sin α cos 2χ) 2 , ∂r * A ∂δ = −ie iφ cos α sin α(sin γ + sin α cos 2χ) cos γ(1 + sin γ sin α cos 2χ) 2 . FIG. 3 . 3Ratio of the pumped current (left) and the conductance (right) of the NIS junction and the NIN junction in graphene versus χ for ǫ = 0 and U0 = 0. . M Büttiker, H Thomas, A Prêtre, Z. Phys. B. 94133M. Büttiker, H. Thomas, and A. Prêtre, Z. Phys. 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We remark that ǫ = 0 is the point of highest interest for observing quantum pumping. since it corresponds to the situation in which no bias voltage is appliedWe remark that ǫ = 0 is the point of highest interest for observing quantum pumping, since it corresponds to the situation in which no bias voltage is applied. The conductance is then given by G = (4/3)G0 at zero bias eV = 0 and decreases to the value G = (4 − π)G0 for eV ≫ ∆0. See also Ref. [13], where it was assumed that U0 ≫ EF. in which case the Fermi level mismatch is minimalSee also Ref. [13], where it was assumed that U0 ≫ EF . The conductance is then given by G = (4/3)G0 at zero bias eV = 0 and decreases to the value G = (4 − π)G0 for eV ≫ ∆0, in which case the Fermi level mismatch is minimal. Mesoscopic quantum physics. C W J Beenakker, E. Akkermans, G. Montambaux, J.-L. Pichard, and J. Zinn-JustinNorth-Holland, Amsterdamsee also cond-mat/9406083C. W. J. Beenakker, Mesoscopic quantum physics, edited by E. Akkermans, G. 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[ "royalsocietypublishing.org/journal/rsif Parameter estimation and uncertainty quantification using information geometry", "royalsocietypublishing.org/journal/rsif Parameter estimation and uncertainty quantification using information geometry" ]	[ "Jesse A Sharp [email protected] \nSchool of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n\nARC Centre of Excellence for Mathematical and Statistical Frontiers\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n", "Alexander P Browning \nSchool of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n\nARC Centre of Excellence for Mathematical and Statistical Frontiers\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n", "Kevin Burrage \nSchool of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n\nARC Centre of Excellence for Mathematical and Statistical Frontiers\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n\nDepartment of Computer Science\nUniversity of Oxford\nOxfordUK\n", "Matthew J Simpson \nSchool of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n\nCentre for Data Science\nQueensland University of Technology\nBrisbaneQueenslandAustralia\n" ]	[ "School of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "ARC Centre of Excellence for Mathematical and Statistical Frontiers\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "ARC Centre of Excellence for Mathematical and Statistical Frontiers\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "School of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "ARC Centre of Excellence for Mathematical and Statistical Frontiers\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "Department of Computer Science\nUniversity of Oxford\nOxfordUK", "School of Mathematical Sciences\nQueensland University of Technology\nBrisbaneQueenslandAustralia", "Centre for Data Science\nQueensland University of Technology\nBrisbaneQueenslandAustralia" ]	[]	In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide dataindependent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.	10.1098/rsif.2021.0940	[ "https://arxiv.org/pdf/2111.12201v3.pdf" ]	244,527,104	2111.12201	0fa385883e110467dface5aa9841ec696d0aa10c	royalsocietypublishing.org/journal/rsif Parameter estimation and uncertainty quantification using information geometry Jesse A Sharp [email protected] School of Mathematical Sciences Queensland University of Technology BrisbaneQueenslandAustralia ARC Centre of Excellence for Mathematical and Statistical Frontiers Queensland University of Technology BrisbaneQueenslandAustralia Alexander P Browning School of Mathematical Sciences Queensland University of Technology BrisbaneQueenslandAustralia ARC Centre of Excellence for Mathematical and Statistical Frontiers Queensland University of Technology BrisbaneQueenslandAustralia Kevin Burrage School of Mathematical Sciences Queensland University of Technology BrisbaneQueenslandAustralia ARC Centre of Excellence for Mathematical and Statistical Frontiers Queensland University of Technology BrisbaneQueenslandAustralia Department of Computer Science University of Oxford OxfordUK Matthew J Simpson School of Mathematical Sciences Queensland University of Technology BrisbaneQueenslandAustralia Centre for Data Science Queensland University of Technology BrisbaneQueenslandAustralia royalsocietypublishing.org/journal/rsif Parameter estimation and uncertainty quantification using information geometry 10.1098/rsif.2021.0940Received: 20 December 2021 Accepted: 30 March 2022Review Cite this article: Sharp JA, Browning AP, Burrage K, Simpson MJ. 2022 Parameter estimation and uncertainty quantification using information geometry. J. R. Soc. Interface 19: 20210940. Author for correspondence: Jesse A. SharpSubject Category: Life Sciences-Mathematics interface Subject Areas: systems biologycomputational biology Keywords: inferencelikelihoodpopulation modelslogistic growthepidemic models In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide dataindependent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub. Introduction Computational and mathematical models are versatile tools, providing valuable insight into complex processes in the life sciences. Models can further our understanding of mechanisms and processes, facilitate development and testing of hypotheses, guide experimentation and data collection and aid design of targeted interventions [1][2][3][4][5]. However, there are considerable challenges associated with calibrating these models to data. For example, models need to be sufficiently sophisticated to adequately reflect the behaviour of the underlying system, while ideally admitting identifiable parameters that are interpretable and that can be estimated from available or obtainable data [6,7]. Further, available data can be limited and often are not collected for the express purpose of parameter estimation; data may be noisy or incomplete, or may not provide the level of detail or sample size required to obtain precise parameter estimates [8][9][10][11][12]. Owing to the challenges associated with parameter estimation, we are often interested in not only point estimates, but also the associated uncertainty [13][14][15]. Quantifying and interpreting this uncertainty establishes a level of confidence in parameter estimates and, by extension, in the insights derived from the model. Further, this uncertainty quantification can give insights into identifiability: whether the information in a dataset can be used to infer unique or sufficiently precise parameter estimates for a given model [16]. Often we are concerned with both structural identifiability and practical identifiability [17][18][19][20][21]. Structural identifiability can be thought of as a property of the underlying model structure and parametrization, and refers to whether it is theoretically possible to determine unique parameter values, given an infinite amount of perfect noise-free data [16,22,23]. Structural identifiability requires that unique parameter combinations precipitate distinct model outputs. Structural identifiability occurs if and only if the Fisher information matrix, which we soon discuss, is of full rank [24]. By contrast, practical identifiability is less well defined, and depends on the quality and quantity of data available and existing knowledge of the parameters [22]. In the context of profile-likelihood methods, practical non-identifiability can manifest as contours of the log-likelihood function that do not admit closed levels; the log-likelihood does not reach a predetermined statistical threshold within the physical parameter regime [25]. If a model is not structurally identifiable, it cannot be practically identifiable. Practical non-identifiability may be addressed through improving either data quantity or data quality [19,22]. Data quantity can be improved by increasing the number of observations, such as by making additional observations at different time points. Data quality may be improved through reducing the noise present in the data; for example, by obtaining a dataset with reduced measurement error or repeating measurements across experiments [26,27]. It is also possible to resolve practical non-identifiability through incorporating existing knowledge about parameters, such as physical constraints or information established in previous studies; or specifically in the Bayesian inferential framework, through informative priors [28]. Addressing structural non-identifiability is more challenging; for example, this may necessitate a change to the underlying model structure [20,27,29]. Uncertainty quantification takes many forms, with common examples, including Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping [8,12,[30][31][32]. Bayesian methods are widely used for parameter estimation and uncertainty quantification, with Bayesian computation being employed throughout the mathematical biology and systems biology literature. Broadly, these methods involve repeated sampling of parameter values from a prior distribution and invoking Bayes' theorem to approximate the posterior distribution; the posterior distribution describes knowledge about the probability of parameter combinations after taking into account the observed data and any prior information [22,32]. Well-known approaches include rejection sampling, Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) or particle filtering. In rejection sampling, parameters drawn from a prior distribution are used to simulate the model. Simulated data are compared with the observed data based on some distance metric; if this metric is within a prescribed tolerance, the parameters are accepted as a sample from the approximate posterior distribution, otherwise they are rejected [30,33]. Rejection sampling can be computationally expensive as the rejection rate can be significant with an uninformative prior [34,35]. In MCMC, the parameter space is sampled following a Markov chain-a memoryless stochastic process where the probability of the next state depends only on the previous state [36]-with a stationary distribution corresponding to the posterior distribution. Samples are accepted or rejected based on the relative likelihood between the current configuration and proposed sample [11,32,37,38]. For SMC, rejection sampling can be used to produce an initial coarse approximation of the posterior distribution. This coarse approximation informs further (sequential) sampling efforts in the region of parameter space corresponding to high likelihood, reducing the rejection rate when compared with rejection sampling alone [11,34,39]. MCMC and SMC approaches can offer significantly improved efficiency in comparison with rejection sampling [32,34], but both involve specifying hyperparameters and these choices are not always obvious. In situations where the likelihood function is intractable or not easily evaluated, approximate Bayesian computation (ABC) provides a range of related likelihoodfree methods for estimating posterior distributions [40]. Popular approaches include ABC rejection sampling [35,[39][40][41][42], ABC MCMC [43][44][45] and ABC SMC [11,34]; we do not focus on ABC methods here, as the approaches we explore in this work are applied to problems with tractable likelihoods. We direct interested readers to the wealth of information in the references provided. For Bayesian inference methods, uncertainty can be quantified based on features such as the coefficient of variation and probability intervals of the posterior distribution [12]. There are a variety of approaches for uncertainty quantification for frequentist inference methods. In profile likelihood, a parameter of interest is varied over a fixed set of values, while re-estimating the other parameters; this provides insight into identifiability and uncertainty [1]. In asymptotic analysis, confidence regions can be constructed based on local information via a Taylor expansion of the Fisher information about the maximum likelihood estimate (MLE) [8,25]. In bootstrapping, data are repeatedly sampled and parameter estimates are computed from the samples; these estimates are used to construct confidence intervals [31]. Through the geometric approaches we review in this work, more akin to traditional approaches for sensitivity analysis [14,46,47], we explore the curvature of the parameter space through an information metric induced by the likelihood function. Whereas likelihood-based approximate confidence regions provide insight into specific level curves of the likelihood function-the levels of which depend on an asymptotic large sample argument [36]-this geometric approach provides insight into the shape and sensitivity of the parameter space. For example, we compute geodesic curves that describe the geometric relationship between distributions with different parameters [48]; and explore the scalar curvature throughout parameter spaces. We review ideas from information geometry in the context of inference and uncertainty quantification; not with a view to replacing established methods such as profile likelihood, asymptotic analysis, bootstrapping and Bayesian methods [8,12,31,32], but rather to supplement them where additional insight may prove useful. Information geometry is a branch of mathematics connecting aspects of information theory including probability theory and statistics with concepts and techniques in differential geometry [49]. In this exposition, we seek to outline only the key concepts required to understand the information geometric analysis in this work. However, we note that more thorough and rigorous treatments of the concepts introduced in this section, and mathematical foundations of information geometry, can be found in texts and surveys such as [49][50][51]. Central to the information geometry ideas explored in this work is the concept of a statistical manifold; an abstract geometric representation of a distribution space, or a Riemannian manifold consisting of points that correspond to probability distributions, with properties that we later discuss. For example, the set of normal distributions parametrized by mean, μ, and standard deviation, σ > 0, pðx; m, sÞ ¼ 1 s ffiffiffiffiffiffi 2p p exp À ðx À mÞ 2 2s 2 " # , x [ R,ð1:1Þ can be thought of as a two-dimensional surface with coordinates (μ, σ) [50]. In this work, we will use θ to refer to the parameters of interest that we seek to estimate; i.e. θ = (μ, σ) for the univariate normal distribution with unknown mean and standard deviation. In §3, we consider various combinations of components of θ, including model parameters, variability in observations characterized by a separate royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 observation noise model, and initial conditions associated with a differential equation-based process model. When referring to all possible parameters, rather than solely the unknown parameters to be estimated, we denote this Q. In applications where we consider multiple datasets, or different candidate models or candidate parameter values, we are interested in methods of comparing distributions. A well-known measure for comparing a probability distribution, P, with another, Q, is the Kullback-Leibler (KL) divergence from P to Q, denoted D KL ðP, QÞ [52]. The KL divergence, or relative entropy, can be computed as [52] D KL ðP, QÞ ¼ ð pðxÞ log pðxÞ qðxÞ dx ¼ E p log pðxÞ qðxÞ ! , ð1:2Þ where p(x) and q(x) are the respective probability density functions of P and Q. Consider two sets of parameters, θ* andû; let log( p(x)) = log( p(x\|θ)) = ℓ(θ) and logðqðxÞÞ ¼ logðpðxjûÞÞ ¼ 'ðûÞ, where ℓ( · ) denotes the log-likelihood, discussed in detail in §2. If p(x\|θ*) is the true distribution and pðxjûÞ is our estimate, then (1.2) is the expected log-likelihood ratio and the relationship between MLE and KL divergence becomes evident; maximizing the likelihood is equivalent to minimizing the KL divergence [53]. An issue with the KL divergence is asymmetry; D KL ðP, QÞ = D KL ðQ, PÞ. It is not necessarily obvious in a given situation which orientation of the KL divergence will most appropriately inform decisions such as model selection [54]. Owing to the aforementioned asymmetry, and its failure to satisfy the triangle inequality, the KL divergence is not a metric-it is not a measure of distance in a differential geometric sense-on a given manifold [50]. One means of addressing this asymmetry is through devising various symmetrized forms of the KL divergence to inform model selection criteria [54]. Alternatively, we may approach the issue from a geometric perspective. It is natural to think of geometry in terms of objects or shapes in Euclidean, or flat, space. Euclidean space is characterized by orthonormal basis vectors; the standard basis in three dimensions being e 1 = (1, 0, 0) T , e 2 = (0, 1, 0) T , e 3 = (0, 0, 1) T , where superscript T denotes the transpose. In the n-dimensional orthonormal basis, we can compute the squared infinitesimal distance between the points S and S þ ds, where ds has components ds i , as [55] kdsk 2 ¼ X n i¼1 ðds i Þ 2 : ð1:3Þ Differential geometry extends ideas from Euclidean geometry to manifolds. Manifolds are topological spaces that resemble flat space about each individual point in the space; they can be considered locally flat, but have a different topology globally. The sphere is a classic example, whereby points on the surface are locally topologically equivalent to two-dimensional Euclidean space, but globally the sphere is curved and has a compact topology; it is bounded and closed [56]. In particular, we are interested in Riemannian manifolds; differentiable manifolds-sufficiently locally smooth that our typical notions of calculus remain valid-upon which we are able to measure geometric quantities such as distance, through the existence of a Riemannian metric on the tangent space of the manifold, that generalizes the notion of an inner product from Euclidean geometry [57]. A Riemannian metric is a smooth covariant 2-tensor field; on a differentiable manifold M, the Riemannian metric is given by an inner product on the tangent space of the manifold, T p M, which depends smoothly on the base point p [57,58]. A tangent space can be thought of as a multidimensional generalization of a tangent plane to a three-dimensional surface. Each point p on a manifold is associated with a distinct tangent space. An n-dimensional manifold has infinitely many ndimensional tangent spaces; the collection of these tangent spaces is referred to as the tangent bundle of the manifold. On a manifold each tangent space can have different basis vectors, in contrast to Euclidean geometry, where tangent vectors at any point have the same basis vectors. A consequence of the distinct basis vectors of tangent spaces on manifolds is that tangent vectors at different points on the manifold cannot be directly added or subtracted. Introducing an affine connection on the manifold connects nearby tangent spaces, such that the manifold looks infinitesimally like Euclidean space, which facilitates differentiation of tangent vectors [59]. Formally, we introduce the unique, torsion-free Levi-Civita connection, r; this is an affine connection on the Riemannian manifold that yields isometric parallel transport, such that inner products between tangent vectors, defined by the metric, are preserved [60]. The coefficients of this connection are the Christoffel symbols, which we discuss further in §2. Readers are directed to [59][60][61] for further detail regarding the Levi-Civita connection and how it relates to other concepts discussed in this work. A manifold equipped with such a connection and a Riemann metric is a Riemann manifold. Metric tensors can be thought of as functions that facilitate computation of quantities of interest such as distances on a manifold. A metric matrix with elements g ij , G = [g ij ], is positive definite and symmetric [57]. The metric matrix defines an inner product between u and v as 〈u, v〉 G = u T Gv, and the length of u as kuk G ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi hu, ui G p [62]. On a Riemannian manifold, we consider a generalization of the square of the infinitesimal distance element (1.3), appropriate for non-orthonormal bases [55], given by kdsk 2 ¼ X n i,j¼1 g ij ds i ds j : Foundational to information geometry is the notion that the Fisher information matrix defines a Riemannian metric on a statistical manifold [63]. observation process, given a model, m = m(θ), where J(θ) is the Jacobian of the model with respect to the parameters. The number of independent, identically distributed (iid) observations in the likelihood is given by N; with statistical independence, the Fisher information is additive [65]. Expression (1.4) highlights a link between sensitivity analysis, structural identifiability and practical identifiability [66]. For sensitivity analysis and structural identifiability, only the curvature of the model space is studied through J(θ). In practical identifiability analysis, the sensitivity of the model is linked to the data through an observation process, and the curvature of the parameter space is studied through, for example, I ðuÞ. In this review, we present and explore fundamental techniques in inference and information geometry, including confidence regions, geodesic curves and scalar curvature. Through application to standard distributions and canonical models in the life sciences, including population growth processes and epidemic transmission, we demonstrate how these techniques can be combined to provide additional insights into parameter estimation and uncertainty quantification. Starting with parameter estimates inferred from real data, we use mathematical models to generate synthetic data with different numbers of observations and at varying points in time, to explore the impact that these aspects have on the inference and information geometry results. Specifically, we consider univariate and multivariate normally distributed observation processes; linear, exponential and logistic models of population growth; and the classical susceptible, infectious, recovered (SIR) model of epidemic transmission [67,68]. Although the examples considered in this work are based on ordinary differential equation (ODE) process models drawn from the life sciences, the techniques we consider are general and can be applied in the context of parameter estimation and uncertainty quantification in any discipline and for other model formulations. By considering standard distributions and canonical models, we are able to explore the inference and information geometry techniques through a series of examples with incremental increases in complexity. Through this approach, we consider the techniques as applied to both linear and nonlinear ODE models, coupled nonlinear ODE systems and data with both one and many observed variables. We consider cases where model parameters, initial conditions and the standard deviation of the data are to be estimated from data. The inference and information geometry techniques considered in this work are general, and can be applied far more widely than the examples we consider here. To improve the accessibility of these methods, code used to implement the inference and information geometry techniques applied in this work is written in the open source Julia language [69] and is available on GitHub. In §2, we describe the inference and information geometry methods implemented in this work, including maximum likelihood estimation, profile-likelihood-based approaches, geodesic curves and scalar curvature calculations. Results of applying these techniques to univariate and multivariate normal distributions, linear, exponential and logistic growth models and the SIR model are presented in §3. We discuss the utility of these techniques and identify opportunities for extension and further consideration in §4. Methods Here we describe the parameter inference and information geometry methods used to produce results in this work. We also describe the numerical methods used to implement these techniques. The techniques we discuss in this section readily generalize to parameter spaces with an arbitrary number of dimensions, so we discuss the techniques here for arbitrary dimensions. However, for the sake of exploring the techniques through visualization in §3, we restrict ourselves to two-dimensional manifolds. In context, this means we consider only two parameters to be inferred in any given example, treating other parameters as known and fixed; for example, as if they are drawn from prior knowledge or pre-estimated. Although we consider deterministic mathematical models, data used to estimate parameters can exhibit significant variability. We follow a standard approach and assume that the mathematical models describe the expected behaviour, and that our observations are normally distributed about this expected behaviour [18]. This allows us to think about a statistical model, m(θ, t), in terms of its expected behaviour, μ, and the standard deviation of the observations, σ, mðu, tÞ ¼ ðmðu, tÞ, sðu, tÞÞ: We restrict the examples in this work to cases where σ is constant, setting σ(θ, t) = σ. In this work we focus on the most commonly employed additive noise model [5,11,18,19,27]. Additive noise implies that the variance of the data is independent of the mean behaviour. In cases where variance scales with mean behaviour, multiplicative noise may be more appropriate. The information geometric methods presented here are applicable in cases where the Fisher information can be obtained, including models with multiplicative noise and parameter-or timedependent standard deviation. However, obtaining the Fisher information is a separate challenge, and can be difficult when considering different process and noise models. Parameter inference In this work, parameter estimates are inferred from data following a standard maximum log-likelihood-based approach. We make observations at L time points, T = (t 1 , t 2 , …, t L ). At each time point, we make N observations, X ¼ ðx 1 ðTÞ, x 2 ðTÞ, . . . , x N ðTÞÞ. With this notation, the log-likelihood function is 'ðu; X Þ ¼ X L j¼1 X N i¼1 log f(x i ðt j Þ; mðu, t j Þ, s 2 ),ð2:1Þ where f(x; μ, σ 2 ) is the probability density function associated with our observation process. In this work, we hold N constant across time points, though non-constant N is easily incorporated into equation (2.1) as N j . The likelihood function can be thought of as the joint probability density of all the data for a given set of parameters. In examples where σ is unknown, we treat σ as an element of θ, but note that the expected model behaviour is independent of σ. The MLE is the point estimate,û, that satisfieŝ u ¼ arg max u 'ðu; XÞ,ð2:2Þ where arg max( · ) returns the argument, θ, that maximizes 'ðu; XÞ in (2.2). The associated maximum log-likelihood is 'ðûÞ. MLEs of the parameters of interest are obtained by solving (2.2) numerically as outlined later in §2. For an iid sample from a univariate normal distribution, N ðm, s 2 Þ, maximizing the likelihood function of μ is equivalent to performing least-squares estimation [22], although having access to the likelihood function facilitates uncertainty quantification. Presenting confidence regions alongside MLEs enhances our interpretation of the likelihood function, while still acknowledging that the estimates carry uncertainty [36]. We apply a probability-based log-likelihood approach when constructing confidence regions for model parameters. From Wilks' theorem [36], asymptotically as N → ∞, an approximate α-level confidence royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 region is given by u : 'ðuÞ ! 'ðûÞ À D n,a 2 & ' , ð2:3Þ where Δ ν,α is the αth-quantile of the χ 2 (ν) distribution, with ν degrees of freedom [1]. In this work, the degrees of freedom correspond to the number of parameters of interest, i.e. ν = dim(θ). To enable comparison between different datasets and models in §3, we consider the normalized log-likelihood, 'ðuÞ ¼ 'ðuÞ À 'ðûÞ. This forms the basis for log-likelihood ratiobased hypothesis tests [36]. The normalized log-likelihood is zero at the MLE:'ðûÞ ; 0. Information geometry As outlined in §1, the Fisher information describes the curvature of the log-likelihood function. It describes how much information a random variable, X, contains about a parameter, θ. For unbiased estimators, the inverse of the Fisher information provides a lower bound on the covariance matrix, via the Cramér-Rao inequality [70]. Formally, the Fisher information is the covariance of the score, where the score is defined as the partial derivative of the log-likelihood with respect to θ [36,64]. The Fisher information matrix can be written as [36,71] ½I ðuÞ ij ¼ E X @ @u i log f ðX; uÞ @ @u j log f ðX; uÞ ! : ð2:4Þ We can recover our expression for the Fisher information in equation (1.4) from equation (2.4), by considering how equation (2.4) changes under reparametrization and via application of the chain-rule for differentiation [64]. With observations at L unique times, T = (t 1 , t 2 , …, t L ), we can think of a model as a mapping between the parameters and the outputs that we can observe, mðuÞ : u ! À m 1 ðu, t 1 Þ, s Á , À m 2 ðu, t 2 Þ, s Á , . . . , À m L ðu, t L Þ, s Á : ð2:5Þ We consider some examples where σ is unknown and is estimated as a part of the analysis; in these instances σ ∈ θ, however we express σ explicitly in the mapping presented in (2.5) to emphasize that it behaves somewhat differently from a model parameter. The expected behaviour of the model does not depend on σ, and variability in the data maps directly to σ. In all the examples we consider, σ is constant. This could be extended to incorporate variability dependent on the expected behaviour; for example, logistic growth with standard deviation that depends on the population density [72]. In the mapping, this could be expressed as σ(μ(θ, t)). Following equation (1.4), we can form the Fisher information as a combination of the Fisher information matrix of the observation process, OðmÞ, and the Jacobian of the model with respect to the parameters, J(θ). From (2.5), with ν unknown parameters (dimðuÞ ¼ n), we can view the model Jacobian as JðuÞ ¼ @m 1 @u1 @m 1 @u2 . . . @m 1 @un @s @u1 @s @u2 . . . @s @un @m 2 @u1 @m 2 @u2 . . . @m 2 @un @s @u1 @s @u2 . . . @s @un . . . . . . . . . @m j @u1 @m j @u2 . . . @m j @un @s @u1 @s @u2 . . . @s @un 0 B B B B B B B B B B B B B @ 1 C C C C C C C C C C C C C A : ð2:6Þ Noting that we are taking σ to be independent of model parameters, all of the partial derivatives of σ in (2.6) are zero, except the case where θ i = σ, for some i ∈ {1, 2, …, ν}, whereby the corresponding partial derivative is unity. Given a set of N normally distributed observations at a single point in time, we have an observation process characterized by a mean, μ, and standard deviation, σ. The Fisher information for such an observation is given by I ðm, sÞ ¼ N s 2 D, where D ¼ diagð1, 2Þ: ð2:7Þ This can be verified by applying equation (2.4) to (1.1). For data at L time points with N 1 , N 2 , …, N L observations at each time, with constant standard deviation, the Fisher information for the observation process is a 2L × 2L (block) diagonal matrix, I ðm, sÞ ¼ diag N 1 s 2 D, N 2 s 2 D, . . . , N L s 2 D : ð2:8Þ Similarly, for a model with M species, where we have observations of all M species at only one time point we recover Fisher information in the form of (2.8). For observations of M species at L time points we form a 2LM × 2LM (block) diagonal matrix from (2.8). Assuming a constant standard deviation, for the computations in this work we could more simply express (2.8) as the diagonal matrix diagðN 1 =s 2 , N 2 =s 2 , . . . , N L =s 2 , 2 P N i =s 2 Þ, where P N i is the total number of observations contributing to our information regarding the standard deviation, and the factor of 2 comes from (2.7). In this case, the model Jacobian as presented in (2.6) is modified such that only the final row includes the partial derivatives with respect to the standard deviation. Before outlining specific techniques of information geometry, we present a conceptual example to develop some intuition for information geometric concepts. Consider the manifold corresponding to the family of univariate normal distributions parametrized by mean, μ, and standard deviation, σ > 0. Let P N ðm 1 , sÞ and Q N ðm 2 , sÞ be two normal distributions. Geometrically speaking, increasing σ reduces the distance between P and Q; this corresponds to a contraction of the space. Conversely, decreasing the variance dilates the space; as σ → 0, the Fisher information, diag(1/σ 2 , 1/σ 2 ), is degenerate and the distance between P and Q tends to infinity. Equipped with the Fisher information, we may begin to explain some foundational ideas from information geometry, including geodesic curves, geodesic distances between distributions for statistical models and scalar curvature [49]. We denote the elements of the Fisher information as IðuÞ ¼ ½g ij ðuÞ, and its inverse I ðuÞ À1 ¼ ½g ij ðuÞ, where u ¼ ðu 1 , u 2 , . . . , u n Þ are the coordinates of the manifold. While uncertainty in estimates is typically characterized by the Fisher information at only a single point, based on the Cramér-Rao inequality [70], information geometry uses the Fisher information throughout the parameter space. A Riemann geodesic is a curve forming the shortest path between two points in a Riemannian manifold [73]. The length of this shortest curve is referred to as the Fisher or Fisher-Rao distance [74]. We soon discuss a relationship between confidence regions and the length of geodesic curves. Informally, with greater information supporting an MLE, coinciding with an increase in its relative likelihood, confidence regions tighten. This also corresponds to a dilation of the parameter manifold; thereby increasing the geodesic distance between the MLE and other parameter combinations, reflecting their relatively reduced likelihood. A curve z(s), parameterized by s, connecting the points z 1 = z(s 1 ) and z 2 = z(s 2 ) on a Riemannian manifold, has length [58] LðzÞ ¼ ð s2 s1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X n i,j¼1 g ij ðuðzðsÞÞÞ du i ðzðsÞÞ ds du j ðzðsÞÞ ds v u u t ds: ð2:9Þ A Riemann geodesic is a curve that minimizes L(z) (2.9), such that the distance between two points on a Riemannian manifold royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 is given by the curve that satisfies dðz 1 , z 2 Þ ¼ minfLðzÞ : zðs 1 Þ ¼ z 1 , zðs 2 Þ ¼ z 2 g: For Gaussian likelihoods, there is an asymptotic relationship between the geodesic distance between the MLE,û, and a point u a that corresponds to an α-level confidence region on the manifold [75]. The geodesic distance betweenû and u a : dðû, u a Þ can be written in terms of the αth quantile of the χ 2 (ν) distribution dðû, u a Þ ¼ ffiffiffiffiffiffiffiffi ffi D n,a p : ð2:10Þ Pairing equations (2.3) and (2.10) yields an asymptotic relationship between confidence regions and geodesic length [75] 2('ðûÞ À 'ðuÞ) dðû, u a Þ 2 as N ! 1: ð2:11Þ In §3, we present likelihood-based confidence regions alongside geodesic curves of the corresponding length, as characterized by (2.10), and comment on the validity of equation (2.11) in a range of scenarios. Geodesic curves satisfy the following system of differential equations in n dimensions [76]: d 2 u m ds 2 þ X n i,j¼1 G m ij du i ds du j ds ¼ 0, m ¼ 1, . . . , n, ð2:12Þ where s is the parametrization of the geodesic curve, in accordance with equation (2.9), and G m ij are the Christoffel symbols of the second kind [50], defined as G m ij ¼ 1 2 X n l¼1 g ml @g lj @u i þ @g li @u j À @g ij @u l : ð2:13Þ We can convert from Christoffel symbols of the second kind to Christoffel symbols of the first kind by lowering the contravariant (upper) index through multiplication by the metric: G kij ¼ g km G m ij [77]. Here, repeated indices, in this case m, imply that a summation is to be performed over the repeated index, following the Einstein summation convention [56]. Conversely, we can recover Christoffel symbols of the first kind from Christoffel symbols of the second kind via the inverse metric: g km G kij ¼ G m ij . Christoffel symbols of the second kind are the connection coefficients of the Levi-Civita connection; the Christoffel symbols are symmetric in the covariant (lower) indices [60]. On an n-dimensional manifold, the Christoffel symbol is of dimension n × n × n. Geodesics can be used to construct theoretical confidence regions, to measure the geometric distance between probability distributions and to perform hypothesis testing; for example, to test equality of parameters [48,51,78]. Under certain conditions, analytical expressions can be obtained for the solutions of the geodesic equations, and the corresponding Fisher-Rao distances, for example, in the case of the univariate (1.1) and multivariate (3.2) normal distributions [74,79]. However, we solve equation (2.12) numerically, after converting the second-order ODE to a first-order system of ODEs using standard techniques. We are also interested in exploring the scalar curvature, also known as the Ricci scalar, of our manifolds. To compute the scalar curvature, we must first construct the Riemann tensor, and subsequently the Ricci tensor. As we only require these tensors for computation of the scalar curvature, and do not attempt to interpret these tensors directly in this work, we provide only a limited outline of their interpretation. The Riemann curvature tensor is constructed from the Christoffel symbols and their first partial derivatives. Here, it is convenient to think about these partial derivatives as being with respect to the parameters of interest. Owing to the possibility of raising or lowering indices of Christoffel symbols and tensors via the metric, there are several equivalent expressions for computing the Riemann curvature tensor [77]. The elements of the Riemann tensor of the first kind can be written as R ijkl ¼ @G jli @k À @G jki @l þ G ilr G r jk À G ikr G r jl : ð2:14Þ The Riemann tensor of the first kind is a (0, 4) tensor (with no contravariant indices and four covariant indices), and can be converted to the (1, 3) Riemann tensor of the second kind via the inverse of the metric: g im R ijkl ¼ R m jkl . On an n-dimensional manifold, the Riemann tensor is of dimension n × n × n × n; owing to various symmetries, however, there are far fewer independent elements [80]. The Riemann tensor provides information about the intrinsic curvature of the manifold. A geometric interpretation is that a vector from a point on the manifold, parallel transported around a parallelogram, will be identical to its original value when it returns to its starting point if the manifold is flat. In this case, the Riemann tensor vanishes. If the manifold is not flat, the Riemann tensor can be used to quantify how the vector differs following this parallel transport [81]. From the Riemann tensor of the second kind, we can compute the Ricci tensor of the first kind. The Ricci tensor, R ij , is obtained by contracting the contravariant index with the third covariant index of the Riemann tensor of the second kind; that is, R ij ¼ R m ijm : ð2:15Þ On an n-dimensional manifold, the Ricci tensor is of dimension n × n and is symmetric [81]. The Ricci tensor can quantify the changes to a volume element as it moves through the manifold, relative to Euclidean space [81]. The scalar curvature, Sc, can be obtained as a contraction of the Ricci tensor Sc ¼ g ij R ij : ð2:16Þ The scalar curvature is invariant; it does not change under a change of coordinates (re-parametrization). For Gaussian likelihoods, the corresponding manifold is flat, characterized by zero scalar curvature everywhere. As such, the scalar curvature provides a measure of how the likelihood of the underlying statistical model deviates from being Gaussian-often referred to as non-Gaussianity in the physics and cosmology literature-irrespective of the parametrization [60]. As we will explore in §3, it can also provide insights into parameter identifiability. Hypothesis testing Here we outline the approach for performing likelihood-ratiobased hypothesis tests, and hypothesis tests based on geodesic distance. As we consider synthetic data in this work, we know the true parameter values, θ t . In practical applications this is not the case. As such, we may seek to test whether some previously held notion about the true parameters, θ t = θ 0 , is supported by the data, based on the computed MLE. This could be investigated via the following hypothesis test: H 0 : u t ¼ u 0 and H 1 : u t = u 0 : ' ð2:17Þ From equation (2.3), the test statistic for such a likelihood-ratiobased hypothesis test can be expressed as l LR ¼ À2ð'ðu 0 Þ À 'ðûÞÞ,ð2:18Þ where asymptotically as N → ∞, λ LR ∼ χ 2 (ν), following Wilk's theorem [36]. From the asymptotic relationship given in equation ( l GD ¼ dðu 0 ,ûÞ 2 : ð2:19Þ The likelihood values required to compute equation (2.18) can be obtained directly by evaluating equation (2.1). To compute the geodesic distance between two specific points in parameter space, as required by equation (2.19), it is necessary to solve a boundary value problem to obtain the geodesic curve between θ 0 andû. Approximate p-values can be computed from these test statistics as 1 À F x 2 ðnÞ ðl LR Þ and 1 À F x 2 ðnÞ ðl GD Þ, respectively, where F x 2 ðnÞ is the cumulative distribution function of χ 2 (ν) [1]. We provide practical examples of each of these approaches to hypothesis testing in §3. Numerical implementation All numerical techniques used to produce the results in this work are implemented in the open source Julia language [69]; we use a combination of existing Julia packages and bespoke implementations. There are several aspects of numerical computation in this work, including approximate solutions to systems of ODEs, differentiation with both finite differences and forward mode automatic differentiation, likelihood computation and nonlinear optimization. Nonlinear optimization for obtaining MLEs and parameter combinations corresponding to particular confidence levels is performed with the Julia package NLopt.jl, using the Bound Optimization by Quadratic Approximation (BOBYQA) algorithm. BOBYQA is a derivative-free algorithm for solving bound constrained optimization problems [82]. Approximate solutions to ODEs are obtained using the Julia package DifferentialEquations.jl [83]. The second-order Heun's method [84], a two-stage Runge-Kutta method, is used for obtaining contours of the log-likelihood function to form approximate likelihood-based confidence regions [1]. Heun's method is implemented as Heun() in DifferentialEquations.jl. Approximate solutions to geodesic differential equations are obtained using the Tsitouras implementation of the Runge-Kutta method, which employs Runge-Kutta pairs of orders 5 and 4 [85], implemented as Tsit5() in DifferentialEquations.jl. Boundary value problems for geodesic-distance-based hypothesis tests are solved using the DifferentialEquations.jl implementation of a shooting method, using Tsit5(). Code for reproducing all examples in this work is available on GitHub. Results In this section, we present results combining likelihood-based parameter inference and uncertainty quantification with ideas from information geometry, including geodesic curves and scalar curvature. We apply these techniques to univariate and multivariate normal distributions, linear, exponential and logistic population growth models and the SIR model. Through these canonical examples, we explore pedagogically differences in the inference and information geometry results that arise as we consider parameter estimation and uncertainty for increasingly complex systems. Synthetic data for the univariate and multivariate normal distributions are generated by sampling from the respective distributions given in equation (3.1). For simplicity, in this work we consider synthetic data from uncorrelated observation processes with constant standard deviation in both time and parameter space. However, we note that the techniques presented in this work can be generalized to handle data with non-constant variance and for other distributions where the Fisher information is available [72]. Univariate : x i N ðm, s 2 Þ, Multivariate : x i MVNðm, SÞ, ð3:1Þ where S ¼ diagðs 2 Þ is the covariance matrix. For the population growth and SIR models considered in this work, synthetic data are generated by drawing from a normal distribution with mean described by the model solution and a prescribed standard deviation, effectively substituting μ = μ(θ, t) in equation (3.1) for observation processes with a single variable and μ = μ(θ, t) for observation processes with several variables. When σ is one of the parameters to be estimated, σ ∈ θ, but μ does not depend on σ. Parameter values that we use to generate synthetic data correspond to parameter estimates inferred from field data in the literature [2,16]. We present a series of figures in this section visualizing the normalized log-likelihood,', and scalar curvature, Sc, as heatmaps, with likelihood-based 95% confidence regions and geodesics with a length corresponding to a 95% confidence distance superimposed. All results are computed numerically, as outlined in §2, with code available on GitHub. Unless otherwise indicated, each set of geodesics includes 20 geodesics with initial velocities corresponding to equidistant points uniformly distributed on the circumference of a unit circle. As such, the apparent clustering of geodesics in some examples highlights differences in the scaling and stretching of parameter spaces. Each scalar curvature and log-likelihood heatmap is computed on a uniformly discretized 100 × 100 grid. Normal distributions We first consider parameter inference and information geometry techniques applied to observations drawn directly from univariate and bivariate normal distributions, with no underlying process model. In figure 1, we present results for the univariate normal distribution (1.1), estimating θ = (μ, σ). The true mean and standard deviation used to generate data are (μ, σ) = (0.7, 0.5). Estimates are obtained via maximum likelihood estimation. MLEs of normal variance are known to provide biased underestimates [36], and the derivation of the Fisher information assumes an unbiased estimator [86]. This may partially explain the particular differences observed between the likelihood-based confidence region and the endpoints of the geodesics in figure 1, wherein the geodesics not only appear to suggest a tighter confidence region but also appear to be biased towards parameter space with smaller standard deviation. As the number of observations increases from N = 10 to N = 100, we observe not only that the MLE more precisely estimates the true parameter values, but also that the endpoints of the geodesic curves more closely correspond to the likelihood-based confidence regions. This is consistent with both the theoretical asymptotic relationship between geodesic length and likelihood-based confidence regions given in equation (2.11), and also the bias of the MLE for standard deviation decreasing, as N increases. The manifold representing the family of normal distributions parametrized by θ = (μ, σ) has constant scalar curvature Sc = −1. The probability density function for the multivariate normal distribution with two independent variables, x, y [ R, with constant standard deviation σ is pðx, y; m 1 , m 2 , sÞ ¼ 1 2ps 2 exp À ðx À m 1 Þ 2 þ ðy À m 2 Þ 2 2s 2 ! ! : ð3:2Þ In equation (3.2), there are three parameters that we could estimate from data: Q ¼ ðm 1 , m 2 , sÞ. As we estimated the mean and standard deviation for the univariate normal distribution in figure 1, we consider inference of both means for the multivariate normal, θ = (μ 1 , μ 2 ). Results are presented in figure 2. Even with a small number of observations (N = 10), we observe an excellent match between the likelihood-based confidence regions and geodesics when only estimating means. As expected, increasing N results in an MLE closer to the true values, and tighter confidence regions. We also observe that the confidence regions are symmetric with respect to each mean parameter. When estimating only the mean parameters of the multivariate normal distribution, we see that the scalar curvature is zero everywhere. This is to be expected, as the Fisher information for normally distributed observation processes, equation (2.7), depends only on the standard deviation and not the mean. As such all of the partial derivatives used to construct the Christoffel symbols (2.13) are zero; this vanishing of the Christoffel symbols translates to zero scalar curvature through equations (2.14)-(2.16). We also observe that, in contrast to the evident curvature of the geodesics for the univariate normal case presented in figure 1, the geodesic curves in figure 2 appear perfectly straight when plotted in Euclidean geometry. The Riemann tensor (2.14) is zero everywhere when inferring multivariate normal means. This suggests that the manifold is flat. Results presented in this work predominantly feature 95% confidence regions. We note that, although this choice is common [87], it is also arbitrary, and equivalent analysis could be performed at different confidence levels. In examples where the geodesic endpoints approximately align with the likelihood-based confidence regions at the 95% level, we expect intermediate points along the geodesics to also approximately align with corresponding likelihood contours, in accordance with equation (2.11). However, in examples where we observe a mismatch between geodesic endpoints and likelihood-based confidence intervals at the 95% level, we do not expect intermediate points along geodesics to correspond to likelihood contours. This is demonstrated in figure 3. Having considered the techniques as applied directly to distributions, we now incorporate ODE-based process models, such that our observations are normally distributed about the solution of a mathematical model. Population growth models The canonical logistic growth model, alongside generalizations and related sigmoid models such as Gompertz and royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 Richards' models, have been extensively applied to study population growth dynamics in the life sciences [16,88]. In figure 4, we present data from the literature describing the area covered by hard corals in a region as they regrow following an adverse event. This can be modelled as a logistic growth process [16]. Logistic growth of a population with density C(t) is characterized by a growth rate r > 0, initial condition C(0) > 0 and carrying capacity K > 0. Treating parameter values ðr, Cð0Þ, KÞ ¼ ð0:9131 ðyr À1 Þ, 0:7237%, 79:74%Þ, and standard deviation s ¼ 2:301%, inferred in the literature from this field data as the true values, we generate various synthetic datasets with multiple observations at various time points. The logistic growth model is well approximated by the exponential growth model when C(t) ≪ K [89], and early time exponential growth is approximately linear. Before considering the inference and information geometry techniques as applied to the logistic model, we first consider the more fundamental linear and exponential growth models. In figure 5, we present example synthetic linear and exponential data, and in figure 6 synthetic logistic data. In the context of population growth models, the presence of variability in observations at a single time point could reflect, for example, measurement error, variability in population estimates or expert judgement [90]. Linear growth Linear growth describes growth at a constant rate, independent of the population density. The linear growth model and solution are given by dC dt ¼ a and CðtÞ ¼ at þ Cð0Þ: With parameters Q ¼ ða, Cð0Þ, sÞ, mðQ, tÞ ¼ at þ Cð0Þ describes the expected model behaviour. In figure 7a-f, we present inference results for the linear model for all pairwise combinations of Q. The partial derivatives of the linear model with respect to the parameters a and C(0), required to form the Jacobians, J(θ), are @mðQ, tÞ @a ¼ t and @mðQ, tÞ @Cð0Þ ¼ 1: Recall from equation (2.6) that we only require the partial derivatives corresponding to unknown parameters in any given example. When estimating θ = (a, C(0)) we find that, similar to the multivariate normal case where we estimate means, the scalar curvature is zero everywhere. We also observe that the endpoints of the geodesics align with the likelihood-based confidence region. We stress that this arises through the relationship in equation (2.11), and is not forced to occur via 2), are marked with green discs, with the MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with geodesic lengths corresponding to a theoretical 95% confidence region. Increasing the number of data points, N, tightens the confidence regions. In contrast to the univariate case where we infer standard deviation in figure 1, when only inferring the mean parameters of the multivariate normal distribution, we see that even with few observations, N = 10, the geodesics and likelihood-based confidence regions match closely. As we are estimating means only, and there is no model-induced curvature, the scalar curvature is zero everywhere. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 termination of the numerical solution of the ODE once it reaches the likelihood-based confidence region. However, due to the relationship between a and C(0), we find that the confidence regions in this case are not symmetric about the MLE with respect to each parameter. Rather, we see that for a given normalized log-likelihood value a larger growth rate corresponds to a smaller initial condition, and vice versa. This aligns with our intuition when considering fitting a straight line through data, as presented in figure 5a; lines with a greater slope (a) must start lower (C(0)) to fit the data. When one of the parameters to be estimated is σ, we observe similar results to the univariate normal case; geodesic endpoints are offset in the direction of decreasing σ relative to the likelihood-based confidence regions, and there is constant Exponential growth Exponential growth describes growth at a rate proportional to the size of the population. The exponential growth model and solution are dC dt ¼ aC and CðtÞ ¼ Cð0Þ expðatÞ: With parameters Q ¼ ða, Cð0Þ, sÞ, mðQ, tÞ ¼ Cð0Þ expðatÞ describes the expected model behaviour. The partial derivatives of the exponential model with respect to the parameters a and C(0), required to form the Jacobians, J(θ), are @mðQ, tÞ @a ¼ tCð0Þ expðatÞ and @mðQ, tÞ @Cð0Þ ¼ expðatÞ: By construction, as detailed in figure 5, the linear and exponential models with identical parameters and initial conditions produce very similar behaviours over a sufficiently small time scale. This is seen when comparing the inference results for the exponential model, presented in figure 7g-l, with the corresponding linear results in figure 7a-f. When inferring θ = (a, σ), deviations from the corresponding linear results are minimal. The likelihood-based confidence region and corresponding geodesic endpoints for θ = (a, C(0)) are marginally tighter and less elliptical. When inferring θ = (C(0), σ), we find that the confidence region for the exponential model is narrower with respect to C(0) than that of the linear model, though near-identical with respect to σ. As for the linear case, the scalar curvature is Sc = −1/N everywhere when σ is one of the unknown parameters, and zero everywhere otherwise. Logistic growth Logistic growth describes growth at a rate dependent on the size of the population, with growth ceasing once the population reaches a carrying capacity. For sufficiently small populations relative to the carrying capacity, logistic growth is approximately exponential [89]. As the population approaches the carrying capacity, the rate of growth slows. The logistic growth model is dCðtÞ dt ¼ rCðtÞ 1 À CðtÞ K , with solution CðtÞ ¼ Cð0ÞK Cð0Þ þ ðK À Cð0ÞÞ exp ( À rt) : ð3:3Þ The long-time limit of equation (3.3) is lim t→∞ C(t) = K. The behaviour of the logistic model can be described by the three model parameters and standard deviation: Q ¼ ðr, Cð0Þ, K, sÞ. We can compute the partial derivatives required to form the Jacobian matrices, J(θ), analytically, mðQ, tÞ ¼ Cðr, Cð0Þ, K, tÞ ¼ Cð0ÞK Cð0Þ þ ðK À Cð0ÞÞ exp ( À rt) , @mðQ, tÞ @r ¼ Cð0ÞKtðK À Cð0ÞÞ exp ( À rt) ððK À Cð0ÞÞ exp ( À rt) þ Cð0ÞÞ 2 , @mðQ, tÞ @Cð0Þ ð3:4Þ ¼ K 2 exp (rt)ðCð0Þðexp Recall that θ includes only the unknown parameters to be estimated, so the components required from equation (3.4) to form J(θ) depend on the specific example. Example synthetic logistic data are presented in figure 6, demonstrating the model fits for θ = (r, C(0)), θ = (r, K ) and θ = (r, σ). With data at early, mid-and late time, T = (t 1 , t 2 , t 3 ) = (2.74, 6.84, 10.95) yr, we observe an excellent model fit in all cases. The fit is best when θ = (r, σ), as only one model parameter is unknown. Comparing θ = (r, C(0)) and θ = (r, K) we observe a marginally better fit at late time when K is known, and at early time when C is known, as expected. We present inference results for the logistic model for θ = (r, C(0)) in figure 8a-f and for θ = (r, K ) in figure 8g-l. We do not present further results of inferring σ for the logistic model, as little insight is gained beyond what we glean from the linear and exponential growth results. For θ = (r, C(0)), the normalized log-likelihood reflects the same relationship between growth rate and initial condition as for the linear and exponential cases. With early-mid time data and earlymid-late time data, we are able to infer θ = (r, C(0)). With only mid-late time data, we find that the parameters are not practically identifiable. This can be seen from figure 8c; the normalized log-likelihood remains above the threshold prescribed in equation (2.3), and a closed likelihood-based 95% confidence region cannot be constructed. This is also reflected in figure 8f alongside zero scalar curvature, such that the plot appears empty. Comparing figure 8a,b, and noting that they each rely on the same total number of observations, the importance of early and mid-time data when inferring θ = (r, C(0)) is reinforced. The confidence region is tighter with only early-mid data than with the same amount of data spread across early, mid-and late times. Inferring θ = (r, K ) reflects similar behaviour. In figure 8j and the associated zoomed-in view (figure 8g), inferring the carrying capacity from only early-mid time data results in an extremely wide confidence region, though the parameters remain identifiable. The geodesics emanating from the MLE match the likelihood-based confidence region very well in directions where the normalized log-likelihood is steep; however, they do not quite reach the true parameter value in the direction where the normalized log-likelihood is relatively flat. Comparing figure 8g,j with figure 8h,i, the MLE for θ = (r, K ) appears to be relatively poor when only early-mid time data are used. When considering θ = (r, C(0)), we see that, with earlymid time data and mid-late time data, the scalar curvature is zero everywhere. However, introducing a third time point (early-mid-late data) results in a non-constant negative scalar curvature. We expect that this relates to the royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 relationships between the parameters, and the difference between a mapping (where we have two pieces of information and two parameters to estimate) and a fit (where we have three pieces of information and two parameters to estimate). We do not observe similar behaviour for θ = (r, K ) with data at three time points; the scalar curvature still appears to be zero everywhere. One explanation for this is that data at t 1 , where C(t) ≪ K, may be effectively independent of K, providing no information about K [15]. This may effectively reduce the problem to a mapping. Given that the scalar curvature is a feature of the manifold rather than the data, it is of interest to investigate what would happen were the true parameters to lie within this region of non-constant scalar curvature. To address this, we generate an alternate set of synthetic logistic growth data using parameter values from within the high curvature region, (r, C(0)) = (0.9, 0.2), with (K, σ) = (79.74, 2.301) as before. Inference results are presented in figure 9. We still observe correspondence between the endpoints of the geodesics and the likelihood-based confidence region; however, the confidence region is now significantly narrower and reflects a more hyperbolic shaped relationship between r and C(0) in terms of the normalized log-likelihood. Increasing the number of observations, as depicted in figure 9c, has the expected effects of tightening the confidence region and reducing the scalar curvature. This reduces the apparent curvature of the confidence region. SIR epidemic model The SIR model describes the dynamics of epidemic transmission through a population [2]. Populations are assumed to be composed of susceptible, s(t), infected, i(t), and Figure 8. Logistic growth model with inferred growth rate, r, and initial condition, C(0) (a-f ), and with inferred growth rate, r, and carrying capacity, K (g-l ). True parameters are as noted in figure 6, with known standard deviation, σ = 2.301. Heatmaps visualize the normalized log-likelihood (a-c, g-j ) and the scalar curvature (d-f, k-l ). The true parameter values are marked with green discs, with the MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with lengths corresponding to a theoretical 95% confidence distance. Columns of the figure correspond to observations from early-mid time (T = (t 1 , t 2 )), early-mid-late time (T = (t 1 , t 2 , t 3 )) and mid-late time (T = (t 2 , t 3 )), where (t 1 , t 2 , sufficient for our purposes in this work; however, numerous extensions to the SIR model are considered in the literature. These extensions incorporate factors such as age structure, birth and death, exposed but not yet infected individuals, seasonality, competition between infectious strains, waning immunity, vaccination and spatial structure [2,91-93]. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 Data pertaining to the proportion of a population infected during an influenza outbreak in a boarding school are presented in figure 10. Observations in the original data record the number of infected individuals over a 14-day period [2], in a population of N ¼ 763, with initial populations (s(0), i(0), r(0)) = (762, 1, 0). These data are used in [2] to estimate parameters for the SIR model, which, after scaling such that S + I + R = 1, are β = 1.6633 and γ = 0.44036. We treat these values as the true parameters when generating synthetic data, examples of which are presented in figure 11. In the context of an SIR model, the presence of multiple observations at a single time point could reflect, for example, reporting errors, uncertainty in test accuracy or expert judgement [94,95]. In the boarding school data considered in [2], observations pertain only to the number of infected individuals. Given that the SIR model features multiple populations, data could in theory contain observations of the other populations also. Example synthetic data with observations on all three populations are presented in figure 11b. The SIR model as described in equation (3.5) does not admit a closed form analytical solution, so we apply numerical techniques to solve the system. This becomes somewhat computationally expensive, as the Fisher information computations rely on partial derivatives of the model solution with respect to the parameters to form the model Jacobian, and the information geometry computations require partial derivatives of the Fisher information up to second order. Approximating these partial derivatives using numerical techniques entails solving the system of ODEs several times. Some computational cost may be spared through taking advantage of the known relationship that S + I + R = 1. For brevity, we restrict our investigation of the SIR model to the cases where θ = (β, γ) and θ = (β, σ). Results in figure 12 correspond to the case where observations pertain only to the number of infected individuals, while those in figure 13 are produced from data containing observations of all three populations. In both cases, the results for θ = (β, σ) align with those observed in previous results; the geodesics appear to define a marginally smaller area and are offset from the likelihood-based confidence regions in the direction of decreasing σ and the scalar curvature is the constant Sc = −1/N. Regardless of whether we observe only the infected population or all populations, inferring θ = (β, γ) produces a non-constant positive scalar curvature. In figure 12b, where only I is observed, we see that the geodesics emanating from the MLE extend beyond the likelihood-based confidence region. This also occurs in figure 13b, where all three populations are observed, however it is difficult to perceive at this scale. Based on this result, and the observations involving negative scalar curvature when inferring σ, it might seem that positive scalar curvature produces geodesics that extend beyond corresponding likelihood-based confidence regions, whereas negative scalar curvature has the opposite effect. However, repeating the analysis with different synthetic datasets-generated from a different random seed-suggests that in some cases the geodesics will extend beyond the likelihood-based confidence regions, and in some cases they will fall short, however the scalar curvature remains positive in all cases. Hypothesis testing In figure 14, we present several example hypothesis tests, using both likelihood-ratio-based and geodesic-distancebased approaches, as outlined in §2. Test statistics and corresponding p-values for each hypothesis test are provided in table 1. For the multivariate normal distribution, where we observe that the endpoints of geodesics corresponding to a theoretical 95% confidence distance align closely with the likelihood-based 95% confidence regions, we find that the results of the hypothesis tests are near-identical. Further, the hypothesis test results are consistent with our interpretation of the 95% confidence regions; test points within the confidence regions have p-values greater than 0.05, while test points outside the confidence regions have p-values less than 0.05. We also perform hypothesis tests for the logistic model in the high curvature region of parameter space. Like before, results are comparable for different numbers of observations at each time point, N = (10, 10, 10) and N = (50, 50, 50), as considered in figure 9. Even in this high curvature region, we find that the endpoints of geodesics corresponding to a theoretical 95% confidence distance very closely match the likelihood-based 95% confidence regions. This is again reflected in the results of the hypothesis tests, where very similar results are obtained from the likelihood-ratio-based hypothesis tests and the geodesic-distance-based hypothesis tests, even for relatively extreme θ 0 . As the number of observations increases, we observe for each θ 0 considered that, in accordance with the confidence regions tightening, the test statistics increase and accordingly p-values decrease. As we are using synthetic data and know the true parameters, we can use hypothesis testing to pedagogically investigate Wilks' theorem [36] and the asymptotic relationship given in (2.11). We generate 1000 synthetic datasets and for each dataset perform a hypothesis test for the true parameters. This is repeated for the univariate and multivariate normal distributions with N = 10 and N = 1000 observations. In figure 15, we present densities for both the likelihood-ratio-based and geodesic-distance-based test statistics, alongside the probability density of x 2 2 . For the multivariate normal distribution with θ = (μ 1 , μ 2 ), the density profiles for λ LR and λ GD are near-identical, as expected following the results in figure 14 and table 1. We also observe a good match between these profiles and x 2 2 , even with royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 just N = 10. For the univariate normal distribution with θ = (μ, σ), when N = 10 we observe differences between λ LR and λ GD . Both profiles are similar to x 2 2 , though there appears to be a higher density in the tails of the distributions of the test statistics. As the number of observations increases to N = 1000, the difference between λ LR and λ GD reduces significantly, and both closely match x 2 2 . From Wilks' theorem [36] and (2.11), asymptotically 95% of the 95% confidence regions we construct should contain the true parameter values. We can determine what proportion of the likelihood-based and geodesic-distance-based 95% confidence regions that we construct contain the true parameter values using the information presented in figure 15. This is done by comparing the test statistics with the critical value, Discussion Parameter estimation is wrought with challenges relating to the availability and quality of experimental or field data [8,9,11,12]. This prompts a strong consideration of uncertainty quantification to support point estimation of model parameters [13]. In this section, we discuss the results presented in §3. We highlight opportunities for application of information geometry techniques, including geodesic curves and scalar curvature, to supplement traditional maximum-likelihood-based parameter inference and uncertainty quantification. We conclude by outlining areas for further investigation. Even for relatively small sample sizes, we observe good correspondence between the likelihood-based 95% confidence regions and the endpoints of geodesic curves corresponding to a theoretical 95% confidence distance, in accordance with the asymptotic relationship described in equation (2.11), particularly when estimating model parameters. When estimating standard deviation, as outlined in §3, geodesics appear to suggest a tighter confidence region and appear to be biased towards parameter space with smaller standard deviation. We observe this effect decreasing as the number of observations increases, in line with the known underestimation bias of minimum-likelihood estimates of variance [36]. The misalignment of likelihoodbased confidence regions and geodesic endpoints appears to occur more frequently in examples with non-zero scalar curvature, although we observe a good match in figure 9 despite the non-constant scalar curvature. Visualizing the scalar curvature throughout a parameter space can indicate areas where there may be issues with identifiability. Areas with significant non-constant scalar curvature can suggest a complicated relationship between parameters in terms of the normalized log-likelihood, such as the hyperbolic confidence region observed in figure 9. However, it is possible to produce examples, such as figure 8c,f, where there is practical non-identifiability despite zero scalar curvature everywhere. Although we do not show it here, for the logistic model with θ = (r, K) in the region of parameter space where C(0) ≈ K, computation of the scalar curvature breaks down as the Fisher information matrix becomes singular. Here, it may be obvious that we cannot identify the growth rate, r, from a process that is initialized at its steady state (C(0) = K). However, observing this behaviour in general may help to detect issues with identifiability, particularly for models without analytical solutions. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 The information geometry techniques we discuss are primarily implemented numerically; as such there is a computational cost to consider. For the normal distributions and population growth models in this work, where analytical solutions are available, the information geometry techniques are not disproportionately more computationally expensive than the traditional likelihood-based inference and confidence regions. Examples such as the SIR model, where no analytical solution is available, represent a significantly greater computational burden. However, this impacts both the likelihood-based inference and information geometry techniques as the underlying system of ODEs, for example equation (3.5), must be solved numerous times. The computational cost associated with the information geometry techniques depends significantly on the desired resolution for the scalar curvature surface, and on the number of geodesic curves. A suitable approach may be to first compute the scalar curvature on a coarse grid to identify areas of interest to investigate with a refined grid. Further, the geodesic curves and scalar curvature computations are highly amenable to parallelization, which can significantly reduce computation time. This computational cost will generally pale in comparison with the costs associated with collecting experimental or field data, and may be easily justified if the information geometry techniques are used to guide data collection. If information geometric analysis identifies a region of parameter space with significant non-constant scalar curvature for a model, such as in figure 9, and practitioners have a prior expectation that the true parameter values fall somewhere within this region, this may indicate that a greater quantity or quality of data is needed to improve identifiability for that particular model. Alternatively, such analysis may guide practitioners in choosing favourable experimental conditions; for example in cell culture experiments, where it is possible to vary the initial cell seeding density [1]. Experimental design is a process wherein experiments are performed or simulated iteratively with perturbations, such that some measure of information is maximized. Through this process, the most informative experiments are identified, facilitating design of optimal experimental protocols [96][97][98]. Common to these approaches is the importance of quantifying and comparing information. While we do not consider optimal experimental design in this work, there is potential to incorporate information geometric techniques in the experimental design process as a means of comparing information between experimental perturbations. This is an area for further investigation. Although we focus on how information geometry can supplement traditional maximum-likelihoodbased inference and uncertainty quantification, primarily through visualization, it should be noted that concepts from information geometry have also found application in the inference context from a computational efficiency standpoint. For example in Bayesian inference, by defining Monte Carlo sampling methods on a Riemann manifold, the geometric structure of the parameter space can be exploited [99]. Simulated paths across the manifold automatically adapt to local structure, facilitating efficient convergence, even in higher dimensions and in the presence of strong correlation Figure 14. Example hypothesis tests for the: (a) univariate normal distribution, with θ = (μ, σ),û ¼ ð0:5050, 0:4846Þ; (b) multivariate normal distribution, with θ = (μ 1 , μ 2 ),û ¼ ð0:7109, 1:1498Þ; logistic model with θ = (r, C(0)) in the high curvature region as considered in figure 9, with (c) N = (10, 10, 10), u ¼ ð0:9195, 0:1723Þ, and (d ) N = (50, 50, 50),û ¼ ð0:9287, 0:1682Þ. In each case, we test several example hypotheses, θ 0 , marked by coloured discs. Geodesics between the MLEs (red discs) and each θ 0 are shown in red. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with lengths corresponding to a theoretical 95% confidence distance. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 [99,100]. Concepts from information geometry, including geodesic curves, are also implemented in methods for model reduction [101]. These applications of information geometry techniques to improve computational algorithms highlight further utility of geometric concepts for inference in higher dimensions, beyond that which we demonstrate through visualization in this work. Geodesics can be used to measure the distance between probability distributions. As demonstrated in §3, it is possible to perform hypothesis tests based on geodesic distance [48,51,78]. The approach for performing a hypothesis test is to solve a boundary value problem to find the geodesic connecting two points in parameter space, and use the corresponding geodesic distance to compute a test statistic. For the examples considered in this work, such boundary value problems are readily solved numerically using standard techniques, such as those included in the Julia package DifferentialEquations.jl [83]. Careful numerical handling may be required for geodesic curves close to boundaries of parameter space. For more complicated examples, particularly those in high-dimensional manifolds, achieving converging solutions to geodesic boundary value problems can prove challenging. There is scope for a review of the different numerical methods for solving boundary value problems, with a particular focus on their applicability to solving geodesic boundary value problems for hypothesis testing in high-dimensional manifolds. In this work, we only consider models that admit unimodal likelihoods. In cases where the likelihood is multimodal, provided that we are able to obtain the Fisher information required to compute the Christoffel symbols, we are still able to compute the scalar curvature and perform hypothesis tests based on geodesic distance. With multimodal likelihoods, it would not be possible to construct confidence regions from geodesics emanating from the MLE. Although, we note that constructing confidence regions for multimodal likelihoods is also problematic with traditional likelihood-based inference methods. There are several avenues for future research in this area. Here, we consider two-dimensional manifolds to facilitate convenient visualization; however, the inference and information geometry techniques are general, and can be readily applied to higher dimensional manifolds [36,59], albeit with increased computational cost. Extending this analysis to three dimensions would enable consideration of situations where there is scalar curvature associated both with the variability of the observation process, σ, and also with interactions between model parameters; for example, it may be insightful to consider θ = (β, γ, σ) for the SIR model, where we associate a constant negative scalar curvature with σ and non-constant positive scalar curvature due to interactions between β and γ. In three dimensions, likelihood-based confidence regions can be visualized as a series of two-dimensional slices oriented in three-dimensional space [1]; this technique could be applied to visualize slices of the scalar curvature in three dimensions. One approach for visualization in higher dimensions is to produce an ensemble of these two-or three-dimensional confidence regions for various combinations of parameters of interest, with other parameters fixed at their MLEs. Alternatively, in higher dimensions it may be more appropriate to use non-visual techniques, such as hypothesis testing. While we have considered ODE models, there is appetite in the literature for parameter estimation, uncertainty quantification and identifiability analysis for more complicated models, including partial differential equations, stochastic differential equations (SDEs) and delay differential equations [19,102,103]. This appetite extends to non-differential-equation-based models, including agent-based models [104] and network models [105]. A natural extension of this work is to present examples demonstrating how the information geometry techniques can be applied to these more complicated models. This will introduce new challenges, though it may be possible to leverage existing techniques; for example, linear noise approximation may be used to obtain a representation of the Fisher information matrix for SDEs [24]. Further, we fix σ across observation times, model parameters and populations. However, the techniques presented in this work can be generalized to handle data with non-constant variance [72]; the expression for the Fisher information matrix given in equation (1.4) can be extended to account for a parameter-dependent covariance matrix [106]. Investigation of examples paralleling those in §3, but with non-constant standard deviation, may prove insightful. Here, the Fisher information defines a Riemann metric on the statistical manifold. For some inference problems, it is not practical to obtain the Fisher information. Where the Fisher information is not available, the sample-based observed information-computed as negative the Hessian of the log-likelihood function, or via Monte Carlo methods-may be available [107,108]. The observed information has been demonstrated to equip a manifold with an observed geometric structure akin to the expected geometric structure associated with the Fisher information [109]. Further work could identify the viability of the techniques presented here in situations where only the observed information is available, particularly for local approximation about the MLE. Owing to the additive nature of the Fisher information, having N observations results in a constant scalar curvature of Sc = −1/N, as presented in royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 figure 1c,d. It is straightforward, although tedious, to verify this result through combining equations (1.4), (2.13)-(2.16). Figure 1 . 1Univariate normal distribution with inferred mean, μ, and standard deviation, σ. Heatmaps visualize the normalized log-likelihood,' (a,b), and the scalar curvature, Sc (c,d ). True parameter values, (μ, σ) = (0.7, 0.5), are marked with green discs, with the MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with a geodesic length corresponding to a theoretical 95% confidence region. Increasing the number of data points, N, tightens the confidence regions, improves the correspondence between geodesic curves and likelihoodbased confidence regions and reduces the scalar curvature. Figure 2 . 2Multivariate normal distribution with inferred means, μ 1 and μ 2 , with known constant standard deviation, σ = 0.3. Heatmaps visualize the normalized log-likelihood,' (a,b), and the scalar curvature, Sc (c,d ). True parameter values, (μ 1 , μ 2 ) = (0.8, 1. Figure 3 . 3Comparison of confidence regions at intermediate-likelihood values and geodesic distances. Results correspond to (a) univariate normal distribution with inferred mean, μ, and standard deviation, σ, as considered in figure 1a, and (b) multivariate normal distribution with inferred means, μ 1 and μ 2 , as considered in figure 2a. MLEs are indicated using red discs. Dashed curves correspond to likelihood-based 50% (green), 90% (blue) and 95% (orange) confidence regions. Solid lines are geodesic curves emanating from the MLEs, with geodesic lengths within a theoretical 50% (green), 90% (blue) and 95% (orange) confidence distance. Figure 4 . 4Markers correspond to data from field studies, representing the percentage of area in a region covered by hard corals, as the coral population regrows following depletion by an external event[16]. Data originally extracted from the Australian Institute of Marine Science (AIMS) Long Term Monitoring Program (LTMP) eAtlas (eatlas.org.au/gbr/ltmp-data). A logistic model is fitted to the data in[16], with inferred parameters: r = 0.9131 (yr −1 ), Cð0Þ ¼ 0:7237%, K ¼ 79:74% and standard deviation σ = 2.301; this is reproduced here as the green curve.royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 scalar curvature of Sc = −1/N. The geodesics and confidence regions appear symmetric with respect to the model parameter, about the MLE. Figure 5 . 5Example synthetic data generated from the linear and exponential models with comparison of early time linear and exponential model fits, inferring a and C(0). N = 10 observations per time point, with time points T = (0.1, 0.25, 0.5). True parameter values are a = 0.9131, C(0) = 0.7237, with known standard deviation, σ = 0.2301. For generating synthetic early time linear and exponential data, we reduce the standard deviation relative to the σ = 2.301 computed from the logistic model, as early time data produced with C(0) = 0.7237 and σ = 2.301 produce negative population density observations. Inference produces MLEs of ðâ,Ĉð0ÞÞ ¼ ð0:8988, 0:6642Þ for the linear model and ðâ,Ĉð0ÞÞ ¼ ð0:9412, 0:6695Þ for the exponential model. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 Figure 6 . 6Example synthetic data generated from the logistic growth model. The logistic model is fitted to the synthetic data, inferring pairwise combinations of r with C(0), K and σ. Observations are made at T = (2.74, 6.84, 10.95) years, with N = 10 observations per time point. True parameter values are r = 0.9131, C(0) = 0.7237, K = 79.74 and σ = 2.301. Figure 7 . 7Linear (a-f ) and exponential (g-l ) models with inferred pairwise combinations of growth rate, a, initial condition, C(0), and standard deviation, σ. Heatmaps visualize the normalized log-likelihood,' (a-c, g-i), and the scalar curvature, Sc (d-f, j-l ). Observations are made at T = (0.1, 0.25, 0.5), with 10 observations per time point, corresponding to the example data presented infigure 5. The true parameter values are marked with green discs, with the MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with lengths corresponding to a theoretical 95% confidence distance. True values of model parameters correspond to the logistic growth parameters; a = 0.9131, C(0) = 0.7237, with reduced standard deviation σ = 0.2301. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 13 recovered, r(t), individuals. The total population, N, is held constant. When analysing the SIR model in this work, we consider each population as a proportion of the total population, such that SðtÞ ¼ sðtÞ=N, IðtÞ ¼ iðtÞ=N and RðtÞ ¼ rðtÞ=N. Quantities N, s(t), i(t) and r(t) are dimensional with dimensions of number of individuals, whereas S(t) ∈ [0, 1], I(t) ∈ [0, 1] and R(t) ∈ [0, 1] are dimensionless quantities with the property that S(t) + I(t) + R(t) = 1. While the coral re-growth process considered in the population model examples takes place over many years, epidemics occur over a time scale of days or weeks. As such, we now take t to represent time as measured in days, rather than years. The parameters of the SIR model are the infection rate, β (d −1 ), and the rate at which infected individuals are removed, γ (d −1 ), for example, via recovery from the > > > > > ;ð3:5ÞAlongside β and γ we could also treat the initial conditions, S(0), I(0) and R(0), as unknown parameters to be estimated. The standard SIR model presented in equation ( Figure 9 . 9Logistic growth model with inferred growth rate, r, and initial condition, C(0), with known standard deviation, σ = 2.301, and carrying capacity, K = 79.74. Heatmaps visualize the normalized log-likelihood (a) and the scalar curvature (b,c). Data are observed at T = (2.74, 6.84, 10.95), with 10 (a,b) and 50 (c) observations per time point. The true parameter values are marked with green discs, with MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are 100 geodesic curves emanating from the MLEs, with lengths corresponding to a theoretical 95% confidence distance. Figure 10 . 10Data marked with red discs represent the number of infected individuals during an influenza outbreak in a boarding school[2]. Susceptible, S(t), infected, I(t), and recovered, R(t), populations are modelled according to equation (3.5) based on parameters inferred in[2], β = 1.6633, γ = 0.44036; we treat these as the true parameters when generating synthetic data. Initial population proportions are S(0) = 762/763, I(0) = 1/763 and R(0) = 0. Δ 2,0.05 , from(2.3). For the multivariate normal distribution with N = 10 we find that 95.7% of the likelihood-based and geodesic-distance-based confidence regions contain the true parameter values. With N = 1000 we find that 94.observing S, I and R, inferring b, g Figure 11 .Figure 12 . 1112Example synthetic data generated from the SIR model under the scenarios where: (a) only the number of infected individuals is observed and (b) we have observations pertaining to all three populations. Observations are marked with discs. Populations are modelled according to equation (3.5) based on parameters inferred in [2]. Initial population proportions are S(0) = 762/763, I(0) = 1/763 and R(0) = 0. In (a) there are N = 10 observations at each time point, and we prescribe σ = 0.05; in (b) there are three observations per time point, per population, with prescribed σ = 0.03. The choices of σ are sufficiently small that the data generated consist only of positive observed population proportions. Inferring θ = (β, γ) in (a,b) and θ = (β, σ) in (c,d ) for the SIR model with observations only on the number of infected individuals. Observations in the synthetic data occur at T = (4, 7, 10), with N = 10 observations per time point. True parameters, (β, γ, σ) = (1.66334, 0.44036, 0.05), are marked with green discs, with MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with lengths corresponding to a theoretical 95% confidence distance. Initial populations are as described infigure 11. Figure 13 . 13Inferring θ = (β, γ) in (a,b) and θ = (β, σ) in (c,d ), with observations on all three variables, S, I and R. Observations in the synthetic data occur at T = (4, 7, 10), with three observations of each population at each time point; 27 observations in total, as depicted in figure 11b. True parameters, (β, γ, σ) = (1.66334, 0.44036, 0.03), are marked with green discs, with MLEs indicated using red discs. Magenta curves correspond to likelihood-based 95% confidence regions. Black lines are geodesic curves emanating from the MLEs, with lengths corresponding to a theoretical 95% confidence distance. Initial populations as described infigure 11. Figure 15 . 15Step histograms show the density of the distribution of test statistics for each hypothesis testing approach, for (a,b): the univariate normal distribution with θ = (μ, σ), and (c,d ): the multivariate normal distributions with θ = (μ 1 , μ 2 ). Test statistics are computed from the true parameter values and the MLE, for 1000 sets of synthetic data. Datasets represented in (a,c) contain N = 10 observations, while in (b,d) N = 1000. Purple curves correspond to the density of the x 2 2 distribution, while blue dotted lines represent the likelihood-ratio-based test statistics and orange dashed lines represent the geodesic-distance-based test statistics. 2.11), it follows that under the same asymptotic relationship the test statistic for a hypothesis test based on geodesic distance royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940is [78] (rt) À 1Þ þ KÞ 2and @mðQ, tÞ @K ¼ Cð0Þ 2 exp (rt)ðexp (rt) À 1Þ ðCð0Þðexp (rt) À 1Þ þ KÞ 2 : 9 > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > ; royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 the true parameters, approaching the theoretical 95%. For the univariate normal distribution with N = 10 we find that 93.2% of the likelihood-based confidence regions contain the true parameter, while only 88.0% of the geodesic-distancebased confidence regions contain the true parameters. With N = 1000, we find that 95.2% of the likelihood-based confidence regions and 95.1% of the geodesic confidence regions contain the true parameters. Table 1 . 1Hypothesis test results. royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940model θ 0 λ LR λ GD p LR p GD multivariate normal (0.8,1.0) 3.3737 3.3737 0.1851 0.1851 (0.9,1.4) 10.9297 10.9297 0.0042 0.0042 univariate normal (0.6,0.3) 7.5051 5.1954 0.0235 0.0744 (0.6,0.6) 1.0460 1.2201 0.5927 0.5433 (0.6,0.85) 4.6134 6.5226 0.0996 0.0383 (0.6,1.0) 6.9271 10.6665 0.0313 0.0048 logistic (10, 10, 10) (1.0,0.095) 2.7086 2.5798 0.2581 0.2753 (0.87,0.25) 1.6387 1.5626 0.4407 0.4578 (0.92,0.21) 30.0130 29.9821 3.0391×10 −7 3.0865×10 −7 (0.9,0.15) 56.2776 56.5328 6.0185×10 −13 5.2969×10 −13 logistic (50, 50, 50) (1.0,0.095) 31.3038 31.0222 1.5939×10 −7 1.8349×10 −7 (0.87,0.25) 4.2062 4.2276 0.1221 0.1208 (0.92,0.21) 97.6247 97.5164 <10 −16 <10 −16 (0.9,0.15) 368.1479 368.7335 <10 −16 <10 −16 t 3 ) = (2.74, 6.84, 10.95) yr. Each plot reflects a total of 30 observations, distributed equally between the specified time points. The red outline in ( j ) corresponds to the (zoomed in) region (g), also outlined in red. In (g,j ), we plot 1000 geodesics to observe the geodesic near the true parameter values. We do not present Sc corresponding to (g,j ); however, it is zero everywhere.royalsocietypublishing.org/journal/rsif J. R. Soc. Interface 19: 20210940 Data accessibility. Data and code are made available on GitHub. Authors' contributions. J.A.S.: conceptualization, formal analysis, investigation, methodology, software, visualization, writing-original draft, writing-review and editing; A.P.B.: conceptualization, investigation, methodology, software, writing-review and editing; K.B.: conceptualization, methodology, supervision, writing-review and editing; M.J.S.: conceptualization, funding acquisition, methodology, project administration, supervision, writing-review and editing.All authors gave final approval for publication and agreed to be held accountable for the work performed herein.Conflict of interest declaration. We declare we have no competing interests. Acknowledgements. 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[ "Silicon Photonics Wavelength-Independent C-Band Tunable Optical Filter With Feasible Thermal Tuning Requirements", "Silicon Photonics Wavelength-Independent C-Band Tunable Optical Filter With Feasible Thermal Tuning Requirements" ]	[ "Saif Alnairat [email protected] \nADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany\n", "Benjamin Wohlfeil \nADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany\n", "Stevan Djordjevic \nADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany\n", "Bernhard Schmauss \nADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany\n" ]	[ "ADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany", "ADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany", "ADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany", "ADVA Optical Networking SE\nMärzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany" ]	[]	A filter design based on Vernier microrings and wideband directional couplers is proposed for ASE noise suppression in next generation DCI applications. We demonstrate a ∼40 nm FSR-free filter with > 20.5 dB average ER and 3dB-BW of ∼75 GHz, achieving wavelength-independent performance and full tunability with a maximum tuning temperature of ∼75 K.	null	[ "https://export.arxiv.org/pdf/2304.11096v1.pdf" ]	254,930,131	2304.11096	82568b0e6c1ce2c70b42e0c07a691851dd07bb39	Silicon Photonics Wavelength-Independent C-Band Tunable Optical Filter With Feasible Thermal Tuning Requirements Saif Alnairat [email protected] ADVA Optical Networking SE Märzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany Benjamin Wohlfeil ADVA Optical Networking SE Märzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany Stevan Djordjevic ADVA Optical Networking SE Märzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany Bernhard Schmauss ADVA Optical Networking SE Märzenquelle 1, 2) LHFT, Friedrich-Alexander Universität Erlangen-Nürnberg98617, 91058Meiningen, ErlangenGermany, Germany Silicon Photonics Wavelength-Independent C-Band Tunable Optical Filter With Feasible Thermal Tuning Requirements A filter design based on Vernier microrings and wideband directional couplers is proposed for ASE noise suppression in next generation DCI applications. We demonstrate a ∼40 nm FSR-free filter with > 20.5 dB average ER and 3dB-BW of ∼75 GHz, achieving wavelength-independent performance and full tunability with a maximum tuning temperature of ∼75 K. Introduction In order to deploy > 25 Tbit/s transmission systems in next-generation Data Center Interconnect (DCI) networks, performance challenges have to be resolved [1], [2] . One of the most critical challenges facing high-speed, filterless, i.e., with no Arrayed Waveguide Grating (AWG) filter, Dense Wavelength-Division Multiplexing (DWDM) transceivers is Optical Signal to Noise Ratio (OSNR) degradation, due to accumulated Amplified Spontaneous Emission (ASE) noise originating from optical amplifiers. Thus, tunable ASE optical filters are essential components for advanced optical transceivers and transmission systems to achieve the OSNR tolerance threshold. Micro-Ring Resonators (MRRs) have been widely used in communication applications due to thier small footprint and scalability [3], [4] . Over the past decade, various integrated silicon MRRsbased optical filters have been explored and different types of configurations have been analyzed. Tunable optical filters based on series coupled and cascaded MRRs in addition to Vernier filters with a tuning range of up to 124 nm were fabricated and reported [5]- [10] . However, designs tunable over > 35 nm range either require very high tuning temperature ∆T = (125 -400) K, or exhibit insufficient 3-dB bandwidth (3dB-BW), low extension ratio (ER) and high insertion loss (IL) to pass through single wavelength channels while simultaneously suppressing the remaining C-band. Furthermore, the performance of these designs varies across their tuning range due to chromatic dispersion and wavelength dependency of the MRRs couplings strength across the tuning range [6], [10] . In addition to varying performance, under-coupling increases the complexity of resonance alignment and automatic tuning due to resonance splitting [6]- [8] . In this work, we propose a novel filter design based on Vernier MRRs and wideband directional couplers, to overcome thermal challenges and wavelength-dependent performance. A two-stage coupled MRR Vernier wavelengthindependent filter example design suitable for dual-polarization (DP) 64-Quadrature Amplitude Modulation (QAM)/64-GBd systems is presented, and its performance is compared to a conventional Vernier filter design. The filter is automatically tuned achieving (73 -75) GHz 3dB-BW and 20.5 dB average ER over the C-band with a maximum ∆T = ∼75 K, realizing wavelengthindependent performance. Design optimization The required properties of the filter depends on the application and the requirements of the system where it is employed. The design approach was first to model an AWG-less DWDM optical communication system which utilizes DP-64-QAM/64-GBd transmitters to investigate and analyze the required properties of the ASE filter, including performance in the aspect of OSNR requirement of the system. The model shows that for a 66-channel system, an average ER of > 17 dB is required to achieve the OSNR threshold for the mentioned system. Thus, the design targets 20 dB average ER and 80 GHz 3dB-BW to increase spectral misalignment tolerance. The design configuration is chosen to achieve the desired characteristics taking into account design limitations, including tunability, thermal challenges, complexity, power consumption and tolerance to fabrication variations. Fig. 1(a) shows a schematic for the design configuration with first and second stages being composed of two different second-order Vernier filters. Considering a silicon-on-insulator (SOI) platform for this design and assuming thermo-optic coefficient of 1.86 × 10 −4 K −1 with a maximum ∆T heater = ∼75 K, which would give a ∆T wg = ∼69 K based on thermal simulation for the designed phase shifters, the minimum resonatorlength (L min ) allowed to achieve 2π phase shift is 121 µm. Giving the group index at 1.55 µm is 3.96, this gives a free spectral range (FSR) of ∼ 5 nm for L min . The lengths of different resonators on both filters are chosen such that the total required FSR, i.e. 40 nm, is the least common multiple of the FSR of individual resonators, and the secondary resonances of the cascaded filter output are mutually suppressed, as shown in Fig. 1(b) To simplify the design, we choose (κ out = κ 11 = κ 13 = κ 21 = κ 23 ) and (κ in = κ 12 = κ 22 ). In this case, the coupling strengths required to achieve the targeted 3dB-BW and average ER with a flat bandpass at 1.55 µm were found to be κ in = 0.18 and κ out = 0.61. Due to chromatic dispersion and wavelength dependency of the power couplings for conventional couplers designs, under and over coupling usually occur, which leads to narrower 3dB-BW with higher ER and larger 3dB-BW with lower ER at the lower and upper end of the spectrum, re-spectively, as shown in Fig. 1(b) and Fig. 1(c). To realize uniform coupling strength over the entire tuning range, inner (ring-ring) and outer (bus-ring) wideband couplers are used. Fig. 1(d) shows simulated cross coupling splitting ratios for the designed conventional and wideband outer and inner ring-couplers, where conventional couplers based on symmetric straight couplers show ±14% variation across the tuning range. The wideband couplers are designed using asymmetric-waveguide based phase control sections to introduce a small phase shift between the two symmetric couplers and compensate for the wavelength-dependent power coupling [11] . In order to realize inner wideband ring-coupler, two different outer wideband ring-couplers designs are needed. Fig. 1(e) shows the layout of the fabricated wideband filter and the schematic of the wideband coupler. All couplers have similar bend radius R = 10 µm, taper length = 1 µm, W 1 = 400 nm, W 2 = 600 nm and W 3 = 500 nm. The rest of the design parameters are summarized in Tab. 1. The resonator lengths of the conventional design are slightly adjusted to take into consideration the change of the effective index in the phase section. To increase tuning flexibility, a power-tap coupler is added at the drop port of the first stage. Measurement, Results and Discussion To measure the cross splitting ratio of the fabricated conventional and wideband directional couplers, a continuous tunable C-band laser and a power meter were coupled to the couplers test structures using a fiber array and grating couplers. The laser wavelength was swept with 0.1 nm step across (1530 nm -1565 nm) wavelength range and the power reading was normalized to the grating coupling loss. Fig. 2(a) shows the cross splitting ratio of conventional and wideband couplers. The inner and outer ring-couplers measured splitting ratios at 1550 nm for conventional and wideband designs are {0.205, 0.58} and {0.163, 0.62}, respectively, which means that the conventional filter design is under-coupled while the wideband filter design is slightly over-coupled. Conventional couplers show ∼ ±15% coupling variation across the tuning range, while wideband couplers show only ±1% variation. To measure and tune the filter spectra, the input port was connected to the laser, and the drop ports for the first stage, i.e., the power tap, and the cascaded filter were connected to photodetectors and feedback control circuitry. The tunable laser source was set to the desired resonance wavelength and an automatic resonance alignment algorithm was applied to tune the optical filter. A combination of modified derivative-based and derivative-free optimization algorithms were adapted based on thermal characterization of the device. An additional step was required to fine tune the conventional filter spectrum due to reso-nance splitting. The IL of conventional power-tap coupler was subtracted from the measured transmission. Fig. 2(b) shows the measured drop port peak at several resonance wavelengths for the conventional filter design. The filter has a ∼40 nm FSR, an average ER of 21 dB and a minimum outof-band suppression of 14 dB. However, the difference in 3dB-BW between the upper and lower end of the C-band is around 15 GHz, while IL varies between 0.9 and 1.4 dB. Furthermore, the resonance splitting due to under-coupling is evident at the lower end of the tuning range. On the other hand, Fig. 2(c) shows the measured drop port peak for the wideband filter design, where the difference in 3dB-BW across the tuning range is less than < 1.5 GHz. The IL is around 1 dB while the average ER is 20.5 dB. The resonance alignment requires only 250 iterations for each tuning process, which is much faster than previously reported for similar device configurations [6] . The power required to tune each thermal phase shifter by 2π is around 80 mW. To increase tolerance to fabrication variations, outer wideband couplers could be designed such that they have similar phase section but different bend radii as well, while keeping the current design for the inner ring-coupler. Conclusions In this paper, we have proposed and experimentally demonstrated a two-stage secondorder MRR Vernier wavelength-independent filter based on wideband directional couplers. The device was automatically tuned across the Cband, achieving < 1.5 GHz 3-dB bandwidth variation and realizing wavelength-independent performance with feasible thermal tuning requirements. Fig. 1 : 1(a) Schematic of a two-stage coupled MRR Vernier filter. Thermal Phase shifters (φ i ) and ring-couplers (κ i ) for both stages are highlighted. (b) Comparison of the simulated drop port transmission spectra of each stage of the Vernier filter and the spectra of the cascaded filter with conventional and wideband directional couplers. (c) Close-up comparison of the drop port peak at 1.53 and 1.57 µm for the conventional cascaded filter design. (d) Simulated cross splitting ratio of conventional and wideband directional couplers for the inner and outer ring-couplers. (e) The layout of the fabricated wavelength-independent filter and a schematic of the wideband ring-couplers. Fig. 2 : 2(a) Measured cross splitting ratio of conventional and wideband directional couplers for the inner and outer ring-couplers. (b) Close-up comparison of the measured drop port peak at 1.53, 1.55 and 1.57 µm for the conventional cascaded filter design.(c) Close-up comparison of the measured drop port peak at 1.53, 1.55 and 1.57 µm for the wideband cascaded filter design. AcknowledgementsThis work has been partially funded by the German Ministry of Education and Research under the grant agreement 13N14937 (PEARLS). The future of optical interconnects for data centers: A review of technology trends. S Aleksic, 10.23919/ConTEL.2017.8000037DOI: 10 . 23919/ConTEL.2017.80000372017 14th International Conference on Telecommunications (Con-TEL). Zagreb, CroatiaS. Aleksic, "The future of optical interconnects for data centers: A review of technology trends", in 2017 14th International Conference on Telecommunications (Con- TEL), Zagreb, Croatia, 2017, pp. 41-46. 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[ "Optimal Procedures for Multiple Testing Problems", "Optimal Procedures for Multiple Testing Problems" ]	[ "Saharon Rosset ", "Ruth Heller ", "Amichai Painsky ", "Ehud Aharoni " ]	[]	[]	Multiple testing problems are a staple of modern statistical analysis. The fundamental objective of multiple testing procedures is to reject as many false null hypotheses as possible (that is, maximize some notion of power), subject to controlling an overall measure of false discovery, like family-wise error rate (FWER) or false discovery rate (FDR). In this paper we formulate multiple testing of simple hypotheses as an infinite-dimensional optimization problem, seeking the most powerful rejection policy which guarantees strong control of the selected measure. In that sense, our approach is a generalization of the optimal Neyman-Pearson test for a single hypothesis. We show that for exchangeable hypotheses, for both FWER and FDR and relevant notions of power, these problems can be formulated as infinite linear programs and can in principle be solved for any number of hypotheses. We apply our results to derive explicit optimal tests for FWER or FDR control for three independent normal means. We find that the power gain over natural competitors is substantial in all settings examined. We also characterize maximin rules for complex alternatives, and demonstrate that such rules can be found in practice, leading to improved practical procedures compared to existing alternatives.f T (u) = Π T k =1 g(u k ) , ∀u ∈ [0, 1] K ,	10.1111/rssb.12507	[ "https://arxiv.org/pdf/1804.10256v2.pdf" ]	88,524,095	1804.10256	dbd65703bccce3f90c6eeabd59dc9bab0f89f809	Optimal Procedures for Multiple Testing Problems April 30, 2018 Saharon Rosset Ruth Heller Amichai Painsky Ehud Aharoni Optimal Procedures for Multiple Testing Problems April 30, 2018 Multiple testing problems are a staple of modern statistical analysis. The fundamental objective of multiple testing procedures is to reject as many false null hypotheses as possible (that is, maximize some notion of power), subject to controlling an overall measure of false discovery, like family-wise error rate (FWER) or false discovery rate (FDR). In this paper we formulate multiple testing of simple hypotheses as an infinite-dimensional optimization problem, seeking the most powerful rejection policy which guarantees strong control of the selected measure. In that sense, our approach is a generalization of the optimal Neyman-Pearson test for a single hypothesis. We show that for exchangeable hypotheses, for both FWER and FDR and relevant notions of power, these problems can be formulated as infinite linear programs and can in principle be solved for any number of hypotheses. We apply our results to derive explicit optimal tests for FWER or FDR control for three independent normal means. We find that the power gain over natural competitors is substantial in all settings examined. We also characterize maximin rules for complex alternatives, and demonstrate that such rules can be found in practice, leading to improved practical procedures compared to existing alternatives.f T (u) = Π T k =1 g(u k ) , ∀u ∈ [0, 1] K , Introduction In a classic hypothesis testing problem, we are given null and alternative hypotheses, and we wish to find good statistical tests for this problem. A good test is expected to be valid and have the desired probability of rejection under the null model, while being powerful and having a high probability of rejection under the alternative. When the hypotheses are both simple and fully specify the distribution of the data, the Neyman-Pearson (NP) Lemma characterizes the most powerful test at every given level, as rejecting for high values of the likelihood ratio, which is simply the ratio of the probability of the data under the alternative hypothesis, to its probability under the null. This "most powerful test" problem can be viewed as an optimization problem, where every point in sample space has to be assigned to reject or non-reject regions, in a manner that maximizes the expected rejection under the alternative distribution, subject to a constraint on its expectation under the null. When the sample space is infinite (such as a Euclidean space), this is an infinite dimensional integer optimization problem, whose optimal solution happens to have the simple structure characterized by NP. When moving from testing a single hypothesis to multiple testing scenarios, several complications are added. First, there is no longer a single universally accepted definition of validity and false discovery. Given a rejection policy, denote the (random) number of rejected hypotheses by R, and the number of falsely rejected hypotheses (true nulls) by V . Two commonly used measures of 1 arXiv:1804.10256v1 [stat.ME] 26 Apr 2018 false discovery, which we denote generically by Err, are: FWER: P(V > 0); FDR: E V R ; R > 0 Furthermore, even once selecting a criterion, different notions of validity exist like only requiring "weak control" under the global null vs "strong control" under all true null configurations. In this paper we concentrate on FWER and FDR under strong control. Second, there is no longer a single notion of power. For example, we may seek a test which maximizes the expected number of rejections if all nulls are false, or one which maximizes our chance of correctly rejecting a single false null, or we may want to maximize the expected number of true rejections under some (prior, estimated or known) distribution on the percentage of false nulls, as in the Bayesian approach to FDR [12,11]. The chosen definition should capture the true "scientific" goal of the testing procedure and the type of discoveries we wish to make. However, once we choose a validity criterion and a power criterion, we can write the problem of finding the optimal test as an optimization problem. Our focus in this paper is in developing theory and algorithms for such Optimal Multiple Testing (OMT) procedures, and investigating the implications that our results have on design of practical multiple testing procedures. Our main result, presented in § 2, is that the goal of finding OMT procedures is attainable in theory for any multiple testing problem, by using calculus of variations results and specific properties of these problems to characterize the optimal solution and devise algorithms for finding it. In § 2.2 we derive one detailed algorithm, and in § 3 we apply our algorithms to find OMT procedures for three normal means with FWER or FDR control. The resulting OMT procedures are much more powerful than relevant alternatives. In § 4 we tackle the more complex -and more realistic -setup where the alternative considered is complex. For this setting we formulate sufficient criteria for optimality and maximin solutions, and demostrate that these sufficient conditions hold in interesting examples, allowing us to find OMT procedures for complex alternatives as well. Problem formulation and notation We assume that our testing problems are already formulated in terms of p-values, properly derived from the data. For a single testing problem, this amounts to assuming: H 0 : U ∼ U (0, 1); H A : U ∼ G, where G(·) is a continuous distribution with density g(·) that is monotone decreasing in its argument. Consequently, the Neyman-Pearson most powerful test is simply to reject for R * (α) = {U ≤ α}. The power of this test is Π = G(α). As an example, consider the elementary test of a normal observation X ∼ N (µ, 1), with H 0 : µ = 0; H A : µ = C < 0. It is easy to see that setting U = Φ(X), where Φ is the standard normal cumulative distribution, gives the desired representation: In a multiple testing problem, we have not one pair of hypotheses, but K > 1 pairs, each representing a separate testing problem. As above, denote test k by: H 0k : U k ∼ U (0, 1); H Ak : U k ∼ G k . We denote the true states of all K tests by the fixed (yet unknown) vector T ∈ {0, 1} K , where, T k = 1 ⇔ H Ak holds. We denote the joint distribution of the K p-values when the true configuration is T by F T , with density f T . We denote by T L , 0 ≤ L ≤ K the special outcomes vector with the first L nulls being false, and the rest true: T L = (1, 1, ..., 1 L , 0, ...., 0 K−L ) t . Throughout this paper, we make the following assumptions: Assumption 1.1. Exchangeability: The K tests are exchangeable, that is: F T (u) = F σ(T ) [σ(u)] , ∀u ∈ [0, 1] K , σ ∈ S K , where S K is the permutation group for K elements. As a consequence, we necessarily have G k ≡ G fixed, and we denote the density of G by g. Assumption 1.2. Arrangement decreasing: For all possible vectors of true outcomes T , denote 1 T = {1 ≤ i ≤ K : T i = 1} the set of false null hypotheses, and 0 T the complementary set. Assume u and the permutation σ are such that if u k ≤ σ(u) k , ∀k ∈ 1 T , then: f T (u) ≥ f T (σ(u)). In words, if the non-null p-values in u are smaller than those in σ(u) (and consequently, the opposite holds for the null p-values), then u has higher density. We call Assumption 1.2 arrangement decreasing because if (u i − u j )(T i − T j ) ≤ 0, then by interchanging entries i and j in the vector u to form the vector σ ij (u), we have the relation f T (u) ≥ f T (σ ij (u)). For testing K normal means, Assumption 1.1 is satisfied if the test statistics have a common variance and pairwise correlation ρ T i ,T j = ρ T j ,T i , where the correlation between pairs of null teststatistics can be different from that of pairs of non-null test-statistics or of pairs with exactly one null test statistic. Assumption 1.2 was studied in [13], where it was shown that it is satisfied for a multivariate normal with a fixed pairwise correlation ρ, as well as the multivariate F distribution and others. However it is easy to see that Assumption 1.2 can be violated for normal means if ρ 0,0 , ρ 1,0 , and ρ 1,1 differ. Hence the two assumptions are not redundant, even for testing normal means. A common assumption for deriving theoretical results in the FDR literature is the assumption that given T 1 , . . . , T K , the p-values u 1 , . . . , u K are independent observations, from U (0, 1) if T i = 0 and from G if T i = 1 [30,11]. With this assumption, the tests are clearly exchangeable. The additional independence assumption gives rise to the following simple characterization of f T : and the Assumption 1.2 simply reduces to requiring monotonicity of g. Hence all our results apply to the indendent case with monotone g, although we only make Assumptions 1.1-1.2. We can now think of a multiple testing procedure as a decision problem on the hypercube [0, 1] K , where at each point we have to make a decision which hypotheses are rejected with the binary decision function D : [0, 1] K → {0, 1} K . Given the exchangeability assumption, we limit our interest to functions that are symmetric, i.e., whose decision does not depend on the order of the hypotheses 1 : Definition 1.1. A decision function D : [0, 1] K → {0, 1} K is symmetric if σ (D(u)) = D (σ(u)) , ∀u ∈ [0, 1] K , σ ∈ S K . Given this requirement, we can in fact limit the definition of D to consider only the "lower corner" set Q = {u : 0 ≤ u 1 ≤ u 2 ≤ . . . ≤ u K ≤ 1}, and extend it to [0, 1] K through the symmetry. Throughout our discussion we limit our attention to functions D that are Lebesgue measurable on Q or [0, 1] K (note that D is also bounded by definition and so integrable). To formulate the OMT problem as an optimization problem we need to select the false discovery criterion we wish to control, and the power function we wish to optimize. For power we consider the following options: Π any (D) = P T K (R(D) > 0) (1.1) Π L (D) = E T L [D 1 (U ) + ... + D L (U )]/L , 1 ≤ L ≤ K. (1.2) In words, Π any is the probability of making any discoveries if all alternatives are true, and it was discussed for example in [18,7]. Π L is the average power (also known as total power, [31]), and it seeks to maximize the expected number of true rejections given that L nulls are false. Note that although calculated assuming the density is f T L , due to the exchangeability assumption and symmetry requirement on D, the value of Π L would be the same if the expectation is calculated relative to any other configuration of L false nulls. Given a selected power measure Π and false discovery measure to control Err, we can write the OMT problem of finding the optimal test subject to strong control as an infinite dimensional integer program, where the optimization is over the value of the function D at every point in the cube: max D:[0,1] K →{0,1} K ,symmetric Π(D) (1.3) s.t. Err T L (D) ≤ α , 0 ≤ L < K. We denote the optimal solution to this problem (assuming it exists) by D * . Note that we have only K constraints and not 2 K − 1 due to exchangeability and symmetry. Several aspects of this optimization problem appear to make it exceedingly difficult to solve: 1. The optimization is over an infinite number of variables (recall that D defines a binary K dimensional decision vector at every point in [0, 1] K ). In § 2 we prove that these problems can be overcome and consequently that we can solve the OMT problem. Previous work on optimal FWER control The simplest class of FWER controlling procedures is that of single-step procedures, where the decision whether to reject a hypothesis is only based on the test statistic (or p-value) for that hypothesis. For the weighted Bonferroni procedures, weights to maximize the average power have been considered, e.g., in [27,32,9]. Optimality results are also available for a more general class of FWER controlling procedures, which requires the selection rules to be monotone. Let D : [0, 1] K → {0, 1} K be the decision function, and u 1 , . . . , u K the p-values for the family of K hypotheses. The ith coordinate D i (u 1 , . . . , u K ) receives the value of one if the ith null hypothesis is rejected, and zero otherwise. Definition 1.2 ([18]). A decision rule D : [0, 1] K → {0, 1} K is said to be monotone if u i ≤ u i for D i (u 1 , . . . , u K ) = 1 but u i > u i for D i (u 1 , . . . , u K ) = 0 implies that D(u 1 , . . . , u K ) = D(u 1 , . . . , u K ). In words, if the value of the rejected p-values is decreased, and the value of the non-rejected p-values is increased, the set of rejections remains unchanged. If restricted to monotone decision rules, the optimal procedure is in the family of stepwise procedures [18] 2 . The restriction to monotone decision rules excludes closed testing procedures [19] that are based on the combined p-value (e.g., based on the sum of the z-scores) of the intersection hypothesis. Such procedures have been shown to have better power than stepwise procedures, unless there is a single strong signal among a group of otherwise null or very weak signals (in which case step-down tests are best), see e.g., [16,7]. A direction that is most similar to ours, of pursuing optimal power with strong FWER control, with no restriction on the form of regions generated, was explored in [23], and optimal rejection regions were presented for K = 2 which are clearly not in the family of monotone selection rules. While the derivation of optimal monotone selection rules in [18] is relatively easy, the derivation of the optimal rules as suggested in [23] is computationally difficult. Their optimization technique is based on discrete approximation of the relevant probabilities, which is computationally feasible with two hypotheses, but may be infeasible for more hypotheses. They leave the extension to more than two hypotheses for future research. Our objective of maximizing power with strong FWER control is similar to that of [23]. However, we address the optimization of the continuous problem in a general framework, which is different from their approach. From the equations of the optimal solution, we demonstrate how we can gain insight into the nature of the rejection region. We demonstrate for K = 2, 3 hypotheses the significantly higher power that can be obtained over the stepwise procedures of [18], and we show that the optimal rejection regions are not monotone. Previous work on optimal FDR control The most common FDR controlling procedure is the Benjamini-Hochberg (BH) procedure [5], which has been shown to perform nearly optimally for various loss functions assuming the hypotheses are exchangeable, when the fraction of null hypotheses is close to one [12]. Asymptotically, as the number of hypotheses grows to infinity, [2] showed that the BH procedure is optimal in some sense. A class of FDR controlling procedures that has gained much interest in recent years concerns thresholding the local FDR [10], which is equivalent to thresholding the individual likelihood ratios. The local FDR can be used for optimal power with control of the marginal FDR (i.e., the ratio of expected number of false positives and expected number of discoveries), assuming T 1 , . . . , T K are iid Bernoulli and conditional on T 1 , . . . , T K , the p-values u 1 , . . . , u K are independent observations from U (0, 1) if T i = 0 and from G if T i = 1 [30]. The marginal FDR is asymptotically equivalent to the FDR under some conditions [12]. Assuming exact knowledge of the sum of alternative densities i:T i =1 g i , a procedure aimed at maximizing the expected number of true rejections with marginal FDR control is suggested in [29]. In [30] and [29], the optimal oracle procedure rejects the hypotheses with a test statistic above (or a p-value below) a fixed threshold which depends only on the nominal level α assuming 1.1, independence of the test statistics, and monotonicity of the alternative density g. So the oracle procedure is a monotone procedure (Definition 1.2). Our objective of maximizing power with strong FDR control stands apart from the work of [30] and [29] in several important ways: we do not assume knowledge of the percentage of true null hypotheses; our FDR control guarantee is non-asymptotic and it is strong control; we relax the independence assumption. We present the optimization of the continuous problem, and show from the equations of the optimal solution how we can gain insight into the nature of the rejection region. We demonstrate for K = 2, 3 hypotheses the significantly higher power that we can obtain over the BH procedure as well as over the procedure of [26] which provides a small but uniform improvement over the BH procedure. We further show that the optimal rejection region is not monotone. Main result: OMT procedures for exchangeable hypotheses To prove that the problem (1.3) can be solved in our settings of interest, we present a collection of results, which address each of the problems discussed above. Lemma 2.1. Under Assumptions 1.1,1.2, for any of our considered power and level criteria, there is an optimal symmetric solution D * that is weakly monotone, i.e., it rejects the smallest p-values at every u ∈ [0, 1] K : u i ≤ u j ⇒ D * i (u) ≥ D * j (u). Proofs or all our lemmas are supplied in Appendix A. This result implies that on the "lower corner" set Q, D * can be characterized via k * (u) = max{k ≤ K : D * k (u) = 1}, the "last and largest" p-value rejected by D * at u ∈ Q. Given this solution on Q we can extend it to [0, 1] K using the symmetry property: D * (u) = σ −1 u (D * (σ u (u))), where σ u is the sorting permutation for u, so that σ u (u) is the order statistic of u. Once we limit our discussion to functions D that have this structure, we can simplify the mathematical description of the objective and constraints of our optimization problem, as follows. First, taking into account exchangeability and symmetry, the L-expected power can be written as a linear functional of D on Q: Π L (D) = [0,1] K f T L (u) L k=1 D k (u)du/L = L!(K − L)! Q i∈( K L ) f i (u) k∈i D k (u)du/L, (2.1) where i indexes the set of all subsets of size L. The notation k ∈ i is shorthand that the kth null is set to false by the ith configuration. Similarly f i (·) is the density when the configuration of L false nulls is the one indexed by i. Similarly, weak monotonicity of the optimal solution (as guaranteed in Lemma 2.1) is sufficient to simplify Π any to a linear functional too: Π any (D) = K! Q D 1 (u)f T K (u)du. (2.2) Moving to the constraints, symmetry and exchangeability allow us to write the constraints of Problem (1.3) in the form : F W ER T L (D) = u∈[0,1] K I {V (D(u)) > 0} f T L (u)du ≤ α, 0 ≤ L < K A similar expression can be written for FDR. By Lemma 2.1, we can rewrite these K constraints as linear functionals of the decision function D on Q for both FWER and FDR: F W ER L (D) = L!(K − L)! Q k D k (u) i∈( K L ),īmin=k f i (u)du, (2.3) F DR L (D) = L!(K − L)! Q k D k (u) i∈( K L ) f i (u)r ki du. (2.4) whereī min is the minimal element not in the i'th configuration of false nulls (that is, the true null hypothesis with the smallest index), and r ki is the difference in false discovery proportion (FDP) if we reject the k versus k − 1 smallest p-values, i.e., r ki = \|{1, . . . k} ∩ i c \| k − \|{1, . . . k − 1} ∩ i c \| k − 1 where i c denotes the actual set of true nulls in the configuration indexed by i (and we also assume 0/0 = 0). See Appendix B for detailed derivation of F DR L (D). Taking all of these together we conclude that for any combination of objective of the form Π any , Π L and strong control of FWER or FDR, we can rewrite Problem (1.3) as an infinite integer program on the set Q, with a linear objective and K linear constraints. Our next step is to replace this integer program with its relaxed infinite linear program, by replacing the function D : Q → {0, 1} K with a continuous version D : Q → [0, 1] K Let us write the resulting linear program in a generic form which unifies the expressions in Eqs. (2.1-2.4): To solve this problem we appeal to calculus of variations to derive the Euler-Lagrange (EL) conditions for an optimal solution to this problem [15]. The EL coditions are similar to the Karush-Kuhn-Tucker (KKT) conditions for finite optimization problems, but EL are only necessary and not also sufficient like KKT. We derive the EL conditions for this problem in Appendix C, and also show there that they can be rephrased in the following KKT-like manner, requiring the following to hold almost everywhere for optimality, in addition to the (primal feasibility) constraints of Problem (2.5): max D:Q→[0,1] K Q K i=1 a i (u)D i (u) du (2.5) s.t. Q K i=1 b L,i (u)D i (u) du ≤ α , 0 ≤ L < K. 0 ≤ D K (u) ≤ . . . ≤ D 1 (u) ≤ 1 , ∀u ∈ Q,a i (u) − K−1 L=0 µ L b L,i (u) − λ i (u) + λ i+1 (u) = 0, i = 1, . . . , K. (2.6) µ L Q K i=1 b L,i (u)D i (u) du − α = 0, L = 0, . . . , K − 1 (2.7) λ K+1 (u)D K (u) = 0 ∀u ∈ Q (2.8) λ j (u)(D j−1 (u) − D j (u)) = 0 , ∀u ∈ Q, j = 2, . . . , K (2.9) λ 1 (u)(D 1 (u) − 1) = 0 , ∀u ∈ Q., (2.10) where µ L , L = 0, . . . , K − 1 and λ j (u), j = 1, . . . , K + 1, u ∈ Q are non-negative Lagrange multiplies, condition (2.6) is the stationarity condition, and conditions (2.7-2.10) are the complementary slackness conditions. We now prove two important properties of any solution to the above conditions. Our first result is that under mild "non-redundancy" conditions, a solution to the EL conditions is integer almost everywhere in [0, 1] K . We make the following assumption: Assumption 2.1. The set of density functions f T : T ∈ {0, 1} K , is non-redundant, i.e. there is no non-trivial set of 2 K constants γ T ∈ R : T ∈ {0, 1} K such that: 1. γ T are not all zero 2. P U T γ T f T (U ) = 0 > 0, where P U is the Lebesgue measure on [0, 1] K . This assumption is mild given the highly non-linear nature of the functions f T in typical applications (as in our examples below). Lemma 2.2. Under Assumption 2.1, the optimal solution D * to Problem (2.5) is integer almost everywhere on [0, 1] K . Our next result is that for this problem, solving the EL conditions in fact leads to an optimal solution, so they are also sufficient. The result relies on explicit derivation of the dual to the infinite linear program (2.5) (see [1] for details on derivation of dual to infinite linear programs): 2. It achieves strong duality: min µ,λ α K−1 L=0 µ L + Q λ 1 (u)du (2.11) s.t. a k (u) − K−1 L=0 µ L b L,k (u) + λ k+1 (u) − λ k (u) ≤ 0, ∀k, u λ k (u) ≥ 0, ∀k, u ; µ L ≥ 0, L = 0, . . . , K − 1.α K−1 L=0 µ * L + Q λ * 1 (u)du = Q K i=1 a i (u)D * i (u) du. Consequently, any solution to the EL conditions above is an optimal solution to Problem (2.5). Putting together all our lemmas and their implications gives our main result: Theorem 2.1. Under Assumptions 1.1, 1.2, 2.1, for any choice of power function from Π any , Π L and error measure FWER or FDR, the OMT procedure can be explicitly found by finding an integer solution which is feasible for Problem (2.5) and complies with the optimality conditions (2.6)-(2.10) . Resulting algorithm Our main result can be directly used to find the solution, by searching over the space of K Lagrange multipliers of the integral constraints, as follows. Given a set of candidate multipliers µ L , ; L = 0, . . . , K − 1, for i = 1, . . . , K, define: R i (u) = a i (u) − K−1 L=0 µ L b L,i (u). Denote by D µ (u) a solution which complies with the λ complementary slackness Conditions (2.8)-(2.10) for this value of µ. It is easy to confirm that under the non-redundancy Assumption 2.1, this dictates that almost surely: D µ 1 (u) = I ∪ K l=1 l k=1 R k (u) > 0 D µ i (u) = I D µ i−1 ∩ ∪ K l=i l k=i R k (u) > 0 , i = 2, . . . , K. In the way it is constructed, this solution guarantees that Conditions (2.6),(2.8)-(2.10) hold. Now we have to ensure that primal feasibility and complementary slackness for the µ's hold, in other words find µ * ∈ R K such that the following holds for L = 0, . . . , K − 1: µ * L ≥ 0 ∩ Q K i=1 b L,i (u)D µ * i (u) du = α ∪ (2.12) µ * L = 0 ∩ Q K i=1 b L,i (u)D µ * i (u) du ≤ α . It is easy to confirm that if we find such a solution, then it is feasible, it complies with Conditions (2.6)-(2.10), and it is obviously binary. Thus, a complete algorithm for solving our OMT problems involves: 1. An approach for searching the space (R + ∪ {0}) K of possible µ vectors for a solution µ * . 2. An approach for integration (exact or numerical), to calculate Q K i=1 b L,i (u)D µ i (u) du for any given µ vector and asses the error relative to the conditions on µ * . Given these two components, we have a fully characterized algorithm for solving Problem (1.3). Maximizing Π 3 for K = 3 independent tests under FDR control We now choose a specific instance of the general problem above, to demonstrate a detailed derivation of the formulas and the resulting algorithm. We use the power function Π L=3 , that is, maximizing expected rejections in case all nulls are in fact false. Recall that g denotes the density of each coordinate of u under the alternative. Putting together the objective and constraints F DR L (D) from Eq. (2.4) for the relaxed infinite linear program on Q gives: max D Q (D(u) t 1 3 )g(u 1 )g(u 2 )g(u 3 )du s.t. F DR 0 (D) = 6 Q D 1 (u)du ≤ α, F DR 1 (D) = 2 Q D 1 (u) (g(u 2 ) + g(u 3 )) + D 2 (u) g(u 1 ) − g(u 2 ) 2 + D 3 (u) g(u 1 ) + g(u 2 ) − 2g(u 3 ) 6 du ≤ α, F DR 2 (D) = 2 Q D 1 (u) (g(u 2 )g(u 3 )) + D 2 (u) g(u 1 )g(u 3 ) − g(u 2 )g(u 3 ) 2 + D 3 (u) 2g(u 1 )g(u 2 ) − g(u 1 )g(u 3 ) − g(u 2 )g(u 3 ) 6 du ≤ α, 0 ≤ D 3 (u) ≤ D 2 (u) ≤ D 1 (u) ≤ 1, ∀u ∈ Q, Applying our results we denote: R 1 (u) = g(u 1 )g(u 2 )g(u 3 ) − 6µ 0 − 2µ 1 (g(u 2 ) + g(u 3 )) − 2µ 2 g(u 2 )g(u 3 ) R 2 (u) = g(u 1 )g(u 2 )g(u 3 ) − 2µ 1 g(u 1 ) − g(u 2 ) 2 − 2µ 2 g(u 1 )g(u 3 ) − g(u 2 )g(u 3 ) 2 R 3 (u) = g(u 1 )g(u 2 )g(u 3 ) − 2µ 1 g(u 1 ) + g(u 2 ) − 2g(u 3 ) 6 − 2µ 2 2g(u 1 )g(u 2 ) − g(u 1 )g(u 3 ) − g(u 2 )g(u 3 ) 6 , and use these to define the corresponding D functions: D µ 1 (u) = I {R 1 (u) > 0 ∪ R 1 (u) + R 2 (u) > 0 ∪ R 1 (u) + R 2 (u) + R 3 (u) > 0} D µ 2 (u) = I {D(u) µ 1 ∩ (R 2 (u) > 0 ∪ R 2 (u) + R 3 (u) > 0} D µ 3 (u) = I {D(u) µ 2 ∩ R 3 (u) > 0} , and use these to search for µ * complying with Eq. (2.12). Example applications: K = independent normal means We now demonstrate applications of our results and algorithms for strong FWER and strong FDR control, which illustrate the potential power gain from using OMT procedures as well as the potential insight gained from examining which constraints are tight for the optimal solution. We consider tests of the form H 0k : X ∼ N (0, 1) vs H Ak : X ∼ N (θ, 1) for k = 1, 2, 3 with θ < 0, where all test statistics are independent. The power functions we consider are Π θ,3 (D) the average power when all three nulls are false, and Π θ,any (D), the probability of making at least one rejection when all three nulls are false, which we term minimal power. 11 Optimal FWER controlling procedure For strong FWER control, we compare and contrast the two objectives for maximization: Π θ,3 (D) and Π θ,any (D). We demonstrate the potential power gain over the popular sequentially rejective procedure of Holm [14], henceforth Bonferroni-Holm. Among all monotone rejection policies, the following procedure was shown in [18] to maximize certain aspects of power 3 for K = 3 independent and exchangeable test statistics, with sorted p-values u 1 ≤ u 2 ≤ u 3 : (1) if u 1 > 1−(1−α) 1/3 reject no hypotheses, otherwise reject its null hypothesis and proceed to next step; (2) if u 2 > 1 − (1 − α) 1/2 make no further rejections, otherwise reject its null hypothesis and proceed to next step; (3) if u 3 > α make no further rejections, otherwise reject all null hypotheses. Since for α = 0.05, 1 − (1 − α) 1/i ≈ α/i for i = 2, 3, the difference of this policy (which performs at each step Sidak's test [25]) from Bonferroni-Holm is negligible (less than 10 −5 in average or minimal power for our experiments below), and therefore we only compare henceforth with Bonferroni-Holm. Table 1 shows the power comparison for various values of θ. The power of Bonferroni-Holm is smaller than the power of the OMT policy for Π θ,3 by more than 20%. However, the average power of the OMT policy for Π θ,any can be lower than that of Bonferroni-Holm. Of course, the minimal power of the OMT policy for Π θ,any is much higher than that of Bonferroni-Holm. Insight into the reasons for the power gaps is obtained by examining the rejection regions of the different procedures (Figure 1). For a 2-dimensional display, we selected slices of the 3-dimensional rejection region that are fixed by the minimum p-value. We show the slices with a very small minimum p-value, the largest minimum p-value for which Bonferroni-Holm still makes rejections (i.e., 0.05/3), a minimal p-value slightly below the nominal level, and a minimal p-value above the nominal level. The boundaries between one, two, or three rejections are necessarily parallel to the axes for Bonferroni-Holm but not parallel to the axes for the OMT rejection policies. Therefore, OMT policies are not monotone policies (definition 1.2), and the non-monotonicity is manifest in the negative slopes: the values of all p-values are taken into account in the decision on the number of hypotheses to reject. For OMT policies, rejections of hypotheses with p-values greater than the nominal level are possible. This is due to the structure of the optimization problem: the likelihood for three false nulls is small if u 3 is close to one, and therefore to maximize the objective it is preferred to reject some of the p-values near the diagonal rather than include rejection regions where u 3 is close to one (unless u 1 is very small), while maintaining strong FWER control. The OMT policy for Π θ,any rejects only the minimal p-value, since there is no gain in the objective function for rejecting more than one hypothesis. Interestingly, for a large range of θs the only tight constraint for the OMT policy is the global null constraint. The optimal global test statistic is K i=1 Φ −1 (u i )/ √ K. Therefore, the level α OMT policy for K false nulls when the only tight constraint is the global null constraint is to reject the hypothesis with minimal p-value if K i=1 Φ −1 (u i )/ √ K < z α , where z α is the αth quantile of the standard normal distribution. In Figure 1 this rejection region is shown in row 6; the OMT rejection region for a value of θ for which both the global null constraint and the constraint of FWER control when there is one false null are tight, is shown in row 5. Interestingly, for K = 2 the OMT policy for Π θ,any is to the reject the hypothesis with minimal p-value if 2 i=1 Φ −1 (u i )/ √ 2 < z α for any θ < 0. This was also noted in [22] in a similar setting for two hypotheses. However, for K = 3, such a policy is no longer valid for all θ < 0, since the FWER when there is one false null will be inflated for θ ∈ (−1.6, −0.8). Optimal FDR controlling procedure The following procedure, which is uniformly more powerful than the BH procedure, was suggested in [26]. If there exists an i ∈ {1, . . . , K} such that u i ≤ iα/K, then reject all hypotheses up to max{i : u i ≤ iα/(K − 1)}. They called this procedure minimally adaptive Benjamini-Hochberg (MABH). Both BH and MABH have rejection regions in parallel to the axes (monotone decision rules, as defined in [18]), and therefore may have reduced power in comparison to procedures that violate the monotonicity property, in particular OMT procedures. For K = 3 hypotheses, Table 2 shows the power comparison for various values of θ. We see that OMT policies optimized for Π θ,3 are significantly more powerful than BH and MABH: they offer more than three-fold power for the lower-power settings θ ∈ {−0.35, −.5} and about 25% more power in the higher power setting. From a comparison of rejection regions in Figure 2, we see that as in § 3.1, the rejection region near the diagonal is larger than with MABH, and rejections of hypotheses with p-values greater than α are possible. As expected, the OMT rejection regions are not monotone in the sense of [18]. The non-monotone behaviour separating rejections from no rejections is reasonable. However, a less reasonable non-monotone behaviour of the optimal rejection regions is the positive slope separating the red and green regions for θ = −2. Such a rejection region is counter-intuitive, since it contradicts the reasonable principle that if (u 1 , u 2 , u 3 ) ≤ (u 1 , u 2 , u 3 ) and (u 1 , u 2 , u 3 ) result in rejection of all null hypotheses, then (u 1 , u 2 , u 3 ) will also result in rejection of all null hypotheses. We return to this issue in our Discussion. For weak signal (θ = −0.35), the only tight constraint is the global null constraint, and we are maximizing the average power. Therefore, the OMT procedure is to reject all three hypotheses if 3 i=1 Φ −1 (u i )/ √ 3 ≤ z α . We can find the range of values of θ for which the rejection policy that rejects all three hypotheses if the global null is rejected is valid. It satisfies F DR L = 3−L 3 Φ(z α − Lθ/ √ 3) for L = 1, 2. For α = 0.05, F DR 1 ≤ α if θ ≥ −0.356, and F DR 2 ≤ α if θ ≥ −0.527. Therefore, this policy achieves strong FDR control whenever θ ≥ −0.356. This policy is clearly optimal for θ = −0.35 since it is the optimal policy if the only constraint is the global null constraint. The following proposition provides the general result. Proposition 3.1. Let θ * = min{θ : K−L K Φ(z α − Lθ/ √ K) ≤ α ∀ L = 1, . . . , K − 1}. Then for θ ≥ θ * , the OMT policy for Π θ,K with strong FDR control at level α is to reject all K hypotheses if K i=1 Φ −1 (u i )/ √ K < z α . For K = 3 and α = 0.05, the optimal rejection policy for θ > −0.356 is to reject all three hypotheses, since only the global null constraint is tight. For θ < −0.356, the optimal rejection policy is driven by the non-global constraints. For θ = −0.5, the only tight constraint is F DR 1 = 0.05, and the optimal rejection region includes either three or one rejections. For θ = −2, the tight constraints are F DR 1 = F DR 2 = 0.05 (so the global null constraint is loose), and either one, two or three rejections can occur. Beyond simple hypotheses: dealing with complex alternatives In practical multiple testing scenarios, it is often more realistic to assume that there is no specific known alternative distribution, but that there is a family of relevant alternatives indexed by a parameter θ ∈ Θ A [17]. Hence, it is important to expand our results to dealing with complex alternatives. Requiring strong control under a range of alternatives translates to requiring that the constraints in (1.3) hold for every alternative distribution (note that in multiple testing, unlike the single hypothesis case, the constraints do depend on the alternative). To formulate our objective, we define the following additional notations: when the alternative distribution has parameter θ, denote the density of a vector u ∈ [0, 1] K with null configuration T ∈ {0, 1} K by f T,θ (u), and correspondingly the integrated error Err T,θ . Π θ (D) is the power of the policy D when the parameter is θ (and the power can be any of Π θ,any , Π θ,L as before). We consider two objectives: 1. Single objective. Assume we have a specific alternative that is of special interest, denote it θ 0 , and wish to optimize the power for this selected alternative, while maintaining validity for all considered alternatives: max D:[0,1] K →{0,1} K Π θ 0 (D) (4.1) s.t. Err T,θ (D) ≤ α , ∀T ∈ {0, 1} K , θ ∈ Θ A . 2. Maximin. In this case, we aim to maximize the minimal power among all alternatives of interest θ ∈ Θ B ⊆ Θ A , under the same set of constraints: max D:[0,1] K →{0,1} K min θ∈Θ B Π θ (D) (4.2) s.t. Err T,θ (D) ≤ α , ∀T ∈ {0, 1} K , θ ∈ Θ A . We note that to maintain exchangeability, all our formulations still assume that the parameter θ is the same for all alternatives, even under the complex alternative setting. These optimization problems now have, in addition to an infinite number of variables, also an infinite number of integral constraints (assuming Θ A is an infinite set). A simpler but related problem is the "Generalized Neyman Pearson" setting of [8,24], where a single hypothesis test with complex null and/or alternative hypotheses was considered. In these papers, the authors were able to use convex duality arguments to prove that under some conditions, the intuitive solution of choosing the "closest distributions" from the null and alternative set and finding the Neyman Pearson test for the simple testing problem they imply, is indeed optimal. However, even for the single test case they consider, there are complexities relating to convexity and closure arguments, implying that these extremal closest distributions may not actually belong to the set of distributions in each hypothesis, and there is no general simple algorithm for finding them. The multiple testing scenario is more complex due to the existence of different types of constraints, some of them depending on the alternative, as discussed before. We are therefore unable to offer similar guarantees on existence and sparsity of the optimal solutions to the problems we pose. Instead, we offer an approach that assumes existence of an optimal solution that can be characterized using a single value of the parameter. If the assumption holds, our proposed approach is able to find this OMT solution and -importantly -confirm its optimality. Let D * (θ 0 , θ) be the optimal solution of the optimization problem that uses parameter θ 0 in the objective and θ ∈ Θ A in the constraints: D * (θ 0 , θ) = arg max D:[0,1] K →{0,1} K Π θ 0 (D) (4.3) s.t. Err T,θ (D) ≤ α , ∀T ∈ {0, 1} K . The following result states a sufficient condition for an optimal solution to problem (4.1). Proposition 4.1. Assume that we find a parameter value θ A ∈ Θ A such that the solution D * (θ 0 , θ A ) controls Err at level α at all parameter values θ ∈ Θ A : Err T,θ (D * (θ 0 , θ A )) ≤ α , ∀T ∈ {0, 1} K , θ ∈ Θ A , then D * (θ 0 , θ A ) is the optimal solution to the complex alternative problem (4.1). The following corollary simplifies the use of this result for finding θ A . Corollary 4.1. If θ A in the above Proposition exists, then we have: Π θ 0 (D * (θ 0 , θ A )) ≤ Π θ 0 (D * (θ 0 , θ)) ∀θ ∈ Θ A . In words: the power of the optimal solution for constraints at θ A is minimal among all optimal solutions D * (θ 0 , θ). With this corollary, we have a simple policy for trying to solve (4.1): 1. Search over Θ A to find θ A = arg min θ Π θ 0 (D * (θ 0 , θ)). Check whether the control condition in Proposition 4.1 holds. This approach requires solving problems of the form (4.3), which are equivalent to problems we solve in § 2, where the parameter θ can be different in the power objective and in the constraints. Next, we derive a similar sufficient condition for existence of a maximin solution, and corresponding approach for finding it. Proposition 4.2. Assume that we can find two values θ 0 , θ A ∈ Θ A such that: 1. D * (θ 0 , θ A ) is the optimal solution of the single objective problem (4.1) at θ 0 . The power of this solution at other values is higher: Π θ 0 (D * (θ 0 , θ A )) ≤ Π θ (D * (θ 0 , θ A )) ∀θ ∈ Θ B . Then D * (θ 0 , θ A ) is the solution to the maximin problem (4.2). The usefulness of this last result is not immediately evident, since the conditions seem harsh. As we show below, it can be practically useful when the problem is such that Θ B has a minimal element, and there exists inherent monotonicity in the problem such that when θ 0 is taken as the minimal element, the conditions hold. Example: testing two independent normal means We seek the maximin optimal rejection policy with the objective function of average power for two false nulls, Π θ,2 . By solving the optimization problem for a single θ < 0 constraint at a time, we compute series of rejection policies D * (θ 0 , θ). We identify the value of θ with minimal power, θ A , so that Π θ 0 ,2 (D * (θ 0 , θ A )) ≤ Π θ 0 ,2 (D * (θ 0 , θ)) for all θ < 0. Once we find this, we can check if for all θ < θ 0 , we have: Π θ,2 (D * (θ 0 , θ A )) ≥ Π θ 0 ,2 (D * (θ 0 , θ A )), in which case by Proposition 4.1, the computed solution is the maximin solution for Θ B = (−∞, θ 0 ]. This turned out to be the case for all θ 0 values in the two independent normal means example with FWER or FDR control. Table 3 shows the power comparison for various values of θ 0 . The maximin power was higher than that of the monotone rejection rule, but to a lesser extent than the OMT policy at a single θ 0 constraint. The power gap between the maximin and OMT solutions decreased with θ 0 , and it was negligible at θ 0 = −2, while both were still better than the competitors in terms of power. Figure 3 shows the optimal and maximin rejection policies. A problematic non-monotone behavior of the maximin optimal rejection policies is manifest for both error controls: the boundary between the regions where one versus two hypotheses are rejected has a positive slope for θ 0 ≤ −1; for θ 0 = −0.5 there is a gap between the regions where two hypotheses versus one hypothesis are rejected. These regions contradict the reasonable principle that if (u 1 , u 2 ) < (u 1 , u 2 ) and the policy for (u 1 , u 2 ) is rejection of both hypotheses, then with (u 1 , u 2 ) both hypotheses should also be rejected. As the signal strength of the objective function \|θ 0 \| increases, this undesired behaviour is less pronounced, as manifested by the slope on the boundary between red and blue regions being steeper. The counter-intuitive rejection regions are due to the fact that it is possible to add pieces to the rejection region without violating the error control. For example, the maximin optimal rejection policy for θ 0 = −0.5 occurs at θ A = −1.29 for strong FWER control and at θ A = −1.36 for strong FDR control. The chance of both p-values being of similar value and not very small is negligible if exactly one hypothesis is at θ A , hence the penalty for the red regions at θ A is negligible, but the added power at θ 0 θ A is non-negligible. For a discussion of the advantages and disadvantages of such counter-intuitive rejection regions, see [20]. Figure 3: Rejection regions optimized for Π θ0,2 , subject to level 0.05 error control. Rows 1 and 2: strong FWER control with optimal and maximin procedures, respectively. Rows 3 and 4: strong FDR control with optimal and maximin procedures, respectively. In red: reject both hypotheses; in blue: reject only one hypothesis. Discussion Our development in this paper has focused on establishing solvability of OMT problems, and solving relatively low dimensional instances numerically, up to K = 3. In modern problems, K can often be in hunderds, thousands or even millions (like in Genome Wide Association Studies). To address feasibility of solution for larger K, we need to consider the computational complexity of numerical solution, and in particular its dependence on K. There are three components to the computation: 1. Searching in parameter space for the K Lagrange multipliers which solve the problem. 2. For each set of multipliers considered, performing numerical integration over the set Q in the K-dimensional hypercube. 3. For each evaluation of the integrand in the integration, calculating the coefficients in (2.3,2.4). The complexity of the first two items depends on the specific algorithms used for search and integration, of which there is a large variety [21], and identifying the best approaches for our type of problems is a topic for future research. For the third item -calculation of coefficients for the linear constraints -we can make some progress. The representation in (2.3,2.4) appears to be exponential in K, however it is easy to see that these coefficients can be calculated in complexity O(K 2 ) using a dynamic programming approach, for independent hypotheses (details omitted). Hence by combining state of the art approaches for parameter optimization and numerical integration, with efficient calculation of the coefficients at each integration point, problems of dimension much higher than K = 3 can be solved exactly and efficiently. It seems quite clear, however, that to go to dimensions in the thousands or higher, approximations would be required. One direction for such approximations is the use of hierarchical controlling procedures, where hypotheses are divided to groups, within each group an optimal testing procedure is employed, and the results are summed up using group-aggregation techniques. For example, for a multiple testing problem with N × K hypotheses, if we have optimal rejection policies for K hypotheses, we can adjust the level of testing within each of N groups of K hypotheses in order to solve the bigger problem with the same error guarantee. Specifically, for FWER control at level α, we can apply the optimal rejection policy at level α/N for each group of K hypotheses, and this procedure will clearly be far more powerful than the Bonferroni procedure on the N × K hypotheses p-values. For FDR control, the level of the optimal rejection policy within each group of K hypotheses may be closer to α than to α/N [10,4]. The gain over the BH procedure on all N × K hypotheses may be substantial. Following the introduction of the FDR in [5], other related notions of error rate have been suggested, including the empirical Bayes FDR which has gained much attention [11]. The vast majority of these procedures are monotone procedures (Definition 1.2). In this work we demonstrated that the optimal procedure for maximizing the expected number of rejections if L hypotheses are false is not monotone, and that there can be substantial power gain in using the optimal decision which depends on the values of all test statistics (Figure 2 and Table 2). As far as we know, this is the first work that shows that the objective and constraints are linear in the decision function, thus enabling the computation of optimal rejection regions for FDR control. Similar steps can be followed to establish that other objectives that are of interest with FDR control, such as expected weighted loss minimization [30], are also linear in the decision function. If in addition the number of (or a lower bound on) true nulls K − L out of the K hypotheses is known (or can be estimated), then this knowledge can further be exploited to define an easier optimization problem with at most L constraints. Such a procedure will naturally have higher power than the optimal procedure with K constraints, and it can be interesting to examine its power gain over adaptive methods for FDR control available in the literature, e.g., [28,6]. For FWER control, hierarchical procedures that base the decision for a hypothesis on the values of all test statistics have been advocated on an intuitive basis for the setting of non-sparse signals, but without the justification by an optimality theory [19,16]. Lehmann et al. [18] developed an optimality theory for procedures restricted to be monotone (which exclude procedures suggested in [19,16]). Under this requirement, optimal testing procedures can take simple forms like being limited to "step-down" rules [18], and these can be derived relatively easily, with no need for complex methodology we develop here. However, the OMT regions we obtain for all problems we consider are far from being monotone (Figure 1 ). The reward is significantly higher power than step-down procedures can supply for such problems (Table 1). We may still ask what constitutes a "reasonable" test, and what happens when optimal tests do not comply with reasonableness expectations. This issue has been raised in other contexts in [20]. Although we believe that the type of monotonicity in rejection rules discussed above is not a reasonable requirement, it seems that other requirements do make sense. In particular, a weaker form of monotonicity requires that if u v, where is the coordinate-wise partial order, then rejected hypotheses at v are a subset of rejected hypotheses in u. Surprisingly, some of our derived optimal procedures do not comply with this seemingly sensible requirement, including FDR regions in Figure 2, and more pronounced, maximin regions in Figure 3. Requiring "sensible" behavior as additional constraints in our optimization does not appear to be a solvable problem, and in our view is also not the correct approach. Rather, we believe that the resulting optimal solutions represent relevant properties of the problems solved and the power criteria and constraints used. Hence, non-intuitive optimal solutions indicate interesting or problematic aspects of the problem solved rather than of the solution. We leave for future research consideration of other power functions, such as linear combinations of the functions we propose here, which can also be solved within our framework. We also leave for future research consideration of other error measures (see Table 1 in [3] for a list of measures which can be viewed as generalization of the FWER and FDR). All our solutions were derived under the exchangeability Assumption 1.1. While this assumption (or more commonly, the stronger assumption that also includes independence) is common to the vast majority of work on non-trivial multiple testing procedures, it is not necessarily appropriate for many problems, where different alternatives might be relevant for different tests, or where dependence structures are not symmetric. We note that at least for FWER, it is possible to write the optimization problem (1.3) as an integer linear program without assuming exchangeability. Since the results in § 2 do not apply, this entails maintaining all 2 K constraints, and also adding 2 K additional variables, hence potential practical utility of this approach is unclear. Details are omitted for brevity. j − 1 in the stationarity condition (2.6): j−1 i=l a i (u) − K−1 L=0 µ L b L,i (u) − λ i (u) + λ i+1 (u) = a l (u) − a j−1 (u) − K−1 L=0 µ L b L,l (u) + K−1 L=0 µ L b L,j−1 (u) = 0, where all the λ terms have cancelled out due to the telescopic nature of the sum, and λ l = λ j = 0. Hence we have concluded that having any non-binary value in D(u) implies a l (u) − a j−1 (u) − K−1 L=0 µ L b L,l (u) + K−1 L=0 µ L b L,j−1 (u) = T γ T f T (u) = 0, where the last equality relies on the definition of a i , b L,i functions as linear combinations of the f T functions. Hence, by the assumption of non redundancy, for any pair l, j: P U a l (U ) − a j−1 (U ) − K−1 L=0 µ L b L,l (U ) + K−1 L=0 µ L b L,j−1 (U ) = 0. Proof of Lemma 2.3 Feasibility of dual solution holds by construction: µ, λ are non-negative Largange multipliers by definition, and the EL conditions require that a i (u) − K−1 L=0 µ * L b L,i (u) − λ * i (u) + λ * i+1 (u) = 0 , ∀i, u. To calculate the dual objective, we explicitly derive the value of λ * 1 (u) as a function of the other variables. If D * K (u) = 1, then λ * K+1 (u) = 0 and it is easy to see from (2.6)-(2.10) that λ * 1 (u) is equal to K i=1 a i (u) − K−1 L=0 µ * L b L,i (u) . Similarly, if D * j−1 (u) − D * j (u) = 1 for j ∈ {2, . . . , K − 1}, then λ * j (u) = 0 and λ * 1 (u) is equal to j−1 i=1 a i (u) − K−1 L=0 µ * L b L,i (u) . It thus follows that λ * 1 (u) = K j=1 D * j (u) a j (u) − K−1 L=0 µ * L b L,j (u) . Therefore, Q λ * 1 (u)du = Q   K j=1 a j (u)D * j (u)   du − K−1 L=0 µ * L Q   K j=1 b L,j (u)D * j (u)   du. 26 Therefore the dual objective is equal to the primal objective: K−1 L=0 µ * L α + Q   K j=1 a j (u)D * j (u)   du − K−1 L=0 µ * L Q   K j=1 b L,j (u)D * j (u)   du = K−1 L=0 µ * L    α − Q   K j=1 b L,j (u)D * j (u)   du    + Q   K j=1 a j (u)D * j (u)   du = Q   K j=1 a j (u)D * j (u)   du, where we have used the complementary slackness conditions for the µ * s in the last equality. Proof of Proposition 4.1 Among all D : [0, 1] K → {0, 1} K that satisfy Err T,θ A (D) ≤ α, D * (θ 0 , θ A ) has the highest power (by definition): Π θ 0 (D * (θ 0 , θ A )) > Π θ 0 (D). Therefore, if in addition Err T,θ (D * (θ 0 , θ A )) ≤ α , ∀T ∈ {0, 1} K , θ ∈ Θ A , D * (θ 0 , θ A ) has the highest power among all potential solutions to (4.1), i.e., it is the optimal solution of the single objective optimization problem. Proof of Corollary 4.1 Suppose by contradiction that there exists another θ = θ 0 such that Π θ 0 (D * (θ 0 , θ A )) > Π θ 0 (D * (θ 0 , θ)). Then Err T,θ (D * (θ 0 , θ A )) > α for at least one T ∈ {0, 1} K , otherwise the definition of D * (θ 0 , θ) as the optimal solution to (4.3) is violated. But this contradicts the fact that Err T,θ (D * (θ 0 , θ A )) ≤ α , ∀T ∈ {0, 1} K , θ ∈ Θ A , thus proving the corollary. Proof of Proposition 4.2 Define a feasible solution D as one that satisfies Err T,θ (D) ≤ α , ∀T ∈ {0, 1} K , θ ∈ Θ A . Since D * (θ 0 , θ A ) is the optimal solution to (4.1) by Assumption 1, then for any feasible D min θ∈Θ B Π θ (D) ≤ Π θ 0 (D) ≤ π θ 0 (D * (θ 0 , θ A )). Moreover, since by Assumption 2 min θ∈Θ B Π θ (D * (θ 0 , θ A )) = Π θ 0 (D * (θ 0 , θ A )), it follows that min θ∈Θ B Π θ (D) is upper bounded by Π θ 0 (D * (θ 0 , θ A )). So: If L = 0, then F DR 0 (D) = F W ER 0 (D) = 3! Q D 1 (u)du. Let F DP k denote the false discovery proportion if the k smallest p-values are rejected, i.e., if D k (u) − D k+1 (u) = 1. For L > 0, the expression (2.4) follows since: F DR L (D) = E K−1 k=1 [D k (u) − D k+1 (u)]F DP k + D M F DP M = E K k=1 D k (u)F DP k − K−1 k=1 D k+1 (u)F DP k = E K k=1 D k (u)F DP k − K k=2 D k (u)F DP k−1 = E D 1 (u)F DP 1 + K k=2 D k (u)[F DP k − F DP k−1 ] . C Calculus of variations optimality conditions Our optimization problem is: max Q k a k (u)D k (u)du s.t. Q k b Lk (u)D k (u)du ≤ α ∀0 ≤ L < K 0 ≤ D K (u) ≤ · · · ≤ D j (u) ≤ D i (u) ≤ · · · ≤ D 1 (u) ≤ 1 ∀u ∈ [0, 1] K . We eliminate the inequality constraints, by introducing non-negative auxiliary variables, and then square those variables to also eliminate non-negativity constraints: The Euler-Lagrange (EL) necessary conditions for a solution to this optimization problem may be obtained through calculus of variations [15]. Let y 1 (x), y 2 (x), . . . , y n (x) : R → R be a set of n functions and I = max Q k A k (u)D k (u)du s.t. x F x 0 F (y 1 (x), y 2 (x), . . . , y n (x); y 1 (x), y 2 (x), . . . , y n (x); x)dx (C.2) be a definite integral over fixed boundaries x 0 , x F . Every set of y 1 (x), y 2 (x), . . . , y n (x) which maximize or minimize (C.2) must satisfy a set of n equations d dx ∂F ∂y i − ∂F ∂y i = 0 i = 1, . . . , n. (C. 3) In addition, let ϕ j 1 (y 1 (x), y 2 (x), . . . , y n (x); x) = 0 j 1 = 1, . . . , m 1 < n, (C.4) be a set of m 1 < n point-wise equality constraints on y 1 (x), y 2 (x), . . . , y n (x) and x F x 0 Ψ j 2 (y 1 (x), y 2 (x), . . . , y n (x); y 1 (x), y 2 (x), . . . , y n (x); x) = C j 2 j 2 = 1, . . . , m 2 , (C.5) be a set of m 2 integral equality constraints on y 1 (x), y 2 (x), . . . , y n (x). Then, every set of n functions y 1 (x), y 2 (x), . . . , y n (x) which maximize (C.2), subject to the constraints (C.4, C.5) must satisfy the EL equations, d dx ∂Φ ∂y i − ∂Φ ∂y i = 0 i = 1, . . . , n, (C.6) where Φ = F − m 1 j 1 =1 λ j 1 (x)ϕ j 1 − m 2 j 2 =1 µ j 2 Ψ j 2 . (C.7) The unknown functions λ j 1 (x) and constants µ j 2 are called the Lagrange multipliers. The differential equations in (C.6) are necessary conditions for a maximum, provided that all the quantities on the left hand side of (C.6) exist and are continuous. Hence, the set of y 1 (x), y 2 (x), . . . , y n (x) which maximize (C.2) subject to the constraints (C.4,C.5), is to be determined, together with unknown Lagrange multipliers, from (C.4,C.5,C.6). This derivation may also be extended to a higher dimensional case, x, y 1 (x), y 2 (x), . . . , y n (x) ∈ R d , as appears in [15]. In this case the EL equations are where y i,k ∂y i ∂x k and Φ follows the same definition as in (C.7), with Ψ j 2 (y 1 (x), y 2 (x), . . . , y n (x); y 1,1 (x), y 1,2 (x), . . . , y 1,d (x), . . . , y n,1 (x), y n,2 (x), . . . , y n,d (x); x) = C j 2 j 2 = 1, . . . , m 2 . Therefore, the Lagrangian Φ for our optimization problem (C.1) is Φ = k a k (u)D k (u) − K−1 L=0 µ L k b Lk (u)D k (u) + E 2 L (u) − (C.9) λ K (u) e 2 K (u) − D K (u) − K−1 k=1 λ k (u) e 2 k + D k+1 (u) − D k (u) − λ 0 (u) D 1 (u) + e 2 0 (u) − 1 . The necessary conditions for the minimizers of (C.1) are that the original constraints are met with equality, and additionally 1. ∂Φ ∂D k = a k (u) − K−1 L=0 µ L b Lk (u) + λ k (u) − λ k−1 (u) = 0 ∀1 ≤ k ≤ K 2. ∂Φ ∂e k = 2e k (u)λ k (u) = 0 ∀0 ≤ k ≤ K 3. ∂Φ ∂E L = 2µ L E L (u) = 0 ∀0 ≤ L < K It is interesting to notice that these condition are exactly the KKT conditions for the discrete optimization case, where u is over a finite grid. Specifically, the first condition corresponds to the derivatives of the Lagrangian, while conditions (2), (3), are equivalent to the complementary slackness property. H 0 : 0U ∼ U (0, 1); H A : U ∼ Φ Φ −1 (.) − C ,and the corresponding Neyman Pearson region:R * (α) = {X : Φ(X) ≤ α} = {X : X ≤ Φ −1 (α)}. where a i , i = 1, . . . , K and b L,i , i = 1, . . . , K, L = 0, . . . , K − 1 are fixed non-negative real functions over Q. In fact, the functions a i , b L,i are all linear combinations of the density functions f T : T ∈ {0, 1} K , with non-negative coefficients that depend on the specific choice of Π, Err. Lemma 2. 3 . 3Consider a solution D * , µ * , λ * which complies with conditions (2.6)-(2.10) and is feasible for Problem (2.5), then 1. This solution is feasible for the dual Problem (2.11). Figure 1 : 1For fixed values of the minimum p-value, the 2-dimensional slices of the following 3-dimensional rejection regions for strong FWER control at level 0.05: Bonferroni-Holm (row 1); OMT procedure for Π θ,3 at θ = −0.5 (row 2), at θ = −1.33 (row 3), and at θ = −2 (row 4); OMT procedure for Π θ,any at θ = −1.33 (row 5), and for any θ > −0.75 or θ < −1.6 (row 6). In green: reject all three hypotheses; in red: reject exactly two hypotheses; in blue: reject only one hypothesis. Since Bonferroni-Holm makes no rejections if all p-values are greater than 0.05/3, the top two right panels are empty. For each panel, the rejection region is in the top right quadrant of the partition of the plane by the point (u 1 , u 1 ). Figure 2 : 2For fixed values of the minimum p-value, the 2-dimensional slices of the following 3-dimensional rejection regions for strong FDR control at level 0.05: MABH (row 1); OMT rejection regions for Π θ,3 at θ = −0.35 (row 2), at θ = −0.5 (row 3), and at θ = −2 (row 4). In green: reject all three hypotheses; in red: reject exactly two hypotheses; in blue: reject only one hypothesis. Since MABH makes no rejections if all p-values are greater than 0.05, the top right panel is empty. For each panel, the rejection region is in the top right quadrant of the partition of the plane by the point (u 1 , u 1 ). Π 27 B 27θ (D) = Π θ 0 (D * (θ 0 , θ A )). Derivation of the expression for F DR L (D) Table 1 : 1Average power (columns 2-4) and minimal power (columns 5-7) when all null hypotheses are false, for different discovery policies with strong FWER control at level 0.05. Π θ,3 Π θ,any Bonferroni-OMT policy OMT policy Bonferroni-OMT policy OMT policy θ Holm for Π θ,3 for Π θ,any Holm for Π θ,3 for Π θ,any -0.5 0.0547 0.111 0.073 0.149 0.194 0.218 -1.33 0.241 0.363 0.247 0.515 0.660 0.742 -2 0.530 0.633 0.323 0.837 0.931 0.968 Table 2 : 2Average power (i.e., probability of rejecting a single hypothesis) for various policies with strong FDR control guarantee at the 0.05 level.θ BH MABH OMT policy for Π θ,3 -0.35 0.042 0.045 0.150 -0.5 0.059 0.064 0.196 -2 0.574 0.633 0.799 Table 3 : 3Average power, Π θ 0 ,2 , for the following rejection policies. For strong FWER control: Bonferroni-Holm (column 2); OMT policy for Π θ 0 ,2 (column 3); maximin optimal for Π θ,2 with Θ B = {θ : θ ≤ θ 0 } and Θ A = {θ : θ ≤ 0} (column 4). For strong FDR control: MABH (column 5); OMT policy for Π θ 0 ,2 (column 6); maximin optimal for Π θ,2 with Θ B = {θ : θ ≤ θ 0 } and Θ A = {θ : θ ≤ 0} (column 7). Strong FWER control Strong FDR control θ 0 Bonferroni-Holm OMT maximin MABH OMT maximin -0.5 0.076 0.118 0.099 0.086 0.174 0.129 -1 0.184 0.251 0.237 0.214 0.326 0.296 -2 0.581 0.637 0.636 0.660 0.734 0.733 . This is a discrete optimization problem, which can be impractical to solve even in finite dimensional cases.1 Given the definitions that follow, we can in fact prove that optimal procedures are indeed symmetric without assuming this form, if we allow randomized policies.For simplicity we choose to state this as an assumption here. The decision on whether to reject in stepwise procedures depends on the rank of the p-value: step-down procedures begin by looking at whether the most significant p-value should be rejected; step-up procedures begin by looking at the least significant p-value . The specific aspect of power considered in[18] was the following: the minimal probability of at least one rejection, among all configurations with L ≥ 1 exchangeable non-null hypotheses with signal θ ≥ . AcknowledgementsThe authors thank Yoav Benjamini, Marina Bogomolov, Jelle Goeman and Ryan Tibshirani for useful comments. This research was supported by Israeli Science Foundation grants 1804/16 (SR) and 1049/16 (RH).A ProofsProof of Lemma 2.1 Given a candidate symmetric solution D, we prove the lemma by constructing an alternative solutionD that complies with the condition and has no lower objective and no higher constraints than D. For every pair of indexes 1 ≤ i < j ≤ K, define:By the symmetry requirement of D, there is a symmetric setwhere u ∈ A ij ⇔ σ ij (u) ∈ A ji , and σ ij is the permutation that switches coordinates i, j only. We will now examine the solutionD which is equal to D everywhere, except on the sets A ij , A ji , where it switches the value of coordinates i, j:We now show the following:1. For all power functions we consider, Π(D) ≥ Π(D).For all constraints we consider ErrThereforeD is an improved solution compared to D. This can be done for all i, j pairs, and we end up withD which has the desired property and is superior to D. This holds in particular for D = D * the optimal solution, hence we are guaranteed to have a weakly monotone optimal solution.It remains to prove properties 1,2 above. For the power, notice that we can write Eqs. (1.1,1.2) as:where r is arrangement increasing as a function of D(u) in the spirit of Assumption 1.2, that is: if we permute the coordinates of D(u) such that the first L associated with non-nulls increase, and the last K − L associated with nulls decrease, then r(D(u)) increases. Considering a specific pair of indexes i < j above, if i > L (both nulls) or j ≤ L (both non-nulls), then by symmetry of D and exchangeability of f T L , we have that Π(D) = Π(D). If i ≤ L but j > L, we have that ∀u ∈ A ij :where the last relation is the arrangement increasing property. We also have ∀u ∈ A ij , f T L (u) ≥ f T L (σ ij (u)) , because of Assumption 1.2. Hence we get:where the inequality (*) uses the simple inequality for non-negative numbers:Since D,D differ only on A ij ∪ A ji and we can repeat this operation for all i, j pairs, this proves property 1. For property 2, the proof is very similar, if we notice that every constraint for both FWER and FDR can be written as:where s is now arrangement decreasing (because it captures false discovery as opposed to power). The same exact steps replicate the result above, and confirm that property 2 holds.Proof of Lemma 2.2Assume that for some u ∈ Q and index j we have that 0 < D j (u) < 1. 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[ "TESTS OF DIRECT AND INDIRECT CPT VIOLATION AT A B FACTORY", "TESTS OF DIRECT AND INDIRECT CPT VIOLATION AT A B FACTORY" ]	[ "Don Colladay \nPhysics Department\nIndiana University Bloomington\n47405INU.S.A\n", "V Alan Kostelecký \nPhysics Department\nIndiana University Bloomington\n47405INU.S.A\n" ]	[ "Physics Department\nIndiana University Bloomington\n47405INU.S.A", "Physics Department\nIndiana University Bloomington\n47405INU.S.A" ]	[]	The issue of testing CPT invariance at a B factory is considered. We present asymmetries that permit a clean extraction of quantities parametrizing direct and indirect CPT violation, using information from Υ(4S) decay via coherent B d B d pairs into various final states. Estimates are given of the bounds on CPT violation that could be obtained in present and planned machines.	10.1016/0370-2693(94)01600-h	[ "https://export.arxiv.org/pdf/hep-ph/9501372v1.pdf" ]	55,105,341	hep-ph/9501372	8246583777dae820ab7b6046fbb6dfcab993d3b6	TESTS OF DIRECT AND INDIRECT CPT VIOLATION AT A B FACTORY 25 Jan 1995 October 1994 Don Colladay Physics Department Indiana University Bloomington 47405INU.S.A V Alan Kostelecký Physics Department Indiana University Bloomington 47405INU.S.A TESTS OF DIRECT AND INDIRECT CPT VIOLATION AT A B FACTORY 25 Jan 1995 October 1994Accepted for publication in Physics Letters B The issue of testing CPT invariance at a B factory is considered. We present asymmetries that permit a clean extraction of quantities parametrizing direct and indirect CPT violation, using information from Υ(4S) decay via coherent B d B d pairs into various final states. Estimates are given of the bounds on CPT violation that could be obtained in present and planned machines. Introduction. Invariance under the combined discrete symmetry CPT is known to be a feature of local relativistic point-particle field theories [1,2,3,4,5,6,7], including the standard model. The current bound on CPT violation is obtained from experiments in the kaon system [8,9], where one figure of merit is a few parts in 10 18 . The basic reason underlying the feasibility of high precision tests of CPT is the interferometric nature of the kaon system, in which a small mass difference and the possibility of strangeness oscillations amplify effects that would otherwise be unobservable. The B 0 -B 0 system is also an interferometric system, but one with detailed properties different from the kaon system. In addition to its use as a probe of standard-model features and parameters, it is also possible at least in principle to use a B factory to further test CPT invariance. At present, several efforts are underway to develop B factories, for example, at Cornell and SLAC in the U.S. and at KEK in Japan. 1 These machines are designed to create relatively high fluxes of Υ(4S) and hence of correlated B 0 d -B 0 d pairs. Other than intrinsic interest, additional motivation for testing CPT comes from the possibility that it is violated in the context of string theory, ultimately due to the nonlocal nature of strings [11]. It is natural to expect any effects of this type to be strongly suppressed at accessible energy scales. However, the interferometric sensitivity of neutral-meson systems may make such effects observable [12,13]. In any neutral-meson system, the string scenario suggests that direct CPT violation would be too small to measure, so all complex direct-violation parameters are effectively zero. In contrast, for each meson system the real and imaginary parts of the complex indirect-violation parameter satisfy a certain condition (see Eq. (2) below), while the magnitude of this parameter can in principle take values attainable in the present or next generation of machines. Experimentally testing these ideas evidently requires isolating the various quantities parametrizing CPT violation. In the decay of a vector meson to a correlated neutral-meson pair, a CP-violating effect (together with the associated T or CPT violation) can be extracted through consideration of selected asymmetries in decay rates 1 A discussion of some of the many extant proposals can be found in ref. [10]. to various final states. For the B system, the literature contains some asymmetries that vanish if CPT is preserved [13,14,15]. However, the issue of a clean extraction of CPT-violation parameters has been an open problem because T and CPT effects have not yet been disentangled. One purpose of this paper is to fill this gap. We present a means of separating quantities parametrizing both direct and indirect CPT violation by identifying certain asymmetries that isolate these parameters. We also obtain estimates for the bounds on CPT violation that could be obtained in present and planned B factories using these asymmetries. Preliminaries. In this section, we present key definitions and equations used later in the paper. Generally, our conventions [13] are analogues of standard ones in the kaon system. Throughout, we take all CP violation (and hence T and CPT violation) to be small, and we neglect terms that are higher-order in small quantities. The eigenvectors of the effective hamiltonian for the B 0 -B 0 system are given by \|B S = 1 √ 2 [(1 + ǫ B + δ B )\|B 0 + (1 − ǫ B − δ B )\|B 0 ] , \|B L = 1 √ 2 [(1 + ǫ B − δ B )\|B 0 − (1 − ǫ B + δ B )\|B 0 ] .(1) The CP-violating complex parameters ǫ B and δ B are measures of indirect T and indirect CPT violation, respectively. For completeness, we note here that in the string-inspired scenario for CPT violation the quantity δ B satisfies Im δ B = ± ∆γ 2∆m Re δ B ,(2) where ∆γ = γ S − γ L and ∆m = m L − m S denote lifetime and mass differences, respectively, of the physical particles B S and B L . Except where stated, we do not impose the condition (2) in what follows. The initial state \|i of the B d -B d pair arising from the Υ(4S) decay has J P C = 1 −− . Taking the direction of the B-meson momenta to be along the z axis, this state can be written as \|i = 1 √ 2 [\|B S (ẑ)B L (−ẑ) − \|B L (ẑ)B S (−ẑ) ] ,(3) where the argument (±ẑ) denotes a meson moving in the ±ẑ direction. In what follows, we label the two mesons by α = 1, 2 and take them to decay into final states \|f α at times t α as measured in the rest frame of the Υ(4S) decay. Defining the transition amplitudes a αS = f α \|T \|B S , a αL = f α \|T \|B L(4) and their ratios η α = a αL /a αS , it follows that the amplitude A 12 (t 1 , t 2 ) for the decay is A 12 (t 1 , t 2 ) = 1 √ 2 a 1S a 2S (η 2 e −i(m S t 1 +m L t 2 )− 1 2 (γ S t 1 +γ L t 2 ) − η 1 e −i(m L t 1 +m S t 2 )− 1 2 (γ L t 1 +γ S t 2 ) ).(5) Experiments observe integrated rates. It is useful to introduce first the onceintegrated decay rate I(f 1 , f 2 , ±v) = 1 2 ∞ v dt \|A 12 (t 1 , t 2 )\| 2 ,(6) where t = t 1 + t 2 is the sum of the decay times and v = \|t 2 − t 1 \| is the magnitude of their difference. Calculation gives [13] I(f 1 , f 2 , ±v) = \|a 1S a 2S η 1 \| 2 2γ e − 1 2 γv e ∓ 1 2 ∆γv + \|r 21 \| 2 e ± 1 2 ∆γv − 2\|r 21 \| cos(∆mv ± ∆φ) ,(7) where γ = γ S + γ L , r 21 = η 2 /η 1 , and ∆φ = φ 1 − φ 2 , with φ α given by η α ≡ \|η α \|e iφα . It is also useful to define symbols for two frequently occurring combinations of the basic parameters ∆m, γ, and ∆γ: a 2 = ∆m 2 + 1 4 ∆γ 2 , b 2 = ∆m 2 + 1 4 γ 2 .(8) In subsequent sections, we use the following twice-integrated decay rates: Γ(f 1 , f 2 ) = ∞ −∞ dv I(f 1 , f 2 , v) = 1 2γ S γ L \|a 1S a 2L \| 2 + \|a 1L a 2S \| 2 − γ S γ L b 2 (a * 1S a * 2L a 1L a 2S + c.c.) , (9) Γ + (f 1 , f 2 ) = ∞ 0 dv I(f 1 , f 2 , v) = 1 2γ \|a 1S a 2L \| 2 γ L + \|a 1L a 2S \| 2 γ S − ( a * 1S a * 2L a 1L a 2S 1 2 γ − i∆m + c.c.) ,(10)Γ − (f 1 , f 2 ) = 0 −∞ dv I(f 1 , f 2 , v) = 1 2γ \|a 1S a 2L \| 2 γ S + \|a 1L a 2S \| 2 γ L − ( a * 1S a * 2L a 1L a 2S 1 2 γ + i∆m + c.c.) ,(11)Γ + incl (f 1 ) = f 2 Γ + (f 1 , f 2 ) = 1 2γ \|a 1S \| 2 + \|a 1L \| 2 − 2[a * 1S a 1L (Re ǫ B + iIm δ B ) + c.c.] .(12) Note that we do not need the quantity Γ − incl (f 1 ) ≡ f 2 Γ − (f 1 , f 2 ) = Γ + incl (f 1 ) in what follows. 3. Direct CPT Violation. In this section we consider certain special B decays, called semileptonic-type decays, for each of which we present an asymmetry that extracts a corresponding parameter measuring direct CPT violation. These decays include the usual semileptonic decays, along with a special class of other modes B 0 → f for which there is no lowest-order weak process that would allow a significant contamination of either B 0 → f or B 0 → f . Among the decays observed to date [16], the semileptonic- type ones include B 0 → D − D + s , B 0 → J/ψK + π − , B 0 → J/ψK * 0 (892), B 0 → ψ(2S)K * 0 (892) , and similar decays into excited states. Note that the mode B 0 → J/ψK 0 is excluded because the K 0 is not a directly observable final state. For the other modes that have been seen, a CKM-suppressed process exists contributing to the contaminating transitions. The various transition amplitudes associated with the decay of the neutral B meson to a semileptonic-type final state f can be parametrized as follows [17,18]: f \|T \|B 0 = F f (1 − y f ) , f \|T \|B 0 = x f F f (1 − y f ) , f \|T \|B 0 = F * f (1 + y * f ) , f \|T \|B 0 = x * f F * f (1 + y * f ) .(13) In these expressions, the parameters on the right-hand side are all complex. The Each individual final state f offers the opportunity to test direct CPT violation through a measurement of the corresponding Re y f . To extract Re y f from rate information, we first determine the amplitudes introduced in Eq. (4). Using the above definitions, to first order in small quantities we find a f S = 1 √ 2 F f (1 + ǫ B + δ B − y f + x f ) , a f L = 1 √ 2 F f (1 + ǫ B − δ B − y f − x f ) , a f S = 1 √ 2 F * f (1 − ǫ B − δ B + y * f + x * f ) , a f L = − 1 √ 2 F * f (1 − ǫ B + δ B + y * f − x * f ) .(14) With these amplitudes, we can calculate for the correlated B pairs the inclusive decay rates Γ + incl (f ) of the type in Eq. (12), where one final state is required to be f while the other is arbitrary and where the decay into f occurs first. We can then extract the asymmetry A + f between decays into f and f . This asymmetry is proportional to Re y f . Explicit calculation gives 2 A + f ≡ Γ + incl (f ) − Γ + incl (f ) Γ + incl (f ) + Γ + incl (f ) = −2Re y f .(15) This asymmetry provides a clean way of extracting an effect from direct CPT violation, independently of any T or indirect CPT violation. In section 5 below, we comment on the experimental feasibility of using this asymmetry and we provide an estimate of the bound attainable on Re y f using Eq. (15). Although not central to the purpose of the present paper it is worth noting that, once a bound (or value) on Re y f has been extracted, the quantity Re ǫ B measuring T violation can be determined without making the assumption of CPT invariance. Consider the total integrated rate asymmetry A tot f,f , given by A tot f,f ≡ Γ(f, f ) − Γ(f , f ) Γ(f, f ) + Γ(f , f ) = 4Re (ǫ B − y f ) .(16) We see that a nonzero value of the combination 1 4 (A tot f,f − 2A + f ) ≡ Re ǫ B for any given final state f of the semileptonic type can be unambiguously attributed to the T-violation parameter Re ǫ B without making the assumption of CPT invariance. We also note that the derivation of Eqs. (15) and (16) applies also to the K 0 -K 0 system, when the final state is f = π − l + ν l . The quantities Re y l and Re ǫ K for this system can therefore also be obtained in this way. Indeed, one can show that, in the absence of CPT invariance and without resorting to a fit to a ∆t-dependent quantity, this method is the only way to extract Re ǫ K from integrated asymmetries in φ decay. The final products of the Υ(4S) decay involve K S and K L rather than K 0 and K 0 . The ratios of matrix elements useful for asymmetry determination therefore involve the former states. Using the definitions (17), we obtain: η J/ψK S ≡ J/ψK S \|T \|B L J/ψK S \|T \|B S = ǫ * K + ǫ B + δ * K − δ B − Re (F J/ψ y J/ψ ) Re F J/ψ − 1 2Re F J/ψ (x J/ψ F J/ψ − x * J/ψ F * J/ψ ) + i Im F J/ψ Re F J/ψ 1 − i Im F J/ψ Re F J/ψ (ǫ * K + ǫ B + δ * K + δ B ) − 1 2Re F J/ψ (x J/ψ F J/ψ + x * J/ψ F * J/ψ ) + i Im (F J/ψ y J/ψ ) Re F J/ψ , η J/ψK L ≡ J/ψK L \|T \|B S J/ψK L \|T \|B L = ǫ * K + ǫ B − δ * K + δ B − Re (F J/ψ y J/ψ ) Re F J/ψ + 1 2Re F J/ψ (x J/ψ F J/ψ − x * J/ψ F * J/ψ ) + i Im F J/ψ Re F J/ψ 1 − i Im F J/ψ Re F J/ψ (ǫ * K + ǫ B − δ * K − δ B ) + 1 2Re F J/ψ (x J/ψ F J/ψ + x * J/ψ F * J/ψ ) + i Im (F J/ψ y J/ψ ) Re F J/ψ .(18) In these equations, the parameters ǫ K and δ K are quantities parametrizing indirect T and CPT violation in the kaon system. The goal is to identify an asymmetry or combination of asymmetries permitting the extraction of Re δ B . To this end, we introduce the following two rate asymmetries: A f,K S ≡ Γ(f, J/ψK S ) − Γ(f , J/ψK S ) Γ(f, J/ψK S ) + Γ(f , J/ψK S ) = 2Re (ǫ B − y f − δ B ) − 2γ S γ L b 2 Re (η J/ψK S ) ,(19) and A f,K L ≡ Γ(f, J/ψK L ) − Γ(f , J/ψK L ) Γ(f, J/ψK L ) + Γ(f , J/ψK L ) = 2Re (ǫ B − y f + δ B ) − 2γ S γ L b 2 Re (η J/ψ K L ) .(20) In deriving the explicit form of these two asymmetries, we have assumed that violations of the ∆B = ∆Q rule are independent of violations of CPT invariance, so that x f = x f and x J/ψ = x J/ψ . Since Im F J/ψ controls the direct T violation in these processes, we have also treated it as a small quantity. The difference between the asymmetries in Eqs. (19) and (20) is a function of CPT-violating parameters: 3 A L,S ≡ A f,K L − A f,K S = 4 b 2 [a 2 Re δ B + γ S γ L Re δ K ] .(21) The measurement of A L,S provides a means of obtaining a fairly stringent bound on Re δ B . The point is that the parameter Re δ K can be bounded using rate information from semileptonic K decays. This is discussed further in the next section. Note also that this result is independent of the state f , which makes the statistics more favorable by allowing a sum over the class of semileptonic-type final states. We emphasize that the asymmetry combination A L,S permits the extraction of Re δ B independently of effects from direct CPT violation and direct or indirect T violation arising in either the B or the K systems. Once a bound on Re δ B is established, Im δ B can in turn be obtained from a measurement of double semileptonic decay rates of the Υ(4S). The quantity to be measured is [13] A f,f ≡ Γ + (f, f ) − Γ − (f, f ) Γ + (f, f ) + Γ − (f, f ) = 4 b 2 ∆γRe δ B + 2∆mγ S γ L Im δ B γ(b 2 + γ S γ L ) .(22) In obtaining the explicit form of this asymmetry, it is assumed that any violation of the ∆B = ∆Q rule is independent of CP violation, so that x f = x * f . Estimates of bounds attainable. In this section, we investigate the bounds attainable from the above analyses on the quantities parametrizing direct and indirect CPT violation in the B system. Following the methods of ref. [19], for each of the relevant quantities we provide an estimate of the number of Υ(4S) events required to reduce the error in the associated asymmetry to one standard deviation. In general, for an asymmetry A = (N + −N − )/(N + +N − ), the binomial distribution implies that the expected number of events N + required to observe a nonzero A at the Nσ level is N 2 (1 + A )(1 − A 2 )/2 A 2 . To convert this to Υ(4S) events, this number must be multiplied by two to account for the branching ratio of Υ(4S) into two neutral B mesons and by the inverse branching ratio for the latter into the relevant final states. The assumption that any T and CPT violations are small implies that interference effects in the correlated decays can be neglected. We first consider bounds on the various Re y f , which provide measures of direct CPT violation. The relevant asymmetry is A + f , given by Eq. (15). Since the second final state is unrestricted, it is sufficient to multiply only by the inverse branching ratio for the process B 0 → f . An additional multiplicative factor appears because the asymmetry involves only those events for which the decay into f occurs first. This factor is two because in the B system γ S ≈ γ L , which makes ∆t > 0 events about as likely as ∆t < 0 events. Combining this information, we find that the number N Υ(4S) (Re y f ) of Υ(4S) events needed to reduce the error in Re y f to within one standard deviation σ is N Υ(4S) (Re y f ) ≃ 1 2σ 2 BR(B 0 → f ) .(23) Next, we consider the bound on Re δ B , parametrizing indirect CPT violation. For the combination A L,S of asymmetries given by Eq. (21), the errors in A f,K S and A f,K L must be combined in quadrature. Also, an estimate is needed of the size of the coefficients of Re δ B and Re δ K in the equation. For the latter, we take [16] γ S ≈ γ L and 4 x = 2∆m/γ ≃ ±0.71. With these values, Eq. (21) becomes A L,S ≈ 1.3Re δ B + 2.7Re δ K .(24) Since A L,S is independent of the specific semileptonic-type final state f , the corresponding branching ratios can be summed. This gives f BR(B 0 → f ) ≃ 15%. Since the K 0 is roughly 50% K S and 50% K L , we take [16] BR (B 0 → J/ψK S ) ≈ BR(B 0 → J/ψK L ) ≈ 1 2 BR(B 0 → J/ψK 0 ) ≃ 3.8 × 10 −4 .(25) From the limit cited in ref. [20], the current bound on Re δ K lies at the 10 −3 level. For simplicity, take Re δ K to be zero, i.e., sufficiently well bounded by K-decay experiments. We also take the errors in the asymmetries A f,K S and A f,K L to be roughly equal. Then, we find that the number N Υ(4S) (Re δ B ) of Υ(4S) events needed to reduce the error in Re δ B to within one standard deviation σ is N Υ(4S) (Re δ B ) ≃ 1.8 × 10 4 σ 2 .(26)N Υ(4S) (Im δ B ) ≃ 5 σ 2 .(27) The ease with which experimental information can be obtained differs for the above quantities. In particular, the asymmetries involved in bounding Re y f and Im δ B require knowledge of the sign of ∆t. However, at a symmetric B factory the distance between decay vertices of the two B mesons is only about 60µm, and information about the location of the Υ(4S) decay is difficult to acquire. A symmetric B factory is therefore best suited to measure or bound Re δ B . The situation is improved at an asymmetric B factory, where the boost alters the topology of the events (see, for example, ref. [21].) This creates a greatly decreased angular separation and hence an easier determination of the sign of ∆t. 6. Summary. We have presented asymmetries that allow the independent extraction of quantities parametrizing both direct and indirect CPT violation in the B system. These asymmetries are given in Eq. (15) for Re y f , in Eq. system is worthwhile since CPT invariance is a fundamental symmetry of the standard model. If any violation is uncovered, the possibility of stringy effects in the system can be tested and the source can be isolated by the methods presented above. independent complex quantities x f and x f are included to allow for the possibility of a violation in the ∆B = ∆Q rule. They vanish if the rule is exact, so in what follows we treat them as small. If T invariance holds, all the quantities x f , x f , F f , and y f are real. If CPT invariance holds, x f = x f and y f = 0. The parameter y f is therefore a measure of direct CPT violation in the decay to the final state f . Its real part Re y f is the present focus of our attention. 4 . 4Indirect CPT violation. The complex parameter δ B is a measure of indirect CPT violation. We first consider a means of obtaining its real part and subsequently address the issue of the imaginary part.Consider decays of the correlated B pair into either J/ψK S or J/ψK L in one decay channel and a semileptonic-type state f in the other. In analogy with Eq.(14), we define the transition amplitudes to the states involving K 0 and K 0 as follows:J/ψK 0 \|T \|B 0 = F J/ψ (1 − y J/ψ ) , J/ψK 0 \|T \|B 0 = x J/ψ F J/ψ (1 − y J/ψ ) , J/ψK 0 \|T \|B 0 = F * J/ψ (1 + y * J/ψ ) , J/ψK 0 \|T \|B 0 = x * J/ψ F * J/ψ (1 + y * J/ψ ) .(17)As before, this allows for possible violation of the ∆B = ∆Q rule via x J/ψ and x J/ψ , while the complex parameter y J/ψ characterizes direct CPT violation. These parameters are assumed small in what follows. (21) for Re δ B , and in Eq. (22) for Im δ B . We have also shown that, once direct violation is measured or bounded, the quantity Re ǫ B parametrizing indirect T violation can be obtained from the asymmetry (16) without assumptions regarding CPT invariance.Assuming no severe acceptance or background effects, it appears experimentally feasible to put bounds on both direct and indirect CPT violation. Estimates of the bounds attainable are given in Eq. (23) for Re y f , in Eq. (26) for Re δ B , and in Eq. (27) for Im δ B . Bounding Re δ B is possible at either a symmetric or an antisymmetric B factory. This can be performed by comparing B d decays into J/ψK S with decays into J/ψK L , without the need for information about ∆t. Measurements of Re y f and Im δ B require a knowledge of the sign of ∆t, which is more easily obtained at an asymmetric factory. Accumulation of about 10 7 or 10 8 correlated B d -B d pairs, which could result from about one running year at a B factory meeting typical design luminosities, should permit the determination of bounds on the various quantities to approximately the 10 −2 level. Independent examination of the different possible types of CPT violation in the B Finally, given a bound on Re δ B , Eq. (22) can be used to provide an estimate of the number N Υ(4S) (Im δ B ) of Υ(4S) events needed to reduce the error in Im δ B to within one standard deviation σ. For example, if the string-inspired relation (2) is valid, a similar calculation to those above gives[13] An analysis keeping terms to all orders in x f and x f shows that Eq.(15) has no linear corrections in these quantities. A derivation relaxing the constraint of small direct T violation shows that Eq. (21) is correct up to terms simultaneously quadratic in Im F J/ψ and linear in δ B or δ K . 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[ "Solving the liar detection problem using the four-qubit singlet state", "Solving the liar detection problem using the four-qubit singlet state" ]	[ "Adán Cabello \nDepartamento de Física Aplicada II\nUniversidad de Sevilla\n41012SevillaSpain\n" ]	[ "Departamento de Física Aplicada II\nUniversidad de Sevilla\n41012SevillaSpain" ]	[]	A method for solving the Byzantine agreement problem [M. Fitzi, N. Gisin, and U. Maurer, Phys. Rev. Lett. 87, 217901 (2001)] and the liar detection problem [A. Cabello, Phys. Rev. Lett. 89, 100402 (2002)] is introduced. The main advantages of this protocol are that it is simpler and is based on a four-qubit singlet state already prepared in the laboratory.	10.1103/physreva.68.012304	[ "https://export.arxiv.org/pdf/quant-ph/0210079v3.pdf" ]	55,165,236	quant-ph/0210079	e4188d37712913187e43019fea31d050c323ff5e	Solving the liar detection problem using the four-qubit singlet state Jul 2003 Adán Cabello Departamento de Física Aplicada II Universidad de Sevilla 41012SevillaSpain Solving the liar detection problem using the four-qubit singlet state Jul 2003(Dated: March 31, 2022)arXiv:quant-ph/0210079v3 15numbers: 0367Hk0250Le0365Ud0365Ta A method for solving the Byzantine agreement problem [M. Fitzi, N. Gisin, and U. Maurer, Phys. Rev. Lett. 87, 217901 (2001)] and the liar detection problem [A. Cabello, Phys. Rev. Lett. 89, 100402 (2002)] is introduced. The main advantages of this protocol are that it is simpler and is based on a four-qubit singlet state already prepared in the laboratory. = 1 √ 6 (\|012 − \|021 − \|102 + \|120 + \|201 − \|210 )(1) (the lower and upper indexes refer, respectively, to the number of constituents and the dimension of the Hilbert space of any of the constituents of the composite system) was shown to be useful for solving the "liar detection problem" [1], which is a simplification of a classically unsolvable problem in quantum computing called the "Byzantine generals' problem" or the "Byzantine agreement problem" introduced in two seminal papers by Lamport, Pease, and Shostak [2,3], and inspired by a pseudohistorical scenario [4]. A version of this problem was previously solved by Fitzi, Gisin, and Maurer [5] also using the three three-level singlet state. Ref. [1] ended by remarking that the preparation of the required states would pose a formidable challenge. Meanwhile, Gisin has proposed a method for preparing the three three-level singlet state [6]. To my knowledge, no experimental results have as yet been reported. On the other hand, the Munich group [7] (see also Ref. [8]) has prepared, using parametric down-converted photons entangled in polarization, a four-qubit entangled state that can be expressed as S (2) 4 = 1 2 √ 3 (2\|0011 − \|0101 − \|0110 −\|1001 − \|1010 + 2\|1100 ).(2) This state belongs to a family of states of N (even) qubits that generalizes the familiar two-qubit singlet state S (2) 2 = 1 √ 2 (\|01 − \|10 ). (3) * Electronic address: [email protected] Any member of this family can be expressed as S (2) N = 1 N 2 ! N 2 + 1 permutations of 0...01...1 z! N 2 − z !(−1) N 2 −z × \|ij . . . n ,(4) where the sum is extended to all of the states obtained by permuting the state \|0 . . . 01 . . . 1 , which contains the same number of 0's and 1's; z is the number of 0's in the first N/2 positions (for example, in \|01 , z = 1; in \|1100 , z = 0; in \|010110 , z = 2). Expression (4) is similar to that introduced in Ref. [1] for the N -particle N -level singlet states (which, according to the notation introduced here, should be denoted as \|S (N ) N ). The interesting point is that the N -qubit "singlet" [9] states given by Eq. (4) share some properties with the N -particle N -level singlet states introduced in Ref. [1]. Both are nonseparable and N -lateral unitary invariant. Nonseparability means that no local model can mimic the predictions of quantum mechanics for these states [10] and implies that the outcomes of the measurements do not reveal predefined results. N -lateral unitary invariance means that, if we act on any state with the tensor product of N equal unitary operators, the result will be to reproduce the same state: U N \|ψ = \|ψ ,(5)U N being U ⊗ . . . ⊗ U , where U is a unitary operator. This implies that the same correlations between the outcomes of measurements on the N particles occur for different sets of measurements. Both properties were essential to the solutions to the problems proposed in Refs. [1,5]. The question is whether the \|S (2) N states can be used to perform tasks that, so far, were specific to the \|S (N ) N states. The \|S (2) N states can be used to distribute cryptographic keys [11,12], encode quantum information in decoherence-free subspaces [13,14,15,16], perform secret sharing, and teleclone quantum states (telecloning is a process combining quantum teleportation and optimal quantum cloning from 1 input to M outputs) [17]. In fact, for N ≥ 4, the \|S Ref. [17]. In this paper, I shall show that the state \|S (2) 4 can also be used to solve the Byzantine generals' and liar detection problems. The Byzantine generals' problem is a classical problem in distributed computing defined as follows. N generals are connected by secure pairwise classical channels. A commanding general must send an order to his N −1 lieutenant generals such that: (a) All loyal lieutenant generals obey the same order; (b) if the commanding general is loyal, then every loyal lieutenant general obeys his order. If the commanding general is loyal, (a) follows from (b). However, the commanding general may be a traitor. The Byzantine generals' problem is a metaphor for distributed processes, some of which may be faulty. For N = 3 generals and 1 traitor the problem has no classical solution, as proved in Refs. [2,3]. However, a version thereof can be solved with the aid of quantum mechanics. The protocol proposed in Ref. [5] solves the Byzantine generals' problem for N = 3 in the following sense: if all generals are loyal then, after the protocol, all the generals would obey the commanding general's order. If one general is a traitor then, after the protocol, either the two loyal generals would obey the commanding general's order or abort the process. As can be easily seen, the Byzantine generals' problem for N = 3 generals and 1 traitor is equivalent to the liar detection problem defined as follows [1]: three parties A(lice), B(ob), and C(arol) are connected by secure pairwise classical channels. A sends a message m to B and C, and B forwards the message to C. If both A and B are honest, then C should receive the same message from A and B. However, A could be dishonest and send different messages to B and C, m AB = m AC , or, alternatively, B could be dishonest and send a message that is different from that he has received, m BC = m AB . For C the problem is to ascertain without a shadow of a doubt who is being dishonest. In Sec. III I shall introduce a protocol based on the \|S (2) 4 state for solving the liar detection problem (and thus also the Byzantine generals' problem) in the following sense: if both A and B are honest then C will receive the same message from A and B. If one of A and B is dishonest then, after the protocol, either C ascertain who is being dishonest or C does not accept the message from one of the parties A and B. Before getting into the details of the protocol, it would be useful to give a rough idea of how it works: besides their messages, A and B must send some additional information. For instance, A must also send B some private information correlated with the message. B and C must be able to check the authenticity of the information they have received by using private information. In addition, to convince C that the message B is sending her is actually that B received from A, B would also need to send C the information B has received from A. In this scenario, being a liar means that she/he did not distribute the appropriate additional information. The advantage derives from the fact that C can detect the origin of the inap-propriate information. II. DISTRIBUTE AND TEST PROTOCOL The protocol for solving the liar detection problem described in Sec. III is based on the assumption that, at the beginning of the protocol, A, B, and C share a large number L of four-qubit systems in the \|S (2) 4 state described by Eq. (2) such that, for each four-qubit system j (from 1 to L), A possesses two qubits (qubits j 1 and j 2 , or qubits j 1 and j 3 , but she does not know which two), B possesses one qubit (j 3 or j 2 , but he does not know which one), and C possesses the fourth qubit j 4 and knows which qubits A and B have in their possession. The purpose of this distribute and test protocol, preceding the protocol for liar detection, is to prepare the initial scenario required for the second protocol and test whether or not said initial scenario has been achieved. The distribute and test protocol has only two possible outcomes: success or failure. In case of success, C would assume that the required initial scenario has been achieved and then the liar detection can be reliably accomplished. In case of failure, C would conclude that something went wrong and would abort any subsequent action. The initial assumptions are that there are only three parties involved: A, B, and C, and that they are connected by secure pairwise classical channels and by noiseless quantum channels. The distribute and test protocol is as follows. (i) C prepares a large number M > L of four-qubit systems in the \|S state. For each four-qubit system j (from 1 to M ), C sends two qubits (qubits j 1 and j 2 , or qubits j 1 and j 3 ) to A, one qubit to B (j 3 or j 2 ), and keeps the fourth qubit for herself. (ii) For each four-qubit system j sent, C checks whether or not A and B have received the right number of qubits (two qubits for A and one for B). (iii) C randomly chooses two large enough subsets S 1 and S 2 of the distributed four-qubit systems. These subsets are used only to test whether or not the initial scenario required for the protocol for liar detection has been achieved. Qubits belonging to S 1 and S 2 will be discarded at the end of the distribute and test protocol. If N i is the number of four-qubit systems in S i , then M = N 1 + N 2 + L. (iv) For each four-qubit system j of S 1 , C asks A to send B her two qubits. (v) C checks whether or not B has received A's qubits. (vi) C asks B to perform some measurements on each of the three qubits (the two qubits B has received from A plus the one that has been in B's possession from the start), and report his results to her by using the secure pairwise classical channel between them. For instance, C can ask B to perform a measurement of the spin along the same direction on all three qubits, M j . (vii) C performs a measurement on her corresponding qubit. For instance, C can measure M j . The outcomes of B and C's measurements must be correlated. For instance, if they have measured M j on the four qubits, two of them must be "0" and the other two must be "1"; due to unitary invariance, this occurs for any M j . Due to the fact that A's qubits are entangled with B and C's qubits, the outcomes can be proved [10] to be genuinely unpredictable (if one does not know the results of the other measurements). C checks whether the expected correlations occur. (viii) For each four-qubit system j of S 2 , C follows the same steps as in (iv)-(vii), only exchanging the roles of A and B. (ix) Since C's choice of subsets S 1 and S 2 is known by A and B only after all qubits have been distributed among them, then, if all the outcomes are correctly correlated, C would assume that the remaining L distributed four-qubit systems are in the \|S (2) 4 state and use them for the protocol described below. If not, C would conclude that something went wrong and abort any subsequent action. III. PROTOCOL FOR LIAR DETECTION Let us suppose that the message m that A sends to B and C, and B sends to C is a bit value "0" or "1" and that all three parties agree to use the following protocol. (I) For each four-qubit system, C asks A and B to perform the same measurement on their individual qubits. After a large number of these measurements, both B and C (A) are in possession of a long list of (pairs of) 0's and 1's l B , and l C (l A ) such as the following. Each of these lists has the following properties. (a) It is random (i.e., generated by an intrinsically unrepeatable method that gives each possible number the same probability of occurring). (b) It is correlated to the other two lists. If "00" ("11") is in position j in l A , then "1" ("0") is in position j in l B and l C . (c) Each party knows only his/her own list. (II) The message A sends B is denoted as m AB . In addition to m AB , A must also send B the list l Requirements (II) and (III) force A to send B information which is correct but perhaps incomplete. Otherwise, if A sends a list containing n erroneous data, then the probability that B does not accept the message m AB would be at last (2 n − 1)/2 n . (IV) The message B sends C is denoted as m BC . B must also send C a list l (mBC) A , which is supposedly the same l (mAB) A that B has received from A. (V) The message A sends C is denoted as m AC . A must also send C a list l AC , which is supposedly l A . (VI) When C finds that m AC = m BC , she has three lists l C , l (mBC ) A , and l AC , to help her find out whether it is A or B who is being dishonest. C must first check whether l AC is consistent with l C . If not, then A is the liar. If yes, then C must check whether l (mBC ) A has an appropriate length and is consistent with l AC . If not, then B is the liar. At this stage, this is the only possibility. IV. CONCLUSIONS The main practical advantage of the introduced protocol over those presented in Refs. [1,5] is that it requires a simpler quantum state (a four-qubit state instead of a three-qutrit state) which has been already prepared in a laboratory [7]. This would make the new protocol immediately applicable for solving both the Byzantine generals' and liar detection problems. The main theoretical advantage is that the introduced protocol seems to be fundamental in a greater degree than those in Refs. [1,5], in the sense that it requires that only one party uses trit values, instead of all three parties. Although both the Byzantine generals' and liar detection problems assume that the parties are connected by secure pairwise classical channels (secure messengers in Ref. [4]), this is an unrealistic constraint for real applications. Note, however, that the protocol described above also works if the classical channels between A and C, and between B and C are not secure but public and unjamable (i.e., which cannot be tampered with). Obviously, such a channel cannot be used to distribute delicate information (like, for instance, whether or not the generals will attack, or the lists l (mBC) A and l AC ), but can be used, together with an additional quantum channel, to implement a standard quantum key distribution protocol [11,12] to send this delicate information. The final picture gives us a quantum solution to a problem without classical solution: the liar detection problem without secure classical channels. state, the only twoqubit state of total spin zero, was coined, together with the name "triplet" (which denotes three orthogonal states of total spin one). In this paper, I use the name "singlet" for any of the \|S (2) N states because, as Eq. (4) shows, they are a generalization of \|S (2) 2 . However, for N > 2, with N even, the dimension of the subspace of total spin zero is N !/[(N/2)!(N/2 + 1)!] (see Ref. [15]). [10] For instance, one can check that \|S states are the N -lateral unitary invariant version of the telecloning states introduced in . . . . . . . . . . . . l A in which m AB appears twice. For instance, if A sends B the message m AB = 0, then she must also send B the list l(mAB=0) A = {1, 3, 6, . . .} since, in l A , "00" appears in positions 1, 3, 6,. . . Note that, if the sequences are random and long enough, then the length of l (mAB=0) A must be about one-quarter of the total length L of the list l A . (III) B would not accept the message if the received list l (mAB) A is not compatible with l B or the length of l (mAB ) A ≪ L/4. For instance, if l (mAB =0) A = {1, 2, 3, 6, . . .}, then B would not accept the message because "0" is in position 2 in l B , so A cannot have "00" in this position. . A Cabello, Phys. Rev. Lett. 89100402A. Cabello, Phys. Rev. Lett. 89, 100402 (2002). . M Pease, R Shostak, L Lamport, J. ACM. 27228M. Pease, R. Shostak, and L. Lamport, J. ACM 27, 228 (1980). . L Lamport, R Shostak, M Pease, ACM Trans. Programming Languages and Syst. 4382L. Lamport, R. Shostak, and M. Pease, ACM Trans. Pro- gramming Languages and Syst. 4, 382 (1982). Sitting in their respective camps, the generals are meditating. Because of the redoubtable fortifications, no battalion by itself can succeed; the attack must be carried out by several of them together or otherwise they would be thrusted back and incur heavy losses that would infuriate the Grand Sultan. Worse, that would jeopardize the prospects of a defeated general to become Vizier. The generals can agree on a common plan of action by communicating thanks to the messenger service of the Ottoman Army, which can deliver messages within an hour, certifying the identity of the sender and preserving the content of the message. Some of the generals, however, are secretly conspiring against the others. Their aim is to confuse their peers so that an insufficient number of generals is deceived into attacking. The resulting defeat will enhance their own status in the eyes of the Grand Sultan. The generals start shuffling messages around. A Taken, Panconesi, Byzantium, 1453 AD. The city of Constantinople, the last remnants of the hoary Roman Empire, is under siege. Powerful Ottoman battalions are camped around the city on both sides of the Bosphorus, poised to launch the next, perhaps final, attack. those trying to agree on a time to launch the offensive, the others trying to split their ranks. . . " To my knowledge, however, no historical account of the fall of Constantinople mentions these treacherous generalsTaken from A. Panconesi (unpublished): "Byzantium, 1453 AD. The city of Constantinople, the last remnants of the hoary Roman Empire, is under siege. Powerful Ot- toman battalions are camped around the city on both sides of the Bosphorus, poised to launch the next, per- haps final, attack. Sitting in their respective camps, the generals are meditating. Because of the redoubtable for- tifications, no battalion by itself can succeed; the attack must be carried out by several of them together or other- wise they would be thrusted back and incur heavy losses that would infuriate the Grand Sultan. Worse, that would jeopardize the prospects of a defeated general to become Vizier. The generals can agree on a common plan of ac- tion by communicating thanks to the messenger service of the Ottoman Army, which can deliver messages within an hour, certifying the identity of the sender and pre- serving the content of the message. Some of the gener- als, however, are secretly conspiring against the others. Their aim is to confuse their peers so that an insufficient number of generals is deceived into attacking. The re- sulting defeat will enhance their own status in the eyes of the Grand Sultan. The generals start shuffling mes- sages around, those trying to agree on a time to launch the offensive, the others trying to split their ranks. . . 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Brassard, in Proceedings of the In- ternational Conference on Computers, Systems, and Sig- nal Processing, Bangalore, India, 1984 (IEEE, New York, 1984), p. 175. . A K Ekert, Phys. Rev. Lett. 67661A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991). . G M Palma, K Suominen, A K Ekert, Proc. R. Soc. London, Ser. A. 452567G.M. Palma, K. Suominen, and A.K. Ekert, Proc. R. Soc. London, Ser. A 452, 567 (1996). . L.-M Duan, G.-C Guo, Phys. Rev. Lett. 791953L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953 (1997). . P Zanardi, M Rasetti, Phys. Rev. Lett. 793306P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). . J Kempe, D Bacon, D A Lidar, K B Whaley, Phys. Rev. A. 6342307J. Kempe, D. Bacon, D.A. Lidar, and K.B. Whaley, Phys. Rev. A 63, 042307 (2001). . M Murao, D Jonathan, M B Plenio, V Vedral, Phys. Rev. A. 59156M. Murao, D. Jonathan, M.B. Plenio, and V. Vedral, Phys. Rev. A 59, 156 (1999).	[]
[ "A biological sequence comparison algorithm using quantum computers", "A biological sequence comparison algorithm using quantum computers" ]	[ "Büsra Kösoglu- ", "Christian Bernecker ", "Jody M Burks ", "Rüdiger Buchkremer ", "\nInternational Business Machines Corporation (IBM)\nFOM University of Applied Sciences\nGermany, Germany\n", "\nInternational Business Machines Corporation (IBM)\nFOM University of Applied Sciences\nArmonkNYGermany\n" ]	[ "International Business Machines Corporation (IBM)\nFOM University of Applied Sciences\nGermany, Germany", "International Business Machines Corporation (IBM)\nFOM University of Applied Sciences\nArmonkNYGermany" ]	[]	Genetic information is encoded in a linear sequence of nucleotides, represented by letters ranging from thousands to billions. Mutations refer to changes in the DNA or RNA nucleotide sequence. Thus, mutation detection is vital in all areas of biology and medicine.Careful monitoring of virulence-enhancing mutations is essential. However, an enormous amount of classical computing power is required to analyze genetic sequences of this size. Inspired by human perception of vision and pixel representation of images on quantum computers, we leverage these techniques to implement a pairwise sequence analysis. The methodology has a potential advantage over classical approaches and can be further applied to identify mutations and other modifications in genetic sequences. We present a method to display and analyze the similarity between two genome sequences on a quantum computer where a similarity score is calculated to determine the similarity between nucleotides.ResultsUnderstanding the abbreviated language of biological information. Biological information, such as that in genetic sequences, is abbreviated by letters representing each chemical residue in the order of its relative appearance in the biochemical polymer chain. In nucleic acid (DNA, RNA) chain sequences, the individual nucleotides are represented by the phosphate attached to the deoxyribose or ribose sugar's 5' carbon (the phosphate is the "5' end") to the unmodified hydroxyl group attached to the sugar's 3' carbon (the hydroxyl is the "3' end"), also known as 5' to 3' directionality. In protein sequences, the sequence representation begins with the amino acid with a free amino group ("free" being not involved in a peptide bond). It ends with the amino acid with a free carboxyl group (or N to C direction). Each type of biological sequence has its letter code representation sets: for example, A, G, C, and T for DNA representing nucleotides with nitrogenous bases Adenine, Guanine, Cytosine, and Thymine; amino acid codes such as W for Tryptophan, G for Glycine, P for Proline, etc.) 14 .Biochemical identity or similarity. The most common method for comparing biological sequences involves comparative studies using sequence alignments. In alignments, sequences are arranged so that regions of importance can be compared (seeFigure 1)and evolutionary, functional, structural, or other features can be identified and analyzed 15 . At least two sequences are required for the simplest of alignments (pairwise alignments comparing two sequences), and multiple sequence alignments compare three or more sequences. Common themes analyzed for various reasons (mutational analyses, phylogenetic relationships, comparative sequence analysis for structurefunction relationships 16 include identity and similarity. Identity between sequences is where, in an alignment, two or more sequences contain the same exact residue (e.g., adenine and adenine) at the same relative position in that residue sequence. Sequence similarity means that the residues are from the same biochemical family. In nucleotides, purines are similar (adenine and guanine), and pyrimidines are similar (cytosine and thymine). In amino acids, similarity can be informed by the makeup of the amino acids R groups. Identity and similarity scores for a given alignment are calculated using similarity matrices such as BLOSUM or PAM 17 . Because we mention two different contexts for identity and similarity (biochemical versus frequency), sequence identity and similarity about biochemical information (Adenine, Guanine, etc.) will be referred to as "biochemical" identity or similarity.	null	[ "https://export.arxiv.org/pdf/2303.13608v4.pdf" ]	257,757,452	2303.13608	90b50c406392dc5f69987f545757684c993534d2	A biological sequence comparison algorithm using quantum computers Büsra Kösoglu- Christian Bernecker Jody M Burks Rüdiger Buchkremer International Business Machines Corporation (IBM) FOM University of Applied Sciences Germany, Germany International Business Machines Corporation (IBM) FOM University of Applied Sciences ArmonkNYGermany A biological sequence comparison algorithm using quantum computers Robert Loredo, Quantum, Yorktown Heights, NY Michele Grossi, European Organization for Nuclear Research (CERN), Geneva Genetic information is encoded in a linear sequence of nucleotides, represented by letters ranging from thousands to billions. Mutations refer to changes in the DNA or RNA nucleotide sequence. Thus, mutation detection is vital in all areas of biology and medicine.Careful monitoring of virulence-enhancing mutations is essential. However, an enormous amount of classical computing power is required to analyze genetic sequences of this size. Inspired by human perception of vision and pixel representation of images on quantum computers, we leverage these techniques to implement a pairwise sequence analysis. The methodology has a potential advantage over classical approaches and can be further applied to identify mutations and other modifications in genetic sequences. We present a method to display and analyze the similarity between two genome sequences on a quantum computer where a similarity score is calculated to determine the similarity between nucleotides.ResultsUnderstanding the abbreviated language of biological information. Biological information, such as that in genetic sequences, is abbreviated by letters representing each chemical residue in the order of its relative appearance in the biochemical polymer chain. In nucleic acid (DNA, RNA) chain sequences, the individual nucleotides are represented by the phosphate attached to the deoxyribose or ribose sugar's 5' carbon (the phosphate is the "5' end") to the unmodified hydroxyl group attached to the sugar's 3' carbon (the hydroxyl is the "3' end"), also known as 5' to 3' directionality. In protein sequences, the sequence representation begins with the amino acid with a free amino group ("free" being not involved in a peptide bond). It ends with the amino acid with a free carboxyl group (or N to C direction). Each type of biological sequence has its letter code representation sets: for example, A, G, C, and T for DNA representing nucleotides with nitrogenous bases Adenine, Guanine, Cytosine, and Thymine; amino acid codes such as W for Tryptophan, G for Glycine, P for Proline, etc.) 14 .Biochemical identity or similarity. The most common method for comparing biological sequences involves comparative studies using sequence alignments. In alignments, sequences are arranged so that regions of importance can be compared (seeFigure 1)and evolutionary, functional, structural, or other features can be identified and analyzed 15 . At least two sequences are required for the simplest of alignments (pairwise alignments comparing two sequences), and multiple sequence alignments compare three or more sequences. Common themes analyzed for various reasons (mutational analyses, phylogenetic relationships, comparative sequence analysis for structurefunction relationships 16 include identity and similarity. Identity between sequences is where, in an alignment, two or more sequences contain the same exact residue (e.g., adenine and adenine) at the same relative position in that residue sequence. Sequence similarity means that the residues are from the same biochemical family. In nucleotides, purines are similar (adenine and guanine), and pyrimidines are similar (cytosine and thymine). In amino acids, similarity can be informed by the makeup of the amino acids R groups. Identity and similarity scores for a given alignment are calculated using similarity matrices such as BLOSUM or PAM 17 . Because we mention two different contexts for identity and similarity (biochemical versus frequency), sequence identity and similarity about biochemical information (Adenine, Guanine, etc.) will be referred to as "biochemical" identity or similarity. Introduction In recent years, substantial progress has been made by scientists in researching our genetic makeup down to the smallest detail to identify genetic diseases and potential targets for therapeutic interventions. The genome and the epigenome analysis can provide many clues for diagnosing and treating diseases 1 . In bioinformatics, a genome's biochemical information is represented as sequences of letters. These text strings can be compared in sequence analyses to identify differences and commonalities that may be important for functional or structural reasons in RNA and protein gene products. Thus, new revolutionary technologies such as quantum computers promise a new era in medicine and biology to expand genome research further. After all, identifying genomic variations and their physiological implications are critical first steps to enhancing the development of novel therapeutic tools and strategies. Genome analysis is also of great importance in molecular diagnostics and optimizing industrial processes for drug or food production 2 . Genome analyses for diagnostic purposes were unimaginable not so long ago, but today, numerous technologies exist to perform genome analyses. Depending on whether the whole genome or only a part of the genome is to be analyzed, such a genome analysis can take weeks or months using state-of-the-art workstations 3 . Particularly in pandemics such as COVID-19, genome sequencing and mutations play an important role. Viruses constantly mutate to evade the host population's immune responses. More extended sequencing processing and analysis times result in further delays in identifying methods for controlling the pandemic, such as appropriate pharmaceutical interventions, biological treatments, and vaccine development, production, and deployment 4 . With acceleration from quantum computing, broader complexity classes of problems can potentially be solved more efficiently 5 , such as mutation searches and pattern recognition in gene sequences. This is because quantum computers are based on the interactions of quantum mechanical states. Classical computers compute with single bits that always assume one of exactly two possible states, zero or one. A quantum computer, on the other hand, calculates using quantum computing principles such as superposition, entanglement, and interference to find solutions to complex problems. These principles have illustrated various speed-ups in many issues 6,7 . A quantum computer has a fundamental unit called a quantum bit, qubit for short. It is like a bit on a classical computer. Current quantum systems are all in their nascent phases of their development. They have not yet reached the level of quantum advantage, a term commonly used to refer to the point where a quantum system can solve a classically intractable real-world problem. Limitations of the current near-term systems include but are not limited to decoherence, gate fidelity, connectivity, and lack of error correction 8 . Due to the nature of these early near-term devices, where the slightest changes in the state due to errors can alter the computational processes of the qubits, there is still much work that can be done to advance the fields of study. Much research on quantum circuits optimizes the mapping from the digital circuit to the analog circuit, which helps reduce errors by applying software and hardware transpiration techniques 9 . These techniques can make transmitting information less prone to decoherence and maintain the quantum state during the computational process. Hardware techniques are applied to ensure that the qubits are isolated from environmental interferences such as disturbances and the noise emitting from the electronic controllers. Minimizing such sources of error is currently an exciting scientific challenge all to its own. There are many potentials in which a quantum computer can be leveraged. It is not only life sciences that will benefit from quantum computers but other domains and industries such as finance, manufacturing, and artificial intelligence (AI). In this paper, we will identify the similarity between sequences using a quantum computer based on existing encoding patterns and techniques 11,12 . Image processing is one of the most researched areas in computer science. This approach was chosen for a mutation search on the quantum computer due to its similarity in identifying sequential digital information. This will be illustrated and implemented based on a practical example using a similarity method to compare quantum images on an IBM quantum computer called Flexible Representation of Quantum Images (FRQI) 13 . This method was selected based on the similarity between the information set and differences among the data, which can be encoded using quantum states. Mapping genetic sequences onto quantum computers using Toffoli and basis gates. In current quantum computing algorithms, information is generally mapped to a quantum state that represents the data in a way to be able to run on a quantum computer. In this case, the quantum state will be encoded via gate-based operations provided by the quantum computer. To map a genetic sequence on a quantum computer, we use a set of gates to represent two pieces of information: the position of the biological sequence and the value at that position. We use multi-control gates to entangle the information together on a quantum computer. One of the most common multi-control gates is the Toffoli gate. In Figure 2 below, a Toffoli gate 22 , which in our experiment entangles three qubits together where the first two represent the position and the third is the value at the position. A Toffoli gate is a 3-qubit gate with two control qubits and one target qubit. In figure 2, the Toffoli gate is shown with three qubits. q0 and q1 represent the control qubits and influence the target qubit q2. The control, identified by the solid sphere, triggers the action of the target qubit identified by the larger sphere with a symbol indicating the type of control; in this case, it is a NOT gate. When both q0 and q1 are enabled (set), the action at the target qubit is performed. If q0 and q1 are not enabled, then no action is performed on the target qubit, q2. Quantum gate-based systems do have one similarity to classical techniques in that they use what is referred to as basis gates to construct more enormous complex gates, such as the Toffoli gate. In classical systems, these basis gates are often referred to as universal gates, such as the AND, NOT, and NAND gates. Quantum systems also have basis gates, U and CNOT gates are two examples of single-qubit and multi-qubit gates, respectively. Therefore, to construct a Toffoli gate using the basis gates available on a quantum computer, you will need a combination of nine universal gates, also called U-gates, and six multiqubit gates (CNOT), which when combined, as illustrated in Figure 2, a circuit depth of 11. A qubit can execute several such gates, which initially do not involve restrictions. But the more gates applied on a qubit, the deeper the quantum circuit becomes. Over time will begin to experience some of the effects of noise, such as decoherence, which could then introduce errors to the results of your experiment. Despite its simple ideal circuit representation, the Toffoli gate and its multi-dimension extension (MCX) are largely adopted in this work, resulting in long circuit decomposition. Representing DNA residues on a quantum circuit through parameterized rotations. Representing the information onto a quantum computer involves a few steps. The first step is to determine to represent the states of our sequence. The table below shows four DNA residues and, thus, a gene sequence defined on a quantum circuit where the theta function will represent the four parameterized variations. Rotations around the axis, i.e., π, are used as a qubit state. This allows us to represent each state by an angle, represented in degrees or radians. These angles determine the position in which the respective qubits are rotated so that they can be recognized directly based on the angle definition, whether the respective qubit is encoded as A, C, T, or G. The angle definition is used to determine the state of the qubit. The qubit represents adenine (A) by setting the parameter to π. The Multi-control qubit gate definition of π/2 is derived from mapping the state to the basis gate on the device. Basis gates are native to the quantum hardware commonly used to create more complex gates, like universal gates in classical computing. To better illustrate this, Figure 3 represents the Adenine in a circuit. The five wires represent the qubits on which the multi-control gates and operators are applied. The first qubit, titled "strip," illustrates how we will indicate which sequences we wish to compare; this will be explained in more detail in the next section. The following figure represents the encoding of the sequence at position (1,1) with a value of 'A' (Adenine), which is represented as a quantum state of /2, which is parameterized as a /4 rotation. According to the encoding strategy adopted, Adenine representation as a quantum state represents the multi-control gate CCCROT decomposition. In this work, in comparison to the Adenine representation in Figure 3, Thymine is represented in Figure 4 with a rotation of π/6. It is defined on the circuit as π/24 because of the multi-qubit gate definition. After encoding all values, we add a Hadamard gate to the strip qubit, followed by a measurement operator that will measure the value of the strip qubit, which we will then use to calculate the similarity. Sequence comparison with Quantum. This section introduces and describes a quantum algorithm This section introduces and describes a quantum algorithm implementation for sequence comparison. Sequence comparison is vital for identifying functional regions, mutations such as polymorphisms, determining different forms of genes such as alleles that result in specific traits or diseases, and many other techniques 23 . In the population, different alleles exist that lead to different expressions in the individual's phenotype and ultimately result in, e.g., brown or blond hair. According to the Human Genome Project 24 , it is easier to identify mutations that cause a particular disease, leading to improved diagnoses, prevention, or therapies. One of the latest and most promising techniques is the CRISPR method of the two Nobel Prize winners, Jennifer Doudna and Emmanuelle Charpentier, also called gene scissors, that promises new possibilities against cancer, AIDS, and several hereditary diseases 25 . We compare two sequences in a pairwise alignment at the quantum implementation to detect patterns and mutations as in classical algorithms such as Needleman-Wunsch and Smith-Waterman; the two sequences are compared position by position 26,27 . The "similarity" approach and the technique described by Fei Yan et al. 28 are used for comparing two gene sequences, which analyzes the sequence information using the strip qubit to identify which sequence pairs to reach. The method includes an evaluation of the similarities between the encoded quantum sequence representation of the same size, here replaced by the four nucleotides. A similarity value is estimated based on the probability distribution of the readouts from quantum measurements. The proposed method provides a significant speed-up compared to traditional computers as it requires less computational power 28 . This is due to the use of various quantum gates to transform all the information-encoded sequences into the strip. This is done by first preparing the sequences into quantum states where each value contains the index (position) in the sequence and is assigned a variable that includes A, C, T, and G. In this experiment. We use a 2x2 matrix to represent a subset of the gene sequences, which are then compared to each other using a single qubit which we will refer to as a strip qubit 28 . The reference sequence we will use to compare against different sequences will be specified by the strip0 labeled qubit. In this example, the Adenine is represented on all four entries on the circuit, as illustrated in Figure 5. A possible encoding strategy adopted here to map letters, the nucleic bases, is to represent all four possible positions 00, 01, 10, and 11 on the quantum circuit. Each letter has its position, which is identified by the numbers below. We defined for each letter a quantum position. The letter "A" represents Adenine. At strip1, the comparison sequence to the reference sequence strip0, the nucleic base T is taken, with the four possible positions, as illustrated in Figure 6. The process flow of the similarity search between the two gene sequences is implemented in this experiment with a similar approach as in the publication of Yan et al. 28 . Figure 7 shows the scheme for parallel comparison of quantum images, which reflects the process flow of the experiment in this work. Figure 6. A parallel comparison of two sequences. The process is divided into three steps. They start with pre-processing to represent a sequence on a quantum computer. In the following quantum process, two sequences run a similarity search per letter. The measurement provides at the end in a histogram the result between these two sequences are similar or not. First, as shown in Figure 7, the pre-processing step generates a quantum circuit representing each gene sequence using amplitude estimation techniques. Then, both gene sequences are compared using the pairwise comparison method, which determines the rotation difference between each sequence. This process is completed by measuring the strip qubit, which generates a snapshot, or shot, of the resulting comparison. These shots are taken 8,000 times in the experiment. Finally, we view the result counts on a histogram. The result we are most concerned with is the probability of 1. This is the value we add as a parameter to determine the similarity score of the two sequences. To determine the similarity between the sequences, we must first extrapolate the probability results of 1, P1. We then use the P1 result value as a parameter to determine the similarity score between sequence1 and sequence2 as shown in the following similarity equation: sim (sequence1, sequence2) = 1 -2P1 Table 2 shows two gene sequences, columns 1 and 2, respectively, where each entry contains each specific value and their represented phase rotation angles. The differences between the four are shown in sequential order. The third column indicates the probability results of the state \|1, P1: Identifying mutations by the similarity of the phase angles. This section provides evidence and the experimental results obtained with the proposed quantum algorithm identifying mutations by the similarity of the phase angles. The experiment results for the expectation value P1 come from the probability output by measuring the strip qubit that connects the sequences to be compared. The expectation value P1 in the above table represents the differences between the gene sequences for each position. An important role is played by the distance between the phases, defined by the phase angles in each position. The closer the two-phase angles in each position are, the smaller the expected value or P1. Here 1 is the expected value, sequence1 is the reference sequence, and sequence2 is the comparison sequence. After the measurement is made, the first thing that is determined is the probability of P1. Using the probability of P1, the similarity score (sequence1, sequence2) between the reference and comparison sequences is determined. The result from the equation indicates whether a change is present. If the similarity value =1, then the two sequences are the same, whereas if the similarity value is less than 1, there is a difference between the two sequences. In this case, the probability of getting a \|1 state result is 0.378, as shown in Figure 8. We use this result to include in the similarity equation to determine the similarity between the two sequences. This results in the following calculation for the similarity score: sim (sequence1, sequence2) = 1 -(2 * 0.378) = 0. 246 It results in the following interpretation: A similarity score of approx. 24.6% that both gene sequences are identical, indicating a differentiation between the two sequences. Discussion The goal of this work was two-fold. First, to represent a gene sequence as a quantum state based on FRQI. Second, to perform a comparison, utilizing the differences between phase angles of the two gene sequences and calculating the similarity score between them. It allowed us to illustrate that a quantum system makes sequential search possible. However, due to noise and errors, current quantum systems' limitations made it impossible to perform a similarity score with an entire gene sequence that can contain millions upon millions of values. It is known that the human genome has about 3 billion base pairs, which is not currently possible to map completely as a quantum state. Nevertheless, this project has demonstrated that quantum computers have the potential to solve complex problems such as similarity scoring faster and with less memory in principle than classical computers. Because comparing two sequences of the size 100.000 with a 4-bit Integer will end in a memory consumption of 37 gigabytes 26,29 . The Hirschberg algorithm reduces the memory consumption to a linear space of O(n) 30,31 . With Quantum memory, information can be stored as a superposition of a Qubit. The entanglement of qubits results in exponential growth of memory. A one-dimensional sequence requires only ([log2 n]) qubits and N bits in a classical system 13,32,33 . The memory consumption is ([log2n + log2m]) qubits for comparing two sequences. The approach in this work could be used on larger quantum computers in the future, expanded with more base pairs, and even analyzed with multiple gene sequences in parallel. Moreover, this approach is just one of many others which could be used for a mutation search. Some of the methods could be string comparison using hamming distance 34 , string comparison using Grover's search algorithm 35 , or, as described in the article by Niroula, P., and Nam, Y., a quantum pattern-matching algorithm that matches a search string (pattern) of length M inside a longer text of length N 36 . Quantum computers are still in the early stages and are subject to several challenges. Quantum systems have potential use in various applications in life sciences 37 or healthcare, where breakthroughs are expected soon. Quantum mechanical calculations should make it possible to quantitatively predict molecules' properties 38 . Another essential use case is the application in genomics. Big Data analytics can analyze everlarger data generated by wearables, inside content, and eHealth apps. In addition, genetic testing is also increasingly in demand. Quantum computing and faster DNA sequencing would enable more comprehensive analyses of this data and lead to a speedier diagnosis. Because of the complexity, we see particular potential in using quantum technologies in systems medicine 39 . Diseases are complex, as we have painfully discovered in the recent pandemic. Not only the genetic code but also the microbiome, the proteome, the metabolome, or the virome may play a crucial role. All systems interact and generate a higher degree of complexity that we can hardly manage with classical computers 40 . Thus, we propose "quantum systems medicine" for further medical research. The challenges of developing a powerful quantum computer are immense because it outperforms classical computers. Quantum computer architectures must increase the number of qubits, improve connectivity between qubits, decrease error rates in operations and storage, and extend algorithm development to all areas where classical computer science faces inherent bottlenecks. In conclusion, the quantum advantage would be achieved when a quantum processor outperforms classical computers by solving problems faster, cheaper, or more accurately. Materials, methods, and limitations Limitations of quantum computers. The conducted investigation in this domain is subject to certain restrictions due to the limitations of the current quantum computers of today. Data analysis with a real dataset could not be performed due to the circuit depth necessary to represent the data. Therefore, the similarity comparison in this work is limited to two short gene sequences. Each gene sequence has four nucleotides, primarily used as an example that can, over time, scale as the technology of quantum systems continues to evolve. There are three main challenges: 1. Scalability: at the time of this writing, there are 433-qubit machines with short decoherence times. We used the 5-qubit machine in this study because only a subset of a gene sequence containing four nucleotides was examined. Error rates: if the circuit is too deep, that is, too many quantum operators are used and the coherence time of the qubits is exceeded, the results will contain some noise which affects the precision of the results. This is where a higher quantum volume may help in the future as we get to error correction and, eventually, logical qubits. 3. Data complexity: because a single gene already contains many nucleotides (sequences of letters), only a part of a gene section can be replicated due to hardware limitations. This is because there are several quantum operators behind a base pair, which represent the complexity. In that case, the whole gene could be mapped on the quantum computer, and a similarity comparison for multiple gene sequences could be used with more powerful quantum computers. Future work. Gene sequencing and analysis is a critical yet complex step in medical research, where the genes are still not fully understood. Scientists are researching and assigning the respective functions of the individual genetic building blocks. An analysis of genes represents only a tiny part of the overall complexity of a disease. Many different components are involved in holistic systems medicine, such as studying proteins, small molecules, chemical reactions, bacteria, viruses, or even the social network in which people interact. Although complexity is increased many times 40 , combining natural language processing (NLP) and quantum computing could lead to new insights. NLP, part of artificial intelligence, can be applied to image and text analysis. NLP offers the possibility of performing an otherwise very time-consuming investigation much faster and examining more significant amounts of textual data. An example application is the Artificial Intelligence Double Funnel by Buchkremer et al. 41 . And it is at this point that quantum computers could be of great help in analyzing these complex systems. As the increase in performance of quantum systems become available, more complex tasks could be solved with the help of quantum computers, which could include decoding the human genome. The processor architectures and performance, such as hardware quality and continuing development in error mitigation and error correction, are continuously improving. IBM plans a 1,121-qubit quantum computer for 2023 and many error suppression and mitigation techniques, which will significantly advance and enable high scaling 42 . Even though we are still at the beginning of a very long road, quantum computers will open many possibilities for us in the future that will help solve currently intractable problems. Materials and Methods. In this section, we want to provide more details about implementing the algorithm described in this work. The comparison process consists of the following steps: Step 1: Assign each element a rotation angle; in this case, we used those defined in Table 1. Step 2: Create a quantum circuit that includes several qubits that will represent the index number of the characters in each sequence string as a value of logN, where N is the number of characters in each sequence. In this example, we use N=4. Therefore, log4 = 2 qubits. Then add the strip qubit; in this case, we will use a single qubit to represent the strip since we are comparing two sequences. And finally, a single qubit encodes the value of the sequence element. Finally, add a classical register to read the results of the strip qubit. Table 3 illustrates the assignment labels. Basic Circuit Represents the four nuclear bases, each with phase angles Step 3: Set the strip and index qubits into a superposition state using Hadamard gates, as illustrated in figure 3. Step 4: Encode each strip, index, and sequence value using a multi-control, single target gate, as illustrated in figures 3 and 4. In this case, the gate will consist of three controls, and one target, where the controls align with the strip and index qubits, and the target gate is a rotation gate aligned with the value of the sequence (either A, C, T, G), labeled DNA in the circuit. Step 5: Set the control to capture each index position of each strip, denoted in Figs 3 and 4. Step 6: Add another Hadamard gate to the strip qubit to complete the circuit. Step 7: Measure the strip qubit; this will be used to determine the difference between sequences based on the result of P1. The basic circuit is defined as follows and represented in Figures 3 and 4. We now present an application of a quantum algorithm based on the Flexible Representation of Quantum Images (FRQI) which we applied to biological sequences. In its seminal definition, this algorithm provides a quantum representation of images that allows efficient encoding of classical data into a quantum state, i.e., color information and pixel position. Within this algorithm, encoding classical data into a quantum state requires a polynomial number of simple gates. The idea here is to leverage the quantum encoding techniques to represent the varying nucleotide (4) (or amino acid -20) representations using the Multi-Control-RY gate (MCRY). Here we define the quantum state \|Sequence (θ)ñ as a normalized state that encodes with this formulation the genetic sequences to compare, as a function of q: \|Sequence(θ)⟩ = 1 2 n ∑ (cos θ i \|0⟩ + sin θ i \|1⟩) ⊗ \|i⟩ 2 2n −1 i=0 A simple example of a 4-character nucleotide sequence, where the quantum registers represent the strip, index (reference in the gene sequence), and the nucleotide basis value, is given below, with corresponding θ angles (nitrogen bases) and associated Kets (position encoding): \|Sequence1⟩ = 1 2 [(cos θ 0 \|0⟩ + sin θ 0 \|1⟩) ⊗ \|000⟩ + (cos θ 1 \|0⟩ + sin θ 1 \|1⟩) ⊗ \|100⟩ + (cos θ 2 \|0⟩ + sin θ 2 \|1⟩) ⊗ \|101⟩ + (cos θ 3 \|0⟩ + sin θ 3 \|1⟩) ⊗ \|111⟩ + (cos θ 4 \|0⟩ + sin θ 4 \|1⟩) ⊗ \|010⟩ + (cos θ 5 \|0⟩ + sin θ 5 \|1⟩) ⊗ \|011⟩ + (cos θ 6 \|0⟩ + sin θ 6 \|1⟩) ⊗ \|001⟩ + (cos θ 7 \|0⟩ + sin θ 7 \|1⟩) ⊗ \|110⟩ ] In this example, we have set all units of the first sequence (theta 1) to p/4 (A) and the second sequence (theta2) to p/24 (T) and obtained a probability of P1 to be 0.378, which resulted in a similarity score of 0.246, or 24.6%. Figure 1 . 1Types of sequence alignments. (a) Pairwise sequence analysis showing residues biochemically identical between alignment positions. Relative positions are compared vertically by column, with individual sequences on each row. Biochemically Identical residues are indicated with "\|." Position numbers on either side of each sequence indicate position numbers in the original sequence record. This is similar to the output of the BLAST algorithm 18 . A Gap, or section of the sequence for which there is no information, is indicated by a. "-." The gap may signal a deletion mutation for homologous or evolutionarily related sequences. (b) A simple multiple sequence alignment from Clustal19 . Sequence positions are compared vertically, with gaps introduced (with a penalty) to move segments of the sequence string in a manner that maximizes the alignment score calculated by a given algorithm. (c) A comparative sequence alignment for identifying secondary structure features in RNAs 20,21 ; Sequences are arranged in phylogenetic or evolutionary order. Then a proposed location marker indicating features such as base pairing (e.g., a pairing "mask") is placed at the top of the alignment. In this example, proposed canonical Watson-Crick and G-U "wobble" pairs are indicated in upper case, and mismatches or non-pairs are indicated in lower case. The 2:1 Larsen-Zwieb rule is the benchmark for identifying compensatory base changes or changes between sequences resulting in a base pair. The base pairing supported ("S") by this analysis is reported at the bottom of the alignment for each position of the helix and shown on the right with position numbers in a secondary structure diagram. Figure 2 . 2A Toffoli gate or CCX (left) and its decomposition is represented in hardware native 1-qubit, 2-qubit basis gates (right). Figure 3 . 3Figure 3. According to the encoding strategy adopted, Adenine representation as a quantum state represents the multi-control gate CCCROT decomposition. Figure 3 . 3Thymine representation as a quantum circuit, representing the decomposed circuit. Figure 4 . 4A sequence built from four letters. Figure 5 . 5The letter "T" represents Thymine, which has the same numbers as Adenine. Figure 8 . 8Measured results of the strip qubit; after running 8000 shots, the value for \|1 P1 is 0.378. Table 1 : 1Quantum state representations of DNA residues and parameterized U gate rotations used in this study. Table 2 : 2Results of the two compared sequences; comparing the two sequences on a quantum computer illustrate the varying probability results of P1x. Table 3 . 3Circuit labels and definitions. Each qubit(s) is identified to illustrate whether the qubit applies to the position or value.Definition Strip 0 Idx 0 Idx 1 Dna 0 References the gene sequence Position (X-axes) Position (Y-axes) AcknowledgmentsM.G. is supported by CERN through CERN Quantum Technology Initiative.Data Availability StatementWe used the 5-qubit IBM cloud machine in this study because only a subset of a gene sequence containing four nucleotides was examined. This work was performed based on the Qiskit textbook chapter on quantum image processing (available at https://qiskit.org/textbook/ch-applications/image-processing-frqi-neqr.html).FundingThis project was not funded.Author contributionsThe authors of this paper contributed the following: R. B. conceived the idea of searching genetic sequences as letters on quantum computers. R. B. and B.K.K. conceived the study. R. L. identified the initial concept to implement this work based on the Qiskit textbook chapter on quantum image processing for a bioinformatic application. M. G. contributed to the coding of the sequences using the methods described above. J. B. contributed her research in the field of bioinformatics, providing the context of gene sequencing and biological sequence analysis. C. B. contributed his research in DNA sequencing with a particular focus on the performance and memory consumption of traditional alignment methods such as BLAST and FASTA. B.K.K., who authored her graduate thesis on this topic and served as the motivation to write this paper. R. B. served as B.K.K.´s graduate advisor on her graduate thesis at the FOM University of Applied Sciences. 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[]	[ "Patrick Côt \nDominion Astrophysical Observatory\nAT&T Bell Laboratories\nDepartment of Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA\n\nDepartment of Physics and Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA\n\nMcMaster University Hamilton\nL8S 4M1ONCanada\n", "Douglas L Welch [email protected] \nDominion Astrophysical Observatory\nAT&T Bell Laboratories\nDepartment of Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA\n\nDepartment of Physics and Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA\n\nMcMaster University Hamilton\nL8S 4M1ONCanada\n", "Philippe Fischer \nDominion Astrophysical Observatory\nAT&T Bell Laboratories\nDepartment of Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA\n\nDepartment of Physics and Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA\n\nMcMaster University Hamilton\nL8S 4M1ONCanada\n", "K Gebhardt [email protected] \nPresent Address: Department of Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA\n", "Patrick Côt ", "Philippe Fischer ", "Karl Gebhardt ", "Douglas L Welch ", "\nDepartment of Physics and Astronomy\nDominion Astrophysical Observatory\nMcMaster University Hamilton\nL8S 4M1ONCanada\n", "\nDepartment of Physics and Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich RoadV8X 4M6VictoriaBCCanada\n", "\nDepartment of Physics and Astronomy\nMcMaster University Hamilton\nL8S 4M1ONCanada\n", "\nDepartment of Physics and Astronomy\nand AT&T Bell Laboratories\nMcMaster University Hamilton\n600 Mountain Avenue, 1D-316, Murray HillL8S 4M1, 07974ON, NJCanada\n", "\nRutgers University\n08855-0849PiscatawayNJ\n" ]	[ "Dominion Astrophysical Observatory\nAT&T Bell Laboratories\nDepartment of Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA", "Department of Physics and Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA", "McMaster University Hamilton\nL8S 4M1ONCanada", "Dominion Astrophysical Observatory\nAT&T Bell Laboratories\nDepartment of Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA", "Department of Physics and Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA", "McMaster University Hamilton\nL8S 4M1ONCanada", "Dominion Astrophysical Observatory\nAT&T Bell Laboratories\nDepartment of Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA", "Department of Physics and Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA", "McMaster University Hamilton\nL8S 4M1ONCanada", "Present Address: Department of Astronomy\nUniversity of Michigan\nDennison Building, Ann Arbor48109-1090MIUSA", "Department of Physics and Astronomy\nDominion Astrophysical Observatory\nMcMaster University Hamilton\nL8S 4M1ONCanada", "Department of Physics and Astronomy\nHerzberg Institute of Astrophysics National Research Council\n5071 West Saanich RoadV8X 4M6VictoriaBCCanada", "Department of Physics and Astronomy\nMcMaster University Hamilton\nL8S 4M1ONCanada", "Department of Physics and Astronomy\nand AT&T Bell Laboratories\nMcMaster University Hamilton\n600 Mountain Avenue, 1D-316, Murray HillL8S 4M1, 07974ON, NJCanada", "Rutgers University\n08855-0849PiscatawayNJ" ]	[ "Astrophysical Journal" ]	BV CCD frames have been used to derive surface brightness pro les for NGC 3201 which extend out to approximately 18 0 . A total of 857 radial velocities with median precision ' 1 km s 1 for 399 member giants have been used to trace the velocity dispersion pro le out to 32.1 0 (the approximate tidal radius determined from ts of single-mass, isotropic King-Michie models to the cluster surface brightness pro les). The median difference in radial velocity for stars on either side of an imaginary axis stepped through the cluster in 1 increments shows a statistically signi cant maximum amplitude of 1.22 0.25 km s 1 . We discuss several possible explanations of this result, including: (1) cluster rotation; (2) preferential stripping of stars on prograde orbits near the limiting radius; (3) the projection of the cluster space velocity onto the plane of the sky and; (4) a slight drift in the velocity zero point. It is di cult to unambiguously identify the primary cause of the observed structure in the velocity eld, however, and we suspect that all of the above processes may play a role. The BV surface brightness pro les and radial velocities have been modeled with both single-and multi-mass King-Michie models and nonparametric techniques. The corresponding density-and M/L-pro les show good agreement over the interval 1:5 < R < 10 pc, and both approaches suggest a steady rise in M/L with distance from the cluster center. Due to the low cluster luminosity, we are unable to place useful constraints on the anisotropy of the velocity dispersion pro le, though the global mass-tolight ratio is well-constrained by the models: M/L B ' M/L V ' 2.0 0.2 for the multi-mass and nonparametric models, compared to ' 1.65 0.15 for models having equal-mass stars. Our best-t, multi-mass models have mass function slopes of x ' 0:75 0:25, consistent with recent ndings that the form of the mass function depends on the position relative to the potential of the Galaxy.	10.1086/176532	[ "https://export.arxiv.org/pdf/astro-ph/9506042v1.pdf" ]	17,198,184	astro-ph/9506042	81747257f9fcb46d0f5cd428e3443c2e3c5e505c	November 20 1995 Patrick Côt Dominion Astrophysical Observatory AT&T Bell Laboratories Department of Astronomy Herzberg Institute of Astrophysics National Research Council 5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA Department of Physics and Astronomy University of Michigan Dennison Building, Ann Arbor48109-1090MIUSA McMaster University Hamilton L8S 4M1ONCanada Douglas L Welch [email protected] Dominion Astrophysical Observatory AT&T Bell Laboratories Department of Astronomy Herzberg Institute of Astrophysics National Research Council 5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA Department of Physics and Astronomy University of Michigan Dennison Building, Ann Arbor48109-1090MIUSA McMaster University Hamilton L8S 4M1ONCanada Philippe Fischer Dominion Astrophysical Observatory AT&T Bell Laboratories Department of Astronomy Herzberg Institute of Astrophysics National Research Council 5071 West Saanich Road, Victoria, 600 Mountain Ave., 1D-316 Murray HillV8X 4M6, 07974BC, NJCanada, USA Department of Physics and Astronomy University of Michigan Dennison Building, Ann Arbor48109-1090MIUSA McMaster University Hamilton L8S 4M1ONCanada K Gebhardt [email protected] Present Address: Department of Astronomy University of Michigan Dennison Building, Ann Arbor48109-1090MIUSA Patrick Côt Philippe Fischer Karl Gebhardt Douglas L Welch Department of Physics and Astronomy Dominion Astrophysical Observatory McMaster University Hamilton L8S 4M1ONCanada Department of Physics and Astronomy Herzberg Institute of Astrophysics National Research Council 5071 West Saanich RoadV8X 4M6VictoriaBCCanada Department of Physics and Astronomy McMaster University Hamilton L8S 4M1ONCanada Department of Physics and Astronomy and AT&T Bell Laboratories McMaster University Hamilton 600 Mountain Avenue, 1D-316, Murray HillL8S 4M1, 07974ON, NJCanada Rutgers University 08855-0849PiscatawayNJ Astrophysical Journal November 20 1995astro-ph/9506042 6 Jun 95 DYNAMICS OF THE GALACTIC GLOBULAR CLUSTER NGC 3201 BV CCD frames have been used to derive surface brightness pro les for NGC 3201 which extend out to approximately 18 0 . A total of 857 radial velocities with median precision ' 1 km s 1 for 399 member giants have been used to trace the velocity dispersion pro le out to 32.1 0 (the approximate tidal radius determined from ts of single-mass, isotropic King-Michie models to the cluster surface brightness pro les). The median difference in radial velocity for stars on either side of an imaginary axis stepped through the cluster in 1 increments shows a statistically signi cant maximum amplitude of 1.22 0.25 km s 1 . We discuss several possible explanations of this result, including: (1) cluster rotation; (2) preferential stripping of stars on prograde orbits near the limiting radius; (3) the projection of the cluster space velocity onto the plane of the sky and; (4) a slight drift in the velocity zero point. It is di cult to unambiguously identify the primary cause of the observed structure in the velocity eld, however, and we suspect that all of the above processes may play a role. The BV surface brightness pro les and radial velocities have been modeled with both single-and multi-mass King-Michie models and nonparametric techniques. The corresponding density-and M/L-pro les show good agreement over the interval 1:5 < R < 10 pc, and both approaches suggest a steady rise in M/L with distance from the cluster center. Due to the low cluster luminosity, we are unable to place useful constraints on the anisotropy of the velocity dispersion pro le, though the global mass-tolight ratio is well-constrained by the models: M/L B ' M/L V ' 2.0 0.2 for the multi-mass and nonparametric models, compared to ' 1.65 0.15 for models having equal-mass stars. Our best-t, multi-mass models have mass function slopes of x ' 0:75 0:25, consistent with recent ndings that the form of the mass function depends on the position relative to the potential of the Galaxy. INTRODUCTION Improved observational constraints on the internal dynamics of globular clusters are demanded by many of the most fundamental questions regarding their formation and evolution. For instance, does the velocity dispersion pro le (VDP) fall o with projected distance from the cluster center in the manner predicted by multi-mass, King-Michie models (Da Costa and Freeman 1976;Gunn and Gri n 1979) or does an appreciable amount of dark matter reside in the envelope of some clusters, giving rise to a at VDP? Do global M/Ls vary from cluster to cluster and how does the M/L change with radius in a given cluster? What is the form of the cluster mass function and how signi cant are the observed correlations of mass function slope with cluster position in the Galaxy (Capaccioli, Piotto and Stiavelli 1993)? How abundant are primordial binaries in these Population II systems (Hut et al. 1992;Côt e et al. 1994) and what is their radial distribution? How common are central velocity dispersion cusps (Peterson, Seitzer and Cudworth 1989) and do they re ect post core-collapse evolution (Spitzer 1985;Grabhorn et al. 1992) or the presence of massive central bodies (Newell, Da Costa and Norris 1976)? Are stellar orbits in the outer regions of the cluster predominantly radial or has the tidal eld of the Galaxy induced isotropy near the tidal radius, as suggested by the three-body/Fokker-Planck models of Oh and Lin (1992)? And to what extent do the underlying dynamics a ect the mix of stellar populations (see Trimble and Leonard 1994 for a recent review)? Clearly, answers to many of these questions require an understanding of how the velocity dispersion varies from the cluster core to the tidal radius. Early measurements of globular cluster velocity dispersions were based on the broadening of stellar absorption lines in long-slit spectra of the integrated cluster light (Illingworth 1976). However, since the requisite measurements are possible only in the cluster core (and are complicated by the presence of central binaries which tend to produce overestimates of the dispersion and luminous giants which often dominate the measured spectrum; Zaggia et al. 1992), this technique is capable of providing little more than a central M/L for a given cluster. An alternative approach is to use proper motions of individual stars to determine the run of velocity dispersion. Though potentially very powerful (these observations contain the two components of the VDP needed to solve the non-rotating Jeans equation; Leonard et al. 1992), proper motions of the requisite precision are exceedingly di cult to measure for stars in such crowded elds (e:g: Cudworth and Monet 1979). Most work on cluster dynamics has therefore made use of individual stellar radial velocities, since a precision of ' 1 km s 1 is often attainable using large telescopes, high-QE detectors and cross-correlation techniques. Nevertheless, progress has been slow, since the measurement of a hundred or more such velocities with a single-channel spectrograph or a radial velocity scanner is tremendously time-consuming. As a consequence, only a handful of dynamical studies based on large radial velocity samples (N > 100) have appeared in print (e:g:, M3, Gunn and Gri n 1979;M2, Pryor et al. 1986; ! Cen and 47 Tuc, Meylan and Mayor 1986;M13, Lupton, Gunn and Gri n 1987;NGC 6397, Meylan, Mayor andDubath 1991 andNGC 362, Fischer et al. 1993). In addition, the sequential nature of the observations has restricted work (with one notable exception; Seitzer 1983) primarily to the inner cluster regions where the probability of observing member stars is highest. Once radial velocities are in hand, cluster membership is more easily established, though for many clusters, the velocity-space distributions of eld and cluster stars show considerable overlap. In these cases, even with kinematic information, assigning cluster membership remains a rather dubious business. With the introduction of multi-object spectrographs on many 4.0m-class telescopes, surveys to trace VDPs over the full range in cluster radius have become feasible. In this paper, we present a dynamical study of the Galactic globular cluster NGC 3201 based on 857 radial velocities for 399 member stars which have been used to derive a projected VDP which extends from the core to the approximate tidal radius. NGC 3201 is the logical cluster for such an endeavor, since its systemic radial velocity of 494 km s 1 ensures no overlap with the eld star population (see Table 1 for a summary of general cluster properties). The radial velocities used in this analysis have been presented in a companion paper (Côt e et al. 1994) and were accumulated primarily with ARGUS, the ber-fed, bench-mounted, multi-object spectrograph on the CTIO 4.0m telescope. ARGUS is ideally suited for a complete sampling of the VDP since it o ers (1) high velocity precision with the echelle grating, (2) the ability to acquire spectra for 24 stars simultaneously, (3) a minimum ber separation of 10 00 (an important consideration for the crowded cores of most globular clusters) and (4) a 50 0 eld of view which allows the simultaneous observation of both core and envelope stars. The resulting VDP has been combined with BV surface brightness pro les (SBPs) based on CCD photometry to investigate the cluster dynamics using both single-and multi-mass King-Michie models (Michie 1963;King 1966a;Da Costa and Freeman 1976;Gunn and Gri n 1979) and nonparametric models (Merritt and Tremblay 1994;Gebhardt and Fischer 1995). OBSERVATIONS AND REDUCTIONS Realistic models of globular clusters require a knowledge of not only the light pro le but also the radial variation in velocity dispersion (see Lupton, Gunn and Gri n 1985). In this section we describe the data upon which our SBPs and VDP for NGC 3201 are based. Surface Photometry and Star Counts SBPs for NGC 3201 were constructed from BV CCD frames collected with the 1.0m Swope telescope at Las Campanas Observatory on 21/22 January and 22/23 February 1991. The detector used was the 1024 1024 Tek2 CCD (readnoise = 7e , gain = 2e /ADU and scale = 0.609 00 /pixel) so that each image measures 10.4 0 10.4 0 . Exposure times were 120s for V and 180s for B. BV frame pairs were obtained for ve separate elds on the night on 22/23 January 1991: one centered on the cluster core and four o set by roughly 7 0 toward the NE, SE, SW, and NW directions ( Figure 1 shows the relative positioning of these ve elds). Another sequence of BV frames was obtained on the night of 22/23 February 1991, this time for elds o set from the cluster center by 14 0 , 23 0 , 32 0 , 41 0 and 50 0 in both the N and S directions. Frames were bias-subtracted, overscan-corrected, trimmed and at-elded with the usual IRAF 4 tasks. Instrumental magnitudes were determined with DoPHOT (Schechter et al. 1993) and calibrated using nine unsaturated, on-frame photoelectric standards chosen from the lists of Alcaino and Liller (1984) and Lee (1977). A comparison of our photometry with that of Brewer et al. (1993) showed excellent agreement in V and a slight (but systematic) di erence in (B V ) in the sense that our inferred colors are ' 0.04 mag redder than those of Brewer et al: (1993). Due to its low Galactic latitude (see Table 1) and the fact that it is a sparse cluster (concentration class = X; Shapley 1930), determining a reliable background level for NGC 3201 is somewhat problematic. Previous work based on visual star counts made on photographic plates (Peterson and King 1975;King et al. 1968) placed the cluster tidal radius at r t ' 36 0 . We therefore used our DoPHOT photometry for all elds out to ' 50 0 to perform stars counts in concentric annuli positioned on the cluster center found by Shawl and White (1986). Only main-sequence turno stars and evolved giants were used to construct the surface density pro les, ensuring that the measured SBPs correspond to stars of almost identical mass. Of course, crowding in the cluster core reduces the completeness of the star counts \| in this region we performed surface photometry in the manner described by Fischer et al. (1993). The central CCD images were divided into concentric annuli positioned on the cluster center. These annuli were then divided into eight azimuthal sections; the mean pixel value for each of these sectors was then determined and the median of these eight measurements was adopted as the surface brightness for the annulus (at the area-weighted mean radius). The uncertainty in the surface photometry was taken to be the standard error in the median of the eight sectors; Poisson statistics were used to determine the corresponding uncertainties in the star counts. The surface photometry and star count surface densities were then merged by matching (via least-squares) the two datasets in the range 4 < R < 9 pc (where incompleteness in the star counts was negligible). The 4 IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under contract to the National Science Foundation. nal, background-subtracted BV SBPs for NGC 3201 are given in Table 2 which records the projected radius, the adopted surface brightness and its source. In converting R and to pc and L pc 2 , we have adopted M V; = 4.83, M B; = 5.48 (Binney and Tremaine 1987), an apparent distance modulus of (m M) V = 14:20 0:15 and a cluster reddening of E(B V ) = 0:21 0:02 (Brewer et al. 1993) so that 1pc = 40.24 00 at NGC 3201. Radial Velocities As previously noted, the number of dynamical studies of globular clusters based on large radial velocity samples is rather small. Moreover, the observed VDPs for these clusters generally extend to only r=r t 0:25. Our reasons for observing cluster members at large projected radii in NGC 3201 were twofold: (1) to trace the cluster VDP out to ' r t and (2) to search for primordial binaries and nd their radial distribution. The results of our search for binaries in NGC 3201 have already been published, along with the entire sample of NGC 3201 radial velocities (Côt e et al. 1994). 5 The reader is referred to the above reference for a more complete discussion of the spectroscopic observations and reductions. Spectra were accumulated during several observing runs with telescopes at both Las Campanas and CTIO. Photon-counting echelle spectrographs on the Las Campanas 2.5m and CTIO 4.0m telescopes were used to measure 267 radial velocities for 189 stars (chosen from the nder charts of Lee 1977) within ' 5 0 of the cluster center during observing runs in January/February 1991. Object spectra in the range 5120 { 5460 A were crosscorrelated against template spectra for a variety of IAU radial velocity standard stars to give heliocentric radial velocities with precision 1.3 { 1.7 km s 1 . The bulk of the spectra were obtained during two observing runs (February 15 { 16 and March 15 { 16 1992) with ARGUS: the bench-mounted, ber-fed, multi-object spectrograph on the CTIO 4.0m telescope. Object spectra in the range 5090 { 5160 A were cross-correlated against high S/N spectra of the twilight/dawn sky. Repeat observations suggest that the ARGUS velocities have a median accuracy of 1 km s 1 . This sample of 1730 radial velocities for 1316 stars was then combined with the 129 radial velocities (92 member stars) used in the lone previous dynamical study of NGC 3201 (Da Costa et al. 1993) and also published in Côt e et al. (1994). The complete survey therefore consists of 1859 radial velocities for 1318 stars within 36 0 of the cluster center. As pointed out earlier, the high systemic radial velocity of 494 km s 1 for NGC 3201 (see Figure 1 of Côt e et al. 1994 and below) ensures the unambiguous identi cation of all eld stars (a total of 889 radial velocities for 879 eld stars were accumulated). Tables 2 and 3 of Côt e et al. (1994) list all 970 radial velocities for 439 cluster members. Any radial velocity variables in the sample such as RR Lyraes or binary stars will lead to overestimates of the velocity dispersion and must be removed from the nal sample; the 19 known photometric variables in our survey (Fourcade and Laborde 1966;Sawyer-Hogg 1973) were therefore omitted along with the 21 candidate binaries listed in Côt e et al. (1994). The nal sample therefore consists of (weighted) mean velocities for 399 cluster members in the range 0:08 0 R 32:1 0 based on 857 radial velocities. Absolute positions with precision 1 00 for all 399 program stars, derived from our CCD frames and APM scans using the HST Guide Star Catalog, are recorded in Côt e et al. (1994). Estimates of the mean velocity v 0 and intrinsic velocity dispersion 0 of NGC 3201 can be obtained with the formulae of Armandro and Da Costa (1986). However, as noted by Suntze et al. (1993), care must be taken in applying these forumlae since the Armandro and Da Costa (1986) estimator of the error in the intrinsic variance assumes an unweighted variance, not the weighted variance given by their formulae. We have therefore followed the prescription of Suntze et al. (1993) in deriving the mean velocity and intrisic velocity dispersion, although for the present sample the di erence amounts to less than a few percent. Based on the above sample of 420 velocities (i:e: excluding only the known photometric variables), we nd v 0 = 494:0 0:2 km s 1 and 0 = 3:70 0:13 km s 1 . Removing the 21 binary candidates listed in Côt e et al. (1994) changes these numbers only slightly: v 0 = 494:0 0:2 km s 1 and 0 = 3:66 0:13 km s 1 . Considering only the 93 stars within 1.46 0 (i:e: one core radius) of the cluster center gives v 0 = 493:2 0:4 km s 1 and 0 = 3:88 0:28 km s 1 . For all three samples, these estimates of v 0 and 0 are virtually identical to those obtained using the technique of Peterson and Latham (1986). Using the maximum-likelihood approach of Pryor and Meylan (1993) yields v 0 = 494:4 0:2 km s 1 and 0 = 3:77 0:16 km s 1 for all 420 stars, v 0 = 494:4 0:2 km s 1 and 0 = 3:69 0:13 km s 1 for the restricted sample of 399 stars and, v 0 = 494:5 0:4 km s 1 and 0 = 3:64 0:25 km s 1 for the 93 stars within one core radius. In x3.1.2, we review the maximum-likelihood estimators of the systemic velocity and velocity dispersion devised by Gunn and Gri n (1979). Although these estimates are model-dependent, we nd v 0 494:2 km s 1 and 0 4:3 km s 1 using this approach, in good agreement with the above results (note that both the nonparametric and binned VDPs show central dispersions which are slightly lower than that seen at intermediate radii; the maximum-likelihood scaling of the single-and multi-mass model VDPs makes use of all of the velocities and therefore leads to slightly larger estimates for the central dispersion). The dispersion pro le is discussed in more detail in x3.1. Possible Structure in the Velocity Field The relationship between heliocentric radial velocity and both radius and position angle is shown in the upper and middle panels of Figure 2. The latter of these plots suggests some dependence of the observed velocity on position angle, a trend which is more apparent in the lower panel of Figure 2, where we have plotted the median radial velocity versus position angle for eight azimuthal bins of equal width. The results are summarized in Table 3 which records the bin number, the number of stars in each sector, the range in position angle, the mean position angle for the sector and the median radial velocity. Another way of identifying an azimuthal dependence of radial velocity, and one commonly used to search for rotation in globular clusters, is to step an imaginary axis through the cluster in small, angular increments and compute the median velocity di erence on either side of this line, V r;med , for each angle. The results of such a procedure are shown in the upper panel of Figure 3 which shows the dependence of V r;med on axis angle . Also shown is the best-t sine curve which has an amplitude of 1.22 0.25 km s 1 and a phase shift of 277 12 , implying a position angle for the axis of = 7 12 . How signi cant is this detection? To answer this question, we have generated 1000 arti cial datasets (i:e: 399 radial velocities at the corresponding locations of our program stars). The simulated radial velocity for each star has been chosen by adopting the mean dispersion for a star at the projected radius of the program object (estimated from the best-t, single-mass, isotropic King-Michie model). For each star, a realistic amount of observational noise (typical velocity uncertainty ' 1 km s 1 ) has been included. Each simulated dataset was then analyzed in a manner identical to that used for the real data. The histogram of the resulting amplitudes is given in the lower panel of Figure 3. Only eight times in 1000 trials did the best-t sine wave have an amplitude of 1.22 km s 1 or greater; we therefore conclude that the observed signal is signi cant at the 99.2% level. Of course, such simulations do not account for possible di erences in the velocity zero points from di erent runs. For example, since the bulk of the radial velocities were accumulated during a pair of two-night observing runs with Argus, it is possible that a drift in the velocity zero point could give rise to the observed trend, provided the eld of view studied on a given night is appreciably smaller than the total eld of view. As discussed in Côt e et al. (1994), small zero point corrections were applied to the velocities accumulated during di erent observing runs in order to bring them onto a common system. As a result, the mean velocities for two Argus runs show good agreement: 494.38 km s 1 and 494.37 km s 1 for the rst and second runs, respectively. On the other hand, the mean velocities for the rst and second nights of the second Argus run show an o set of 1.4 km s 1 , suggesting that a shift in the velocity zero point may be to blame. However, it is unlikely that such a shift is solely responsible for the observed trend, since the sample of stars observed on March 15/16 1992 and March 16/17 1992 have almost indentical distributions with respect to the total eld of view of the survey. We now discuss a number of other possible origins of the observed dependence of radial velocity on position angle: cluster rotation, the stripping of stars near the tidal radius by encounters with the Galactic disk and the projection of the cluster space velocity onto the plane of the sky. Rotation Since the lone previous dynamical study of NGC 3201 (based on mean radial velocities for 92 stars; Da Costa et al. 1993) found appreciable rotation in the range 1.3 0 r 3.2 0 where 51 stars showed a formally signi cant rotation amplitude of 0.7 0.2 km s 1 , it would not be surprising if a small amount of rotation was observed in our sample of velocities. It is therefore natural to ask whether or not the velocity di erence of 1.22 km s 1 evident in Figure 3 can be due solely to rotation. If we assume that the observed amplitude is, in fact, purely a consequence of cluster rotation, we have V rot = ' 1.22/3.67 = 0:33 0:08 for the ratio of ordered (V rot ) to random ( ) motions, consistent with the theoretical ratio for the purely rotationally-attened case. 6 The location of NGC 3201 in the -V rot = plane is given in Figure 4. For comparison, we also show the ( , V rot = )relation for rotationally-attened oblate spheroids with isotropic velocity dispersion tensors (Binney 1978;Binney and Tremaine 1987). Further evidence that the observed velocity di erence is at least partly caused by rotation is provided by the orientation of the axis which maximizes V r;med \| the kinematically determined rotation axis is located at a (projected) position angle of 7 12 , in excellent agreement with the position angle of the cluster's photometric minor axis (2 7 according to White and Shawl 1987). That is, both the amplitude and position of the NGC 3201 rotation axis are in good agreement with that expected for a rotationally-attened oblate spheroid. Nevertheless, we believe that it is unlikely that rotation alone is the cause of the observed dependence on position angle. Although the good agreement between (1) the assumed rotation axis and the photometric minor axis and (2) the observed and expected cluster ellipticity lends support to the notion that rotation is partly responsible for observed trend in velocity, the amplitudes of the besttting sine-curves t to increasingly distant samples of radial velocities show an unexpected increase with radius (see Table 4), suggesting that some other e ect may also be at work. Tidal Stripping An increase in apparent rotation at large radii has been predicted by Oh and Lin (1992), who carried out an investigation of the tidal evolution of globular clusters using a Fokker Planck/three-body integration approach. They con rmed earlier ndings (Keenan and Innanen 1975;Je erys 1976;Keenan 1981) that stars on direct orbits are less stable than their retrograde counterparts. Prolonged interaction with the Galactic tidal eld therefore results in preferential stripping of such stars and can lead to an apparent rotation of the cluster. Oh and Lin (1992) also note that such an apparent rotation can be extended into relatively small radii for clusters with appreciable velocity anisotropy \| not inconsistent with the results of our dynamical modeling (see x4.4). It is therefore possible that such a process is at work in NGC 3201 and has contributed to the apparent cluster rotation at large radii. Motion Across the Line of Sight Finally, we note that another, albeit more speculative, explanation of the observed trend is possible: if NGC 3201 has a substantial component of its systemic velocity directed across the line of sight, then the observed dependence of radial velocity on position relative to the cluster core may be a result of slightly di erent projections of the cluster space velocity along the line of sight (since the radial velocities are scattered over an area of nearly one square degree; see Figure 5). If it is assumed that the velocity variations are solely due to systemic motion across the line of sight, the cluster space velocity can be computed using only the observed radial velocities. In a rectangular coordinate system with X toward = 0 , = 0 , Y toward = 90 , = 0 and Z toward the north celestial pole, we can neglect the radial velocity dispersion of the cluster and write (Feast et al. 1961) v i = X cos i cos i + Y sin i cos i + Z sin i (1) where v i is the observed radial velocity of a cluster member, i ; i are its coordinates and XY Z are the components of the cluster space velocity. The velocity components which minimize the 2 of the above equation are: X = 409:9 25:0 km s 1 , Y = 23:0 23:2 km s 1 and Z = 340:0 23:2 km s 1 . All 399 cluster members have been used in the t, with the radial velocities weighted by i;w = ( i 2 +v s 2 i 2 ) 1=2 where i;w is the adopted uncertainty, i is the observational error associated with the ith radial velocity and v s i is the local radial velocity dispersion of the cluster according to the best-t singlemass, isotropic King-Michie model (see x3). Of course, in order to convert XY Z into the Galactic rest frame, we must correct for solar motion. To do this, we adopt a correction of T i = 108:1 cos i cos i + 112:4 sin i cos i 172:1 sin i : ( 2) to the radial velocity of an object at i ; i (epoch 2000.0 coordinates). In deriving this correction we have adopted a basic solar motion of 16.5 km s 1 toward l = 53 ; b = 25 (Binney and Tremaine 1987) and an LSR motion of 220 km s 1 toward l = 90 ; b = 0 (Kerr and Lynden-Bell 1986). The best-t space velocity for NGC 3201, corrected for solar motion and Galactic rotation, is then v i;c = 302( 25) cos i cos i 135( 23) sin i cos i 168( 23) sin i which implies a cluster velocity in the Galactic rest frame of ( ; ; Z) = ( 216 23; 214 24; 212 25). The magnitude of the velocity, jV s j = 370 41 km s 1 , is well below the local Galactic escape velocity of 475 km s 1 (Carney et al. 1988;Cudworth 1990). We emphasize, however, that any space velocity derived in this fashion must be regarded as extremely uncertain since we have completely neglected rotation and tidal stripping, both of which are likely to play a role in explaining the large-scale trends in the velocity eld. Nevertheless, a proper motion study of NGC 3201 is clearly desirable since it would provide an direct test of our spectroscopically derived space velocity. Regardless of the exact cause (or causes) of the observed structure in the velocity eld, we have chosen to neglect it in modeling the cluster dynamics. This decision can be justi ed by assuming that the observed dependence of velocity on position agle is due entirely to rotation. The low ellipticity of NGC 3201 (like those of most other globular clusters, 95% of which have 0:20; White and Shawl 1987) suggests that ordered motions are dynamically unimportant. For instance, although neglecting a rotation of 1 km s 1 in the dynamical analysis will lead to overestimates of the cluster mass (Fischer et al. 1992a), the resulting errors will be at most a few percent (see xIVa of Pryor et al. 1986), DYNAMICAL MODELS In order to determine the form of the cluster mass function, the anisotropy of the VDP and several other interesting parameters including cluster mass, luminosity and M/L, we have t single-and multi-mass King-Michie models to the observed BV SBPs and radial velocities. Since these models have seen widespread use in the study of globular clusters, the dynamical parameters derived from these models will be directly comparable to those of other clusters. In x3.2 we describe the results of modeling the observed SBPs and radial velocities using a nonparametric technique in which the form of the distribution function is not assumed a priori (Gebhardt and Fischer 1995). Single-and Multi-Mass Models Anisotropic, single-mass King-Michie models (King 1966;Michie 1963) assume a dis-tribution function of the form f(E; J) / e J 2 (e E 1) (4) where E and J refer to the energy and angular momentum of the cluster stars. Similarly, anisotropic, multi-mass models (Da Costa and Freeman 1976;Gunn and Gri n 1979) have, for each mass class, the distribution function f i (E; J) / e J 2 (e A i E 1):(5) It is assumed that equipartition of energy in the cluster core has produced a dependence of the form A i / m i , where m i is the mean mass of the ith mass class. For each single-mass model, the anisotropy radius r a (the radius beyond which the velocity dispersion tensor is mostly radial) is held constant and the dimensionless central potential (King 1966), W 0 , is varied until the best-t values of the scale (or core) radius, r s , the scale luminosity and the cluster concentration parameter, c = log(r t =r s ), are obtained. In this way, we t a grid of models with varying amounts of anisotropy to the cluster SBP. For each model, we determine the scale velocity, v s , which gives the best match between the projected model VDP and the observed VDP (see x 3.1.2 for details of the tting procedure). For the multi-mass models, we include another parameter, x, the global slope of the cluster mass function. Both r a and x are then held constant for each model and the best-tting scale radius, scale luminosity and concentration parameter are computed. The dimensionless, projected model VDP for the cluster giants is then scaled via maximum-likelihood to the measured velocities to yield v s . The BV SBPs for NGC 3201 are shown in Figure 6 along with the best-t single-mass King-Michie models (with r a =r s = 1; 10; 5 and 3). In the upper panel of Figure 7 we show the resulting VDP for NGC 3201; the solid line represents the best-t, isotropic, single-mass model VDP, scaled by v s to the measured velocities. The LOWESS estimate of the velocity dispersion (see Gebhardt et al. 1994) used in the nonparametric modeling is indicated in the lower panel by the solid line. The velocity dispersion pro le computed in annular bins is given in Table 5, whose columns record the bin number, sample size, radial range, median radius and intrinsic velocity dispersion estimated using the approach of Suntze et al. (1993) as well as that of Pryor and Meylan (1993). Both estimates of the binned dispersion pro le are plotted in the lower pannel of Figure 7; the lled triangles indicate the Suntze et al. (1993) estimates of the dispersion while the lled squares represent those found using the Pryor and Meylan (1993) approach. For each bin, both the central location (\mean") and the scale (\dispersion") have been treated as free parameters. A summary of the mass classes adopted for the multi-mass models is given in Table 6 which records, from left to right, the bin number, the lower bin boundary, the upper bin boundary and a description of the bin contents. We have followed the prescription of Pryor et al. (1989) in accounting for the evolved stars. Stars more massive for the mainsequence turno are assumed to have become cluster white dwarfs with objects having main-sequence masses in the ranges 8 { 4M , 4 { 1.5M and 1.5 { 0.826M (the mass at the tip of the red giant branch is taken to be 0.826M after Bergbusch andVandenBerg 1992 andBrewer et al. 1993) assumed to have resulted in cluster white dwarfs with masses of 1.2, 0.7 and 0.5M , respectively. The neutron stars produced from higher mass stars are assumed to have been expelled from the cluster potential well, though the millisecond pulsars (Phinney 1992;Hut et al. 1992) and bright X-ray sources (Forman et al. 1978) seen in several clusters suggest that at least some of these objects contain neutron stars. Given the high space velocities observed for pulsars in the Galactic disk (' 210 km s 1 ; Lyne et al. 1982), it is unclear how neutron stars can remain bound to their respective clusters (e:g: the central escape velocity in NGC 3201 is < 10 km s 1 according to our models). Moreover, neutron stars produced via Type II supernovae (which occurred approximately 10 10 years ago in globular clusters) should have evolved to pulse periods in excess of ' 1 second (Bailyn 1993). Models in which millisecond pulsars are produced by the accretioninduced collapse of cluster white dwarfs (Michel 1987;Bailyn and Grindlay 1990) avoid these di culties, so we have chosen to assume that neutron stars produced through Type II supernovae have been expelled from the cluster. We have adopted a modi ed mass function (Pryor et al. 1989(Pryor et al. , 1991 of the form (M) = M (1+x) dM; for M 0:3M(6) (M) = MdM; for M < 0:3M (7) which is similar to that observed for local disk stars (Miller and Scalo 1979;Scalo 1986). The mass function is taken to have both high-and low-mass cuto s, for which we adopt M H = 8.0M and M L = 0.16M . As pointed out by Gunn and Gri n (1979), the choice of M L is somewhat arbitrary \| reducing the low-mass cuto leads to models with enhanced numbers of low-mass stars at large radii and, consequently, to a higher inferred cluster masses. For stars fainter than the upper main sequence, we have used the isochrones of Bergbusch and VandenBerg (1992) 3.1.1 Luminosity-to-Mass Ratios For each tted model, we wish to compute two \Population" M/Ls \| a global massto-light ratio (M/L) and a central mass-to-light ratio (M/L) 0 . In order to derive the population M/Ls corresponding to our adopted mass function, we require a mean luminosityto-mass ratio, L/M, for the component stars in each of our adopted mass bins. Some of the pitfalls involved in this rather uncertain process have been discussed by Pryor et al. (1986). Brie y, the mass bin containing the evolved cluster stars (red giants, subgiants, horizontal branch stars) and stars near the main-sequence turno contributes virtually all of the cluster light; cluster population M/Ls therefore depend sensitively on the L/M adopted for this bin. A photometrically and spatially complete luminosity function for these stars is therefore required since, for NGC 3201, the L V /M of stars in this bin varies from ' 840 at the tip of the red giant branch to ' 2 at the main-sequence turno (Bergbusch and VandenBerg 1992). We have therefore used our wide-eld BV CCD images to perform star counts in NGC 3201 of stars brighter than V = 19:97 (corresponding to a mass of 0.75M ). Our counts are photometrically complete for stars of this brightness (the majority of which are expected to fall within the CCD elds shown in Figure 1). Each star brighter than this limiting magnitude was assigned a position (based on its location in the cluster color-magnitude diagram) on either the 16 Gyr, Fe/H] = {1.26, O/Fe] = +0.55 isochrone of Bergbusch and VandenBerg (1992) or the corresponding horizontal branch evolutionary sequence of Dorman (1992); probable eld stars were rejected from the analysis. Both the luminosity and mass of the individual stars were added to derive a mean L/M for the stars in the bin \| based on counts of 7660 upper main-sequence, subgiant, red giant branch stars and 237 horizontal branch stars, we adopted mean L/Ms of L V /M = 10.03 and L B /M = 9.33 for stars in the range 0.75 { 0.826M . (Throughout this paper, we give mass-to-light ratios in solar units.) Since our photometry is not deep enough to derive a reliable luminosity function for the fainter main sequence stars, we used the same Bergbusch and VandenBerg (1992) isochrone to derive a mean L/M for each of the remaining bins. Although these L/Ms vary with the adopted mass function slope, the dependence is very weak, amounting to a 3% decrease in the mean L/M of the main-sequence stars in the bin as x increases from 0.0 to 2.0. Fitting the Models For each model, we have minimized 2 = N X j=1 1 j 2 L s (r j =r s ) j ] 2 in order to get the best-t scale radius, r s , and scale luminosity, L s . Here j are the measured surface brightnesses and (r j =r s ) are the projected model surface brightnesses at radii r j . The corresponding uncertainties in j are given by j . The 2 goodness-of-t statistic is computed for each tted model and the reduced gravitational potential, W 0 , is varied until the computed 2 is minimized. The maximum-likelihood estimators for the scale velocity, v s , and the cluster systemic velocity, v 0 , are then found by solving (Gunn and Gri n 1979) N X i=1 v i (v 2 s 2 i + 2 i ) v 0 N X i=1 1 (v 2 s 2 i + 2 i ) = 0 (9) N X i=1 (v i v 0 ) 2 (v 2 s 2 i + 2 i ) N X i=1 1 (v 2 s 2 i + 2 i ) = 0(10) where v i and i are the measured radial velocities and corresponding uncertainties; i refers to the projected, dimensionless (model) velocity dispersion at the radius corresponding to v i . According to our models, v 0 ranges from 494.1 { 494.3 km s 1 . We then compute two \Dynamical" M/Ls (once again, a global and a central M/L) using our tted model and the maximum-likelihood estimator for the scale velocity. In order for the model to be considered acceptable, both dynamical M/Ls should match the population M/Ls computed with the assumed mass function (of course, the best models should also have relatively low 2 values). For each model, we then compute a number of cluster parameters which are summarized in Tables 7 and 9 (for the V -band) and Tables 8 and 10 (for the B-band). Fitted cluster parameters are given in Tables 7 and 8 which record, from left to right, the anisotropy radius in units of the scale radius, the mass function slope, the cluster concentration parameter, the dimensionless central potential W 0 , the scale radius in pc, the central surface brightness 0 in L pc 2 , the reduced 2 for the t to the SBP, the probability of meeting or exceeding this 2 , the scale velocity in km s 1 , the central and global population M/Ls and the central and global dynamical M/Ls. Derived cluster parameters are recorded in Tables 9 and 10 whose columns contain, from left to right, the anisotropy radius in units of the scale radius, the mass function slope, the scale radius in pc, the half-mass radius r h in pc, the tidal radius r t in pc, the model central velocity dispersion v s 0 in km s 1 , the total cluster luminosity L in L , the central luminosity density 0 in L pc 3 , the total cluster mass M in M , the central mass density 0 , the mean density inside the half-mass radius h , the mean density inside the tidal radius t (all in M pc 3 ), the logarithm of the half-mass relaxation time t r0 in years (Lightman and Shapiro 1978) and the logarithm of the half-mass relaxation time t rh in years (Spitzer and Hart 1971). 7 3.1.3 Monte Carlo Simulations Also recorded in Tables 7 { 10 are the one sigma uncertainties for each of the above parameters, determined through Monte Carlo experiments like those described by Pryor et al. (1989) andFischer et al. (1992b). Brie y, 1000 datasets were generated from the best-t model using the estimated uncertainties in the actual SBP. Both the arti cial SBPs and the simulated radial velocities have points at the identical distance (and, for the velocities, identical position angle) as the actual data. For the radial velocity simulations, we have followed the prescription of Fischer et al. (1992b) and have generated random three-dimensional positions as well as radial and tangential velocities for each of the measured stars. The velocities were then projected onto the plane of the sky and a random measurement error (based on the actual uncertainty) included. Based on model ts to these 1000 simulated datasets, the rms dispersion about the mean of each parameter has been taken as the one sigma uncertainty. Of course, the uncertainty derived in this manner represents only the internal error in the tted parameter. The large spread in the best-t parameters computed from the various models (for example, the cluster mass ranges from 1.1 10 5 M to 5.4 10 5 M based on ts to the V -band SBP) demonstrates that the true errors are likely to be much larger. For example, it is now recognized that vastly di erent density pro les are capable of providing equally impressive ts to the SBPs of most globular clusters (e:g: Merritt 1993), so that cluster parameters derived using the King-Michie formalism need not re ect the true physical state of the cluster. To investigate this possibility, we have modeled the observed SBPs and radial velocities using nonparametric techniques which make no a priori assumption about the cluster distribution function. Nonparametric Models Since a complete discussion of the nonparametric technique may be found in Gebhardt and Fischer (1995), only a brief description is given here. In all cases, we have assumed that the stellar velocities are isotropic \| the extension to anisotropic velocities will be reported in the near future (Gebhardt and Merritt 1995). The technique requires both a cluster SBP and VDP (the latter is estimated using a LOWESS t to the data; see Figure 7). We then estimate the deprojected quantities through the Abel integrals (r) = 1 Z r t r d dR dR p R 2 r 2 (11) (r)v 2 r (r) = 1 Z r t r d( 2 p ) dR dR p R 2 r 2(12) where (r) is the luminosity density, is the surface brightness, and p and v r are the projected and deprojected velocity dispersions, respectively. In practice, the above integrals cannot be evaluated out to the tidal radius since the cluster surface brightness and velocity dispersion near the tidal radius are poorly known. For this reason, the point where the velocity dispersion is last measured has been taken as the upper limit (although we do not consider the contributions from beyond this point, the e ect is non-negligible only near the tidal radius Since these equations involve two and one half derivatives of both the surface brightness and the projected velocity dispersion, a certain amount of smoothing of the (noisy) data is required. We use a spline smoother with the smoothing parameter chosen by generalized cross validation (Wahba 1990). All calculations are performed in logarithmic space to avoid enhanced weighting of the higher values when using the spline tter. We then compute the cluster mass density and M/L pro les. Observational biases and con dence bands are determined through Monte Carlo simulations in which arti cial datasets are generated by randomly choosing a velocity from a Gaussian distribution with the standard deviation given by the dispersion pro le at the radius of each observation and the uncertainty of each velocity measurement. The procedure described above is then used to compute, in a completely analogous fashion, the mass density and M/L pro les for each simulation. By generating 1000 simulations, we have a distribution in mass density and M/L at each point in our pro le which we use to measure both the mode and the 95% con dence band. The central location of the simulation distribution minus the initial estimate of the mass density is then adopted as the estimate for the bias. Once determined, we must correct for the bias by adding it back into the original estimate. The con dence bands are correspondingly shifted for the bias as well, though the con dence bands require twice the bias to be added since the simulations have a bias from both the technique and from the original estimate. We have assumed that the velocity distribution at each radius is Gaussian. The tidal cuto ensures that this is not the case, and a fully nonparametric technique would need to include a proper estimation of the velocity distribution at each radius. Nevertheless, we feel that deviations from the assumed Gaussian distribution are likely to be small enough to have negligible e ect on our results. Figure 8 shows the mass density pro le and M/L pro le of NGC 3201 computed in this fashion. The solid lines are the bias-corrected mass density and M/L estimates while the dotted lines indicate the 95% con dence bands. Although the VDP and SBP extend to both smaller and larger radii than are plotted, the con dence bands become so large that the estimates of mass density are essentially meaningless in these regions. The dashed lines in Figure 8 indicate the cluster mass density and M/L pro les according to one of the best-t, multi-mass King-Michie models (r a =r s = 1, x = 1:0) from x3.1. In general, the pro les determined using the di erent approaches show very good agreement \| the mass density and M/L pro les determined with the the multi-mass models fall within the 95% con dence bands of the nonparametric pro les for virtually all radii. The M/L pro le determined via the King-Michie approach shows a systematic rise in the outer regions of the cluster (a consequence of the assumption of energy equipartition among the various mass species), whereas that derived from the nonparametric models shows a rather low central value of M/L V 1 and a steady rise to M/L V 4 at 10 pc. RESULTS Previous Work on NGC 3201 How do the results of our modeling compare to previous studies? The rst attempt to measure the cluster SBP was that of King et al. (1968) who performed star counts on photographic plates, tracing the SBP out to a radius of approximately 20 0 . Their best-t isotropic, single-mass model was found to have c = 1:56. The lone previous dynamical study of NGC 4.2 Mass-to-Light Ratios Using spectra of the integrated cluster light, Illingworth (1976) found the M/Ls of ten centrally concentrated globular clusters to fall in the range 0.9 to 2.6 with a mean of 1.6, in excellent agreement with the NGC 3201 mass-to-light ratio computed from singlemass King models. For a sample of 32 clusters with reliable central velocity dispersions, Mandushev et al. (1991) derived a mean M/L of ' 1.2 using single-mass King models, whereas Pryor and Meylan (1993) compiled velocity dispersion data for 56 galactic globular clusters and modeled the available SBPs and velocities with multi-mass King models, reporting a mean of 1.7 0.9 for the entire sample. (Since this value is sensitive to outliers, they also report a mean of 2.3 1.1 based on biweight estimators). Imposing the criterion that an acceptable model must t have similar (1) Mass Function Slope Inspection of Tables 7 -10 shows that models with mass function slopes in the range 0:5 < x < 1:0 provide the best match between the central and global dynamical and population M/Ls. A grid of models with stepsize x = 0:1 t over this interval showed best agreement for slopes of x = 0:7 and 0:8 (for the respective V -and B-band SBPs); we therefore adopt a best-t value of x = 0:75 0:25. How sensitive is this value to the form of the adopted mass function? For mass functions with no transition at M = 0.3M (i:e: single exponent mass functions), models with x = 0:4 0:5 show the best agreement between the population and dynamical M/Ls. Similarly, experiments with a variety lower mass cuto s con rmed the ndings of earlier workers (e:g: Gunn and Gri n 1979) who noted that changing M L has relatively little e ect on the observable properties of the models \| for instance, reducing M L from 0.16M to 0.05M lowers our best-t value of x = 0:75 0:25 by 0.25. In both cases, models having x > 1:0 yield dynamical M/Ls which are unacceptably large compared to those implied by the adopted mass function. NGC 3201 has recently been the subject of a photometric study by Brewer et al. (1993) who used BV I CCD images for a eld located approximately seven core radii from the cluster center to study the cluster luminosity and mass functions. Only for the I-band did their data extend faint enough to reliably estimate the mass function exponent. For stars in the approximate range 0.40 { 0.22M , their mass function was found to rise sharply with index x = 2:0 0:3 (though they point out that if their lowest mass data point is excluded, this estimate drops to 1.5 0.4). In either case, their measured mass function exponent is somewhat larger than our dynamically determined value of x = 0:75 0:25. Given the di erences in the respective forms of the adopted mass functions (and the fact that our best-t value applies to a considerably di erent mass range: 8.0{0.3M ) such a discrepancy is perhaps not suprising. Since possible correlations of mass function slope with other cluster parameters (Piotto 1991;Richer et al. 1991;Capaccioli et al. 1993) have important implications for the formation and dynamical evolution of not only the Galactic globular cluster system but also the Galactic halo, it is natural to ask whether or not this slope agrees with previously suggested trends. The left panel of Figure 9 shows the global mass function slope plotted against the distance from the Galactic plane, jZ G j, for the 17 clusters studied by Capaccioli et al. (1993) with two new additions: NGC 362 (Fischer et al. 1993) and NGC 3201 (open square). Global mass function slope versus Galactocentric distance, jR G j, is given in the second panel of Figure 9 for the same sample of 19 clusters (all jZ G j and jR G j are taken from the compilation of Djorgovski 1993 who assumes a solar Galactocentric distance of 8 kpc, as opposed to Capaccioli et al. 1993who adopt 8.8 kpc following Harris 1976. In both cases, NGC 3201 agrees with the previously identi ed trends (which Capaccioli et al. 1993 interpreted as evidence for the dynamical evolution of a universal globular cluster mass function due to Galactic disk-shocking). Figure 10 shows the global mass function slope for the same cluster sample plotted against the logarithm of the half-mass relaxation time, t rh , taken from Djorgovski (1993) for all clusters except NGC 3201 (for which we adopt log t rh = 8.775; see Tables 8 and 10) and the cluster disruption time, t d , de ned by Richer et al. (1991) as the inverse of the cluster destruction rate calculated by Aguilar et al. (1988). Richer et al. (1991) found a correlation between t d and the mass function slope below 0.4M based on a sample of six clusters with deep luminosity functions. However, with NGC 362 and NGC 3201 added to the Capaccioli et al. (1993) sample, no such correlation between x and t d (or t rh ) is evident in Figure 10, as previously noted by Capaccioli et al. 1993. 4.4 Anisotropy Unlike most previous studies on cluster dynamics which have been able to place only rather weak limits on the cluster M/L and mass function slope, our large sample of radial velocities has allowed us to put more stringent constraints on these parameters. However, NGC 3201 was targeted for study principally on the basis of its high systemic radial velocity and without consideration to its overall luminosity \| since it is an intrinsically sparse cluster with a SBP extending over only ' three orders of magnitude in luminosity (compared to the ve or more decades available for some clusters such as M15, O'Neill 1978 andM3, Da Costa andFreeman 1976) the SBPs provide relatively little information on the anisotropy of the stellar velocities. As previously mentioned, mass function slopes in the range 0:5 < x < 1:0 provide the best ts to the observed SBPs and VDP. In general, for values of x in this range, only those models with very strong anisotropy (r a =r s = 3) are ruled out by the observed SBP. Although isotropic models usually provide the best match to the data, models having anisotropy radii in the range 5 < r a =r s < 10 often provide acceptable ts. (Models with r a =r s = 5 are weakly ruled out if M L is reduced from 0.16M to 0.05M .) While de nite conclusions about the velocity anisotropy in NGC 3201 would therefore be premature, it appears that both isotropic and weakly anisotropic orbits provide the most impressive ts to the cluster SBPs. Because of the low cluster velocity dispersion, the radial velocities provides no further discrimination 8 \| the agreement between the observed and theoretical VDPs is excellent (see Figure 7) for each of the r a =r s = 1; 10 and 5 models (the t to the r a =r s = 3 models is marginally inferior). The excellent match of the observed VDP to the isotropic and weakly anisotropic models may be a consequence of two-body relaxation in the core and the in uence of the Galactic tidal eld near the cluster boundary \| Oh and Lin (1992) recently constructed Fokker-Planck/three-body integration models for the tidal regions of globular clusters and found that the Galactic tidal torque induces isotropy near the limiting radius, consistent with our ndings for NGC 3201. SUMMARY We have carried out a dynamical analysis of the nearby globular cluster NGC 3201 based on B-and V -band CCD measures of the cluster surface brightness pro le and 857 radial velocities for 399 cluster giants. The observed VDP extends over the full range in cluster radius with member giants detected as far as 32 0 ( r t ) from the cluster core. The median di erence in radial velocity for stars on either side of an imaginary axis stepped through the cluster in 1 increments shows a maximum amplitude of 1.22 0.25 km s 1 . Monte Carlo experiments suggest that this observed amplitude is signi cant at the 99.2 % level. Possible explanations of this observation include: (1) cluster rotation (supported by the good agreement between the observed cluster ellipticity and photometric minor axis orientation and that expected for a rotationally-attened oblate spheroid); (2) preferential 8 Of course, high-quality proper motions for the cluster members studied here would provide a better diagnostic of the anisotropy of the VDP. Unfortunately, proper motions of the requisite precision do not yet exist for NGC 3201. stripping of stars on prograde orbits near the limiting radius; (3) the projection of the cluster space velocity onto the plane of the sky and; (4) a slight drift in the Argus velocity zero point. It is di cult to identify which of these processes is the dominat one and we suspect that each may play a role in explaining the observed structure in the velocity eld. Capaccioli et al. (1993). Due to the low cluster luminosity, we are able to place only weak constraints on the anisotropy of VDP \| isotropic orbits generally provide the best ts to the observations though models with anisotropy radii as small as r a =r s = 5 are still capable of providing impressive ts. An obvious extension of the present work is the measurement of proper motions for the radial velocity members observed in this study. Unfortunately, the requisite observations are exceedingly di cult since the centermost stars in NGC 3201 are expected to show proper motions of only 20 milliarcseconds per century. Such observations would not only yield the two components of the VDP needed to solve the non-rotating Jeans equation but would also obviate the assumption of an isotropic VDP implicit in our nonparametric models. The authors thank the TACs of both the Observatories of the Carnegie Institute of Washington and the Cerro Tololo Inter-American Observatory for the allocation of telescope time. Thanks also to Mike Irwin for providing the APM/UK-Schmidt plate scans used in this study, and to Ruth Peterson for several useful comments. This research was funded in part by the Natural Sciences and Engineering Council of Canada. The radial velocities and surface brightness pro les used in this analysis are available in machinereadable form \| contact the rst author for details. Table 3). Armandro and Da Costa (1986) technique ( lled triangles) as well as that found using the Pryor and Meylan (1993) maxmimum-likelihood estimators ( lled squares). The dispersion in each bin has been computed using the associated average bin velocity, rather than mean velocity of the whole sample. Galactic disk, jZ G j, for the 17 clusters ( lled circles) of Capaccioli et al. (1993) and NGC 362 (Fischer et al. 1993 (Fischer et al. 1993) and NGC 3201. Relaxation timescales for all clusters except NGC 3201 are taken from Djorgovski (1993). For NGC 3201 (open square), we have adopted log t rh = 8.775, the average of the determinations based on single-mass, isotropic King-Model model ts to the B-and V -band surface brightness pro les, in excellent agreement with the value of 8.79 found by Djorgovski (1993). (Right Panel) Global mass function slope versus the \disruption timescale" according to Aguilar et al. (1988) (1987) (1993) estimator; int has been computed using the entire sample of 399 cluster members, whereas 0;int refers to the dispersion obtained using only those 93 stars within one core radius of the cluster center. Figure Captions to estimate the L/M of stars in the various mass bins (see below). Since their 16 Gyr, Fe/H] = {1.26 and O/Fe] = +0.55 isochrone ends at 0.1596M , we have chosen to truncate our mass function at 0.16M . In x 4 we discuss some of the consequences of adopting di erent low-mass cuto s. 3201, that of Da Costa et al. (1993), combined the King et al. (1968) star counts with more recent photoelectric aperture photometry and CCD surface photometry to derive a somewhat lower concentration of c = 1:38, suggesting that King et al. (1968) underestimated the cluster background (since NGC 3201 is a low latitude cluster, background contamination is rather severe). NGC 3201 was also included in the CCD survey of globular cluster structural parameters of Trager et al. (1995), who found c = 1:31 and r c = 1:45 0 , in excellent agreement with our values of c = 1:26 and r c = 1:46 0 (V -band SBP) and c = 1:33 and r c = 1:38 0 (B-band SBP). Da Costa et al. (1993) combined their SBP with mean radial velocities for 92 cluster giants (included in the present sample and published in Côt e et al. 1994) to derive a cluster M/L of 1.6 0.5 using isotropic, single-mass King-Michie models. (For comparison, they found v 0 = 493:0 1:0 km s 1 and 0 = 4:4 0:5 km s 1 .) This is in good agreement with our values of M/L V = 1.62 0.11 and M/L B = 1.66 0.11 (also for isotropic, single-mass models). However, Da Costa et al. central population and dynamical M/Ls and (2) global population and dynamical M/Ls, we nd good agreement (for both the B and V SBPs) for models having global mass-to-light ratios in the range 1:8 2:5. The best-tting models (which have x = 0:75; see below) have M/L V ' M/L B = 2.0 0.2 and (M/L V ) 0 ' (M/L B ) 0 = 1.75 0.09. It therefore appears that the M/L of NGC 3201 does not di er signi cantly from mean of the Galactic globular cluster population. Single-mass model ts to the observed SBPs show good agreement with those of Da Costa et al. (1993) and Trager et al. (1995). The cluster M/Ls derived from these single component models (M/L V ' M/L B ' 1.7 0.1) are also in good agreement with that found in the only previous dynamical study of NGC 3201 based on radial velocities of individual member stars (Da Costa et al. 1993). Multi-mass and nonparametric models yield slightly higher values, M/L V ' M/L B ' 2.0 0.2, and both approaches suggest the cluster M/L increases monotonically in the range 1.5 { 10 pc. The best-t, multi-mass models have mass function slopes of x ' 0:75 0:25, consistent with the x(jZ G j) and x(R G ) correlations observed by Figure 1 { 1The location with respect to the cluster center (indicated by the dot) of our ve innermost CCD elds. The circle represents the NGC 3201 core radius (r s = 1:46 0 ) determined from single-mass, isotropic King-Michie models ts to the V -band SBP. Each CCD frame measures 10.4 0 on a side with east to the top and north to the left. A series of partly overlapping elds (extending out to about 50 0 from the NGC 3201 center in both of the north and south directions) have been omitted for clarity. Figure 2 { 2(Upper Panel) Heliocentric radial velocity versus distance from cluster center for the same sample of cluster members. The dashed line at 494.2 km s 1 indicates the mean cluster velocity according to the maximum-likelihood technique of Gunn and Gri n (1979). (Middle Panel) Heliocentric radial velocity versus position angle for 399 NGC 3201 members. (Lower Panel) Annular bins of median radial velocity versus position angle for the same sample of 399 stars (see Figure 3 { 3(Upper Panel) The di erence in median radial velocity for stars on either side of an axis at position angle . Also shown is the best-t sine curve which indicates a position angle of 7 12 for the cluster rotation axis. (Lower Panel) Histogram of amplitudes of the best-t sine curves for 1000 Monte Carlo simulations of the data with no rotation. The observed amplitude of 1:22 0:25 km s 1 (indicated by the arrow) is signi cant at the 99.2% level. Figure 4 { 4The V rot = versus ellipticity ( = 1 b=a) for a rotationally-attened, oblate spheroid (dashed line) with an isotropic velocity dispersion tensor. NGC 3201 is indicated by the lled circle. Projection e ects tend to move points on the curve in the approximate direction of the origin. Figure 5 { 5The location of the 200 outermost NGC 3201 members on the plane of the sky. Objects indicated by open circles have (v v 0 ) < 0 while those shown as open squares have (v v 0 ) > 0 (where v 0 = 494:2 km s 1 ). In all cases, the size of the point is proportional to the magnitude of velocity residual. A cluster tidal radius of 26.8 0 (V -band, single-mass models) is indicated by the dotted circle, though this parameter is very poorly constrained by the observations. Figure 6 { 6(Upper Panel) The V -band surface brightness pro le for NGC 3201.Circles indicate the results of CCD surface photometry while the square indicate points determined by CCD star counts. The best-tting singlemass, King-Michie models are also shown as the solid (isotropic), dotted (r a =r s = 10), short-dashed (r a =r s = 5) and long-dashed (r a =r s = 3) lines. (Lower Panel) Same as above except for the B-band. Figure 7 { 7(Upper Panel) The velocity dispersion pro le for NGC 3201. The solid line represents the pro le expected on the basis of the best-t, V -band, isotropic, single-mass model. Each point represents the absolute di erence between the stellar velocity and the tted mean cluster velocity for our 399 program objects. (Lower Panel) The LOWESS estimate of the velocity dispersion (solid line) and the corresponding 90% con dence bands (dotted lines). For comparison, we also show the binned velocity dispersion pro le derived using theSuntze et al. (1993) variant of the The outermost bin contains 39 stars; all others contain 40. For ease of comparison, the King-Michie pro le shown in the upper panel is indicated by the dashed line. Figure 8 { 8(Upper Panel) The mass density pro le of NGC 3201 according to one of the best-t multi-mass, King-Michie models (r a =r s = 1, x = 1:0; dashed line). The variation in mass density determined from nonparametric modeling of the individual velocities is given by the solid line (the dotted lines indicate 95% con dence bands). (Lower Panel) The variation in M/L V computed with the same King-Michie model (dashed line). The same pro le determined with nonparametric models is shown as the solid line; the dotted lines represent 95% con dence bands. Figure 9 { 9(Left Panel) Global mass function slope, x, versus distance from the Figure 10 { 10(Left Panel) Global mass function slope, x, versus the logarithm of the half-mass relaxation timescale for the 17 clusters ( lled circles) of Capaccioli et al. (1993), NGC 362 systemic cluster velocity according to the estimator of Suntze et al: (1993) using the entire sample of 399 cluster members (i.e. candidate binaries excluded). The quoted error in the mean velocity refers to the internal uncertainty and neglects the zero point uncertainty of about 1.0 km s 1 . b { Intrinsic, one-dimensional velocity dispersions according to the Suntze et al: ). Once the deprojected quantities are in hand, we can use the Jeans equation to estimate the mass and mass density(Binney and Tremaine 1987):M(r) = rv 2 r G d ln d lnr + d lnv 2 r d lnr (13) and (r) = 1 4 r 2 dM dr : ). NGC 3201 is indicated by the open square. (Right Panel) Global mass function slope versus distance from the Galactic center, R G , for the same 19 clusters. Once again, NGC 3201 has been included as the open square. In all cases, we have taken jZ G j and R G from the compilation ofDjorgovski (1993). for NGC 362, NGC 3201 and 16 of the clusters ( lled circles) in the Capaccioli et al. (1993) sample (NGC 5053 was omitted by Aguilar et al. 1988 in their study of Galactic globular cluster system). NGC 3201 is shown as the open square.Mailing Addresses: TABLE 1 1General Cluster Parameters Parameter NGC 3201 Source (2000.0) 10 h 17 m 36.75 s Shawl and White (1986) (2000.0) 46 24 0 40.2 00 Shawl and White (1986) l I I 277 13 0 40.8 00 Shawl and White (1986) b I I 8 38 00 29.0 00 Shawl and White (1986) E(B V ) 0.21 0.02 mag Lee (1977) (m M) V 14.20 0.15 mag Brewer et al. (1993) R 5.1 0:4 kpc Brewer et al. (1993) Age 15 2 Gyr Brewer et al. (1993) Fe/H] 1.3 0.1 Brewer et al. (1993) b=a 0.88 0.01 White and Shawl TABLE 2 2BV Surface Brightness Pro les R V Type a R B Type a (pc) (L V pc 2 ) (pc) (L B pc 2 ) 0.25 2361.8 1590:6 SP 0.25 2566.0 1276:4 SP 0.59 1604.2 414:3 SP 0.59 1876.1 371:9 SP 0.96 1928.6 479:3 SP 0.96 2076.1 391:8 SP 1.33 1517.9 401:3 SP 1.33 1634.0 349:0 SP 1.71 1002.9 164:6 SP 1.71 1162.1 151:7 SP 2.09 906.2 162:4 SP 2.09 975.5 160:1 SP 2.46 609.9 139:7 SP 2.46 681.2 98:9 SP 2.84 566.6 98:4 SP 2.84 668.7 116:1 SP 3.20 454.2 20:9 SC 3.22 516.7 72:3 SP 3.22 462.6 94:4 SP 3.22 512.0 26:1 SC 3.59 410.4 18:9 SC 3.60 450.0 24:9 SC 3.60 395.6 68:7 SP 3.60 475.6 75:5 SP 3.97 347.7 16:0 SC 3.97 384.8 21:3 SC 3.98 341.4 88:2 SP 3.98 381.3 72:0 SP 4.35 254.2 27:1 SP 4.34 323.0 17:9 SC 4.35 308.5 15:7 SC 4.35 291.9 33:8 SP 4.73 263.8 13:4 SC 4.72 302.9 16:8 SC 4.73 273.7 58:5 SP 4.73 326.0 56:6 SP 5.10 219.4 12:1 SC 5.10 238.4 15:4 SC 5.11 171.1 33:0 SP 5.11 196.4 38:9 SP 5.47 180.8 10:9 SC 5.48 182.5 12:7 SC 5.49 238.4 121:2 SP 5.49 236.2 113:8 SP 5.87 175.9 43:7 SP 5.87 187.6 43:0 SP 5.87 154.6 10:0 SC 5.87 160.4 11:1 SC 6.24 111.0 5:1 SP 6.24 171.1 11:0 SC TABLE 2 2cont'd BV Surface Brightness Pro les R V Type a R B Type a (pc) (L V pc 2 ) (pc) (L B pc 2 ) 6.25 151.8 9:1 SC 6.24 131.0 6:0 SP 6.62 156.1 34:8 SP 6.62 184.2 36:5 SP 6.62 133.4 8:6 SC 6.63 133.4 10:5 SC 6.99 112.0 7:8 SC 7.00 131.0 9:7 SC 7.00 124.0 18:5 SP 7.00 145.0 16:8 SP 7.37 96.7 7:1 SC 7.38 107.0 8:9 SC 7.38 88.2 8:6 SP 7.38 100.3 9:8 SP 7.75 72.7 21:1 SP 7.75 85.8 34:5 SP 7.76 83.4 6:9 SC 7.76 101.2 8:9 SC 8.12 81.1 15:6 SP 8.12 90.6 17:4 SP 8.13 72.0 7:0 SC 8.13 79.7 8:5 SC 8.49 65.7 6:4 SC 8.51 86.5 37:9 SP 8.51 89.0 49:4 SP 8.51 62.7 7:6 SC 8.88 66.9 27:4 SP 8.88 64.5 28:7 SP 8.90 63.3 6:2 SC 8.90 68.1 7:6 SC 9.27 53.6 5:7 SC 9.26 59.3 7:2 SC 9.66 43.0 5:2 SC 9.65 55.6 6:7 SC 10.02 48.0 5:4 SC 10.03 59.9 6:7 SC 10.41 36.4 4:8 SC 10.41 36.8 5:7 SC 10.79 28.7 4:5 SC 10.78 34.1 5:6 SC 11.16 22.4 4:2 SC 11.14 28.1 5:2 SC 11.56 25.0 4:3 SC 11.56 30.0 5:2 SC 11.92 29.7 4:3 SC 11.92 32.6 5:0 SC 12.30 27.4 4:1 SC 12.29 36.1 5:0 SC 12.68 23.8 3:9 SC 12.67 27.1 4:7 SC TABLE 2 2Note: a { SP = surface photometry; SC = star countscont'd BV Surface Brightness Pro les R V Type a R B Type a (pc) (L V pc 2 ) (pc) (L B pc 2 ) 13.04 21.5 3:9 SC 13.04 28.9 4:8 SC 13.42 11.3 3:8 SC 13.80 27.4 4:8 SC 13.81 16.0 3:8 SC 14.19 25.0 4:7 SC 14.19 17.4 3:9 SC 14.56 22.8 4:7 SC 14.56 11.9 3:9 SC 14.94 7.1 5:4 SC 14.94 8.9 4:0 SC 15.31 6.8 5:5 SC 15.31 4.8 6:0 SC 15.71 12.9 4:7 SC 15.72 8.1 4:1 SC 16.05 18.4 4:8 SC 16.07 12.6 4:0 SC 16.46 15.3 4:7 SC 16.47 5.1 5:9 SC 16.84 7.8 5:4 SC 16.83 3.3 8:3 SC 17.22 17.4 4:8 SC 17.21 8.9 4:2 SC 17.59 3.4 4:3 SC 18.27 4.5 2:5 SC 18.18 6.2 3:6 SC 19.48 10.3 6:8 SC 18.37 8.6 5:7 SC 23.53 3.0 2:6 SC 18.71 7.8 6:9 SC 27.37 2.1 3:4 SC 19.10 4.9 5:6 SC 23.65 4.1 4:0 SC 27.37 3.5 4:3 SC TABLE 3 3Dependence of Radial Velocity on Position AngleBin N Sector < > Median V r (deg.) (deg.) (km s 1 ) 1 41 0{ 45 19.7 2.0 495.18 0.63 2 40 45{ 90 68.4 2.2 495.60 0.80 3 55 90{135 113.8 1.7 495.26 0.81 4 50 135{180 155.8 1.7 495.00 0.66 5 44 180{225 203.2 1.9 494.35 0.63 6 57 225{270 247.2 1.7 494.50 0.57 7 65 270{315 290.6 1.7 492.91 0.66 8 47 315{360 336.3 1.9 494.12 0.76 TABLE 4 4Dependence of Apparent Rotation on Sample SizeN R min A (arcmin) (km s 1 ) (deg) 399 0.0 1.22 0.25 277 12 300 1.5 1.42 0.25 262 10 200 3.1 1.76 0.38 274 13 100 6.2 1.79 0.49 274 17 50 10.0 2.11 0.46 307 13 25 14.3 1.99 1.34 273 38 TABLE 5 Binned 5Velocity Dispersion Pro le for NGC 3201Bin N Range R med S PM (pc) (pc) (km s 1 ) (km s 1 ) 1 40 0.1{ 1.3 0.960 3.35 0.38 3.35 0.40 2 40 1.3{ 1.9 1.51 3.84 0.43 4.32 0.44 3 40 1.9{ 2.6 2.24 2.99 0.34 2.99 0.44 4 40 2.6{ 3.5 3.04 4.23 0.47 4.52 0.50 5 40 3.5{ 4.6 4.13 3.42 0.39 4.27 0.44 6 40 4.6{ 5.9 5.27 3.47 0.40 3.93 0.72 7 40 5.9{ 7.7 6.80 3.09 0.39 3.76 0.40 8 40 7.9{10.9 9.29 3.59 0.42 3.61 0.45 9 40 10.9{15.9 12.9 3.02 0.35 3.03 0.37 10 39 16.0{47.9 22.5 1.91 0.25 2.00 0.26 TABLE 6 6Notes: MS = main sequence stars; WD = white dwarfs; RG = red giants; HB = horizontal branch stars = L V /M and L B /M determined semi-empirically from CCD star counts. See text for details.Adopted Mass Bins Bin M min M max Contents (M ) (M ) 1 0.160 0.250 MS 2 0.250 0.350 MS 3 0.350 0.450 MS 4 0.450 0.550 MS & WD 5 0.550 0.650 MS 6 0.650 0.750 MS & WD 7 0.750 0.826 MS & HB & RG 8 0.826 8.000 WD TABLE 7 V 7-Band Fitted ParametersPopulation Dynamical TABLE 9 V 9-Band Derived Parametersr a =r s x r s r h r t v s 0 L V log 0 M log 0 log h log t log t r0 log t rh (pc) (pc) (pc) (km s 1 ) (10 4 L V ) (L V pc 3 ) (10 5 M ) (M pc 3 ) (M pc 3 ) (M pc 3 ) (yr) (yr) 1 2.18 0.19 6.2 0.6 39.9 2.3 4.22 0.17 8.00 0.22 2.64 0.09 1.30 0.08 2.85 0.08 1.83 0.08 -0.31 0.07 8.01 0.16 8.79 0.05 10 2.27 0.19 6.2 0.6 43.7 3.4 4.25 0.18 8.02 0.23 2.61 0.08 1.30 0.08 2.82 0.07 1.82 0.09 -0.43 0.10 8.04 0.16 8.80 0.05 5 2.50 0.18 6.3 0.8 50.8 7.6 4.30 0.18 8.01 0.27 2.53 0.07 1.34 0.09 2.75 0.07 1.81 0.10 -0.61 0.18 8.14 0.12 8.84 0.04 3 2.77 0.17 6.5 0.8 70.1 31.1 4.40 0.19 8.15 0.36 2.47 0.06 1.38 0.10 2.70 0.06 1.78 0.11 -1.02 0.41 8.24 0.09 8.89 0.03 1 0.0 1.43 0.23 6.9 0.5 35.0 1.0 5.04 0.17 7.87 0.22 2.81 0.14 1.14 0.08 3.15 0.16 1.63 0.10 -0.20 0.04 7.53 0.11 8.83 0.04 10 0.0 1.51 0.20 5.7 0.5 38.7 2.0 5.09 0.17 7.98 0.24 2.78 0.12 1.16 0.08 3.11 0.11 1.87 0.10 -0.32 0.07 7.58 0.09 8.75 0.05 5 0.0 1.82 0.18 6.9 0.6 45.8 5.0 5.19 0.17 8.01 0.27 2.66 0.09 1.25 0.08 2.99 0.08 1.66 0.10 -0.51 0.14 7.76 0.07 8.84 0.05 3 0.0 2.15 0.16 8.1 0.8 57.6 11.1 5.37 0.19 8.12 0.30 2.57 0.07 1.35 0.09 2.91 0.06 1.48 0.14 -0.77 0.22 7.93 0.06 8.93 0.05 1 0.5 1.73 0.24 6.5 0.6 37.4 1.3 4.66 0.17 7.84 0.22 2.70 0.12 1.41 0.09 2.98 0.12 1.78 0.10 -0.20 0.03 7.76 0.09 8.94 0.05 10 0.5 1.79 0.21 6.8 0.6 41.6 2.6 4.73 0.17 7.95 0.24 2.68 0.11 1.45 0.09 2.96 0.10 1.75 0.09 -0.32 0.07 7.80 0.08 8.96 0.05 5 0.5 2.00 0.18 7.6 0.6 52.5 7.9 4.87 0.17 8.08 0.28 2.61 0.08 1.55 0.10 2.90 0.08 1.63 0.09 -0.59 0.17 7.91 0.07 9.02 0.04 3 0.5 2.18 0.15 8.0 0.8 98.5 43.8 5.11 0.19 8.61 0.40 2.59 0.07 1.72 0.12 2.88 0.06 1.60 0.11 -1.37 0.41 8.01 0.06 9.06 0.06 1 1.0 2.04 0.21 9.8 0.7 45.0 2.1 4.28 0.16 8.04 0.23 2.64 0.09 1.90 0.13 2.84 0.09 1.39 0.09 -0.30 0.04 7.98 0.07 9.24 0.04 10 1.0 2.09 0.20 10.0 0.7 52.2 4.8 4.33 0.17 8.14 0.25 2.62 0.09 1.98 0.15 2.83 0.09 1.37 0.09 -0.48 0.10 8.01 0.07 9.26 0.04 5 1.0 2.21 0.19 10.6 1.0 84.1 36.6 4.50 0.17 8.41 0.37 2.58 0.08 2.16 0.20 2.80 0.08 1.34 0.10 -1.06 0.37 8.06 0.07 9.30 0.06 3 1.0 2.48 0.12 11.9 1.1 195.2 98.6 4.75 0.18 8.84 0.29 2.51 0.05 2.37 0.20 2.76 0.04 1.23 0.09 -2.12 0.47 8.19 0.05 9.37 0.06 1 1.5 2.45 0.18 11.7 0.9 64.8 4.8 3.95 0.17 8.39 0.24 2.58 0.07 2.91 0.26 2.71 0.06 1.33 0.08 -0.59 0.07 8.20 0.06 9.45 0.05 10 1.5 2.46 0.21 14.2 1.4 86.5 18.7 4.01 0.17 8.51 0.31 2.56 0.07 3.13 0.35 2.71 0.07 1.11 0.09 -0.94 0.21 8.21 0.07 9.55 0.06 5 1.5 2.58 0.11 14.9 1.7 216.5 86.7 4.20 0.17 8.74 0.22 2.50 0.04 3.38 0.29 2.68 0.04 1.08 0.11 -2.10 0.39 8.25 0.05 9.59 0.07 3 1.5 3.30 0.08 15.8 2.1 377.2 148.2 4.34 0.19 8.51 0.16 2.32 0.05 3.71 0.32 2.56 0.03 1.05 0.13 -2.80 0.44 8.51 0.05 9.63 0.08 1 2.0 3.16 0.19 18.2 1.9 97.9 12.0 3.79 0.18 8.40 0.28 2.41 0.05 4.90 0.61 2.56 0.05 0.98 0.09 -0.90 0.11 8.49 0.07 9.78 0.07 10 2.0 3.41 0.18 16.3 2.0 110.6 22.6 3.85 0.19 8.15 0.22 2.34 0.05 4.95 0.59 2.52 0.05 1.13 0.11 -1.06 0.20 8.57 0.06 9.73 0.08 5 2.0 3.69 0.15 21.3 1.6 199.7 47.5 3.97 0.20 8.13 0.16 2.29 0.04 5.38 0.44 2.50 0.04 0.82 0.09 -1.79 0.26 8.67 0.05 9.87 0.05 3 2.0 4.07 0.15 19.4 1.7 244.3 42.0 4.17 0.22 8.02 0.14 2.24 0.04 5.12 0.34 2.49 0.03 0.92 0.12 -2.08 0.20 8.79 0.04 9.82 0.05 Table 3of Côt e et al. 1994 is also available on the AAS CD-ROM Series, Volume 5, 1995. 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[ "FLASH LIGHTENS GRAY PIXELS", "FLASH LIGHTENS GRAY PIXELS" ]	[ "Yanlin Qian ", "Song Yan ", "Joni-Kristian Kämäräinen ", "Jiri Matas \nCenter for Machine Perception\nCzech Technical University\nPrague\n", "\nComputing Sciences\nTampere University\n\n" ]	[ "Center for Machine Perception\nCzech Technical University\nPrague", "Computing Sciences\nTampere University\n" ]	[]	In the real world, a scene is usually cast by multiple illuminants and herein we address the problem of spatial illumination estimation. Our solution is based on detecting gray pixels with the help of flash photography. We show that flash photography significantly improves the performance of gray pixel detection without illuminant prior, training data or calibration of the flash. We also introduce a novel flash photography dataset generated from the MIT intrinsic dataset.	10.1109/icip.2019.8803468	[ "https://arxiv.org/pdf/1902.10466v1.pdf" ]	67,856,032	1902.10466	83ef4e43ffe3b6e00906fb424c59bb3b2c97acf3	FLASH LIGHTENS GRAY PIXELS Yanlin Qian Song Yan Joni-Kristian Kämäräinen Jiri Matas Center for Machine Perception Czech Technical University Prague Computing Sciences Tampere University FLASH LIGHTENS GRAY PIXELS Index Terms-spatial illumination estimationgray pixelflash photographycolor constancy In the real world, a scene is usually cast by multiple illuminants and herein we address the problem of spatial illumination estimation. Our solution is based on detecting gray pixels with the help of flash photography. We show that flash photography significantly improves the performance of gray pixel detection without illuminant prior, training data or calibration of the flash. We also introduce a novel flash photography dataset generated from the MIT intrinsic dataset. INTRODUCTION We address the illumination estimation problem which aims to measure the chroma of illumination in order to remove the color-bias from a captured image [1]. Illumination estimation can help in high-level vision tasks, e.g. object recognition, tracking [2] and intrinsic image decomposition. There exists a large number of related works, from the traditional non-learning approaches [3] to recent deep learning based approaches [4,5,6]. However, the vast majority of these works concentrate on the case of a single global illumination which is often an invalid assumption [7]. In this paper, we explore a more-complex less-optimistic setting -mixed illumination 1 . Spatially-varying illumination refers to that on a captured scene, each pixel captures different number of light phantoms when the camera shutter is on. In other words, all pixels do not share the same configuration of lights [7], which is the default assumption for single-illumination estimation. Compared to the single global illumination setting, the mixed illumination setting better corresponds to the real world [8], but is more challenging clearly, as it extends the ill-posed problem from one point to a spatial map [7], without extra knowledge or input. To circumvent the hardness that spatially-varying illumination brings, efforts are put as follows: user guidance or human interaction is given as a supervisory signal [9,10]; small patch is assumed to be cast by only one light [11]; light color 1 We traverse related works and there exists multiple terms referring to the same thing (which may confuse readers), i.e. mixed illumination, spatiallyvarying illumination, multiple illumination, mixed lighting condition. ambi +flash img pred flash pred GT corr img Fig. 1. A multi-illuminant image (ambi), Gray Pixel [13] outputs an erroneous prediction (single value, pred) compared to the groundtruth (GT). By virtue of the flash image (+flash) the proposed flash photography gray pixel provides an accurate spatial estimate (pred) and faithful corrected image (corr img). and number need to be known before experiments [12]. Unlike these methods, we make use of flash photography. Flash photography refers to image processing techniques which use non-flash/flash image pairs. This technique is well adopted to spatial illumination estimation [14,15], which assume each patch illuminated by one light and obtain decent results. Hui et al. [7] proposed a closed-form solution of spatial illumination for the case of a calibrated flash. In essence, flash calibration in [7] equals to knowing the "groundtruth" surface albedo in a flash-only image. What's more, flash may appear in other forms, i.e. varying sunlight, cast shadow, which may be hard to calibrate. In this paper, we propose a novel spatial illumination estimation method, using flash photography, without need of flash calibration and any other illuminant prior. Our method relies on gray pixel detection [13]. The original work assumes Lambertian surfaces, and then revisited and improved by [16,17]. The original and extended gray pixel methods however fail in the case of mixed illumination (the top row in Fig. 1), but we analytically show how flash photography circumvents the problem and "lightens" gray pixel photometrically and in performance (bottom row in Fig. 1). The interplay of flash photography and gray pixel enables us to achieve a largely increased performance on our synthesized laboratory dataset and some real-world images, than running gray pixel methods alone, without knowing the flash color. Our contributions are three-fold: • We revisit and revise the gray pixel methods for the case of mixed illumination. • Leveraging flash photography and the gray pixel method, we propose a novel learning-free and well-performing method for spatial illumination estimation. • We propose a novel flash-photography dataset for benchmarking multi-illuminant methods. The dataset is based on the MIT intrinsic dataset. The rest of this paper is organized as follows. Section 2 revisits gray pixel and its variants before we introduce the flash-photography gray pixel in Section 3. In Section 4 we describe the MIT intrinsic based dataset for our task. Section 5 covers the experiments and results. We conclude in Section 6. GRAY PIXEL Assuming one light source and narrow sensor response, Gray Pixel [13] is derived from the Lambertian model, given as: I c (p) = R c (p) max(n(p) s, 0) l c ,(1) which shows the color channel c at the location p in image I is a function of a surface albedo R, surface normal n, light direction s and illumination color l. Following the procedure in [13], applying log and a Mexican hat filter δ on Eq. 1 yields: δ log I c (p) = δ log R c (p) + δ log max(n(p) s, 0) + δ log l c .(2) A single light casting a small local neighborhood (the same color and direction), Eq. 2 simplifies to: δ log I c (p) = δ log R c (p),(3) which is the core of gray pixel. δ log I c (p) = δ log I c (p), ∀c, c ∈ {R, G, B} defines a "pure gray pixel". To rank pixels w.r.t. "grayness", [13] defines the following grayness function, up to a scale: g(p) = c∈R,G,B (δ log I c (p) −δ log I(p)) 2 /δ log I(p),(4) whereδ log I(p) is the mean value of δ log I c (p). This method works robustly with single-illumination scenes where diffuse reflection (the Lambertian assumption) is dominant. Then [16] augments the above Gray Pixel by replacing Eq. 4 with the following luminance-independent function: g (p) = cos −1 1 √ 3 δ log I(p) 1 δ log I(p) 2 ,(5) where · n refers to the n norm. g (p) = 0 refers to pure gray pixel. To remove spurious color pixels, [16] applies a mean shift clustering to choose the strongest mode -dominant illumination vector. The mechanism to detect gray pixel is further improved in [17]. Different to [13,16], which are based on the Lambertian model, Qian et al. [17] uses the dichromatic reflection model [18] to derive a set of more strict constraints for gray pixels: g (p) = δ(log(I R ) − log(\|I\|)), δ(log(I G ) − log(\|I\|)) 2 ,(6) where \|I\| refers to (I R + I G + I B ). We refer readers to the original paper for more details. In Section 5 we report their performance in a mixed-lighting dataset and show how flash photography improves them. FLASH GRAY PIXEL In the sequel, we first investigate what will happen to the existing gray pixel methods in the case of mixed illumination. Then we propose a novel gray pixel method using flash photography, termed as Flash Gray Pixel. Here we do analysis on the original Gray Pixel [13], but similar conclusion for [16,17] can be inferred in an analogue manner. A scene is illuminated by N light sources or arbitrary type and color. To describe the image formation process in this case, Eq. 1 is modified to: I c (p) = R c (p) i λ i (p) l c i ,(7) where λ i (p) represents the shading term max(n(p) s i , 0). Eq. 3 changes to: δ log I c (p) = δ log R c (p) + δ log( i λ i (p) l c i ) . (8) Since that is a mixed illumination image, the light configuration varies from pixel to pixel, making the right most term in Eq. 8 non-zero and therefore Eq. 3 fails. This finding explains that mixed illumination hinders the performance of the gray pixel method, which motivates us to leverage flash photography. When a flash light is present, the flash image I f is expressed as: I c f (p) = R c (p)( i λ i (p) l c i + λ f (p) l c f ),(9) where λ f (p) is the shading term of flash light at the position p and l f the unit-norm chroma vector. Subtracting Eq. 7 from Eq. 9, we get a flash-only image I fo : I c fo (p) = R c (p)λ f (p) l c f ,(10) which is a more solid ground for searching gray pixels. In other terms, flash image helps to remove the negative effect of spatially-varying light configurations and gray pixels are now flash gray pixels. Flash gray pixels can be found from the grayness map of a residual flash/no-flash image. To select flash gray pixels robustly, we follow [13] to compute flash-only illumination component for each pixel: K-means is used to cluster the top N % gray pixels into preset M clusters; then the illumination at location p is computed using: L c f o (p) = M m=1 ω m L c m ,(11) where L m refers to the average illumination for the cluster m and ω m controls the connection between the pixel I(x, y) to the cluster m, unfolded as: ω m = e − Dm 2σ 2 M n=1 e − Dn 2σ 2 ,(12) where D m is the Euclidean distance from the pixel to the centroid of the cluster m. Eq. 12 encourages nearby pixels to share a similar illumination. Combining the flash-only illumination L c f o (p) with Eq. 9 allows to color-corrected the image I f o by I c gray (p) = I c fo (p)/L c fo (p),(13) and the mixed illumination is: L c (p) = I c (p)/I c gray (p) .(14) Flash gray pixel can be filled by more advanced gray pixel methods for further improvement. DATASET We adapted the MIT Intrinsic benchmark [19] for our task 2 . This dataset was originally collected for the intrinsic image decomposition task, containing 20 single objects illuminated by uncalibrated whitish light sources from 10 different directions. This property allows us to render each image in the combinations of 1 − 9 directed lights with arbitrary chroma to compose a new no-flash image I. The 6-th direction is always roughly frontal and was thus used as the flash source and together with I forms the flash image I f . We generated I and I f with varying number N of light sources, from 2 to 8. Note that even for the easiest case (N = 2), a pixel may be simultaneously affected by two light sources, violating the global illumination assumption and breaking the original gray pixel as demonstrated in Fig. 2. Considering the fact that gray pixel methods are designed for realistic consumer images "in the wild" [13] that contain at least a few gray pixels, we left out the 5 chromatic objects 3 . In total, the new dataset contains 105 flash/no-flash image pairs with spatial illumination map ground truth ((15 objects and 7 choices of N ). EXPERIMENTS Our setup is the following: we run the three variants of Gray Pixel methods with flash/no-flash image pairs. On each image the top 10% gray pixels are selected to cover enough area. The cluster number is set to M = N , which is the number of illuminants. Evaluation metric is the standard average angular error [13]. Results are summarized in Table 1 which shows the performance of flash gray pixel on the dataset in Section 4. "GP+f " refers to flash gray pixel on the basis of the original gray pixel method "GP". Results show that flash photography extensions of all gray pixel variants [13,16,17] systematically improve the results. For all three methods, the improvement over the Fig. 3. Qualitative comparison on estimated illumination of flash gray pixels, with N light sources (2,3,5,8). ambi +flash GP GP+f Fig. 4. Qualitative comparison on color-corrected image of flash gray pixels in real-world images. Macbeth Col-orChecker is excluded for illumination estimation. whole dataset is over 40% in median and 30% in mean. The flash photography variants achieve the sufficient color constancy accuracy (≤ 3.0 • ) in almost all cases. Fig. 3 illustrates predicted illumination between gray pixel methods and their flash variants. It is clear that the original versions cannot find the fine-grained details of mixed illumination. For example, the frog back and stomach are cast with different colors that confuses the original GP methods. Among all gray pixel methods, the original GP [13] suffers from the mixed illumination the most. MSGP [16] and DGP [17] perform slightly better, but not to a satisfying degree without the flash. With flash gray pixel, all three methods perform similarly thus verifying the efficiency of flash photography. DGP [17] does not perform better than the original GP which is due to the fact that our dataset does not contain specular reflectance components. As the number of diverse illumiants increases, all GP methods performs better. This can be explained by the fact that a large number of lights with various colors additively mix toward a whitish color. Flash gray pixel variants are not affected by this, obtaining consistent results. Images from real-world scenes (Fig. 4, images retrieved from [7]), show that the proposed methods also produce high quality results (e.g. the white wall is white). CONCLUSION In this paper we reconsider gray pixel detection through the medium of flash photography. We find that computing a residual map from the flash/no-flash pair, gray pixel methods can effectively measure the grayness of each pixel and provide spatial color constancy. We find that computing a residual map from the flash/no-flash pair, gray pixel methods can effectively measure the grayness of each pixel, allowing a large margin improvement in spatially-varying illumination estimation. The method is pragmatic -it is lightweight, does not need any illuminant prior, training, flash light calibration or user input. Fig. 2 . 2Example images from the generated flash photography dataset. Each sample is a triplet: {no-flash image I, flash-on image I f , illumination ground truth map}. Images are generated from the MIT intrinsic dataset by changing the colors and mixing the original directional illumination images. Table 1 . 1Results for the gray pixel (GP) variants and their flash photography versions with the mixed illumination dataset. The values are angular errors averaged over image between the estimated and ground truth illumination maps (lower is better). Gray denotes the median and white the mean error. N is the number of light sources. The "all" column shows the mean statistics over all choices of N .Method/N 2 3 4 5 6 7 8 all GP [13] 5.86 6.15 5.33 6.65 4.01 5.74 3.73 5.34 3.04 4.93 3.59 4.80 3.37 4.80 4.07 5.49 GP+f 2.37 3.49 2.32 3.85 2.35 3.79 2.39 3.94 2.39 3.80 2.39 3.85 2.39 3.88 2.37 3.80 MSGP [16] 4.91 5.66 5.25 6.03 4.10 5.23 3.52 4.81 3.17 4.42 3.54 4.35 2.87 4.52 4.10 5.00 MSGP +f 2.34 3.17 2.29 3.32 2.34 3.31 2.34 3.40 2.33 3.38 2.50 3.72 2.30 3.46 2.34 3.40 DGP [17] 6.03 6.26 5.26 6.72 4.52 5.89 4.08 5.47 3.42 5.09 3.97 55.07 3.50 4.99 4.13 5.64 DGP +f 2.37 3.73 2.34 4.00 2.37 3.96 2.39 4.01 2.40 3.93 2.40 4.02 2.39 3.99 2.37 3.93 ambi +flash GP GP+f MSGP MSGP+f DGP DGP+f GT Method/N 2 3 5 8 There are recent datasets that provide flash/no-flash image pairs[20], but these are unsuitable for our purposes due to unknown illumination number. "apple, pear, frog2, potato, turtle" Colour and illumination in computer vision. 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[ "Offline Imitation Learning with Suboptimal Demonstrations via Relaxed Distribution Matching", "Offline Imitation Learning with Suboptimal Demonstrations via Relaxed Distribution Matching" ]	[ "Lantao Yu [email protected] \nComputer Science Department\nStanford University\n\n", "Tianhe Yu [email protected] \nComputer Science Department\nStanford University\n\n", "Jiaming Song [email protected] \nComputer Science Department\nStanford University\n\n", "Willie Neiswanger [email protected] \nComputer Science Department\nStanford University\n\n", "Stefano Ermon [email protected] \nComputer Science Department\nStanford University\n\n" ]	[ "Computer Science Department\nStanford University\n", "Computer Science Department\nStanford University\n", "Computer Science Department\nStanford University\n", "Computer Science Department\nStanford University\n", "Computer Science Department\nStanford University\n" ]	[]	Offline imitation learning (IL) promises the ability to learn performant policies from pre-collected demonstrations without interactions with the environment. However, imitating behaviors fully offline typically requires numerous expert data. To tackle this issue, we study the setting where we have limited expert data and supplementary suboptimal data. In this case, a well-known issue is the distribution shift between the learned policy and the behavior policy that collects the offline data. Prior works mitigate this issue by regularizing the KL divergence between the stationary state-action distributions of the learned policy and the behavior policy. We argue that such constraints based on exact distribution matching can be overly conservative and hamper policy learning, especially when the imperfect offline data is highly suboptimal. To resolve this issue, we present RelaxDICE, which employs an asymmetrically-relaxed f -divergence for explicit support regularization. Specifically, instead of driving the learned policy to exactly match the behavior policy, we impose little penalty whenever the density ratio between their stationary state-action distributions is upper bounded by a constant. Note that such formulation leads to a nested min-max optimization problem, which causes instability in practice. Re-laxDICE addresses this challenge by supporting a closedform solution for the inner maximization problem. Extensive empirical study shows that our method significantly outperforms the best prior offline IL method in six standard continuous control environments with over 30% performance gain on average, across 22 settings where the imperfect dataset is highly suboptimal.	10.48550/arxiv.2303.02569	[ "https://export.arxiv.org/pdf/2303.02569v1.pdf" ]	257,364,973	2303.02569	d05a0088e6ab6ba367650527c7e1cc46524da3dc	Offline Imitation Learning with Suboptimal Demonstrations via Relaxed Distribution Matching Lantao Yu [email protected] Computer Science Department Stanford University Tianhe Yu [email protected] Computer Science Department Stanford University Jiaming Song [email protected] Computer Science Department Stanford University Willie Neiswanger [email protected] Computer Science Department Stanford University Stefano Ermon [email protected] Computer Science Department Stanford University Offline Imitation Learning with Suboptimal Demonstrations via Relaxed Distribution Matching 2 NVIDIA (Work done while at Stanford) Offline imitation learning (IL) promises the ability to learn performant policies from pre-collected demonstrations without interactions with the environment. However, imitating behaviors fully offline typically requires numerous expert data. To tackle this issue, we study the setting where we have limited expert data and supplementary suboptimal data. In this case, a well-known issue is the distribution shift between the learned policy and the behavior policy that collects the offline data. Prior works mitigate this issue by regularizing the KL divergence between the stationary state-action distributions of the learned policy and the behavior policy. We argue that such constraints based on exact distribution matching can be overly conservative and hamper policy learning, especially when the imperfect offline data is highly suboptimal. To resolve this issue, we present RelaxDICE, which employs an asymmetrically-relaxed f -divergence for explicit support regularization. Specifically, instead of driving the learned policy to exactly match the behavior policy, we impose little penalty whenever the density ratio between their stationary state-action distributions is upper bounded by a constant. Note that such formulation leads to a nested min-max optimization problem, which causes instability in practice. Re-laxDICE addresses this challenge by supporting a closedform solution for the inner maximization problem. Extensive empirical study shows that our method significantly outperforms the best prior offline IL method in six standard continuous control environments with over 30% performance gain on average, across 22 settings where the imperfect dataset is highly suboptimal. Introduction Imitation learning (IL) (Pomerleau 1988;Ho and Ermon 2016a;Ross, Gordon, and Bagnell 2011) studies the problem of programming agents directly with expert demonstrations. However, successful IL usually demands a large amount of optimal trajectories, and many adversarial IL methods (Ho and Ermon 2016a;Fu, Luo, and Levine 2018;Ke et al. 2020;Kostrikov et al. 2018) require online interactions with the environment to get samples from intermediate policies for policy improvement. Considering these limitations, we focus on the setting of offline imitation learning with supplementary imperfect demonstrations , which holds the promise of addressing these challenges (i.e. no large collection of expert data and no online interactions with the environment during training). Specifically, we aim to learn a policy using a small amount of expert demonstrations and a large collection of trajectories with unknown level of optimality that are typically cheaper to obtain. As in prior offline reinforcement learning (RL) and offline policy evaluation works, offline IL also has the distribution shift problem Kumar et al. 2019;Fujimoto, Meger, and Precup 2018): the agent performs poorly during evaluation because the learned policy deviates from the behavior policy used for collecting the offline data. To mitigate this problem, prior works based on distribution correction estimation (the "DICE" family) (Nachum et al. 2019a,b;Lee et al. 2021;Kim et al. 2021;Kostrikov, Nachum, and Tompson 2020;Zhang, Liu, and Whiteson 2020;Yang et al. 2020) collectively use a distribution divergence measure (e.g. fdivergence) to regularize the learned policy to be similar to the behavior policy. However, such regularization schemes based on exact distribution matching can be overly conservative. For example, in settings where the offline data is highly suboptimal, such an approach will require careful tuning of the regularization strength (denoted as α) in order to find the delicate balance between policy optimization on limited expert data and policy regularization to the behavior policy. Otherwise, the resulting policy will either suffer from large distribution shift because of small α or behave too similarly to the suboptimal behavior policy due to large α. We argue that a more appropriate regularization for offline imitation learning with limited expert data and diverse supplementary data is indispensable, which is the goal of this work. Towards this end, we draw inspiration from domain adaptation theory (Wu et al. 2019a) and present RelaxDICE, which employs an asymmetrically-relaxed f -divergence for explicit support regularization instead of exact distribution matching between the learned policy and the suboptimal behavior policy. On one hand, we still encourage the learned policy to stay within the support of the pre-collected dataset such that policy evaluation/improvement is stable and reliable. On the other hand, we will not drive the learned policy to exactly match the behavior policy since the offline demon- Figure 1: Illustration of regularizations based on relaxed distribution alignment (left) and exact distribution matching (right). The curves represent trajectories sampled from the expert policy (green), the behavior policy that collects the suboptimal data (blue), and the learned policy (red and purple) under different kinds of regularization. Dashed lines represent the support of these distributions. strations have unknown level of optimality (see Figure 1 for illustration). Different from (Wu, Tucker, and Nachum 2019;Levine et al. 2020) which tried to directly regularize the policies and observed little benefits in the context of offline RL, we enforce such a regularization over stationary stateaction distributions to effectively reflect the diversity in both states and actions (rather than enforce constraints only on policies/action distributions). However, this leads to a nested min-max optimization problem that causes instability during training. We surprisingly found that our new formulation enjoys a closed-form solution for the inner maximization problem, thus preserving the key advantage of previous state-ofthe-art DICE methods Kim et al. 2021). Furthermore, the stationary state-action distribution of the suboptimal behavior policy can be potentially modified to be closer to that of the expert policy, by leveraging an approximate density ratio obtained from expert and suboptimal data. Thus we further propose RelaxDICE-DRC, an extension of RelaxDICE by penalizing the relaxed f -divergence between the stationary state-action distributions of the learned policy and the density-ratio-corrected behavior policy. This method also enjoys a desirable closed-form solution for the inner maximization and a potential for better policy improvement. We empirically evaluate our method on a variety of continuous control tasks using environments and datasets from the offline RL benchmark D4RL ). We construct datasets where there are a small amount of expert demonstrations and a large collection of imperfect demonstrations with different levels of optimality following the design choice in ). More importantly, for each environment, we design up to four different settings that are much more challenging than the ones in , in the sense that the supplementary imperfect data are highly suboptimal. Extensive experimental results show that our method outperforms the most competitive prior offline IL method across all 22 tasks by an average margin over 30%. Furthermore, RelaxDICE is much more performant and robust with respect to hyperparameter changes than prior works ) in our challenging settings, demonstrating the superiority of our relaxed distribution matching scheme for offline imitation learning. Background Markov Decision Process. A Markov decision process (MDP) is defined by M = S, A, T, r, p 0 , γ , where S is a set of states; A is a set of actions; T : S × A → ∆(S) is the transition distribution and T (s t+1 \|s t , a t ) specifies the probability of transitioning from state s t to state s t+1 by executing action a t ; p 0 ∈ ∆(S) is the initial state distribution; R : S × A → R is the reward function; and γ ∈ [0, 1] is the discount factor. A policy π : S → ∆(A) maps from states to distributions over actions, which together with the MDP M, induces a stationary state-action distribution d π (s, a) (also called occupancy measure): d π (s, a) = (1 − γ) ∞ t=0 γ t Pr(st = s, at = a\|s0 ∼ p0, at ∼ π(·\|st), st+1 ∼ T (·\|st, at)). Here 1−γ is a normalization factor such that the occupancy measure is a normalized distribution over S × A. Because of the one-to-one correspondence described in the following theorem, a policy optimization problem can be equivalently formulated as an occupancy measure optimization problem. (1) Define π d (a\|s) := d(s,a) a d(s,a ) . Then d is the occupancy measure for π d . Conversely if π is a policy such that d is its occupancy measure, then π = π d and d satisfies Eq. (1). The Bellman flow constraints in Eq. (1) essentially characterize all possible occupancy measures consistent with the MDP, such that they can be induced by some policies. Therefore it is necessary to enforce these constraints when we design optimization problems over occupancy measures. IL with Expert Data. We can learn performant policies via imitation learning when a set of expert demonstrations D E is provided. The expert dataset D E = {(s, a, s )} is generated according to (s, a) ∼ d E , s ∼ T (·\|s, a), where d E is the occupancy measure of the expert policy. A classical IL approach is behavior cloning (BC), which optimizes a policy π by minimizing the expected KL between π E (·\|s) and π(·\|s) for s ∼ d E (s) (the state marginal of expert occupancy measure): arg min π E d E (s) DKL π E (·\|s) π(·\|s) = − E d E (s,a) [log π(a\|s)]. Alternatively, IL can be formulated as minimizing the fdivergence between occupancy measures: min d D f (d d E ) (Ho and Ermon 2016b;Kostrikov, Nachum, and Tompson 2020;Ke et al. 2020;Ghasemipour, Zemel, and Gu 2020). However, since estimating and minimizing f -divergence requires the unknown density ratio d/d E , which can be obtained only through variational estimation using samples from d E and d (all intermediate policies), these IL methods are not offline and have to use adversarial training. Offline IL with Expert and Non-Expert Data. The standard IL setting above typically requires a large amount of optimal demonstrations from experts, and sometimes require online interactions with the MDP. To address these limitations, researchers proposed to study offline IL with limited expert data and supplementary imperfect data ), a meaningful yet challenging setting where no interaction with the environment is allowed, and we only have a small amount of expert demonstrations D E and an additional collection of suboptimal demonstrations D U with unknown level of optimality. The pre-collected dataset D U = {(s, a, s )} is generated according to (s, a) ∼ d U , s ∼ T (·\|s, a) with d U being the occupancy measure of some unknown behavior policy. In this setting, the key is to study how to leverage the additional imperfect dataset D U to provide proper regularization to help the policy/occupancy measure optimization on D E . Towards this end, DemoDICE ) extends the offline RL method OptiDICE ) and uses D KL (d d U ) to realize the regularization. Moreover, we note that a key to their success is both OptiDICE and DemoDICE avoid the nested min-max optimization (Nachum et al. 2019b) by supporting a closed-form solution for their inner maximization problem. Density Ratio Estimation via Classification. Thanks to the connection between density ratio estimation and classification (Menon and Ong 2016;Yu, Jin, and Ermon 2021), given samples from two distributions p and q, we can use any strictly proper scoring rule and a link function ψ dr to recover the density ratio p/q. For example, we can use logistic regression to approximately recover d E /d U : c * = arg max c:S×A→(0,1) E d E (s,a) [log c(s, a)]+ E d U (s,a) [log(1 − c(s, a))](2) Since c * (s, a) = d E (s,a) d E (s,a)+d U (s,a) , the optimal density ratio can be recovered as: r * (s, a) = ψ dr (c * (s, a)) = c * (s, a) 1 − c * (s, a) = d E (s, a) d U (s, a)(3) Offline IL with Suboptimal Demonstrations via RelaxDICE In this section, we present RelaxDICE, a novel method for offline imitation learning with expert and supplementary non-expert demonstrations. A key question to study in this meaningful yet challenging setting is how to derive offline algorithms with appropriate regularization Ω(d, d U ) to effectively leverage the additional imperfect dataset D U . Formally, we begin with the following constrained optimization problem over the occupancy measure: where Eq. (5) is the Bellman flow constraints introduced in Theorem 1 that any valid occupancy measure must satisfy, and α > 0 is a weight factor balancing between minimizing KL divergence with d E (estimated with the limited expert data) and preventing deviation from d U . For example, a popular regularization choice in prior offline IL and offline RL works is the f -divergence D f (d d E ), which was originally designed for exact distribution matching between a model distribution and a target distribution (Nowozin, Cseke, and Tomioka 2016). Although this choice can indeed enforce d to be close to d U , we think that divergences or distances for exact distribution matching can be overly conservative and may lead to undesired effects when d U is highly suboptimal. In this case, even the true optimal occupancy measure (corresponding to the true optimal policy) will incur a high penalty from Ω(d, d U ). Although we can reduce α to mitigate the negative effect, we cannot remove the bias unless α approaches zero, which will then leave us at risk of exploring out-of-support state-actions because of a too small regularization strength. Moreover, prior theoretical work on offline RL (Zhan et al. 2022) also suggests that a smaller α will lead to a worse sample complexity and a higher error floor. Proofs for this section can be found in the appendix. max d≥0 − D KL (d d E ) − αΩ(d, d U ) (4) s.t. An Optimistic Fix to the Pessimistic Regularization To ensure the suboptimal dataset contains useful information about the optimal policy π * , theoretical studies typically make some assumptions about D U . As a motivating example, a minimal assumption adopted in (Zhan et al. 2022) is the following π * -concentrability 1 (where d * is the occupancy measure of π * ): Assumption 1. d U (s, a) > 0 and there exists a constant B such that d * (s, a)/d U (s, a) ≤ B, ∀s, a. Under this assumption, we argue that an ideal regularization Ω(d, d U ) would aim to bound the density ratio d/d U by a constant, instead of driving towards d ≡ d U . In other words, we still want to regularize d to stay in the support of d U so that policy evaluation/improvement is stable and reliable under a small distribution shift, but different from a divergence like D f (d d U ), we will impose little penalty on d if d/d U ≤ B, so that we will not enforce d to exactly match d U and the optimal policy can be preserved under the regularization (i.e., d * ∈ arg min d Ω(d, d U )). Towards this end, we draw inspiration from domain adaptation theory (Wu et al. 2019a) and propose to use the following relaxed f -divergence to realize Ω(d, d U ): Definition 1 (Asymmetrically-relaxed f -divergence). Given a constant β > 1 and a strictly convex and continuous function f : R + → R satisfying f (1) = 0, the asymmetrically-relaxed f -divergence between two distributions p and q (defined over domain X ) is defined as: D f β (p q) = X q(x) f β p(x) q(x) dx,(6) where f β is a partially linearized function of f defined as: f β (u) = f (u) + C f,β if u ≥ β f (β)u − f (β) if u < β (7) where the constant C f,β : = −f (β) + f (β)(β − 1). It is worth noting that f β is also continuous, convex (but not strictly convex) and satisfies f β (1) = 0. More importantly, D f β (p q) = 0 if and only if p(x)/q(x) ≤ β, ∀x ∈ X (proof can be found in the appendix). This property is valuable for IL with suboptimal demonstrations: Proposition 1. Under Assumption 1, for any strictly convex function f , let Ω 1 (d, d U ) = D f (d d U ) and Ω 2 (d, d U ) = D f β (d d U ) with β = B. When the behavior policy is not optimal (d U = d * ), then Ω 1 is biased while Ω 2 preserves the optimal policy (i.e. d * / ∈ arg min d Ω 1 (d, d U ) and d * ∈ arg min d Ω 2 (d, d U )) Thus we propose to use the relaxed f -divergence to realize the regularization. Let Ω(d, d U ) = D f β (d d U ) and we aim to solve the constrained optimization problem in Eq. (4)-(5) in an offline fashion. Apply a change of variable ω(s, a) = d(s,a) d U (s,a) to the Lagrangian of above constrained optimization, we can get the following optimization problem over ω and v (with v(s) being the Lagrange multipliers) (derivations can be found in the appendix): max ω≥0 min v L α,β (ω, v) := (1 − γ)E p 0 (s) [v(s)]+ (8) E d U (s,a) ω(s, a)ev(s, a) − ω(s, a) log(ω(s, a)) − α f β (ω(s, a)) Here, e v (s, a) : (3) and (T v)(s, a) := s T (s \|s, a)v(s ). Note that Eq. (8) can be estimated only using offline datasets D E and D U (assuming D U contains a set of initial states sampled from p 0 ). = log d E (s,a) d U (s,a) + γ(T v)(s, a) − v(s), where the density ratio d E /d U can be estimated via Eq. (2)- However, the nested min-max optimization in Eq. (8) usually results in unstable training in practice. To avoid this issue, we follow Kim et al. 2021) to assume that every state s ∈ S is reachable for the given MDP M and thus there exists a strictly feasible ω such that ω(s, a) = d(s, a)/d U (s, a) > 0, ∀s, a. Since Eq. (8) is a convex optimization problem with strict feasibility, due to strong duality via Slater's condition (Boyd and Vandenberghe 2004), we know that: max ω≥0 min v L α,β (ω, v) = min v max ω≥0 L α,β (ω, v)(9) By changing the max-min problem to min-max problem and using a particular convex function to instantiate the relaxed f -divergence, we can obtain the following closed-form solution for the inner maximization problem: Theorem 2. Let D f β be the relaxed f -divergence in Def- inition 1, with the associated convex function defined as f (u) = u log u. Then the closed-form solution ω * v (s, a) = arg max ω≥0 L α,β (ω, v) is: ω * v (s, a) = (10)    exp e v (s, a) 1 + α − 1 if A(s, a) exp (e v (s, a) − 1 − α(log β + 1)) otherwise where A(s, a) denotes the event: ev(s,a) α+1 > log β + 1. De- fine h(ω(s, a)) := ω(s, a)e v (s, a) − ω(s, a) log(ω(s, a)) − α f β (ω(s, a)) such that L α,β (ω, v) = E d U (s,a) [h(ω(s, a))]+ (1 − γ)E p0(s) [v(s)]. Then we have: h(ω * v (s, a)) =    (1 + α) exp e v (s, a) 1 + α − 1 + C 1 if A(s, a) exp (e v (s, a) − 1 − α(log β + 1)) + C 2 otherwise where C 1 = −αC f,β and C 2 = α(log β + 1) are constants w.r.t. ω and v. Based on Theorem 2, RelaxDICE solvesv * = arg min v L α,β (v) = L α,β (ω * v , v) , which provides us a tractable way to leverage a less conservative support regularization to effectively learn from potentially highly suboptimal offline data. RelaxDICE with Density Ratio Correction As discussed before, given datasets D E and D U , we can obtain an approximate density ratior(s, a) ≈ d E (s,a) d U (s,a) . Although we should not expect such an approximate density ratio to be accurate given limited samples, it is likely that the density-ratio-corrected occupancy measurer · d U is closer to the expert occupancy measure d E than d U . Thus another rational choice for realizing the regularization Ω(d, d U ) in Eq. (4) is the relaxed f -divergence between d andr · d U . With this goal, we derive an extension of our method, Re-laxDICE with Density Ratio Correction (RelaxDICE-DRC). Let Ω(d, d U ) = D f β (d r · d U ) . Similar to the derivation of RelaxDICE, we apply a change of variable ω(s, a) = d(s,a) d U (s,a) to the Lagrangian of the constrained optimization problem in Eq.(4)-(5) and with strong duality, we can obtain the following min-max optimization problem (derivations in the appendix): min v max ω≥0 L † α,β (ω, v) := (1 − γ)E p 0 (s) [v(s)]+ (11) E d U (s,a) ω(s, a) (ev(s, a) − log(ω(s, a))) − αr(s, a) f β ω(s, a) r(s, a) where v(s) is the Lagrange multiplier and e v (s, a) is defined same as before. Similar to RelaxDICE, we then introduce the following theorem to characterize the closed-form solution of the inner maximization problem in Eq. (11) to avoid nested min-max optimization: Theorem 3. Let D f β be the relaxed f -divergence in Defini- tion 1, with f (u) = u log u. Then the closed-form solution ω * v (s, a) = arg max ω≥0 L † α,β (ω, v) is: ω * v (s, a) = (12)    exp e v (s, a) + α logr(s, a) 1 + α − 1 if B(s, a) exp (e v (s, a) − 1 − α(log β + 1)) otherwise where B(s, a) denotes the event: ev(s,a)−logr(s,a) α+1 > log β + 1. Define h † (ω(s, a)) := ω(s, a)e v (s, a) − ω(s, a) log(ω(s, a)) − αr(s, a) f β ω(s,a) r(s,a) such that L † α,β (ω, v) = E d U (s,a) [h † (ω(s, a))] + (1 − γ)E p0(s) [v(s)]. Then we have: h † (ω * v (s, a)) =    (1 + α) exp ev(s, a) + α logr(s, a) 1 + α − 1 + C3 if B(s, a) exp (ev(s, a) − 1 − α(log β + 1)) + C4 otherwise where C 3 = −αC f,βr (s, a) and C 4 = α(log β + 1)r(s, a) are constants w.r.t. ω and v. Based on Theorem 3, RelaxDICE-DRC solvesv * = arg min v L † α,β (v) = L † α,β (ω * v , v) , which has the potential for better policy learning because of a more well-behaved regularization. Policy Extraction Given v * , the corresponding density ratioω * can be recovered according to Eq. (10) (for RelaxDICE) and Eq. (12) (for RelaxDICE-DRC) respectively. We can then use the following weighted BC objective (importance sampling or selfnormalized importance sampling) for policy extraction: max π E d U (s,a) [ω * (s, a) log π(a\|s)] or (13) max π E d U (s,a) [ω * (s, a) log π(a\|s)] E d U (s,a) [ω * (s, a)] In practice, we use samples from D U to estimate the expectations and we find that the latter one (using the selfnormalized weight) tends to perform better, which we employ in our experiments. Practical Considerations Since we only have samples (s, a, s ) from the dataset D U , similar to (Kostrikov, Nachum, and Tompson 2020;Lee et al. 2021;Kim et al. 2021), we have to use a single-point estimationê v (s, a, s ) = log d E (s,a) d U (s,a) + γv(s ) − v(s) for e v (s, a) . This estimation is generally biased (when the MDP is stochastic) due to the non-linear exponential function outside of e v . However, similar to the observation in (Kostrikov, Nachum, and Tompson 2020), we found this simple approach was enough to achieve good empirical performance on the standard benchmark domains we considered, thus we do not further use the Fenchel conjugate to remove the bias (Nachum et al. 2019a). We use multilayer perceptron (MLP) networks to parametrize the classifier c θ in Eq. (2), the Lagrange multiplier v φ , and the policy π ψ in Eq. (13). Since the objectives L α,β (v) and L † α,β (v) contain exponential terms, we use gradient penalty (Gulrajani et al. 2017) to enforce Lipschitz constraints on networks c θ and v φ , which can effectively stabilize the training. The required density ratio d E /d U inê v will be estimated via Eq. (3) asr θ = c θ 1−c θ . Regarding the hyperparameter β, in RelaxDICE, since ideally β should be around the upper bound of d E /d U , we can automatically set it using the approximate density ratiô r θ (e.g., by setting β to be the running average of the maximum estimated density ratio of each minibatch); while in RelaxDICE-DRC, β should characterize the upper bound of the density ratio d E /(r θ · d U ), which we expect to be small (e.g.1.5 or 2) asr θ · d U is a density-ratio-corrected occupancy measure. In summary, RelaxDICE does not introduce new hyperparameter that requires tuning by automatically setting β according to the data, while RelaxDICE-DRC has the potential for better policy learning with the requirement of manually specifying β. More details of the practical implementations can be found in the appendix. Related Work Learning from imperfect demonstrations. Imitation learning (Pomerleau 1988;Ross, Gordon, and Bagnell 2011;Ho and Ermon 2016a;Spencer et al. 2021) typically requires many optimal demonstrations, which could be expensive and time-consuming to collect. To address this limitation, imitation learning from imperfect demonstrations (Wu et al. 2019b;Brown et al. 2019;Brown, Goo, and Niekum 2020;Brantley, Sun, and Henaff 2019;Tangkaratt et al. 2020;Wang et al. 2018) arises as a promising alternative. To do so, prior works assume that the imperfect demonstrations consist of a mixture of expert data and suboptimal data and have considered two-step importance weighted IL (Wu et al. 2019b), learning imperfect demonstrations with adversarial training (Wu et al. 2019b;Wang et al. 2021) and training an ensemble of policies with weighted BC objectives (Sasaki and Yamashina 2020). Note that (Wu et al. 2019b) assumes the access to the optimality labels in the imperfect demonstrations whereas (Wang et al. 2021) and (Sasaki and Yamashina 2020) remove this strong assumption, which is followed in our work. Moreover, (Wu et al. 2019b) and (Wang et al. 2021) require online data collection for policy improvement while our work focuses on learning from offline data. The closest work to our setting is DemoDICE , which performs offline imitation learning with a KL constraint to regularize the learned policy to stay close to the behavior policy. Such constraint can mitigate the distribution shift issue when learning from offline data Kumar et al. 2019;Fujimoto, Meger, and Precup 2018), but could be overly conservative due to the exact distribution matching regularization especially when the imperfect data is highly suboptimal (see Proposition 1). Our method mitigates this issue by instead using a support regularization. Although (Wu, Tucker, and Nachum 2019) discussed a brief empirical exploration that using support regularization over policies offers little benefits, we instead formulate it as a constrained optimization over occupancy measures to take into consideration the diversity in both states and actions and observed clear practical benefits. Moreover, we surprisingly found that the increased complexity in minimax optimization can be resolved by the convenient closedform solutions of the inner maximization problems. Offline learning with stationary distribution correction. Prior works in offline RL / IL have used distribution correction to mitigate distribtuion shift. AlgaeDICE (Nachum et al. 2019b) leverages a dual formulation of f -divergence (Nachum et al. 2019a) to regularize the stationary distribution besides the policy improvement objective in offline RL. ValueDICE (Kostrikov, Nachum, and Tompson 2020) uses a similar formulation to AlgaeDICE for off-policy distribution matching with expert demonstrations. However, both Val-ueDICE and AlgaeDICE need to solve the nested min-max optimization problem, which is usually unstable in practice. OptiDICE and DemoDICE ) resolve this issue via deriving a closed-form solution of their inner optimization problem. Our method also enjoys the same desired property while using an asymmetricallyrelaxed f -divergence (Wu et al. 2019a) as a more appropriate regularization in face of highly suboptimal offline data. Experiments In our experiments, we aim to answer the following three questions: (1) how do RelaxDICE and RelaxDICE-DRC compare to prior works on standard continuous-control tasks using limited expert data and suboptimal offline data? (2) can RelaxDICE remain superior performance compared to prior methods as the quality of the suboptimal offline dataset decreases? (3) can RelaxDICE behave more robustly with respect to different hyperparameter choices compared to prior methods? Environments, Datasets and Task Construction. In order to answer these questions, we consider offline datasets of four MuJoCo (Todorov, Erez, and Tassa 2012) locomotion environments (hopper, halfcheetah, walker2d and ant) and two Adroit robotic manipulation environments (hammer and relocate) from the standard offline RL benchmark D4RL . To construct settings where we have varying data quality of the suboptimal offline dataset, for MuJoCo tasks, we use 1 trajectory from the expert-v2 datasets as D E for each environment and create the suboptimal offline data D U by mixing N E transitions from expert-v2 datasets and N R transitions from random-v2 datasets with 4 different ratios. We denote these settings as L1 (Level 1), L2 (Level 2), L3 (Level 3) and L4 (Level 4), which correspond to N E N R ≈ 0.2, 0.15, 0.1, 0.05 respectively. The higher the level, the more challenging the setting is. Note that all of the four settings are of much more suboptimal data composition compared to the data configuration used for D U adopted in , where D U in the most imperfect setting can have N E N R > 10.0, e.g. on walker2d. The rationale of constructing such challenging datasets is that in practice, it is much cheaper to generate suboptimal and even random data and therefore a successful offline IL method should be equipped with the capacity of tackling these suboptimal offline datasets. In order to excel at L1, L2, L3 and L4, a successful algorithm must effectively leverage D U to provide proper regularization for policy optimization. For Adroit tasks, following similar design choice, we construct three levels of data compositions, i.e. L1, L2 and L3. Please see the appendix for details. Comparisons. To answer these questions, we first consider the following prior approaches. We compare Re-laxDICE to DemoDICE , which performs offline imitation learning with supplementary imperfect demonstrations via applying a KL constraint between the occupancy measure of the learned policy and that of the behavior policy. We also consider BCND (Sasaki and Yamashina 2020) as a baseline, which learns an ensemble of policies via a weighted BC objective on noisy demonstrations. Moreover, we compare to BC(η) , where η ∈ {0, 0.5, 1.0} corresponds to a weight factor that balances between minimizing the negative log-likelihood on expert data D E and minimizing the negative log-likelihood on suboptimal offline data D U : min π L BC(η) (π) := −η 1 \|D E \| (s,a)∈D E log π(a\|s) − (1 − η) 1 \|D U \| (s,a)∈D U log π(a\|s). Finally, we consider the importance-weighted BC(η) denoted as BC-DRC(η), i.e. BC with density ratio correction, where we train a classifier to approximate the density ratio d E /d U asr via Eq. (2)-(3), and perform weighted BC(η) usingr as the importance weights: min π L BC-DRC(η) (π) := −η 1 \|D E \| (s,a)∈D E log π(a\|s) − (1 − η) 1 \|D U \| (s,a)∈D Ur(s, a) log π(a\|s). For all the tasks, we use α = 0.2 for RelaxDICE and use α = 0.05 for DemoDICE as suggested in , which is also verified in our experiments. We pick α and β for RelaxDICE-DRC via grid search, which we will discuss in the appendix. For more details of the experiment set-ups, evaluation protocols, hyperparameters and practical implementations, please see the appendix. Results of Empirical Evaluations To answer question (1) and (2), we evaluate RelaxDICE, RelaxDICE-DRC and other approaches discussed above on 6 D4RL environments (4 MuJoCo locomotion tasks and 2 Adroit robotic manipulation tasks) with 22 different settings in total. We present the full results in Table 1. As shown in Table 1, RelaxDICE-DRC achieves the best performance in 18 out of 22 tasks whereas RelaxDICE excels in the remaining 4 settings. It is also worth noting that RelaxDICE outperforms the strongest baseline De-moDICE in 20 out of 22 settings, without requiring tuning two hyperparameters as in RelaxDICE-DRC. Overall, we observe that the best performing method (either RelaxDICE or RelaxDICE-DRC) achieves over 30% performance im- . Numbers are averaged across 5 seeds, ± the 95%-confidence interval. We bold the top 2 highest performances. Either RelaxDICE or RelaxDICE-DRC achieves the best performance in each of 22 settings and outperforms the strongest baseline DemoDICE by a large margin in L3 and L4 settings where the offline data is highly suboptimal, suggesting the importance of using a relaxed distribution matching regularization. provement on average over DemoDICE. Moreover, in settings where the offline data is highly suboptimal, e.g. L3 and L4, both RelaxDICE and RelaxDICE-DRC can significantly outperform DemoDICE except on walker2d-L4 where RelaxDICE is a bit worse than DemoDICE but RelaxDICE-DRC prevails. In particular, on high-dimensional locomotion tasks such as ant and complex manipulation tasks such as hammer, RelaxDICE and RelaxDICE-DRC outperform DemoDICE by a significant margin on hard datasets such as L3 and L4. These suggest that using a less conservative support regularization can be crucial in cases with extremely low-quality offline data, supporting our theoretical analysis. Sensitivity of Hyperparameters To answer question (3), we perform an ablation study on the sensitivity of the hyperparameter α in RelaxDICE and DemoDICE , which controls the strength of the regularization between the learned policy and the behavior policy. We pick two continuous-control tasks halfcheetah and walker2d and evaluate the performance of RelaxDICE and DemoDICE using α ∈ {0.05, 0.1, 0.2, 0.3, 0.4, 0.5} on all four settings in each of the two tasks. As shown in Figure 2 in the appendix, Re-laxDICE is much more robust w.r.t. α compared to De-moDICE in all of the 8 scenarios as RelaxDICE remains roughly a flat line in all eight plots and the performance of DemoDICE drops significantly as α increases. We think the reason is that DemoDICE employs a conservative exact distribution matching constraint and therefore requires dif-ferent values of α on datasets with different data quality to find the delicate balance between policy optimization based on limited D E and regularization from suboptimal D U , e.g. higher α when the data quality is high and lower α when the data is highly suboptimal. In contrast, RelaxDICE imposes a relaxed support regularization, which is less conservative and therefore less sensitive w.r.t. data quality. Since tuning hyperparameters for offline IL / RL in a fully offline manner remains an open problem and often requires expensive online samples (Monier et al. 2020;Kumar et al. 2021;Kurenkov and Kolesnikov 2021), we believe Re-laxDICE's robustness w.r.t. the hyperparameters should significantly benefit practitioners. Conclusion and Discussion We present RelaxDICE, a novel offline imitation learning methods for learning policies from limited expert data and supplementary imperfect data. Different from prior works using regularizations originally designed for exact distribution matching, we employ an asymmetrically relaxed fdivergence as a more forgiving regularization that proves effective even for settings where the imperfect data is highly suboptimal. Both RelaxDICE and its extension RelaxDICE-DRC can avoid unstable min-max optimization of the regularized stationary state-action distribution matching problem by supporting a closed-form solution of the inner maximization problem, and show superior performance to strong baselines in our extensive empirical study. Derivation for RelaxDICE We propose to use the relaxed f -divergence to realize the regularization Ω(d, d U ) and aim to solve the following constrained optimization problem in an offline fashion: max d≥0 − D KL (d d E ) − αD f β (d d U )(14max d≥0 min v L α,β (d, v) := − D KL (d d E ) − αD f β (d d U ) + s v(s)((1 − γ)p 0 (s) + γ(T d)(s) − (Bd)(s))(16) Plugging in the definitions of KL divergence and relaxed f -divergence in Definition 1, L α,β (d, v) in Eq. (16) can be written as: L α,β (d, v) = E d(s,a) log d E (s, a) · d U (s, a) d U (s, a) · d(s, a) − αE d U (s,a) f β d(s, a) d U (s, a) + (1 − γ)E p0(s) [v(s)] + E d(s,a) [γ(T v)(s, a) − v(s)] (17) = E d U (s,a) d(s, a) d U (s, a) log d E (s, a) d U (s, a) + γ(T v)(s, a) − v(s) − log d(s, a) d U (s, a) − αE d U (s,a) f β d(s, a) d U (s, a) + (1 − γ)E p0(s) [v(s)](18) where Eq. (17) + (1 − γ)E p0(s) [v(s)](19) which can be estimated only using offline datasets D E and D U (assuming D U additionally contains a set of initial states sampled from p 0 ). Remark. We note that DemoDICE ) is a special case of RelaxDICE when β → 0, which can be verified according to Definition 1 and Theorem 2 (we will always be in the first condition of the piecewise function as log β + 1 → −∞ when β → 0). Derivation for RelaxDICE-DRC As discussed before, another attractive choice for realizing the regularization is the relaxed f -divergence between d and the density-ratio-corrected behavior occupancy measurer · d U . Let Ω(d, d U ) = D f β (d r · d U ) and we aim to solve the following constrained optimization problem in an offline fashion: max d≥0 − D KL (d d E ) − αD f β (d r · d U )(20) s.t. Similar to the derivation of RelaxDICE, we can obtain the following Lagrangian for the constrained optimization problem in Eq.(20)-(21) (with v(s) being the Lagrange multipliers): L † α,β (d, v) = E d U (s,a) d(s, a) d U (s, a) e v (s, a) − log d(s, a) d U (s, a) − αE d U (s,a) r(s, a) · f β d(s, a) r(s, a) · d U (s, a) + (1 − γ)E p0(s) [v(s)](22) Similarly, we use a change of variable ω(s, a) = d(s,a) d U (s,a) and apply strong duality to obtain the following min-max problem over ω: min v max ω≥0 L † α,β (ω, v) := E d U (s,+ (1 − γ)E p0(s) [v(s)](23) Proofs Lemma 1. For distributions p and q defined on domain X , if p(x) q(x) < β, ∀x ∈ X , then the relaxed f -divergence D f β (p q) = 0. Proof. According to Definition 1, if p/q < β everywhere, we have: D f β (p, q) = X q(x) f β p(x) q(x) dx = X q(x) f (β) p(x) q(x) − f (β) dx = f (β) X p(x)dx − f (β) X q(x)dx = 0 Theorem 2. Let D f β be the relaxed f -divergence in Definition 1, with the associated convex function defined as f (u) = u log u. Then the closed-form solution ω * v (s, a) = arg max ω≥0 L α,β (ω, v) is: ω * v (s, a) = (10)    exp e v (s, a) 1 + α − 1 if A(s, a) exp (e v (s, a) − 1 − α(log β + 1)) otherwise where A(s,h(ω * v (s, a)) =    (1 + α) exp e v (s, a) 1 + α − 1 + C 1 if A(s, a) exp (e v (s, a) − 1 − α(log β + 1)) + C 2 otherwise where C 1 = −αC f,β and C 2 = α(log β + 1) are constants w.r.t. ω and v. Proof. Since f β is a continuous piecewise function, h(ω(s, a)) is also a continuous piecewise function: h(ω(s, a)) = e v (s, a)ω(s, a) − ω(s, a) log(ω(s, a)) − αf (ω(s, a)) − αC f,β if ω(s, a) ≥ β e v (s, a)ω(s, a) − ω(s, a) log(ω(s, a)) − αf (β)ω(s, a) + αf (β) if β ≥ ω(s, a) ≥ 0 When f (u) = u log u and f (u) = log u + 1, the gradient of h(ω(s, a)) is given by: and the overall maximum will be either ω * ≤β (s, a) or ω * ≥β (s, a) depending on whose function value is larger (the global maximum must be one of the local maximums). h (ω(s, a)) = e v (s, a) − (α + 1)(log(ω(s, a)) + 1) if ω(s, a) ≥ β e v (s, a) − log(ω(s, a)) − 1 − α(log β + 1) if β ≥ ω(s, a) ≥ 0 In the following, we will use the fact that h is strictly decreasing due to the strict concavity of h (an affine function plus a strictly concave function). (1) When e v (s, a) > (α + 1)(log β + 1) (or equivalently h (β) > 0): For β ≥ ω(s, a) ≥ 0, we know that h (ω(s, a)) ≥ h (β) = e v (s, a) − (α + 1)(log β + 1) > 0, so ω * ≤β (s, a) = β. For ω(s, a) ≥ β, since h is strictly decreasing and h (β) > 0, we know that ω * ≥β (s, a) is attained at h (ω(s, a)) = e v (s, a) − (α + 1)(log(ω(s, a)) + 1) = 0. Thus ω * ≥β (s, a) = exp ev(s,a) 1+α − 1 . Moreover, because h (ω(s, a)) > 0 when β ≤ ω(s, a) < ω * ≥β (s, a), we know that h(ω * ≤β (s, a)) = h(β) < h(ω * ≥β (s, a)), and the overall maximum is ω * (s, a) = exp ev(s,a) 1+α − 1 > β. In this case, the maximum function value of h is: h(ω * (s, a)) = e v (s, a)ω * (s, a) − ω * (s, a) log(ω * (s, a)) − αf (ω * (s, a)) − αC f,β = ω * (s, a) (e v (s, a) − (α + 1) log(ω * (s, a))) − αC f,β = (α + 1)ω * (s, a) − αC f,β = (α + 1) exp e v (s, a) 1 + α − 1 − αC f,β (2) When e v (s, a) ≤ (α + 1)(log β + 1) (or equivalently h (β) ≤ 0) : Since lim ω(s,a)→0 h (ω(s, a)) = +∞ and h (β) ≤ 0, ω * ≤β (s, a) is attained at h (ω(s, a)) = e v (s, a) − log(ω(s, a)) − 1 − α(log β + 1) = 0. Thus ω * ≤β (s, a) = exp (e v (s, a) − 1 − α(log β + 1)). For ω(s, a) ≥ β, since h is strictly decreasing and h (β) ≤ 0, so h (ω(s, a)) ≤ 0 and ω * ≥β (s, a) = β. Moreover, because h(ω * ≤β (s, a)) ≥ h(β) = h(ω * ≥β (s, a)), the overall maximum is ω * (s, a) = exp (e v (s, a) − 1 − α(log β + 1)) ≤ β. In this case, the maximum function value of h is: h(ω * (s, a)) = e v (s, a)ω * (s, a) − ω * (s, a) log(ω * (s, a)) − αf (β)ω * (s, a) + αf (β) = ω * (s, a) (e v (s, a) − log(ω * (s, a)) − α(log β + 1)) + α(log β + 1) = ω * (s, a) + α(log β + 1) = exp (e v (s, a) − 1 − α(log β + 1)) + α(log β + 1) Theorem 3. Let D f β be the relaxed f -divergence in Definition 1, with f (u) = u log u. Then the closed-form solution ω * v (s, a) = arg max ω≥0 L † α,β (ω, v) is: ω * v (s, a) =(h † (ω * v (s, a)) =    (1 + α) exp ev(s, a) + α logr(s, a) 1 + α − 1 + C3 if B(s, a) exp (ev(s, a) − 1 − α(log β + 1)) + C4 otherwise where C 3 = −αC f,βr (s, a) and C 4 = α(log β + 1)r(s, a) are constants w.r.t. ω and v. Proof. Since f β is a continuous differentiable piecewise function, h † (ω(s, a)) is also a continuous differentiable piecewise function. When f (u) = u log u and f (u) = log u + 1, we have: The gradient of h † (ω(s, a)) is given by: h(ω(s, a)) and the overall maximum will be either ω * ≤βr (s, a) or ω * ≥βr (s, a) depending on whose function value is larger (the global maximum must be one of the local maximums). h † (ω(s, a)) =        e v ( In the following, we will use the fact that (h † ) is strictly decreasing due to the strict concavity of h † (an affine function plus a strictly concave function). (1) When e v (s, a) > (α + 1)(log β + 1) + logr(s, a) (or equivalently (h † ) (βr(s, a)) > 0): For βr(s, a) ≥ ω(s, a) ≥ 0, we know that (h † ) (ω(s, a)) ≥ (h † ) (βr(s, a)) = e v (s, a)−(α+1)(log β +1)−logr(s, a) > 0, so ω * ≤βr (s, a) = βr(s, a). For ω(s, a) ≥ βr(s, a), since (h † ) is strictly decreasing and (h † ) (βr(s, a)) > 0, we know that ω * ≥βr (s, a) is attained at (h † ) (ω(s, a)) = e v (s, a) − (α + 1)(log(ω(s, a)) + 1) + α logr(s, a) = 0. Thus ω * ≥βr (s, a) = exp ev(s,a)+α logr(s,a) 1+α − 1 . Moreover, because h(ω * ≤βr (s, a)) = h(βr(s, a)) < h(ω * ≥βr (s, a)), and the overall maximum is ω * (s, a) = exp ev(s,a)+α logr(s,a) 1+α − 1 > βr(s, a). In this case, the maximum function value of h is: h(ω * (s, a)) = e v (s, a)ω * (s, a) − ω * (s, a) log(ω * (s, a)) − αω * (s, a) log ω * (s, a) r(s, a) − αC f,βr (s, a) = ω * (s, a) (e v (s, a) − (α + 1) log(ω * (s, a)) + α log(r(s, a))) − αC f,βr (s, a) = (α + 1)ω * (s, a) − αC f,βr (s, a) = (α + 1) exp e v (s, a) + α logr(s, a) 1 + α − 1 − αC f,βr (s, a) (2) When e v (s, a) ≤ (α + 1)(log β + 1) + logr(s, a) (or equivalently (h † ) (βr(s, a)) ≤ 0) : Since lim ω(s,a)→0 (h † ) (ω) = +∞ and (h † ) (βr(s, a)) ≤ 0, ω * ≤βr (s, a) is attained at (h † ) (ω(s, a)) = e v (s, a) − log(ω(s, a)) − 1 − α(log β + 1) = 0. Thus ω * ≤βr (s, a) = exp (e v (s, a) − 1 − α(log β + 1)). For ω(s, a) ≥ βr(s, a), since (h † ) is strictly decreasing and (h † ) (βr(s, a)) ≤ 0, so (h † ) (ω(s, a)) ≤ 0 and ω * ≥βr (s, a) = βr(s, a). Moreover, because h(ω * ≤βr (s, a)) ≥ h(βr(s, a)) = h(ω * ≥βr (s, a)), the overall maximum is ω * (s, a) = exp (e v (s, a) − 1 − α(log β + 1)). In this case, the maximum function value of h is: h(ω * (s, a)) = e v (s, a)ω * (s, a) − ω * (s, a) log(ω * (s, a)) − αf (β)ω * (s, a) + αf (β)r(s, a) = ω * (s, a) (e v (s, a) − log(ω * (s, a)) − α(log β + 1)) + α(log β + 1)r(s, a) = ω * (s, a) + α(log β + 1)r(s, a) = exp (e v (s, a) − 1 − α(log β + 1)) + α(log β + 1)r(s, a) Additional Experimental Results and Details Sensitivity of Hyperparameters for Controlling Regularization Strength Task Description We consider offline datasets of four MuJoCo (Todorov, Erez, and Tassa 2012) locomotion environments (hopper, halfcheetah, walker2d and ant) and two Adroit robotic manipulation environments (hammer and relocate) from the standard offline RL benchmark D4RL . For each environment, we construct different settings where there is a limited amount of expert demonstrations (denoted as D E ) and a relatively large collection of suboptimal trajectories (denoted as D U ) by mixing N E transitions from expert datasets and N R transitions from extremely low-quality datasets 2 . We denote these settings as L1 (Level 1), L2 (Level 2), L3 (Level 3) and L4 (Level 4), where a higher level means a more challenging setting. Note that for Adroit environments, we use three settings L1, L2 and L3 instead of four due to the complexity of the high-dimensional tasks. It's also worth noting that all the considered tasks here are much more challenging than the settings in , in the sense that these settings are of much more suboptimal data composition (even tasks L1 have lower N E /N R ratios than the most challenging tasks in ). We summarize the details of these tasks in Table 2. Evaluation Protocols For all the methods except BCND, we run 1M training iterations (gradient steps) and we report the average performance of the last 50k (5%) steps to capture their asymptotic performance at convergence. For BCND, we follow the implementation in Appendix E.1 in , which has predefined number of iterations according to dataset statistics and we also report the average performance of the last 50k steps for consistent evaluation. For four MuJoCo locomotion environments halfcheetah, hopper, walker2d and ant, we follow to compute the normalized score as: normalized score = 100 × score − random score expert score − random score where the expert score and random score corresponds to the average return of trajectories in expert-v2 and random-v2 respectively. For two Adroit environments hammer and relocate, we use the recommended reference score in D4RL 3 to compute the normalized score. Hyperparameters Algorithm Hyperparameters We use γ = 0.99 as the discount factor of the MDP. All methods use a batch size of 256. The hyperparameters for each of the compared algorithm are summarized below. • BC: η ∈ {0.0, 0.5, 1.0}. • BC-DRC: η ∈ {0.0, 0.5}. • BCND: We follow the hyperparameter configurations in Appendix E.1 in ). • DemoDICE 4 : α = 0.05 across all tasks, as suggested in ) and verified in our experiments. • RelaxDICE: α = 0.2 across all tasks. As shown in Figure 2, RelaxDICE can achieve a potentially better performance if using different values for different tasks. Nevertheless, we fix α = 0.2 for all tasks to demonstrate the robustness of RelaxDICE across different tasks. Moreover, we automatically set β to be the running average of the maximum estimated density ratio (r θ = c θ 1−c θ ) of each minibatch. • RelaxDICE-DRC: We perform a grid search for α ∈ {0.05, 0.1, 0.2, 0.5, 1.0} and β ∈ {1.5, 2.0}. Note that here we can potentially use a larger regularization strength α because we employ a relaxed support regularization based on the densityratio-corrected occupancy measurer · d U , which has the potential for better policy improvement. β should characterize the upper bound of the density ratio d E /(r θ · d U ), which we expect to be close to 1 (e.g.1.5 or 2) sincer θ · d U is a density-ratiocorrected occupancy measure. For full reproducibility, we summarize the used hyperparameters in Table 3. Implementation Details We follow all the other hyperparameters from Computation Resources We train RelaxDICE and RelaxDICE-DRC on a single NVIDIA GeForce RTX 2080 Ti with 5 random seeds for at most 3 hours for all the tasks. Dataset License All datasets in our experiments are from the open-sourced D4RL 5 benchmark . All datasets there are licensed under the Creative Commons Attribution 4.0 License (CC BY). Theorem 1 ((Feinberg and Shwartz 2012; Syed, Bowling, and Schapire 2008)). Suppose d satisfies the following Bellman flow constraints: a d(s, a) = (1 − γ)p0(s) + γ s ,a T (s\|s , a )d(s , a ), ∀s. and d(s, a) ≥ 0, ∀s, a. T (s\|s , a )d(s , a ), ∀s ∈ S. (5) simplicity, we define (Bd)(s) := a d(s, a), (T d)(s) := s ,a T (s\|s , a )d(s , a ) and (T v)(s, a) := s T (s \|s, a)v(s ). First, we can obtain the following Lagrangian for above constrained optimization problem (with v(s) being the Lagrange multipliers): uses the fact that s v(s)(T d)(s) = s,a d(s, a)(T v)(s, a), and the density ratio d E /d U can be estimated via Eq. (2)-(3); and Eq. (18) uses importance sampling to change the expectation w.r.t. d to an expectation w.r.t. d U for offline learning. Define e v (s, a) := log d E (s,a) d U (s,a) + γ(T v)(s, a) − v(s) and use a change of variable ω(s, a) = d(s,a) d U (s,a) , we obtain the following optimization problem: max ω≥0 min v L α,β (ω, v) := E d U (s,a) ω(s, a)e v (s, a) − ω(s, a) log(ω(s, a)) − α f β (ω(s, a)) T (s\|s , a )d(s , a ), ∀s ∈ S. a) denotes the event: ev(s,a) α+1 > log β+1. Define h(ω(s, a)) := ω(s, a)e v (s, a)−ω(s, a) log(ω(s, a))−α f β (ω(s, a)) such that L α,β (ω, v) = E d U (s,a) [h(ω(s, a))] + (1 − γ)E p0(s) [v(s)]. Then we have: Define ω * ≤β (s, a) and ω * ≥β (s, a) as the local maximum for[0, β] and [β, +∞) respectively: ω * ≤β (s, a) := arg max β≥ω(s,a)≥0h(ω(s, a)) and ω * ≥β (s, a) v (s, a) + α logr(s, a) 1 + α − 1 if B(s, a) exp (e v (s, a) − 1 − α(log β + 1)) otherwise where B(s, a) denotes the event: ev(s,a)−logr(s,a) α+1 > log β + 1. Define h † (ω(s, a)) := ω(s, a)e v (s, a) − ω(s, a) log(ω(s, a)) s,a) such that L † α,β (ω, v) = E d U (s,a) [h † (ω(s, a))] + (1 − γ)E p0(s) [v(s)]. Then we have: s, a)ω(s, a) − ω(s, a) log(ω(s, a)) − αω(s, a) log ω(s, a) r(s, a) − αC f,βr (s, a) if ω(s, a) r(s, a) ≥ β e v (s, a)ω(s, a) − ω(s, a) log(ω(s, a)) − αf (β)ω(s, a) + αf (β)r(s, a) if β ≥ ω(s, a) r(s, a) ≥ 0 (h † ) (ω(s, a)) = e v (s, a) − (α + 1)(log(ω(s, a)) + 1) + α logr(s, a) if ω(s, a) ≥ βr(s, a) e v (s, a) − log(ω(s, a)) − 1 − α(log β + 1) if βr(s, a) ≥ ω(s, a) ≥ 0Define ω * ≤βr (s, a) and ω * ≥βr (s, a) as the local maximum for [0, βr(s, a)] and [βr(s, a), Figure 2 : 2Ablation study on the sensitivity of α for RelaxDICE and DemoDICE. RelaxDICE is much more robust w.r.t. different values of α compared to DemoDICE across different data compositions. Table 1: Results for four MuJoCo environments halfcheetah, hopper, walker2d and ant and two Adroit environments hammer and relocate from D4RLBC-DRC BC Envs Tasks RelaxDICE RelaxDICE-DRC DemoDICE BCND η = 0.0 η = 0.5 η = 0.0 η = 0.5 η = 1.0 L1 74.6±9.1 73.6±6.3 70.9±9.0 6.6±2.9 1.4±1.1 2.9±3.9 1.8±1.2 7.6±8.0 17.8±11.7 L2 64.2±8.7 70.0±13.7 54.4±6.4 4.8±4.2 2.4±1.2 4.9 ± 2.7 2.9±2.1 1.6±1.5 17.8±11.7 hopper L3 36.2±5.6 41.5±4.3 31.4±9.7 2.3±2.3 1.8±0.5 1.5 ± 0.6 5.0±3.3 1.4±0.9 17.8±11.7 L4 38.7±8.2 40.2±6.9 34.9±5.6 0.9±0.3 0.7±0.2 1.5±0.7 0.8±0.4 0.8±0.3 17.8±11.7 L1 59.1±8.6 66.7±5.1 58.6±8.0 2.5±0.1 2.6 ±0.0 2.6±0.0 2.6±0.0 2.6±0.0 0.9 ± 1.1 L2 49.3±4.7 52.1±2.0 48.3±3.9 2.5±0.1 2.6 ±0.0 2.6±0.0 2.6±0.0 2.6±0.0 0.9 ± 1.1 halfcheetah L3 35.0±6.6 37.9±4.0 32.9±2.1 2.5±0.0 2.6 ±0.0 2.6±0.0 2.6±0.0 2.6±0.0 0.9 ± 1.1 L4 13.3±2.1 16.1±4.1 10.5±0.5 2.6±0.0 2.6 ±0.0 2.6±0.0 2.6±0.0 2.6±0.0 0.9 ± 1.1 L1 92.6±7.6 99.4±1.9 98.8±1.6 3.0±3.9 0.6±0.6 0.3±0.2 2.2±1.1 0.2±0.0 8.5±3.6 L2 69.7±20.3 57.7±12.5 42.1±23.9 0.1±0.2 0.2±0.0 0.5±0.3 0.2±0.0 0.3±0.1 8.5±3.6 walker2d L3 41.9±23.1 56.7±26.2 23.4±20.6 0.7±0.6 0.8±1.1 0.4±0.2 0.2±0.0 0.3±0.1 8.5±3.6 L4 26.3±17.2 49.5±17.4 39.8±22.4 0.2±0.2 0.2±0.1 0.5±0.4 0.2±0.1 0.2±0.1 8.5±3.6 L1 91.9±3.6 89.0±5.3 77.9±8.7 12.2±2.4 61.7±4.9 21.3±1.2 66.2±11.2 21.3±1.1 -8.3±4.3 L2 75.2±5.8 82.1±7.1 70.5±4.2 15.6±2.3 50.8±6.1 20.6±1.8 54.4±4.9 19.9±1.6 -8.3±4.3 ant L3 58.7±7.1 59.6±9.0 49.9±2.9 17.0±0.9 38.1±5.2 18.8±4.7 37.6±3.0 22.6±0.1 -8.3±4.3 L4 43.2±7.2 41.3±4.0 -5.3±41.4 13.6±1.7 29.0±3.9 22.4±0.2 28.0±3.1 22.3±0.3 -8.3±4.3 L1 24.2±17.6 27.3±13.9 4.1±3.6 0.2±0.0 0.4±0.1 0.3±0.0 0.4±0.0 0.3±0.0 4.8±2.9 hammer L2 18.3±12.0 18.4±13.6 17.3±8.9 0.2±0.0 0.3±0.0 0.5±0.4 0.3±0.1 0.3±0.0 4.8±2.9 L3 19.5±15.5 20.6±12.8 14.1±10.2 0.2±0.0 0.4±0.0 0.3±0.0 0.4±0.0 0.3±0.0 4.8±2.9 L1 48.0±2.3 50.9±5.3 40.9±13.7 -0.1±0.0 -0.2±0.0 6.4±2.0 -0.2±0.0 7.3±2.2 0.2±0.3 relocate L2 45.1±5.7 52.4±7.7 43.0±8.2 -0.2±0.0 5.0±2.4 -0.2±0.0 -0.2±0.0 4.5±2.1 0.2±0.3 L3 39.0±4.5 43.3±7.8 27.1±6.9 -0.2±0.0 -0.2±0.0 2.9±1.9 -0.2±0.0 3.3±3.0 0.2±0.3 Expert Dataset D E Suboptimal Dataset D U EnvsTasks # of transitions from expert-v2 # of transitions from expert-v2 # of transitions from random-v2Expert Dataset D E Suboptimal Dataset D U EnvsTasks # of transitions from expert-v0 # of transitions from expert-v0 # of transitions from cloned-v0L1 1k 200k 1000k L2 1k 150k 1000k halfcheetah L3 1k 100k 1000k L4 1k 50k 1000k L1 1k 14k 22k L2 1k 10k 22k hopper L3 1k 5k 22k L4 1k 2k 22k L1 1k 10k 20k L2 1k 5k 20k walker2d L3 1k 3k 20k L4 1k 2k 20k L1 1k 30k 180k L2 1k 20k 180k ant L3 1k 10k 180k L4 1k 5k 180k L1 2k 1000k 1000k hammer L2 2k 790k 1000k L3 2k 590k 1000k L1 10k 1000k 1000k relocate L2 10k 790k 1000k L3 10k 590k 1000k Table 2 : 2Dataset statistics for four MuJoCo environments halfcheetah, hopper, walker2d and ant and two Adroit environments hammer and relocate from D4RL. listed as follows: • Policy network π ψ (for BC, BC-DRC, BCND, DemoDICE, RelaxDICE and RelaxDICE-DRC): three-layer MLP with 256 hidden units, learning rate 3 × 10 −5 . • Lagrange multiplier network v φ (for DemoDICE, RelaxDICE and RelaxDICE-DRC): three-layer MLP with 256 hidden units, learning rate 3 × 10 −4 , gradient penalty coefficient 1 × 10 −4 . • Classifier network c θ (for DemoDICE, RelaxDICE and RelaxDICE-DRC): three-layer MLP with 256 hidden units, learning rate 3 × 10 −4 , gradient penalty coefficient 10.Envs Tasks α β L1 1.0 1.5 L2 1.0 1.5 hopper L3 0.5 2.0 L4 0.2 1.5 L1 1.0 1.5 L2 0.5 1.5 halfcheetah L3 0.2 2.0 L4 0.2 2.0 L1 0.2 2.0 L2 0.5 2.0 walker2d L3 0.1 1.5 L4 0.05 2.0 L1 0.1 1.5 L2 0.2 1.5 ant L3 0.5 2.0 L4 0.5 2.0 L1 0.5 1.5 hammer L2 0.05 1.5 L3 0.5 2.0 L1 0.5 1.5 relocate L2 0.5 2.0 L3 0.05 1.5 Table 3 : 3Hypeparameters used for RelaxDICE-DRC. 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[ "Published as a conference paper at ICLR 2023 LEARNING TO REASON OVER VISUAL OBJECTS", "Published as a conference paper at ICLR 2023 LEARNING TO REASON OVER VISUAL OBJECTS" ]	[ "Shanka Subhra Mondal ", "Taylor W Webb [email protected] ", "Jonathan D Cohen ", "\nPrinceton University\nPrincetonNJ\n", "\nUniversity of California\nLos Angeles Los AngelesCA\n", "\nPrinceton University\nPrincetonNJ\n" ]	[ "Princeton University\nPrincetonNJ", "University of California\nLos Angeles Los AngelesCA", "Princeton University\nPrincetonNJ" ]	[]	A core component of human intelligence is the ability to identify abstract patterns inherent in complex, high-dimensional perceptual data, as exemplified by visual reasoning tasks such as Raven's Progressive Matrices (RPM). Motivated by the goal of designing AI systems with this capacity, recent work has focused on evaluating whether neural networks can learn to solve RPM-like problems. Previous work has generally found that strong performance on these problems requires the incorporation of inductive biases that are specific to the RPM problem format, raising the question of whether such models might be more broadly useful. Here, we investigated the extent to which a general-purpose mechanism for processing visual scenes in terms of objects might help promote abstract visual reasoning. We found that a simple model, consisting only of an object-centric encoder and a transformer reasoning module, achieved state-of-the-art results on both of two challenging RPM-like benchmarks (PGM and I-RAVEN), as well as a novel benchmark with greater visual complexity (CLEVR-Matrices). These results suggest that an inductive bias for object-centric processing may be a key component of abstract visual reasoning, obviating the need for problem-specific inductive biases.	10.48550/arxiv.2303.02260	[ "https://export.arxiv.org/pdf/2303.02260v1.pdf" ]	257,365,083	2303.02260	29d5577b5350dcb88e95416c11f14a4a92ce9178	Published as a conference paper at ICLR 2023 LEARNING TO REASON OVER VISUAL OBJECTS Shanka Subhra Mondal Taylor W Webb [email protected] Jonathan D Cohen Princeton University PrincetonNJ University of California Los Angeles Los AngelesCA Princeton University PrincetonNJ Published as a conference paper at ICLR 2023 LEARNING TO REASON OVER VISUAL OBJECTS * Equal Contribution A core component of human intelligence is the ability to identify abstract patterns inherent in complex, high-dimensional perceptual data, as exemplified by visual reasoning tasks such as Raven's Progressive Matrices (RPM). Motivated by the goal of designing AI systems with this capacity, recent work has focused on evaluating whether neural networks can learn to solve RPM-like problems. Previous work has generally found that strong performance on these problems requires the incorporation of inductive biases that are specific to the RPM problem format, raising the question of whether such models might be more broadly useful. Here, we investigated the extent to which a general-purpose mechanism for processing visual scenes in terms of objects might help promote abstract visual reasoning. We found that a simple model, consisting only of an object-centric encoder and a transformer reasoning module, achieved state-of-the-art results on both of two challenging RPM-like benchmarks (PGM and I-RAVEN), as well as a novel benchmark with greater visual complexity (CLEVR-Matrices). These results suggest that an inductive bias for object-centric processing may be a key component of abstract visual reasoning, obviating the need for problem-specific inductive biases. INTRODUCTION Human reasoning is driven by a capacity to extract simple, low-dimensional abstractions from complex, high-dimensional inputs. We perceive the world around us in terms of objects, relations, and higher order patterns, allowing us to generalize beyond the sensory details of our experiences, and make powerful inferences about novel situations Spearman (1923); Gick & Holyoak (1983); Lake et al. (2017). This capacity for abstraction is particularly well captured by visual analogy problems, in which the reasoner must abstract over the superficial details of visual inputs, in order to identify a common higher order pattern (Gentner, 1983;Holyoak, 2012). A particularly challenging example of these kinds of problems are the Raven's Progressive Matrices (RPM) problem sets (Raven, 1938), which have been found to be especially diagnostic of human reasoning abilities (Snow et al., 1984). A growing body of recent work has aimed to build learning algorithms that capture this capacity for abstract visual reasoning. Much of this previous work has revolved around two recently developed benchmarks -the Procedurally Generated Matrices (PGM) (Barrett et al., 2018), and the RAVEN dataset (Zhang et al., 2019a) -consisting of a large number of automatically generated RPM-like problems. As in RPM, each problem consists of a 3 × 3 matrix populated with geometric forms, in which the bottom right cell is blank. The challenge is to infer the abstract pattern that governs the relationship along the first two columns and/or rows of the matrix, and use that inferred pattern to 'fill in the blank', by selecting from a set of choices. As can be seen in Figure 1, these problems can be quite complex, with potentially many objects per cell, and multiple rules per problem, yielding a highly challenging visual reasoning task. . STSN combines slot attention, an objectcentric encoding method, and a transformer reasoning module. Slot attention decomposes each image panel into a set of K slots, which are randomly initialized and iteratively updated through competitive attention over the image. STSN assigns a score to each of the 8 potential answers, by independently evaluating the combination of each answer choice together with the 8 context panels. For each answer choice, slots are extracted from that choice, and the context panels, and these slots are concatenated to form a sequence that is passed to the transformer, which then generates a score. The scores for all answer choices are passed through a softmax in order to compute the task loss L task . Additionally, the slots for each image panel are passed through a slot decoder, yielding a reconstruction of that image panel, from which the reconstruction loss L recon is computed. There is substantial evidence that human visual reasoning is fundamentally organized around the decomposition of visual scenes into objects (Duncan, 1984;Pylyshyn, 1989;Peters & Kriegeskorte, 2021). Objects offer a simple, yet powerful, low-dimensional abstraction that captures the inherent compositionality underlying visual scenes. Despite the centrality of objects in visual reasoning, previous works have so far not explored the use of object-centric representations in abstract visual reasoning tasks such as RAVEN and PGM, or at best have employed an imprecise approximation to object representations based on spatial location. Recently, a number of methods have been proposed for the extraction of precise object-centric representations directly from pixel-level inputs, without the need for veridical segmentation data (Greff et al., 2019;Locatello et al., 2020;Engelcke et al., 2021). While these methods have been shown to improve performance in some visual reasoning tasks, including question answering from video (Ding et al., 2021) and prediction of physical interactions from video Wu et al. (2022), previous work has not addressed whether this approach is useful in the domain of abstract visual reasoning (i.e., visual analogy). To address this, we developed a model that combines an object-centric encoding method, slot attention (Locatello et al., 2020), with a generic transformer-based reasoning module (Vaswani et al., 2017). The combined system, termed the Slot Transformer Scoring Network (STSN, Figure 1) achieves state-of-the-art performance on both PGM and I-RAVEN (a more challenging variant of RAVEN), despite its general-purpose architecture, and lack of task-specific augmentations. Furthermore, we developed a novel benchmark, the CLEVR-Matrices (Figure 2), using a similar RPM-like problem structure, but with greater visual complexity, and found that STSN also achieves state-of-the-art performance on this task. These results suggest that object-centric encoding is an essential component for achieving strong abstract visual reasoning, and indeed may be even more important than some task-specific inductive biases. RELATED WORK Since the introduction of the PGM (Barrett et al., 2018) and RAVEN (Zhang et al., 2019a) datasets, a number of methods have been proposed for learning to solve RPM-like problems Barrett et al. (2018) Zhuo & Kankanhalli (2022). Though significant progress has been made, the best performing methods generally rely on inductive biases that are specifically tailored to the RPM problem format. For instance, the Scattering Compositional Learner (SCL) (Wu et al., 2020), arguably the best current model (achieving strong performance on both PGM and I-RAVEN), assumes that rules are independently applied in each feature dimension, with no interaction between features. Similarly, the Multi-Scale Relation Network (MRNet) (Benny et al., 2021), which achieves strong performance on PGM, explicitly builds the row-wise and column-wise structure of RPM problems into its architecture. These approaches raise the question of whether problemspecific inductive biases are necessary to achieve strong performance on these problems. Here, we explore the utility of a more general-purpose inductive bias -a mechanism for processing visual scenes in terms of objects. In contrast, most previous approaches to solving RPM-like problems have operated over embeddings of entire image panels, and thus likely fail to capture the compositional structure of such multi-object visual inputs. Some work has attempted to approximate object-centric representations, for instance by treating spatial location as a proxy for objects (Wu et al., 2020), or by employing encodings at different spatial scales (Benny et al., 2021) (therefore preferentially capturing larger vs. smaller objects), but it is not clear that these approximations extract precise object-centric representations, especially in problems with many overlapping objects, such as PGM. Recently, a number of methods have been proposed to address the challenging task of annotationfree object segmentation (Greff et al., 2019;Locatello et al., 2020;Engelcke et al., 2021). In this approach, the decomposition of a visual scene into objects is treated as a latent variable to be inferred in the service of a downstream objective, such as autoencoding, without access to any explicit segmentation data. Here, we used the slot attention method (Locatello et al., 2020), but our approach should be compatible with other object-centric encoding methods. Our method employs a generic transformer (Vaswani et al., 2017) to perform reasoning over the object-centric representations extracted by slot attention. This approach allows the natural permutation invariance of objects to be preserved in the reasoning process. A few other recent efforts have employed systems that provide object-centric representations as the input to a transformer network (Ding et al., 2021;Wu et al., 2022), most notably ALOE (Attention over Learned Object Embeddings (Ding et al., 2021)), which used a different object encoding method (MONet (Burgess et al., 2019)). Such systems have exhibited strong visual reasoning performance in some tasks, such as question answering from video, that require processing of relational information. Here, we go beyond this work, to test: a) the extent to which object-centric processing can subserve more abstract visual reasoning, involving the processing of higher-order relations, as required for visual analogy tasks such as PGM and I-RAVEN; and b) whether this approach obviates the need for problemspecific inductive biases that have previously been proposed for these tasks. APPROACH PROBLEM DEFINITION Each RPM problem consists of a 3 × 3 matrix of panels in which each panel is an image consisting of varying numbers of objects with attributes like size, shape, and color. The figures in each row or column obey a common set of abstract rules. The last panel (in the third row and column), is missing and must be filled from a set of eight candidate panels so as to best complete the matrix according to the abstract rules. Formally, each RPM problem consists of 16 image panels X = {x i } 16 i=1 , in which the first 8 image panels are context images X c = {x i } 8 i=1 (i.e., all panels in the 3 × 3 problem matrix except the final blank panel), and the last 8 image panels are candidate answer images X a = {x i } 16 i=9 . The task is to select y, the index for the correct answer image. Published as a conference paper at ICLR 2023 3.2 OBJECT-CENTRIC ENCODER STSN employs slot attention (Locatello et al., 2020) to extract object-centric representations. Slot attention first performs some initial processing of the images using a convolutional encoder, producing a feature map, which is flattened to produce inputs ∈ R N ×Dinputs , where N = H × W (the height and width of the feature map), and D inputs is the number of channels. Then, the slots slots ∈ R K×D slot are initialized, to form a set of K slot embeddings, each with dimensionality D slot . We set the value of K to be equal to the maximum number of objects possible in a given image panel (based on the particular dataset). For each image, the slots are randomly initialized from a distribution N (µ, diag(σ)) ∈ R K×D slot with shared mean µ ∈ R D slot and variance σ ∈ R D slot (each of which are learned). The slots are then iteratively updated based on a transformer-style attention operation. Specifically, each slot emits a query q(slots) ∈ R K×D slot through a linear projection, and each location in the feature map emits a key k(inputs) ∈ R N ×D slot and value v(inputs) ∈ R N ×D slot . A dot product query-key attention operation followed by softmax is then used to generate the attention weights attn = softmax( 1 √ D slot k(inputs) · q(slots) ), and a weighted mean of the values updates = attn · v(inputs) is used to update the slot representations using a Gated Recurrent Unit (Cho et al., 2014), followed by a residual MLP with ReLU activations. More details can be found in Locatello et al. (2020). After T iterations of slot attention, the resulting slots are passed through a reasoning module, that we describe in the following section. In order to encourage the model to make use of slot attention in an object-centric manner, we also included a slot decoder to generate reconstructions of the original input images. To generate reconstructions, we first used a spatial broadcast decoder to generate both a reconstructed imagex k , and a mask m k , for each slot. We then generated a combined reconstruction, by normalizing the masks across slots using a softmax, and using the normalized masks to compute a weighted average of the slot-specific reconstructions. REASONING MODULE After object representations are extracted by slot attention, they are then passed to a transformer (Vaswani et al., 2017). For each candidate answer choice x a ∈ {x i } 16 i=9 , the transformer operates over the slots obtained from the 8 context images slots x1..8 , and the image for that answer choice slots xa . We flattened the slots over the dimensions representing the number of slots and images, such that, for each candidate answer, the transformer operated over flatten(slots x1..8 , slots xa ) ∈ R 9K×D slot . We then applied Temporal Context Normalization (TCN) (Webb et al., 2020), which has been shown to significantly improve out-of-distribution generalization in relational tasks, over the flattened sequence of slots. To give the model knowledge about which slot representation corresponded to which row and column of the matrix, we added a learnable linear projection R 6 → R D slot from one-hot encodings of the row and column indices (after applying TCN). We concatenated a learned CLS token (analogous to CLS token in Devlin et al. (2018)) of dimension D slot , before passing it through a transformer with L layers and H self-attention heads. The transformed value of the CLS token was passed through a linear output unit to generate a score for each candidate answer image, and the scores for all answers were passed through a softmax to generate a predictionŷ. OPTIMIZATION The entire model was trained end-to-end to optimize two objectives. First, we computed a reconstruction loss L recon , the mean squared error between the 16 image panels and their reconstructed outputs. Second, we computed a task loss L task , the cross entropy loss between the target answer index and the softmax-normalized scores for each of the candidate answers. These two losses were combined to form the final loss L = λ * L recon + L task , where λ is a hyperparameter that controls the relative strength of the reconstruction loss. Problems are governed by RPM-like problem structure, but with greater visual complexity (rendered using approach similar to CLEVR dataset (Johnson et al., 2017)). This particular problem is an example of the 'Location' problem type. The reader is encouraged to identify the correct answer, and rule for each attribute. EXPERIMENTS DATASETS PGM. The PGM dataset was introduced by Barrett et al. (2018), and consists of problems belonging to eight different regimes with different generalization difficulty. Each matrix problem in PGM is defined by the abstract structure S = {[r, o, a] : r ∈ R, o ∈ O, a ∈ A}, where R = {progression, XOR, AND, OR, consistent union} are the set of rules (note that 'consistent union' is also referred to as 'distribution-of-3'), O = {shape, line} are the set of objects, and A = {size, type, position, color, number} are the set of attributes. Each regime consists of 1.2M training problems, 20K validation problems, and 200K testing problems. Due to the enormous size of the dataset we focused on the neutral, interpolation, and extrapolation regimes. In the neutral regime, the training and test sets are sampled from the same underlying distribution, whereas the interpolation and extrapolation regimes both involve out-of-distribution generalization. Given the set of feature values for each attribute, the interpolation regime involves training on all even-indexed feature values and testing on all oddindexed values, and the extrapolation regime involves training on the lower half of feature values and testing on the upper half of feature values. More details can be found in Barrett et al. (2018). I-RAVEN. The RAVEN dataset was introduced by Zhang et al. (2019a), with problems belonging to seven different configurations. These configurations are defined by the spatial layout of the elements in each panel, ranging from low visual complexity (e.g., the 'Center' configuration, in which each panel contains just a single object in the center of the image), to high visual complexity (e.g., the 'O-IG' configuration, in which each panel contains an outer object surrounding an inner grid of objects). Some configurations have multiple components C to which separate rules can be bound. Thus, each problem in RAVEN is defined by the abstract structure S = {[r, c, a] : r ∈ R, c ∈ C, a ∈ A}, where R = {constant, progression, arithmetic, distribution-of-3} are the set of rules, C are the set of components (depending on the particular configuration), and A = {number, position, size, type, color} are the set of attributes. There are a total of 42K training problems, 14K validation problems, and 14K testing problems. We trained STSN jointly on all configurations in RAVEN. It was subsequently discovered that the original RAVEN dataset employed a biased method for generating candidate answers, that could be exploited so as to achieve near perfect performance by only viewing these candidate answers (i.e., ignoring the problem itself) (Hu et al., 2021). To address this, Hu et al. (2021) proposed the Impartial RAVEN (I-RAVEN) dataset, with an unbiased procedure for generating candidate answers. As with most recent work in this domain, we performed our evaluation on I-RAVEN. CLEVR-Matrices. We created a novel dataset of RPM-like problems using realistically rendered 3D shapes, based on source code from CLEVR (a popular visual-question-answering dataset) (Johnson et al., 2017). Problems were formed from objects of three shapes (cube, sphere, and cylinder), three sizes (small, medium, and large), and eight colors (gray, red, blue, green, brown, purple, cyan, and yellow). Objects were placed on a 3 × 3 grid of locations (such that there was a maximum of 9 objects in each panel), which was oriented randomly in each problem. Lighting was varied randomly between each panel, and objects were randomly assigned one of two textures (metal or rubber). Rules were independently sampled for shape, color, and size, from the set R = {null, constant, distribution-of-3}. Location was determined based on three different problem types. In the first problem type ('Logic'), locations were determined based on a logical rule sampled from R = {AND, OR, XOR}. In the second problem type ('Location'), locations were determined based on a rule sampled from R = {constant, distribution-of-3, progression}. In the third problem type ('Count'), the count of objects in each panel was determined based on a rule sampled from R = {constant, distribution-of-3, progression}, and locations were randomly sampled to instantiate that count. Example problems are shown in Figure 2 and Section A.5. Answer choices were generated using the attribute bisection tree algorithm proposed by Hu et al. (2021), which was used to generate answer choices for I-RAVEN. Our dataset thus does not contain the biases identified in the original RAVEN dataset. We generated 20K problems for each type, including 16K for training, 2K for validation, and 2K for testing. We trained STSN jointly on all three problems types. BASELINES We compared our model to several baselines, as detailed in Tables 1-3. To the best of our knowledge, these baselines include the current best performing models on the I-RAVEN and PGM benchmarks. We didn't use any auxiliary information (i.e., training to explicitly label the underlying rules), and hence for fair comparison we only compared to baselines that didn't use auxiliary loss. There are too many baselines to describe them each in detail, but here we briefly describe the best performing baselines. The baseline that achieved the best overall performance was the Scattering Compositional Learner (SCL) (Wu et al., 2020). SCL employs an approximate form of object segmentation based on fixed spatial locations in a convolutional feature map, followed by a dual parameter-sharing scheme, in which a shared MLP (shared across 'objects') is used to generate object embeddings, and another shared MLP (shared across attributes) is used to classify rules for each attribute. We also compare against the Multi-Layer Relation Network (MLRN) (Jahrens & Martinetz, 2020) and the Multi-scale Relation Network (MRNet) (Benny et al., 2021), both of which achieved strong results on PGM. MLRN builds on the Relation Network (Santoro et al., 2017), which uses a shared MLP to compute learned relation vectors for all pairwise comparisons of a set (in this case, the set of embeddings for all image panels in a problem). MLRN passes the output of one RN to another RN, thus allowing second-order relations to be modeled. MRNet creates image embeddings at different spatial scales, allowing it to approximate segmentation of larger vs. smaller objects, and then computes both row-wise and column-wise rule embeddings, which are aggregated across both rows/columns and spatial scales. EXPERIMENTAL DETAILS We give a detailed characterization of all hyperparameters and training details for our models in Section A.2. We employed both online image augmentations (random rotations, flips, and brightness changes) and dropout (in the transformer), when training on I-RAVEN (details in Section A.2). We also trained both SCL and MLRN on CLEVR-Matrices, and compared to two alternative versions of SCL on I-RAVEN, one that employed the same image augmentations, TCN, and dropout employed by our model, and another that combined SCL with slot attention (also with image augmentations, TCN and dropout) referred to as 'Slot-SCL' For I-RAVEN, to be consistent with previous work (Wu et al., 2020), we report results from the best out of 5 trained models. Similarly, for CLEVR-Matrices, we report results from the best out of 3 trained models for STSN, SCL, and MLRN. For PGM, we only trained 1 model on the neutral regime, 1 model on the interpolation regime, and 1 model on the extrapolation regime, due to the computational cost of training models on such a large dataset. For the PGM neutral regime, we pretrained the convolutional encoder, slot attention, and slot decoder on the reconstruction objective with the neutral training set, and fine-tuned while training on the primary task. For the PGM interpolation regime, all model components were trained end-to-end from scratch. For the the PGM extrapolation regime, we employed a simultaneous dual-training scheme, in which the convolutional encoder, slot attention, and slot decoder were trained on recon- Test Accuracy (%) Model Neutral Interpolation Extrapolation CNN+MLP (Barrett et al., 2018) 33.0 --CNN+LSTM (Barrett et al., 2018) 35.8 --ResNet-50 (Barrett et al., 2018) 42.0 --Wild-ResNet (Barrett et al., 2018) 48.0 --CoPINet (Zhang et al., 2019b) 56.4 --WReN (β = 0) ( Barrett et al., 2018) 62.6 64.4 17.2 VAE-WReN (Steenbrugge et al., 2018) 64.2 --MXGNet (β = 0) (Wang et al., 2020) 66.7 65.4 18.9 LEN (β = 0) (Zheng et al., 2019) 68.1 --DCNet (Zhuo & Kankanhalli, 2022) 68.6 59.7 17.8 T-LEN (β = 0) (Zheng et al., 2019) 70.3 --SRAN (Hu et al., 2021) 71.3 --Rel-Base (Spratley et al., 2020) 85.5 -22.1 SCL (Wu et al., 2020) 88.9 --MRNet (Benny et al., 2021) 93.4 68.1 19.2 MLRN (Jahrens & Martinetz, 2020) 98.0 57.8 14.9 STSN (ours) 98.2 78.5 20.4 struction for both the neutral and extrapolation training sets (thus giving these components of the model exposure to a broader range of shapes and feature values), while the transformer reasoning module was trained on the primary task using only the extrapolation training set. . The next best model on I-RAVEN, SCL (95%) performed worse on PGM (88.9%), perhaps due to its more limited objectencoding methods (PGM includes a large number of spatially overlapping objects). We evaluated the next best model on PGM, MLRN (98%), on I-RAVEN (using code from the authors' publicly available respository), and found that it displayed very poor performance (29.8%), suggesting that some aspect of its architecture may be overfit to the PGM dataset. Thus, STSN achieved a ∼ 5% increase in average performance across both of the two datasets relative to the next best overall model (97.0% average performance on PGM Neutral and I-RAVEN for STSN vs. 92.0% for SCL), despite incorporating fewer problem-specific inductive biases. RESULTS To further investigate the utility of STSN's object-centric encoding mechanism, we evaluated STSN, SCL, and MLRN on our newly developed CLEVR-Matrices dataset (Table 3). STSN displayed very strong performance (99.6% average test accuracy), whereas both SCL (70.5% average test accuracy) and MLRN (30.8% average test accuracy) performed considerably worse. This is likely due to the fact that these models lack a precise object-centric encoding mechanism, and were not able to cope with the increased visual complexity of this dataset. Finally, we also evaluated both STSN and SCL on a dataset involving analogies between feature dimensions (e.g., a progression rule applied to color in one row, and size in another row) (Hill et al., 2019). STSN outperformed SCL on this dataset as well (Table 12), likely due to the fact that SCL assumes that rules will be applied independently within each feature dimension. This result highlights the limitation of employing inductive biases that are overly specific to certain datasets. ABLATION STUDY We analyzed the importance of the different components of STSN in ablation studies using the I-RAVEN dataset (Table 4). For I-RAVEN, our primary STSN implementation employed dropout, which we found yielded a modest improvement in generalization, but our ablation studies were performed without dropout. Thus, the relevant baseline for evaluating the isolated effect of each ablation is the version of STSN without dropout. First, we removed the slot attention module from STSN, by averaging the value embeddings from the input feature vectors over the image space (i.e., using only a single slot per panel). The average test accuracy decreased by more than 20%, suggesting that object-centric representations play a critical role in the model's performance. The effect was particularly pronounced in the 'O-IG' (a large outer object surrounding an inner grid of smaller objects) and '3Grid' (a 3 × 3 grid of objects) configurations, likely due to the large number of objects per panel in these problems. Next, we performed an ablation on TCN, resulting in a test accuracy decrease of around 7%, in line with previous findings demonstrating a role of TCN in improved generalization (Webb et al., 2020). We also performed an ablation on the size of the reasoning module, finding that a smaller transformer (L = 4 layers) did not perform as well. Finally, we performed an ablation on the image augmentations performed during training, resulting in a test accuracy decrease of more than 3%, suggesting that the augmentations also helped to improve generalization. Overall, these results show that the use of object-centric representations was the most important factor explaining STSN's performance on this task. Figure 3: Slot-specific reconstructions generated by STSN. 3 problems were chosen at random from the PGM neutral test set. The first two images for each problem show the original image and the combined reconstruction. The following images show the slot-specific reconstruction for each of the slots. In general, STSN's slot attention module implemented a nearly perfect objectbased segmentation of its input images, despite receiving no veridical segmentation information during training or test. STSN used 16 slots per image for this dataset, but generally left the slots not assigned to objects unused. Only 8 slots are pictured for these example problems since the remaining slot-specific reconstructions were completely blank. VISUALIZATION OF OBJECT MASKS We also visually inspected the attention behavior of STSN's slot attention module (Figure 3). We found that STSN's slot-specific reconstructions conformed nearly perfectly to the individual objects in the image panels of PGM, with the remaining slots left unused. This confirms that STSN was engaged in object-centric processing. We also evaluated STSN on I-RAVEN with a range of values for λ (the parameter that governs the relative emphasis placed on the reconstruction loss), and found that with lower values of λ, STSN's reconstructions were no longer object-centric. With a value of λ = 100, STSN's reconstructions were blurrier, and multiple objects tended to be combined into a single slot ( Figure 5 in Section A.4). With a value of λ = 1, STSN's reconstructions completely failed to capture the content of the original image ( Figure 6). Interestingly, these changes in reconstruction quality were mirrored by changes in performance on the reasoning task, with an average test accuracy of 90.1% for λ = 100 and 74.2% for λ = 1 (relative to 95.7% for λ = 1000, Figure 4). This is consistent with our hypothesis that encouraging high-quality reconstructions (through a sufficiently high weight on L recon ) would encourage object-centric encoding behavior, which would in turn promote more generalizable visual reasoning strategies. Thus, for STSN to fully exploit its object-centric encoding mechanisms, it is important to use a high enough value of λ so as to ensure high-quality reconstructions. CONCLUSION AND FUTURE DIRECTIONS We have presented a simple, general-purpose visual reasoning model, organized around the principle of object-centric processing. Our proposed model, STSN, displayed state-of-the-art performance on both of two challenging visual reasoning benchmarks, PGM and I-RAVEN, as well a novel reasoning benchmark with greater visual complexity, CLEVR-Matrices, despite the relative lack of problemspecific inductive biases. These results suggest that object-centric processing is a powerful inductive bias for abstract visual reasoning problems such as RPM. Some previous work has proposed novel relational inductive biases for the purposes of achieving strong out-of-distribution generalization in visual reasoning problems (Webb et al., 2021;Zhang et al., 2021;Kerg et al., 2022). This work has often assumed (i.e., hand-coded) object-centric representations. We view our approach as complementary with these previous approaches, and suggest that a fruitful avenue for future work will be to pursue the integration of object-centric and relational inductive biases. A APPENDIX A.1 CODE AND DATA AVAILABILITY All code can be downloaded from https://github.com/Shanka123/STSN. The CLEVR-Matrices dataset can be downloaded from https://dataspace.princeton.edu/handle/88435/dsp01fq977z011. A.2 HYPERPARAMETERS AND TRAINING DETAILS Images were resized to 80 × 80 for I-RAVEN and PGM, and 128 × 128 for CLEVR-Matrices. Pixels were normalized to the range [0, 1]. For I-RAVEN, we also applied random online augmentations during training, including horizontal and vertical flips, rotations by multiples of 90 • , and brightness changes by a factor in the range [0.5,1.5]. Note that we applied the same augmentations to all the image panels in a given problem, so that the abstract rule remained the same. Table 5 describes the hyperparameters for the convolutional encoder used on the I-RAVEN and PGM datasets, and Table 6 describes the hyperparameters used on the CLEVR-Matrices dataset. These encoders employed a positional embedding scheme consisting of 4 channels, each of which coded for the position of a pixel along one of the 4 cardinal directions (top→bottom, bottom→top, left→right, right→left), normalized to the range [0, 1]. These positional embeddings were projected through a fully-connected layer to match the number of channels in the convolutional feature maps, and then added to these feature maps. Feature maps were flattened along the spatial dimension, followed by layer normalization (Ba et al., 2016), and two 1D convolutional layers. For slot attention, we used K = 9 slots for I-RAVEN and CLEVR-Matrices, and K = 16 slots for PGM. The number of slot attention iterations was set to T = 3, and the dimensionality of the slots was set to D slot = 32 for I-RAVEN and PGM, and D slot = 64 for CLEVR-Matrices. The GRU had a hidden layer of size D slot . The residual MLP had a single hidden layer with a ReLU activation, followed by a linear output layer, both of size D slot . Table 7 describes the hyperparameters for the slot decoder used on the I-RAVEN and PGM datasets, and Table 8 describes the hyperparameters used on the CLEVR-Matrices dataset. Each of the K slots was passed through the decoder, yielding a slot-specific reconstructed imagex k and mask m k . Each slot was first converted to a feature map using a spatial broadcast operation , in which the slot was tiled to match the spatial dimensions of the original input image (80 × 80 for I-RAVEN and PGM, 128×128 for CLEVR-Matrices). Positional embeddings were then added (using the same positional embedding scheme as in the encoder), and the resulting feature map was passed through a series of convolutional layers. The output of the decoder had 2 channels for I-RAVEN and PGM, and 4 channels for CLEVR-Matrices. One of these channels corresponded to the mask m k , and the other channels corresponded to the reconstructed imagex k (grayscale for I-RAVEN and PGM, RGB for CLEVR-Matrices). The slot-specific reconstructions were combined by applying a softmax to the masks (over slots) and computing a weighted average of the reconstructions, weighted by the masks. Table 9 gives the hyperparameters for the transformer reasoning module, and Table 10 gives the training details for all datasets. We used a reconstruction loss weight of λ = 1000 for all datasets. We used the ADAM optimizer (Kingma & Ba, 2014) and all experiments were performed using the Pytorch library (Paszke et al., 2017). Hardware specifications are described in Table 11. To make the comparison between STSN and SCL as fair as possible, we trained a version of SCL on I-RAVEN for 500 epochs using image augmentations (the same as used for STSN), TCN, and I-RAVEN 1 A100, 40GB RAM PGM-Neutral 6 A100, 40GB RAM PGM-Interpolation 6 A100, 40GB RAM PGM-Extrapolation 6 A100, 40GB RAM CLEVR-Matrices 8 A100, 80GB RAM dropout. TCN was applied following the last feedforward residual block of the first scattering transformation N a . Dropout with a probability of 0.1 was applied during training to all layers in the second scattering transformation N r . We also compared STSN to a hybrid model that combined SCL and slot attention, termed 'Slot-SCL'. For this model, we replaced SCL's object and attribute transformations (N o and N a ) with the slot attention module, using the same hyperparameters as STSN. The outputs of slot attention were concatenated and passed to SCL's rule module (N r ). This model also employed the same image augmentations, dropout, and TCN as our model. The model was trained for 400 epochs. The results for all comparisons with SCL on I-RAVEN reflect the best out of 5 trained models. A.3 ANALOGIES BETWEEN FEATURE DIMENSIONS Figure 1 : 1Slot Transformer Scoring Network (STSN) ; Steenbrugge et al. (2018); Van Steenkiste et al. (2019); Zhang et al. (2019b); Zheng et al. (2019); Spratley et al. (2020); Jahrens & Martinetz (2020); Wang et al. (2020); Wu et al. (2020); Benny et al. (2021); Hu et al. (2021); Figure 2 : 2Example problem from our proposed CLEVR-Matrices dataset. Figure 4 :Figure 5 :Figure 6 : 456Slot-specific reconstructions generated by STSN for λ = 1000 on I-RAVEN. Slot-specific reconstructions generated by STSN for λ = 100 on I-RAVEN. Slot-specific reconstructions generated by STSN for λ = 1 on I-RAVEN.A.5 CLEVR-MATRICES EXAMPLESFigures 7-12 show some additional example problems from the CLEVR-Matrices dataset, along with annotations describing their problem type and the rules for each attribute. Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 : 789101112Problem type: location. Rules: [color: constant], [shape: null], [size: distribution-of-Problem type: location. Rules: [color: constant], [shape: constant], [size: distribution-Problem type: logic. Rules: [color: distribution-of-3], [shape: distribution-of-3], [size: distribution-of-3], [location: AND]. Problem type: logic. Rules: [color: distribution-of-3], [shape: null], [size: distributionof-3], [location: XOR]. Problem type: count. Rules: [color: constant], [shape: constant], [size: constant], Problem type: count. Rules: [color: distribution-of-3], [shape: distribution-of-3], [size: distribution-of-3], [count: progression]. Table 1 : 1Results on I-RAVEN.Test Accuracy (%) Table 2 : 2Results on PGM. Table 3 : 3Results on CLEVR-Matrices.Test Accuracy (%) Model Average Logic Location Count MLRN (Jahrens & Martinetz, 2020) 30.8 47.4 21.4 23.6 SCL (Wu et al., 2020) 70.5 80.9 65.8 64.9 STSN (ours) 99.6 99.2 100.0 99.6 Table 1 shows the results on the I-RAVEN dataset. STSN achieved state-of-the-art accuracy when averaging across all configurations (95.7%), and on two out of seven configurations ('U-D' and 'O- Table 4 : 4Ablation study on the I-RAVEN dataset.Test Accuracy (%) Table 5 : 5CNN Encoder for I-RAVEN and PGM.Type Channels Activation Kernel Size Stride Padding 2D Conv 32 ReLU 5 × 5 1 2 2D Conv 32 ReLU 5 × 5 1 2 2D Conv 32 ReLU 5 × 5 1 2 2D Conv 32 ReLU 5 × 5 1 2 Position Embedding - - - - - Flatten - - - - - Layer Norm - - - - - 1D Conv 32 ReLU 1 1 0 1D Conv 32 - 1 1 0 Table 6 : 6CNN Encoder for CLEVR-Matrices.Type Channels Activation Kernel Size Stride Padding 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 Position Embedding - - - - - Flatten - - - - - Layer Norm - - - - - 1D Conv 64 ReLU 1 1 0 1D Conv 64 - 1 1 0 Table 7 : 7Slot Decoder for I-RAVEN and PGM.Type Channels Activation Kernel Size Stride Padding Spatial Broadcast - - - - - Position Embedding - - - - - 2D Conv 32 ReLU 5 × 5 1 2 2D Conv 32 ReLU 5 × 5 1 2 2D Conv 32 ReLU 5 × 5 1 2 2D Conv 2 - 3 × 3 1 1 Table 8 : 8Slot Decoder for CLEVR-Matrices.Type Channels Activation Kernel Size Stride Padding Spatial Broadcast - - - - - Position Embedding - - - - - 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 64 ReLU 5 × 5 1 2 2D Conv 4 - 3 × 3 1 1 Table 9 : 9Hyperparameters for Transformer Reasoning Module. H is the number of heads, L is the number of layers, D head is the dimensionality of each head, and D M LP is the dimensionality of the MLP hidden layer.I-RAVEN PGM CLEVR-Matrices Neutral Interpolation Extrapolation H 8 8 8 8 8 L 6 24 24 6 24 D head 32 32 32 32 32 D M LP 512 512 512 512 512 Dropout 0.1 0 0 0 0 Table 10 : 10Training details for all datasets.I-RAVEN PGM CLEVR-Matrices Neutral Interpolation Extrapolation Batch size 16 96 96 96 64 Learning rate 4e − 4 8e − 5 8e − 5 8e − 5 8e − 5 LR warmup steps 75k 10k 10k 10k 10k Epochs 500 161 83 71 200 Table 11 : 11Hardware specifications for all datasets. 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[ "Energy-Efficient Building HVAC Control Using Hybrid System LBMPC", "Energy-Efficient Building HVAC Control Using Hybrid System LBMPC" ]	[ "Anil Aswani \nElectrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA\n", "Neal Master [email protected] \nElectrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA\n", "Jay Taneja [email protected] \nElectrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA\n", "Andrew Krioukov [email protected] \nElectrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA\n", "David Culler [email protected] \nElectrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA\n", "Claire Tomlin [email protected] \nElectrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA\n" ]	[ "Electrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA", "Electrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA", "Electrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA", "Electrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA", "Electrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA", "Electrical Engineering and Computer Sciences\nUniversity of California\n94720BerkeleyCAUSA" ]	[]	Improving the energy-efficiency of heating, ventilation, and air-conditioning (HVAC) systems has the potential to realize large economic and societal benefits. This paper concerns the system identification of a hybrid system model of a building-wide HVAC system and its subsequent control using a hybrid system formulation of learning-based model predictive control (LBMPC). Here, the learning refers to model updates to the hybrid system model that incorporate the heating effects due to occupancy, solar effects, outside air temperature (OAT), and equipment, in addition to integrator dynamics inherently present in low-level control. Though we make significant modeling simplifications, our corresponding controller that uses this model is able to experimentally achieve a large reduction in energy usage without any degradations in occupant comfort. It is in this way that we justify the modeling simplifications that we have made. We conclude by presenting results from experiments on our building HVAC testbed, which show an average of 1.5MWh of energy savings per day (p = 0.002) with a 95% confidence interval of 1.0MWh to 2.1MWh of energy savings.	10.3182/20120823-5-nl-3013.00069	[ "https://arxiv.org/pdf/1204.4717v1.pdf" ]	1,611,278	1204.4717	a6d4988acf7f4ca6fa44a542ba380d110216e163	Energy-Efficient Building HVAC Control Using Hybrid System LBMPC Anil Aswani Electrical Engineering and Computer Sciences University of California 94720BerkeleyCAUSA Neal Master [email protected] Electrical Engineering and Computer Sciences University of California 94720BerkeleyCAUSA Jay Taneja [email protected] Electrical Engineering and Computer Sciences University of California 94720BerkeleyCAUSA Andrew Krioukov [email protected] Electrical Engineering and Computer Sciences University of California 94720BerkeleyCAUSA David Culler [email protected] Electrical Engineering and Computer Sciences University of California 94720BerkeleyCAUSA Claire Tomlin [email protected] Electrical Engineering and Computer Sciences University of California 94720BerkeleyCAUSA Energy-Efficient Building HVAC Control Using Hybrid System LBMPC Model-based controladaptive controlmethodologyevaluation Improving the energy-efficiency of heating, ventilation, and air-conditioning (HVAC) systems has the potential to realize large economic and societal benefits. This paper concerns the system identification of a hybrid system model of a building-wide HVAC system and its subsequent control using a hybrid system formulation of learning-based model predictive control (LBMPC). Here, the learning refers to model updates to the hybrid system model that incorporate the heating effects due to occupancy, solar effects, outside air temperature (OAT), and equipment, in addition to integrator dynamics inherently present in low-level control. Though we make significant modeling simplifications, our corresponding controller that uses this model is able to experimentally achieve a large reduction in energy usage without any degradations in occupant comfort. It is in this way that we justify the modeling simplifications that we have made. We conclude by presenting results from experiments on our building HVAC testbed, which show an average of 1.5MWh of energy savings per day (p = 0.002) with a 95% confidence interval of 1.0MWh to 2.1MWh of energy savings. INTRODUCTION Nearly 10% of greenhouse gas emissions and 25% of the electricity used in the United States is due to heating, ventilation, and air-conditioning (HVAC) systems in buildings (U.S. DoE, 2009;McQuade, 2009). This has driven research into better control methods (e.g., (Nghiem and Pappas, 2011;Kelman and Borrelli, 2011;Oldewurtel et al., 2012;Liao et al., 2012;Aswani et al., 2011b;Ma et al., 2012)) that can help mitigate the negative externalities due to the large energy consumption of HVAC, while still ensuring the comfort of building occupants. But the heterogeneity of HVAC equipment with respect to their physical modalities makes it difficult to develop a control design methodology that scales to many types of equipment. Even when new HVAC controllers are designed, experimentally comparing their efficiency and comfort in relation to existing controllers is difficult because of the large temporal variability in weather and occupancy conditions. Identifying energy models of HVAC equipment can be difficult because some equipment is designed to operate most efficiently at certain temperatures or settings, which requires extensive measurement to characterize. Moreover, This material is based upon work supported by the National Science Foundation under Grants CNS-0931843 (CPS-ActionWebs) and CNS-0932209 (CPS-LoCal). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the National Science Foundation. not all buildings have equipment to directly measure the energy consumption of only HVAC equipment. Siroky et al. (2011) showed that a linear model predictive controller (MPC) could provide a 10-20% reduction in energy usage of a ceiling radiant heating system, as compared to the default, manufacturer-provided controller. The temperature difference between a water set point and the outside air temperature (OAT) was used to approximate the energy usage of each controller. Aswani et al. (2011b) designed a new controller for an air-conditioner on a single-room testbed, which achieved up to 30% savings on warm days and up to 70% savings on cool days. Mathematical models of the temperature dynamics of different control schemes and their energy characteristics were constructed in order to allow comparisons between different controllers of experimentally measured HVAC energy usage to simulations over identical weather and occupancy conditions. A 20% improvement in performance of thermal storage for campus-wide building cooling was achieved by Ma et al. (2012) using better control methods. Direct energy measurements of the equipment along with a regression model of baseline performance were used to compare controllers. This analysis approach requires direct measurements, because it can only statistically distinguish large differences. and is part of our BRITE-S testbed. Experiments with HVAC Controllers Overview This paper describes our design methodology for an energy-efficient controller of a building-wide HVAC system that is able to maintain comfortable occupant conditions. We begin by describing our HVAC testbed, before explaining the modeling procedure (cf. (Aswani et al., 2012b)) that was used to identify the thermal dynamics of the building and the HVAC system. Next, we describe a hybrid system (Tomlin et al., 2000) version of learning-based model predictive control (LBMPC) (Aswani et al., 2011a). LBMPC is a robust form of adaptive MPC. Compared to linear parameter-varying MPC (Kothare et al., 2000;Falcone et al., 2008), LBMPC differs in that it provides robustness to model changes using tube MPC (e.g., (Chisci et al., 2001)). Furthermore, the robust, adaptive MPC in (Fukushima et al., 2007;Adetola and Guay, 2011) use an adaptive model with an uncertainty measure to ensure robustness, while LBMPC uses an adaptive model to improve performance and a nominal model with an uncertainty measure to provide robustness. We conclude by experimentally comparing, on a buildingwide HVAC system, our hybrid system LBMPC controller to the default controller. This is done using the comparison methodology described in (Aswani et al., 2012a), which uses nonparametric methods that can compute and compare quantitative metrics of energy usage and occupant comfort for different HVAC controllers. BRITE-S TESTBED The Berkeley Retrofitted and Inexpensive HVAC Testbed for Energy Efficiency in Sutardja Dai Hall (BRITE-S) platform (Krioukov et al., 2011;Aswani et al., 2012b) is a building-wide HVAC system that maintains the indoor environment of a 141,000-square-foot building, shown in Fig. 1, that is divided between a four-floor nanofabrication laboratory (NanoLab) and seven floors of general space (including office space, classrooms, and a coffee shop). The building automation equipment can be measured and actuated through a BACnet protocol interface. The HVAC system uses a 650-ton chiller to cool water. Airhandler units (AHUs) with variable-frequency drive fans distribute air cooled by the water to variable air volume (VAV) boxes throughout the building. Since the NanoLab must operate within tight tolerances, our control design can only modify the operation of the general space AHUs and VAV boxes, with no modification of chiller settings that are shared between the NanoLab and general space. The default, manufacturer-provided controller in BRITE-S uses PID loops to actuate the VAV boxes and keeps a constant supply air temperature (SAT) within the AHUs. Conventional SAT reset control is not possible because several VAV boxes provide maximum air ow rates throughout the day for nearly the entire range of SATs. These zones are often dominated by heating from computer equipment. IDENTIFYING THERMAL DYNAMICS For the purpose of modeling and control, we will assume that data is sampled every 15 minutes; angle brackets (i.e., · ) denote measurements sampled at this rate. Let T s k and T o k be the SAT and OAT, respectively, at time k. Similarly, T j k is the temperature in the j-th zone of the building at time k, for j = 1, . . . , Z zones. The VAV box in the j-th zone controls the zone temperature T j by modulating the amount of cool air sent to the zone F j k and the amount that the air is reheated R j k . In general, the thermal dynamics of each zone are T j k + 1 = f j T 1 k , . . . , T Z k , F 1 k , . . . , F Z k , R 1 k , . . . , R Z k , T o k , T s k , O, X ,(1) where f j (·) is some unknown nonlinear function, O are variables related to occupancy, and X are variables related to other effects like the use of equipment, solar heating, etc. Some simplifying assumptions are typically made by (a) considering the physical adjacency of different zones (Kelman and Borrelli, 2011;Oldewurtel et al., 2012;Liao et al., 2012), and (b) assuming that the effect of occupancy and other factors enters additively into the dynamics (Aswani et al., 2011b(Aswani et al., , 2012b. After these assumptions, the model is T j k + 1 = f j T n1 k , . . . , T nq k , F j k , R j k , T o k , T s k + O + X,(2) where {n 1 , . . . , n q } is the set of zones adjacent to j. Additional assumptions allow further modeling simplifications. We assume that the zone temperatures do not vary significantly throughout the day, since their temperature is in principle being controlled by the HVAC system. Furthermore, we assume that the additive influence of the occupancy and other effects can be modeled by a single term q j k . Even after making these assumptions, the model to be identified is nonlinear since the SAT T s affects the thermal dynamics in a bilinear form (Kelman and Borrelli, 2011;Oldewurtel et al., 2012). We take a hybrid system approach by forcing the SATs to belong to a finite set of values M = {T s1 , . . . , T sp }, where p is the number of modes (Aswani et al., 2012b). This allows us to consider multiple linearizations of (2). In our application to the BRITE-S testbed, we took p = 3 and M = {52 • F, 58 • F, 62 • F}. For the m i -th mode (for m i ∈ {1, . . . , p}) of fixed SAT T sm i , the model is given by T j k + 1 = a mi n1 T n1 k + . . . a mi nq T nq k + b mi j F j k + c mi j R j k + d mi j T o k + q j k , (3) where the coefficients are unknown scalars and q j is an unknown function of time k. The purpose of the system identification is to compute these unknown values. Experiments for System Identification In (Aswani et al., 2012b), we used one week of data with a fixed SAT to identify a model. Identifying a hybrid system model where the SAT is able to change is more challenging, because identifying a model of equal fidelity would require three weeks (since we have three modes) of experimental data. As a result, we modified our modeling approach. We conducted experiments in which a small amount of data was gathered by cycling through all of the SATs in M so as to cover all of the modes of our hybrid system. Our experiment was as follows: Starting at midnight, we set the SAT to T s1 = 52 • F for two hours. We next set the SAT to T s2 = 58 • F for two hours. After this, the SAT was changed to T s3 = 64 • F for two hours. During these six, consecutive hours, the other HVAC configuration was kept fixed. The reason for picking a relatively quick horizon for all three experiments is that the heating load is roughly constant over a short time span. Initial Parameter Identification We used a small amount of training data to construct an approximate initial model that was used to do control. To improve the controller performance, we have the option to re-identify the model using the semiparametric regression approach discussed in (Aswani et al., 2012b). The approximate initial model was constructed as follows: We begin by making additional modeling simplifications. Specifically, the exogenous heating load term q j k was changed to also include the effects of OAT and adjacent zone temperatures. For modeling purposes, we assume that the heating load term does not change significantly over a short time. This was ensured by conducting the experiments for modeling within a quick time span. We know a priori that this last assumption is only approximate, but it serves to provide an initial model for which additional measurements can then be used to improve it. Suppose we have (a) measurements for the m i -th mode at times k such that L mi ≤ k ≤ U mi ; and (b) prior distributions for the coefficients a mi j ∼ N (a mi j ,ã mi j ), b mi j ∼ N (b mi j ,b mi j ), c mi j ∼ N (c mi j ,c mi j ), where the notation N (µ, Σ) denotes a jointly Gaussian random variable with mean µ and covariance Σ. Our initial model is given by T j k + 1 = a mi j T j k + b mi j F j k + c mi j R j k + q j k ,(4) and the coefficients can be identified by solving the following Bayesian, constrained least squares problem min p mi=1 Um i −1 k=Lm i (T j k + 1 − a mi j T j k (5) − b mi j F j k − c mi j R j k − q j k ) 2 + (a mi j − a mi j ) 2 /ã mi j + (b mi j − b mi j ) 2 /b mi j + (c mi j − c mi j ) 2 /c mi j s.t. a r j = a s j , ∀r, s ∈ {1, . . . , p} (6) b r+1 j < (T sr+1 /T sr ) · b r j , ∀r ∈ {1, . . . , p − 1} (7) c r j = c s j , ∀r, s ∈ {1, . . . , p} (8) q j k = q j q , ∀k, q ∈ r∈{1,...,p} [L r , U r ](9) The constraints in the optimization problem reflect constraints between different hybrid modes of the HVAC system. Constraint (6) ensures that the time-constants of the thermal dynamics are constant for each mode of the hybrid system, while constraint (7) encodes the fact that cooler temperatures will provide greater amounts of cooling in each zone. Constraint (8) reflects that the reheating capability of each VAV box is relatively constant across different SATs. Lastly, constraint (9) represents the approximation that the occupancy is fixed over a short window of time. Modeling the VAV Box Control Each VAV box uses air flow F j and reheat amount R j to modulate the temperature of the j-th zone. The VAV boxes use a PID controller (note that the use of a PID controller in zone control is typical for building HVAC systems (Honeywell, 1997)), and so we approximate this as a proportional controller. This is not restrictive because the "learning" portion of our LBMPC controller can compensate for the unmodeled integrator portion of this control. We will discuss this in the next section. Let e j k = T j k − S j k be the difference between the zone temperature and the temperature set point for the j-th zone (S j ). Then, we use the following model for the control of reheat amount R j k =    100, if e j k < −1 • F −100 · e j k , if − 1 • F ≤ e j k < 0 • F 0, if e j k ≥ 0 • F .(10) We assumed that each VAV box has the same controller, and so this controller model was identified by fitting a piecewise linear model to the observed data points (e j k , R j k ) over all zones. A comparison of the data points for a single zone and the fitted model is shown in Fig. 2. A similar model is used for the air flow amount F j k =    α j , if e j k < 0 • F (ω j − α j ) · e j k + α j , if 0 • F ≤ e j k < 1 • F ω j , if e j k ≥ 1 • F ,(11) where α j and ω j are the minimum and maximum amount of air flow allowed in the j-th zone. These values are configured by the building manager. THE HYBRID SYSTEM LBMPC CONTROLLER Several inputs can be used for control: The reheat R j and air flow F j in each zone can be explicitly actuated, or they Fig. 2. A scatter plot of data taken from a single zone of temperature error and reheat amount in percentage is shown, along with the model (solid line) that we use for the VAV box controller. The discrepancy is largely due to the integrator term that we leave unmodeled; our approach is to "learn" the value of the integrator term as we are doing control. can be implicitly actuated by varying the set point S j . Here, we only actuate the SAT; our methodology can be extended to also utilize the other inputs. For the sake of argument, suppose that the sequence of SATs is fixed over a horizon of length N and starting at time k = 1. So if the mode sequence of SATs is M = m 1 , . . . , m N , then the corresponding SATs are T s 1 = T sm 1 , . . . , T s N = T sm N . Consequently, the temperature in each zone at time k can be computed by just "running" or simulating the model forward in time; no optimization is needed to compute the values of F j ,R j ,T j , etc. As a result, minimizing our cost function subject to the thermal dynamics and system constraints requires optimization over only the set of possible sequences. In other words, our optimization problem is an integer program. Fortunately, there is a constraint that greatly reduces the computational complexity: The SAT can only change once every hour; the reason for this constraint is that large, frequent changes to the SAT can damage the HVAC equipment. So if the horizon is N = 16, which corresponds to four hours when sampling at 15 minute intervals, then there are 3621 different combinations. In order to further simplify the computational complexity, we make use of a heuristic that further reduces the computation. We specify that the mode is fixed over the span of every hour. For our setup -where we have three modes and four possible mode changes -this means that we have to compute the cost for 3 4 = 81 different combinations. This is a low number of combinations that can be computed under one second on a desktop computer because, as mentioned earlier, we do not need to optimize over other variables. For future extensions where other variables are used to do control, we would only need to solve 81 convex optimization problems (specifically quadratic programs), which can be reasonably solved. Energy Modeling Before we present the optimization formulation of the controller, we provide some intuition into the form of the cost function we use. A building HVAC system typically has several individual pieces of equipment that contribute to the overall energy usage. Within BRITE-S, most of the energy consumption is due to three elements: the fans in the AHU, the chiller that cools water and indirectly cools air, and the reheating that occurs each zone's VAV box. Even though parameterizations of the energy usage of the equipment are known (Kelman and Borrelli, 2011;Oldewurtel et al., 2012), modeling these features is difficult because individual energy measurements are usually not available. There is another subtle point regarding the energy models and their relationship to our cost function. Let E 1 (·), E 2 (·), E 3 (·) be the energy due to fans, chiller, and reheating. One possible cost is j (T j −S j ) 2 +λE 1 (·)+ µE 2 (·) + γE 3 (·), where λ, µ, γ ∈ R are constants. There are two reasons for weighting energy usage based on type: First, the energy usage of the equipment is often related to the mechanical and physical stress on the equipment, and so differential weighting allows a finer level of regulation with respect to such considerations. Second, the energy usage of equipment sometimes comes from different energy sources. For example, heating is provided by steam while the fans are powered by electricity in BRITE-S. Because the cost function weights each energy usage function, we really only need to know the energy model up to a constant, unknown scaling factor, because this gets subsumed into the scaling in the cost function. This simplifies the energy modeling that we do for the purpose of control. We use the following energy models: The fan energy usage is E 1 ∼ ( j F j ) 3 and the chiller energy is Kelman and Borrelli, 2011;Oldewurtel et al., 2012), while the reheat energy is E 3 ∼ j R j . E 2 ∼ (T o − T s ) · j F j ( Optimization Formulation We can now present the optimization formulation of the hybrid system LBMPC controller. The basic intuition behind LBMPC (Aswani et al., 2011a) is that two models of the system are kept. The first is a nominal model that is used with respect to the constraints, and the second model is used in the cost function and is updated using data gathered during control. This maintains robustness while improving performance through model updates. Without loss of generality, we assume that the control action is being computed for time k = 1; also recall that we do control at a rate of every 15 minutes. If T j 1 , R j 1 , F j 1 are the predictions of the linear model from time k = 0, then letq j i = T j 1 − T j 1 ,f j i = F j 1 − F j 1 ,r j i = R j 1 − R j 1 . The intuition is that these terms with hats represent corrections to the predictions of the MPC and provide the adaptation inherent in the controller (Aswani et al., 2011b). Specifically,q j represents an estimate of the heating load due to occupants, weather, solar heating, and equipment;f j represents the integrator term in the PID control of air flow in the j-th VAV box; andr j represents the integrator term in the PID control of reheat amount in the j-th VAV box. The control action is given by the minimizer to min m1,...,m N N i=1 j (T j i + 1 − S j i + 1 ) 2 (12) + λ( jF j i ) 3 + γ jR j i + µ(T o i − T s i ) · jF j i s.t. (4), (10), (11) F j = F j +f j ;R j = R j +r j T j i + 1 = a mi jT j i + b mi jF j i + c mi jR j i +q j i T s i = T mi T s 4q + r = T s 4q + s , ∀r, s ∈ {1, . . . , 4} (13) ∧ q ∈ {0, . . . , N/4 − 1} T j i ∈ [66 • F, 78 • F], ∀j ∈ {1, . . . , Z} Note that constraint (13) allows the SAT T s to switch value only once an hour, which reduces the computational complexity of the controller as discussed previously. A desktop computer took an average of under one second to solve this optimization problem for BRITE-S. Furthermore, we used values of λ = 6.7e4/( j α j ) 3 , µ = 1.3e−3, and γ = 6.7. These values were picked using an iterative process in which (a) the control was computed but not used for actuation; (b) the control and predictions were analyzed for if SAT stayed at higher values, the reheat amount was low, and the air flow remained at moderate levels; (c) the coefficients λ, µ, γ were changed and the process starting at (a) was repeated until (b) was satisfied. MEASURING EFFICIENCY Separate energy measurements of the general space HVAC system are not available in BRITE-S. Instead, we only have access to measurements of both the general space HVAC and the NanoLab HVAC. We denote these energy measurements as E[i] for i = 1, . . . , D, where the index i is over hourly intervals. Furthermore, we have measurements of the OAT that correspond to the energy usage measurements: T o [i] for i = 1, . . . , D. The general model describing the relationship between energy usage, OAT, occupancy O, and other factors (e.g., solar effects, equipment, etc.) X is E = f (T o , O, X), where f (·, ·, ·) is an unknown, nonlinear relationship. But because occupancy and other factors are generally not directly measured, it is typically only possible to consider the energy usage averaged over occupancy and other factors E o,x (T o ) = E f (T o , O, X) T o . This can be estimated using nonparametric regression (Györfi, 2002), and it is a curve that describes the relationship between the OAT and the average energy consumption. Intuitively, if the data points ( The average amount of energy used in one hour is therefore E g = E g E o,x (T o ) = E g E f (T o , O, X) T o , where g(T o ) is a probability distribution of OATs. This notation allows us to define the average amount of energy used in one day as E day = 24 i=1 E gi , where g i (T o ) is the probability distribution of OAT during the i-th hour of the day. For simplicity, we assume a uniform distribution for the OAT. This allows us to compare the energy efficiency of two controllers. We can compute the quantities defined above for each controller, and a nonparametric methodology (Vilar-Fernndez et al., 2007) can be used to determine whether the differences are statistically significant. MEASURING OCCUPANT COMFORT In order to define a quantification of comfort that is both tractable and will scale to many buildings, we focus on a measure that is only dependent on temperature. We assume that the average temperature for the j-th zone is measured at hourly intervals T j [i] for i = 1, . . . , D. Let (x) + be the thresholding function, which is defined so that (x) + = 0 if x < 0 and (x) + = x otherwise. We define our quantification of comfort using soft thresholding as C = 1/Z · Z j=1 1 0 (\|T j − S j \| − B j ) + dt, where the integral with respect to dt is over one hour of time, S j is the set point of the zone, and B j is the amount of temperature deviation for which the building has been configured. The intuition is that this quantity increases whenever T j exceeds S j by more than B j , and the amount of increase in this quantity is proportional to the amount and duration of temperature deviation. The BRITE-S building is configured for B j ≡ 1 • F for all zones and times. EXPERIMENTAL RESULTS The hybrid system LBMPC controller was used to control the SAT in BRITE-S for 8 days that spanned both weekdays and weekends, and this was compared to 22 days in which the default, manufacturer-provided controller was used. Recall that our comparison methods implicitly account for variations in energy usage due to occupancy, and this is made explicit in the 95% confidence interval for estimated values. The building and HVAC configurations were kept identical for when both controllers were used, and this configuration has been in use for over one year. For notational reasons, we use a superscript 1 to refer to the default controller, and superscript 2 denotes the hybrid system LBMPC controller. Also, we define the quantity ∆Ê 2,1 =Ê 2 day −Ê 1 day to be the estimated difference between the average energy usage over a day of the LBMPC controller and that of the default controller. The value ∆Ĉ 2,1 =Ĉ 2 day −Ĉ 1 day is an analogous quantity of the estimated difference in average comfort over a day. The estimated energy characteristicsÊ 1 o,x (T 0 ),Ê 2 o,x (T 0 ) are shown in Fig. 3, and their differences are statistically significant (p = 0.002). The estimated difference in average energy usage over one day ∆Ê 2,1 = −1.53MWh is statistically significant (p = 0.002). And the 95% confidence interval is ∆Ê 2,1 ∈ [−2.07, −1.02]MWh. The estimated comfort characteristicsĈ 1 o,x (T 0 ),Ĉ 2 o,x (T 0 ) are shown in Fig. 4, and their differences are not statistically significant (p = 0.8). Furthermore, the estimated difference in average comfort over one day ∆Ĉ 2,1 = −0.75 • F is not statistically significant (p = 0.5), meaning that there is not enough evidence to exclude that ∆C 2,1 = 0 • F. The LBMPC controller provides modest energy savings at most OATs, which sum up to significant savings over a day. And because the difference in comfort characteristics is not statistically significant, this suggests that the LBMPC and default controllers provides comparable levels of comfort. CONCLUSION We have presented a hybrid model of building HVAC and described its control using hybrid system LBMPC. Experiments show substantial savings, and future directions for further energy savings were discussed. More broadly speaking, our experiments on BRITE-S, and previously on BRITE, show that the LBMPC methodology can provide significant energy savings for a wide variety of HVAC systems operating using different physical modalities. Fig. 1 . 1Sutardja Dai Hall is 141,000-square-foot building, T o [i], E[i]) for i = 1, . . . , D represent a scatter plot of energy usage versus OAT; thenÊ o,x (T o ) represents the smoothed version of the scatter plot. Fig. 3 .Fig. 4 . 34The cross marks and solid line (points and dashed line) denote the energy characteristics of the LBMPC (default) controller. 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[ "Modified Skellam, Poisson and Gaussian distributions in semi-open systems at charge-like conservation law", "Modified Skellam, Poisson and Gaussian distributions in semi-open systems at charge-like conservation law" ]	[ "Yu M Sinyukov \nBogolyubov Institute for Theoretical Physics\n03680KievUkraine\n" ]	[ "Bogolyubov Institute for Theoretical Physics\n03680KievUkraine" ]	[]	A modification of the Skellam and Poisson distributions is proposed for subsystems when the constraints imposed by the charge conservation law in the complete system are taken into account.Such distributions can be applied, for example, for an analysis of the fluctuations of baryon and net baryon numbers in certain pseudo-rapidity interval in A + A and p + p collisions with high multiplicities. The presented modified Skellam, Poisson and Gaussian distributions can be utilized also in various branches of science, when one studies the fluctuations of the two variables related to a subsystem, as well as the distribution of the difference of these variables, while the mentioned difference in the total system is fixed.	null	[ "https://arxiv.org/pdf/1805.03884v1.pdf" ]	56,330,944	1805.03884	b3c69fbe9ab7af4f4a347184bd2dc68ef7ad7b90	Modified Skellam, Poisson and Gaussian distributions in semi-open systems at charge-like conservation law May 2018 Yu M Sinyukov Bogolyubov Institute for Theoretical Physics 03680KievUkraine Modified Skellam, Poisson and Gaussian distributions in semi-open systems at charge-like conservation law May 20181numbers: 052013852410Pa2460Ky2575Dw2575Gz A modification of the Skellam and Poisson distributions is proposed for subsystems when the constraints imposed by the charge conservation law in the complete system are taken into account.Such distributions can be applied, for example, for an analysis of the fluctuations of baryon and net baryon numbers in certain pseudo-rapidity interval in A + A and p + p collisions with high multiplicities. The presented modified Skellam, Poisson and Gaussian distributions can be utilized also in various branches of science, when one studies the fluctuations of the two variables related to a subsystem, as well as the distribution of the difference of these variables, while the mentioned difference in the total system is fixed. I. INTRODUCTION The famous Poisson distribution in statistics appears in 1837 [1]. More than one century later J. G. Skellam published the paper "The frequency distribution of the difference between two Poisson variates belonging to different populations" [2]. Both the Poisson and Skellam distributions are widely utilized for an analysis of very different phenomena, in various branches of sciences. Despite our approach is quite general, to be concrete, this note we address to the actual and relatively new field of science -relativistic nucleus-nucleus and proton-proton collisions with high multiplicity at, particularly, CERN LHC and BNL RHIC. In these experiments an extremely high energy of colliding nucleons converts into multiparticle and multicomponent systems. The successful analysis of the particle production for different species within the statistical models [3][4][5], where a minimal number of parameters, such as the temperature and chemical potentials are almost enough to describe and predict the data, indicates a thermal nature of the final state formed in multiple elementary processes accompanied the collisions. Therefore the methods of statistical physics, that utilized micro-canonical, canonical and grand canonical ensembles to described the formed systems are widely used. Typically, only a part of the system is acceptable for the direct data analysis at current LHC or RHIC detectors. It is clear, that in the case of full acceptance, the so-called 4π geometry, the net baryon number in the final state that is equal to initial one -number of nucleons in the two colliding nuclei or just value 2 in p + p collisions -is fixed and not fluctuated. In small subsystems of the total final system, the distributions of baryons and anti-baryons can be both approximated by the Poissonian ones, then difference between the corresponding particle numbers (the net baryon number) are distributed according to related Skellam function [2]. It was checked by the ALICE Collaboration [6] that such a situation really takes place in relatively small pseudo-rapidity interval in P b + P b collisions. In the case, when subsystem of baryons + anti-baryons, that is available for analysis, is not small and comparable to the total system -such a situation takes place in nucleus-nucleus collisions with relatively low energies (BES at RHIC, FAIR and NICA planning experiments) -the baryon charge conservation law will certainly deform both the Poisson and Skellam distributions for baryon/anti-baryon numbers and net baryon charge in the subsystem. Despite the fact that in some publications [7][8][9][10], the baryon number conservation law in such semi-open systems is discussed to some extend, the problem as whole is not solved up to now. In this note we start from generalization of the Skellam distribution; the corresponding modification of the Poisson one will follow from the former in straightforward way. II. THE STATEMENT OF THE PROBLEM Suppose that independent discrete values n i are distributed according to Poisson, P (N i , n i ), with the mean values N i : P (N i , n i ) = N n i i n i ! exp(−N i )(1) If there are several uncorrelated subsystems, then the sum of discrete values, n i , can be presented again as the Poisson distribution P ( N i , n i ). Let us consider a system, consisting of particles of two species: say, baryons (N, n) and anti-barions (N , n). Since the distributions are of the Poissonian type and mutually independent, one can calculate distribution p(k) of difference k between particle numbers in both components, in our case, baryon minus antibaryon numbers k = n − n: p(k; N, N ) = ∞ n P (N, n + k)P (N , n). The latter is expressed by the Skellam distribution [2]: p(k; N, N ) = exp(−N − N ) N N k/2 I k (2 N N )(2) Here I k is the modified Bessel function of the first kind. The distribution is normalized: ∞ k=−∞ p(k; N, N ) = 1(3) The mean value m is m = ∞ k=−∞ kp(k; N, N ) = N − N .(4) Let us input constraint for this baryon-antibaryon system and associate it with the baryon number conservation in the isolated complete system. Namely, we consider the ensemble of i n i ), brings the condition for the two components {n 1 , n 1 } and {n 2 , n 2 }, as well as for the differences k 1 = n 1 − n 1 and k 2 = n 2 − n 2 , in both subsystems: B = n 1 + n 2 − n 1 − n 2 = k 1 + k 2 = N 1 + N 2 − N 1 − N 2(5) Our aim is to generalize the Skellam distribution (2) for the semi-open i-systems, i = 1 or 2. One of the possiblities to do this, is just to construct some function F by multiplying the product of independent distributions by the Kronecker δ-function, e.g., F = P (N 1 , n 1 )P (N 2 , n 2 )P (N 1 ,n 1 )P (N 2 ,n 2 )δ B n−n , or F = P ( N i , n i )P (N 1 ,n 1 )P (N 2 ,n 2 )δ B n−n , or F = P (N 1 , n 1 )P (N 2 , n 2 )P ( N i , n i )δ B n−n , or half-sum of the last two expressions, etc. 1 . After that a Skellam-like distrubution is presented for subsystem i as p(k i ; N i , N i ) = n j ,n l F({n j ,n l })δ k i n i −n i(6) The resulting distribution will depend on the choice of the initial form F. It cannot be derived unambiguously without a microscopic model/theory that includes mechanisms ensuring the charge conservation law in the total system. Our goal is to find the simplest analytic approximation to this problem which is based on general constraints required for such semi-open systems. We will compare our results with the model based on the binomial distribution (see [9], [8]) with probabilities q and q for baryon and anti-baryon to belong to certain subsystem, say "1", when the total numbers n and n in complete system are given. Let now these numbers n and n fluctuate with the Poisson distributions having the mean values D and D correspondingly. In such a model F in Eq. (6) is F({n j ,n l }) = C n,n δ B n−n δ n n 1 +n 2 δ n n 1 +n 2 P (D, n)P (D, n) × q n 1 (1 − q) n 2 q n 1 (1 − q) n 2 n! n 1 ! n 2 ! n! n 1 ! n 2 ! (7) where C is the normalization constant and q = n 1 / n = N 1 /N, q = n 1 / n = N 1 /N .(8) Only the numerical calculations are possible to provide summation of infinite series of hypergeometric functions in order to find from (7) the distribution over k 1 = n 1 − n 1 according to Eq. (6). Suddenly imposed constraint δ B n−n dramatically changes the initial Poissonianbased "bare" distributions for n j orn l . Correspondingly, the mean values N = n , N = n are differ from the "bare" mean values (D, D in Eq. (7)), and their connection is expressed through the sum of hypergeometric functions. Nevertheless, to compare our results with results based, for certainty, on the binomial-Poissonian distribution (7) we provide such calculations. III. GENERALIZATION OF THE SKELLAM DISTRIBUTION In this note we propose the natural generalization of the Skellam and Poisson distributions that not deal with the "bare" distributions and corresponding "bare" mean values but are expressed analytically directly through experimentally observed particle numbers and its mean values for the system and subsystems. Pursuing this aim let us present the modified Skellam distribution for subsystem i in the following generalized form: p(k i ; N i , N i ) = e −(M i +M i ) M i M i (k i −k 0 i )/2 I k i −k 0 i (2 M i M i )(9) To fix the expressions for M i , M i , k 0 i in the ansatz (9) let us list the conditions for the modified Skellam distribution: 1. The normalization condition (3) with substitutions p →p, k → k i , and {N, N } → {N i , N i } must be satisfied for p(k i ; N i , N i ). 2. The equality (4) for the mean value must be satisfied under the above substitutions. 3. Because of the symmetry of the conservation law constraint (5) with respect to the mutual permutations N 1 ↔ N 2 , N 1 ↔ N 2 and k 1 ↔ k 2 , the analytical expressions for modified Skellam distributions for the two subsystems should transform to each other p(k 1 ; N 1 , N 1 ) ↔ p(k 2 ; N 2 , N 2 ) under these permutations. 4. If one of the subsystems (say, "1") is much smaller than the other one, N 1 + N 1 N 2 , N 2 , then fluctuations of the two components, baryon and anti-baryon, in the subsystem "1" are uncorrelated Poissonian ones (the second subsystem just plays the role of a "thermal bath"). So the distribution (9) tends to the Skellam one (2), p(k 1 ; N 1 , N 1 ) → p(k 1 ; N 1 , N 1 ), N 1 + N 1 N 2 , N 2 ,(10) see Fig. 2. The situation is wise-verse as for the permutation "1 ↔ "2 . 5. When one of the subsystems (say, "1") vanishes, N 1 , N 1 → 0, the subsystem "2" occupies, in fact, the total system, N 2 − N 2 = B, and then, according to the net baryon charge conservation law, p(k 2 ; N 2 , N 2 ) → δ B k 2 , N 2 → N, N 2 → N .(11) The situation must be, of course, wise-verse when one permutes the systems, "1 ↔ "2 . 6. One more restricting condition appears if the total system has only one-component, e.g. when N i → 0 for both i = 1, 2, see Fig. 3. M i = N/8. On the other hand, if N i → N , it must be: p → δ B k i =n i . As we will find, these six obvious conditions are enough to define a simplest expression for the modified Skellam distribution (9). In what follows we will refer to these conditions are satisfied when G i , G i ≥ 0 and k 0 i is integer. This can be seen immediately when one changes the summation variable: k i → q i = k i − k 0 i . In what follows, using the concrete expressions for k 0 i , we always imply the nearest integer numbers to the corresponding values. With the same changing of the summation variable, one can find that condition (C2) brings us the expression for k 0 i : k 0 i = N i + G i − N i − G i(12) To guarantee condition (C3) -the invariant form of the modified distribution under permutation "1 ↔ "2 -let us transform p(k 1 ; N 1 , N 1 ) into p(k 2 ; N 2 , N 2 ) by means of Eqs. (5) and (12). Then one has k 1 − k 0 1 = −(k 2 − k 0 2 ) + G 1 − G 1 + G 2 − G 2 . Putting G 1 + G 2 = G 1 + G 2 one gets k 1 − k 0 1 = −(k 2 − k 0 2 ) and also transforms the exponent e −G 1 −G 1 into e −G 2 −G 2 in (9). Then, accounting for Bessel function property I n (z) = I −n (z) and negative sign before the expression (k 2 − k 0 2 ) after transformation of k 1 − k 0 1 , one gets the final result to fulfill the condition (C3): G 1 = G 2 , G 2 = G 1(13) Now let us present G i , G i in the form G i = α i N i and G i = α i N i . The limits described by the condition (C4) in the situation when, say subsystem N 1 + N 1 is much smaller than both of components, N 2 and N 2 , of the subsystem "2 , require that α 1 → 1 and α 1 → 1 in this case. Then the modified Skellam distribution tends to standard one according to Eq. (10). Similarly for the second system. To satisfy the condition (C5) when system "2" tends to be the total system, and so N 1 → 0 and N 1 → 0, one must to put G 2 → 0 and G 2 → 0. Then k 2 = k 0 2 = N 2 − N 2 and the Bessel function in Eq. (9) is zero at all orders except zero, when it is unity. So the equation (11) and condition (C5) are satisfied. To guaranty the symmetries (13) and above discussed limiting values, one has to put α 1 = N 2 /(N 1 + N 2 ), α 2 = N 1 /(N 1 + N 2 ) and α 2 = N 1 /(N 2 + N 1 ), α 1 = N 2 /(N 2 + N 1 ). Finally G 1 = G 2 = N 1 N 2 N 1 + N 2 , G 2 = G 1 = N 2 N 1 N 2 + N 1 (14) Note, a common function Q(N i , N i ), which is symmetric under permutation "1 ↔ "2 and vanish in the limits discussed in (C4),(C5), can be add to G i , G i without violation of all the properties discussed in (C1)-(C5) and values for k 0 i (12). To fix it let us take into account the condition (C6). Then the simplest function Q that guaranties all the requirements is: Q = k 0 1 k 0 2 /2(N + N ).(15) So, finally M i = G i + Q, M i = G i + Q.(16) The modified Skellam distribution (9) In Fig. 4 we present, just for illustration, the comparison between modified Skellam distribution (2), Skellam-like binomial-based distribution (7) σ 2 i = ∞ k=−∞ (k − m i ) 2 p(k; N i , N i ) = (M i + M i ) (17) S i = ∞ k=−∞ k − m i σ 3 p(k; N i , N i ) = M i − M i M i + M i 3/2 (18) K i = ∞ k=−∞ k − m i σ 4 p(k; N i , N i ) − 3 = 1 M i + M i(19) Note, that the above equalities are exact, strictly speaking, when k 0 i are integer numbers. Nevertheless, one can utilize the above expressions as the corresponding analytical continuations. Note, if i-system is a fairly small part of the total system (each of its components), then M i → N i , M i → N i and all the moments of the modified Skellam distribution are coincided with the known ones. If the subsystem, say "2", tends to the total system: In the special case q = q in (8) the ratio of σ 2 i /σ 2 i,Skellam coincides (deviations not exceed 0.5%) with the result [6] for this ratio, 1 − q, obtained in the binomial-based model (7), (6). In Fig. 5 we demonstrate the ratios of variances σ 2 bin , obtained from Eqs. (6), (7) for the binomial-Poissonian distribution, and our result (17) for σ 2 , in the cases when q = q and also when q = q + 1 2 . Both situations are considered in the two limits: when the total net baryon numbers in the total system are B = 1 and B = 150. The maximal deviation between the models is reached 7% and achieved at small B when q = q + 1 2 . Let us consider the case when the subsystem "2" has also only one component, namely, for the charge number conservation law in the system such as in Fig. 7: N 2 → N, N 2 → N ,P (n; N ) = e −M M (n−k 0 ) (n − k 0 )! , where M = N N N + N , k 0 = N − M.(20) As this was marked earlier, k 0 is a nearest integer number of the corresponding value. At the systems under the constraint that all the systems in the ensemble have exactly the same net baryon number B, for other observables the averaged values are fixed only. Such an ensemble can be realized, for instance, in very central nucleus-nucleus collisions. Then the total final multiplicity can fluctuate, but fluctuations of the net baryon number in the total system are completely suppressed. Let us divide the total system into the two i-subsystems, FIG. 1. The total system. Any selected sybsystem i, with the mean numbers {N 1 , N 1 } and {N 2 , N 2 }, Fig. 1. The fluctuations of particle numbers, baryon n i and anti-baryonn i , in the two subsystems are not independent anymore because the conservation law constraint, n − n = N − N = B = const (n = i n i , n = FIG. 2 . 2Illustration to items (C4), (C5) and Eqs.(10), (11). FIG. 3 . 3Illustration to items (C6).Then the fluctuations in k i = n i − n i = n i inside the selected i-subsystem arise only because of the fluctuations of the baryons between the subsystems "1" and "2". It is obviously, that when N 1 = N 2 = N/2, the relative fluctuations, σ i /N i , in any single subsystem will be twice suppressed as compare to the SkellamN i =0 → Poisson one in an independent subsystems. The fluctuation in the single i−subsystem is extended to the entire system N : it enforces the same fluctuation (with opposite sign) in the other subsystem because of the charge conservation law. The dispersion of fluctuations related to (9) is defined, similar as in the Skellam case, by σ = √ M 1 + M 2 (see below, Eq. (17)) and in independent Poisson subsystems, when M i = N/2, is σ ind = √ N . So,to get σ = σ ind /2 when the conservation law constraint is imposed, one should put as (C1), (C2), (C3), (C4), (C5), (C6). For the compactness of the subsequent presentation in an analysis of the items (C1)-(C5), we temporary rename the notations: M i , M i → G i , G i . The normalization condition (C1) with the expressions (16) for M i , M i and (12) for k 0 i generalizes the original distribution (2) for semi-open (sub)systems N i ,N i with the constraint for the total system N −N = B = const. The generalization is satisfied the obvious and necessary physical conditions (C1)-(C6). , (6) and just Skellam distributions (2) at the same average values N i , N i . The probabilities (8) are q = 0.6, q = 0.5, and B = 80. One can see that the uncorrected for charge conservation law Skellam distribution is much wider than the ones accounting for this restriction. Let us find for the semi-open i-subsystem the variance σ 2 , skewness S -the measure of lack of symmetry of the probability distribution, and excess kurtosis K -the measure of the original Skellam distribution. "tailedness". The calculations with modified Skellam distribution (9), (12), (16) accounting for the corresponding mean values m i = N i − N i , see (C2), give the results then M 1 and M 2 go to zero, so the dispersion σ and all the other central moments, describing the fluctuations of the net baryon number, tends to zero. FIG. 5 . 5The lines are related to the total net charge B = 1 and B = 150 correspondingly at q = q; blue and green lines -to B = 150 and B = 1 correspondingly at q = q + 1/2. IV. THE MODIFIED POISSON AND GAUSSIAN DISTRIBUTION The analogy of the Poisson distribution P 1+2 for semi-open, say, "1"-subsystem containing only single component, say, baryons N 1 , is described by Eq. (9) with N 1 = 0. See Fig. 6.So that P 1+2 (n) = p(k 1 = n 1 − n 1 = n 1 ≡ n; N 1 , N 2 , N 2 , N 1 = 0).FIG. 6. The cartoon for the situation when the only one component in the selected subsystem "1" exists: N 1 = n 1 = 0. Then one has k 1 = n 1 − n 1 = n 1 ≡ n, and the Poisson-like distribution P 1+2 (n) at the total charge conservation law takes place. N 2 2= 0, so the only subsystems N 1 , N 2 = 0 compose the total system, seeFig. 7. Then the modified Poisson distribution is P (n) = p(k 1 = n 1 − n 1 = n 1 ≡ n; N 1 , N 2 , N 1 = 0, N 2 = 0). A significant analytical simplification in this case can be achieved if one neglects the term QFIG. 7. The cartoon for the situation when in the selected subsystem "1" the component N 1 = n 1 = 0, as well as the component N 2 = n 2 = 0 in the subsystem "2" . Then one has k 1 = n 1 − n 1 = n 1 ≡ n, and the modified Poisson distribution P (n) takes place. (15) (it is, at least, one order of the value (1/8) less than M 1 ) in Eq.(16) for such a reduced system. Later we shall check such an approximation, see Figs. 8, 9. Then M i = G i . Denoting the mean numbers of baryons N 1 = N and antibaryons N 2 = N , one has M 2 ≈ G 2 = 0, M 1 ≈ G 1 ≡ M = N N N +N , k 0 1 ≡ k 0 = N − M , and using the passage to the zeroth argument in Bessel function in Eq. (9), one can write the modified Poisson distribution, P , accounting α ≡ N N +N → 1 the Eq. (20) coincides with the Poisson distribution (1), P (n; N ) = P (n; N ), at α 1 , P → δ N =B n . In Fig. 8 we demonstrate comparison of modified Poisson distribution P (n; N ) (20) with some other distributions at the same mean values N, N and B = 1. One can see that the results in our simple approximation (20) and the binomial-Poissonian model (7), (6) are fairly close, and at the same time the standard Poisson distribution (1) is much wider because does not take into account the charge conservation law. FIG. 8. A comparision of the modified Poisson distribution (20) with the Poisson-like one based on binomial-Poissonian distribution (7), (6), and also with the standard Poisson distribution. The net baryon number B = 1. The modified Poisson distribution is normalized as easy to check. The mean value m = n = N . In the approximation (20) the variance σ 2 , skewness S and kurtosis K are defined by the formulas (17)-(19) with M 1 = M and M 2 = 0. In Fig. 9 we present the ratio of variances obtained in the modified Poisson function P (20) and in the corresponding binomial-Poissonian distribution (7) with probabilities q = 1 and q = 0, for a wide interval of net baryon charge B. We see a good agreement within 1% between these two models. As for the Poisson-like distribution, P 1+2 , the deviation for the corresponding results can reach 14% at relatively small B. The transition to the correspondent Gaussian distribution is straightforward by means of the Stirling approximation and standard procedure: x = n = M (1 + δ); N, M 1, δ 1. Then one get from (20): P (n; N ) the Poisson-like one P 1+2 (see Fig. 6) -red points, as the function of net baryon number B. The limits are obvious: if N N , then M → N in (21), if N N (N → B), then P (n; N ) → δ(x − N ). V. SUMMARY A simple analytical generalization of the Skellam distribution for an arbitrary twocomponent subsystem accounting for a charge-like conservation law in the total system is proposed. It is compared with the numerically evaluated binomial-Poissonian model, a very good agreement with previously found in such a model 2 [9] the variation of the net baryon charge is observed. The results coinside within less than 0.5%. The same concerns the case when the number of baryons is much more than anti-baryons, and wise-verse. Being consider in full region of baryon and antibaryon probabilities to belong the subsystem, the deviation in results of these two models do not exceed 15%. The extremely simple approximations for the Poisson and the corresponding Gaussian distribution for considered type of systems are obtained based on the modified Skellam distribution. The presented formulas for the modified Poisson distribution are in a good agreement with numerical calculations in the binomial-Poissonian model, and so can be considered as a good analytic approximations for the later. It is worthy noting that despite the closeness of the results, the analytic approach proposed in the note is fully independent and the simplest among the possible models generalizing the Skellam and Poisson distributions for semi-open subsystems under the total charge-like conservation constraint. The analytic expressions for variation, skewness and kurtosis generated by the modified distributions are presented. The work is planning to apply for an analysis of different baryon & anti-baryon observables in pp and AA collisions at high and intermediate energies. e.g., in Ref.[7] the Kronecker-delta constraint is imposed for the product of the partition functions of initially independent subsystems. for the particular case of equal probabilities to find baryon and anti-baryon in the selected sybsystem. ACKNOWLEDGMENTSAuthor is grateful to P. Braun-Munzinger and A. Rustamov for initializing discussions.The research was carried out within the scope of the EUREA: European Ultra Relativistic Energies Agreement (European Research Network "Heavy ions at ultrarelativistic energies") and support by NASU, Agreement F-2018. The work is partially supported by NAS of Ukraine Targeted research program "Fundamental research on high-energy physics and nuclear physics (international cooperation)". Poisson's Sorbonne lectures on probability and decision theory. S.-D Poisson, 1837ParisS.-D. Poisson, Poisson's Sorbonne lectures on probability and decision theory, Paris, 1837. . J G Skellam, Journal of the Royal Statistical Society, A. 109296J. G. Skellam, Journal of the Royal Statistical Society, A 109 (1946) 296. . P Braun-Munzinger, J Stachel, J P Wessels, N Xu, Phys. Lett. B. 34443P. Braun-Munzinger, J. Stachel, J. P. Wessels and N. Xu, Phys. Lett. B 344 (1995) 43; . P , P. . V Braun-Munzinger, T Koch, J Schafer, Stachel, Physics Reports. 62176Braun-Munzinger, V. Koch, T. Schafer, J. Stachel, Physics Reports 621 (2016) 76. . F Becattini, M Gazdzicki, J Sollfrank, Eur. J. Phys. C. 5143F. Becattini, M. Gazdzicki and J. Sollfrank, Eur. J. Phys. C 5 (1998) 143. . J Cleymans, K Redlich, Phys. Rev. C. 6054908J. Cleymans and K. Redlich, Phys. Rev. C 60 (1999) 054908. . A Rustamov, for the ALICE CollaborationarXiv:1704.05329nucl-exA. Rustamov (for the ALICE Collaboration), arXiv:1704.05329 [nucl-ex], 2017. . V V Begun, M Gazdzicki, M I Gorenstein, O S Zozulya, Phys.Rev. 7034901V.V. Begun, M. Gazdzicki, M.I. Gorenstein, O.S. Zozulya, Phys.Rev. C70 (2004) 034901. . J Steinheimer, V Koch, Phys. Rev. C. 9634907J. Steinheimer, V. Koch, Phys. Rev. C 96 (2017) 034907. . P Braun-Munzinger, A Rustamov, J Stachel, Nucl. Phys. A. 960114P. Braun-Munzinger, A. Rustamov, and J. Stachel Nucl. Phys. A 960 (2017) 114. . 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[ "On the information entropy of matter-waves in quasi-disorder potentials", "On the information entropy of matter-waves in quasi-disorder potentials" ]	[ "Krishna Kajal ", "Dey \nDepartment of Physics (UG & PG)\nBanwarilal Bhalotia College\nAsansol-713303India\n", "Sudipta Das \nDepartment of Physics\nGovernment General Degree College\nChapra ShikraNadia-741123India\n", "Golam Ali Sekh \nDepartment of Physics (UG & PG)\nBanwarilal Bhalotia College\nAsansol-713303India\n" ]	[ "Department of Physics (UG & PG)\nBanwarilal Bhalotia College\nAsansol-713303India", "Department of Physics\nGovernment General Degree College\nChapra ShikraNadia-741123India", "Department of Physics (UG & PG)\nBanwarilal Bhalotia College\nAsansol-713303India" ]	[]	We consider ultracold Bose gases in quasi-random potentials and quantify localization of matter waves by means of Shannon information entropy. We explicitly examine the role of quasi-random potentials in producing localized states in the linear and nonlinear regimes. It is seen that the information entropic-based approach can be more useful to quantify localization of different types of states observed in Bose-Einstein condensates.	10.1140/epjd/e2018-90259-7	[ "https://arxiv.org/pdf/1710.05632v4.pdf" ]	119,094,157	1710.05632	7d112145261271307ecd86def947f44eb1e1e69d	On the information entropy of matter-waves in quasi-disorder potentials 25 Jan 2018 Krishna Kajal Dey Department of Physics (UG & PG) Banwarilal Bhalotia College Asansol-713303India Sudipta Das Department of Physics Government General Degree College Chapra ShikraNadia-741123India Golam Ali Sekh Department of Physics (UG & PG) Banwarilal Bhalotia College Asansol-713303India On the information entropy of matter-waves in quasi-disorder potentials 25 Jan 2018numbers: 0375Lm, 0545Yv Keywords: Bose-Einstein condensateQuasi-disorder potentialShannon entropyLocalization We consider ultracold Bose gases in quasi-random potentials and quantify localization of matter waves by means of Shannon information entropy. We explicitly examine the role of quasi-random potentials in producing localized states in the linear and nonlinear regimes. It is seen that the information entropic-based approach can be more useful to quantify localization of different types of states observed in Bose-Einstein condensates. I. INTRODUCTION An important problem in the context of many-body quantum systems is understanding the effects of interactions. Ultracold bosonic atoms have been proved to be a powerful tool to reproduce several interacting physical systems with great flexibility over different parameters [1][2][3][4][5][6][7]. A disorder potential in the ultra-cold atomic system can naturally be induced either from imperfection in magnetic-trap wire [8][9][10][11][12] or during the fabrication of device [13]. In the presence of disorder potentials the interacting bosons leads to the so-called dirty boson problems. It offers a platform to understand the interplay between the disorder potential and the mean-field interaction in producing localized and super-fluid states. Understanding the interplay between localized and super-fluid phases is a long standing problem in condensed matter physics [14,15]. In the negligible interaction limit, the weak disorder leads to exponential localized one-body wave-function, often called the Anderson localized state [16]. It occurs due to interference of the repeatedly reflected quantum matter waves in the random potential. The scenario gets changed in the presence of strong disorder potential. Here the competition between inter-atomic interaction and disorder potentials becomes important. Specifically, the bosons undergo a phase transition from super-fluid to Bose glass (BG) due to localization in the minima of the random potential. The BG phase is a many-body insulating phase which is characterized by finite compressibility, absence of gap in the spectrum and infinite super-fluid susceptibility [17][18][19][20][21][22][23]. The effect of disorder potential has been studied experimentally in the ultracold atomic systems either by the use of laser speckle patterns [24,25] or by applying incommensurable optical lattices [26][27][28][29]. In the presence of lattice disorder Anderson localization has directly be observed in one-dimensional configuratiion [25,29]. In compositionally disorder platform Anderson localization * Electronic address: [email protected] and photonic band-tail states have recently been confirmed experimentally [30]. Several theoretical investigations including the finite temperature phase transition from super-fluid to insulator [31], localized state with exponential tail [32], disorder induced phase coherence [33] and distribution of bosons localized at the minimum of the disorder potential [34] show current interest in this direction. A natural framework to study localization phenomena can be provided by the information theoretic approach [35][36][37][38]. It is due to Shannon who establishes a link between physical effects and information entropy. Shannon information entropy depends on the probability density corresponding to changes in some observables. Quantifying observable physical effects using information entropy is useful in many applicative contexts. For example, it is shown that the Shannon information entropy can give better information on the effect of electronic correlation in many-electron systems [39][40][41]. Time evolution of the sum of entropies in position and momentum spaces helps in understanding collapse and revival phenomena of quantum wave packets [42]. Our objective in this paper is to study effects of quasidisorder potentials on the density of matter waves in Bose-Einstein condensates (BECs). Localization of quantum matter waves results from combined effects of kinetic energy, inter-atomic interaction and disorder potentials. The disorder potential interplays with kinetic energy to determine the phase of the condensates. Non-linearity due to inter-atomic interaction further introduces complexity in the system. An average amount of information from the event can, however, be obtained by the measure of Shannon entropy. For a continuous probability distribution ρ(x) = \|φ(x)\| 2 , Shannon information entropy in coordinate space is defined by S ρ = − ρ(x) ln ρ(x) dx(1) with ρ(x)dx = 1. Let ψ(p) be the Fourier transform of φ(x), the Shannon entropy corresponding to γ(p) = \|ψ(p)\| 2 is given by S γ = − γ(p) ln γ(p) dp.(2) In one-dimension, the values of S ρ and S γ are shown to satisfy S ρ + S γ ≥ (1 + ln π).(3) This inequality is often referred to as Bialynicki-Birula-Mycielski (BBM) uncertainty relation [45]. It implies that if a distribution in coordinate space is localized then the distribution in momentum space is diffused and vice versa. We know that disorderness augments entropy. Therefore, a larger value of S ρ is associated with the smaller value of S γ [38]. We know that localization of ultra-cold atoms in pseudo-disorder potential is affected (i) with the change of interaction strength (γ) and/or (ii) by the variation of relative strength, wavelength and phase of the lattices constituting the pseudo-disorder potential. Understandably, atoms in BEC loaded in optical lattices are distributed in different lattice sites. With the variation of interaction the number of occupied sites gets modified. Therefore, for a particular quasi-random potential the Shannon entropy of density distributions in coordinateand momentum-spaces are likely to be affected. We see that, for a particular state with fixed width and quasidisorder lattice parameter, S ρ changes with the variation of γ and attains a minimum value at γ = γ m for maximally localized states. At the same time momentum spaces entropy tries to take lower and lower values. However, the entropic uncertainty relation restricts the lowest values of S ρ and S γ . Therefore, S ρ along with BBM inequality can serve as the measure of localization of matter waves in Bose-Einstein condensates. The paper is organized as follows. In Section 2, we introduce an appropriate model and present a variational study to estimate optimum values of parameters. More specifically, we calculate energy of the system and find the values of different parameters for energetically stable condensates. The energy variation with inter-atomic interaction suggests the existence of two different types of states in the system. In section 3, a numerical study is presented with a view to examine the nature of the localized states. We calculate Shannon entropy to quantify the localization in the system. Finally, we conclude by noting the main outcome of the work in section 4. II. THEORETICAL FORMULATION The theoretical prediction on the phenomenon of localization of electro-magnetic waves in disorder potential has recently been realized experimentally in ultracold atomic systems by Roati et al. [29]. It was performed by the use of one-dimensional quasi-periodic potential created by suerposing two standing waves of slightly different wave lengths, namely, λ 1 = 1032 nm and λ 2 = 862 nm such that the value of λ r = λ 2 /λ 1 maintains incommensurable condition. Understandably, this is a bichromatic lattice potential where the effects of secondary lattice changes the periodicity and magnitude of the principal lattice in a irregular manner. The quasi-periodic potential can be written as [29] V QD (x) = 2 i=1 2s i E i cos 2 (k i x).(4) Clearly, 2s i measures the strength of the quasi-periodic potential in the unit of recoil energy E i = 2 k 2 i 2m with k i = 2π λi , the wave number. It is shown in [29] that the value of s i must be limited by s 1 ≤ 10 and s 2 ≤ 3. Recently, insulating phases due to the action of weak disorder lattice have been studied by loading ultracold atoms in quasi-periodic potentials. Some differences in the properties of ultracold gases in pure-disorder and quasi-disorder potentials are observed. However, most of the properties are shown to be almost identical in both the cases [46]. In view of the above, we consider a cylindrical shaped BECs of N atoms in quasi-disorder potential given in Eq. (4). The geometry of the condensate can be achieved experimentally by making the frequency of the transverse harmonic trap (ω ⊥ ) much greater than that of longitudinal component (ω x ) of the trap such that ω ⊥ ≫ ω x [5]. The dynamics of trapped BECs is satisfactorily described by means of an effective mean-field equation, often called Gross-Pitaevskii equation (GPE). The GPE in quasi-onedimension(Q1D) is given by i ∂ψ ∂t = − 2 2m ∂ 2 ψ ∂x 2 + 1 2 mω 2 x x 2 ψ + V QD (x)ψ + 2a s N ω ⊥ \|ψ\| 2 ψ(5) with \|ψ\| 2 dx = 1. Here a s stands for the atomic scattering length. Understandably, Eq.(5) models the BECs in quasi-disorder potentials. Rescaling length, time and energy in the units of a ⊥ = /(mω ⊥ ), ω −1 ⊥ and ω ⊥ we get i ∂φ ∂t = − 1 2 ∂ 2 φ ∂x 2 + 1 2 λ 2 x x 2 φ + V QD (x)φ + γ\|φ\| 2 φ (6) with V QD (x) = 2 i=1 s i k 2 i cos 2 (k i x) ,(7) where λ x = ω x /ω ⊥ , γ = 2a s N/(N a ⊥ ) and φ = √ a ⊥ ψ with the norm N = \|φ(x, t)\| 2 dx. We introduce, for convenience, k 1 = k p and k 2 = k s and represent V p = s 1 k 2 p and V s = s 2 k 2 s as the strengths of primary and secondary lattices respectively. Different approximation methods are usually used to solve the GP equation in (5) [47][48][49][50]. One of the widely used methods to treat the problem analytically is the variational approach [51][52][53][54]. To begin with we restate the initial-boundary value problem in Eq.(6) by writing the Gross-Pitaevskii energy functional E[φ, φ * ]= +∞ −∞ 1 2 \|φ x \| 2 +V D \|φ\| 2 + 1 2 λ 2 x x 2 \|φ\| 2 + 1 2 γ \|φ\| 4 dx.(8) Here, the first and last terms represent respectively kinetic and interaction energies while the second and third terms give contributions of quasi-periodic potential and harmonic trap. In order to understand the dependence of localized states on different parameters of the system, we consider φ(x) = N √ πa e −x 2 /2a 2 ,(9) as a trial solution. Here a is the variational parameter. This is a reasonable assumption of localized states since in the absence of inter-atomic interaction and V QD = 0, the solution of Eq. (6) is a Gaussian function. The interatomic interaction changes only size of the condensate cloud. We demand that this change will be reflected in the change of variational parameter. To determine values of the parameters for V QD = 0 and γ = 0 we insert Eq. (9) in Eq. (8) and calculate E N = 1 4a 2 + 1 4 λ 2 x a 2 + N γ 2 √ 2πa + 1 2 V p 1 + e −k 2 p a 2 + 1 2 V s 1 + e −k 2 s a 2 .(10) We know that the necessary condition for optimization i.e., ∂E/∂a = 0 allows us to determine the effective values of a. Let a 0 be the value of a which satisfies the sufficient condition ∂ 2 E/∂a 2 \| a=a0 > 0 and, thus permits energetically stable condensates. However, it is an interesting curiosity to find effective energy for fixed quasi-periodic lattice parameters but different interaction strengths or vice versa. We expect that the interplay between the quasi-periodic and nonlinear potentials in creating localized/extended state can change the shape of the effective potential. With a view to examine the interplay between nonlinear and quasi-disorder potentials, we choose different parameters within the experimental limit. More explicitly, we have chosen to work with λ p = 10 and λ s = 8.38 which give actual wavelengths of the bi-chromatic lattices in the unit of a ⊥ ≈ 1µm. The ratio of secondary and primary lattices thus consistents with the experinemtal value λ r = 0.835 [29]. The choice of λ p is not unique. Indeed, one can work with different values of λ p keeping λ r at 0.835 [55]. Strengths of the lattices can be calculated from V p = 4π 2 s 1 /λ 2 p and V s = 4π 2 s 2 /λ 2 s for s 1 ≤ 10 and s 2 ≤ 3. Thus fixing the values of lattice parameters, we plot in Fig. 1 effective energy E as a function of a for different interaction strengths. Specifically, top and bottom panels give E versus a for attractive and repulsive interactions respectively. The existence of negative minima at certain values of a indicates that a BEC can hold stable bound states in both repulsive and attractive cases. However, the value of a of a stable condensate decreases as attractive interaction increases while a state becomes extended as the repulsive interaction increases. It is worth mentioning that negative minimum appears in the effective potential if the quasi-periodic pattern arises from two lattices which are out-of-phase. However, one can get localized states by superposing two in-phase lattices. In this case minimum energy becomes positive and localization takes place only if \|s 2 \|/\|s 1 \| ≪ 1 [55]. III. INFORMATION ENTROPY OF BEC MATTER WAVES We know that different types of localized states are observed in the BECs in the presence of periodic and pseudo-random potentials. The interplay between lattice potential and nonlinear interaction gives rise spatially localized waves [56]. We have seen that analytical approximation method can help in understanding the interplay between different parameter of the system and allow us to estimate width/size of the condensate distribution. However, exact shape or nature of the distribution plays a crucial role in the entropic based approach. In this context we note that the Gaussian type states having different widths obtained from the energy optimization condition for different γ cannot detect the variation of entropic uncertainty in the Shannon formalism. This leads us to consider a numerical based approach to get more detail description of the density profile. A. Information entropy in periodic potentials We consider both non-interacting and interacting BECs in periodic potentials. Depending on the type of interaction (attractive and repulsive) the system can be responsible for different types of localized states e.g., bright and gap solitons, the physical origin of which are quite different. To begin with we perform the numerical simulation [57] of Eq.(6) with potential given in Eq. (7) for V s = 0. The result of which is displayed in Fig. 2. Particularly, it gives density distributions both in co- ordinate (top panel) and momentum (bottom panel)spaces for V p = −6 and λ p = 10. Top panel shows that the peak (spatial distribution) of density distribution of a bound state decreases (broadens) as the interaction changes from attractive to repulsive. As expected, the momentum space shows the opposite trends. Entropies of different states are shown in Table 1. We see that the S ρ gradually decreases (increases) with the increase of attractive(repulsive) interaction. An opposite trend in entropy variation is found in the momentum-space distribution. Therefore, a clear indication is that the spatial extension of BEC matter waves in co-ordinate space reduces if the interaction is tuned from repulsive to attractive. The maximum localization is, however, limited by the entropic uncertainty in Eq. (3). This entropic uncertainty led us to demand that the sum of S ρ + S γ tends towards the lowest value for the most feasible state. Looking carefully Table 1 we see that the width of a state reduces gradually with the tuning of inter-atomic interaction from repulsive to attractive through γ = 0. The entropic uncertainty obtained from the direct numerical simulation of Eq.(6) gradually increases except for weakly interacting case. More specifically, entropic uncertainty for γ = 0 becomes larger than those of the interacting cases. This can be understood by noting that the central peak of the distribution for γ = 0 decreases due to appearance of new smaller peaks on either side of central maximum resulting an effective broadening of the state. It may be remembered that the localized matter waves for γ < 0 and γ > 0 can be identified as bright and gap solitons respectively [56]. For brevity we say that the high information entropy at γ = 0 causes spatial broadening of the profile. B. Information entropy in quasi-periodic potentials We have noted that two lattices with different wavenumbers, strengths and phases are superposed to create a quasi-periodic potential. It is seen that the effective energy of BEC can either be positive or negative depending on lattice parameter (Fig. 1). The BEC in quasi-periodic potential with positive effective energy exhibits Anderson type localization in the γ → 0 limit due to interplay between kinetic energy and optical lattices. An interacting BEC with negative effective energy, on the other hand, can support localized states. In this case, the interplay between pseudo-random and nonlinear potentials in conjunction with kinetic energy comes into play to control nature of the states having negative effective energy. To quantify the localization of those states in terms of information entropy we calculate density profile from numerical simulation of the Eq.(6) with the potential given in Eq. (7). The result on the density distribution \|φ(x)\| 2 is displayed in Fig.3. It is seen that the localization of states supported by quasi-disorder lattice becomes narrower in coordinate space as the attractive interaction increases. As expected momentum space distribution exhibits opposite trends. We know that both momentum and coordinate space distributions, in principle, tend to be more localized to attain minimum entropy. The minimum entropic uncertainty relation, however, limits the lowest value of entropy for localization. We find that the physically acceptable most localized state attains lowest value of the entropic uncertainty condition. The listed data in Table 2 clearly implies the fact that the state with γ = −1 should be more physically acceptable localized states. Interestingly enough, the co-ordinate space distribution shows more localization with the increase of attractive interaction resulting smaller entropy. In that case sum of entropies increases due to faster increase of momentum-space entropies leading to less feasible states. It is interesting to note that the localized state with negative energy shows different behaviour in the presence of repulsive interaction due to quasi-disorder lattices. Here the quasi-disorder potential interplays with kinetic energy in the mechanism of localization / delo- calization phenomena. We notice that the distribution of density profile is maximally localized in the coordinate space and diffusely distributed in the momentum-spaces with γ = 0 (see Fig. 4). More specifically, with the increase of γ the central narrow peak diminishes which results weak localization in coordinate spaces and strong localization in momentum space. The entropies in coordinate and momentum spaces show opposite trends for γ > 1. However, the sum of the two entropies increases for γ > 0. The most feasible state is the one which has minimum entropic uncertainty. A localized state with positive energy can be created in a BEC with repulsive inter-atomic interaction in the presence of a bi-chromatic lattice which results from the superposition of in-phase optical lattices. In this case, the central peak gets localized in the two nearby sites of the central lattice maximum (Fig. 5). Understandably, physical origin of this type of localised state is different from that of a bound state. Here localization takes place due to multiple reflections from lattice disorder and there is no role of interaction (see curve for γ = 0 in Fig. 5). We have seen that the entropic uncertainty is minimum for γ = 0 and it increases for γ = 0. We demand that stronger localization of the state causes entropy to decrease. However, maximally localized state corresponds to that which gives minimum entropic uncertainty (Fig. 6). IV. CONCLUSION We have investigated the phenomenon of spatial localization of Bose-Einstein condensates in periodic and quasi-periodic potentials separately within the framework of Gross-Pitaevskii equation. In particular, we have studied the role of nonlinear interaction and quasidisorder potential in creating localized or extended bound states. Our investigations on the existence of different types of states for interacting and weaklyinteracting BECs are based on both analytical and numerical approaches. It is found that the BECs can support a family of states, the spatial extension of which can be varied by changing the interaction strength and with the variation of relative strength, wave-number and phases of the constituents of bi-chromatic lattices. We have quantified localization of a state by the measure of Shannon information entropy. It is seen that the information entropy of Shannon in position-space (S ρ ) is a useful physical quantity to measure localization of any kind of states. Smaller value of S ρ infers that the localization is more in coordinate space while converse is true for the entropy S γ in momentum space. Both the entropies try to get smaller and smaller values to produce more and more localized density profile. Their lower values are, however, restricted by the entropic uncertainty relation. More specifically, uncertainty sum attains lowest value for the most feasible localized state. We have seen that with the increase of attractive (repulsive) interaction localization of BEC distribution gradually increases (decreases) without any upper (lower) bound in coordinate-space. The entropy based approach can allow us to identify a critical value of interaction strength or lattice parameter for maximally localized BEC wave-packets. We have studied information entropy of noninteracting BECs in quasi-disorder and periodic potentials. Our study shows that the entropic uncertainty for the bound state embedded in quasi-disorder potential is smaller than that in periodic potential. The noninteracting BECs in presence of in-phase bi-chromatic lattice can also support state with positive energy. We have verified that the entropic uncertainty of all possible states in the presence of interaction either attractive or repulsive becomes larger than that of the state in noninteracting BECs. This investigation clearly reveals the fact that disordering in the lattice is responsible for localization. FIG. 1 : 1Top panel. It shows energy (E) as a function of a for γ < 0. Bottom panel. It gives E versus a for γ ≥ 0. In both the panels we consider s1 = −6.0 , s2 = 0.9 , λp = 10, λs = 0.835λp, N = 1. Also we assume that the trap is sufficiently flat and thus the effects of λx can be negligible at the central region of the harmonic confined. FIG. 2 : 2Color(online) Top panel shows density distribution (\|φ\| 2 ) as a function of x for different interaction strength γ at Vp = −6.0, Vs = 0, λp = 10 and λr = 0.835 in coordinate spaces. A similar variation in momentum spaces fitted with Gaussian function is shown in the bottom panel. Solid, dashed and dotted curve give density distributions for γ = 0, 2 and −1. entropies and width of different states obtained by the variation of γ of BECs in a periodic potential. FIG. 3 : 3Color(online) Top panel gives density distribution (\|φ(x)\| 2 ) in coordinate space for attractive inter-atomic interaction. Bottom panel displays density distribution (\|ψ(p)\| 2 ) obtained from Gaussian fit in momentum-space. In both the panels solid, dashed and dotted curve represent states for different values of attractive interaction strength, namely, γ = −1, −2 and −3 respectively with Vp = −6.0 and Vs = 0.9. FIG. 4 : 4: Shannon entropies and width of different states in BECs in quasi-periodic potential for different values of γ. Other parameters are kept same as those used in Fig. Color(online) Top Panel : Density distribution (\|φ(x)\| 2 ) in coordinate space. Bottom panel : The variation of (\|ψ(p)\| 2 ) obtained from Gaussian fit in momentum-space for the values of different parameters similar to those used in the top panel. In both the panel solid, dashed and dotted curve represent states for different values of repulsive interaction strength, namely, γ = 0, 1 and 3 respectively with Vp = −6.0, Vs = 0.9 and λp = 10. FIG. 5 : 5Color(online). Top Panel gives density distribution (\|φ(x)\| 2 ). Bottom panel shows density distribution (\|ψ(p)\| 2 ) in momentum space for same values of γ as those used in the Bottom panel. In both the panels solid, dashed and dotted curves represent the states for interaction strengths γ = 0, 1.5, 2 with Vp = 2.0 and Vs = 0.4, λp = 4. Momentum space distributions displayed here are obtained from Gaussian fit to visualize localization effect clearly. FIG. 6 : 6Color(online) Entropic uncertainty S = Sρ + Sγ of a particular state with the variation of interaction strength γ. TABLE III : IIIShannon entropy and entropic sum of different states of BECs in quasi-periodic potential for different values of repulsive interaction. Other parameter are same as those used inFig. 4. 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[ "laser with master oscillator power amplifiers", "laser with master oscillator power amplifiers", "laser with master oscillator power amplifiers", "laser with master oscillator power amplifiers" ]	[ "Fuyong Wang ", "Zhipeng Qin ", "Guoqiang Xie ", "Peng Yuan ", "Liejia Qian ", "Xiaodong Xu ", "Jun Xu ", "Fuyong Wang ", "Zhipeng Qin ", "Guoqiang Xie ", "Peng Yuan ", "Liejia Qian ", "Xiaodong Xu ", "Jun Xu " ]	[]	[]	We report on a diode-pumped passively mode-locked Yb:Lu1.5Y1.5Al5O12 (Yb:LuYAG) laser for the first time to our knowledge. With the mixed crystal of Yb:LuYAG as gain medium, the mode-locked laser generated 2.2 W of average output power with a repetition rate of 83.9 MHz and pulse duration of 2.2 ps at the wavelength of 1030 nm. In order to obtain higher output power, the output from the mode-locked oscillator was further amplified to 8.5 W by two-stage single-pass amplifiers. The high-power picosecond laser is very useful for applications such as pumping of mid-infrared optical parametric oscillators, material micro-processing, and UV light generation, etc.Index Terms-mode locked lasers, laser amplifiers, ultrafast optics.	10.1364/josab.35.000231	[ "https://arxiv.org/pdf/1410.2088v2.pdf" ]	118,390,257	1410.2088	b45edc492482cb8a12697bf125d036ccc8864fdd	laser with master oscillator power amplifiers Fuyong Wang Zhipeng Qin Guoqiang Xie Peng Yuan Liejia Qian Xiaodong Xu Jun Xu laser with master oscillator power amplifiers Index Terms-mode locked laserslaser amplifiersultrafast optics We report on a diode-pumped passively mode-locked Yb:Lu1.5Y1.5Al5O12 (Yb:LuYAG) laser for the first time to our knowledge. With the mixed crystal of Yb:LuYAG as gain medium, the mode-locked laser generated 2.2 W of average output power with a repetition rate of 83.9 MHz and pulse duration of 2.2 ps at the wavelength of 1030 nm. In order to obtain higher output power, the output from the mode-locked oscillator was further amplified to 8.5 W by two-stage single-pass amplifiers. The high-power picosecond laser is very useful for applications such as pumping of mid-infrared optical parametric oscillators, material micro-processing, and UV light generation, etc.Index Terms-mode locked lasers, laser amplifiers, ultrafast optics. I. INTRODUCTION Passively mode-locked picosecond lasers at 1 µm wavelength with high average output powers are desired for materials micro-processing, coherent UV light generation and synchronously pumped optical parametric oscillators (SPOPO's), etc [1], [2], [3], [4]. With high-power picosecond pulses pumping at 1 µm wavelength, ultrashort pulses can be easily extended to mid-infrared spectral region by SPOPO [5], [6]. In passively mode-locked lasers, Nd 3+ or Yb 3+ ion-doped gain media have been widely investigated and used for generation of high-power picosecond laser pulses [7], [8], [9], [10], [11], [12], [13], [14]. In general, Yb 3+ ion has a lower intrinsic quantum defect and wider emission spectrum than Nd 3+ ion, so it is easier to emit shorter pulses with higher laser efficiency. Yb 3+ -doped material has been considered to be a very attractive gain media for efficient and high power lasers around 1 µm. In the past decade, many interesting results with high average power have been reported for mode-locking operations based on diode pumped Yb 3+ -doped crystals, such as Yb:YAG [15], [16], Yb:KGW [17] and Yb:GSO [13]. Usually direct generation of high-power picosecond laser from oscillators need some special design, such as thin disk laser [18]. In addition, high-average-power picosecond laser can be generated by master oscillator power amplifiers (MOPA), in which picosecond pulses emitted from master oscillator serve as seeding, and then are amplified by one or multiple stage amplifiers. The Yb-doped garnet crystals generally possess high thermal conductivity, excellent physical and chemical property, and mature fabrication process, which is a good candidate for high-average-power picosecond lasers. Among Yb 3+ -doped garnet crystals, Yb:YAG and Yb:LuAG have been widely investigated in mode-locked lasers [15], [16], [19]. While as their mixed crystal, Yb:Lu 1.5 Y 1.5 Al 5 O 12 (Yb:LuYAG) has never been investigated in mode-locking operation. Since Yb:LuYAG has wider emission linewidth and inherits the superior property of Yb:YAG and Yb:LuAG, it is expected that Yb:LuYAG is a potential gain medium for ultrashort pulse lasers [20]. In this paper, we experimentally demonstrated a passively mode-locked Yb:LuYAG laser for the first time to our knowledge. With a semiconductor saturable absorber mirror (SESAM) as mode locker, an average output power of 2.2 W has been generated from the mode-locked oscillator. In order to achieve higher average output power, the output from the oscillator was further amplified by two-stage singlepass amplifiers. With the two-stage amplifiers, 8.5 W average output power was finally achieved with pulse duration of 2.2 ps and repetition rate of 83.9 MHz. The built MOPA laser system provides an ideal pumping source for mid-infrared SPOPO. II. EXPERIMENT SETUP The experimental setup of the diode-pumped mode-locked Yb:LuYAG laser is shown in Fig. 1. A commercial fibercoupled laser diode with fiber core diameter of 105 µm, NA of 0.22 and maximum output power of 25 W (FOCUSLIGHT FL-FCMSE55-25-940) was used as the pump source and its emission wavelength was 940 nm. Through the lens F 1 and F 2 , the pump light was imaged into the crystal with waist spot size of 126 µm in diameter. The laser mode at waist spot was calculated to be ∼ 100 µm in diameter based on the ABCD propagation matrix method. The mixed crystal of Yb:LuYAG with 8% Yb-doping has a cross section of 3 × 3 mm 2 and a length of 2.5 mm. In order to remove the generated heat when pumping, the laser crystal was wrapped with indium foil and tightly mounted in a water-cooled copper holder, and the circulating water temperature was sustained at 15 • C. A single SF10 prism was used in the cavity to maintain horizontal polarization of laser beam. M 1 , M 2 and M 3 are all plano-concave mirrors with different radii of curvature (M 1 =-100 mm, M 2 =M 3 =-200 mm) and coated with high reflectivity from 1030 nm to 1090 nm (R> 99.7%). Besides, M 1 was also anti-reflectively coated for pump wavelength (T> 95%). The output coupler had a transmission of 35%. The semiconductor saturable absorber mirror (BATOP GmbH) with saturation fluence of 90 µJ/cm 2 , modulation depth of 1.2% and relaxation time of 10 ps was employed to initiate and sustain the mode-locking operation. The calculated laser mode size on the SESAM was 61 µm in radium. The output beam from the oscillator was isolated by an optical isolator (CONOPTICS 700 series) for preventing feedback from the amplifiers. In the amplifiers section, all the optical elements of first-stage amplifier were as same as that of second stage. In order to achieve high gain in the active medium, we used high-brightness single emitter LDs (nLIGHT NL-CN-8.0-940-F-5) instead of the fiber-coupled LDs as the pump source. The pump beams were imaged into the crystal by two lenses with focal lengths of 100 mm and 80 mm, respectively. Two Yb:YAG crystals were used as the active media in the amplifier stages. Considering the reabsorption of Yb:YAG crystal, an optimal crystal length of 3.5 mm was employed with Yb-doping concentration of 10%. Fig. 1 © £ " # $ % ¦ & ' " ( ¥ ) ¡ 0 § ¡ 1 2 3 4 ¡ ¢ £ ¤ ¢ § ¨ © ! © 5 ¥ ) ¡ ¡ ¢ £ ¤ ¢ § ¨ © ! © 5 ¥ ) ¡ III. EXPERIMENTAL RESULTS AND DISCUSSION After carefully aligning the cavity, the stable continuouswave mode locking could be achieved. However, the single pulse tended to break up into synchronous or asynchronous multiple pulses due to over saturation on the SESAM. By increasing output coupling and enlarging the laser mode on the SESAM, the multi-pulse phenomenon was suppressed effectively in the experiment. The stable mode-locked pulse trains were captured by a high speed detector (New Focus, Model: 1611Fs-AC) and displayed by a digital oscillator with 500-MHz bandwidth (Tektronix, DPO3054), as shown in Fig. 2. The radio frequency (RF) spectrum of the mode-locked pulses was measured with the RF spectrum analyzer (Agilent E4402B). The mode-locking stability and the pulse repetition rate were confirmed by the radio frequency spectrum shown in Fig. 3. The fundamental RF frequency was 83.9 MHz with a signal-to-noise ratio of more than 70 dB. The RF spectrum proved clean CW mode-locking operation without any Qswitching instability in the Yb:LuYAG laser. The CW mode locking was very stable and could be sustained for some hours. Even though the mode locking was broken, it could be initiated spontaneously. The autocorrelation trace and spectrum of the mode-locked pulses are shown in Fig. 4. The mode-locked pulse duration was measured in a noncollinear way with a commercial autocorrelator (APE Pulse Check 50). Assuming a sech 2 profile, the mode-locked pulses had a pulse duration of 2.2 ps. The corresponding mode-locked pulse spectrum had a FWHM of 2 nm centered at 1030 nm, measured by the optical spectrum analyzer (Ocean Optics, USB4000). The output laser beam had a round TEM 00 mode. Figure 5 illustrates the average output power versus absorbed pump power in the CW and CW mode locking regimes for the Yb:LuYAG oscillator. In the CW regime, the SESAM was replaced by a high-reflective mirror in the laser. The maximum CW output power reached 2.4 W with a slope efficiency of 61% and no roll-off appeared in the experiment. When the SESAM was employed, the CW laser switched to mode-locking operation. Beyond 4.2 W of absorbed pump power, CW mode locking could be achieved. The maximum mode-locking average output power reached 2.2 W with a slope efficiency of 59%. In order to obtain higher average output power while keeping single pulse operation, two-stage power amplifiers were adopted following the oscillator. In each stage amplifier, the seeding laser passed the active medium only once. Figure 6 demonstrates the average output power dependence on incident pump power for the first-stage and second-stage amplifier, respectively. After optical isolator, the remained maximum average power was 2 W. Seeded with this laser power, the first-stage amplifier emitted a maximum average output power of 5.3 W with an incident pump power of 6.5 W, with an optical-optical efficiency as high as 51%. Seeded with 5.3 W of average power, the second-stage amplifier generated a maximum average power of 8.5 W under an incident pump power of 7 W, with an optical-optical efficiency of 46%. In both stage amplifiers, there was no roll-off of output power. The higher average output power was only limited by the available pump power of LDs. After amplification, the pulse duration kept almost constant and the laser beam had a round TEM 00 mode. With the MOPAs, the mode-locked pulses have a pulse energy of 101 nJ and peak power of 46 KW, which will be an ideal ultrashort laser source for mid-infrared OPO pumping, coherent UV light generation, and materials microprocessing, etc. IV. CONCLUSION In conclusion, we have experimentally demonstrated what we believe the first operation of diode-pumped mode-locked Yb:LuYAG laser. The mode-locked laser generated pulses with a pulse duration of 2.2 ps, repetition rate of 83.9 MHz and average output power of 2.2 W. The Yb:LuYAG shows excellent mode-locking performance as active medium of highaverage-power ultrashort pulse lasers. With two-stage singlepass power amplifiers, a maximum average output power of 8.5 W was achieved, with optical-optical efficiency as high as 51% and 46% in the first-stage and second-stage amplifiers, respectively. The research results show that Yb active mediumbased MOPA is an efficient way for realizing the high-averagepower ultrashort pulse generation. The work is partially supported by the National Natural Science Foundation of China (Grant No. 61008018 and 11121504) and the National Basic Research Program of China (Grant No. 2013CBA01505). 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[ "Black hole mass and Hamilton-Jacobi counterterms", "Black hole mass and Hamilton-Jacobi counterterms" ]	[ "A Batrachenko \nMichigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA\n\nIntroduction\n\n", "James T Liu [email protected] \nMichigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA\n\nIntroduction\n\n", "R Mcnees [email protected] \nMichigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA\n\nIntroduction\n\n", "W A Sabra \nCenter for Advanced Mathematical Sciences (CAMS) and Physics Department\nAmerican University of Beirut\nLebanon\n", "W Y Wen [email protected] \nMichigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA\n\nIntroduction\n\n" ]	[ "Michigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA", "Introduction\n", "Michigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA", "Introduction\n", "Michigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA", "Introduction\n", "Center for Advanced Mathematical Sciences (CAMS) and Physics Department\nAmerican University of Beirut\nLebanon", "Michigan Center for Theoretical Physics\nRandall Laboratory of Physics\nThe University of Michigan\n48109-1120Ann ArborMIUSA", "Introduction\n" ]	[]	We apply the method of holographic renormalization to computing black hole masses in asymptotically anti-de Sitter spaces. In particular, we demonstrate that the Hamilton-Jacobi approach to obtaining the boundary action yields a set of counterterms sufficient to render the masses finite for four, five, six and seven-dimensional R-charged black holes in gauged supergravities. In addition, we prove that the familiar black hole thermodynamical expressions and in particular the first law continues to holds in general in the presence of arbitrary matter couplings to gravity.	10.1088/1126-6708/2005/05/034	[ "https://export.arxiv.org/pdf/hep-th/0408205v1.pdf" ]	15,423,648	hep-th/0408205	c0ecc4ba6ff06122c8e9be5c4783de6217eb07c0	Black hole mass and Hamilton-Jacobi counterterms 26 Aug 2004 A Batrachenko Michigan Center for Theoretical Physics Randall Laboratory of Physics The University of Michigan 48109-1120Ann ArborMIUSA Introduction James T Liu [email protected] Michigan Center for Theoretical Physics Randall Laboratory of Physics The University of Michigan 48109-1120Ann ArborMIUSA Introduction R Mcnees [email protected] Michigan Center for Theoretical Physics Randall Laboratory of Physics The University of Michigan 48109-1120Ann ArborMIUSA Introduction W A Sabra Center for Advanced Mathematical Sciences (CAMS) and Physics Department American University of Beirut Lebanon W Y Wen [email protected] Michigan Center for Theoretical Physics Randall Laboratory of Physics The University of Michigan 48109-1120Ann ArborMIUSA Introduction Black hole mass and Hamilton-Jacobi counterterms 26 Aug 2004* Emails: [email protected], We apply the method of holographic renormalization to computing black hole masses in asymptotically anti-de Sitter spaces. In particular, we demonstrate that the Hamilton-Jacobi approach to obtaining the boundary action yields a set of counterterms sufficient to render the masses finite for four, five, six and seven-dimensional R-charged black holes in gauged supergravities. In addition, we prove that the familiar black hole thermodynamical expressions and in particular the first law continues to holds in general in the presence of arbitrary matter couplings to gravity. Introduction The AdS/CFT conjecture, where gravity in anti-de Sitter space is holographically dual to a conformal field theory on the boundary, has led to additional interest in black hole thermodynamics. In this context the thermal properties of an AdS black hole configuration are dual to that of the finite temperature CFT. A particularly well studied example of this is the Hawking-Page phase transition [1] for black holes in AdS, corresponding to a deconfinement transition in the dual field theory [2]. A common approach to extracting thermodynamic quantities from the black hole background is to evaluate the on-shell gravitational action, I, as well as the boundary stress tensor T ab , given by T ab = 2 √ −h δI δh ab , (1.1) where h ab is the boundary metric. According to black hole thermodynamics, the on-shell value of the action may be identified with the thermodynamic potential Ω according to I = βΩ. For static backgrounds with the time-like Killing vector ∂/∂t, the energy E is given by the ADM mass, extracted from the tt component of the boundary stress tensor. Black hole thermodynamics has been widely explored in the context of pure Einstein gravity with a cosmological constant. In this case it is well known that the first law of thermodynamics, dE = T dS, holds rather generally. Furthermore, the thermodynamic potential Ω is equivalent to the Helmholtz free energy F , so that F = E − T S is also satisfied. The fact that these features of black hole thermodynamics closely parallel those of ordinary thermodynamics has been the motivation behind the study of AdS/CFT at finite temperature. For AdS/CFT, however, it is necessary to extend the results of black hole thermodynamics to encompass gauged supergravities or more general systems of matter coupled to gravity. In these systems it is possible to turn on conserved R-charges in addition to the temperature. It is then appropriate to work with the grand canonical ensemble, as discussed in [3]. The thermodynamic potential is related to the energy according to Ω = E − T S − Φ I Q I ,(1.2) where Q I are the set of conserved R-charges and Φ I are the corresponding electric potentials (which play the role of chemical potentials). Although it is expected that (1.2) would be satisfied in general, a slight complication arises in that both Ω and E (extracted from the on-shell Euclidean action and the boundary stress-tensor, respectively) are divergent quantities and require renormalization. One approach to dealing with this problem, as suggested by Brown and York [4], is to subtract the divergent action of a reference spacetime from the action for the spacetime of interest. In many cases this technique is sufficient, but it suffers from two main drawbacks. First, it requires that we embed a boundary with intrinsic metric h ab in the reference spacetime, which is often not possible. Second, the procedure is not intrinsic to the spacetime of interest, and all physical quantities are defined with respect to a particular reference spacetime. This becomes problematic when the appropriate reference background is unknown or ambiguous. These problems can be avoided by using the boundary counterterm approach for removing divergences from the action [5,6]. There are two common prescriptions for calculating the boundary counterterms. The first of these involves the asymptotic expansion of bulk fields near the boundary of spacetime. This approach is clearly defined and rigorous; it provides a complete set of covariant counterterms that remove all divergences from the on-shell action [7][8][9][10][11][12][13][14]. The second method, which we employ in this paper, is based on the Hamilton-Jacobi formalism [15][16][17][18][19][20]. The Hamilton-Jacobi formalism, which has found many applications in semi-classical gravity, was first applied in the AdS/CFT context by de Boer, Verlinde, and Verlinde [15]. We will not discuss the motivations and subtleties of this approach; instead we refer the reader to the excellent review by Mück and Martelli in [18]. In this paper we reexamine the familiar asymptotically AdS d black hole solutions of gauged supergravities in 4, 5, 6 and 7 dimensions, and demonstrate how divergences are renormalized through the addition of appropriate Hamilton-Jacobi counterterms. Given a well-defined renormalization scheme, we are able to prove that the relation (1.2) is automatically satisfied for all such black hole solutions. This proof is, in fact, quite general, and is anticipated to remain valid for more general asymptotically AdS d backgrounds. General Charged Black Hole Solutions While details of the various supergravity theories depend crucially on dimension, general features of the bosonic sector can be treated in a dimension independent manner. We may thus consider a general bosonic action for gravity coupled to a set of scalar and vector fields given in the form I[g µν , φ i , A I µ ] = − 1 16πG d M d d x √ −g [R − 1 2 G ij (φ)∂ µ φ i ∂ µ φ j − 1 4 G IJ (φ)F I µν F µν J − V (φ)] + 1 8πG d ∂M d d−1 x √ −h Θ. (2.1) This action is appropriate to a d-dimensional spacetime M with a (d − 1)-dimensional boundary ∂M. The Gibbons-Hawking surface term is given in terms of the trace of the extrinsic curvature Θ µν of the boundary Θ µν = − 1 2 (∇ µ n ν + ∇ ν n µ ), (2.2) where n µ is the outward-pointing normal on ∂M, and h µν is the induced metric. The equations of motion derived from (2.1) are R µν = 1 2 G ij (φ)∂ µ φ i ∂ ν φ j + 1 2 G IJ F I µλ F ν λ J − 1 2(d − 2) g µν F I ρσ F ρσ J + 1 d − 2 g µν V, ∇ µ (G IJ F J µν ) = 0, ∇ µ (G ij ∇ µ φ j ) = 1 2 (∂ φ i G jk )∂ µ φ j ∂ µ φ k + 1 4 (∂ φ i G IJ )F I µν F µν J + ∂ φ i V. (2.3) Since we are interested in spherically symmetric black holes carrying electric charge, in much of the following we choose to work with a field ansatz of the form ds 2 = −e −2(d−3)B(r) f (r)dt 2 + e 2B(r) dr 2 f (r) + r 2 dΩ 2 d−2 , φ i = φ i (r), A I t = A I t (r). (2.4) Anticipating the explicit solutions of interest, we have specialized to a black hole ansatz where the g tt and g rr warp factors are appropriately related. Doing so simplifies some of the intermediate expressions below. However this condition will be relaxed when exploring thermodynamic considerations more generally in section 5. Before proceeding, note that the (d − 2)-sphere may be parametrized as dΩ 2 d−2 = dψ 2 + sin 2 ψdΩ 2 d−3 ,(2.2R ψ ψ = − 1 2(d−2) G IJ F I µν F µν J + 2 d−2 V. (2.6) This expression will prove useful below when evaluating the on-shell action. Stationary R-charged black holes Although the explicit form of the matter sector depends on the theory of interest, the stationary R-charged black holes share a common gravitational description. In particular, the metric of (2.4) has the form ds 2 = −H(r) −(d−3)/(d−2) f (r) dt 2 + H(r) 1/(d−2) dr 2 f (r) + r 2 dΩ 2 d−2 ,(2.7) where f (r) = 1 − µ r d−3 + g 2 r 2 H(r). (2.8) The function H(r) remains to be determined via the equations of motion, and will be influenced by the set of matter fields and charges that are turned on. Nevertheless, in general H(r) admits an expansion in inverse powers of r: H(r) = i H i (r) = 1 + α 1 r d−3 + α 2 r 2(d−3) + α 3 r 3(d−3) + · · · . (2.9) For the solutions considered below the function H(r) may be given explicitly as a product of harmonic functions: H(r) = i H i (r) = i 1 + q i r d−3 . (2.10) In this case, the expansion coefficients in (2.9) are related to the charges q i according to α 1 = i q i , α 2 = i<j q i q j , α 3 = i<j<k q i q j q k , etc.. (2.11) Note that, in the notation of (2.4), the warp factor B(r) is given simply by B(r) = 1 2(d−2) log H(r). We will examine these black holes in more detail in section 4. However, we first turn to the evaluation of the on-shell action, corresponding to the thermodynamic potential Ω. The regulated action and energy We now proceed to evaluate the on-shell action for spherically symmetric configurations of the form (2.4). It is well known that the action diverges due to the behavior of the metric and matter fields near the boundary of an asymptotically AdS d spacetime. Anticipating these infrared divergences, a natural (but non-covariant) way of regulating the calculation is to 'cut-off' the spacetime at a large but finite value of the AdS radial coordinate, r = r 0 . The result is a truncated spacetime, which we denote M 0 , whose 'boundary' ∂M 0 is located at the cut-off. We may now calculate the regulated action on the truncated spacetime. To do so, we consider the bulk and boundary terms separately. For the bulk contribution, we first take the trace of the Einstein equation and substitute it into (2.1) to obtain I bulk = − 1 16πG d M 0 d d x √ −g − 1 2(d − 2) G IJ F I µν F µν J + 2 d − 2 V . (2.12) Using the equation of motion (2.6), this expression may be rewritten as I bulk = − 1 8πG d M 0 d d x √ −gR ψ ψ . (2.13) This may now be evaluated by explicit computation of R ψψ for the black hole metric (2.4). The result turns out to be a total derivative √ −gR ψ ψ = − d dr (r d−2 f (r)B ′ (r) + r d−3 (f (r) − 1)). (2.14) Hence I bulk = βω d−2 8πG d r d−2 f (r)B ′ (r) + r d−3 (f (r) − 1) r 0 r + = βω d−2 8πG d r d−2 0 f (r 0 )B ′ (r 0 ) + r d−3 0 (f (r 0 ) − 1) + r d−3 + ,(2.15) where r + is the location of the horizon, given by f (r + ) = 0. The factor β = 2π/T is the periodicity along the (Euclidean) time direction, and ω d−2 is the volume of the unit (d − 2)-sphere. Turning to the Gibbons-Hawking surface term, we start by noting that the unit normal in the r direction is given by n r = e −B(r) f (r) 1 2 . Using the definition (2.2) the components of the extrinsic curvature tensor are: Θ tt = −h tt e −B(r) f (r) 1 2 −(d − 3)B ′ (r) + f ′ (r) 2f (r) , Θ αβ = −h αβ e −B(r) f (r) 1 2 B ′ (r) + 1 r , (2.16) where indices α, β denote coordinates on the (d − 2)-sphere. The trace of the extrinsic curvature is then given by: Θ = −e −B(r) f (r) 1 2 B ′ (r) + f ′ (r) 2f (r) + d − 2 r . (2.17) The Gibbons-Hawking term, evaluated at the boundary of the regulated spacetime, is: I GH = − βω d−2 8πG d r d−2 0 f (r 0 )B ′ (r 0 ) + 1 2 r d−2 0 f ′ (r 0 ) + (d − 2)r d−3 0 f (r 0 ) . (2.18) Assembling these terms, the regulated value of the on-shell action (2.1) is given by I reg = I bulk + I GH = βω d−2 8πG d −(d − 3)r d−3 0 f (r 0 ) − 1 2 r d−2 0 f ′ (r 0 ) − r d−3 0 + r d−3 + . (2.19) This expression diverges as the cut-off is removed, r 0 → ∞, and must be renormalized by an appropriate counterterm prescription. Before addressing the counterterms, however, we first derive an expression for the unrenormalized ADM energy. To do so, we start with the unrenormalized boundary stress tensor, given by T ab = 2 √ −h δI δh ab = − 1 8πG d (Θ ab − Θh ab ). (2.20) Making use of (2.16), the time-time component of the stress tensor has the form √ −hT tt = − 1 8πG d h tt (d − 2)(r d−2 f (r)B ′ (r) + r d−3 f (r)) (2.21) so that E reg = − ω d−2 8πG d (d − 2)(r d−2 0 f (r 0 )B ′ (r 0 ) + r d−3 f (r 0 )) (2.22) While this ADM energy also diverges as the cut-off is removed, the difference (I reg −βE reg ) is finite in this limit. In other words, the difference between the thermodynamic potential and the energy is a priori finite, and does not need renormalization. Nevertheless, one is often interested in understanding the energy of the system on its own, and in this case a proper choice of counterterms must be made. We now turn to a Hamilton-Jacobi analysis in order to fix the counterterm action. Hamilton-Jacobi counterterms As we have seen above, the on-shell action for gravity on an asymptotically AdS spacetime typically contains infrared divergences related to the behavior of the metric (and any other fields) near the boundary. We now review the calculation of boundary counterterms and demonstrate that the Hamilton-Jacobi method generates appropriate counterterms for canceling all power-law divergences in the on-shell action. In order to facilitate the Hamiltonian analysis it is convenient to foliate this spacetime with constant r hypersurfaces, orthogonal to a spacelike unit normal n µ . The hypersurface defined by the cut-off r = r 0 can be thought of as the 'boundary' of the regulated spacetime, with lim r 0 →∞ ∂M 0 = ∂M. (3.1) Using the Gauss-Codacci equations, the action (2.1) can be rewritten in terms of the intrinsic curvature R of the hypersurfaces and the extrinsic curvature Θ ab describing their embedding in M 0 . Note that, now that we have fixed the normal to point in the r direction, we will use indices a, b, . . . for tensors defined on the constant r hypersurfaces of the foliation. The regulated action is then given by: I = − 1 16πG d M 0 d d x √ −g R + Θ 2 − Θ ab Θ ab − 1 2 G ij (φ)n µ ∂ µ φ i n ν ∂ ν φ j − 1 2 G ij (φ)h ab ∂ a φ i ∂ b φ j − 1 4 G IJ (φ) F ab I F J ab − 1 2 G IJ (φ)h ab n µ F I µa n ν F J νb − V (φ) . (3.2) The action (3.2) is explicitly quadratic in first derivatives of the fields φ i , A I µ , and h µν . Taking into account the holographic principle of flows in the radial direction, we define conjugate momenta and the Hamiltonian with respect to the AdS radial coordinate r, as opposed to the usual choice of a time coordinate. In this case, the momenta conjugate to these fields are given by: π i = 1 16πG d G ij (φ) n µ ∂ µ φ j , π a I = 1 16πG d G IJ (φ) h ab n µ F J µb , π ab = 1 16πG d h ab Θ − Θ ab . (3.3) Using these momenta, the Hamiltonian density obtained from (3.2) is: H = 16πG d 1 2 G ij (φ)π i π j + π ab π ab − 1 d − 2 π a a π b b + 1 2 G IJ h ab π I a π J b + 1 16πG d R − 1 2 G ij (φ)h ab ∂ a φ i ∂ b φ j − V (φ) − 1 4 G IJ F I ab F ab J +G IJ h ab π I a n µ ∂ b A J µ . (3.4) Diffeomorphism invariance of the theory constrains the Hamiltonian (and other generators of coordinate transformations) to vanish. in other words, H[π i , φ i , π a I , A I a , π ab , h ab ] = 0. (3.5) To obtain the Hamilton-Jacobi equation we must rewrite the Hamiltonian constraint in terms of functional derivatives of the on-shell action. The on-shell action is a functional of the bulk fields evaluated at the boundary ∂M 0 . According to Hamilton-Jacobi theory the variational derivative of the on-shell action with respect to a field's boundary value gives the momenta conjugate to that field, evaluated at ∂M 0 . Thus, the momenta (3.3) can be written as functional derivatives of the on-shell action: π i = 1 √ −h δI δφ i , π a I = 1 √ −h δI δA I a , π ab = 1 √ −h δI δh ab , (3.6) where the fields in (3.6) are evaluated at r 0 . Finally, replacing the momenta appearing in the Hamiltonian with functional derivatives of the on-shell action, we obtain the Hamilton-Jacobi equation: H δI δφ i , φ i , δI δA I a , A I a , δI δh ab , h ab = 0. (3.7) The Hamilton-Jacobi equation is a functional differential equation for the on-shell action in terms of the boundary values of the bulk fields. Derivation of the counterterm action Using the Hamilton-Jacobi equation, we can obtain a set of counterterms that will remove power-law divergences from the on-shell action. We first write the regulated on-shell action as: I reg = Γ − I ct (3.8) The first term, Γ, represents the part of the action which is finite 1 upon removing the cut-off. The second term, I ct , represents the power-law divergences appearing in the action. The terms appearing in I ct are conveniently organized in terms of an inverse metric expansion, as described in [18]. A sufficient counterterm action for the gauged supergravity black hole solutions we consider is given by: I ct = 1 8πG d ∂M 0 d d−1 x √ −h W (φ) + C(φ) R + D(φ)R 2 + E(φ)R ab R ab . (3.9) The first two terms contain the divergences that appear in four and five dimensions, while in six and seven dimensions it is necessary to include the remaining terms. In constructing this action we have discarded a number of possible gradient counterterms of the form M ij (φ)∂ a φ i ∂ a φ j , etc., because the scalar fields only depend on the AdS radial coordinate r. In addition, since the counterterm action should respect any residual bulk symmetries, the U(1) gauge fields should only appear in terms of gauge-invariant field strengths F I ab . These terms do not contribute to (3.9) for the electrically charged configurations given by the ansatz (2.4). It is important to note, however, that if one wishes to study fluctuations around the black hole backgrounds then such counterterms must be included in (3.9), since the fluctuations may depend on the transverse coordinates. For such cases, the counterterm action (3.9) alone is not sufficient for calculations of correlators in the field theory duals of these solutions. The momenta can be decomposed into contributions from the terms in (3.8), schematically of the form: π = π Γ − P. (3.10) The contributions P due to the counterterm action are given by functional derivatives of (3.9) with respect to the fields on ∂M 0 : P i = 1 √ −h δI ct δφ i = 1 8πG d ∂ W ∂φ i + ∂ C ∂φ i R + ∂ D ∂φ i R 2 + ∂ E ∂φ i R ab R ab , P ab = 1 √ −h δI ct δh ab = 1 8πG d 1 2 h ab W − C G ab + 1 2 h ab D R 2 + 1 2 h ab E R cd R cd −2D RR ab + 2ER dba c R c d − E ∇ c ∇ c R ab . (3.11) The term G ab appearing in the expression for P ab is the boundary Einstein tensor, given by: G ab = R ab − 1 2 h ab R (3.12) The counterterms W (φ), C(φ), . . . are now determined by substituting these momenta into the Hamilton-Jacobi equation (3.7) and solving it order-by-order in the expansion (3.9). We denote the various terms in the Hamiltonian by H = H (0) + H (1) + H (2) + . . . + H Γ . (3.13) The terms H (i) represent contributions from I ct , with the index counting the number of inverse metrics appearing in that term. For the backgrounds we are interested in this is an adequate measure of the degree of divergence these terms represent. Evaluating these terms leads to differential equations for the functions appearing in (3.9). The most illuminating of these is the equation for W (φ) that comes from the term H (0) in the Hamiltonian constraint: H (0) = 1 16πG d 2 G ij (φ) ∂ W ∂φ i ∂ W ∂φ j − d − 1 d − 2 W 2 − V .(3.14) Setting H (0) = 0 recasts (3.14) as the familiar relation for the potential V (φ) in terms of the superpotential W (φ): V = 2G ij (φ) ∂W ∂φ i ∂W ∂φ j − d − 1 d − 2 W 2 . (3.15) The conclusion is that the leading term in the counterterm action (3.9) is simply proportional to the superpotential W (φ) [15]. We obtain similar equations for the functions C(φ), D(φ), and E(φ) by evaluating the remaining terms in (3.13). The equation derived from H (1) = 0 determines C(φ) in terms of W (φ): 1 2 + 2G ij (φ) ∂ W ∂φ i ∂ C ∂φ j − d − 3 d − 2 C W = 0. (3.16) The counterterms W (φ) and C(φ), determined by equations (3.15) and (3.16), completely characterize the power-law divergences in four and five dimensions. For the six and seven dimensional supergravities there are two additional counterterms whose coefficients D(φ) and E(φ) are determined by two equations obtained from functionally independent terms in the equation H (2) = 0: − G ij (φ) ∂ C ∂φ i ∂ C ∂φ j − 2G ij (φ) ∂ W ∂φ i ∂ D ∂φ j + d − 5 d − 2 D W + d − 1 2(d − 2) C 2 = 0, −2G ij (φ) ∂W ∂φ i ∂ E ∂φ j + d − 5 d − 2 E W − 2C 2 = 0. (3.17) In five dimensions, where D(φ) and E(φ) are not included in the counterterm action (3.9), H (2) actually represents a potentially non-vanishing term in the expansion (3.13) for H: H (2) = 1 8πG d G ij (φ) ∂ C ∂φ i ∂ C ∂φ j R 2 + 2 C 2 (R ab R ab − 1 3 R 2 ) . (3.18) In principle such a term might signal the presence of a logarithmic divergence in the on-shell action, corresponding to a Weyl anomaly in the dual field theory. However, for the solutions we are interested in the terms appearing in (3.18) either vanish due to the S 1 × S d−2 topology of the boundary, or vanish sufficiently rapidly near the boundary so as to not contribute any additional divergences to the effective action. While we have shown, in equation (3.15), that the leading counterterm is simply the superpotential W (φ), we have not provided explicit solutions for the remaining terms. For the gauged supergravity solutions we are interested in it is sufficient to solve for the functions C(φ), D(φ), and E(φ) as a power series in φ i , out to order O(φ 2 ). However, rather than writing general solutions, which would depend on the choice of basis for the gauged supergravity scalars, we will specialize to an appropriate expansion for each of the d-dimensional black holes that we consider in the next section. Finally, it should be noted that the functions C(φ), D(φ), and E(φ) can be written in terms of integrals of the superpotential and its derivatives, but these expressions are not particularly illuminating. Counterterm renormalization of the energy Since the counterterm W (φ) is simply related to the superpotential according to (3.15), its form is already determined. For the remaining counterterms, C(φ), D(φ) and E(φ), their solutions as power series expressions may be motivated by noting that the large r asymptotics of the black hole solution (2.4) generically has the form f (r) ∼ g 2 r 2 , B(r) ∼ 1 r d−3 , φ i (r) ∼ 1 r d−3 , A I t (r) ∼ 1 r d−3 . (3.19) To cancel divergences, and to provide possibly finite counterterms, the series solution to C(φ) must be determined to O(1/r d−3 ) while the series solutions to D(φ) and E(φ) must be determined to O(1/r d−5 ). As a result, only the leading terms will be important C(φ) = c 0 + c i φ i + unimportant, D(φ) = d 0 + unimportant, E(φ) = e 0 + unimportant. (3.20) Assuming, from symmetry, that the linear term c i is absent in C(φ), we only need to compute the constant pieces c 0 , d 0 and e 0 from (3.16) and (3.17). As a result, we find that the relevant contribution of the counterterm action has the form I ct = 1 8πG d d d−1 x √ −h W (φ) + 1 2(d − 3)g R + 1 2(d − 5)(d − 3) 2 g 3 R ab R ab − d − 1 4(d − 2) R 2 + · · · . (3.21) This can be compared with similar expressions for pure gravitational backgrounds, as found in [5,6]. Note that g is also the inverse of the AdS length scale, ℓ −1 , which is given in terms of the constant term V 0 in the scalar potential V (φ) by: ℓ = − (d − 1)(d − 2) V 0 . (3.22) Corresponding to the counterterms in (3.21), the regulated boundary stress tensor picks up an additional contribution T ab ct = 1 8πG d h ab W (φ) − 1 2(d − 3)g (2R ab − Rh ab ) + 1 2(d − 5)(d − 3) 2 g 3 4R acbd R cd − d − 1 d − 2 R ab R + h ab R cd R cd − d − 1 4(d − 2) R 2 . (3.23) Some terms proportional to derivatives of R along the boundary have been omitted, as they vanish for the spherically symmetric solutions of interest. For black hole metrics of the form (2.4), the boundary Ricci tensor is given by R tt = 0, R αβ = (d − 3)h αβ e −2B r −2 . (3.24) Thus the counterterm action and contribution to the energy may be expressed as E ct = I ct β = ω d−2 8πG d e B f 1 2 r d−2 W (φ) + (d − 2) 2 g e −2B r d−4 − (d − 2) 8 g 3 e −4B r d−6 + · · · , (3.25) where E ct = ω d−2 √ −h h tt T ct tt (3.26) is the counterterm contribution to the ADM energy. The relation I ct = βE ct demonstrates that, while the counterterms are necessary to render both the action and the energy finite, the validity of the thermodynamic relation Ω = E − T S − Q I Φ I is unaffected by any finite shift in the counterterm action. The renormalized action and mass Given the regulated action (2.19) and energy (2.22), as well as the corresponding counterterm expressions (3.21) and (3.25), we are now in a position to examine the various R-charged black holes. In each case we calculate the renormalized action Γ and energy E ren , and show that they are finite in the r 0 → ∞ limit. D = 4 black holes In four dimensions, the N = 2 truncation of gauged N = 8 supergravity yields a system with three complex scalars and four U(1) gauge fields. For simplicity, we consider a truncation of the scalar sector by setting the axionic components to zero. While this is in principle an inconsistent truncation, this is nevertheless a valid procedure when applied to the non-rotating electrically charged black holes. In this case, the three dilatonic scalars may be parametrized by a constrained set of real fields X i satisfying X 1 X 2 X 3 X 4 = 1. The potential and superpotential are then given by V = −g 2 i<j X i X j , W = 1 2 g i X i . (4.1) In addition to the metric, (2.7), the four-charge black holes have gauge potentials and scalars given by [21][22][23] A i (1) = q i + µ q i 1 − 1 H i dt, X i = H 1/4 H i . (4.2) As a result, the regulated action integral, (2.19), becomes I reg = βω 2 8πG 4 −2g 2 r 3 0 − 3 2 g 2 α 1 r 2 0 − (2 + g 2 α 2 )r 0 + 1 2 µ − 1 2 g 2 α 3 + r + (4.3) Note that the first three terms are divergent as r 0 → ∞. Of course, the boundary counterterm remains to be evaluated. To do so, we simply insert the form of the superpotential, given by (4.1), into (3.25) to obtain βE ct = I ct = βω 2 8πG 4 2g 2 r 3 0 + 3 2 g 2 α 1 r 2 0 + (2 + g 2 α 2 )r 0 − µ + 1 2 g 2 α 3 . (4.4) We now see explicitly that the divergent terms in the regulated action are canceled by the counterterms. Furthermore, the nonlinear charge term, proportional to α 3 , vanishes in the renormalized action Γ = I reg + I ct = βω 2 8πG 4 − 1 2 µ + r + . (4.5) Turning to the ADM energy, we first evaluate (2.22) in four dimensions to obtain E reg = ω 2 8πG 4 −2g 2 r 3 0 − 3 2 g 2 α 1 r 2 0 − (2 + g 2 α 2 )r 0 + 2µ + 1 2 α 1 − 1 2 g 2 α 3 . (4.6) Combining this with E ct yields the linear mass/charge relation E ren = ω 2 8πG 4 µ + 1 2 α 1 = ω 2 8πG 4 µ + 1 2 (q 1 + q 2 + q 3 + q 4 ) . (4.7) D = 5 black holes As in the ungauged case, gauged D = 5, N = 2 supergravity coupled to an arbitrary number of vector multiplets has a natural description in terms of special geometry. Here we consider only the particular case of the STU model, corresponding to the U(1) 3 truncation of maximal gauged supergravity. The black holes in this model, which may carry up to three charges, have been well studied [3]. The counterterm renormalization prescription for black holes in the STU model was recently examined in [24] for single-charge black holes and in [25] for three-charge black holes. In this model, the potential and superpotential are given by V = −4g 2 i<j X i X j = −4g 2 i 1 X i , W = g i X i ,(4.8) where the two real scalars are encoded in the constrained fields X 1 X 2 X 3 = 1. Furthermore, the gauge potentials and scalars have the form [26,27] A i (1) = q i + µ q i 1 − 1 H i dt, X i = H 1/3 H i . (4.9) Working out the regulated action and energy, we find the similar expressions I reg = βω 3 8πG 5 −3g 2 r 4 0 − (3 + 2g 2 α 1 )r 2 0 + µ − g 2 α 2 + r 2 + , E reg = ω 3 8πG 5 −3g 2 r 4 0 − (3 + 2g 2 α 1 )r 2 0 + 3µ + α 1 − g 2 α 2 . (4.10) The divergences are renormalized by the counterterm action βE ct = I ct = βω 3 8πG 5 3g 2 r 4 0 + (3 + 2g 2 α 1 )r 2 0 − 3 2 µ + 3 8g 2 + g 2 α 2 . (4.11) Consequently, we find the familiar results Γ = βω 3 8πG 5 − 1 2 µ + r 2 + + 3 8g 2 ,(4.12) and E ren = ω 3 8πG 5 3 2 µ + α 1 + 3 8g 2 = ω 3 8πG 5 3 2 µ + q 1 + q 2 + q 3 + 3 8g 2 . (4.13) D = 6 black holes In six dimensions, the gauged N = (1, 1) supergravity admits two inequivalent AdS vacua [28], only one which is supersymmetric. It is this one that we consider. The bosonic components of the supergravity multiplet consists of a graviton g µν , antisymmetric tensor B µν , SU(2) × U(1) gauge fields A I µ , A µ and a dilaton φ. The potential and superpotential have the form V = −g 2 9X 2 + 12 X 2 − 1 X 6 , W = g 3X + 1 X 3 (4.14) where X = e − 1 2 √ 2 φ . The gauging of [28] which leads to an AdS 6 vacuum also turns on a mass for the antisymmetric tensor. More directly, the abelian vector A µ is absorbed by B µν for mass generation. Thus we only consider abelian black holes charged under the U(1) subgroup of SU (2). The gauge potential and dilaton are given by A 3 (1) = q + µ q 1 − 1 H dt, X = H −1/4 . (4.15) Note, however, that H = H 2 , so that α 1 = 2q and α 2 = q 2 . The regulated six-dimensional action is I reg = βω 4 8πG 6 −4g 2 r 5 0 − 4r 3 0 − 5g 2 qr 2 0 + 3 2 µ + r 3 + ,(4.16) The regulated ADM energy is similarly This is the first case when the curvature-squared counterterms turn out to be important. We end up with simple expressions for the regulated action and ADM energy E reg = ω 4 8πG 6 −4g 2 r 5 0 − 4r 3 0 − 5g 2 qr 2 0 + 3q + 4µ .Γ = βω 4 8πG 6 − 1 2 µ + r 3 + , E ren = ω 4 8πG 6 (2µ + 3q) . (4.19) Note the absence of any Casimir energy for the odd-dimensional boundary theory. D = 7 black holes Maximal gauged supergravity in seven dimensions involves the gauging of an SO(5) Rsymmetry, as can be deduced from the S 4 reduction of eleven-dimensional supergravity. This can be truncated to half-maximal supergravity (with SU(2) gauging) coupled to an abelian vector multiplet. For simplicity, however, we consider a further truncation to two abelian vectors and two scalars. In general, this is no longer a consistent supergravity theory. However, it is consistent to consider a subset of solutions, including the electrically charged black holes of present interest. Because of the slightly unusual nature of the truncated theory, the potential has a more complicated structure [23,29] V = −2g 2 8X 1 X 2 + 4 X 2 1 X 2 + 4 X 1 X 2 2 − 1 X 4 1 X 4 2 ,(4.20) where X 1 and X 2 are unconstrained fields. In terms of canonically normalized scalars, we may take the representation X 1 = e 1 √ 10 ϕ 1 + 1 √ 2 ϕ 2 , X 2 = e 1 √ 10 ϕ 1 − 1 √ 2 ϕ 2 . (4.21) The superpotential has the form W = g 2X 1 + 2X 2 + 1 X 2 1 X 2 2 . (4.22) For the R-charged black holes, the two gauge potentials and two scalars are given in terms of the harmonic functions H i by [23,29] A i (1) = q i + µ q i 1 − 1 H i dt, X i = H 2/5 H i . (4.23) This yields the expression for the superpotential W = gH 2/5 2 H 1 + 2 H 2 + 1 . (4.24) The resulting regulated on-shell action is 25) and the regulated ADM energy is I reg = βω 5 8πG 7 −5g 2 r 6 0 − 5r 4 0 − 3g 2 α 1 r 2 0 + 2µ + r 4 + ,(4.E reg = ω 5 8πG 7 −5g 2 r 6 0 − 5r 4 0 − 3g 2 α 1 r 2 0 + 5µ + 2α 1 . (4.26) Note that these expressions are already at most linear in the charges. In six or higher dimensions, the asymptotic scalar behavior falls off sufficiently rapidly so that the scalars do not contribute to the boundary counterterm. We find βE ct = I ct = βω 5 8πG 7 5g 2 r 6 0 + 5r 4 0 + 3g 2 α 1 r 2 0 − 5 2 µ − 5 16g 4 ,(4.27) so that the renormalized values are Γ = βω 5 8πG 7 − 1 2 µ + r 4 + − 5 16g 4 ,(4.28) and E ren = ω 5 8πG 7 5 2 µ + 2α 1 − 5 16g 4 . (4.29) Black hole energy and thermodynamics In the previous sections we demonstrated explicitly that the on-shell action and the ADM energy may be renormalized by introducing an appropriate counterterm action given by a Hamilton-Jacobi analysis. Turning to the dual field theory, the renormalized on-shell action is to be identified with the thermodynamic potential Ω according to Γ = β Ω. Likewise, the ADM energy E ren ought to be identified with the energy (including Casimir energy) of the field theory. For backgrounds with non-trivial R-charge, the thermodynamic potential may be related to the energy according to Ω = E − T S − Φ I Q I ,(5.T tt = 1 8πG d (Θ tt − Θh tt ),(5.11) so that the ADM energy is E reg = ω d−2 8πG d √ −h(−Θ t t + Θ). (5.12) In addition, the entropy and temperature are given by S = 1 4G d A r + = ω d−2 8πG d 2πe (d−2)B r d−2 r + , T = 1 4π e A−B df dr r + . (5.13) Hence T S = ω d−2 8πG d 1 2 e A+(d−3)B r d−2 df dr r + = − ω d−2 8πG d √ −hΘ t t r + (5.14) Here we have used the expression √ −hΘ t t = −e A+(d−3)B r d−2 f dA dr + 1 2 r d−2 df dr , (5.15) which is valid for the metric (5.2). Note that the first term in Θ t t vanishes at the horizon (at least for a regular horizon). Combining the above expressions, and substituting into (5.9), we finally obtain the expected relation Ω reg = E reg − T S − Φ I Q I . (5.16) Note that the unrenormalized quantities Ω reg and E reg both diverge as we remove the regulator, r 0 → ∞. However, this divergence is cancelled by a counterterm action (3.9) which contributes equally with Ω ct = E ct . Hence, the thermodynamic relation (5.1) always holds identically, with or without counterterm insertion, at least for the counterterm structures that we are interested in. Conclusions In general, the notion of mass or energy in a gravitational system can be rather difficult to define in a precise and useful manner. Nevertheless, rigorous definitions of energy and conserved charges are essential in the application of black hole thermodynamics. Here we have highlighted a holographic approach, based on the Hamilton-Jacobi formalism, to dealing with black holes in asymptotically AdS spacetimes. In this approach, conserved quantities (including the mass) may be extracted from the boundary stress tensor, so long as the gravitational action itself is regulated in an appropriate manner. We demonstrate, in particular, that the Hamilton-Jacobi method generates the appropriate boundary counterterms for removing all divergences of the on-shell action pertaining to stationary R-charged AdS black holes in four, five, six and seven-dimensional gauged supergravities. Although the importance of the boundary stress tensor method has been realized for some time, and the notion of holographic renormalization has been well developed, less attention has been given to systems with a non-trivial matter sector. In this paper, we have focused on gravitational systems with long-range scalars, and have shown that they may be treated in a uniform manner, regardless of spacetime dimension or specific matter content. Of the actual black holes we have investigated, we note that the nontrivial scalar counterterms (namely the non-constant parts of the superpotential W ) are divergent in four, finite in five, and vanishing in higher dimensions. And yet they all have a common origin, namely the Hamiltonian constraint (3.14) arising from the Hamilton-Jacobi analysis of the counterterms, It would of course be natural to apply the Hamilton-Jacobi counterterm prescription developed here to the study of thermodynamics of other interesting systems with nontrivial matter fields. For example, masses of rotating supersymmetric AdS 5 black holes [30,31] was recently considered in [25]. It may also be of interest to apply the above methods in examining the properties of the five-dimensional black ring solutions [32][33][34][35]. Finally, note that the boundary stress tensor contains information not just on the energy of the system but also on general conserved quantities corresponding to additional Killing symmetries. In particular, angular momentum along with the thermodynamics of rotating solutions has been explored in [36][37][38][39] (see also [40][41][42][43][44]). For stationary solutions, the analysis of the previous section indicates that any suitably chosen regulator will preserve the thermodynamic relation (1.2). However, the introduction of angular momentum yields additional complications meriting further study [39]. The full resolution of the first law of black hole thermodynamics in the AdS/CFT context with rotation will certainly be an important accomplishment with widespread implications. ( 4 . 17 ) 417At the same time, evaluation of the boundary counterterm, (3.25), yields βE ct = I ct = βω 4 8πG 6 4g 2 r 5 0 + 4r 3 0 + 5g 2 qr 2 0 − 2µ(4.18) 1 ) 1where Q I are the set of conserved R-charges, and Φ I are the corresponding horizon values of the electric potential. Here we prove that the relation (5.1) is automatically satisfied for the black hole solutions of the previous section.We start with a static, stationary metric in d dimensions, of the form (2.4) temperature with entropy, and the last term gives directly the product Φ I Q I up to charge More explicitly, the tt component of the regulated boundary stress tensor,(2.20), isds 2 d = −e 2A f dt 2 + e 2B dr 2 f + r 2 dΩ 2 d−2 . (5.2) normalization Q I = βω d−2 16πG d q I . (5.10) In general Γ might contain logarithmic divergences. These divergences, which are related to the Weyl anomaly in the dual field theory, can be addressed using the Hamilton-Jacobi approach. However, the gauged supergravity solutions we consider are free of such divergences. t (r + ). It is now clear that the first term, proportional to (−Θ t t +Θ), may be related to the ADM energy, the second term may be related to the product of AcknowledgmentsThis material is based upon work supported by the National Science Foundation under grant PHY-0313416 and by the US Department of Energy under grant DE-FG02-95ER40899. JTL and WS wish to acknowledge the hospitality of the Khuri lab at the Rockefeller University, where part of this work was completed.Note, however, that here we allow independent warp factors for the time and space directions. This choice of coordinates is specialized so that the boundary of AdS is located at r → ∞ and also so that ∂/∂t is a natural time-like Killing vector. We further assume that the matter sector preserves the time translation symmetry, so that in particular all matter fields are independent of t.As in (2.1) the unrenormalized action integral is composed of two pieces, the bulk integral and the surface term. 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[ "Hierarchical Reinforcement Learning Based Video Semantic Coding for Segmentation 1 nd", "Hierarchical Reinforcement Learning Based Video Semantic Coding for Segmentation 1 nd" ]	[ "Guangqi Xie ", "Xin Li ", "Shiqi Lin ", "Zhibo Chen [email protected] ", "Cn ", "Li Zhang ", "Kai Zhang ", "Yue Li [email protected] ", "\nUnviersity of Science and Technology of China\nHefeiChina\n", "\nByteDance Inc\n\n" ]	[ "Unviersity of Science and Technology of China\nHefeiChina", "ByteDance Inc\n" ]	[]	The rapid development of intelligent tasks, e.g., segmentation, detection, and classification, etc, has brought an urgent need for semantic compression, which aims to reduce the compression cost while maintaining the original semantic information. However, it is impractical to directly integrate the semantic metric into the traditional codecs since they cannot be optimized in an end-to-end manner. To solve this problem, some pioneering works have applied reinforcement learning to implement image-wise semantic compression. Nevertheless, the video semantic compression has not been explored since its complex reference architectures and compression modes. In this paper, we take a step forward to video semantic compression and propose the Hierarchical Reinforcement Learning based taskdriven Video Semantic Coding, named as HRLVSC. Specifically, to simplify the complex mode decision of video semantic coding, we divided the action space into frame-level and CTU-level spaces in a hierarchical manner, and then explore the best mode selection for them progressively with the cooperation of framelevel and CTU-level agents. Moreover, since the modes of video semantic coding will exponentially increase with the number of frames in a Group of Pictures (GOP), we carefully investigate the effects of different mode selections for video semantic coding, and design a simple but effective mode simplification strategy for it. We have validated our HRLVSC on video segmentation task with HEVC reference software HM16.19. Extensive experimental results demonstrated that our HRLVSC can achieve over 39% BD-rate saving for video semantic coding under the Low Delay P configuration.	10.1109/vcip56404.2022.10008806	[ "https://export.arxiv.org/pdf/2208.11529v1.pdf" ]	251,765,191	2208.11529	7031c450840395e171d99212bbc59d124a2ee798	Hierarchical Reinforcement Learning Based Video Semantic Coding for Segmentation 1 nd Guangqi Xie Xin Li Shiqi Lin Zhibo Chen [email protected] Cn Li Zhang Kai Zhang Yue Li [email protected] Unviersity of Science and Technology of China HefeiChina ByteDance Inc Hierarchical Reinforcement Learning Based Video Semantic Coding for Segmentation 1 nd Index Terms-semantic video coding for segmentationrate- distortion optimizationhierarchical reinforcement learning The rapid development of intelligent tasks, e.g., segmentation, detection, and classification, etc, has brought an urgent need for semantic compression, which aims to reduce the compression cost while maintaining the original semantic information. However, it is impractical to directly integrate the semantic metric into the traditional codecs since they cannot be optimized in an end-to-end manner. To solve this problem, some pioneering works have applied reinforcement learning to implement image-wise semantic compression. Nevertheless, the video semantic compression has not been explored since its complex reference architectures and compression modes. In this paper, we take a step forward to video semantic compression and propose the Hierarchical Reinforcement Learning based taskdriven Video Semantic Coding, named as HRLVSC. Specifically, to simplify the complex mode decision of video semantic coding, we divided the action space into frame-level and CTU-level spaces in a hierarchical manner, and then explore the best mode selection for them progressively with the cooperation of framelevel and CTU-level agents. Moreover, since the modes of video semantic coding will exponentially increase with the number of frames in a Group of Pictures (GOP), we carefully investigate the effects of different mode selections for video semantic coding, and design a simple but effective mode simplification strategy for it. We have validated our HRLVSC on video segmentation task with HEVC reference software HM16.19. Extensive experimental results demonstrated that our HRLVSC can achieve over 39% BD-rate saving for video semantic coding under the Low Delay P configuration. I. INTRODUCTION Deep learning has brought a technological revolution in visual intelligent tasks, such as image/video object detection [1]- [4], segmentation [5]- [8], face detection [9], [10], and person reidentification [11]- [13], etc, which enables the intelligent tasks to be greatly developed and broadly applied in people's lives. Meanwhile, billions of images/videos for intelligent tasks have caused a heavy burden for transmission and storage. However, the commonly-used image/video codecs, such as JPEG, AVC [14], HEVC [15], VVC [16], and related techniques [17], [18] are usually designed and optimized with pixel-level metrics, such as PSNR [19], and perceptual metrics [20], [21], which are not optimal for semantic compression. It is crucial to investigate how to reduce the compression cost while maintaining the semantic information * Corresponding author of images/videos according to specific intelligent tasks (i.e., task-driven semantic coding). Different from the pixel-wise fidelity (e.g., PSNR ) and perceptual fidelity (e.g., MS-SSIM), task-driven semantic fidelity is hard to be integrated into traditional codecs directly. The reason lies in that traditional codecs cannot be optimized in an end-to-end manner such as learning-based coding(e.g., LBHIC [22]). Furthermore, traditional codecs conduct image/video compression on pixel level, while semantic information cannot be represented with simple pixel-wise metrics like PSNR. To tackle the above challenges, some works [23], [24] take a step forward to task-driven semantic coding, and then introduce the reinforcement learning to implement the Rate-Distortion Optimization (RDO) of semantic coding. Despite that these studies have achieved great performances on many image intelligent tasks, (including image segmentation, detection and classification), their methods are only designed for the all-intra mode of semantic coding instead of unified video semantic coding. Therefore, task-driven video semantic coding is still under-explored. Compared with image coding, where the compression of different coding tree units (CTUs) are relatively independent [25], video compression contains amounts of compression modes and reference architectures. It is complex and timeconsuming to find the optimal mode selection for task-driven video semantic coding. To simplify the mode decision for video compression, previous works [26]- [36] have investigated the effects of the coding modes of current frame on the following frames in one GOP, and then, model the distortion or rate propagation with linear function [26]- [32], deep learning model [33] or reinforcement learning agent [34]- [36]. However, the above methods only explored the relationship between different mode selections and pixel-wise fidelity, which is not suitable for task-driven semantic fidelity. In this paper, we take a step forward to task-driven video semantic coding. To integrate the video semantic fidelity to the rate-distortion optimization of traditional codecs and solve the challenges brought by tremendous amounts of mode selections, we propose the hierarchical reinforcement learning based task-driven video semantic coding, named as HRLVSC. Specifically, we divide the mode selections of video compression into frame level and CTU level in a hierarchical manner, and then find the best mode selection for task-driven The hierarchically structured network generates optimal task-driven compression modes from parent level to child level progressively. In the first stage, the parent-level agent takes three consecutive frames and corresponding masks as input, extracts deep features and then predicts the quantization parameters(QP) for each frame. Finer control is at the next stage. The child-level agent extracts corresponding features and outputs the relative QP around the central frame-level QP for each CTU. Finally, traditional codec takes selected modes from both levels and conducts the optimal task-driven compression. The actual rate and semantic fidelity of each level are used as rewards to train the agents in a coarse-to-fine manner. video semantic coding progressively with the cooperation of frame-level and CTU-level agents. Moreover, to simplify the mode decision for video semantic coding, we also carefully investigate the effects of different compression modes for taskdriven video semantic coding. Based on our exploration, we design an efficient but effective mode simplification strategy for video semantic coding. To validate the effectiveness of our HRLVSC, we select video segmentation as target task. Extensive experiments under the Low Delay P configuration have demonstrated that our HRLVSC can achieve over 39% BD-rate saving for video semantic coding. The main contributions of our work can be summarized as follows: • As the pioneering work, we propose the hierarchical reinforcement learning based video semantic coding(i.e., HRLVSC) for segmentation, where the complex mode space is simplified into frame-level and CTU-level, and the optimal mode selection is learned with the cooperation of frame-and CTU-level agents in a progressive manner. • We carefully explore the correlation between different mode selections and the semantic fidelity, and propose an efficient but effective mode simplification strategy for task-driven video semantic coding. • Extensive experiments on video segmentation task under low-delay P configuration have demonstrated the superiority of our proposed HRLVSC, which exceeds the standard software HM16.19 1 by a BD-rate saving of 39%. The rest of the paper is organized as follows. In sec. II, we clarify our task-driven video semantic coding scheme HRLVSC in detail. Sec. III describes our experimental setting 1 Available:https://hevc.hhi.fraunhofer.de/svn/svn HEVCSoftware/tags/HM-16.19/ and validates the effectiveness of our proposed HRLVSC by comparing it with the state-of-the-art codecs and a series of ablation studies. Finally, we conclude this paper in Section IV. II. PROPOSED METHOD In this section, we will introduce our hierarchical reinforcement learning based video semantic coding scheme (i.e., HRLVSC) from the perspective of problem formulation, technique details and mode simplification. A. Problem formulation Task-driven video semantic coding aims to reduce the computation cost while maintaining the semantic information existed in videos, which can be formulated as: min J s , J s = D(M) + λ s T f t=1 N i=1 R t,i (M),(1) where J s , D(M) and R t,i (M) are rate-distortion performance, the semantic distortion of whole video and the rate of the i th CTU in the t th frame, respectively. M represents the selected mode for compression, and λ s is the hyperparameter to adjust the importance of rate and distortion. T f and N are the number of frames and CTUs in one frame, respectively. To find the best mode M * for Eq. 1, a straightforward method is to utilize a reinforcement learning agent to explore the optimal mode adaptively like the work [23]. However, the optional modes for video semantic compression will exponentially increase with the number of frames and CTUs, which inevitably prevents the learning of RL agent. To simplify the exploration space and enables the RL agent to learn the optimal mode effectively and efficiently, we divide the exploration space into frame level and CTU level, and then, introduce the hierarchical reinforcement learning to solve the Eq. 1. Specifically, as shown in Eq. 2, we set two-step goals respectively for parentlevel agent and child-level agent. The parent-level agent RL p aims to explore the best frame-level mode M f * in pursuit of minimizing the frame-level rate-distortion J sf . And the childlevel agent RL c is devoted to minimizing the final semantic rate-distortion performance J s by finding the best CTU-level mode M c * while cooperating with parent-level agent. J sf = D(M f ) + λ s T f t=1 R t (M f ) J s = D(M c \|M * f ) + λ s T f t=1 N i=1 R t,i (M c \|M * f ) M * f = RL p (min J sf ), M * c = RL c (min J s )(2) With hierarchical reinforcement learning in Eq. 2, we can obtain the best mode pair {M f * , M c * } for task-driven semantic coding effectively and efficiently. In this paper, the framelevel mode M f and CTU-level mode M c are the Quantization Parameter (QP) of one frame and the relative QP (i.e., ∆QP ) around the central frame-level QP for each CTU. B. Hierarchical Reinforcement Learning Based Video Semantic Coding for Segmentation We aim to utilize the parent-level agent and child-level agent to explore the optimal frame-level mode and CTU-level mode based on the former, respectively. Therefore, we model this decision-making problem as a hierarchical MDP process that aligns well with the nature of hierarchical reinforcement learning (HRL). A one-level RL agent commonly models the policy learning problem as a Markov decision process (MDP) represented with (S, A, P, R, γ, T ). The RL agent observes the environment state s ∈ S and relies on the learnable policy π(a \|s) : S × A → [0, 1] to take an action a ∈ A. Then, the RL agent receives a step-wise reward r : S × A → R. The environment moves to next state with a transition function denoted as P : S × A × S → [0, 1]. γ ∈ (0, 1] is a discount factor and T is a time horizon. We aim to learn an optimal policy π * which can maximize the accumulative reward R. Furthermore, HRL contains two-level RL agents, i.e., the parent-level (frame-level) agent and the child-level (CTUlevel) agent, so as to learn a parent policy π P (a P s P ) and a child policy π C (a C s C , a P ). The parent policy outputs a parent action a P ∈ A P , which is taken as the condition for the following decision by the child policy. Thanks to the parentlevel agent which first makes decisions at frame-level, the action space of child-level agent can be effectively reduced. With a smaller action space, it's easier for the CTU-level policy to find a more effective strategy. Here, we give the detailed design of HRL for task-driven video semantic coding. State: Both the parent-level policy and the child-level policy need to perceive the content of the frames. Therefore, given three consecutive frames and corresponding masks, we concatenate them and utilize the deep features extracted by convolution network as the parentlevel state and the child-level state. Action: The parent-level agent is responsible for assigning QP for a frame in a coarse way. QP is a central value within a range. Based on the decision of the parent-level agent, the child-level agent further determines QP around the central value for each CTU in the frame. Reward: The parent-level agent and the child-level agent target at minimizing the impact of compression on the performance of downstream network. We define the reward functions of parent-level and child-level policies as: Rw f = M s (M f ) − λ s T f t=1 R t (M f ) − α f Rw c = M s M c \| P * f − λ s T f t=1 N i=1 R t,i M c \| M * f − α c (3) where the frame-level reward Rw f is expressed as the sum of two terms: the task-related fidelity M s (M f ), which is measured by the Mean Intersection over Union (mIOU) for video segmentation, and the negative sum of rate R t (M f ) of each frame. α f and α c are hyperparameters to keep the initial reward close to zero, and then stimulus the agent to explore more optimal actions. λ s is an adjustable semantic coding parameter that balances the semantic distortion against rate. The CTU-level reward Rw c is similar to the frame-level reward except for the following difference. First, the distortion term is estimated by the task accuracy M s after all of the CTUlevel parameters are selected, and the rate term is measured by the negative sum of the bitrate R t,i of all CTUs. Second, considering that the CTU-level action M c is restricted by frame-level action M f , the CTU-level reward is consequently conditioned on M f . The two reward functions are consistent with the objective of the hierarchical policy in Eq. 2. Given that the actions/modes QP and QP subject to discrete distribution, therefore, we adopt the widely used Advantage Actor-Critic (A2C) algorithm [37] where the actor network aims to learn a discrete control policy while the critic network focuses on estimating the value of state V πθ ϕ (s). The parent-level and child-level agents are detailed in Fig. 1. C. Mode simplification Although the HRL has reduced the action space greatly in some content, the optional modes for video semantic coding are still severely unaffordable for training since the expensive coding time cost. To further cut down the training cost, we aim to find an effective but efficient mode simplification strategy for task-driven video semantic coding. Specifically, we design the mode simplification strategy from two perspectives, i.e., frame-level simplification and CTU-level simplification. For frame-level actions, deciding one specific QP value for each frame in one GOP is impractical. The complexity will increase exponentially with the number of frames in one GOP. To reduce the complexity while keeping the characteristic of original traditional codecs, we simplify the action space by only deciding the QPs for the first two frames in one GOP. The QPs of other frames in this GOP are set with the offset used in traditional codecs. For CTU-level action space, we simplify the action space based on the characteristic of semantic task, i.e., the accuracy of semantic task are mainly associated with the semantic-related region in one frame. Therefore, we utilize the semantic mask, which is generated with corresponding task, to divide the region of one frame into two parts, i.e., semanticrelated region and semantic-unrelated region. Then, we can allocate two CTU-level QPs respectively for these two regions, without requiring to decide one QP value for each CTU. III. EXPERIMENTS A. Dataset and Implementation Details To validate the effectiveness of our proposed HRLVSC, we conduct experiments on video segmentation task under the commonly-used low-delay P configuration with the reference software HM 16.19. Dataset: We construct a video semantic coding dataset to optimize and validate our HRLVSC framework, named as TVSC dataset. The dataset contains one video semantic task, i.e., video segmentation task. Our dataset is based on the commonly-used dataset DAVIS2017 [38] for video segmentation task. For each video, we resized them as 960 × 544, and compressed them with the modes used in our optional action spaces. Implementation Details: The HRL model is implemented with Pytorch platform. We train our HRL model with one NVIDIA 1080Ti GPU for 10000 iterations. The batchsize is 30 and the learning rate is 1e-3 for parent-level agent and 1e-4 for child-level agent. B. Performance Analysis 1) Compared with HEVC Anchor: To verify the effectiveness of our HRLVSC scheme, we compare our HRLVSC with standard HEVC codec HM 16.19 under Low-delay P configuration. For HEVC, we select the QP from 22 to 37 for compression as our baseline. For our proposed HRLVSC, we set λ as 0, 0.05, 0.1, 0.15, 0.2, 0.4, 0.6, 0.8, 1.0, respectively. The result is shown in Table I. From the table, we can observe that our proposed HRLVSC scheme outperforms HEVC anchor by a bit saving of 39.93%. Even compared with HEVC anchor with rate control, our HRLVSC can still achieve a bit rate saving of 20.47%, which demonstrates the effectiveness of our HRLVSC for task-driven video semantic coding. 2) Ablation Study: We conduct the ablation studies from two perspectives: 1) Replacing the hierarchical reinforcement learning agents with one reinforcement learning agent used in [23]. 2) Substituting our HRLVSC with hand-crafted methods i.e., assigning the higher QP value for semantic-unrelated CTUs and lower QP value for semantic-related CTUs. The experimental results are as follows: Ablation on HRL: As shown in 2, the performance of applying only one reinforcement learning agent i.e., RL in Fig. 2, for both frame-level and CTU-level QPs decision, is lower than employing hierarchical reinforcement learning. The reasons for that are: 1) The hierarchical structure restricts the child-level action space, which makes the training procedure more stable. 2) The action spaces are complex, which is hard for RL to make decision. Comparison with hand-crafted schemes In this part, we will discuss the overwhelming advantage of proposed HRL against hand-crafted schemes. For video segmentation task, we can obtain the segmentation masks for frames in a video. Therefore, the hand-crafted methods can assign different QP values for each CTU based on the semantic importance i.e., the segmentation mask ratio S for each CTU. In other words, when the mask ratio is higher, the CTU will be assigned lower QP value for better coding quality. For frame-level QP i.e., QP f , we keep the original QP value in standard software HM 16.19 for hand-crafted scheme. For CTU-level QP value in hand-crafted scheme, we attempt to adopt different functions to establish the relationship between QP value and semantic importance i.e., mask ration S, respectively as linear, exponential, square, log, and square root functions. The experimental results are shown in Fig 2, we can find that handcrafted schemes are far from our proposed HRLVSC, since our scheme can capture the optimal relationship between QP value and semantic importance. 3) Complexity Analysis: In this section, we compare our HRLVSC with standard software HM 16.19 from the perspective of time complexity. As shown in Table II, our HRLVSC does not increase any decoding time. For encoding time, our HRL agents only take about 1.52 seconds for the QP decision of the whole video with the size of 960x544. It is efficient and effective to apply our algorithm in the task-driven video semantic coding. IV. CONCLUSION In this paper, we are the first to investigate the taskdriven video semantic coding, and propose the hierarchical reinforcement learning based scheme HRLVSC for it. Unlike the image semantic coding, task-driven video semantic coding contains tremendous reference architectures and coding modes, which inevitably prevents its development. To tackle this challenge, we divide the complex coding modes into frame level and CTU level, and then, introduce the hierarchical RL agents for them. To further reduce the time complexity for training, we carefully design a simple but effective mode simplification strategy for task-driven video semantic coding. Extensive experiments on video segmentation task under lowdelay P configuration have validated the effectiveness of our scheme. We will extend our HRLVSC to more video semantic tasks and more configuration of video coding in future work. Fig. 1 . 1Illustration of our proposed HRLVSC. 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[ "Noise induced transitions in semiclassical cosmology", "Noise induced transitions in semiclassical cosmology" ]	[ "Esteban Calzetta ", "Enric Verdaguer \nDepartament de Física Fonamental and Institut de Física d'Altes Energies (IFAE)\nUniversitat de Barcelona\nAv. Diagonal 647E-08028BarcelonaSpain\n", "\nde Astronomía y Física del Espacio (IAFE)\nDepartamento de Física\nUniversidad de Buenos Aires\nCiudad Universitaria1428Buenos AiresArgentina\n" ]	[ "Departament de Física Fonamental and Institut de Física d'Altes Energies (IFAE)\nUniversitat de Barcelona\nAv. Diagonal 647E-08028BarcelonaSpain", "de Astronomía y Física del Espacio (IAFE)\nDepartamento de Física\nUniversidad de Buenos Aires\nCiudad Universitaria1428Buenos AiresArgentina" ]	[]	A semiclassical cosmological model is considered which consists of a closed Friedmann-Robertson-Walker in the presence of a cosmological constant, which mimics the effect of an inflaton field, and a massless, non-conformally coupled quantum scalar field. We show that the back-reaction of the quantum field, which consists basically of a non local term due to gravitational particle creation and a noise term induced by the quantum fluctuations of the field, are able to drive the cosmological scale factor over the barrier of the classical potential so that if the universe starts near zero scale factor (initial singularity) it can make the transition to an exponentially expanding de Sitter phase. We compute the probability of this transition and it turns out to be comparable with the probability that the universe tunnels from "nothing" into an inflationary stage in quantum cosmology. This suggests that in the presence of matter fields the back-reaction on the spacetime should not be neglected in quantum cosmology.	10.1103/physrevd.59.083513	[ "https://arxiv.org/pdf/gr-qc/9807024v2.pdf" ]	59,405,211	gr-qc/9807024	9fdd83a6fbb7c9d1ecf2fbd907c4af0562781a01	Noise induced transitions in semiclassical cosmology 25 Nov 1998 Esteban Calzetta Enric Verdaguer Departament de Física Fonamental and Institut de Física d'Altes Energies (IFAE) Universitat de Barcelona Av. Diagonal 647E-08028BarcelonaSpain de Astronomía y Física del Espacio (IAFE) Departamento de Física Universidad de Buenos Aires Ciudad Universitaria1428Buenos AiresArgentina Noise induced transitions in semiclassical cosmology 25 Nov 1998arXiv:gr-qc/9807024v2 A semiclassical cosmological model is considered which consists of a closed Friedmann-Robertson-Walker in the presence of a cosmological constant, which mimics the effect of an inflaton field, and a massless, non-conformally coupled quantum scalar field. We show that the back-reaction of the quantum field, which consists basically of a non local term due to gravitational particle creation and a noise term induced by the quantum fluctuations of the field, are able to drive the cosmological scale factor over the barrier of the classical potential so that if the universe starts near zero scale factor (initial singularity) it can make the transition to an exponentially expanding de Sitter phase. We compute the probability of this transition and it turns out to be comparable with the probability that the universe tunnels from "nothing" into an inflationary stage in quantum cosmology. This suggests that in the presence of matter fields the back-reaction on the spacetime should not be neglected in quantum cosmology. I. INTRODUCTION A possible scenario for the creation of an inflationary universe is provided by cosmological models in which the universe is created by quantum tunneling from "nothing" into a de Sitter space. This creation is either based on an instanton solution or in a wave function solution which describes the tunneling in a simple minisuperspace model of quantum cosmology [1,2]. In the inflationary context one of the simplest cosmological models one may construct is a closed Friedmann-Robertson-Walker (FRW) model with a cosmological constant. The cosmological constant is introduced to reproduce the effect of the inflation field at a stationary point of the inflaton potential [1]. The dynamics of this universe is described by a potential with a barrier which separates the region where the scale factor of the universe is zero, where the potential has a local minimum, from the region where the universe scale factor grows exponentially, the de Sitter or inflationary phase. The classical dynamics of this homogeneous and isotropic model is thus very simple: the universe either stays in the minimum of the potential or it inflates. The classical dynamics of the preinflationary era in such cosmological models may be quite complicated, however, if one introduces anisotropies, inhomogeneities or other fields. Thus, for instance, all anisotropic Bianchi models, except Bianchi IX, are bound to inflate in the presence of a cosmological constant [3]. Also in the previous model but with an inhomogeneous scalar radiation field the universe may get around the barrier [4] and emerge into the inflationary stage even if initially it was not. The emergence of an inflationary stage of the universe also seems to be aided by semiclassical effects such as particle creation which enhances the radiation energy density of the preinflationary era and thus enlarges the set of inflating initial conditions [5,6]. In this paper we consider a semiclassical model consisting of a closed FRW cosmology with a cosmological constant in the presence of a quantum massless scalar field. This quantum field may be seen as linear perturbations of the inflaton field at its stationary point or as some other independent linear field. Because the field is free the semiclassical theory is one loop exact. The expectation value in a quantum state of the stress-energy tensor of this scalar field influences by back-reaction the dynamics of the cosmological scale factor. There are here two main effects at play: on the one hand, since the field is not conformaly coupled particle creation will occur and, on the other hand, the quantum fluctuations of this stress-energy tensor induce stochastic classical fluctuations in the scale factor [7,8]. Thus the cosmological scale factor is subject to a history dependent term due to gravitational particle creation and also to noise due to these quantum fluctuations. We examine the possibility that a universe starting near the local minimum may cross the barrier and emerge into the inflationary region by the back-reaction of the quantum field on the scale factor. This is, in some sense, the semiclassical version of tunneling from nothing in quantum cosmology. It is important to stress the difference between this calculation and the usual approach to quantum tunneling. The usual approach [9,10,1,11,12] begins with the calculation of an instanton or tunneling solution, which is a solution to the Euclidean classical (or sometimes semiclassical, see [13]) equations of motion. Because of symmetry, the scalar field is set to zero from the start. Its effect, if at all, is considered as a contribution to the prefactor of the tunneling amplitude [11], which is usually computed to one loop accuracy in the test field approximation. The effect of dissipation [14] or even of particle creation [15] on quantum tunneling has been considered in some quantum mechanical systems but it ought to be noticed that to this date the effects of stress-energy fluctuations on the tunneling amplitude has not been considered in the literature, to the best of our knowledge. Even when the instanton is sought as a solution to semiclassical equations [13] this is done under approximations that effectively downplay the role of particle creation, and back-reaction fluctuations are not considered at all. To underlie that the mechanism for barrier penetration to be investigated here is a different physical process than that computed from instantons in the test field approximation, we have chosen to ignore the quantum aspects of the gravitational field, so that in the absense of back reaction fluctuations the tunneling rate would be zero. From the point of view of the usual approximation, it could be said that our calculation amounts to a nonperturbative calculation of the tunneling amplitude, since the key element is that we go beyond the test field approximation, and consider the full effect of back reaction on the universe. At least in principle, it ought to be possible to combine both the usual and our approach. The whole scheme would ressemble the derivation of the Hu-Paz-Zhang equations [16,17], once the subtleties of quantum cosmological path integrals are factored in [18]. In this paper, we follow the methodology of Langer's classic paper [19], namely, we shall consider an ensemble of Universes whose evolution is rendered stationary by the device that, every time a member of the ensemble escapes the barrier it is captured and reemitted within the barrier. This fictitious stationary solution has a nonzero flux accross the barrier, and the activation probability is derived from this flux. Since semiclassical cosmology distinguishes a particular time (that when the quantum to classical transition takes place), it is meaningful to ask whether the stationary solution is relevant to the behavior of a solution with arbitrary initial data at the "absolute zero of time". The answer is that the stationary solution is indeed relevant, because the relaxation time which brings an arbitrary solution to the steady one is exponentially shorter than the time it takes to escape the barrier. We discuss this issue in detail in Appendix F. The fact itself of assuming a semiclassical theory, i.e., where no gravitational fluctuations are included, indicates that our model must be invalid very close to the cosmological singularity. Therefore, we are forced to assume that some mechanism forces the universe to avoid this region, while being too weak to affect significatively the behavior of larger universes. For example, if we take the cosmological constant, in natural units, to be about 10 −12 (which corresponds to GUT scale inflation), then the presence of classical radiation with an energy density of order one (while the amount necessary to avoid recollapse in the classical theory is 10 12 ), would be sufficient. A more sophisticated possibility would be to appeal to some quantum gravitational effect, which could be as simple as Heisenberg's uncertainty principle, to make it impossible for the universe to linger for long times too close to the singularity. Even with this simple setting, it is impossible to make progress without further simplifications, and we would like to give here a summary of the most significative ones. The most basic simplifying assumption is that the deviation from conformal coupling, measured by the parameter ν to be introduced below (see Eq. (2)), is small. This will allow us to set up the problem as a perturbative expansion in ν, whereby we shall stick to the lowest nontrivial order, namely O(ν 2 ). Of course, the quantity of highest interest, the escape probability itself, will turn out to be nonperturbative in ν; however, our procedure ought to capture its leading behavior. Even to second order in ν, the Closed Time Path (CTP) effective action, whose variation yields the semiclassical equations for the universe scale factor, involves the calculation of several kernels. We have formal exact expressions for these kernels, but the results are too involved for further manipulation. This suggests a second simplification, namely, to substitute the exact kernels for their analogs as computed in an spatially flat universe with the same scale factor. Technically, this amounts to making a continuous approximation in the mode decomposition of the field. This is clearly justified when the separation between the frequencies for different modes is small, for example, as compared with the characteristic rate of the universe espansion. This condition holds for most orbits within the barrier, excepting maybe those where the universe never grows much larger than Planck's scales, a case which we shall not discuss, for the reasons given above. The semiclassical evolution equations emerging from the CTP effective action differ from the usual Einstein equations in three main respects: 1) the polarization of the scalar field vacuum induces an effective potential, beyond the usual terms associated to spatial curvature and the cosmological constant; also the gravitational constants are renormalized by quantum fluctuations; 2) there appears a memory dependent term, asociated to the stress-energy of particles created along the evolution; and 3) there appears a stochastic term associated to the quantum fluctuations of the scalar field. We shall focus our attention in the last two aspects, neglecting the one loop effective gravitational potential. It ought to be noted that, lacking a theory of what the bare potential is exactly like, the semiclassical theory does not uniquely determine the renormalized potential either. Moreover, the presence of stochasticity and memory are aspects where the semiclassical physics is qualitatively different from the classical one, not so for the modified effective potential. In any case, these corrections are very small unless very close to the cosmological singularity (where in any case the one loop approximation is unreliable, as implied by the logarithmic divergence of the quantum corrections). So, assuming again that some mechanism will make it impossible for the universe to stay very close to the singularity, the neglect of the renormalized potential is justified. Even after the neglect of the renormalized potential, the equations deriving from the CTP effective action are higher than second order, therefore do not admit a Cauchy problem in the usual terms, and also lead to possibly unphysical solutions. In order to reduce them to second order equations, and to ensure that the solutions obtained are physical, it is necessary to implement an order reduction procedure as discussed by several authors [20]. This order reduction means that higher derivatives are expressed in terms of lower ones as required by the classical equations of motion. In this spirit, in the memory term, we substitute the history of a given state of the universe by the classical trajectory leading up to the same endpoint. Because the classical trajectory is determined by this endpoint, in practice this reduces the equations of motion to a local form, although no longer Hamiltonian. The equations of motion for the model, after all these simplifications have been carried out, have the property that they do not become singular when the universe scale factor vanishes. As a consequence, the universe goes accross the cosmic singularity and emerges in a new "cosmic cycle". Because the escape time is generally much larger than the recollapse time, we may expect that this will happen many times in the evolution of a single trajectory. For this reason, our model describes a cyclic universe, being created and destroyed many times (but keeping the memory of the total amount of radiation and extrinsic curvature at the end of the previous cycle), and eventually escaping from this fate to become an inflationary universe. It should be noted that this does not detract from the rigor of our derivations, since it is after all a feature of the mathematical model, it being a matter of opinion whether it affects the application of our studies to the physical universe. For comparison we have studied a different, also mathematically consistent, model in which the universe undergoes a single cosmic cycle and obtain similar results (see Appendix G). After this enumeration of the main simplifying assumptions to be made below, let us briefly review what we actually do. Our first concern is to derive the semiclassical equations of motion for the cosmological scale factor, by means of the CTP effective action. The imaginary terms in this action can be shown to carry the information about the stochastic noises which simulate the effect on the geometry of quantum fluctuations of the matter field [21][22][23][24][25][26][27]. After this noises have been identified, the semiclassical equation is upgraded to a Langevin equation. We then transform this Langevin equation into a Fokker-Planck equation, and further simplify it by averaging along classical trajectories. In this way, we find an evolution equation for the probability density of the universe being placed within a given classical trajectory. The actual universe jumps between classical trajectories, as it is subject to the non Hamiltonian nonlocal terms and forcing from the random noises. Finding the above equation of evolution requires a careful analysis of both effects. Finally, we investigate the steady solutions of this equation, and derive the escape probability therein. Again we are forced to consider the problem of very small universes, as the nontrivial steady solutions are nonintegrable in this limit. However, the solutions to the Wheeler-DeWitt equation associated to our model, which in this limit is essentially the Schrödinger operator for an harmonic oscillator, shows no singular behavior for small universes. Thus we shall assume that this divergence will be cured in a more complete model, and accept the nontrivial solution as physical. The main conclusion of this paper is that the probability that the universe will be carried over the barrier by the sheer effect of random forcing from matter stress-energy fluctuations is comparable to the tunneling probability computed from gravitational instantons. This effect demonstrates the relevance of quantum fluctuations in the early evolution of the universe. Besides its relevance to the birth of the universe as a whole, this result also may be used to estimate the probability of the creation of inflationary bubbles within a larger universe. We shall report on this issue in a further communication. The plan of the paper is the following. In section II we compute the effective action for the cosmological scale factor and derive the stochastic semiclassical back-reaction equation for such scale factor. In section III we construct the Fokker-Planck equation for the probability distribution function of the cosmological scale factor which corresponds to the stochastic equation. In section IV we use the analogy with Kramers' problem to compute the probability that the scale factor crosses the barrier and reaches the de Sitter phase. In the concluding section V we compare our results with the quantum tunneling probability. Some computational details are included in the different sections of the Appendix. A short summary of this long Appendix is the following: Appendix A gives some details of the renormalization of the CTP effective action; Appendix B explains how to handle the diffusion terms when the Fokker-Planck equation is constructed; in Appendix C we formulate and discuss Kramers problem in action-angle variables; the short Appendix D gives the exact classical solutions for the cosmological scale factor; in Appendix E the averaged diffusion and dissipation coefficients for the averaged Fokker-Planck equation are derived; in Appendix F the relaxation time is computed in detail; and finally in Appendix G the calculation of the escape probability for the scale factor is made for a model which undergoes a single cosmic cycle. II. SEMICLASSICAL EFFECTIVE ACTION In this section we compute the effective action for the scale factor of a spatially closed FRW cosmological model, with a cosmological constant in the presence of a quantum massless field coupled non-conformally to the spacetime curvature. The semiclassical cosmological model we consider is described by the spacetime metric, the classical source, which in this case is a cosmological constant, and the quantum matter sources. A. Scalar fields in a closed universe The metric for a closed FRW model is given by, ds 2 = a 2 (t) −dt 2 +g ij (x k )dx i dx j , i, j, k = 1, ..., n − 1,(1) where a(t) is the cosmological scale factor, t is the conformal time, andg ij (x k ) is the metric of an (n − 1)-sphere of unit radius. Since we will use dimensional regularization we work, for the time being, in n-dimensions. Let us assume that we have a quantum scalar field Φ(x µ ), where the Greek indices run from 0 to n− 1. The classical action for this scalar field in the spacetime background described by the above metric is S m = − dx n √ −g g µν ∂ µ Φ * ∂ ν Φ + n − 2 4(n − 1) + ν RΦ * Φ ,(2) where g 00 = a 2 , g 0i = 0, g ij = a 2g ij , g is the metric determinant, ν is a dimensionless parameter coupling the field to the spacetime curvature (ν = 0 corresponds to conformal coupling), R is the curvature scalar which is given by R = 2(n − 1)ä a 3 + (n − 1)(n − 4)ȧ 2 a 4 + (n − 1)(n − 2) 1 a 2 ,(3) where an over dot means derivative with respect to conformal time t. Let us now introduce a conformally related field Ψ Ψ = Φa n−2 2 ,(4) and the action S m becomes, S m = dtdx 1 . . . dx n−1 g Ψ * Ψ − (n − 2) 2 4 Ψ * Ψ − νa 2 RΨ * Ψ + Ψ * ∆ (n−1) Ψ ,(5) where ∆ (n−1) is the (n − 1)-Laplacian on the (n − 1)-sphere, ∆ (n−1) Ψ ≡ 1 √g ∂ i gg ij ∂ j Ψ .(6) Let us introduce the time dependent function U (t) U (t) = −νa 2 (t)R(t),(7) and the d'Alambertian ✷ = −∂ 2 t + ∆ (n−1) of the static metricds 2 = a −2 ds 2 . The action (5) may be written then as, S m = dtdx 1 . . . dx n−1 g Ψ * ✷Ψ − (n − 2) 2 4 Ψ * Ψ + U (t)Ψ * Ψ .(8) When ν = 0 this is the action of a scalar field Ψ in a background of constant curvature. The quantization of this field in that background is trivial in the sense that a unique natural vacuum may be introduced, the "in" and "out" vacuum coincide and there is no particle creation [28]. This vacuum is, of course, conformally related to the physical vacuum, see (4). The time dependent function U (t) will be considered as an interaction term and will be treated perturbativelly. Thus we will make perturbation theory with the parameter ν which we will assume small. To carry on the quantization we will proceede by mode separation expanding Ψ(x µ ) in terms of the (n − 1)dimensional spherical harmonics Y l k (x i ), which satisfy [29] ∆ (n−1) Y l k (x i ) = −l(l + n − 2)Y l k (x i ),(9) where l = 0, 1, 2, ...; l ≥ k 1 ≥ k 2 ≥ ... ≥ k n−2 ≥ 0; k = (k 1 , ..., ±k n−2 ). These generalized spherical harmonics form an orthonormal basis of functions on the (n − 1)-sphere, gdx 1 ...dx n−1 Y l * k (x i )Y l ′ k ′ (x i ) = δ ll ′ δ k k ′ ,(10) and we may write, Ψ(x µ ) = l=0 k Ψ l k (t)Y l k (x i ).(11) When Ψ is a real field, the coefficients Ψ l k are not all independent, for instance in three dimensions we simply have Ψ l * k = Ψ l − k . Now let us substitute (11) into (5), use (9) and note that (n − 2) 2 /4 + l(l + n − 2) = (l + 1 + (n − 4)/2) 2 . If we also introduce a new index k instead of l by k = l + 1, so that k = 1, 2, ... we obtain S m = dt ∞ k=1 k Ψ l * kΨ l k − M 2 k Ψ l * k Ψ l k + U Ψ l * k Ψ l k(12) where M k ≡ k + n − 4 2 .(13) Note that the coefficients of (11), Ψ l k (t) are just functions of t (1-dimensional fields), and for each set (l, k) we may introduce two real functions φ l k (t) andφ l k (t) defined by Ψ l k ≡ 1 √ 2 φ l k + iφ l k ,(14) then the action (12) becomes the sum of the actions of two independent sets formed by an infinite collection of decoupled time dependent harmonic oscillators S m = 1 2 dt ∞ k=1 k φ l k 2 − M 2 k φ l k 2 + U (t) φ l k 2 + ... ,(15) where the dots stand for an identical action for the real 1-dimensional fieldsφ l k (t). We will consider, from now on, the action for the 1-dimensional fields φ l k only. If our starting field Φ in (2) is real the results from this "half" action, i.e. the written term in (15), are enough, if Φ is complex we simply have to double the number of degrees of freedom. Since M k depends on k but not on k, there is no dependence in the action on the vector k and we can substitute k by k 1 which gives the degeneracy of the mode k. This is given by [11], k 1 = (2k + n − 4)(k + n − 4)! (k − 1)!(n − 2)! .(16) Note that for n = 2, i.e. when the space section is a circle k 1 = 2; when n = 3, which corresponds to the case of the ordinary spherical harmonics k 1 = 2k − 1 (or 2l + 1 in the usual notation); and for n = 4, which is the case of interest here, the space section of the spacetime are 3-spheres and we have k 1 = k 2 . The field equation for the 1-dimensional fields φ l k (t) are, from (15), φ l k + M 2 k φ l k = U (t)φ l k ,(17) which in accordance with our previous remarks will be solved perturbatibely, being U (t) the perturbative term. The solutions of the unperturbed equation can be written as linear combinations of the normalized positive and negative frequency modes, f k and f * k respectively, where f k (t) = 1 √ 2M k exp(−iM k t).(18) B. Closed time path effective action Let us now derive the semiclassical closed time path (CTP) effective action Γ CT P for the cosmological scalar factor due to the presence of the quantum scalar field Φ. The computation of the CTP effective action is similar to the computation of the ordinary (in-out) effective action, except that now we have to introduce two fields, the plus and minus fields φ ± , and use appropriate "in" boundary conditions. These two fields basically represent the field φ propagating forward and backward in time. This action was introduced by Schwinger [30] to derive expectation values rather than matrix elements as in the ordinary effective action, and it has been used recently in connexion with the back-reaction problem in semiclasical gravity [31,21,32]. Here we follow the notations and conventions of refs. [7,8] Note that since we are considering the interaction of the scale factor a with the quantum field φ, in the CTP effective action we have now two scalar fields φ ± and also two scale factors a ± . The kinetic operators for our 1-dimensional fields φ l k are given by A k = diag(−∂ 2 t − M 2 k + U + (t), ∂ 2 t + M 2 k − U − (t)) . The propagators per each mode k, G k (t, t ′ ) are defined as usual by A k G k = δ, and are 2 × 2 matrices with components (G k ) ± ± . To one loop order in the quantum fields φ ± and at three level in the classical fields a ± the CTP effective action for a ± may be written as, Γ CT P [a ± ] = S g [a + ] − S g [a − ] + S cl m [a + ] − S cl m [a − ] − i 2 k=1 ∞ k Tr(ln G k ),(19) where S g is the pure gravitational action, S cl m is the action of classical matter which in our case will include the cosmological constant term only, and G k is the propagator for the mode k which solves (17). In principle the Γ CT P depends on the expectation value in the quantum state of interest, the "in" vacuum here, of both a ± (the classical field) and of φ ± . To get the previous expression we have substituted the solution of the dynamical equation for the expectation value of the scalar field which is 0, in\|φ\|0, in = 0, so that there is no dependence on the expectation values of φ ± in the effective action. Because of the interaction term U (t) in (17) the propagator G k cannot be found exactly and we treat it perturbatively. Thus we can write G k = G 0 k (1 − U G 0 k + U G 0 k U G 0 k + . . .) where the unperturbed propagator is (G 0 k ) −1 = diag(−∂ 2 t − M 2 k , ∂ 2 t + M 2 k ) . This unperturbed propagator has four components (G 0 k ) + + = ∆ kF , (G 0 k ) − − = −∆ kD , (G 0 k ) + − = −∆ + k and (G 0 k ) − + = ∆ − k , where ∆ kF , ∆ kD and ∆ ± k are the Feynman, Dyson and Wightman propagators for the mode k. This is a consequence of the boundary conditions which guarantee that our quantum state is the "in" vacuum \|0, in . These propagators are defined with the usual iǫ prescription by ∆ kF (t − t ′ ) = 1 2π ∞ −∞ exp(−iω(t − t ′ )) ω 2 − (M 2 k − iǫ) dω (20) = −i [f k (t)f ⋆ k (t ′ )θ(t − t ′ ) + f ⋆ k (t)f k (t ′ )θ(t ′ − t)] , ∆ kD (t − t ′ ) = 1 2π ∞ −∞ exp(−iω(t − t ′ )) ω 2 − (M 2 k + iǫ) dω (21) = i [f ⋆ k (t)f k (t ′ )θ(t − t ′ ) + f k (t)f ⋆ k (t ′ )θ(t ′ − t)] , ∆ + k (t − t ′ ) = if ⋆ k (t)f k (t ′ ), ∆ − k (t − t ′ ) = −if k (t)f ⋆ k (t ′ ).(22) The trace term in the effective action (19) will now be expanded up to order ν 2 . The linear terms in ν are tadpoles which are zero in dimensional regularization. Thus we can write the effective action as Γ CT P [a ± ] ≃ S g [a + ] − S g [a − ] + S cl m [a + ] − S cl m [a − ] + T + + T − + T,(23) where T ± = − i 4 ∞ k=1 k Tr U ± (G 0 k ) ± ± U ± (G 0 k ) ± ± , T = i 2 ∞ k=1 k Tr U + (G 0 k ) + − U − (G 0 k ) − + .(24) The pure gravitational part of the action, S g , includes the Einstein-Hilbert action and a quadratic counterterm which is needed for regularization of the divergences of (24), S g = 1 l 2 P d n x √ −gR + ν 2 µ n−4 c 32π 2 (n − 4) d n x √ −gR 2 ,(25) where µ c is an arbitrary mass scale which gives the correct dimension to the counterterm, and l 2 P = 16πG, the square of the Planck length. To regularize the divergencies in T ± we need to expand the action (25) in powers of n − 4. Using our metric (1), we can perform the space integration in (25) which leads to the volume to the (n−1)-sphere. Expanding now in powers of n − 4, and recalling that the volume of the three-sphere is 2π 2 we may write S g = S EH g + S div g , where the first term stands for the Einstein-Hilbert action in four dimensions and the second term is the first order correction in this expansion, S g [a] = 2π 2 l 2 P dt 6a 2 ä a + 1 ,(26)S div g [a, µ c ] = 1 16 1 n − 4 dtU 2 1 (t) + dt U 2 1 (t) ln(aµ c ) + 2U 1 (t)U 2 (t) .(27) Here U 1 (t) and U 2 (t) are defined by the expansion of U in powers of n − 4. That is, from (7) and (3) we can write U (t) = U 1 (t) + (n − 4) U 2 (t), where U 1 = −6ν ä a + 1 , U 2 = −ν 2ä a + 3ȧ 2 a 2 + 5 .(28) The classical matter term S cl m includes in our case the cosmological constant Λ ⋆ only. It can be understood as the term which gives the effect of the inflaton field at the stationary point of the inflaton potential [1]: S cl m [a] = −2π 2 dta 4 Λ ⋆ .(29) C. Computation of T an T ± Let us first compute T in (24), which may be written as T = − i 2 ∞ k=1 k dtdt ′ U + (t)∆ + k (t − t ′ )U − (t ′ )∆ − k (t ′ − t).(30) Since this term will not diverge we can perform the computation directly in n = 4 dimensions. In this case k 1 = k 2 and M k = k, thus using (22) and (18) we have T = − i 2 dtdt ′ ∞ k=1 U + (t)k 2 f * 2 k (t)f 2 k (t ′ )U − (t ′ ) = − dtdt ′ U + (t)D(t − t ′ )U − (t ′ ) − i dtdt ′ U + (t)N (t − t ′ )U − (t ′ ),(31) where we have introduced the kernels D and N as, D(t − t ′ ) ≡ − 1 8 ∞ k=1 sin 2k(t − t ′ ) = − 1 16 PV cos(t − t ′ ) sin(t − t ′ ) (32) N (t − t ′ ) ≡ 1 8 ∞ k=1 cos 2k(t − t ′ ) = 1 16 π ∞ n=∞ δ(t − t ′ − nπ) − 1 ,(33) and we have computed the corresponding series. The kernels D and N are called dissipation and noise kernel, respectively, using the definitions of [8]. It is interesting to compare with Ref. [7,8] where a spatially flat universe was considered. Our results may be formaly obtained from that reference if we change vol ∞ 0 dk there, where vol is the volume of the space section (assume for instance a finite box), by 2π 2 ∞ k=1 . In the spatially flat case the noise is a simple delta function (white noise), whereas here we have a train of deltas. Note also that we have, in practice, considered a real scalar field only since we considered only half of the action, i.e. the written part of (15). Thus for the complex scalar field we need to multiply these kernels by two, i.e. the dissipation kernel is 2D and the noise kernel is 2N . Note also that the definition of the dissipation kernel here and in Ref. [21] differ by a sign. Let us now perform the more complicated calculation of T ± . Since these integrals diverge in n = 4 we work here in arbitrary n (dimensional regularization). From (24) and the symmetries of ∆ kF and ∆ kD we have T ± = − i 4 dtdt ′ U ± (t)∆ 2 F/D (t − t ′ )U ± (t ′ ),(34) where we have introduced, ∆ 2 F/D (t − t ′ ) ≡ ∞ k=1   k 1   ∆ 2 kF/D (t − t ′ ) = ∞ −∞ dω 2π e −iω(t−t ′ ) I(ω),(35) where I(ω) is defined after having made an integral in ω with appropriate contour, recall the definitions (20) and (21). After using (16) and the definition (13) of M k , I(ω) is given by I(ω) = ± i 2(n − 2)! ∞ k=1 (k + n − 4)! (k − 1)! [(k + (n − 4)/2) 2 − (ω/2) 2 ± i0 + ] ≡ ± i 2(n − 2)! ∞ k=1 a k (ω),(36) where we have introduced the coefficients a k in the last series expression. In Appendix A we prove that this series diverges like 1/(n − 4), and thus we can regularize it using (27). Furthermore its imaginary part is finite and leads to the noise kernel N defined above. Thus according to (112), (114) and (119) from the Appendix we can write (35) as ∆ 2 F/D (t − t ′ ) = ∓ i 4 δ(t − t ′ ) n − 4 − 1 2π 2 K ± (t − t ′ ) ,(37) where, we have defined K ± (t − t ′ ) ≡ 16π 2 [A(t − t ′ ) ± iN (t − t ′ )] .(38) Here A(t − t ′ ) is a finite kernel which will be discussed below. We can now substitute (37) into (34) and use the expansion of U (t) in powers of n − 4 given in (28) to get T ± = ∓ 1 n − 4 dt(U ± 1 ) 2 + 2 dtU ± 1 U ± 2 − 1 2π 2 dtdt ′ U ± 1 (t)K ± (t − t ′ )U ± 1 (t ′ ) .(39) D. The regularized CTP effective action We are now in the position to compute the regularized semiclassical CTP effective action. Let us substitute in (23) the actions (26), (27) and (29), and the results (31) for T and (39) for T ± . It is clear that the divergent term in (39), i.e. the term proportional to 1/(n − 4), will be cancelled by the divergent counterterm in (27). Also the terms dtU 1 U 2 in these equations will cancel. Thus, we finally get the regularized semiclassical action Γ CT P [a ± ] = S R g,m [a + ] − S R g,m [a − ] + S R IF [a ± ],(40) where the regularized gravitational and classical matter actions are, S R g,m [a] = 2π 2 l 2 P dt 6a 2 ä a + 1 − 2π 2 dt a 4 Λ ⋆ + 1 16 dt U 2 1 (t) ln(aµ c ).(41) To write the remaining part, S R IF , we note that the kernels A and N in (38), satisfy the symmetries A(t−t ′ ) = A(t ′ −t) and N (t − t ′ ) = N (t ′ − t). Taking into account also that D(t − t ′ ) = −D(t ′ − t) we obtain S R IF [a ± ] = 1 2 dtdt ′ ∆U (t)H(t − t ′ ){U (t ′ )} + i 2 dtdt ′ ∆U (t)N (t − t ′ )∆U (t ′ ),(42) where we have defined H(t − t ′ ) = A(t − t ′ ; µ c ) − D(t − t ′ ),(43)∆U = U + − U − , {U } = U + + U − .(44) In (43) we have explicitly written that the kernel A depends on the renormalization parameter µ c . We note that this effective action has an imaginary part which involves the noise kernel N . However, because of the quadratic dependence of this term in ∆U it will not contribute to the field equations if we derive such equations from δΓ CT P /δa + \| a ± =a = 0. This, in fact, gives the dynamical equations for expectation values of the field a(t). However, we recall that we are dealing with the interaccion of a "system", our classical (one dimensional) field a(t), with an "environment" formed by the degrees of freedom of the quantum system and that we have integrated out the degrees of freedom of the environment (note that in the effective action we have substituted the solutions of the field equations for the expectation value of the quantum field). In this case the regularized action S R IF can be understood as the influence action of the system-environment interaction, which describes the effect of the environment on the system of interest [33,34]. The imaginary part of the influence action is known [21][22][23][24][25][26][27] to give the effect of a stochastic force on the system, and we can introduce an improved semiclassical effective action, S ef f [a ± ; ξ] = S R g,m [a + ] − S R g,m [a − ] + 1 2 dtdt ′ ∆U (t)H(t − t ′ ){U (t ′ )} + dtξ(t)∆U (t),(45) where ξ(t) is a Gaussian stochastic field defined by the following statistical averages ξ(t) = 0, ξ(t)ξ(t ′ ) = N (t − t ′ ).(46) The kernel H in the effective action gives a non local effect (due to particle creation), whereas the source ξ gives the reaction of the environment into the system in terms of a stochastic force. The formal derivation of the last term of (45) can be seen as follows. The Feynman and Vernon influence functional [33] of the system-environment interaction is defined from the influence action S IF by F IF = exp(iS IF ). Note now that by using a simple path integral Gaussian identity, the imaginary part of (42) can be formally recovered in F IF with the following functional Fourier transform F IF = DξP [ξ] exp i[Re(S IF ) + dtξ(t)∆U (t)] , where P [ξ] = exp[− 1 2 dtdt ′ ξ(t)N −1 (t − t ′ )ξ(t ′ )] Dξ exp − 1 2 dtdt ′ ξ(t)N −1 (t − t ′ )ξ(t ′ ) , can be interpreted as a Gaussian probability distribution for the field ξ. That is, the influence funcional may be seen as the statistical average of ξ dependent influence functionals constructed with the "effective" influence action Re(S IF ) + dtξ(t)∆U (t). The physical interpretation of this result, namely, that the semiclassical equations are now the stochastic equations derived from such effective action may be seen, for instance, in Ref. [22]. E. Stochastic semiclassical back-reaction equation The dynamical equation for the scale factor a(t) can now be found from the effective action (45) in the usual way, that is by functional derivation with respect to a + (t) and then equating a + = a − ≡ a. These equations include the back-reaction of the quantum field on the scale factor. It is convenient to use a rescaled scale factor b and cosmological constant Λ defined by b(t) ≡ √ 24π l P a(t), Λ ≡ l 4 P 12π 2 Λ ⋆ .(47) The regularized action S R g,m becomes after one integration by parts, S R g,m [b] = − 1 2 dt  ḃ 2 − b 2 + 1 12 Λb 4 − 9 2 ν 2 b b + 1 2 ln(bμ),  (48) where we have also rescaled the renormalization parameterμ. The remaining term in (45) does not change with this rescaling except that now U (t) should be written in terms of b, thus according to (28) we have U (t) = −6ν b b + 1 .(49) The dynamical equation for b(t) is: δS ef f [b ± ; ξ] δb + b ± =b = 0.(50) This equation improves the semiclassical equation by taking into account the fluctuations of the stress-energy tensor of the quantum field [35][36][37]. When averaged over ξ the equation leads to the usual semiclassical equation for the expectation value of b(t). Now this equation leads to the typical non physical runaway solutions due to the higher order time derivatives involved in the quantum correction terms. To avoid such spurious solutions we use the method of order reduction [20] into the equations (50). In this method one asumes that equation (50) are perturbative equations in which the perturbations are the quantum corrections. To leading order the equation reduces to the classical equation b + b 1 − 1 6 Λb 2 = O(ν).(51) The terms withb or with higher time derivatives in the quantum corrections of the equation (50) are then substituted using recurrently the classical equation (51). In this form the solutions to the semiclassical equations are also perturbations of the classical solutions. Thus by functional derivation of (45), using (48), we can write the stochastic semiclassical back-reaction equation (50) aṡ p = −V ′ (b) − δV ′ (b) + F (b, p, t) + J(ξ, b, p),(52) where a prime means a derivative with respect to b, and we have introduced p ≡ḃ. The classical potential V (b) is V (b) = 1 2 b 2 − Λ 24 b 4 ,(53) and its local quantum correction is δV (b) = − 3ν 2 Λ 4 1 2 b 2 − Λ 48 b 4 − p 2 ln(bμ) ,(54) where we have implemented order reduction in this term. On the other hand the term F (b, p, t) involves nonlocal contributions and may be written as, F (b, p, t) = − ∂U ∂b I − d 2 dt 2 ∂U ∂b I = 6ν d 2 dt 2 1 b −b b 2 I,(55) where I(b, p, t) is defined by I(b, p, t) ≡ ∞ −∞ dt ′ H(t − t ′ )U (t ′ ).(56) After order reduction, U (t ′ ) must be evaluated on the classical orbit with Cauchy data b (t) = b, p (t) = p, whereby it reduces to U = −Λνb 2 . The function J is the noise given by J (ξ, b) = 6ν d 2 dt 2 ξ b −b ξ b 2 and, after order reduction, by J(ξ, b, p) = 6ν ξ b − 2ξp b 2 + 2ξV ′ (b) b 2 + 2ξp 2 b 3 ,(57) with ξ(t) defined in (46) in terms of the noise kernel. F. Approximate kernels N and H To simplify the nonlocal term F (b, p, t) and the noise J(ξ, b, p) we will approximate the kernel H and the noise kernel N keeping only the first delta function, i.e. n = 0, in the train of deltas which define the noise kernel N . This amounts to take the continuous limit in k in the definition (33) of N . In fact, we take the sum in k as an integral and we get N (u) = ∞ 0 dk cos 2ku = π 16 δ(u).(58) This is equivalent to assume that the spacetime spatial sections are flat and of volume 2π 2 , see Ref. [7]. Similarly the dissipation kernel D defined in (32) becomes D(u) = − 1 8 ∞ 0 dk sin 2ku = − 1 16 PV 1 u .(59) The same approximation may be used to compute the kernel A defined in (36)- (38). The computation of this kernel can be read directly from (118), see also Ref. [7] A(u) = − 1 8 ∞ −∞ dω 2π e −iωu ln \|ω\| \|µ c \| = 1 16 Pf 1 \|u\| + 1 8 (γ + ln µ c )δ(u),(60) where γ is Euler's number and Pf means the Hadamard principal function whose meaning will be recalled shortly. To perform this last Fourier tranform we write ln \|ω\| = lim ǫ→0 + [exp(−ǫ\|ω\|) ln \|ω\|], use the integrals ∞ 0 dω ln ω cos(ωu) exp(−ǫω) and ∞ 0 dω cos(ωu) exp(−ǫω) which can be found in [40], and take into account that [2x tan −1 (u/ǫ) + ǫ ln(u 2 + ǫ 2 )]/(u 2 + ǫ 2 ) = d[ln(u 2 + ǫ 2 ) tan −1 (u/ǫ)]/du. When ǫ → 0 + the lastH(u) = 1 8 Pf θ(u) u + γ + ln µ c 8 δ(u).(61) The distribution Pf(θ(u)/u) should be understood as follows. Let f (u) be an arbitrary tempered function, then ∞ −∞ duPf θ(u) u f (u) = lim ǫ→0 + ∞ ǫ du f (u) u + f (0) ln ǫ .(62) The approximation of substituting the exact kernels by their flat space counterparts is clearly justified when the radius of the universe is large, which is when the semiclassical approximation works best. Once the local approximation for the noise kernel follows, the corresponding expression for D can be obtained by demanding that their Fourier transforms be related by the same fluctuation-dissipation relation as in the exact formula. III. THE FOKKER-PLANCK EQUATION Now we want to determine the probability that a universe starting at the potential well goes over the potential barrier into the inflationary stage. In statistical mechanics this problem is known as Kramers' problem. To describe such process we have the semiclassical back-reaction equation (52), which is a stochastic differential equation (a Langevin type of equation). As it is well known [38] to study this problem it is better to construct a Fokker-Planck equation, which is an ordinary differential equation for a distribution function. Thus, the first step will be to derive the Fokker-Planck equation corresponding to the stochastic equation (52). The key features of this stochastic equation are: a potential given by the local potentials (53) and (54), a non local term given by the function F and a noise term J. The classical part of the potential has a local minimum at b = 0 then reaches a maximum and decreases continuously after that. The inflationary stage corresponds to the classical values of b beyond this potential barrier. If we start near b = 0 the noise term will take the scale factor eventually over the barrier, but if we want to compute the escape probability we need to consider both noise and non locality. It should do no harm if we disregard the local quantum correction to the potential, δV (b), the reason is the following. This term is a consequence of renormalization, but in semiclassical gravity there is a two parameter ambiguity in terms which are quadratic in the curvature in the gravitational part of the action. This ambiguity is seen here only in the parameterμ because we have simply ignored the other possible parameter which was not essential in the renormalization scheme. Furthermore we should not trust the semiclassical results too close to b = 0, since the semiclassical theory should break down here. Thus the possible divergence at b = 0 may be disregarded and we should think of this renormalized term as just a small correction to the classical potential, as it is indeed for all radii of the universe unless b ≪ 1. Thus the classical potential V (b) should contain the main qualitative features of the local renormalized potential. To construct the Fokker-Planck equation let us introduce the distribution function f (b, p, t) = δ (b (t) − b) δ (p (t) − p) ,(63) where b(t) and p(t) are solutions of equation (52) for a given realization of ξ(t), b and p are points in the phase space, and the average is taken both with respect to the initial conditions and to the history of the noise as follows. One starts by considering the ensemble of systems in phase space obeying equation (52) for a given realization of ξ(t) and different initial conditions. This ensemble is described by the density ρ(b, p, t) = δ(b(t) − b)δ(p(t) − p) , where the average is over initial conditions. Next one defines the probability density f (b, p, t) as the statistical average over the realizations of ξ(t), that is f (b, p, t) = ρ(b, p, t) ξ The next manipulations are standard [39], we take the time derivative of f , ∂ t f = ḃ (t) ∂ b(t) δ (b (t) − b) δ (p (t) − p) + δ (b (t) − b)ṗ (t) ∂ p(t) δ (p (t) − p) , and note that ∂ b(t) δ (b (t) − b) = −∂ b δ (b (t) − b) , and that p (t) δ (b (t) − b) δ (p (t) − p) = pf (b, p, t) . Performing similar manipulations for the other terms and using the equations of motion (52) we find ∂f ∂t = {H, f } − ∂ ∂p [F (b, p, t)f ] − ∂ ∂p Φ,(64) where we have defined H(b, p) = 1 2 p 2 + V (b),(65) thus disregarding the potential δV (b) in (52), the curly brackets are Poisson brackets, i.e. {H, f } = −p(∂f /∂b) + V ′ (b)(∂f /∂p), and Φ = J (ξ, b, p) δ (b (t) − b) δ (p (t) − p) .(66) Equation (64) is not yet a Fokker-Planck equation, to make it one we need to write Φ in terms of the distribution function f . This term will be called the diffusion term since it depends on the stochastic field ξ(t). From (66) and (57) we may write Φ = 6ν C 2 b − 2C 1 p b 2 + 2V ′ b 2 + 2p 2 b 3 C 0(67) where C n = d n dt n ξ(t) δ(b(t) − b)δ(p(t) − p) ,(68) for n = 0, 1, 2. To manipulate the difussion term of (64) we will make use of the functional formula for Gaussian averages [41], ξ(t)R [b (t) , p (t)] = dt ′ N (t − t ′ ) δ δξ (t ′ ) R [b (t) , p (t)] ,(69) where R is an arbitrary functional of ξ(t). Under the approximation (58) for the noise kernel C 0 = π 16 δ δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) t ′ →t = − πνΛ 8 b ∂ ∂p f (b, p, t) ,(70) where we have used (122) in the last step. The expressions for C 1 and C 2 are similarly obtained, first one uses the time translation invariance of the noise kernel to perform integration by parts, then the problem reduces to taking time derivatives of (70). The results are (see Appendix B for details) C 1 = (πνΛ/8)(b∂ b f − p∂ p f ),(71)C 2 = (πνΛ/8)(2p∂ b f + V ′ ∂ p f + bV ′′ ∂ p f ).(72) Finally, after substitution in (67) and using the equation of motion to lowest order we have Φ = − πν 2 Λ 2 4 b 2 ∂f ∂p ,(73) which by (64) leads to the final form of the Fokker-Planck equation ∂f ∂t = {H, f } − ∂ ∂p [F (b, p, t)f ] + πν 2 Λ 2 4 b 2 ∂ 2 f ∂p 2(74) We also notice that in the absense of a cosmological constant, we get no diffusion. This makes sense, because in that case the classical trajectories describe a radiation filled universe. Such universe would have no scalar curvature, and so it should be insensitive to the value of ν as well. A. Averaging over angles We want to compute the probability that a classical universe trapped in the potential well of V (b) goes over the potential barrier as a consequence of the noise and non locality produced by the interacction with the quantum field, and end up in the de Sitter phase. A universe that crosses this potential barrier will reach the de Sitter phase with some energy which one would expect will correspond to the energy of the quantum particles created in the previous stage. Note that this differs from the quantum tunneling from nothing approach in which the universe gets to the de Sitter stage tunneling from the potential minimum b = 0 with zero energy. In practice, this difference will not be so important because as the universe inflates any amount of energy density will be diluted away. For this computation we will follow closely the solution of Kramers' problem [42] reviewed in Appendix C. The three key features of such computation are: first, the introduction of action-angle canonical variables (J, θ); second, the asumption that f depends on J only, i.e. f (J); and, third, the use of the averaged Fokker-Planck equation over the angle variable θ. Of course, the Fokker-Planck equation in Kramers' problem, (127), is much simpler than our equation (74) due to the non local character of the latter; thus we need to take care of this problem, and it is quite remarkable that a relatively simple solution can be found. Thus, let us consider equation (74), introduce (J, θ) and assume that f (J), in the Appendix we have seen that the dissipation term which involves ∂ 2 f /∂p 2 can be written in terms of derivatives with respect to J, see equation (130). Since we now have {H, f } = 0 we can write ∂f ∂t = πν 2 Λ 2 4 b 2 1 Ω ∂f ∂J + p 2 Ω ∂ ∂J 1 Ω ∂f ∂J − ∂ ∂p F (b, p, t) f − pF Ω ∂f ∂J .(75) Next we take the average of (75) with respect to the angle θ. The averaged equation involves the two pairs of integrals dθb 2 p 2 , dθb 2 , and dθpF , dθ∂ p F . The components of each pair are related by a derivative with respect to J. In fact, let us introduce D(J) = 1 2πΩ 2π 0 dθb 2 p 2 ,(76) changing the integration variable to b, see Appendix C, this integral may be written as Ω dbb 2 p, and using that ∂ J p\| b = Ω/p we have dD dJ = 1 2π dθb 2 .(77) Similarly, let us introduce S(J) = 1 2πΩ 2π 0 dθpF (b, p, t),(78) again by a change of integration variable this integral may be written as Ω dbF and by derivation with respect to J we get dS dJ = 1 2π 2π 0 dθ ∂ ∂p F (b, p, t).(79) Finally, the average of the Fokker-Planck equation (75) becomes, ∂f ∂t = πν 2 Λ 2 4 ∂ ∂J D(J) Ω ∂f ∂J − ∂ ∂J (Sf ).(80) This equation may be written as a continuity equation ∂ t f + ∂ J K = 0, where the probability flux K may be identified directly from (80). We see that, as in Kramers' problem, stationary solutions with positive flux K 0 should satisfy πν 2 Λ 2 4 D(J) Ω ∂f ∂J − S(J)f = −K 0 .(81) B. The non local contribution S(J) We need to handle now the term S(J), defined in (78). The problem here lies in the non local term F (b, p, t) defined in (55)-(56), with U (t) given by (49). Since this term gives a quantum correction to a classical equation we will adopt the order reduction prescription. Thus let us assume that b(t ′ ) and p(t ′ ) in the integral which defines F are solutions to the classical equations of motion with Cauchy data b(t) = b and p(t) = p, then the integrand in F (b, p, t) will depend explicitly on time only through b and p. This means that the time dependence of U (t ′ ) may be written as U (b, p, t ′ − t). If we now write the Cauchy data in terms of the action-angle variables (J, θ), since the equation of motion for the angle variable is simplyθ = Ω we may write b[B(θ, J), P (θ, J), t] = b(θ + Ωt, J) and similarly for p. This means that we may substitute the time derivative operator d/dt by Ω∂/∂θ in F (b, p, t). Thus substituting (55) and (56) into (78), using Ωd/dθ instead of d/dt, integrating by parts and using the expression for U (t) given by (49) we get S = 6ν 2πΩ 2π 0 dθ d dt ṗ b I(t).(82) This may be simplified using the equation of motion (51) to lowest order, then changing dθ by Ωdt we have S = − ν 2 Λ 2 2π 2π/Ω 0 dt d dt b 2 (t) ∞ −∞ dt ′ H(t − t ′ )b 2 (t ′ ).(83) Note that this term is of order ν 2 Λ 2 as the diffusion term (75). Thus it is convenient to introduce S by S(J) = πν 2 Λ 2 4 S(J).(84) Now we can make use of (61) for the kernel H (note that the local delta term does not contribute), and introduce a new variable u = t − t ′ , instead of t ′ to write S as S(J) = −1 4π 2 2π/Ω 0 dt d dt b 2 (t) Pf ∞ 0 du u b 2 (t − u).(85) The equation for the stationary flux (81) becomes D(J) Ω ∂f ∂J − S(J)f = − 4 πν 2 Λ 2 K 0 .(86) All that remains now is to find appropriate expressions for D and S in this equation and follow Kramers' problem in the Appendix to compute K 0 . From now on, however, it is more convenient to use the energy E as a variable instead of J, where E = H(J) and thus we will compute D(E) and S(E) in what follows. C. Evaluating S and D Let us begin by recalling the basic features of the classical orbits. The most important feature of the classical dynamics is the presence of two unstable fixed points at p = 0, b = ±2 √ E s , where E s = 3/(2Λ) is also the corresponding value of the "energy" E = p 2 /2 + V (b). These fixed points are joined by a heteroclinic orbit or separatrix. Motion for energies greater than E s is unbounded. For E ≤ E s , we have outer unbound orbits and inner orbits confined within the potential well. These periodical orbits shall be our present concern. As it happens, the orbits describing periodic motion may be described in terms of elliptic functions (see Appendix D). The exact expression for the orbits leads to corresponding expressions for D and S (see Appendix E). Introducing a variable k k 2 = 1 − 1 − E Es 1 + 1 − E Es ,(87) so that k 2 ∼ E/4E s for low energy, while k 2 → 1 as we approach the separatrix, we find D (E) = 8E 2 15π 1 + k 2 3/2 k 4 2 1 − k 2 + k 4 E [k] − 2 − 3k 2 + k 4 K [k] ,(88) where K and E are the complete elliptic integrals of the first and second kind (see [44], [45]) K [k] = 1 0 dx (1 − x 2 ) (1 − k 2 x 2 ) , E [k] = 1 0 dx (1 − k 2 x 2 ) (1 − x 2 ) .(89) The corresponding expression for S is S (E) = 8E 2 π 2 1 + k 2 2 k 4 {α [k] E [k] − γ [k] K [k]} ,(90) where α [k] = ∞ 0 du u 2 sn 2 u 1 − 1 + k 2 3 sn 2 u − u sn u 1 − 1 + k 2 sn 2 u + k 2 sn 4 u 1/2 ,(91) sn u being the Jacobi elliptic function, and γ [k] = ∞ 0 du u 2 sn 2 u 1 − 1 + 2k 2 3 sn 2 u − E [u, k] sn u 1 − 1 + k 2 sn 2 u + k 2 sn 4 u 1/2 ,(92) where E [u, k] is the incomplete elliptic integral of the second kind E [u, k] = sn u 0 dx (1 − k 2 x 2 ) (1 − x 2 ) .(93) The conclusion of all this is that, while D and S individually behave as E 2 times a smooth function of E/E s , their ratio is relatively slowly varying. At low energy, we find D ∼E 2 /2 and S ∼E 2 /4. As we approach the separatrix, D →0.96 E 2 s and S →1.18 E 2 s . Meanwhile, the ratio of the two goes from 0.5 to 1. 23. This means that we can write the equation for stationary distributions as ∂f ∂E − β (E) f = − 4 πν 2 Λ 2 g (E) K 0 E 2 ,(94) where β and g are smooth order one functions. There is a fundamental difference with respect to Kramers' problem, namely the sign of the second term in the left hand side. In the cosmological problem, the effect of nonlocality is to favour diffussion rather than hindering it. We may understand this as arising from a feedback effect associated with particle creation (see [46]). IV. THE TUNNELING AMPLITUDE Having found the reduced Fokker-Planck equation Eq. (94), we must analyze its solutions in order to identify the range of the flux K 0 . We shall first consider the behavior of the solutions for E ≤ E s , and then discuss the distribution function beyond the separatrix. Since our derivation is not valid there, for this later part we will have to return to an analysis from the equations of motion. For concreteness, in what follows it is convenient to choose the order of magnitude of the cosmological constant. We shall assume a model geared to produce GUT scale inflation, thus Λ ∼ 10 −12 , and correspondingly E s ∼ 10 12 is very large in natural units. A. Distribution function inside the potential well As we have already discussed, the approximations used in building our model break down at the cosmological singularity, and therefore Eq. (94) cannot be assumed to hold in a neighborhood of E = 0. Thus it is best to express the solution for f in terms of its value at E = E s f (E) = 4K 0 πν 2 Λ 2 σe E dE ′ β(E ′ ) + f p (E) ,(95) where σ is an arbitrary constant and the particular solution f p (E) is chosen to vanish at E = E s f p (E) = e E dE ′ β(E ′ ) Es E dE ′ g(E ′ )E ′2 e − E ′ dE ′′ β(E ′′ ) ,(96) so that f (E s ) ∼ 4K 0 πν 2 Λ 2 σe β(Es)Es .(97) Because of the exponential suppression, the particular solution is dominated by the lower limit in the integral, leading to f p (E) ∼ 1 g(E)E 2 [β(E) + 2/E] − e −β(Es)(Es−E) g(E s )E 2 s [β(E s ) + 2/E s ] .(98) For E ≪ 1 we see that f p ∼ E −1 , but this behavior cannot be extrapolated all the way to zero as it would make f non integrable. However we must notice that neither our treatment (i.e., the neglect of logarithmic potential corrections) nor semiclassical theory generally is supposed to be valid arbitrarily close to the singularity. Thus we shall assume that the pathological behavior of Eq. (94) near the origin will be absent in a more complete theory, and apply it only from some lowest energy E δ ∼ 1 on. There are still 12 orders of magnitude between E δ and E s . Since we lack a theory to fix the value of the constant σ, we shall require it to be generic in the following sense. We already know that f p vanishes at E s , by design, and then from the transport equation (94) we derive df p /dE = −[g(E s )E 2 s ] −1 there. So unless σ ≤ [β(E s )g(E s )E 2 s ] −1 exp[−β(E s )E s ] ∼ 10 −24 exp(−10 12 ), f has a positive slope as it approaches the separatrix from below. We shall assume a generic σ as one much above this borderline value, so that for E ≥ 1 the right hand side of the reduced Fokker-Planck equation may be neglected, and f grows exponentially f (E) ∼ 4K 0 σ πν 2 Λ 2 e β(E)E (99) B. Outside the well Beyond the separatrix, all motion is unbounded and there is no analog of action-angle variables, so we must return to the original variables b, p. Also note that we are only interested in the regime when E ≥ E s , that is, we shall not consider unbound motion below the top of the potential. Let us first consider the behavior of classical orbits in the (b, p) plane. Our first observation is that as the universe gets unboundedly large, the effects of spatial curvature become irrelevant. This means that we may approximate U ∼ −6νb/b, and accordingly the classical equation of motion asb ∼ Λb 3 /6. In this regime, classical orbits are quickly drawn to a de Sitter type expansion, whereby they can be parametrized as b (t ′ ) = b (t) 1 + Λ 12 b (t) (t − t ′ ) . (100) After substituting U ∝ b 2 , it is easily seen that the nonlocal term I is proportional to b 2 (t), and that therefore the nonlocal force F vanishes (see Eq. (55)). Therefore what we are dealing with are the local quantum fluctuations of the metric, which one would not expect to act in a definite direction, but rather to provide a sort of diffussive effect. To see this, let us observe that if we look at the Fokker-Planck equation as a continuity equation, then we may write it as Rather thanb andp, however, it is convenient to use the components ofK along and orthogonal to a classical trajectory. Since the energy E is constant along trajectories, ∇E lies in the orthogonal direction, so the orthogonal component is simply K E , or, since E = H(J), K J . Our whole calculation so far amounts to computing the mean value of K J (see eq. (80)); indeed the first term acts as diffussion, opposing the gradients of f . The big surprise is the second term being positive, forcing a positive flux towards larger energies. Observe that, in particular, the mean flux across the separatrix is positive. Since for a stationary solution the flux is conserved, the flux must be positive accross any trajectory. Now beyond the separatrix the term S of (80) is absent because F vanishes and, as we shall see, D remains positive. So, to obtain a positive flux, it is necessary that ∂f /∂E < 0, as we will now show. To compute D beyond the separatrix, we observe that although there are no longer action-angle variables, we may still introduce a new pair of canonical variables (E, τ ), where E labels the different trajectories and τ increases along classical trayectories, withτ = 1. It works as follows. The relationship between p and b, p = 2E + Λ 12 b 4 , becomes, for low energy p = Λ 12 b 2 + 12 Λ E b 2 .(101) This same relationship corresponds to a canonical transformation with generating functional W , W (b, E) = Λ 12 b 3 3 − 12 Λ E b , and the new canonical coordinate τ follows from τ = ∂W ∂E = − 12 Λ 1 b . (102) Comparing with Eq. (100), this is just τ = − 12 Λ 1 b(t 0 ) 1 + Λ 12 b(t 0 )(t 0 − t) , for some constant of integration t 0 . Indeedτ = 1, as it must. Writing the Fokker-Planck equation (74) in the new variables (E, τ ) is an exercise in Poisson brackets, simplified by the approximation ∂b/∂E ∼ 0 (to see that this approximation is justified we may go to one more order in E in the expressions for p, W and τ and we find that for large b, ∂b/∂E ∼ −12/(5Λb 3 )). Thus from Eq. (74) with F = 0, we get ∂f ∂t = − ∂f ∂τ + πν 2 Λ 3 48 b 6 ∂ 2 f ∂E 2 ,(103) so that K τ = f (that is, the universe moves along the classical trajectory withτ = 1), and K E = − πν 2 Λ 3 48 b 6 ∂f ∂E . with only the normal diffussive term present, as it was expected. Since K E must be positive (at least in the average) f must decreases beyond the separatrix, as we wanted to show. This result, in fact, can be made more quantitative if we note that Eq. (103) for a stationary distribution function f is essentially a heat equation which can be solved in the usual way. For this it is convenient to change to a new variable s = −1/(5τ 5 ) which is positive semidefinite since the conformal time τ is negative in the de Sitter region. The equation then can be written as, ∂f ∂s = d ∂ 2 f ∂E 2 ,(104) where d ≡ 36πν 2 . Its solution can be written as f (E) = 1 √ 4πds dE ′ e (E−E ′ ) 2 4sd h(E ′ ),(105) where h(E ′ ) is a function which determines the value of f at τ = −∞. It is easy to compute 0 −∞ dτ f (E, τ ) ∝ ∞ 0 dE ′ h(E ′ ) (E − E ′ ) 7/5 ,(106) which shows that for large E, f in fact decreases as E −7/5 . C. The tunneling amplitude After the two previous subsections, we gather that the stationary solutions to the Fokker-Planck equation display a marked peak at E = E s . We may now estimate the flux by requesting, as we do for Kramers' problem in the Appendix, that the total area below the distribution function should not exceed unity. Unless the lower cutoff E δ is very small (it ought to be exponentially small on E s to invalidate our argument) the integral is dominated by that peak, and we obtain K 0 ≤ (prefactor) exp [−β(E s )E s ] .(107) The prefactor depends on Λ, ν, g (1), β (1), σ and the details of the peak shape. Using E s = 3/(2Λ), β (E s ) = 1.23, we get K 0 ≤ (prefactor) exp − 1.84 Λ .(108) In the last section of the Appendix we have computed the flux when one considers a cosmological model with a single cosmic cycle. The result (193) is qualitatively similar to this one, it just gives a sligthly lower probability. This semiclassical result must now be compared against the instanton calculations. V. CONCLUSIONS In this paper we have studied the possibility that a closed isotropic universe trapped in the potential well produced by a cosmological constant may go over the potential barrier as a consequence of back-reaction to the quantum effects of a non conformally coupled quantum scalar field. The quantum fluctuations of this field act on geometry through the stress-energy tensor, which has a deterministic part, associated to vacuum polarization and particle creation, and also a fluctuating part, related to the fluctuation of the stress-energy itself. The result is that the scale factor of the classical universe is subject to forcing due to particle creation and also to a stochastic force due to these fluctuations. We compute the Fokker-Planck equation for the probability distribution of the cosmological scale factor and compute the probability that the scale factor crosses the barrier and ends up in the de Sitter stage where b ∼ 12/Λ cosh( Λ/12t ′ ), where t ′ is cosmological time bdt = dt ′ , if it was initially near b ∼ 0. The result displayed in (108) is that such probability is K 0 ∼ exp − 1.8 Λ ,(109) or a similar result, displayed in (193), if we consider a cosmological model undergoing a single cosmic cycle. This result is comparable with the probability that the universe tunnels quantum mechanically into the de Sitter phase from nothing [1]. In this case from the classical action (48) S R g,m [b], i.e. neglecting the terms of order ν, one constructs the Euclidean action S E , after changing the time t = iτ , S E [b] = 1 2 dτ ḃ 2 + b 2 − 1 12 Λb 4 .(110) The Euclidean trajectory is b = 12/Λ cos( Λ/12τ ′ ), where τ ′ is Euclidean cosmological time (this is the instanton solution). This trajectory gives an Euclidean action S E = 4/Λ. The tunneling probability is then p ∼ exp − 8 Λ .(111) This result, which in itself is a semiclassical result, is comparable to ours, (109), but it is of a very different nature. We have ignored the quantum effects of the cosmological scale factor but we have included the back-reaction of the quantum fields on this scale factor. Also our universe reaches the de Sitter stage with some energy due to the particles that have been created. In the instanton solution only the tunneling amplitude of the scale factor is considered and the universe reaches the de Sitter phase with zero energy. Taken at face value, our results seem to imply that the nonlocality and randomness induced by particle creation are actually as important as the purely quantum effects. This conclusion may be premature since after all Eq. (109) is only an upper bound on the flux. Nevertheless, our results show that ignoring back-reaction of matter fields in quantum cosmology may not be entirely justified. We expect to delve further on this subject in future contributions. VI. APPENDIX To facilitate the reading of this Appendix we repeat here the summary of its contents given in the Introduction: Appendix A gives some details of the renormalization of the CTP effective action; Appendix B explains how to handle the diffusion terms when the Fokker-Planck equation is constructed; in Appendix C we formulate and discuss Kramers problem in action-angle variables; the short Appendix D gives the exact classical solutions for the cosmological scale factor; in Appendix E the averaged diffusion and dissipation coefficients for the averaged Fokker-Planck equation are derived; in Appendix F the relaxation time is computed in detail; and finally in Appendix G the calculation of the escape probability for the scale factor is made for a model which undergoes a single cosmic cycle. A. Divergences of T Here we compute the finite imaginary part of the series defined in (36) and prove that the real part diverges like 1/(n − 4). The finite real part of the series will not be found explicitly, its exact form is not needed in the calculation of this paper. Let us now call ε ≡ n − 4, and call F (ω) the series (36) which we can write in terms of the Gamma functions as, F (ω) ≡ ∞ k=1 a k (ω) = ∞ k=1 Γ(k + ε + 1) Γ(k) 1 (k + ε/2) 2 − (ω/2) 2 + i0 + = ∞ k=1 Γ(k + ε + 1) Γ(k) PV 1 (k + ε/2) 2 − (ω/2) 2 − iπδ[(k + ε/2) 2 − (ω/2) 2 ] ≡ F R + iF I ,(112) where we have used that (x ± i0 + ) −1 = PV(1/x) ∓ iπδ(x). Let us first concentrate in the imaginary part F I and compute, according to (35), its Fourier transform F I ≡ ∞ −∞ dω 2π e −iω(t−t ′ ) F I = 1 2 ∞ k=1 Γ(k + ε + 1) Γ(k) cos(k + ε/2)(t − t ′ ) k + ε/2 ,(113) where we have used that 2(k + ε/2)δ[(k + ε/2) 2 − (ω/2) 2 ] = δ(k + ε/2 + ω/2) + δ(k + ε/2 − ω/2). Now the last expression is clearly convergent when ε = 0, thus we get F I = 1 2 ∞ k=1 cos k(t − t ′ ),(114) this series can be summed up and we get the train of deltas of (33), thus recovering the noise kernel, fromF I = 8N (t − t ′ ). Let us now see that the real part of the series diverges like 1/ε. Using that Γ(x + 1) = xΓ(x) the principal part of a k can also be written as a k (ω) = Γ(k + ε) Γ(k) k + ε (k + ε/2) 2 − (ω/2) 2 .(115) It is clear from this expression that the divergences when ε = 0 come from the ratio of gamma functions in (115) when k is large. Let us now separate the sum ∞ k=1 a k = N −1 k=1 a k + ∞ k=N a k where N ≫ 1. We can use now that for large x, Γ(x) = √ 2πx x−1/2 e −x (1 + O(1/x)) and the definition of e, e = lim n→∞ (1 + 1/n) n , to prove that Γ(k + ε)/Γ(k) = k ε (1 + O(1/k)). Substituting a k byā k , defined bȳ a k = k ε 1 + O 1 k k + ε (k + ε/2) 2 − (ω/2) 2 ,(116) in the second sum of the previous separation and we can write ∞ k=1 a k = N −1 k=1 a k + ∞ k=Nā k . Now we can use the Euler-Maclaurin summation formula [40] to write ∞ k=Nā k = ∞ N dkā k + . . ., where the dots stand for terms which are finite since they depend on succesive derivatives ofā k at the integration limits. Thus we may write ∞ k=1 a k = N −1 k=1 a k − N 0 dkā k + ∞ 0 dkā k . The first sum and first integral of this last equation are finite for all ε, thus we can take ε = 0, in which case a k =ā k = k/(k 2 − (ω/2) 2 ). The sum and integral may then be performed (writing 2a k = 1/(k + ω/2) + 1/(k − ω/2)) and the ln N which appears in both expressions cancel, the next to leading order terms differ by order O(1/N ). Therefore the divergence is in the last integral ∞ 0 dkā k = ∞ 0 dk k ε+1 (k + ε/2) 2 − (ω/2) 2 ,(117) where here ε is an arbitrary parameter. This integral is easily computed [40], and when it is expanded in powers of ε we get ∞ 0 dkā k = − 1 n − 4 + 1 2 ln(ω/2) 2 .(118) Thus, according to (35), (36) and (112) we compute the Fourier transform of F R , F R = − δ(t − t ′ ) n − 4 − 8A(t − t ′ ),(119) where A(t − t ′ ) stands for a finite kernel (see Eq. (60) ). B. The diffussion terms We want to compute (66) which can be written as (67) in terms of the functions C n (n = 0, 1, 2) of (68). The simplest function C 0 can be written after using (69) as, C 0 = dt ′ N (t − t ′ ) δ δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) , whereas to write the other two functions we observe that the noise kernel is translation invariant, so integrating by parts (in a distribution sense) C n = dt ′ N (t − t ′ ) ∂ n ∂t ′n δ δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) . We now use the local approximation for the noise kernel to get C 0 = π 16 δ δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) t ′ →t , and similarly C 1 and C 2 . As we know, this reduces to C 0 = −π 16 ∂ ∂b δb (t) δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) + ∂ ∂p δp (t) δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) . A functional derivative of the equations of motion leads to d dt δb (t) δξ (t ′ ) = δp (t) δξ (t ′ ) , d dt δp (t) δξ (t ′ ) = −V ′′ [b (t)] δb (t) δξ (t ′ ) + 6ν 1 b (t ′ ) d 2 dt 2 δ (t − t ′ ) + V ′ [b (t ′ )] b 2 (t ′ ) δ (t − t ′ ) , where actually we are computing the right hand side only to lowest order in ν. This suggests writing δp (t) δξ (t ′ ) = G (t − t ′ ) θ (t − t ′ ) + 6ν b (t ′ ) d dt δ (t − t ′ ) ,(120)δb (t) δξ (t ′ ) = R (t − t ′ ) θ (t − t ′ ) + 6ν b (t ′ ) δ (t − t ′ ) , which works provided dR dt = G, R (0) = 0, dG dt = −V ′′ [b (t)] R, G (0) = 6ν V ′ [b (t ′ )] b 2 (t ′ ) − V ′′ [b (t ′ )] b (t ′ ) = 2νΛb(t ′ ). In the coincidence limit δb (t) δξ (t ′ ) t ′ →t = 0, δp (t) δξ (t ′ ) t ′ →t = 2νΛb,(121) which leads to δ δξ (t ′ ) δ (b (t) − b) δ (p (t) − p) t ′ →t = −2νΛb ∂ ∂p f (b, p, t) .(122) The diffusive terms also involve the first and second derivatives of the propagators with respect to t ′ . To find them, we make the following reasoning. We have just seen that, for example, R(t, t) ≡ 0, therefore ∂ ∂t ′ R (t, t ′ ) t ′ →t = − ∂ ∂t R (t, t ′ ) t ′ →t = −G (t, t) = −2νΛb.(123) With a slight adaptation, we also get ∂ ∂t ′ G (t, t ′ ) t ′ →t = ∂ ∂t [ G (t, t ′ )\| t ′ →t ] − ∂ ∂t G (t, t ′ ) t ′ →t , so that we have ∂ ∂t ′ G (t, t ′ ) t ′ →t = 2νΛp.(124) Iterating this argument, we find ∂ 2 ∂t ′2 R (t, t ′ ) t ′ →t = − ∂ ∂t G (t, t) − ∂ ∂t ′ G (t, t ′ ) t ′ →t , where we have permutted a t and a t ′ derivative and used the equations of motion. From this we thus get ∂ 2 ∂t ′2 R (t, t ′ ) t ′ →t = −4νΛp.(125) The last formula of this type that we need is ∂ 2 ∂t ′2 G (t, t ′ ) t ′ →t = ∂ ∂t ∂ ∂t ′ G (t, t ′ ) t ′ →t − ∂ ∂t ′ ∂ ∂t G (t, t ′ ) t ′ →t , which from the equations of motion leads to ∂ 2 ∂t ′2 G (t, t ′ ) t ′ →t = −2νΛV ′ (b) + ∂ ∂t ′ V ′′ [b (t)] R (t, t ′ ) (126) = −2νΛV ′ (b) − V ′′ [b] 2νΛb = −2νΛ ∂ ∂b (bV ′ [b]) . C. Kramers problem For our purposes in this paper we call Kramers problem [42] the computation of the "tunneling amplitude" or, more properly, the escape probability of a particle confined in a potential V (b), such as (53) for instance, which has a maximum and a separatrix with an energy E s . The particle is subject to a damping force γp (p =ḃ) and white noise with amplitude γkT , according to the fluctuation-dissipation relation, where γ is a friction coefficient, k Boltzmann constant and T the temperature. The Fokker-Planck equation in this case is [38] ∂f ∂t = {H, f } + γ ∂ ∂p pf + kT ∂f ∂p ,(127) where H is given by (65). Since the particle is trapped in the potential it undergoes periodic motion, in this case it is convenient to introduce action-angle variables [43] (J, θ) as canonical variables instead of (b, p), thus making a canonical transformation b = B(θ, J), p = P (θ, J). The action variable J is defined by J = 1 2π pdb.(128) Since p can be written in terms of b and H, substitution in (128) and inversion implies that H = H(J), and ∂H ∂J = Ω(J),(129) is the frequency of the motion. The other canonical variable, the angle variable θ, satisfies a very simple equation of motionθ = Ω and changes from 0 to 2π. At high energies, that is, near the separatrix when J → J s , the motion ceases to be periodic and Ω → 0. At low energies, let as assume that b = 0 is a stable minimum of the potential, near this minimum the potential approaches the potential of a harmonic oscillator with frequency ω, V (b) ∼ ωb 2 /2 (in our case we simply have ω = 1), then J → 0, H ∼ ωJ and Ω ∼ ω. ∂f ∂t + γ n =0 ∂c n ∂t (J, t) e in(θ−Ωt) = γ f + p 2 Ω ∂f ∂J + kT 1 Ω ∂f ∂J + p 2 Ω ∂ ∂J 1 Ω ∂f ∂J .(130) Fourier expanding the coefficients in the right hand side we obtain a set of equations for the c n coefficients. The equation for f itself follows from the average of this equation over the angle variable θ. Let us change the integration variable in the definition (128) of J, db = ∂ θ b\| J dθ, taking into account that over a classical trajectory J is constant, and thatθ = Ω we have ∂ θ b\| J = p/Ω(J). Thus, we can write (128) as, 1 2π 2π 0 dθp 2 = JΩ(J).(131) Using this result we can now take the average of equation (130) over θ. This average reads simply, ∂f ∂t = γ ∂ ∂J J f + kT Ω ∂f ∂J .(132) As one would expect exp(−E/kT ) is a solution of this equation. Let us now see whether this equation, which is a transport equation, admits stationary solutions with positive probability flux [19]. Note that we may write this equation as a continuity equation ∂ t f + ∂ J K = 0, where the flux K can be read directly from (132). Therefore a stationary solution with positive flux K 0 should satisfy kT Ω ∂f ∂J + f = − K 0 γJ ,(133) which can be integrated to give, f = K 0 γkT e −E/kT J0 J dξ ξ Ω(ξ)e E(ξ)/kT .(134) For any K 0 , f diverges logarithmically when J → 0, however this is an integrable singularity in J and this is not a problem as we will see shortly. In our problem the action variable J satisfies that J ≤ J s and equation (134) proves that there is a real and positive solution for any J in such a range, which corresponds to choosing J 0 = J s . Given a solution we may determine the flux K 0 imposing the condition that the probability of finding the particle trapped in the potential well should not be greater than unity [19], i.e. Js 0 f (J)dJ ≤ 1. This is equivalent to 1 ≥ K 0 γkT Js 0 dξ ξ Ω(ξ)e E(ξ)/kT ξ 0 dJe −E/kT(135) Since the integral is regular at zero it is dominated by the contribution from the upper limit, and the integral may be evaluated approximately. One gets K 0 ≤ γ ωJ s kT exp − E s kT ,(136) where we have used that near the separatrix H ∼ ωJ s . Typically the flux is very small so that the probability of finding the particle in the potential well is nearly one, therefore the value of K 0 approaches the right hand side of (136). We should remark here that in the order reduction scheme that we are following, to compute the noise and the non local terms we use the classical equations of motion. In fact, these terms have a quantum origin in our case and its computation is one of the tasks we have to perform in order to define our particular Kramers problem. Thus the use of the action-angle variables, which is convenient for the classical equations of motion, is also convenient (after order reduction) in our approach to the Kramers problem. D. A look at the orbits In what follows, we shall quote extensively from Abramowitz and Stegun, Ref. [44] (from now on, AS), and Whittaker and Watson, Ref. [45] (henceforth, WW). The motion is described by the Hamiltonian H = 1 2 p 2 + b 2 − Λ 24 b 4 .(137) The energy is conserved, and on an energy surface H = E, the momentum is p 2 = 2E − b 2 + Λb 4 /12. The classical turning points correspond to p = 0. Introducing the separatrix energy E s = 3/(2Λ), we can write the four turning points as b 2 ± = 4E s 1 ± 1 − E E s ,(138) two of them ±b − are inside the barrier, and two ±b + are outside it. The momentum can now be written as p 2 = 2E 1 − k 2 b 2 b 2 − 1 − b 2 b 2 − ,(139) where we have introduced k 2 = (b − /b + ) 2 , see Eq. (87). The equation for the orbit is b = b − x(t), where x = sn b + t √ 8E s , k ,(140) where sn is the Jacobi Elliptic Function (we follow the notation from WW 22.11; to convert to AS, put m = k 2 , and see AS 16.1.5 Ω = πb + 2 √ 8E s K [k] . (141) E. The D and S functions The function D is given by D(J) = 1 2π 2π/Ω 0 dt b 2 p 2 , which by introducing b = b − x can be written as D(J) = 1 2π 4 1 0 (b − dx) b 2 − x 2 2E (1 − k 2 x 2 ) (1 − x 2 ) = 2 π √ 2Eb 3 − σ [k] ,(142) where σ [k] = 1 0 dx x 2 (1 − k 2 x 2 ) (1 − x 2 ).(143) Following a suggestion in WW 22.72, this can be reduced to complete elliptic integrals of the first and second kinds (we will need the third kind for the S function), to get the result quoted in the main text. The function S is given by S = −1 4π 2 T 0 dt d dt b 2 (t) Pf ∞ 0 du u b 2 (t − u). Let us consider an orbit beginning at b (0) = 0, and divide the time interval in four quarters: I) 0 ≤ t ≤ T /4, II) T /4 ≤ t ≤ T /2, III) T /2 ≤ t ≤ 3T /4, IV) 3T /4 ≤ t ≤ T . We have the following relationships: I) in the first quarter, b I = b (t) , p I = p (t), II) in the second quarter, b II (t) = b I T 2 − t , p II = −p I T 2 − t , III) in the third quarter, b III (t) = −b I t − T 2 , p III = −p I t − T 2 , IV) in the fourth quarter, b IV (t) = −b I (T − t), p II = p I (T − t) . This suggests parametrizing time in terms of a unique variable τ , 0 ≤ τ ≤ T /4, as follows: I) In the first quarter, t = τ , II) In the second quarter, t = T /2 − τ , III) In the third quarter, t = T /2 + τ , IV) In the fourth quarter, t = T − τ . We can then write S = −1 2π 2 T /4 0 dτ b(τ )p (τ ) Pf ∞ 0 du u b 2 (τ − u) − b 2 T 2 − τ − u + b 2 T 2 + τ − u − b 2 (T − τ − u) . Since b 2 is an even function of t with period T /2, we have S = −1 π 2 T /4 0 dτ b(τ )p (τ ) Pf ∞ 0 du u b 2 (τ − u) − b 2 (τ + u) , and since the second integrand is obviously even S = 1 π 2 T /4 0 dτ b(τ )p (τ ) Pf ∞ −∞ du u b 2 (τ + u) .(144) To proceed, we must appeal to the addition theorem for elliptic functions (AS 16.17.1). Next we use the differential equation for Jacobi elliptic functions (AS 16.16.1) and integrate by parts to get S = 32EE s π 2 k 2 Pf ∞ 0 du u 2 sn 2 (u) ρ k 2 sn 2 (u) ,(145) where ρ [n] = 1 0 dx x 2 (1 − k 2 x 2 ) (1 − x 2 ) [1 − nx 2 ] ,(146) which can be expressed in terms of complete elliptic integrals ρ = −k 2 c ′ + a k 2 K [k] − a k 2 E [k] − cΠ [n, k] ,(147) where the last term is the complete elliptic integral of the third kind (AS 17.7.2), with sin α = k, a = 1 n 1 n − 1 + k 2 3k 2 , c ′ = 1 n 2 3k 2 + 1 n 1 n − 1 + k 2 k 2 , c = 1 n 1 k 2 + 1 n 1 n − 1 + k 2 k 2 . Since in our application we allways have n ≤ k 2 , we may use formulae AS (17.7.6) and (17. 4.28) to get the result in the text (recall that E/E s = 4k 2 / 1 + k 2 2 ) F. Relaxation time The aim of this section is to estimate the time on which a solution to the transport equation with arbitrary initial conditions relaxes to a steady solution as discussed in the main body of the paper, in section IV. The way this kind of problem is usually handled [47] is to write the Fokker-Planck equation (74) in a way ressembling a (Euclidean) Schrödinger equation ∂f ∂t = Lf .(148) Then if a complete basis of eigenfunctions of the L operator can be found Lf n (p, q) = E n f n (p, q) ,(149) a generic solution to Eq. (148) reads f (p, q, t) = c n f n (p, q) e Ent .(150) Therefore, provided no eigenvalue has a positive real part, the relaxation time is the inverse of the real part of the largest nonzero eigenvalue. The L operator may have purely imaginary eigenvalues, in which case it does not relax towards any steady solution. This problem differs from the ordinary quantum mechanical one in several aspects, the most important being that the L operator does not have to be either Hermitian or anti-Hermitian. That is why the eigenvalues will be generally complex, rather than just real or imaginary. Also, it is important to notice that the "right" eigenvalue problem Eq. (149) is different from the "left" eigenvalue problem: g n ← − L = E ′ n g n . For example, for any L of the form L = ∂ i K i , where the K's are themselves operators, g 0 ≡ 1 is a solution to this (left) equation (with zero eigenvalue), while it may not be a solution to Eq. (149) at all. Our problem In our case, the L operator can be read from Eq. (74). Since we are taking ν as a small parameter, it is natural to write L = L 0 + L 1 , where L 0 f = {H, f } ,(151)L 1 f = − ∂ ∂p [F f ] + πν 2 Λ 2 4 b 2 ∂ 2 f ∂p 2 .(152) The spectral decomposition of L 0 is very simple. In action-angle variables L 0 f = −Ω (J) ∂f ∂θ .(153) Imposing periodicity in θ we find the following eigenvalues: 0 and E 0 n,χ = −inΩ (χ) ,(154) with n integer (note that L 0 is anti-Hermitian). The eigenvalue 0 is infinitely degenerate: any function of J alone is an eigenvector with zero eigenvalue. The E 0 n,χ have eigenfunctions f 0 n,χ (J, θ) = e inθ √ 2π δ (J − χ) ,(155) and, barring accidental degeneracy (the ratio of frequencies for two different actions being rational) are non degenerate. These eigenfunctions are normalized with the Hilbert product (g \|f ) = Js 0 2π 0 dJdθ g * f as (0nξ \|0nχ ) = δ (ξ − χ), where here and in the rest of this section we use Dirac's notation. Having solved the eigenvalue problem for L 0 , it is only natural to see that of L as an exercise in time independent perturbation theory. There are three differences with the ordinary textbook problem: 1) L 1 is neither Hermitian nor anti-Hermitian; 2) one of the eigenvalues of L 0 is degenerate; 3) the eigenfunctions of L 0 are not normalizable. In spite of this, the basic routine from quantum mechanics textbooks still works. Perturbations to nonzero eigenvalues Let us seek the first order correction to E 0 n,χ . We write the exact eigenvalue as E n,χ = E 0 n,χ +E 1 n,χ +... corresponding to the exact eigenfunction f n,χ = f 0 n,χ + f 1 n,χ + ..., and obtain L 1 f 0 n,χ + L 0 f 1 n,χ = E 1 n,χ f 0 n,χ + E 0 n,χ f 1 n,χ .(156) For m = n we multiply both sides of the equation by f 0 * m,ξ , use that L 0 is anti-Hermitian and integrate over J and θ, to get (0mξ \|1nχ ) = 0mξ L 1 0nχ E 0 n,χ − E 0 m,ξ .(157) In the m = n = 0 case, the same operation yields E 1 n,χ (0nξ \|0nχ ) = 0nξ L 1 0nχ − E 0 n,χ − E 0 n,ξ (0nξ \|1nχ ) ,(158) and we may write L 1 f 0 n,χ = e inθ √ 2π [R + iI] ,(159) where R = L 1 δ (J − ξ) − n 2 πν 2 Λ 2 4 b 2 ∂θ ∂p b 2 δ (J − ξ) .(160) Whatever the imaginary part I is, it is not relevant to the relaxation time; in a similar way, the average of the first term in Eq. (160) yields no term proportional to (0nξ \|0nχ ) Therefore, we conclude that Re E 1 n,χ = −n 2 πν 2 Λ 2 8π 2π 0 dθ b 2 ∂θ ∂p b 2 J=χ .(161) We see on dimensional grounds alone that the relaxation time (the inverse of this equation) will be of order E 2 s (recall that E s = 3/(2Λ), much shorter than the average tunneling time, which is proportional to the inverse of (108). The expression (161) may be slightly simplified by using the identity (∂θ/∂p)\| b = −(∂b/∂J)\| θ , which follows from the transformation from one set of variables to the other being canonical. We may write Re E 1 n,χ = −n 2 πν 2 Λ 2 32π 2π 0 dθ ∂b 2 ∂J θ 2 J=χ .(162) We may Fourier transform b 2 as a function of θ, derive term by term, and use Parseval's identity, to conclude that in any case Re E 1 n,χ ≥ n 2 πν 2 Λ 2 (8π) 2 d dJ 2π 0 dθ b 2 2 J=χ .(163) The integral in this expression can be performed, recall that b = b − x(t) where x(t) is given in (140). We recall also that Ω = θt with Ω given in (141), and then use as integration variable 2θK[k]/π, where K[k] is the elliptic integral defined in (89), to get finally Re E 1 n,χ ≥ n 2 πν 2 Λ 2 (4π) d dJ b 2 − k 2 1 − E [k] K [k] 2 .(164) Rather than a general formula, let us investigate the limiting cases. For J → 0, we have: J ∼ E, k 2 ∼ E/(4E s ), b 2 − ∼ 2E, E[k] ∼ (π/2)(1 − k 2 /4), and K[k] ∼ (π/2)(1 + k 2 /4). In this limit we thus get Re E 1 n,χ ≥ n 2 ν 2 Λ 2 4 .(165) For J → J s (near the separatrix) we can use the following approximations: b 2 − ∼ 4E s 1 − 1 − E/E s , k 2 ∼ 1 − 2 1 − E/E s , K[k] ∼ (1/2) ln[16/(1 − k 2 )] ∼ (1/4) ln[64/(1 − E/E s )], E[k] ∼ 1 + (1/4) 1 − E/E s {ln[64/(1 − E/E s )] − 1}, and dE/dJ = Ω ∼ π/(2K[k] ). Thus the correction to the eigenvalue diverges. In both cases, we get that the relaxation time is much smaller than the tunneling time. Perturbation of the zero eigenvalue We now confront the harder problem of finding the first order correction to the zero eigenvalue. The idea, as in quantum mechanics, is that the first order eigenvalues shall be the eigenvalues of the restriction of L 1 to the proper subspace of the zero eigenvalue, namely, the infinite dimensional space of all θ independent functions. If f 0,χ corresponds to an eigenfunction with null eigenvalue, the first order secular equation becomes L 1 f 0 0,χ + L 0 f 1 0,χ = E 1 0,χ f 0 0,χ .(166) We elliminate the second term in the left hand side of this equation by projecting back on θ independent functions, by averaging over θ. Fortunately the average over θ of L 1 acting on a θ independent function is precisely what we did in section III, so using (80) and (84) we can write down the eigenvalue problem πν 2 Λ 2 4 d dJ D Ω d dJ − S f = λf,(167) where we call λ the eigenvalue, to avoid confussion with the energy. The left hand side of this equation is a sum of two terms, the first one being Hermitian, and the second undefined. However, if we introduce a new function Ψ by f = Ψ exp 1 2 E dE ′ β (E ′ ) where β = S/D we can write πν 2 Λ 2 4 d dJ D Ω d dJ − 1 2 dS dJ − ΩS 2 4D Ψ = λΨ.(168) Recall that we have seen in section III that S is an increasing function of E (or J). Therefore, multiplying by Ψ * and integrating, we see that λ must be real and negative. This is an important result. Let us introduce a new non negative parameter α, λ = − πν 2 Λ 2 8 α,(169) and write equation (168) using E as independent variable instead of J (dE/dJ = Ω), and then introduce a new function ψ by Ψ = ψ/ √ D. Finally (168) becomes − 1 2 ψ ′′ + V α (E) ψ = 0 (170) where V α (E) = 1 4D dS dE + S 2 2D + D ′′ − D ′2 2D − α Ω ,(171) which looks like a Schrödinger equation with a weird potential. We have therefore transformed the problem of finding the eigenvalues of equation (168) into the question of for which values of α a particle of zero energy has a bound state in the potential V α (E). To get an idea of what is going on, let us make the approximation D ∼ cE 2 , S ∼ βD, where c and β are constant, then V α (E) = β 4E 2 2E + β 2 E 2 − α cβΩ(172) When α = 0, we should get back some results of section IV. Indeed, in this case the solutions for large E go like exp(±βE/2), which, after the equation relating f with Ψ, means that the solutions either are exponentially growing or bounded. The first ones correspond to steady solutions with non zero flux (those in section IV), while the second ones are the stationary solutions with no flux. Note that the change from Ψ to ψ, that we made previously, enforces the pathological E −1 low energy behavior we found in section IV. For α = 0, the effective potential V α has two classical turning points, i.e. points where V α (E) = 0. For small E we find E 1 ∼ α/(2cβ) (we use that Ω(E 1 ) ∼ 1), and for large E we find E 2 given by Ω −1 (E 2 ) ∼ cβ 2 E 2 s /(2α), which under the asymptotic form Ω −1 (E) ∼ ln[64/(1 − E/E s )]/( √ 2π), is E 2 ∼ E s {1 − 64 exp[−πcβ 2 E 2 s /( √ 2α)]}. The first classically allowed region sits precisely where the theory is unreliable, and we ought to disregard it as an artifact. Therefore the low α eigenstates must be related to the presence of the second allowed region, near the separatrix. This is consistent with the fact that the zeroth order eigenvalues are −inΩ (see Eq. (154)), and so they tend to accumulate around 0 as we approach the separatrix. In the second classically allowed region (large E) we may approximate V α (E) ∼ α 4cE 2 s 1 Ω (E 2 ) − 1 Ω (E) .(173) As an estimate, we may look for values of α such as V α satisfies a Bohr-Sommerfeld condition Es E2 dE −2V α (E) ∼ nπ,(174) The relevant value of c being 0.96 near the separatrix, see the end of section III, thus β ∼ 1. 23. Taking the log of Eq. (175), we find the lowest eigenvalue α 1 = √ 2πcβ 2 E 2 s ln 128 √ 2β 2 E 2 s /π 1 + O ln ln E s ln E s .(176) This is the result we were looking for. Going back to the beginning, we translate this into eigenvalues of the Fokker-Planck operator, see Eqs. (149) and (167), λ ∼ − 9πν 2 32 √ 2πcβ 2 ln 128 √ 2β 2 E 2 s /π ,(177) where we have used (169) and that E s = 3/(2Λ). Thus we conclude that the relaxation time grows logarithmically with E s , while the tunneling time grows exponentially. In fact, the tunneling time is proportional to the inverse of (108), and so it goes like ∼ exp(1.23 E s ). Therefore it is totally justified to analyze tunneling under the assumption that all transient solution have died out, and we only have the steady solutions discussed in section IV. G. A single cosmic cycle The purpose of this section is to discuss whether it is possible to generalize the discussion of the paper to models with a single cosmic cycle. The basic problem is that an universe emerging from the singularity with a finite expansion rate is bound to lead to infinite particle production [48]. Therefore, in order to make sense, it is unavoidable to modify the behavior of the model close to the singularity, and there is no unique way to do this. Of course, a possibility is to assume that the singularity behaves as a perfectly reflecting boundary, which is equivalent to what we have done so far. Another possibility, to be discussed here, is that the evolution is modified for very smal universes, so that p vanishes as b → 0. For example, if the initial stages of expansion (and the final stages of collapse) are replaced by an inflationary (deflationary) period, then p ∼ b 2 ,ṗ ∼ b 3 , etc. We shall assume such an evolution in what follows. In these models, the singularity is literally pushed to the edge of time. The D and S functions The D function is given by Eq. (76), where now we average over a half period only. However, the periodicity of the integrand is precisely T /2, so the average over a half period is the same as the full average. Therefore, D ∼ E 2 /2 at low energy, and 0.96 E 2 close to the separatrix as we had in the many cycles model. For the function S, let us begin from Eq. (78), modified to represent average over a half period S (J) = 1 π T /2 0 dt pF (b, p, t),(178) then use Eq. (55) for F and integrate by parts twice to get S (J) = 6ν π p d dt I b T /2 0 − 6ν π ṗ I b T /2 0 + 6ν π T /2 0 dt p b − pb b 2 I(b, p, t).(179) The discussion above on the approach to the singularity means that the integrated terms vanish. In the remaining term we use the equations of motionb =ṗ = −V ′ (b) and get where we have truncated the u integral to restrict it to the range where the equations of motion hold. Instead of looking for a general expression, we shall only consider the low energy limit and the behavior close to the separatrix. Low energy limit For low energy, b = ( √ 2E/Ω) sin Ωt. Substituting this into Eq. (181), changing the order of integration and performing some simple integrations we obtain, S = E 2 2π 2 Ω 4 Pf T /2 0 du u [1 − cos 2Ωu + π sin 2Ωu] = 6.89 E 2 2π 2 Ω 4 ,(182) where the last integration has been performed numerically. Thus, S retains the main features as in the previous case, the most important being the sign and energy dependence. Close to the separatrix Close to the separatrix, we must make allowance for the fact that the orbit spends an increasing amount of time near the turning point b − . It is thus convenient to isolate the central portion of the orbit. Let us rewrite Eq. (181) as S = − 1 2π 2 T /4 0 dt db 2 dt t 0 du u b 2 (t − u) + T /2 T /4 dt db 2 dt t 0 du u b 2 (t − u) .(183) Divide the u integral by quarter orbits, write t = T /2 − t ′ in some of these integrals, and use the periodicity and parity of b 2 and db 2 /dt. We can then rewrite S as Observe that the factor db 2 /dt effectively cuts off the t integrals at times much shorter than T /4. So we can take the limit T → ∞, whereby A converges to the expression for S of the previous case, i.e. Eq. (144). Here, our problem is to estimate B. To evaluate C, we integrate by parts and take the limit T → ∞, S = A + B(184)C = b 4 − 2π 2 ln T 4 − 1 2π 2 ∞ 0 dt db 2 dt ∞ 0 dv ln (t + v) db 2 dv + O 1 T .(185) Let us use the same argument in D, take the limit and add C to get B. The final result is S = expression gives a representation of πPf(1/\|u\|). Finally, using (59) and (60) the kernel of interest H(u) = A(u) − D(u) can be written as, ∂f ∂t = − ∇ K, and this allows us to identify the flux. For example, if the Fokker-Planck equation reads ∂f ∂t = ∂A ∂b + ∂B ∂p , then whatever A and B are, K = −Ab − Bp, where a circunflex denotes an unit vector in the corresponding direction. If γ = 0, then the solution to the Fokker-Planck equation is an arbitrary function of J and θ − Ωt. Stationary solutions are therefore functions of J alone. We may seek a general solution as f (J, t) + γ n =0 c n (J, t) e in(θ−Ωt) , in this case we have ∂ p f = ∂ p J\| b ∂ J f . From (65) we have that p = [2(H − V (b))] 1/2 and, consequently, ∂ J p\| b = Ω/p whose inverse is ∂ p J\| b = p/Ω. This can be used to write ∂ 2 f /∂p 2 in terms of derivatives with respect to J, and since now {H, f (J)} = 0 we can write the Fokker-Planck equation (127) in the new variables as, keeping only first order terms, ( this only makes sense if we treat the separatrix as a turning point). To perform the integral, we introduce a new variable x = ln[(1 − E 2 /E s )/(1 − E/E s )]. The integral turns out to be nπ ∼ √ α(1 − E 2 /E s ) ∞ 0 dx √ xe −x / 2 √ 2πc, and so the eigenvalues are the roots of 2 dt I(b, p, t).(180)Next use Eqs. (56), (61), (49), (51) and the redefinition (84) to write Let us write B = C + D, [1] A. Vilenkin, Phys. Lett. 117B,25 (1982); Phys. Rev.D 27, 2848 (1983);30, 509 (1984). ACKNOWLEDGMENTSWe are grateful to Leticia Cugliandolo, Miquel Dorca, Larry Ford, Jaume Garriga, Bei-Lok Hu, Jorge Kurchan, Diego Mazzitelli, Pasquale Nardone, Juan Pablo Paz, Josep Porrà, Albert Roura and Alex Vilenkin for very helpful suggestions and discussions. This work has been partially supported by the European project CI1-CT94-0004 and by the CICYT contracts AEN95-0590, Universidad de Buenos Aires, CONICET and Fundación Antorchas.Using that at the separatrix b = √ 4E s tanh(t/ √ 2), the double integral in the above expression gives 13.89 E 2 s /(2π 2 ), and we finally have,The fluxWe shall now show that, in spite of the divergence in S, f itself remains finite as we approach the separatrix. Basically, the arguments in section IV still hold, so the equation to solve iswhich is well behaved as x → ∞.In order to estimate the flux, we now need the integral of f in a neighborhood of the separatrix, namely K −1 ∼ dE f . With the same change of variables as above, we getThe integral peaks when 64 E s (β + α ln x)e −x = 1, which defines x 0 = ln(64 E s ) + ln(β + α ln x 0 ), and thusIn order to get back the old result when α = 0, we must assume a lower limit for the integral at x ∼ ln 64 ∼ 4.16, which corresponds to E ∼ 0. This limit is high enough that the integral is dominated by the lower limit (e −x ln x peaks below e), so we finally obtain,This result should be compared to our previous result (108), or (109). In spite of everything, we are still above the quantum tunneling probability (111). Thus, considering a cosmological model which undergoes a single cosmic cycle does not qualitatively change our conclusions. . J B Hartle, S W Hawking, Phys. Rev. D. 282960J.B. Hartle and S.W. Hawking, Phys. Rev. D 28, 2960 (1983); . A Vilenkin, 333560A. Vilenkin, ibid. 33, 3560 (1986). . R M Wald, Phys. Rev. D. 282118R.M. Wald, Phys. Rev. D 28, 2118 (1983). . E Calzetta, C El Hasi, Phys. Rev. D. 512713E. 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[ "Can lies be faked? Comparing low-stakes and high-stakes deception video datasets from a Machine Learning perspective", "Can lies be faked? Comparing low-stakes and high-stakes deception video datasets from a Machine Learning perspective" ]	[ "Mateus Karvat Camara \nAdriana Postal Universidade Estadual do Oeste do Paraná\nUniversidade Estadual do Oeste do Paraná\nUniversity of Nottingham\nUniversidade Tecnológica Federal do Paraná\n\n", "Tomas Henrique Maul [email protected] \nAdriana Postal Universidade Estadual do Oeste do Paraná\nUniversidade Estadual do Oeste do Paraná\nUniversity of Nottingham\nUniversidade Tecnológica Federal do Paraná\n\n", "Gustavo Paetzold [email protected] \nAdriana Postal Universidade Estadual do Oeste do Paraná\nUniversidade Estadual do Oeste do Paraná\nUniversity of Nottingham\nUniversidade Tecnológica Federal do Paraná\n\n" ]	[ "Adriana Postal Universidade Estadual do Oeste do Paraná\nUniversidade Estadual do Oeste do Paraná\nUniversity of Nottingham\nUniversidade Tecnológica Federal do Paraná\n", "Adriana Postal Universidade Estadual do Oeste do Paraná\nUniversidade Estadual do Oeste do Paraná\nUniversity of Nottingham\nUniversidade Tecnológica Federal do Paraná\n", "Adriana Postal Universidade Estadual do Oeste do Paraná\nUniversidade Estadual do Oeste do Paraná\nUniversity of Nottingham\nUniversidade Tecnológica Federal do Paraná\n" ]	[]	Despite the great impact of lies in human societies and a meager 54% human accuracy for Deception Detection (DD), Machine Learning systems that perform automated DD are still not viable for proper application in real-life settings due to data scarcity. Few publicly available DD datasets exist and the creation of new datasets is hindered by the conceptual distinction between low-stakes and highstakes lies. Theoretically, the two kinds of lies are so distinct that a dataset of one kind could not be used for applications for the other kind. Even though it is easier to acquire data on low-stakes deception since it can be simulated (faked) in controlled settings, these lies do not hold the same significance or depth as genuine high-stakes lies, which are much harder to obtain and hold the practical interest of automated DD systems. To investigate whether this distinction holds true from a practical perspective, we design several experiments comparing a high-stakes DD dataset and a low-stakes DD dataset evaluating their results on a Deep Learning classifier working exclusively from video data. In our experiments, a network trained in low-stakes lies had better accuracy classifying high-stakes deception than lowstakes, although using low-stakes lies as an augmentation strategy for the high-stakes dataset decreased its accuracy.	10.48550/arxiv.2211.13035	[ "https://export.arxiv.org/pdf/2211.13035v1.pdf" ]	253,802,003	2211.13035	53dfec9325a471d7ae722cf569ae57fc338d945f	Can lies be faked? Comparing low-stakes and high-stakes deception video datasets from a Machine Learning perspective Mateus Karvat Camara Adriana Postal Universidade Estadual do Oeste do Paraná Universidade Estadual do Oeste do Paraná University of Nottingham Universidade Tecnológica Federal do Paraná Tomas Henrique Maul [email protected] Adriana Postal Universidade Estadual do Oeste do Paraná Universidade Estadual do Oeste do Paraná University of Nottingham Universidade Tecnológica Federal do Paraná Gustavo Paetzold [email protected] Adriana Postal Universidade Estadual do Oeste do Paraná Universidade Estadual do Oeste do Paraná University of Nottingham Universidade Tecnológica Federal do Paraná Can lies be faked? Comparing low-stakes and high-stakes deception video datasets from a Machine Learning perspective Despite the great impact of lies in human societies and a meager 54% human accuracy for Deception Detection (DD), Machine Learning systems that perform automated DD are still not viable for proper application in real-life settings due to data scarcity. Few publicly available DD datasets exist and the creation of new datasets is hindered by the conceptual distinction between low-stakes and highstakes lies. Theoretically, the two kinds of lies are so distinct that a dataset of one kind could not be used for applications for the other kind. Even though it is easier to acquire data on low-stakes deception since it can be simulated (faked) in controlled settings, these lies do not hold the same significance or depth as genuine high-stakes lies, which are much harder to obtain and hold the practical interest of automated DD systems. To investigate whether this distinction holds true from a practical perspective, we design several experiments comparing a high-stakes DD dataset and a low-stakes DD dataset evaluating their results on a Deep Learning classifier working exclusively from video data. In our experiments, a network trained in low-stakes lies had better accuracy classifying high-stakes deception than lowstakes, although using low-stakes lies as an augmentation strategy for the high-stakes dataset decreased its accuracy. Introduction Lies are widespread throughout human societies and it is estimated that humans lie more than twice a day [13]. In contexts such as Justice, lies can have meaningful consequences, with 53% of the exonerations in the United States between 1989 and 2012 involving deception [24]. Therefore, researchers have been interested in Deception Detec-tion (DD) for centuries [39], and, since human accuracy for this task is merely 54% [8], attempts have been made to build automated systems for DD, the most successful being the polygraph. However, despite its widespread use in courts, the polygraph has proven to be an unreliable system [17,32] and its usage in official settings is currently not recommended. The quest for automated DD systems has been reinvigorated by the success of Machine Learning (ML) applications and many researchers have directed efforts toward this field. Even though some papers achieved impressive accuracies above the 90% mark [9,11,14,20,36,40,58,61], the proper application of ML systems to DD in real-life is still not viable due to data scarcity [39,47]. Few video-based DD datasets have been created and even fewer made available for public use, hindering the development of proper techniques and experiments in the area. But most importantly, the existing datasets lack sufficient data for use in real-life settings, with no public DD dataset having more than 400 videos, which is far from enough to properly train a system capable of correctly identifying lies outside of experimental settings [39]. Aside from the inherent challenges of building datasets based on human subjects such as ethics and biases, DD has a conceptual barrier that complicates matters further: the distinction between low-stakes and high-stakes lies [59]. According to the literature [27], there is a significant distinction between lies told in different contexts since cues for deception will emerge in high-stakes situations in which the liars develop a stronger emotional response, a phenomenon known as non-verbal leakage. According to Frank and Ekman [19], "it is the presence of these emotions, such as guilt, fear of being caught, and disgust, that can betray the liar's deception when they are leaked through nonverbal behaviors such as facial expressions [...] or voice tone". In this sense, trials, interviews or negotiations could be considered as high-stakes scenarios, while contexts that do not elicit such strong reactions could be seen as low-stakes: truth games, roleplaying or even situations in which participants are instructed to lie by researchers. Even though low-stakes lies are more frequent in human lives and can be easily simulated (faked) in controlled settings, the practical interest for automated DD falls on highstakes lies due to their potential impact. Based on that, solving the issue of data scarcity for DD would mean creating a large and diverse public dataset of high-stakes lies. However, the acquisition of this type of data poses ethical and privacy issues and, most importantly, given that human accuracy for DD is so low, any labeling of these samples should be based on unassailable proof that testimonies were either truthful or deceptive. Ideally, ML systems would be trained on fake lies easily acquired in controlled settings (low-stakes lies) and applied to high-stakes DD scenarios, though the conceptual distinction between these two kinds of lies has blocked such possibility. We then arrive at an impasse since, on one hand, high-stakes datasets are needed for real-life applications but prove difficult to be built, and, on the other hand, low-stakes datasets can be created more easily from lies told in fake scenarios yet have limited practical relevance. Despite being a possible solution to such an impasse, to the best of our knowledge, no study has evaluated if there is, indeed, a significant distinction between low-stakes and high-stakes deception from an ML perspective, from which two questions of practical significance arise: "Can an ML system trained in a low-stakes deception dataset be used to classify high-stakes lies?" and "Can low-stakes lies be used for data augmentation in high-stakes deception datasets?". Hence, this paper investigates this distinction by comparing the performance of a Deep Learning system with two datasets: Real-life Trial (RLT) [53], a high-stakes dataset with footage from trials and currently considered the standard video-based DD dataset [48]; and Box of Lies (BoL) [55], a low-stakes dataset with videos taken from a truth game on the television show "The Tonight Show Starring Jimmy Fallon ®". Focusing exclusively on video data from each dataset and performing binary classification, a network based on the Slowfast architecture [16] is created and several experiments are performed to evaluate if the aforementioned distinction does indeed hold true based on experimental results. We hope that, by examining this distinction, researchers may redirect their efforts toward building largescale deception datasets while making the best use of the limited data available in this important application area. Thus, this paper is organized as follows: Sec. 2 discusses the existing public video-based DD datasets and ML papers that use them, Sec. 3 describes our methodology, Sec. 4 presents and discusses the results from our experiments and Sec. 5 summarizes our findings. Deception detection datasets Back in 2012, Gokhman et al. [21] and Fitzpatrick and Bachenko [18] discussed the development of a standard Deception Detection (DD) dataset, since none was available at the time, which required every paper to build its own dataset, limiting comparisons between studies and hindering the advancement of the field. Since then, many papers have been published with experiments conducted in video-based datasets which were not publicly available [6,7,39,51,56] due to the sensitive nature of the task. However, four video-based public datasets (presented in Tab. 1) have been created and are discussed here. We focus on datasets with videos for their simplicity of being deployed in real-life situations, in contrast to Electroencephalogram (EEG) or Functional near-infrared spectroscopy (fNIRS) which require special equipment to acquire data, and for the fact that they present the best results, in relation to audio and text, in papers that compare the performance of unimodal classification [40,48,61]. Yet, several papers have worked with DD datasets from other modalities (some of which are publicly available), such as EEG [1,3,15], fNIRS [30], audio [22,49,62] and text [10,12,29,31]. Low-stakes datasets To the best of our knowledge, there are currently three publicly available video-based low-stakes DD datasets: Bag-of-Lies [26], Miami University Deception Detection Dataset (MU3D) [46] and Box of Lies (BoL) [55]. Bag-of-Lies [26] is described by its authors as a "casual deception" dataset, containing 325 videos of 35 volunteers that had to describe images from a selected set being free to be truthful or deceptive. Being a multimodal dataset, it has visual, audio and gaze data (acquired with an eye tracker) for all testimonies, and EEG data for 201 videos. The MU3D [46] was created with the goal of being a standardized, unbiased and balanced dataset, being made of 320 videos from 80 subjects (20 Black female, 20 Black male, 20 White female, and 20 White male), each one having 2 lies and 2 truths. In each video, participants describe people they like and people they dislike, with all videos having audio and accompanying transcriptions. The BoL dataset [55] features videos from participants playing the "Box of Lies" game in the "The Tonight Show Starring Jimmy Fallon ®" television show. Played between a celebrity guest and the show's host, the game consists of multiple rounds in which players describe (deceptively or truthfully) an object which is hidden from the other player, with the listener having to guess whether the description was a truth or a lie. Sample frames are presented in Fig. 1. The BoL dataset provides transcriptions, annotations on gestures and facial displays, as well as labels for small segments (called utterances) within each round played, with the videos being available on YouTube in an official playlist of the television show. When the Box of Lies paper [55] was published, 68 rounds were available, though currently this number has gone up to 93 rounds. Using these 68 rounds, the original Box of Lies paper achieves an accuracy of 65% with a Random Forest classifier on multimodal data, while Zhang et al. [64], also using Random Forest, achieves 73% accuracy for multimodal classification and 67% for classification solely on videos. High-stakes datasets To the best of our knowledge, the only publicly available video-based high-stakes DD dataset is the Real-life Trial dataset [53] (RLT), which makes it the standard dataset for papers that apply ML techniques to DD [48]. It originally consists of 121 videos from 56 unique individuals taken from trials in which truthful or deceptive testimonies were verified by the police as such, thus allowing the dataset creators to objectively label each video. It is available on one of its authors' page [50] and accompanies transcriptions for each video and annotations on non-verbal behavior such as facial displays and hand gestures. Sample frames from the dataset are presented in Fig. 2. Several papers have used the RLT dataset for ML ex- periments, which have their results and techniques summarized in Tab. 2. Among these, data augmentation strategies, classifiers, feature selection and preprocessing techniques vary greatly. With a single exception [11], all papers that combine video, audio and transcripts achieve better results in multimodal classification than in unimodal classification from videos, showcasing the strength of combining multiple types of data for this task. But most importantly, all papers achieve results above human accuracy for DD (54% [8]) and the accuracy measured for this specific set of videos by the creators of RLT in a set of experiments with human volunteers (last row of Tab. 2). However, according to Belavadi et al. [6], "there is insufficient evidence that AI systems for detecting deception are likely to achieve adequate accuracy in real-world use" and, given that the RLT dataset is small, such impressive experimental results do not provide such evidence. Moreover, Mambreyan et al. [47] have shown that the RLT dataset has significant gender bias which can be exploited by an ML classifier, further highlighting the need for bigger, more diverse and less biased publicly available video-based highstakes DD datasets. Methodology To evaluate a possible distinction between low-stakes and high-stakes deception from a Machine Learning (ML) perspective, the Box of Lies (BoL) [55] and Real-life Trial (RLT) [53] datasets were selected. While RLT is, to the best of our knowledge, the only publicly available video-based high-stakes Deception Detection (DD) dataset, BoL is the low-stakes dataset that has the closest number of videos to RLT (considering its current number of 93 rounds), making it a fairer comparison than other low-stakes datasets which have almost thrice the number of videos of RLT. Dataset preparation Following the steps taken by other papers that work with RLT [9,14,33,48,52,61], we remove videos from the dataset which are deemed unsuitable for classification based solely on visual data, such as those in which the speaker's face is hidden or out of focus during most of the video, or where there are multiple people in the foreground. After that, the dataset was reduced to 110 videos portraying 51 individuals. Among the remaining videos, some had noisy frames or frames showing other people instead of the speaker, so they were edited with Kdenlive [38] and these frames were removed. The full list of removed and edited videos is avail-able in the Supplementary Material. While the BoL dataset was originally labeled by utterance, the RLT dataset is labeled by video, which required a change in labels in BoL. Therefore, each round was taken as a single video labeled according to the veracity of the object's description and, for each video, the frames not showing the speaker were removed with Kdenlive [38]. Despite assigning a single label to whole videos (which contain truthful and deceptive segments), such labeling scheme follows the scheme from RLT (allowing for better comparison between datasets) and removes subjectivity from the process. By contrast, the original labeling by utterance categorized segments with unclear veracity as deceptive, which resulted in 82.2% of the samples being labeled as deceptive. After this dataset preparation step, the resulting BoL and RLT datasets were such as presented in Tab. 3, from which it can be inferred that an ML system trained in RLT should have better results than a similar one trained in BoL, since RLT has more data while also being more diverse (more individuals) and balanced. To illustrate such balance, the individual with the highest number of videos in RLT is depicted in 21 videos ( shown in Tab. 4. Despite not being a desirable property of the dataset, RLT has a significant gender bias, which should also make ML systems trained on it have better results due to bias exploitation. As a final preparation step, both datasets are enriched with copies of their videos flipped horizontally, doubling the number of videos in each dataset. Network training Focusing solely on DD from videos (as discussed in Sec. 2), we use the Slowfast [16] architecture to perform binary classification. This architecture was chosen for its balance between classification accuracy and low computational cost, achieving better results than other Deep Learning video recognition architectures that have similar computational cost, such as Hidden TSN [65], TSM [44], TEINet [45], MSNet [42], TEA [43], STM [34] and CSN [57]. Such criterion was used due to limited computational power and time allocation of resources for our experiments, which were performed in a shared computer with an Intel Core i3-10100F 3.6 GHz CPU, 16 GB 2666 MHz RAM and a GTX 1650 4 GB GPU. These limitations also prompted us to use the Slowfast implementation available in the Glu-onCV framework [25] for its optimizations and ease of implementation, as well as performing non-exhaustive hyperparameter search prior to k-fold testing. For each network trained, hyperparameter search was performed independently, with the best 5 hyperparameter combinations later being used with 5-fold testing. Since non-exhaustive search was performed, the 5 best combinations were used for testing as a means of overcoming a possible deficit from the non-exhaustive search. An 80/20 split was performed for hyperparameter search, with an initial set of values being tested, and the following combinations being chosen according to the previous combinations' accuracy on the validation set. Therefore, hyperparameter values that did not perform well were assessed but soon discarded. Such strategy meant not all hyperparameter combinations were evaluated, but results on the validation set were considered satisfactory given that each combination had its results thoroughly analyzed. Hyperparameters evaluated and their corresponding values were: • Slowfast configuration: 4x16 and 8x8; • Optimizer: SGD, Adam and RMSProp; • Learning rate: ranging from 10 −1 to 10 −6 in exponential increments; • Weight decay: 10 −2 , 10 −4 , 10 −6 and no weight decay; • Learning rate decay strategy: reducing learning rate by a factor of 10 every 40 epochs, every 10 epochs, or not reducing it at all; • Momentum: 0.98, 0.9 and 0.5. For most trials, 100 epochs were used, but for combinations that had not converged by 100 epochs, greater numbers were also evaluated. The batch size was limited to 1 since there was not enough memory available for greater values, a limitation which also inhibited the use of different Slowfast backbones apart from ResNet50 [28]. The list of all hyperparameter combinations evaluated is available in the code repository for this paper. Experimental setup Our experimental setup is presented in Fig. 3. As a first step, hyperparameter search is performed considering each dataset independently. The 5 hyperparameter combinations with the highest validation accuracies are then used for 5fold testing with their respective datasets. Even though both use the Slowfast architecture, the combination that yields the best accuracy for RLT is named Net 1 while the one which yields the best accuracy for BoL is named Net 2. Both are considered optimized for their respective datasets. Figure 3. Experimental setup used to evaluate differences between a low-stakes deception dataset (Box of Lies -BoL [55]) and a highstakes one (Real-life Trial -RLT [53]). The network used was Slowfast [16], with three distinct training configurations based on the hyperparameter searches performed for the distinct training sets (RLT, BoL and RLT+BoL). Letter tags are used in each Experiment for later reference. Trained on the RLT dataset, Net 1 is then used for inference on the BoL dataset while Net 2, having been trained on BoL, is used for inference on RLT (Step 2 in Fig. 3). Since both Net 1 and Net 2 were optimized for their datasets, this step allows us to evaluate whether each kind of deception dataset (high-stakes and low-stakes) can be used as training data for inference on the other kind. Finally, both datasets are combined (Step 3 in Fig. 3) and hyperparameter search is performed for the mixed dataset, to evaluate the efficacy of combining the two kinds of deception datasets. The 5 best hyperparameter combinations are used for 5-fold testing and the best one is named Net 3. Aiming to evaluate possible data augmentation strategies, both Nets 1 and 2 are retrained (using their previous set of hyperparameters) on the mixed dataset and tested with their original datasets. Results and discussion Given that 5-fold testing was performed for all experiments except cross-testing, a total of 27 runs were conducted for testing. Hyperparameter search for Nets 1, 2 and 3 required 44, 176 and 77 runs, respectively. Therefore, a total of 324 runs were performed. If an exhaustive hyperparameter search was performed along with 5-fold (exhaustive search being done for each fold), the number of runs would be above 10000, which would not be viable given our previously described computing power limitations. Even though our approach does not guarantee optimal results, it achieves adequate accuracies with a fraction of the resources needed. The results from hyperparameter search runs are available on the code repository of this paper. Network optimization The hyperparameter search for Nets 1, 2 and 3 was performed and the 5 best combinations of each were later used for 5-fold testing. The combination which had the best result was considered the one that better optimized its respective network. Tab. 5 presents the hyperparameter values for each of these combinations. The results of the optimization step for each of the networks are presented in Tab. 6, from which the assumptions raised in Sec. 3.1 hold true: the RLT dataset (Exp. A) had better results than BoL (Exp. B). Both networks had results above human accuracy for DD (54%), however both were below the results achieved by most works presented in Sec. 2, which can be attributed to experimental limitations (such as a batch size of 1) and the fact that the Slowfast architecture was originally created for the Kinetics-400 [37] dataset, which has considerably more data with 306245 videos, and often showed overfitting in our experiments. The combination of both datasets (Exp. E), however, had its results closer to those from BoL (Exp. B) than those from RLT (Exp. A), indicating that an increase in training data, by itself, is not enough to achieve better results. It is unclear to us whether these results occurred due to an improper combination of data with a significant semantic difference (low-stakes lies and high-stakes lies) or because of properties of these specific datasets (gender bias, category bias, imbalance of the number of videos by individual). Real-life Trial (Net 1) results The results from experiments done with Net 1 are presented in Tab. 7, which highlights the previous considerations on the difficulty of each dataset. While the best results are seen when RLT is trained and tested on its own (Exp. A), cross-testing on BoL (Exp. C) had results below human accuracy, and adding BoL videos as an augmentation strategy (Exp. F) ends up reducing the original accuracy (Exp. A). Even though this cross-testing scenario does not hold much practical purpose since pragmatic interest lies in testing being performed with high-stakes lies and training with low-stakes lies, the evaluation of the augmentation strategy meets such interests. That being so, in our experiments, using low-stakes lies to augment a high-stakes deception dataset resulted in worse performance than working with the high-stakes dataset on its own. The results suggest that this behavior occurred due to semantic differences between these datasets (high-stakes and low-stakes deception). However, it is important to point out that the reduction in performance might have been caused by the aforementioned properties within the datasets which are unrelated to these semantic differences. Box of Lies (Net 2) results Experiments performed with Net 2 are shown in Tab. 8, which presents noteworthy results. Even though training and testing in BoL reached a low accuracy (Exp. B), crosstesting with RLT (Exp. D) increased this accuracy and using RLT to augment data on BoL (Exp. G) significantly improved results. In view of the considerations previously presented on the difficulty of each dataset, the results from Exp. G can be understood from the perspective that videos from RLT make training easier, allowing the network to better identify patterns within the testing dataset (BoL). From another point of view, it might be argued that high-stakes deception are more easily identified than low-stakes, making them suitable for data augmentation. However, Exp. D's results are not entirely conclusive. It might be argued that, since training was performed in a difficult dataset, testing in an easier one would improve results. Despite that, such an experimental scenario perhaps holds the greatest practical interest due to the ease of acquiring low-stakes deception data and the difficulty of doing so for high-stakes data (which gets the most practical interest for DD systems). From our experiments alone, it could be said that an ML system trained with low-stakes lies could be acceptably applied to a set of high-stakes lies. However, due to our experimental limitations and limited data, such findings cannot be generalized for all DD datasets. Conclusion We set out to tackle the issue of data scarcity in Deception Detection (DD), aiming to shed light on the conceptual distinction between high-stakes and low-stakes lies by presenting numerical data that might support researchers on the task of building new datasets for such an important application area. Currently, datasets for DD are either built with high-stakes or low-stakes lies and applications trained in one kind theoretically can only be used for inference on lies of that same kind. Through our investigation, we evaluated this distinction by experimentally comparing two datasets: Real-life Trial (RLT) [53], a high-stakes DD dataset, and Box of Lies (BoL) [55], a low-stakes DD dataset. Different experiments were performed with the Slowfast [16] architecture and their results were used to analyze whether such a distinction holds true. From these experiments we found that: 1. The network trained in high-stakes lies performed better than the network trained in low-stakes lies; 2. Combining both datasets into a single one had worse results than working with the high-stakes dataset on its own; 3. Using the network trained in high-stakes deception for inference on low-stakes deception had results below human accuracy; 4. Using low-stakes deception as a data augmentation strategy for the high-stakes dataset did not improve results; 5. The network trained in low-stakes lies had better accuracy classifying high-stakes deception than low-stakes; 6. Using high-stakes lies as a data augmentation strategy for the low-stakes dataset showed a significant improvement in results. Given these findings, we conclude that there is a clear distinction between the RLT and BoL datasets. However, similar experiments should be performed with different datasets to assess whether these results stem from differences in these datasets' incidental properties (mainly bias and amount of data) or from a deep semantic difference in their data (low-stakes and high-stakes lies). Nonetheless, to the best of our knowledge, we have shown, for the first time, that a low-stakes DD dataset can be acceptably used to train a Machine Learning (ML) classifier created for inference on high-stakes deception data. Finally, we highlight the need for bigger, less biased and more balanced publicly available video-based DD datasets. Despite impressive results in experimental settings, ML DD systems are not yet ready for deployment in real-life settings given the lack of sufficient data for their proper training. Figure 1 . 1Sample frames from the Box of Lies dataset (BoL)[55] which contains videos of low-stakes deception from a television show. The top two frames are lies and the bottom two are truths. Figure 2 . 2Sample frames from the Real-life Trial dataset (RLT)[53] which contains videos of high-stakes deception from trials. The top two frames are lies and the bottom two are truths. Table 1 . 1Publicly available video datasets for Deception Detection. Numbers in parentheses are presented for reference and correspond to the numbers used in this paper for the respective datasets.Dataset Category Videos Individuals Observations Real-life Trial [53] High-stakes 121 (110) 56 (51) Standard dataset for Deception Detection [48] Box of Lies [55] Low-stakes 68 (93) 26 (33) - Miami University Deception Detection Dataset [46] Low-stakes 320 80 Balanced dataset Bag-of-Lies [26] Low-stakes 325 35 Includes EEG and Gaze information Table 2. Comparison of papers that use the Real-life Trial dataset[53] for Deception Detection with Machine Learning. Accuracy for classification exclusively from videos (Acc. V) is presented, as well as multimodal classification with Video, Audio and Transcripts (Acc. V+A+T). Marked accuracies () are actually AUC. Papers that use exclusively Deep Learning classifiers are grouped on top for convenience. All classifiers achieve better results than the measured human performance for this dataset presented on the bottom row.Paper Classifier Details Acc. V Acc. V+A+T [58] CNN + LSTM Vague methodology 100 - [36] CNN Preprocessing with Local Binary Patterns 97.35 97.33 [14] CNN ResNet [28], GANs [23] 93.61 97.00 [40] 3DCNN Fusion by Hadamard product 93.08 96.14 [20] 3DCNN Fusion by concatenation 78.57 96.42 [52] CNN + RNN Facial reconstruction 72.8 - [9] SVM Features from AlexNet [41], Multiview learning 99 99 [11] kNN Manually annotated behavioral cues 94 78 [61] Logistic Regression Features: Improved Dense Trajectory (IDT) [60] 89.88 92.21* [66] NN Features: Facial displays and hand gestures 78.53 84.18 [2] RBF-SVM Features: Action Units (AUs) [4] 76.84 - [48] SVM Features from OpenFace [5], Affect [54] 76 84 [35] LMNN Features from CNN + LSTM network 75 84.16 [63] RF, SVM Emotion classification 71.15 87.59 [53] DT Features: Facial displays and hand gestures 68.59 75.20 [33] SVM Features: AUs [4] 67.2 78.95 [47] Linear SVM Features: IDT [60], Gender classifier 64.6 - [47] Linear SVM Features: IDT [60] 57.4 - [53] Human performance Average between 3 annotators 46.50 56.47 19.1% of the videos) while for BoL, this number goes up to 34 videos (36.6% of the videos).Also, considering the findings from Mambreyan et al.[47], we consider biases present in each dataset, which areReal-life Trial Box of Lies Videos 110 93 Individuals 51 33 Average number of videos by individual 2.2 2.8 Standard deviation of videos by individual 3.5 5.6 Average video length 26.9 s 20.8 s Table 3. Comparison between the Real-life Trial (RLT) [53] and Box of Lies (BoL) [55] datasets after dataset preparation. Each video portrays a single individual. Not only does RLT have more videos depicting a greater number of individuals with a longer av- erage length, but it also is more balanced in regards to the number of videos by individual (a higher standard deviation of videos by individual indicating a less balanced dataset). Real-life Trial Box of Lies Truth ratio 51.8% 46.3% Women ratio 61.8% 48.4% Truth ratio for women 35.3% 53.3% Truth ratio for men 78.6% 39.6% Table 4. A comparison of biases in the Real-life Trial (RLT) [53] and Box of Lies (BoL) [55] datasets after dataset preparation. Even though BoL has a slight bias toward lies, RLT has a sig- nificant gender bias. Table 8 . 8Comparison of results obtained for experiments (Exp.) on Net 2. Cross-testing improved results, indicating that RLT is an easier dataset than BoL. 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[ "Transport equation in generalized Campanato spaces", "Transport equation in generalized Campanato spaces" ]	[ "Dongho Chae ", "Jörg Wolf \nSchool of Mathematics ( * )\nChung-Ang University Dongjak-gu Heukseok-ro 84\n06974SeoulRepublic of Korea\n\nKorea Institute for Advanced Study Dongdaemun-gu Hoegi-ro 85\n02455SeoulRepublic of Korea\n", "\nDepartment of Mathematics ( * )\n\n" ]	[ "School of Mathematics ( * )\nChung-Ang University Dongjak-gu Heukseok-ro 84\n06974SeoulRepublic of Korea", "Korea Institute for Advanced Study Dongdaemun-gu Hoegi-ro 85\n02455SeoulRepublic of Korea", "Department of Mathematics ( * )\n" ]	[]	In this paper we study the transport equation in R n × (0, T ), T > 0,The critical case is particularly interesting, and is applied to the local well-posedness problem in a space close to the Lipschitz space in our companion paper[6]. More specifically, in the critical case s = q = N = 1 we have the embedding relations,	10.4171/rmi/1394	[ "https://arxiv.org/pdf/1904.08215v2.pdf" ]	118,697,245	1904.08215	8806ce4a3e355af58db85569196f786825a8f90e	Transport equation in generalized Campanato spaces 18 Apr 2019 Dongho Chae Jörg Wolf School of Mathematics ( * ) Chung-Ang University Dongjak-gu Heukseok-ro 84 06974SeoulRepublic of Korea Korea Institute for Advanced Study Dongdaemun-gu Hoegi-ro 85 02455SeoulRepublic of Korea Department of Mathematics ( * ) Transport equation in generalized Campanato spaces 18 Apr 2019arXiv:1904.08215v2 [math.AP] In this paper we study the transport equation in R n × (0, T ), T > 0,The critical case is particularly interesting, and is applied to the local well-posedness problem in a space close to the Lipschitz space in our companion paper[6]. More specifically, in the critical case s = q = N = 1 we have the embedding relations, Introduction Let 0 < T < +∞ and Q = R n × R + with n ∈ N, n ≥ 2. We consider the transport equation (1.1)    ∂ t f + (v · ∇)f = g in Q, v = v 0 on R n × {0}, where f = f (x 1 , . . . , x n ) is unknown, while v = (v 1 , · · · , v n ) = v(x, t) represents a given drift velocity and g = g(x 1 , . . . , x n ) given function. Our aim in this paper is to obtain estimates of solutions to (1.1) in generalized Campanato spaces. The proof relies on a key estimate in terms of local oscillation. As byproduct we get existence of solutions in Besov spases and Tribel-Lizorkin spases, which can be estimated by the data belonging to these spaces. One of the main motivations to study the transport equation in such generalized Campanato spaces is to apply it to prove local well-posedness of the incompressible Euler equations in function space embedded in the Lipschitz space, which includes linearly growing functions at spatial infinity. For recent developments of the local well-posedness/ill-posedness of the Euler equations in various critical function spaces embedded in C 0,1 (R n ) we refer [3,4,11,13,16,17,1,7,12]). We would also like to refer [8] for the study of transport equation with drift velocity in less regular space. For the application of our new function spaces in the critical case to the Euler equations please see our companion paper [6]. Let us introduce the function spaces we will use throughout the paper. Let N ∈ N ∪ {0, −1}. By P N (Ṗ N respectively) we denote the space of all polynomial (all homogenous polynomials respectively) of degree less or equal N. We equip the space P N with the norm P (p) = P L p (B(1)) . Note that since dim(P N ) < +∞ all norms · (p) , 1 ≤ p ≤ ∞, are equivalent. For notational convenience, in case N = −1 we use the convention P −1 = {0}, which consists of the trivial polynomial P ≡ 0. Let f ∈ L p loc (R n ), 1 ≤ p ≤ +∞. For x 0 ∈ R n and 0 < r < ∞ we define the oscillation We note that from our convention above in the case N = −1 we have osc p,−1 (f ; x 0 , r) := \|B(r)\| − 1 p f L p (B(x 0 ,r)) . Then, we define for 1 ≤ q, p ≤ +∞ and s ∈ (−∞, N + 1] the spaces L s q(p,N ) (R n ) =    f ∈ L p loc (R n ) \|f \| L s q(p,N) := j∈Z 2 −sj osc p,N (f ; ·, 2 j ) q 1 q L ∞ < +∞    . Furthermore, by L k,s q(p,N ) (R n ), k ∈ N, we denote the space of all f ∈ W k, p loc (R n ) such that D k f ∈ L s q(p,N ) (R n ). The space L k,s q(p,N ) (R n ) will be equiped with the norm f L k,s q(p,N) = \|D k f \| L s q(p,N) + f L p (B(1)) , f ∈ L k,s q(p,N ) (R n ). According to the characterization theorem of the Triebel-Lizorkin spaces in terms of oscillation, we have          f ∈ F s r,q (R n ) ⇔ f L min{r,q} + 0 j=−∞ 2 −sj osc p,N (f ; ·, 2 j ) q 1 q L r < +∞. 0 < r < +∞, 0 < q ≤ ∞, s > , and we could regard the spaces L s q(p,N ) (R n ) as an extension of the limit case of F s r,q (R n ) as r → +∞. In fact in case q = +∞ and s > 0 we get the usual Campanato spaces with the isomorphism relation(cf. [5,10]) L n+ps,p N (R n ) ∼ = L s ∞(p,N ) (R n ). Furthermore, in the case N = 0, s = 0 and q = ∞ we get the space of bounded mean oscillation, i.e., L 0 ∞(p,0) (R n ) ∼ = BMO. In case N = −1 and s ∈ (− n p , 0) the above space coincides with the usual Morrey space M n+ps (R n ). We note that the oscillation introduced in (1.2) is attained by a unique polynomial P * ∈ P N . According to Theorem 3.6 (see Section 3 below), for the spaces L 1 1(p.1) (R n ) we have the following embedding properties ֒→ L 1 1(p,1) (R n ) ֒→ L 0,1 1(p,0) (R n ) ֒→ C 0,1 (R n ). Accordingly, (1.4) ∇u ∞ ≤ c u L 1 1(p,1) . Furthermore, for every f ∈ L k 1(p,k) (R n ), k ∈ {0, 1}, there exists a uniqueṖ k ∞ (f ) ∈Ṗ 1 , such that for all x 0 ∈ R n f converge asymptotically toṖ k ∞ (f ) as \|x\| → +∞. The precise meaning of this asymptotic limit will be given in Section 3 below. We are now in a position to present our first main result. Theorem 1.1 (The case N = 0). Let 0 < T < +∞. Let s ∈ (− n q , 0), 1 < p < +∞, 1 ≤ q ≤ +∞. Let v ∈ L 1 (0, T ; L p loc (R n )), with (1.5) T 0 ∇v(τ ) ∞ dτ < +∞. Then for every f 0 ∈ L s q(p,0) (R n ) and g ∈ L 1 (0, T ; L s q(p,0) (R n )) there exists a unique solution f ∈ L ∞ (0, T ; L s q(p,0) (R n )) to the transport equation (1.1). Furthermore, it holds for almost all t ∈ (0, T ) \|f (t)\| L s q(p,0) ≤ c \|f 0 \| L s q(p,0) + T 0 \|g(τ )\| L s q(p,0) dτ exp c T 0 ∇v(τ ) ∞ dτ . (1. 6) In case N = 1 we get Theorem 1.2 (The case N = 1 and s = 1). Let 0 < T < +∞ and 1 < p < +∞, 1 ≤ q ≤ +∞. Let v ∈ L 1 (0, T ; L 1 q(p,1) (R n )) with (1.5) and (1.7) (v(τ ); x 0 , 2 j ) 1 q dτ < +∞ Let f 0 ∈ L 1 q(p,1) (R n ) and g ∈ L 1 (0, T ; L 1 q(p,1) (R n )) satisfying the condition = \|z\| L 1 q(p,1) + sup x 0 ∈R n \|∇Ṗ 1 x 0 ,1 (z)\|. Our third main result concerns the case s > 1. Theorem 1.3 (The case N ≥ 1 and s > 1). Let 0 < T < +∞, N ∈ N, 1 < s < +∞, 1 < p < +∞, and 1 ≤ q ≤ +∞. Let v ∈ L 1 (0, T ; L s q(p,N ) (R n )) with (1.5) and Let f 0 ∈ L s q(p,N ) (R n ) and g ∈ L 1 (0, T ; L s q(p,N ) (R n )) satisfying the condition ∇f 0 ∞ + T 0 ∇g(τ ) ∞ dτ < +∞. (1.10) Then, there exists a unique solution f ∈ L ∞ (0, T ; L 1 q(p,1) (R n )) to the transport equation (1.1) together with the estimate \|f (t)\| L s q(p,0) ≤ c \|f 0 \|L s q(p,0) + T 0 \|g(τ )\|L s q(p,0) exp c T 0 v(τ ) L s q(p,0) dτ , (1.11) where \|z\|L s q(p,0) stands for the semi norm defined by \|z\|L s q(p,0) = \|z\| L s q(p,0) + ∇z ∞ . From Theorem 1.2 we get the following corollary for the special case s = q = N = 1, which will be useful for our future application to the Euler equations in the critical spaces. Corollary 1.4. Let 0 < T < +∞, 1 < p < +∞. Let v ∈ L 1 (0, T ; L 1 1(p,1) (R n )), f 0 ∈ L 1 1(p,1) (R n ) and g ∈ L 1 (0, T ; L 1 1(p,1) (R n )) . Then there exists a unique solution f ∈ L ∞ (0, T ; L 1 1(p,1) (R n )) to the transport equation (1.1). Furthermore, it holds for all t ∈ (0, T ) f (t) L 1 1(p,1) ≤ C 1 + T 0 \|v(τ )\| L 1 1(p,1) dτ exp c T 0 ∇v(τ ) ∞ dτ . (1.12) where C = c f 0 L 1 1(p,1) + T 0 g(τ ) L 1 1(p,1) dτ , while c = const > 0 depending on n and p. Remark 1.5. Using the well-known characterization of B 1 ∞,1 (R n ) in terms of oscillation, we easily verify the embeddings (1.13) B 1 ∞,1 (R n ) ֒→ L 1 1(p,1) (R n ) ∩ L ∞ (R n ) ֒→ L 1 1(p,1) (R n ). Indeed, referring to [15, Theorem, Chap.1.7.3]), we see that v ∈ B 1 ∞,1 (R n ) ⇔ 0 j=−∞ 2 −j osc p,1 (v; ·, 2 j ) L ∞ + v L ∞ < +∞. This shows that for x ∈ R n it holds j∈Z 2 −j osc p,1 (v; x, 2 j ) ≤ 0 j=−∞ 2 −j osc p,1 (v; ·, 2 j ) L ∞ + ∞ j=1 2 −j osc p,1 (v; x, 2 j ) + v L ∞ . On the other hand, it is readily seen that osc p,1 (v; x, 2 j ) ≤ 2 v L ∞ . Accordingly, the second sum on the right-hand side is bounded by v L ∞ . This yields v L 1 1(p,1) = j∈Z 2 −j osc p,1 (v; x, 2 j ) + v L 2 (B(1)) ≤ j∈Z 2 −j osc p,1 (v; x, 2 j ) + c v B 1 ∞,1 ≤ c 0 j=−∞ 2 −j osc p,1 (v; ·, 2 j ) L ∞ + c v L ∞ ≤ c v B 1 ∞,1 . Secondly, according to [14, p. 85] (see also [1]) we have the embedding B 1 ∞,1 (R n ) ֒→ C 1 (R n ) ∩ L ∞ (R n ) . On the other hand, there exists a function f ∈ L 1 1(p,1) (R n ) which is not in C 1 (R n ) (see Appendix B). This clearly shows that L 1 1(p,1) (R n ) contains less regular functions then B 1 ∞,1 (R n ). Thirdly, since L 1 1(p,1) (R n ) contains linearly growing function at infinity, in particular polynomials of of degree less or equal one, L 1 1(p,1) (R n ) is strictly bigger than B 1 ∞,1 (R n ) in terms of asymptotic behaviors as infinity. We also note that the use of our generalized Campanato spaces to handle the bounded domain problem is quite convenient as in the case of usual Campanato spaces. Preliminariy lemmas Let X = {X j } j∈Z be a sequence of non-negative real numbers. Given s ∈ R and 0 < q < +∞, we denote {2 js } · X := {2 js X j } j∈Z , X q := {X q j } j∈Z respectively. We define S α,q : X = {X j } j∈Z → Y = {Y j } j∈Z , where Y j = (S α,q (X)) j = 2 jα ∞ i=j (2 −iα X i ) q 1 q , j ∈ Z. From the above definition, in case of α = 0, it follows that (2.1) S 0,q (X) ℓ ∞ = X ℓ ∞ ≤ X ℓ q ∀ X ∈ ℓ q . Clearly, for all α, β ∈ R it holds (2.2) 2 βj (S α,q (X)) j = S α+β,q ({2 βi X i }) j , j ∈ Z. Given X = {X j } j∈Z , Y = {Y j } j∈Z , we denote X ≤ Y if X j ≤ Y j for all j ∈ Z. Throughout this paper, we frequently make use of the following lemma, which could be regarded as a generalization of the result in [2]. Lemma 2.1. For all β < α and 0 < p ≤ q ≤ +∞ it holds (2.3) S β,q (S α,p (X)) ≤ 1 1 − 2 −(α−β) S β,q (X). Proof: We first observe (S β,q (S α,p X)) j = 2 jβ ∞ i=j 2 −iβq (S α,p X) q i 1 q = 2 jβ    ∞ i=j 2 −iβq   2 iα ∞ l=i (2 −αl X l ) p 1 p   q    1 q = 2 jβ    ∞ i=j 2 i(α−β)q ∞ l=i 2 −(α−β)pl 2 −βpl X p l q p    1 q = (S 0,q (S α−β,p ({2 −βpi X p i })) j . (2.4) 1. The case p = 1, β = 0. Let X be sequence with X j = 0 except finite j ∈ {m, m + 1, . . .}. By the aid of Hölder's inequality, we get (S 0,q (S α,1 (X))) q j = ∞ i=j 2 iα ∞ l=i 2 −αl X l q = ∞ i=j 2 iqα ∞ l=i 2 −αl X l ∞ l=i 2 −αl X l q−1 = ∞ i=j 2 iqα ∞ l=0 2 −α(l+i) X l+i ∞ l=i 2 −αl X l q−1 = ∞ l=0 2 −αl ∞ i=j X l+i 2 iα ∞ l=i 2 −αl X l q−1 = ∞ l=0 2 −αl ∞ i=j X l+i S α,1 (X) q−1 i ≤ ∞ l=0 2 −αl ∞ i=j X q l+i 1 q ∞ i=j (S α,1 (X)) q i q−1 q ≤ 1 1 − 2 −α (S 0,q (X)) j (S 0,q (S α,1 (X))) q−1 j , where we used the fact ( ∞ i=j X q l+i ) 1 q ≤ ( ∞ i=j X q i ) 1 q = (S 0,q X) j for all l ≥ 0. Dividing both sides by (S 0,q (S α,1 (X))) q−1 j , we get (2.3). In the general case S 0,q (X) j < +∞ we obtain from (2.3) for the truncated sequence the property S 0,q (S α,1 (X)) j < +∞. This shows (2.3) for the general case. where < ·, · > denotes the dual pairing. Below we use the notation N 0 = N ∪ {0}. Then, f * ϕ ∈ C ∞ (R n ) and for every multi index α ∈ N n 0 it holds D α (f * ϕ) = f * (D α ϕ) = (D α f ) * ϕ. Given x 0 ∈ R n , 0 < r < +∞ and f ∈ S ′ we define the mean [f ] α x,r = f * D α ϕ r (x). where ϕ r (y) = r −n ϕ(r −1 (y)), and ϕ ∈ C ∞ c (B(1)) stands for the standard mollifier, such that R n ϕdx = 1. Note that in case f ∈ L 1 loc (R n ) we get [f ] 0 x,r = R n f (x − y)ϕ r (y)dy = B(x,r) f (y)ϕ x,r (−y)dy, where ϕ x,r = ϕ r (· + x). Furthermore, from the above definition it follows that (3.1) [f ] α x,r = (D α f ) * ϕ r (x) = [D α f ] 0 x,r . For f ∈ L 1 loc (R n ) and α ∈ N n 0 we immediately get (3.2) [f ] α x,r ≤ cr −\|α\|−n f L 1 (B(x,r)) ∀x ∈ R n , r > 0. Lemma 3.1. Let x 0 ∈ R n , 0 < r < +∞ and N ∈ N 0 . For every f ∈ S ′ there exists a unique polynomial P N x 0 ,r (f ) ∈ P N such that (3.3) [f − P N x 0 ,r (f )] α x 0 ,r = 0 ∀ \|α\| ≤ N. Proof: Set L = n+N N . Clearly, dim P N = L. We define the mapping T N : P N → R L , by (T N Q) α = [Q] α x 0 ,r , \|α\| ≤ N, Q ∈ P N . In order to prove the assertion of the lemma it will be sufficient to show that T N is injective, since by P N = L this implies, T N is also surjective. In fact, this can be proved by induction over N. In case N = 0 we see this by the fact that (T 0 1) 0 = [1] 0 x 0 ,r = 1. This T 0 stands for the identity in P 0 ∼ = R. Assume T N −1 is injective. Let Q = \|α\|≤N a α x α ∈ P N such that T N (Q) = 0. Using (3.1), this implies for \|α\| = N 0 = [Q] α x 0 ,r = \|β\|≤N a β D α x β 0 x 0 ,r = [α!a α ] 0 x 0 ,r = α!a α . Here, we used the formula D α x β = α!δ αβ for all \|β\| ≤ N. Accordingly, Q ∈ P N −1 , and it holds T N −1 (Q) = T N (Q) = 0. By our assumption it follows Q = 0. This proves that T N is injective and thus surjective. (3.4) P N −\|β\| x 0 ,r (D β f ) = D β P N x 0 ,r (f ). 2. The mapping P N x 0 ,r : L p (B(x 0 , r)) → P N , 1 ≤ p ≤ +∞, defines a projection, i.e. P N x 0 ,r (Q) = Q ∀ Q ∈ P N , (3.5) P N x 0 ,r (f ) L p (B(x 0 ,4r)) ≤ c P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ c P N 0,1 p f L p (B(x 0 ,r)) . (3.6) where P N 0,1 p = sup g∈L p (B(1)) g =0 P N 0,1 (g) L p (B(1)) g L p (B(1)) = sup g∈L p (B(x 0 ,r)) g =0 P N x 0 ,r (g) L p (B(x 0 ,r)) g L p (B(x 0 , 3. For all f ∈ W p, j (B(x 0 , r)), 1 ≤ p < +∞, 1 ≤ j ≤ N + 1, it holds (3.8) f − P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ cr j \|α\|=j D α f − D α P N x 0 ,r (f ) L p (B(x 0 ,r)) . Proof: 1. Let γ ∈ N n 0 be a multi index with \|γ\| ≤ N − \|β\|. Obviously, \|β + γ\| ≤ N. From the definition of P N x 0 ,r , observing (3.3), and employing (3.1) we find [P N −\|β\| x 0 ,r (D β f )] γ x 0 ,r = [D β f ] γ x 0 ,r = D β f * D γ ϕ r (x 0 ) = f * D β+γ ϕ r (x 0 ) = [f ] β+γ x 0 ,r = [P N x 0 ,r (f )] β+γ x 0 ,r = [D β P N x 0 ,r (f )] γ x 0 ,r . As we have seen in the proof of Lemma 3.1, the mapping T N −\|β\| : P N −\|β\| → P N −\|β\| is injective. This yields (3.4). 2. We show that P N x 0 ,r is a projection, i.e. P N x 0 ,r (Q) = Q for all Q ∈ P N . Indeed, given Q ∈ P N , by the definition of P N x 0 ,r (3.3) it follows that [Q − P N x 0 ,r (Q)] α x 0 ,r = 0 ∀ \|α\| ≤ N. Consequently, T N (Q − P N x 0 ,r (Q)) = 0. Since T N is injective we get P N x 0 ,r (Q) = Q. The inequality (3.6) can be verified by a standard scaling and translation argument. (3.9) f − P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ cr j−1 \|α\|=j−1 D α f − D α P N x 0 ,r (f ) L p (B(x 0 ,r)) . Thanks to (3.5) for all \|α\| = j − 1 it holds, D α P N x 0 ,r (f ) = P N −j+1 x 0 ,r (D α f ). Hence, [D α f − D α P N x 0 ,r (f )] 0 x 0 ,r = 0. An application of the Poincaré inequality gives (3.10) D α f − D α P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ cr DD α f − DD α P N x 0 ,r (f ) L p (B(x 0 ,r)) . Combining (3.9) and (3.10), we get (3.8).    f − P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ cr N +1 D N +1 f L p (B(x 0 ,r)) ∀ f ∈ W N +1, p (B(x 0 , r)). Corollary 3.4. For all x 0 ∈ R n , 0 < r < +∞, N ∈ N 0 , and 1 ≤ p < +∞ it holds (3.12) f − P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ c inf Q∈P N f − Q L p (B(x 0 ,r)) = cr n p osc p,N (f ; x 0 , r). Proof: Let Q ∈ P N be arbitrarily chosen. In view of (3.5) we find f − P N x 0 ,r (f ) = f − Q − P N x 0 ,r (f − Q). Hence, applying triangle inequality, along with (3.6) we get f − P N x 0 ,r (f ) L p (B(x 0 ,r)) ≤ f − Q L p (B(x 0 ,r)) + P N x 0 ,r (f − Q) L p (B(x 0 ,r)) ≤ c f − Q L p (B(x 0 ,r)) . This shows the validity of (3.12). In our discussion below and in the sequel of the paper it will be convenient to work with smooth functions. Using the standard mollifier we get the following estimate in L k,s q(p,N ) (R n ) for the mollification. Lemma 3.5. Let ε > 0. Given f ∈ S ′ , we define the mollification f ε (x) = [f ] 0 x,ε = f * ϕ ε (x), x ∈ R n . 1. For all f ∈ L k,s q(p,N ) (R n ), and all ε > 0 it holds (3.13) \|f ε \| L k,s q(p,N) ≤ c\|f \| L k,s q(p,N) . 2. Let f ∈ L p loc (R n ) such that for all 0 < ε < 1, (3.14) \|f ε \| L k,s q(p,N) ≤ c 0 , then f ∈ L k,s q(p,N ) (R n ) and it holds \|f \| L k,s q(p,N) ≤ c 0 . Proof: 1. We may restrict ourself to the case k = 0. Let x 0 ∈ R n and j ∈ Z. Set 0 < r < +∞. By the definition of P N x 0 ,r (f ) (cf. (3.3) ) together with (3.1) it follows that for all \|α\| ≤ N and for almost all y ∈ R n , f * D α ϕ r (x 0 − y) = [f ] α x 0 −y,r = [P N x 0 −y,r (f )] α x 0 −y,r = P N x 0 −y,r (f ) * D α ϕ r (x 0 − y). Multiplying both sides by ϕ 0,ε (y), integrate the result over R n and apply Fubini's theorem, we get for all \|α\| ≤ N [f ε ] α x 0 ,r = (f * ϕ ε * D α ϕ r )(x 0 ) = (f * D α ϕ r * ϕ ε )(x 0 ) = R n (f * D α ϕ r )(x 0 − y)ϕ ε (y)dy = R n P N x 0 −y,r (f ) * D α ϕ r (x 0 − y)ϕ ε (y)dy = R n R n P N x 0 −y,r (f )(x)D α ϕ r (x 0 − y − x)ϕ ε (y)dxdy = R n R n P N x 0 −y,r (f )(x − y)D α ϕ r (x 0 − x)ϕ ε (y)dxdy = R n R n P N x 0 −y,r (f )(x − y)ϕ ε (y)dyD α ϕ r (x 0 − x)dx = R n P N x 0 −y,r (f )(x − y)ϕ ε (y)dy α x 0 ,r . This shows that P N x 0 ,r (f ε )(x) = R n P N x 0 −y,r (f )(x − y)ϕ ε (y)dy, x ∈ R n , (3.15) = R n P N x 0 −εy,r (f )(x − εy)ϕ(y)dy, x ∈ R n . (3.16) Accordingly, \|f ε (x) − P N x 0 ,2 j (f ε )(x)\| p ≤ R n \|f (x − εy) − P N x 0 −εy,2 j (f )(x − εy)\|ϕ(y)dy p . Integration of both sides over B(x 0 , 2 j ) and multiplication with 1 \|B(2 j )\| , using Jensen's inequality with respect to the probability measure ϕdy, we find osc p,N (f ε ; x 0 , 2 j ) ≤ − B(x 0 ,2 j ) R n \|f (x − εy) − P x 0 −εy,2 j (f )(x − εy)\|ϕ(y)dy p dx 1 p = R n − B(x 0 −εy,2 j ) \|f (x) − P x 0 −εy,2 j (f )(x)\| p dx 1 p ϕ(y)dy ≤ c B(1) osc p,N (f ; x 0 − εy; 2 j )ϕ(y)dy. Multiplying both sides by 2 −js applying the ℓ q norm to both sides of the resultant inequality, and using Minkowski's inequality, we are led to j∈Z (2 −js osc p,N (f ε ; x 0 , 2 j )) q 1 q ≤ c B(1) j∈Z (2 −js osc p,N (f ; x 0 − εy; 2 j )) q 1 q ϕ(y)dy ≤ c\|f \| L s q(p,N) . Taking the supremum over all x 0 ∈ R n in the above inequality shows (3.13). 2. Let f ∈ L p loc (R n ) satifying (3.14). This implies that f ∈ W k, p loc (R n ). Let x 0 ∈ R n and l, m ∈ Z, l < m. According to the absolutely continuity of the Lebesgue measure together with (3.14) it follows m j=l (2 −js osc p,N (D k f ; x 0 , 2 j )) q = lim εց0 m j=l (2 −js osc p,N (D k f ε ; x 0 , 2 j )) q ≤ c q 0 . This shows that {2 −sj osc p,N (D k f ; x 0 , 2 j )} j∈Z ∈ ℓ q , and its sum is bounded by c 0 . Accord- ingly, f ∈ L k,s q(p,N ) (R n ), and it holds \|f \| L k,s q(p,N) ≤ c 0 . We are are now in a position to prove the following embedding properties. First, let us introduce the definition of the projection to the space of homogenous polynomial P N x 0 ,r : S ′ →Ṗ N defined by means oḟ P N x 0 ,r (f )(x) = \|α\|=N 1 α! [f ] α x 0 ,r x α , x ∈ R n . Clearly, for all f ∈ S ′ it holds (3.17) D αṖ N x 0 ,r (f ) =Ṗ N −\|α\| x 0 ,r (D α f ) ∀ \|α\| ≤ k. Theorem 3.6. 1. For every N ∈ N 0 the following embedding holds true (3.18)    L N 1(p,N ) (R n ) ֒→ C N −1,1 (R n ) if N ≥ 1 L 0 1(p,0) (R n ) ֒→ L ∞ (R n ) if N = 0. 2. For every f ∈ L N 1(p,N ) (R n ) there exists a uniqueṖ N ∞ ∈Ṗ N , such that for all x 0 ∈ R n lim r→∞Ṗ x 0 ,r (f ) →Ṗ N ∞ (f ) in P N . Furthermore,Ṗ N ∞ : L N 1(p,N ) (R n ) →Ṗ N is a projection, with the property (3.19) D αṖ N ∞ (f ) =Ṗ N −\|α\| ∞ (D α f ) ∀ \|α\| ≤ N. 3. For all g, f ∈ L 1 1(p,1) (R n ) it holdṡ P 1 ∞ (g∂ k f ) =Ṗ 1 ∞ (g)∂ kṖ 1 ∞ (f ) =Ṗ 1 ∞ (g)Ṗ 0 ∞ (∂ k f ), k = 1, . . . , n. (3.20) In addition, for g ∈ C 0,1 (R n ; R n ), and for all f ∈ L 0 1(p,0) (R n ) it holdṡ P 0 ∞ (g∂ k f ) := lim r→∞ P 0 0,r (g∂ k f ) = 0, k = 1, . . . , n, (3.21) where g∂ k f = ∂ k (gf ) − ∂ k gf ∈ S ′ . 4. For all v ∈ L 1 1(p,1) (R n ; R n ) with ∇ · v = 0 almost everywhere in R n and f ∈ L 1 1(p,1) (R n ) it holdṡ P 0 ∞ (∇v · ∇f ) =Ṗ 0 ∞ (∇v) ·Ṗ 0 ∞ (∇f ). (3.22) Proof: 1. Let ε > 0 be arbitrarily chosen. Let f ∈ L N 1(p,N ) (R n ). Set f ε = f * ϕ ε . By Lemma 3.5 we get f ε ∈ L N 1(p,N ) (R n ) and it holds (3.23) \|f ε \| L N 1(p,N) ≤ c\|f \| L N 1(p,N) . Let x 0 ∈ R n be fixed. Let j ∈ Z. Clearly, f ε ∈ C ∞ (R n ). Let α ∈ N n 0 be a multi index with \|α\| = N. Then D α P N x 0 ,2 j (f ε ) = P 0 x 0 ,2 j (D α (f ε )) = [D α (f ε )] 0 x 0 ,2 j = D αṖ N x 0 ,2 j (f ε ). Let m ∈ Z. Since D α f ε is continuous we have D α f ε (x) = lim j→−∞ [D α f ε ] 0 x,2 j ∀x ∈ R n . Using triangle inequality along with (3.5) and (3.13), and using (3.2), we get \|D α f ε (x) − [D α f ε ] 0 x,2 m \| = m j=−∞ [D α f ε ] 0 x,2 j−1 − [D α f ε ] 0 x,2 j ≤ m j=−∞ [D α f ε ] 0 x,2 j−1 − [D α f ε ] 0 x,2 j = m j=−∞ [f ε ] α x,2 j−1 − [f ε ] α x,2 j = m j=−∞ [f ε − P N x,2 j (f ε )] α x,2 j−1 − [f ε − P N x,2 j (f ε )] α x,2 j = m j=−∞ [f ε − P N x,2 j (f ε )] α x,2 j−1 ≤ c m j=−∞ 2 −jN osc p,N (f ε ; x, 2 j ) ≤ c\|f ε \| L N 1(p,N) ≤ c\|f \| L N 1(p,N) . (3.24) Thus, {D N f ε } is bounded in L ∞ (B(r) ) for all 0 < r < +∞. By means of Banach-Alaoglu's theorem and Cantor's diagonalization principle we get a sequence ε k ց 0 as k → +∞ and f ∈ W N, ∞ loc (R n ), such that for all 0 < r < +∞ D N f ε k → D N f weakly− * in L ∞ (B(r)) as k → +∞. Furthermore, from (3.24) we get for almost all x ∈ R n and all m ∈ Z, (3.25) \|D N f (x)\| ≤ c\|f \| L N 1(p,N) + \|α\|=N \|[f ] α x,2 m \|. Let x 0 ∈ R n be fixed. We now choose m ∈ Z such that 2 m−1 ≤ \|x 0 \| < 2 m . Then noting B(x 0 , 2 m ) ⊂ B(2 m+1 ), employing (3.5) and (3.2), we get \|[f ] α x 0 ,2 m \| − \|[f ] α 0,2 m \| ≤ [f ] α x 0 ,2 m − [f ] α 0,2 m = [f − P N 0,2 m+1 ] α x 0 ,2 m − [f − P N 0,2 m+1 ] α 0,2 m ≤ c2 −mN osc p,N (f ; 0, 2 m+1 ) ≤ c\|f \| L N 1(p,N) . Similarly, we get for all j ∈ Z \|[f ] α 0,2 m \| − \|[f ] α 0,2 j \| ≤ c\|f \| L N 1(p,N) . Thus, combining the two inequalities we have just obtained, using triangle inequality, we find for all j ∈ Z, \|α\|=N \|[f ] α x,2 m \| ≤ c\|f \| L N 1(p,N) + \|α\|=N \|[f ] α 0,2 j \|. This together with (3.25), we infer for all j ∈ Z D N f ∞ ≤ c\|f \| L N 1(p,N) + c \|α\|=N \|[f ] α 0,2 j \| ≤ c\|f \| L N 1(p,N) + c Ṗ N 0,2 j (f ) . (3.26) This completes the proof of (3.18). Let x 0 ∈ R n . Let m, l ∈ Z, l < m. Noting thatṖ N x 0 ,2 j (Q) = Q for all Q ∈Ṗ N anḋ P N x 0 ,2 j (Q) = 0 for all Q ∈ P N −1 , we get the following identity for all j, k ∈ Ż P N x 0 ,2 j (P N x 0 ,2 k (f )) =Ṗ N x 0 ,2 k (f ). Using triangle inequality together with the above identity, (3.2) and (3.12) we estimate Ṗ N x 0 ,2 l (f ) −Ṗ N x 0 ,2 m (f ) ≤ m j=l+1 Ṗ N x 0 ,2 j−1 (f ) −Ṗ N x 0 ,2 j (f ) ≤ c m j=l+1 2 −jN −j n p Ṗ N x 0 ,2 j−1 (f − P N x 0 ,2 j (f )) −Ṗ N x 0 ,2 j (f − P N x 0 ,2 j (f )) L p (B(x 0 ,2 j )) ≤ c m j=l+1 2 −jN osc p,N (f ; x 0 , 2 j ). Owing to f ∈ L N 1(p,N ) (R n ) the right-hand side of the above inequality tends to zero as m, l → +∞. This shows that {Ṗ N x 0 ,2 m (f )} is a Cauchy sequence inṖ N and converges to a unique limitṖ N ∞,x 0 . We claim that (3.27)Ṗ N ∞,x 0 =Ṗ N ∞,0 =:Ṗ N ∞ (f ). In fact, for m ∈ Z such that \|x 0 \| ≤ 2 m , we obtain Ṗ N x 0 ,2 m (f ) −Ṗ N 0,2 m (f ) ≤ c2 −mN −m n p Ṗ N x 0 ,2 m (f − P N 0,2 m+1 (f )) −Ṗ N 0,2 m (f − P N 0,2 m+1 (f )) L p (B(x 0 ,2 m )) ≤ c2 −mN osc p,N (f ; 0, 2 m+1 ) → 0 as m → +∞. Consequently, (3.27) must hold. The identity (3.19) is an immediate consequence of (3.17). 3. Now, let g, f ∈ L 1 1(p,1) (R n ). Let x 0 ∈ R n . Let α ∈ N n 0 with \|α\| = 1. We first show that {[g∂ k f ] α x 0 ,2 j } j∈N , k ∈ {1, . . . , n}, is a Cauchy sequence. Let j ∈ N be fixed. We easily calculate, [g∂ k f ] α x 0 ,2 j−1 − [g∂ k f ] α x 0 ,2 j = g∂ k f − P 1 x 0 ,2 j (g)[∂ k f ] 0 x 0 ,2 j α x 0 ,2 j−1 − g∂ k f − P 1 x 0 ,2 j (g)[∂ k f ] 0 x 0 ,2 j α x 0 ,2 j . Furthermore, applying integration by parts, we get, g∂ k f − P 1 x 0 ,2 j (g)[∂ k f ] 0 x 0 ,2 j α x 0 ,2 j−1 = (g − P 1 x 0 ,2 j (g)([∂ k f ] 0 x 0 ,2 j ) α x 0 ,2 j−1 + g · (∂ k f − [∂ k f ] 0 x 0 ,2 j ) α x 0 ,2 j−1 = − R n (g − P 1 x 0 ,2 j (g))([∂ k f ] 0 x 0 ,2 j )D α ϕ x 0 ,2 j−1 dx − R n g(∂ k f − [∂ k f ] 0 x 0 ,2 j )D α ϕ x 0 ,2 j−1 dx = − R n (g − P 1 x 0 ,2 j (g))([∂ k f ] 0 x 0 ,2 j )D α ϕ x 0 ,2 j−1 dx + R n ∂ k g(f − P 1 x 0 ,2 j (f ) − [f − P 1 x 0 ,2 j (f )] 1 x 0 ,2 j )D α ϕ x 0 ,2 j−1 dx + R n g(f − P 1 x 0 ,2 j (f ) − [f − P 1 x 0 ,2 j (f )] 1 x 0 ,2 j )∂ k D α ϕ x 0 ,2 j−1 dx. This together with (3.12) yields g∂ k f − P 1 x 0 ,2 j (g)[∂ k f ] 0 x 0 ,2 j α x 0 ,2 j−1 ≤ c ∇f ∞ 2 −j osc p,1 (v; x 0 , 2 j ) + c ∇v ∞ 2 −j osc p,1 (f ; x 0 , 2 j ). By an analogous reasoning we find g∂ k f − P 1 x 0 ,2 j (g)[∂ k f ] 0 x 0 ,2 j α x 0 ,2 j ≤ c ∇f ∞ 2 −j osc p,1 (v; x 0 , 2 j ) + c ∇v ∞ 2 −j osc p,1 (f ; x 0 , 2 j ). Let l, m ∈ Z with l < m be arbitrarily chosen. Using triangle inequality together with the two estimates we have just obtained, we estimate [g∂ k f ] α x 0 ,2 l − [g∂ k f ] α x 0 ,2 m = m j=l+1 [g∂ k f ] α x 0 ,2 j−1 − [g∂ k f ] α x 0 ,2 j ≤ c ∇f ∞ m j=l+1 2 −j osc p,1 (g; x 0 , 2 j ) + c ∇g ∞ m j=l+1 2 −j osc p,1 (f ; x 0 , 2 j ). Since g, f ∈ L 1 1(p,1) (R n ) the right-hand side converges to zero as l, m → +∞. Thus, {[g∂ k f ] α x 0 ,2 l } is a Cauchy sequence, and has a unique limit say a x 0 . Let j ∈ N such that 2 j ≥ \|x 0 \|. Thus, B(x 0 , 2 j ) ⊂ B(2 j+1 ). By the same reasoning as above we estimate [g∂ k f ] 0 x 0 ,2 j − [g∂ k f ] α 0,2 j+1 = c ∇f ∞ 2 −j osc p,1 (g; 0, 2 j+1 ) + c ∇g ∞ 2 −j osc p,1 (f ; 0, 2 j+1 ). Since the right-hand side converges to zero as j → +∞ we get a x 0 = a 0 . Setting [g∂ k f ] α ∞ = a 0 , we complete the proof of (3.20). Next, we prove (3.21). Let g ∈ C 0,1 (R n ) and f ∈ L 0 1(p,0) (R n ). Applying integration by parts and product rule, we calculate [g∂ k f ] 0 x 0 ,r = − B(x 0 ,r) ∂ k g(y)(f (y) − [f ] 0 x 0 ,r )ϕ r (x 0 − y)dy + B(x 0 ,r) g(y)(f (y) − [f ] 0 x 0 ,r )∂ k ϕ r (x 0 − y)dy. Applying Hölder's inequality, we easily get \|[g∂ k f ] 0 x 0 ,r \| ≤ c ∇g ∞ osc p,0 (f ; x 0 , r) + cr −1 g L ∞ (B(x 0 ,r)) osc p,0 (f ; x 0 , r). Noting that r −1 g L ∞ (B(x 0 ,r)) ≤ c\|g(x 0 )\|+c ∇g ∞ , and using the fact that osc p,0 (f ; x 0 , r) → 0 as r → r + ∞, we obtain (3.21). It remains to show the identity (3.22). Let v ∈ L 1 1(p,1) (R n ; R n ) with ∇ · v = 0 and f ∈ L 1 1(p,1) (R n ). Using (3.19) together with ∇ · v = 0 and (3.20), we obtain [∂ k v · ∇f ] 0 x 0 ,r = ∂ j P 1 x 0 ,r ((∂ k v j )f ) = ∂ jṖ 1 x 0 ,r ((∂ k v j )f ) → ∂ jṖ 1 ∞ (∂ k v j )Ṗ 1 ∞ (f ) =Ṗ 0 ∞ (∂ k v) ·Ṗ 0 ∞ (∇f ) as r → +∞. This shows thaṫ P 0 ∞ (∂ k v · ∇f ) = lim r→∞ [∂ k v · ∇f ] 0 x 0 ,r =Ṗ 0 ∞ (∂ k v) ·Ṗ 0 ∞ (∇f ). This completes the proof of the Lemma. Next, we prove the following norm equivalence which is similar to the properties of the known Campanato space. Lemma 3.7. Let 1 ≤ p < +∞, 1 ≤ q ≤ +∞, and N, N ′ ∈ N 0 , N < N ′ , s ∈ [− n p , N +1). If f ∈ L k,s q(p,N ′ ) (R n ), and satisfies (3.28) lim m→∞Ṗ L 0,2 m (D k f ) = 0 ∀ L = N + 1, . . . , N ′ . then f ∈ L k,s q(p,N ) (R n ) and it holds, (3.29) \|f \| L k,s q(p,N ′ ) ≤ \|f \| L k,s q(p,N) ≤ c\|f \| L k,s q(p,N ′ ) . Proof: We may restrict ourself to the case k = 0. First, let us prove that for all s ∈ [− n p , N) and for all f ∈ L s q(p,N ) (R n ) such that (3.30) lim m→∞Ṗ N 0,2 m (f ) = 0. it follows that f ∈ L s q(p,N −1) (R n ), together with the estimate (3.31) \|f \| L s q(p,N−1) ≤ c\|f \| L s q(p,N) . Let x 0 ∈ R n , 0 < r < +∞. Noting that P N x 0 ,2r (f ) −Ṗ N x 0 ,2r (f ) ∈ P N −1 , we see thaṫ P N x 0 ,r (P N x 0 ,2r (f )) =Ṗ N x 0 ,r (P N x 0 ,2r (f ) −Ṗ N x 0 ,2r (f )) +Ṗ N x 0 ,r (Ṗ N x 0 ,2r (f )) =Ṗ N x 0 ,2r (f ) . By a scaling argument and triangle inequality we infer r −N − n p Ṗ N x 0 ,r (f ) L p (B(x 0 ,r)) − (2r) −N − n p Ṗ N x 0 ,2r (f ) L p (B(x 0 ,2r)) = Ṗ N x 0 ,r (f ) − Ṗ N x 0 ,2r (f ) ≤ Ṗ N x 0 r (f ) −Ṗ N x 0 ,2r (f ) = Ṗ N x 0 ,r (f − P N x 0 ,2r (f ) ≤ c(2r) −N − n p f − P N x 0 ,2r (f ) L p (B(x 0 ,2r)) ≤ cr −N osc p,N (f ; x 0 , 2r). Let j, m ∈ Z, j < m. Using the above estimate we deduce that 2 −jN −j n p Ṗ N x 0 ,2 j (f ) L p (B(x 0 ,2 j )) − 2 −mN −m n p Ṗ N x 0 ,2 N (f ) L p (B(x 0 ,2 m )) ≤ c m−1 i=j 2 −iN osc p,N (f ; x 0 , 2 i+1 ) ≤ c2 N m−1 i=j 2 −iN osc p,N (f ; x 0 , 2 i ). Observing (3.36), we see that lim m→∞ 2 −mN −m n p Ṗ N x 0 ,2 N (f ) L p (B(x 0 ,2 m )) = lim m→∞ Ṗ N x 0 ,2 m (f ) = 0. Thus, letting m → +∞ in the above estimate, we arrive at 2 jN Ṗ N x 0 ,2 j (f ) = 2 −j n p Ṗ N x 0 ,2 j (f ) L p (B(x 0 ,2 j )) ≤ c2 jN ∞ i=j 2 −iN osc p,N (f ; x 0 , 2 i ) = c S N,1 (osc p,N (f ; x 0 )) j , (3.32) where osc p,N (f ; x 0 ) stands for a sequence defined as osc p,N (f ; x 0 ) i = osc p,N (f ; x 0 , 2 i ), i ∈ Z. Using triangle inequality together with (3.32), we obtain osc p,N −1 (f ; x 0 , 2 j ) = 2 −j n p inf P ∈P N−1 f − P L p (B(x 0 ,2 j )) ≤ c2 −j n p f − P N x 0 ,2 j (f ) +Ṗ N x 0 ,2 j (f ) L p (B(x 0 ,2 j )) ≤ c2 −j n p f − P N x 0 ,2 j (f ) L p (B(x 0 ,2 j )) + c2 −j n p Ṗ N x 0 ,2 j (f ) L p (B(x 0 ,2 j )) ≤ c osc p,N (f ; x 0 , 2 j ) + 2 jN Ṗ N x 0 ,2 j (f ) ≤ c osc p,N (f ; x 0 , 2 j ) + c S N,1 (osc p,N (f ; x 0 )) j . (3.33) Noting that osc p,N (f ; x 0 , 2 j ) ≤ S N,1 (osc p,N (f ; x 0 , 2 j )), we infer from (3.33) (3.34) osc p,N −1 (f ; x 0 ) j = osc p,N −1 (f ; x 0 , 2 j ) ≤ c S N,1 (osc p,N (f ; x 0 )) j , j ∈ Z. Applying S s,q to both sides of (3.34), and using Lemma 2.1, we get the inequality \|f \| L s q(p,N−1) = sup x 0 ∈R n S s,q ( osc p,N −1 (f ; x 0 )) ≤ c sup x 0 ∈R n S s,q (osc p,N (f ; x 0 )) = \|f \| L s q(p,N ′ ) , which implies (3.31). We are now in a position to apply (3.31) iteratively, replacing N by N + 1 to get N ′ ) . This completes the proof of the lemma. Then for L ∈ N, L > N, we estimate for multi index α with \|α\| = L \|f \| L s q(p,N) ≤ c\|f \| L s q(p,N+1) ≤ . . . ≤ c\|f \| L s q(p,\|D αṖ L 0,2 m (f )\| = \|D αṖ L 0,2 m ((f − P N 0,2 m ))\| ≤ c2 −Lm osc p,N (f, 0, 2 m ) ≤ c2 m(N −L) \|f \| L s q(p,N) → 0 as m → +∞. Hence, (3.28) is fulfilled. A careful inspection of the proof of Lemma 3.7 gives the following. Corollary 3.10. Let N, N ′ ∈ N 0 , N < N ′ . Let f ∈ L p loc (R n ) satisfy (3.28) with k = 0. Then, for all x 0 ∈ R n and j ∈ Z it holds, (3.36) osc p,N (f ; x 0 , 2 j ) ≤ c(S N +1,1 (osc p,N ′ (f ; x 0 ))) j . Proof: Set k = N ′ − N. Using (3.34) with N ′ in place of N, we find (3.37) osc p,N ′ −1 (f ; x 0 , 2 j ) ≤ c(S N ′ ,1 (osc p,N ′ (f ; x 0 ))) j , j ∈ Z. Iterating this inequality k-times and applying Lemma 2.1, we arrive at osc p,N (f ; x 0 ) = osc p,N ′ −k (f ; x 0 ) ≤ cS N +1,1 (S N +2,1 . . . S N ′ ,1 (osc p,N ′ (f ; x 0 ))) ≤ cS N +1,1 (osc p,N ′ (f ; x 0 )). Whence, (3.36). We also have the following growth properties of functions in L s q(p,N ) (R n ) as \|x\| → +∞ Lemma 3.11. Let N ∈ N 0 . Let f ∈ L s q(p,N ) (R n ), 1 ≤ q ≤ +∞, 1 ≤ p < +∞, s ∈ [N, N + 1). 1. In case s ∈ (N, N + 1) it holds (3.38) \|f (x)\| ≤ c(1 + \|x\| s ) f L s q(p,N) ∀x ∈ R n . 2. In case s = N it holds (3.39) \|f (x)\| ≤ c(1 + log(1 + \|x\|) 1 q ′ \|x\| N ) f L N q(p,N) ∀x ∈ R n . Here q ′ = q q−1 , c = const > 0, depending on q, p, s, N and n. Proof: 1. The case s ∈ (N, N + 1). Let x 0 ∈ R n . Let j ∈ N 0 such that 2 j ≤ 1 + \|x 0 \| ≤ 2 j+1 . Let α be a multi index with \|α\| = N. Verifying that D α f (x 0 ) = lim i→−∞ D αṖ N x 0 ,2 i (f ), using triangle inequality we find \|D α f (x 0 )\| ≤ j i=−∞ \|D αṖ N x 0 ,2 i (f ) − D αṖ N x 0 ,2 i−1 (f )\| + \|D αṖ N x 0 ,2 j (f )\| ≤ c j i=−∞ 2 −iN osc p,N (f ; x 0 , 2 i ) + \|D αṖ N x 0 ,2 j (f )\|. By the aid of Hölder's inequality we find j i=−∞ 2 −iN osc p,N (f ; x 0 , 2 i ) = j i=−∞ 2 −i(N −s) 2 −is osc p,N (f ; x 0 , 2 i ) ≤ c2 j(s−N ) \|f \| L s q(p,N) ≤ c(1 + \|x 0 \| s−N )\|f \| L s q(p,N) . On the other hand, \|D αṖ N x 0 ,2 j (f )\| = \|D αṖ N x 0 ,2 j (f − P N 0,2 j+1 (f )\| + \|D α (P N 0,2 j+1 (f ) − P N 0,1 (f ))\| + \|D α P N 0,1 (f )\| ≤ 2 −jN − n p f − P N 0,2 j+1 (f ) L p (x 0 ,2 j+1 ) + c j i=0 2 −i(N −s) 2 −is osc p,N (f ; 0, 2 i ) + c f L p (B(1)) ≤ osc p,N (f ; 0, 2 j+1 ) + c j i=0 2 −i(N −s) 2 −is osc p,N (f ; 0, 2 i ) + c f L p (B(1)) ≤ c(1 + \|x 0 \| s−N ) f L s q(p,N) . Accordingly, (3.40) D N f (x) ≤ c(1 + \|x\| s−N ) f L s q(p,N) . This implies (3.38). 2. The case s = N. Let x 0 ∈ R n . As above we choose j ∈ N 0 such that 2 j ≤ 1 + \|x 0 \| < 2 j+1 . In this case we first claim D NṖ N x 0 ,1 (f ) ≤ (log(1 + \|x 0 \|)) 1 q ′ f L N q(p,N) . (3.41) Indeed, arguing as above using triangle inequality along with Hölder's inequality, we get D NṖ N x 0 ,1 (f ) ≤ j i=1 D NṖ N x 0 ,2 i (f ) − D NṖ N x 0 ,2 i−1 (f ) + D NṖ N x 0 ,2 j (f ) ≤ j i=1 2 −N i osc p,N (f ; x 0 , 2 i ) + D NṖ N x 0 ,2 j (f ) ≤ j+1 i=1 2 −N i osc p,N (f ; x 0 , 2 i ) + D NṖ N 0,2 j+1 (f ) ≤ cj 1 q ′ \|f \| L N q(p,N) + D NṖ N 0,2 j+1 (f ) . Similarly, D NṖ N 0,2 j+1 (f ) ≤ cj 1 q ′ \|f \| L N q(p,N) + D NṖ N 0,1 (f ) . Combining the two inequalities we have just obtained, we get (3.41). Let i ∈ Z. Then by triangle inequality together with (3.41) we find 2 − n p −iN Ṗ N x 0 ,2 i (f ) L p (x 0 ,2 i ) ≤ c D NṖ N x 0 ,2 i (f ) ≤ c 1 l=i D NṖ N x 0 ,2 l (f ) − D NṖ N x 0 ,2 l−1 (f ) + c D NṖ N x 0 ,1 (f ) ≤ c 1 l=i D NṖ N x 0 ,2 l (f ) − D NṖ N x 0 ,2 l−1 (f ) + c D NṖ N x 0 ,1 (f ) ≤ c 1 l=i 2 −N l osc p,N (f ; x 0 , 2 l ) + c D NṖ N x 0 ,1 (f ) ≤ c\|i\| 1 q ′ \|f \| L N q(p,N) + c(log(1 + \|x 0 \|)) 1 q ′ f L N q(p,N) . This shows that N) . N) . 2 −i(N −1) osc p,N −1 (f ; x 0 , 2 i ) ≤ 2 −i(N −1) osc p,N (f ; x 0 , 2 i ) + 2 − n p Ṗ N x 0 ,2 i (f ) L p (x 0 ,2 i ) ≤ 2 −i(N −1) osc p,N (f ; x 0 , 2 i ) + c2 i \|i\| 1 q ′ + (log(1 + \|x 0 \|)) 1 q ′ f L N q(p,(f ; x 0 , 2 i ) ≤ c 1 + (log(1 + \|x 0 \|)) 1 q ′ f L N q(p, Let α be a multi index with \|α\| = N −1. Noting that D α f (x 0 ) = lim i→−∞ D αṖ N −1 x 0 ,2 i (f ), using triangle inequality together with (3.43), we infer \|D α f (x 0 )\| ≤ \|D αṖ N −1 x 0 ,2 i (f )\| + c 1 i=−∞ \|D αṖ N −1 x 0 ,2 i (f ) − D αṖ N −1 x 0 ,2 i−1 (f )\| ≤ \|D αṖ N −1 x 0 ,2 i (f )\| + c 1 i=−∞ 2 −(N −1) osc p,N −1 (f ; x 0 , 2 i ) ≤ D N −1Ṗ N −1 x 0 ,2 i (f ) + c 1 + (log(1 + \|x 0 \|)) 1 q ′ f L N q(p,N) . Arguing as above using triangle inequality, using (3.42), we find D N −1Ṗ N −1 x 0 ,2 i (f ) ≤ c j i=0 2 −(N −1)i osc p,N −1 (f, x 0 , 2 i ) + D N −1Ṗ N −1 x 0 ,2 j (f ) ≤ c j i=0 2 −(N −1)i osc p,N −1 (f, x 0 , 2 i ) + j+1 i=0 2 −(N −1)i osc p,N −1 (f, 0, 2 i ) + D N −1Ṗ N −1 0,1 (f ) ≤ c2 j j 1 q ′ f L N q(p,N) ≤ c(1 + log(1 + \|x 0 \|) 1 q ′ \|x 0 \|) f L N q(p,N) . Combining the above inequalities we obtain \|D N −1 f (x 0 )\| ≤ (1 + log(1 + \|x 0 \|) 1 q ′ \|x 0 \|) f L N q(p,N) . This yields (3.39). Using the Poincaré's inequality and Lemma 3.7, we get the following embedding. Lemma 3.12. Let N ∈ N 0 , k ∈ N 0 , 1 < p < +∞, 1 ≤ q ≤ +∞, s ∈ [N, N + 1). 1. In case q = ∞ and s / ∈ N it holds (3.44) L k,s ∞(p,N ) (R n ) ∼ = C k+N,s−N (R n ). 2. In case q = ∞ and s ∈ N it holds (3.45) L k,s ∞(p,N ) (R n ) ∼ = BMO k+s (R n ). where BMO N = f ∈ L 1 loc (R n ) sup j∈Z 2 −N j osc 1,N (f ; x 0 , 2 j ) < +∞ . 3. In case 1 ≤ q < ∞ it holds (3.46) L k,s q(p,N ) (R n ) ֒→ L k+s q(p,N +k) (R n ) ֒→ L k,s ∞(p,N ) (R n ). Proof: 1. In case k = 0 the space L s ∞(p,N ) (R n ) coincides with the Campanato space L p,N n+p(s−N ) N (R n ) which is isomorphic to C N,s−N (R n ), (cf. [10, Chap. III 1.], [5]). In case, k ≥ 1. For f ∈ L k,s ∞(p,N ) (R n ) we get D k f ∈ L s ∞(p,N ) (R n ) ∼ = C N,s−N (R n ), which shows (3.44). 2. In case k = 0, and s = N the space L s ∞(p,N ) (R n ) coincides with the Campanato space L p,N n N (R n ). According to [10, Chap. III,1.] this space coincides with the space BMO N . In case k ≥ 1 we argue as above to verify (3.45). 3. Let L k,s q(p,N ) (R n ). Using Poincaré inequality (3.8) with j = k, we find osc p,N +k (f ; x 0 , 2 j )≤ c2 jk osc p,N (D k f ; x 0 , 2 j ). Accordingly, {2 −(s+k)j osc p,N +k (f ; x 0 , 2 j )} j∈Z ℓ q ≤ c {2 −sj osc p,N (D k f ; x 0 , 2 j )} j∈Z ℓ q , where osc p,N (f ; x 0 ) = {osc p,N (f ; x 0 , 2 j )} j∈Z . Taking the supremum over all x 0 ∈ R n on both sides of the above estimate, we get the first embedding. It remains to show the second embedding. To see this we first notice that L k+s q(p,N +k) (R n ) ֒→ L k+s ∞(p,N +k) (R n ). Indeed, 2 −(s+k)j osc p,N +k (f ; x 0 , 2 j ) ≤ 2 −(s+k)j S k+s,q ( osc p,N +k (f ; x 0 )) j ≤ \|f \| L k+s q(p,N+k) . Taking the supremum over all j ∈ Z and x 0 ∈ R n , we get the embedding L k+s q(p,N +k) (R n ) ֒→ L k+s ∞(p,N +k) (R n ). On the other hand, in case s ∈ (N, N + 1), from (3.44) it follows L k+s ∞(p,N +k) (R n ) ∼ = C k+N,s−N (R n ) ∼ = L k,s ∞(p,N ) (R n ). In case s = N using (3.45), we also get L k+N ∞(p,N +k) (R n ) ∼ = L k,N ∞(p,N ) (R n ). This shows desired embedding. Using Gagliardo-Nirenberg's inequalities, we can get the interpolation properties. First let us recall the Gagliardo-Nirenberg inequalities. Lemma 3.13. Let j, N ∈ N 0 , 0 ≤ j < k. Let 1 ≤ p, p 0 , p 1 ≤ +∞, and θ ∈ j N , 1 , satisfying (3.47) 1 p = j n + 1 − θ p 0 + 1 p 1 − k n θ. Then, for all f ∈ L p 0 (B(1)) ∩ W k, p 1 (B(1)) it holds (3.48) D j f L p (B(1)) ≤ c f 1−θ L p 0 (B(1)) f θ W k, p 1 (B(1)) . Notice that, using the generalized Poincaré inequality, under the assumption of Lemma 3.13, for all f ∈ L p 0 (B(1)) ∩ W k, p 1 (B(1)), and N ∈ N 0 , N ≥ k − 1 the following inequality holds (3.49) D j (f − P N 0,1 (f )) L p (B(1)) ≤ c f − P N 0,1 (f )) 1−θ L p 0 (B(1)) D k f − D k P N 0,1 (f ) θ L p 1 (B(1)) . By a standard scaling and translation argument, we deduce from (3.49) that for all x 0 ∈ R n , 0 < r < +∞, N ∈ N 0 , N ≥ k − 1, and for all f ∈ L p 0 (B(x 0 , r)) ∩ W k, p 1 (B(x 0 , r)) the following inequality holds D j f − P N −j x 0 ,r (D j f ) L p (B(x 0 ,r)) = D j (f − P N x 0 ,r (f )) L p (B(x 0 ,r)) ≤ c f − P N x 0 ,r (f )) 1−θ L p 0 (B(x 0 ,r)) D k f − P N −k x 0 ,r (D k f ) θ L p 1 (B(x 0 ,r)) . (3.50) Theorem 3.14. Let j, k, N ∈ N 0 , 0 ≤ j < k ≤ N + 1. Let 1 ≤ p, p 0 , p 1 < +∞, 1 ≤ q, q 0 , q 1 ≤ +∞, −∞ < s, s 0 , s 1 < N + 1, and θ ∈ j N , 1 , satisfying 1 p = j n + 1 − θ p 0 + 1 p 1 − k n θ, (3.51) 1 q = 1 − θ q 0 + θ q 1 , (3.52) s + j = (1 − θ)s 0 + θ(s 1 + k). (3.53) Then, for all L s 0 q 0 (p 0 ,N ) (R n ) ∩ L k,s 1 q 1 (p 1 ,N ) (R n ) it holds . Proof: Observing (3.51) and (3.52), thanks to (3.50) we find 2 −sl osc p,N −j (D j f ; x 0 , 2 l ) ≤ c2 −ls 0 (1−θ)−ls 1 θ osc p 0 ,N (f ; x 0 , 2 l ) 1−θ osc p 1 ,N −k (D k f ; x 0 , 2 l )] θ = c[2 −ls 0 osc p 0 ,N (f ; x 0 , 2 l )] 1−θ [2 −ls 1 osc p 1 ,N −k (D k f ; x 0 , 2 l )] θ . According to (3.53), we may apply ℓ q norm to both sides of the above inequality and use Hölder's inequality. This gives l∈Z (2 −sl osc p,N −j (D j f ; x 0 , 2 l )) q 1 q ≤ c l∈Z (2 −ls 0 osc p 0 ,N (f ; x 0 , 2 l )) q 0 1−θ q 0 l∈Z (2 −ls 1 osc p 1 ,N −k (D k f ; x 0 , 2 l )) q 1 θ q 1 . Taking the supremum over all x 0 ∈ R n , we get the assertion (3.54). Remark 3.15. Consider the special case    N = k, p = p 0 = p 1 , θ = j k , s = s 0 = s 1 = 0, 1 ≤ q < +∞, q 0 = +∞, q 1 = qk j . (3.55) Then, (3.54) reads (3.56) f L j,0 q(p,k−j) ≤ c f 1− j k L 0 ∞(p,k) f j k L k,0 qk j (p,0) ≤ c f 1− j k BM O f j k L k,0 q(p,0) . Under the assumption that (3.57) lim m→∞Ṗ L 0,2 m (D j u) = 0 ∀ L = 1, . . . , k − j, we estimate the term on the left hand side by the aid of (3.29) with N = 0 and N ′ = k − j. This yields (3.58) f L j,0 q(p,0) ≤ c f 1− j k BM O f j k L k,0 q(p,0) . We are now in a position to prove the following product estimate. Theorem 3.16. Let 1 < p < +∞. Let N ∈ N 0 and s ∈ (−∞, N + 1). Then for all f, g ∈ L k,s q(p,N ) (R n ) ∩ L ∞ (R n ), it holds (3.59) f g L k,s q(p,N) ≤ c f ∞ g L k,s q(p,N) + g ∞ f L k,s q(p,N) . Proof: Let α, β ∈ N n 0 two multi index both are not zero with \|α + β\| = k. Set \|α\| = j. Using triangle inequality, we see that D α f D β g − P N +k−j x 0 ,r (D α f )P N +j x 0 ,r (D β g) L p (B(x 0 ,r)) ≤ c (D α f − P N +k−j x 0 ,r (D α f ))(D β g − P N +j x 0 ,r (D β g)) L p (B(x 0 ,r)) + c (D α f − P N +k−j x 0 ,r (D α f ))P N +j x 0 ,r (D β g) L p (B(x 0 ,r)) + c P N +k−j x 0 ,r (D α f ))(D β g − P N +j x 0 ,r (D β g)) L p (B(x 0 ,r)) = I + II + III. (3.60) Using Hölder's inequality together with Gaglirdo-Nirenberg's inequality (3.50), we estimate (B(x 0 ,r)) . I ≤ c D α f − P N +k−j x 0 ,r (D α f ) L k j p (B(x 0 ,r)) D β g − P N +j x 0 ,r (D β g) L k k−j p (B(x 0 ,r)) = c D α f − D α P N +k x 0 ,r (f ) L k j p (B(x 0 ,r)) D β g − D β P N +k x 0 ,r (g) L k k−j p (B(x 0 ,r)) ≤ c D j (f − P N +k x 0 ,r (f )) L k j p (B(x 0 ,r)) D k−j (g − P N +k x 0 ,r (g)) L k k−j p (B(x 0 ,r)) ≤ c f − P N +k x 0 ,r (f ) 1− j k L ∞ (B(x 0 ,r)) D k (f − P N +k x 0 ,r (f )) j k L p (B(x 0 ,r)) × × c g − P N +k x 0 ,r (g) j k L ∞ (B(x 0 ,r)) D k (g − P N +k x 0 ,r (g)) 1− j k L p (B(x 0 ,r)) ≤ c f 1− j k L ∞ (B(x 0 ,r)) D k f − P N x 0 ,r (D k f )) j k L p (B(x 0 ,r)) × × g j k L ∞ (B(x 0 ,r)) D k g − P N x 0 ,r (D k g)) 1− j k L p Applying Young's inequality, we obtain (B(x 0 ,r)) . In order to estimate II we make use of the inequality (B(x 0 ,r)) , which can be proved by a standard scaling argument. Together with Poincaré's inequality we find (B(x 0 ,r)) . By an analogous reasoning we get I ≤ c f L ∞ (B(x 0 ,r)) D k g − P N x 0 ,r (D k g) L p (B(x 0 ,r)) + c g L ∞ (B(x 0 ,r)) D k f − P N x 0 ,r (D k f ) L pP N +j x 0 ,r (D β g) L ∞ (B(x 0 ,r)) ≤ cr −(k−j) g L ∞II ≤ cr k−j D k−j (D α f − P N +k−j x 0 ,r (D α f )) L p (B(x 0 ,r)) r −(k−j) g L ∞ (B(x 0 ,r)) ≤ c g L ∞ (B(x 0 ,r)) D k f − P N x 0 ,r (D k f ) L pIII ≤ c f L ∞ (B(x 0 ,r)) D k g − P N x 0 ,r (D k g) L p (B(x 0 ,r)) . Inserting the estimates of I, II and III into the right-hand side of (3.60), we arrive at D α f D β g − P N +k−j x 0 ,r (D α f )P N +j x 0 ,r (D β g) L p (B(x 0 ,r)) ≤ c f L ∞ (B(x 0 ,r)) D k g − P N x 0 ,r (D k g) L p (B(x 0 ,r)) + c g L ∞ (B(x 0 ,r)) D k f − P N x 0 ,r (D k f ) L p (B(x 0 ,r)) . (3.61) Let γ ∈ N 0 be a multi index with \|γ\| = k. Using Leibniz formula, we compute D γ (f g) = α+β=γ γ! α!β! D α f D β g. Thus, employing Corollary 3.4, using triangle inequality together with (3.61), we obtain (B(x 0 ,r)) . This yields the product estimate D γ (f g) − P 2N +k x 0 ,r (D γ (f g)) L p (B(x 0 ,r)) ≤ c inf Q∈P 2N+k D γ (f g) − Q L p (B(x 0 ,r)) ≤ c D γ (f g) − α+β=γ γ! α!β! P N +k−j x 0 ,r (D α f )P N +j x 0 ,r (D β g) L p (B(x 0 ,r)) = c α+β=γ γ! α!β! (D α f D β g − P N +k−j x 0 ,r (D α f )P N +j x 0 ,r (D β g)) L p (B(x 0 ,r)) ≤ c f ∞ D k g − P N x 0 ,r (D k g) L p (B(x 0 ,r)) + c g ∞ D k f − P N x 0 ,r (D k f ) L p(3.62) osc p,2N +k (D k (f g); x 0 , r) ≤ c f ∞ osc p,N (D k g; x 0 , r) + c g ∞ osc p,N (D k f ; x 0 , r). Into (3.62) we insert r = 2 j , j ∈ Z, and multiply this by 2 −sj . Then, applying the ℓ q norm to both sides of (3.62), we are led to (3.63) f g L k,s q(p,2N+k) ≤ c f ∞ g L k,s q(p,N) + g ∞ f L k,s q(p,N) . Verifying (3.28) holds for N ′ = 2N + k, we are in a position to apply Lemma 3.7 with N ′ = 2N + k. This gives (3.59). Proof of the main theorems We start with the following energy identity for solutions to the transport equation. Let 1 < p < +∞, x 0 ∈ R and 0 < r < +∞. We denote ϕ x 0 ,r = ϕ(r −1 (x 0 − ·)). We define the following minimal polynomial P N, * x 0 ,r (f ), f ∈ L p (B(x 0 , r)), by (4.1) (f − P N, * x 0 ,r (f ))ϕ x 0 ,r p = min Q∈P N (f − Q)ϕ x 0 ,r p . The existence and uniqueness of such polynomial is shown in appendix of the paper. We recall the notation ϕ x 0 ,r = r −n ϕ(r −1 (x 0 − ·)). We have the following. Lemma 4.1. Given v ∈ L 1 (0, T ; C 0,1 (R n ; R n )), and g ∈ L 1 (0, T ; L p loc (R n )), let f ∈ L ∞ (0, T ; C 0,1 (R n )) ∩ C([0, T ]; L p loc (R n )) be a weak solution to the transport equation (4.2) ∂ t f + (v · ∇)f = g in Q T . Let N ∈ N 0 . Define, L =    2N − 1 if N ≥ 1 0 if N = 0. Then for all t ∈ [0, T ] it holds e(t) = e(0) + t 0 v · ∇ϕ x 0 ,r \|f − P L, * x 0 ,r (f )\| p ϕ p−1 x 0 ,r e(τ ) 1−p dxdτ + 1 p t 0 B(x 0 ,r) ∇ · v\|f − P L, * x 0 ,r (f )\| p ϕ p x 0 ,r e(τ ) 1−p dxdτ + t 0 B(x 0 ,r) v · ∇P L, * x 0 ,r (f (τ )) · \|f − P L, * x 0 ,r (f (τ ))\| p−2 (f − P L, * x 0 ,r (f ))ϕ p x 0 ,r e(τ ) 1−p dxdτ + t 0 B(x 0 ,r) (g − P N x 0 ,r (g))\|f − P L, * x 0 ,r (f )\| p−2 (f − P L, * x 0 ,r (f ))ϕ p x 0 ,r e(τ ) 1−p dxdτ = e(0) + I + II + III + IV, (4.3) where e(τ ) = (f (τ ) − P L, * x 0 ,r (f (τ )))ϕ x 0 ,r p , τ ∈ [0, T ]. In addition, the following inequality holds for all t ∈ [0, T ] osc p,L f (t); x 0 , r 2 ≤ c osc p,L (f (0); x 0 , r) + cr −1 t 0 v(τ ) L ∞ (B(x 0 ,r)) osc p,N (f (τ ); x 0 , 2r)dτ + c t 0 ∇ · v(τ ) L ∞ (B(x 0 ,r)) osc p,N (f (τ ); x 0 , 2r)dτ + δ N 0 c t 0 osc p,N (v(τ ); x 0 , r) ∇P N x 0 ,r (f (τ )) L ∞ (B(x 0 ,r)) dτ + c t 0 osc p,N (g(τ ); x 0 , r)dτ, (4.4) where δ N 0 = 0 if N = 0 and 1 otherwise. Proof: Let x 0 ∈ R n , 0 < r < +∞ be fixed. Let δ ≥ 0 we define F δ (z) = (δ + \|z\| 2 ) p−2 2 z, z ∈ R n . Let N ∈ N 0 . Set L = 0 if N = 0 and L = 2N − 1 if L ≥ 1. For δ > 0 by P L,δ x 0 ,r (f (τ )) ∈ P L , 0 ≤ τ ≤ T ,F δ (f (τ ) − P L,δ x 0 ,r (f (τ )) · Qϕ p x 0 ,r dx = 0 ∀ τ ∈ [0, T ], ∀ Q ∈ P L . Furthermore, for all τ ∈ [0, T ] it holds (4.6) P L,δ x 0 ,r (f (τ )) → P L, * x 0 ,r (f (τ )) in L p (B(x 0 , r)) as δ ց 0. According to (A.7) the function s → P L,δ x 0 ,r (f (s)) is differentiable for δ > 0, and from (4.2) we get ∂ t (f − P L,δ x 0 ,r (f )) + (v · ∇)(f − P L,δ x 0 ,r (f )) + (v · ∇)P L,δ x 0 ,r (f ) = g − ∂ t P L,δ x 0 ,r (f ) in Q T . (4.7) First let us verify that ∂ t P L,δ x 0 ,r (f (τ )) ∈ P L for all τ ∈ [0, T ]. In fact, for any multi index α ∈ N 0 with \|α\| = L+1, recalling P L,δ x 0 ,r (f ) ∈ P L , we get D α ∂ t P L,δ x 0 ,r (f ) = ∂ t D α P L,δ x 0 ,r (f ) = 0. This shows the claim. We multiply (4.7) by F δ (f (τ ) − P L,δ x 0 ,r (f (τ ))ϕ p x 0 ,r , integrate over B(x 0 , r) and apply integration by parts. This together with (4.5) yields 2. Using the triangle inequality, we estimate I ≤ c t 0 ∇ϕ x 0 ,r · v(τ ) ∞ f (τ ) − P L, * x 0 ,r (f (τ )) L p (B(x 0 ,r)) e(τ ) p−1 e(τ ) 1−p dτ ≤ c t 0 ∇ϕ x 0 ,r · v(τ ) ∞ f (τ ) − P L, * x 0 ,r (f (τ )) L p (B(x 0 ,r)) dτ ≤ c t 0 ∇ϕ x 0 ,r · v(τ ) ∞ (f (τ ) − P L, * x 0 ,2r (f (τ )))ϕ x 0 ,2r p dτ + c t 0 ∇ϕ x 0 ,r · v(τ ) ∞ P L, * x 0 ,r (f (τ )) − P L, * x 0 ,2r (f (τ )) L p (B(x 0 ,r)) dτ = I 1 + I 2 . Thanks to the minimizing property (4.1) we get I 1 ≤ cr −1 t 0 v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P L x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) dτ. On the other hand, for estimating I 2 , making use of (A.12), we see that for all τ ∈ [0, T ], P L, * x 0 ,r (f (τ )) − P L, * x 0 ,2r (f (τ )) = P L, * x 0 ,r (f (τ ) − P L x 0 ,2r (f (τ ))) − P L, * x 0 ,2r (f (τ ) − P L x 0 ,2r f (τ )). This, together with (A.8) and (A.1), yields P L, * x 0 ,r (f (τ )) − P L, * x 0 ,2r (f (τ )) L p (B(x 0 ,r)) ≤ P L, * x 0 ,r (f (τ ) − P L x 0 ,2r f (τ )) L p (B(x 0 ,r)) + P N, * x 0 ,2r (f (τ ) − P L x 0 ,2r (f (τ ))) L p (B(x 0 ,r)) ≤ c f (τ ) − P L x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) . Consequently, I 2 enjoys the same estimate as I 1 , which gives I ≤ cr −1 t 0 v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P L x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) dτ. Using (A.1), we immediately get II ≤ c t 0 ∇ · v(τ ) L ∞ (B(x 0 ,r)) (f (τ ) − P L, * x 0 ,r (f (τ )))ϕ x 0 ,r L p (B(x 0 ,r)) dτ ≤ c t 0 ∇ · v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P L x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) dτ. We proceed with the estimation of III. Clearly, in case N = 0, since P L, * x 0 ,r (f (τ )) = const for all τ ∈ [0, T ], the integral III vanishes. Thus, it only remains the case N > 0. Let τ ∈ [0, T ] be fixed. Making use of (4.5) with δ = 0, we find B(x 0 ,r) v(τ ) · ∇P L, * x 0 ,r (f (τ )) · \|f (τ ) − P L, * x 0 ,r (f (τ ))\| p−2 (f (τ ) − P L, * x 0 ,r (f (τ )))ϕ p x 0 ,r dx = B(x 0 ,r) v(τ ) · ∇(P L, * x 0 ,r (f (τ )) − P N x 0 ,r (f (τ ))) · F 0 f (τ ) − P L, * x 0 ,r (f (τ )) ϕ p x 0 ,r dx + B(x 0 ,r) v(τ ) · ∇P N x 0 ,r (f (τ )) · F 0 f (τ ) − P L, * x 0 ,r (f (τ )) ϕ p x 0 ,r dx = B(x 0 ,r) v(τ ) · ∇(P L, * x 0 ,r (f (τ )) − P N x 0 ,r (f (τ ))) · F 0 f (τ ) − P L, * x 0 ,r (f (τ )) ϕ p x 0 ,r dx + B(x 0 ,r) (v(τ ) − P N x 0 ,r (v(τ ))) · ∇P N x 0 ,r (f (τ )) · F 0 f (τ ) − P L, * x 0 ,r (f (τ )) ϕ p x 0 ,r dx = J 1 + J 2 . Using the fact that P L, * x 0 ,r (Q) = P N x 0 ,r (Q) = Q for all Q ∈ P N , we get with Q = P N x 0 ,r (f (τ )) for all τ ∈ (0, t) ∇(P L, * x 0 ,r (f (τ )) − P N x 0 ,r (f (τ )) L p (B(x 0 ,r)) ≤ cr −1 f (τ ) − P N x 0 ,r (f (τ )) L p (B(x 0 ,r)) . Then Hölder's inequality yields J 1 ≤ cr −1 v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P N x 0 ,r (f (τ )) L p (B(x 0 ,r)) e(τ ) p−1 . Similarly, J 2 ≤ c v(τ ) − P N x 0 ,r (v(τ )) L p (B(x 0 ,r)) ∇P N x 0 ,r (f (τ )) L ∞ (B(x 0 ,r)) e(τ ) p−1 . Inserting the estimates of J 1 and J 2 into the integral of III, we obtain III ≤ cr −1 t 0 v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P N x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) dτ + c t 0 v(τ ) − P N x 0 ,r (v(τ )) L p (B(x 0 ,r)) ∇P N x 0 ,r (f (τ )) L ∞ (B(x 0 ,r)) dτ. To estimate IV , we use Hölder's inequality. This leads to IV ≤ t 0 g(τ ) − P N x 0 ,r (g(τ )) L p (B(x 0 ,r)) dτ. Inserting the estimates of I, II, III and IV into the right-hand side of (4.3), we find e(t) ≤ e(0) + cr −1 t 0 v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P N x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) dτ + c t 0 ∇ · v(τ ) L ∞ (B(x 0 ,r)) f (τ ) − P N x 0 ,2r (f (τ )) L p (B(x 0 ,2r)) dτ + c t 0 v(τ ) − P N x 0 ,r (v(τ )) L p (B(x 0 ,r)) ∇P N x 0 ,r (f (τ )) L ∞ (B(x 0 ,r)) dτ + c t 0 g(τ ) − P N x 0 ,r (g(τ )) L p (B(x 0 ,r)) dτ. (4.8) Noting that f (t) − P L x 0 , r 2 (f (t)) L p (B(x 0 , r 2 )) ≤ c (f (t) − P L, * x 0 ,r (f (t)))ϕ x 0 ,r p = ce(t), and using (A.1), recalling that L = 2N − 1, the inequality (4.4) follows from (4.8). Remark 4.2. Given v ∈ L 1 (0, T ; C 0,1 (R n ; R n )), and π ∈ L 1 (0, T ; W 1, 2 loc (R n ; R n )), let f ∈ L ∞ (0, T ; C 0,1 (R n ; R n )) with ∇ · f = 0 be a weak solution to the system (4.9) ∂ t f + (v · ∇)f = −∇π in Q T . Then, repeating the proof of Lemma 4.1 for the case p = 2 and N = 1 in the vector valued case, we find e(t) = e(0) + t 0 v · ∇ϕ x 0 ,r \|f − P 1, * x 0 ,r (f )\| 2 ϕ x 0 ,r e(τ ) −1 dxdτ + 1 2 t 0 B(x 0 ,r) ∇ · v\|f − P 1, * x 0 ,r (f )\| 2 ϕ 2 x 0 ,r e(τ ) −1 dxdτ + t 0 B(x 0 ,r) v · ∇P 1, * x 0 ,r (f ) · (f − P 1, * x 0 ,r (f ))ϕ 2 x 0 ,r e(τ ) −1 dxdτ + t 0 B(x 0 ,r) (∇π − P 1 x 0 ,r (∇π))(f − P 1, * x 0 ,r (f ))ϕ 2 x 0 ,r e(τ ) −1 dxdτ = e(0) + I + II + III + IV, (4.10) where e(τ ) = (f (τ ) − P 1, * x 0 ,r (f (τ )))ϕ x 0 ,r 2 , τ ∈ [0, T ]. The integrals I, II and III can be estimated as in the proof of Lemma 4.1. For the estimation of IV we proceed as follows. Assume that the mollifier ϕ ∈ C ∞ c (B(1)) is radial symmetric. Let u ∈ L 1 (B(x 0 , r)). It can be checked easily that the minimal polynomial P 1, * x 0 ,r (u) is given by P 1, * x 0 ,r (u)(x) = 1 R n ϕ 2 x 0 ,r dy R n uϕ 2 x 0 ,r dy+ n R n ϕ 2 x 0 ,r \|x 0 − y\| 2 dy R n uϕ 2 x 0 ,r (y i −x 0,i )dy(x i −x 0,i ). In case u = (u 1 , . . . , u n ) with ∇ · u = 0 almost everywhere in B(x 0 , r), recalling that ϕ is radialsymmetric, by Gauss' theorem we get ∇ · P 1, * x 0 ,r (u)(x) = n R n ϕ 2 x 0 ,r \|x 0 − y\| 2 dy B(x 0 ,r) u · (y − x 0 )ϕ 2 x 0 ,r dy = 0. Using integration by parts together with ∇ · P 1, * x 0 ,r (f (τ )) = 0, and applying Sobolev-Poincaré inequality, we get B(x 0 ,r) (∇π(τ ) − P 1 x 0 ,r (∇π(τ )))(f (τ ) − P 1, * x 0 ,r (f (τ )))ϕ 2 x 0 ,r e(τ ) −1 dx = −2 B(x 0 ,r) (π(τ ) − P 2 x 0 ,r (π(τ )))(f (τ ) − P 1, * x 0 ,r (f (τ )))ϕ x 0 ,r · ∇ϕ x 0 ,r e(τ ) −1 dx ≤ cr −1 B(x 0 ,r) \|∇π(τ ) − P 1 x 0 ,r (∇π(τ )))\| 2n n+2 dx n+2 2n ≤ cr (∇π(τ ); x 0 , r)dτ. Inserting the estimates of I, II, III and IV into the right-hand side of (4.10), and arguing as in the proof of Lemma 4.1, we arrive at osc 2,1 f (t); x 0 , r 2 ≤ c osc 2,1 (f (0); x 0 , r) + cr −1 t 0 v(τ ) L ∞ (B(x 0 ,r)) osc 2,1 (f (τ ); x 0 , 2r)dτ + c t 0 ∇ · v(τ ) L ∞ (B(x 0 ,r)) osc 2,1 (f (τ ); x 0 , 2r)dτ + c t 0 osc 2,1 (v(τ ); x 0 , r)\|∇P 1 x 0 ,r (f (τ )\|dτ + c t 0 osc 2n n+2 ,1 (∇π(τ ); x 0 , r)dτ. (4.11) Proof of the main theorems 1. Existence and uniqueness in terms of particle trajectories. Assume f 0 ∈ L s q(p,N ) (R n ), g ∈ L 1 (0, T ; L s q(p,N ) (R n )), and ∇v ∈ L 1 (0, T ; L ∞ (R n )). Let (x, t) ∈ Q T be fixed. By X t (x, ·) we denote the unique solution to the ODE (4.12) d dτ X t (x, τ ) = v(X t (x, τ ), τ ), τ ∈ [0, T ], X t (x, t) = x, which is ensured by Carathéodory's theorem. We define the flow map Φ t,τ : R n → R n by means of Φ t,τ (x) = X t (x, τ ), x ∈ R n , τ, t ∈ [0, T ]. By the uniqueness of this flow we get the inverse formula Φ −1 t,τ (x) = Φ τ,t (x). Furthermore, from (4.12) we deduce that (4.13) d dτ Φ t,τ (x) = v(Φ t,τ (x), τ ), τ ∈ [0, T ], Φ t,t (x) = x. Let (x, t) ∈ Q T . We set y = Φ t,0 (x), which is equivalent to x = Φ 0,t (y). We define f by means of (4.14) f (x, t) = f 0 (y) + t 0 g(Φ 0,s (y), s)ds. Recalling that f (t) is Lipschitz for almost all t ∈ (0, T ), we see that f is differentiable with respect to time almost everywhere in (0, T ). Recalling the inverse formula, it holds x = Φ 0,t (y). Consequently, for y ∈ R n fixed we get from (4.14) (4.15) f (Φ 0,t (y), t) = f 0 (y) + t 0 g(Φ 0,s (y), s)ds ∀ t ∈ (0, T ). Differentiating (4.15) with respect to t, and observing (4.13), we obtain (4.16) ∂ t f (Φ 0,t (y), t) + (v(Φ 0,t (y), t) · ∇)f (Φ 0,t (y), t) = g(Φ 0,t (y), t). This shows that f solves (1.1) in Q T . In addition, verifying that Φ 0,0 (x) = x, we get from (4.15) f (x, 0) = f 0 (x) ∀ x ∈ R n . This solution is also unique. In fact, assume there is another solution f solves (1.1). Setting w = f − f , then w solves (1.1) with homogenous data. In other words for every y ∈ R n the function Y (t) = w(Φ 0,t (y), t) solves the ODĖ Y = 0, Y (0) = 0, which implies Y ≡ 0, and thus w(Φ 0,t (y), t) = 0. With y = Φ t,0 (x) we get w(x, t) = 0 for all (x, t) ∈ Q T . 2. Growth of the solution as \|x\| → +∞. Applying ∇ x to both sides of (4.13), and using the chain rule, we find that (4.17) d dτ ∇Φ s,τ (x) = ∇v(Φ s,τ (x), τ ) · ∇Φ s,τ (x). Integration with respect to τ over (s, t) yields ∇Φ s,t (x) = I + t s ∇v(Φ s,τ (x), τ ) · ∇Φ s,τ (x)dτ, where I stands for the unit matrix. Thus, for all s, t ∈ (0, T ), \|∇Φ s,t (x)\| ≤ 1 + t s ∇v(τ ) ∞ \|∇Φ s,τ (x)\|dτ. By means of Gronwall's lemma it follows that for all s, t ∈ (0, T ) (4.18) \|∇Φ s,t (x)\| ≤ exp t s ∇v(τ ) ∞ dτ . From the definition (4.12) we deduce that ∇f (x, t) = ∇f 0 (Φ t,0 (x)) · ∇Φ t,0 (x) + t 0 ∇ x g(Φ 0,τ (Φ t,0 (x)), τ )dτ = ∇f 0 (Φ t,0 (x)) · ∇Φ t,0 (x) + t 0 ∇g(Φ 0,τ (Φ t,0 (x)), τ ) · ∇Φ 0,τ (Φ t,0 (x)) · ∇Φ t,0 (x)dτ. Thus, in case ∇f 0 ∈ L ∞ (R n ) and g ∈ L 1 (0, T ; L ∞ (R n )), in view of (4.18) we get for all t ∈ (0, T ) (4.19) ∇f (t) ∞ ≤ ∇f 0 ∞ + T 0 ∇g(τ ) ∞ exp 2 T 0 ∇v(τ ) ∞ dτ . Using integration by parts, from (4.13) we get for all s, t ∈ (0, T ) Φ s,t (x) − x = Φ s,t − Φ s,s (x) = t s v(Φ s,τ (y), τ ) − v(0, τ )dτ + t s v(0, τ )dτ. This leads to the inequality \|Φ s,t (x)\| ≤ \|x\| + T 0 \|v(0, τ )\|dτ + t s ∇v(τ ) ∞ \|Φ s,τ (y)\|dτ. By means of Gronwall's lemma we find for all s, t ∈ (0, T ) (4.20) \|Φ s,t (x)\| ≤ \|x\| + T 0 \|v(0, τ )\|dτ exp T 0 ∇v(τ ) ∞ dτ ≤ c(1 + \|x\|). Let x ∈ R n and t ∈ (0, T ). In case N = 0, s ∈ [0, 1), using Lemma 3.11, we get \|f 0 (x)\| ≤ c(1 + \|x\| s ) f 0 L s q(p,N) , (4.21) \|g(x, τ )\| ≤ c(1 + \|x\| s ) g(τ ) L s q(p,N) . (4.22) In case N = 1, s = 1 and 1 < q ≤ ∞ we get by Lemma 3.11 \|f 0 (x)\| ≤ c(1 + log(1 + \|x\|) 1 q ′ \|x\|) f 0 L s q(p,N) , (4.23) \|g(x, τ )\| ≤ c(1 + log(1 + \|x\|) 1 q ′ \|x\|) g(τ ) L s q(p,N) (4.24) with q ′ = q q−1 . In the remaining cases having ∇f 0 ∈ L ∞ (R n ) and ∇g ∈ L 1 (0, T ; L ∞ (R n )), we find, \|f 0 (x)\| ≤ c(1 + \|x\|)( f 0 L s q(p,N) + ∇f 0 ∞ ), (4.25) \|g(x, τ )\| ≤ c(1 + \|x\|)( g(τ ) L s q(p,N) + ∇g(τ ) ∞ ). (4.26) Setting y = Φ t,0 (x), we get from (4.15) \|f (x, t)\| ≤ \|f 0 (y)\| + (1 + log(1 + \|x\|) 1 q ′ \|x\|) if s = 1, where c stands for a constant depending on s, q, p, N, n and f 0 , g and v. 3. Local energy estimation. Let x 0 ∈ R n . Let ξ ∈ C 2 ([0, T ]; R n ) be a solution to the ODE (4.28)ξ(τ ) = v(x 0 + ξ(τ ), τ ) τ ∈ [0, T ]. We set F (x, τ ) = f (x + ξ(τ ), τ ), V (x, τ ) = v(x + ξ(τ ), τ ) −ξ(τ ), G(x, s) = g(x + ξ(τ ), τ ), (x, s) ∈ Q T . It is readily seen that V solves the transport equation (4.29) ∂ t F + (V · ∇)F = G in Q T . In particular, from (4.28) we infer (4.31) where δ N 0 = 0 if N = 0 and 1 otherwise. −j V (τ ) L ∞ (B(x 0 ,2 j+1 )) ≤ c ∇v(τ ) ∞ , we find osc p,L (F (t); x 0 , 2 j ) ≤ c osc p,L (f 0 (· + ξ(0)); x 0 , 2 j+1 ) + c t 0 ∇v(τ ) ∞ osc p,N (F (τ ); x 0 , 2 j+2 )dτ + δ N 0 c t 0 osc p,N (V (τ ); x 0 , 2 j+1 ) ∇P N x 0 ,r (F (τ )) L ∞ (B(x 0 ,2 j+1 )) dτ + c t 0 osc p,N (G(τ ); x 0 , 2 j+1 )dτ, Proof of (1.6) in Theorem 1.1 Inequality (4.31) gives osc p,0 (F (t); x 0 , 2 j ) ≤ c osc p,0 (F (0); x 0 , 2 j+1 ) + c t 0 ∇v(τ ) ∞ osc p,0 (F (τ ); x 0 , 2 j+2 )dτ + c t 0 osc p,0 (G(τ ); x 0 , 2 j+1 )dτ. (4.32) Observing (4.27), since s < 1, we get S 1,1 (osc p,0 (f (τ ); x 0 )) < +∞. Thus, applying S 1,1 to both sides of (4.37), we obtain S 1,1 (osc p,0 (F (t); x 0 )) ≤ cS 1,1 (osc p,0 (F (0); x 0 )) + c t 0 ∇v(τ ) ∞ S 1,1 (osc p,0 (F (τ ); x 0 ))dτ + c t 0 S 1,1 (osc p,0 (G(τ ); x 0 ))dτ. (4.33) Applying Gronwall's lemma, we deduce from (4.40) osc p,0 (F (t); x 0 ) ≤ S 1,1 (osc p,0 (F (t); x 0 )) ≤ c S 1,1 (osc p,0 (F (0); x 0 ) + t 0 S 1,1 (osc p,0 (G(τ ); x 0 ))dτ exp c t 0 ∇v(τ ) ∞ dτ . (4.34) Let t ∈ [0, T ]. Clearly, the constant in (4.34) is independent of the choice of the characteristic for ξ. Therefore, we may choose ξ such that ξ(t) = 0, which implies F (t) = f (t). Hence, we may replace F (t) by f (t) on the left-hand side of (4.34). Afterwards, with the help of Lemma 2.1 we are in a position to operate S s,q to both sides of (4.34), verifying F (0) = f 0 (· − ξ(0)), that yields (S s,q (osc p,0 (f (t); x 0 ))) j ≤ c (S s,q (osc p,0 (f 0 (· − ξ(0); x 0 )))) j + t 0 (S s,q (osc p,0 (G(τ ); x 0 )))) j dτ exp c t 0 ∇v(τ ) ∞ dτ . Multiplying both sides by 2 −js , we get Passing j → −∞ and taking the supremum over x 0 ∈ R n in (4.36), we get (1.6). ∞ i=j (2 −si osc p,1 (f (t); x 0 ; 2 i )) q 1 q ≤ c \|f 0 \| L s q(p,0) + t 0 \|G(τ )\| L s q(p,0) dτ exp c t 0 ∇v(τ ) ∞ dτ . Proof of (1.9) in Theorem 1.2. Recalling that V (x 0 , τ ) = 0 for all τ ∈ [0, T ], we see that 2 −j V (τ ) L ∞ (B(x 0 ,2 j+1 )) ≤ c ∇v(τ ) ∞ and 2 −j osc p,0 (V (τ ); x 0 , 2 j+1 ) ≤ c ∇v(τ ) ∞ . Thus, (4.15) leads to osc p,1 (F (t); x 0 , 2 j ) ≤ c osc p,1 (F (0); x 0 , 2 j+1 ) + c t 0 ∇v(τ ) ∞ osc p,1 (F (τ ); x 0 , 2 j+2 )dτ + c t 0 osc p,1 (V (τ ); x 0 , 2 j+1 )\|∇Ṗ 1 x 0 ,2 j+1 (F (τ ))\|dτ + c t 0 osc p,1 (G(τ ); x 0 , 2 j+1 )dτ. (4.37) In case j ≥ 0, using triangle inequality, we get \|∇Ṗ 1 x 0 ,2 j (F (τ ))\| ≤ c j i=0 2 −i osc p,1 (F (τ ); x 0 , 2 i ) + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| ≤ c2 −j (S 3,1 (osc p,1 (F (τ ); x 0 ))) j + \|∇Ṗ 1 x 0 ,1 (F (τ ))\|. In case j < 0, using triangle inequality along with Hölder's inequality, we find \|∇Ṗ 1 x 0 ,2 j (F (τ ))\| ≤ c j i=0 2 −i osc p,1 (F (τ ); x 0 , 2 i ) + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| ≤ (−j) 1 q ′ 0 i=j 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (τ ))\|. Summing up the above estimates, we arrive at osc p,1 (V (τ ); x 0 , 2 j+1 )\|∇Ṗ 1 x 0 ,2 j+1 (F (τ ))\| ≤ 2 −j osc p,1 (V (τ ); x 0 , 2 j+1 )(S 3,1 (osc p,1 (F (τ ); x 0 ))) j + c(j − ) 1 q ′ osc p,1 (V (τ ); x 0 , 2 j+1 ) 0 i=−∞ 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| , (4.38) where j − = − min{j, 0}. Applying the operator S 2,1 to the both sides of the above inequality, and making use of Lemma 2.1, with p = q = 1, α = 3 and β = 2, we obtain S 2,1 osc p,1 (V (τ ); x 0 , 2 i+1 )\|∇Ṗ 1 x 0 ,2 i+1 (F (τ ))\| ≤ c\|v(τ )\| L 1 q(p,1) S 2,1 (osc p,1 (F (τ ); x 0 )) + cS 2,1 (i − ) 1 q ′ osc p,1 (V (τ ); x 0 , 2 i ) 0 i=−∞ 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| . (4.39) Observing (4.27), all sum in the above estimates are finite. Again appealing to (4.20) we are in a position to apply S 2,1 to both sides of (4.37) to get S 2,1 (osc p,1 (F (t); x 0 )) ≤ cS 2,1 (osc p,1 (F (0); x 0 )) + c t 0 ( ∇v(τ ) ∞ + \|v(τ )\| L 1 q(p,1) )S 2,1 (osc p,1 (F (τ ); x 0 ))dτ + c t 0 S 2,1 (i − ) 1 q ′ osc p,1 (V (τ ); x 0 , 2 i ) 0 i=−∞ 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| dτ + c t 0 S 2,1 (osc p,1 (G(τ ); x 0 ))dτ. (4.40) Applying Gronwall's lemma, we are led to osc p,1 (F (t); x 0 ) ≤ S 2,1 (osc p,1 (F (t); x 0 )) ≤ cS 2,1 (osc p,1 (f 0 (· + ξ(0); x 0 )) + c t 0 S 2,1 (i − ) 1 q ′ osc p,1 (V (τ ); x 0 , 2 i ) 0 i=−∞ 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + c\|∇Ṗ 1 x 0 ,1 (F (τ ))\| dτ + c t 0 S 2,1 (osc p,1 (G(τ ); x 0 ))dτ exp T 0 ( ∇v(τ ) ∞ + \|v(τ )\| L 1 q(p,1) )dτ. (4.41) Observing (1.7), using Lemma 2.1, we may apply S 1,q to both sides of (4.41). Accordingly, sup t∈[0,T ] S 1,q (osc p,1 (F (t); x 0 )) < +∞. For given t ∈ [0, T ] we may choose ξ such ξ(t) = 0. Thus, the same holds for f (t) in place of F (t). Now, we are able to apply S 1,q to both sides of (4.38), which yields S 1,q osc p,1 (V (τ ); x 0 , 2 i+1 )\|∇Ṗ 1 x 0 ,2 i+1 (F (τ ))\| ≤ c\|v(τ )\| L 1 q(p,1) S 1,q (osc p,1 (F (τ ); x 0 )) + cS 1,q (i − ) 1 q ′ osc p,1 (V (τ ); x 0 , 2 i ) ∞ i=−∞ 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| . (4.42) Applying S 1,q to both sides of (4.37) multiplying the result by 2 −j and letting j → −∞, we infer ∞ i=−∞ 2 −iq (osc p,1 (F (t); x 0 , 2 i )) q 1 q ≤ c\|f 0 \| L 1 q(p,1) + c t 0 ∇v(τ ) ∞ Integrating this inequality over (0, t) and applying integration by parts, we obtain \|∇Ṗ 1 x 0 ,1 (F (t))\| ≤ \|∇Ṗ 1 x 0 ,1 (F (0))\| + c t 0 ∇v(τ ) ∞ \|∇Ṗ 1 x 0 ,1 (F (τ ))\|dτ + t 0 ∇v(τ ) ∞ osc p,1 (F (τ ), x 0 ; 1)dτ + t 0 \|∇Ṗ 1 x 0 ,1 (G(τ ))\|dτ ≤ f 0 L 1 q(p,1) + c t 0 ∇v(τ ) ∞ osc p,1 (F (τ ), x 0 ; 1)dτ + t 0 g(τ ) L 1 q(p,1) dτ, (4.46) where \|z\|L 1 q(p,0) stands for the semi norm \|z\|L 1 q(p,1) = \|z\| L 1 q(p,1) + sup x 0 ∈R n \|∇Ṗ 1 x 0 ,1 (z)\|. Combining (4.43) and (4.46), we arrive at ∞ i=−∞ 2 −iq (osc p,1 (F (t); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (t))\| ≤ c\|f 0 \|L 1 q(p,1) + c t 0 ∇v(τ ) ∞ ∞ i=−∞ 2 −iq (osc p,1 (F (t); x 0 , 2 i )) q 1 q dτ + c t 0 ∞ i=−∞ (i − ) q−1 (2 −i osc p,1 (V (τ ); x 0 , 2 i )) q 1 q ∞ i=−∞ 2 −iq (osc p,1 (F (τ ); x 0 , 2 i )) q 1 q + \|∇Ṗ 1 x 0 ,1 (F (τ ))\| dτ + c t 0 \|g(τ )\|L 1 q(p,1) dτ. (4.47) Applying Gronwall's lemma and for given t ∈ [0, T ] choosing ξ such that ξ(t) = 0, and taking the supremum over x 0 ∈ R n , we obtain the desired estimate (1.9). Proof of (1.11) in Theorem 1.3. We first define χ(x 0 , t) = sup j∈Z 2 −j osc p,0 (F (t); x 0 , 2 j ), (x 0 , t) ∈ R n × [0, T ]. Clearly, thanks to (4.27) χ(x 0 , t) is finite. Noting that ∇P N x 0 ,2 j+1 (F (τ )) L ∞ (B(x 0 ,2 j+1 )) ≤ cχ(x 0 , τ ), we get from (4.31) with L = 2N − 1 First let us estimate the term osc p,0 (F (t); x 0 , 2 j+1 ). In view of (4.37) with j + 1 in place of j, and recalling that ∇f 0 ∈ L ∞ (R n ), g ∈ L 1 (0, T ; L ∞ (R n )), we see that Multiplying both sides of (4.49) by 2 −j and taking the supremum over all j ∈ Z, using the triangle inequality, we obtain Next, once more using (4.27) we see that S s,q (osc p,N (F (t); x 0 )) < +∞, for all t ∈ [0, T ]. Thus, we apply S s,q to both sides of (4.51) and use Lemma 2.1. This combined with Proof of (1.12) in Corollary 1.4. In view of Theorem 3.6 we have ∇f 0 ∈ L ∞ (R n ), ∇g ∈ L 1 (0, T ; L ∞ (R n )). More precisely, (3.26) yields ∇f 0 ∞ ≤ c f 0 L 1 1(p,1) , T 0 ∇g(τ ) ∞ dτ ≤ c g L 1 (0,T ;L 1 1(p,1) ) . osc p,2N −1 (F (t); x 0 , 2 j ) ≤ c osc p,N (F (0); x 0 , 2 j+1 ) + c t 0 ∇v(τ ) ∞ osc p,N (F (τ ); x 0 , 2 j+2 )dτ + c2 −j In particular, this shows that condition (1.8) of Theorem 1.2 is fulfilled. Furthermore, since v ∈ L 1 (0, T ; L 1 1(p,1) (R n )), condition of Theorem 1.2 (1.7) is also satisfied. Now, we are in a position to apply of Theorem 1.2, which yields f ∈ L ∞ (0, T ; L 1 1(p,1) (R n )). This allows to apply S 1,1 to both sides of (4.31). This together with Gronwall's Lemma and the inequality \|∇Ṗ 1 x 0 ,2 j+1 (F (τ ))\| ≤ c2 −j ∇f (τ ) ∞ yields (S 1,1 (osc p,1 (F (t); x 0 ))) j ≤ c (S 1,1 (osc p,1 (F (0); x 0 )) j + Choosing ξ so that ξ(t) = 0, multiplying both sides by 2 −j and letting j → −∞ taking the supremum over x 0 ∈ R n , we deduce from (4.54) \|f (t)\| L 1 1(p,1) ≤ c f 0 L 1 1(p,1) + t 0 g(τ ) L 1 1(p,1) dτ + t 0 v(τ ) L 1 1(p,1) ∇f (τ ) ∞ dτ exp c t 0 ∇v(τ ) ∞ dτ . (4.55) Combining (4.55) and (4.19) along with (4.27) in order to estimate f (t) L p (B(1)) , we get the desired estimate (1.12). Below we prove the uniqueness parts of Theorem 1.1, Theorem1.2, Theorem1.3 and Corollary1.4. In fact we prove the stronger version of it, namely the strong-weak uniqueness. Strong-weak uniqueness. Let f ∈ L 2 loc (R n ) be a weak solutions to (1.1). Then w = f − f solves the transport equation with homogenous data (4.56) ∂ t w + (v · ∇)w = 0 in Q T , w = 0 on R n × {0} in a weak sense, i.e. for all t ∈ (0, T ), and for all ϕ ∈ L ∞ (0, t; W 1,2 (R n ))∩W 1,1 (0, t; L 2 (R n )) with supp(ϕ) ⋐ R n × [0, t], it holds − t 0 R n w∂ t ϕ + (v · ∇)ϕw + ∇ · vϕwdxds = − R n w(t)ϕ(t)dx. (4.57) Let ψ ∈ C ∞ c (R n ) be a given function. Using the method of characteristics, for every ε > 0 we get a solution ϕ ε ∈ L ∞ (0, t; W 1,2 (R n )) ∩W 1,1 (0, t; L 2 (R n )) of the the following dual problem (4.58) ∂ t ϕ ε + v · ∇ϕ ε + ∇ · v ε ϕ ε = 0 in Q t , ϕ ε (t) = ψ in R n . Noting that ∇v ε (τ ) ∞ ≤ ∇v(τ ) ∞ , using Gronwall's lemma we see that ϕ ε 1 + ϕ ε ∞ ≤ c with a constant c > 0 independent of ε > 0. Since v(0, ·), ∇v(·) ∞ ∈ L 1 (0, T ) using (4.20), we get a number 0 < R < +∞ such that supp(ϕ ε ) ⊂ B(R)×[0, t]. In (4.57) putting ϕ = ϕ ε , and using (4.58), we infer R n w(t)ψdx = t 0 R n w∂ t ϕ ε + (v · ∇)ϕ ε w + ∇ · vϕ ε wdxds = t 0 B(R) ∇ · (v − v ε )ϕ ε wdxds. ∇ · (v − v ε )ϕ ε wdxds → 0 as ε ց 0. Letting ε ց 0 in (4.59), we deduce that R n w(t)ψdx = 0. Whence, w ≡ 0. This shows the uniqueness. A Minimal polynomials Let < p < +∞. Let x 0 ∈ R n and 0 < r < +∞ be fixed. Set φ = ϕ(r −1 (x 0 − ·)), where ϕ ∈ C ∞ c (B(1)), being radial symmetric, stands for the standard mollifier. For δ ≥ 0 we define the following functional J δ : L p (B(x 0 , r)) → R by J δ (f ) = B(x 0 ,) (δ + \|f \| 2 ) p 2 φ p dx, f ∈ L p (B(x 0 , r)). Recall P N , N ∈ N 0 , denotes the space of all polynomial of degree less or equal N. Since J δ is strict convex and lower semi continuous with J δ (f ) → +∞ as f L p (B(x 0 ,r)) → +∞. For each f ∈ L p (B(x 0 , r)) there exists a unique P N,δ x 0 ,r (f ) ∈ P N with (A.1) J δ (P N,δ x 0 ,r (f ) − f ) = min P ∈P N J δ (P − f ) Clearly, the mapping J δ,f : P → J δ (P − f ) is differentiable as a function from P N into R. Since the first variation must vanish at each minimizer, we get (A.2) DJ δ,f (P N,δ x 0 ,r (f ), P ) = 0 ∀ P ∈ P N . This shows that (A.3) B(x 0 ,r) F δ (P N,δ x 0 ,r (f ) − f ) · P φ p dx = 0 ∀ P ∈ P N . where F δ (u) = (δ + \|u\| 2 ) p−2 2 u, u ∈ R n . It is well known that F δ is monotone and continuously differentiable for each δ > 0 . Then, there exists a unique solution f ∈ L ∞ (0, T ; L 1 q(p,1) (R n )) to the transport equation (1.1). Furthermore, it holds for all t ∈ (0, T )\|f (t)\|L 1 q(p,1) ≤ c \|f 0 \|L 1 . 1 . 1Let f ∈ S ′ . Then for all \|β\| ≤ N it holds 3 . 3We prove (3.8) by induction over j. For j = 1 (3.8) follows from the usual Poincaré inequality, since [f − P N x 0 ,r (f )] 0 x 0 ,r = 0. Assume (3.8) holds for j − 1. Thus, Remark 3. 8 . 8For all f ∈ L s q(p,N ) (R n ), 1 ≤ p < +∞, 1 ≤ q ≤ +∞, s ∈ [− n p , N + 1), the condition (3.28) is fulfilled, and therefore(3.29) holds for all f ∈ L s q(p,N ) (R n ) under the assumptions on p, q, s, N and N ′ of Lemma 3.7. To verify this fact we observe for f ∈ L s q(p,N ) Remark 3. 9 . 9In case q = ∞, since L s ∞(p,N ) (R n ) coincides with the usual Campanato space, and Lemma 3.7 is well known (cf.[10, p. 75]). sides over i = −∞ to i = 1 and applying Hölder's inequality, we get (3.43) 1 i=−∞ 2 −i(N −1) osc p,N −1 we denote the minimal polynomial, defined in the Appendix A. (cf. Lemma A. Φ 0,s (y), s)\|ds Employing (4.21)-(4.26) together with (4.20), we see that for all (x, t) ∈ Q T \|x\| min{s,1} ) if s = 1, ( 4 . 430) V (x 0 , τ ) = 0 ∀ τ ∈ [0, T ]. Set L = 2N − 1 if N > 0 and L = 0 if N = 0.According to (4.4) of Lemma 4.1 with r = 2 j+1 , j ∈ Z, noting that in view of (4.30) it holds 2 ∇g(τ ) ∞ dτ. (4.27) we have S N +1,1 (osc p,N F (t); x 0 ) < +∞ for all t ∈ [0, T ]. Applying S N +1,1 to both sides of (4.48), and using Corollary 3.10 with N ′ = 2N − 1 ( V (τ ); x 0 ))) j ∇f (τ ) ∞ dτ exp c t 0 ∇v(τ ) ∞ dτ . Furthermore, there exists a constant c > 0 independent of δ such that for all u, v ∈ R m ,(F δ (u) − F δ (v))(u − v) ≥ c(p − 1)(δ + \|u\| + \|u − v\|) p−2 \|u − v\| 2 , (A.4) \|F δ (u) − F δ (v)\| ≤ cp(δ + \|u\| + \|u − v\|) AcknowledgementsChae was partially supported by NRF grants 2016R1A2B3011647, while Wolf has been supported supported by NRF grants 2017R1E1A1A01074536. The authors declare that they have no conflict of interest.∂ t (δ + \|f (τ ) − P L,δ x 0 ,r (f (τ ))\| 2 ) 1 2 ϕ x 0 ,r p (δ + \|f (τ ) − P L,δ x 0 ,r (f (τ ))\| 2 ) 1 2 ϕ x 0 ,r p−1 p = 1 p ∂ t (δ + \|f (τ ) − P L,δ x 0 ,r (f (τ ))\| 2 ) 1 2 ϕ x 0 ,r p p = B(x 0 ,r) v(τ ) · ∇ϕ x 0 ,r (δ + \|f (τ ) − P L,δ x 0 ,r (f (τ ))\| 2 ) p 2 ϕ p−1 x 0 ,r dxv(τ ) · ∇P L,δ x 0 ,r (f (τ ))F δ (f (τ ) − P L,δ x 0 ,r (f (τ )))ϕ p x 0 ,r dx + B(x 0 ,r) (g(τ ) − P N x 0 ,r (g(τ )))F δ (f (τ ) − P L,δ x 0 ,r (f (τ )))ϕ p x 0 ,r dx.In the last line we used identity (4.5) for Q = P N x 0 ,r (g(τ )). Multiplying both sides of the above identity by e δ (τ )1, with respect to τ , and applying integration by parts, we findIn the above identity, letting δ → 0 and making use of (4.6), we obtain (4.4).Next, we require to estimate \|∇Ṗ 1x 0 ,1 (F (τ ))\| by the initial data f 0 and g. We applẏ P 1x 0 ,1 to both sides (4.29). This givesNoting thatṖ 1x 0 ,1 (F ), and applying ∇ to both sides of (4.44), we inferOn the other hand,Inserting this identity into the right-hand side of (4.45), multiplying the result byx 0 ,1 (F )\| , we get the following differential inequalityWe obtain the following properties of G δ .strictly monotone, bijective, and in case δ > 0 stronly monotone and is a C 1 diffeomorphism.2. In case δ > 0, the mapping f → P N,δ x 0 ,r (f ) : L p (B(x 0 , r)) → P N is Fréchet differentiable, and its derivative is given by3. For all f ∈ L p (B(x 0 , r)) it holdswhere P N, * x 0 ,r (f ) = P N,0 x 0 ,r (f ).Proof: 1. Observing (A.4), we get for all f ∈ L p (B(x 0 , r)), and P, Q ∈ P NThis immediately shows that G δ (f, ·) is strictly monotone and in case δ > 0 strongly monotone. Here we have used the fact that P L 2 (B(x 0 ,r)) defines an equivalent norm on P N . Furthermore, if δ > 0 we see that G δ (f, ·) : P N into (P N ) ′ is continuously differentiable and coercitive, i.e.G δ (f, P ), P P → 0 as P → +∞.Applying the theory of monotone operators, we see that G δ (f, ·) is bijective, and is a C 1 diffeomophism.2. Let δ > 0 and f ∈ L p (B(x 0 , r)). Let P N,δ x 0 ,r (f ) ∈ P N denote the minimizer of the functional J δ (· − f ) in P N . In view of (A.6) we have G δ (f, P N,δ x 0 ,r (f )) = 0. Since D 2 G δ is an isomorphism from P N into (P N ) ′ , by the implicit function theorem we infer that the mapping P N,δ x 0 ,r : L p (B(x 0 , r)) → P N is Fréchet, differentiable, and it holds (A.7).Proof of (A.8). Since J δ is convex and recalling the minimizing property of P N,δ x 0 ,r (f ), we getThis shows thatWhence, (A.8)3. Now, let δ k ց 0 as k → +∞. By (A.8) we see that {P N,δ k x 0 ,r (f )} is bounded. Thus, there exists a subsequence, and P N,x 0 ,r (f )) as j → +∞ for all x ∈ B(x 0 , r) by Lebesgue's theorem of dominated convergence it follows 0 = G δ k j (P N,δ k j x 0 ,r (f )) → G 0 (f, P N, * x 0 ,r ). Since G 0 (f, ·) is strictly monotone, the zero is unique, and thus P N, *x 0 ,r (f ) = P N,0 x 0 ,r (f ). Thus, convergence property (A.9) is verified. Furthermore, in (A.8) letting δ ց 0, we see thatr)) . This completes the proof of the lemma.Remark A.2. The mapping P N,δ x 0 ,r : L p (B(x 0 , r)) → P N fulfills the projection propertyIn fact, this follows immediately from (A.1) by setting f = Q therein.B Example of a function in L 1 1(p,1) (R n ) \ C 1 (R n )The following example shows that L 1 1(p,1) (R n ) is not in C 1 (R n ). For simplicity we only consider the case n = 1 since general case n ∈ N can be reduced to n = 1. We defineProof of f ∈ L 1 1(p,1) (R): Thanks to (3.46) it will be sufficient to show that u ∈ L 0 1(p,0) (R). In what follows we estimate osc p,0 (u; x, r) for x ∈ [0, 1] and 0 < r < +∞. We start with the case x = 0. For 2 −m−1 < r ≤ 2 −m we get osc p,0This yields,We consider the following three cases. 1. First, in case 2 −m−1 < r < +∞ by triangle inequality we get osc p,0 (u; x, r) ≤ c osc p,0 (u; 0, 8r).2. In case 2 −2m < r ≤ 2 −m−1 , again by triangle inequality we find osc p,03. In case 0 < r ≤ 2 −2m , using Poincaré's inequality, we obtain osc p,0where p ′ = p p−1 .Using the the estimates above together with (B.1), we obtain In case x < 0 there exists m ∈ Z such that −2 m+1 < x ≤ −2 m . Using triangle inequality together with (B.1), we easily see that Similarly by the aid of (B.2) we get j∈Z osc p,0 (u; x, 2 j ) ≤ c j∈Z osc p,0 (u; 1, 2 j ) ≤ c for all x ≥ 1. This shows that u ∈ L 0 1(p,0) (R n ), and thus f ∈ L 1 1(p,1) (R n ) but u / ∈ C 1 (R). ) − v ε (s)) → 0 in L 2 (B(R)) as ε ց 0 for almost all s ∈ (0, t), by the aid of Vitali's convergence theorem. 180Noting that ∇ · (v(s. it follows that We now define the mapping G δ : L p (B(x 0 , r)) × P N → (P N ) ′ by G δ (f, P ), Q = B(x 0 ,rNoting that ∇ · (v(s) − v ε (s)) → 0 in L 2 (B(R)) as ε ց 0 for almost all s ∈ (0, t), by the aid of Vitali's convergence theorem ([9, p. 180]) it follows that We now define the mapping G δ : L p (B(x 0 , r)) × P N → (P N ) ′ by G δ (f, P ), Q = B(x 0 ,r) H Bahouri, J. -Y Chemin, R Danchin, Fourier Analysis and Nonlinear Partial Differential equations. SpringerH. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential equations, Springer (2011). L p -estimates for certain non-regular pseudo-differential operators. G Bourdaud, Comm. Partial Diff. Eqs. 7G. Bourdaud, L p -estimates for certain non-regular pseudo-differential operators, Comm. Partial Diff. Eqs. 7, 1023-1033 (1982). Strong illposedness of the incompressible Euler equations in integer C m spaces. J Bourgain, D Li, Geom. Funct. Anal. 25J. Bourgain and D. Li, Strong illposedness of the incompressible Euler equations in integer C m spaces, Geom. Funct. Anal. 25, 1-86 (2015). Strong illposedness of the incompressible Euler equations in Borderline Sobolev spaces. J Bourgain, D Li, Invent. Math. 201J. Bourgain and D. Li, Strong illposedness of the incompressible Euler equations in Borderline Sobolev spaces, Invent. Math. 201, 97-157 (2015). Proprieti di una famiglia di spazi funzionali. S Campanato, Ann. Sc. Norm. Sup. Pisa. 18S. Campanato, Proprieti di una famiglia di spazi funzionali, Ann. Sc. Norm. Sup. Pisa 18, 137-160 (1964). The Euler equations in a critical case of the generalized Campanato space. D Chae, J Wolf, arxiv preprintD. Chae and J. Wolf, The Euler equations in a critical case of the generalized Campanato space, arxiv preprint (2019). J.-Y Chemin, Perfect incompressible fluids. OxfordClarendon pressJ.-Y. Chemin, Perfect incompressible fluids. Clarendon press, Oxford (1998). Ordinary differential equations, transport theory and Sobolev spaces. R J Diperna, P L Lions, Invent. Math. 98R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547 (1989). G B Folland, Real analysis. Pure and Applied Mathematics. New YorkJohn WileySecond ed.G.B. Folland, Real analysis. Pure and Applied Mathematics (Second ed.), New York: John Wiley, (1999). Multiple integrals in the calculus of variations and nonlinear elliptic systems. M Giaquinta, Ann. Math. Stud. 105Princeton Univ. PressM. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Stud. No. 105, Princeton Univ. Press, Princeton (1983). . P G Lemarié-Rieusset, Euler Equations and Real Harmonic Analysis, Arch. Rational Mech. Anal. 204P.G. Lemarié-Rieusset, Euler Equations and Real Harmonic Analysis, Arch. Ra- tional Mech. Anal. 204, 355-386 (2012). . P L Lions, Mathematical topics in fluid mechanics. 1Oxford University PressP. L. Lions, Mathematical topics in fluid mechanics Volume 1, Oxford University Press, New York, (1996). Existence of solution for the Euler equations in a critical Besov space B 1 ∞,1 (R n ). H C Pak, Y J Park, Comm. P.D.E. 29H.C. Pak and Y. J. Park, Existence of solution for the Euler equations in a critical Besov space B 1 ∞,1 (R n ), Comm. P.D.E. 29,1149-1166 (2004). M E Taylor, Pseudodifferential operators, paradifferential operators, and layer potentials. AMS81M.E. Taylor, Pseudodifferential operators, paradifferential operators, and layer potentials, AMS, vol 81 (2000). Theory of function spaces. H Triebel, Monographs in mathematics. 84H. Triebel, Theory of function spaces. Monographs in mathematics Vol 84, Birkhäuser, Basel 1992. Hydrodynamics in Besov spaces. M Vishik, Arch. Rat. Mech. Anal. 145M. Vishik, Hydrodynamics in Besov spaces, Arch. Rat. Mech. Anal. 145, 197-214 (1998). Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. M Vishik, Ann. Sci.École Norm. Sup. 326M. Vishik, Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci.École Norm. Sup. 32(6), 769-812 (1999).	[]
[ "Users and Contemporary SERPs: A (Re-)Investigation Examining User Interactions and Experiences", "Users and Contemporary SERPs: A (Re-)Investigation Examining User Interactions and Experiences" ]	[ "Nirmal Roy [email protected] \nDelft University of Technology Delft\nThe Netherlands\n", "David Maxwell [email protected] \nDelft University of Technology Delft\nThe Netherlands\n", "Claudia Hauff [email protected] \nDelft University of Technology Delft\nThe Netherlands\n" ]	[ "Delft University of Technology Delft\nThe Netherlands", "Delft University of Technology Delft\nThe Netherlands", "Delft University of Technology Delft\nThe Netherlands" ]	[ "Proceedings of the 45th International ACM SIGIR Confer-ence on Research and Development in Information Retrieval (SIGIR '22)" ]	The Search Engine Results Page (SERP) has evolved significantly over the last two decades, moving away from the simple ten blue links paradigm to considerably more complex presentations that contain results from multiple verticals and granularities of textual information. Prior works have investigated how user interactions on the SERP are influenced by the presence or absence of heterogeneous content (e.g., images, videos, or news content), the layout of the SERP (list vs. grid layout), and task complexity. In this paper, we reproduce the user studies conducted in prior works-specifically those of Arguello et al.[4]and Siu and Chaparro [29]-to explore to what extent the findings from research conducted five to ten years ago still hold today as the average web user has become accustomed to SERPs with ever-increasing presentational complexity. To this end, we designed and ran a user study with four different SERP interfaces: (i) a heterogeneous grid; (ii) a heterogeneous list; (iii) a simple grid; and (iv) a simple list. We collected the interactions of 41 study participants over 12 search tasks for our analyses. We observed that SERP types and task complexity affect user interactions with search results. We also find evidence to support most (6 out of 8) observations from[4,29]indicating that user interactions with different interfaces and to solve tasks of different complexity have remained mostly similar over time.	10.1145/3477495.3531719	[ "https://export.arxiv.org/pdf/2207.13117v1.pdf" ]	250,340,406	2207.13117	155cf5ff2a2fcee6474bb71f68de02821c905111	Users and Contemporary SERPs: A (Re-)Investigation Examining User Interactions and Experiences ACMCopyright ACMJuly 11-15, 2022. July 11-15, 2022 Nirmal Roy [email protected] Delft University of Technology Delft The Netherlands David Maxwell [email protected] Delft University of Technology Delft The Netherlands Claudia Hauff [email protected] Delft University of Technology Delft The Netherlands Users and Contemporary SERPs: A (Re-)Investigation Examining User Interactions and Experiences Proceedings of the 45th International ACM SIGIR Confer-ence on Research and Development in Information Retrieval (SIGIR '22) the 45th International ACM SIGIR Confer-ence on Research and Development in Information Retrieval (SIGIR '22)Madrid, Spain; New York, NY, USA; Madrid, SpainACM11July 11-15, 2022. July 11-15, 202210.1145/3477495.3531719This research has been supported by NWO VIDI project SearchX (639.022.722) and NWO project Aspasia (015.013.027). This work is licensed under a Creative Commons Attribution International 4.0 License.CCS CONCEPTS • Information systems → Search interfaces• Human-centered computing → Empirical studies in HCI KEYWORDS Human Computer InteractionInteractive Information RetrievalSearch InterfacesSearch TasksReproducibility ACM Reference Format: Nirmal Roy, David Maxwell, and Claudia Hauff 2022 Users and Con- temporary SERPs: A (Re-)Investigation: Examining User Interactions and The Search Engine Results Page (SERP) has evolved significantly over the last two decades, moving away from the simple ten blue links paradigm to considerably more complex presentations that contain results from multiple verticals and granularities of textual information. Prior works have investigated how user interactions on the SERP are influenced by the presence or absence of heterogeneous content (e.g., images, videos, or news content), the layout of the SERP (list vs. grid layout), and task complexity. In this paper, we reproduce the user studies conducted in prior works-specifically those of Arguello et al.[4]and Siu and Chaparro [29]-to explore to what extent the findings from research conducted five to ten years ago still hold today as the average web user has become accustomed to SERPs with ever-increasing presentational complexity. To this end, we designed and ran a user study with four different SERP interfaces: (i) a heterogeneous grid; (ii) a heterogeneous list; (iii) a simple grid; and (iv) a simple list. We collected the interactions of 41 study participants over 12 search tasks for our analyses. We observed that SERP types and task complexity affect user interactions with search results. We also find evidence to support most (6 out of 8) observations from[4,29]indicating that user interactions with different interfaces and to solve tasks of different complexity have remained mostly similar over time. INTRODUCTION The Search Engine Results Page (SERP) has evolved significantly over the last two decades, moving away from the ten blue links paradigm, to considerably more complex presentations that contain results from multiple verticals and multiple granularities of textual information (snippets, direct answers, entity cards, etc.)-all interleaved within one page. The incorporation of heterogeneous content in a SERP has been shown to change how users interact with web results [2,3,7,10,17,21,27,36]. How (and where) content is displayed in a SERP affects user interactions as well [4,28,30]. While contemporary web SERPs maintain the original idea of a list of items that are ranked in decreasing order of relevance, alternative presentations such as a grid layout-as also recently (again) popularised by You.com-have also been explored [14,20,29,37]. In addition, past research [4,29,30,36] has shown that user behaviour on the SERP does not only depend on the presentation of information, but also on the search task at hand. For a navigational task such as 'find and access the homepage of SIGIR 2022', a user-in the ideal case-requires a single query and a single click. Contrast this to an informational task, such as 'good restaurants near the venue of SIGIR 2022'. This requires the scanning of multiple results, and likely results in further query reformulations to learn more about specific suggestions. As commercial web search engine SERPs have evolved over time (and thus end users have become accustomed to different types of SERPs), we explore in this paper to what extent user study findings from 5 − 10 years ago still hold today. Specifically, we focus our attention on reproducing the experimental setup of two prior studies: Arguello et al. [4] (published in 2012) as well as Siu and Chaparro [29] (published in 2014)-these both investigated how user interactions on the SERP are influenced by the presence or absence of heterogeneous content, the layout of the SERP (list vs. grid), and task complexity. Inspired by the two papers we reproduce, our study is guided by the following research questions. RQ1 How does a user's interactions with a SERP differ when results are presented in a list and grid layout? RQ2 How does task complexity affect user interactions with a SERP? RQ3 What is the interplay between task complexity and SERP layout on user interactions? RQ4 How do users perceive the different SERP layouts? To this end, we conducted a user study with = 41 participants that each were given 12 search tasks of varying complexity (ranging from search tasks of type Remember to Analyse) to solve with one of four different SERP interfaces: (i) heterogeneous grid; (ii) heterogeneous list; (iii) simple grid; and (iv) simple list. We explore whether the following eight observations from [4,29] about users and their interactions with list vs. grid layouts-and heterogeneous vs. simple results-across different task complexities hold today. O1 Users fixated significantly more on the grid layout SERP compared to the list layout SERP for completing more complex tasks [29]. O2 On the grid layout SERP, users fixated on search results significantly more for completing more complex tasks compared to simple tasks. A similar observation was found for the list layout SERP [29]. O3 On the list layout SERP, users fixated significantly longer for completing more complex tasks compared to simple tasks. For the grid SERP, there were no significant differences in fixation duration between varying task complexities [29]. O4 In the list layout SERP, more complex tasks required significantly greater levels of search interaction: longer search sessions, more clicks on SERP, and more web pages visited [4]. O5 In a SERP where web results are arranged in a list layout, users clicked on significantly more vertical results when they were present on the main page of the SERP (blended, heterogeneous display) compared to when they were only present as tabs (non-blended, simple display) [4]. O6 Task complexity did not have a significant effect on user interaction with vertical results in the list layout SERP [4]. O7 The interplay between task complexity and display of verticals (blended, heterogeneous display vs. non-blended, simple display) did not have a significant effect on user interaction with vertical results in the list layout SERP [4]. O8 Neither study [4,29] found significant differences in user evaluation of the different SERP types, list vs grid layout for the former and blended vs non-blended display for the latter, in their experiments. In our user study, we observed that SERP types and task complexity affect user interactions with search results. We also find evidence to support most-6 out of 8-observations from [4,29]. RELATED WORK 2.1 Task Complexity and User Interactions A number of works have focused on the effect of task types on user interactions on SERPs. Buscher et al. [8] performed a large-scale analysis using query logs to understand how individual and task differences might affect search behaviour. Their findings show that there are cohorts of users who examine search results in a similar way. They also showed that the type of task has a pronounced impact on how users engage with the SERP. Arguello et al. [4] observed that the more complex the task, the more users would interact with various components on the SERP whereas Thomas et al. [32] found that users tended to examine the result list deeper and more quickly when facing complex tasks. Jiang et al. [12] compared user interactions in relatively long search sessions (10 minutes; about 5 queries) for search tasks of four different types. Wu et al. [36] also observed differences in user interactions with the SERP based on whether they had to look for answers to a factoid question or a non-factoid question. In these studies, the SERPs were composed of web search results in the de facto list format. SERP Presentation and User Interactions Sushmita et al. [30] observed that positioning (top, middle, bottom) of different verticals on a SERP affects clickthrough rates of users when the verticals (news, image and video) were presented in a blended manner with the web search results. Arguello et al. [4] also looked into how task complexity affects user interactions and usage of aggregated vertical results when they are interleaved with web results, versus when they are presented as tabs. On a similar note, they observed that for more complex tasks, users clicked on more vertical results when they were interleaved with web search results. Bota et al. [7] conducted a crowdsourced online user study to investigate the effects of entity cards given ambiguous search topics. They found that the presence of entity cards has a strong effect on both the way users interact with search results and their perceived task workload. Furthermore, Levi et al. [16] performed a comprehensive analysis of the presentation of results from seven different verticals (including a community question answering vertical) based on the logs of a commercial web search engine. They observed that the community question answering vertical receives on average the highest number of clicks compared to other verticals. Wu et al. [36] studied how the presence of answer modules on SERPs affect user behaviour and whether that varies with question types (factoid vs. non-factoid). They found that the answer module helps users complete search tasks more quickly, and reduces user effort. In the presence of answer modules, users' clicks on web search results were significantly reduced while answering factoid questions. Shao et al. [28] conducted a user study to understand how user interaction is affected by the presence of results in the right rail of a heterogeneous SERP in addition to the traditional web results in the left-rail. They found that users have more interactions with the SERPs, appear to struggle more, and feel less satisfied if they examine the right-rail results. Overall, findings observed that the presence of verticals and other heterogeneous modalities of results and their position on the SERP affect user interactions. In these studies, results were also presented in the de facto list format. Kammerer and Gerjets [14] observed that when web search results are presented in a grid layout, the impact of search result positioning on selecting trustworthy sources is drastically reduced in comparison to the more traditional list approach. Users typically follow a top-down approach when scanning lists, and are more susceptible to select untrustworthy sources if they appear high up in the list. This effect is reduced for a grid-based presentation. However, the authors do not compare different types of tasks, nor do they explore user behaviours when results from various vertical features of the search engines are present on the SERP. Siu and Chaparro [29] compared the eye-tracking data of grid and list SERP layouts with two types of tasks (informational vs. navigational), and investigated potential differences in gaze patterns. The 'F-shaped' pattern was less prominent on the grid in comparison to the list layout. These two studies explore how user interactions change when web results are presented in a grid vs. a list layout. They do not, however, include vertical results in their study. METHODOLOGY To address our four overarching research questions as outlined in §1, we conducted a user study with = 41 participants. Each participant was assigned to one of four experimental search interface conditions (interfaces: between-subjects), and completed 12 search tasks (tasks: within-subjects). Our four experimental search interface conditions considered the layout type (list-vs. grid-based layout) and the verticals present on the SERP (heterogeneous content vs. homogeneous content). These combinations result in the interface conditions outlined below, with examples of the two layout types presented in Figure 1, with further details provided in §3.1. SL Simple List Considered as our baseline interface condition (the standard and widely used ten blue links [11]), this interface presents results in a list, with each result presented one under the other. All results are web results, and as such are homogeneous in terms of presented content. SG Simple Grid The same homogeneous approach to content is taken as for SL, but with results presented in a grid-based approach. Instead of scrolling along the vertical, participants subjected to this interface scroll along the horizontal. HL Heterogeneous List Similar in approach to SL, HL presents results in a list. However, different verticals are mixed in with the standard web results. Beyond web results, heterogeneous content used in this study includes image and video results. HG Heterogeneous Grid Similar to HL but now the content is displayed in grid form, with web-based results appearing in a grid, before additional image and video content. Search Interface Design and System Given the above, our goal is to find out to what extent the observations from prior studies by both Arguello et al. [4] and Siu and Chaparro [29] are valid after almost a decade of SERP design evolutions (and additions). To operationalise our four experimental interface conditions, we first needed to create a SERP template design that closely mirrors the design of a contemporary web search engine. For this study, we selected the Google SERP as it presents information recognisably, and commands approximately 92% market share. 1 A replica template was created with particular attention paid to the colour schemes, fonts, width and height of componentsas well as the spacing between them. The end result was a highly realistic template of a contemporary SERP, on which we based all of our study's results pages. 2 Figure 1 presents the SERP template used, presenting results for the query 'how do dams generate electricity?'. Present are examples for both list- (Figure 1(a)) and grid-based (Figure 1 SERP Template Overview (b)) interfaces. A query box is provided 1 . However, this is disabled and provides no functionality for this study. It does however display the query terms that were used to derive the presented results a priori (see §3.2). This is presented next to the information need for a given task 2 , alongside which there is a button to take the participant to the next stage of the experiment. The SERP template also provides links to additional results pages, namely Images and Videos 3 , emulating the setup of the study by Arguello et al. [4]. As shown at 4 , a grid-based layout is shown for both image and video pages, as is the norm in commercial web search engines such as Google and Bing. For images, a total of 16 were displayed (in a 4 × 4 grid); for videos, a total of nine were shown (in a 3 × 3 grid). On the SERP, standard web results are presented 5 , with the de facto 10 results per page (RPP) provided. For list-based interfaces 5 (SL and HL), web results are displayed in the standard way, with one result following the other down the left rail of the SERP. Gridbased interfaces 5 (SG and HG) present the results in a carousel user interface component (emulating the setup of Siu and Chaparro [29]), where the 10 results are arranged in the form of a 5 × 2 grid. A total of six (3 × 2) results were visible above-the-(vertical)-fold; access to the remaining four results (2 × 2) was made available through use of a button to scroll across. As denoted by the red dashed boxes in Figure 1 6 , heterogeneous content is also added to the SERP template. Present only in experimental search interface conditions HL and HG, these components provided inline image and video results to the participants. Like web results in the grid-based interfaces, these were also scrollable, mimicking the behaviour of contemporary web search engine SERPs. Sufficient content was placed within these components to ensure two complete scrolls could be completed; the number of images displayed varied as their widths were variable. On interface condition SG, image and video components were placed under the third web result; on interface condition HG, they were placed directly underneath the web results grid. The SERP template was also fully interactive-participants could click on links of web results 7 , with a new browser tab then opening to present the page at the linked URL. In addition, images and videos within the SERP could also be interacted with. Clicking on an image 8 took the participant to the webpage containing the image (again, in a new tab). Videos, all sourced from YouTube, could be played on the SERP itself 9 , with the necessary infrastructure in place to enable such functionality. If the participant wished to view the video on the YouTube website itself, they could click the link underneath 10 to do so. Again, YouTube links opened a new tab in the participant's browser. SERP Definition Note that for each query, there are three unique results pages to replicate the study of Arguello et al. [4]. These are the results 'landing', or All page, containing the web results (and additional components, for interface conditions HL and HG)-as shown in Figure 1, as well as the Images and Videos pages. From hereon in, we refer to a SERP as the 'landing' page, containing web results. To replicate the study of Siu and Chaparro [29], the 'landing' page itself is sufficient. Capturing Interactions and Experiences Integrated with the SERP template was LogUI [19], a framework-agnostic JavaScript library for capturing different interactions and other events within a web-based environment. LogUI was configured to capture a series of mouse events (including hovers, clicks, and scrolls) over the various components of the SERP. Interactions on components included (but were not limited to) web results, images and video contents (including the capturing of the playback, pausing, and completion of YouTube videos). We also recorded interactions to (and on) the supplementary Images and Videos pages. Browser-wide events All (as shown), Images, and Videos result pages. Heterogeneous content is displayed in red boxes, and is not present in the two homogeneous content interface conditions (SG and SL). Circled numbers correspond to the narrative of Section 3.1. were also captured, and included the ability for us to compute the time spent away from the SERP-when participants would click on a document/image/video link, which would open a new tab. Experience data was captured via a number of Qualtrics 3 surveys; a pre-and post-experiment survey were completed by each participant, in addition to the small post-task summary the participants had to write. More details on these questions and the flow of the experiment can be found in §3.4. Our setup ensured that participants would jump between the Qualtrics surveys and SERPs as and when required. Static SERPs As alluded to with the disabled query box 1 , our experimental setup featured no programmable backend or search functionality. This meant that there was no additional querying functionality. We served manually curated SERPs that we produced a priori for each of the 12 search tasks we asked participants to undertake. This setup ensured that all study participants viewed the same results (a setup also chosen in prior studies, such as those by Sushmita et al. [30] and Wu et al. [36]). While making the search experience somewhat less realistic, it did provide us with the benefit of not having to deal with participants submitting diverse queries. 3 https://www.qualtrics.com/ The design also removed a confounding variable and allowed us to address our four RQs by calculating the user interaction measures on a fixed set of web results, images, and videos. Search Tasks For this study, we used four different types of information need: a Navigational type (where individuals seek to find particular websites), and three different informational categories, belonging to the Remember (involving the retrieval, recognition, and recalling of relevant knowledge), Understand (constructing meaning from information sources), and Analyse (involving the breakdown of information info constituent parts, and determining how they related to one another) categories. For each category, we produced three unique information needs. This led to a total of 12 information needs which are listed in Table 1. Particular attention was paid to designing tasks that enticed participants to not only look at web results but also at image and video search results as well. The choice of our information needs is based on the study designs used by both Arguello et al. [4] and Siu and Chaparro [29]. More specifically, Arguello et al. [4] designed a series of tasks that required different levels of diversity of information to complete-as well as different amounts of search effort. Tasks were grounded in the revised Bloom taxonomy, as outlined by Anderson and Krathwohl [1]. 4 Search tasks were informational and belong to the Remember, Understand, and Analyse categories. In addition, Siu and Chaparro [29] employed just two categories of tasks for their studynavigational and informational. Our design thus combines the setups of both prior works that we wish to examine. Query Selection and SERP Curation To ensure that participants received helpful search results, we required a common search query for each. To this end, we ran a small crowdsourced pilot study on the Prolific platform 5 . This pilot had = 25 workers, with the design largely inspired by the study reported by Bailey et al. [6]. Workers took approximately 10 minutes to complete the task and were paid at the hourly rate of GBP8.00 for their time. All 12 information needs were presented to the workers. They were instructed to type the query terms that they would issue to their web search engine of choice if they were seeking information to address the information need. Collected queries were then normalised (case normalisation, stripped punctuation, whitespace cleanup) and passed through the Bing Spell Check API to generate a final canonical form of each submitted query. Subsequently, we determined the most frequently occurring query variation for the 12 tasks, taking this query forward as the one to use for the next stage of our study. These are listed in the parentheses in Table 1. 6 Curating SERPs We then used a combination of the Bing Web Search API, Bing Image Search API, and Bing Video Search API to curate a collection of: web (title, snippet text, and target URL); image (source image and document URL); and video (video source URL) results for each of the 12 queries. Snippet text was truncated to the equivalent of two sentences/lines, as this has been previously shown to be a good trade-off in terms of providing a sufficient information scent and encouraging interaction (i.e., clicks) [18]. Video links were filtered to YouTube only, as utilising only one video content provider reduced complexity for playback on our SERPs. Any URLs that proved non-functional or redirected to a 404 page were also removed. The content was then placed on our SERP templates, allowing us to construct SERPs, an image results page, and a video results page for each query. SERP variations for all four search interface conditions were produced. Experimental Procedure The 12 search tasks undertaken by each participant were preceded and followed by pre-and post-experiment surveys. We first performed screen and browser viewport resolution checks, requiring that all participants use a maximised browser window with a resolution of 1920 × 1080 or greater. This ensured that we could guarantee the SERPs displayed to the participants could be viewed without scrollbars along the horizontal. If the checks were successful, participants began the experiment by providing basic demographic information and were also asked minor questions on their search engine usage, specifically on what components on a contemporary SERP they often make use of. In addition, we asked what their preferred search engine is. They were then randomly assigned to one of the four search interface conditions (SL, SG, HL, or HG). Participants were primed to summarise their findings after each search task (in no more than 50 words). Upon acceptance of this instruction, the first search task began, with a SERP similar to the one presented in Figure 1(a) or Figure 1(b). With the selected query 1 and information need 2 present, participants then began to examine the content. Participants were not given a minimum or maximum amount of time to search. We reiterate that they were also not given the opportunity to issue their queries. Once they were satisfied with what they had found, they clicked the Answer button at the top of the SERP 2 , and entered their summary. Once complete, the next task began. This process was repeated for the remaining 11 tasks which were displayed to them in random order. Other researchers have also employed randomisation for condition allocation to minimise topic ordering effects [15,36]. After the search tasks had been completed, participants then moved on to the post-experiment survey. We used the sub-scales from O'Brien's Engagement Scale [22,23] as was done by Arguello et al. [4]. These are aimed at eliciting their evaluation of the interface they used on the following aspects of engagement: focused attention; perceived usability; experience; aesthetics; and felt involvement. The engagement scale was originally designed to evaluate shopping websites, and hence we modified/removed the statements pertaining to shopping to suit our needs. For example, we changed the original statement (belonging to aesthetics sub-scale) "This shopping website was aesthetically appealing" to "The layout of the results page is aesthetically appealing". For all statements in the sub-scales, participants indicated their level of agreement (1=strongly agree; 5=strongly agree). We also used the search effectiveness sub-scale used by Arguello et al. [4] to evaluate how effective the interfaces were in helping participants find information. In total, we used 26 statements from the six sub-scales to elicit user evaluation of the search interfaces. The reliability scores (Cronbach's Alpha) for the sub-scales are reported in Table 3. They were also asked to rate the perceived usefulness of web, image and video results. Study Participants Like our pilot, we recruited participants from the Prolific platform. Our = 41 participants were native English speakers from the United Kingdom, with a 95% approval rate on the platform, and had a minimum of 250 prior successful task submissions. From our participants, 32.5% identified as female, and 67.5% as male. The mean age of our participants was 36.5 ± 9.7, with a minimum age of 22 and a maximum of 68. 92% of participants listed Google as their preferred search engine, with the remaining 8% identified as DuckDuckGo users. With respect to the highest completed education level, 51.2% possess a Bachelors (or equivalent), 24.4% have a Masters (or equivalent), 19.5% have a high school degree, and 4.9% have an Associate (or equivalent). 95% of participants cited using web results on a contemporary SERP, 78% made use of image results, with 37% citing that they used video results. In our random assignment, 11 participants were assigned to HG, with ten participants each assigned to HL, SL, and SG. The experiment lasted on average 40 minutes for the 41 participants. Like our pilot participants, they were compensated at the rate of Table 1: Overview of information needs and their type. The rightmost column shows the most popular query obtained from our query selection pilot study, outlined in Section 3.3. Numbers in parentheses indicate how many crowdworkers ( = 25) submitted the most popular query variation. Here, Nav.=Navigation, Remem.=Remember, and Underst.=Understand. Type Information Need Most Popular Query Variation Nav. You want to find the homepage of Andrew Zimmern, the chef. andrew zimmern chef (14) You want to find the page of Air Jordan on the Nike website. nike air jordan (14) You want to find the page displaying the Flixbus route map in Europe. flixbus europe route map (6) Remem. You want to know where is the pituitary gland located in the body. where is the pituitary gland (9) You want to find out what clothes the famous cartoon character Mickey Mouse typically wears. what clothes does mickey mouse wear (5) You want to find out how to calculate the volume of an ellipsoid. ellipsoid volume formula (4) Underst. You want to find out the steps required to make a paper airplane. how to make a paper airplane (10) You want to briefly explain how dams generate electricity. how do dams generate electricity (17) You want to find out how to prevent shower mirrors from fogging. stop shower mirror fogging (3) Analyse You want to get into martial arts, but you have no fighting experience. Which form of martial arts is more suitable for beginners? best martial arts for beginners (3) You want to find out the main things to look for while installing solar panels on the roof of a house. things to consider before installing rooftop solar panels (1) You want to buy a new camera lens for taking professional pictures of your friend. Which camera lenses are best for portrait photography? best camera lenses for portrait photography (3) Table 2: Results of a factorial mixed ANOVA, where interface is between-subjects, and task is within-subjects variable. A ✓ indicates significant effect ( < 0.05) on the particular user interaction and ✗ indicates no significant effect. GBP 8.00 per hour. All participants who registered completed the study; post-hoc checks confirmed that they had provided sensible answers for each task, and as such we approved all who took part for payment. As such, our base analyses are reported over 41 * 12 = 492 search sessions and their corresponding interaction logs. RESULTS AND DISCUSSION For our analyses 7 , we conduct a series of mixed factorial ANOVA tests to observe if task complexity, SERP types or the interplay between them have a significant effect on interactions. We follow up the ANOVA with post-hoc -tests with Bonferroni correction ( < 0.05) to observe where significant differences occur. We evaluate if observations O4-O7 also hold in grid SERPs, HG and SG. Table 2 presents results that are relevant to our first three research questions. Here, a ✓ indicates a significant effect ( < 0.05) on the particular user interaction, and a ✗ indicates no significant effect. As seen in Tables 2 and 3, different SERP types do not have a significant effect on user interactions except for: (i) the number of web results clicked (row I, Table 3); and (ii) the number of hovers on videos present in the video results page (XV, Table 3). Post-hoc 7 All data and code pertaining to our analyses are available here. tests reveal that participants in the HL condition have significantly more web result clicks than their HG and SG counterparts (I, Table 3). HL participants also have longer web result reading times compared to participants in any of the other SERP conditions (II , Table 3)-albeit not significant. Furthermore, participants with the list interfaces (HL and SL) have a greater number of hovers over web results compared to HG and SG. As a result, we cannot confirm O1 where Siu and Chaparro [29] found significantly more fixation counts on the grid interface than on the list interface. We note that, since we did not record eye gaze data, we are approximating fixation counts by user interactions such as web result clicks and snippet text hovers, as mouse position has been shown to correlate with gaze positions in prior studies [21,24,25]. One of the possible reasons for the difference in observation with O1 can be that our participants are more familiar with the standard list layout of web results, as a majority use Google as their main search engine. RQ1: SERP Type and User Interactions Arguello et al. [4] do not compare user interactions with web results on heterogeneous SERPs vs. simple SERPs. However, we observe that participants using the simple SERP interfaces (SG and SL) scan web search results to lower depths than those of their heterogeneous interface counterparts (IV, Table 3). The lack of information (i.e., fewer verticals) on the SERP requires participants to scan web results to a greater depth in the ranked list. Table 3 (VII-X), we find that on average HL participants interact more with image and video results that are present on the SERP compared to their HG counterparts. SG and SL participants interact more with vertical results present in the image and video results page than those of their heterogeneous counterparts (XI-XIV, Table 3). Post-hoc tests also reveal that SL participants have significantly more hovers on video results on the video results page than participants in the other SERP conditions (XIV, Table 3). The lack of vertical results on the SERP makes the participants interact with them in the respective vertical results pages which shows that our informational needs indeed require participants to seek out image and video search results as well. HG and HL participants seem to be satisfied with vertical results present on the SERP and the former barely interacted with vertical results present in the respective results pages (XI-XIV, Table 3). Looking at overall interactions with vertical results (adding interactions with vertical results present on the SERP and the vertical results pages for HG and HL), we see that HG and HL have slightly more interactions than SG and SL respectively. This difference is not significant, but we do see a trend in the line of O5 where Arguello et al. [4] observed a higher number of vertical result clicks when they were blended with the web results in the SERP. This is compared to when they were only present on the respective vertical results page. On a side note, the higher interactions with vertical results present on the SERP by HL participants compared to HG participants (VII-X, Table 3, also depicted in Figure 2(c)) can be attributed to the fact that images and videos in HL SERPs appear in the middle of the web results (between rank 3 and 4) whereas they appear below the web results in HG SERPs. Participants in the latter interface expend comparatively more effort to access the vertical results, thereby reducing their interaction. We leave further analysis on the effect of positioning of vertical results on user interaction for future work. Based on Addressing RQ1, we found that the interface has a significant main effect on the clicks on web results and hovers on videos on the video results-page but not on other user interactions. Table 4 shows that the informatioon needs of the Analyse type, which are the most complex among our information needs, warrant most web result clicks (I), web result dwell time (II) and session duration from participants (III). Table 2 shows that the main effect of task complexity on these interactions is significant. Participants reach greater web result click depth (IV, Table 4) for Analyse tasks, albeit not significant. Post-hoc tests reveal that (i) Analyse tasks receive significantly more web result clicks than Remember tasks; (ii) Analyse and Understand tasks lead to significantly higher web result dwell times than Navigation tasks; and (iii) the session duration for Analyse and Understand tasks are significantly higher than for Navigational tasks, while the session duration for Analyse tasks is also significantly greater than that for Remember tasks. Overall, we find that user interactions on web search results increase as the complexity of information needs increase which is inline with the observations of Arguello et al. [4] and we can partially confirm O4. Arguello et al. [4] did not include Navigational tasks in their experiments. We argue that they can be considered as tasks requiring the lowest level of cognition, and as such follow the trend of O4-they receive the least interaction among all task categories. The only exception to this was web result clicks-the nature of the task requires participants to click web result links to ascertain that they found the correct page. RQ2: Task Complexity and User Interactions We approximate fixation duration in O3 by observing hover duration over the web results, akin to fixation count in §4.1. Although participants hover longer over web results (V, Table 4) and snippet text (VI) for Remember tasks compared to other tasks, the difference across tasks is not significant. Moreover, the mean hover duration on web results (snippet and title) for participants belonging to the grid SERP types (HG and SG) is longer than for those belonging to the list SERP types (VI, Table 3). As seen from Table 2, the interplay between SERP type and task complexity do not have a significant effect on hover duration over web results. As a result, we can only partially confirm O3 where Siu and Chaparro [29] also do not find significant differences in fixation duration for grid layout for the tasks but they did find significantly longer fixation duration on the list layout for more complex tasks. Among interactions with vertical results present on the SERP (VII-X, Table 4), we observe that Remember tasks receive the most interactions on average. Post-hoc tests reveal that: (i) participants click significantly more on images present on the SERP (VIII, Table 4) for Remember and Navigational tasks compared to Analyse; and (ii) they hover significantly more on images present on the SERP (X, Table 4) for Remember tasks compared to all other task categories. For images present on image results page, we again observe significantly more image clicks (XII, Table 4) and hovers (XIV, Table 4) for Remember tasks compared to the more complex Understand or Analyse tasks. Findings regarding user interactions with vertical results (present on the SERP and the vertical results pages) and their relationship with task complexity is contrary to the observations of Arguello et al. [4], and hence we cannot confirm O6. The high interaction with vertical results for Remember tasks together with the fact that participants hover over web results and snippet text longer (on average) for the same task (V & VI, Table 4) shows that participants prefer to address information needs of the Remember type by either hovering over web results and interacting with verticals rather than clicking the link. Arguello et al. [4] do not observe hover duration in their analysis. To answer RQ2, we find that task complexity does have a significant effect on several user interactions. With participants interacting more with web results as tasks get more complex, we observe significantly more interactions with image results for Remember tasks compared to more complex Analyse/Understand tasks. RQ3: Task Complexity, SERP Type and User Interactions As seen in Table 2, we do not observe a significant effect of the interplay between SERP types and task complexity on user interaction with web results or verticals which is similar to what Arguello et al. [4] found. Hence, we can confirm O7. From Figure 2(a), we observe that participants across all SERP types click the most web results for Analyse tasks (in line with O4). For each task type, HG participants click the least number of web results and for most tasks participants with grid SERPs click lower ranked web results than those with list SERPs (in contrary to O1). Approximating fixation count by web result clicks, as done in §4.2, we see for each SERP type, the complex Analyse tasks receive more interaction than the less complex Remember or Understand tasks. Although pairwise comparisons do not show a significant difference in web result clicks between different tasks for each SERP, we observe a trend similar to O2-more complex tasks requiring higher document clicks. From Figure 2(b), we see that participants across all SERP types take longest to finish Analyse tasks and least amount of time to finish Navigational tasks (also in line with O4). Finally, Figure 2(c) corroborates our findings from §4.2, as we see that participants across both SERP types interact most with image results for Remember tasks compared to other tasks. As mentioned earlier, this observation is contrary to what Arguello et al. [4] observed in their study (O6). In Figure 3, we plot the distribution of where (which rank) participants made their first click of web results for Navigation and Analyse tasks. The HL SERP is the only one where the web results are "broken" by vertical results at rank three, and as a result, we observe that most of the first clicks for both tasks appear before rank four (subplot (b) of Figure 3). For the other SERPs, the first click distribution for the tasks is more uniform. This is especially prominent for Analyse tasks, where we see due to the absence of verticals on SL SERP (comparing Analyse HL and Analyse SL in Figure 3) participants are willing to go further down the list before their first click. We also expect a peak around the first result for Navigation tasks, which is true for all SERP types except Navigation HG in subplot (a). Either the participants using that SERP type prefer to not click a lot as is evident from Table 3 (fewest web result clicks by HG participants), or they chose to explore more before their first click. It has been observed in earlier works [13,14] that participants have a trust bias for list SERPs (they click on web results appearing higher up the ranked order). The trust bias had been found previously to be less prevalent in grid SERPs [14]. We also find evidence of similar user interaction in subplot (b) compared to subplot (a) where participants are more open to exploration before their first click. To conclude, we find for RQ3, that the interplay between SERP types and task complexity does not have a significant effect on user interactions. RQ4: Perceived Experience of SERPs Turning our attention to the post-experiment surveys, we observe little difference in participant ratings of the systems (XVIII-XXIII, Table 3). This is in line with both Arguello et al. [4] and Siu and Chaparro [29], who also did not find significant differences in user ratings for different interfaces. Therefore, we can confirm O8. We also observe that web search results on average are perceived to be more useful (XVII, Table 3) than image or video results (XV-XVI, Table 3). This is in line with the click behaviour of participants. Across all SERP types, they clicked on more web results than they did on images or videos. Arguello et al. [4] also found the overall number of vertical clicks to be lower than that on web results. Image results were perceived to be more useful by SL participants followed by their HL counterparts (XV, Table 3), which is reflected in their behaviour as well. While the former has the most interactions with images present on the image results page (XI-XIII, Table 3) compared to participants in other cohorts, the latter interacted most with images present on the SERP (VII-IX, Table 3). CONCLUSION Summary In this work, we set out to answer the question of how four different types of SERP and four different types of tasks of varying levels of complexity affect user interaction with web, image and video results. We also explore whether observations about users and their interactions from the studies of Arguello et al. [4] and Siu and Chaparro [29] hold with contemporary SERPs. We observed the following findings with respect to our research questions. RQ1 The SERP has a significant main effect on the number of clicks on web results and the number of hovers on videos on the video results page, but not on other user interactions. RQ2 Task complexity has a significant effect on user interactions. While participants interact more with web results as the task becomes more complex, we observe significantly more interactions with image results for Remember tasks compared to the more complex Analyse or Understand tasks. RQ3 The interplay between SERP types and task complexity does not have a significant effect on user interactions. RQ4 There is little difference in the evaluation of the four SERP types by participants. Out of eight observations, we found evidence to confirm two (O7, O8), with partial evidence for a further four (O2, O3, O4, O5). These findings indicate that the user interactions over different interfaces for solving tasks of varying complexity have remained mostly similar over time. However, we employed different information needs-and recruited different participants-from the prior studies. Nevertheless, the evidence contrary to O1 and O6 has interesting implications-introducing SERPs that users are not familiar with might result in a decrease in interaction. Although the grid layout can present search results in a condensed format (displaying more items in a given screen space compared to the list layout), users might still end up exploring more in the familiar list layout. Additionally, interactions with vertical results are not only dependent on the complexity of the tasks, but also the type of information need. As we observed, certain simpler tasks might warrant more interaction with vertical results than more complex tasks [30]. Reproducing IIR studies Several variables exist that might affect the observations of an IIR study. An unexhaustive list includes the selection of users, interfaces, and task types. Although both Arguello et al. [4] and Siu and Chaparro [29] described how their respective interfaces looked, they did not point to any resources which would help us replicate them. Moreover, we believe that the more users become familiar with a particular interface, the more important it is to present a similar interface to them during a study examining their behaviours. As mentioned in Section 3.1, we have created templates of SERPs that resemble google.com and you.com, and released them for further use. We believe our templates will be useful for the community to eliminate confounding variables in IIR studies that might arise due to SERP presentation. Secondly, Arguello et al. [4] and Siu and Chaparro [29] did not mention the entire set of tasks used in their studies. As a result, we came up with our tasks of different complexity, as presented in Table 1. Two studies by Urgo et al. [33,34] both list examples of tasks pertaining to different complexities which also offer useful resources for future IIR studies. Our tasks differ with respect to the fact that we designed tasks that specifically enticed participants to not only look at web results, but also to image and video search results as well. It is important to have a fixed set of tasks and similar interfaces to reproduce and enable reliable comparison of observations (e.g., the number of queries, documents opened, etc.) with prior IIR studies. Lastly, in most cases, it will not be possible to have the same participants while reproducing IIR studies. Crowdsourcing provides a solution for capturing user interactions as it has been shown that there is little difference in the quality between crowdsourced and lab-based studies [39]. Power analysis can be used to determine the number of participants required given the experimental conditions of a particular study. It also might be useful to release experimental logs from these studies, after careful ethical checks and considerations. This will permit future researchers to examine them closely, and use them to develop, for example, models of user interaction and search behaviour. Limitations and Future Work There are several areas with scope for future refinement. First, although we tried to select information needs that cover a broad range of topics, we cannot be certain that the results generalise to information needs with other characteristics. Second, we did not provide querying functionality to users-and hence it will be worthwhile to explore if that has an overall effect on user interactions. Thirdly, the positions of vertical results on the main page of the SERP were fixed, and we know from previous work [28,30] that user interactions with verticals is affected by where they are displayed on the SERP. In the future, we aim to investigate varying the position of verticals on list and grid interfaces, and their effect on user interaction. The findings from this study can be further applied to designing and evaluating SERP presentations and the placement of heterogeneous content. Understanding and modelling user interactions will also help us work on methodologies for interface optimisation [35] and SERP evaluation, along the same veins of prior studies [5,9,26,31,38]. Figure 1 : 1Examples of both the (a) list-based and (b) grid-based interfaces trialled. Note the inclusion of links for the separate clicks ✓ (F = 4.27, = 0.01) ✓ (F = 4.18, = 0.01) ✗ Mean web result reading time (s) ✗ ✓ (F = 3.97, = 0.01) ✗ Mean session duration (s) ✗ ✓ (F = 12.72, < 0.0001) ✗ Mean web result hover duration (s) ✗ ✗ ✗ Image clicks (SERP) ✗ ✓ (F = 7.24, = 0.004) ✗ Video clicks (SERP) ✗ ✗ ✗ Image hovers (SERP) ✗ ✓ (F = 6.98, = 0.009) ✗ Video hovers (SERP) ✗ ✗ ✗ Image clicks (image results page) ✗ ✓ (F = 4.66, = 0.01) ✗ Video clicks (video results page) ✗ ✗ ✗ Image hovers (image results page) ✗ ✓ (F = 5.39, = 0.01) ✗ Video hovers (video results page) ✓ (F = 3.36, = 0.02) ✗ ✗ Figure 2 : 2Interaction plots, showing effects of SERP types and task complexity over: (a) clicking on web results; (b) the mean session duration (in seconds); and (c) clicks on images presented on the SERP. 8 subplot (b) of Figure 3 : 3Distribution of ranks of the first clicked web results for participants over both grid-based interfaces SL and HL (a), and list-based interfaces SG and HG (b). Table 3 : 3User interactions for different interfaces across all tasks. † indicates that there is a significant main effect of SERP layout on that particular user interaction. H G , H L , S G , S L indicate significant difference with HG, HL, SG and SL respectively. Maximum values for each interaction is highlighted in bold.Interface Condition Table 4 : 4User interactions for different task complexity across all search interfaces. † indicates that there is a significant main effect of task complexity on that particular user interaction. N , R , U , A indicate significant difference with navigational, remember, understand and analyse tasks. Maximum values for each interaction is highlighted in bold.Row Interactions Navigational Remember Understand Analyze I Web result clicks † 4.00(±2.65) 3.76(±3.25) A 4.12(±2.55) 5.34(±4.07) R II Mean web result reading time (s) † 14.99(±11.62) A,U 20.57(±21.18) 24.05(±22.64) N 26.91(±29.18) N III Mean session duration (s) † 69.95(±52.66) A,U 97.11(±76.00) A 106.90(±62.22) N 134.75(±73.98) N,R IV Maximum web result click depth 3.32(±2.59) 3.88(±2.27) 3.85(±2.37) 4.27(±2.77) V Mean web result hover duration (s) 0.86(±1.98) 3.26(±14.43) 2.05(±8.28) 2.31(±9.89) VI Mean snip. text hover duration (s) 0.08(±0.22) 0.12(±0.24) 0.07(±0.11) 0.08(±0.15) VII Image clicks (SERP) † 0.17(±0.38) A 0.44(±1.00) A 0.12(±0.40) 0.00(±0.00) N,R VIII Video clicks (SERP) 0.00(±0.00) 1.15(±5.59) 0.12(±0.64) 1.88(±11.40) IX Image hovers (SERP) † 1.10(±2.45) R 4.83(±10.20) N,U,A 1.07(±2.53) R 0.54(±1.98) R X Video hovers (SERP) 0.88(±2.61) 4.49(±15.71) 1.90(±5.51) 1.29(±4.47) XI Image clicks (image results page) † 0.07(±0.35) 0.32(±0.65) U,A 0.12(±0.40) R 0.02(±0.16) R XII Video clicks (video results page) 0.00(±0.00) 0.00(±0.00) 0.15(±0.69) 0.02(±0.16) XIII Image hovers (image results page) † 1.15(±4.11) 4.37(±10.92) A 0.73(±2.55) 0.07(±0.35) R XIV Video hovers (video results page) 0.02(±0.16) 0.00(±0.00) 0.56(±2.21) 0.12(±0.64) https://gs.statcounter.com/search-engine-market-share. 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[ "On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models", "On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models" ]	[ "Sven Gowal [email protected] \nUniversity of Oxford\n\n", "Krishnamurthy ( Dj \nUniversity of Oxford\n\n", ") Dvijotham \nUniversity of Oxford\n\n", "Robert Stanforth [email protected] \nUniversity of Oxford\n\n", "Rudy Bunel \nUniversity of Oxford\n\n", "Chongli Qin Deepmind [email protected] \nUniversity of Oxford\n\n", "Jonathan Uesato Deepmind \nUniversity of Oxford\n\n", "Relja Arandjelović Deepmind \nUniversity of Oxford\n\n", "Timothy Mann Deepmind [email protected] \nUniversity of Oxford\n\n", "Pushmeet Kohli Deepmind [email protected] \nUniversity of Oxford\n\n" ]	[ "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n", "University of Oxford\n" ]	[]	Recent works have shown that it is possible to train models that are verifiably robust to norm-bounded adversarial perturbations. While these recent methods show promise, they remain hard to scale and difficult to tune. This paper investigates how interval bound propagation (IBP) using simple interval arithmetic can be exploited to train verifiably robust neural networks that are surprisingly effective. While IBP itself has been studied in prior work, our contribution is in showing that, with an appropriate loss and careful tuning of hyper-parameters, verified training with IBP leads to a fast and stable learning algorithm. We compare our approach with recent techniques, and train classifiers that improve on the state-of-the-art in single-model adversarial robustness: we reduce the verified error rate from 3.67% to 2.23% on MNIST (with ∞ perturbations of = 0.1), from 19.32% to 8.05% on MNIST (at = 0.3), and from 78.22% to 72.91% on CIFAR-10 (at = 8/255).	null	[ "https://arxiv.org/pdf/1810.12715v2.pdf" ]	53,112,003	1810.12715	246c1f44332cb28b28f5d847236bdfbd4ba26bae	On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models Sven Gowal [email protected] University of Oxford Krishnamurthy ( Dj University of Oxford ) Dvijotham University of Oxford Robert Stanforth [email protected] University of Oxford Rudy Bunel University of Oxford Chongli Qin Deepmind [email protected] University of Oxford Jonathan Uesato Deepmind University of Oxford Relja Arandjelović Deepmind University of Oxford Timothy Mann Deepmind [email protected] University of Oxford Pushmeet Kohli Deepmind [email protected] University of Oxford On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models Recent works have shown that it is possible to train models that are verifiably robust to norm-bounded adversarial perturbations. While these recent methods show promise, they remain hard to scale and difficult to tune. This paper investigates how interval bound propagation (IBP) using simple interval arithmetic can be exploited to train verifiably robust neural networks that are surprisingly effective. While IBP itself has been studied in prior work, our contribution is in showing that, with an appropriate loss and careful tuning of hyper-parameters, verified training with IBP leads to a fast and stable learning algorithm. We compare our approach with recent techniques, and train classifiers that improve on the state-of-the-art in single-model adversarial robustness: we reduce the verified error rate from 3.67% to 2.23% on MNIST (with ∞ perturbations of = 0.1), from 19.32% to 8.05% on MNIST (at = 0.3), and from 78.22% to 72.91% on CIFAR-10 (at = 8/255). Introduction Despite the successes of deep learning, it is well-known that neural networks are not robust. In particular, it has been shown that the addition of small but carefully chosen deviations to the input, called adversarial perturbations, can cause the neural network to make incorrect predictions with high confidence [2,3,10,13,16]. Robust optimization techniques, like the one developed by Madry et al. [14], overcome this problem by trying to find the worst case adversarial examples at each training step and adding it to the training data. While the resulting models show strong empirical evidence that they are robust against many attacks, we cannot yet guarantee that a stronger adversary (for example, one that does brute-force enumeration to compute adversarial perturbations) cannot find inputs that cause the model to predict incorrectly. In Appendix E, we provide an example that motivates why Projected Gradient Descent (PGD) -the technique at the core of Madry et al.'s method -is not always capable of finding the worst case attack. This has driven the need for formal verification: a provable guarantee that neural networks are consistent with a specification for all possible inputs to the network. Substantial progress has been made: from complete methods that use Satisfiability Modulo Theory (SMT) [4,8,11] or Mixed-Integer Programming (MIP) [1,5,17] to incomplete methods that rely on solving a convex relaxation of the verification problem [6,7,9,15,18,19,20]. Incomplete methods scale to larger models than complete methods and, as such, can be used inside the training loop to build models that are not only robust, but also intrinsically easier to verify [6,15,19,21]. In this paper, we study interval bound propagation (IBP), a simple algorithm for training verifiably robust classifiers. IBP allows to define a loss by computing an upper bound on the maximum difference between any pair of logits when the input can be perturbed within an ∞ norm-bounded ball. We show that this approach can achieve strong results, outperforming state-of-the art. The core approach behind IBP has been studied in previous papers -it is equivalent to the constant verifier used by Dvijotham et al. [6] and to the interval domain from Mirman et al. [15]. Beyond providing new baseline results, the contributions of this paper are as follows: • We propose several enhancements that improve the performance of IBP for verified training. In particular, we differentiate ourselves from [15] by using a different loss function, and by eliding the last linear layer of the neural network, thereby improving our estimate of the worst case logits. We explain our training methodology by detailing how key hyper-parameters are scheduled throughout training. • We compare our trained models to those from other approaches in terms of robustness to Projected Gradient Descent (PGD) attacks [3] and show that they are competitive against Madry et al. [14] and Wong et al. [20] across a wide range of ∞ perturbation radii (hereafter denoted by ). We also compare IBP to Wong Methodology Neural network. We focus on feed-forward neural networks trained for classification tasks. The input to the network is denoted x 0 and its output is a vector of raw un-normalized predictions (hereafter logits) corresponding to its beliefs about which class x 0 belongs to. During training, the network is fed pairs of input x 0 and correct output label y true , and trained to minimize a misclassification loss, such as cross-entropy. For clarity of presentation, we assume that the neural network is defined by a sequence of transformations h k for each of its K layers. That is, for an input z 0 (which we define formally in the next paragraph), we have z k = h k (z k−1 ) k = 1, . . . , K(1) Verification problem. We are interested in verifying that neural networks satisfy a specification by generating a proof that this specification holds. We consider specifications that require that for all inputs in some set X (x 0 ) around x 0 , the network output satisfies a linear relationship c T z K + d ≤ 0 ∀z 0 ∈ X (x 0 )(2) where c and d are a vector and a scalar that may depend on the nominal input x 0 and label y true . As shown by Dvijotham et al. [7], many useful verification problems fit this definition. In this paper, however, we focus on the robustness to adversarial perturbations within some ∞ norm-bounded ball around the nominal input x 0 . Adversarial robustness is achieved if no adversary can succeed in changing the classification outcome away from the true label y true , i.e., argmax z K = y true for all elements z 0 ∈ X (x 0 ). Formally, we want to verify that for each class y: (e y − e ytrue ) T z K ≤ 0 ∀z 0 ∈ X (x 0 ) = {x \| x − x 0 ∞ < }(3) where e i is the standard i th basis vector and is the perturbation radius. Ultimately, verifying a specification like (2) can be done by searching for a counter-example: maximize z0∈X (x0) c T z K + d subject to z k = h k (z k−1 ) k = 1, . . . , K(4) If the optimal value of the above optimization problem is smaller than 0, the specification (2) is satisfied. Interval bound propagation. IBP's goal is to find an upper bound to the optimization problem (4). In this context, the simplest approach is to bound the activation z k of each layer by an axis-aligned bounding box (i.e., z k ( ) ≤ z k ≤ z k ( )) using interval arithmetic. For ∞ adversarial perturbations of size , we have for each element z k,i of z k : z k,i ( ) = minimize z k−1 ( )≤z k−1 ≤z k−1 ( ) e T i h k (z k−1 ) and z k,i ( ) = maximize z k−1 ( )≤z k−1 ≤z k−1 ( ) e T i h k (z k−1 )(5) where z 0 ( ) = x 0 − 1 and z 0 ( ) = x 0 + 1. As shown in Appendix A, the above optimization problems can be solved quickly for linear and ReLU layers. Finally, the upper and lower bounds of the output logits z K can be used to construct an upper bound on the solution of (4): maximize z K ( )≤z K ≤z K ( ) c T z K + d(6) As shown in Appendix A, a tighter upper bound can be computed by eliding the final linear layer with the specification. Overall, the adversarial specification (3) is upper-bounded by z K,y ( ) − z K,ytrue ( ). It corresponds to an upper bound on the worst-case logit difference between the true class y true and any other class y. Loss. In the context of classification under adversarial perturbation, the above specification corresponds to having the logit corresponding to the true class equal to its lower bound and the other logits equal to their upper bound:ẑ K,y ( ) = z K,y ( ) if y = y true z K,ytrue ( ) otherwise(7) We can then formulate our training loss as L = κ (z K , y true ) Lfit +(1 − κ) (ẑ K ( ), y true ) Lspec(8) where is the supervised cross-entropy loss and κ is a hyperparameter that governs the relative weight of satisfying the specification (L spec ) versus fitting the data (L fit ). If = 0 then z K =ẑ K ( ), and thus (8) becomes equivalent to a standard classification loss. We note that L spec can also be used as a standard regularizer within other techniques (like adversarial training). Training procedure. To stabilize the training process and get a good trade-off between nominal and adversarial accuracy, we create a learning curriculum by scheduling the values of κ and when computing the loss (8). Starting with κ = 1 and slowly reducing it throughout training helps get more balanced models that reach higher nominal accuracy (under no perturbation). In practice, we found that using a final value of κ = 1/2 works well on MNIST and CIFAR-10. More importantly, starting with = 0 and slowly raising it up to a target perturbation radius train is necessary. We note that train does not need to be equal to the perturbation radius used during testing, as using higher values creates robust models that generalize better. For more details on the training procedure, see Appendix B. Results We demonstrate that IBP can train verifiable networks and compare its performance to the state-ofthe-art on MNIST and CIFAR-10. For both datasets, we compare IBP to three alternative approaches: the nominal method, which corresponds to standard non-robust training with cross-entropy loss; Table 1: Comparison with the state-of-the-art. Comparison of the nominal test error, empirical PGD error rate, and verified bound on the error rate, along with the best known results from the literature. The test error corresponds to the test set error rate when there is no adversarial perturbation. The empirical PGD error rate is calculated using 200 iterations of PGD and 10 random restarts. When computationally feasible, we calculate the verified adversarial error rate using an exact verifier that relies on solving a MIP system [1] using the commercial Gurobi Solver with a timeout of 10 minutes (dashes "-" indicate that we were unable to verify these networks beyond the trivial 100% error rate bound within the imparted time limit). * Results reported from the literature (these results may use different architectures and different verification procedures). On these rows a dash "-" indicates that the corresponding result in not available. Across the full spectrum, IBP acheives higher empirical accuracy (computed using PGD attacks) and higher provable accuracy (computed by an exact verifier). Table 1 provides punctual results (for more results see Table 3 in Appendix C Dataset Conclusion We have presented an approach for training verifiable models and provided baseline results for MNIST and CIFAR-10. Our experiments have shown that the proposed approach outperforms competing techniques in terms of verified bounds on adversarial error rates in image classification problems, while also training faster. In the future, we hope that these results can serve as a useful baseline. We believe that this is an important step towards the vision of specification-driven ML. A Algorithmic details Bound propagation for affine layers. As explained in Section 2, given bounds on the inputs z k−1 ≤ z k−1 ≤ z k−1 of layer k, it is useful to be able to efficiently compute bounds on the outputs z k ≤ h k (z k−1 ) ≤ z k . In other words, we are interested in solving the following optimization of all i: z k,i / z k,i = min / max z k−1 ≤z k−1 ≤z k−1 e T i h k (z k−1 )(9) For the affine layers (e.g., fully connected layers, convolutions) of the form h k (z k−1 ) = W z k−1 + b solving this optimization problem can be done efficiently with only two matrix multiplies: µ k−1 = z k−1 + z k−1 2 r k−1 = z k−1 − z k−1 2 µ k = W µ k−1 + b r k = \|W \|r k−1 z k = µ k − r k z k = µ k + r k(10) Bound propagation for element-wise monotonic activations. Propagating bounds through any element-wise monotonic activation function (e.g., ReLU, tanh, sigmoid) is trivial. Assuming an increasing function h k , we have: z k = h k (z k−1 ) z k = h k (z k−1 )(11) Notice how for element-wise non-linearities the (z k , z k ) formulation is better, while for affine transformations (µ k , r k ) is more efficient. Switching between parametrizations depending on h k incurs a slight computational overhead, but since affine layers are typically more computationally intensive , the formulation (10) is worth it. Elision of last layer. Bound propagation is not necessary for the last linear layer of the network. Indeed, we can find an upper bound to the solution of (4) that is tighter than proposed by (6). Assuming h K (z K−1 ) = W z K−1 + b, we have: maximize z K ≤z K ≤z K c T z K + d ≥ maximize z K−1 ≤z K−1 ≤z K−1 c T h K (z K−1 ) + d = maximize z K−1 ≤z K−1 ≤z K−1 c T W z K−1 + c T b + d = maximize z K−1 ≤z K−1 ≤z K−1 c T z K−1 + d(12) with c = W T c and d = c T b + d. B Network sizes and training parameters Architectures. The architectures of the three models used in this paper are presented in Table 2. The first two models (i.e., small and medium) are equivalent to the small and large models in Wong et al. [20] 5 . To the best of our knowledge, the large model is significantly larger (in terms of number of hidden units) than any other verified model presented in the literature. Table 2: Architecture of the three models used in this paper. All layers are followed by RELU activations. The last fully connected layer is omitted. "CONV k w×h+s" corresponds to a 2D convolutional layer with k filters of size w×h using a stride of s in both dimensions. "FC n" corresponds to a fully connected layer with n outputs. Training procedure. We train for 100 and 350 epochs with batch sizes of 100 and 50 on MNIST and CIFAR-10 respectively (using the train and valid set). Hence, the total number of training steps is 60K for MNIST and 350K for CIFAR-10. The networks were trained using the Adam [12] algorithm with an initial learning rate of 10 −3 . For MNIST, we decay the learning rate by 10× at steps 15K and 25K. For CIFAR-10, we decay the learning rate by 10× at steps 200K, 250K and 300K. To stabilize the training process and get a good trade-off between nominal and adversarial accuracy, we create a curriculum using κ and (see (8)): • κ controls the relative weight of satisfying the specification versus fitting the data. Hence, we found that starting with κ = 1 and slowly reducing it throughout training helps get more balanced models with higher nominal accuracy. In practice, we found that using a final value of κ = 1/2 works well on MNIST and CIFAR-10, although we have not particularly tuned this value. • More importantly, we found that starting with = 0 and slowly raising it up to a target perturbation radius train was necessary. We note that train does not need to be equal to the perturbation radius used during testing, using higher values creates robust models that generalize better. Overall, after a warm-up period of 2K and 10K steps for MNIST and CIFAR-10 respectively, we linearly anneal both hyperparameters for 10K and 150K steps. The networks trained using Wong et al. [20] were trained using the schedule and learning rate proposed by the authors. For Madry et al. [14], we used a learning rate schedule identical to IBP and, for the inner optimization, adversarial examples are generated by 7 steps of PGD with Adam [12] and a learning rate of 10 −1 . Note that our reported results for these two methods closely match or beat published results, giving us confidence that we performed a fair comparison. . This plot shows the median performance (along with the 25 th and 75 th percentiles across 10 independent training processes) and confirms that IBP is stable and produces consistent results. Additionally, for IBP, we clearly see the effect of ramping the value of up during training (which happens between steps 2K and 12K). Table 3 reports the verified robustness for a larger set of perturbation radii than Table 1 and also provides additional results from the literature. We always report results with respect to the complete test set of 10K images for both MNIST and CIFAR-10. The test error corresponds to the test set error rate when there is no adversarial perturbation. For self-trained models, the PGD error rate is calculated using 200 iterations of PGD and 10 random restarts. We calculate the verified adversarial error rate using an exact verifier that relies on solving a MIP system [1] using the commercial Gurobi Solver with a timeout of 10 minutes 6 . Upon timeout, we fallback to solving a relaxation of the verification problem with an LP [8] (using Gurobi again). When both approaches fail to provide a solution within the imparted time, we count the example as attackable. Thus, the verified error rate reported in this table may be over-estimating the exact verified error rate. All methods use the same model architectures (except results from the literature). For clarity, we do not report the results for all train values (i.e., train ∈ {0.1, 0.2, 0.3, 0.4} for MNIST and train ∈ {2/255, 8/255} for CIFAR-10). Figure 1a shows the effect of train in an easily digestible form. C Additional results Overall, IBP is not only competitive with the state-of-the-art under PGD attack, but also demonstrates better verified robustness across the board -except for CIFAR-10 with = 2/255 where it performs similarly to Xiao et al. [21] and is worse than Wong et al. [20]. This effect remains to be investigated in more details: from our experience, the method from Wong et al. provides tighter bounds when the perturbation radius is small, thus giving better feedback when training on CIFAR-10 at = 2/255. We also note that, although IBP is a subset of Dvijotham et al. [6], it performs on par (or even better), which highlights the difficulty of training robust model with more complicated methods. For completeness, Figure 3 shows the empirical accuracy against PGD attacks of varying intensity for all models. We observe that IBP tends to be slightly worse than Madry et al. for similar network sizes -except for the large model where Madry et al. is likely overfitting (as it performs worse than the medium-sized model). [21], the reported verified bound on the error rate is not computed with an exact solver and may be over-estimated. For this model, [21] only provides estimates computed from 1000 samples (rather than the full 10K images). * [15] only provides estimates computed from 500 samples (rather than the full 10K images). Additionally, they use a slightly smaller = 0.007 = 1. D Convolutional filters Figure 4 shows the first layer convolutional filters resulting from training a small robust model on MNIST against a perturbation radius of = 0.1. Overall, the filters tend to be extremely sparse -at least when compared to the filters obtained by training a nominal non-robust model (this observation is consistent with [20]). We can qualitatively observe that Wong et al. produces the sparsest set of filters. Similarly, as shown in Figure 5, robust models trained on CIFAR-10 exhibit high levels of sparsity in their convolutional filters. Madry et al. seems to produce more meaningful filters, but they remain sparse compared to the non-robust model. E When Projected Gradient Descent is not enough For a given example in MNIST, this section compares the worst-case attack found by PGD with the one found using a complete solver. The underlying model is a medium sized network trained using IBP with = 0.1. The nominal image, visible in Figure 6a, has the label "eight", and corresponds to the 1365 th image of the test set. The worst-case perturbation of size = 0.1 found using 200 PGD iterations and 10 random restarts is shown in Figure 6b. In this particular case, the robust network is still able to successfully classify the attack as an "eight". Without any verifiable proof, we could wrongly assume that our network is robust to ∞ perturbation on that image. However, when running a complete solver (using a MIP formulation), we are able to find a counter-example that successfully induces the model to misclassify the "eight" as a "two" (as shown in Figure 6c). Figure 7 shows the untargeted adversarial loss (optimized by PGD) around the nominal image. In these loss landscapes, we vary the input along a linear space defined by the worse perturbations found by PGD and the MIP solver. The u and v axes represent the magnitude of the perturbation added in each of these directions respectively and the z axis represents the loss. Typical cases where PGD is not optimal are often a combination of two factors that are qualitively visible in this figure: • We can observe that the MIP attack only exists in a corner of the projected ∞ -bounded ball around the nominal image. Indeed, since PGD is a gradient-based method, it relies on taking gradient steps of a given magnitude (that depends on the learning rate) at each iteration. That is, unless we allow the learning rate to decay to a sufficiently small value, the reprojection on the norm-bounded ball at each iteration will force the PGD solution to bounce between the edges of that ball without hitting its corner. • The second, more subtle, effect concerns the gradient direction. Figure 7b, which shows a top-view of the loss landscape, indicates that a large portion of ∞ ball around the nominal image pushes the PGD solution towards the right (rather than the bottom). In other words, gradients cannot always be trusted to point towards the true worst-case attack. : Loss landscapes around the nominal image of an "eight". It is generated by varying the input to the model, starting from the original input image toward either the worst attack found using PGD (u direction) or the one found using a complete solver (v direction). In (a), the z axis represents the loss and the orange and blue colors on the surface represent the classification predicted by the model. We observe that while the PGD attack (blue dot) is correctly classified as an "eight", the MIP attack (red dot) is misclassified as a "two". Panel (b) shows a top-view of the same landscape with the decision boundary in black. For both panels, the diamond-shape represents the projected ∞ ball of size = 0.1 around the nominal image. Figure 1 compares 1IBP to Wong et al. on MNIST for all perturbation radii between 0 and 0.45 across all models. Figure 1 : 1Accuracy against different adversarial perturbations: (a) shows the verified/provable worst-case accuracy, and (b) shows the empirical adversarial accuracy computed by running PGD. Faded lines show individual models, while bold lines show the best accuracy across all models. In (a), for Wong et al., the dots correspond to exact bounds computed using a MIP solver, while the black bold line corresponds to a lower bound computed using [20] without random projections. Figure 2 : 2Median evolution of the nominal (no attacks) and empirical PGD accuracy (under perturbations of = 0.3) as training progresses for 10 independently trained large models on MNIST. The shaded areas indicate the 25 th and 75 th percentiles. Figure 2 2shows how the empirical PGD accuracy (on the test set) increases as training progresses for IBP and Madry et al. 6Figure 3 : 3As an example, for Wong et al., there are 3 timeouts at = 0.1, 18 timeouts at = 0.2 and 58 timeouts at = 0.3 for the 10K examples of the test set. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Perturbation radius ( Empirical adversarial accuracy computed by running PGD. Faded lines show individual models, while bold lines show the best accuracy across all models. Figure 4 : 4First layer convolutional filters resulting from training a small robust model on MNIST against a perturbation radius of = 0.1 for all methods. Figure 5 : 5First layer convolutional filters resulting from training a small robust model on CIFAR-10 against a perturbation radius of = 2/255 for all methods. Figure 6 : 6Attacks of size = 0.1 found on the 1365 th image of the MNIST test set. For (b) and (c), the left pane shows the adversarial image, while the right pane shows the perturbation rescaled for clarity. Figure 7 7Figure 7: Loss landscapes around the nominal image of an "eight". It is generated by varying the input to the model, starting from the original input image toward either the worst attack found using PGD (u direction) or the one found using a complete solver (v direction). In (a), the z axis represents the loss and the orange and blue colors on the surface represent the classification predicted by the model. We observe that while the PGD attack (blue dot) is correctly classified as an "eight", the MIP attack (red dot) is misclassified as a "two". Panel (b) shows a top-view of the same landscape with the decision boundary in black. For both panels, the diamond-shape represents the projected ∞ ball of size = 0.1 around the nominal image. et al. in terms of verified robustness by using a complete Mixed-Integer Programming (MIP) solver. • We demonstrate that IBP is not only computationally cheaper, but achieves the state-of-theart in single-model 2 verified robustness on MNIST (with verified upper bounds on the error rate of 2.23% and 8.05% at = 0.1 and = 0.3 respectively) and CIFAR-10 (with a verified error rate of 72.91% at = 8/255). adversarial training, following Madry et al.[14], which generates adversarial examples on the fly during training; and Wong et al.[20], which trains models that are provably robust. As detailed in Appendix B, we train three different model sizes (i.e., small, medium and large) for each of the four methods. On MNIST, for each model and each method, we trained models that are robust to a wide range of perturbation radii by setting train to 0.1, 0.2, 0.3 or 0.4. On CIFAR-10, we train the same models using IBP and Madry et al. with train ∈ {2/255, 8/255} 3 .Epsilon Method Test error PGD Verified MNIST = 0.1 Nominal 0.65% 27.72% - Madry et al. 0.59% 1.34% - Wong et al. 1.08% 2.89% 3.01% IBP 1.06% 2.11% 2.23% Wong et al. [20]* 1.08% - 3.67% MNIST = 0.3 Nominal 0.65% 99.63% - Madry et al. 0.70% 3.73% - Wong et al. 13.52% 26.16% 26.92% IBP 1.66% 6.12% 8.05% Xiao et al. [21]* 2.67% 7.95% 19.32% CIFAR-10 = 8/255 Nominal 16.66% 100.00% 100.00% Madry et al. 20.33% 72.02% - IBP 47.91% 64.53% 72.91% Wong et al. [20]* 71.33% - 78.22% ). Compared to Wong et al., IBP achieves lower error rates under normal and adversarial conditions, as well as better verifiable bounds, settings the state-of-the-art in verified robustness to adversarial attacks. Additionally, IBP remains competitive against Madry et al. by achieving a lower PGD error rate on CIFAR-10 (albeit at the cost of an increased nominal error rate) 4 . Finally, we note that when training the small network on MNIST with a Titan Xp GPU (where standard training takes 1.5 seconds per epoch), IBP only takes 3.5 seconds per epoch compared to 8.5 seconds for Madry et al. and 2 minutes for Wong et al. (using random projection of 50 dimensions). Indeed, IBP creates only two additional passes through the network (as detailed in Appendix A), compared to Madry et al. for which we used seven PGD steps. Table 3 : 3Comparison with the state-of-the-art.Comparison of the nominal test error (under no perturbation), The use of ensembles or cascades (as done by Wong et al.[20]) is orthogonal to the work presented here. During testing, we can verity robustness against different values (not necessarily the ones trained on).4 This result only holds for our constrained set of network sizes. The best known empirical adversarial error rate for CIFAR-10 at = 8/255 using Madry et al. is 52.96%. 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[ "Quantum-like modeling of the order effect in decision making: POVM viewpoint on the Wang-Busemeyer QQ-equality", "Quantum-like modeling of the order effect in decision making: POVM viewpoint on the Wang-Busemeyer QQ-equality" ]	[ "Aleksandr Lebedev \nInternational Center for Mathematical Modeling in Physics\nCognitive Sciences Linnaeus University\nSE-351 95VäxjöSweden\n", "Andrei Khrennikov \nInternational Center for Mathematical Modeling in Physics\nCognitive Sciences Linnaeus University\nSE-351 95VäxjöSweden\n" ]	[ "International Center for Mathematical Modeling in Physics\nCognitive Sciences Linnaeus University\nSE-351 95VäxjöSweden", "International Center for Mathematical Modeling in Physics\nCognitive Sciences Linnaeus University\nSE-351 95VäxjöSweden" ]	[]	In recent years, quantum mechanics has been actively used in areas outside of physics, such as psychology, sociology, theory of decision-making, game theory, and others. In particular, quantum mechanics is used to explain the paradoxes arising in cognitive psychology and decision making. Wang and Busemeyer invented a quantum model and approach as well as nonparametric equality (so-called QQ-equality), explaining the questions order effect. The primary objective of this note is to test the possibility to expand the Wang-Busemeyer model by considering questions which are mathematically represented by positive operator valued measures. We found that, for such observables, the QQ-equality can be violated. But, we also showed that, in principle, it is possible to reduce expanded model to the original Wang-Busemeyer model by expanding the context of the questions. This version of preprint is aimed to point out to annoying miscalculation in version 1. This miscalculation might mislead a reader who is not experienced in operating with POVMs. Otherwise the main line of construction and reasoning presented in version 1 is right and it can be easily completed by the reader on the basis of version 1 and the correction remark in version 2.	null	[ "https://export.arxiv.org/pdf/1811.00045v2.pdf" ]	53,636,959	1811.00045	1de4acff548390768248b00e3cc5b0a2842c2a1e	Quantum-like modeling of the order effect in decision making: POVM viewpoint on the Wang-Busemeyer QQ-equality Aleksandr Lebedev International Center for Mathematical Modeling in Physics Cognitive Sciences Linnaeus University SE-351 95VäxjöSweden Andrei Khrennikov International Center for Mathematical Modeling in Physics Cognitive Sciences Linnaeus University SE-351 95VäxjöSweden Quantum-like modeling of the order effect in decision making: POVM viewpoint on the Wang-Busemeyer QQ-equality arXiv:1811.00045v2 [quant-ph] 1 Apr 2023 In recent years, quantum mechanics has been actively used in areas outside of physics, such as psychology, sociology, theory of decision-making, game theory, and others. In particular, quantum mechanics is used to explain the paradoxes arising in cognitive psychology and decision making. Wang and Busemeyer invented a quantum model and approach as well as nonparametric equality (so-called QQ-equality), explaining the questions order effect. The primary objective of this note is to test the possibility to expand the Wang-Busemeyer model by considering questions which are mathematically represented by positive operator valued measures. We found that, for such observables, the QQ-equality can be violated. But, we also showed that, in principle, it is possible to reduce expanded model to the original Wang-Busemeyer model by expanding the context of the questions. This version of preprint is aimed to point out to annoying miscalculation in version 1. This miscalculation might mislead a reader who is not experienced in operating with POVMs. Otherwise the main line of construction and reasoning presented in version 1 is right and it can be easily completed by the reader on the basis of version 1 and the correction remark in version 2. Introduction This version is aimed to point out to annoying miscalculation in version 1 [1]. Quantum mechanics was originally created to explain the paradoxes arising in classical physics. At the same time, a powerful mathematical apparatus of quantum probability theory was created, which was later effectively used to explain the paradoxes not only in physics, but also in other fields, such as cognitive psychology, decision making, and social sciences, see, for example, monographs [3] - [8] and a few recent representative papers [9] - [19]. In particular, Wang and Busemeyer [2]used the quantum formalism and methodology of experiment to explain order effects in question answering. Wang and Busemeyer established a non-parametric inequality (known as QQ-inequality) to which the probabilities of an experiment must satisfy in order for a quantum model to exist for them, as follows: p(AyBn) + p(AnBy) − p(ByAn) − p(BnAy) = 0, where A and B correspond to questions with two possible outcomes "Yes" and "No". The joint probabilities are the probabilities of receiving given answers to questions A and B in the same order as they appear, e.g. P (AyBn) means the probability to obtain negative answer to the question B before obtaining affirmative answer to the question A. The quantumlike model assumes that questions are represented by Hermitian operators; therefore the answers Ay, An, By, Bn are represented by orthogonal projectors. The following questions naturally arise: 1. Is it imperative to require these operators to be projectors? 2. Is it possible to expand the context of the questions in such a way that, although the original operators are not projectors, the extended questions would already correspond to the projectors? The paper presents examples of measurement operators corresponding to POVM for which the QQ-inequality does not hold. The dependence of the left part of the QQ-inequality on the state is discussed. Further, using the Naimark's theorem [20], lifting of such operators are constructed for which this equality does hold. The example of violation QQ-inequality for POVM-observables POVM stands for 'Positive operator-valued measure'. More precisely, it is the set of measurement operators {E a } (called effects) that form a complete set of Hermitian non-negative operators. It means that it has the following properties: 1. E a = E † ; 2. φ\| E a \|φ ≥ 0 for all vectors \|φ 3. a E a = 1 For POVMs represented as E a = M a M † a ,(1) the state update generated by a measurement is given by the formula ρ → ρ a = M a ρM † a tr(E a ρ) .(2) In particular, for a pure state \|ψ , \|ψ → \|ψ a = M a \|ψ \|\|M a \|ψ \|\| .(3) We remark that if M a is projection, then the effect E a is also projection. Such POVMs are called projection observables. The inverse is not correct. An effect E a that is projection can be decomposed with operators M a (see (1)) that are not projections. In particular, POVM does not determine the state update operation. The right theory for the description of observations and the corresponding state updates is quantum instrument theory (see [21] for its simple presentation). However, in this note we do not apply this theory and we proceed with formulas (1), (2). Within quantum instrument theory, these formulas determine so called atomic instruments. The operators corresponding to two measurement procedures are An + Ay = I, Bn + By = I,(4) where labels n,y denote the answers "no" and "yes". For the first equality, we select the operators: 5/6 1/2 √ 3 1/2 √ 3 1/2 + 1/6 −1/2 √ 3 −1/2 √ 3 1/2 = I.(5) Moreover, the second equality, we select the operators: 1/6 1/2 √ 3 1/2 √ 3 1/2 + 5/6 −1/2 √ 3 −1/2 √ 3 1/2 = I.(6) We need to calculate the following quantity: qq = p(AyBn) + p(AnBy) − p(ByAn) − p(BnAy).(7) where A represents the first measurement and B represents the second one and vice verse. For the update, we should use, e.g., square roots from the (Hermitian) operators An,Ay, Bn,By (cf. version 1 [1] in that we misleadingly used not square roots, but the question-operators), Q 1 = √ An, Q 2 = Ay, P 1 = √ Bn, P 2 = By.(8) In our terms, (7) has the form: qq(ρ) = T r(P 2 Q 1 ρQ 1 P 2 + P 1 Q 2 ρQ 2 P 1 − Q 2 P 1 ρP 1 Q 2 − Q 1 P 2 ρP 2 Q 1 ) = = T r(P 2 Q 1 ρQ 1 P 2 + P 1 Q 2 ρQ 2 − Q 2 P 1 ρP 1 − Q 1 P 2 ρP 2 Q 1 ) = T r(Q 1 P 2 P 2 Q 1 ρ + Q 2 P 1 Q 2 ρ − P 1 Q 2 P 1 ρ − P 2 Q 1 Q 1 P 2 ρ). It is easy to check that e.g. for the pure state ψ = (1,1)/ √ 2, qq(ρ ψ ) = 0, where ρ ψ = \|ψ ψ\|. This is the good place to emphasize that the Wang-Bussemeyer's QQeffect is state dependent. This is not just the property of questions, but questions asked to people in the special mental states. The state dependence plays the crucial role in quantum-like modeling. Recently its role was highlighted in article [22] advertising quantum logic of human mind. Lifting of POVMs to projections: Naimark's dilation theorem Naimark's theorem states that, for any given POVM, the Hilbert space, can be extended to a larger space that POVM can be realized as performing orthogonal measurements in that larger Hilbert space. We denote this process of obtaining the mapping from POVM to orthogonal measurements as 'lifting'. We can now consider 3-dimensional space and new operators A ′ n + A ′ y = I, B ′ n + B ′ y = I.(9) For the first equality, we select the operators:    5 6 1 2 √ 3 1 3 √ 2 1 2 √ 3 1 2 − 1 √ 6 1 3 √ 2 − 1 √ 6 2 3    +    1 √ 6 − 1 2 √ 3 − 1 3 √ 2 − 1 2 √ 3 1 2 1 √ 6 − 1 3 √ 2 1 √ 6 1 3    = 1 For the second equality, we select the operators:    1 √ 6 1 2 √ 3 − 1 3 √ 2 1 2 √ 3 1 2 − 1 √ 6 − 1 3 √ 2 − 1 √ 6 1 3    +    5 6 − 1 2 √ 3 1 3 √ 2 − 1 2 √ 3 1 2 1 √ 6 1 3 √ 2 1 √ 6 2 3    = 1 Note that these operators are self-adjoint. Furthermore, they are projectors. They are lifting of the operators considered in section 2 in equations (5), (6). Since now questions are mathematically represented by projections, the QQ-equality for them holds true for any state. Interpretation of POVM-questions and experimenting POVMs of non-projection type represent fuzzy measurements. "Fuzziness" is in the procedure of the quantum state update. If in (2) the operator M a is not projection, then generally, for a pure state \|ψ , updated states \|ψ a , a = n,y, are not orthogonal. Alice by answering "no" to say question A does not wash out the possibility to answer "yes", if the same question will be asked immediately. Questions represented by projection observables are characterized by repeatability of answers, so by answering "no" to the A-question, Alice will answer again "no" to this question with probability one. This difference between projection and non-projection questions can be tested experimentally. The indirect test is based on testing the combination of question order and response replicability effects [21]. However, it would be interesting to perform direct testing. Another really challenging problem is to check experimentally the consequences of the Naimark's dilation theorem. To find a question A that cannot be represented by projective measurement, but can be represented by an atomic POVM A. Then lift A to a higher dimensional state space as projection observable A ′ and realize the latter as a question completing the original question A. AcknowledgmentThe authors would like to thank Jerome Bussemeyer who pointed out to the aforementioned miscalculation in version 1 and stimulated us to submit this version. Quantum-like modeling of the order effect in decision making: POVM viewpoint on the Wang-Busemeyer QQ-equality. A Lebedev, A Khrennikov, Lebedev, A., Khrennikov, A. 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Ubiquitous quantum structure: from psychology to finances. A Khrennikov, SpringerBerlin-Heidelberg-New YorkKhrennikov, A. Ubiquitous quantum structure: from psychology to finances; Springer: Berlin-Heidelberg-New York, 2010. Quantum models of cognition and decision. J R Busemeyer, P D Bruza, Cambridge PressCambridge, UKBusemeyer, J. R.; Bruza, P. D. Quantum models of cognition and decision; Cambridge Press: Cambridge, UK, 2012. Quantum dynamics for classical systems: with ap-plications of the Number operator. F Bagarello, WileyEd; New York, USABagarello, F. Quantum dynamics for classical systems: with ap-pli- cations of the Number operator; Wiley Ed.: New York, USA, 2012. Quantum social science. E Haven, A Khrennikov, Haven, E.; Khrennikov, A. Quantum social science; M Asano, A Khrennikov, M Ohya, Y Tanaka, I Yamato, Quantum adaptivity in biology: from genetics to cognition. Heidelberg-Berlin-New YorkSpringerAsano, M.; Khrennikov, A.; Ohya, M.; Tanaka, Y.; Yamato, I. Quantum adaptivity in biology: from genetics to cognition, Springer: Heidelberg-Berlin-New York, 2015. . E Haven, A Khrennikov, T R Robinson, Quantum Methods in Social Science: A First Course. WSPHaven, E.; Khrennikov, A.; Robinson, T. R. Quantum Methods in Social Science: A First Course; WSP: Singapore, 2017. A quantum probability expla-nation for violation of rational decision theory. E M Pothos, J R Busemeyer, Proc. Royal. Soc. B. 276Pothos, E. M.; Busemeyer, J. R. A quantum probability expla-nation for violation of rational decision theory. Proc. Royal. Soc. B 2009, 276, 2171-2178. Can quantum probability pro-vide a new direction for cognitive modeling?. E M Pothos, J R Busemeyer, Behav. Brain Sc. 36Pothos, E. M.; Busemeyer, J. R. Can quantum probability pro-vide a new direction for cognitive modeling? Behav. Brain Sc. 2013, 36, 255-327. A quantum theoretical explanation for probability judgment errors. J R Busemeyer, E M Pothos, R Franco, J Trueblood, Psych. Rev. 118Busemeyer, J. R.; Pothos, E. M.; Franco, R.; Trueblood, J. A quantum theoretical explanation for probability judgment errors. Psych. Rev. 2011, 118, 193-218. Quantum-like models cannot account for the conjunction fallacy. T Boyer-Kassem, S Duchene, E Guerci, 10Theor. Decis.Boyer-Kassem, T.; Duchene, S.; Guerci, E. Quantum-like models can- not account for the conjunction fallacy. Theor. Decis. 2015, 10, 1-32. The Relationship between quantum and clas-sical correlation in games. A Brandenburger, Game Econ. Behav. 69Brandenburger, A. The Relationship between quantum and clas-sical correlation in games. Game Econ. Behav. 2010, 69, 175-183. Quantum dynamics applied to cognition: A consideration of available op-tions. J Broekaert, I Basieva, P Blasiak, E M Pothos, Phil. Trans. R. Soc. A. 2017Broekaert, J.; Basieva, I.; Blasiak, P.; Pothos, E. M. Quantum dy- namics applied to cognition: A consideration of available op-tions. Phil. Trans. R. Soc. A 2017, 375(2106), 375-405. A phenomenological operator descrip-tion of interactions between populations with applications to mi-gration. F Bagarello, F Oliveri, Mathematical Models and Methods in Applied Sciences. 23Bagarello, F.; Oliveri, F. A phenomenological operator descrip-tion of interactions between populations with applications to mi-gration. Mathematical Models and Methods in Applied Sciences 2013, 23 (03), 471-492. Are quantum-mechanical-like models possible, or necessary, outside quantum physics? Phys. Scripta. A Plotnitsky, 014011. 33163Plotnitsky, A. Are quantum-mechanical-like models possible, or nec- essary, outside quantum physics? Phys. Scripta 2014, T163, 014011. 33 Quantum Bayesianism as the basis of general theory of decision-making. A Khrennikov, Phil. Trans. R. Soc. A. 374Khrennikov, A. Quantum Bayesianism as the basis of general theory of decision-making. Phil. Trans. R. Soc. A 2016, 374, 20150245. J Preskill, Lecture notes quantum information and computation. Preskill, J. Lecture notes quantum information and computation, http://www.theory.caltech.edu/people/preskill/ph229/notes/chap3.pdf (1998). Modeling combination of question order effect, response replicability effect, and QQ-equality with quantum instruments. M Ozawa, A Khrennikov, J. Math. Psych. 100102491Ozawa, M. and Khrennikov, A.: Modeling combination of question or- der effect, response replicability effect, and QQ-equality with quantum instruments. J. Math. Psych. 100, 102491 (2021) Nondistributivity of human logic and violation of response replicability effect in cognitive psychology. M Ozawa, A Khrennikov, J. Math. Psych. 112102739Ozawa, M. and Khrennikov, A.: Nondistributivity of human logic and violation of response replicability effect in cognitive psychology. J. Math. Psych. 112, 102739 (2023).	[]
[ "NODAL DECOMPOSITIONS OF A SYMMETRIC MATRIX", "NODAL DECOMPOSITIONS OF A SYMMETRIC MATRIX" ]	[ "Theo Mckenzie ", "John Urschel " ]	[]	[]	Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is Ω(n/ log n) and o(n). regularity assumptions [1], among many others.Courant's theorem, and nodal domains in general, has also been studied in the discrete setting of graphs. This setting poses a number of unique challenges, as eigenvectors may vanish (e.g., equal zero) at some entries, while in Courant's setting the nodal set is of measure zero[14]. In this setting, a nodal domain of an eigenvector ϕ of the generalized Laplacian of a graph is a maximal connected component on which the eigenvector entries do not change sign, e.g. ϕ(i)ϕ(j) ≤ 0 for all i = j in the domain. More generally, for a signed graph, a nodal domain requires that M ij ϕ(i)ϕ(j) ≤ 0 for i = j in the domain. This definition is based on the convention that off-diagonal entries are "typically" negative, as is the case for the Laplacian 1 .The earliest known result in this setting is due to Gantmacher and Krein, who studied the sign properties of eigenvectors of generalized Laplacians of the path graph and proved a tight	null	[ "https://export.arxiv.org/pdf/2305.10598v2.pdf" ]	258,762,223	2305.10598	35f1908dd7a93ec0bc787498a177727e184f5fe9	NODAL DECOMPOSITIONS OF A SYMMETRIC MATRIX Theo Mckenzie John Urschel NODAL DECOMPOSITIONS OF A SYMMETRIC MATRIX Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of a signed Erdős-Rényi graph is Ω(n/ log n) and o(n). regularity assumptions [1], among many others.Courant's theorem, and nodal domains in general, has also been studied in the discrete setting of graphs. This setting poses a number of unique challenges, as eigenvectors may vanish (e.g., equal zero) at some entries, while in Courant's setting the nodal set is of measure zero[14]. In this setting, a nodal domain of an eigenvector ϕ of the generalized Laplacian of a graph is a maximal connected component on which the eigenvector entries do not change sign, e.g. ϕ(i)ϕ(j) ≤ 0 for all i = j in the domain. More generally, for a signed graph, a nodal domain requires that M ij ϕ(i)ϕ(j) ≤ 0 for i = j in the domain. This definition is based on the convention that off-diagonal entries are "typically" negative, as is the case for the Laplacian 1 .The earliest known result in this setting is due to Gantmacher and Krein, who studied the sign properties of eigenvectors of generalized Laplacians of the path graph and proved a tight Introduction Courant's nodal domain theorem states that the zero-level set (i.e., the set of points where the eigenfunction equals zero) of the k th lowest energy eigenfunction of a Laplacian on a smooth bounded domain in R d with Dirichlet boundary conditions divides the domain into at most k subdomains (see [16], and his text co-authored with Hilbert in the following year [17]). The zero-level set is commonly referred to as the nodal set, the resulting subdomains are referred to as the nodal domains, and the number of subdomains is referred to as the nodal count. Results of this type have been of great interest in spectral geometry and mathematical physics (see, e.g. [54]), with refinements in dimension two (e.g., Pleijel's nodal domain theorem) [12,41,42], and extensions to p-Laplacians [18,23], Riemannian manifolds [40], and domains with low estimate for the nodal count in this setting (see [29] for a revised English edition of the original 1950 Russian text). Fiedler's tree theorem proves exact nodal count estimates for trees, generalizing the work of Gantmacher and Krein. Namely he showed that the number of nodal domains of a non-vanishing eigenvector of a symmetric, acyclic, irreducible matrix is exactly the index of the corresponding eigenvalue indexed in increasing order [26] (in fact, both Gantmacher and Krein's result and Fiedler's result extend to the signed case, described below). These results can be thought of as a discrete version of Sturm's oscillation theorem for ordinary differential equations [48,49], of which Courant's theorem is a generalization. For discrete generalized Laplacians, the nodal count of a non-vanishing eigenvector corresponding to the k th eigenvalue is at most k. However, when an eigenvector vanishes on some vertices, complications arise, as the vertex sets of nodal domains no longer forms a partition of the vertex set. In this setting, there are a number of competing versions of nodal domains and nodal theorems. Most notably, many authors have considered the concept of weak and strong nodal domains: a strong nodal domain of an eigenvector ϕ of a symmetric matrix M is simply a nodal domain as defined above, i.e., a maximally connected induced subgraph for which M ij ϕ(i)ϕ(j) < 0 for all edges M ij = 0, and a weak nodal domain is a maximally connected induced subgraph for which M ij ϕ(i)ϕ(j) ≤ 0 for all edges M ij = 0. Davies, Gladwell, Leydold, and Stadler proved a weak and strong nodal count theorem for generalized Laplacians: given an eigenpair (λ, ϕ) of an irreducible generalized Laplacian M , the weak and strong nodal count of ϕ are at most k and k + r − 1, respectively, where k and r are the index (in increasing order) and multiplicity of λ, respectively [10,19,20]. There are a number of other proofs of various versions of this statement [21,24,27,44,51]; see [19,Sec. 2] for a discussion of the results (and the correctness of some of the statements and associated proofs) in these works. The second author of the current paper proved a decomposition version of the Davies-Gladwell-Leydold-Stadler theorem, that for the k th eigenvalue of a symmetric generalized Laplacian of a discrete graph, a positive proportion of eigenvectors ϕ in the corresponding eigenspace can be decomposed into at most k signed nodal domains (i.e., there exists a signing ε satisfying ε(i)ϕ(i) ≥ 0 for all i with classical nodal count at most k) [50]. In addition to discrete versions of Courant's nodal theorem, measuring the gap between the actual nodal count and Courant's bound has also been of interest. Most notably, Berkolaiko showed that a non-vanishing eigenvector of an irreducible generalized Laplacian has nodal count at least k−ν, where k is the index of the corresponding eigenvalue and ν is the cyclomatic number of the associated graph [9] 2 (see [15] for an alternate proof). This result was later strengthened by Xu and S.T. Yau, producing a lower bound for an arbitrary eigenvector ϕ of k + r − 1 − ν − \|i 0 (ϕ)\|, where k and r are the index and multiplicity of the corresponding eigenvalue, ν is the cyclomatic number of the associated graph, and i 0 (ϕ) = {i \| ϕ(i) = 0} [53]. The related study of nodal surplus (stemming from [9,15]) is an active area of research, see [2,3,4,5] for details. In this work, we consider nodal count theorems for the generalized Laplacian of arbitrary signed graphs, e.g., arbitrary symmetric matrices. Such estimates are important for a number of reasons. First, when dealing with a finite element approximation of an elliptic operator, the resulting stiffness matrix is not always a generalized Laplacian and Courant's bound may fail to hold (e.g., a 2D triangularization with some obtuse angles); see, for instance, [31] for details. More generally, nodal domains give us an idea of relationship between the structure of a matrix and that of its eigenvector, and signed graphs are used frequently in practice (e.g., Ising models, correlation clustering, etc). However, as the setting of discrete Laplacians added barriers (in the form of vanishing vertices) to a direct version of the classical Courant nodal domain theorem, the extension from generalized Laplacian matrices to arbitrary symmetric matrices brings unique challenges, and ambiguity as to what constitutes a nodal domain in this setting. Path nodal domains for symmetric matrices have been studied by Mohammadian [38] and, recently, by Ge and Liu [30]. In this model, we count the number of connected components of the graph induced on "good" edges, namely edges for which the product of eigenvector entries of vertices in the edge respects the sign of the edge. Namely, given a symmetric irreducible matrix M ∈ R n×n and an eigenvector ϕ corresponding to an eigenvalue of index k and multiplicity r, let G < ϕ = ({1, ..., n}, E < ϕ ) and G ≤ ϕ = ({1, ..., n}\i 0 (ϕ), E ≤ ϕ ), where E < ϕ = {(i, j) ∈ E(G) \| M ij ϕ(i)ϕ(j) < 0} and E ≤ ϕ = {(i, j) ∈ E(G) \| M ij ϕ(i)ϕ(j) ≤ 0} (with G ≥ ϕ and G > ϕ defined analogously). Let κ(·) be the number of connected components of a graph. Mohammadian proved that κ(G ≤ ϕ ) ≤ k, κ(G < ϕ ) ≤ k + (r − 1), and that, if i 0 (ϕ) = ∅, then κ(G < ϕ ) ≥ k − ν [38]. Ge and Liu expanded upon the analysis of Mohammadian by proving lower bounds for κ(G < ϕ ), and showing that κ(G < ϕ ) ≤ k for any eigenvector of minimal support [30]. In addition, they produced an upper bound on a formulation of weak nodal domain slightly different from G ≤ ϕ , and proved a number of estimates for acyclic matrices. There is a peculiarity specific to the path nodal domain, in that it is possible that i, j in the same nodal domain could satisfy M ij ϕ(i)ϕ(j) > 0, whereas this is impossible in un-signed graphs. Namely, i and j could be identifiably negatively related, but still be a part of the same nodal domain, if there is a path of good edges from i to j. Here, "bad" edges are treated equivalently to as if there were no edge at all. In an effort to have our decomposition depend on all edges, we take a different approach from that of previous authors in a number of ways. Rather than study edges or walks with a classical nodal-type property and ignore bad edges (i.e. consider only M ij ϕ(i)ϕ(j) < 0), we aim to prove bounds for induced subgraphs ("nodal" subsets) with the nodal-type property, thus producing subsets of the domain for which the eigenvector does not change sign and the matrix, restricted to this subset, is a generalized Laplacian (both up to a sign transformation M → DM D, ϕ → Dϕ, for some involutory diagonal matrix). In this work, we focus on the minimal size of decompositions of the domain into nodal subsets. This is a stricter definition than that of path nodal domains; therefore, we expect more nodal domains in this case. In what follows, we prove upper bounds regarding nodal decompositions of a symmetric matrix. In particular, we prove a natural analogue of Courant's nodal domain theorem for any matrix and eigenvector pair that depends only on the energy level of the eigenvector and how "close" to a generalized Laplacian the given matrix is (Theorem 1.3). One large appeal of our formulation is in deducing the structure of eigenvectors of random graphs. Studying the structure of eigenvectors of random graphs is a well known problem with applications in both computer science and mathematical physics, see, for example, [37,43]. On dense Erdős-Rényi graphs, eigenvectors that do not correspond to the highest eigenvalue have exactly two nodal domains [7,22], which are approximately the same size [33], and all vertices are on the boundary of their nodal domains [46]. This follows the general notion of quantum ergodicity, that roughly, the distribution of entries in an eigenvector of these graphs should be close to a joint Gaussian distribution with individual entries close to independent [11]. All of these results show that Erdős-Rényi graphs have "trivial" eigenvector structure, in that the graph is too dense for these nodal domain statistics to detect different structure within the graph (this is not the case for random regular graphs, for which eigenvectors of the most negative eigenvalues have many nodal domains [28]). In our setting, we consider an Erdős-Rényi signed graph, denoted by G(n, p, q) for n ∈ N, 0 < p, q < 1. Here, we randomly sample an n vertex graph, where each of the n 2 possible edges has a p probability of being a positive edge, a q probability of being a negative edge, and 1−p−q of not existing. Signed Erdős-Rényi graphs are relevant to community detection and mathematical physics, as well as elsewhere. For example, in the Sherrington-Kirkpatrick model, positive/negative edges in an Erdős-Rényi graph correspond to as the presence of ferromagnetic/antiferromagnetic bonds in the discrete Hamiltonian. Using the same argument as [22], with high probability, every eigenvector ϕ has κ(G < ϕ ) = κ(G > ϕ ) = 1 (we prove this in Appendix A). However, as we show in Section 4, our stronger notion of a nodal domain shows the nontrivial structure of the eigenvectors of a G(n, p, q), as the number of nodal domains scales sublinearly. Specifically, we show that with high probability, there are o(n) nodal domains in this signed random graph. A simple lower bound of Ω(n/ log 1 1−(p∨q) n) is given in Proposition 1.7, which follows from the size of the maximum clique in a graph. 1.1. Definitions, Notation, and Results. Let M be an n×n symmetric, irreducible matrix with eigenpair (λ, ϕ). We denote the index of λ (the number of eigenvalues strictly less than λ, plus one) by k, and the multiplicity by r. We can associate with M a signed graph . We denote by ν the cyclomatic number (dimension of the unsigned cycle space) of G. Given a subset of vertices S, we denote by M S the \|S\| × \|S\| principal submatrix of M corresponding to S. Similarly, we write ϕ(S) as the subvector of ϕ corresponding to S. We denote by − → 1 S the vector that equals one on S and zero elsewhere; its dimension is always clear from context. For a vertex v ∈ V , we write ϕ(v) = ϕ({v}) and − → 1 v = − − → 1 {v} . For two vectors x, y of the same length, x • y denotes the entrywise product of x and y. Let p ∨ q = max{p, q} and p ∧ q = min{p, q}. A key ingredient in the worst-case analysis that follows is a notion of how "far" a signed graph is from being equivalent to an all positive graph, e.g., how far a matrix is from being a generalized Laplacian. We make use of the notion of frustrated edges and frustration index. Definition 1.1. Given a signed graph G = ([n], E, σ) and a state ε ∈ {±1} n , an edge {i, j} ∈ E is said to be frustrated if σ i,j ε(i)ε(j) < 0. The frustration index f of G is the minimum number of frustrated edges over all states ε ∈ {±1} n . Computing the frustration index of a signed graph is NP-hard, via a reduction from the maxcut problem. In what follows, we often assume that the number of positive off-diagonal pairs of M is exactly f; this can be done by performing an involutory diagonal transformation DM D for a diagonal matrix D corresponding to a state ε ∈ {±1} n that achieves the frustration index. Given a vector x, we denote by i 0 (x) the set of indices where x equals zero, i.e., i 0 (x) := {j \| x(j) = 0}. Definition 1.2. Given a symmetric matrix M ∈ R n×n and a non-vanishing vector x ∈ R n (i.e., i 0 (x) = ∅), we denote by N(x) the minimal quantity s for which there exists a partition [n] = s =1 V with G[V ] connected and M ij x(i)x(j) < 0 for all {i, j} ∈ E[V ], = 1, . .., s, i.e., the minimal decomposition of the domain into nodal subsets. Computing N(ϕ) is NP-hard (as is often the case for problems involving signed graphs), as, given a signed graph G = ([n], E, σ), it is NP-hard to compute the smallest s such that there exists a partition [n] = s =1 V with G[V ] connected and σ ij = +1 for all (i, j) ∈ E[V ], = 1, ..., s, via a reduction from the clique-cover problem. This hardness holds not just for adversarial ϕ, but for eigenvectors ϕ, as, for any signed graph G = ([n], E, σ), one can always produce a matrix M with sparsity structure G and an eigenvector with constant sign. We provide a short proof of the bounds k + (r − 1) − ν ≤ N(ϕ) ≤ k + f (1.1) for non-vanishing eigenvectors in Section 2 (Proposition 2.1). This follows quickly from Mohammadian's aforementioned results and an analysis of frustrated edges, but we provide a short proof of independent interest. Let N s (ϕ) be the minimal nodal decomposition of ϕ restricted to non-vanishing entries (i.e., N s (ϕ) = N(ϕ\| ϕ(i) =0 )). We prove that for any symmetric matrix M there exists an orthonormal eigenbasis ϕ 1 , ..., ϕ n ordered by increasing energy, such that N s (ϕ k ) ≤ k for all k (Proposition 2.3), extending a result of Gladwell and Zhu for generalized Laplacians [31]. However, the restriction to non-vanishing entries leads to a number of limitations in the above result. For instance, the eigenbases satisfying the proposition statement may be of measure zero. For this reason, we prove a more robust version of the above statement, focusing on nodal decompositions of the entire domain. In particular, in Section 3, we prove that there exists a subset of orthonormal eigenbases, of positive measure, such that for every basis ϕ 1 , ..., ϕ n in this subset ordered by increasing energy, there exist corresponding signings ε 1 , ..., ε n ∈ {±1} n satisfying ϕ k (i)ε k (i) ≥ 0 for all i ∈ [n] and N(ε k ) ≤ k + f for all k. Theorem 1.3. Let M be an n × n symmetric, irreducible matrix, and let B be the set of corresponding orthonormal eigenbases of R n . Then there exists a subset Φ ⊂ B of co-dimension zero and an ordering of basis elements by increasing energy such that, for every {ϕ 1 , ..., ϕ n } ∈ Φ, there exists signings ε 1 , ..., ε n ∈ {±1} n satisfying ϕ k (i)ε k (i) ≥ 0 for all i, k ∈ [n] and N(ε k ) ≤ k + f, k = 1, ..., n, (1.2) where f is the frustration index of the signed graph of M . Informally, the above theorem tells us that if we choose an arbitrary orthonormal eigenbasis, there is a positive probability that there is a signing of vanishing entries of our eigenbasis so that the resulting vectors satisfy the upper bounds of Inequality 1.1. This can be viewed as a stronger version of "weak nodal bounds," as vanishing vertices cannot be used to connect both positively and negatively signed vertices. However, we can only hope for such a result for a positive proportion of bases; the unsigned star graph is an illustrative example of this limitation (here, the proportion of bases satisfying Theorem 1.3 is exponentially small). The positive probability portion of the above theorem is quite important; this property forces the resulting partitions in the basis to represent the "dynamics" of the eigenspace, e.g., the theorem statement does not require the artificial vanishing of vertices (as in the proof of Proposition 2.3). Our proof of Theorem 1.3 proceeds as follows: (1) Characterize the structure of an eigenspace E λ with eigenvectors that simultaneously vanish, i.e., i 0 (λ) := {j \| ϕ(j) = 0 for all ϕ ∈ E λ } = ∅, and parameterize E λ using eigenvectors of the connected components of M restricted to [n]\i 0 (λ). (2) Algorithmically define the sign vector ε associated with any eigenvector ϕ that is nonvanishing on [n]\i 0 (λ) and restrict the elements of the orthonormal basis ϕ 1 , ..., ϕ r of E λ to certain half-spaces so that N(ε s ) ≤ k + (s − 1) + f. In Section 4, we analyze the nodal count of the adjacency operator of the Erdős-Rényi signed graph. A lower bound is given by the combinatorial properties of a G(n, p, q), that a nodal domain is similar to a clique, and that that all cliques in a random graph are of size O(log n). Therefore there are Ω(n/ log n) nodal domains. An upper bound proves to be a tougher challenge; however, we show the following in Section 4. We do this by showing that for any fixed k, we can partition almost the entire graph into nodal domains of size k. In order to show that there are many nodal domains in our graph of size k, we use quantum ergodicity [34] to show that eigenvector statistics emulate those of random Gaussians. Therefore, fix k, and consider ϕ the ith eigenvector of A, where A is the adjacency matrix of the graph G ∼ G(n, p, q). We proceed as follows. (1) For a set of vertices S, create a function f s (A S , ϕ(S)) on the induced subgraph on S that confirms there is a nodal domain inside S. (4) Use quantum ergodicity to compare E(p(A S , ϕ)) with E(p(A S , g)), where g is a multivariate standard normal Gaussian. (5) Show that with Gaussian inputs there are many nodal domains. This general method was used in [33] to show the two nodal domains of an unsigned Erdős-Rényi graph are approximately the same size in the bulk of the spectrum. However, there are key challenges specific to our question. (i) Our function f does not rely solely on ϕ, but on A as well. (ii) The polynomial approximation in [33] is found using the closure of univariate polynomials in a specific Sobolev norm. Our function testing for nodal domains must be multivariate, as we need to control the sign of k 2 edges at once. (iii) We need to worry about the overlap of different nodal domains. For example, merely checking the number of size k nodal domains is not sufficient, as all of these could overlap on one vertex. To solve (i), we have the added step of showing that A and ϕ are close to independent in f . Because ϕ is delocalized with high probability, perturbing a small number of entries does not significantly change ϕ or p s . We do this by Taylor expanding products of eigenvector entries. For (ii) we use results concerning the density of univariate polynomials in Sobolev norms of Rodríguez [45]. These results do not generalize to all multivariate polynomials, but nevertheless, we can interpret our function f as a composition of univariate functions, and show that approximation of each of these univariate functions is sufficient. For (iii), rather than count the number of sets of size k that are nodal domains, we count the number of sets of size s that contain a nodal domain, for s k. There is some delicacy needed in that we require a set size s that is large enough such that almost all sets contain a nodal domain, but small enough that when no sets of size s that have a nodal domain are left, there are only few vertices left. This tightness is shown by Janson's Inequality [35]. Results in quantum ergodicity concern finite degree polynomials. Therefore we must consider constant k rather than k up to log n, where we would expect these results to remain true. Moreover, the high probability statement is not strong enough to union bound over all indices at once. We suspect that the eigenvector entries are independent enough such that N (ϕ) will emulate the chromatic number of a G(n, p) graph and will be Θ(n/ log n). Conjecture 1.5. Fix constants 0 < , p, q < 1. With high probability, a G(n, p, q) graph has all eigenvectors ϕ with index i ∈ [ n, (1 − )n] with N(ϕ) = Θ(n/ log n). Note that we require our eigenvalue is in the "bulk" of the spectrum, in order to use quantum ergodicity. In Appendix A, we compare this result to the path nodal domains of Mohammadian and Ge and Liu, and show that counting path nodal domains does not deduce graph structure. Proposition 1.6. For fixed 0 < p, q < 1, consider the adjacency matrix of the graph G ∼ G(n, p, q). With probability n −ω(1) , every eigenvector ϕ satisfies κ(G < ϕ ) = 1. This follows using the same proof as [22], using more recent eigenvector delocalization results, as is done in [46]. We finish this discussion by giving a proof of the lower bound for G(n, p, q). Proposition 1.7. With high probability, for fixed 0 < p, q < 1, any eigenvector ϕ of the adjacency matrix of a G ∼ G(n, p, q) has N(ϕ) = Ω(n/ log 1 1−(p∧q) n). Proof. Consider any set of vertices S of size k. We will show that typically, there is no signing of eigenvectors that makes S a nodal domain. With high probability, ϕ is nonzero [39]. Take the signs of ϕ to be arbitrarily fixed. In order for a set of vertices S to be a nodal domain, we must have A ij ϕ(i)ϕ(j) ≥ 0. Each edge satisfies this with probability at most ((1−p)∨(1−q)). Therefore the probability that S forms a nodal domain is at most ((1 − p) ∨ (1 − q)) ( k 2 ) . Union bounding over all possible signings, if we assume without loss of generality that p ≥ q, the probability that S forms a nodal domain is at most 2 k (1 − q) ( k 2 ) = exp(k log 2 + k 2 log(1 − q)). Therefore the probability that there exists any such set is at most n k exp k log 2 + k 2 log(1 − q) . As n k ≤ n k , if, say, k = 3 log Classical Nodal Bounds In this section, we provide tight upper and lower bounds on the nodal count of a non-vanishing eigenvector of a symmetric matrix in terms of the corresponding eigenvalue index and multiplicity, and a number of graph invariants. In addition, for vanishing eigenvectors, we prove the existence of orthonormal eigenbases with strong nodal count satisfying Courant-type nodal upper bounds. Finally, we illustrate that the set of eigenbases satisfying such conditions may be of measure zero. 2.1. Non-Vanishing Nodal Count. Here, we prove tight bounds on the nodal count of a non-vanishing eigenvector. This result follows from [38] combined with an argument regarding the number of additional domains created by frustrated edges. However, we provide a direct proof by modifying a technique of Fiedler [26] and making use of known results regarding the inertia of signed Laplacian matrices. We have the following result. Proposition 2.1. Let M be a symmetric irreducible matrix and ϕ be a non-vanishing eigenvector corresponding to an eigenvalue of index k and multiplicity r. Then k + (r − 1) − ν ≤ N(ϕ) ≤ k + f, (2.1) where ν and f are the cyclomatic number and frustration index of the signed graph of M . Proof. Consider a symmetric, irreducible n × n matrix M with eigenpair (λ, ϕ), where λ has index k and multiplicity r, and ϕ is non-vanishing (ϕ(i) = 0 for all i). The matrix We make use of the following result regarding the inertia of a signed Laplacian: let κ + and κ − be the number of connected components of the graph of a n × n signed Laplacian L restricted to positive and negative entries, respectively; then B = D ϕ (M − λI)D ϕ , where D ϕ is a diagonal matrix with ϕ on the diagonal,κ + − 1 ≤ λ + ≤ n − κ − and κ − − 1 ≤ λ − ≤ n − κ + , where λ + and λ − are the number of positive and negative eigenvalues of L, respectively [13, Thm. 2.10]. Therefore, for B = D ϕ (M − λI)D ϕ , we have κ + − 1 ≤ n − k − r + 1 ≤ n − κ − and κ − − 1 ≤ k − 1 ≤ n − κ + . Let e + and e − be the number of pairs of off-diagonal entries of B that are positive and negative, respectively. Then κ + ≥ n − e + and κ − ≥ n − e − , giving our desired lower bound k + (r − 1) − ν ≤ n − κ + + 1 − ν ≤ κ − + e − + e + − (n + ν − 1) = κ − ≤ N M (ϕ).(ϕ) ≤ s ≤ κ − + f ≤ k + f, completing the proof. The proof actually produces a lower bound of k + (r − 1) − ν + ν + + ν − , where ν + and ν − are the cyclomatic numbers of the graphs of M restricted to entries where M ij ϕ(i)ϕ(j) > 0 and M ij ϕ(i)ϕ(j) < 0, respectively. However, in this work we attempt to focus on bounds in terms of graph invariants rather than quantities depending on the eigenvector itself. Below we give a simple example illustrating the tightness of the above bounds in general. Example 2.2. Let M be the negative adjacency matrix of the path on n vertices, where n + 1 is an odd prime, and consider B = M + C, for some < 2/(n + 1) 5 and symmetric matrix C with \|C ij \| ≤ 1, i, j ∈ [n]. The minimal eigenvalue gap of M is bounded below by min λ,λ ∈Λ(M ) \|λ − λ \| ≥ 3π 2 /4(n + 1) 2 , and, because n + 1 is an odd prime, every entry of each eigenvector of M is bounded away from zero, namely \|ϕ k (i)\| = 2/n sin[ikπ/(n + 1)] ≥ π/ √ 2(n + 1) 3/2 for all i, k ∈ [n]. Let {(µ k , φ k )} n k=1 be the eigenpairs of B. By [32, Cor. 8.1.6], \|λ k − µ k \| ≤ n, and so the spectrum of B is simple and interlaces with that of M . In addition, by [32, Thm. 8.1.12], ϕ k − φ k < π/ √ 2(n + 1) 3/2 , and so the eigenvectors of B are also non-vanishing and have the same sign pattern as the eigenvectors of M . By Fielder's tree theorem [26,Corollary 2.5], N M (ϕ k ) = k. To illustrate the tightness of the lower bound, we note that, for any ν ≤ k − 2, we may add ν edges to M (through the matrix C), connecting ν pairs of non-adjacent nodal domains, and resulting in N B (φ k ) = k − ν. For the upper bound, we note that when k, f n, each nodal domain is large and we may add non-crossing frustrated edges (with respect to the path ordering) within nodal domains, giving N B (φ k ) = k + f. Orthonormal Eigenbases with Classical Strong Nodal Count. When an eigenvector has some vanishing entries, the above bounds no longer hold in general. By slightly modifying an argument of Mohammadian [38], it is not hard to show that an upper bound of N s (ϕ) ≤ k + (r − 1) + f holds and is tight in general. However, by using a well-chosen orthonormal eigenbasis, we can obtain improved upper bounds, using a variation on a wellknown technique (see [10,Sec. 3.2]). Proposition 2.3. Let M be an n × n symmetric matrix. There exists an orthonormal eigenbasis ϕ 1 , ..., ϕ n ordered by increasing energy, satisfying N s (ϕ k ) ≤ k + f, k = 1, ..., n, (2.2) where f is the frustration index of the signed graph of M . Proof. It suffices to consider an n×n symmetric matrix M with an eigenvalue λ of index k and multiplicity r, and produce an orthonormal basis ϕ 1 , ..., ϕ r of the corresponding eigenspace satisfying N s (ϕ ) ≤ k+( −1)+f for = 1, ..., r. In addition, suppose, without loss of generality, that M is the matrix in the equivalence class {DM D \| D involutory diagonal matrix} that minimizes the number of positive off-diagonal entries, i.e., the number of positive off-diagonal entries of M equals f. We proceed by induction. Suppose that we have orthonormal eigenvectors ϕ 1 , ..., ϕ , < r, satisfying our desired nodal count N s (ϕ t ) ≤ k + (t − 1) + f for t = 1, ..., (if = 0, we have no such vectors). Consider an arbitrary eigenvector ϕ orthogonal to ϕ 1 , ..., ϕ . Consider a minimal nodal decomposition V 1 , ..., V t of [n]\i 0 (ϕ) for which i and j are in the same nodal domain only if M ij ≤ 0 (i.e., a nodal decomposition satisfying this condition of minimal size). Suppose that t > k + + f, otherwise we are already done. Let x p (i) = ϕ(i) i ∈ V p 0 otherwise , p = 1, ..., t and consider the set of vectors x α = t p=1 α p x p in their span, parameterized by α 1 , ..., α t . By the orthogonality of {x 1 , ..., x t }, this set is a subspace of R n of dimension t. Let α(i) := α p for i ∈ V p , p = 1, ..., t. If x α is a unit vector, then x T α M x α = λ − (i,j)∈E M ij ϕ(i)ϕ(j) (α(i) − α(j)) 2 . (2.3) If vertices i and j are in the same nodal domain, then α(i) = α(j). So, the only pairs (i, j) ∈ E for which M ij ϕ(i)ϕ(j) (α(i) − α(j)) 2 is strictly negative are those for which i and j are in different nodal domains, say i ∈ V p and j ∈ V q , and there exists some i * ∈ V p and j * ∈ V q such that M i * ,j * > 0. There are f edges with M i,j > 0, and so the subspace of vectors x α for which α(i) = α(j) for all M ij > 0 is of dimension at least t − f > k + . In addition, in this subspace, the Rayleigh quotient of all vectors x α is at most λ. By restricting our x α further to be orthogonal to ϕ 1 , ..., ϕ and the eigenspaces of the k − 1 eigenvalues strictly less than λ, we are left with a subspace of dimension of at least t − f − (k − 1) − > 1, which consists solely of eigenvectors of λ orthogonal to ϕ 1 , ..., ϕ . This implies that there exists an α in this subspace with α 1 = ... = α t−f−k− = 0, and therefore x α has N s (x α ) ≤ t − (t − f − k − ) = k + + f, completing the proof. Finally, by simply analyzing the Laplacian of a star graph, we note that the eigenbases satisfying the above proposition may be of measure zero. Example 2.4. Let M = n i=2 (e 1 − e i )(e 1 − e i ) T , e.g. , the graph Laplacian of a star. This matrix has eigenvalues 0, 1, and n, of multiplicity 1, n − 2, and 1, respectively. The eigenvalue λ = 1 has eigenspace E 1 = {x \| n i=2 x(i) = 0, x(1) = 0}. The x ∈ E 1 satisfying N s (x) ≤ 2 must have all but two entries equal to zero in an eigenspace of dimension n − 2. In the following section, we address this limitation by proving a more robust theorem regarding signings of eigenvectors. Orthonormal Eigenbases Satisfying Non-Vanishing Nodal Bounds In this section, we prove Theorem 1.3, breaking our analysis into three parts (as detailed in Section 1). First, we analyze the structure of repeated eigenvalues whose corresponding eigenvectors all vanish on some set of coordinates, Then, we place sign restrictions on an orthonormal basis of such an eigenspace so that, if such a basis exists, our desired nodal counts will be satisfied. Finally, we show that such non-vanishing orthonormal bases do exist and constitute a positive proportion of all orthonormal eigenbases. The proof of Theorem 1.3 is algorithmic in nature. For illustrative purpose, an example of this algorithm applied to a small matrix is given in Appendix B. 3.1. Part I: Structure of Eigenspaces with Vanishing Entries. Let M be a symmetric, irreducible n × n matrix with eigenvalue λ of index k and multiplicity r, and corresponding eigenspace E λ . Recall that i 0 (x) := {j ∈ [n] \| x(j) = 0}, and let i 0 (λ) = {j ∈ [n] \| ∀ϕ ∈ E λ , ϕ(j) = 0}. We note that i 0 (ϕ) = i 0 (λ) for all but a set of positive co-dimension of ϕ ∈ E λ . Suppose G[i 0 (λ)] has connected components on p vertex sets X 1 , ..., X p ⊂ i 0 (λ) and G [n]\i 0 (λ) has connected components on q vertex sets Y 1 , ..., Y q ⊂ [n]\i 0 (λ). Let us write the matrix M and an arbitrary eigenvector ϕ ∈ E λ in block notation with respect to the partition X 1 , ..., X p , Y 1 , ..., Y q : M = N A A TM , ϕ = 0 ϕ , where N =       N (1) 0 . . . 0 0 N (2) . . . . . . . . . . . . . . . 0 0 . . . 0 N (p)       ,M =       M (1) 0 . . . 0 0 M (2) . . . . . . . . . . . . . . . 0 0 . . . 0 M (q)       , are block diagonal matrices, N (i) ∈ R \|X i \|×\|X i \| for i = 1, ..., p, and M (j) ∈ R \|Y j \|×\|Y j \| for j = 1, ..., q, and A =    A (1,1) . . . A (1,q) . . . . . . . . . A (p,1) . . . A (p,q)    ,φ =    ϕ (1) . . . ϕ (q)    , A (i,j) ∈ R \|X i \|×\|Y j \| for i = 1, ..., p, j = 1, ..., q, and ϕ (j) ∈ R \|Y j \| for j = 1, ..., q. Next, we define H = (X, Y, E H ) to be the bipartite graph, with bipartition X = {x 1 , ..., x p } and Y = {y 1 , ..., y q }, representing the connectivity between the elements of {X i } p i=1 and {Y j } q j=1 , i.e., E H = (x i , y j ) \| A (i,j) = 0 . Let us define u(i) := smallest index j ∈ [q] such that x i ∼ H y j , v(j) := smallest index i ∈ [p] such that x i ∼ H y j . We order the elements of X and Y so that, for any i 1 > 1, d H (x i 1 , x i 2 ) = 2 for some i 2 < i 1 , and v(j 1 ) ≤ v(j 2 ) if j 1 < j 2 . In addition, we suppose that our original ordering of vertex sets X 1 , ..., X p and Y 1 , ..., Y q corresponds to the aforementioned ordering of x 1 , ..., x p and y 1 , ..., y q in H. See Figure 1 for an example of such a bipartite graph for a small matrix (a full example illustrating our procedure for this matrix is provided in Appendix B. From analysis of the eigenvalue-eigenvector equation, we note that E λ can be equivalently represented as the set of vectors ϕ (1) , ..., ϕ (q) satisfying q j=1 A (i,j) ϕ (j) = 0, i = 1, ..., p, and M (j) ϕ (j) = λ ϕ (j) , j = 1, ..., q. Let φ (j) 1 , ..., φ (j) r j be a non-vanishing orthonormal basis for the orthogonal projection of E λ to the indices of Y j , j = 1, ..., q. This projection (restricted to the indices of Y j ) is a subspace of the eigenspace of the matrix M (j) and eigenvalue λ. LetÊ λ = span{φ (j) σ } j=1,...,q σ=1,...,r j ,k be the index of λ with respect toM , andr be the dimension ofÊ λ . By eigenvalue interlacing,k +r ≤ k + r. Each eigenvector ϕ ∈ E λ can be represented uniquely in the basis φ (j) σ j=1,...,q σ=1,...,r j , say, ϕ = q j=1 r j σ=1 α j σ φ (j) σ . In fact, we can associate E λ with the subspace of {α j σ } j=1,...,q σ=1,...,r j ∼ = Rr for some constants c j ,σ ∈ R, = 1, ..., γ, j = 1, ..., q, σ = 1, ..., r j . satisfying q j=1 r j σ=1 α j σ A (i,j) φ (j) σ = 0, i = 1, ..., p, Recall our eigenspace E λ corresponds to the subspace of {α j σ } j=1,...,q σ=1,...,r j ∼ = Rr satisfying the γ homogeneous equations h {α j σ } j=1,...,q σ=1,...,r j = 0. Consider the matrix associated with these γ equations, where row i corresponds to the linear form h i , i = 1, ..., γ, and the columns {1, ...,r} correspond to the variables {α j σ } j=1,...,q σ=1,...,r j listed in reverse order: α q rq , ..., α q 1 , α q−1 r q−1 , ..., α q−1 1 , ..., α 1 r 1 , ..., α 1 1 (e.g., column one corresponds to α q rq , columnr to α 1 1 ). Let us further suppose that the system of equations h {α j σ } j=1,...,q σ=1,...,r j = 0 is in reduced row echelon form with respect to the aforementioned ordering of equations and variables. In this case, the pivot for h [·] is given by η , σ = argmax j,σ (j + σ/r j )1{c α η σ = − (j,σ) ∈Σ c j ,σ α j σ , = 1, ..., γ. With the values of pivot variables fixed, our eigenspace E λ is parameterized by the coefficients {α j σ } j=1,...,q σ=1,...,r j \ {α η σ } γ =1 , and in what follows we work directly with this formulation: E λ = span φ (j) σ − γ =1 c j ,σ φ (η ) σ (j, σ) = Σ . Finally, we note that, for two eigenvectors ϕ 1 , ϕ 2 ∈ E λ with coefficients { 1 α j σ } and { 2 α j σ }, respectively, ϕ 1 , ϕ 2 = (j,σ) ∈Σ 1 α j σ 2 α j σ + (ι,ω) ∈Σ 2 α ι ω γ =1 c j ,σ c ι ,ω . (3.1) 3.2. Part II: Restrictions That Produce Classical Nodal Bounds. Now that we have sufficiently characterized the structure of an eigenspace with vanishing entries, we are now prepared to restrict the choices of an orthonormal basis ϕ 1 , ..., ϕ r of E λ and the choices of signings of vanishing vertices (given by ε 1 , ..., ε r ) so that N(ε s ) ≤ k + (s − 1) + f for s = 1, ..., r, where k is the index of λ and f is the frustration index of the signed graph of M . Here we make use of the notation introduced in Subsection 3.1, and we assume M minimizes the number of pairs of positive off-diagonal entries over the set {DM D \| D involutory diagonal matrix }. Therefore where a ∈ X i for some i and b ∈ X i ∪ Y u(i) . By applying Proposition 2.1 to each of the q connected components of G[[n]\i 0 (λ)] and using the inequalityk +r ≤ k +r, any non-vanishing eigenvector ofφ ∈Ê λ satisfies (for an arbitrary integer s ≥ 0) In what follows, we always assume that any nodal decomposition ofφ under consideration also satisfies this property. Let us denote the coefficients corresponding to the eigenvector ϕ s by s α j σ . For each (ϕ s , ε s ), we aim to fix the signs ε s (j) of vanishing entries j ∈ i 0 (λ) and restrict the values of the elements s α j σ so that the original nodal count is decreased by at least q − γ − s − (f −f) (using q − γ − s + 1 elements) and ϕ s is orthogonal to ϕ t for all t < s (using s − 1 elements). N(φ) ≤k + (q − 1) +f ≤ k + (s − 1) + f + q − γ − s − (f −f) . First, we describe our signing of i 0 (λ) for each vector. Consider an arbitrary non-vanishing vectorφ ∈Ê λ . We restrict our signing so that ε is constant over each X i , i = 1, ..., p. In particular, we set: ε(a) = sgn(ϕ(b)) for all a ∈ X i and some fixed b ∈ Y u(i) in the neighborhood of X i . Ignoring edges between X and Y that are not between X i and Y u(i) for some i (i.e., each X i is connected to only one Y j ), the above signing of i 0 (λ) increases the nodal count by at most f, already giving the bound N(ε) ≤ N(φ) +f ≤ k + (s − 1) + f + q − γ − s − (f −f −f) .(3.3) Next we will choose the signs of some of our coefficients s α j σ so that the nodal count decreases further by using edges between X and Y that are not between X i and Y u(i) for some i. In particular, we restrict our basis coefficients s α j σ so that the nodal count of the s th eigenvector is at least q − γ − s − (f −f −f) less than the bound of Inequality 3.3. We need only consider s < q − γ, otherwise our desired bound already holds. Let us partition the variables s α j σ into four sets, based on their function in the analysis that follows: Π s E = variables fully restricted so that ϕ s ∈ E λ , Π s S = variables restricted in sign so that N(ε s ) ≤ k + (s − 1) + f, Π s O = variables fully restricted so that ϕ s , ϕ t = 0 for all t < s, Π s F = unrestricted variables. LetŶ = {y j \| j = η , = 1, ..., γ}. We note that \| Y \| ≥ q − γ, and denote the indices of the first q − γ elements of Y in Y by j 1 < ... < j q−γ . For s < q − γ, we can set Π s E = s α j σ \| (j, σ) ∈ Σ , Π s S = s α jm σ \| m = 2, ..., q − γ − s + 1, σ = 1, ..., r jm , Π s O = s α jm r jm \| m = q − γ − s + 2, ..., q − γ , Π s F = { s α j σ } j=1,...,q σ=1,...,r j \ Π s E Π s S Π s O . By definition, when s = 1, Π s O = ∅. We aim to show that the nodal count is at least q − γ − s − (f −f −f) less than Inequality 3.3 for each ε s by traversing the elements y j 1 , ..., y j q−γ , and restricting the signs of the variables in Π s S in some way. . We claim that these restrictions are consistent (e.g., there exist vectors simultaneously satisfying all conditions), and sufficient to produce our desired nodal bounds. Claim 3.1. Suppose an eigenvector ϕ = (j,σ) =Σ α j σ φ (j) σ − γ =1 c j ,σ φ (η ) σ ∈ E λ and signing ε ∈ {±1} n satisfy (1) ϕ(i)ε(i) ≥ 0 for all i ∈ [n], (2) ϕ(i) = 0 for all i ∈ [n]\i 0 (λ), (3) ε(a) = sgn(ϕ(b)) for all a ∈ X i and some fixed b ∈ Y u(i) in the neighborhood of X i ,(4)sgn r jm σ=1 α jm σ φ (jm) σ (a m ) = ε(b m ) for m = 2, ..., q − γ − s + 1, for some s > 0. Then N ε ≤ k + (s − 1) + f. Furthermore, there exists some (ϕ, ε) satisfying the above conditions for s = 1. Proof. We break our proof of the desired claim into two parts: first, we show the existence of (ϕ, ε) satisfying the conditions of the claim for s = 1, and then we show that satisfying Properties (1)-(4) for an arbitrary s implies N ε ≤ k + (s − 1) + f. We first aim to show that Properties (3) and (4) are consistent with each other, e.g., we can choose α j σ , (j, σ) ∈ Σ, and ε so that both properties simultaneously hold. By the construction of our pivots, any α η σ is a linear function of variables of lower index, e.g., variables α j σ with j ≤ η , and, if j = η , then σ < σ . Therefore, ε\| X i is a function of α j σ with j ≤ u(i). Property (4) is a sign restriction on the variables {α jm σ } r jm σ=1 corresponding to Y jm , depending on the quantity ε\| X v(jm) , which depends only on α j σ with j ≤ u(v(j m )). Therefore, it suffices to show that j m > u(v(j m )) for m > 1. Recall that d H (x i 1 , x i 2 ) = 2 for all i 1 > 1 and some i 2 < i 1 , and that v( j 1 ) ≤ v(j 2 ) if j 1 < j 2 . Because m > 1, we may consider j m−1 < j m , and note that v(j m−1 ) ≤ v(j m ). If v(j m−1 ) = v(j m ), then u(v(j m )) ≤ j m−1 < j m , proving the desired result. If v(j m−1 ) < v(j m ), then v(j m ) = 1, and so there exists some i < v(j m ) with d H (x i , x v(jm) ) = 2. This implies that there is some j with y j ∼ H x i and y j ∼ H x v(jm) , giving v(j) ≤ i < v(j m ), implying j < j m and so u(v(j m )) ≤ j < j m . Therefore, we may indeed choose α j σ , (j, σ) ∈ Σ, and ε so that both Properties (3) and (4) simultaneously hold. What remains is to formally choose ϕ so that Properties (2) and (4) hold. Each entry ϕ(i), i ∈ i 0 (λ), is a non-trivial linear function in α: ϕ(i) = (j,σ) =Σ α j σ φ (j) σ (i) − γ =1 c j ,σ φ (η ) σ (i) , and has a lexicographically largest pair (j, σ) for which the coefficient corresponding to α j σ is non-zero. If α j σ is not this variable for any i ∈ i 0 (λ), then we simply set α j σ = 0. We set the values of the remaining α j σ iteratively, starting with the smallest values of j (and, conditional on j, the smallest values of σ). Consider some α j σ not yet set, with all variables corresponding to smaller pairs (j , σ ) already set to some fixed value. If σ = r j and j = j m for some 1 < m < q − γ, then α j σ is restricted to the half-line defined by Property (4). Let us consider the set of linear functions ϕ(i) only in α j σ and variables corresponding to smaller pairs (j , σ ). With all variables of lower index set to fixed values, each function is a non-trivial linear function of only α j σ . By avoiding the at most n − \|i 0 (λ)\| choices of α j σ for which any of these linear functions can be zero (also choosing α j σ to satisfy Property (4) if σ = r j and j = j m for some 1 < m < q − γ), and recursing, we have constructed a pair (ϕ, ε) satisfying the conditions of the claim. What remains is to show that Properties (1)-(4) (for s arbitrary) implies that N ε ≤ k + (s − 1) + f. Recall, by Proposition 2.1 and eigenvalue interlacing, that N ε\| [n]\i 0 (λ) = N ϕ\| [n]\i 0 (λ) ≤ k + (s − 1) + f + q − γ − s − (f −f) . (3.4) Because ε is constant on each X i (by Property (3)), N ε\| X i is at most one plus the number of positive pairs of off-diagonal entries of M on the indices of X i . In addition, because ε\| X i has the same sign as ϕ(b) for some b ∈ Y u(i) in the neighborhood of X i (again, by Property (3)), either b is in the same nodal domain as some a ∈ X i , or there is a positive edge between X i and Y u(i) . Let G = ([n], E ) be the subgraph of G with E = E \ (a, b) ∈ E \| a ∈ X i , b ∈ Y j , j = u(i) . The subgraph of H corresponding to G , denoted by H , is a forest consisting of trees, each with root in Y and some number of leaves in X. Recall thatf equals the number of pairs of positive off-diagonal entries either within some X i or between some X i and Y u(i) . The nodal count N G (ε) is then at most N ε\| [n]\i 0 (λ) +f, as N ε\| X i is at most one plus the number of positive pairs of off-diagonal entries within X i , and either some vertex of X i is in the same nodal domain as a vertex of Y u(i) , or there is a positive off-diagonal entry between X i and Y u(i) . Combining this observation with Inequality 3.4 gives What's left is to show that the eigenvectors can simultaneously satisfy the conditions of Claim 3.1 and be orthogonal to each other, and that there is a set of co-dimension zero of such orthonormal bases. We break the remainder of the argument into two parts: First, we show that we can build orthogonal eigenvectors ϕ 1 , ..., ϕ q−γ−1 so that, for ϕ s , the conditions of Claim 3.1 are satisfied for vertices i≤v(j q−γ−s+1 ) X i and j≤j q−γ−s+1 Y j , but possibly not for the remaining vertices (Claim 3.2). Then, we show that such a set of orthogonal eigenvectors can be extended to an orthonormal basis, rotated so that the conditions of Claim 3.1 hold for all vertices, and that these conditions are maintained under sufficiently small rotations of the orthonormal basis (Claim 3.3). N G (ε) ≤ N ε\| [n]\i 0 (λ) +f ≤ k + (s − 1) + f + q − γ − s − (f −f −f) .N G (ε) ≤ N G (ε) − q − γ − s − (f −f −f) ≤ k + (s − 1) + f, Claim 3.2. There exists eigenvectors ϕ 1 , ..., ϕ q−γ−1 , ϕ s = (j,σ) =Σ s α j σ φ (j) σ − γ =1 c j ,σ φ (η ) σ ∈ E λ , and signings ε 1 , ..., ε q−γ−1 ∈ {±1} n such that (ϕ 1 , ε 1 ) satisfy the conditions of Claim 3.1 for s = 1, and (ϕ s , ε s ), s = 2, ..., q − γ − 1, satisfy (1) ϕ s (i)ε s (i) ≥ 0 for all i ∈ [n], (2) ϕ s (i) = 0 for all i ∈ j≤j q−γ−s+1 Y j , (3) ε s (a) = sgn(ϕ s (b)) for all a ∈ X i and some fixed b ∈ Y u(i) in the neighborhood of X i , i = 1, ..., v(j q−γ−s+1 ),(4)sgn r jm σ=1 α jm σ φ (jm) σ (a m ) = ε(b m ) for m = 2, ..., q − γ − s + 1, (5) ϕ s , ϕ t = 0 for all t < s. Proof. Claim 3.1 guarantees the existence of our desired pair (ϕ 1 , ε 1 ). To prove this claim, we repeat a version of the proof of Claim 3.1 in which the eigenvector ϕ s needs only satisfy half-space and non-vanishing conditions for Y j , j ≤ j q−γ−s+1 , has α j σ = 0 for all elements of Π s F with j > j q−γ−s+1 , and coefficients in Π s O chosen so that ϕ s is orthogonal to ϕ t , t < s. We proceed by induction on the invertibility of the matrices associated with this orthogonalization procedure. In particular, given ϕ 1 , ..., ϕ s−1 , and some fixed choice of values for s α j σ ∈ Π s O ∪ Π s E , orthogonality of ϕ s to ϕ t , t < s, is equivalent (by Equation 3.1) to the s − 1 elements s α j σ ∈ Π s O satisfying the s − 1 linear equations q−γ m=q−γ−s+2 s α jm r jm t α jm r jm + (ι,ω) ∈Σ t α ι ω γ =1 c jm ,r jm c ι ,ω = − sα j σ ∈Π s O ∪Π s E s α j σ t α j σ + (ι,ω) ∈Σ t α ι ω γ =1 c j ,σ c ι ,ω for t = 1, ..., s − 1. If the restrictions of ϕ 1 , ..., ϕ s−1 to Π s O are linearly independent, then the above system has a solution, as this restriction is a basis on Π s O , and so we may choose ϕ s Π s O so that each ϕ t Π s O , ϕ s Π s O , t < s, is equal to any quantity we desire. Therefore, it suffices to show that, at each step, the eigenvectors ϕ 1 , ..., ϕ s−1 restricted to Π s O are linearly independent. We begin with our base case of s = 2. Our matrix is a scalar; we simply require that 1 α j q−γ r j q−γ + (ι,ω) ∈Σ 1 α ι ω γ =1 c j q−γ ,r j q−γ c ι ,ω = 0, and note that the coefficient corresponding to 1 α j q−γ r j q−γ in the above linear function must be positive, as the eigenvector with α Then, for a sufficiently small choice of δ, sgn ϕ s−1 + δx equals sgn ϕ s−1 on Y j , j ≤ j q−γ−s+1 . By simply replacing ϕ s−1 by ϕ s−1 + δx, we now have our desired property, while maintaining all sign conditions. Claim 3.3. Let B λ be the set of orthonormal bases of E λ . Then there exists a manifold Φ λ ⊂ B λ of co-dimension zero, an ordering of basis elements {ϕ 1 , ..., ϕ r } ∈ Φ λ , and signings ε 1 , ..., ε r ∈ {±1} n satisfying ϕ s (i)ε s (i) ≥ 0 for all i ∈ [n], s = 1, ..., r, such that N(ε s ) ≤ k + (s − 1) + f, s = 1, ..., r. Proof. By Claim 3.2, there are eigenvectors ϕ 1 , ..., ϕ q−γ−1 that almost satisfy the conditions of Claim 3.1, but may vanish on [n]\ i 0 (λ) j≤j q−γ−s+1 Y j . Consider an arbitrary extension and re-normalization of ϕ 1 , ..., ϕ q−γ−1 to an orthonormal basis for E λ : ϕ 1 , ..., ϕ r . The vector ϕ 1 is non-vanishing on [n]\i 0 (λ), and using small rotations involving this vector, we may make every eigenvector in the basis non-vanishing on [n]\i 0 (λ) without changing the sign of any non-zero entry. To do so, we make use of Givens rotations G i,j (θ) ∈ R n×n , e.g., the orthogonal matrix with non-zero entries G i,j (θ) kk = 1 for k = i, j, G i,j (θ) kk = cos θ for k = i, j, and G i,j (θ) ij = − G i,j (θ) ji = sin θ. Let ρ be the magnitude of the smallest non-zero entry of ϕ 1 , ..., ϕ r , and consider the following sequence of Givens rotations: ϕ 1 ... ϕ r = r i=2 G 1,i (ρ/2 r ) ϕ 1 ... ϕ r . Any non-zero entry ϕ s (j) has the same sign as ϕ s (j), as \|ϕ s (j)\| − \|ϕ s (j)\| ≤ \|ϕ s (j)\| 1 − cos r−1 (ρ/2 r ) + (r − 1) sin(ρ/2 r ) ≤ 1 − (1 − (ρ/2 r ) 2 /2) r−1 + (r − 1)(ρ/2 r ) ≤ 2 r−1 (ρ/2 r ) 2 + (r − 1)(ρ/2 r ) < ρ, and any zero entry ϕ s (j), j ∈ i 0 (λ) results in a non-zero ϕ s (j), as each s = 1 has only one Givens rotation applied to it, and every entry ϕ 1 (j), j ∈ i 0 (λ), is non-vanishing and remains so after each rotation. Therefore, there exists a choice of {ϕ 1 , ..., ϕ r } ∈ B λ and a compatible set of signings ε 1 , ..., ε r ∈ {±1} n such that each eigenvector vanishes only on i 0 (λ), and (ϕ s , ε s ), s < q − γ, satisfy the conditions of Claim 3.2. Again, we recall that, for s ≥ q−γ, by simply setting ε(a) = sgn(ϕ(b)) for all a ∈ X i and some fixed b ∈ Y u(i) in the neighborhood of X i , i = 1, ..., p, we automatically have To complete the proof, we simply expand the point ϕ 1 , ..., ϕ r ∈ B λ using rotations of arbitrarily small angle. In particular, we set N(ε s ) ≤ k − γ + q − 1 + f ≤ k + (s − 1) + f. For s < q − γ, (ϕ s , ε s ),Φ λ = ( r 2 ) i=1 G p i ,q i (θ i ) ϕ 1 ... ϕ r p i , q i ∈ [r], θ i ∈ [−υ, υ], i = 1, ..., r , where υ < 2 −( r 2 ) min s∈1,...,r j∈[n]\i 0 (λ) \|ϕ s (j)\|. Considering an arbitrary entry ϕ s (j) > 0, we note that the composition of r 2 Givens rotations with angles in [−υ, υ] can change this entry by at most ϕ s (j) − cos ( r 2 ) (υ)ϕ s (j) − ( r 2 ) i=1 sin(υ) cos i−1 (υ) ≤ \|ϕ s (j)\|(1 − cos ( r 2 ) (υ)) + r 2 sin(υ) ≤ \|ϕ s (j)\|(1 − (1 − υ 2 /2) ( r 2 ) ) + r 2 υ ≤ 2 ( r 2 ) υ 2 + r 2 υ < ϕ s (j) , and so every basis in Φ λ has the same sign pattern as {ϕ 1 , ..., ϕ r }, and, therefore, the same ε 1 , ..., ε r . This completes the proof. Nodal Count of Signed Erdős-Rényi Graphs In this section, we prove Theorem 1.4. We consider an Erdős-Rényi random signed graph G(n, p, q), with 0 < p, q < 1 fixed constants. We give ϕ the index i. We prove the following. There is a constant γ > 0 such that for every k > 0, there is some N such that for n > N , the probability that N(ϕ) < 2n/k is at least 1 − n −γ . Therefore, by taking an infinite increasing sequence of k, the number of nodal domains is o(n) with probability 1 − O(n −γ ). In order to be consistent with the literature and reason about the spectrum, we work with both the adjacency matrix A and a normalized versionÃ := 1 √ (p+q)n A, so that E(Ã 2 u,v = 1/n). We proceed with the steps as listed in the introduction. and will show this is sufficient. To quantify whether a set \|S\| = s contains a nodal domain of size k, we consider the function f s : R ( s+1 2 ) → R, defined as f s (A S , √ nϕ(S)) := 1 >0    B∈( S k ) (u,v)∈( B 2 ) 1 >0 (nϕ(u)ϕ(v)A uv )    . Therefore f s asks whether S contains a nodal domain that is a clique. We call these clique domains. This is more specific than a general nodal domain, but is analytically easier to deal with. Moreover, we expect these form a constant portion of all nodal domains, so our asymptotic result does not change. Similar analysis would work with a slightly more complicated function that counts nodal domains of any type, namely 1 ≥0    B∈( S k ) 1 ≥0   T ∈T S uv∈T A 2 uv   (u,v)∈( B 2 ) 1 ≥0 (nϕ(u)ϕ(v)A uv )    where T S is the set of spanning trees of S. Note that we have renormalized f s so that, typically, its eigenvector inputs are Θ(1). 4.2. Part II: Polynomial Approximation. We approximate f s with a finite degree polynomial. We localize the distribution of A to A S by defining M ⊂ Mat sym (s) as the set of feasible assignments of A S . Specifically, M is the set of matrices M such that for any pair of indices u, v M uv ∈ {0} u = v {0, ±1} u = v. We equip M with the distribution of A S . For M ∈ M we will writeM := 1 √ (p+q)n M . From now on, we denote by g a length s vector of i.i.d. standard normal Gaussians of length s. Abusing notation, we will also denote the s dimensional probability measure of g by g. Lemma 4.2. For any 0 < δ < 1, C > 1, and M ∈ M, there is a finite degree polynomial p s : R ( s+1 2 ) → R, such that the following are true. We consider y ∈ R s to stand in for the contribution of ϕ. Then Before we prove this lemma, we give an idea of the method. We will work in a weighted Sobolev normed space. Our weight function µ is a probability measure on R defined as dµ(x) := 1 2 e −\|x\| . (4.5) We then define the Sobolev norm as f W k,p = ∞ −∞ k i=0 \|f (x) (i) \| p dµ(x) 1/p . The key fact is that under this norm, f s can be approximated by polynomials. This, by a result of Rodríguez, is implied by the fact that polynomials are dense in L p (R, µ) (see [36] Page 170). We now show that bounding the Sobolev norm also bounds the infinity norm on intervals. Lemma 4.4. For µ defined in (4.5) and F W 1,p ≤ , F 1 [−C,C] ∞ ≤ 4e 2C . Proof. We will show that because the derivative is small, if the infinity norm is large, then this breaks the norm bound. Assume that \|F (x)\| ≥ ζ for some x ∈ [−C, C]. For y ∈ [−2C, 2C], as dµ(y) ≥ 1 2 e −2C , \|F (x) − F (y)\| = y x F (t)dt ≤ 2e 2C y x F (t) dµ(t) ≤ 2e 2C y x F (t) p dµ(t) 1/p y x 1dµ(t) p−1 p ≤ 2e 2C \|x − y\| p−1 p . Therefore, if \|x − y\| < ζe −2C /(4 ) However, x+c x−c \|F (y)\|dµ ≤ ∞ −∞ \|F \| p dµ 1/p ∞ −∞ 1dµ p−1 p ≤ . Combining these gives ζ ≤ 2e 2C /c. The definition of c gives the result. Proof of Lemma 4.2. We first approximate f s (M, y) with a differentiable version. Therefore define the function η δ : R → R η δ (x) =    0 x ≤ 0 3x 2 /δ 2 − 2x 3 /δ 3 x ∈ [0, δ] 1 x ≥ δ . From now on, we write r := s k . Our differentiable approximation of f s is η 1/2    α∈[r] uv∈( α 2 ) η δ (M uv g u g v )    . The outer function receives parameter 1/2, considering we just want it to distinguish 0 from 1 or more. The inner function needs parameter δ, as it takes the Gaussian input. We approximate η 1/2 and η δ separately with polynomials P and Q. We take our outer approximation first. Consider C 2 := 2 s k . We approximate η 1/2 on the compact interval [−C 2 , C 2 ] with a polynomial P so that (P − η 1/2 )1 [−C 2 ,C 2 ] ∞ ≤ for some to be determined. We denote by d the degree of P . We choose Q by having it approximate η δ in W 1,p for p := k 2 4d. Using Lemma 4.3 and Lemma 4.4, we can approximate η δ to such accuracy that (Q − η δ )1 [−2C,2C] ∞ ≤ , Q Lp ≤ 1. (4.6) for some to be determined. This second inequality is possible as η δ Lp < 2 − 1 p . We can now show the properties of P ( i∈[r] Q(x i )). Denote by 1 C the event that g u g v ∈ [−C, C] for each pair of vertices u, v ∈ S. For the infinity norm, we consider   η 1/2    α∈[r] uv∈( α 2 ) η δ (M uv g u g v )    − P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )       1 C ∞ . (4.7) By (4.6), and the fact that η δ ∞ ≤ 1, we have that under 1 C , \| α∈[r] uv∈( α 2 ) η δ − α∈[r] uv∈( α 2 ) Q \| ≤ r (1 + ) ( k 2 ) − 1 . Using the approximation (1 + ) ( k 2 ) − 1 ≤ k 2 for small , (4.7) ≤ max x∈[0,r] \|x−y\|≤ k 2 r \|η 1/2 (x) − P (y)\| ≤ max x∈[0,r] \|x−y\|≤ k 2 r \|η 1/2 (y) − P (y)\| + \|η 1/2 (x) − η 1/2 (y)\| ≤ + 3 k 2 r. For sufficiently small , this satisfies (4.2) and (4.3). Now we will show (4.4). E   P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    2    ≤ E   P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    2 1 C    + E   P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    2 1 C    For the first term on the right, we have by the infinity norm bound, P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    2 1 C dg ≤ + 3 k 2 r For the other term, we use E   P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    2 1 C    ≤ E   P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    4    1/2 Pr(1 C ) 1/2 We can write P (x) 4 = 4d m=0 c m x m . This gives R s P    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    4 dg = 4d m=0 R s c m    α∈[r] uv∈( α 2 ) Q(M uv g u g v )    m dg ≤ 4d m=0 \|c m \|r m−1 α∈[r] R s uv∈( α 2 ) \|Q(M uv g u g v )\| m dg ≤ k 2 −1 4d m=0 \|c m \|r m−1 α∈[r] uv∈( α 2 ) R s \|Q(g u g v )\| m( k 2 ) dg. where at the last line we use the AM GM inequality and the symmetry of the measure. Recall that the product of two i.i.d. standard normal Gaussians has the tail bound Pr(\|g u g v \| ≥ t) ≤ 2e −t (see [52,Lemma 2.7.7]). Therefore, (1) For any fixed unit vector w ⊥ − → 1 [n] and any c > 0 there is a constant γ > 0 such that with probability 1 − O(n −γ ), R \|Q(g u g v )\| m( k 2 ) dg ≤ 4 ∞ −∞ \|Q(x)\| m( k 2 ) dµj =i 1 \|λ j − λ i \| ≤ n 1+c (4.8) and for every eigenvector ϕ j ofÃ, \| w, ϕ j \| ≤ n −1/2+c . (4.9) (2) With probability 1 − n −ω(1) ϕ ∞ ≤ log 4 n/ √ n. (4.10) We define Ω 4.5 to be the high probability event that (4.8),(4.9),(4.10) are true for some c < 1/20 and a finite set of vectors w to be determined throughout the course of the proof. We wish to use these delocalization results to control the change in ϕ(S) while changing A S . Therefore, consider the normalized block adjacency matrix of the G(n, p, q) graph Note that ψÃ(Ã S ) = ϕ(S). We can now decouple the dependence of A and ϕ. . Therefore taking the Taylor expansion at x = A S , \|ψÃ uv (M ) − ϕ(u)ϕ(v)\| ≤ ∞ k=1 2 (p + q)n k x 1 ,...x k ∈[( s 2 )] 1 k 1 !k 2 ! · · · k ( s 2 ) ! ∂ k ∂x 1 , . . . x k ψÃ uv (A S ) . (4.11) where k m is the number of times the mth edge is chosen. In order to calculate the partial derivative, we proceed as per [11,Section 4]. Define V x to be the matrix with 1's in the off-diagonal coordinates corresponding to x, but 0's elsewhere. Moreover, we denote the Green's function by G(z) := (A − z) −1 . For a vector w, taking a contour integral around only the ith eigenvalue gives, by the Cauchy residue formula, w, ϕ 2 = 1 2πi w, G(z)w dz. Therefore, by the Cauchy residue formula again, ∂ k ∂x 1 , . . . x k w, ϕ 2 = (−1) k k! 2πi w, ∈[k] (G(z)V x )G(z)w dz = k(−1) k k! j∈([n]\{i}) k w, ϕ j 1 w, ϕ (ϕ T j k V x k ϕ) ∈[k−1] (ϕ T j V x ϕ j +1 ) ∈[k] (λ j − λ i ) . Therefore, assuming Ω 4.5 , we have that for unit w ⊥ − → 1 [n] and ignoring a coefficient only depending on k, \|∂ (k) ab w, ϕ 2 \| ≤ k(k!) j∈([n]\{i}) k w, ϕ j 1 w, ϕ (ϕ T j k V ϕ) ∈[k−1] (ϕ T j V ϕ j +1 ) ∈[k] (λ j − λ i ) . ≤ k(k!)2 k (log n) 8k n −k−1+2c   j∈[n]\{i} 1 \|λ j − λ i \|   k ≤ k(k!)2 k (log n) 8k n −1+2c+kc . Here we use the three assumptions of Ω 4.5 . Notice that it is key that c < 1/2, which means we cannot consider all bulk eigenvectors at once. In order to use this on ϕ(u)ϕ(v), we write ϕ(u)ϕ(v) = 1 2 − → 1 u + − → 1 v , ϕ 2 − − → 1 u , ϕ 2 − − → 1 v , ϕ 2 . Therefore we orthogonalize each of these to − → 1 [n] with the three vectors w 1 = 1 2 + 4 n−2 ( − → 1 u + − → 1 v − 2 n − 2 − → 1 [n]\{u,v} ) w 2 = 1 1 + 1 n−1 ( − → 1 u − 1 n − 1 − → 1 [n]\{u} ) w 3 = 1 1 + 1 n−1 ( − → 1 v − 1 n − 1 − → 1 [n]\{v} ). We denote the target vectors 1 √ 2 ( − → 1 u + − → 1 v ), − → 1 u , − → 1 v by v 1 , v 2 , v 3 . We have for i = 1, 2, 3, v i , ϕ 2 − w i , ϕ 2 = v i + w i , ϕ · v i − w i , ϕ = (v i − w i ) + 2w i , ϕ · v i − w i , ϕ . Therefore, if we define y i (ϕ) = (v i − w i ) + 2w i , ϕ · (v i − w i ), ϕ , then ϕ(u)ϕ(v) = 1 2 2 w 1 , ϕ 2 − w 2 , ϕ 2 − w 3 , ϕ 2 + 1 2 (2y 1 (ϕ) − y 2 (ϕ) − y 3 (ϕ)). (4.12) Similar to before, we can write, k(k!)\|∂ (k) ab y i (ϕ)\| = j∈([n]\{i}) k v i − w i , ϕ j 1 (v i − w i ) + 2w i , ϕ (ϕ T j k V ϕ) ∈[k−1] (ϕ T j V ϕ j +1 ) ∈[k] (λ j − λ i ) . Under the assumption of Ω 4.5 , and the fact that v i − w i = O(n −1/2 ), (v i − w i ) + 2w i , ϕ · (v i − w i ), ϕ j 1 = O(n −1+c ). Therefore, by the same argument as before, \|∂ (k) ab y i (ϕ)\| ≤ k(k!)2 k (log n) 8k n −1+2c+kc . By (4.12) and the previous derivative bounds, \|ψÃ uv (M ) − ϕ(u)ϕ(v)\| ≤ 4 ∞ k=1 ( 2 (p + q)n ) k x 1 ,...,x k ∈[( s 2 )] 1 k 1 !k 2 ! · · · k ( s 2 ) ! ∂ k ∂x 1 , . . . x k ψÃ uv (Ã S ) ≤ 4 s 2 n −1+2c ∞ k=1 ( 2 (p + q)n ) k k2 k (log n) 8k n ck x 1 ,...,x k ∈[( s 2 )] k! k 1 !k 2 ! · · · k ( s 2 ) ! ≤ 4 s 2 n −1+2c ∞ k=1 ( 2 (p + q)n ) k k2 k (log n) 8k n ck s 2 k as we are summing over all multinomials. Therefore, we have \|ψÃ uv (M ) − ϕ(u)ϕ(v)\| = O(n −3/2+3c ) As F is a polynomial, assuming Ω 4.5 , \|(∂ k F )(A S , √ nϕ(S)))\| ≤ log C n for some fixed constant C and any direction. Therefore, assuming Ω 4.5 , we can once again expand according to the partial derivatives in the direction ψ uv . F (A S , √ nψÃ(M )) − F (A S , √ nϕ(S)) ≤ deg(F ) k=1 ∂ (k) (F (A S , √ nϕ(S))) n k/2 \|ψÃ uv (M ) − ϕ(u)ϕ(v)\| k k! = O(n −1+3c ). 4.4. Part IV: Quantum Unique Ergodicity. In this section, we utilize the powerful quantum ergodicity theorem to reduce our problem to one on polynomials of Gaussians. To do this, we introduce some new notation. Definition 4.7. We consider the following different expectations. (1) E A refers to the expectation across the choice of the random adjacency matrix A. (2) E S refers to the expectation over a randomly selected s × s principal submatrix of a fixed matrix A, for s constant. Here, we present a reduced form of the quantum unique ergodicity theorem for eigenvectors. ϕ of A of index i ∈ [ n, (1 − )n], there is a constant ν > 0 such that \|E A [F (n ϕ, w 2 )] − E(F (g 2 ))\| = O(n −ν ) where g 2 is a vector of squared normalized independent Gaussians. The following is an application of this lemma as vectors supported on a small subset S are close to orthogonal to − → 1 [n] . Proof of Lemma 4.10. Assume that m = 1. The proof for m = 2 is identical. We claim that in fact it is enough to show that there is some constant ν such that with probability 1 − O(n −ν ), E S (p s (A S , ϕ(S))) = E A [E M (p S (M, ϕ(S)))] + O(n −ν ). (4.14) To see why this is sufficient, assuming (4.14), then by Lemma 4.9 with F = E M (p s (M, √ nϕ(S))), with probability 1 − O(n −ν ∧ν 1 ), E S (p s (A S , ϕ(S))) = E(E M (p s (M, g))) + O(n −ν ∧ν 1 ) as desired. We will prove (4.14) through a Chebyshev inequality. Therefore we need to calculate the expectation and variance. First, for M ∈ M, we define E A (X M 1 Ω4.5 ) = E A ([E S (p s (A S , ϕ(S))1(A S = M ))] 1 Ω4.5 ) = Pr(A S = M ) E A E S (E M 1 (p s (A S , ψÃ(M 1 )))1 Ω4.5 + O(n −1+3c ) = Pr(A S = M ) E A (E S (E M 1 (p s (M 1 , ϕ(S))))1 Ω4.5 ) + O(n −1+3c ) = Pr(A S = M ) E A (E M 1 (p s (M 1 , ϕ(S)))1 Ω4.5 ) + O(n −1+3c ) for any fixed S. Therefore, summing over all M gives M ∈M E A (X M 1 Ω4.5 ) = E A (E M 1 (p s (M 1 , ϕ(S)))1 Ω4.5 ) + O(n −1+3c ). (4.16) Next we will deal with the second moment. We once again shift p s according to Lemma 4.6, We have Pr( E A (( M ∈M X M ) 2 1 Ω4.5 ) = M,M ∈M E A (E S [p s (M, ϕ(S))1(A S = M )]E S [p s (M , ϕ(S))1(A S = M )]1 Ω4.5 ) = M,M ∈M E A    1 n s 2 S 1 ,S 2 ∈( [n] s ) (p s (M, ϕ(S 1 ))(p s (M , ϕ(S 2 ))1(A S 1 = M, A S 2 = M ))1 Ω4A S 1 = M, A S 2 = M ) = Pr(A S 1 = M )Pr(A S 2 = M ) if S 1 ∩ S 2 = ∅. There are at most n s s n s−1 sets S 1 , S 2 such that S 1 ∩ S 2 = ∅. By (4.10), assuming Ω 4.5 , p s (M, ϕ(S)) = (log n) O (1) . Therefore We are almost done; we just need to remove the dependence on Ω 4.5 on the right hand side of (4.18). Therefore, we use (4.13) for F := E M [p s (M, ϕ(S)) 2 ] to obtain that for some constant E A (( M ∈M X M ) 2 1 Ω4.5 ) = M,M ∈M E A 1 n s 2 S 1 ,S 2 ∈( [n] s ) (E M 1 ,M 2 (p s (M 1 , ϕ(S 1 ))p s (M 2 , ϕ(S 2 )))) × Pr(A S 1 = M )Pr(A S 2 = M )1 Ω4.5 + O(n −1+3c ). which is E A [E M (p S (M, ϕ(S)))] 2 + O(n −1+3c ).ν 2 , \|E A E M (p S (M, ϕ(S)))1 Ω4.5 \| ≤ \|E A E M (p S (M, ϕ(S))) 2 \| 1/2 Pr(1 Ω4.5 ) 1/2 ≤ \|E A E M (p S (M, ϕ(S)) 2 ) \| 1/2 Pr(1 Ω4.5 ) 1/2 ≤ \|E(E M (p s (M, g) 2 ))\| 1/2 (1 + O(n −ν 2 ))Pr(1 Ω4.5 ) 1/2 = O(n −γ/2 ) (4.19) where the second to last line follows from Lemma 4.9 and the last line follows from (4.4) and Lemma 4.5. (4.14) follows from combining (4.18) and (4.19). 4.5. Part V: Counting Domains. We are now ready to think more directly about counting domains. We will show in this section, that with Gaussian inputs, the expectation is large. First, we show that E(E A [f s (A S , g)]) is large. Lemma 4.11. For any set \|S\| = (p ∧ q) −k , E[E A (f s (A S , g))] ≥ 1 − exp(−(p ∧ q) −k 2 (1/2−o k (1)) ). Proof. Without loss of generality, assume that p ≥ q. Fix some set S. Define I α to be the event that the set of k vertices α forms a k-clique nodal domain. We then define X := α∈( [S] k ) I α . In order to quantify the dependence across different α, we define ∆ := E M (X) + α∼β E M (I α I β ), where α ∼ β if E M (I α I β ) > E M (I α )E M (I β ) . By Janson's inequality, ([35], see [6] In order to utilize this, we compute the overlap, α∼β β∼α E M (I α I β ) ≤ α∼β k r=2 k r s − k k − r q −( r 2 ) E M (I α ) 2 ≤ 2 α∈( S k ) k 2 s − k k − 2 q −1 E M (I α ) 2 = o k (1)E M (X) 2 . Therefore by (4.20), Pr(X = 0) ≤ exp [−(1 − o k (1))E M (X)] . (4.21) In order to calculate E M (X), consider a specific α ∈ [S] k . For a vertex pair (u, v) ∈ α, Pr(A uv g u g v > 0) ≥ q. Therefore, the probability that all edges in α are positive is at least q ( k 2 ) , giving E M (X) ≥ q ( k 2 ) s k . Plugging this into (4.21) gives Pr(X = 0) ≤ exp −(1 − o k (1))q ( k 2 ) s k . We chose s = q −k in (4.1). By then using the approximation m j ≥ m j j j , we have Pr(X = 0) ≤ exp(−q −k 2 (1/2−o k (1)) ) and therefore E(E M (f s (g))) ≥ 1 − exp(−q −k 2 (1/2−o k (1)) ). Before we prove Theorem 1.4, we need one more eigenvector structure result. for all I ⊂ [n], \|I\| ≥ n. Note that this implies that with probability 1 − n −ω(1) , for some constant Given the statement of Lemma 4.11 concerning E A (f s (A S , g)), we translate this to a bound on the polynomial p s . For fixed C, δ > 0, define the event F C,δ as the event that ∀u, v ∈ S, g u g v ∈ [−C, 0] ∪ [δ, C]. We then have the decomposition c > 0, v : \|ϕ(v)\| ≤ δ n ≤ c (log 1 δ ) 2 n.\|E(p s (g)) − E(f s (g))\| ≤ E(p s (g)1 F C,δ ) − E(f s (g)1 F C,δ ) + E(p s (g)1 F C,δ ) + E(f s (g)1 F C ,δ ) . By (4.2), E(p s (g)1 F C,δ ) − E(f s (g)1 F C,δ ) ≤ δ. For the next term, we use (4.3). To bound the probability of F C,δ , note the maximum of the PDF of the univariate standard normal is 1 √ 2π , Therefore, by (4.4), E(p s (g)1 F C,δ ) ≤ E(p s (g) 2 ) 1/2 Pr(F C,δ ) 1/2 ≤ √ 2(s δ 2π + 2s 2 e −C ) 1/2 . The last error term is E(f s (g)1 F C,δ ) ≤ s 2 e −C . Combining this estimation with Lemma 4.10 and Lemma 4.11 gives that, for sufficiently small δ and sufficiently large C, there is some ν > 0 such that with probability 1 − O(n −ν ), (1))). Now, we want to limit the contribution of the bad sets in ϕ(S). Here, we set F C,δ (ϕ S ) to be the event that for each pair of vertices u, v ∈ S, nϕ(u)ϕ(v) ∈ ([−C, 0] ∪ [δ, C]). To give an lower bound on this probability, we use (4.22) and the trivial bound that the number of vertices of value at least C/n is at most n/C. By Lemma 4.10, E S (p s (A S , ϕ(S))) ≥ 1 − δ − exp(−q −k 2 (1/2 − o k\|E S (p s (A S , ϕ(S))1 F C,δ (ϕ S ) )\| ≤ \|E S (p s (A S , ϕ(S)) 2 )\| 1/2 Pr(F C,δ (ϕ S )) 1/2 ≤ (1 + O(n −ν ))\|E(E M (p s (M, g) 2 )\| 1/2 Pr(F C,δ (ϕ S )) 1/2 ≤ (1 + O(n −ν ))(1 + δ)( c (log 1 δ ) 2 + 1 C ) 1/2 . for some constant c. Therefore, for sufficiently large C, \|E S (p s (A S , ϕ(S))1 F C,δ )\| ≥ 1 − for := δ + exp(−q −k 2 (1/2−o k (1)) ) + c 2 (log 1 δ ) (4.23) where c 2 is some constant. We set δ such that the second term is the dominant term. Specifically, δ ≤ exp − exp(q −k 2 ) suffices. We now interpret p s on the set F C,δ . By 4.2, if S has a k-clique domain, then p s ≤ 1 + δ, and if S does not, p s ≤ δ. Therefore, the fraction of S that contains a clique domain is at least ζ, where ζ satisfies (1 + δ)ζ + (1 − ζ)δ = 1 − . This gives ζ = 1 − δ − ≥ 1 − 2 . Therefore, in our graph, at least (1 − 2 ) n s of all sets of size s contain a k clique nodal domain. In order to show that we can partition our graph into 2n/k nodal domains, we proceed greedily. We start with our entire vertex set [n]. We arbitrarily select an s-set S in our vertex set that contains a clique domain. We remove an arbitrary k-clique domain inside S from the vertex set. We repeat this process with our new vertex set, and continue until there are no sets s that contain a clique domain. As at least (1 − 2 ) n s s-sets contain a k-clique domain, this continues until there are at most r vertices left, for r s ≤ 2 n s . Once again using the approximation m j j j ≤ m j ≤ m j e j j j , we have that r s ≤ 2 n s e s . (4.24) By our definition of in (4.23) and s in (4.1), r ≤ n exp(−q −k 2 (1/2−o k (1) /s + 1) ≤ n exp(−q −k 2 (1/2−o k (1)+k + 1). Therefore, by this process, for large k we have at most n k + ne 1/s ≤ 2n/k nodal domains. Proposition A.1. Let 0 < p, q < 1 be constant, and A be the adjacency matrix of an Erdős-Rényi signed graph G(n, p, q). Then, with probability 1−n −ω(1) , κ(G > ϕ ) = 1 for all eigenvectors ϕ of A. As mentioned in Section 1, in the study of nodal counts of adjacency matrices, it is standard convention to consider domains where A ij ϕ(i)ϕ(j) > 0, as the negative of the adjacency matrix is a generalized Laplacian, which we follow below. Given that both p and q are constant and the result is independent of eigenvalue indexing, this convention has no impact on the result (e.g., the same result holds for κ(G < ϕ )). We first recall a number of known results from random matrix theory. Lemma A.2. Let A be the adjacency matrix of G(n, p, q), where 0 < p, q < 1 are fixed. The following are true. (1) (Nonzero vector [39]) With probability 1 − n −ω(1) , every eigenvector ϕ is nonzero − n ω(1) , A − (p − q)11 T ≤ 2 + o n (1) (p + q)n. To prove the desired result, we start with two weaker statements. Proof. Suppose κ(G > ϕ ) ≥ k. We denote by S an arbitrary set of vertices with exactly one vertex taken from each connected component of G > ϕ . The induced subgraph on S cannot contain an edge (u, v) satisfying A uv ϕ(u)ϕ(v) < 0. We now "re-sign" the vertices, by multiplying A on the left and the right by a diagonal matrix D, which has diagonal entry 1 in all entries that are not a vertex in S, and sgn(ϕ(u)) for vertices in S. According to this signing, (DAD) i,j ≤ 0, for all vertices u, v ∈ S. For fixed D and random A, the probability that (DAD) u,v has the correct sign is at most (1 − p ∨ q). Therefore, by union bounding over all possible D and sets of vertices S, we have the probability of κ(G > ϕ ) ≥ k is at most n k 2 k (1 − p ∨ q) ( k 2 ) This probability is n −ω(1) for k ≥ 3 log 1 1−p∨q n. Lemma A.4. With probability 1 − n −ω(1) , the second largest connected component of G > ϕ is at most one vertex. Proof. Denote by U the vertex set of the largest connected component of G > ϕ , which, by the previous lemma, must be of size at least n/(3 log 1 1−p∨q n). Any connected component V = U of size k must have all edges (u, v) ⊂ U × V satisfy A uv ϕ(u)ϕ(v) ≤ 0. We now consider a different re-signing, by multiplying A on the left and the right by a diagonal matrix D, which has diagonal entry 1 in all entries that are not in V , and sgn(ϕ(v)) for v ∈ V . For u ∈ U and v ∈ V , we have sgn(A uv ϕ(u)ϕ(v)) = sgn((DAD) uv ϕ(u)) ≤ 0. Assume that \|V \| ≥ 2. For u ∈ U and v 1 , v 2 ∈ V , if (u, v 1 ), (u, v 2 ) are edges, then sgn((DAD) uv 1 ) = sgn((DAD) uv 2 ). (A.2) With probability 1 − n ω(1) , the number of vertices in U in the shared neighborhood of v 1 , v 2 is (1 + o n (1))(p + q) 2 \|U \|. For any fixed signing D, the probability of (u, v 1 ), (u, v 2 ) having the same sign is at most p 2 + q 2 . Set k = \|V \|. Considering \|U \| ≥ n/(3 log 1 1−p∨q n), if we union bound over all n k possible sets V and 2 k signings, the probability of there being a signing D that satisfies (A.2) is at most n k 2 k (p 2 + q 2 ) (p+q) 2 n/(3 log 1/(1−p∨q) n)( k 2 ) . For k ≥ 2, this probability is n −ω(1) . We are now prepared to prove Proposition A.1. Proof of Proposition A.1. We break our analysis into two cases, treating the spectral radius separately from the rest of the spectrum. Suppose ϕ is a unit eigenvector corresponding to the spectral radius of A. By Fact (3) of Lemma A.2, with probability 1 − n −ω(1) , A has one eigenvalue equal to 1 + o n (1) (p − q)n and all other eigenvalues of magnitude at most 2 + o n (1) (p + q)n. Furthermore, with probability 1 − n −ω(1) , − → 1 [n] T A − → 1 [n] = n i,j=1 A ij = 1 + o n (1) (p − q)n(n − 1). Now, let us consider the representation of − → 1 [n] in the eigenbasis of A. We have − → 1 [n] T A − → 1 [n] ≤ 1 + o n (1) \|p − q\|n ( − → 1 [n] T ϕ i ) 2 + 2 + o n (1) (p + q)n n − ( − → 1 [n] T ϕ i ) 2 . Therefore, with probability 1 − n −ω(1) , ( − → 1 [n] T ϕ) 2 = 1 − o n (1) n, and so ϕ − sgn( − → 1 [n] T ϕ) − → 1 [n] / √ n = o n (1). This implies that ϕ has constant sign, say ϕ(u) > 0, on 1 − o n (1) n of its vertices. Now, consider an isolated vertex v in G > ϕ . With probability 1 − n −ω (1) , v has edges to (1 + o n (1))pn different vertices u satisfying ϕ(u) > 0, A u,v = +1 and A u,v ϕ(u)ϕ(v) < 0, and edges to (1 + o n (1))qn different vertices w satisfying ϕ(w) > 0, A w,v = −1 and A w,v ϕ(w)ϕ(v) < 0, a contradiction. Therefore, κ(G > ϕ ) > 1 with probability n −ω(1) . Now, consider an arbitrary unit eigenvector ϕ corresponding to an eigenvalue in the rest of the spectrum. Suppose there is a v that is an isolated vertex in G > ϕ . With probability 1 − n −ω (1) , v has edges to (1 + o n (1))(p + q)n different vertices, u, all of which satisfy A u,v ϕ(u)ϕ(v) < 0. The eigenvector equation at v gives λϕ(v) = u∼v A uv ϕ(u). Therefore, by the eigenvector equation at v and noting that A u,v ϕ(u)ϕ(v) < 0 for u ∼ v, we have \|λϕ(v)\| = Consider the following non-vanishing orthogonal 3 bases for the orthogonal projections of E λ onto Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , Y 6 and Y 7 : φ (1) 1 Y 1 , φ(1)2 Y 1 = φ (5) 1 Y 5 , φ(5)2 Y 5 =      1 2 −3   ,   −5 4 1      , φ(2)1 Y 2 = φ (3) 1 Y 3 = φ (4) 1 Y 4 = φ (6) 1 Y 6 = φ(7) 1 Y 7 = (1) , and let us consider a vector in their span: ϕ = α 1 1 φ (1) 1 + α 1 2 φ (1) 2 + α 2 1 φ (2) 1 + α 3 1 φ(3) 1 + α 4 1 φ 1 + α 5 1 φ 1 + α 5 2 φ 2 + α 6 1 φ 1 + α 7 1 φ 1 . The vector ϕ ∈ E λ if and only if ϕ(5) + ϕ(7) + ϕ(11) = ϕ(5) + ϕ(10) + ϕ(13) + ϕ(14) = ϕ(3) + ϕ(4) + ϕ(11) = 0, which, in terms of α's, produces our γ = 3 homogeneous equations: −3α 1 1 + α 1 2 + α 2 1 + 2α 5 1 + 4α 5 2 = 0, −3α 1 1 + α 1 2 + α 3 1 + α 4 1 + α 5 1 − 5α 5 2 = 0, 2α 5 1 + 4α 5 2 + α 6 1 + α 7 1 = 0. Performing Gaussian elimination, and solving for the pivots, we obtain: Each ϕ(i), i ∈ i 0 (λ), is a linear function of {α j σ } j=1,...,q σ=1,...,r j \ {α η σ } γ =1 . For i ∈ Y j for some y j ∈Ŷ , the linear function is obvious. We write the explicit function for each i ∈ Y 5 ∪ Y 7 below: ϕ(4) = α 7 1 = −α 6 1 + α 2 1 + α 1 2 − 3α 1 1 , ϕ(10) = α 5 1 − 5α 5 2 = −α 4 1 − α 3 1 − α 1 2 + 3α 1 1 , ϕ(11) = 2α 5 1 + 4α 5 2 = −α 2 1 − α 1 2 + 3α 1 1 , ϕ(12) = −3α 5 1 + α 5 2 = α 4 1 + α 3 1 + α 2 1 + 2α 1 2 − 6α 1 1 . We are now prepared to produce an eigenbasis {ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 , ϕ 5 , ϕ 6 } and corresponding signings {ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 } of E λ satisfying Theorem 1.3. For i 0 (λ), we set ε s (6) = ε s (9) = sgn ϕ s (5) , ε s (15) = ε s (16) = sgn ϕ s (10) , and ε s (8) = sgn ϕ s (11) , s = 1, 2, 3, 4, 5, 6. We need only consider ϕ s , s < q−γ; an arbitrary non-vanishing extension suffices for ϕ 4 , ϕ 5 , ϕ 6 . We have ϕ 1 : Π 1 E = { 1 α 5 1 , 1 α 5 2 , 1 α 7 1 }, Π 1 S = { 1 α 2 1 , 1 α 3 1 , 1 α 4 1 }, Π 1 O = ∅, Π 1 F = { 1 α 1 1 , 1 α 1 2 , 1 α 6 1 }, ϕ 2 : Π 2 E = { 2 α 5 1 , 2 α 5 2 , 2 α 7 1 }, Π 2 S = { 2 α 2 1 , 2 α 3 1 }, Π 2 O = { 2 α 4 1 }, Π 2 F = { 2 α 1 1 , 2 α 1 2 , 2 α 6 1 }, ϕ 3 : Π 3 E = { 3 α 5 1 , 3 α 5 2 , 3 α 7 1 }, Π 3 S = { 3 α 2 1 }, Π 3 O = { 3 α 3 1 , 3 α 4 1 }, Π 3 F = { 3 α 1 1 , 3 α 1 2 , 3 α 6 1 }. We begin with ϕ 1 , and follow the proof of Claim 3.1. We set 1 α 1 1 = 0. The variable 1 α 1 2 cannot be zero, but is otherwise unconstrained. We set 1 α 1 2 = 1, and so ε 1 (6) = ε 1 (9) = sgn(ϕ 1 (5)) = sgn(1) = +1. The variable 1 α 2 1 cannot be zero, and must satisfy sgn( 1 α 2 1 φ (2) 1 (7)) = sgn( 1 α 2 1 ) = ε 1 (6) = +1 and ϕ 1 (11) = − 1 α 2 1 − 1 = 0. We set 1 α 2 1 = 1, implying that sgn(ϕ 1 (11)) = −1, and so ε 1 (8) = −1. The variable 1 α 3 1 cannot be zero, and must satisfy sgn( 1 α 3 1 φ 1 (13)) = sgn( 1 α 3 1 ) = ε 1 (9) = +1. We set 1 α 3 1 = 1. As in the previous two cases, 1 α 4 1 must be positive, but must also satisfy ϕ 1 (10) = − 1 α 4 1 − 1 − 1 = 0 and ϕ 1 (12) = 1 α 4 1 + 1 + 1 + 2 = 0. We set 1 α 4 1 = 1, implying that sgn(ϕ 1 (10)) = −1, and so ε 1 (15) = ε 1 (16) = −1. All that remains is to set 1 α 6 1 . It must satisfy ϕ 1 (4) = − 1 α 6 1 + 1 + 1 = 0. We set 1 α 6 1 = 1, giving ϕ 1 = (−5, 4, 1, 1, 1, 0, 1, 0, 0, −3, −2, 5, 1, 1, 0, 0) T , ε 1 = (−1, +1, +1, +1, +1, +1, +1, −1, +1, −1, −1, +1, +1, +1, −1, −1) T . Next, we consider ϕ 2 , and follow the proof of Claim 3.2. Here, we are only concerned with half-space and orthogonality conditions, as vanishing entries can be addressed using ϕ 1 and an appropriate Givens rotation (see Claim 3.3). We set 2 α 1 1 = 0. The variable 2 α 1 2 cannot be zero, but is otherwise unconstrained. We set 2 α 1 2 = 1, and so ε 2 (6) = ε 2 (9) = sgn(ϕ 2 (5)) = sgn(1) = +1. The variable 2 α 2 1 must satisfy sgn( 2 α 2 1 φ (2) 2 (7)) = sgn( 2 α 2 1 ) = ε 2 (6) = +1. We set 2 α 2 1 = 1. The variable 2 α 3 1 must satisfy sgn( 2 α 3 1 φ 2 (13)) = sgn( 2 α 3 1 ) = ε 2 (9) = +1. We set 2 α 3 1 = 1. We also set 2 α 6 1 = 1. What remains is to set 2 α 4 1 so that ϕ 1 , ϕ 2 = ϕ 1 , (−5, 4, 1, 1, 1, 0, 1, 0, 0, − 2 α 4 1 − 2, −2, 2 α 4 1 + 4, 1, 2 α 4 1 , 0, 0) T = 9 2 α 4 1 + 76 = 0. We set 2 α 4 1 = −76/9, giving ϕ 2 = (−5, 4, 1, 1, 1, 0, 1, 0, 0, 58/9, −2, −40/9, 1, −76/9, 0, 0) T , ε 2 = (−1, +1, +1, +1, +1, +1, +1, −1, +1, +1, −1, −1, +1, −1, +1, +1) T . Finally, we consider ϕ 3 . We set 3 α 1 1 = 0 and 3 α 1 2 = 1, implying that ε 3 (6) = ε 3 (9) = +1, and so 3 α 2 1 must be positive. We set 3 α 2 1 = 1, and also set 3 α 6 1 = 1. We set 3 α 3 1 and 3 α 4 1 so that ϕ 1 , ϕ 3 = ϕ 1 , (−5, 4, 1, 1, 1, 0, 1, 0, 0, − 3 α 3 1 − 3 α 4 1 − 1, −2, 3 α 3 1 + 3 α 4 1 + 3, 3 α 3 1 , 3 α 4 1 , 0, 0) T = 9 3 α 3 1 + 9 3 α 4 1 + 67 = 0, ϕ 2 , ϕ 3 = ϕ 2 , (−5, 4, 1, 1, 1, 0, 1, 0, 0, − 3 α 3 1 − 3 α 4 1 − 1, −2, 3 α 3 1 + 3 α 4 1 + 3, 3 α 3 1 , 3 α 4 1 , 0, 0) T = −(89/9) 3 α 3 1 − (58/3) 3 α 4 1 + (263/9) = 0. We set 3 α 3 1 = 55/3 and 3 α 4 1 = −98/9, giving ϕ 3 = (−5, 4, 1, 1, 1, 0, 1, 0, 0, −76/9, −2, 94/9, 55/3, −98/9, 0, 0) T , ε 3 = (−1, +1, +1, +1, +1, +1, +1, −1, +1, −1, −1, +1, +1, −1, −1, −1) T . Below we provide a nodal decomposition for each ε s , s = 1, 2, 3, that satisfies the bound of Theorem 1.3: G = ([n], E, σ), where [n] := {1, ..., n}, E = {(i, j) \| M ij = 0, i = j}, and σ : E → {±1} is defined by σ ij = −sgn(M ij ). Let G[S] be the induced subgraph of G on vertex set S ⊂ [n], with corresponding edge set E[S] ( 3 ) 3Prove that a positive proportion of orthonormal bases of E λ are non-vanishing on [n]\i 0 (λ) and satisfy the half-space conditions of Step (3). Theorem 1. 4 . 4For any 0 < , p, q < 1, there is some constant γ > 0 such that for any fixed index i ∈ [ n, (1 − )n], with probability 1 − O(n −γ ), the ith eigenvector ϕ has N(ϕ) = o(n) nodal domains. ( 2 ) 2Approximate f with a finite degree polynomial p s of entries of the matrix and entries of the eigenvector.(3) Show that the matrix entries and eigenvector are close to independent in p. then with probability n −Ω(log n) there are no nodal domains of size k. As no nodal domain can have size k, there are at least Ω(n/ log 1 1−q n) domains. The desired result follows from a union bound over all ϕ. is a signed Laplacian matrix, and, by Sylvester's law of inertia [32, Thm. 8.1.17], has n − k − r + 1 positive and k − 1 negative eigenvalues. Suppose (w.l.o.g.) that M has exactly f pairs of positive off-diagonal entries. Then, by choosing a nodal decomposition [n] = s =1 V where entries i and j are in the same nodal subset only if M ij ≤ 0 and s is as small as possible, we have N M Figure 1 . 1The signed graph G = ([16], E, σ) associated with the example matrix M used in Appendix B, and the corresponding bipartite graph for the eigenspace corresponding to λ = 0. The matrix has diagonal entries equal to −1 for i = 1, 2, 5, 10, 11, 12, has off-diagonal entries equal to +1 for dashed edges and −1 for solid edges, and zeros otherwise. The vertices6,8,9,15, 16 vanish on the eigenspace of λ = 0. See Appendix B for the full analysis associated with this matrix. or, by Gaussian elimination, γ :=r − r homogeneous equations h {α j σ } j=1,...,q σ=1,... ..., γ, and we denote the set of pivots by Σ := {(η , σ ) \| = 1, ..., γ}. Let us fix the values of the variables corresponding to the γ pivots so that the equations h [{α j σ }] = 0, = 1, ..., γ, are satisfied: there are exactly f positive off-diagonal entries. Let us define: f = number of pairs of positive off-diagonal entries ofM , f = number of pairs of positive off-diagonal entries {M a,b , M b,a } as noted in the proof of Proposition 2.1, each eigenvectorφ has a nodal decomposition satisfying (3.2) where vertices i and j are in the same nodal subset only if M ij ≤ 0. For each pair j m , v(j m ), m = 2, ..., q − γ, let a m , b m ∈ [n] be a pair of vertices satisfying a m ∈ Y jm , b m ∈ X v(jm) , and a m ∼ G b m . For every s α m ) = ε s (b m ), resulting in a m and b m being in the same nodal domain if there is no frustrated edge between Y jm and X v(jm) We have j m > u(v(j m )) for m > 1, and so j m and v(j m ) are in different trees of H . Let H t be the graph resulting from the addition of edges {(j m , v(j m )) \| m = 2, ...t} to H . More generally, j t+1 and v(j t+1 ) are in different connected components of H t for any t = 2, ..., q − γ − s. Now, let us consider the effect of the addition of each of the edges (j m , v(j m )), m = 2, ..., q − γ − s + 1, on our nodal bound (e.g., consider the nodal count of the sequence of graphs H , H 2 , ...., H q−γ+s−1 ). Property (4) implies that, for m = 2, ..., q − γ − s + 1, either a m and b m are in the same nodal domain (implying that H m has one less nodal domain than H m−1 ), or there is a frustrated edge of M between Y jm and X v(jm) , and so . Part III: Many Orthonormal Bases Satisfy the Conditions of Part II. Claim 3.1 shows that the conditions for our eigenvectors ϕ 1 , ..., ϕ r (and corresponding signings) are satisfiable and, if these conditions are satisfied, then the desired nodal bound is achieved. all other variables zero must have non-zero norm. By adding this single linear constraint to ϕ 1 , we have satisfied our base case of s = 2. Now consider an arbitrary s > 2. After selecting ϕ s−1 , if ϕ 1 , ..., ϕ s−1 restricted to Π s O are linearly independent, then we simply choose our coefficients for ϕ s as in the proof of Claim 3.1 for Y j , j ≤ j q−γ−s+1 , and choose Π s O to satisfy the above system. If the eigenvectors are linearly dependent, then, by induction ϕ 1 Π s O , ..., ϕ s−2 Π s O are linearly independent, and so ϕ s−1 Π s O is in the span of the other vectors. Let x ∈ E λ be a vector orthogonal to ϕ 1 , ..., ϕ s−2 such that x satisfying the conditions of Claim 3.2 and not vanishing on [n]\i 0 (λ) implies, by Claim 3.1, that N(ε s ) ≤ k + (s − 1) + f in this case as well. 4. 1 . 1Part I: Function Definition. In order to count nodal domains, we choose some s such that almost all sets of s vertices contain a nodal domain of size k. We set s := (p ∧ q) −k (4.1) ( 1 ) 1For any M ∈ M, p s satisfies the following bounds.\|p s (M, y) − f s (M, y)\| ≤ δ y : ∀i, j ∈ [s] 2 , y i y j ∈ [−C, 0] ∪ [δ, C] (4.2) \|p s (M, y)\| ≤ 1 + δ y : ∀i, j ∈ [s] 2 , y i y j ∈ [−C, C] (4.3)(2)For any M ∈ M, p s (M, y) is even in y and E p s (M, g) 2 ≤ 2. (4.4) Lemma 4. 3 . 3[[45] Proposition 4.2] For µ defined in (4.5), k ∈ N and 1 ≤ p < ∞, polynomials are dense in the W k,p among all functions that have bounded W k,p norm. , then \|F (y)\| ≥ ζ/2. Set c := min{C, ζe −2C /(4 ) p p−1 }. Then, once again using the bound dµ(y) ≥ 1 2 e −2C , x+c x−c \|F (y)\|dµ ≥ cζe −2C /2. Q Pr(1 C ) ≤ 2re −\|C\| , for sufficiently large C(M uv g u g v )    L 2 (g) ≤ 1 + + Pr(1 C ) 1/2 8 m \|c m \|r m ≤ 2. 4.3. Part III: Independence. We use the following structural laws concerning the spectrum and eigenvectors. These structural results are a combination of [34, Equation 4.11], [11, Proposition 4.3], and [25, Corollary 3.2].Lemma 4.5. Consider eigenvalue λ i ofÃ with eigenvector ϕ. Fixing the rest ofÃ, we can replaceÃ S withM to create a new adjacency matrix. Namely, givenÃ, we define a function ψÃ(M ) : Mat sym (s) → R s , where ψÃ(M ) = ϕM (S), for ϕM the ith eigenvector of ÃSÃS,SA S,SM . Lemma 4. 6 . 6Assume Ω 4.5 . Then for any finite degree even polynomial F : R s → R and M ∈ M,\|F ( √ nψÃ(M )) − F ( √ nϕ(S))\| = O(n −1+3c ).Proof. We track the change in ϕ(u)ϕ(v) for u, v ∈ S. Therefore we consider the function ψÃ uv (x) : R ( s 2 ) → R defined asψÃ uv (M ) := [ψÃ(M )](u)[ψÃ(M )](v). For M, M ∈ M, ( 3 ) 3E M refers to the expectation, given a matrix A and a subset S ⊂ V , over replacing A S with M ∈ M, according to the distribution of M. (4) E without a subscript refers to the expectation over some function of g. Lemma 4.8. [[11] Theorem 1.5] Fix > 0, a finite degree polynomial F : R → R, and any unit vector w ⊥ − → 1 [n] . For an eigenvector Lemma 4. 9 . 9[[33] Lemma 2.1] For any finite degree even polynomial F : R s → R, and finite set of vertices S, there is a constant ν 1 > 0 such that\|E A (F ( √ nϕ(S))) − E(F (g))\| = O(n −ν 1 ). (4.13)Here, F does not take A S as an input. However, because of the near independence of A S and ϕ(S) in p s shown in Lemma 4.6, we can translate this into a similar bound on p s (A S , ϕ(S)). Lemma 4 . 10 . 410For any C, δ > 0 and p s defined in Lemma 4.2, there is a constant ν > 0 such that with probability 1 − O(n −ν ), for m ∈ {1, 2}, \|E S (p s (A S , ϕ(S)) m ) − E(E M (p s (M, g) m ))\| = O(n −ν ). X M := E S (p s (A S , ϕ(S))1(A S = M )). (4.15) By Lemma 4.6, under Ω 4.5 , p s (A S , ϕ(S)) = E M 1 (p s (A S , ψÃ(M 1 ))) + O(n −1+3c ), for M 1 distributed according to M. Therefore, E M 1 ,M 2 (p s (M, ψÃ(M 1 ))p s (M , ψÃ(M 2 ))) + O(n −1+3c ) ×Pr(A S 1 = M, A S 2 = M )1 Ω4.5 \|E S (p s (A S , ϕ(S))) − E A [E S (E M (p S (M, ϕ(S))))1 Ω4.5 ]\| ≥ n −c ) = O(n −((1+c)∧γ)).(4.18) ) ≤ exp(−E M (X) 2 /∆).(4.20) Lemma 4 . 412 ([47] Theorem 1.5). There is some constant c 2 such that for any > 0, with probability 1 − n −ω(1) , every eigenvector of A satisfies v(I) 2 ≥ ( · e −c 2 / √ ) 7 v 2 . Proof of Theorem 1.4. Throughout this proof, when we write p s (·), namely with one input, we mean the function p s : R s → R defined as p s (x) := E M (p s (M, x)). ( 2 ) 2(Combination of Lemma 4.12 and (4.10)) For any fixed c, with probability 1 − n ω(1) , every unit eigenvector ϕ is such that any set of vertices of size \|S\| ≥ cn satisfies Yin Theorem[8], see[52, Theorem 4.4.5]) With probability 1 Lemma A. 3 . 3With probability 1 − n −ω(1) , every eigenvector ϕ of the adjacency matrix has κ(G > ϕ ) ϕ( 6 ) 6However, considering the infinity norm bound on the eigenvector from (4.10) and the Bai Yin theorem (Fact (3) of Lemma A.2), this happens with probability at most n −ω(1) .Appendix B. An illustrative exampleConsider the 16 by 16 symmetric matrix M with M ii = −1 for i = 1{i, j} = {6, 9}, {15, 16} −1 for {i, j} = {1, 2}, {1, 5}, {2, 5}, {3, 8}, {4, 8}, {5, 6}, {5, 9}, {6, 7}, {6, 11}, {8, 11}, {9, 10}, {9, 13}, {9, 14}, {10, 11}, {10, 12}, {10, 15}, {10, 16}, {11, 12}, {11, 15}, {11, 16}, {12, 15}, {12, 16} 0 otherwise for i = j. See Figure 1 for the signed sparsity graph G = ([16], E, σ) of M . Consider the eigenvalue λ = 0 of M , with index k = 7, multiplicity r = 6, and corresponding eigenspace = ϕ(8) = ϕ(9) = ϕ(15) = ϕ(16) = ϕ(1) + ϕ(2) + ϕ(5) = ϕ(5) + ϕ(7) + ϕ(11) = ϕ(5) + ϕ(10) + ϕ(13) + ϕ(14) = ϕ(10) + ϕ(11) + ϕ(12) = ϕ(3) + ϕ(4)We haveX 1 = {6, 9}, X 2 = {15, 16}, X 3 = {8}, Y 1 = {1, 2, 5}, Y 2 = {7}, Y 3 = {13}, Y 4 = {14}, Y 5 = {10, 11, 12}, Y 6 = {3}, and Y 7 = {4}. The corresponding bipartite graph H on vertices X = {x 1 , x 2 , x 3 } and Y = {y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 } is inFigure 1. = {(η 1 , σ 1 ), (η 2 , σ 2 ), (η 3 , σ 3 )} = {(5, 1), (5, 2), (7, 1)},Ŷ = {y 1 , y 2 , y 3 , y 4 , y 6 }, q − γ = 7 − 3 = 4, and j i = i for i = 1, ..., q − γ. ε 1 : 1{1}, {2, 5, 6, 7}, {3}, {4}, {8, 10, 11, 15}, {9, 13, 14}, {12}, {16}, ε 2 : {1}, {2, 5, 6, 7}, {3}, {4}, {8, 11, 12}, {9, 10, 13, 15}, {14}, {16}, ε 3 : {1}, {2, 5, 6, 7}, {3}, {4}, {8, 10, 11, 15}, {9, 13}, {12}, {14}, {16}. Berkolaiko stated a slightly weaker version of the aforementioned result, but the associated proof also proves the aforementioned version. For simplicity in this example, we use an orthogonal basis rather than an orthonormal one. However, as a result, Equation 3.1 does not apply, and the norms of eigenvectors must be taken into account. AcknowledgementsThe authors would like to thank Lior Alon and Sergey Lototsky for helpful discussions.Appendix A. Path nodal domains of signed Erdős-Rényi graphsConsider an Erdős-Rényi signed graph G(n, p, q), with 0 < p, q < 1 fixed, and its associated adjacency matrix A, e.g., A is a n × n symmetric matrix with diagonal entries equal to zero and i.i.d. off-diagonal pairs, each equal to +1 with probability p, −1 with probability q, and 0 with probability 1 − p − q. We prove the following:The required bounds for ε s , s > 3, hold for any non-vanishing vector ϕ in E λ with ε defined as above. On Courant's nodal domain theorem. 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[ "Revisiting Long-term Time Series Forecasting: An Investigation on Linear Mapping", "Revisiting Long-term Time Series Forecasting: An Investigation on Linear Mapping" ]	[ "Zhe Li \nHarbin Institute of Technology\nShenzhen\n", "Shiyi Qi \nHarbin Institute of Technology\nShenzhen\n", "Yiduo Li [email protected] \nHarbin Institute of Technology\nShenzhen\n", "Zenglin Xu [email protected] \nHarbin Institute of Technology\nShenzhen\n" ]	[ "Harbin Institute of Technology\nShenzhen", "Harbin Institute of Technology\nShenzhen", "Harbin Institute of Technology\nShenzhen", "Harbin Institute of Technology\nShenzhen" ]	[]	Long-term time series forecasting has gained significant attention in recent years. While there are various specialized designs for capturing temporal dependency, previous studies have demonstrated that a single linear layer can achieve competitive forecasting performance compared to other complex architectures. In this paper, we thoroughly investigate the intrinsic effectiveness of recent approaches and make three key observations: 1) linear mapping is critical to prior long-term time series forecasting efforts; 2) RevIN (reversible normalization) and CI (Channel Independent) play a vital role in improving overall forecasting performance; and 3) linear mapping can effectively capture periodic features in time series and has robustness for different periods across channels when increasing the input horizon. We provide theoretical and experimental explanations to support our findings and also discuss the limitations and future works. Our framework's code is available at https://github.com/plumprc/RTSF.Preprint. Under review.	10.48550/arxiv.2305.10721	[ "https://export.arxiv.org/pdf/2305.10721v1.pdf" ]	258,762,346	2305.10721	7df8166259163b96b7fe7bab90da5c355cc00cfc	Revisiting Long-term Time Series Forecasting: An Investigation on Linear Mapping Zhe Li Harbin Institute of Technology Shenzhen Shiyi Qi Harbin Institute of Technology Shenzhen Yiduo Li [email protected] Harbin Institute of Technology Shenzhen Zenglin Xu [email protected] Harbin Institute of Technology Shenzhen Revisiting Long-term Time Series Forecasting: An Investigation on Linear Mapping Long-term time series forecasting has gained significant attention in recent years. While there are various specialized designs for capturing temporal dependency, previous studies have demonstrated that a single linear layer can achieve competitive forecasting performance compared to other complex architectures. In this paper, we thoroughly investigate the intrinsic effectiveness of recent approaches and make three key observations: 1) linear mapping is critical to prior long-term time series forecasting efforts; 2) RevIN (reversible normalization) and CI (Channel Independent) play a vital role in improving overall forecasting performance; and 3) linear mapping can effectively capture periodic features in time series and has robustness for different periods across channels when increasing the input horizon. We provide theoretical and experimental explanations to support our findings and also discuss the limitations and future works. Our framework's code is available at https://github.com/plumprc/RTSF.Preprint. Under review. Introduction Time series forecasting has become increasingly popular in recent years due to its applicability in various fields, such as electricity forecasting [9], weather forecasting [2], and traffic flow estimation [5]. With the advances in computing resources, data volume, and model architectures, deep learning techniques, such as RNN-based [11,20] and CNN-based models [4,22], have outperformed traditional statistical methods [1,3] in terms of higher capacity and robustness. Recently, there have been increasing interests in using Transformer-based methods to capture longterm temporal correlations in time series forecasting [31,26,33,15,21,30]. These methods have demonstrated promising results through various attention mechanisms and non-autoregressive generation (NAR) techniques. However, a recent work LTSF-Linear family [27] has shown that these Transformer-based methods may not be as effective as previously thought and found that their reported forecasting results may mainly rely on one-pass prediction compared to autoregressive generation. Instead, the LTSF-Linear, which uses only a single linear layer, surprisingly outperformed existing complex architectures by a large margin. Built on this work, subsequent approaches [14,18,13,25] discarded the encoder-decoder architecture and focused on developing temporal feature extractors and modeling the mapping between historical inputs and predictions. While these methods have achieved improved forecasting performance, they are still not significantly better than linear models. Additionally, they often require a large number of adjustable hyper-parameters and specific training tricks, such as normalization and channel-specific processing, which may potentially affect the fairness of comparison. Based on these observations, we raise the following questions: (1) Are temporal feature extractors effective for long-term time series forecasting? (2) What are the underlying mechanisms explaining the effectiveness of linear mapping in time series forecasting? and (3) What are the limits of linear models and how can we improve them? In the following sections, after introducing problem definition and experimental setup, we conduct a comprehensive investigation with intensive experiments and analysis into the inner working mechanisms of recent time series forecasting models, aiming to answer these questions raised above through extensive experiments and theoretical analysis. The main contributions of this paper are: • We investigate the efficacy of different components in recent time series forecasting models, finding that linear mapping is critical to their forecasting performance, as shown in Sec. 3. • We demonstrate the effectiveness of linear mapping for learning periodicity in long-term time series forecasting tasks with both theoretical and experimental evidence and propose simple yet effective baselines for a fairer comparison in the future (as shown in Table 3). • We examine the limitations of the linear mapping when dealing with multivariate time series with different periodic channels, and analysis the impact of the input horizon and a remedial technique called Channel-Independent, as shown in Figures 10 General framework. Figure 1 illustrates the general framework of recent works [14,32,6,13,18,25] for time series forecasting, comprising three core components: RevIN [10], a reversible normalization layer; a temporal feature extractor such as attention, MLP, or convolutional layers; and a linear projection layer that projects the final prediction results. Given the potential impact of hyper-parameter adjustment and various training tricks on comparison fairness, we first examine the effectiveness of different temporal feature extractors. Without loss of generality, we meticulously select four notable recent developments: PatchTST [18] (attention), MTS-Mixers [13] (MLP), TimesNet [25] and SCINet [14] (convolution). All of these methods follow this common framework and have achieved state-of-the-art forecasting performance, as claimed. Considering the fact that their reported forecasting accuracy is not significantly better than a single linear layer, we conduct new experiments using the ETT benchmark to check the contribution of each part in their proposed approaches. Figure 2 demonstrates the forecasting performance of different models on ETTh1 over 4 different prediction lengths. The baseline "RLinear" refers to a linear projection layer with RevIN. The fixed random extractor means that we only initialize the temporal feature extractor randomly and do not update its parameters in the training phase. It is worth noting that the RevIN significantly improves the forecasting accuracy of these approaches. Thus, comparing one method with others which do not use RevIN may lead to unfair results due to its advantage. With the aid of RevIN, even a simple linear layer can outperform current state-of-the-art baseline PatchTST. Remarkably, our findings suggest that even using a randomly initialized temporal feature extractor with untrained parameters can induce competitive, even better forecasting results. It is necessary to consider what these feature extractors have learned from time series data. Figure 3 illustrates the weights of the final linear projection layer and different temporal feature extractors, such as MLP and attention on ETTh1. Interestingly, when the temporal feature extractor is a MLP, both MLP and projection layer learn chaotic weights, whereas the product of the two remains consistent with the weights learned from a single linear layer. On the other hand, when the temporal feature extractor is attention, it also learns about messy weights, but the weight learned by the projection layer is similar to that of a single linear layer, implying the importance of linear projection in time series forecasting. To mitigate any potential dataset-specific bias, we conducted more experiments on the full ETT benchmarks, using the same comparison protocol as [18]. Table 1 demonstrates the forecasting results of RLinear and selected models. Interestingly, the simple baseline RLinear has comparable or even better performance in most cases compared to carefully designed methods. Sometimes, these delicate models using temporal feature extractors even perform worse than an untrained prototype. It is important to note that models with a fixed random temporal feature extractor generally exhibit similar forecasting performance, and approach a single linear layer. These intriguing observations prompt us to question whether temporal feature extractors are necessary, and why linear mapping is so effective in long-term time series forecasting. Theoretical and Empirical Study on the Linear Mapping Roles of Linear Mapping in Forecasting Linear mapping learns periodicity. Consider a single linear layer as Y = XW + b,(1) where W ∈ R n×m is the weight, also termed as the transition matrix, and b ∈ R 1×m is the bias. [8,28], denoted as x(t) = s(t) + f (t) + . Numerous methods [26,33,23,24,27] have been developed to decompose time series into seasonal and trend terms, leveraging neural networks to capture periodicity and supplement trend prediction. However, it's worth noting that a single linear layer can also effectively learn periodic patterns. Theorem 1. Given a seasonal time series satisfying x(t) = s(t) = s(t − p) where p ≤ n is the period, there always exists an analytical solution for the linear model as [x 1 , x 2 , . . . , x n ] · W + b = [x n+1 , x n+2 , . . . , x n+m ],(2)W (k) ij = 1, if i = n − kp + (j mod p) 0, otherwise , 1 ≤ k ∈ Z ≤ n/p , b i = 0.(3) Equation 3 indicates that linear mapping can predict periodic signals when the length of the input historical sequence is not less than the period, but that is not a unique solution. Since the values corresponding to each timestamp in s(t) are almost impossible to be linearly independent, the solution space for the parameters of W is extensive. In particular, it is possible to obtain a closed-form solution for more potential values of W (k) with different factor k when n p. The linear combination of [W (1) , . . . , W (k) ] with proper scaling factor also satisfies the solution of Equation 2. Corollary 1.1. When the given time series satisfies x(t) = ax(t − p) + c where a, c are scaling and translation factors, the linear model still has a closed-form solution to Equation 2 as W (k) ij = a k , if i = n − kp + (j mod p) 0, otherwise , 1 ≤ k ∈ Z ≤ n/p , b i = k−1 l=0 a l · c.(4) Now we know that a single linear layer can effectively capture periodicity in time series. Weights visualized in Figure 3 also support our viewpoint where the transition matrix from input to output shows significant periodicity (24 time steps per period). However, in practice, time series generally follow the Assumption 1 so that the trend term may affect the learning of linear models. Figure 4 illustrates the forecasting results of a linear layer on simulated seasonal and trend signals, including a sine wave, a linear function, and their sum. As expected, the linear model fits seasonality well but performs poorly on the trend, regardless of whether it has a bias term or not. Chen et al. [6] have also studied similar issues and provided an upper bound on the performance of linear models when forecasting time series with seasonal and trend components. Based on their work, we have adjusted their conclusion and derived the following theorem. Theorem 2. Let x(t) = s(t) + f (t) where s(t) is a seasonal signal with period p and f (t) satisfies K-Lipschitz continuous. Then there exists a linear model as Equation 2 with input horizon size n = p + τ, τ ≥ 0 such that \|x(n + j) −x(n + j)\| ≤ K(p + j), j = 1, . . . , m. Proof. To simplify the proof process, we assume that the timestamp of historical data is 1 to n. Then for the j-th true value x(n + j) to be predicted, we have x(n + j) = x(p + τ + j) = s(τ + j) + f (p + τ + j).(5) Supposing that the linear model can only learn periodic patterns, we can directly use Equation 3 as an approximate solution where we choose k = 1. Thus, the prediction for x(n + j) iŝ x(n + j) = XW + b = x(n − p + (j mod p)) = s(τ + j) + f (τ + (j mod p)). Leveraging properties of K-Lipschitz continuous we can get \|x(n + j) −x(n + j)\| = \|f (p + τ + j) − f (τ + (j mod p))\| ≤ K\|p + j − (j mod p)\| ≤ K(p + j).(7) Although the forecasting error of linear models for trend terms is bounded, it can still impact prediction results as the timestamp accumulates or the trend term becomes more significant. This could potentially be why linear models prone to perform poorly in trend prediction. Disentanglement and Normalization Problems in Disentanglement. If the trend term can be eliminated or separated from the seasonal term, forecasting performance can be improved. Previous works [8,28,26,23,24,33,29,27] have focused on disentangling time series into seasonal and trend components to predict them individually. In general, they utilized the moving average implemented by an average pooling layer with a sliding window of a proper size to get trend information from the input time series. Then, they identified seasonal features from periodic signals obtained by subtracting trend items from the original data. However, these disentanglement methods have some problems as reported in [12]. Firstly, the sliding window size should be larger than the maximum period of seasonality parts, or the decoupling will be inadequate. Secondly, due to the usage of the average pooling layer, alignment requires padding on both ends of the input time series, which inevitably distorts the sequence at the head and tail. Besides, even if the signals are completely disentangled, or they only have trend terms, the issue of under-fitting trend terms persists. Therefore, while disentanglement may improve forecasting performance, it still has a gap with some recent advanced models. Turning trend into seasonality. The key to disentanglement is subtracting the moving average from the original time series, which is related to normalization. Kim et al. [10] recognized that some statistical information of time series, such as mean and variance, continuously change over time due to the distribution shift problem. To address this challenge, they developed RevIN, a method that first normalizes the input historical time series and feeds them into forecasting modules before denormalizing the prediction for final results. Previous works [16,7,25,18] have attributed the effectiveness of RevIN more to normalization for alleviating distribution shift problems. However, the range and size of values in time series are also meaningful in real-world scenarios. Directly applying normalization to input data may erase this statistical information and lead to poor predictions. Figure 5 illustrates the forecasting results on the simulated trend signal with two channels using different normalization methods. It is challenging to fit trend changes solely using a linear layer. Applying batch normalization even induces worse results, and layer normalization results in meaningless prediction close to zero. Disentangling the simulated time series also does not work. However, with the help of RevIN, a single linear layer can accurately predict trend terms. The core of reversible normalization lies in reversibility. It eliminates trend changes caused by moment statistics while preserving statistical information that can be used to restore final forecasting results. Figure 6 illustrates how RevIN affects seasonal and trend terms. For the seasonal signal, RevIN scales the range but does not change the periodicity. For the trend signal, RevIN scales each segment into the same range and exhibits periodic patterns. RevIN is capable of turning some trends into seasonality, making models better learn or memorize trend terms. Figure 7 showcases forecasting results of the linear model with RevIN on simulated time series with seasonal and trend terms. RevIN converts continuously changing trends into multiple segments with a fixed and similar trend, demonstrating periodic characteristics. As a result, errors in trend prediction caused by accumulated timesteps in the past can be alleviated, leading to more accurate forecasting results. Experimental Evaluation In this section, we first evaluate the performance of different models on real-world datasets, and then examine the scenarios with multiple periods among various channels. Table 2 provides statistical information of those six real-world datasets. We perform experiments using three latest competitive baselines: PatchTST [18] (ICLR 2023), TimesNet [25] (ICLR 2023), and DLinear [27] (AAAI 2023). Given that RevIN significantly improves the forecasting performance, we add two simple baselines, RLinear and RMLP with two linear layers and a ReLU activation, for a fairer comparison. Table 3 provides an overview of forecasting results for all benchmarks. However, these well-designed models are not better than our proposed two simple baselines. It is likely that the success of these models is due to the learning of periodicity via linear mapping and efficiency of reversible normalization. Interestingly, we have noticed that RLinear does not perform significantly better than complex models on datasets with a large number of channels, such as Weather and ECL, which will be studied in the next section. Comparison on Real-world Datasets When Linear Meets Multiple Periods among Channels Although linear mapping is capable of learning periodicity in time series, it faces challenges when dealing with multi-channel datasets. To address this issue, a possible solution is to use Channel Independent [18] (CI) modeling, which treats each channel in the time series independently. While this approach can improve forecasting accuracy, it also significantly increases computational overhead. Figure 8 illustrates the forecasting results of different models applied to simulated time series with three distinct periodic channels. It is observed that RLinear-CI and RMLP are able to fit curves, while RLinear fails. This suggests that a single linear layer may struggle to learn different periods within channels. Nonlinear units or CI modeling may be useful in enhancing the robustness of the model for multivariate time series with different periodic channels. Table 4 provides forecasting results on Weather and ECL of RLinear using CI, which achieves comparable performance with RMLP, confirming that a single linear layer may be vulnerable to varying periods among channels. To further investigate the effect of linear mapping on multivariate time series, we conduct simulations using a series of sine waves with angular frequencies ranging from 1/30 to 1/3 and the length of 3000. Figure 10 demonstrates the forecasting results under different settings. Our findings indicate that the linear model consistently performs well on time series with two channels, regardless of whether the difference in periodicity is small or large. However, as the number of channels with different periods increases, the linear model gradually performs worse, while models with nonlinear units or CI continue to perform well. Additionally, increasing the input horizon can effectively alleviate the forecasting performance of the linear model on multi-channel datasets. These observations suggest that existing models may focus on learning seasonality, and that the differences in periodicity among different channels in multivariate time series are key factors that constrain forecasting performance. Theorem 3 provides an explanation of linear models in forecasting multivariate time series. ( × 150) As shown in Figure 11, increasing the input horizon can lead to a significant improvement in forecasting performance. This is because a longer input horizon covers more potential periods, minimizing the performance gap between linear models and those with nonlinear units. However, it is worth noting that RLinear-CI and RMLP perform worse on the ETTh1 dataset when the input horizon is longer, which may be due to the small volume of this particular dataset. Furthermore, it should be noted that there is an upper limit to the performance improvement achieved by increasing the input horizon. This limit may be highly dependent on the periodic patterns present in datasets. Conclusion This paper systematically investigate the effect of linear mapping in long-term time series forecasting, with the following important takeaways: (1) linear mapping is critical to prior long-term time series forecasting methods, where they generally prone to learn similar affine transform, which corresponds to specific periodic patterns, from input historical observation to output prediction; (2) RevIN (reversible normalization) and CI (Channel Independent) improve overall forecasting performance via simplifying learning about periodicity; and (3) linear mapping has robustness to fit multivariate time series with different periodic channels when increasing input horizon, while it may induce under-fitting of short period features. We provide theoretical explanations and conduct extensive experiments on both simulated and real-world datasets to support our findings. Limitations and future work. Long-term time series benchmarks often display consistent seasonal patterns. To improve the model's generalization ability, it is worthwhile to study how it performs when the seasonality changes. It would also be valuable to explore the applicability of our theories to other tasks, such as short-term time series forecasting. We acknowledge that these explorations will be left for future work. A Proofs For better readability, we have re-listed the unproven theorems as follows. Theorem 1. Given a seasonal time series satisfying x(t) = s(t) = s(t − p) where p ≤ n is the period, there always exists an analytical solution for the linear model as [x 1 , x 2 , . . . , x n ] · W + b = [x n+1 , x n+2 , . . . , x n+m ],(8)W (k) ij = 1, if i = n − kp + (j mod p) 0, otherwise , 1 ≤ k ∈ Z ≤ n/p , b i = 0.(9) Proof. ∀x n+j , 1 ≤ j ≤ m, z ∈ Z + , x n+j = x n−zp+(j mod p) according to x(t) = x(t−p), thus we have [x 1 , x 2 , . . . , x n ] · W (k) = [x n−kp+1 , x n−kp+2 , . . . , x n−kp+m ] = [x n+1 , x n+2 , . . . , x n+m ]. Proof. ∀x n+j , 1 ≤ j ≤ m, z ∈ Z + , x n+j = a z x n−zp+(j mod p) + z−1 l=0 a l · c according to x(t) = ax(t−p)+c, thus we have [x 1 , . . . , x n ]·W (k) +b = [a k x n−kp+1 + k−1 l=0 a l ·c, . . . , a k x n−kp+m + k−1 l=0 a l · c] = [x n+1 , . . . , x n+m ]. Theorem 3. Let X = [s 1 , s 2 , . . . , s c ] ∈ R c×n be the input historical multivariate time series with c channels and the length of n. If each signal s i has a corresponding period p i , there must be a linear model Y = XW + b that can predict the next m time steps when n ≥ lcm(p 1 , p 2 . . . , p c ). Proof. Apparently p = lcm(p 1 , p 2 . . . , p c ) is the least common period for all channels. According to Equation 9, there must be a linear model satisfying Equation 8 for each channel s c ∈ R 1×n . B Experimental details Reproduction. We implemented the reversible normalization module using the default setting from RevIN [10]. Our baseline RLinear model consists of ReVIN and a single linear layer that maps the input to the output time series. The RMLP model, on the other hand, comprises RevIN, an MLP for temporal interaction, and a linear projection layer. The MLP includes two linear layers with ReLU activation, and we set the number of hidden states to 512. The baseline RLinear-CI includes c RLinears for modeling c channels individually. Details on benchmarks and baselines. We adopt the same pre-processing protocol in [18]. Across all benchmarks, we set the initial learning rate to 0.005 and the batch size to 128. The results of PatchTST, MTS-Mixers, TimesNet, SCINet, and DLinear are based on our reproduction. For models with a fixed random temporal feature extractor setting, we initialize these models with the fixed random seed 1024. We use the hyper-parameters suggested in their original papers for each model. Figure 1 : 1The general framework for time series forecasting, comprising of RevIN[10], a temporal feature extractor, and a linear projection layer. Figure 2 : 2Forecasting results of selected models on ETTh1 [31] dataset. MSE and MAE results are averaged from 4 different prediction lengths {96, 192, 336, 720}. The lower MSE and MAE indicate the better forecasting performance. Figure 3 : 3Weights visualization on ETTh1 where the input and output length are set as 96. Figure 4 : 4Forecasting visualization of a linear model on simulated seasonal and trend signals. Figure 5 : 5Forecasting results on the trend signal with different normalization methods. Figure 6 : 6The effect of RevIN applied to seasonal and trend signals. Each segment separated by a dashed line contains the input historical time series X and the predicted sequence Y . Figure 7 : 7Forecasting results of a linear layer with RevIN on simulated time series with seasonal and trend terms. Each segment separated by a dashed line consists of historical and prediction sequences. Figure 8 : 8Forecasting results on simulated time series with three channels of different periods. Figure 9 : 9Simulated sine waves with angular frequency ranges from 1/30 to 1/3 and the length of 200. Figure 10 : 10Left: Forecasting resutls on simulated 2-variate time series. ∆ω denotes the difference in angular frequency between channels. Middle: Forecasting results of different models on simulated datasets with different periodic channels. Right: Impact of input horizon on forecasting performance.Theorem 3. Let X = [s 1 , s 2 , . . . , s c ] ∈ R c×n be the input historical multivariate time series with c channels and the length of n. If each signal s i has a corresponding period p i , there must be a linear model Y = XW + b that can predict the next m time steps when n ≥ lcm(p 1 , p 2 . . . , p c ). Figure 11 : 11Impact of input horizon on forecasting results. Lower MSE indicates better performance. Corollary 1. 1 . 1When the given time series satisfies x(t) = ax(t − p) + c where a, c are scaling and translation factors, the linear model still has a closed-form solution to Equation 8 asW (k) ij = a k , if i = n − kp + (j mod p) 0, otherwise , 1 ≤ k ∈ Z ≤ n/p , b i = and 11.Problem definition. Given a historical time series observation X = [x 1 , x 2 , . . . , x n ] ∈ R c×n with c channels and n time steps, forecasting tasks aim to predict the next m time steps Y = [x n+1 , x n+2 , . . . , x n+m ] ∈ R c×m where m denotes forecasting horizon. We need to learn a map F : X c×n → Y c×m where X and Y are consecutive in the original time series data. Electricity Transformer Temperature) with four datasets with different granularity records six power load features and oil temperature from electricity transformers; (2) Weather 1 contains 21 meteorological indicators in the 2020 year from nearly 1600 locations in the U.S.; and (3) ECL 2 records the hourly electricity consumption of 321 customers from 2012 to 2014. For a fair comparison, we follow the same evaluation protocol in[18] and split all datasets into training, validation, and test sets. Our proposed baselines are trained using the L2 loss and the Adam[17] optimizer. The training process is early stopped within 20 epochs. MSE (Mean Squared Error) and MAE (Mean Absolute Error) are adopted as evaluation metrics for comparison. R-squared score is used for empirical study as it can eliminate the impact of data scale. All the models are implemented in PyTorch[19] and tested on a single Nvidia V100 32GB GPU for three times.2 Problem Definition and Experimental Setup Experimental setup. Our experiments are conducted on simulated time series and six public real- world datasets: (1) ETT [31] ( Table 1 : 1Forecasting results on the full ETT benchmarks. The length of the historical horizon and the prediction length are set as 336. † indicates the temporal feature extractor with fixed random weights.Dataset ETTh1 ETTm1 ETTh2 ETTm2 Method MSE MAE MSE MAE MSE MAE MSE MAE RLinear 0.420 0.423 0.370 0.383 0.325 0.386 0.273 0.326 PatchTST 0.431 0.436 0.366 0.392 0.331 0.380 0.276 0.332 †PatchTST 0.429 0.435 0.371 0.389 0.328 0.384 0.280 0.331 MTS-Mixers 0.414 0.425 0.378 0.399 0.353 0.407 0.291 0.337 †MTS-Mixers 0.423 0.424 0.377 0.392 0.351 0.405 0.282 0.334 TimesNet 0.493 0.468 0.406 0.418 0.358 0.420 0.304 0.353 †TimesNet 0.428 0.439 0.384 0.400 0.342 0.406 0.306 0.351 SCINet 0.467 0.469 0.404 0.423 0.365 0.414 0.329 0.369 †SCINet 0.428 0.431 0.386 0.398 0.349 0.403 0.299 0.345 Assumption 1. A general time series x(t) can be disentangled into seasonality part s(t) and trend part f (t) with tolerable noise Table 2 : 2Statistical information of all datasets for time series forecasting.Dataset ETTh1/h2 ETTm1/m2 Weather ECL #Channel 7 7 21 321 Timesteps 17,420 69,680 52,696 26,304 Granularity 1 hour 15 minutes 10 minutes 1 hour Data Partition 6:2:2 (month) 7:2:1 Table 3 : 3Time series forecasting results. The length of the historical horizon is 336 and prediction lengths are {96, 192, 336, 720}. The best results are in bold and the second one is underlined.Method RLinear RMLP PatchTST TimesNet DLinear Metric MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE Table 4 : 4Forecasting results on Weather and ECL of RLinear using CI where the input horizon is 336.Dataset Weather ECL Method Metric 96 192 336 720 96 192 336 720 RLinear MSE 0.175 0.218 0.265 0.329 0.140 0.154 0.171 0.209 MAE 0.225 0.260 0.294 0.339 0.235 0.248 0.264 0.297 RLinear-CI MSE 0.146 0.189 0.241 0.314 0.134 0.149 0.166 0.202 MAE 0.194 0.235 0.275 0.327 0.232 0.246 0.265 0.293 = 50/150 = 45/150 = 40/150 = 35/150 = 30/150 = 25/150 = 20/150 = 15/150 = 10/150 = 5/150 https://www.ncei.noaa.gov/data/local-climatological-data/ 2 https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014 Time series analysis: Forecasting and control. Oliver D Anderson, George E P Box, Gwilym M Jenkins, The Statistician. 27265Oliver D. Anderson, George E. P. Box, and Gwilym M. Jenkins. 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[ "When Search Meets Recommendation: Learning Disentangled Search Representation for Recommendation CCS CONCEPTS • Information systems → Recommender systems. KEYWORDS Recommendation; Search; Contrastive Learning; Disentanglement Learning ACM Reference Format", "When Search Meets Recommendation: Learning Disentangled Search Representation for Recommendation CCS CONCEPTS • Information systems → Recommender systems. KEYWORDS Recommendation; Search; Contrastive Learning; Disentanglement Learning ACM Reference Format" ]	[ "Zihua Si [email protected] ", "Zhongxiang Sun [email protected] ", "Xiao Zhang [email protected] ", "Jun Xu [email protected] ", "Xiaoxue Zang [email protected] ", "Yang Song [email protected] ", "Kun Gai [email protected] ", "Ji-Rong Wen [email protected] ", "Zihua Si ", "Zhongxiang Sun ", "Xiao Zhang ", "Jun Xu ", "Xiaoxue Zang ", "Yang Song ", "Kun Gai ", "Ji-Rong Wen ", "\nGaoling School of Artificial Intelligence\nGaoling School of Artificial Intelligence Renmin\nRenmin University of China\nBeijingChina\n", "\nGaoling School of Artificial Intelligence Renmin\nUniversity of China\nBeijingChina\n", "\nUniversity of China\nBeijingChina\n", "\nGaoling School of Artificial Intelligence Renmin\nKuaishou Technology Co., Ltd\nBeijing, BeijingChina, China\n", "\nUniversity of China\nBeijingChina\n" ]	[ "Gaoling School of Artificial Intelligence\nGaoling School of Artificial Intelligence Renmin\nRenmin University of China\nBeijingChina", "Gaoling School of Artificial Intelligence Renmin\nUniversity of China\nBeijingChina", "University of China\nBeijingChina", "Gaoling School of Artificial Intelligence Renmin\nKuaishou Technology Co., Ltd\nBeijing, BeijingChina, China", "University of China\nBeijingChina" ]	[ "Taipei, Taiwan.ceedings of the 46th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '23)" ]	Modern online service providers such as online shopping platforms often provide both search and recommendation (S&R) services to meet different user needs. Rarely has there been any effective means of incorporating user behavior data from both S&R services. Most existing approaches either simply treat S&R behaviors separately, or jointly optimize them by aggregating data from both services, ignoring the fact that user intents in S&R can be distinctively different. In our paper, we propose a Search-Enhanced framework for the Sequential Recommendation (SESRec) that leverages users' search interests for recommendation, by disentangling similar and dissimilar representations within S&R behaviors. Specifically, SESRec first aligns query and item embeddings based on users' query-item interactions for the computations of their similarities. Two transformer encoders are used to learn the contextual representations of S&R behaviors independently. Then a contrastive learning task is designed to supervise the disentanglement of similar and dissimilar representations from behavior sequences of S&R. Finally, we extract user interests by the attention mechanism from three perspectives, i.e., the contextual representations, the two separated behaviors containing similar and dissimilar interests. Extensive experiments on both industrial and public datasets demonstrate that SESRec consistently outperforms state-of-the-art models.	10.1145/3539618.3591786	[ "https://export.arxiv.org/pdf/2305.10822v1.pdf" ]	258,762,362	2305.10822	d72c48147a795ccc713df3595f3422b9cff37e8a	When Search Meets Recommendation: Learning Disentangled Search Representation for Recommendation CCS CONCEPTS • Information systems → Recommender systems. KEYWORDS Recommendation; Search; Contrastive Learning; Disentanglement Learning ACM Reference Format ACMCopyright ACMJuly 23-27, 2023. July 23-27, 2023 Zihua Si [email protected] Zhongxiang Sun [email protected] Xiao Zhang [email protected] Jun Xu [email protected] Xiaoxue Zang [email protected] Yang Song [email protected] Kun Gai [email protected] Ji-Rong Wen [email protected] Zihua Si Zhongxiang Sun Xiao Zhang Jun Xu Xiaoxue Zang Yang Song Kun Gai Ji-Rong Wen Gaoling School of Artificial Intelligence Gaoling School of Artificial Intelligence Renmin Renmin University of China BeijingChina Gaoling School of Artificial Intelligence Renmin University of China BeijingChina University of China BeijingChina Gaoling School of Artificial Intelligence Renmin Kuaishou Technology Co., Ltd Beijing, BeijingChina, China University of China BeijingChina When Search Meets Recommendation: Learning Disentangled Search Representation for Recommendation CCS CONCEPTS • Information systems → Recommender systems. KEYWORDS Recommendation; Search; Contrastive Learning; Disentanglement Learning ACM Reference Format Taipei, Taiwan.ceedings of the 46th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '23) Taipei, Tai-wan; New York, NY, USAACM11July 23-27, 2023. July 23-27, 2023Empirical studies * Corresponding authors. Work partially done at Engineering Research Center of Next-Generation Intelligent Search and Recommendation, Ministry of Education. Work done when Zihua Si and Zhongxiang Sun were interns at Kuaishou. ACM ISBN 978-1-4503-9408-6/23/07. . . $15.00 further validate that SESRec successfully disentangle similar and dissimilar user interests from their S&R behaviors.. 2023. When Search Meets Recommendation: Learning Disentangled Search Representation for Recommendation. In Pro- Modern online service providers such as online shopping platforms often provide both search and recommendation (S&R) services to meet different user needs. Rarely has there been any effective means of incorporating user behavior data from both S&R services. Most existing approaches either simply treat S&R behaviors separately, or jointly optimize them by aggregating data from both services, ignoring the fact that user intents in S&R can be distinctively different. In our paper, we propose a Search-Enhanced framework for the Sequential Recommendation (SESRec) that leverages users' search interests for recommendation, by disentangling similar and dissimilar representations within S&R behaviors. Specifically, SESRec first aligns query and item embeddings based on users' query-item interactions for the computations of their similarities. Two transformer encoders are used to learn the contextual representations of S&R behaviors independently. Then a contrastive learning task is designed to supervise the disentanglement of similar and dissimilar representations from behavior sequences of S&R. Finally, we extract user interests by the attention mechanism from three perspectives, i.e., the contextual representations, the two separated behaviors containing similar and dissimilar interests. Extensive experiments on both industrial and public datasets demonstrate that SESRec consistently outperforms state-of-the-art models. INTRODUCTION Recommender systems and search engines have been widely deployed in online platforms to help users alleviate information overload. Currently, with the vast increase of data on the Internet, solely using one of the recommender systems or the search engines cannot meet users' information needs. Hence, many social media platforms, e.g., YouTube and TikTok, provide both search and recommendation (S&R) services for users to obtain information. As users express their diverse interests in both scenarios, it is feasible to enhance the recommendation system by jointly modeling the behaviors of both, and the core challenge is how to effectively leverage users' search interests for capturing accurate recommendation interests 1 . Early studies [28,29] have demonstrated that jointly optimizing the S&R models benefits both performances. Recently, several works [5,17,24,27,31] have been proposed to boost the recommendation using search data. As such, devising a searchenhanced framework is a promising research area in the recommendation field. Due to that recommendation with search data is still a nascent research area in both academia and industry, recent works [5,24,27,31] usually only incorporate users' S&R behaviors by feeding them into one encoder to mine users' interests. Despite their effectiveness, most previous works ignore the differences between users' interests in S&R behaviors by modeling them without considering their correlations. However, in real-world applications, search behaviors may strengthen or be complementary to the interests revealed in the recommendation behaviors. For example, Figure 1(a) illustrates partial behavior histories of a user in the short-video scenario. While users browsing the items/video suggested by the recommendation system, they may spontaneously start searching by typing queries, which usually differ from the video content in the recommendation feed. We refer to such case as spontaneous search. In contrast, users may also start searching by clicking on the suggested query that is related to the current item/video being played, which we denote as passive search. To verify the universality of this phenomenon, we conducted data analysis from the real-world data collected from the Kuaishou 2 app, shown in Figure 1(b). The data analysis is based on behaviors of millions of users. For each search behavior, if the categories of the items exist in the set of categories of the items interacted by this user in the past seven days, this search behavior is similar to recent recommendation behaviors, and otherwise dissimilar. The similar search behaviors reflect users' strong interests overlapped in the recommendation behaviors and should be strengthened. The dissimilar behaviors may be undiscovered interests, which are probably newly emerging and unfulfilled in the recommendation feed. As a result, it is critical to disentangle the similar and dissimilar representations between S&R behaviors. To address such a problem, we devise a search-enhanced framework for sequential recommendation, namely SESRec, to learn the disentangled search representation for the recommendation. In details, in order to disentangle the similar and dissimilar interests between two behaviors, we propose to decompose each history sequence into two sub-sequences that respectively represent the similar and dissimilar interests so that we can extract user interests from multiple aspects. To learn the similarity between two behaviors, we first align query-item embeddings with an InfoNCE loss based on users' query-item interactions. Then two separate encoders are utilized to model S&R behaviors and generate their contextual representations. Due to the lack of the labels denoting the similarity of interests between the obtained contextual representations, we propose to leverage self-supervision to guide learning the similar and dissimilar interests. Specifically, we exploit the co-attention mechanism to learn the correlation between S&R's contextual representations. Based on the co-attention score, for both of the contextual representations, we not only aggregate them to generate anchors which are considered to maintain the shared interests of S&R, but also partition them into two sub-sequences, which are considered to represent the similar and dissimilar behaviors between S&R (respectively referred to as the positives and the negatives). Then following contrastive learning, we define a 2 https://www.kuaishou.com/en triplet loss to push the anchors closer to the positives than the negatives. Finally, we employ the attention mechanism to extract user interests from three aspects, i.e., the contextual representations, the positives, and the negatives of S&R. In this way, the disentangled interests of S&R behaviors enhance the prediction for the next interaction. The contributions of this paper are summarized as follows: • To the best of our knowledge, it is the first time that users' S&R interests are jointly considered and disentangled into similar and dissimilar representations for user modeling. We pioneer the practice of learning disentangled search representations for the recommendation. • We propose a search-enhanced framework for the sequential recommendation. By jointly considering users' S&R behaviors, we extract users' interests from multiple aspects by decomposing both S&R behaviors into two parts based on their co-attention scores: one for behaviors containing similar interests and the other for behaviors containing dissimilar interests. Moreover, we also utilize self-supervised learning to guide the decomposition. • We conduct extensive experiments on two datasets. The experimental results validate the effectiveness of our proposed SESRec. In particular, SESRec outperforms traditional sequential models which do not leverage search data, as well as other search-aware models, which neglect the correlation between users' interests in S&R behaviors. RELATED WORK Recommendation with Search Data. In both academia and industry, research that enhances recommendation using search data is relatively rare. Only a few works involve in this area. Zamani and Croft [28,29] assume that S&R models could potentially benefit from each other by jointly training with both S&R data. Yao et al. [27] design an approach called USER that mines user interests from the integrated user behavior sequences and accomplishes these two tasks in a unified way. NRHUB [24] exploits heterogeneous user behaviors, i.e., webpage browsing and search queries, to enhance the recommendation model. IV4REC [17] and IV4REC+ [18] leverages search queries as instrumental variables to reconstruct user and item embeddings and boost recommendation performance in a causal learning manner. Query-SeqRec [5] is a query-aware model, which incorporates issued queries and browsing items to capture users' interests and intents. SRJGraph [31] is a GNN-based method which incorporates queries into user-item interaction edges as attributes. In this work, we also develop a framework to learn disentangled search representation for recommendation. Sequential Recommendation. Sequential recommendation methods mine user interests by modeling sequential relationships of user behaviors. An early work [6] first utilizes the GRU mechanism to model user preferences. And attention mechanisms are introduced to capture sequential patterns, such as STAMP [12]. There are works using CNN architectures, e.g., Caser [20] treats the historical item sequence as an "image" and adopts a CNN for user modeling. For other neural network architectures, several models employ GNN [2,26] which construct a graph for historical sequences. Currently, lots of models leverage the transformer architecture, e.g., SASRec [8] and BERT4Rec [19]. Several works [33,34] devise an attention mechanism to adaptively learn user interests from behaviors. FMLP-Rec [36] is a state-of-the-art that leverages an all-MLP architecture with learnable filters for sequential recommendation. Unlike these works, this work incorporates users' search activities into the sequential recommendation task. Contrastive Learning for Recommendation. With the successful development of contrastive learning, this technique has been widely adopted in recommendation [7,13,32,35]. As for sequential recommendation, Zhou et al. [35] first devise auxiliary selfsupervised objectives to enhance data representations via pre-training methods. Ma et al. [13] propose a sequence-to-sequence training strategy by performing self-supervised learning in the latent space. Recently, several works [30,32] are proposed for learning users' diverse interests. For example, Zhang et al. [30] propose a contrastive learning framework to disentangle long and short-term interests for recommendation with self-supervision. In this work, we propose to disentangle the similar and dissimilar interests between S&R behaviors with self-supervision signals. PROBLEM FORMULATION Assume that the sets of users, items, and queries are denoted by U, I, and Q respectively, where ∈ U denotes a user, ∈ I denotes an item, and ∈ Q denotes a query. For the recommendation history, a user has a context of chronologically ordered item interactions: = [ 1 , 2 , . . . , ], where is 's interacted item sequence, is the number of items that user has interacted till timestamp , and is the -th interacted item. For the search history, a user has a context of chronologically ordered issued queries: = [ 1 , 2 , . . . , ], where is 's issued query sequence, is the number of queries issued before , and is the -th issued query. When using the search service, user also clicks items after issuing queries: = (1) 1 ,(2)1 ,(1) 2 , . . . , (1) , (2) ,(3) , where is the 's clicked item sequence corresponding to , and ( ) is the -th clicked item under query . The number of the clicked items corresponding to each query can be different and at minimum 0. Based on the above notations, we define the task of sequential recommendation with search data. Given the contextual sequences , and of a user's recommendation and search histories, the sequential recommendation with search data task aims to predict the next item that the user is likely to interact with at timestamp +1, i.e., ( +1 \| , , ). Note that it differs from the conventional sequential recommendation task, which predicts ( +1 \| ). OUR APPROACH: SESREC In this section, we elaborate on the proposed SESRec. Overview The overview of SESRec is illustrated in Figure 2. First, we embed the sparse input data into dense representations. Then, we leverage the transformer layers [23] to learn the contextual representations of historical behaviors. For disentangling interests, we separate two behavior sequences into sub-sequences that respectively represent similar and dissimilar interests. And we aggregate behavior sequences into vectors to represent user interests w.r.t. the candidate item. Finally, we concatenate all the vectors together to get the overall representation vector, which is followed by an Multilayer Perceptron (MLP) to generate the ultimate prediction. Specifically, we design several components to disentangle user interests with self-supervision and aggregate user interests from all aspects. These designed components are shown in Figure 2 within colored boxes. We align query and item representations into the same semantic space with the InfoNCE loss. Then we separate S&R behavior sequences into sub-sequences respectively. We leverage self-supervision signals to guide the separation based on the triplet loss. Last, we introduce an interest extraction module that aggregates the original sequences and constructed sub-sequences to form aggregated, similar and dissimilar interest representations of both behaviors. Encoding Sequential Behaviors Embedding Layer. We maintain separate look-up tables for IDs and attributes of users, items, and queries. As for users and items, given a user (an item), we concatenate his (its) ID and attribute embeddings to form a user (item) representation: e = e ID ∥e 1 ∥ · · · ∥e (e = e ID ∥e 1 ∥ · · · ∥e ), where ∥ denotes concatenation, 1 , . . . , and 1 , . . . , denote the attributes of users and items, respectively. As for queries, each query contains several terms ( 1 , 2 , . . . , \| \| ). We obtain the query embedding by concatenating query ID embedding and the mean pooling of term embeddings: e = e ID ∥MEAN(e 1 , e 2 , . . . , e \| \| ), where e ID is Queries and Clicked Items (Search) …… Query …… Interacted Items (Recommendation) Query Candidate Item v User u …… …… …… e 8 E e D D b E 8 b E @ b E 2 e 8 E e D D b E 8 b E @ b E 2 1 e 8 E e D D b E 8 b E @ b E 2 1 e 8 E e D D b E 8 b E @ b E 2 1 + + Query-item Alignment Bias Encoding E B E A E B E A 1 Transformer Layer MHA & FFN MHA & FFN H B H A H B H A 1 Interest Contrast Multi- interest Extraction P B N B P A N A P B N B P A N A P B N B P A N A P B N B P A N A 1 e 8 E e D D b E 8 b E @ b E 2 1 Interest Contrast Multi- interest Extraction u B u A u B u A MLP PredictionˆC +1 D,E Multi-interestH B H A H B H A Co-attention P B N B P A N A 1 P B N B P A N A 1 partition partition P B N B P A N A P B N B P A N A 1 A a B a A A a B a A 1 A a B a A 1 Pulling Closer Pushing Away H B H A 1 P B N B P A N A 1 P B N B P A N A 1 e 8 E e D D b E 8 b E @ b E 2 attention attention attention u B u A 1 attention Value Key Query Outputs Figure 2: The architecture of SESRec. From left to right is the process of modeling S&R histories. On the far right is the process of ultimate prediction. The three colored modules with dashed lines conduct interest disentanglement. the query ID embedding, e is the ID embedding of the -th term, and MEAN() denotes mean pooling. Many queries occur repeatedly in search data, so query ID is informative. And most queries consist of less than five terms and lack strong sequential patterns. So the average pooling operation, following the bag-of-words paradigm, is effective and efficient. For a user , given the context , , and , we obtain embedding matrices of historically interacted items, issued queries, and clicked items, denoted as E = [e 1 , e 2 , . . . , e ] ⊺ ∈ R × , E = [e 1 , e 2 , . . . , e ] ⊺ ∈ R × , and E = [e 1 , e 2 , . . . , e \| \| ] ⊺ ∈ R \| \|× , respectively, where and are dimensions of item and query embeddings. Besides, we incorporate learnable position encoding matrices to model the sequential order of behaviors, denoted as P ∈ R × and P ∈ R × , respectively, where is dimension. As for search behaviors, we also adopt type embedding for queries of different sources, e.g., user-typed queries, user-specific historical queries, and queries related to the current item. The search type embedding matrix is denoted by M ∈ R × , where is the total number of all possible search sources. Because it is challenging to model user interests with query and item representations in unaligned vector spaces, we transform the item and query embeddings into latent vector spaces with the same dimension. The transformed embedding matrices of interacted items, issued queries, and clicked items are calculated as: E = E W , E = E W , E = E W ,(1) where E ∈ R × , E ∈ R × and E ∈ R \| \|× are transformed matrices, W ∈ R × and W ∈ R × are trainable parameters for linear projection. Bias Encoding. We incorporate position encodings for S&R behaviors to make use of the order relations of the sequence. The recommendation sequence matrix E ∈ R × is obtained by summing the interacted item matrix and the position matrix: E = E + P(2) As for search behaviors, we additionally introduce type encodings along with position encodings. The search sequence matrix E ∈ R × is defined as: E = E + E + P + M(3) where E ∈ R × is the matrix of the mean pooling of all clicked items' matrices under each query, and M ∈ R × is the type matrix. To obtain E , we first group clicked items E by the issued queries, i.e., several items clicked by the same query are divided into the same group. Then we apply a mean pooling operation on each group to get the matrix E ∈ R × . The type matrix M is defined as a sequence of type embeddings for each query in the search history, where each element m ∈ R in M is obtained from the look-up table M . We add the type matrix to model the correlation between search behaviors and search sources. Considering clicked items contain identical users' interests as their corresponding queries, we fuse the issued query sequence and clicked item sequence to form a unified search sequence by adding them up in Equation (3). Transformer Layer. To learn an enhanced contextual representation for each element in a given sequence, we use the transformer layer [23] to capture the relations between each element with other elements in the S&R sequences. The transformer layer generally consists of two sub-layers, i.e., a multi-head self-attention layer and a point-wise feed-forward network. We apply the transformer layers for S&R sequences, respectively: F = MHA (E ), F = MHA (E ),(4)H = FFN (F ), H = FFN (F ),(5) where H ∈ R × and H ∈ R × denote enhanced matrices of S&R sequences respectively, the multi-head self-attention is abbreviated to "MHA", and the two-layer feed-forward network is abbreviated to "FFN". Self-supervised Interest Disentanglement As mentioned before, user interests between S&R behaviors have overlaps and differences. Since there does not exist any annotated label of user interests, we leverage contrastive learning techniques to disentangle the S&R behaviors with self-supervision and then extract user interests from three aspects, i.e., the aggregated behaviors, the two separated behaviors containing similar and dissimilar interests. Query-item Alignment. It is challenging for the behavior encoders to jointly learn user interests from S&R behaviors that have unaligned embeddings. Also, it is unfeasible to disentangle user interests from S&R behaviors without knowing the semantic similarities between queries and items. Thus, we align the embeddings of queries and items as follows before further extracting user interests from them. Because items and queries have different forms of features, their original embeddings are unaligned in different vector spaces. As shown in Equation (1), we first transform the item and query embeddings into a latent vector space. Then, inspired by works [10,16] for multi-model learning, we leverage a contrastive learning loss to teach the model which queries and items are similar or different. Given issued query and clicked item sequence matrices E = [ê 1 ,ê 2 , . . . ,ê ] ⊺ ∈ R × and E = [ê 1 ,ê 2 , . . . ,ê \| \| ] ⊺ ∈ R \| \|× , we minimize the sum of two InfoNCE [21] losses: one for query-toitem alignment L , A q2i = − ∑︁ =1 \| \| ∑︁ =1 log exp( (ê ,ê )/ ) ℎ ∈I neg exp( (ê ,ê ℎ )/ ) ,(6) and the other for item-to-query alignment L , A i2q = − ∑︁ =1 \| \| ∑︁ =1 log exp( (ê ,ê )/ ) ∈ Q neg exp( (ê ,ê )/ ) ,(7) where is a learnable temperature parameter, \| \| denotes the number of clicked items of query which satisfies =1 \| \| = \| \|, I neg and Q neg denote the sets of randomly sampled items and queries respectively, and is a similarity function. The function is defined as: (p, q) = tanh(p ⊺ W A q), where tanh denotes the activation function and the introduction of W A ∈ R × ensures the query-item correlation estimation can be different with the criterion used in the ultimate prediction. Finally, the query-item alignment loss is obtained by: L , ali = 1 2 (L , A q2i + L , A i2q ),(8) Interest Contrast. To conduct interest disentanglement, we employ a contrastive learning mechanism to distinguish similar and dissimilar interests between the contextual representations of behaviors H and H . After the transformer layers, given the matrices H and H , we construct a co-dependant representation matrix of both behaviors, which generates the similarity scores of two sequences. Inspired by recent works [15,25] for question answering, we leverage the coattention technique. We first compute an affinity matrix A ∈ R × as follows: A = tanh(H W (H ) T ),(9) where W ∈ R × is a learnable weight matrix. The affinity matrix A contains affinity scores corresponding to all pairs of recommendation behaviors and search behaviors. We multiply the affinity matrix A and the search matrix H (or the recommendation matrix H ), and then normalize the multiplication results to get similarity scores for each element in one sequence across all the elements in the other sequence: a = softmax(W H T A T ), a = softmax(W H T A),(10) where a ∈ R and a ∈ R are similarity scores, W , W ∈ R 1× are trainable parameters for linear projection. Next, we exploit a triplet loss to self-supervise the disentanglement of similar and dissimilar interests between two behaviors. Given similarity scores a and a , elements in H and H with higher scores can be interpreted as representative ones for similar interests, while elements with lower scores can be interpreted as representative ones for dissimilar interests. Let P (N ) denote the set of elements containing similar (dissimilar) interests of S&R behaviors. As such, we perform hard selection to separate S&R sequences into two subsequences as follows: P = {h \| a > }, N = {h \| a ≤ },(11)P = {h \| a > }, N = {h \| a ≤ },(12) where h , h ∈ R are the -th vectors in matrices H and H , a and a are similarity scores for h and h respectively, and are selection thresholds. Since a and a are normalized after softmax, we empirically set the thresholds and to the uniform values 1 and 1 . The positives with similarity scores larger than the thresholds can be interpreted as similar interests with aboveaverage similarities. The negatives, as the counterparts of positives, are with below-average similarities. Then we design the anchors, positives and negatives of the triplet loss. To guide learning the disentanglement, we utilize the original sequences H and H to form anchors, and leverage the separated subsequences P , P and N , N to serve as positives and negatives. The anchors, positives, and negatives can be calculated as: i = ∑︁ =1 a h , i = MEAN(P ), i = MEAN(N ),(13)i = ∑︁ =1 a h , i = MEAN(P ), i = MEAN(N ),(14) where i , ∈ R are anchors, i , i ∈ R are positives, i , i ∈ R are negatives. Then we perform contrastive learning, which requires the anchors to be similar with positives, and to be different from negatives. Based on these vectors, We implement triplet losses for S&R behaviors, respectively. Formally, the loss function is computed as follows: L tri ( , , ) = max{ ( , ) − ( , ) + , 0},(15) where denotes distance function which is implemented as euclidean distance, denotes a positive margin value, , and denote anchors, positives and negatives, respectively. Finally, the interest contrast loss can be obtained by summing up two triplet losses, one for recommendation behaviors, the other for search behaviors: L , con = L tri (i , i , i ) + L tri (i , i , i ),(16) Remark. In most cases, users use S&R services at different frequencies. The lengths and update frequencies of two behaviors are different since they are collected from different services. That is why we employ triplet losses for the two behaviors, respectively. We update the model parameters of each behavior with its own constructed interest representations, which ensures the consistency of model training. Besides, considering that similar and dissimilar interests usually overlap with each other to some extent, there is no clear distinction between them. The triplet loss performs pairwise comparisons, which reduces the differences between similar things and increases the differences between different things. That is why we use the triplet loss instead of other contrastive loss functions, e.g., InfoNCE [21], which imposes too strong punishment on the similarity between positives and negatives. Multi-interest Extraction. Based on the original behaviors and separated behaviors containing similar and dissimilar interests, we extract user interests from three aspects, i.e., aggregated, similar, and dissimilar interests. Given a candidate item , we utilize an attention mechanism to reallocate the user interests w.r.t. the candidate item. For recommendation behaviors, interests can be extracted from three aspects as follows: u all = ∑︁ =1 all h , all = exp((h ) T W e ) =1 exp((h ) T W e ) ,(17)u sim = ∑︁ h ∈ P sim h , sim = exp((h ) T W e ) h ∈ P exp((h ) T W e ) ,(18)u diff = ∑︁ h ∈N diff h , diff = exp((h ) T W e ) h ∈N exp((h ) T W e ) ,(19) where u all , u sim , u diff ∈ R are representative vectors for aggregated, similar, and dissimilar interests, W is the trainable parameters to model correlation between recommendation behaviors and u = u all ∥u sim ∥u diff ,(20) where u ∈ R 3 . Similarly, we can obtain the representation of search interests in the same way, i.e., u ∈ R 3 . Prediction and Model Training Prediction. To predict the interaction, we utilize the widely adopted two-layer MLP [33,34] to model feature interaction and make predictions. Given a user and an item at timestamp + 1, the prediction score can be calculated as follows: +1 , = MLP(u ∥u ∥e ∥e ),(21) whereˆ+ 1 , denotes the prediction score, e and e are embeddings of the item and the user , respectively. Model Training. Following the existing works' settings [33,34], we adopt the negative log-likelihood function to supervise the final prediction: L , rec = − 1 ∑︁ ∈ O +1 , log(ˆ+ 1 , ) + (1 − +1 , ) log(1 −ˆ+ 1 , ),(22) where O is the set composed of training pairs of one positive item and −1 negative items. In order to apply additional self-supervised signals about query-item alignment and interest disentanglement, we train our model in an end-to-end manner under a multi-task learning schema. The overall loss function is formulated as: L = \|U \| ∑︁ =1 ∑︁ =1 (L , rec + L , ali + L , con ) + \|\|Θ\|\| 2 .(23) where \|U\| is the number of users, denotes the timestamp of the user 's latest interaction, and are hyper-parameters for additional tasks, and \|\|Θ\|\| 2 denotes the 2 regularization to avoid over-fitting. EXPERIMENT 5.1 Experimental Setup 5.1.1 Dataset. SESRec needs user S&R behavior logs simultaneously. In the following experiments, we evaluated models on two datasets: one is collected from logs of a short-video app, and the other is based on a widely used public Amazon dataset [4,14]. Table 1 reports statistics of both datasets. Kuaishou Dataset: This dataset is constructed based on behavior logs of 35,721 users who elected to use both S&R services on the short-video app named Kuaishou over one month in 2022. The historical S&R behaviors have been recorded. For dataset preprocessing, following the common practice in [8,19,36], we group interaction records by users, sort them by timestamp ascendingly and filter unpopular items and users with fewer than five interaction records. To the best of our knowledge, there doesn't exist a public dataset that contains both S&R behaviors. Following a standard approach for product search [1], we enhance a recommendation dataset, the Amazon dataset [4,14], by generating search behaviors. We adopt the "Kindle Store" subset of the five-core Amazon dataset that covers data in which all users and items have at least five reviews. The detailed generation process of search behaviors 4 can be found in [1]. Please note that this automatically constructed search dataset has been widely used by product search community [1,3,22,29]. Following [3], we randomly select one query for items with multiple constructed queries to model the sequential behaviors. Following previous works [8,19,36], we adopt the leave-oneout strategy to split both datasets. For each user, we hold out the most recent action for testing, the second most recent action for validation, and all the remaining actions for training. Besides, to ensure data quality, we filter interactions in which the user doesn't have historical S&R history simultaneously. Evaluation Metrics. Following [19,36], we employ several widely used ranking metrics, including Hit Ratio (HIT), Normalized Discounted Cumulative Gain (NDCG), and Mean Reciprocal Rank (MRR). We report HIT with = 1, 5, 10 and NDCG with = 5, 10. We pair the ground-truth item with 99 randomly sampled items that the user has never interacted with. For all metrics, we calculate them according to the ranking of items and report average results. Baseline Models. In this work, we compare SESRec with state-of-the-art methods. For sequential recommendation methods without leveraging search data, we include following sequential models: (1) STAMP [12]: It captures users' general interests from the long-term memory and short-term memory; (2) DIN [34]: It uses an attention mechanism to model user interest from historical behaviors w.r.t. a target item; (3) GRU4Rec [6]: It is the first work to apply RNN to sessionbased recommendation with a ranking based loss; (4) SASRec [8]: It is a unidirectional transformer-based sequential model, which uses self-attention to capture sequential preferences; (5) DIEN [33]: It enhances DIN by combining attention with GRU units to take interests evolution into consideration; (6) FMLPRec [36]: It is an all-MLP model with learnable filters which can adaptively attenuate the noise information in historical sequences. For methods using search data, we include following searchaware models: (7) NRHUB [24]: It is a news recommendation model leveraging heterogeneous user behaviors; (8) JSR [28]: It is a general framework which optimizes a joint loss. We implement it following [17] to ensure a fair comparison with other sequential models; (9) IV4REC [17]: It is a model-agnostic framework exploiting search queries as instrumental variables to enhance the recommendation model. Following the original paper, we apply this framework over DIN; (10) Query-SeqRec [5]: It is a query-aware sequential model which incorporates queries into user behaviors using a transformer-based model. (11) SRJGraph [31]: It is a GNNbased model which exploits a heterogeneous graph to model the user-item and user-query-item interactions for S&R. Implementation Details. All hyper-parameters of baselines are searched following suggestions from the original papers. For all models, the maximum sequence length of recommendation (search) history is set to 150 (25) on the Kuaishou dataset and 15 (15) on the Amazon dataset. , and are set as 48, 64 and 48 (32, 32, and 32) on the Kuaishou dataset (Amazon dataset). For the fair competition, we deploy the same setting of item embeddings on all models. For query embeddings, we also randomly initialize term embeddings for all search-aware models. The batch size is set as 256. The hyper-parameters and are set as 0.1 and 0.001, respectively. The margin is set as 0.1. We use the Adam [9] with a learning rate of 0.001, and adopt early-stopped training to avoid over-fitting. More details can be found in the open source codes 5 . Table 2 reports the recommendation results on the two datasets. Overall Performance We have the following observations: • Search-aware models do not always bring performance gains. SRJGraph is the SOTA approach that mines both S&R behaviors. However, the SOTA sequential model FMLP-Rec can obtain compatible or even better performance than SRJGraph. Besides, Query-SeqRec shares a similar architecture as SASRec but achieves slightly poorer performance than SASRec. These phenomenons indicate • Compared to baseline models, SESRec achieves the best performance on both datasets in most cases, significantly outperforming the SOTA sequential and search-aware methods FMLP-Rec and SRJGraph by a large margin (paired t-test at -value < 0.01). The relative improvements over conventional sequential models reveal that leveraging search behaviors can boost recommendation models. Furthermore, the substantial performance gains over search-aware models validate the effectiveness of interest disentanglement in S&R behaviors. • Comparing SESRec on two datasets, SESRec achieves fewer relative improvements on the Amazon dataset. Considering that search data of the Amazon dataset was automatically constructed, many items share the same queries, and the number of queries is sparse compared with the number of items, as shown in Table 1. Thus, the query-item alignment and interest contrast modules play a minor role in boosting recommendation performance on this dataset. Detailed Empirical Analysis In this section, we conducted more detailed experiments on the real-world Kuaishou dataset, providing in-depth analyses of how and why SESRec achieved state-of-the-art performance. Ablation Study. SESRec consists of several key components, including alignment for queries and items, disentanglement for user S&R interests with self-supervised signals, and the multi-interest extraction module. To investigate how different components affect the performance of SESRec, we conducted ablation studies by progressively adding three components to the base model. We added these modules one by one because each module depends on previous modules. The base model solely processes S&R behaviors with transformer layers and the interest extraction module of aggregated interests. Table 3 shows the results on the Kuaishou dataset. Next, we give a detailed discussion about each component: • L , ali : denotes the loss function of query-item alignment, which guarantees that model captures the correlation between queries and items. We observed that adding L , ali leads to consistent performance gain. The results demonstrate that understanding the interactions between queries and items is beneficial to jointly model S&R behaviors. • L , con : refers to the loss function of interest contrast, which is designed to disentangle similar and dissimilar interests between S&R behaviors. The interest disentanglement leads to performance improvement, which indicates the necessity of disentanglement. We attribute the improvement to the fact that L , con helps the model capture more accurate representations of user interests. • MIE: is short for the multi-interest extraction module, which is designed to extract interests from three perspectives, i.e., aggregated, similar, and dissimilar interests. We found that MIE contributes to the final prediction, validating the importance of MIE. Though L , con has distinguished interests into similar and dissimilar ones, the model cannot explicitly leverage similar or dissimilar interests for prediction without MIE. Analysis of Disentangled Interests. The core operation of interest disentanglement is the separation of similar and dissimilar interests. To illustrate whether the interests are disentangled, we visualize and interpret the similarity between constructed sets of similar and dissimilar interests in Figure 3. Behaviors P , P containing similar interests and N , N containing dissimilar interests are separated in Equation (11) and (12). For each user, we calculated the Jensen-Shannon (JS) divergence [11] between sets with similar interests, i.e., (P ∥P ), which is colored in red in Figure 3. The same calculation also was done for sets with dissimilar interests, i.e., (N ∥N ), which is colored in blue in Figure 3. As discussed in section 1, we can use the categories of items (interacted in recommendation history) and (clicked in search history) to estimate the similarities between two behaviors. The calculation of JS divergence is based on the distributions of item categories corresponding to P , P and N , N for each user. Figure 3 illustrates the results for all users. We observed that similar interests tend to have smaller values of JS divergence than dissimilar interests, where red data has more counts smaller than 0.6 compared with blue data. This phenomenon indicates P and P are more similar than N and N , verifying the capability of SESRec to disentangle user interests. (11) and (12). We investigated the impacts of different positive/negative selection thresholds , . In practice, we utilize an adaptive way, i.e., setting thresholds as mean values ( = 1 , = 1 ), to choose the positives and negatives. To show how , affect the performance of SESRec, we conducted experiments where , are set as constants, e.g., 1 8 and 1 16 , and we also investigated another adaptive way that sets , as the median values of given similarity scores. Table 4 presents the results with different settings of , . We can observe that constant settings ( , = 1 8 or 1 16 ) yield inferior performances compared with the two adaptive strategies. We postulate that the constant settings can not handle behaviors with different lengths consistently because longer behaviors lead to smaller mean values of scores normalized by softmax. We also find that the adopted mean value strategy achieves better performance than the median value strategy in most cases. The median value strategy separates behavior sequences into two parts of the same length. However, the similar and dissimilar parts of users' behaviors do not satisfy this distribution in most cases. That is why the adopted mean value strategy achieves the best performance. Effect of Query-item Alignment. We conducted experiments to explore how the query-item alignment facilitates representation learning and whether SESRec understands the similarity of queries and items. Toward this end, we tested the relevance between queries and their clicked items. For search behaviors, we split queries and their clicked items into pairs, where each pair consists of a query and its corresponding item. And we obtained their embeddings learned by SESRec and a variation of SESRec, which removes the query-item alignment loss L , ali . We calculated the cosine similarity of each query-item pair based on the embeddings and plotted the distribution of similarity scores in Figure 4. From the results, we can observe that embeddings learned by SESRec have smaller similarity scores than those learned without L , ali . These results indicate that the query-item alignment module ensures that queries are closed to their corresponding items in correlation. Impact of Hyper-parameters. Since we design two additional tasks, hyper-parameters and are introduced to balance the objectives in the final loss function, as defined in Equation (23). To investigate the impacts of these hyper-parameters, we conducted experiments with varying and respectively. When varying one parameter, the other is set as a constant, where 1e-1 for and 1e-3 for . From the results in Figure 5, we found that the performance peaks when is 1e-1 and is 1e-3. With a further increase of hyperparameters, the recommendation performances become worse. We attribute it to the fact that the recommendation prediction task becomes less important with larger and , which verifies the necessity of hyper-parameters to balance different tasks in the multi-task learning schema. CONCLUSION In this paper, we propose to learn disentangled search representation for recommendation with a search-enhanced framework, namely SESRec. SESRec exploits the query-item interactions to help the recommendation model to learn better representations of queries and items. With the help of self-supervision, SESRec disentangles the similar and dissimilar representations between users' search and recommendation behaviors to capture users' interests from multiple aspects. Besides, SESRec provides an end-to-end multi-task learning framework for estimating the parameters. Extensive experiments on industrial and public datasets demonstrate that SESRec consistently outperforms state-of-the-art baselines. Moreover, we empirically validate that SESRec successfully learn the disentangled representations of user interests. Figure 3 : 3Visualization of similarity between similar and dissimilar interests in S&R behaviors for all users based on a histogram. We use the JS divergence of item category distributions to estimate similarities between S&R behaviors. Similar interests are more similar than dissimilar interests with smaller values of JS divergence. 5.3.3 Analysis of , in Interest Disentanglement. The separation of similar and dissimilar interests depends on the positive/negative Figure 4 : 4Distribution of cosine similarity between representations of queries and corresponding items based on box plots. Rectangles denote mean values. With query-item alignment loss L , ali , embeddings of query-item pairs are more similar with higher cosine values. Figure 5 : 5Effects of hyper-parameters and in terms of NDCG@10 and MRR. video about dogs, the user chooses to click on the suggested query (passive search) to explore more. Later, after watching a food video, the user searches "world cup 2022", a spontaneous search unrelated to the watched video. (b) Statistics of search behaviors collected from the Kuaishou app. 57% of the search behaviors are spontaneous and 43% are passive. 23% of the spontaneous searches have dissimilar interests to the recommendation interests.Click Watch Passive Search Spontaneous Search Recommendation World Cup 2022 Recommendation Mother dog and puppies (a) An example of user S&R behaviors. 34 43 23 0 10 20 30 40 50 60 Spontaneous Search Passive Search Percentage(%) No Yes Search interests are similar to recommendation interests (b) Distribution of spontaneous and passive search w.r.t. recommendation interests. Figure 1: S&R behaviors in the short-video scenario. (a) Af- ter watching a Table 1 : 1Statistics of datasets used in this paper. 'S' and 'R' denote search and recommendation, respectively.Dataset Users Items Queries Actions-S Actions-R Kuaishou 35,721 822,832 398,924 922,531 11,381,172 Amazon 68,223 61,934 4,298 934,664 989,618 the candidate item. By concatenating these three vectors, we can get the representation of recommendation interests: Table 2 : 2Overall performance comparisons on both datasets. The best and the second-best performance methods are denoted in bold and underlined fonts respectively. * means improvements over the second-best methods are significant (p-value < 0.01).Dataset Kuaishou Amazon (Kindle Store) Category Method NDCG@5 NDCG@10 HIT@1 HIT@5 HIT@10 MRR NDCG@5 NDCG@10 HIT@1 HIT@5 HIT@10 MRR Sequential STAMP 0.2544 0.2981 0.1413 0.3616 0.4970 0.2569 0.2612 0.3103 0.1336 0.3833 0.5352 0.2608 DIN 0.2969 0.3418 0.1792 0.4092 0.5484 0.2976 0.2999 0.3495 0.1591 0.4340 0.5871 0.2942 GRU4Rec 0.3247 0.3688 0.1890 0.4517 0.5881 0.3180 0.3099 0.3662 0.1479 0.4648 0.6388 0.2993 SASRec 0.3252 0.3693 0.1904 0.4501 0.5864 0.3187 0.3822 0.4312 0.2187 0.5324 0.6838 0.3675 DIEN 0.3217 0.3704 0.1914 0.4463 0.5969 0.3192 0.3336 0.3803 0.1871 0.4706 0.6150 0.3246 FMLP-Rec 0.3354 0.3787 0.1953 0.4651 0.5988 0.3270 0.4073 0.4550 0.2349 0.5651 0.7121 0.3883 Search-aware NRHUB 0.2964 0.3431 0.1665 0.4199 0.5647 0.2933 0.2744 0.3265 0.1329 0.4099 0.5708 0.2704 JSR 0.3015 0.3513 0.1738 0.4241 0.5783 0.3004 0.3221 0.3722 0.2057 0.4386 0.5937 0.3224 IV4REC 0.3114 0.3591 0.1877 0.4282 0.5761 0.3116 0.3473 0.3960 0.1853 0.4985 0.6258 0.3331 Query-SeqRec 0.3117 0.3581 0.1740 0.4412 0.5844 0.3055 0.3692 0.4142 0.2187 0.5083 0.6470 0.3572 SRJGraph 0.3297 0.3762 0.2046 0.4479 0.5917 0.3277 0.3670 0.4043 0.2760 0.4898 0.6242 0.3708 SESRec 0.3541* 0.4054* 0.2173* 0.4848* 0.6436* 0.3490* 0.4224* 0.4663* 0.2580 0.5723* 0.7074 0.4046* Amazon Dataset 3 : Table 3 : 3Ablation studies by progressively adding proposed modules to the base model. MIE is short for the multiinterest extraction module.Model N@5 N@10 H@1 H@5 H@10 MRR Base 0.3394 0.3770 0.2027 0.4618 0.6094 0.3294 +L , ali 0.3464 0.3982 0.2106 0.4762 0.6308 0.3421 +L , con 0.3507 0.4021 0.2139 0.4812 0.6406 0.3459 +MIE 0.3541 0.4054 0.2173 0.4848 0.6436 0.3490 that blindly incorporating S&R histories may be insufficient to cap- ture users' diverse interests because users' interests in essence are entangled. Table 4 : 4The analysis of the positive/negative selection thresholds , in interest disentanglement, as defined in Equation(11)and(12)., N@5 N@10 H@1 H@5 H@10 MRR 1/16 0.3429 0.3927 0.2125 0.4681 0.6224 0.3429 1/8 0.3449 0.3940 0.2148 0.4691 0.6211 0.3415 Median 0.3510 0.4035 0.2135 0.4828 0.6453 0.3462 Mean 0.3541 0.4054 0.2173 0.4848 0.6436 0.3490 behaviors selection, as defined in Equation In this paper, we use recommendation interests to refer to user interests captured by the recommendation system, and search interests to refer to users' interests revealed in their search history. The Amazon review dataset can be found at http://jmcauley.ucsd.edu/data/amazon/.4 The constructed search data is available at https://github.com/QingyaoAi/Amazon-Product-Search-Datasets. https://github.com/Ethan00Si/SESREC-SIGIR-2023 ACKNOWLEDGMENTS Learning a Hierarchical Embedding Model for Personalized Product Search (SIGIR '17). 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