diff --git a/-dE5T4oBgHgl3EQfRg6t/content/tmp_files/2301.05522v1.pdf.txt b/-dE5T4oBgHgl3EQfRg6t/content/tmp_files/2301.05522v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb999732a26fc097d268f37f27e17b27039672fd --- /dev/null +++ b/-dE5T4oBgHgl3EQfRg6t/content/tmp_files/2301.05522v1.pdf.txt @@ -0,0 +1,286 @@ +Hyperparameter Optimization as a Service on INFN +Cloud +Matteo Barbetti1,2 and Lucio Anderlini2 +1 Department of Information Engineering, University of Florence, +via Santa Marta 3, Firenze, Italy +2 Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, +via G. Sansonse 1, Sesto Fiorentino (FI), Italy +E-mail: Matteo.Barbetti@fi.infn.it +Abstract. +The simplest and often most effective way of parallelizing the training of complex +machine learning models is to execute several training instances on multiple machines, possibly +scanning the hyperparameter space to optimize the underlying statistical model and the learning +procedure. Often, such a meta learning procedure is limited by the ability of accessing securely +a common database organizing the knowledge of the previous and ongoing trials. Exploiting +opportunistic GPUs provided in different environments represents a further challenge when +designing such optimization campaigns. In this contribution we discuss how a set of RestAPIs +can be used to access a dedicated service based on INFN Cloud to monitor and possibly +coordinate multiple training instances, with gradient-less optimization techniques, via simple +HTTP requests. The service, named Hopaas (Hyperparameter OPtimization As A Service), is +made of web interface and sets of APIs implemented with a FastAPI back-end running through +Uvicorn and NGINX in a virtual instance of INFN Cloud. The optimization algorithms are +currently based on Bayesian techniques as provided by Optuna. A Python front-end is also +made available for quick prototyping. We present applications to hyperparameter optimization +campaigns performed combining private, INFN Cloud and CINECA resources. +1. Introduction +In the last decade, machine learning (ML) has become an incredibly valuable tool in practically +every field of application, from scientific research to industry. +Increasingly complex models +achieve surprising results in a wide range of applications, such as image generation [1], language +modelling [2] or medical diagnosis [3]. Most of the ML techniques rely on the optimization of +an objective function with respect to some internal parameters, describing the performance of +the algorithm. Usually, when the optimum of the objective function is a minimum, the names +cost or loss function are adopted. +The fastest iterative optimization techniques rely on the +(Stochastic) Gradient Descent technique [4]. Unfortunately, for a wide class of optimization +problems the gradient of the loss function with respect to the model parameter is extremely +expensive to compute or cannot be defined at all. For example, optimization problems involving +noisy loss functions in contexts where analytical derivatives cannot be computed cannot rely on +gradient-descent techniques, requiring the adoption of slower, often heuristic, methods. A widely +adopted option is to define a surrogate model describing the variations of the loss function across +the parameter space together with its uncertainty, driving the optimization algorithm to explore +those regions where improvements were not statistically excluded from previous evaluations. +arXiv:2301.05522v1 [cs.DC] 13 Jan 2023 + +Techniques adopting this approach are referred to as Bayesian optimization (BO) methods and +have been an active area of research in ML for the last decade [5, 6, 7, 8, 9]. +Tuning the performance of ML models may benefit from the optimization of the +hyperparameters, defined as all those parameters that are not learned during the model training +procedure but encode some arbitrariness in the architecture of the model itself or in the procedure +to train it [5]. In practice, hyperparameter optimization (HPO) studies require training the +model multiple times to explore the hyperparameter space. +Since training ML models is +computationally expensive, HPO campaigns should focus as much as possible on those regions +of the hyperparameter space where the model performs better to reduce the time needed for +finding the best configuration. On the other hand, the loss is often a noisy function of the +hyperparameters as multiple training procedures may result in different performance because of +the intrinsic randomness of the stochastic gradient-descent techniques. +Exploring the hyperparameter space requires many independent trainings, +or trials, +that can run in parallel on different computing resources. +In general, accessing more +resources enables the exploration of larger hyperparameter spaces, possibly resulting in better +models. +Opportunistic access to compute resources may provide valuable contribution to +HPO campagins. +Unfortunately, coordinating studies on resources from different providers, +restrictions and regulations challenges the adoption of existing HPO services. +In this document, we propose Hopaas (Hyperparameter OPtimization As A Service), +implementing a set of RestAPIs to orchestrate HPO studies across multiple computing instances. +Computing nodes from multiple HPC centers can concur dynamically to the same optimization +study, requesting to the Hopaas server a set of hyperparameters to test and then sending +back the outcome of the training procedure. +Several trials of one or more studies can be +tracked and monitored through the web interface provided by the Hopaas service. A reference +implementation, with a server instance1 deployed on INFN Cloud resources and a simple client +package [10] wrapping the RestAPIs to Python functions, is also discussed. +2. Hopaas API specification +We refer to a trial as a single training attempt with a specific set of hyperparameters to test. A +study represents an optimization session and includes a collection of trials. In practice, a study +is unambiguously defined by the set of hyperparameters to optimize, the range of values where +searching the optimum, and the modality in which this search is carried out (e.g., grid search, +Bayesian methods [5], or evolutionary algorithms [11]). +The core activity of the Hopaas service is to manage distributed optimization studies. A set +of RestAPIs is designed to create trials, finalize them, and update the service on intermediate +values of the objective function to enable early termination of the trial before the conclusion +of the training. The APIs named ask, tell and should prune implement these actions upon +POST HTTP requests, with user authentication based on an API token in the request path. +A computing node ready to test a set of hyperparameters will query the Hopaas server via +the ask API, including in the request body all the information needed to define an optimization +study unambiguously. The Hopaas server will define a new trial, possibly assigning it to an +existing study, or creating a new one, and providing a unique identifier of the new trial and the +set of hyperparameters to evaluate as part of the HTTP response. +Usually, the evaluation of the set of hyperparameters consists of training a model defined by +those hyperparameters aiming at the resulting value of the objective function. The evaluated +performance metric may correspond to the loss function computed during the training procedure +but, in general, it can be any numerical score obtained processing a given set of hyperparameters. +Once the evaluation is completed, the computing node will finalize the trial using the tell API, +1 Visit https://hopaas.cloud.infn.it for additional details. + +POST REQUEST +ASK +POST REQUEST +TELL +Hopaas server +Processing Trial #1 +Study A +Computing node +loss +trials +Computing node +Study B +Processing Trial #1 +Study A +Study B +Trial #1 +Trial #2 +Trial #1 +Trial #2 +Trial #3 +Computing node +Study B +Processing Trial #3 +/ +Computing node +Study B +Processing Trial #2 +Processing Trial #2 +Study A +Computing node +On-prem +Figure 1. +A Hopaas server orchestrating multiple studies across multiple sites. +whose body will include the unique identifier of the trial and the final evaluation of the objective +function. +The Hopaas server may serve multiple ask requests from different sources, assigning them +to one or different studies, while updating the surrogate model each time a new evaluation is +made available by querying the tell API. A schematic representation of the orchestration of +studies in multiple sites is reported in Figure 1. +Depending on the specific ML algorithm, intermediate evaluations of the objective function +can be accessed during the training procedure and used to abort non-promising trials (pruning) +without wasting computing power to take the training procedure to an end. Optionally, the +computing node may update the Hopaas server with intermediate evaluations of the objective +function by querying the should prune API for monitoring and pruning purposes. The body of +a should prune request will contain the unique identifier of the trial, the intermediate value of +the loss function and an integer number encoding the progress of the training procedure, named +the step. The HTTP response will indicate whether the study should be early terminated, or it +is sufficiently likely to result in an improvement over the previous tests. +A reference Python front-end was developed aiming at a facilitated access to the Hopaas +service from Python applications [10]. While Python is a primary choice for many scientific +applications, it should be noticed that the client simply wraps the RestAPIs into classes and +functions, as the Hopaas protocol is designed to be language-agnostic, relying on widely adopted +web communication standards. In addition, the Hopaas client is also framework-agnostic since +the evaluation of the objective function for a given set of hyperparameters can be implemented +with any framework and environment. +3. Implementation +The reference implementation for the Hopaas service running on INFN Cloud relies on +containerized applications orchestrated with docker-compose. The web server implementing +the RestAPIs is a scalable set of Uvicorn instances running an application based on the FastAPI +framework. The BO algorithms are provided by integrating the back-end with Optuna, while + +INPN +CLOUDCINECAawsCERNINPN +CLOUDhttp://hopaas.cloud.infn.it/api/ask/user_token +POST +http://hopaas.cloud.infn.it/api/tell/user_token +POST +200 +empty HTTP response +Set-up of the optimization study +(e.g. title, min/max, sampler, search space) +1. +Retrieving the trial parameters +(both constant and optimizable parameters) +4. +Objective function computation +(e.g. closed formula, machine learning score) +5. +Creation/loading of the optimization study +(same set-up allows to load existing optimization study) +2. +Parameters sampled according to study set-up +(the prepared set of parameters defines an optimization trial) +3. +Optimization study updated with trial results +(parallel tests enabled by gradientless optimization strategies) +6. +Processing... +Ready for testing a new trial set +(both from the same or a new optimization study) +7a. +Ready for providing a new trial set +(score-driven suggestions enabled by default) +7b. +Framework-agnostic optimization campaign +Computing instance +Hopaas server ++ +Authorized users only! +200 +HTTP response with trial parameters +Figure 2. +Workflow of an optimization study with a client-server approach based on RestAPIs. +future extensions to additional frameworks are planned. The access to the Uvicorn instances +from the Internet is mediated by an NGINX reverse proxy accessed via the encrypted HTTPS +protocol. A PostgreSQL instance is part of the docker-compose configuration to provide shared +persistency to the multiple instances of the web application back-end. +The workflow of the +interaction between the Hopaas server and computing nodes is depicted in Figure 2. +The same Hopaas server is designed to serve web-based user access. A web application, +developed in HTML, CSS and JavaScript, is shipped to the client browser as defined by a set +of web-specific APIs in Uvicorn. The web pages of the front-end provide dynamic visualizations +by fetching data from specialized APIs at regular intervals. Plots showing the evolution of the +loss reported by different studies and trials are obtained with the Chartist library. +The user authentication and authorization procedure of the web application is managed +relying on access tokens as defined by the OAuth2 standard, using the INFN GitLab instance +as identity provider. +Support for INDIGO IAM is also planned for the future [12]. +Once +authenticated, users can generate multiple API tokens through the web application. Each API +token has a validity period defined at generation and can be revoked at any time. Tokens with +shorter validity are more appropriate for usage in public or untrusted contexts. +4. Tuning the LHCb ultra-fast simulation models with Hopaas and Marconi 100 +Machine Learning is an important research area in High Energy Physics (HEP), with first +applications dating back to the 1990s. Recent years have witnessed an explosion in the use of +ML techniques in HEP to face the computational challenges raised by the upcoming and future +runs of the Large Hadron Collider (LHC) [13]. With an increasing number, complexity and +range of applications of the ML models, HPO is becoming popular in HEP [14], and specialized +frameworks targeting distributed computing are being developed [7]. +The reference implementation of the Hopaas service presented here has been successfully used +for HEP applications and, in particular, to optimize the parameterizations for Lamarr [15], a +novel LHCb ultra-fast simulation framework. Most of the parameterizations of Lamarr rely + +scikitdmlc +XGBoostA +wwWINPN +CLOUDuvicornFastAPion Generative Adversarial Networks (GANs) [16], advanced algorithms taken from Computer +Vision that were demonstrated to be able to well reproduce the distributions obtained from +standard simulation techniques [17, 18]. Adversarial models are particularly sensitive to the +choice of the hyperparameter configuration and require intensive optimization campaigns to +model accurately the target distributions. +Several optimization studies have been orchestrated by the Hopaas service using diverse +computing instances, from scientific providers (like INFN, CERN and CINECA) and from +commercial cloud provider (like GCP or AWS). Most of the resources have been provided by +the CINECA supercomputer Marconi 100, with a custom network configuration to enable +the communication with the Hopaas server [19]. Hopaas was able to coordinate dozens of +optimization studies with hundreds of trials on each study from more than twenty concurrent +and diverse computing nodes. +5. Conclusion and future work +Hyperparameter tuning and Bayesian methods for gradient-less optimization provide an effective +and simple mean of exploiting opportunistic compute resources to improve ML models. +Unfortunately, environment variability and constraints set by different resource providers make +the application of existing HPO services challenging. +With Hopaas, we propose a solution +designed to require the addition of the thinnest possible layer in the model training application, +querying a central service via HTTPS and minimal RestAPIs. A reference implementation with +a server instance running on INFN Cloud and a Python client was presented and tested in a real- +world application to coordinate hyperparameter optimization campaigns on multiple resource +providers including INFN, CERN, and CINECA. In the future we will improve the quality of +the Web User Interface, for example enabling custom model documentation and sharing among +multiple users, and introduce support to multi-objective optimizations. +Acknowledgments +We would like to thank Doina Cristina Duma and the rest of the INFN Cloud group for the +technical support in the deployment and test of Hopaas. We acknowledge enlightening and +motivating discussions with Diego Ciangottini, Stefano Dal Pra, Piergiulio Lenzi and Daniele +Spiga, especially on future applications and developments. +References +[1] Ramesh A et al. 2021 PMLR’21 pp 8821–8831 +[2] Brown T et al. 2020 NeurIPS’20 pp 1877–1901 +[3] Richens J G, Lee C M and Johri S 2020 Nat. Comm. 11 3923 +[4] Orr G and M¨uller K 2003 Neural Networks: Tricks of the Trade (Springer Berlin Heidelberg) +[5] Bergstra J et al. 2011 NeurIPS’11 pp 2546–2554 +[6] Bergstra J, Yamins D and Cox D 2013 PMLR’13 pp 115–123 +[7] Liaw R et al. 2018 (Preprint 1807.05118) +[8] Akiba T et al. 2019 KDD’19 pp 2623–2631 (Preprint 1907.10902) +[9] Head T et al. 2021 URL https://doi.org/10.5281/zenodo.5565057 +[10] Barbetti M and Anderlini L 2023 URL https://doi.org/10.5281/zenodo.7528502 +[11] Tani L et al. 2021 EPJ C 81 170 +[12] Spiga D et al. 2020 EPJ Web Conf. 245 07020 +[13] Albertsson K et al. 2018 J. Phys.: Conf. Ser. 1085 022008 +[14] Wulff E, Girone M and Pata J 2022 (Preprint 2203.01112) +[15] Anderlini L et al. 2022 PoS ICHEP2022 233 +[16] Goodfellow I et al. 2014 NeurIPS’14 pp 2672–2680 (Preprint 1406.2661) +[17] Anderlini L et al. (LHCb) 2022 (Preprint 2204.09947) +[18] Ratnikov F et al. 2023 Nucl. Instrum. Meth. A 1046 167591 +[19] Mariotti M, Spiga D and Boccali T 2021 PoS ISGC2021 002 + diff --git a/-dE5T4oBgHgl3EQfRg6t/content/tmp_files/load_file.txt b/-dE5T4oBgHgl3EQfRg6t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf6535e642031764538b7e90beb51c960e15d759 --- /dev/null +++ b/-dE5T4oBgHgl3EQfRg6t/content/tmp_files/load_file.txt @@ -0,0 +1,159 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf,len=158 +page_content='Hyperparameter Optimization as a Service on INFN Cloud Matteo Barbetti1,2 and Lucio Anderlini2 1 Department of Information Engineering, University of Florence, via Santa Marta 3, Firenze, Italy 2 Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Sansonse 1, Sesto Fiorentino (FI), Italy E-mail: Matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='Barbetti@fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='it Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The simplest and often most effective way of parallelizing the training of complex machine learning models is to execute several training instances on multiple machines, possibly scanning the hyperparameter space to optimize the underlying statistical model and the learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Often, such a meta learning procedure is limited by the ability of accessing securely a common database organizing the knowledge of the previous and ongoing trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Exploiting opportunistic GPUs provided in different environments represents a further challenge when designing such optimization campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In this contribution we discuss how a set of RestAPIs can be used to access a dedicated service based on INFN Cloud to monitor and possibly coordinate multiple training instances, with gradient-less optimization techniques, via simple HTTP requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The service, named Hopaas (Hyperparameter OPtimization As A Service), is made of web interface and sets of APIs implemented with a FastAPI back-end running through Uvicorn and NGINX in a virtual instance of INFN Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The optimization algorithms are currently based on Bayesian techniques as provided by Optuna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A Python front-end is also made available for quick prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' We present applications to hyperparameter optimization campaigns performed combining private, INFN Cloud and CINECA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Introduction In the last decade, machine learning (ML) has become an incredibly valuable tool in practically every field of application, from scientific research to industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Increasingly complex models achieve surprising results in a wide range of applications, such as image generation [1], language modelling [2] or medical diagnosis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Most of the ML techniques rely on the optimization of an objective function with respect to some internal parameters, describing the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Usually, when the optimum of the objective function is a minimum, the names cost or loss function are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The fastest iterative optimization techniques rely on the (Stochastic) Gradient Descent technique [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Unfortunately, for a wide class of optimization problems the gradient of the loss function with respect to the model parameter is extremely expensive to compute or cannot be defined at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' For example, optimization problems involving noisy loss functions in contexts where analytical derivatives cannot be computed cannot rely on gradient-descent techniques, requiring the adoption of slower, often heuristic, methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A widely adopted option is to define a surrogate model describing the variations of the loss function across the parameter space together with its uncertainty, driving the optimization algorithm to explore those regions where improvements were not statistically excluded from previous evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='05522v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='DC] 13 Jan 2023 Techniques adopting this approach are referred to as Bayesian optimization (BO) methods and have been an active area of research in ML for the last decade [5, 6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Tuning the performance of ML models may benefit from the optimization of the hyperparameters, defined as all those parameters that are not learned during the model training procedure but encode some arbitrariness in the architecture of the model itself or in the procedure to train it [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In practice, hyperparameter optimization (HPO) studies require training the model multiple times to explore the hyperparameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Since training ML models is computationally expensive, HPO campaigns should focus as much as possible on those regions of the hyperparameter space where the model performs better to reduce the time needed for finding the best configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' On the other hand, the loss is often a noisy function of the hyperparameters as multiple training procedures may result in different performance because of the intrinsic randomness of the stochastic gradient-descent techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Exploring the hyperparameter space requires many independent trainings, or trials, that can run in parallel on different computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In general, accessing more resources enables the exploration of larger hyperparameter spaces, possibly resulting in better models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Opportunistic access to compute resources may provide valuable contribution to HPO campagins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Unfortunately, coordinating studies on resources from different providers, restrictions and regulations challenges the adoption of existing HPO services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In this document, we propose Hopaas (Hyperparameter OPtimization As A Service), implementing a set of RestAPIs to orchestrate HPO studies across multiple computing instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Computing nodes from multiple HPC centers can concur dynamically to the same optimization study, requesting to the Hopaas server a set of hyperparameters to test and then sending back the outcome of the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Several trials of one or more studies can be tracked and monitored through the web interface provided by the Hopaas service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A reference implementation, with a server instance1 deployed on INFN Cloud resources and a simple client package [10] wrapping the RestAPIs to Python functions, is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Hopaas API specification We refer to a trial as a single training attempt with a specific set of hyperparameters to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A study represents an optimization session and includes a collection of trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In practice, a study is unambiguously defined by the set of hyperparameters to optimize, the range of values where searching the optimum, and the modality in which this search is carried out (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=', grid search, Bayesian methods [5], or evolutionary algorithms [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The core activity of the Hopaas service is to manage distributed optimization studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A set of RestAPIs is designed to create trials, finalize them, and update the service on intermediate values of the objective function to enable early termination of the trial before the conclusion of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The APIs named ask, tell and should prune implement these actions upon POST HTTP requests, with user authentication based on an API token in the request path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A computing node ready to test a set of hyperparameters will query the Hopaas server via the ask API, including in the request body all the information needed to define an optimization study unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The Hopaas server will define a new trial, possibly assigning it to an existing study, or creating a new one, and providing a unique identifier of the new trial and the set of hyperparameters to evaluate as part of the HTTP response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Usually, the evaluation of the set of hyperparameters consists of training a model defined by those hyperparameters aiming at the resulting value of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The evaluated performance metric may correspond to the loss function computed during the training procedure but, in general, it can be any numerical score obtained processing a given set of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Once the evaluation is completed, the computing node will finalize the trial using the tell API, 1 Visit https://hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='it for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' POST REQUEST ASK POST REQUEST TELL Hopaas server Processing Trial #1 Study A Computing node loss trials Computing node Study B Processing Trial #1 Study A Study B Trial #1 Trial #2 Trial #1 Trial #2 Trial #3 Computing node Study B Processing Trial #3 / Computing node Study B Processing Trial #2 Processing Trial #2 Study A Computing node On-prem Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A Hopaas server orchestrating multiple studies across multiple sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' whose body will include the unique identifier of the trial and the final evaluation of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The Hopaas server may serve multiple ask requests from different sources, assigning them to one or different studies, while updating the surrogate model each time a new evaluation is made available by querying the tell API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A schematic representation of the orchestration of studies in multiple sites is reported in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Depending on the specific ML algorithm, intermediate evaluations of the objective function can be accessed during the training procedure and used to abort non-promising trials (pruning) without wasting computing power to take the training procedure to an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Optionally, the computing node may update the Hopaas server with intermediate evaluations of the objective function by querying the should prune API for monitoring and pruning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The body of a should prune request will contain the unique identifier of the trial, the intermediate value of the loss function and an integer number encoding the progress of the training procedure, named the step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The HTTP response will indicate whether the study should be early terminated, or it is sufficiently likely to result in an improvement over the previous tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A reference Python front-end was developed aiming at a facilitated access to the Hopaas service from Python applications [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' While Python is a primary choice for many scientific applications, it should be noticed that the client simply wraps the RestAPIs into classes and functions, as the Hopaas protocol is designed to be language-agnostic, relying on widely adopted web communication standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In addition, the Hopaas client is also framework-agnostic since the evaluation of the objective function for a given set of hyperparameters can be implemented with any framework and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Implementation The reference implementation for the Hopaas service running on INFN Cloud relies on containerized applications orchestrated with docker-compose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The web server implementing the RestAPIs is a scalable set of Uvicorn instances running an application based on the FastAPI framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The BO algorithms are provided by integrating the back-end with Optuna, while INPN CLOUDCINECAawsCERNINPN CLOUDhttp://hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='it/api/ask/user_token POST http://hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='it/api/tell/user_token POST 200 empty HTTP response Set-up of the optimization study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' title, min/max, sampler, search space) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Retrieving the trial parameters (both constant and optimizable parameters) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Objective function computation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' closed formula, machine learning score) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Creation/loading of the optimization study (same set-up allows to load existing optimization study) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Parameters sampled according to study set-up (the prepared set of parameters defines an optimization trial) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Optimization study updated with trial results (parallel tests enabled by gradientless optimization strategies) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Ready for testing a new trial set (both from the same or a new optimization study) 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Ready for providing a new trial set (score-driven suggestions enabled by default) 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Framework-agnostic optimization campaign Computing instance Hopaas server + Authorized users only!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 200 HTTP response with trial parameters Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Workflow of an optimization study with a client-server approach based on RestAPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' future extensions to additional frameworks are planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The access to the Uvicorn instances from the Internet is mediated by an NGINX reverse proxy accessed via the encrypted HTTPS protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A PostgreSQL instance is part of the docker-compose configuration to provide shared persistency to the multiple instances of the web application back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The workflow of the interaction between the Hopaas server and computing nodes is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The same Hopaas server is designed to serve web-based user access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A web application, developed in HTML, CSS and JavaScript, is shipped to the client browser as defined by a set of web-specific APIs in Uvicorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The web pages of the front-end provide dynamic visualizations by fetching data from specialized APIs at regular intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Plots showing the evolution of the loss reported by different studies and trials are obtained with the Chartist library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The user authentication and authorization procedure of the web application is managed relying on access tokens as defined by the OAuth2 standard, using the INFN GitLab instance as identity provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Support for INDIGO IAM is also planned for the future [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Once authenticated, users can generate multiple API tokens through the web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Each API token has a validity period defined at generation and can be revoked at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Tokens with shorter validity are more appropriate for usage in public or untrusted contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Tuning the LHCb ultra-fast simulation models with Hopaas and Marconi 100 Machine Learning is an important research area in High Energy Physics (HEP), with first applications dating back to the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Recent years have witnessed an explosion in the use of ML techniques in HEP to face the computational challenges raised by the upcoming and future runs of the Large Hadron Collider (LHC) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' With an increasing number, complexity and range of applications of the ML models, HPO is becoming popular in HEP [14], and specialized frameworks targeting distributed computing are being developed [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' The reference implementation of the Hopaas service presented here has been successfully used for HEP applications and, in particular, to optimize the parameterizations for Lamarr [15], a novel LHCb ultra-fast simulation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Most of the parameterizations of Lamarr rely scikitdmlc XGBoostA wwWINPN CLOUDuvicornFastAPion Generative Adversarial Networks (GANs) [16], advanced algorithms taken from Computer Vision that were demonstrated to be able to well reproduce the distributions obtained from standard simulation techniques [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Adversarial models are particularly sensitive to the choice of the hyperparameter configuration and require intensive optimization campaigns to model accurately the target distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Several optimization studies have been orchestrated by the Hopaas service using diverse computing instances, from scientific providers (like INFN, CERN and CINECA) and from commercial cloud provider (like GCP or AWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Most of the resources have been provided by the CINECA supercomputer Marconi 100, with a custom network configuration to enable the communication with the Hopaas server [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Hopaas was able to coordinate dozens of optimization studies with hundreds of trials on each study from more than twenty concurrent and diverse computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Conclusion and future work Hyperparameter tuning and Bayesian methods for gradient-less optimization provide an effective and simple mean of exploiting opportunistic compute resources to improve ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Unfortunately, environment variability and constraints set by different resource providers make the application of existing HPO services challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' With Hopaas, we propose a solution designed to require the addition of the thinnest possible layer in the model training application, querying a central service via HTTPS and minimal RestAPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' A reference implementation with a server instance running on INFN Cloud and a Python client was presented and tested in a real- world application to coordinate hyperparameter optimization campaigns on multiple resource providers including INFN, CERN, and CINECA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' In the future we will improve the quality of the Web User Interface, for example enabling custom model documentation and sharing among multiple users, and introduce support to multi-objective optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' Acknowledgments We would like to thank Doina Cristina Duma and the rest of the INFN Cloud group for the technical support in the deployment and test of Hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' We acknowledge enlightening and motivating discussions with Diego Ciangottini, Stefano Dal Pra, Piergiulio Lenzi and Daniele Spiga, especially on future applications and developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' References [1] Ramesh A et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 2021 PMLR’21 pp 8821–8831 [2] Brown T et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'} +page_content=' 2020 NeurIPS’20 pp 1877–1901 [3] Richens J G, Lee C M and Johri S 2020 Nat.' metadata={'source': 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b/4dE2T4oBgHgl3EQfOQZ5/content/tmp_files/2301.03746v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1e1fec756360e6a109f0ddb765e7ff526bb40d5 --- /dev/null +++ b/4dE2T4oBgHgl3EQfOQZ5/content/tmp_files/2301.03746v1.pdf.txt @@ -0,0 +1,2602 @@ +Total energy-shaping control for mechanical systems via +Control-by-Interconnection +Joel Ferguson1 +Abstract— Application of IDA-PBC to mechanical systems +has received much attention in recent decades, but its ap- +plication is still limited by the solvability of the so-called +matching conditions. In this work, it is shown that total energy- +shaping control of under-actuated mechanical systems has a +control-by-interconnection interpretation. Using this interpreta- +tion, alternate matching conditions are formulated that defines +constraints on the added energy, rather then the total closed- +loop energy. It is additionally shown that, for systems that are +under-actuated degree one with the mass matrix depending +on a single coordinate, the kinetic energy matching conditions +resolve to ODEs which can be evaluated numerically. Using this +approach controllers are proposed for the benchmark cart-pole +and acrobot systems. +I. INTRODUCTION +Energy-based methods for controlling nonlinear physical +systems have been shown to be effective in a variety of +physical domains [1]. Such methods consider the energy +and structure of the system to be controlled to derive +control strategies that exploit the natural system behaviours. +Interconnection and Damping Assignment, Passivity-Based +Control (IDA-PBC) is one such control methodology where +the control input is designed such that the closed-loop can be +interpreted as an alternate physical system with a different +energy, interconnection and damping structure [2]. +While IDA-PBC has been applied to a broad range of +systems, particular attention has been given to mechanical +systems which exhibit a rich canonical structure [3], [4], [5]. +In the case of fully-actuated systems, IDA-PBC allows a +user to arbitrarily modify the potential and kinetic energy +of the closed-loop system [6], a process known as total +energy shaping [3]. For under-actuated systems, however, +application of IDA-PBC is limited by solutions that satisfy +a set of PDEs, the so-called matching conditions. Much re- +search effort has been committed to solving these equations, +with solutions posed in several special cases [4], [5], [6]. +This design methodology has been applied to a number of +benchmark examples such as the cart-pole, acrobot, spider +crane, amongst others. +Control-by-Interconnection (CbI) describes a sub-class of +energy-based control methods that falls under the umbrella +of IDA-PBC [7], [8]. Under this scheme, the controller is +assumed to be a passive system that is interconnected with +a passive plant to be controlled via the passive input-output +pair. Casimirs, conserved quantities between the control sub- +system and the plant, can be constructed to help shape the +1Joel +Ferguson +is +with +the +School +of +Engineer- +ing, +The +University +of +Newcastle, +Australia +Email: +joel.ferguson@newcastle.edu.au +energy of the closed-loop system. It is known that potential +energy shaping of fully-actuated mechanical systems falls +into the class of CbI [7], [9]. Control of underactuated +mechanical systems has been explored in the context of CbI +by applying nonlinear PID controllers to both the standard +and alternate passive outputs [10], [11], [12]. The idea of +using PID for stabilisation of passive systems was formalised +in [13] and a general characterisation of all passive outputs +from a given system was characterised. +In this work, the connection between IDA-PBC and CbI +for under-actuated mechanical systems is explored. Using +the bond graph formalism (see [14] for introduction), a +control sub-system is proposed that allows for shaping of +the kinetic and potential energies of the closed-loop system. +By representing the controller as a passive interconnection, +the requisite matching conditions are reformulated in terms +of the added mass and added potential energy. Equivalence +between the CbI and IDA-PBC is then established in the +case of mechanical systems by identifying Casimirs relating +the controller states to those of the plant. Finally, using the +reformulated matching conditions, it is shown that in the case +that the mass matrix depends on only one coordinate that the +kinetic energy matching conditions can be formulated as an +ODE that can be evaluated numerically for implementation. +Notation. Function arguments are declared upon definition +and are omitted for subsequent use. 0n×m denotes a n × m +zeros matrix whereas In denotes a n×n identity matrix. For +mappings H : Rn → R, we denote the transposed gradient as +∇H := +� ∂H +∂x +�⊤. For P = P ⊤ ∈ Rn×n, λmin [P] , λmax [P] +denotes the minimum and maximum (real) eigenvalues of P, +respectively. +II. BACKGROUND AND PROBLEM +FORMULATION +In this section a number of key concepts necessary for the +subsequent developments are briefly revised. +A. Control-by-interconnection +In +this +work +we +consider +input-state-output +port- +Hamiltonian systems (ISO-PHS) of the form +˙xp = Fp(xp)∇xpHp(xp) + Gp(xp)up +yp = G⊤ +p (xp)∇xpHp(xp) +(1) +where xp ∈ Rp is the state of the plant, Fp(xp) ∈ Rp×p is +the combined interconnection and damping matrix satisfying +Fp(xp) + F ⊤ +p (xp) ≤ 0, Hp(xp) ∈ R is the Hamiltonian, +up ∈ Rm is the input, Gp(xp) ∈ Rp×m is the input +arXiv:2301.03746v1 [eess.SY] 10 Jan 2023 + +mapping matrix and yp ∈ Rm is the natural passive output +corresponding to the input up. +CbI assumes that the controller is a passive system that is +interconnected with the plant (1) via a passive interconnec- +tion. In this work, we consider a controller subsystem with +two input-output ports, described by the ISO-PHS +� +� +˙xc +−yc1 +−yc2 +� +� = +� +� +K11(xc) +K12(xc) +K13(xc) +K21(xc) +K22(xc) +K23(xc) +K31(xc) +K32(xc) +K33(xc) +� +� +� +�� +� +:=K(xc) +� +� +∇xcHc +uc1 +uc2 +� +� +(2) +where xc ∈ Rc is the state of the controller, Hc(xc) ∈ +R is the controller Hamiltonian, uc1, yc1 +∈ +Rm and +uc2, yc2 ∈ Rr are passive input-output pairs and K(xc) ∈ +R(p+m+r)×(p+m+r) satisfies K(xc) + K⊤(xc) ≤ 0 [15]. +The controller system (2) can be interconnected with the +plant (1) via the passive interconnection +up = −yc1 +uc1 = yp, +(3) +resulting in the closed-loop dynamics +� +� +˙xp +˙xc +−yc2 +� +� = +� +� +Fp + GpK22G⊤ +p +GpK21 +GpK23 +K12G⊤ +p +K11 +K13 +K32G⊤ +p +K31 +K33 +� +� +� +�� +� +Fcl +� +� +∇xpHp +∇xcHc +uc2 +� +� +(4) +where uc2, yc2 is a passive input-output pair to the inter- +connected system. Noting that K + K⊤ ≤ 0, the closed- +loop interconnection and damping structure Fcl satisfies +Fcl + F ⊤ +cl ≤ 0 also. +In the case of stabilisation, the objective is to construct +the plant functions Hc(xc), K(xc) to ensure the existence +of Casimirs which statically relate the controller states to +functions of the plant states +xc = fc(xp), +(5) +for fc(x) ∈ Rc. The Casimir functions and controller initial +conditions are then designed to assign a desirable minimum +to the total energy function +W(xp) = H(xp) + Hc(xc)|xc=fc(xp). +(6) +It is noted that the Lyapunov candidate W(xp) can be gen- +eralised to a function of H, Hc and the Casimirs xc −fc(xp) +[15]. Methods to ensure the existence of and constructing +Casimirs have been reported in [7] and the references therein. +B. Underactuated mechanical systems +The primary objective of this work is to apply CbI to the +class of underactuated mechanical systems, described by the +dynamics � ˙q +˙p +� += +�0n×n +In +−In +0n×n +� �∇qH +∇pH +� ++ +�0n×m +G +� +u +H(q, p) = 1 +2p⊤M −1(q)p +� +�� +� +:=T (q,p) ++V (q) +y = G⊤∇pH, +(7) +where q ∈ Rn, p ∈ Rn are the configuration and momen- +tum vectors, respectively, u ∈ Rm in the input, M(q) = +M ⊤(q) > 0 is the inertia matrix and y is the natural passive +output corresponding the the input u. The input mapping +matrix G is assumed to be constant and have the structure +G = +� +Im +0(n−m)×m +� +, +(8) +where n − m < n is the degree of underactuation of the +system1. The Hamiltonian H(q, p) is the sum of the kinetic +energy T(q, p) and the potential energy V (q), which allows +the gradient of H with respect to q to be written as +∇qH(q, p) = ∇qT(q, p) + ∇qV (q). +(9) +A full-rank left-annihilator for the input mapping matrix (8) +is defined as +G⊥ = +�0(n−m)×m +I(n−m) +� +(10) +which satisfies G⊥G = 0(n−m)×m. +In the subsequent development, we will require an alter- +nate representation of the gradient of the kinetic energy with +respect to configuration ∇qT(q, p). Noting that the kinetic +energy is quadratic in p, the gradient ∇qT(q, p) can always +be factored into the form +∇qT(q, p) = E(q, p)M −1(q)p, +(11) +for some matrix E(q, p) ∈ Rn×n. This has been previously +noted in [16] using the Christoffel symbols. Note, however, +that the matrix E(q, p) is non-unique and in this work we +will use the representation given by +∇qT(q, p) = 1 +2 +∂⊤ +∂q +� +M −1(q)p +� +p += 1 +2 +∂⊤ +∂q +� +M −1(q)p +� +M(q) +� +�� +� +:=E(q,p) +M −1(q)p. +(12) +In constructing a CbI interpretation to total energy shaping +it is useful to define a virtual input-output pair for the system +(7) by defining the input +uv = Gu, +(13) +which allows the system to written similarly to a fully- +actuated system as +� ˙q +˙p +� += +�0n×n +In +−In +0n×n +� �∇qH +∇pH +� ++ +�0n×n +In +� +uv +yv = ∇pH, +(14) +where uv, yv ∈ Rn. From the definition (13), it is clear that +any input uv must satisfy +G⊥uv = 0m×1, +(15) +1The assumed structure of G requires that the first m configuration +coordinates are chosen to be collocated with the actuators. This class of +dynamics falls into the broader class of ISO-PHS (1). For a more general +input mapping matrix ¯G(q) ∈ Rn×m, there exists a change of coordinates +recovering the structure (8) if the columns of ¯G(q) are involute. + +which will be ensured in subsequent control design. Assum- +ing that (15) holds, the input u can be described as a function +of uv by +u = G⊤uv. +(16) +The advantage of constructing the virtual input-output pair is +that the virtual output now describes the full velocity vector +yv = M −1(q)p = ˙q, +(17) +a property that will be exploited when shaping the potential +energy. +C. IDA-PBC for underactuated mechanical systems +IDA-PBC is a control design methodology whereby the +control signal is designed such that the closed-loop dynamics +have a port-Hamiltonian (pH) structure. When applied to +underactuated mechanical systems, the target closed-loop +dynamics have the structure +� ˙q +˙p +� += +� +0n×n +M −1(q)Md(q) +−Md(q)M −1(q) +J2(q, p) − GKdG⊤ +� �∇qHd +∇pHd +� +Hd(q, p) = 1 +2p⊤M −1 +d (q)p + Vd(q), +(18) +where Md(q) = M ⊤ +d (q) > 0, Vd(q) are the desired closed- +loop inertia matrix and potential energies, respectively, +J2(q, p) = −J⊤ +2 (q, p) is skew-symmetric and Kd = K⊤ +d ≥ 0 +is a tuning parameter used for damping injection. If Vd +is minimised at the target configuration, Hd qualifies as a +Lyapunov function for the closed-loop system [4]. +The complexity of applying IDA-PBC to underactuated +system is satisfying the so-called matching conditions. This +conditions requires that the dynamics of the open-loop and +closed-loop systems must agree on the spaces perpendicular +to the control signal. The structure chosen fo the closed-loop +system in (18) ensures that the dynamics of q agree with (7). +Comparing the dynamics of p results in the condition +G⊥ � +∇qH − Md(q)M −1(q)∇qHd − J2(q, p)∇pHd +� += 0(n−m)×1, +(19) +which +defines +a +PDE +that +should +be +solved +for +Md(q), Vd(q), J2(q, p). +Noting +the +structure +of +the +Hamiltonians, this PDE can be separated into the components +involving p, and those that do not by +G⊥ � +∇qT − Md(q)M −1(q)∇qTd − J2(q, p)M −1 +d (q)p +� += 0(n−m)×1 +G⊥ � +∇qV − Md(q)M −1(q)∇qVd +� += 0(n−m)×1. +(20) +These expressions are known as the kinetic energy and +potential energy matching equations, respectively. +D. Contributions +The objective of this work is to construct a CbI interpreta- +tion of IDA-PBC when applied to underactuated mechanical +systems. The contributions of this work are threefold: +C.1 ISO-PHS with Casimirs that statically relate states are +considered and a closed-form solution to remove the +Casimirs by reducing the dimension of the state vector +is proposed. This solution can be applied to the closed- +loop dynamics of CbI implementations of the form (4) +to describe the resulting dynamics as a function of xp +only. +C.2 A CbI controller for underactuated mechanical system +of the form (2) is proposed and the resulting closed- +loop is shown to be equivalent to the well-known dy- +namics (18). The CbI interpretation generates alternate +matching conditions to the expressions (20), describing +constraints on the added mass and added potential +energy. +C.3 Using the alternate matching conditions, it is shown that +the kinetic energy matching equations reduce to ODEs +in the special case of underactuation degree one where +the mass matrix is a function of only one configuration +coordinate. It is demonstrated that numerical methods +can be utilised in such cases to avoid solving these +expressions analytically. +E. Related works +Significant attention has been given to solving the match- +ing equations (20) in recent decades. In [3], [17] it was +shown that is the system is under-actuated degree one and +the mass matrix depends on a single un-actuated coordinate, +the kinetic energy matching condition can be simplified to an +ODE. Using a novel parametrisation of J2(q, p), a general so- +lution for under-actuated degree one system was proposed in +[4] under the assumption that the mass matrix depends only +on the actuated coordinates. This approach was extended +in [18] using a momentum transformation to simplify the +matching equations. More recently, solutions to the potential +energy matching equations were considered in [19] under +the assumption that the mass matrix and potential energy +functions were dependent on only one variable. Finally, a +general solution to the special case of 2 degree-of-freedom +system was proposed in [6]. The studies [20], [21] considered +the effects of friction on the closed-loop stability using IDA- +PBC. +In recent works, alternate approaches to constructing so- +lutions to the matching equations have been explored. The +existence of conservative forces that cannot be factorised +into a skew-symmetric matrix J2(q, p) was investigated in +[22], which resulted in alternate matching equations. Im- +plicit system representations were used in [23] to construct +solutions in an over-parameterised space where the closed- +loop dynamics were subject to constraints. By working in +the larger dimension, a solution to a under-actuated degree +2 crane system was proposed. Pfaffian differential equations +were utilised in [24] which resulted in the kinetic energy + +PDEs being converted to an alternate form which admits +simpler solutions. +Some authors have investigated the possibility of avoiding +the matching equations altogether by considering the control +signal to be a CbI. The work [10] relied on a Lagrangian +structure and several technical assumptions to verify the +existence of a second passive output corresponding to the +input u. Using this second output, a stabilising control was +designed that ensured stability without requiring a solution +to the matching PDEs. A similar approach was proposed in +[12], [11] where a second passive output was utilised and +the control assumed to have a PID structure. +III. CASIMIR REDUCTION +In this section, a method for reducing the state dimension +of ISO-PHS with Casimirs is derived. The reduction method +applies to general ISO-PHS with Casimirs and can be directly +applied to the resulting closed-loop dynamics of CbI schemes +of the form (4) to describe the system as a function of xp +only. In the sequel, the reduction method will be used to show +equivalence between the CbI controller for underactuated +mechanical systems and the IDA-PBC dynamics (18). Before +introducing the state reduction solution, a useful lemma is +required. +Lemma 1: Consider a square block matrix of arbitrary +dimension +A = +�A11 +A12 +A21 +A22 +� +(21) +and assume that A22 is invertible. If the symmetric com- +ponent of A is negative semi-definite, A + A⊤ ≤ 0, the +symmetric component of the Schur complement +A11 − A12A−1 +22 A21 +(22) +is negative semi-definite also. +Proof: First note that the Schur complement of X can +be computed by +� +I +−A⊤ +21A−⊤ +22 +� +A +� +I +−A−1 +22 A21 +� += A11 − A12A−1 +22 A21. +(23) +The symmetric component of this expression is negative +semi-definite as A + A⊤ ≤ 0. +The solution for reducing the dimension of ISO-PHS +which exhibit Casimirs is now introduced. This development +applies to systems of the form +� +� +˙x1 +˙x2 +−y +� +� = +� +� +F11(x) +F12(x) +F13(x) +F21(x) +F22(x) +F23(x) +F31(x) +F32(x) +F33(x) +� +� +� +�� +� +F (x) +� +� +∇x1H +∇x2H +u +� +� +(24) +where x ∈ Rp+c is the state of the system which has been +partitioned into x1 ∈ Rp, x2 ∈ Rc, H(x1, x2) ∈ R is the +Hamiltonian, F(x) ∈ R(p+c)×(p+c) is the full-rank intercon- +nection and damping matrix satisfying F(x) + F ⊤(x) ≤ 0, +u ∈ Rm is the input and y ∈ Rm is the corresponding passive +output. It is assumed that the system contains a Casimir and +the states have been partitioned such that the Casimir can be +written as +x2 = fc(x1), +(25) +where fc(x1) ∈ Rc is differentiable. +The first step in constructing a minimal system represen- +tation is defining a new set of coordinates given by +w = x2 − fc(x1) = 0c×1, +(26) +which is identically equal to zero by construction. The +system (24) can be described in the coordinates (x1, w) by +� +� +˙x1 +−y +˙w +� +� = +� +� +Ip +0p×c +0p×m +0m×p +0m×c +Im +− ∂fc +∂x1 +Ic +0c×m +� +� +� +� +˙x1 +˙x2 +−y +� +� += +� +� +Ip +0p×c +0p×m +0m×p +0m×c +Im +− ∂fc +∂x1 +Ic +0c×m +� +� F +× +� +� +Ip +0p×m +− ∂⊤fc +∂x1 +0c×p +0c×m +Ic +0m×p +Im +0m×c +� +� +� +� +∇x1Hr +u +∇wHr +� +� += +� +� +¯F11 +¯F12 +¯F13 +¯F21 +¯F22 +¯F23 +¯F31 +¯F32 +¯F33 +� +� +� +�� +� +¯ +F +� +� +∇x1Hr +u +∇wHr +� +� , +(27) +where +Hr(x1, w) :=H(x1, w + fc(x1)) +¯F11(x1) =F11(x)|x2=fc(x1) +¯F12(x1) =F13(x)|x2=fc(x1) +¯F13(x1) =F12(x) − F11(x)∂⊤fc +∂x1 +���� +x2=fc(x1) +¯F21(x1) =F31(x)|x2=fc(x1) +¯F22(x1) =F33(x)|x2=fc(x1) +¯F23(x1) =F32(x) − F31(x)∂⊤fc +∂x1 +���� +x2=fc(x1) +¯F31(x1) =F21(x) − ∂fc +∂x1 +F11(x) +���� +x2=fc(x1) +¯F32(x1) =F23(x) − ∂fc +∂x1 +F13(x) +���� +x2=fc(x1) +¯F33(x1) =F22(x) − F21(x)∂⊤fc +∂x1 +− ∂fc +∂x1 +F12(x) ++ ∂fc +∂x1 +F11(x)∂⊤fc +∂x1 +���� +x2=fc(x1) +. +(28) +Recalling that w is identically equal to zero, ˙w is also +equal to zero. Consequently, the final row of (27) is a +constraint that needs to be resolved to construct a minimal +system representation. There are two methods of reduction +that will be considered. Firstly, in the transformed coordi- +nates (27) it can occur that one or more columns of ¯F⋆3 is +identically zero. Without loss of generality, it is assumed that + +the first d columns of ¯F⋆3 are equal to zero. To remove the +zero rows, the full-rank matrix B is defined as +B = +�0d×(c−d) +I(c−d) +� +, +(29) +which acts to select the non-zero columns of ¯F⋆3. The zero +columns and corresponding rows are removed from (27) by +� +� +˙x1 +−y +B⊤ ˙w +� +� = +� +� +¯F11 +¯F12 +¯F13B +¯F21 +¯F22 +¯F23B +B⊤ ¯F31 +B⊤ ¯F32 +B⊤ ¯F33B +� +� +� +�� +� +:= ¯ +FB +� +� +∇x1Hr +u +B⊤∇wHr +� +� , +(30) +which does not modify the system dynamics. Note that as +¯F + ¯F ⊤ ≤ 0, ¯FB + ¯F ⊤ +B ≤ 0 also. Using this representation, +a method for resolving the remaining constraint equations is +presented under the assumption that B⊤ ¯F33B is full rank. +Proposition 1: Consider the pH system (24) with Casimir +(25). If the matrix B⊤ ¯F33B is full-rank for all x1 the system +can be described by a reduced-order model +� ˙x1 +−y +� += Fr(x1) +�∇x1Hr +u +� +, +(31) +where +Hr(x1) =H (x1, fc(x1)) +Fr(x1) = +� ¯F11 +¯F12 +¯F21 +¯F22 +� +− +� ¯F13B +¯F23B +� +(B⊤ ¯F33B)−1 � +B⊤ ¯F31 +B⊤ ¯F32 +� +(32) +and ¯F(x1) satisfies Fr(x1) + F ⊤ +r (x1) ≤ 0. +Proof: The expression B⊤ ˙w, defined in (30), is iden- +tically equal to 0(c−d)×1 by construction. Note, however, +that the gradient B⊤∇wHr is not necessarily equal to +zero. Assuming that B⊤ ¯F33B is full rank, the expression +B⊤∇wHr can be described as +B⊤∇wHr = − (B⊤ ¯F33B)−1 � +B⊤ ¯F31 +B⊤ ¯F32 +� � +∇x1Hr +u +� +(33) +Substituting this expression into the dynamics (30) resolves +to the reduced dynamics (31). +To verify that Fr + F ⊤ +r +≤ 0, note that Fr is the Schur +complement of ¯FB which satisfies ¯FB + ¯F ⊤ +B ≤ 0. It follows +that Fr + F ⊤ +r ≤ 0 by application of Lemma 1. +Proposition 1 showed that an ISO-PHS that exhibits a +Casimir function can be described in a reduced state-space. +The class of dynamics that are derived from application of +CbI (4) that result in a Casimir of the form (5) falls into the +class of systems (24). The following Corollary tailors the +Casimir reduction for this important sub-class of dynamics. +Corollary 1: If the closed-loop dynamics of a CbI scheme +(4) exhibit a Casimir of the form (5), the system can be +equivalently expressed in the form (31) where +x1 =xp +Hr(xp) =Hp(xp) + Hc (fc(xp)) +¯F11 =Fp + GpK22G⊤ +p +¯F12 =GpK23 +¯F13 =GpK21 − +� +Fp + GpK22G⊤ +p +� ∂⊤fc +∂xp +¯F21 =K32G⊤ +p +¯F22 =K33 +¯F23 =K31 − K32G⊤ +p +∂⊤fc +∂xp +¯F31 =K12G⊤ +p − ∂fc +∂xp +� +Fp + GpK22G⊤ +p +� +¯F32 =K13 − ∂fc +∂xp +GpK23 +¯F33 =K11 − K12G⊤ +p +∂⊤fc +∂xp +− ∂fc +∂xp +GpK21 ++ ∂fc +∂xp +� +Fp + GpK22G⊤ +p +� ∂⊤fc +∂xp +(34) +and B is suitably chosen as per (29) using the expressions +¯F⋆3. The arguments have been dropped from the definitions +of ¯F⋆⋆(xp) for the sake of readability. +Proof: +The result follows from direct application of +Proposition 1 to the dynamics (4). +IV. CONTROL-BY-INTERCONNECTION FOR +MECHANICAL SYSTEMS +In this section, a control-by-interconnection scheme for +under-actuated mechanical systems is presented. A dynamic +2-port control system is introduced with the intention that it +will be interconnected to the plant (14) via one of the ports. +The controller states are constructed to be statically related to +the plant states after interconnection, resulting in Casimirs. +By applying Proposition 1, the closed-loop dynamic are +defined in a reduced space in which the dynamics coincide +with standard total energy-shaping control (18). +The proposed CbI scheme is shown in Figure 1. The +intention of this control subsystem is to interconnect with the +plant (14) via the uc1, yc1 power port. The second input uc2 +is available for subsequent control design, such as damping +injection. The terms M(qa2), E(qa2, pa) are the plant mass +matrix (7) and factorisation of the kinetic energy gradient +(12) evaluated at the controller states whereas J(qa2, pa) = +−J(qa2, pa)⊤ ∈ Rn×n is a skew-symmetric matrix to be +chosen. The three-port storage element Ha(qa1, qa2, pa) has +states qa1, qa2, pa ∈ Rn and energy function similar to +mechanical systems, +Ha(qa1, qa2, pa) = 1 +2p⊤ +a M −1 +a (qa2)pa +� +�� +� +:=Ta(qa2,pa) ++ Vd(qa2) − V (qa1) +� +�� +� +:=Va(qa1,qa2) +, +(35) + +Fig. 1: Mass shaping as CbI for under-actuated mechanical +systems. +where M −1 +a (qa2) is the inverse added mass, Ta(qa2, pa) is +the added kinetic energy, Vd(qa2) is the desired closed-loop +potential energy, V (qa1) is the plant potential energy function +(7) evaluated at the plant state qa1 and Va(qa1, qa2) is the +total added potential energy. It is important to note that, +although M −1 +a (qa2) is represented as a matrix inverse, it +need not be invertible nor positive. Indeed, it will be shown +in subsequent developments that the key requirement is that +M −1 +d (q) := M −1(q) + M −1 +a (q) +(36) +should be positive definite. +In subsequent analysis it will be shown that the intercon- +nection of the control system with the plant (14) via the +interconnection +uv = −yc1 +uc1 = yv, +(37) +yields Casimirs +� +� +qa1 +qa2 +pa +� +� = +� +� +In +0n×n +In +0n×n +0n×n +In +� +� +� +q +p +� +� +�� +� +fc(xp) +. +(38) +Assuming the Casimirs exist, some intuition regarding the +construction of the control system in Figure 1 can be +provided. Both qa1, qa2 were constructed to be equal to q. +Firstly ˙qa1 is equal to ˙q by interconnection with the plant +virtual output yv via a 1-junction. To verify a similar relation +for qa2, assume that qa2 = q, pa = p holds which results +in ∇paHa = M −1 +a (q)p and uc1 + ∇paHa = M −1 +d p. With +this in mind, the transformer can be seen to reconstruct the +velocity ˙q = M −1(q)p for the bottom 1-junction, resulting +in ˙qa1 = ˙q. +To construct a Casimir pa = p, first note from (7) and +(12) that the plant momentum dynamics can be expressed as +˙p = −∇qV − E(q, p)M −1(q)p + uv. +(39) +The control structure acts to remove these forces from the +plant via the right side of the control structure and re- +introduce them via the top 0-junction where they are shared +with the dynamics of pa. The ˙qa1 bond acts to cancel the +gravity term from the plant −∇qV . Recalling that the bottom +1-junction has flow equal to M −1(q)p, the right-side gyrator +cancels the term −E(q, p)M −1(q)p from the plant. The left- +side gyrator then re-introduces the force −E(q, p)M −1(q)p +via the top 0-junction where it is shared between ˙p and ˙pa, +establishing the desired Casimir. +The claimed Casimir (38) is now formalised in the follow- +ing Proposition. For this development, note that the gradients +of the added energy Ha(·) satisfy +∇qa1Ha = −∇qa1V +∇qa2Ha = ∇qa2Ta + ∇qa2Vd +∇qa2Ta = 1 +2 +∂⊤ +∂qa2 +� +M −1 +a (qa2)pa +� +pa +∇paHa = M −1 +a (qa2)pa +(40) +and the expressions A(·), B(·), C(·) in Figure 1 can be +evaluated as +A(qa2, pa, uc1) =M −1(qa2)Md(qa2) [uc1 + ∇paHa] +B(qa2, pa, uc1) =∇qa2Ha − E⊤(qa2, pa)∇paHa +− M(qa2)J(qa2, pa)Md(qa2) +× [uc1 + ∇paHa] +C(qa2, pa, uc1) =Md(qa2)M −1(qa2)∇qa2Ha +− Md(qa2)M −1(qa2)E⊤(qa2, pa)∇paHa +− Md(qa2)J(qa2, pa)Md(qa2) +× [uc1 + ∇paHa] , +(41) +with Md(·) defined in (36). To ensure that the Casimir exists, +a number of requirements are imposed on the selection of +the added inverse mass M −1 +a (q) and closed-loop potential +energy Vd(q) which are equivalent of the standard matching +conditions used in IDA-PBC (20). +Proposition 2: Consider the control system in Figure 1 +and assume that it is interconnected to the plant (14) via the +interconnection (37). If M −1 +a (qa), Vd(qa) are chosen such +that +G⊥C(qa2, pa, uc1)|qa2=q,pa=p = G⊥∇qV +(42) +and the controller states are initialised as qa1(0) = qa2(0) = +q(0), pa(0) = p(0), the Casimir (38) holds for all time. +Proof: Consider that at some time instant T (38) holds, +implying that +qa1(T) = qa2(T) = q(T), pa(T) = p(T). +(43) +It is shown that if (42) is satisfied, then the derivatives of +the states also agree +˙qa1(T) = ˙qa2(T) = ˙q(T), ˙pa(T) = ˙p(T), +(44) + +establishing the existence of a Casimir for all future time. +We proceed by first establishing the relationship for the +configuration vector. From (37) and (14) uc = M −1(q)p +which establishes ˙qa1(T) = ˙q(T). The input uc = M −1(q)p +is substituted into A(·) (41) to find +A|t=T = M −1(qa2)Md(qa2) +� +M −1(q)p + M −1 +a (qa2)pa +� +|t=T += M −1(q)p|t=T += ˙q|t=T , +(45) +confirming that ˙qa2(T) = ˙q(T). +Next we consider the behaviour of the momentum states. +First note that, from the bond graph in Figure 1 and the +definition (16), the plant input uv is given by +u = G⊤uv += G⊤ [Guc2 − C(qa2, pa, uc1) + ∇qa1V (qa1)] += uc2 − G⊤ [C(qa2, pa, uc1) − ∇qa1V (qa1)] . +(46) +Using the control definition (46) and the condition (42), the +plant dynamics (7) can be expanded as +˙p = − ∇qT(q, p) − +�G⊤∇qV (q) +G⊥∇qV (q) +� ++ Gu += − ∇qT(q, p) − +� +G⊤∇qV (q) +G⊥∇qV (q) +� ++ G +� +uc2 − G⊤ [C(qa2, pa, uc1) − ∇qa1V (qa1)] +� += − ∇qT(q, p) + Guc2 +− +�G⊤ {C(qa2, pa, uc1) + ∇qV (q) − ∇qa1V (qa1)} +G⊥∇qV (q) +� +(47) +Note that at time T, ∇qV (q)|t=T += ∇qa1V (qa1)|t=T . +Additionally recall the assumption (42) which allows the +simplification +˙p|t=T = − ∇qT(q, p) − C(qa2, pa, uc1) + Guc2. +(48) +Recalling the identity (45), the dynamics of pa at time T can +be expanded to +˙pa = Gv − E(qa2, pa)A(·)|t=T − C(qa2, pa, uc1)|t=T += Gv − ∇qT(q, p)|t=T − C(qa2, pa, uc1)|t=T , +(49) +which agrees with (48). As (48) and (49) agree at time T, +(44) is verified for the momentum states. If at the initial time +t = 0 we have qa(0) = q(0), pa(0) = p(0), it follows that +qa(t) = q(t) and pa(t) = p(t) for all time via integration, +completing the proof. +Proposition 2 has established that the Casimir (38) holds +under some technical assumptions that will be verified in +subsequent design. Before proceeding, we note that the +control subsystem in Figure 1 can be written in the form +(2) with +xc = +� +q⊤ +a1 +q⊤ +a2 +p⊤ +a +�⊤ +Hc(qa1, qa2, pa) = Ha(qa1, qa2, pa) +K11(qa2, pa) = +� +� +0n×n +0n×n +0n×n +0n×n +0n×n +M −1Md +0n×n +−MdM −1 +D − D⊤ + MdJMd +� +� +K12(qa2, pa) = +� +� +In +M −1Md +MdJMd − D⊤ +� +� +K13 = +� +� +0n×m +0n×m +G +� +� +K21(qa2, pa) = +� +−In +−MdM −1 +D + MdJMd +� +K31 = +� +0m×n +0m×n +−G⊤� +K22(qa2, pa) = MdJMd +K23 = G +K32 = −G⊤ +K33 = 0m×m +D(qa2, pa) = MdM −1E⊤. +(50) +In the subsequent developments it is assumed that the +requisite (42) of Proposition 2 holds, implying qa1(t) = +qa2(t) = q(t), pa(t) = p(t). Condition (42) will be verified +by choice of M −1 +a +and Vd. Assuming the Casimir holds, the +expressions for A(·), B(·), C(·) in (41) can be simplified to +A(q, p) =M −1(q)p +B(q, p) =∇qTa(q, p) + ∇qVd(q, p) − E⊤(q, p)M −1 +a (q)p +− M(q)J(q, p)p +C(q, p) =Md(q) +� +M −1(q)∇qTa(q, p) + M −1(q)∇qVd(q, p) +−M −1(q)E⊤(q, p)M −1 +a (q)p − M −1(q)J(q, p) +� +(51) +Recalling the definition of ∇qaTa in (40), it is noted that +C(·) contains some terms which are quadratic in p and some +that are functions of q only. The function C(·) is divided into +C(q, p) =CKE(q, p) + CP E(q) +CKE(q, p) = +� +M −1 +a (q) + M −1(q) +�−1 +� +�� +� +Md(q) +{Y (q, p) − J(q, p)} p +CP E(q) = +� +M −1 +a (q) + M −1(q) +�−1 M −1(q)∇qVd, +(52) +where KE represents kinetic energy, PE represents poten- +tial energy and Y is defined as +Y (q, p) =1 +2M −1(q)∂⊤ +∂q +� +M −1 +a (q)p +� +− 1 +2 +∂ +∂q +� +M −1(q)p +� +M −1 +a (q). +(53) +As Y is linear in p it can be written as +Y (q, p) = +n +� +i=1 +piY i(q), +(54) + +where +Y i(q) =1 +2M −1(q)∂⊤ +∂q +� +M −1 +a (q)ei +� +− 1 +2 +∂ +∂q +� +M −1(q)ei +� +M −1 +a (q). +(55) +The +key +constraint +for +control +design +is +choosing +M −1 +a (q), Vd(q) satisfying the matching condition (42). From +the definition of C(·) in (51), the constraint equation is +a function of both M −1 +a (q) and +� +M −1 +a (q) + M −1(q) +�−1, +making direct design of this matrix difficult. To simplify +the design process, an alternate characterisation of (42) is +introduced. +In the following proposition, the inverse mass matrix, +inverse added mass matrix, interconnection matrix and Y (·) +are partitioned as +� +m11(q) +m⊤ +21(q) +m21(q) +m22(q) +� += +�G⊤ +G⊥ +� +M −1(q) +� +G +G⊥⊤� +� +ma11(q) +m⊤ +a21(q) +ma21(q) +ma22(q) +� += +�G⊤ +G⊥ +� +M −1 +a (q) +� +G +G⊥⊤� +� +J11(q, p) +−J⊤ +21(q, p) +J21(q, p) +J22(q, p) +� += +�G⊤ +G⊥ +� +J(q, p) +� +G +G⊥⊤� +�Y11(q, p) +Y12(q, p) +Y21(q, p) +Y22(q, p) +� += +�G⊤ +G⊥ +� +Y (q, p) +� +G +G⊥⊤� +. +(56) +Using the above definitions, an alternate characterisation of +(42) is presented. +Proposition 3: The matching condition (42) is satisfied if: +• The added mass matrix M −1 +a (q) is chosen such that +D(q) +� +Y i(q) + Y i⊤(q) +� +D⊤(q) = 0(n−m)×(n−m) +(57) +for all i ∈ {1, . . . , n} where +D(q) = +�(m21 + ma21)(m11 + ma11)−1 +−In−m +� +. +(58) +• The desired potential energy Vd(q) satisfies +s1(q)G⊥∇qV = −s2(q)G⊤∇qVd − s3(q)G⊥∇qVd += −D(q)M −1(q)∇qVd, +(59) +where +s1(q) = (m22 + ma22) +− (m21 + ma21)(m11 + ma11)−1(m⊤ +21 + m⊤ +a21) +(60) +is the Schur complement of M −1(q) + M −1 +a (q) and +s2(q) = (m21 + ma21)(m11 + ma11)−1m11 − m21 +s3(q) = (m21 + ma21)(m11 + ma11)−1m⊤ +21 − m22. +(61) +Proof: +From the graph in Figure 1 and the intercon- +nection (37), the virtual input uv is given by +uv = Gu =Gv − C + ∇qV. +(62) +Recalling the definition of G, (62) is equivalent to (42). +Collecting the terms u, v and left multiplying by M −1 +a +M −1 +results in +� +M −1 +a ++ M −1� +G(u − v) += +� +M −1 +a ++ M −1� +{−CKE − CP E + ∇qV } +(63) +Due to the structure of G in (8), (63) has the left annihilator +D(·), defined in (58). +Left multiplying (63) by D(·) and separating the compo- +nents into those relating to the kinetic and potential energies +result in +0(n−m)×1 = −D +� +M −1 +a ++ M −1� +CKE +(64) +0(n−m)×1 = −D +� +M −1 +a ++ M −1� +{CP E − ∇qV } . +(65) +Using (52), (65) is expanded to +0(n−m)×1 =D +� +M −1 +a ++ M −1� +∇qV +− DM −1∇qVa, +(66) +which can be seen to agree with (59) after expanding. +Now considering the constraint on the kinetic energy +expression (64), the definition (52) is substituted to find +0(n−m)×n =D {−Y + J} . +(67) +Using the relevant definitions, the first component of (67) +can be solved for J21(q, p) as +J21(q, p) =(m21 + ma21)(m11 + ma11)−1(−Y11 + J11) ++ Y21. +(68) +Substituting this expression back into the second component +of (67) reveals the constraint +0(n−m)×(n−m) +=(m21 + ma21)(m11 + ma11)−1(−Y12 − J⊤ +21) +− (−Y22 + J22) +=(m21 + ma21)(m11 + ma11)−1 � +−Y12 − Y ⊤ +21 +� +− (m21 + ma21)(m11 + ma11)−1(−Y11 + J11)⊤ +× (m11 + ma11)−1(m⊤ +21 + m⊤ +a21) − (−Y22 + J22) += − D +� +−Y11 + J⊤ +11 +−Y12 − Y ⊤ +21 +0(n−m)×m +−Y22 + J22 +� +D⊤. +(69) +The term J22 is taken as below to solve the skew-symmetric +part of this expression, +J22 += −1 +2D +� +−Y11 + Y ⊤ +11 + J⊤ +11 − J11 +−Y12 − Y ⊤ +21 +Y ⊤ +12 + Y21 +−Y22 + Y ⊤ +22 +� +D⊤, +(70) +where J11 ∈ Rm×m is a free skew-symmetric term. The +symmetric part of (69) must also be equal to zero, implying +that +D +� +Y + Y ⊤� +D⊤ = 0(n−m)×(n−m). +(71) +Finally, noting that this must be true for each pi, the +condition (57) follows. + +Remark 1: The expression (57) implicitly defines a set of +PDEs that must be satisfied by any choice of M −1 +a (q). From +the definition of Y i in (55), the first m equations are describe +partial differential equations involving the partial derivatives +of ma11, ma21. The remaining n − m equations describe +partial differential equations involving the partial derivatives +of ma21, ma22. This structure can be useful for resolving the +equations into a standard representation for solving. +Corollary 2: In the special case of under-actuation degree +1, if M −1, M −1 +a +is a function of only 1 configuration variable +qi, the kinetic energy matching equations (57) can be reduced +to a set of ODEs. +d +dqi +� +m⊤ +a21 +ma22 +� += g +� +ma11, d +dqi +ma11, M −1, d +dqi +M −1 +� +, (72) +where g(·) ∈ Rn is a function implicitly defined by the +matching conditions (57) and ma11(qi) can be chosen freely. +Proof: Assuming that the mass matrix M −1 is function +only of a single configuration variable qi, we will also impose +that the added mass M −1 +a +is a function only of the same +variable. As a consequence, the matching expression (57) is +now only a function in the single variable qi. Notably, all +partial derivatives of M −1 +a +with respect to qk, where k ̸= i, +are equal to zero. +Noting Remark 1, the first n − 1 expressions of (57) pro- +duce differential equations involving the partial derivatives +of ma11, ma21. The dimension of ma21 is 1 × (n − 1), so +the first n − 1 equations can be solved simultaneously to +find an expression for +d +dqi m⊤ +a21. The nth expressions of (57) +can then be resolved for an expression for +d +dqi ma22, which +has dimension 1. Combining these expressions, the matching +equations (57) can be resolved into an ODE of the form (72). +Remark 2: Corollary 2 describes situations in which the +kinetic energy matching equations can be reduced to an +ODE. The solution, however, will depend on the choice of +ma11(qi) and may not be globally defined. This poses the +question of how should the function ma11(qi) be chosen to +ensure an appropriate solution M −1 +a (qi)—a nonlinear control +problem! +The results of Corollary 1 describe the degrees of freedom +that exist when constructing a solution to the added inverse +mass matrix. Similar degrees of freedom exist in the defini- +tion of the closed-loop potential energy that can be exploited +to ensure positivity of the chosen function. The following +Corollary defines a free function that can be utilised to this +effect. +Corollary 3: Suppose that there exists a full rank matrix- +valued function K(q) ∈ Rm×m such that the integral +Γ(q) = +� +K(q)G⊤ � +M −1 +a (q) + M −1(q) +� +M(q) dq, +(73) +exists. The desired closed-loop potential energy can be +chosen as +Vd(q) = Vm(q) + Vf(Γ(q)), +(74) +where Vm(·) must be chosen to satisfy the potential energy +matching conditions (59) and Vf(·) is a free function that +does not impact the matching equations. Consequently, the +matching equation (59) can be equivalently written as +s1(q)G⊥∇qV = −s2(q)G⊤∇qVm − s3(q)G⊥∇qVm += −D(q)M −1(q)∇qVm. +(75) +Proof: Computing the gradient of Vd results in +∇qVd = ∇qVm + ∂⊤Γ +∂q ∇ΓVf += ∇qVm + M +� +M −1 +a ++ M −1� +GK⊤∇ΓVf. +(76) +From the definition of D(q) in (58), we have the identity +D(q)M −1M(q) +� +M −1 +a (q) + M −1(q) +� +G = +�Im +⋆� +G += 0(n−m)×m +(77) +Substituting the expression (76) into (59) and noting the +above expression results in the simplified matching equation +(75). +Remark 3: Vf(·) is a free function precisely because Γ is +an integral of the passive output yc2. The potential energy +Vf could be alternatively constructed as a capacitor element +added to the input uc2 in Figure 1. +Now we arrive at one of the key results of this work, the +equivalence of the proposed CbI scheme and total energy- +shaping control of underactuated mechanical systems. As- +suming that the CbI scheme has been constructed to satisfy +the required matching conditions to ensure the existence of a +Casimir of the form qa = q, pa = p, Proposition 1 is applied +to reconstruct the reduced closed-loop structure (18). +Proposition 4: Consider the underactuated mechanical +system with virtual input (14) and assume that Ma(q), Vd(q) +are chosen such that the conditions of Proposition 3 are sat- +isfied in some neighbourhood of a point (q, p) = (q⋆, 0n×1). +If the control signal is chosen as +u(q, p) =v − G⊤ � +Md(q)M −1(q) +� +−E⊤(q, p)M −1 +a (q)p ++∇qaHa(qa, pa) − M(q)J(q, p)p] − ∇qV } +(78) +where +Md(q) = +� +M −1 +a (q) + M −1(q) +�−1 , +(79) +the following hold: +• The closed-loop dynamics have the form +� +˙q +˙p +� += +� +0n×n +M −1(q)Md(q) +−Md(q)M −1(q) +J2(q, p) +� �∇qHd +∇pHd +� ++ +�0n×m +G +� +v +Hd(q, p) = 1 +2p⊤M −1 +d (q)p + Vd(q) +y = G⊤∇pHd, +(80) +where +J2(q, p) =Md(q) +� +J(q, p) + M −1(q) +� +E(q, p) − E⊤(q, p) +� +×M −1(q) +� +Md(q) + Md(q)M −1(q)E⊤(q, p) +− E(q, p)M −1(q)Md(q) +(81) + +• If Md(q), Vd(q) satisfy +Md(q) > 0, +Vd(q) > 0 +(82) +in some neighbourhood of (q, p) += +(q⋆, 0n×1), +(q⋆, 0n×1) is a stable equilibrium of the closed-loop +system for v = 0m×1. +• If the input signal v is used for damping injection +v = −KdG⊤y +(83) +for some positive Kd ∈ Rm×m and the equilibrium +(q, p) = (q⋆, 0n×1), (q⋆, 0n×1) is locally detectable +from the output y, the point (q⋆, 0n×1) is asymptotically +stable. +Proof: Interconnection of the mechanical system with +the control subsystem results in a closed-loop of the form +(4), where xc, Hc and K⋆⋆ are defined in (50) and +xp = +�q +p +� +Fp = +� +0n×n +In +−In +0n×n +� +Gp = +�0n×n +In +� +. +(84) +From (38), we have that +∂fc +∂xp += +� +� +In +0n×n +In +0n×n +0n×n +In +� +� . +(85) +To verify the claim, Corollary 1 is applied which requires +a suitable definition of B. Expanding the definitions of ¯F⋆3 +from (34) reveals +¯F13 = +�0n×n +0n×n +−In +0n×n +In − MdM −1 +D +� +¯F23 = +�0n×n +0n×n +0n×n +� +¯F33 = +� +� +0n×n +0n×n +0n×n +0n×n +0n×n +In +0n×n +−In +0n×n +� +� , +(86) +resulting in the choice +B = +� +� +0n×n +0n×n +In +0n×n +0n×n +In +� +� . +(87) +Expanding the expression B⊤ ¯F33B results in +B⊤ ¯F33B = +�0n×n +−In +In +0n×n +� +(88) +which is invertible, ensuring that Corollary 1 can be applied. +Expanding the definitions of Fr in (32) results in the reduced +dynamics +� +� +˙q +˙p +−y +� +� = +� +� +0n×n +M −1Md +0n×n +−MdM −1 +¯J2 +G +0n×n +−G⊤ +0n×n +� +� +� +�� +� +Fr +� +� +∇qHd +∇pHd +v +� +� +¯J2 = MdJMd + D − D⊤ + MdM −1D⊤ − DM −1Md, +(89) +which agrees with (80) when substituting in the definition +for D in (50). Stability and asymptotic stability of the point +(q⋆, 0n×1) follows from Proposition 1 of [17]. +Remark 4: From Proposition 4 it is clear that M −1 +a (q) +does not need to be a positive matrix. Rather, the closed- +loop mass M −1 +d +must be positive to ensure stability fo the +system. In cases that M −1 +a +is positive, the control sub-system +in Figure 1 is passive. +V. EXAMPLE APPLICATIONS +In this section the matching conditions derived in Propo- +sition 3 are used to construct stabilising control laws for +the cart-pole and acrobot systems. In both cases, the mass +matrix depends on only one configuration variable, so the +kinetic energy matching conditions can be reduced to ODEs +as detailed in Corollary 2. This enables the solutions to be +constructed numerically, removing the need to analytically +solve the equations. +Both +examples +were +prepared +in +Matlab +2022a +and +the +source +code +is +available +via +https://github.com/JoelFerguson/Underactuated Mechanical CbI. +A. Cart-pole example +The cart-pole system, shown in Figure 2, attempts to +balance the pole of length ℓ and mass mp in the upright +position by applying a force F to the cart with mass mc. +The state q1 describes the horizontal displacement of the cart +whereas q2 describes the angle of the pole from vertical in +the clockwise direction. The cart-pole system can be written +Fig. 2: The cart-pole system attempts to balance the pole in +the upright position by regulating the force F. +as a pH system of the form (7) with +q = +�q1 +q2 +� +M(q) = +� mc + mp +mpl cos q2 +mpl cos q2 +mpl2 +� +V (q) = mpgl cos q2 +G = +�1 +0 +� +. +(90) +In the subsequent control design, the parameters mc = mp = +l = 1, g = 9.8 have been used. + +dw +mcThe mass matrix of the cart-pole system depends only on +q2, the unactuated coordinate. The added inverse mass is +assumed to also be a function of q2 also, allowing it to be +written as +M −1 +a (q2) = +� +ma11(q2) +m⊤ +a21(q2) +ma21(q2) +ma22(q2) +� +. +(91) +As noted in Corollary 2, the kinetic energy matching equa- +tions (57) can be reduced to an ODE as both M −1, M −1 +a +are a function of only one variable. The associate ODE is of +the form (72) for qi = q2 where ma11(q2) is a free function +to be chosen. The ODE can be evaluated using numerical +solvers. +Before solving the ODE associated with the kinetic energy +matching equations, consideration should be given to how +the resulting mass matrix impacts the closed-loop potential +energy Vd. Recalling (74), the closed-loop potential energy +is composed of a free term Γ(·) and a term Vm(q) which +must satisfy (75), where s1, s2, s3 are defined in (60), (61). +As the potential V, M −1, M −1 +a +are all functions of only q2, +Vm is also assumed to be a function of q2 only, reducing +(75) to the ODE +∇q2Vm = −s1(q2) +s3(q2)∇q2V, +(92) +which can be evaluated numerically once a solution for +M −1 +a (q2), and hence s1(·), s3(·), are found. noting that the +vector field ∇q2V is divergent from the point q2 = 0, the +closed-loop vector field ∇q2Vm should reverse the direction +locally. This is ensured if the ratio s1(q) +s3(q) is positive in some +neighbourhood of the origin. Recalling that s1(q) is the Schur +complement of M −1 + M −1 +a , which is necessarily positive, +it is required that s3(q) be positive in some neighbourhood +of q2 = 0. The values +ma11(0) = 0 +ma21(0) = −2 +ma22(0) = 8, +(93) +where chosen which result in s1(0) = 1, s3(0) = 1 and +λmin +� +M −1(0) + M −1 +a (0) +� += 0.917 > 0. +The added inverse mass matrix can now be found by +numerically evaluating the ODE (72). The term ma11(q2) is +a free function that was chosen to be constant ma11(q2) = +0, +∂ +∂q2 ma11 = 0 for this example. The resulting functions for +ma21(q2), ma22(q2) were found to exist on the interval q2 ∈ +[−0.48, 0.48] and are shown in Figure (3). From Proposition +4, M −1(q2)+M −1 +a (q2) should be positive to ensure stability, +so the minimum eigenvalue of this expression is shown in +the same figure. +The closed-loop potential energy Vm(q2) can now be +obtained by numerically by evaluating the ODE (92). The +terms s1(·), s3(·) are evaluated using the solutions to M −1 +a +shown in Figure (3). The resulting function Vm(q2) is shown +in Figure (4). As expected, the function is positive in some +neighbourhood of q2 = 0 due to the choice of the added +mass at q2 = 0 in (93). +-0.5 +0 +0.5 +q2 +-2 +0 +2 +4 +6 +8 +ma11 +ma12 +ma22 +-0.5 +0 +0.5 +q2 +0.1 +0.15 +0.2 +min[M-1+Ma +-1] +Fig. 3: A solution for the inverse added mass M −1 +a +was found +to exist for the cart-pole system on the domain q2 between +±0.48 radians. +-0.5 +0 +0.5 +q2 +0 +1 +2 +3 +4 +5 +Vm +Fig. 4: Solution for the added potential energy Vm(q2) for +the cart-pole system. +The proposed functions of M −1 +a , Vm can be used to +construct a controller to stabilise the pendulum in the upright +position. To ensure stability of q1 = 0 also, the free term +Vf(Γ(q)), defined in (74), is constructed. The function Γ(·) +defined by the integral (73), where K(q) is a free function +chosen to ensure solvability. Noting that M −1, M −1 +a +are +functions of q2 only, the parametrisation +�β1(q2) +β2(q2)� += G⊤ � +M −1 +a (q2) + M −1(q2) +� +M(q2) +(94) +is introduced. The free function is chosen as K(q2) = +1 +β1(q2), +resulting in +Γ(q) = +� � +1 +β2(q2) +β1(q2) +� +dq += q1 + +� β2(q2) +β1(q2) dq2, +(95) +which can be solved numerically from the initial condition +Γ(02×1) = 0. The function Vf(·) was taken as Vf(Γ(q)) = +1 +2κΓ(q)2 with κ = 5 for simulation. A contour plot of the +resulting closed-loop potential energy is shown in Figure (5). +Note that a minimum has been assigned to q = 02×1. As a +final control design stage, damping is injected via the new +passive input/output pair with +v = −5G⊤(M −1 +a ++ M −1)p. +(96) +The complete control signal is defined by the expression (46). +The cart-pole system was simulated for 5 seconds from ini- +tial conditions q(0) = (0, 0.3), p(0) = (0, 0). The resulting +state evolution and closed-loop energy Hd is shown in Figure + +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +q2 +-0.5 +0 +0.5 +q1 +log(Vd) +Fig. 5: Contour plot of the closed-loop potential energy +Vd(q) = Vm(q2) + Vf(Γ(q)) for the cart-pole system on +log scale. +6. As expected, the proposed controller stabilises the origin +and the closed-loop energy Hd decreases monotonically. +0 +1 +2 +3 +4 +5 +time (s) +-2 +0 +2 +q1 +q2 +p1 +p2 +0 +1 +2 +3 +4 +5 +time (s) +0 +1 +2 +Hd +Fig. 6: Numerical simulation of cart-pole system in closed- +loop with CbI scheme. +B. Acrobot example +The acrobot system, shown in Figure 7, consists of 2 links +with an actuator supplying a input torque τ fixed between +the base and second links. The base link has displacement of +q2, measured from vertical, length ℓ2, mass m2, moment of +inertia Jℓ1 and centre of mass ℓc2 from the base pivot point. +The actuated link has displacement of q1 measured relative +to the base link, length ℓ1, mass m1, moment of inertia Jℓ1 +and centre of mass ℓc1 from the actuated pivot point. The +control objective of this system is to stabilise the upright +equilibrium position (q1, q2) = (0, 0). The acrobot system +Fig. 7: The acrobot system attempts to balance in the vertical +position by manipulating the torque generated by an actuator +between the two links. +can be written as a pH system of the form (7) with +M(q) = +� +c2 +c2 + c3 cos q1 +c2 + c3 cos q1 +c1 + c2 + 2c3 cos q1 +� +V (q) = c4g cos q2 + c5g cos(q1 + q2) +G = +�1 +0 +� +, +(97) +where +c1 = m2ℓ2 +c2 + m1ℓ2 +2 + Jℓ2 +c2 = m1ℓ2 +c1 + Jℓ1 +c3 = m1ℓ2ℓc1 +c4 = m2ℓc2 + m1ℓ1 +c5 = m1ℓc1. +(98) +For the purposes of simulation, we take the values g = 9.8, +c1 = 2.3333, c2 = 5.3333, c3 = 2, c4 = 3, c5 = 2 which +were previously used in [5], [25]. +In this example, the total energy-shaping controller pro- +posed in [5] is reconstructed as a CbI control scheme by +solving the matching conditions of Proposition 3. In that +work, the closed-loop mass matrix was chosen to be the +constant matrix +M −1 +d += +� 0.3385 +−0.9997 +−0.9997 +5.9058 +� +(99) +which will be recovered in subsequent computations. +The mass matrix of the acrobot system depends only on +q1, the actuated coordinate. The added inverse mass matrix +is assumed to be a function of only q1 also, resulting in the +structure +M −1 +a (q1) = +� +ma11(q1) +m⊤ +a21(q1) +ma21(q1) +ma22(q1) +� +. +(100) +As the system is underactuated degree 1 and the mass matrix +is a function of only one variable, the kinetic energy match- +ing equations can be reduced to an ODE as per Corollary +2. The resulting ODE has the form (72) with qi = q1 and +where ma11(q1) is a free function. + +In order to recover the result (99), this free function +ma11(q1) is chosen as +ma11(q1) = G⊤ � +M −1 +d +− M −1(q1) +� +G += 0.3385 − c1 + c2 + 2c3 cos q1 +c1c2 − c2 +3 cos2(q1) . +(101) +The initial conditions ma12(0), ma22(0) are similarly defined +as +ma12(0) = G⊥ � +M −1 +d +− M −1(0) +� +G = −0.1313 +ma12(0) = G⊥ � +M −1 +d +− M −1(0) +� +G⊥⊤ = 5.2743. +(102) +The added inverse mass was evaluated numerically and the +results are shown in Figure 8. As the previously reported +solution (99) is globally defined, it is unsurprising that the +inverse added mas is also globally defined. As expected, the +minimum eigenvalue of M −1 + M −1 +a +is constant also. +-2 +0 +2 +q1 +-2 +0 +2 +4 +6 +8 +ma11 +ma12 +ma22 +-2 +0 +2 +q1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +min[M-1+Ma +-1] +Fig. 8: A solution for the added mass M −1 +a +for the acrobot +was found to exist globally. +Solving the potential energy PDE (75) is difficult due to +the open-loop potential energy being a function of both q1 +and q2. This dependence implies that Vm cannot be resolved +directly using an ODE solver. Considering the structure of +V in (97), it is proposed that the closed-loop energy Vm has +the structure +Vm(q) = f1(q1) sin(q2) + f2(q1) cos(q2), +(103) +which has derivatives +∇q1Va =∂f1 +∂q1 +sin(q2) + ∂f2 +∂q1 +cos(q2) +∇q2Va =f1(q1) cos(q2) − f2(q1) sin(q2). +(104) +The open-loop potential energy has gradients +∇q1V = − c5g sin(q1) cos(q2) − c5g cos(q1) sin(q2) +∇q2V = − c4g sin(q2) − c5g sin(q1) cos(q2) +− c5g cos(q1) sin(q2). +(105) +Substituting the expressions (104) and (105) into (59) and +matching coefficients results in the system of equations +� ∂f1 +∂q1 +∂f2 +∂q1 +� += +1 +s2(q1) +× +�c4gs1(q1) + c5gs1(q1) cos(q1) + s3(q1)f2(q1) +c5gs1(q1) sin(q1) − s3(q1)f1(q1) +� +, +(106) +which can be evaluated numerically. The values of f1, f2 +at the origin should be chosen to ensure that the origin is +an equilibrium point and Vm is positive in q2. Considering +the expressions (105), (106), the origin is an equilibrium for +f1(0) = 0. The energy function (103) is locally positive +with respect to q1 for f2(0) negative. For the purpose of +simulation, f2(0) = −50 was used. The resulting function +Vm is shown in Figure 9. +Fig. 9: Solution for the added potential energy Vm(q2). +Considering Figure 9, it is clear that Vm is not positive +definite with respect to the origin. Note, however, that q2 = 0 +has been stabilised. To ensure stability of q1 = 0 also, +the free term Vf(Γ(q)), defined in (74), is constructed. The +function Γ(·) defined by the integral (73), where K(q) is +a free function chosen to ensure solvability. Noting that +M −1, M −1 +a +are functions of q1 only, the parametrisation +� +β1(q1) +β2(q1) +� += G⊤ � +M −1 +a (q1) + M −1(q1) +� +M(q1) +(107) +is introduced. The free function is chosen as K(q1) = +1 +β2(q1), +resulting in +Γ(q) = +� � +β1(q1) +β2(q1) +1 +� +dq += +� β1(q1) +β2(q1) dq1 + q2, +(108) +which can be solved numerically from the initial condition +Γ(02×1) = 0. The function Vf(·) was taken as Vf(Γ(q)) = +1 +2κΓ(q)2 with κ = 250 for simulation. A contour plot of +the resulting closed-loop potential energy on a log scale is +shown in Figure 10. Note that a minimum has been assigned +to q = 02×1. As a final control design stage, damping is +injected via the new passive input/output pair with +v = −5G⊤(M −1 +a ++ M −1)p. +(109) +The complete control signal is defined by the expression (46). +The acrobot system was simulated for 20 seconds from ini- +tial conditions q(0) = (0, 0.5), p(0) = (0, 0). The resulting + +100 +m +0 +-100 +-200 +4 +2 +0 +-2 +a2 +44 +2 +0 +-2 +q1-3 +-2 +-1 +0 +1 +2 +3 +q2 +-3 +-2 +-1 +0 +1 +2 +3 +q1 +log(Vd) +Fig. 10: Contour plot of the closed-loop potential energy +Vd(q) = Vm(q) + Vf(Γ(q)) on log scale for the acrobot +system. +state evolution and closed-loop energy Hd is shown in Figure +11. As expected, the proposed controller stabilises the origin +and the closed-loop energy Hd decreases monotonically. +0 +5 +10 +15 +20 +time (s) +-1 +0 +1 +2 +q1 +q2 +0 +5 +10 +15 +20 +time (s) +-10 +0 +10 +p1 +p2 +0 +5 +10 +15 +20 +time (s) +0 +20 +40 +Hd +Fig. 11: Numerical simulation of acrobot system in closed- +loop with CbI scheme. +VI. CONCLUSIONS AND FUTURE WORKS +In this work total energy shaping has been shown to +have a CbI interpretation which results in alternate matching +equations related to the added inverse mass. These equations +were utilised to construct controllers for the cart-pole and +acrobot systems, both of which have the property that the +mass matrix depends on only one variable, using numerical +methods. While the proposed approach is effective, a number +of technical aspects of this approach require further investi- +gation. In particular: +• As detailed in Corollary 2, The kinetic energy matching +equations can be posed as ODEs in the special case +that the mass matrix depends on only one configuration +variable. This property allows the matching equations +to be evaluated numerical using ODE solvers. Further +investigation into solving the matching equations in the +case that the mass matrix is a function of multiple +configuration variables is required. In some cases it +may be possible to decouple the dependence on each +coordinate, recovering equivalent ODEs. Alternatively, +the numerical evaluation of the matching PDEs should +be investigated. +• When evaluating the kinetic energy matching equations +in (72), the term ma11(qi) is a free function that can +be used to control the resulting added inverse mass. +As seen in the cart-pole example of Section V-A, poor +choice of this function results in the solution only being +defined on a small domain. Conversely, in the acrobot +example of Section V-A this term was chosen to ensure +a global solution to the matching equations. Choice of +this function defines a nonlinear control problem that +should be investigated to ensure desirable behaviour of +the result. +• In both examples of Section V the controllers were +designed to stabilise the origin of the respective sys- +tems. While this was achieved and verified numerically, +asymptotic stability was not established. Asymptotic +stability requires that the passive output of the closed- +loop system is zero-state detectable, a task that is non- +trivial for underactuated systems. Further investigation +into methods for injecting damping into the unactuated +momentum channels of the closed-loop system is re- +quired. It is hoped that the CbI interpretation of the +controller shown in Figure 1 may provide new insight +into how this might be achieved. +REFERENCES +[1] R. Ortega and E. 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Reger, “IDA-PBC for underactuated mechanical +systems in implicit port-hamiltonian representation,” 2019 18th Euro- +pean Control Conference, ECC 2019, pp. 614–619, 2019. +[24] M. R. J. Harandi and H. D. Taghirad, “Solution of matching equations +of +IDA-PBC +by +Pfaffian +differential +equations,” +International +Journal of Control, pp. 1–11, 2021. [Online]. Available: https: +//doi.org/10.1080/00207179.2021.1972345 +[25] A. Donaire, J. G. Romero, R. Ortega, B. Siciliano, and M. Crespo, +“Robust IDA-PBC for underactuated mechanical systems subject to +matched disturbances,” International Journal of Robust and Nonlinear +Control, vol. 27, no. 6, pp. 1000–1016, 2017. + diff --git a/4dE2T4oBgHgl3EQfOQZ5/content/tmp_files/load_file.txt b/4dE2T4oBgHgl3EQfOQZ5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98d9215a0a7d83f6d25b98cb69e25f08586f4e32 --- /dev/null +++ b/4dE2T4oBgHgl3EQfOQZ5/content/tmp_files/load_file.txt @@ -0,0 +1,716 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf,len=715 +page_content='Total energy-shaping control for mechanical systems via Control-by-Interconnection Joel Ferguson1 Abstract— Application of IDA-PBC to mechanical systems has received much attention in recent decades, but its ap- plication is still limited by the solvability of the so-called matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In this work, it is shown that total energy- shaping control of under-actuated mechanical systems has a control-by-interconnection interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Using this interpreta- tion, alternate matching conditions are formulated that defines constraints on the added energy, rather then the total closed- loop energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' It is additionally shown that, for systems that are under-actuated degree one with the mass matrix depending on a single coordinate, the kinetic energy matching conditions resolve to ODEs which can be evaluated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Using this approach controllers are proposed for the benchmark cart-pole and acrobot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' INTRODUCTION Energy-based methods for controlling nonlinear physical systems have been shown to be effective in a variety of physical domains [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Such methods consider the energy and structure of the system to be controlled to derive control strategies that exploit the natural system behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Interconnection and Damping Assignment, Passivity-Based Control (IDA-PBC) is one such control methodology where the control input is designed such that the closed-loop can be interpreted as an alternate physical system with a different energy, interconnection and damping structure [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' While IDA-PBC has been applied to a broad range of systems, particular attention has been given to mechanical systems which exhibit a rich canonical structure [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In the case of fully-actuated systems, IDA-PBC allows a user to arbitrarily modify the potential and kinetic energy of the closed-loop system [6], a process known as total energy shaping [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' For under-actuated systems, however, application of IDA-PBC is limited by solutions that satisfy a set of PDEs, the so-called matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Much re- search effort has been committed to solving these equations, with solutions posed in several special cases [4], [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This design methodology has been applied to a number of benchmark examples such as the cart-pole, acrobot, spider crane, amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Control-by-Interconnection (CbI) describes a sub-class of energy-based control methods that falls under the umbrella of IDA-PBC [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Under this scheme, the controller is assumed to be a passive system that is interconnected with a passive plant to be controlled via the passive input-output pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Casimirs, conserved quantities between the control sub- system and the plant, can be constructed to help shape the 1Joel Ferguson is with the School of Engineer- ing, The University of Newcastle, Australia Email: joel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='ferguson@newcastle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='au energy of the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' It is known that potential energy shaping of fully-actuated mechanical systems falls into the class of CbI [7], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Control of underactuated mechanical systems has been explored in the context of CbI by applying nonlinear PID controllers to both the standard and alternate passive outputs [10], [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The idea of using PID for stabilisation of passive systems was formalised in [13] and a general characterisation of all passive outputs from a given system was characterised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In this work, the connection between IDA-PBC and CbI for under-actuated mechanical systems is explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Using the bond graph formalism (see [14] for introduction), a control sub-system is proposed that allows for shaping of the kinetic and potential energies of the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' By representing the controller as a passive interconnection, the requisite matching conditions are reformulated in terms of the added mass and added potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Equivalence between the CbI and IDA-PBC is then established in the case of mechanical systems by identifying Casimirs relating the controller states to those of the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Finally, using the reformulated matching conditions, it is shown that in the case that the mass matrix depends on only one coordinate that the kinetic energy matching conditions can be formulated as an ODE that can be evaluated numerically for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Function arguments are declared upon definition and are omitted for subsequent use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0n×m denotes a n × m zeros matrix whereas In denotes a n×n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' For mappings H : Rn → R, we denote the transposed gradient as ∇H := � ∂H ∂x �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' For P = P ⊤ ∈ Rn×n, λmin [P] , λmax [P] denotes the minimum and maximum (real) eigenvalues of P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' BACKGROUND AND PROBLEM FORMULATION In this section a number of key concepts necessary for the subsequent developments are briefly revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Control-by-interconnection In this work we consider input-state-output port- Hamiltonian systems (ISO-PHS) of the form ˙xp = Fp(xp)∇xpHp(xp) + Gp(xp)up yp = G⊤ p (xp)∇xpHp(xp) (1) where xp ∈ Rp is the state of the plant, Fp(xp) ∈ Rp×p is the combined interconnection and damping matrix satisfying Fp(xp) + F ⊤ p (xp) ≤ 0, Hp(xp) ∈ R is the Hamiltonian, up ∈ Rm is the input, Gp(xp) ∈ Rp×m is the input arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='03746v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='SY] 10 Jan 2023 mapping matrix and yp ∈ Rm is the natural passive output corresponding to the input up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' CbI assumes that the controller is a passive system that is interconnected with the plant (1) via a passive interconnec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' we consider a controller subsystem with two input-output ports,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' described by the ISO-PHS � � ˙xc −yc1 −yc2 � � = � � K11(xc) K12(xc) K13(xc) K21(xc) K22(xc) K23(xc) K31(xc) K32(xc) K33(xc) � � � �� � :=K(xc) � � ∇xcHc uc1 uc2 � � (2) where xc ∈ Rc is the state of the controller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Hc(xc) ∈ R is the controller Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' uc1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' yc1 ∈ Rm and uc2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' yc2 ∈ Rr are passive input-output pairs and K(xc) ∈ R(p+m+r)×(p+m+r) satisfies K(xc) + K⊤(xc) ≤ 0 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The controller system (2) can be interconnected with the plant (1) via the passive interconnection up = −yc1 uc1 = yp, (3) resulting in the closed-loop dynamics � � ˙xp ˙xc −yc2 � � = � � Fp + GpK22G⊤ p GpK21 GpK23 K12G⊤ p K11 K13 K32G⊤ p K31 K33 � � � �� � Fcl � � ∇xpHp ∇xcHc uc2 � � (4) where uc2, yc2 is a passive input-output pair to the inter- connected system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Noting that K + K⊤ ≤ 0, the closed- loop interconnection and damping structure Fcl satisfies Fcl + F ⊤ cl ≤ 0 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In the case of stabilisation, the objective is to construct the plant functions Hc(xc), K(xc) to ensure the existence of Casimirs which statically relate the controller states to functions of the plant states xc = fc(xp), (5) for fc(x) ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The Casimir functions and controller initial conditions are then designed to assign a desirable minimum to the total energy function W(xp) = H(xp) + Hc(xc)|xc=fc(xp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (6) It is noted that the Lyapunov candidate W(xp) can be gen- eralised to a function of H, Hc and the Casimirs xc −fc(xp) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Methods to ensure the existence of and constructing Casimirs have been reported in [7] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Underactuated mechanical systems The primary objective of this work is to apply CbI to the class of underactuated mechanical systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' described by the dynamics � ˙q ˙p � = �0n×n In −In 0n×n � �∇qH ∇pH � + �0n×m G � u H(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) = 1 2p⊤M −1(q)p � �� � :=T (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='p) +V (q) y = G⊤∇pH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (7) where q ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p ∈ Rn are the configuration and momen- tum vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' u ∈ Rm in the input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' M(q) = M ⊤(q) > 0 is the inertia matrix and y is the natural passive output corresponding the the input u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The input mapping matrix G is assumed to be constant and have the structure G = � Im 0(n−m)×m � , (8) where n − m < n is the degree of underactuation of the system1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The Hamiltonian H(q, p) is the sum of the kinetic energy T(q, p) and the potential energy V (q), which allows the gradient of H with respect to q to be written as ∇qH(q, p) = ∇qT(q, p) + ∇qV (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (9) A full-rank left-annihilator for the input mapping matrix (8) is defined as G⊥ = �0(n−m)×m I(n−m) � (10) which satisfies G⊥G = 0(n−m)×m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In the subsequent development, we will require an alter- nate representation of the gradient of the kinetic energy with respect to configuration ∇qT(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Noting that the kinetic energy is quadratic in p, the gradient ∇qT(q, p) can always be factored into the form ∇qT(q, p) = E(q, p)M −1(q)p, (11) for some matrix E(q, p) ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This has been previously noted in [16] using the Christoffel symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Note, however, that the matrix E(q, p) is non-unique and in this work we will use the representation given by ∇qT(q, p) = 1 2 ∂⊤ ∂q � M −1(q)p � p = 1 2 ∂⊤ ∂q � M −1(q)p � M(q) � �� � :=E(q,p) M −1(q)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (12) In constructing a CbI interpretation to total energy shaping it is useful to define a virtual input-output pair for the system (7) by defining the input uv = Gu, (13) which allows the system to written similarly to a fully- actuated system as � ˙q ˙p � = �0n×n In −In 0n×n � �∇qH ∇pH � + �0n×n In � uv yv = ∇pH, (14) where uv, yv ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' From the definition (13), it is clear that any input uv must satisfy G⊥uv = 0m×1, (15) 1The assumed structure of G requires that the first m configuration coordinates are chosen to be collocated with the actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This class of dynamics falls into the broader class of ISO-PHS (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' For a more general input mapping matrix ¯G(q) ∈ Rn×m, there exists a change of coordinates recovering the structure (8) if the columns of ¯G(q) are involute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' which will be ensured in subsequent control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Assum- ing that (15) holds, the input u can be described as a function of uv by u = G⊤uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (16) The advantage of constructing the virtual input-output pair is that the virtual output now describes the full velocity vector yv = M −1(q)p = ˙q, (17) a property that will be exploited when shaping the potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' IDA-PBC for underactuated mechanical systems IDA-PBC is a control design methodology whereby the control signal is designed such that the closed-loop dynamics have a port-Hamiltonian (pH) structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' When applied to underactuated mechanical systems, the target closed-loop dynamics have the structure � ˙q ˙p � = � 0n×n M −1(q)Md(q) −Md(q)M −1(q) J2(q, p) − GKdG⊤ � �∇qHd ∇pHd � Hd(q, p) = 1 2p⊤M −1 d (q)p + Vd(q), (18) where Md(q) = M ⊤ d (q) > 0, Vd(q) are the desired closed- loop inertia matrix and potential energies, respectively, J2(q, p) = −J⊤ 2 (q, p) is skew-symmetric and Kd = K⊤ d ≥ 0 is a tuning parameter used for damping injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If Vd is minimised at the target configuration, Hd qualifies as a Lyapunov function for the closed-loop system [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The complexity of applying IDA-PBC to underactuated system is satisfying the so-called matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This conditions requires that the dynamics of the open-loop and closed-loop systems must agree on the spaces perpendicular to the control signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The structure chosen fo the closed-loop system in (18) ensures that the dynamics of q agree with (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Comparing the dynamics of p results in the condition G⊥ � ∇qH − Md(q)M −1(q)∇qHd − J2(q, p)∇pHd � = 0(n−m)×1, (19) which defines a PDE that should be solved for Md(q), Vd(q), J2(q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Noting the structure of the Hamiltonians, this PDE can be separated into the components involving p, and those that do not by G⊥ � ∇qT − Md(q)M −1(q)∇qTd − J2(q, p)M −1 d (q)p � = 0(n−m)×1 G⊥ � ∇qV − Md(q)M −1(q)∇qVd � = 0(n−m)×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (20) These expressions are known as the kinetic energy and potential energy matching equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Contributions The objective of this work is to construct a CbI interpreta- tion of IDA-PBC when applied to underactuated mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The contributions of this work are threefold: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='1 ISO-PHS with Casimirs that statically relate states are considered and a closed-form solution to remove the Casimirs by reducing the dimension of the state vector is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This solution can be applied to the closed- loop dynamics of CbI implementations of the form (4) to describe the resulting dynamics as a function of xp only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='2 A CbI controller for underactuated mechanical system of the form (2) is proposed and the resulting closed- loop is shown to be equivalent to the well-known dy- namics (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The CbI interpretation generates alternate matching conditions to the expressions (20), describing constraints on the added mass and added potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3 Using the alternate matching conditions, it is shown that the kinetic energy matching equations reduce to ODEs in the special case of underactuation degree one where the mass matrix is a function of only one configuration coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' It is demonstrated that numerical methods can be utilised in such cases to avoid solving these expressions analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Related works Significant attention has been given to solving the match- ing equations (20) in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In [3], [17] it was shown that is the system is under-actuated degree one and the mass matrix depends on a single un-actuated coordinate, the kinetic energy matching condition can be simplified to an ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Using a novel parametrisation of J2(q, p), a general so- lution for under-actuated degree one system was proposed in [4] under the assumption that the mass matrix depends only on the actuated coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This approach was extended in [18] using a momentum transformation to simplify the matching equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' More recently, solutions to the potential energy matching equations were considered in [19] under the assumption that the mass matrix and potential energy functions were dependent on only one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Finally, a general solution to the special case of 2 degree-of-freedom system was proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The studies [20], [21] considered the effects of friction on the closed-loop stability using IDA- PBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In recent works, alternate approaches to constructing so- lutions to the matching equations have been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The existence of conservative forces that cannot be factorised into a skew-symmetric matrix J2(q, p) was investigated in [22], which resulted in alternate matching equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Im- plicit system representations were used in [23] to construct solutions in an over-parameterised space where the closed- loop dynamics were subject to constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' By working in the larger dimension, a solution to a under-actuated degree 2 crane system was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Pfaffian differential equations were utilised in [24] which resulted in the kinetic energy PDEs being converted to an alternate form which admits simpler solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Some authors have investigated the possibility of avoiding the matching equations altogether by considering the control signal to be a CbI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The work [10] relied on a Lagrangian structure and several technical assumptions to verify the existence of a second passive output corresponding to the input u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Using this second output, a stabilising control was designed that ensured stability without requiring a solution to the matching PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' A similar approach was proposed in [12], [11] where a second passive output was utilised and the control assumed to have a PID structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' CASIMIR REDUCTION In this section, a method for reducing the state dimension of ISO-PHS with Casimirs is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The reduction method applies to general ISO-PHS with Casimirs and can be directly applied to the resulting closed-loop dynamics of CbI schemes of the form (4) to describe the system as a function of xp only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In the sequel, the reduction method will be used to show equivalence between the CbI controller for underactuated mechanical systems and the IDA-PBC dynamics (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Before introducing the state reduction solution, a useful lemma is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Lemma 1: Consider a square block matrix of arbitrary dimension A = �A11 A12 A21 A22 � (21) and assume that A22 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If the symmetric com- ponent of A is negative semi-definite, A + A⊤ ≤ 0, the symmetric component of the Schur complement A11 − A12A−1 22 A21 (22) is negative semi-definite also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proof: First note that the Schur complement of X can be computed by � I −A⊤ 21A−⊤ 22 � A � I −A−1 22 A21 � = A11 − A12A−1 22 A21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (23) The symmetric component of this expression is negative semi-definite as A + A⊤ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The solution for reducing the dimension of ISO-PHS which exhibit Casimirs is now introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This development applies to systems of the form � � ˙x1 ˙x2 −y � � = � � F11(x) F12(x) F13(x) F21(x) F22(x) F23(x) F31(x) F32(x) F33(x) � � � �� � F (x) � � ∇x1H ∇x2H u � � (24) where x ∈ Rp+c is the state of the system which has been partitioned into x1 ∈ Rp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' x2 ∈ Rc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' H(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' x2) ∈ R is the Hamiltonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' F(x) ∈ R(p+c)×(p+c) is the full-rank intercon- nection and damping matrix satisfying F(x) + F ⊤(x) ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' u ∈ Rm is the input and y ∈ Rm is the corresponding passive output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' It is assumed that the system contains a Casimir and the states have been partitioned such that the Casimir can be written as x2 = fc(x1), (25) where fc(x1) ∈ Rc is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The first step in constructing a minimal system represen- tation is defining a new set of coordinates given by w = x2 − fc(x1) = 0c×1, (26) which is identically equal to zero by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The system (24) can be described in the coordinates (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' w) by � � ˙x1 −y ˙w � � = � � Ip 0p×c 0p×m 0m×p 0m×c Im − ∂fc ∂x1 Ic 0c×m � � � � ˙x1 ˙x2 −y � � = � � Ip 0p×c 0p×m 0m×p 0m×c Im − ∂fc ∂x1 Ic 0c×m � � F × � � Ip 0p×m − ∂⊤fc ∂x1 0c×p 0c×m Ic 0m×p Im 0m×c � � � � ∇x1Hr u ∇wHr � � = � � ¯F11 ¯F12 ¯F13 ¯F21 ¯F22 ¯F23 ¯F31 ¯F32 ¯F33 � � � �� � ¯ F � � ∇x1Hr u ∇wHr � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (27) where Hr(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' w) :=H(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' w + fc(x1)) ¯F11(x1) =F11(x)|x2=fc(x1) ¯F12(x1) =F13(x)|x2=fc(x1) ¯F13(x1) =F12(x) − F11(x)∂⊤fc ∂x1 ���� x2=fc(x1) ¯F21(x1) =F31(x)|x2=fc(x1) ¯F22(x1) =F33(x)|x2=fc(x1) ¯F23(x1) =F32(x) − F31(x)∂⊤fc ∂x1 ���� x2=fc(x1) ¯F31(x1) =F21(x) − ∂fc ∂x1 F11(x) ���� x2=fc(x1) ¯F32(x1) =F23(x) − ∂fc ∂x1 F13(x) ���� x2=fc(x1) ¯F33(x1) =F22(x) − F21(x)∂⊤fc ∂x1 − ∂fc ∂x1 F12(x) + ∂fc ∂x1 F11(x)∂⊤fc ∂x1 ���� x2=fc(x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (28) Recalling that w is identically equal to zero, ˙w is also equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Consequently, the final row of (27) is a constraint that needs to be resolved to construct a minimal system representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' There are two methods of reduction that will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Firstly, in the transformed coordi- nates (27) it can occur that one or more columns of ¯F⋆3 is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Without loss of generality, it is assumed that the first d columns of ¯F⋆3 are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To remove the zero rows, the full-rank matrix B is defined as B = �0d×(c−d) I(c−d) � , (29) which acts to select the non-zero columns of ¯F⋆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The zero columns and corresponding rows are removed from (27) by � � ˙x1 −y B⊤ ˙w � � = � � ¯F11 ¯F12 ¯F13B ¯F21 ¯F22 ¯F23B B⊤ ¯F31 B⊤ ¯F32 B⊤ ¯F33B � � � �� � := ¯ FB � � ∇x1Hr u B⊤∇wHr � � , (30) which does not modify the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Note that as ¯F + ¯F ⊤ ≤ 0, ¯FB + ¯F ⊤ B ≤ 0 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Using this representation, a method for resolving the remaining constraint equations is presented under the assumption that B⊤ ¯F33B is full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proposition 1: Consider the pH system (24) with Casimir (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If the matrix B⊤ ¯F33B is full-rank for all x1 the system can be described by a reduced-order model � ˙x1 −y � = Fr(x1) �∇x1Hr u � , (31) where Hr(x1) =H (x1, fc(x1)) Fr(x1) = � ¯F11 ¯F12 ¯F21 ¯F22 � − � ¯F13B ¯F23B � (B⊤ ¯F33B)−1 � B⊤ ¯F31 B⊤ ¯F32 � (32) and ¯F(x1) satisfies Fr(x1) + F ⊤ r (x1) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proof: The expression B⊤ ˙w, defined in (30), is iden- tically equal to 0(c−d)×1 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Note, however, that the gradient B⊤∇wHr is not necessarily equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Assuming that B⊤ ¯F33B is full rank, the expression B⊤∇wHr can be described as B⊤∇wHr = − (B⊤ ¯F33B)−1 � B⊤ ¯F31 B⊤ ¯F32 � � ∇x1Hr u � (33) Substituting this expression into the dynamics (30) resolves to the reduced dynamics (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To verify that Fr + F ⊤ r ≤ 0, note that Fr is the Schur complement of ¯FB which satisfies ¯FB + ¯F ⊤ B ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' It follows that Fr + F ⊤ r ≤ 0 by application of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proposition 1 showed that an ISO-PHS that exhibits a Casimir function can be described in a reduced state-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The class of dynamics that are derived from application of CbI (4) that result in a Casimir of the form (5) falls into the class of systems (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The following Corollary tailors the Casimir reduction for this important sub-class of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Corollary 1: If the closed-loop dynamics of a CbI scheme (4) exhibit a Casimir of the form (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' the system can be equivalently expressed in the form (31) where x1 =xp Hr(xp) =Hp(xp) + Hc (fc(xp)) ¯F11 =Fp + GpK22G⊤ p ¯F12 =GpK23 ¯F13 =GpK21 − � Fp + GpK22G⊤ p � ∂⊤fc ∂xp ¯F21 =K32G⊤ p ¯F22 =K33 ¯F23 =K31 − K32G⊤ p ∂⊤fc ∂xp ¯F31 =K12G⊤ p − ∂fc ∂xp � Fp + GpK22G⊤ p � ¯F32 =K13 − ∂fc ∂xp GpK23 ¯F33 =K11 − K12G⊤ p ∂⊤fc ∂xp − ∂fc ∂xp GpK21 + ∂fc ∂xp � Fp + GpK22G⊤ p � ∂⊤fc ∂xp (34) and B is suitably chosen as per (29) using the expressions ¯F⋆3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The arguments have been dropped from the definitions of ¯F⋆⋆(xp) for the sake of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proof: The result follows from direct application of Proposition 1 to the dynamics (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' CONTROL-BY-INTERCONNECTION FOR MECHANICAL SYSTEMS In this section, a control-by-interconnection scheme for under-actuated mechanical systems is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' A dynamic 2-port control system is introduced with the intention that it will be interconnected to the plant (14) via one of the ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The controller states are constructed to be statically related to the plant states after interconnection, resulting in Casimirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' By applying Proposition 1, the closed-loop dynamic are defined in a reduced space in which the dynamics coincide with standard total energy-shaping control (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The proposed CbI scheme is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The intention of this control subsystem is to interconnect with the plant (14) via the uc1, yc1 power port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The second input uc2 is available for subsequent control design, such as damping injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The terms M(qa2), E(qa2, pa) are the plant mass matrix (7) and factorisation of the kinetic energy gradient (12) evaluated at the controller states whereas J(qa2, pa) = −J(qa2, pa)⊤ ∈ Rn×n is a skew-symmetric matrix to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The three-port storage element Ha(qa1, qa2, pa) has states qa1, qa2, pa ∈ Rn and energy function similar to mechanical systems, Ha(qa1, qa2, pa) = 1 2p⊤ a M −1 a (qa2)pa � �� � :=Ta(qa2,pa) + Vd(qa2) − V (qa1) � �� � :=Va(qa1,qa2) , (35) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 1: Mass shaping as CbI for under-actuated mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' where M −1 a (qa2) is the inverse added mass, Ta(qa2, pa) is the added kinetic energy, Vd(qa2) is the desired closed-loop potential energy, V (qa1) is the plant potential energy function (7) evaluated at the plant state qa1 and Va(qa1, qa2) is the total added potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' It is important to note that, although M −1 a (qa2) is represented as a matrix inverse, it need not be invertible nor positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Indeed, it will be shown in subsequent developments that the key requirement is that M −1 d (q) := M −1(q) + M −1 a (q) (36) should be positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In subsequent analysis it will be shown that the intercon- nection of the control system with the plant (14) via the interconnection uv = −yc1 uc1 = yv, (37) yields Casimirs � � qa1 qa2 pa � � = � � In 0n×n In 0n×n 0n×n In � � � q p � � �� � fc(xp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (38) Assuming the Casimirs exist, some intuition regarding the construction of the control system in Figure 1 can be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Both qa1, qa2 were constructed to be equal to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Firstly ˙qa1 is equal to ˙q by interconnection with the plant virtual output yv via a 1-junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To verify a similar relation for qa2, assume that qa2 = q, pa = p holds which results in ∇paHa = M −1 a (q)p and uc1 + ∇paHa = M −1 d p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' With this in mind, the transformer can be seen to reconstruct the velocity ˙q = M −1(q)p for the bottom 1-junction, resulting in ˙qa1 = ˙q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To construct a Casimir pa = p, first note from (7) and (12) that the plant momentum dynamics can be expressed as ˙p = −∇qV − E(q, p)M −1(q)p + uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (39) The control structure acts to remove these forces from the plant via the right side of the control structure and re- introduce them via the top 0-junction where they are shared with the dynamics of pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The ˙qa1 bond acts to cancel the gravity term from the plant −∇qV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Recalling that the bottom 1-junction has flow equal to M −1(q)p, the right-side gyrator cancels the term −E(q, p)M −1(q)p from the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The left- side gyrator then re-introduces the force −E(q, p)M −1(q)p via the top 0-junction where it is shared between ˙p and ˙pa, establishing the desired Casimir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The claimed Casimir (38) is now formalised in the follow- ing Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' For this development,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' note that the gradients of the added energy Ha(·) satisfy ∇qa1Ha = −∇qa1V ∇qa2Ha = ∇qa2Ta + ∇qa2Vd ∇qa2Ta = 1 2 ∂⊤ ∂qa2 � M −1 a (qa2)pa � pa ∇paHa = M −1 a (qa2)pa (40) and the expressions A(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' B(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' C(·) in Figure 1 can be evaluated as A(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' uc1) =M −1(qa2)Md(qa2) [uc1 + ∇paHa] B(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' uc1) =∇qa2Ha − E⊤(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa)∇paHa − M(qa2)J(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa)Md(qa2) × [uc1 + ∇paHa] C(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' uc1) =Md(qa2)M −1(qa2)∇qa2Ha − Md(qa2)M −1(qa2)E⊤(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa)∇paHa − Md(qa2)J(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa)Md(qa2) × [uc1 + ∇paHa] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (41) with Md(·) defined in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To ensure that the Casimir exists, a number of requirements are imposed on the selection of the added inverse mass M −1 a (q) and closed-loop potential energy Vd(q) which are equivalent of the standard matching conditions used in IDA-PBC (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proposition 2: Consider the control system in Figure 1 and assume that it is interconnected to the plant (14) via the interconnection (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If M −1 a (qa), Vd(qa) are chosen such that G⊥C(qa2, pa, uc1)|qa2=q,pa=p = G⊥∇qV (42) and the controller states are initialised as qa1(0) = qa2(0) = q(0), pa(0) = p(0), the Casimir (38) holds for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proof: Consider that at some time instant T (38) holds, implying that qa1(T) = qa2(T) = q(T), pa(T) = p(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (43) It is shown that if (42) is satisfied, then the derivatives of the states also agree ˙qa1(T) = ˙qa2(T) = ˙q(T), ˙pa(T) = ˙p(T), (44) establishing the existence of a Casimir for all future time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' We proceed by first establishing the relationship for the configuration vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' From (37) and (14) uc = M −1(q)p which establishes ˙qa1(T) = ˙q(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The input uc = M −1(q)p is substituted into A(·) (41) to find A|t=T = M −1(qa2)Md(qa2) � M −1(q)p + M −1 a (qa2)pa � |t=T = M −1(q)p|t=T = ˙q|t=T , (45) confirming that ˙qa2(T) = ˙q(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Next we consider the behaviour of the momentum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' First note that, from the bond graph in Figure 1 and the definition (16), the plant input uv is given by u = G⊤uv = G⊤ [Guc2 − C(qa2, pa, uc1) + ∇qa1V (qa1)] = uc2 − G⊤ [C(qa2, pa, uc1) − ∇qa1V (qa1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (46) Using the control definition (46) and the condition (42), the plant dynamics (7) can be expanded as ˙p = − ∇qT(q, p) − �G⊤∇qV (q) G⊥∇qV (q) � + Gu = − ∇qT(q, p) − � G⊤∇qV (q) G⊥∇qV (q) � + G � uc2 − G⊤ [C(qa2, pa, uc1) − ∇qa1V (qa1)] � = − ∇qT(q, p) + Guc2 − �G⊤ {C(qa2, pa, uc1) + ∇qV (q) − ∇qa1V (qa1)} G⊥∇qV (q) � (47) Note that at time T, ∇qV (q)|t=T = ∇qa1V (qa1)|t=T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Additionally recall the assumption (42) which allows the simplification ˙p|t=T = − ∇qT(q, p) − C(qa2, pa, uc1) + Guc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (48) Recalling the identity (45), the dynamics of pa at time T can be expanded to ˙pa = Gv − E(qa2, pa)A(·)|t=T − C(qa2, pa, uc1)|t=T = Gv − ∇qT(q, p)|t=T − C(qa2, pa, uc1)|t=T , (49) which agrees with (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As (48) and (49) agree at time T, (44) is verified for the momentum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If at the initial time t = 0 we have qa(0) = q(0), pa(0) = p(0), it follows that qa(t) = q(t) and pa(t) = p(t) for all time via integration, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proposition 2 has established that the Casimir (38) holds under some technical assumptions that will be verified in subsequent design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Before proceeding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' we note that the control subsystem in Figure 1 can be written in the form (2) with xc = � q⊤ a1 q⊤ a2 p⊤ a �⊤ Hc(qa1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) = Ha(qa1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) K11(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) = � � 0n×n 0n×n 0n×n 0n×n 0n×n M −1Md 0n×n −MdM −1 D − D⊤ + MdJMd � � K12(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) = � � In M −1Md MdJMd − D⊤ � � K13 = � � 0n×m 0n×m G � � K21(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) = � −In −MdM −1 D + MdJMd � K31 = � 0m×n 0m×n −G⊤� K22(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) = MdJMd K23 = G K32 = −G⊤ K33 = 0m×m D(qa2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) = MdM −1E⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (50) In the subsequent developments it is assumed that the requisite (42) of Proposition 2 holds, implying qa1(t) = qa2(t) = q(t), pa(t) = p(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Condition (42) will be verified by choice of M −1 a and Vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Assuming the Casimir holds, the expressions for A(·), B(·), C(·) in (41) can be simplified to A(q, p) =M −1(q)p B(q, p) =∇qTa(q, p) + ∇qVd(q, p) − E⊤(q, p)M −1 a (q)p − M(q)J(q, p)p C(q, p) =Md(q) � M −1(q)∇qTa(q, p) + M −1(q)∇qVd(q, p) −M −1(q)E⊤(q, p)M −1 a (q)p − M −1(q)J(q, p) � (51) Recalling the definition of ∇qaTa in (40), it is noted that C(·) contains some terms which are quadratic in p and some that are functions of q only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The function C(·) is divided into C(q, p) =CKE(q, p) + CP E(q) CKE(q, p) = � M −1 a (q) + M −1(q) �−1 � �� � Md(q) {Y (q, p) − J(q, p)} p CP E(q) = � M −1 a (q) + M −1(q) �−1 M −1(q)∇qVd, (52) where KE represents kinetic energy, PE represents poten- tial energy and Y is defined as Y (q, p) =1 2M −1(q)∂⊤ ∂q � M −1 a (q)p � − 1 2 ∂ ∂q � M −1(q)p � M −1 a (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (53) As Y is linear in p it can be written as Y (q, p) = n � i=1 piY i(q), (54) where Y i(q) =1 2M −1(q)∂⊤ ∂q � M −1 a (q)ei � − 1 2 ∂ ∂q � M −1(q)ei � M −1 a (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (55) The key constraint for control design is choosing M −1 a (q), Vd(q) satisfying the matching condition (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' From the definition of C(·) in (51), the constraint equation is a function of both M −1 a (q) and � M −1 a (q) + M −1(q) �−1, making direct design of this matrix difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To simplify the design process, an alternate characterisation of (42) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In the following proposition, the inverse mass matrix, inverse added mass matrix, interconnection matrix and Y (·) are partitioned as � m11(q) m⊤ 21(q) m21(q) m22(q) � = �G⊤ G⊥ � M −1(q) � G G⊥⊤� � ma11(q) m⊤ a21(q) ma21(q) ma22(q) � = �G⊤ G⊥ � M −1 a (q) � G G⊥⊤� � J11(q, p) −J⊤ 21(q, p) J21(q, p) J22(q, p) � = �G⊤ G⊥ � J(q, p) � G G⊥⊤� �Y11(q, p) Y12(q, p) Y21(q, p) Y22(q, p) � = �G⊤ G⊥ � Y (q, p) � G G⊥⊤� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (56) Using the above definitions, an alternate characterisation of (42) is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proposition 3: The matching condition (42) is satisfied if: The added mass matrix M −1 a (q) is chosen such that D(q) � Y i(q) + Y i⊤(q) � D⊤(q) = 0(n−m)×(n−m) (57) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' , n} where D(q) = �(m21 + ma21)(m11 + ma11)−1 −In−m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (58) The desired potential energy Vd(q) satisfies s1(q)G⊥∇qV = −s2(q)G⊤∇qVd − s3(q)G⊥∇qVd = −D(q)M −1(q)∇qVd, (59) where s1(q) = (m22 + ma22) − (m21 + ma21)(m11 + ma11)−1(m⊤ 21 + m⊤ a21) (60) is the Schur complement of M −1(q) + M −1 a (q) and s2(q) = (m21 + ma21)(m11 + ma11)−1m11 − m21 s3(q) = (m21 + ma21)(m11 + ma11)−1m⊤ 21 − m22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (61) Proof: From the graph in Figure 1 and the intercon- nection (37), the virtual input uv is given by uv = Gu =Gv − C + ∇qV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (62) Recalling the definition of G, (62) is equivalent to (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Collecting the terms u, v and left multiplying by M −1 a +M −1 results in � M −1 a + M −1� G(u − v) = � M −1 a + M −1� {−CKE − CP E + ∇qV } (63) Due to the structure of G in (8), (63) has the left annihilator D(·), defined in (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Left multiplying (63) by D(·) and separating the compo- nents into those relating to the kinetic and potential energies result in 0(n−m)×1 = −D � M −1 a + M −1� CKE (64) 0(n−m)×1 = −D � M −1 a + M −1� {CP E − ∇qV } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (65) Using (52), (65) is expanded to 0(n−m)×1 =D � M −1 a + M −1� ∇qV − DM −1∇qVa, (66) which can be seen to agree with (59) after expanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Now considering the constraint on the kinetic energy expression (64), the definition (52) is substituted to find 0(n−m)×n =D {−Y + J} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (67) Using the relevant definitions, the first component of (67) can be solved for J21(q, p) as J21(q, p) =(m21 + ma21)(m11 + ma11)−1(−Y11 + J11) + Y21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (68) Substituting this expression back into the second component of (67) reveals the constraint 0(n−m)×(n−m) =(m21 + ma21)(m11 + ma11)−1(−Y12 − J⊤ 21) − (−Y22 + J22) =(m21 + ma21)(m11 + ma11)−1 � −Y12 − Y ⊤ 21 � − (m21 + ma21)(m11 + ma11)−1(−Y11 + J11)⊤ × (m11 + ma11)−1(m⊤ 21 + m⊤ a21) − (−Y22 + J22) = − D � −Y11 + J⊤ 11 −Y12 − Y ⊤ 21 0(n−m)×m −Y22 + J22 � D⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (69) The term J22 is taken as below to solve the skew-symmetric part of this expression, J22 = −1 2D � −Y11 + Y ⊤ 11 + J⊤ 11 − J11 −Y12 − Y ⊤ 21 Y ⊤ 12 + Y21 −Y22 + Y ⊤ 22 � D⊤, (70) where J11 ∈ Rm×m is a free skew-symmetric term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The symmetric part of (69) must also be equal to zero, implying that D � Y + Y ⊤� D⊤ = 0(n−m)×(n−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (71) Finally, noting that this must be true for each pi, the condition (57) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Remark 1: The expression (57) implicitly defines a set of PDEs that must be satisfied by any choice of M −1 a (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' From the definition of Y i in (55), the first m equations are describe partial differential equations involving the partial derivatives of ma11, ma21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The remaining n − m equations describe partial differential equations involving the partial derivatives of ma21, ma22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This structure can be useful for resolving the equations into a standard representation for solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Corollary 2: In the special case of under-actuation degree 1, if M −1, M −1 a is a function of only 1 configuration variable qi, the kinetic energy matching equations (57) can be reduced to a set of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' d dqi � m⊤ a21 ma22 � = g � ma11, d dqi ma11, M −1, d dqi M −1 � , (72) where g(·) ∈ Rn is a function implicitly defined by the matching conditions (57) and ma11(qi) can be chosen freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proof: Assuming that the mass matrix M −1 is function only of a single configuration variable qi, we will also impose that the added mass M −1 a is a function only of the same variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As a consequence, the matching expression (57) is now only a function in the single variable qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Notably, all partial derivatives of M −1 a with respect to qk, where k ̸= i, are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Noting Remark 1, the first n − 1 expressions of (57) pro- duce differential equations involving the partial derivatives of ma11, ma21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The dimension of ma21 is 1 × (n − 1), so the first n − 1 equations can be solved simultaneously to find an expression for d dqi m⊤ a21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The nth expressions of (57) can then be resolved for an expression for d dqi ma22, which has dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Combining these expressions, the matching equations (57) can be resolved into an ODE of the form (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Remark 2: Corollary 2 describes situations in which the kinetic energy matching equations can be reduced to an ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The solution, however, will depend on the choice of ma11(qi) and may not be globally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This poses the question of how should the function ma11(qi) be chosen to ensure an appropriate solution M −1 a (qi)—a nonlinear control problem!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The results of Corollary 1 describe the degrees of freedom that exist when constructing a solution to the added inverse mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Similar degrees of freedom exist in the defini- tion of the closed-loop potential energy that can be exploited to ensure positivity of the chosen function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The following Corollary defines a free function that can be utilised to this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Corollary 3: Suppose that there exists a full rank matrix- valued function K(q) ∈ Rm×m such that the integral Γ(q) = � K(q)G⊤ � M −1 a (q) + M −1(q) � M(q) dq, (73) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The desired closed-loop potential energy can be chosen as Vd(q) = Vm(q) + Vf(Γ(q)), (74) where Vm(·) must be chosen to satisfy the potential energy matching conditions (59) and Vf(·) is a free function that does not impact the matching equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Consequently, the matching equation (59) can be equivalently written as s1(q)G⊥∇qV = −s2(q)G⊤∇qVm − s3(q)G⊥∇qVm = −D(q)M −1(q)∇qVm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (75) Proof: Computing the gradient of Vd results in ∇qVd = ∇qVm + ∂⊤Γ ∂q ∇ΓVf = ∇qVm + M � M −1 a + M −1� GK⊤∇ΓVf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (76) From the definition of D(q) in (58), we have the identity D(q)M −1M(q) � M −1 a (q) + M −1(q) � G = �Im ⋆� G = 0(n−m)×m (77) Substituting the expression (76) into (59) and noting the above expression results in the simplified matching equation (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Remark 3: Vf(·) is a free function precisely because Γ is an integral of the passive output yc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The potential energy Vf could be alternatively constructed as a capacitor element added to the input uc2 in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Now we arrive at one of the key results of this work, the equivalence of the proposed CbI scheme and total energy- shaping control of underactuated mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As- suming that the CbI scheme has been constructed to satisfy the required matching conditions to ensure the existence of a Casimir of the form qa = q, pa = p, Proposition 1 is applied to reconstruct the reduced closed-loop structure (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proposition 4: Consider the underactuated mechanical system with virtual input (14) and assume that Ma(q), Vd(q) are chosen such that the conditions of Proposition 3 are sat- isfied in some neighbourhood of a point (q, p) = (q⋆, 0n×1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If the control signal is chosen as u(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) =v − G⊤ � Md(q)M −1(q) � −E⊤(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p)M −1 a (q)p +∇qaHa(qa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' pa) − M(q)J(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p)p] − ∇qV } (78) where Md(q) = � M −1 a (q) + M −1(q) �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (79) the following hold: The closed-loop dynamics have the form � ˙q ˙p � = � 0n×n M −1(q)Md(q) −Md(q)M −1(q) J2(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) � �∇qHd ∇pHd � + �0n×m G � v Hd(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) = 1 2p⊤M −1 d (q)p + Vd(q) y = G⊤∇pHd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (80) where J2(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) =Md(q) � J(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) + M −1(q) � E(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) − E⊤(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) � ×M −1(q) � Md(q) + Md(q)M −1(q)E⊤(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) − E(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p)M −1(q)Md(q) (81) If Md(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Vd(q) satisfy Md(q) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Vd(q) > 0 (82) in some neighbourhood of (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' p) = (q⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0n×1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (q⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0n×1) is a stable equilibrium of the closed-loop system for v = 0m×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' If the input signal v is used for damping injection v = −KdG⊤y (83) for some positive Kd ∈ Rm×m and the equilibrium (q, p) = (q⋆, 0n×1), (q⋆, 0n×1) is locally detectable from the output y, the point (q⋆, 0n×1) is asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Proof: Interconnection of the mechanical system with the control subsystem results in a closed-loop of the form (4), where xc, Hc and K⋆⋆ are defined in (50) and xp = �q p � Fp = � 0n×n In −In 0n×n � Gp = �0n×n In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (84) From (38), we have that ∂fc ∂xp = � � In 0n×n In 0n×n 0n×n In � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (85) To verify the claim, Corollary 1 is applied which requires a suitable definition of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Expanding the definitions of ¯F⋆3 from (34) reveals ¯F13 = �0n×n 0n×n −In 0n×n In − MdM −1 D � ¯F23 = �0n×n 0n×n 0n×n � ¯F33 = � � 0n×n 0n×n 0n×n 0n×n 0n×n In 0n×n −In 0n×n � � , (86) resulting in the choice B = � � 0n×n 0n×n In 0n×n 0n×n In � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (87) Expanding the expression B⊤ ¯F33B results in B⊤ ¯F33B = �0n×n −In In 0n×n � (88) which is invertible, ensuring that Corollary 1 can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Expanding the definitions of Fr in (32) results in the reduced dynamics � � ˙q ˙p −y � � = � � 0n×n M −1Md 0n×n −MdM −1 ¯J2 G 0n×n −G⊤ 0n×n � � � �� � Fr � � ∇qHd ∇pHd v � � ¯J2 = MdJMd + D − D⊤ + MdM −1D⊤ − DM −1Md, (89) which agrees with (80) when substituting in the definition for D in (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Stability and asymptotic stability of the point (q⋆, 0n×1) follows from Proposition 1 of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Remark 4: From Proposition 4 it is clear that M −1 a (q) does not need to be a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Rather, the closed- loop mass M −1 d must be positive to ensure stability fo the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In cases that M −1 a is positive, the control sub-system in Figure 1 is passive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' EXAMPLE APPLICATIONS In this section the matching conditions derived in Propo- sition 3 are used to construct stabilising control laws for the cart-pole and acrobot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In both cases, the mass matrix depends on only one configuration variable, so the kinetic energy matching conditions can be reduced to ODEs as detailed in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This enables the solutions to be constructed numerically, removing the need to analytically solve the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Both examples were prepared in Matlab 2022a and the source code is available via https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='com/JoelFerguson/Underactuated Mechanical CbI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Cart-pole example The cart-pole system, shown in Figure 2, attempts to balance the pole of length ℓ and mass mp in the upright position by applying a force F to the cart with mass mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The state q1 describes the horizontal displacement of the cart whereas q2 describes the angle of the pole from vertical in the clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The cart-pole system can be written Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 2: The cart-pole system attempts to balance the pole in the upright position by regulating the force F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' as a pH system of the form (7) with q = �q1 q2 � M(q) = � mc + mp mpl cos q2 mpl cos q2 mpl2 � V (q) = mpgl cos q2 G = �1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (90) In the subsequent control design, the parameters mc = mp = l = 1, g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='8 have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' dw mcThe mass matrix of the cart-pole system depends only on q2, the unactuated coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The added inverse mass is assumed to also be a function of q2 also, allowing it to be written as M −1 a (q2) = � ma11(q2) m⊤ a21(q2) ma21(q2) ma22(q2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (91) As noted in Corollary 2, the kinetic energy matching equa- tions (57) can be reduced to an ODE as both M −1, M −1 a are a function of only one variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The associate ODE is of the form (72) for qi = q2 where ma11(q2) is a free function to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The ODE can be evaluated using numerical solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Before solving the ODE associated with the kinetic energy matching equations, consideration should be given to how the resulting mass matrix impacts the closed-loop potential energy Vd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Recalling (74), the closed-loop potential energy is composed of a free term Γ(·) and a term Vm(q) which must satisfy (75), where s1, s2, s3 are defined in (60), (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As the potential V, M −1, M −1 a are all functions of only q2, Vm is also assumed to be a function of q2 only, reducing (75) to the ODE ∇q2Vm = −s1(q2) s3(q2)∇q2V, (92) which can be evaluated numerically once a solution for M −1 a (q2), and hence s1(·), s3(·), are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' noting that the vector field ∇q2V is divergent from the point q2 = 0, the closed-loop vector field ∇q2Vm should reverse the direction locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This is ensured if the ratio s1(q) s3(q) is positive in some neighbourhood of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Recalling that s1(q) is the Schur complement of M −1 + M −1 a , which is necessarily positive, it is required that s3(q) be positive in some neighbourhood of q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The values ma11(0) = 0 ma21(0) = −2 ma22(0) = 8, (93) where chosen which result in s1(0) = 1, s3(0) = 1 and λmin � M −1(0) + M −1 a (0) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='917 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The added inverse mass matrix can now be found by numerically evaluating the ODE (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The term ma11(q2) is a free function that was chosen to be constant ma11(q2) = 0, ∂ ∂q2 ma11 = 0 for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The resulting functions for ma21(q2), ma22(q2) were found to exist on the interval q2 ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='48, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='48] and are shown in Figure (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' From Proposition 4, M −1(q2)+M −1 a (q2) should be positive to ensure stability, so the minimum eigenvalue of this expression is shown in the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The closed-loop potential energy Vm(q2) can now be obtained by numerically by evaluating the ODE (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The terms s1(·), s3(·) are evaluated using the solutions to M −1 a shown in Figure (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The resulting function Vm(q2) is shown in Figure (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As expected, the function is positive in some neighbourhood of q2 = 0 due to the choice of the added mass at q2 = 0 in (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 q2 2 0 2 4 6 8 ma11 ma12 ma22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 q2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='2 min[M-1+Ma 1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 3: A solution for the inverse added mass M −1 a was found to exist for the cart-pole system on the domain q2 between ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='48 radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 q2 0 1 2 3 4 5 Vm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 4: Solution for the added potential energy Vm(q2) for the cart-pole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The proposed functions of M −1 a , Vm can be used to construct a controller to stabilise the pendulum in the upright position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To ensure stability of q1 = 0 also, the free term Vf(Γ(q)), defined in (74), is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The function Γ(·) defined by the integral (73), where K(q) is a free function chosen to ensure solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Noting that M −1, M −1 a are functions of q2 only, the parametrisation �β1(q2) β2(q2)� = G⊤ � M −1 a (q2) + M −1(q2) � M(q2) (94) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The free function is chosen as K(q2) = 1 β1(q2), resulting in Γ(q) = � � 1 β2(q2) β1(q2) � dq = q1 + � β2(q2) β1(q2) dq2, (95) which can be solved numerically from the initial condition Γ(02×1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The function Vf(·) was taken as Vf(Γ(q)) = 1 2κΓ(q)2 with κ = 5 for simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' A contour plot of the resulting closed-loop potential energy is shown in Figure (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Note that a minimum has been assigned to q = 02×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As a final control design stage, damping is injected via the new passive input/output pair with v = −5G⊤(M −1 a + M −1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (96) The complete control signal is defined by the expression (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The cart-pole system was simulated for 5 seconds from ini- tial conditions q(0) = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3), p(0) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The resulting state evolution and closed-loop energy Hd is shown in Figure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='4 q2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 q1 log(Vd) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 5: Contour plot of the closed-loop potential energy Vd(q) = Vm(q2) + Vf(Γ(q)) for the cart-pole system on log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As expected, the proposed controller stabilises the origin and the closed-loop energy Hd decreases monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0 1 2 3 4 5 time (s) 2 0 2 q1 q2 p1 p2 0 1 2 3 4 5 time (s) 0 1 2 Hd Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 6: Numerical simulation of cart-pole system in closed- loop with CbI scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Acrobot example The acrobot system, shown in Figure 7, consists of 2 links with an actuator supplying a input torque τ fixed between the base and second links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The base link has displacement of q2, measured from vertical, length ℓ2, mass m2, moment of inertia Jℓ1 and centre of mass ℓc2 from the base pivot point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The actuated link has displacement of q1 measured relative to the base link, length ℓ1, mass m1, moment of inertia Jℓ1 and centre of mass ℓc1 from the actuated pivot point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The control objective of this system is to stabilise the upright equilibrium position (q1, q2) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The acrobot system Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 7: The acrobot system attempts to balance in the vertical position by manipulating the torque generated by an actuator between the two links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' can be written as a pH system of the form (7) with M(q) = � c2 c2 + c3 cos q1 c2 + c3 cos q1 c1 + c2 + 2c3 cos q1 � V (q) = c4g cos q2 + c5g cos(q1 + q2) G = �1 0 � , (97) where c1 = m2ℓ2 c2 + m1ℓ2 2 + Jℓ2 c2 = m1ℓ2 c1 + Jℓ1 c3 = m1ℓ2ℓc1 c4 = m2ℓc2 + m1ℓ1 c5 = m1ℓc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (98) For the purposes of simulation, we take the values g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='8, c1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3333, c2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3333, c3 = 2, c4 = 3, c5 = 2 which were previously used in [5], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In this example, the total energy-shaping controller pro- posed in [5] is reconstructed as a CbI control scheme by solving the matching conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In that work, the closed-loop mass matrix was chosen to be the constant matrix M −1 d = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3385 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='9997 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='9997 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='9058 � (99) which will be recovered in subsequent computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The mass matrix of the acrobot system depends only on q1, the actuated coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The added inverse mass matrix is assumed to be a function of only q1 also, resulting in the structure M −1 a (q1) = � ma11(q1) m⊤ a21(q1) ma21(q1) ma22(q1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (100) As the system is underactuated degree 1 and the mass matrix is a function of only one variable, the kinetic energy match- ing equations can be reduced to an ODE as per Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The resulting ODE has the form (72) with qi = q1 and where ma11(q1) is a free function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In order to recover the result (99), this free function ma11(q1) is chosen as ma11(q1) = G⊤ � M −1 d − M −1(q1) � G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3385 − c1 + c2 + 2c3 cos q1 c1c2 − c2 3 cos2(q1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (101) The initial conditions ma12(0), ma22(0) are similarly defined as ma12(0) = G⊥ � M −1 d − M −1(0) � G = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='1313 ma12(0) = G⊥ � M −1 d − M −1(0) � G⊥⊤ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='2743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (102) The added inverse mass was evaluated numerically and the results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As the previously reported solution (99) is globally defined, it is unsurprising that the inverse added mas is also globally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As expected, the minimum eigenvalue of M −1 + M −1 a is constant also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 2 0 2 q1 2 0 2 4 6 8 ma11 ma12 ma22 2 0 2 q1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5 min[M-1+Ma 1] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 8: A solution for the added mass M −1 a for the acrobot was found to exist globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Solving the potential energy PDE (75) is difficult due to the open-loop potential energy being a function of both q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This dependence implies that Vm cannot be resolved directly using an ODE solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Considering the structure of V in (97), it is proposed that the closed-loop energy Vm has the structure Vm(q) = f1(q1) sin(q2) + f2(q1) cos(q2), (103) which has derivatives ∇q1Va =∂f1 ∂q1 sin(q2) + ∂f2 ∂q1 cos(q2) ∇q2Va =f1(q1) cos(q2) − f2(q1) sin(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (104) The open-loop potential energy has gradients ∇q1V = − c5g sin(q1) cos(q2) − c5g cos(q1) sin(q2) ∇q2V = − c4g sin(q2) − c5g sin(q1) cos(q2) − c5g cos(q1) sin(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (105) Substituting the expressions (104) and (105) into (59) and matching coefficients results in the system of equations � ∂f1 ∂q1 ∂f2 ∂q1 � = 1 s2(q1) × �c4gs1(q1) + c5gs1(q1) cos(q1) + s3(q1)f2(q1) c5gs1(q1) sin(q1) − s3(q1)f1(q1) � , (106) which can be evaluated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The values of f1, f2 at the origin should be chosen to ensure that the origin is an equilibrium point and Vm is positive in q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Considering the expressions (105), (106), the origin is an equilibrium for f1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The energy function (103) is locally positive with respect to q1 for f2(0) negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' For the purpose of simulation, f2(0) = −50 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The resulting function Vm is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 9: Solution for the added potential energy Vm(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Considering Figure 9, it is clear that Vm is not positive definite with respect to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Note, however, that q2 = 0 has been stabilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' To ensure stability of q1 = 0 also, the free term Vf(Γ(q)), defined in (74), is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The function Γ(·) defined by the integral (73), where K(q) is a free function chosen to ensure solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Noting that M −1, M −1 a are functions of q1 only, the parametrisation � β1(q1) β2(q1) � = G⊤ � M −1 a (q1) + M −1(q1) � M(q1) (107) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The free function is chosen as K(q1) = 1 β2(q1), resulting in Γ(q) = � � β1(q1) β2(q1) 1 � dq = � β1(q1) β2(q1) dq1 + q2, (108) which can be solved numerically from the initial condition Γ(02×1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The function Vf(·) was taken as Vf(Γ(q)) = 1 2κΓ(q)2 with κ = 250 for simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' A contour plot of the resulting closed-loop potential energy on a log scale is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Note that a minimum has been assigned to q = 02×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As a final control design stage, damping is injected via the new passive input/output pair with v = −5G⊤(M −1 a + M −1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' (109) The complete control signal is defined by the expression (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The acrobot system was simulated for 20 seconds from ini- tial conditions q(0) = (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content='5), p(0) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' The resulting 100 m 0 100 200 4 2 0 2 a2 44 2 0 2 q1-3 2 1 0 1 2 3 q2 3 2 1 0 1 2 3 q1 log(Vd) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 10: Contour plot of the closed-loop potential energy Vd(q) = Vm(q) + Vf(Γ(q)) on log scale for the acrobot system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' state evolution and closed-loop energy Hd is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As expected, the proposed controller stabilises the origin and the closed-loop energy Hd decreases monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 0 5 10 15 20 time (s) 1 0 1 2 q1 q2 0 5 10 15 20 time (s) 10 0 10 p1 p2 0 5 10 15 20 time (s) 0 20 40 Hd Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 11: Numerical simulation of acrobot system in closed- loop with CbI scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORKS In this work total energy shaping has been shown to have a CbI interpretation which results in alternate matching equations related to the added inverse mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' These equations were utilised to construct controllers for the cart-pole and acrobot systems, both of which have the property that the mass matrix depends on only one variable, using numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' While the proposed approach is effective, a number of technical aspects of this approach require further investi- gation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In particular: As detailed in Corollary 2, The kinetic energy matching equations can be posed as ODEs in the special case that the mass matrix depends on only one configuration variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' This property allows the matching equations to be evaluated numerical using ODE solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Further investigation into solving the matching equations in the case that the mass matrix is a function of multiple configuration variables is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In some cases it may be possible to decouple the dependence on each coordinate, recovering equivalent ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Alternatively, the numerical evaluation of the matching PDEs should be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' When evaluating the kinetic energy matching equations in (72), the term ma11(qi) is a free function that can be used to control the resulting added inverse mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' As seen in the cart-pole example of Section V-A, poor choice of this function results in the solution only being defined on a small domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Conversely, in the acrobot example of Section V-A this term was chosen to ensure a global solution to the matching equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' Choice of this function defines a nonlinear control problem that should be investigated to ensure desirable behaviour of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' In both examples of Section V the controllers were designed to stabilise the origin of the respective sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' While this was achieved and verified numerically, asymptotic stability was not established.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} +page_content=' 1000–1016, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE2T4oBgHgl3EQfOQZ5/content/2301.03746v1.pdf'} diff --git a/4dFJT4oBgHgl3EQfjyxF/content/tmp_files/2301.11576v1.pdf.txt b/4dFJT4oBgHgl3EQfjyxF/content/tmp_files/2301.11576v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b77fb7793781612dd45f07aaf1c1ce1c8a5c6e5 --- /dev/null +++ b/4dFJT4oBgHgl3EQfjyxF/content/tmp_files/2301.11576v1.pdf.txt @@ -0,0 +1,3435 @@ +arXiv:2301.11576v1 [math.PR] 27 Jan 2023 +EMPIRICAL PROCESS +SAMPLED ALONG A STATIONARY PROCESS +GUY COHEN AND JEAN-PIERRE CONZE +Abstract. Let (Xℓ)ℓ∈Zd be a real random field (r.f.) +indexed by Zd with common +probability distribution function F. Let (zk)∞ +k=0 be a sequence in Zd. The empirical +process obtained by sampling the random field along (zk) is �n−1 +k=0[1Xzk ≤s − F(s)]. +We give conditions on (zk) implying the Glivenko-Cantelli theorem for the empirical +process sampled along (zk) in different cases (independent, associated or weakly corre- +lated random variables). We consider also the functional central limit theorem when the +Xℓ’s are i.i.d. +These conditions are examined when (zk) is provided by an auxiliary stationary pro- +cess in the framework of “random ergodic theorems”. +Contents +Introduction +2 +1. +General results on the empirical process along a sub-sequence +3 +1.1. +Preliminaries +3 +1.2. +A Glivenko-Cantelli type theorem +10 +1.3. +A sufficient condition for a FCLT for the sampled empirical process +12 +2. +Local times for ergodic sums +15 +2.1. +Auxiliary general results +15 +2.2. +Non centered cocycles +22 +2.3. +Counterexamples +23 +3. +Examples +27 +3.1. +Random walks +27 +3.2. +Extensions of the r.w. case +32 +3.3. +Step functions over rotations +34 +Date: January 30, 2023. +2010 Mathematics Subject Classification. Primary: 60F05, 28D05, 22D40, 60G50; Secondary: 47B15, +37A25, 37A30. +Key words and phrases. Empirical process, sampling along a stationary process, local times, Glivenko- +Cantelli theorem, functional central limit theorem, random walks. +1 + +2 +GUY COHEN AND JEAN-PIERRE CONZE +4. +About limit theorems along ergodic sums +37 +4.1. +Glivenko-Cantelli theorem along ergodic sums +37 +4.2. +Discussion: universal sequences +39 +References +40 +Introduction +For a sequence (Xk) of real i.i.d. random variables with common probability distribution +function F, the empirical process is defined by �n−1 +k=0 [1Xk≤s − F(s)]. Recall two classical +results. +(A) the Glivenko-Cantelli theorem: +a.s. the sequence of empirical distribution functions Fn(s) := +1 +n +�n−1 +k=0 1Xk≤s converges +uniformly to F, i.e. sups |Fn(s) − F(s)| → 0; +(B) a functional central limit theorem (FCLT): if the r.v.s Xk have a common distribution +F over [0, 1], then +the process +1 +√n +�n−1 +k=0 [1Xk≤s − F(s)] converges weakly to a Brownian bridge in the space +of cadlag functions on [0, 1]. +In this paper we study the extension of these results when the process is sampled along a +subsequence, analogously to what is done for limit theorems in random scenery. +In the sequel, for d ≥ 1, (Xℓ)ℓ∈Zd will be a real random field (r.f.) indexed by Zd defined +on a probability space (Ω, F, P) with common probability distribution function F. The +expectation on (Ω, P) is denoted by E. We consider in particular the case of a r.f. of i.i.d. +r.v.’s or of stationary associated r.v.’s. +Let (zk)∞ +k=0 be a sequence in Zd. The process obtained by sampling the random field along +(zk) is Wn(s) := �n−1 +k=0[1Xzk≤s − F(s)]. +We will call Wn(s) “empirical process sampled along (zk)”, or simply “sampled empirical +process”. A general question is whether the above results (A), (B) extend to the sampled +empirical process Wn(s), in particular when (zk) is given by another stationary process +with values in Zd. +In Section 1, we give conditions on (zk) implying that (A) and (B) are still valid for +an empirical process sampled along (zk) in different cases: independent, associated or +weakly correlated random variables. The conditions are expressed in terms of the following +quantities associated to the sequence (zk) in Zd: local time, maximal local time and +number of self-intersections (up to time n) defined, for n ≥ 1, by +Nn(ℓ) := #{0 ≤ k ≤ n − 1 : zk = ℓ}, +Mn := max +ℓ +Nn(ℓ), Vn := #{0 ≤ j, k ≤ n − 1 : zj = zk}. +(1) + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +3 +They satisfy � +ℓ Nn(ℓ) = n and n ≤ Vn = � +ℓ N2 +n(ℓ) ≤ nMn ≤ n2. +In the other sections, (zk) is given by a stationary process (or equivalently by the sequence +(Skf(x))k≥1 of ergodic sums of a function f over a dynamical system). +The conditions found in Section 1 lead to study the local times, maximum number of visits, +number of self-intersections for the sequence (Skf(x)). General remarks are presented in +Section 2. Then in Section 3, we consider two families of examples: random walks and +some ergodic sums over a rotation. +The Glivenko-Cantelli theorem along ergodic sums (extension of (A)) is strongly related +to random ergodic theorems, in particular to results in [23] and [25]. This is discussed in +the last Section 4. +Finally let us mention the quenched FCLT for the 2-parameters process +Wn(s, t) := +[nt]−1 +� +k=0 +[1XZk(x)≤s − F(s)], (s, t) ∈ [0, 1]2. +When (Xℓ) is a r.f. of i.i.d. r.v.’s indexed by Z2 and when the sampling is provided by a +2-dimension centered random walk (Zk) with a moment of order 2, the weak convergence +for a.e. x toward a Kiefer-M¨uller process can be shown. This will be the content of a +forthcoming paper. +Acknowledgements. Part of this research was done during visits of the first author to +the IRMAR at the University of Rennes 1 and of the second author to the Center for +Advanced Studies in Mathematics at Ben Gurion University. The authors are grateful to +their hosts for their support. +1. General results on the empirical process along a sub-sequence +1.1. Preliminaries. In this subsection, results on the empirical process along a sub- +sequence are shown for independent variables, as well for some of them for wider classes +(associated, PDQ and weakly correlated random variables). We start by recalling some +notions and auxiliary results. +1) Associated variables +Definition (cf. [17]): A finite set of real random variables T = (T1, T2, . . . , Tn) is said to +be associated if Cov[f(T), g(T)] ≥ 0, for every coordinate-wise non-decreasing functions +f = f(x1, ..., xn) and g = g(x1, ..., xn) for which E[f(T)], E[g(T)], E[f(T) g(T)] exist. An +infinite set of random variables is associated if any finite subset of it is associated. +Association of random variables is preserved under taking subsets and forming unions of +independent sets (of associated random variables). In particular a family of independent +variables is associated. + +4 +GUY COHEN AND JEAN-PIERRE CONZE +Clearly, orthogonal associated random variables are independent. Examples of (non inde- +pendent) stationary associated processes with absolutely summable series of correlations +are provided by some Ising models. References to such examples of stationary Zd random +fields which satisfies the FKG inequalities and with absolutely summable correlations +can be found in Newman’s paper [26]. Notice that the FKG inequalities expresses the +association property of the r.v.’s. +2) PQD variables +Two r.v.’s X, Y are called (cf. [24]) positively quadrant dependent (PQD) if, +P(X > x, Y > y) ≥ P(X > x) P(Y > y), ∀x, y ∈ R. +The property is preserved by centering. Any pairwise associated r.v.’s are pairwise PQD. +Pairwise independent random variables are pairwise PQD associated. +Two random variables X and Y are PQD if and only if for every non-decreasing functions +f and g, Cov(f(X), g(Y )) ≥ 0 (whenever the covariance exists) ([17, Theorem 4.4]). +3) We will use the following results: +a) Maximal inequality of Newman and Wright [27, Inequality (12)]: +If (Wi) is a sequence of centered associated, square integrable random variables, it holds: +P( max +1≤j≤n | +j +� +i=1 +Wi| ≥ λ ∥ +n +� +i=1 +Wi∥2) ≤ 2P(| +n +� +i=1 +Wi| ≥ (λ − +√ +2) ∥ +n +� +i=1 +Wi∥2), ∀λ ≥ 0. +(2) +b) Hoeffding’s identity (see [2, Theorem 3.1]) +Let X, Y be random variables with finite second moments. For any absolutely continuous +functions f, g on R, such that E[f 2(X) + g2(Y )] < ∞, it holds +Cov(f(X), g(Y )) = +� ∞ +−∞ +� ∞ +−∞ +f ′(x)g′(y)[P(X > x, Y > y) − P(X > x)P(Y > y)]dxdy. +In particular, if X, Y are PDQ random variables, if |f ′|, |g′| ≤ M a.e., we have +|Cov(f(X), g(Y ))| ≤ M2Cov(X, Y ). +4) Uniformity in the analogues of Glivenko-Cantelli theorem will follow from the lemma: +Lemma 1.1. [7, Lemma, p 140] Let Fn, F be a family of right continuous distributions on +R. Assume that, for each point x in a dense countable set Q ⊂ R, we have Fn(x) → F(x). +Let J be the set of jumps of F and assume that Fn(x)−Fn(x−) → F(x)−F(x−) for every +x ∈ J. Then Fn(x) → F(x) uniformly in R. +A strong law of large numbers +First we state a law of large numbers for bounded r.v.’s valid under weak hypotheses. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +5 +Let (Uℓ)ℓ∈Zd be a r.f. indexed by Zd of square integrable r.v’s on a probability space +(Ω, F, P). Let (zk)k≥0 be a sequence in Zd, d ≥ 1, with numbers of self-intersections +Vn, n ≥ 1. The partial sums along (zk) are denoted by Sn := +n−1 +� +k=0 +Uzk. +By the Cauchy-Schwarz inequality, if � +ℓ supr |⟨Ur+ℓ, Ur⟩| < +∞, if holds for a finite +constant C0: +∥ +n−1 +� +i=0 +Uzi∥2 +2 = +� +ℓ +� +r +Nn(r + ℓ)Nn(r)⟨Ur+ℓ, Ur⟩ ≤ Vn +� +ℓ +sup +r |⟨Ur+ℓ, Ur⟩| = C0Vn. +(3) +In particular if the r.f. is stationary and the series of correlations is absolutely summable +(i.e., � +ℓ∈Zd |⟨X0, Xℓ⟩| < +∞), then the spectral density of the r.f. +exists and is the +continuous non-negative function ρ on Td with Fourier coefficients +� +Td e2πi⟨ℓ,t⟩ ρ(t) dt = +⟨X0, Xℓ⟩ and it holds: +∥Sn∥2 +2 = ∥ +n−1 +� +i=0 +Uzi∥2 +2 ≤ Vn +� +ℓ +|⟨Uℓ, U0⟩|. +(4) +Proposition 1.2. Suppose the r.v.’s Uℓ on (Ω, P) centered and uniformly bounded by the +same constant K, ∥Uℓ∥∞ ≤ K, ∀ℓ. Assume that (zk) is such that +Vn ≤ C1 +n2 +(log n)β , for constants C1, β. +(5) +1) Then, if β > 1 and +� +ℓ∈Zd +sup +r∈Zd |⟨Ur+ℓ, Ur⟩| < +∞, +2) or if β > ζ for some ζ ∈ [1, 2] and the r.f. (Uℓ) is stationary with +� +ℓ∈Zd +|⟨Uℓ, U0⟩|ζ < ∞, +the (strong) LLN holds: Sn(ω) +n +→ 0, for P-a.e ω. +Proof. 1) For convenience, if t is in R+, we define St as S[t]. From (3) it follows +� +(|Sn| +n )2 dP ≤ C0 +Vn +n2 ≤ C0C1 +1 +(log n)β . +Therefore, putting β = 1 + η and α = 1 − η/2 (which implies αβ > 1) we have +� +k +� +(|S2kα| +2kα )2 dP ≤ C0C1 +� +k +1 +(log 2kα)β = C′ � +k +1 +kαβ < +∞; +hence: +lim +k→+∞ +S2kα +2kα = 0, a.e. +For n ≥ 1, let kn be such that 2(kn)α ≤ n < 2(kn+1)α (that is: kn = [(log2 n)1/α]). +We put qn := 2(kn+1)α − 2(kn)α and pn = n − 2(kn)α ≤ qn. + +6 +GUY COHEN AND JEAN-PIERRE CONZE +For qn, the following estimate holds: qn = 2(kn)α(2(kn+1)α−(kn)α − 1) ∼ C′′ 2(kn)α +(kn)1−α. +Using the uniform boundedness of the r.v.’s, we can write: +|Sn +n − S2(kn)α +2(kn)α | = |S2(kn)α + �2(kn)α+pn +i=2(kn)α +Uzi +2(kn)α + pn +− S2(kn)α +2(kn)α | = |2(kn)α �2(kn)α+pn +i=2(kn)α +Uzi − pnS2(kn)α +2(kn)α(2(kn)α + pn) +| +≤ 2(kn)α �2(kn)α+pn +i=2(kn)α +|Uzi| + pn|S2(kn)α| +2(kn)α(2(kn)α + pn) +≤ 2(kn)α �2(kn)α+qn +i=2(kn)α |Uzi| + qn|S2(kn)α| +2(kn)α(2(kn)α) +≤ qnK2(kn)α + qn|S2(kn)α| +2(kn)α(2(kn)α) += +qn +2(kn)α (K + |S2(kn)α| +2(kn)α ). ≤ +C +2(kn)1−α(K + |S2(kn)α| +2(kn)α ) → 0. +2) We consider now the stationary case. Since ζ = 1 is special case of 1), we assume +ζ ∈]1, 2]. We put β = ζ + η, where η is > 0 in view of the hypothesis. +First, suppose that ζ = 2. Then under the hypothesis, the r.f. has a spectral measure νϕ +absolutely continuous with respect to the Lebesgue measure λ on the torus with a density +ρ ∈ L2(dt) given by the Fourier series ρ(t) = � +ℓ∈Zd⟨Uℓ, U0⟩ e2iπ⟨ℓ,t⟩. +Using the inequality λ{ρ > Mn} ≤ M−2 +n ∥ρ∥2 +2, we can write: +∥Sn∥2 +2 +n2 += 1 +n2 +� +Td | +n−1 +� +j=0 +e2πi⟨zj,t⟩|2 dνϕ(t) ≤ Mn +n2 +� +Td | +n−1 +� +j=0 +e2πi⟨zj,t⟩|2 dt + +� +ρ>Mn +ρ dt +≤ +Mn +Vn +n2 + (λ{ρ > Mn}) +1 +2∥ρ∥2 ≤ Mn +Vn +n2 + M−1 +n ∥ρ∥2 +2. +Taking Mn = (log n)1+ 1 +2 η, we obtain the bound +1 +n2 ∥ +n−1 +� +j=0 +Uzj∥2 +2 ≤ +C +(log n)1+ 1 +2 η +and then we finish the proof as in 1). +Now, suppose that +� +ℓ∈Zd +|⟨Uℓ, U0⟩|ζ < ∞ with 1 < ζ < 2. The spectral density ρ exists +and is in L2(λ), since +� +ℓ∈Zd +|⟨Uℓ, U0⟩|2 < ∞. Moreover it belongs to Lζ′(λ) where ζ, ζ′ are +conjugate exponents (see: [31], p. 102, or [21] Th. 31.22), and it satisfies: +∥ρ∥ζ′ ≤ ( +� +ℓ∈Zd +|⟨Uℓ, U0⟩|ζ)1/ζ. +H¨older’s inequality implies: +� +ρ>Mn ρ dt ≤ (λ{ρ > Mn})1/ζ∥ρ∥ζ′. As +λ{ρ > Mn} ≤ M−ζ′ +n +� +ρζ′ dt = M−ζ′ +n +∥ρ∥ζ′ +ζ′, + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +7 +it follows: +� +ρ>Mn +ρ dt ≤ M−ζ′/ζ +n +∥ρ∥1+ζ′/ζ +ζ′ +. +Therefore, we obtain +1 +n2 +� +Td | +n−1 +� +j=0 +e2πi⟨zj,t⟩|2 dνϕ(t) ≤ Mn +Vn +n2 + +� +ρ>Mn +ρ dt ≤ Mn +Vn +n2 + M−ζ′/ζ +n +∥ρ∥1+ζ′/ζ +ζ′ +. +Now we take Mn such that : Mn/(log n)β = M−ζ′/ζ +n +, i.e. Mn = (log n)β/ζ′. We get +1 +n2 ∥ +n−1 +� +j=0 +Uzj∥2 +2 ≤ +C +(log n)β(1−1/ζ′) = +C +(log n)β/ζ = +C +(log n)1+η/ζ with η > 0, +and the end of the proof is as above. +□ +Remarks 1.3. 1) Let us give an example of a non stationary r.f. (Uℓ) which satisfies +Condition 1) of the previous proposition. +We take (Uℓ = Vℓ Wℓ, ℓ ∈ Zd), where (Vℓ) and (Wℓ) are two r.f.’s independent from +each other, with (Vℓ) centered stationary and such that � +ℓ∈Zd |⟨Vℓ, V0⟩| < ∞, and (Wℓ) +satisfying supℓ,p |⟨Wℓ+p, Wℓ⟩| < ∞. +The r.f. (Wℓ) can be viewed as a (multiplicative) noise (which can be non stationary) +independent from the r.f. (Uℓ). Clearly the condition in 1) is satisfied. +2) For a stationary r.f. (Uℓ) with a bounded spectral density (but with a series of corre- +lations which may be not absolutely summable), then like in 1) the condition β > 1 is +sufficient for the conclusion of the theorem. +Now, we give a pointwise bound for the sampled sums, first for i.i.d. r.v.’s, then for a +stationary random field (Uℓ)ℓ∈Zd of associated r.v.’s. +Proposition 1.4. 1) Suppose that the r.v.’s Uℓ, ℓ ∈ Zd, are i.i.d., centered, uniformly +bounded by a constant K, ∥U0∥∞ ≤ K, and that E|U0|2 = 1. Then it holds +lim sup +n +|Sn| +√Vn (2 log log n) +1 +2 ≤ K, P-a.e. +(6) +If Vn = o(n2 (log log n)−1), then lim +n +Sn +n = 0, P-a.e. +2) Suppose the random field stationary and the r.v.’s Uℓ centered associated. +a) For all ε > 0, it holds, with σn := ∥ �n−1 +i=0 Uzi∥2: +lim sup +n +|Sn| +σn (log σn) +1 +2+ε ≤ 1, P-a.e. +(7) +b) If moreover the r.f. has a summable series of correlations, then, for all ε > 0, +|Sn| = O( +� +Vn (log n) +1 +2+ε), P-a.e. +(8) + +8 +GUY COHEN AND JEAN-PIERRE CONZE +If Vn ≤ Cn2 (log n)−(1+η)) for some constants C, η > 0, then lim +n +Sn +n = 0, P-a.e. +Proof. +A) Recall that σn = ∥ �n−1 +i=0 Uzi∥2. In case 2) we may assume ∥U0∥2 = 1, and +then in all cases σn ≤ n and by association σn ≥ n +1 +2. We have in case 1) σn = √Vn and +in case 2b), for associated variables, by (4): σn ≤ (� +p⟨Up, U0⟩) +1 +2 √Vn. By association, σn +is non-decreasing and tends to infinity. +For ρ > 1, let nk = nk(ρ) be a strictly increasing sequence of integers such that ρk < +σnk ≤ ρk+1. Since 1 ≤ σ2 +k+1 − σ2 +k ≤ 1 + 2k, such a sequence exists after a certain rank. By +the choice of (nk) we have +ρk < σnk ≤ ρk+1 < σnk+1 ≤ ρk+2. +(9) +Moreover, we have σnk+1/σnk ≤ ρ2 and, since σn ≤ n, nk ≥ ρk. +Let (λn) be a non +decreasing sequence of positive numbers such that +λnk > +√ +2, lim supk λnk+1/λnk ≤ 1, +� +k P +��� �nk−1 +i=0 Uzi +�� ≥ (λnk − +√ +2) ∥ �nk−1 +i=0 Uzi∥2 +� +< ∞. +(10) +By the previous inequalities and by Newman-Wright’s inequality (2) for the sequence of +centered associated random variables 1 (Wi) = (Uzi), we have +� +k +P( +max +0≤j≤nk−1 +�� +j +� +i=0 +Uzi +�� ≥ λnk ∥ +nk−1 +� +i=0 +Uzi∥2) ≤ 2 +� +k +P(| +nk−1 +� +j=0 +Uzj| ≥ (λnk− +√ +2)∥ +nk−1 +� +j=0 +Uzj∥2) < +∞. +By the Borel-Cantelli lemma, it follows: +lim sup +k +max0≤j≤nk+1−1 +�� �j +i=0 Uzi +�� +λnk+1 σnk+1 +≤ 1, P-a.e. +Hence P-a.e. +lim sup +k +max0≤j ρ2λiσi, for some i ∈ [nk, nk+1[, then max0≤j ρ2λnkσnk. +This shows: +{|Sn| > ρ2λnσn, i.o.} ⊂ { max +0≤j ρ2λnkσnk, i.o.}. +By this inclusion and (11) it follows: lim sup +n +| �n−1 +i=0 Uzi| +λnσn +≤ ρ2, P-a.e. +Taking ρ = ρn with ρn ↓ 1, we obtain +lim sup +n +| �n−1 +i=0 Uzi| +λnσn +≤ 1, P-a.e. +(12) +B) Choice of a sequence (λk) such that (10) is satisfied. +1as it is a subset of a set of associated r.v.’s + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +9 +Case 1) +Suppose that the Uk’s are i.i.d. r.v.’s. Recall that if (Wj, j ≥ 1) are centered bounded +sequence of independent random variables on (Ω, P), for any finite sum of the Wj’s it +holds by Hoeffding’s inequality for differences of martingale ([20]), for every ε > 0: +P(| +� +j +Wj| > ε) ≤ 2 exp(−1 +2 +ε2 +� +j ∥Wj∥2∞ +). +(13) +We apply it to the family (Nn(ℓ)Uℓ, ℓ ∈ Zd). From the hypotheses, we have: +� +ℓ +∥Nn(ℓ)Uℓ∥2 +∞ ≤ K2 � +ℓ +N2 +n(ℓ) = K2Vn. +With ε = (λ − +√ +2)√Vn, (13) implies: +P +��� � +ℓ +Nn(ℓ) Uℓ +�� ≥ (λ − +√ +2) +� +Vn +� +≤ 2 exp +� +− 1 +2(λ − +√ +2)2 Vn +K2Vn +� += 2 exp +� +− +1 +2K2(λ − +√ +2)2� +. +Let c, δ be such that c > δ > K2. In the previous inequality, we take +λ = λn = (2c log log n) +1 +2. +Let k(c, δ) be such that λnk > +√ +2 and c(1− +2 +√c log log nk ) ≥ δ > 1, for k ≥ k(c, δ). We have: +∞ +� +k=k(c,δ) +P +��� +nk−1 +� +i=1 +Uzi +�� ≥ (λnk − +√ +2) ∥ +nk−1 +� +i=1 +Uzi∥2 +� +≤ 2 +∞ +� +k=k(c,δ) +exp +� +− +1 +2K2(λnk − +√ +2)2� +≤ +2 +exp K2 +∞ +� +k=k(c,δ) +exp +� +− c +K2 log log nk)(1 − +2 +√c log log nk +) +� +≤ +2 +exp K2 +∞ +� +k=k(c,δ) +1 +(k log ρ) +δ +K2 < ∞. +Now we can apply (12). It follows: +lim sup +n +| �n−1 +i=0 Uzi| +� +2c(log log n)Vn +≤ 1, P-a.e. +Taking c = cn with cn ↓ K2, we get (6). +Case 2) +For general associated r.v.’s, we use simply that P +��� �n−1 +i=0 Uzi +�� ≥ λ ∥ �n−1 +i=0 Uzi∥2 +� +≤ +1 +λ2. +We take λn = (log σn) +1 +2 +ε, with ε > 0. By (9) we have λnk ≥ (k log ρ) +1 +2+ε, and therefore, +for a constant C1: � +k +1 +λ2nk ≤ C1 +� +k k−(1+2ε) < +∞; hence condition (10). + +10 +GUY COHEN AND JEAN-PIERRE CONZE +Moreover we have k log ρ ≤ log σnk ≤ log σnk+1 ≤ (k + 2) log ρ; hence +λnk+1 +λnk += +�log σnk+1 +log σnk) +� 1 +2+ε ≤ (1 + 2 +k) +1 +2+ε → 1. +By (12), this proves (7) in 2a) +For case 2b) we have σ2 +n ≤ Vn +� +p⟨Up, U0⟩ and σn ≤ n, hence it yields (8). The last +conclusion in case 2b) is now clear. Remark that it follows also from Proposition 1.2. +□ +1.2. A Glivenko-Cantelli type theorem. +Empirical process +Let us consider a random field of r.v.’s (Xℓ, ℓ ∈ Zd) on (Ω, F, P) with common distribution +function F. Let (zk) ⊂ Zd be a sequence with self-intersections (Vn). +Notation. We say that (Xℓ, ℓ ∈ Zd) satisfies a Glivenko-Cantelli theorem along a sequence +(zk) in Zd if +lim +n sup +s | 1 +n +n +� +k=1 +1(−∞,s](Xzk(ω)) − F(s)| = 0, for P-a.e.ω. +We show now a Glivenko-Cantelli theorem along a sequence (zk) under various hypotheses +on (zk) and on (Xℓ) (mixing, i.i.d., associated or PQD). +Le (Xℓ, ℓ ∈ Zd) be a r.f. Denoting by σ(Xℓ) the σ-algebra generated by the random +variable Xℓ, we define a coefficient of mixing by +γ(ℓ) := sup +r∈Zd +sup +A∈σ(Xr), B∈σ(Xℓ+r) +|P(A ∩ B) − P(A)P(B)|. +(14) +Observe that for every s, t ∈ R it holds: +sup +r∈Zd |⟨1Xr≤s − P(Xr ≤ s), 1Xℓ+r≤t − P(Xℓ+r ≤ t)⟩| ≤ γ(ℓ), ∀ℓ ∈ Zd. +(15) +By (15) and Proposition 1.2, we get: +Theorem 1.5. Let (zk) be such that Vn ≤ C1 +n2 +(log n)β , for constants C1 > 0, β. +If � +ℓ∈Zd γ(ℓ) < +∞ and β > 1, +or if the r.f. is stationary and � +ℓ∈Zd γ(ℓ)ζ < +∞, for some ζ ∈ [1, 2] and β > ζ, +then (Xℓ, ℓ ∈ Zd) satisfies a Glivenko-Cantelli theorem along (zk). +Using Proposition 1.4, we consider now the i.i.d. and associated cases. +Theorem 1.6. a) If (Xℓ)ℓ∈Zd is a r.f. of i.i.d. r.v.’s, then under the condition Vn = +o(n2 (log log n)−1) it satisfies a Glivenko-Cantelli theorem along (zk). +b) If (Xℓ)ℓ∈Zd is a strictly stationary r.f. of associated r.v.’s such that � +ℓ⟨Xℓ, X0⟩ con- +verges, then, under the condition Vn = O(n2 log−(1+η) n) for some η > 0, for a.e. ω, we + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +11 +have for each continuity point s of F: +lim +n +1 +n +n−1 +� +k=0 +1(−∞,s](Xzk(ω)) = F(s). +(16) +If F is continuous, the convergence is uniform in s. +Proof. +a) Denote by Fn(s)(ω) the means +1 +n +�n−1 +k=0 1(−∞,s](Xzk(ω)). Let Q be a dense +countable set of continuity points of F. +For every s ∈ Q, by the assumption on Vn and Proposition 1.4, there is a null set N(s) +such that, for a sequence εn tending to 0, for every ω ̸∈ N(s), +|Fn(s)(ω) − F(s)| ≤ εn(Vn log log n)− 1 +2| +n−1 +� +k=0 +� +1(−∞,s](Xzk) − F(s) +� +| → 0. +Then Fn(s)(ω) → F(s) for every ω outside the null set N := ∪s∈QN(s) and for s ∈ Q. +Similarly by Proposition 1.4, for every s in the set J of jumps of F, we have Fn(s)(ω) − +Fn(s−)(ω) → F(s) − F(s−) a.e. As J is countable, this convergence holds for every s ∈ J +and ω ̸∈ ˜N, where ˜N is a null set. +Outside the null set N ∪ ˜N, Lemma 1.1 applied with Q and J implies the result. +b) We consider now the case of a strictly stationary random field of associated r.v.’s. Let +s be a continuity point of the common distribution F. For every ǫ > 0 there exists δ > 0, +such that F(s + δ) − F(s − δ) ≤ ǫ. As in [30], for δ > 0 and s, define the approximated +step function hδ,s by hδ,s(x) = 0, if x ≤ s − δ and hδ,s(x) = 1 + x−s +δ +if s − δ ≤ x ≤ s, +otherwise, hδ,s(x) = 1. It is a non decreasing continuous function with h′ +δ,s(x) = 1/δ for +s−δ < x < s. It follows from the above Hoeffding’s identity applied to this approximated +step function (see [2]): +Cov(hδ,s(Xℓ), hδ,s(X0)) ≤ δ−2⟨Xℓ, X0⟩, +Cov(hδ,s+δ(Xℓ), hδ,s+δ(X0)) ≤ δ−2⟨Xℓ, X0⟩. +By association and non decreasing, +� +hδ,s(Xℓ) +� +as well as +� +hδ,s+δ(Xℓ) +� +are stationary r.f.s +of associated r.v.’s, and we may apply Proposition 1.4 to their centered versions (also +associated). The condition simply reads, for τ = s, s + δ: +� +ℓ +Cov(hδ,τ(Xℓ), hδ,τ(X0)) ≤ δ−2 � +ℓ +⟨Xℓ, X0⟩ < ∞. +We put Sn = �n−1 +k=0 hδ,s(Xzk) and Sn = �n−1 +k=0 hδ,s+δ(Xzk). +By hδ,s+δ(x) ≤ 1{x>s} ≤ +hδ,s(x), it holds Sn ≤ �n−1 +k=0 1(s,∞)(Xzk) ≤ Sn. Hence by Proposition 1.4, we have almost +everywhere 1 +nSn → E[hδ,s+δ(X0)] and 1 +nSn → E[hδ,s(X0)]. Since +E[hδ,s(X0)] ≤ F(s) − F(s − δ) + 1 − F(s) ≤ ǫ + 1 − F(s), +E[hδ,s+δ(X0)] ≥ 1 − F(s + δ) = 1 − F(s) − (F(s + δ) − F(s)) ≥ 1 − F(s) − ǫ, + +12 +GUY COHEN AND JEAN-PIERRE CONZE +we conclude +1 − F(s) − ǫ ≤ lim inf +n +1 +n +n−1 +� +k=0 +1(s,∞)(Xzk) ≤ lim sup +n +1 +n +n−1 +� +k=0 +1(s,∞)(Xzk) ≤ 1 − F(s) + ǫ. +Subtracting the 1’s and taking ǫ → 0, we get (16). +□ +PQD variables. +The result shown for associated variables can be extended to the class of PDQ variables, +but with a stronger condition on the local times of the sequence (zk). +Proposition 1.7. Let (Uℓ) be a centered stationary random field of pairwise PQD r.v.’s +such that � +ℓ⟨Uℓ, U0⟩ converges. Let (zk) be a sequence of points with maximal local times +sequence (Mn). If � +n≥1 +Mn +n2 < +∞, then 1 +n(Uz0 + · · · + Uzn−1) converges a.e. to 0. +Proof. We apply the following result of [4]: let (Yj : j ≥ 1) be a sequence of pairwise +centered PQD r.v.’s. +with finite variance. If � +j≥1 j−2Cov(Yj, �j +i=1 Yi) converges and +supj E|Yj| < ∞, then n−1 �n +i=1 Yi → 0 a.e. +Taking for Yj the (still) pairwise PQD r.v.’s Uzj, we get the result, since Cov(Uzj, Uz1 + +· · · + Uzj) ≤ Mj +� +ℓ⟨U0, Uℓ⟩. +□ +Now, we consider the empirical distribution. +Theorem 1.8. Let (Xℓ) be a centered strictly stationary random field of pairwise PQD +r.v.’s with distribution function F such that � +ℓ⟨Xℓ, X0⟩ converges. Let (zk) be a sequence +of points with maximal local times sequence (Mn). If � +n≥1 +Mn +n2 < +∞, then for each +continuity point s of F, we have for a.e. ω: limn +1 +n +�n−1 +k=0 1(−∞,s](Xzk(ω)) = F(s). +In particular, if F is continuous, the above convergence is uniform over s. +Proof. The r.f.s hδ,s(Xℓ) and hδ,s+δ(Xℓ) are still stationary pairwise PQD. The proof is +analogous to the proof of Theorem 1.6. For the last statement, we use Lemma 1.1. +□ +Remark. If Mn = O(n (log n)−(1+η)), then Vn = O(n2 (log n)−(1+η)). +If Vn ≤ C +n2 +(log n)β , with β > 2, then +� +n≥1 +Mn +n2 < +∞. +As shown in Section 3, � +n≥1 +Mn +n2 converges when the sampling is done along random +walks, but diverges in some examples of sampling along “deterministic” random walks. +1.3. A sufficient condition for a FCLT for the sampled empirical process. +After a Glivenko-Cantelli theorem for sampled empirical processes, we consider now the +Functional Central Limit Theorem (FCLT). Let (zk) be in Zd, d ≥ 1, with the associated +quantities Nn(ℓ), Mn and Vn defined by (1). +Before restricting to a r.f. of i.i.d. r.v.’s, first we examine the variance in the more general +situation where the series of correlations is absolutely summable. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +13 +Kernel associated to a sequence (zk) and variance. +Let Kn be the kernel (which is a real even function on Td depending on n ≥ 0) defined +by the equivalent formulas: +Kn(t) += +| +n−1 +� +k=0 +e2πi⟨zk,t⟩|2 = n + 2 +n−1 +� +k=1 +n−k−1 +� +j=0 +cos(2π⟨zk+j − zj, t⟩) = | +� +ℓ∈Zd +Nn(ℓ) e2πi⟨ℓ,t⟩|2 += +n + 2 +� +ℓ +�n−1 +� +k=1 +n−k−1 +� +j=0 +1zk+j−zj=ℓ +� +cos(2π⟨ℓ, t⟩). +(17) +If (Xℓ, ℓ ∈ Zd) is a stationary r.f. such that � +ℓ∈Zd |⟨Xℓ, X0⟩| < +∞, the spectral density +ρ is continuous and we have: +� +| +n−1 +� +k=0 +Xzk|2dP = +� +Td Kn(t) ρ(t) dt ≤ ∥ρ∥∞Vn ≤ ( +� +ℓ∈Zd +|⟨Xℓ, X0⟩|)Vn. +One can ask if there is an asymptotic variance, i.e., a limit for the normalised quantity +V −1 +n +� +| +n−1 +� +k=0 +Xzk|2dP which is bounded if the series of correlations is absolutely summable. +The existence of asymptotic variance is shown in [11] in the case of summation along a +random walk. We will come back to the question of its positivity for transient random +walks in Subsection 3.1. +Functional Central limit Theorem in the i.i.d. case +The following result gives a sufficient condition for a Functional Central limit Theorem +(FCLT) along a sequence (zk) in the i.i.d. case. +The standard Brownian bridge process W 0(s) is the centered Gaussian process W 0(s) := +W(s)−sW(1) in C(0, 1), where W(s) is the Wiener process. It has the properties W 0(0) = +W 0(1) = 0 and E[W 0(s1)W 0(s2)] = s1 ∧ s2 − s1s2. +Let (Xk)k∈Zd be i.i.d. random variables with common probability distribution F in [0, 1]. +We put W 0 +F = W 0 ◦ F. Let Yn(s) be the random element in D[0, 1] defined by +Yn(s) = +1 +√Vn +n−1 +� +k=0 +[1Xzk≤s − F(s)] = +1 +√Vn +� +ℓ∈Zd +Nn(ℓ) [1Xℓ≤s − F(s)]. +Theorem 1.9. Yn(s) →D[0,1] W 0 +F(s), if (zk) satisfies the condition +lim +n +M2 +n +Vn += 0, +(18) +Proof. +The result follows from the Cram´er-Wold device, if we prove convergence of the +finite dimensional distributions and tightness. The variance is +E[Yn(s)]2 = 1 +Vn +� +ℓ +N2 +n(ℓ) E[1Xℓ≤s − F(s)]2 = σ2(s) = F(s)(1 − F(s)). +(19) + +14 +GUY COHEN AND JEAN-PIERRE CONZE +1) Finite dimensional distributions. The convergence follows from Lindeberg’s theorem +for triangular arrays of independent random variables as in [3, thm 7.2]. The Lindeberg’s +condition for the triangular array of independent r.v.’s +�Nn(ℓ)[1Xℓ≤s − F(s)] +√Vn +� +ℓ,n follows +from +1 +Vn +� +ℓ +� +{Nn(ℓ)|1Xℓ≤s−F (s)|≥ ε√Vn} +N2 +n(ℓ) |1Xℓ≤s − F(s)|2dP +≤ +1 +Vn +� +ℓ +N2 +n(ℓ) +� +{supℓ Nn(ℓ)|1X0≤s−F (s)|≥ε√Vn} +|1X0≤s − F(s)|2dP → 0, +for every ε > 0, since Vn = � +ℓ N2 +n(ℓ) and supℓ Nn(ℓ) +√Vn +→ 0, by assumption. +For the correlation of the process taken at s1 and s2, it holds by independence: +E[Yn(s1)Yn(s2)] += +1 +Vn +� +ℓ1,ℓ2 +Nn(ℓ1)Nn(ℓ2)E[(1Xℓ1≤s1 − F(s1))(1Xℓ2≤s2 − F(s2))] += +1 +Vn +� +ℓ +N2 +n(ℓ)(F(s1 ∧ s2) − F(s1)F(s2)) = F(s1 ∧ s2) − F(s1)F(s2). +This proves the convergence in distribution: Yn(s) → W 0 +F(s) for every s. +Let us show now the convergence of the finite dimensional distributions. Starting with +the asymptotic distribution of aYn(s1) + bYn(s2), by the above computation, we have +E[(aYn(s1) + bYn(s2))2] = +a2F(s1)(1 − F(s1)) + b2F(s2)(1 − F(s2)) + 2ab(F(s1 ∧ s2) − F(s1)F(s2)). +(20) +As above, it is easily seen that Lindeberg’s condition is satisfied for the triangular array +defined by aYn(s1)+bYn(s2). It means that the asymptotic distribution of aYn(s1)+bYn(s2) +is centered Gaussian with variance as computed above. +Note that E[(aW 0(s1) + bW 0(s2))2] is also given by (20) above. +Similarly, for every s1 ≤ · · · ≤ sr, it holds +(Yn(s1), · · · , Yn(sr)) →dist (W 0 +F(s1), . . . , W 0 +F(sr)). +Tightness. First we suppose F continuous. Following the method of [3], it is enough to +show that for s ≤ t ≤ r, uniformly in n, +E[(Yn(t) − Yn(s))2(Yn(r) − Yn(t))2] ≤ C(F(r) − F(s))2. +Putting F(u, v) := F(v) − F(u), f(ℓ, u, v) := 1u 0: +E[(Yn(t) − Yn(s))2(Yn(r) − Yn(t))2] ≤ C(F(r) − F(s))2, ∀n. +Hence by [3, Theorem 15.6], for non decreasing continuous F, the sequence of processes +(Yn(s)) is tight in D(0, 1). This proves that, if F is continuous, then Yn →D(0,1) (W 0 ◦ F). +Now, for a general F a classical method is to use a generalized inverse. Let us describe it +briefly. We consider first the uniform empirical process. Let (ζk) be uniformly distributed +i.i.d. r.v.’s. Denote the empirical process along (zk) with respect to (ζk) by Un(s). By +applying what we have just proved for a continuous distribution, Un(s) →D(0,1) W 0(s). +Now let F −1(t) := inf{s : t ≤ F(s)}. We get P(F −1(ζ0) ≤ s) = P(X0 ≤ s) = F(s), so +Yn(s) =dist. Un(F(s)). We may then proceed as in Billingsley ([3, Theorem 5.1]) to deduce +the FCLT for Yn(s) with W 0(F(s)) as limit. +□ +2. Local times for ergodic sums +In the previous section about limit theorems for the empirical process sampled along (zk), +we have found sufficient conditions on the quantities Vn and Mn associated to (zk). When +(zk) is given by a “cocycle”, zk = Skf(x), one can ask if these conditions are satisfied. +We start with some general facts and construct counterexamples for which condition (18) +is not satisfied. In the next section, we will discuss two very different examples of cocycles: +first the case of random walks, then “stationary random walks” generated by a rotation. +2.1. Auxiliary general results. +First we introduce some notation and make general remarks. + +16 +GUY COHEN AND JEAN-PIERRE CONZE +Notation 2.1. Let (X, B, µ) be a probability space and T a measure preserving trans- +formation acting on X such that the dynamical system (X, B, µ, T) is ergodic. +Let f be a measurable function on X with values in Zd, d ≥ 1. Its ergodic sums generated +by the iteration of T, denoted by fk (or Skf), are +fk(x) := +k−1 +� +j=0 +f(T jx), k ≥ 1, f0(x) = 0. +The sequence (fk(x), k ≥ 1) can be viewed as a “stationary random walk” defined on +(X, B, µ). It will be called a “cocycle” and denoted by (µ, T, f) or simply (T, f). +For x ∈ X, we put (cf. (1)) N0(x, ℓ) = 0 and, for n ≥ 1, +Nn(T, f, x, ℓ) +:= +#{1 ≤ k ≤ n : fk(x) = ℓ}, ℓ ∈ Zd, +Mn(T, f, x) +:= +max +ℓ∈Zd Nn(T, f, x, ℓ), +Vn(T, f, x) +:= +#{1 ≤ j, k ≤ n : fj(x) = fk(x)} = +� +ℓ∈Zd +N2 +n(x, ℓ). +Most of the time, we will omit T and f in the notation and write simply Nn(x, ℓ), Mn(x), +Vn(x). We have � +ℓ Nn(x, ℓ) = n and n ≤ Vn(x) ≤ n Mn(x). +A question is to know if the following conditions hold for a.e. x: +Vn(x) = o(n2 (log log n)−1) or Vn(x) ≤ C1 +n2 +(log n)β , with β > 1, +(21) +lim +n +M2 +n(x) +Vn(x) = 0. +(22) +For a random walk this is related to a question studied in [16] and later in [15]: How +many times does the walk revisit the most frequently visited site in the first n steps? +Cylinder map. +We denote by ˜Tf the map (sometimes called cylinder map) (x, ℓ) → +(Tx, ℓ + f(x)) acting on X × Zd, endowed with the infinite invariant measure ˜µ defined +as the product of µ by the counting measure on Zd. +For ϕ : X × Zd → R the ergodic sums for the cylinder map are +˜Snϕ(x, ℓ) := +n−1 +� +k=0 +ϕ( ˜T k +f (x, ℓ)) = +n−1 +� +k=0 +ϕ(T kx, ℓ + fk(x)). +With ϕ0 := 1X×{0}, it holds +˜Snϕ0(x, −ℓ) = +n−1 +� +k=0 +1X×{0}(T kx, −ℓ + fk(x)) = #{0 ≤ k ≤ n − 1 : fk(x) = ℓ}. +Therefore, ˜Snϕ0(x, −ℓ) = Nn(ℓ)(x). + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +17 +Recurrence/transience. It can be shown that a cocycle (µ, T, f) (over an ergodic dynamical +system) is either recurrent or transient. For f with values in Zd, it means that either +Skf(x) = 0 infinitely often for a.e. x, or Skf(x) = 0 finitely often for a.e. x. In the latter +case, we have limk |Skf(x)| = +∞, µ-a.e. +Let Rn(x) = {ℓ ∈ Zd : fk(x) = ℓ for some k ≤ n} be the “range” of the cocycle, i.e., the +set of points visited by fk(x) up to time n. +In [14] the following result is shown (for the general case of a cocycle with values in a +countable group): let G be a countable group and f : X → G. If the cocycle (T, f) is +recurrent, then Card(Rn(x)) = o(n) for a.e. x. If it is transient, there exists c > 0 such +that Card(Rn(x)) ∼ c n for a.e. x. +Using the lemma below, this implies for a.e. x: +lim inf +n +Vn(x) +n +> 0 in the transient case, Vn(x) +n +→ +∞ in the recurrent case. +(23) +To show (23) we use the following inequality valid for a general sequence (zk): +Lemma 2.2. If A is a non empty subset in Zd, we have: +Vn ≥ +��n−1 +k=0 1zk ∈A +�2 +Card(A) +. +(24) +Proof. Cauchy-Schwarz inequality implies: +n−1 +� +k=0 +1zk ∈A = +� +ℓ∈A +n−1 +� +k=0 +1zk =ℓ ≤ ( +� +ℓ∈A +( +n−1 +� +k=0 +1zk =ℓ)2) +1 +2 (Card(A)) +1 +2 ≤ V +1 +2 +n (Card(A)) +1 +2. +□ +If zk = Skf(x), this show (23). Indeed by taking A = Rn(x) we get +Vn(x) ≥ +n2 +Card(Rn(x)). +(25) +Lemma 2.3. The following formulas hold for quantities defined in 2.1. +Vn(x) += +n + 2 +n−1 +� +k=1 +n−k−1 +� +j=0 +(1fk(T jx)=0), +(26) += +2[Nn−1(Tx, 0) + Nn−2(T 2x, 0) + ... + N1(T n−1x, 0)] + n, n ≥ 2, +(27) +Mn(x) += +max[Nn(x, 0), 1 + +max +1≤k≤n−1 Nn−k(T kx, 0)] ≤ 1 + +max +0≤k≤n−1 Nn(T kx, 0), +(28) += +Mn−1(Tx) + 1ℓ(n−1,Tx)=0 ≤ Mn−1(Tx) + 1. +(29) +Proof. a) From fk(x) = f(x) + fk−1(Tx), k ≥ 1, it follows +Nn(x, ℓ) = Nn−1(Tx, ℓ − f(x)) + 1f(x)=ℓ, n ≥ 1. +(30) + +18 +GUY COHEN AND JEAN-PIERRE CONZE +Therefore we have: +� +ℓ∈Zd +N2 +n(x, ℓ) = +� +ℓ∈Zd +[Nn−1(Tx, ℓ − f(x)) + 1f(x)=ℓ]2 = +� +ℓ∈Zd +[Nn−1(Tx, ℓ) + 1ℓ=0]2 += +� +ℓ̸=0 +[Nn−1(Tx, ℓ)]2 + [Nn−1(Tx, 0) + 1]2 = +� +ℓ +[Nn−1(Tx, ℓ)]2 + 2Nn−1(Tx, 0) + 1. +Hence the relation +Vn(x) = Vn−1(Tx) + 2Nn−1(Tx, 0) + 1. +(31) +We have V1(x) = 1 and by the previous relation we get by induction (26) and (27). +b) For x ∈ X, let ℓ(n, x) (a most visited site by Sk(x) up to time n) be defined by +ℓ(n, x) +:= +0, if Nn(x, 0) ≥ Nn(x, ℓ), for all ℓ ̸= 0, +else +:= +ℓ1, if ℓ1 is such that Mn(x) = Nn(x, ℓ1) > Nn(x, 0). +Let pn(x) ∈ [1, n] be the first visit of Sk(x) to ℓ(n, x) for k = 1, ..., n. +By definition +Mn(x) = Nn(x, ℓ(n, x)). +We have Mn(x) = Nn(x, 0) if ℓ(n, x) = 0, else Mn(x) = Nn−pn(x)(T pn(x)x, 0) + 1, by the +cocycle relation Spn(x)+k(x) − Spn(x)(x) = Sk(T pn(x)x). This implies: +Mn(x) ≤ max[Nn(x, 0), Nn−pn(x)(T pn(x)x, 0) + 1] ≤ max[Nn(x, 0), max +1≤k≤n Nn−k(T kx, 0) + 1]. +It follows (noticing that N0(x, 0) = 0): +Mn(x) ≤ 1 + +max +0≤k≤n−1 Nn−k(T kx, 0) ≤ 1 + +max +0≤k≤n−1 Nn(T kx, 0). +(32) +This shows (28). +c) Observe also the following relation: by (30) we have: +Mn(x) += +sup +ℓ +[Nn−1(Tx, ℓ − f(x)) + 1ℓ−f(x)=0] = sup +ℓ +[Nn−1(Tx, ℓ) + 1ℓ=0] += +max [sup +ℓ̸=0 +Nn−1(Tx, ℓ), Nn−1(Tx, 0) + 1]. +If ℓ(n − 1, Tx) = 0, then Nn−1(Tx, 0) ≥ supℓ̸=0 Nn−1(Tx, ℓ). If ℓ(n − 1, Tx) ̸= 0, then +Nn−1(Tx, 0) < supℓ̸=0 Nn−1(Tx, ℓ). This shows (29). +Remark 2.4. By (28), if Kn is a uniform bound over x of Nn(x, 0), then Mn(x) ≤ Kn. +Likewise, if Nn(x, 0) ≤ Kn, for a.e. x, then Mn(x) ≤ Kn, for a.e. x. +Now we show that the set of x ∈ X such that limn +M2 +n(x) +Vn(x) = 0 has measure 0 or 1. +Lemma 2.5. It holds: lim +n [M2 +n(x) +Vn(x) − M2 +n(Tx) +Vn(Tx) ] = 0. +If T is ergodic, there is a constant γ ∈ [0, 1] such that lim sup +n +M2 +n(x) +Vn(x) = γ for a.e. x. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +19 +Proof. We use (31) and (29). Putting ε = 1ℓ(n−1,Tx)=0, we have: +|M2 +n(x) +Vn(x) − M2 +n−1(Tx) +Vn−1(Tx) | = |M2 +n−1(Tx) + ε(2Mn−1(Tx) + 1) +Vn−1(Tx) + 2Nn−1(Tx, 0) + 1 − M2 +n−1(Tx) +Vn−1(Tx) | += |ε(2Mn−1(Tx) + 1) +Vn(x) +− (2Nn−1(Tx, 0) + 1) +Vn(x) +M2 +n−1(Tx) +Vn−1(Tx) | +≤ 2Mn−1(Tx) + 1 +Vn(x) ++ 2Nn−1(Tx, 0) + 1 +Vn(x) +≤ 4Mn−1(Tx) +Vn(x) ++ +2 +Vn(x) ≤ +4 +√n + +2 +Vn(x). +For the last inequality we use that either Mn(x) ≥ √n, hence Mn(x) +Vn(x) ≤ +1 +Mn(x) ≤ +1 +√n, or +Mn(x) < √n, hence Mn(x) +Vn(x) ≤ +√n +n = +1 +√n. Therefore, +|M2 +n(x) +Vn(x) − M2 +n−1(Tx) +Vn−1(Tx) | → 0. +(33) +Observe now that +Mn(x) = Mn−1(x) + εn, where εn = 0 or = 1, +and εn = 1 if and only if there is ℓn such that +Mn−1(x) = #{1 ≤ k ≤ n − 1 : fk(x) = ℓn} and fn(x) = ℓn. +We have +M2 +n(x) = M2 +n−1(x) + cn, with cn = εn(1 + 2Mn−1(x)) +and Nn(x, ℓ) = Nn−1(x, ℓ) + ε′ +n(ℓ), with ε′ +n(ℓ) = 1fn(x)=ℓ and � +ℓ∈Zd ε′ +n(ℓ) = 1. +Therefore, +Vn(x) = +� +ℓ∈Zd +(Nn−1(x, ℓ) + ε′ +n(ℓ))2 = Vn−1(x) + 2 +� +ℓ∈Zd +ε′ +n(ℓ)Nn(x, ℓ)) + 1, +0 ≤ Vn(x) = Vn−1(x) + dn, with dn ≤ 2Mn(x) + 1. +|M2 +n(x) +Vn(x) − M2 +n−1(x) +Vn−1(x) | = |M2 +n−1(x) + cn +Vn−1(x) + dn +− M2 +n−1(x) +Vn−1(x) | ≤ +cn +Vn(x) + +dn +Vn(x) +M2 +n−1(x) +Vn−1(x) +≤ εn(1 + 2Mn−1(x) +Vn(x) ++ 2Mn(x) + 1 +Vn(x) +M2 +n−1(x) +Vn−1(x) ≤ +2 +Vn(x) + 4Mn(x) +Vn(x) +≤ +2 +Vn(x) + 4 +√n → 0. +From (33) and the convergence above, it follows limn [ M2 +n(x) +Vn(x) − M2 +n(Tx) +Vn(Tx) ] = 0. By ergodicity +of T, this shows the lemma +□ +Case of a coboundary +The case when f is coboundary degenerates. Indeed, the following holds: +Proposition 2.6. If f is a coboundary we have: +a) lim infn +Mn(x) +n +> 0, for a.e. x; +b) there is a constant β > 0 such that +1 +n2Vn(x) → β, for a.e. x; +c) for a.e. x, lim infn +M2 +n(x) +Vn(x) > 0. + +20 +GUY COHEN AND JEAN-PIERRE CONZE +Proof. Suppose that f is coboundary, f = TΦ − Φ. Since f has values in Zd and T is +ergodic, for all component Φj of Φ, e2πiΦj is a constant. It follows that Φ has also its +values in Zd up to an additive constant and we can assume that Φ has values in Zd. +a) We have lim infn +Mn(x) +n +≥ lim infn +1 +nNn(x, 0) > 0, for a.e. x. The positivity results the +following simple argument: +For R ≥ 1, let AR denote the set ∪ℓ:∥ℓ∥≤R(Φ = ℓ). Since, for each ℓ, by Birkhoff’s theorem, +limn +1 +n +� +0≤k≤n−1 1Φ(T kx)=ℓ = µ(Φ = ℓ), it holds +1 +nNn(x, 0) ≥ +� +ℓ∈AR +1(Φ=ℓ)(x) 1 +n +n−1 +� +k=0 +1(Φ=ℓ)(T kx) → +� +ℓ∈AR +1(Φ=ℓ)(x) µ(Φ = ℓ). +Therefore, for every R ≥ 1, lim infn +Mn(x) +n +≥ lim infn +Nn(x,0) +n +≥ � +ℓ∈AR 1(Φ=ℓ)(x) µ(Φ = ℓ), +and the limit when R → ∞ at right is > 0, for a.e. x. +b) For Vn, we have: +Vn(f, x) += +� +ℓ∈Zd +N2 +n(x, ℓ) = +� +ℓ∈Zd +#{0 ≤ k ≤ n − 1 : Φ(T kx) − Φ(x) = ℓ}2 += +� +ℓ∈Zd +#{0 ≤ k ≤ n − 1 : Φ(T kx) = ℓ}2 = +� +ℓ∈Zd +� +� +0≤k≤n−1 +1Φ(T kx)=ℓ +�2, +hence: 1 +n2 +� +ℓ∈AR +� +� +0≤k≤n−1 +1Φ(T kx)=ℓ +�2 = +� +ℓ∈AR +�1 +n +� +0≤k≤n−1 +1Φ(T kx)=ℓ +�2 → +� +ℓ∈AR +(µ(Φ = ℓ))2. +This implies, for every R ≥ 1, +lim inf +n +1 +n2 +� +ℓ∈Zd +� +� +0≤k≤n−1 +1Φ(T kx)=ℓ +�2 ≥ lim +n +1 +n2 +� +ℓ∈AR +� +� +0≤k≤n−1 +1Φ(T kx)=ℓ +�2 = +� +ℓ∈AR +(µ(Φ = ℓ))2. +It follows: lim inf +n +1 +n2 +� +ℓ∈Zd +� +� +0≤k≤n−1 +1Φ(T kx)=ℓ +�2 ≥ +� +ℓ∈Zd +µ(Φ = ℓ)2. For the complementary +of AR, it holds: +� +ℓ:∥ℓ∥>R +� � +0≤kR +1Φ(T jx)=ℓ 1Φ(T kx)=ℓ +≤ +� +0≤j,kR +1Φ(T jx)=ℓ) ( +� +ℓ:∥ℓ∥>R +1Φ(T kx)=ℓ) ≤ +� +0≤j,k 0. +c) Follows from a) and b). +□ +Proposition 2.7. There is a constant β ≥ 0 such that, for a.e. x, limn +Vn(x) +n2 += β. We +have β > 0 if and only if the cocycle (T, f) is a coboundary. +Proof. The case of a coboundary follows from Proposition 2.6. +Suppose now that the cocycle is not a coboundary. From (26), we can write +Vn(x) +n2 += 1 +n + 2 +n +n−1 +� +k=1 +1 +n +n−k−1 +� +j=0 +(1fk(T jx)=0) +≤ 1 +n + 2 +n +n−1 +� +k=1 +1 +n +n−1 +� +j=0 +(1fk(T jx)=0) = 1 +n + 2 +n +n−1 +� +j=1 +Nn(T jx, 0) +n +. +We will show that 1 +n +n−1 +� +j=0 +Nn(T jx, 0) +n +tend to 0 a.e. +By the ergodic theorem of Dunford and Schwarz (in the space of infinite measure X × Z) +applied to ˜Tf and φ0 = 1X×{0}, which is bounded and in Lp(X × Z), for every p ≥ 1, we +get a function ˜φ0(x) which is ˜Tf-invariant and in L1(X × Z) and +lim +n +Nn(x, 0) +n += ˜φ0(x), a.s. +As f is not a coboundary, ˜φ0 is zero a.e. (cf. for instance [12].) +Observe that ∥ supn≥L +Nn(x,0) +n +∥2 → 0, as L goes to +∞. Indeed, for every 0 < ε ≤ 1, +letting Aε,L := {x : supn≥L +Nn(x,0) +n +> ε}, we have µ(Aε,L) → 0, when L → +∞. Since +Nn(x,0) +n +≤ 1, it follows, for L big enough: +� � +sup +n≥L +(Nn(x, 0) +n +) +�2 dµ ≤ ε2 + µ(Aε,L) ≤ 2ε. +We put Λn(x) := sup +s≥n +Ns(x, 0) +s +. By the previous observation, we have limn ∥Λn∥2 = 0. +Let us consider the following maximal function for the action of T: +˜Λn(x) = sup +1≤r<∞ +1 +r +r−1 +� +j=0 +Λn(T jx) = sup +1≤r<∞ +1 +r +r−1 +� +j=0 +sup +s≥n +Ns(T jx, 0) +s +. +(34) +From a classical maximal inequality, we have ∥˜Λn∥2 ≤ 2∥Λn∥2 → 0. +Observe also that, from the definition of ˜Λn in (34), the following inequalities hold: +˜Λn(x) ≥ sup +r,s≥n +1 +r +r−1 +� +j=0 +Ns(T jx, 0) +s +≥ 1 +n +n−1 +� +j=0 +Nn(T jx, 0) +n +. + +22 +GUY COHEN AND JEAN-PIERRE CONZE +The sequence sup +r,s≥n +1 +r +r−1 +� +j=0 +Ns(T jx, 0) +s +is non negative and decreasing. Since ∥˜Λn∥2 → 0, +the L2-norm of its limit in (X, µ) is zero. The result follows. +□ +Remark 2.8. (see also section 4 and [25]) +Let (Uℓ)ℓ∈Zd be a r.f. of square integrable r.v.’s on a probability space (Ω, F, P) stationary +in the weak sense and such that � +ℓ |⟨Uℓ, U0⟩| < +∞. By (4) and Proposition 2.7, if f is +not a coboundary, it holds +1 +n2 ∥ +n−1 +� +k=0 +Ufk(x)∥2 +2 ≤ C Vn(x) +n2 +→ 0, for µ-a.e. x. +Another result of norm convergence whose proof is like the proof of Proposition 1.2 is the +following. Suppose that the r.f. is stationary. Let ϕ be an observable on the dynamical +system (Ω, P, θ) with a spectral measure νϕ. We have: +� +Ω +| +n−1 +� +j=0 +ϕ ◦ θzj|2 dP = +� +T1 | +n−1 +� +j=0 +e2πizjt|2 dνϕ(t). +Assume that νϕ is absolutely continuous with respect to the Lebesgue measure on the +torus, and let ρ ∈ L1(dt) such that dνϕ(t) = ρ(t)dt. For ε > 0 there is M such that +� +ρ>M ρ dt < ε. We have +1 +n2 +� +Td | +n−1 +� +j=0 +e2πi⟨zj,t⟩|2 dνϕ(t) +≤ +M +n2 +� +Td | +n−1 +� +j=0 +e2πi⟨zj,t⟩|2 dt + +� +ρ>M +ρ dt ≤ M Vn +n2 + ε. +This shows that Vn +n2 → 0 implies 1 +n2 +� +Ω +| +n−1 +� +j=0 +ϕ ◦ θzj|2 dP → 0. This is satisfied by every +ϕ ∈ L2(P), if the dynamical system has a Lebesgue spectrum. +In particular, taking zk = fk(x), by Proposition 2.7, if f is not a coboundary, it holds +1 +n2 +� +Ω +| +n−1 +� +j=0 +ϕ(θfj(x)ω)|2 dP(ω) → 0, for a.e. x. +When the spectral density is square integrable, as we have seen in Proposition 1.2, the +pointwise convergence holds under quantitative hypothesis on the sequence (zk). +2.2. Non centered cocycles. +In an ergodic dynamical system (X, µ, T), if f : X → R is an integrable function with +µ(f) > 0, by the ergodic theorem for the ergodic sums ST +n f(x) = �n−1 +k=0 f(T kx), it holds +for a.e. x: limn +1 +nSnf(x) > 0 and therefore limn ST +n f(x) = +∞. If f has values in Z, as + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +23 +the process ST +n f(x) visits finitely often each site, one can think there is a chance that the +following condition is satisfied: +lim +n +M2 +n(T, f, x) +Vn(T, f, x) = 0. +(35) +A case where (35) is satisfied is the following: let X be a topological compact space, +T : X → X a continuous map, which is uniquely ergodic with µ as unique invariant +measure. Let f : X → Z be an integrable function such that µ(f) ̸= 0. Assume f to be +Riemann-integrable (i.e. such that, for every ε > 0, there are two continuous functions +ψ0, ψ1 with ψ0 ≤ f ≤ ψ1 and µ(ψ1 − ψ0) ≤ ε). +Then, the ergodic means of f converge uniformly, and this implies the existence of N +such that 1 +n|ST +n f(x)| ≥ 1 +2|µ(f)| > 0 for n ≥ N and every x. It follows that the number of +visits of ST +n f(x) to 0 is ≤ N, for every x. By remark 2.4, Mn(x) ≤ N, for every x, and a +fortiori (35) is satisfied. +Nevertheless, we will see that (35) may fail in non uniform cases: there are dynamical +systems and sets B of positive measure such that, for f = 1B, +lim sup +n +M2 +n(T, f, x) +Vn(T, f, x) = 1. +(36) +2.3. Counterexamples. +In this subsection, we construct a transient counterexample, and also a recurrent coun- +terexample with a function f of null integral such that (36) is satisfied. +To construct these counterexamples, we start by considering a general ergodic dynamical +system (X, µ, T) and a measurable set B ⊂ X of positive measure. Let TB be the induced +map on B, R(x) = RB(x) = inf{k ≥ 1 : T kx ∈ B} the first return time of x in B and +Rn(x) = RB +n (x) := �n−1 +k=0 R(T k +Bx) the n-th return time of x in B. +We take x ∈ B. If f is a function such that f = 0 outside B, the position of the sums +up to time Rn−1(x) are the positions of the ergodic sums STB +n f for the induced map up +to time n, that is: +{f(x), f(x) + f(TBx), ..., f(x) + f(TBx) + ... + f(T n−1 +B +x)}. +For a site ℓ, the number of visits up to time Rn−1(x) of the ergodic sums for T is +NRn−1(x)(x, ℓ) = +n−1 +� +k=0 +RB(T k +Bx) 1STB +k +f(x)=ℓ +and therefore +VRn−1(x)(T, x) = +� +ℓ +[ +n−1 +� +k=0 +RB(T k +Bx) 1S +TB +k +f(x)=ℓ]2. +(37) + +24 +GUY COHEN AND JEAN-PIERRE CONZE +Case f = 1B. Clearly �n−1 +k=0 f(T k +Bx) = n. For the map T, the ergodic sums of f are +incremented by 1 when and only when the iterates T jx visit the set B. Otherwise, they +stay fixed. The times of visits in B, for x ∈ B, are 0, R(x), R(x) + R(TBx), .... We have: +for x ∈ B, +Rn−1(x)+t +� +j=0 +f(T jx) = n, for t = 0, ..., Rn(x) − Rn−1(x) − 1. +For Nn(T, x, ℓ) = Nn(T, f, x, ℓ), it holds: +Nn(T, x, ℓ) += +0, if n < Rℓ(x), += +t, if n = Rℓ(x) + t, with 0 ≤ t < Rℓ+1(x) − Rℓ(x), += +Rℓ+1(x) − Rℓ(x) = R(T ℓ +Bx), if n ≥ Rℓ+1(x). +For L ≥ 1, we have for the time preceding the L-th return to the basis for f = 1B: +MRL(x)−1(T, f, x) = max +ℓ≤L R(T ℓ +Bx), VRL(x)−1(T, f, x) = +� +ℓ≤L +R2(T ℓ +Bx). +(38) +In order to compute an explicit example, it is easier to start from a given map S and +construct a special flow T over this map. +Let ϕ : X → N be integrable and ≥ 1. The (discrete time) special map T = Tϕ is defined +on ˜X := {(x, k), x ∈ X, k = 0, ..., ϕ(x) − 1} ⊂ X × R, +by T(x, k) := (x, k + 1), if 0 ≤ k < ϕ(x) − 1, := (Sx, 0), if k = ϕ(x) − 1. +Let ˜µ be the probability measure defined on ˜X by ˜µ(A × {k}) = µ(ϕ)−1 µ(A), for k ≥ 0 +and A ⊂ {x : k ≤ ϕ(x) − 1}. It is Tϕ-invariant. The space X can be identified with the +subset B = {(x, 0), x ∈ X} of ˜X with normalized measure. The set B is the basis and +ϕ − 1 the roof function of the special map Tϕ. +As for the map S we will take an ergodic rotation, the special flow Tϕ will be also ergodic +for the measure ˜µ on ˜X. +Observe that the recurrence time R(x) = RB(x) for the special flow in the basis B is ϕ(x) +and the n-th return time of x in B is Rn(x) = RB +n (x) = �n−1 +k=0 ϕ(Skx). +For S, let us take a rotation S = Sα on X = T/Z by α mod 1, where α is irrational. We +denote by qn the denominators of α. We will construct the measure preserving transfor- +mation which is the special flow (with discrete time) over Sα with a roof function ϕ such +that, for cocycle generated by 1B in the system ( ˜X, ˜µ, T), lim inf +n +Vn(T, x) +M2 +n(T, x) = 1. +We will use the next lemma with p = pn, q = qn, the numerators and denominators of α. +Lemma 2.9. Let p, q ≥ 1, (p, q) = 1, be such that |α − p/q| < 1/q2. For every x, there is +a value 0 ≤ i < q such that x + iα mod 1 ∈ [0, 2/q]. +More generally, for every interval I of length 2/q, for every x, there is a value 0 ≤ i < q +such that x + iα mod 1 ∈ I. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +25 +Proof. It is well known that there is exactly one value of jα mod 1, for 0 ≤ j < q, in each +interval [ ℓ +q, ℓ+1 +q [, ℓ = 0, ..., q − 1. Let us recall a proof. For j = 0, jα ∈ [0, 1/q[. The map +j → ℓj = jp mod q, which is injective, is a permutation of the set {1, ..., q − 1} onto itself. +We have α = p/q + γ, with |γ| < 1/q2. +Assuming γ > 0, it follows: jα mod 1 ∈ [ ℓj +q , ℓj +q + j +q2] ⊂ [ ℓj +q , ℓj+1 +q [, for j = 1, ..., q − 1. The +case γ < 0 is treated the same way. +Now let us prove the first point. Let x be in [0, 1[. There is i0 ∈ {0, ..., q − 1} such that +x = i0 +q + θ, with 0 ≤ θ < 1/q. By the claim, there is i ∈ [0, q[ such that iα mod 1 ∈ +[ q−i0 +q , q−i0+1 +q +]. Hence x + iα mod 1 ∈ [θ, 1 +q + θ] ⊂ [0, 2 +q]. +□ +Let (λn) be an increasing sequence of positive integers which will be subjected below to +growth conditions. First we assume that it satisfies the condition: +qλn+1 ≥ 3qλn, ∀n ≥ 1. +(39) +Denote by Jn the interval Jn = [ +3 +qλn+1 +, 3 +qλn +]. For the roof function, we take, with εn = +1 +n2, +ϕ = 1 + +� +n≥1 +⌊εnqλn⌋1Jn. +The function ϕ is integrable: +� +ϕdµ ≤ 1 + 3 � +n εn. Observe also that, by (39), the length +of Jn is > 2/qλn and that (εnqλn) is not decreasing for n ≥ 2 . +Let x be in the basis. By construction, the orbit of x under the iteration of Tϕ is that +of the rotation Sα until it enters the set Bc, complementary of B at some time. Then it +stays in this set, until it reaches the roof and comes down to the basis. Then the dynamic +is that of the rotation, until again Sj +αx falls in the set ϕ > 1 and so on. +Let Wn(x) be the first visit of Sjx in Jn. By lemma 2.9, we have Wn(x) ≤ qλn. +Now we choose f to get a transient counterexample and a recurrent one. +Transient counterexample. +We take f = 1 on the basis and 0 outside. +The sequence (λn) is taken such that +qλn ≥ n (qλn−1)2, n ≥ 1. +(40) +By (38), we obtain (recall that now TB, the induced map in the basis B, is the rotation +S = Sα and R(T j +Bx) = ϕ(Sj +αx)): +MRWn(x)(x)−1(T, x) += +max +j≤Wn(x) ϕ(Sjx), +(41) +VRWn(x)(x)−1(T, x) += +� +j≤Wn(x) +ϕ2(Sjx). +(42) +In the above formula, ϕ(Sjx) is either 1 or (for some k ≤ n−1) 1+⌊εkqλk⌋ ≤ 1+εn−1qλn−1, +excepted for the last term which is 1 + ⌊εnqλn⌋. + +26 +GUY COHEN AND JEAN-PIERRE CONZE +The maximum in (41) (given by the first visit to Jn) is 1 + ⌊εnqλn⌋ ≥ εnqλn. As we have +seen, this first visit for the iterates Sjx occurs at a time ≤ qλn. It follows by (40): +VRWn(x)(x)−1(T, x) +M2 +RWn(x)(x)−1(T, x) +≤ +qλn +(εn−1 qλn−1)2 +(εn qλn − 1)2 + 1 ≤ (εn−1 +εn +)2 (qλn−1)2 +qλn +1 +(1 − (εn qλn)−1)2 + 1 +≤ +2 ( +n +n − 1)2 (qλn−1)2 +qλn ++ 1 ≤ 4 +n + 1, for n big enough. +This shows: lim sup +n +M2 +n(T, f, x) +Vn(T, f, x) = 1. The result is proved for x in the basis B, but is +satisfied for a.e. x ∈ ˜X, since lim sup +n +M2 +n(T, f, x) +Vn(T, f, x) is a.e. constant by ergodicity of the +special flow and Lemma 2.5. +Remark that Skf(x) → +∞ for every point x.The sequence (Nn(x, 0)) is bounded for +every x, but not uniformly in x. +Recurrent counterexample. +In order to obtain a recurrent counterexample, we now use a special cocycle over a rotation +by α (with α bpq) studied later (see Subsection 3.3). +Let f defined on the basis by f(x) = 1[0, 1 +2[(x) − 1[ 1 +2,1[(x) and 0 outside, and Skf(x) = +�k−1 +i=0 f(x + iα mod 1). By (37), we have +VRn−1(x)(T, f, x) += +� +ℓ +[ +n−1 +� +k=0 +ϕ(x + kα) 1Skf(x)=ℓ]2 += +� +ℓ +[ +n−1 +� +k=0 +(1 + +� +j +εjqλj1Jj(x + kα)) 1Skf(x)=ℓ]2. +Observe that for a constant C, 1 + � +j 0, for a.e. x, +the inequality |fn(x)| > c1 (n ln ln n) +1 +2 is satisfied only for finitely many values of n. This +implies that, for a.e. x, there is N(x) such that |fn(x)| ≤ (c1 n ln ln n) +1 +2, for n ≥ N(x); +so that, for N(x) ≤ k < n, |fk(x)| ≤ (c1 k ln ln k) +1 +2 ≤ (c1 n ln ln n) +1 +2. +Therefore we have Card(Rn(x)) ≤ c2(x) (n ln ln n) +1 +2, with an a.e. finite constant c2(x). +In dimension 1, by (25), we get that for a.e. x there is c(x) > 0 such that +Vn(x) ≥ C(x) n +3 +2 (ln ln n)− 1 +2. +The case where a LIL is valid includes the case of a 1-dimensional r.w. centered with +finite variance, but also the class of cocycles for which a martingale method can be used. +Random walks. +Now we consider sequences given by a random walk. For random walks in Zd, the quan- +tities Vn(x), Mn(x) have been studied in many papers since the 50’s. Mn(x) is called +“maximal multiplicity of points on a random walk” by Erd¨os and Taylor [16]. Below, we +give a brief survey of several results for r.w.s. First we recall some definitions. +Let (ζi)i≥0 be a sequence of i.i.d. random vectors on a probability space (X, µ) with +values in Zd and common probability distribution ν. The associated random walk (r.w.) +Z = (Zn) in Zd starting from 0 is defined by Z0 := 0, +Zn := ζ0 + ... + ζn−1, n ≥ 1. + +28 +GUY COHEN AND JEAN-PIERRE CONZE +A r.w. can be seen as a special case of cocycle. Indeed, the r.v.’s ζi can be viewed as the +coordinate maps on (X, µ) obtained as (Zd)Z equipped with the product measure ν⊗Z +and with the shift T acting on the coordinates. We have ζi = ζ0 ◦ T i and the cocycle +relation Zn+n′ = Zn + Zn′ ◦ T n, ∀n, n′ ≥ 0. +Let S := {ℓ ∈ Zd : P(ζ0 = ℓ) > 0} be the support of ν and L the sub-lattice of Zd +generated by S. Let D be the sub-lattice of Zd generated by {ℓ − ℓ′, ℓ, ℓ′ ∈ S}. +For simplicity (and without loss of generality) in what follows we will assume that the +random walk Z is aperiodic (L = Zd). We exclude also the “deterministic” case (i.e., +when P(ζ0 = ℓ) = 1 for some ℓ ∈ Zd) in dimension 1 (the deterministic case in higher +dimension is excluded by aperiodicity). +Notice that all the pointwise limits or bounds mentioned now for random walks are +a.s. +statements. +These bounds will show that conditions (22), (21) are satisfied by +Vn(x), Mn(x) a.s. for on random walks under mild assumptions. +Recurrence/transience. +Recall that a r.w. Z = (Zn) is recurrent if �∞ +n=1 µ(Zn = 0) = +∞ and otherwise transient. +Recurrence occurs if and only if µ(Zn = 0 infinitely often) = 1, and transience if and only +if µ(Zn = 0 infinitely often) = 0 (cf. [8], [9]). +For an aperiodic r.w. Z in dimension d with a moment of order 2 (for d = 1, a moment +of order 1 suffices), for d = 1, 2, Z is recurrent if and only if it is centered. For d ≥ 3, it +is always transient. +Variance. +Let (Xℓ, ℓ ∈ Zd) be a stationary centered r.f. with summable correlation and spectral +density ρ. We have +1 +n∥ +n−1 +� +k=1 +XZk(x)∥2 +2 = +� +Td +1 +n| +n−1 +� +k=0 +e2πi⟨Zk(x),t⟩|2 dt = +� +Td +1 +nKn(x, t) ρ(t) dt, +where, using (17) with zk = Zk(x) and Zk(x) − Zj(x) = Zk(T jx), 1 +nKn reads +1 +nKn(x, t) += 1 + 2 +� +ℓ +�n−1 +� +k=1 +1 +n +n−k−1 +� +j=0 +1Zk(T jx)=ℓ +� +e2πi⟨ℓ,t⟩. +(43) +As already recalled, the existence of the asymptotic variance limn Vn(x)−1 � +| �n−1 +k=0 XZk(x)|2dµ +has been shown in [11] and the positivity of the limit has been discussed. +The asymptotic variance may be zero in case a coboundary condition is satisfied. An +interesting situation is that of the sums along a transient (non deterministic) r.w., where +the asymptotic variance is always > 0. Below we will recall briefly a proof. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +29 +Transient case +For a transient random walk we use the following general result (Lemma 3.14 in [11]): +Lemma 3.1. If (X, µ, T) is an ergodic dynamical system and (ϕk)k≥1 a sequence of func- +tions in L1(X, µ) such that � +k≥1 ∥ϕk∥1 < ∞, then +lim +n +1 +n +n−1 +� +k=1 +n−k−1 +� +j=0 +ϕk(T jx) = +∞ +� +k=1 +� +ϕk dµ, for a.e. x. +(44) +Therefore, for a transient random walk, we obtain for µ-a.e. x: +lim +n +Vn(x) +n += 1 + 2 lim +n +n−1 +� +k=1 +1 +n +n−k−1 +� +j=0 +1Zk(T jx)=0 = 1 + 2 +∞ +� +k=1 +µ(Zk = 0) < +∞ +and the normalisation for the variance is by n up to a finite constant factor. +Variance in the non deterministic transient case. +Now we recall the proof of the positivity of the asymptotic variance. +Let Ψ(t) = E[e2πi⟨ζ0,t⟩], t ∈ Td. Observe that Ψ(t) ̸= 1 for t ̸= 0 in Td, when the r.w. is +aperiodic and |Ψ(t)| < 1, for t ̸∈ Γ1, where Γ1 is the closed subgroup {t ∈ Td : e2πi⟨r,t⟩ = +1, ∀r ∈ D}. We put, for t ∈ Td\{0} and 0 ≤ λ < 1, +Φ(t) := 1 − |Ψ(t)|2 +|1 − Ψ(t)|2 = ℜe[1 + Ψ(t) +1 − Ψ(t)], +Φλ(t) := 1 − λ2|Ψ(t)|2 +|1 − λΨ(t)|2 = −1 + 2 +∞ +� +k=0 +λkℜe(Ψ(t)k) = −1 + 2 +∞ +� +k=0 +λkµ(Zk = ℓ) cos(2π⟨ℓ, t⟩), +where the last relation follows from ℜe(Ψ(t)k) = ℜe(E[e2πi⟨Zk,t⟩]) = � +ℓ µ(Zk = ℓ) cos(2π⟨ℓ, t⟩). +We put Φ(0) = 0.The function Φ is even, non-negative and Φ(t) = 0 only on Γ1, which is +̸= Td when the r.w. is non deterministic (if the r.w. is deterministic, µ(ζ0 = ℓ) = 1 for +some ℓ ∈ Zd and this implies |Ψ(t)| ≡ 1, but this case is excluded). Therefore Φ is ̸= 0 +a.e. for the Lebesgue measure on Td. +Proposition 3.2. (cf. [28]) Let Z = (Zn) be a transient aperiodic random walk in Zd. +There is a non-negative constant M such that the Fourier coefficients of 1 +nKn converges +to those of Φ + Mδ0 and lim +n +� 1 +nKn ρ dt > 0. +Proof. We use that, if (Zn) is a transient, for all ℓ ∈ Zd, we have �∞ +k=1 µ(Zk = ℓ) < +∞. +Therefore, the series I(ℓ) := −1ℓ=0 + �∞ +k=0 [µ(Zk = ℓ) + µ(Zk = −ℓ)] converges and by +(43) and Lemma 3.1, the even functions 1 +nKn(x, .) satisfy: +� +Td +1 +nKn(x, .) cos 2π⟨ℓ, .⟩ dt += −1ℓ=0 + +n−1 +� +k=0 +1 +n +n−k−1 +� +j=0 +[1Zk(T jx)=ℓ + 1Zk(T jx)=−ℓ] → +n→∞ I(ℓ). + +30 +GUY COHEN AND JEAN-PIERRE CONZE +Note that above the sum over k is written starting from 0. By letting n tend to infinity +in the relation +−1ℓ=0 + +∞ +� +k=0 +λk[µ(Zk = ℓ) + µ(Zk = −ℓ)] += +� +Td cos 2π⟨ℓ, .⟩ [−1 + 2ℜe( +1 +1 − λΨ(.))] dt = +� +Td cos 2π⟨ℓ, t⟩ Φλ(.) dt, +we get since the left sum tends to I(ℓ): +I(ℓ) = lim +λ↑1 +� +Td cos 2π⟨ℓ, t⟩ Φλ(t) dt. +Taking ℓ = 0 in the previous formula, it follows from Fatou’s lemma: +I(0) = 1 + 2 +∞ +� +k=1 +µ(Zk = 0) = lim +λ↑1 +� +Td Φλ(t) dt ≥ +� +Td lim +λ↑1 Φλ(t) dt = +� +Td Φ(t) dt. +This shows the integrability of Φ on Td and we can write with a constant M ≥ 0 +I(0) = lim +λ↑1 +� +Td Φλ(t) dt = +� +Td lim +λ↑1 Φλ(t) dt + M = +� +Td Φ(t) dt + M. +Let Uη be the ball of radius η > 0 centered at 0. By aperiodicity of the r.w., Ψ(t) ̸= 1 for +t in Uc +η, the complementary in Td of Uη, This implies supt∈Ucη supλ<1 Φλ(t) < +∞. +Therefore, we get: lim +λ↑1 +� +Ucη +cos 2π⟨ℓ, t⟩ Φλ(t) dt = +� +Ucη +cos 2π⟨ℓ, t⟩ Φ(t) dt, hence: +I(ℓ) = +� +Ucη +cos 2π⟨ℓ, t⟩ Φ(t) dt + lim +λ↑1 +� +Uη +cos 2π⟨ℓ, t⟩ Φλ(t) dt, ∀η > 0, +which can be be written: +− +� +Uη +cos 2π⟨ℓ, .⟩ Φ dt = I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt − lim +λ↑1 +� +Uη +cos 2π⟨ℓ, .⟩ Φλ dt. +(45) +Let ε > 0. By positivity of Φλ, we have, for η(ε) small enough: +(1 − ε) +� +Uη(ε) +Φλ dt ≤ +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φλ dt ≤ (1 + ε) +� +Uη(ε) +Φλ dt; +By subtracting +� +Uη(ε) cos 2π⟨ℓ, t⟩ Φ(t) dt in the previous inequalities and (45), we get: +(1 − ε) +� +Uη(ε) +Φλ dt − +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φ dt +≤ I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt − lim +λ↑1 +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φλ dt + +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φλ dt +≤ (1 + ε) +� +Uη(ε) +Φλ dt − +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φ dt; + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +31 +As we can chose λ such that +| − lim +λ↑1 +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φλ dt + +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φλ dt| ≤ ε, +we obtain: +−ε + (1 − ε) +� +Uη(ε) +Φλ dt − +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φ dt +≤ I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt ≤ ε + (1 + ε) +� +Uη(ε) +Φλ dt − +� +Uη(ε) +cos 2π⟨ℓ, .⟩ Φ dt +For ε small enough, +� +Uη(ε) cos 2π⟨ℓ, .⟩ Φ dt can be made arbitrary small, as well as ε supλ<1 +� +Uη Φλ dt, +since Φ is integrable and supλ<1 +� +Td Φλ dt < ∞. +This shows that I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt − +� +Uη(ε) Φλ dt can be made arbitrarily small for +ε > 0 small and λ close to 1. The same is true for ℓ = 0 and also for the difference +[I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt − +� +Uη(ε) Φλ dt] − [I(0) − +� +Td Φ dt − +� +Uη(ε) Φλ dt] += [I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt] − [I(0) − +� +Td Φ dt] = [I(ℓ) − +� +Td cos 2π⟨ℓ, .⟩ Φ dt] − M]. +Therefore I(ℓ) = +� +Td cos 2π⟨ℓ, t⟩ Φ(t) dt + M for all ℓ and the Fourier coefficients of 1 +nKn +converges to those of Φ + Mδ0. +As the non-negative sequence ( 1 +nKn) is bounded in L1-norm and the density ρ is contin- +uous, this proves +� 1 +nKnρ dt → +� +Φρ dt + Mρ(0). Moreover, the limit is > 0 since both +Φ and ρ are not 0 a.e. +It is shown in [28] that M = 0 for d > 1. +□ +Behaviour of Mn(x). +In the transient case (d ≥ 3) (at least for a simple r.w.), Erd¨os and Taylor (1960) proved +that for a constant γ > 0 depending on the dimension, +lim +n +Mn(x) +log n = γ. +Recurrent case +In dimension 1, H. Kesten has shown that lim sup +n +Mn +√ +n ln ln n += +√ +2/σ. +Therefore in +dimension 1, we have the following lower and upper bounds for Vn: +C1(x) n +3 +2 (ln ln n)− 1 +2 ≤ Vn(x) ≤ C2(x) n +3 +2(ln ln n) +1 +2. +Dimension d = 2. +There is a deterministic rate (law of large numbers): for a constant C0. +� +Vn dµ +n log n → C0 and Vn(x) +n log n → C0, for a.e. x. + +32 +GUY COHEN AND JEAN-PIERRE CONZE +For a planar simple random walk, Erd¨os and Taylor [16] have shown: +lim sup +n +Mn(x) +(log n)2 ≤ 1 +π. +(46) +The result has been extended by Dembo, Peres, Rosen and Zeitouni [15], who proved for +an aperiodic centered random walk on Z2 with moments of all orders: +lim +n +Mn(x) +(log n)2 = +1 +2π det(Γ) +1 +2 , +where Γ is the covariance matrix associated to the random walk. +As shown in the proof in [15], it suffices to suppose that the 2-dimensional r.w. is aperiodic. +Moreover, the proof for the upper bound is based on the local limit theorem which uses +only the existence of the moment of order 2. Therefore, assuming the existence of the +moment of order 2, the upper bound (46) holds. +It follows in this case: there exist C(x) a.e finite such that: +M2 +n(x) +Vn(x) ≤ C(x)(log n)3 +n +. +3.2. Extensions of the r.w. case. +1) Consequence of the Local Limit Theorem (LLT). +The Local Limit Theorem, when it is satisfied by the cocycle (T, f), gives some pointwise +information on Vn(x). For example, if d = 2, the following lemma holds: +Lemma 3.3. Suppose that the LLT holds and d = 2. Then, for every ε > 0, there is an +integrable function C (depending on ε), such that: +Nn(x, 0) ≤ C(x)(ln n)2+ε, Vn(x) ≤ C(x) n (log n)2+ε. +(47) +Proof. By the LLT, it holds, for n ≥ 1, +� +Nn(., 0)dµ = +n +� +k=1 +� +1fk=0 dµ ≤ C +n +� +k=1 +1 +k ≤ C ln n. +Let ε be a positive constant. Putting Γ(x) = �∞ +n=1 n−(2+ε)N2n(x, 0), we have: +� +Γ(x) dµ(x) ≤ C +∞ +� +n=1 +n−(2+ε)n = C +∞ +� +n=1 +n−(1+ε) < +∞, +so that N2n(x, 0) ≤ Γ(x) n2+ε, where Γ is integrable. +If 2kn ≤ n < 2(kn+1), then with p = 2 + ε, we have for n big enough, +Nn(x, 0) +(log2 n)p ≤ N2(kn+1)(x, 0) +kp +n +≤ (kn + 1)p +kp +n +N2(kn+1)(x, 0) +(kn + 1)p += (1 + 1/kn)p Γ(x) ≤ 2Γ(x). + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +33 +For Vn(x), by (27) we have: +� +Vn(x) dµ(x) = 2 +n−1 +� +k=1 +� +Nn−k(x, 0) dµ(x) + n = O(n log n). +As above for Nn(x, 0), the pointwise bound (47) follows. +□ +Among example of cocycles satisfying a LLT, there are the r.w.’s (but with more precise +results as recalled above), but also cocycles generated by functions with values in Zd +depending on a finite number of coordinates over a sub-shift of finite type endowed with +a Gibbs measure ([19]), ([18]). +2) Functions depending on a finite number of coordinates on a Bernoulli scheme. +Now we try to bound Mn(x) in situation which extends slightly that of random walks. +Suppose that (X, µ, T) is a Bernoulli scheme with X = IN, where I is a finite set. Let +f : x → f(x1, ..., xr) be a centered function from X to Zd, d ≥ 1, depending on a finite +number of coordinates. +Let us consider the generalized random walk (Zn) defined by the sequence of ergodic sums +Zn(x) = fn(x) = X0(x) + ... + Xn−1(x), where Xk(x) = f(T kx). +Lemma 3.4. For all m ≥ 1 and for constants Cm, C′ +m independent of ℓ, we have: +� +Nm +n (., ℓ) dµ = +� +[ +n +� +k=1 +1fk=ℓ]m dµ +≤ Cmnm/2, for d = 1, +≤ C′ +m(Logn)m, for d = 2. +Proof. We bound the sum � +1≤k1 0, +if d = 1, Mn(x) = o(n +1 +2 +ε) and, if d = 2, Mn(x) = o(nε). +In the transient case, if there is a moment of order η for some η > 0, then Mn(x) = o(nε) +for all ε > 0. For these estimates, in both cases, see [11]. +A question if to extend the previous results to a larger class of functions depending weakly +on the far coordinates. For such an extension is that explicit bounds in the LLT are not +always available. +3.3. Step functions over rotations. +Now we take X = Tr, r ≥ 1 endowed with µ, the uniform measure and we consider +cocycles over rotations. When they are centered, such cocycles are strongly recurrent and +therefore the associated quantities Vn and Mn are big. The difficult part is to bound them +from above. We will give an example where an upper bound can be obtained. +Let Tα be the rotation by an irrational α. For f : X → Zd, recall that the cylinder map +(cf. Subsection 2.1) is ˜Tf,α = ˜Tα : X ×Zd → X ×Zd defined by ˜Tα(x, ℓ) = (x+α, ℓ+f(x)). +Non centered step cocycles over a rotation. +Let f be a non centered function with a finite number of values values in Zd. Suppose +that f is Riemann integrable, which amounts to assume that, for the uniform measure of +the torus, the measure of the set of discontinuity points of f is zero. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +35 +Then by a remark in Subsection 2.2, Mn(x) is bounded uniformly in x and n. Therefore, +for Vn(x), the bounds n ≤ Vn(x) ≤ Cn are satisfied. +Centered step cocycles over a 1-dimensional rotation. +The interesting situation is that of centered functions. We will consider the case r = 1 +and when the irrational number α has bounded partial quotients. +Recall that an irrational α with continued fraction expansion [0; a1, a2, ..., an, ...] is said +to have bounded partial quotients (bpq) if supn an < +∞. The set of bpq numbers has +Lebesgue measure zero and Hausdorff dimension 1. +In the sequel of this subsection, α will be an irrational bpq number (for instance a qua- +dratic irrational) and f a centered function with values in Z and bounded variation. +By Denjoy-Koksma inequality, there is a logarithmic bound for the cocycle (Tα, f): |fn(x)| ≤ +C ln n, for a constant C. +The cocycle is strongly recurrent to 0 (and this is true for d ≥ 1 if f centered has values +in Zd, when its components have bounded variation). +This makes the corresponding +maximum Mn(x) big. Nevertheless, we will see that condition (18) is satisfied, at least +for a special example. +Lower bound. +Lower bound for Vn and variance, case d = 1. +For a general sequence (zk), we can obtain a lower bound for Vn by an elementary method +when there is an upper bound for the variance defined below. +Lemma 3.5. Defining the mean mn and the variance σ2 +n by +mn = 1 +n +n +� +k=1 +zk, σ2 +n = 1 +n +n +� +k=1 +(zk − mn)2, +we have +Vn ≥ 1 +9 +n2 +σn +, if σn > 1. +(48) +Proof. Suppose that σn > 0. For λ > 1, let ∆λ := [−λσn + mn, λσn + mn] � Z. We have: +σ2 +n ≥ 1 +n +n−1 +� +k=0 +(zk − mn)21zk∈∆c +λ ≥ 1 +n +n−1 +� +k=0 +(1zk∈∆c +λ) λ2σ2 +n. +Therefore: �n−1 +k=0 1zk∈∆λ ≥ n(1 − λ−2). As Card(∆λ) ≤ 2λσn + 1. It follows by (24): +Vn ≥ (1 − λ−2)2 +2λσn + 1 n2. +For λ = 2 we get: Vn ≥ +9 +16 +4σn+1 n2 ≥ +9 +80 +n2 +σn, if σn > 1; hence (48). +□ + +36 +GUY COHEN AND JEAN-PIERRE CONZE +If zk is given by ergodic sums, i.e., zk = fk(x) , let +mn(x) := 1 +n +n +� +k=1 +fk(x), σ2 +n(x) = 1 +n +n +� +k=1 +(fk(x) − mn(x))2. +By [5, Proposition 13], for α bpq and f with bounded vartion, it holds σ2 +n(x) ≤ C ln n. +Using (48) and Vn(x) ≤ nMn(x), this gives a lower bound for Vn(x) and Mn(x): +Vn(x) ≥ c +n2 +√ +ln n +, Mn(x) ≥ c +n +√ +ln n +. +(49) +Below we will get an estimate from above in the following example. +Example 3.6. f = 1[0, 1 +2 ) − 1[ 1 +2,1) and α bpq. +Upper bound for the example (3.6). +For f as above and α bpq, we have by [1], for some constant C1 > 0, +∥Nn(·, 0)∥∞ = ∥ ˜Sn(1T1×{0})(·, 0)∥∞ ≤ +C1n +√log n. +(50) +Remark that the bound (50) is obtained in [1] as the limit of ∥Nn(·, 0)∥p, the Lp-norm +of Nn(·, 0), as p goes to ∞. Therefore the bound holds for the norm ∥.∥ess sup, but it can +be easily replaced by the uniform norm as written above. Indeed, for any x, there is a +neighborhood V (x) of x, such that for y ∈ V (x), |Nn(x, 0) − Nn(y, 0)| ≤ 1 (at most one +jump in V (x)). As one can find y ∈ V (x) satisfying Nn(y, 0) ≤ +C1n +√log n, the same inequality +holds for x, with C1 replaced by 2C1. +Using Remark 2.4, it follows: +Mn(x) ≤ C1 +n +√log n. +(51) +By (51) and since Vn(x) ≤ n Mn(x), we obtain +Vn(x) ≤ C1 +n2 +√log n. +(52) +From (49), (51) and (52), it follows: Vn(x) ≍ n2/√log n and Mn(x) ≍ n/√log n, where +an ≍ bn for two sequences (an) and (bn) means c an ≤ bn ≤ C an, ∀n ≥ 1, with two positive +constants c, C. +Therefore we get in this special example 3.6: +M2 +n(x) +Vn(x) ≤ ( C1n +√log n)2/ +cn2 +√log n = C2 +1 +c +1 +√log n → 0. +(53) +Condition (18) of Theorem 1.9 is satisfied in this example, as well as the condition of +Theorem 1.6 a), hence a Glivenko-Cantelli theorem along (Snf(x)) for i.i.d. r.v.’s. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +37 +But the sufficient conditions for the Glivenko-Cantelli theorems 1.5, 1.6 b), 1.8 are not +satisfied by this cocycle and more generally, in view of the lower bound (49), by a cocycle +defined by step functions over a bpq irrational rotation. +4. About limit theorems along ergodic sums +4.1. Glivenko-Cantelli theorem along ergodic sums. +The Glivenko-Cantelli theorem recalled in the introduction is a (pointwise) law of large +numbers uniform over a set of functions (here the indicators of intervals). +When the +r.v.’s Xk are i.i.d., the proof is an easy consequence of the strong law of large numbers +applied to the sequence of i.i.d. bounded r.v.’s (1Xk≤s). Using Birkhoff’s ergodic theorem, +the Glivenko-Cantelli theorem has been extended to the setting of a strictly stationary +sequence (Xk) of random variables. More precisely, formulated in terms of dynamical +systems, the following holds: +Let (Y, A, ν) be a probability space and S an ergodic measure preserving transformation on +Y . For any measurable function ϕ : Y → R, let us consider the strictly stationary sequence +(Xk) defined by Xk = ϕ◦Sk, k ≥ 0. Then the sequence of empirical distribution functions +satisfies: for ν a.e. y ∈ Y, sups | 1 +n +�n−1 +k=0 1X(y)k≤s − F(s)| → 0, where F(s) = ν(ϕ ≤ s). +Observe that the result is an application of Birkhoff’s theorem and Lemma 1.1 recalled +in Section 1. Its extension to the non ergodic case has been formulated by Tucker [29], +the distribution function F(s) being replaced by the conditional distribution function +E(1ϕ≤s|J ), where J is the σ-algebra of S-invariant sets. In others words, we have: +for ν a.e. y ∈ Y, lim +n→∞ sup +s | 1 +n +n−1 +� +k=0 +1ϕ(Sky)≤s − E(1ϕ≤s|J )(y)| = 0. +The above formula relies on the ergodic decomposition which can be used in the proof. +In the previous framework, for a process, a Glivenko-Cantelli like theorem sampled along +a sequence generated by a dynamical system can be obtained as follows: +As in Subsection 2, let T be an ergodic measure preserving transformation on a probability +space (X, B, µ) and f a measurable function on X with values in Zd, d ≥ 1. +Let us take a second system (Ω, P, θ), where θ = (θℓ)ℓ∈Zd is a Zd-action preserving P. +The skew product associated to the cocycle (T, f) and θ is the map: Tθ,f : (x, ω) → +(Tx, θf(x)ω) from X × Ω to itself. By iteration we get: +T k +θ,f(x, ω) = (T kx, θfk(x)ω). +For example, as Zd-action, we can take a Zd-Bernoulli shift (Ω, P, (θℓ)ℓ∈Zd), with P a +product measure and θ the shift on the coordinates. If X0 is the first coordinate map, +then (Xℓ) = (X0 ◦ θℓ) is a family of i.i.d. r.v.’s indexed by Zd. + +38 +GUY COHEN AND JEAN-PIERRE CONZE +In general, let Iθ,f denote the conditional expectation with respect to the σ-algebra of +Tθ,f-invariant sets. The ergodic theorem for Tθ,f shows that, for ψ ∈ L1(µ × P), +lim +n +1 +n +n−1 +� +k=0 +ψ(T kx, θfk(x)ω) = Iθ,f(ψ)(x, ω), for µ × P-a.e.(x, ω). +(54) +If ϕ is a measurable function on Ω, putting ψs(x, ω) = 1Is(ϕ(ω)), where Is is the half-line +] − ∞, s], we have +ψs(T k +θ,f(x, ω)) = 1Is(ϕ(θfk(x)ω)). +By the quoted Tucker’s result, the convergence in (54) for each ψs, s ∈ R, can be strength- +ened into a uniform convergence with respect to s: +for µ × P-a.e (x, ω), 1 +n sups | �n−1 +k=0 1Is(ϕ(θfk(x)ω)) − I(ψs)(x, ω)| → 0. +Therefore, by the Fubini theorem, there is a “sampled” version of the Glivenko-Cantelli +theorem for the empirical process of a stationary sequence: +Proposition 4.1. For µ-a.e x, we have +| sups +1 +n +�n−1 +k=0 1Is(ϕ(θfk(x)ω)) − I(ψs)(x, ω)| → 0, for P-a.e ω. +When Tθ,f is ergodic, if ψ ∈ L1(µ × P), we have Iθ,f(ψ)(x, ω) = +� +ψ dµ dP, for µ × +P-a.e. (x, ω), and the centering I(ψs)(x, ω) is given by the distribution function F(s) = +µ(ϕ ≤ s). In this case, for a.e. x, a Glivenko-Cantelli theorem with the usual centering +holds for the empirical process sampled along the sequence (zn) given by zn = Snf(x) +(with a set of ω’s of P-measure 1 depending on x). +The lemma below shows, as it is known, that ergodicity of the cylinder map ˜Tf implies +ergodicity of the skew map Tθ,f. Let us sketch a proof. +Lemma 4.2. Suppose that the cocycle (T, f) is recurrent and the map ˜Tf ergodic. If the +action of Zd by θ on (Ω, P) is ergodic, then Tθ,f is ergodic on (X × Ω, µ × P). +Proof. : Let Φ be a Tθ,f invariant measurable function on X × Ω: +Φ(Tx, θf(x)ω) = Φ(x, ω), for a.e. (x, ω). +For a.e. x, there is a set Ω0 +x of full P-measure in Ω such that Φ(Tx, θf(x)ω) = Φ(x, ω), for +all ω ∈ Ω0 +x. As Zd is countable, for a.e. x, there is a set Ωx of full measure such that +Φ(Tx, θf(x)θℓω) = Φ(x, θℓω), for all ω ∈ Ωx. +Let ω ∈ Ωx. The function ϕω(x, ℓ) := Φ(x, θℓω) on X × Zd is measurable, ˜Tf-invariant: +ϕω( ˜Tf(x, ℓ)) += +ϕω(Tx, ℓ + f(x)) = Φ(Tx, θℓ+f(x)ω) += +Φ(Tx, θf(x)θℓω) = Φ(x, θℓω) = ϕω(x, ℓ). +It follows from the ergodicity of ˜Tf that there is a constant cω such that ϕω(x, ℓ) = cω for +a.e. x. Therefore Φ coincides a.e. with a function ψ on Ω which is θ-invariant, hence a +constant by the assumption of ergodicity of the action of Zd on Ω. +□ + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +39 +With Fubini’s argument, we get a Glivenko-Cantelli theorem for a.e. x, if we can show +that the skew map Tθ,f is ergodic. +There are many examples cylinder flows ˜Tf which are shown to be ergodic in the literature +and so providing examples via Lemma 4.2. For instance, we can take for T an irrational +rotation and f = 1[0, 1 +2) − 1[ 1 +2,1). The cocycle (T, f) is ergodic and the above version of +Glivenko-Cantelli theorem applies for any stationary sequence (Xk) (with a conditional +distribution if the stationary sequence is not ergodic). See also examples for which the +skew map is ergodic in [25]. +4.2. Discussion: universal sequences. +The weakness in the approach of the previous subsection for a sampled Glivenko-Cantelli +theorem along ergodic sums (Skf(x), k ≥ 0) is that it yields a set of x’s of µ-measure +1 depending on the dynamical system (Ω, P, θ) and on ϕ. One can try to reinforce the +statement by introducing a notion of “universal property”. +In this direction, the LLN for sums sampled along ergodic sums is closely related in the +following way to the random ergodic theorems which have been studied in several papers. +First, let us call “universally good” a sequence (zk) such that, for every dynamical system +(Ω, P, θ), for every ϕ ∈ L1(P), the sequence 1 +n +�n−1 +k=0 ϕ ◦ θzk converges P-a.e. +We say that (T, f) a “(pointwise) good averaging cocycle” (or a universally representative +sampling scheme) if, for µ-a.e. x, the sequence (Skf(x)) is universally good, i.e., for every +dynamical system (Ω, P, θ), for every ϕ ∈ L1(P), 1 +n +�n−1 +k=0 ϕ ◦ θSkf(x) converges P-a.e. +The definition of a “mean good averaging cocycle” is similar, changing the above conver- +gence into convergence in L2(P)-norm, for every ϕ in L2(P). +A question which has been studied is to find mean or pointwise good averaging cocycles. In +the first direction, examples and counterexamples of mean good averaging 1-dimensional +cocycles are studied in [25], +For pointwise convergence, there are 1-dimensional examples given by cocycles with a +drift. In [23], the following result is shown: the cocycle defined by a random walk with a +moment of order 2 is a pointwise good averaging cocycle if and only if it is not centered. +Moreover it is shown that any ergodic integrable integer-valued stochastic process with +nonzero mean is universally representative for bounded stationary processes. The proofs +are based on the recurrence time theorem ([6]). +Notice that a related, but different, notion can be introduced by restricting the dynamical +system (Ω, P, θ) to belong to a given class C of dynamical systems. +Let us call “pointwise good for a class C of dynamical systems”, a sequence (zk) such that, +for every dynamical system (Ω, P, θ) in the class C, for every ϕ ∈ L1(P), limn +1 +n +�n−1 +k=0 ϕ ◦ +θzk = +� +ϕ dP, P-a.e. There is a similar property for the mean convergence. + +40 +GUY COHEN AND JEAN-PIERRE CONZE +This can be also expressed for a class of random fields satisfying a condition on the decay +of correlations. +For example, by Remark 2.8, every cocycle with values in Zd which is not a coboundary is +a mean good averaging cocycle for the stationary r.f.s on Zd such that � +ℓ |⟨Uℓ, U0⟩| < +∞. +If (zk) is pointwise universally good for a class C, clearly we get the Glivenko-Cantelli +property for any dynamical system (Ω, P, θ) in C and every measurable function ϕ, i.e.: +sups | 1 +n +�n−1 +k=0 1Is(ϕ(θzkω)) − P(ϕ ≤ s)| → 0, for P-a.e ω. +(55) +As we see, there are two different approaches of the notion of universal sequences for a +law of large numbers: either we ask for a LLN along such a sequence for every dynamical +system (Ω, P, θ) and all functions in L1(P) or we fix a class of dynamical systems, or a +class of functions in L1(P). In the latter case, the condition on the sequence (zk) may be +expressed in a quantitative way. Let us give a known example and recall the proof. +Proposition 4.3. Let (zk) be a strictly increasing sequence of positive integers. If the +sequence satisfies: for a finite constant C, zk ≤ Ck, ∀k ≥ 1, then (zk) is a pointwise good +averaging sequence for the class C of dynamical systems (Ω, P, θ) with Lebesgue spectrum. +Proof. There is a dense set of functions ϕ ∈ L1(P) such that +1 +n +n−1 +� +k=0 +ϕ(θzkω) converges P-a.e. +(56) +Indeed, by the SLLN for orthogonal random variables, (56) is satisfied by ϕ ∈ L2(P) such +that ⟨ϕ, ϕ ◦ θk⟩ = 0, ∀k. The Lebesgue spectrum property implies that such functions +span a dense linear space in L2(P), hence in L1(P). +Moreover, the space of functions ϕ such that (56) holds is closed by the ergodic maximal +lemma in view of the assumption on (zk). Therefore (56) is satisfied by every ϕ ∈ L1(P). +□ +To finish, we recall the following example which shows that the behaviour may depend +on the properties of the dynamical system (Ω, P, θ) (cf. [13]): +Let (Ω, F, P) be the interval [0, 1] endowed with the Borel σ-algebra and the Lebesgue +measure and take f = 1[0, 1 +2]. Denote by T the class of invertible measure preserving +transformations on this space. It can be shown that there are increasing sequences (zk) +of positive integers satisfying the conditions of the previous proposition such that, for a +dense Gδ of elements in T with continuous spectrum, the ergodic means of f along (zk) +do not converge P-a.e. +References +[1] +Aaronson, J., Bromberg, M. and Nakada, H.: Discrepancy skew products and affine random +walks, Israel J. Math. 221 (2017), no. 2, 973-1010. + +EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS +41 +[2] +Ambrose L.: Functional generalizations of Hoeffding’s covariance lemma and a formula for +Kendall’s tau, Statistics and Probability Letters, 122 (2017), 218-226. +[3] +Billingsley P.: Convergence of probability measures, 1rst ed. John Wiley & Sons, Inc., 1968. +[4] +Birkel, T.: A note on the strong law of large numbers for positively dependent random variables, +Statist. Probab. Lett. 7 (1988), no. 1, 17-20. +[5] +Borda, B.: On the distribution of Sudler products and Birkhoff sums for the irrational rotation, +Arkiv 2021. +[6] +Bourgain J., Furstenberg H., Katznelson Y. and Ornstein D.: Appendix on return-time se- +quences. Publ. Math. IHES, (69) 42-45, 1989. +[7] +Chung, K.L: A course in probability theory, 3rd ed. (2001), Acad. Press, Inc., San Diego, CA. +[8] +Chung, K. L., Fuchs, W. H. J. On the distribution of values of sums of random variables, Mem. +Amer. Math. Soc. 6 (1951), 157-168. +[9] +Chung, K. L., Ornstein, D., On the recurrence of sums of random variables, Bull. Amer. Math. +Soc. 68 (1962), 30-32. +[10] +Cohen, G., Conze, J.-P.: On the quenched functional CLT in random sceneries, to appear in +Studia Matematica. +[11] +Cohen, G., Conze, J.-P.: CLT for random walks of commuting endomorphisms on compact +abelian groups, J. Theoret. Probab. 30 (2017), no. 1, 143-195. +[12] +Conze, J. P., Remarques sur les transformations cylindriques et les ´equations fonctionnelles, +Pub. s´em. math. info. de Rennes, 1976, fasc. 2, http://www.numdam.org/actas/PSMIR/ +[13] +Conze, J.-P., Convergence des moyennes ergodiques pour des sous-suites, Bull. Soc. Math. Fr., +M´emoire 35, (1973), p. 7-15. +[14] +Deligiannidis G., Gouezel, S., Kosloff, Z.: Boundary of the range of a random walk and the +Folner property, Electron. J. Probab. 26: 1-39 (2021). DOI: 10.1214/21-EJP667 +[15] +Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O.: Thick points for planar Brownian motion and +the Erdos-Taylor conjecture on random walk. Acta Math. 186 (2001), no. 2, 239-270. +[16] +Erdos, P.; Taylor, S. J.: Some problems concerning the structure of random walk paths. Acta +Math. Acad. Sci. Hungar. 11 (1960), 137–162. +[17] +Esary, J. D.; Proschan, F.; Walkup, D. W., Association of random variables, with applications, +Ann. Math. Statist. 38 (1967), 1466-1474. +[18] +Gou¨ezel S.: Berry-Esseen theorem and local limit theorem for non uniformly expanding maps, +Ann. Inst. H. H. Poincar´e PR 41 (2005), 997–1024 +[19] +Guivarc’h, Y. et Hardy, J.: Th´eor`emes limites pour une classe de chaˆınes de Markov et appli- +cations aux diff´eomorphismes d’Anosov, Ann. Inst. H. Poincar´e 24 (1) (1988), 73-98. +[20] +Hoeffding W.: Probability Inequalities for Sums of Bounded Random Variables, Journal of the +AMS vol. 58 (1963), no. 301, 13-30. +[21] +Hewitt, E.; Ross, K. A.: Abstract harmonic analysis, Vol. II. Die Grundlehren der mathematis- +chen Wissenschaften, Band 152 Springer-Verlag, New York-Berlin 1970. +[22] +Kesten, H.: An iterated logarithm law for local time. Duke Math. J. 32 (1965), 447-456. +[23] +Lacey, M.; Petersen, K.; Wierdl, M.; Rudolph, D.: Random ergodic theorems with universally +representative sequences., Ann. I. H. Poincar´e Prob. Stat. 30 (1994), no. 3, 353-395. +[24] +Lehmann, E. L.: Some concepts of dependence, Ann. Math. Statist. 37 (1963), 1137-1153. +[25] +Lema´nczyk, M.; Lesigne, E.; Parreau, F.; Voln´y, D.; Wierdl, M.: Random ergodic theorems and +real cocycles, Israel J. Math. 130 (2002), 285-321. +[26] +Newman, C.M.: Normal fluctuations and the FKG inequalities, Comm. Math. Phys. 74 (1980), +no. 2, 119-128. + +42 +GUY COHEN AND JEAN-PIERRE CONZE +[27] +Newman, C.M. and Wright, A.L.: An invariance principle for certain dependent sequences, Ann. +Probab. 9 (1981), no. 4, 671-675. +[28] +Spitzer, F.: Principles of random walk, The University Series in Higher Mathematics D. Van +Nostrand Co., Inc., Princeton, N.J.-Toronto-London (1964). doi: 10.1007/978-1-4757-4229-9 +[29] +Tucker, H.: A generalization of the Glivenko-Cantelli theorem, Annals of Math. Stat., vol. 20, +no 3, p. 828-830 (1959). +[30] +Yu Hao, A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated +sequences, Probab. Theory Related Fields 95 (1993), no. 3, 357-370. +[31] +Zygmund, A.: Trigonometric series, Second edition, Cambridge University Press, London-New +York 1968. +Guy Cohen, +School of Electrical Engineering, +Ben-Gurion University, Israel +Email address: guycohen@bgu.ac.il +Jean-Pierre Conze, +IRMAR, CNRS UMR 6625, +University of Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France +Email address: conze@univ-rennes1.fr + diff --git a/4dFJT4oBgHgl3EQfjyxF/content/tmp_files/load_file.txt b/4dFJT4oBgHgl3EQfjyxF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..830fa59333c0159d68360d1716d97e1f752f85a4 --- /dev/null +++ b/4dFJT4oBgHgl3EQfjyxF/content/tmp_files/load_file.txt @@ -0,0 +1,1605 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf,len=1604 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='11576v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='PR] 27 Jan 2023 EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS GUY COHEN AND JEAN-PIERRE CONZE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Let (Xℓ)ℓ∈Zd be a real random field (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=') indexed by Zd with common probability distribution function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Let (zk)∞ k=0 be a sequence in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The empirical process obtained by sampling the random field along (zk) is �n−1 k=0[1Xzk ≤s − F(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We give conditions on (zk) implying the Glivenko-Cantelli theorem for the empirical process sampled along (zk) in different cases (independent, associated or weakly corre- lated random variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We consider also the functional central limit theorem when the Xℓ’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' These conditions are examined when (zk) is provided by an auxiliary stationary pro- cess in the framework of “random ergodic theorems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Contents Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' General results on the empirical process along a sub-sequence 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Preliminaries 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' A Glivenko-Cantelli type theorem 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' A sufficient condition for a FCLT for the sampled empirical process 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Local times for ergodic sums 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Auxiliary general results 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Non centered cocycles 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Counterexamples 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Examples 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Random walks 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Extensions of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' case 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Step functions over rotations 34 Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Primary: 60F05, 28D05, 22D40, 60G50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Secondary: 47B15, 37A25, 37A30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Empirical process, sampling along a stationary process, local times, Glivenko- Cantelli theorem, functional central limit theorem, random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 1 2 GUY COHEN AND JEAN-PIERRE CONZE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' About limit theorems along ergodic sums 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Glivenko-Cantelli theorem along ergodic sums 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Discussion: universal sequences 39 References 40 Introduction For a sequence (Xk) of real i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' random variables with common probability distribution function F, the empirical process is defined by �n−1 k=0 [1Xk≤s − F(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Recall two classical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (A) the Glivenko-Cantelli theorem: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' the sequence of empirical distribution functions Fn(s) := 1 n �n−1 k=0 1Xk≤s converges uniformly to F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' sups |Fn(s) − F(s)| → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (B) a functional central limit theorem (FCLT): if the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='s Xk have a common distribution F over [0, 1], then the process 1 √n �n−1 k=0 [1Xk≤s − F(s)] converges weakly to a Brownian bridge in the space of cadlag functions on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In this paper we study the extension of these results when the process is sampled along a subsequence, analogously to what is done for limit theorems in random scenery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In the sequel, for d ≥ 1, (Xℓ)ℓ∈Zd will be a real random field (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=') indexed by Zd defined on a probability space (Ω, F, P) with common probability distribution function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The expectation on (Ω, P) is denoted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We consider in particular the case of a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s or of stationary associated r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Let (zk)∞ k=0 be a sequence in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The process obtained by sampling the random field along (zk) is Wn(s) := �n−1 k=0[1Xzk≤s − F(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We will call Wn(s) “empirical process sampled along (zk)”, or simply “sampled empirical process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' A general question is whether the above results (A), (B) extend to the sampled empirical process Wn(s), in particular when (zk) is given by another stationary process with values in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In Section 1, we give conditions on (zk) implying that (A) and (B) are still valid for an empirical process sampled along (zk) in different cases: independent, associated or weakly correlated random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The conditions are expressed in terms of the following quantities associated to the sequence (zk) in Zd: local time, maximal local time and number of self-intersections (up to time n) defined, for n ≥ 1, by Nn(ℓ) := #{0 ≤ k ≤ n − 1 : zk = ℓ}, Mn := max ℓ Nn(ℓ), Vn := #{0 ≤ j, k ≤ n − 1 : zj = zk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (1) EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS 3 They satisfy � ℓ Nn(ℓ) = n and n ≤ Vn = � ℓ N2 n(ℓ) ≤ nMn ≤ n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In the other sections, (zk) is given by a stationary process (or equivalently by the sequence (Skf(x))k≥1 of ergodic sums of a function f over a dynamical system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The conditions found in Section 1 lead to study the local times, maximum number of visits, number of self-intersections for the sequence (Skf(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' General remarks are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Then in Section 3, we consider two families of examples: random walks and some ergodic sums over a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The Glivenko-Cantelli theorem along ergodic sums (extension of (A)) is strongly related to random ergodic theorems, in particular to results in [23] and [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' This is discussed in the last Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Finally let us mention the quenched FCLT for the 2-parameters process Wn(s, t) := [nt]−1 � k=0 [1XZk(x)≤s − F(s)], (s, t) ∈ [0, 1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' When (Xℓ) is a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s indexed by Z2 and when the sampling is provided by a 2-dimension centered random walk (Zk) with a moment of order 2, the weak convergence for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' x toward a Kiefer-M¨uller process can be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' This will be the content of a forthcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Part of this research was done during visits of the first author to the IRMAR at the University of Rennes 1 and of the second author to the Center for Advanced Studies in Mathematics at Ben Gurion University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The authors are grateful to their hosts for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' General results on the empirical process along a sub-sequence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In this subsection, results on the empirical process along a sub- sequence are shown for independent variables, as well for some of them for wider classes (associated, PDQ and weakly correlated random variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We start by recalling some notions and auxiliary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 1) Associated variables Definition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' [17]): A finite set of real random variables T = (T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' , Tn) is said to be associated if Cov[f(T), g(T)] ≥ 0, for every coordinate-wise non-decreasing functions f = f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=', xn) and g = g(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=', xn) for which E[f(T)], E[g(T)], E[f(T) g(T)] exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' An infinite set of random variables is associated if any finite subset of it is associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Association of random variables is preserved under taking subsets and forming unions of independent sets (of associated random variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In particular a family of independent variables is associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 4 GUY COHEN AND JEAN-PIERRE CONZE Clearly, orthogonal associated random variables are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Examples of (non inde- pendent) stationary associated processes with absolutely summable series of correlations are provided by some Ising models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' References to such examples of stationary Zd random fields which satisfies the FKG inequalities and with absolutely summable correlations can be found in Newman’s paper [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Notice that the FKG inequalities expresses the association property of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 2) PQD variables Two r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s X, Y are called (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' [24]) positively quadrant dependent (PQD) if, P(X > x, Y > y) ≥ P(X > x) P(Y > y), ∀x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The property is preserved by centering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Any pairwise associated r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s are pairwise PQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Pairwise independent random variables are pairwise PQD associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Two random variables X and Y are PQD if and only if for every non-decreasing functions f and g, Cov(f(X), g(Y )) ≥ 0 (whenever the covariance exists) ([17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 3) We will use the following results: a) Maximal inequality of Newman and Wright [27, Inequality (12)]: If (Wi) is a sequence of centered associated, square integrable random variables, it holds: P( max 1≤j≤n | j � i=1 Wi| ≥ λ ∥ n � i=1 Wi∥2) ≤ 2P(| n � i=1 Wi| ≥ (λ − √ 2) ∥ n � i=1 Wi∥2), ∀λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (2) b) Hoeffding’s identity (see [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1]) Let X, Y be random variables with finite second moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' For any absolutely continuous functions f, g on R, such that E[f 2(X) + g2(Y )] < ∞, it holds Cov(f(X), g(Y )) = � ∞ −∞ � ∞ −∞ f ′(x)g′(y)[P(X > x, Y > y) − P(X > x)P(Y > y)]dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In particular, if X, Y are PDQ random variables, if |f ′|, |g′| ≤ M a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=', we have |Cov(f(X), g(Y ))| ≤ M2Cov(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 4) Uniformity in the analogues of Glivenko-Cantelli theorem will follow from the lemma: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' [7, Lemma, p 140] Let Fn, F be a family of right continuous distributions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Assume that, for each point x in a dense countable set Q ⊂ R, we have Fn(x) → F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Let J be the set of jumps of F and assume that Fn(x)−Fn(x−) → F(x)−F(x−) for every x ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Then Fn(x) → F(x) uniformly in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' A strong law of large numbers First we state a law of large numbers for bounded r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s valid under weak hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS 5 Let (Uℓ)ℓ∈Zd be a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' indexed by Zd of square integrable r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v’s on a probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Let (zk)k≥0 be a sequence in Zd, d ≥ 1, with numbers of self-intersections Vn, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The partial sums along (zk) are denoted by Sn := n−1 � k=0 Uzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' By the Cauchy-Schwarz inequality, if � ℓ supr |⟨Ur+ℓ, Ur⟩| < +∞, if holds for a finite constant C0: ∥ n−1 � i=0 Uzi∥2 2 = � ℓ � r Nn(r + ℓ)Nn(r)⟨Ur+ℓ, Ur⟩ ≤ Vn � ℓ sup r |⟨Ur+ℓ, Ur⟩| = C0Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (3) In particular if the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' is stationary and the series of correlations is absolutely summable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=', � ℓ∈Zd |⟨X0, Xℓ⟩| < +∞), then the spectral density of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' exists and is the continuous non-negative function ρ on Td with Fourier coefficients � Td e2πi⟨ℓ,t⟩ ρ(t) dt = ⟨X0, Xℓ⟩ and it holds: ∥Sn∥2 2 = ∥ n−1 � i=0 Uzi∥2 2 ≤ Vn � ℓ |⟨Uℓ, U0⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (4) Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Suppose the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s Uℓ on (Ω, P) centered and uniformly bounded by the same constant K, ∥Uℓ∥∞ ≤ K, ∀ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Assume that (zk) is such that Vn ≤ C1 n2 (log n)β , for constants C1, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (5) 1) Then, if β > 1 and � ℓ∈Zd sup r∈Zd |⟨Ur+ℓ, Ur⟩| < +∞, 2) or if β > ζ for some ζ ∈ [1, 2] and the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (Uℓ) is stationary with � ℓ∈Zd |⟨Uℓ, U0⟩|ζ < ∞, the (strong) LLN holds: Sn(ω) n → 0, for P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 1) For convenience, if t is in R+, we define St as S[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' From (3) it follows � (|Sn| n )2 dP ≤ C0 Vn n2 ≤ C0C1 1 (log n)β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Therefore, putting β = 1 + η and α = 1 − η/2 (which implies αβ > 1) we have � k � (|S2kα| 2kα )2 dP ≤ C0C1 � k 1 (log 2kα)β = C′ � k 1 kαβ < +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' hence: lim k→+∞ S2kα 2kα = 0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' For n ≥ 1, let kn be such that 2(kn)α ≤ n < 2(kn+1)α (that is: kn = [(log2 n)1/α]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We put qn := 2(kn+1)α − 2(kn)α and pn = n − 2(kn)α ≤ qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 6 GUY COHEN AND JEAN-PIERRE CONZE For qn, the following estimate holds: qn = 2(kn)α(2(kn+1)α−(kn)α − 1) ∼ C′′ 2(kn)α (kn)1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Using the uniform boundedness of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s, we can write: |Sn n − S2(kn)α 2(kn)α | = |S2(kn)α + �2(kn)α+pn i=2(kn)α Uzi 2(kn)α + pn − S2(kn)α 2(kn)α | = |2(kn)α �2(kn)α+pn i=2(kn)α Uzi − pnS2(kn)α 2(kn)α(2(kn)α + pn) | ≤ 2(kn)α �2(kn)α+pn i=2(kn)α |Uzi| + pn|S2(kn)α| 2(kn)α(2(kn)α + pn) ≤ 2(kn)α �2(kn)α+qn i=2(kn)α |Uzi| + qn|S2(kn)α| 2(kn)α(2(kn)α) ≤ qnK2(kn)α + qn|S2(kn)α| 2(kn)α(2(kn)α) = qn 2(kn)α (K + |S2(kn)α| 2(kn)α ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' ≤ C 2(kn)1−α(K + |S2(kn)α| 2(kn)α ) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 2) We consider now the stationary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Since ζ = 1 is special case of 1), we assume ζ ∈]1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We put β = ζ + η, where η is > 0 in view of the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' First, suppose that ζ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Then under the hypothesis, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' has a spectral measure νϕ absolutely continuous with respect to the Lebesgue measure λ on the torus with a density ρ ∈ L2(dt) given by the Fourier series ρ(t) = � ℓ∈Zd⟨Uℓ, U0⟩ e2iπ⟨ℓ,t⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Using the inequality λ{ρ > Mn} ≤ M−2 n ∥ρ∥2 2, we can write: ∥Sn∥2 2 n2 = 1 n2 � Td | n−1 � j=0 e2πi⟨zj,t⟩|2 dνϕ(t) ≤ Mn n2 � Td | n−1 � j=0 e2πi⟨zj,t⟩|2 dt + � ρ>Mn ρ dt ≤ Mn Vn n2 + (λ{ρ > Mn}) 1 2∥ρ∥2 ≤ Mn Vn n2 + M−1 n ∥ρ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Taking Mn = (log n)1+ 1 2 η, we obtain the bound 1 n2 ∥ n−1 � j=0 Uzj∥2 2 ≤ C (log n)1+ 1 2 η and then we finish the proof as in 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Now, suppose that � ℓ∈Zd |⟨Uℓ, U0⟩|ζ < ∞ with 1 < ζ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The spectral density ρ exists and is in L2(λ), since � ℓ∈Zd |⟨Uℓ, U0⟩|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Moreover it belongs to Lζ′(λ) where ζ, ζ′ are conjugate exponents (see: [31], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 102, or [21] Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='22), and it satisfies: ∥ρ∥ζ′ ≤ ( � ℓ∈Zd |⟨Uℓ, U0⟩|ζ)1/ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' H¨older’s inequality implies: � ρ>Mn ρ dt ≤ (λ{ρ > Mn})1/ζ∥ρ∥ζ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' As λ{ρ > Mn} ≤ M−ζ′ n � ρζ′ dt = M−ζ′ n ∥ρ∥ζ′ ζ′, EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS 7 it follows: � ρ>Mn ρ dt ≤ M−ζ′/ζ n ∥ρ∥1+ζ′/ζ ζ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Therefore, we obtain 1 n2 � Td | n−1 � j=0 e2πi⟨zj,t⟩|2 dνϕ(t) ≤ Mn Vn n2 + � ρ>Mn ρ dt ≤ Mn Vn n2 + M−ζ′/ζ n ∥ρ∥1+ζ′/ζ ζ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Now we take Mn such that : Mn/(log n)β = M−ζ′/ζ n , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Mn = (log n)β/ζ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We get 1 n2 ∥ n−1 � j=0 Uzj∥2 2 ≤ C (log n)β(1−1/ζ′) = C (log n)β/ζ = C (log n)1+η/ζ with η > 0, and the end of the proof is as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' □ Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 1) Let us give an example of a non stationary r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (Uℓ) which satisfies Condition 1) of the previous proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We take (Uℓ = Vℓ Wℓ, ℓ ∈ Zd), where (Vℓ) and (Wℓ) are two r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.’s independent from each other, with (Vℓ) centered stationary and such that � ℓ∈Zd |⟨Vℓ, V0⟩| < ∞, and (Wℓ) satisfying supℓ,p |⟨Wℓ+p, Wℓ⟩| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' The r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (Wℓ) can be viewed as a (multiplicative) noise (which can be non stationary) independent from the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (Uℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Clearly the condition in 1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 2) For a stationary r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (Uℓ) with a bounded spectral density (but with a series of corre- lations which may be not absolutely summable), then like in 1) the condition β > 1 is sufficient for the conclusion of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Now, we give a pointwise bound for the sampled sums, first for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s, then for a stationary random field (Uℓ)ℓ∈Zd of associated r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 1) Suppose that the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s Uℓ, ℓ ∈ Zd, are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=', centered, uniformly bounded by a constant K, ∥U0∥∞ ≤ K, and that E|U0|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Then it holds lim sup n |Sn| √Vn (2 log log n) 1 2 ≤ K, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (6) If Vn = o(n2 (log log n)−1), then lim n Sn n = 0, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' 2) Suppose the random field stationary and the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='v.’s Uℓ centered associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' a) For all ε > 0, it holds, with σn := ∥ �n−1 i=0 Uzi∥2: lim sup n |Sn| σn (log σn) 1 2+ε ≤ 1, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (7) b) If moreover the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' has a summable series of correlations, then, for all ε > 0, |Sn| = O( � Vn (log n) 1 2+ε), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (8) 8 GUY COHEN AND JEAN-PIERRE CONZE If Vn ≤ Cn2 (log n)−(1+η)) for some constants C, η > 0, then lim n Sn n = 0, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' A) Recall that σn = ∥ �n−1 i=0 Uzi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' In case 2) we may assume ∥U0∥2 = 1, and then in all cases σn ≤ n and by association σn ≥ n 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' We have in case 1) σn = √Vn and in case 2b), for associated variables, by (4): σn ≤ (� p⟨Up, U0⟩) 1 2 √Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' By association, σn is non-decreasing and tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' For ρ > 1, let nk = nk(ρ) be a strictly increasing sequence of integers such that ρk < σnk ≤ ρk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Since 1 ≤ σ2 k+1 − σ2 k ≤ 1 + 2k, such a sequence exists after a certain rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' By the choice of (nk) we have ρk < σnk ≤ ρk+1 < σnk+1 ≤ ρk+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (9) Moreover, we have σnk+1/σnk ≤ ρ2 and, since σn ≤ n, nk ≥ ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Let (λn) be a non decreasing sequence of positive numbers such that λnk > √ 2, lim supk λnk+1/λnk ≤ 1, � k P ��� �nk−1 i=0 Uzi �� ≥ (λnk − √ 2) ∥ �nk−1 i=0 Uzi∥2 � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' (10) By the previous inequalities and by Newman-Wright’s inequality (2) for the sequence of centered associated random variables 1 (Wi) = (Uzi), we have � k P( max 0≤j≤nk−1 �� j � i=0 Uzi �� ≥ λnk ∥ nk−1 � i=0 Uzi∥2) ≤ 2 � k P(| nk−1 � j=0 Uzj| ≥ (λnk− √ 2)∥ nk−1 � j=0 Uzj∥2) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' By the Borel-Cantelli lemma, it follows: lim sup k max0≤j≤nk+1−1 �� �j i=0 Uzi �� λnk+1 σnk+1 ≤ 1, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' Hence P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFJT4oBgHgl3EQfjyxF/content/2301.11576v1.pdf'} +page_content=' lim sup k max0≤j 0, q > 0, and k are real constants, and β satisfies +rij = 0, +si = 0. +(1.2) +In this case, F is neither a Berwald nor Landsberg nor a Douglas metric. +It is remarkable that, for a generalized Berwald (α, β)-metric F = αφ(s), s = β/α, on an +n-dimensional manifold M, we show that S = 0 if and only if F is Riemannian or β is a Killing +form with constant length (see Lemma 3.2). +The celebrated Szab´o rigidity theorem states that every 2-dimensional Berwald surface is +either locally Minkowskian or Riemannian (see [13]). +Every Berwald metric has vanishing +S-curvature [22]. Then, Theorem 1.1 is an extension of Szab´o’s result for (α, β)-metrics. +The condition of generalized Berwaldness can not be dropped from the assumption of The- +orem 1.1. There are many two-dimensional (α, β)-metrics with vanishing S-curvature which +are not Riemannian nor locally Minkowskian. For example, let us consider Shen’s Fish-Tank +Randers metric F = α + β on R2 given by following +F = +� +(xv − yu)2 + (u2 + v2)(1 − x2 − y2) +1 − x2 − y2 ++ +yu − xv +1 − x2 − y2. +F has vanishing S-curvature [14] while it is not a generalized Berwald metric (∥β∥α = +� +x2 + y2). +This metric is not Riemannian. Also, F has vanishing flag curvature while it is not locally +Minkowskian. +In Theorem 1.1, the vanishing of S-curvature is necessary. For example, see the following. + +On a Class of Generalized Berwald Manifolds +3 +Example 1. Let us consider G := {(x, y) ∈ R2|y > 0} and define a multiplication on G by +(x1, y1) ∗ (x2, y2) := (x2y1 + x1, y1y2), for (xi, yi) ∈ G, i = 1, 2. (G, ∗) is a Lie group [8]. One +can introduce a Randers metric F on G by +F = +� +2dx2 + 2dxdx + 2dy2 +y ++ dx + dy +y +. +(1.3) +Then +a11 = a22 = 2 +y2, +a21 = a12 = 1 +y2, +b1 = b2 = 1 +y, +a11 = a22 = 2 +3y2, +a12 = a21 = −1 +3y2, +b1 = b2 = 1 +3y. +It is easy to see that α is a positive-definite Riemannian metric. Also, we get +b := ∥β∥α = +� +aijbibj = +� +bibi = +� +b1b1 + b2b2 = +� +2 +3 +It follows that F is a positive-definite Randers metric on G. In [26], Vincze showed that a +Randers metric F = α + β is a generalized Berwald metric if and only if β is of constant length +with respect to α. Then, the Randers metric defined by (1.3) is a generalized Berwald metric. +We have +dβ = 1 +y2dx ∧ dy ̸= 0. +Thus β is not closed which implies that F can not be a Berwald metric. We have +s11 = s22 = 0, +s12 = 1 +y2 = −s21, +s1 = − 1 +3y, +s2 = 1 +3y. +(1.4) +where sij and si defined by (3.1) and (3.2). We claim that F has not vanishing S-curvature. +On contrary, let S = 0. Then by Lemma 3.2.1 in [5], we have rij + bisj + bjsi = 0. Contracting +it with bi yields +rj + b2sj = 0. +By considering ri + si = 0 and b2 = 2/3, we must have si = 0 which contradicts with (1.4). +Then, the Randers metric (1.3) is a generalized Berwald metric with B ̸= 0, L ̸= 0, D ̸= 0, +S ̸= 0 and E ̸= 0. We claim that F is not R-quadratic. On the contrary, let the Finsler metric +F defined by (1.3) be R-quadratic. By Theorem 6.2.1 in [5], if the two-dimensional Randers +metric (1.3) is R-quadratic then it has constant S-curvature S = 3cF, for some real constant +c. In [23], Xu-Deng proved that every homogeneous Finsler metric of isotropic S-curvature has +vanishing S-curvature. Thus, we must have S = 0 which is a contradiction. Therefore, the +Randers metric (1.3) is not R-quadratic. +Theorem 1.1 may not be held for an (α, β)-metric of constant S-curvature. For example, at +a point x = (x1, x2) ∈ R2 and in the direction y = (y1, y2) ∈ TxR2, consider the Riemannian +metric α = α(x, y) and one form β = β(x, y) as follows +α(x, y) := +� +(y1)2 + e2x1(y2)2, +β(x, y) := y1. +(1.5) +Then sij = 0 and rij = b2aij − bibj which yield ri + si = 0. If φ = φ(s) satisfies +Φ = −6k φ∆2 +b2 − s2, +(1.6) + +On a Class of Generalized Berwald Manifolds +4 +for some constant k, then F = αφ(β/α) has constant S-curvature S = 3kF (see [6]). Here, +∆ = ∆(b, s) and Φ = Φ(b, s) defined by (3.3) and (3.4), respectively. The existence of regular +solution of (1.6) for arbitrary k ∈ R, when α and β are given by (1.5), is proved in [10]. It is +easy to see that F is a generalized Berwald metric while it is not a Berwald metric. +We must mention that Theorem 1.1 does not hold for Finsler metrics of dim(M) ≥ 3. +Denote generic tangent vectors on S3 as u∂/∂x + v∂/∂y + w∂/∂z. The Finsler functions for +Bao-Shen’s Randers metrics F = α + β are given by the following +F = +� +K(cu − zv + yw)2 + (zu + cv − xw)2 + (xv + cw − yu)2 +1 + x2 + y2 + z2 +± +√ +K − 1 (cu − zv + yw) +1 + x2 + y2 + z2 +, +where K > 1 is a real constant. For these metrics, we have +b := ∥β∥α = +� +1 − 1 +K +One can see that, the one form β is not closed, and then F can not be a Douglas metric. This +family of Randers metrics is generalized Berwald metrics with S = 0 which are not Berwaldian. +In [4], it is proved that every 3-dimensional closed manifold admits a non-Riemannian +generalized Berwald metric. However, the closeness condition is very restrictive. Indeed, for +every manifold M with dim(M) ≥ 3, one can construct generalized Berwald (α, β)-metrics that +are not Berwaldian. +Example 2. The projective spherical metric on R3 is given by the following +α := +� +(1 + ||X||2)||Y ||2 − ⟨X, Y ⟩2 +1 + ⟨X, X⟩ +, X ∈ R3, Y ∈ TXR3, +(1.7) +where ⟨, ⟩ and ||.|| denote the Euclidean inner product and norm on R3, respectively. +Put +X = (x, y, z) and Y = (u, v, w). Suppose that β = κ(b1u + b2v + b3w) is a Killing 1-form of α, +where 0 < κ < 1. By a simple calculation, we get +b1 = A1 +2y + A1 +3z + C1 +1 + ⟨X, X⟩ +, +b2 = A2 +1x + A2 +3z + C2 +1 + ⟨X, X⟩ +, +b3 = A3 +1x + A3 +2y + C3 +1 + ⟨X, X⟩ +, +(1.8) +where A = (Ai +j) is an antisymmetric real matrix, and C = (Ci) is a constant vector in R3. Let +us put +C = (0, 1, 0), +A1 +2 = A2 +3 = 0, +A1 +3 = 1. +(1.9) +In this case, β is a Killing 1-form with ∥β∥α = κ < 1 such that is not closed. According to +Shen’s theorem in [12], a regular (α, β)-metric is a Berwald metric if and only if β is parallel +with respect to α. Using the Riemannian metric (1.7) and 1-form β satisfying (1.8) and (1.9), +one can construct generalized Berwald (α, β)-metrics which are not Berwaldian. +Homogeneous Finsler manifolds are those Finsler manifolds (M, F) that the orbit of the +natural action of I(M, F) on M at any point of M is the whole M. For the class of homogeneous +generalized Berwald (α, β)-metrics, we prove the following. +Corollary 1.1. Let F = αφ(s), s = β/α, be a two-dimensional homogeneous generalized +Berwald (α, β)-metric on a manifold M. Then F has isotropic S-curvature if and only if it is +a Riemannian metric of constant sectional curvature or a locally Minkowskian metric. + +On a Class of Generalized Berwald Manifolds +5 +Let G be a connected Lie group with a bi-invariant Finsler metric ¯F, and H a closed +subgroup of G. Denote the Lie algebras of G and H as g and h, respectively. Let ρ be the natural +projection from G to M = G/H. Then there exists a uniquely defined G-invariant metric F +on M such that for any g ∈ G, the tangent map ρ∗ : +� +TgG, ¯F(g, .) +� +→ +� +Tπ(g)M, F(π(g), .) +� +is +a submersion (see Lemma 3.1 in [24]). The G-invariant Finsler metric F is called the normal +homogeneous metric induced by ¯F. The pair (M, F) is said the normal homogeneous space +induced by ρ : G → M = G/H and ¯F. For normal homogeneous generalized Berwald metrics, +we have the following. +Corollary 1.2. Every two-dimensional normal homogeneous generalized Berwald (α, β)-metric +is a Riemannian metric of non-negative constant sectional curvature or a locally Minkowskian +metric. +A Finsler metric F on a manifold M is called a generalized normal homogeneous metric +if for every x, y ∈ M, there exists a δ(x)-translation of (M, F) sending x to y (see [30]). In +this paper, we consider two-dimensional homogeneous generalized Berwald Randers metric of +generalized normal-type and prove the following. +Corollary 1.3. Every two-dimensional generalized normal homogeneous generalized Berwald +Randers metric must be a Riemannian metric of non-negative constant sectional curvature or +a locally Minkowskian metric. +Let (G, ·) be a connected Lie group with identity element e, and λg denotes the left trans- +lation by g ∈ G. A Finsler metric F on G is called a left invariant Finsler metric if it satisfies +F ◦ (λg)∗ = F for any g ∈ G. In [2], Aradi proved that left invariant Finsler metrics are +generalized Berwald metrics. For this class of Finsler metrics, we have the following. +Theorem 1.2. Every left invariant Finsler surface has vanishing S-curvature if and only if it +is a Riemannian metric of constant sectional curvature. +By considering Theorems 1.1 and 1.2, it seems that two-dimensional generalized Berwald +(α, β)-metrics with vanishing S-curvature may be only Riemannian. But we do not find a short +proof for this conjecture. +A Finsler metric F on a manifold M is called an isotropic Berwald metric, if its Berwald +curvature is given by +By(u, v, w) = cF −1� +h(u, v) +� +w − gy(w, ℓ)ℓ +� ++ h(v, w) +� +u − gy(u, ℓ)ℓ +� ++h(w, u) +� +v − gy(v, ℓ)ℓ +� ++ 2FCy(u, v, w)ℓ +� +. +(1.10) +where hy(u, v) = gy(u, v)−F −2(y)gy(y, u)gy(y, v) is the angular form in direction y, C denotes +the Cartan torsion of F and c ∈ C∞(M). Every Berwald metric is an isotropic Berwald metric +with c = 0. The Funk metrics are isotropic Berwald metrics with c = 1/2. As a straightforward +conclusion of Theorem 1.2, one can get the following. +Corollary 1.4. Every left invariant isotropic Berwald surface must be a Riemannian surface +of constant sectional curvature. + +On a Class of Generalized Berwald Manifolds +6 +2 +Preliminaries +Let M be an n-dimensional C∞ manifold, TM = � +x∈M TxM the tangent space and TM0 := +TM − {0} the slit tangent space of M. +Let (M, F) be a Finsler manifold. +The following +quadratic form gy : TxM × TxM → R is called fundamental tensor +gy(u, v) := 1 +2 +∂2 +∂s∂t +� +F 2(y + su + tv) +� +s=t=0, +u, v ∈ TxM. +Let x ∈ M and Fx := F|TxM. To measure the non-Euclidean feature of Fx, one can define +Cy : TxM × TxM × TxM → R by +Cy(u, v, w) := 1 +2 +d +dt +� +gy+tw(u, v) +� +t=0 , +u, v, w ∈ TxM. +The family C := {Cy}y∈TM0 is called the Cartan torsion. +For a Finsler manifold (M, F), its induced spray on TM is denoted by G = G(x, y) which +in a standard coordinate (xi, yi) for TM0 is given by G = yi∂/∂xi − 2Gi(x, y)∂/∂yi, where +Gi := 1 +4gil� ∂2F 2 +∂xk∂yl yk − ∂F 2 +∂xl +� +. +For a Finsler metric F on an n-dimensional manifold M, the Busemann-Hausdorff volume form +dVF = σF(x)dx1 · · · dxn is defined by +σF(x) := +Vol +� +Bn(1) +� +Vol +� +(yi) ∈ Rn �� F +� +yi ∂ +∂xi|x +� +< 1 +�. +Let Gi denote the geodesic coefficients of F in the same local coordinate system. Then for +y = yi∂/∂xi|x ∈ TxM, the S-curvature is defined by +S(y) := ∂Gi +∂yi (x, y) − yi ∂ +∂xi +� +ln σF(x) +� +. +(2.1) +For a vector y ∈ TxM0, the Berwald curvature By : TxM × TxM × TxM → TxM is defined +by By(u, v, w) := Bi +jkl(y)ujvkwl∂/∂xi|x, where +Bi +jkl := +∂3Gi +∂yj∂yk∂yl . +F is called a Berwald metric if B = 0. Every Berwald metric satisfies S = 0 (see [22]). +For y ∈ TxM, define the Landsberg curvature Ly : TxM × TxM × TxM → R by +Ly(u, v, w) := −1 +2gy +� +By(u, v, w), y +� +. +A Finsler metric F is called a Landsberg metric if L = 0. +Taking a trace of Berwald curvature B give us the mean of Berwald curvature E which is +defined by Ey : TxM × TxM → R, where +Ey(u, v) := 1 +2 +n +� +i=1 +gij(y)gy +� +By(u, v, ∂i), ∂j +� +. +(2.2) +where {∂i} is a basis for TxM at x ∈ M. In local coordinates, Ey(u, v) := Eij(y)uivj, where +Eij := 1 +2Bm +mij. + +On a Class of Generalized Berwald Manifolds +7 +Taking a horizontal derivation of the mean of Berwald curvature E along Finslerian geodesics +give us the H-curvature H = H(x, y) which is defined by Hy = Hijdxi ⊗ dxj, where +Hij := Eij|mym. +Here, “|” denotes the horizontal covariant differentiation with respect to the Berwald connection +of F. +For a non-zero vector y ∈ TxM0, one can define Dy : TxM × TxM × TxM → TxM by +Dy(u, v, w) := Di +jkl(y)uivjwk∂/∂xi|x, where +Di +jkl := +∂3 +∂yj∂yk∂yl +� +Gi − +2 +n + 1 +∂Gm +∂ym yi +� +. +(2.3) +D is called the Douglas curvature. F is called a Douglas metric if D = 0. +For a non-zero vector y ∈ TxM0, the Riemann curvature is a family of linear transformation +Ry : TxM → TxM which is defined by Ry(u) := Ri +k(y)uk∂/∂xi, where +Ri +k(y) = 2∂Gi +∂xk − +∂2Gi +∂xj∂yk yj + 2Gj ∂2Gi +∂yj∂yk − ∂Gi +∂yj +∂Gj +∂yk . +(2.4) +The family R := {Ry}y∈TM0 is called the Riemann curvature. +For a flag P := span{y, u} ⊂ TxM with the flagpole y, the flag curvature K = K(P, y) is +defined by +K(x, y, P) := +gy +� +u, Ry(u) +� +gy(y, y)gy(u, u) − gy(y, u)2. +(2.5) +The flag curvature K(x, y, P) is a function of tangent planes P = span{y, v} ⊂ TxM. A Finsler +metric F is of scalar flag curvature if K(x, y, P) = K(x, y) is independent of P. Also, F is +called of isotropic and constant flag curvature if K = K(x) and K = constant, respectively. +Throughout this paper, we use the Berwald connection on Finsler manifolds. The pullback +bundle π∗TM admits a unique linear connection, called the Berwald connection. Let (M, F) +be an n-dimensional Finsler manifold. Let {ej} be a local frame for π∗TM, {ωi, ωn+i} be the +corresponding local coframe for T ∗(TM0) and {ωi +j} be the set of local Berwald connection forms +with respect to {ej}. Then the connection forms are characterized by the structure equations +as follows +• Torsion freeness: +dωi = ωj ∧ ωi +j. +(2.6) +• Almost metric compatibility: +dgij − gkjωk +i − gikωk +j = −2Lijkωk + 2Cijkωn+k, +(2.7) +where ωi := dxi and ωn+k := dyk + yjωk +j. +The horizontal and vertical covariant derivations with respect to the Berwald connection re- +spectively are denoted by “|” and “, ”. For more details, one can see [13]. + +On a Class of Generalized Berwald Manifolds +8 +3 +Proof of Theorems 1.1 and 1.2 +For an (α, β)-metric, let us define bi;j by bi;jθj := dbi − bjθj +i , where θi := dxi and θj +i := γj +ikdxk +denote the Levi-Civita connection form of α. Let +rij := 1 +2(bi;j + bj;i), +sij := 1 +2(bi;j − bj;i), +ri0 := rijyj, +r00 := rijyiyj, +rj := birij, +(3.1) +si0 := sijyj, +sj := bisij, +si +j = aimsmj, +si +0 = si +jyj, +r0 := rjyj, +s0 := sjyj. +(3.2) +Put +Q := +φ′ +φ − sφ, +∆ := 1 + sQ + (b2 − s2)Q′, +Θ := Q − sQ′ +2∆ +, +(3.3) +Φ := −(Q − sQ′)(n∆ + 1 + sQ) − (b2 − s2)(1 + sQ)Q′′, +(3.4) +Ψ := +φ′′ +2 +� +(φ − sφ′) + (b2 − s2)φ′′�, +where b := ∥β∥α. Let Gi = Gi(x, y) and ¯Gi +α = ¯Gi +α(x, y) denote the coefficients of F and α, +respectively, in the same coordinate system. By definition, we have +Gi = Gi +α + αQsi +0 + α−1(r00 − 2Qαs0)(Θyi + αΨbi). +(3.5) +Clearly, if β is parallel with respect to α, that is rij = sij = 0, then Gi = Gi +α = γi +jk(x)yjyk are +quadratic in y. In this case, F reduces to a Berwald metric. +For an (α, β)-metric F = αφ(s), s = β/α, on an n-dimensional manifold M, the S-curvature +is given by +S = +� +2Ψ − f ′(b) +bf(b) +� +(r0 + s0) − +Φ +2α∆2(r00 − 2αQs0), +(3.6) +where +f(b) := +� π +0 sinn−2 t T(b cos t)dt +� π +0 sinn−2 tdt +, +T(s) := φ(φ − sφ′)n−2� +(φ − sφ′) + (b2 − s2)φ′′� +. +For more details, see [6]. +To prove Theorem 1.1, we need the following key lemma. +Lemma 3.1. ([4][20]) Let F = αφ(s), s = β/α, be an (α, β)-metric on a manifold M. Then +F is a generalized Berwald metric if and only if β has constant length with respect to α. +Proof. In [20], Tayebi-Barzegari showed that every (α, β)-metric F = αφ(s), s = β/α, with +sign property is a generalized Berwald metric if and only if the dual vector field β♯ is of +constant Riemannian length. In [9], Ivanov proved that a two-dimensional Finsler metric is a +generalized Berwald metric if and only if it is monochromatic, i.e., Finsler metrics such that +each two tangent spaces are isomorphic as normed spaces. In [4], Bartelmeß-Matveev extended +his result for n-dimensional Finsler metric. It follows that an (α, β)-metric is a generalized +Berwald metric if and only if dual vector field β♯ is of constant Riemannian length. +In [6], Cheng-Shen characterized (α, β)-metrics with isotropic S-curvature on a manifold M +of dimension n ≥ 3. Soon, they found that their result holds for the class of (α, β)-metrics with +constant length one-forms, only. Here, we give a characterization of the class of generalized +Berwald metrics with vanishing S-curvature. + +On a Class of Generalized Berwald Manifolds +9 +Lemma 3.2. Let F = αφ(s), s = β/α, be a generalized Berwald (α, β)-metric on an n- +dimensional manifold M. Then S = 0 if and only if F is Riemannian or β is a Killing form +with constant length. +Proof. Let b := ∥β∥α = +� +amjbjbm = √bmbm. Then, the following holds +∂b +∂xi = 1 +bbmbm|i = 1 +b(ri + si). +(3.7) +By Lemma 3.1, we have b = constant. Then, by (3.7) we obtain ri + si = 0. In this case, (3.6) +reduces to following +S = − +Φ +2α∆2(r00 − 2αQs0), +(3.8) +By (3.8), S = 0 if and only if Φ = 0 or β satisfies +r00 = 2αQs0. +(3.9) +In the case of Φ = 0, F reduces to a Riemannian metric (see Proposition 2.2 in [11]). Now, +let (3.9) holds. We are going to simplify (3.9). For this aim, one can change the y-coordinates +(yi), i = 1, · · · , n, at a point to the polar coordinates (s, uA), where A = 2, · · · , n (see [6]). +For an arbitrary and fix point x ∈ M, let us take an orthonormal basis ei at x such that the +Riemannian metric is written as α = +��n +i=1(yi)2 and its related one-form is given by β = by1, +where b := ||β||α. Let us fix an arbitrary number s such that |s| < b. Define +¯α = +� +� +� +� +n +� +A=2 +(yA)2. +Then, by β = sα we get +y1 = +s +√ +b2 − s2 ¯α, +yA = uA. +(3.10) +Also, we have +α = +b +√ +b2 − s2 ¯α, +β = +bs +√ +b2 − s2 ¯α. +(3.11) +Let us put +¯r10 := +n +� +A=2 +r1AyA, +¯s10 := +n +� +A=2 +s1AyA, +¯r00 := +n +� +A,B=2 +rAByAyB, +¯r0 := +n +� +A=2 +rAyA, +¯s0 := +n +� +A=2 +sAyA. +Then we get the following useful relations +r1 = br11, +rA = br1A, +s1 = 0, +sA = bs1A, +(3.12) +r00 = +s2 +b2 − s2 ¯α2r11 + +2s +√ +b2 − s2 ¯α¯r10 + ¯r00, +(3.13) +r10 = +s +√ +b2 − s2 ¯αr11 + ¯r10, +s0 = ¯s0 = b¯s10. +(3.14) +Using (3.11)-(3.14), the equation (3.9) can be written as follows +¯r00 + +s2 +b2 − s2 ¯α2 r11 = +2 +√ +b2 − s2 +� +b2Q¯s10 − s¯r10 +� +¯α. +(3.15) + +On a Class of Generalized Berwald Manifolds +10 +By (3.15), we get two following relations +¯r00 + +s2 +b2 − s2 ¯α2r11 = 0, +(3.16) +b2Q¯s10 − s¯r10 = 0. +(3.17) +On the other hand, (3.11) implies that +s2 +b2 − s2 ¯α2 − 1 +b2β2 = 0. +(3.18) +By (3.16) and (3.18), we get +b2¯r00 + β2r11 = 0. +(3.19) +The following hold +∂¯r00 +∂y1 = 0, +∂β +∂y1 = b. +Then by differentiating (3.19) with respect to y1 we have β/br11 = 0. Thus r11 = 0 and by +putting it in (3.19), we get ¯r00 = 0. Putting these relations in (3.13) and (3.14) imply that +r00 = +2s¯α +√ +b2 − s2 ¯r10 = 2β +b ¯r10, +(3.20) +r10 = ¯r10. +(3.21) +Now, by considering (3.20) and (3.21), we divide the problem into two cases: (a) ¯r10 = 0 and +(b) ¯r10 ̸= 0. +Case (a): +¯r10 = 0. +In this case, by (3.20) we get rij = 0. +Putting it in (3.9) implies +that si = 0. In this case, β reduces to a Killing one-form of constant length with respect to α. +Case (b): ¯r10 ̸= 0. We have ∂¯r10/∂y1 = 0 and ∂¯s10/∂y1 = 0. Thus, differentiating (3.17) with +respect to y1 yields +(s)y1¯r10 = b2(Q)y1¯s10. +(3.22) +Contracting (3.17) with (s)y1 give us +s(s)y1¯r10 = b2Q(s)y1¯s10. +(3.23) +By (3.22) and (3.23), we get +� +(s)y1Q − s(Q)y1 +� +¯s10 = 0. +(3.24) +According to (3.24), we get ¯s10 = 0 or Q(s)y1 = s(Q)y1. Let ¯s10 = 0 holds. Then (3.17) reduces +to s = 0, which is impossible. Then, we have +Q(s)y1 = s(Q)y1, +which is equal to +(Q)y1 +Q += (s)y1 +s +. +(3.25) + +On a Class of Generalized Berwald Manifolds +11 +Using sy1 ̸= 0 and (Q)y1 = sy1(Q)s, then (3.25) give us +(Q)s +Q += 1 +s +which yields +ln(Q) − ln(s) = c, +where c is a real constant. Thus Q = ks, where k is a non-zero real constant. In this case, we +get φ = +√ +1 + ks2 which shows that F is a Riemannian metric. This completes the proof. +Here, we solve an ODE which will appear in the proof of Theorem 1.1. +Lemma 3.3. Let F = αφ(s), s = β/α, be an (α, β)-metric on a manifold M. Suppose that φ +satisfies following +αΘ2 + 2Λ1Θ1 + Λ2Q = 0, +(3.26) +where +Λ1 := biαyi, +Λ2 := bibjαyiyj, +Θ1 := biQyi, +Θ2 := bibjQyiyj. +Then F is a singular Finsler metric given by +φ = c exp +� � s +0 +k1t + k2 +√ +b2 − t2 +1 + t +� +k1t + k2 +√ +b2 − t2�dt +� +, +(3.27) +where k1 and k2 are real constants, and c > 0 is a non-zero constant. +Proof. Let us put +Ajk := α2ajk − yjyk. +Then, the followings hold +αyi = 1 +αyi, +αyjyk = 1 +α3Ajk, +αyjykyl = − 1 +α5 +� +Ajkyl + Ajlyk + Alkyj +� +. +Also, one can obtain the following +Λ1 = s, +(3.28) +Λ2 = 1 +α(b2 − s2), +(3.29) +Θ1 = 1 +α(b2 − s2)Q′, +(3.30) +Θ2 = 1 +α2(b2 − s2) +� +(b2 − s2)Q′′ − 3sQ′� +. +(3.31) +Suppose that (3.26) holds. Putting (3.28)-(3.31) into (3.26) yield +Q′′ − +s +b2 − s2Q′ + +1 +b2 − s2Q = 0, + +On a Class of Generalized Berwald Manifolds +12 +which implies that +Q = k1s + k2 +√ +b2 − s2, +(3.32) +where k1 and k2 are real constants. Considering (3.3), the equation (3.32) is equal to following +� +1 + k1s2 + k2s +√ +b2 − s2 +� +φ′ = +� +k1s + k2 +√ +b2 − s2 +� +φ. +(3.33) +By (3.33), we get (3.27). It is an almost regular (α, β)-metric, namely, it is singular in two +directions y = (±b, 0, 0) ∈ TxM at any point x (for more details, see [12]). +The function φ in (3.27) is specifically has been seen for the first time in Asanov’s paper [3] +as follows +φ = c exp +� � s +0 +k +√ +b2 − t2 +1 + tk +√ +b2 − t2dt +� +. +(3.34) +Then, its more general form (3.27) found by Shen in [12], where he looked for unicorn metrics, +namely the Landsberg metrics which are not Berwaldian. +Shen realized that the function +φ = φ(b, s) in (3.27) can be expressed in terms of elementary functions. See (7.2) in [12]. +Here, we show that the Douglas curvature of Finsler surfaces satisfies a special relation. +More precisely, we prove the following. +Lemma 3.4. The Douglas curvature of any Finsler surface (M, F) satisfies +Di +jkl|sys = Djklyi +(3.35) +for some tensor D = Dijkdxi ⊗ dxj ⊗ dxk which are homogeneous of degree -1 in y. +Proof. By definition, the Douglas curvature of a two-dimensional Finsler metric is given by +Di +jkl = Bi +jkl − 2 +3 +� +Ejkδi +l + Eklδi +j + Eljδi +k + Ejk,lyi� +. +(3.36) +Taking a horizontal derivation of (3.36) along Finslerian geodesics and using yi +|s = 0 give us +the following +Di +jkl|mym = Bi +jkl|mym − 2 +3 +� +Hjkδi +l + Hklδi +j + Hljδi +k + Ejk,l|mymyi� +. +(3.37) +Every Finsler surface is of scalar flag curvature K = K(x, y). Then, by (11.24) in [13] we have +Bi +jml|kyk = 2KCjlmyi − 1 +3 +� +ylδi +m + ymδi +l − 2glmyi� +Kj − 1 +3 +� +yjδi +m + ymδi +j − 2gjmyi� +Kl +−1 +3 +� +yjδi +l + ylδi +j − 2gjlyi� +Km − 1 +3F 2� +Kjmhi +l + Kjlhi +m + Klmhi +j +� +, (3.38) +where yi = FFyi, Kj := Kyj and Kjk := Kyjyk. Taking a trace of (3.38) yields +Hjl = −1 +2 +� +ylKj + yjKl + F 2Kjl +� +. +(3.39) +By putting (3.38) and (3.39) in (3.37) we get +Di +jkl|mym = 1 +3 +� +6KCjkl + 2 +� +gklKj + gkjKl + gjlKk +� ++ +� +yjKkl + ykKjl + ylKkj +� +−2Ejk,l|mym� +yi.(3.40) +(3.40) give us (3.35). + +On a Class of Generalized Berwald Manifolds +13 +Now, we show that the covariant derivative of Berwald curvature of 2-dimensional gener- +alized Berwald (α, β)-metric with vanishing S-curvature satisfies an interesting relation which +will play an important role in the proof of Theorem 1.1. +Lemma 3.5. The Berwald curvature of any non-Riemannian 2-dimensional generalized Berwald +(α, β)-metric with vanishing S-curvature satisfies following +hm +p Bp +jkl|sys = +sm +0|0(αjklQ + αjkQl + αlkQj + αljQk + αQjkl + αlQjk + αjQlk + αkQjl) ++sm +0(αjklQ|0 + αjkQl|0 + αlkQj|0 + αljQk|0 + αQjkl|0 + αlQjk|0 + αjQlk|0 ++αkQjl|0) + Am +lXjk + Am +jXlk + Am +kXjl + Bm +l Yjk + Bm +jYlk + Bm +kYjl = 0, (3.41) +where +Xjk := Q|0αjk + Qk|0αj + Qj|0αk + Qjk|0α, +Yjk := Qαjk + Qkαj + Qjαk + αQjk, +Am +l := sm +l − F −2s0 +lym, +Bm +l := Am +l|sys = sm +l|0 − F −2s0 +l|0ym. +Proof. By Lemma 3.2, we have rij = 0 and sj = 0. In this case, (3.5) reduces to following +Gi = Gi +α + αQsi +0. +(3.42) +Taking three vertical derivations of (3.42) with respect to yj, yl and yk give us +Bi +jkl = +si +0 +� +αQjkl + αlQjk + αjQlk + αkQjl + αjklQ + αjkQl + αlkQj + αljQk +� ++ si +j +� +Qαlk + Qkαl + Qlαk + αQlk +� ++ si +k +� +Qαjl + Qjαl + Qlαj + αQjl +� ++si +l +� +Qαjk + Qkαj + Qjαk + αQjk +� +. +(3.43) +Contracting (3.43) with hm +i implies that +hm +i Bi +jkl = +� +αjklQ + αjkQl + αlkQj + αljQk + αQjkl + αlQjk + αjQlk + αkQjl +� +sm +0 ++ +� +Qαjk + Qkαj + Qjαk + αQjk +� +(sm +l − F −2s0 +lym) ++ +� +Qαlk + Qkαl + Qlαk + αQlk +� +(sm +j − F −2s0 +jym) ++ +� +Qαjl + Qjαl + Qlαj + αQjl +� +(sm +k − F −2s0 +kym). +(3.44) +On the other hand, by taking a horizontal derivation of Douglas curvature along Finslerian +geodesics and contracting the result with hm +i , we get the following +hm +i Di +jkl|sys = hm +i Bi +jkl|sys − 2 +3 +� +Hjkhm +l + Hklhm +j + Hljhm +k +� +. +(3.45) +Contracting (3.35) with hm +i yields +hm +i Di +jkl|sys = 0. +(3.46) +Since S = 0, then by definition we get H = 0. Thus, (3.45) and (3.46) imply that +hm +i Bi +jkl|sys = 0. +(3.47) +We have hm +i|s = 0. Then, by considering (3.47), we have +� +hm +i Bi +jkl +� +|sys = hm +i Bi +jkl|sys = 0. +(3.48) +Therefore, taking a horizontal derivation of (3.44) along Finslerian geodesic and considering +(3.48) give us (3.41). + +On a Class of Generalized Berwald Manifolds +14 +Proof of Theorem 1.1: Taking a horizontal derivation of ymsm +0 = 0 with respect to the +Berwald connection of F implies that +ym|0sm +0 + ymsm +0|0 = 0. +(3.49) +Since ym|0 = 0, then (3.49) reduces to following +ymsm +0|0 = 0. +(3.50) +By contracting (3.41) with ym and considering (3.50), we get +s0 +lXjk + s0 +jXlk + s0 +kXjl + s0 +l|0Yjk + s0 +j|0Ylk + s0 +k|0Yjl = 0. +(3.51) +Since sj = 0, then one can get +0 = (sj)|0 = (rm0 + sm0)sm +j + bmsm +j|0 +which considering rij = 0, it reduces to following +bmsm +j|0 = −sm0sm +j. +(3.52) +Multiplying (3.52) with yj yields +bisi +0|0 + si0si +0 = 0. +(3.53) +Also, contracting (3.52) with bj implies that +bisi +j|0bj = 0. +(3.54) +Multiplying (3.51) with bjbkbl and considering (3.53) and (3.54) give us +(αΘ2 + 2Λ1Θ1 + Λ2Q)sm0sm +0 = 0. +(3.55) +By (3.55), we have two cases: if sm +0sm0 = 0, since α is a positive-definite metric, then we +find that β is closed. Therefore, by (3.43) we conclude that F reduces to a Berwald metric. +By Szabo’s rigidity result for Finsler surfaces, F reduces to a locally Minkowskian metric +or a Riemannian metric. On the other hand, every Finsler surface has scalar flag curvature +K = K(x, y). According to Akbar-Zadeh theorem in [1], a Finsler manifold (M, F) of scalar +flag curvature K = K(x, y) has isotropic flag curvature K = K(x) if and only if it has vanishing +H-curvature H = 0. Thus, the obtained Riemannian metric has isotropic sectional curvature. +Now, suppose that F is not a Riemannian metric nor a locally Minkowskian metric. Then, +by (3.55) we have αΘ2 + 2Λ1Θ1 + Λ2Q = 0. By Lemma 3.3 we obtain (1.1). In this case, since +S = 0, then F can not be a Douglas metric. On the other hand, the Berwald curvature of +2-dimensional Finsler manifold is given by +Bi +jkl = − 2 +F 2Ljklyi + 2 +3 +� +Ejkhi +l + Eklhi +j + Ejlhi +k +� +. +(3.56) +See the relation (15) in [21]. If F is a Landsberg metric then by considering S = 0, (3.56) implies +that F is a Berwald metric. This is a contradiction. Then, (1.1) is a generalized Berwald metric +which is not Berwald, Landsberg nor Douglas metric. +Proof of Corollary 1.1: In [23], Xu-Deng proved that every homogeneous Finsler metric +of isotropic S-curvature has vanishing S-curvature. Then, by assumption we get S = 0. The + +On a Class of Generalized Berwald Manifolds +15 +Akbar-Zadeh theorem in [1] stated that a Finsler manifold (M, F) of scalar flag curvature +K = K(x, y) has isotropic flag curvature K = K(x) if and only if it has vanishing H-curvature +H = 0. On the other hand, every Finsler surface has scalar flag curvature K = K(x, y). Thus, +by Akbar-Zadeh theorem we get K = K(x). Every scalar function on M which is invariant +under isometries of (M, F) is a constant function. The homogeneity of (M, F) and invariancy +of the flag curvature under isometries of F imply that K = constant. Then, by Theorem 1.1 +we get the proof. +Proof of Corollary 1.2: Let F = αφ(s), s = β/α, be a two-dimensional normal homogeneous +generalized Berwald (α, β)-metric. In [24], Xu-Deng proved that every normal homogeneous +manifold has vanishing S-curvature and non-negative flag curvature. By Theorem 1.1 and the +same method used in the proof of Corollary 1.1, it follows that F is a Riemannian metric of +non-negative constant sectional curvature or a locally Minkowskian metric. +Proof of Corollary 1.3: Let (G/H, F) be a generalized normal homogeneous Randers man- +ifold. In [30], Zhang-Deng proved that F has vanishing S-curvature (Corollary 3.11). Also, +they showed that any generalized normal homogeneous Randers metric has non-negative flag +curvature (see Proposition 3.13 in [30]). Then, by Theorem 1.1 we get the proof. +According to Theorem 1.1, every two-dimensional generalized Berwald (α, β)-metric with +vanishing S-curvature is Riemannian or locally Minkowskian. Every left invariant Finsler metric +is a generalized Berwald metric [2]. Here, we prove Theorem 1.2 which states that left invariant +Finsler metrics with vanishing S-curvature reduce to Riemannian metrics, only. The approach +of the proof of Theorem 1.2 is completely different from Theorem 1.1. +Proof of Theorem 1.2: To prove a homogeneous surface with S = 0 is Riemannian, we only +need to consider the nontrivial case, namely, a 2-dimensional non-Abelian Lie group G with a +left invariant Finsler metric F. At each y ∈ g with F(y) = 1, there is a gy orthonormal basis +e1 = y and e2 tangent to F = 1. At almost all non-zero y, the spray vector field η is nonzero, +i.e., η(y) is a nonzero multiple of e2. By the homogenous S-curvature formula, we have +S(y) = −I(η) = −Cy(η, e2, e2) = 0. +Then, Cy(e2, e2, e2) = 0 everywhere. The speciality of 2-dimensional spaces implies C = 0 +everywhere, so F is Riemannian. By the same method used to prove Corollary 1.1, one can +conclude that the Riemannian metric is of constant sectional curvature. This completes the +proof. +Proof of Corollary 1.4: By assumption, F has isotropic Berwald curvature +Bi +jkl = cF −1� +hjkhi +l + hjlhi +k + hklhi +j + 2Cjklyi� +. +(3.57) +where c = c(x) is a scalar function on M. In [22], it is proved that every Finsler surface of +isotropic Berwald curvature (3.57) metric has isotropic S-curvature S = 3cF. By Xu-Deng’s +result in [24], F has vanishing S-curvature S = 0. Then, by Theorem 1.2, F reduces to a +Riemannian metric. + +On a Class of Generalized Berwald Manifolds +16 +4 +Some Examples of Generalized Berwald Manifolds +In this section, we are going to give some important examples of the class of generalized Berwald +manifolds. First, by using trans-Sasakian structure, we construct a family of odd-dimensional +generalized Berwald Randers metrics. +Example 3. +� +Odd-dimensional generalized Berwald Randers metrics +� +Let M be a differen- +tiable manifold of dimension 2n + 1. +Suppose that η = ηi(x)dxi, ξ = ξi∂/∂xi and ϕ = +ϕi +j∂/∂xi ⊗ dxj are a 1-form, a vector field, and a (1, 1)-tensor on M, respectively. The triple +(η, ξ, ϕ) is called an almost contact structure on M if it satisfies +ϕ(ξ) = 0, +η(ξ) = 1, +ϕ2 = −I + η ⊗ ξ. +A differentiable manifold of odd dimension 2n+ 1 with an almost contact structure is called an +almost contact manifold. Let a manifold M with the (η, ξ, ϕ) structure admits a Riemannian +metric g such that +g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ). +Then M is called an almost contact metric structure and g is called a compatible metric. In this +case, (η, ξ, ϕ, g) is called almost contact metric structure. An almost contact metric structure +(η, ξ, ϕ, g) on M is called a trans-Sasakian structure if it satisfies +(∇Xϕ)Y = c1 +� +g(X, Y )ξ − η(Y )X +� ++ c2 +� +g(ϕX, Y )ξ − η(Y )ϕX +� +for some scalar functions c1 = c1(x) and c2 = c2(x) on M, where ∇ denotes the Levi-Civita +connection of g. +Now, let (η, ξ, ϕ, g) be a trans-Sasakian structure on M. Define α, β : TM → [0, ∞) by +∀(x, y) ∈ TM, +α(x, y) := +� +gx(y, y), +β(x, y) := ǫ ηx(y), +(4.1) +where 0 < ǫ < 1 be a constant. Then, for the Randers metric F := α + β, we have +||β||α = ǫ. +It follows that the class of Randers metrics induced by trans-Sasakian manifolds (M, η, ξ, ϕ, g) +are (2n + 1)-dimensional generalized Berwald metrics on M. +Here, we give a two-dimensional Randers metric F = α +β with vanishing S-curvature. We +show that if F is a generalized Berwald metric then it reduces to a Riemannian metric. +Example 4. Let y = u∂/∂x + v∂/∂y ∈ T(x,y)R2. Consider the Randers metric F = α + β, +where α = α(y) and β = β(y) are given by +α := +�� +1 + (1 − ǫ2)(x2 + y2) +� +(u2 + v2) + +� +1 + ǫ2 + x2 + y2 +� +(xv − yu)2 +� +1 + (1 − ǫ2)(x2 + y2) +�� +1 + x2 + y2 +, +β := − +ǫ(xv − yu) +1 + (1 − ǫ2)(x2 + y2), + +On a Class of Generalized Berwald Manifolds +17 +and ǫ is a real constant (see [15]). F is defined on the whole sphere for |ǫ| < 1. It is remarkable +that, one can rewrite F in a polar coordinate system, x = r cos(θ), y = r sin(θ). Express +α = +� +a11µ2 + a12µν + a21νµ + a22ν2, +β = b1µ + b2ν, +where +a11 = +1 +(1 + r2) +� +1 + (1 − ǫ2)r2�, +a12 = a21 = 0, +a22 = +r2(1 + r2) +� +1 + (1 − ǫ2)r2�2, +b1 = 0, +b2 = − +ǫr2 +1 + (1 − ǫ2)r2. +Let us put A := det(aij). Then, we get +a11 = +r2(1 + r2) +A +� +1 + (1 − ǫ2)r2�2, +a22 = +1 +A(1 + r2) +� +1 + (1 − ǫ2)r2�, +a12 = a21 = 0, +b1 = 0, +b2 = − +ǫr2 +A(1 + r2) +� +1 + (1 − ǫ2)r2�2. +Therefore, we obtain +∥β∥2 +α = aijbibj = bibi = +ǫ2r4 +A(1 + r2) +� +1 + (1 − ǫ2)r2�3 ̸= constant. +(4.2) +(4.2) means that F is not a generalized Berwald metric. On the other hand, a direct computa- +tion yields +r11 = r22 = 0, +r12 = r21 = +ǫ3r3 +(1 + r2) +� +1 + (1 − ǫ2)r2�2, +s11 = s22 = 0, +s12 = +ǫr +� +1 + (1 − ǫ2)r2�2 = −s21 +s1 = +ǫ2r +(1 + r2) +� +1 + (1 − ǫ2)r2�, +s2 = 0, +r1 = − +ǫ4r5 +A(1 + r2)2� +1 + (1 − ǫ2)r2�4, +r2 = 0. +It is easy to find that rij + bisj + bjsi = 0. Then by Lemma 3.2.1 in [5], we get S = 0. Also, +one can see that the following holds +ri + si = ǫ2r +� +A(1 + r2)(1 + (1 − ǫ2)r2)3 − ǫ2r4� +A(1 + r2)2� +1 + (1 − ǫ2)r2�4 +. +(4.3) +According to (4.3), F is a generalized Berwald metric (equivalently, ri + si = 0) if and only if +ǫ = 0 or the following holds +� +1 + (1 − ǫ2)r2�4 = 0. +(4.4) +(4.4) contradicts with |ǫ| < 1. Therefore, F is a generalized Berwald metric if and only if ǫ = 0 +or equivalently β = 0. In this case, F reduces to the standard Riemannian metric F = α. + +On a Class of Generalized Berwald Manifolds +18 +Example 5. (Xu) It is proved that a Finsler metric F = F(x, y) is of Randers type F = α+β if +and only if it is a solution of the navigation problem on a Riemannian manifold (M, h) (see [5]). +Zermelo navigation is an efficient method to study of Randers metrics with certain Riemannian +and non-Riemannian curvature properties. More precisely, any Randers metric F = α + β on +a manifold M is a solution of the following Zermelo navigation problem +h +� +x, y +F − Wx +� += 1, +where h = +� +hij(x)yiyj is a Riemannian metric and W = Wi(x)∂/∂xi is a vector field such +that +h(x, −Wx) = +� +hij(x)Wi(x)Wj(x) < 1. +In fact, α and β are given by +α = +√λh2 + W0 +λ +, +β = −W0 +λ , +respectively and moreover, +λ := 1 − ∥W∥2 +h, +W0 := hijWiyj. +For more details, see [5]. Then, F can be written as follows +F = +� +λh2 + W2 +0 +λ +− W0 +λ . +(4.5) +In this case, the pair (h, W) is called the navigation data of F. +Now, let G/H be any homogeneous manifold and g = h + m is its reductive decomposition. +Suppose m = m0 + m1 be an Ad(H)-invariant decomposition, in which m0 is 1-dimensional +and the Ad(H)-action on m0 is trivial. Let h be a G-invariant Riemannian metric on G/H, +such that m0 and m1 are h-orthogonal to each other. Let W be a G-invairant vector field on +G/H, such that W(o) ∈ m0\{0}. Then, the navigation process with the data (h, W) provides +a G-invariant generalized Berwald Randers metric with S = 0 (see [7]). +Example 6. (Xu) As we mentioned in Introduction, the Bao-Shen’s Randers metrics on S3 are +concrete generalized Berwald metrics, namely they are not Berwaldian. Any non-Riemannian +homogeneous Randers sphere S3 = SU(3)/SU(2) (including Bao-Shen’s Randers metrics) sat- +isfies S = 0 with constant pointwise ∥β∥α-norms. Then, every non-Riemannian homogeneous +Randers sphere is a generalized Berwald metric. An S3 × S1, in which S3 = SU(3)/SU(2) and +the navigation field is tangent to the S3-factor, is a 4-dimensional generalized Berwald Randers +metric (see [29]). +Acknowledgments: The authors are so grateful to Ming Xu for his valuable comments on +this manuscript. Likewise, we thank him for providing us with examples 5 and 6 which improve +the quality of our manuscript. Also, we are thankful to Behzad Najafi, Mansoor Barzegari and +Libing Huang for their reading of this manuscript and their comments. + +On a Class of Generalized Berwald Manifolds +19 +References +[1] H. Akbar-Zadeh, Sur les espaces de Finsler ´a courbures sectionnelles constantes, Bull. +Acad. Roy. Bel. Cl, Sci, 5e S´erie - Tome LXXXIV (1988), 281-322. +[2] B. Aradi, Left invariant Finsler manifolds are generalized Berwald, Eur. J. Pure Appl. +Math. 8(1) (2015), 118–125. +[3] G. +S. +Asanov, +Finsleroid-Finsler +space +with +Berwald +and +Landsberg +conditions, +arXiv:math0603472. +[4] N. Bartelmeß and V. S. Matveev, Monochromatic metrics are generalized Berwald, Differ. +Geom. Appl. 58(2018), 264-271. +[5] X. Cheng and Z. Shen, Finsler Geometry- An Approach via Randers Spaces, Springer, +Heidelberg and Science Press, Beijing, 2012. +[6] X. Cheng and Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel J. +Math. 169(2009), 317-340. +[7] Z. Hu and S. 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Shen, Two-dimensional Finsler metrics of constant curvature, Manuscripta Mathemat- +ica. 109(3) (2002), 349-366. +[16] Z. I. Szab´o, Generalized spaces with many isometries, Geometria Dedicata, 11(1981), 369- +383. +[17] Sz. Szak´al and J. Szilasi, A new approach to generalized Berwald manifolds I, SUT J. Math. +37(2001), 19–41. +[18] J. Szilasi and Sz. Szak´al, +A new approach to generalized Berwald manifolds II, +Publ. Math. Debrecen, 60(2002), 429–453. + +On a Class of Generalized Berwald Manifolds +20 +[19] J. Szilasi, R. L. Lovas, and D. Cs. Kert´esz, Connections, Sprays and Finsler structures, +World Scientific, 2014. +[20] A. Tayebi and M. Barzegari, Generalized Berwald spaces with (α, β)-metrics, Indagationes. +Math. (N.S.). 27(2016), 670-683. +[21] A. Tayebi and E. Peyghan, On Douglas surfaces, Bull. Math. Soc. Science. Math. +Roumanie. Tome 55 (103), No 3, (2012), 327-335. +[22] A. Tayebi and M. Rafie Rad, S-curvature of isotropic Berwald metrics, Sci. China. Series +A: Math. 51(2008), 2198-2204. +[23] M. Xu and S. Deng, Killing frames and S-curvature of homogeneous Finsler spaces, Glas- +gow. Math. Journal. 57(2015), 457-464. +[24] M. Xu and S. Deng, Normal homogeneous Finsler spaces, Transformation Groups. +22(2017), 1143-1183. +[25] C. Vincze, On a special type of generalized Berwald manifolds: semi-symmetric linear +connections preserving the Finslerian length of tangent vectors, European Journal of Math. +3(2017), 1098-1171. +[26] C. Vincze, On Randers manifolds with semi-symmetric compatible linear connections, Inda- +gationes. Math. (N.S.). 26(2015), 363-379. +[27] C. Vincze, On generalized Berwald manifolds with semi-symmetric compatible linear con- +nections, Publ. Math. Debrecen. 83(2013), 741-755. +[28] C. Vincze, T. R. Khoshdani, S. M. Z. Gilani, and M. Ol´ah, On compatible linear connections +of two-dimensional generalized Berwald manifolds: a classical approach, Commun. Math. +27(2019), 51-68. +[29] M. Xu, Geodesic orbit spheres and constant curvature in Finsler geometry, Differ. Geom. +Appl. 61(2018), 197-206. +[30] L. Zhang and S. Deng, On generalized normal homogeneous Randers spaces, Publ. Math. +Debrecen. 90(2017), 507-523. +Akbar Tayebi and Faezeh Eslami +Department of Mathematics, Faculty of Science +University of Qom +Qom. Iran +Email: akbar.tayebi@gmail.com +Email: faezeh.eslami70@gmail.com + diff --git a/7tAzT4oBgHgl3EQfE_oc/content/tmp_files/load_file.txt b/7tAzT4oBgHgl3EQfE_oc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ae86d4c2f56c13a2c8bfef5ff81c767befe202d --- /dev/null +++ b/7tAzT4oBgHgl3EQfE_oc/content/tmp_files/load_file.txt @@ -0,0 +1,798 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf,len=797 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='01001v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='DG] 3 Jan 2023 On a Class of Generalized Berwald Manifolds A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Tayebi and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Eslami January 4, 2023 Abstract The class of generalized Berwald metrics contains the class of Berwald metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this paper, we characterize two-dimensional generalized Berwald (α, β)-metrics with vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let F = αφ(s), s = β/α, be a two-dimensional generalized Berwald (α, β)- metric on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Suppose that F has vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We show that one of the following holds: (i) if F is a regular metric, then it reduces to a Riemannian metric of isotropic sectional curvature or a locally Minkowskian metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (ii) if F is an almost regular metric that is not Riemannian nor locally Minkowskian, then we find the explicit form of φ = φ(s) which obtains a generalized Berwald metric that is neither a Berwald nor Lands- berg nor a Douglas metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' This provides a generalization of Szab´o rigidity theorem for the class of (α, β)-metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In the following, we prove that left invariant Finsler surfaces with vanishing S-curvature must be Riemannian surfaces of constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Finally, we construct a family of odd-dimensional generalized Berwald Randers metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Keywords: Generalized Berwald metric, Berwald metric, (α, β)-metric, S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 1 Introduction A Finsler metric F on a manifold M is called a generalized Berwald metric if there exists a covariant derivative ∇ on M such that the parallel translations induced by ∇ preserve the Finsler function F [18][19][20][26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, F is called a generalized Berwald metric on M and (M, F) is called a generalized Berwald manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' If the covariant derivative ∇ is also torsion-free, then F reduces to a Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Therefore, the class of Berwald metrics belongs to the class of generalized Berwald metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The importance of the class of generalized Berwald manifolds lies in the fact that these manifolds may have a rich isometry group [16][17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For some progress about the class of generalized Berwald manifolds, see [2], [4], [20], [25], [26], [27] and [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' To find generalized Berwald metrics, one can consider the class of (α, β)-metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' An (α, β)- metric is a Finsler metric defined by F := αφ(s), s = β/α, where φ = φ(s) is a smooth function on a symmetric interval (−b0, b0) with certain regularity, α = � aij(x)yiyj is a Riemannian metric and β = bi(x)yi is a 1-form on the base manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The simplest (α, β)-metrics are the Randers metrics F = α+β which were discovered by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Randers when he studied 4-dimensional general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' These metrics have been widely applied in many areas of natural sciences, including physics, biology, psychology, etc [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [26], Vincze proved that a Randers metric F = α + β is a generalized Berwald metric if and only if dual vector field β♯ is of constant 12010 Mathematics subject Classification: 53C60, 53C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' 1 On a Class of Generalized Berwald Manifolds 2 Riemannian length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [20], Tayebi-Barzegari generalized Vincze’s result for (α, β)-metrics and showed that an (α, β)-metric satisfying the so-called sign property is a generalized Berwald metric if and only if β♯ is of constant Riemannian length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, Vincze showed that an (α, β)- metric satisfying the regularity property φ′(0) ̸= 0 is a generalized Berwald metric if and only if β♯ is of constant Riemannian length [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [4], Bartelmeß-Matveev proved that a Finsler metric is a generalized Berwald metric if and only if it is monochromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Generally, two-dimensional Finsler metrics have some different and special Riemannian and non-Riemannian curvature properties from the higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For example, Bartelmeß- Matveev showed that except for torus and the Klein bottle, the other closed 2-dimensional manifolds can not have non-Riemannian generalized Berwald metrics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [28], Vincze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' showed that a connected generalized Berwald surface is a Landsberg surface if and only if it is a Berwald surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The S-curvature is constructed by Shen for given comparison theorems on Finsler manifolds [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' An interesting problem is to study generalized Berwald metrics with vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, we characterize the class of two-dimensional generalized Berwald (α, β)-metrics with vanishing S-curvature and prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let F = αφ(s), s = β/α, be a two-dimensional generalized Berwald (α, β)- metric on a connected manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Suppose that F has vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then one of the following holds: (i) If F is a regular metric, then it reduces to a Riemannian metric of isotropic sectional curvature or a locally Minkowskian metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (ii) If F is an almost regular metric that is not Riemannian nor locally Minkowskian, then φ is given by φ = c exp � � s 0 kt + q √ b2 − t2 1 + kt2 + qt √ b2 − t2dt � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1) where c > 0, q > 0, and k are real constants, and β satisfies rij = 0, si = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2) In this case, F is neither a Berwald nor Landsberg nor a Douglas metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It is remarkable that, for a generalized Berwald (α, β)-metric F = αφ(s), s = β/α, on an n-dimensional manifold M, we show that S = 0 if and only if F is Riemannian or β is a Killing form with constant length (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The celebrated Szab´o rigidity theorem states that every 2-dimensional Berwald surface is either locally Minkowskian or Riemannian (see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every Berwald metric has vanishing S-curvature [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 is an extension of Szab´o’s result for (α, β)-metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The condition of generalized Berwaldness can not be dropped from the assumption of The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' There are many two-dimensional (α, β)-metrics with vanishing S-curvature which are not Riemannian nor locally Minkowskian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For example, let us consider Shen’s Fish-Tank Randers metric F = α + β on R2 given by following F = � (xv − yu)2 + (u2 + v2)(1 − x2 − y2) 1 − x2 − y2 + yu − xv 1 − x2 − y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' F has vanishing S-curvature [14] while it is not a generalized Berwald metric (∥β∥α = � x2 + y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' This metric is not Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, F has vanishing flag curvature while it is not locally Minkowskian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1, the vanishing of S-curvature is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For example, see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 3 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let us consider G := {(x, y) ∈ R2|y > 0} and define a multiplication on G by (x1, y1) ∗ (x2, y2) := (x2y1 + x1, y1y2), for (xi, yi) ∈ G, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (G, ∗) is a Lie group [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' One can introduce a Randers metric F on G by F = � 2dx2 + 2dxdx + 2dy2 y + dx + dy y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) Then a11 = a22 = 2 y2, a21 = a12 = 1 y2, b1 = b2 = 1 y, a11 = a22 = 2 3y2, a12 = a21 = −1 3y2, b1 = b2 = 1 3y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It is easy to see that α is a positive-definite Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, we get b := ∥β∥α = � aijbibj = � bibi = � b1b1 + b2b2 = � 2 3 It follows that F is a positive-definite Randers metric on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [26], Vincze showed that a Randers metric F = α + β is a generalized Berwald metric if and only if β is of constant length with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, the Randers metric defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) is a generalized Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We have dβ = 1 y2dx ∧ dy ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus β is not closed which implies that F can not be a Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We have s11 = s22 = 0, s12 = 1 y2 = −s21, s1 = − 1 3y, s2 = 1 3y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4) where sij and si defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We claim that F has not vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On contrary, let S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 in [5], we have rij + bisj + bjsi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Contracting it with bi yields rj + b2sj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By considering ri + si = 0 and b2 = 2/3, we must have si = 0 which contradicts with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, the Randers metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) is a generalized Berwald metric with B ̸= 0, L ̸= 0, D ̸= 0, S ̸= 0 and E ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We claim that F is not R-quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On the contrary, let the Finsler metric F defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) be R-quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 in [5], if the two-dimensional Randers metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) is R-quadratic then it has constant S-curvature S = 3cF, for some real constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [23], Xu-Deng proved that every homogeneous Finsler metric of isotropic S-curvature has vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus, we must have S = 0 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Therefore, the Randers metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) is not R-quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 may not be held for an (α, β)-metric of constant S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For example, at a point x = (x1, x2) ∈ R2 and in the direction y = (y1, y2) ∈ TxR2, consider the Riemannian metric α = α(x, y) and one form β = β(x, y) as follows α(x, y) := � (y1)2 + e2x1(y2)2, β(x, y) := y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5) Then sij = 0 and rij = b2aij − bibj which yield ri + si = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' If φ = φ(s) satisfies Φ = −6k φ∆2 b2 − s2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='6) On a Class of Generalized Berwald Manifolds 4 for some constant k, then F = αφ(β/α) has constant S-curvature S = 3kF (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, ∆ = ∆(b, s) and Φ = Φ(b, s) defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The existence of regular solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='6) for arbitrary k ∈ R, when α and β are given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5), is proved in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It is easy to see that F is a generalized Berwald metric while it is not a Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We must mention that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 does not hold for Finsler metrics of dim(M) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Denote generic tangent vectors on S3 as u∂/∂x + v∂/∂y + w∂/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The Finsler functions for Bao-Shen’s Randers metrics F = α + β are given by the following F = � K(cu − zv + yw)2 + (zu + cv − xw)2 + (xv + cw − yu)2 1 + x2 + y2 + z2 ± √ K − 1 (cu − zv + yw) 1 + x2 + y2 + z2 , where K > 1 is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For these metrics, we have b := ∥β∥α = � 1 − 1 K One can see that, the one form β is not closed, and then F can not be a Douglas metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' This family of Randers metrics is generalized Berwald metrics with S = 0 which are not Berwaldian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [4], it is proved that every 3-dimensional closed manifold admits a non-Riemannian generalized Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' However, the closeness condition is very restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Indeed, for every manifold M with dim(M) ≥ 3, one can construct generalized Berwald (α, β)-metrics that are not Berwaldian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The projective spherical metric on R3 is given by the following α := � (1 + ||X||2)||Y ||2 − ⟨X, Y ⟩2 1 + ⟨X, X⟩ , X ∈ R3, Y ∈ TXR3, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='7) where ⟨, ⟩ and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='|| denote the Euclidean inner product and norm on R3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Put X = (x, y, z) and Y = (u, v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Suppose that β = κ(b1u + b2v + b3w) is a Killing 1-form of α, where 0 < κ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By a simple calculation, we get b1 = A1 2y + A1 3z + C1 1 + ⟨X, X⟩ , b2 = A2 1x + A2 3z + C2 1 + ⟨X, X⟩ , b3 = A3 1x + A3 2y + C3 1 + ⟨X, X⟩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='8) where A = (Ai j) is an antisymmetric real matrix, and C = (Ci) is a constant vector in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let us put C = (0, 1, 0), A1 2 = A2 3 = 0, A1 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9) In this case, β is a Killing 1-form with ∥β∥α = κ < 1 such that is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' According to Shen’s theorem in [12], a regular (α, β)-metric is a Berwald metric if and only if β is parallel with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Using the Riemannian metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='7) and 1-form β satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9), one can construct generalized Berwald (α, β)-metrics which are not Berwaldian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Homogeneous Finsler manifolds are those Finsler manifolds (M, F) that the orbit of the natural action of I(M, F) on M at any point of M is the whole M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For the class of homogeneous generalized Berwald (α, β)-metrics, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let F = αφ(s), s = β/α, be a two-dimensional homogeneous generalized Berwald (α, β)-metric on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then F has isotropic S-curvature if and only if it is a Riemannian metric of constant sectional curvature or a locally Minkowskian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 5 Let G be a connected Lie group with a bi-invariant Finsler metric ¯F, and H a closed subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Denote the Lie algebras of G and H as g and h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let ρ be the natural projection from G to M = G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then there exists a uniquely defined G-invariant metric F on M such that for any g ∈ G, the tangent map ρ∗ : � TgG, ¯F(g, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=') � → � Tπ(g)M, F(π(g), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=') � is a submersion (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The G-invariant Finsler metric F is called the normal homogeneous metric induced by ¯F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The pair (M, F) is said the normal homogeneous space induced by ρ : G → M = G/H and ¯F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For normal homogeneous generalized Berwald metrics, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every two-dimensional normal homogeneous generalized Berwald (α, β)-metric is a Riemannian metric of non-negative constant sectional curvature or a locally Minkowskian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' A Finsler metric F on a manifold M is called a generalized normal homogeneous metric if for every x, y ∈ M, there exists a δ(x)-translation of (M, F) sending x to y (see [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this paper, we consider two-dimensional homogeneous generalized Berwald Randers metric of generalized normal-type and prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every two-dimensional generalized normal homogeneous generalized Berwald Randers metric must be a Riemannian metric of non-negative constant sectional curvature or a locally Minkowskian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let (G, ·) be a connected Lie group with identity element e, and λg denotes the left trans- lation by g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' A Finsler metric F on G is called a left invariant Finsler metric if it satisfies F ◦ (λg)∗ = F for any g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [2], Aradi proved that left invariant Finsler metrics are generalized Berwald metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For this class of Finsler metrics, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every left invariant Finsler surface has vanishing S-curvature if and only if it is a Riemannian metric of constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By considering Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2, it seems that two-dimensional generalized Berwald (α, β)-metrics with vanishing S-curvature may be only Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' But we do not find a short proof for this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' A Finsler metric F on a manifold M is called an isotropic Berwald metric, if its Berwald curvature is given by By(u, v, w) = cF −1� h(u, v) � w − gy(w, ℓ)ℓ � + h(v, w) � u − gy(u, ℓ)ℓ � +h(w, u) � v − gy(v, ℓ)ℓ � + 2FCy(u, v, w)ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='10) where hy(u, v) = gy(u, v)−F −2(y)gy(y, u)gy(y, v) is the angular form in direction y, C denotes the Cartan torsion of F and c ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every Berwald metric is an isotropic Berwald metric with c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The Funk metrics are isotropic Berwald metrics with c = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' As a straightforward conclusion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2, one can get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every left invariant isotropic Berwald surface must be a Riemannian surface of constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 6 2 Preliminaries Let M be an n-dimensional C∞ manifold, TM = � x∈M TxM the tangent space and TM0 := TM − {0} the slit tangent space of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let (M, F) be a Finsler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The following quadratic form gy : TxM × TxM → R is called fundamental tensor gy(u, v) := 1 2 ∂2 ∂s∂t � F 2(y + su + tv) � s=t=0, u, v ∈ TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let x ∈ M and Fx := F|TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' To measure the non-Euclidean feature of Fx, one can define Cy : TxM × TxM × TxM → R by Cy(u, v, w) := 1 2 d dt � gy+tw(u, v) � t=0 , u, v, w ∈ TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The family C := {Cy}y∈TM0 is called the Cartan torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For a Finsler manifold (M, F), its induced spray on TM is denoted by G = G(x, y) which in a standard coordinate (xi, yi) for TM0 is given by G = yi∂/∂xi − 2Gi(x, y)∂/∂yi, where Gi := 1 4gil� ∂2F 2 ∂xk∂yl yk − ∂F 2 ∂xl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For a Finsler metric F on an n-dimensional manifold M, the Busemann-Hausdorff volume form dVF = σF(x)dx1 · · · dxn is defined by σF(x) := Vol � Bn(1) � Vol � (yi) ∈ Rn �� F � yi ∂ ∂xi|x � < 1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let Gi denote the geodesic coefficients of F in the same local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then for y = yi∂/∂xi|x ∈ TxM, the S-curvature is defined by S(y) := ∂Gi ∂yi (x, y) − yi ∂ ∂xi � ln σF(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1) For a vector y ∈ TxM0, the Berwald curvature By : TxM × TxM × TxM → TxM is defined by By(u, v, w) := Bi jkl(y)ujvkwl∂/∂xi|x, where Bi jkl := ∂3Gi ∂yj∂yk∂yl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' F is called a Berwald metric if B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every Berwald metric satisfies S = 0 (see [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For y ∈ TxM, define the Landsberg curvature Ly : TxM × TxM × TxM → R by Ly(u, v, w) := −1 2gy � By(u, v, w), y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' A Finsler metric F is called a Landsberg metric if L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Taking a trace of Berwald curvature B give us the mean of Berwald curvature E which is defined by Ey : TxM × TxM → R, where Ey(u, v) := 1 2 n � i=1 gij(y)gy � By(u, v, ∂i), ∂j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2) where {∂i} is a basis for TxM at x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In local coordinates, Ey(u, v) := Eij(y)uivj, where Eij := 1 2Bm mij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 7 Taking a horizontal derivation of the mean of Berwald curvature E along Finslerian geodesics give us the H-curvature H = H(x, y) which is defined by Hy = Hijdxi ⊗ dxj, where Hij := Eij|mym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, “|” denotes the horizontal covariant differentiation with respect to the Berwald connection of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For a non-zero vector y ∈ TxM0, one can define Dy : TxM × TxM × TxM → TxM by Dy(u, v, w) := Di jkl(y)uivjwk∂/∂xi|x, where Di jkl := ∂3 ∂yj∂yk∂yl � Gi − 2 n + 1 ∂Gm ∂ym yi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) D is called the Douglas curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' F is called a Douglas metric if D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For a non-zero vector y ∈ TxM0, the Riemann curvature is a family of linear transformation Ry : TxM → TxM which is defined by Ry(u) := Ri k(y)uk∂/∂xi, where Ri k(y) = 2∂Gi ∂xk − ∂2Gi ∂xj∂yk yj + 2Gj ∂2Gi ∂yj∂yk − ∂Gi ∂yj ∂Gj ∂yk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4) The family R := {Ry}y∈TM0 is called the Riemann curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For a flag P := span{y, u} ⊂ TxM with the flagpole y, the flag curvature K = K(P, y) is defined by K(x, y, P) := gy � u, Ry(u) � gy(y, y)gy(u, u) − gy(y, u)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5) The flag curvature K(x, y, P) is a function of tangent planes P = span{y, v} ⊂ TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' A Finsler metric F is of scalar flag curvature if K(x, y, P) = K(x, y) is independent of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, F is called of isotropic and constant flag curvature if K = K(x) and K = constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Throughout this paper, we use the Berwald connection on Finsler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The pullback bundle π∗TM admits a unique linear connection, called the Berwald connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let (M, F) be an n-dimensional Finsler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let {ej} be a local frame for π∗TM, {ωi, ωn+i} be the corresponding local coframe for T ∗(TM0) and {ωi j} be the set of local Berwald connection forms with respect to {ej}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then the connection forms are characterized by the structure equations as follows Torsion freeness: dωi = ωj ∧ ωi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='6) Almost metric compatibility: dgij − gkjωk i − gikωk j = −2Lijkωk + 2Cijkωn+k, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='7) where ωi := dxi and ωn+k := dyk + yjωk j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The horizontal and vertical covariant derivations with respect to the Berwald connection re- spectively are denoted by “|” and “, ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For more details, one can see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 8 3 Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2 For an (α, β)-metric, let us define bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='j by bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='jθj := dbi − bjθj i , where θi := dxi and θj i := γj ikdxk denote the Levi-Civita connection form of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let rij := 1 2(bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='j + bj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='i), sij := 1 2(bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='j − bj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='i), ri0 := rijyj, r00 := rijyiyj, rj := birij, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1) si0 := sijyj, sj := bisij, si j = aimsmj, si 0 = si jyj, r0 := rjyj, s0 := sjyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2) Put Q := φ′ φ − sφ, ∆ := 1 + sQ + (b2 − s2)Q′, Θ := Q − sQ′ 2∆ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) Φ := −(Q − sQ′)(n∆ + 1 + sQ) − (b2 − s2)(1 + sQ)Q′′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4) Ψ := φ′′ 2 � (φ − sφ′) + (b2 − s2)φ′′�, where b := ∥β∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let Gi = Gi(x, y) and ¯Gi α = ¯Gi α(x, y) denote the coefficients of F and α, respectively, in the same coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By definition, we have Gi = Gi α + αQsi 0 + α−1(r00 − 2Qαs0)(Θyi + αΨbi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5) Clearly, if β is parallel with respect to α, that is rij = sij = 0, then Gi = Gi α = γi jk(x)yjyk are quadratic in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, F reduces to a Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For an (α, β)-metric F = αφ(s), s = β/α, on an n-dimensional manifold M, the S-curvature is given by S = � 2Ψ − f ′(b) bf(b) � (r0 + s0) − Φ 2α∆2(r00 − 2αQs0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='6) where f(b) := � π 0 sinn−2 t T(b cos t)dt � π 0 sinn−2 tdt , T(s) := φ(φ − sφ′)n−2� (φ − sφ′) + (b2 − s2)φ′′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For more details, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1, we need the following key lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' ([4][20]) Let F = αφ(s), s = β/α, be an (α, β)-metric on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then F is a generalized Berwald metric if and only if β has constant length with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [20], Tayebi-Barzegari showed that every (α, β)-metric F = αφ(s), s = β/α, with sign property is a generalized Berwald metric if and only if the dual vector field β♯ is of constant Riemannian length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [9], Ivanov proved that a two-dimensional Finsler metric is a generalized Berwald metric if and only if it is monochromatic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=', Finsler metrics such that each two tangent spaces are isomorphic as normed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [4], Bartelmeß-Matveev extended his result for n-dimensional Finsler metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It follows that an (α, β)-metric is a generalized Berwald metric if and only if dual vector field β♯ is of constant Riemannian length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [6], Cheng-Shen characterized (α, β)-metrics with isotropic S-curvature on a manifold M of dimension n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Soon, they found that their result holds for the class of (α, β)-metrics with constant length one-forms, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, we give a characterization of the class of generalized Berwald metrics with vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let F = αφ(s), s = β/α, be a generalized Berwald (α, β)-metric on an n- dimensional manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then S = 0 if and only if F is Riemannian or β is a Killing form with constant length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let b := ∥β∥α = � amjbjbm = √bmbm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, the following holds ∂b ∂xi = 1 bbmbm|i = 1 b(ri + si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='7) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1, we have b = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='7) we obtain ri + si = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='6) reduces to following S = − Φ 2α∆2(r00 − 2αQs0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='8) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='8), S = 0 if and only if Φ = 0 or β satisfies r00 = 2αQs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9) In the case of Φ = 0, F reduces to a Riemannian metric (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Now, let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We are going to simplify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For this aim, one can change the y-coordinates (yi), i = 1, · · · , n, at a point to the polar coordinates (s, uA), where A = 2, · · · , n (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For an arbitrary and fix point x ∈ M, let us take an orthonormal basis ei at x such that the Riemannian metric is written as α = ��n i=1(yi)2 and its related one-form is given by β = by1, where b := ||β||α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let us fix an arbitrary number s such that |s| < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Define ¯α = � � � � n � A=2 (yA)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by β = sα we get y1 = s √ b2 − s2 ¯α, yA = uA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='10) Also, we have α = b √ b2 − s2 ¯α, β = bs √ b2 − s2 ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='11) Let us put ¯r10 := n � A=2 r1AyA, ¯s10 := n � A=2 s1AyA, ¯r00 := n � A,B=2 rAByAyB, ¯r0 := n � A=2 rAyA, ¯s0 := n � A=2 sAyA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then we get the following useful relations r1 = br11, rA = br1A, s1 = 0, sA = bs1A, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='12) r00 = s2 b2 − s2 ¯α2r11 + 2s √ b2 − s2 ¯α¯r10 + ¯r00, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='13) r10 = s √ b2 − s2 ¯αr11 + ¯r10, s0 = ¯s0 = b¯s10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='14) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='11)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='14), the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9) can be written as follows ¯r00 + s2 b2 − s2 ¯α2 r11 = 2 √ b2 − s2 � b2Q¯s10 − s¯r10 � ¯α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='15) On a Class of Generalized Berwald Manifolds 10 By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='15), we get two following relations ¯r00 + s2 b2 − s2 ¯α2r11 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='16) b2Q¯s10 − s¯r10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='17) On the other hand, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='11) implies that s2 b2 − s2 ¯α2 − 1 b2β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='18) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='18), we get b2¯r00 + β2r11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='19) The following hold ∂¯r00 ∂y1 = 0, ∂β ∂y1 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then by differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='19) with respect to y1 we have β/br11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus r11 = 0 and by putting it in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='19), we get ¯r00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Putting these relations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='14) imply that r00 = 2s¯α √ b2 − s2 ¯r10 = 2β b ¯r10, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='20) r10 = ¯r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='21) Now, by considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='21), we divide the problem into two cases: (a) ¯r10 = 0 and (b) ¯r10 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Case (a): ¯r10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='20) we get rij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Putting it in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='9) implies that si = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, β reduces to a Killing one-form of constant length with respect to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Case (b): ¯r10 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We have ∂¯r10/∂y1 = 0 and ∂¯s10/∂y1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus, differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='17) with respect to y1 yields (s)y1¯r10 = b2(Q)y1¯s10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='22) Contracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='17) with (s)y1 give us s(s)y1¯r10 = b2Q(s)y1¯s10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='23) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='23), we get � (s)y1Q − s(Q)y1 � ¯s10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='24) According to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='24), we get ¯s10 = 0 or Q(s)y1 = s(Q)y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let ¯s10 = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='17) reduces to s = 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, we have Q(s)y1 = s(Q)y1, which is equal to (Q)y1 Q = (s)y1 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='25) On a Class of Generalized Berwald Manifolds 11 Using sy1 ̸= 0 and (Q)y1 = sy1(Q)s, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='25) give us (Q)s Q = 1 s which yields ln(Q) − ln(s) = c, where c is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus Q = ks, where k is a non-zero real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, we get φ = √ 1 + ks2 which shows that F is a Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, we solve an ODE which will appear in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let F = αφ(s), s = β/α, be an (α, β)-metric on a manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Suppose that φ satisfies following αΘ2 + 2Λ1Θ1 + Λ2Q = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='26) where Λ1 := biαyi, Λ2 := bibjαyiyj, Θ1 := biQyi, Θ2 := bibjQyiyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then F is a singular Finsler metric given by φ = c exp � � s 0 k1t + k2 √ b2 − t2 1 + t � k1t + k2 √ b2 − t2�dt � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='27) where k1 and k2 are real constants, and c > 0 is a non-zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let us put Ajk := α2ajk − yjyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, the followings hold αyi = 1 αyi, αyjyk = 1 α3Ajk, αyjykyl = − 1 α5 � Ajkyl + Ajlyk + Alkyj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, one can obtain the following Λ1 = s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='28) Λ2 = 1 α(b2 − s2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='29) Θ1 = 1 α(b2 − s2)Q′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='30) Θ2 = 1 α2(b2 − s2) � (b2 − s2)Q′′ − 3sQ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='31) Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Putting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='28)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='31) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='26) yield Q′′ − s b2 − s2Q′ + 1 b2 − s2Q = 0, On a Class of Generalized Berwald Manifolds 12 which implies that Q = k1s + k2 √ b2 − s2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='32) where k1 and k2 are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3), the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='32) is equal to following � 1 + k1s2 + k2s √ b2 − s2 � φ′ = � k1s + k2 √ b2 − s2 � φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='33) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='33), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It is an almost regular (α, β)-metric, namely, it is singular in two directions y = (±b, 0, 0) ∈ TxM at any point x (for more details, see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The function φ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='27) is specifically has been seen for the first time in Asanov’s paper [3] as follows φ = c exp � � s 0 k √ b2 − t2 1 + tk √ b2 − t2dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='34) Then, its more general form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='27) found by Shen in [12], where he looked for unicorn metrics, namely the Landsberg metrics which are not Berwaldian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Shen realized that the function φ = φ(b, s) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='27) can be expressed in terms of elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' See (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2) in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, we show that the Douglas curvature of Finsler surfaces satisfies a special relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' More precisely, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The Douglas curvature of any Finsler surface (M, F) satisfies Di jkl|sys = Djklyi (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='35) for some tensor D = Dijkdxi ⊗ dxj ⊗ dxk which are homogeneous of degree -1 in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By definition, the Douglas curvature of a two-dimensional Finsler metric is given by Di jkl = Bi jkl − 2 3 � Ejkδi l + Eklδi j + Eljδi k + Ejk,lyi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='36) Taking a horizontal derivation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='36) along Finslerian geodesics and using yi |s = 0 give us the following Di jkl|mym = Bi jkl|mym − 2 3 � Hjkδi l + Hklδi j + Hljδi k + Ejk,l|mymyi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='37) Every Finsler surface is of scalar flag curvature K = K(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='24) in [13] we have Bi jml|kyk = 2KCjlmyi − 1 3 � ylδi m + ymδi l − 2glmyi� Kj − 1 3 � yjδi m + ymδi j − 2gjmyi� Kl −1 3 � yjδi l + ylδi j − 2gjlyi� Km − 1 3F 2� Kjmhi l + Kjlhi m + Klmhi j � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='38) where yi = FFyi, Kj := Kyj and Kjk := Kyjyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Taking a trace of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='38) yields Hjl = −1 2 � ylKj + yjKl + F 2Kjl � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='39) By putting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='38) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='39) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='37) we get Di jkl|mym = 1 3 � 6KCjkl + 2 � gklKj + gkjKl + gjlKk � + � yjKkl + ykKjl + ylKkj � −2Ejk,l|mym� yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='40) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='40) give us (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 13 Now, we show that the covariant derivative of Berwald curvature of 2-dimensional gener- alized Berwald (α, β)-metric with vanishing S-curvature satisfies an interesting relation which will play an important role in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The Berwald curvature of any non-Riemannian 2-dimensional generalized Berwald (α, β)-metric with vanishing S-curvature satisfies following hm p Bp jkl|sys = sm 0|0(αjklQ + αjkQl + αlkQj + αljQk + αQjkl + αlQjk + αjQlk + αkQjl) +sm 0(αjklQ|0 + αjkQl|0 + αlkQj|0 + αljQk|0 + αQjkl|0 + αlQjk|0 + αjQlk|0 +αkQjl|0) + Am lXjk + Am jXlk + Am kXjl + Bm l Yjk + Bm jYlk + Bm kYjl = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='41) where Xjk := Q|0αjk + Qk|0αj + Qj|0αk + Qjk|0α, Yjk := Qαjk + Qkαj + Qjαk + αQjk, Am l := sm l − F −2s0 lym, Bm l := Am l|sys = sm l|0 − F −2s0 l|0ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2, we have rij = 0 and sj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5) reduces to following Gi = Gi α + αQsi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='42) Taking three vertical derivations of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='42) with respect to yj, yl and yk give us Bi jkl = si 0 � αQjkl + αlQjk + αjQlk + αkQjl + αjklQ + αjkQl + αlkQj + αljQk � + si j � Qαlk + Qkαl + Qlαk + αQlk � + si k � Qαjl + Qjαl + Qlαj + αQjl � +si l � Qαjk + Qkαj + Qjαk + αQjk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='43) Contracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='43) with hm i implies that hm i Bi jkl = � αjklQ + αjkQl + αlkQj + αljQk + αQjkl + αlQjk + αjQlk + αkQjl � sm 0 + � Qαjk + Qkαj + Qjαk + αQjk � (sm l − F −2s0 lym) + � Qαlk + Qkαl + Qlαk + αQlk � (sm j − F −2s0 jym) + � Qαjl + Qjαl + Qlαj + αQjl � (sm k − F −2s0 kym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='44) On the other hand, by taking a horizontal derivation of Douglas curvature along Finslerian geodesics and contracting the result with hm i , we get the following hm i Di jkl|sys = hm i Bi jkl|sys − 2 3 � Hjkhm l + Hklhm j + Hljhm k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='45) Contracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='35) with hm i yields hm i Di jkl|sys = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='46) Since S = 0, then by definition we get H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='46) imply that hm i Bi jkl|sys = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='47) We have hm i|s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='47), we have � hm i Bi jkl � |sys = hm i Bi jkl|sys = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='48) Therefore, taking a horizontal derivation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='44) along Finslerian geodesic and considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='48) give us (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 14 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1: Taking a horizontal derivation of ymsm 0 = 0 with respect to the Berwald connection of F implies that ym|0sm 0 + ymsm 0|0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='49) Since ym|0 = 0, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='49) reduces to following ymsm 0|0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='50) By contracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='41) with ym and considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='50), we get s0 lXjk + s0 jXlk + s0 kXjl + s0 l|0Yjk + s0 j|0Ylk + s0 k|0Yjl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='51) Since sj = 0, then one can get 0 = (sj)|0 = (rm0 + sm0)sm j + bmsm j|0 which considering rij = 0, it reduces to following bmsm j|0 = −sm0sm j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='52) Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='52) with yj yields bisi 0|0 + si0si 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='53) Also, contracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='52) with bj implies that bisi j|0bj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='54) Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='51) with bjbkbl and considering (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='53) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='54) give us (αΘ2 + 2Λ1Θ1 + Λ2Q)sm0sm 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='55) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='55), we have two cases: if sm 0sm0 = 0, since α is a positive-definite metric, then we find that β is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Therefore, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='43) we conclude that F reduces to a Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By Szabo’s rigidity result for Finsler surfaces, F reduces to a locally Minkowskian metric or a Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On the other hand, every Finsler surface has scalar flag curvature K = K(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' According to Akbar-Zadeh theorem in [1], a Finsler manifold (M, F) of scalar flag curvature K = K(x, y) has isotropic flag curvature K = K(x) if and only if it has vanishing H-curvature H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus, the obtained Riemannian metric has isotropic sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Now, suppose that F is not a Riemannian metric nor a locally Minkowskian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='55) we have αΘ2 + 2Λ1Θ1 + Λ2Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3 we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, since S = 0, then F can not be a Douglas metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On the other hand, the Berwald curvature of 2-dimensional Finsler manifold is given by Bi jkl = − 2 F 2Ljklyi + 2 3 � Ejkhi l + Eklhi j + Ejlhi k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='56) See the relation (15) in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' If F is a Landsberg metric then by considering S = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='56) implies that F is a Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1) is a generalized Berwald metric which is not Berwald, Landsberg nor Douglas metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1: In [23], Xu-Deng proved that every homogeneous Finsler metric of isotropic S-curvature has vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by assumption we get S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The On a Class of Generalized Berwald Manifolds 15 Akbar-Zadeh theorem in [1] stated that a Finsler manifold (M, F) of scalar flag curvature K = K(x, y) has isotropic flag curvature K = K(x) if and only if it has vanishing H-curvature H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On the other hand, every Finsler surface has scalar flag curvature K = K(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Thus, by Akbar-Zadeh theorem we get K = K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every scalar function on M which is invariant under isometries of (M, F) is a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The homogeneity of (M, F) and invariancy of the flag curvature under isometries of F imply that K = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 we get the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2: Let F = αφ(s), s = β/α, be a two-dimensional normal homogeneous generalized Berwald (α, β)-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [24], Xu-Deng proved that every normal homogeneous manifold has vanishing S-curvature and non-negative flag curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 and the same method used in the proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1, it follows that F is a Riemannian metric of non-negative constant sectional curvature or a locally Minkowskian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3: Let (G/H, F) be a generalized normal homogeneous Randers man- ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [30], Zhang-Deng proved that F has vanishing S-curvature (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, they showed that any generalized normal homogeneous Randers metric has non-negative flag curvature (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='13 in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 we get the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' According to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1, every two-dimensional generalized Berwald (α, β)-metric with vanishing S-curvature is Riemannian or locally Minkowskian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Every left invariant Finsler metric is a generalized Berwald metric [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2 which states that left invariant Finsler metrics with vanishing S-curvature reduce to Riemannian metrics, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The approach of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2 is completely different from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2: To prove a homogeneous surface with S = 0 is Riemannian, we only need to consider the nontrivial case, namely, a 2-dimensional non-Abelian Lie group G with a left invariant Finsler metric F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' At each y ∈ g with F(y) = 1, there is a gy orthonormal basis e1 = y and e2 tangent to F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' At almost all non-zero y, the spray vector field η is nonzero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=', η(y) is a nonzero multiple of e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By the homogenous S-curvature formula, we have S(y) = −I(η) = −Cy(η, e2, e2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, Cy(e2, e2, e2) = 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The speciality of 2-dimensional spaces implies C = 0 everywhere, so F is Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By the same method used to prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1, one can conclude that the Riemannian metric is of constant sectional curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4: By assumption, F has isotropic Berwald curvature Bi jkl = cF −1� hjkhi l + hjlhi k + hklhi j + 2Cjklyi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='57) where c = c(x) is a scalar function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In [22], it is proved that every Finsler surface of isotropic Berwald curvature (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='57) metric has isotropic S-curvature S = 3cF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' By Xu-Deng’s result in [24], F has vanishing S-curvature S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2, F reduces to a Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 16 4 Some Examples of Generalized Berwald Manifolds In this section, we are going to give some important examples of the class of generalized Berwald manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' First, by using trans-Sasakian structure, we construct a family of odd-dimensional generalized Berwald Randers metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' � Odd-dimensional generalized Berwald Randers metrics � Let M be a differen- tiable manifold of dimension 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Suppose that η = ηi(x)dxi, ξ = ξi∂/∂xi and ϕ = ϕi j∂/∂xi ⊗ dxj are a 1-form, a vector field, and a (1, 1)-tensor on M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' The triple (η, ξ, ϕ) is called an almost contact structure on M if it satisfies ϕ(ξ) = 0, η(ξ) = 1, ϕ2 = −I + η ⊗ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' A differentiable manifold of odd dimension 2n+ 1 with an almost contact structure is called an almost contact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let a manifold M with the (η, ξ, ϕ) structure admits a Riemannian metric g such that g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then M is called an almost contact metric structure and g is called a compatible metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, (η, ξ, ϕ, g) is called almost contact metric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' An almost contact metric structure (η, ξ, ϕ, g) on M is called a trans-Sasakian structure if it satisfies (∇Xϕ)Y = c1 � g(X, Y )ξ − η(Y )X � + c2 � g(ϕX, Y )ξ − η(Y )ϕX � for some scalar functions c1 = c1(x) and c2 = c2(x) on M, where ∇ denotes the Levi-Civita connection of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Now, let (η, ξ, ϕ, g) be a trans-Sasakian structure on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Define α, β : TM → [0, ∞) by ∀(x, y) ∈ TM, α(x, y) := � gx(y, y), β(x, y) := ǫ ηx(y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1) where 0 < ǫ < 1 be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, for the Randers metric F := α + β, we have ||β||α = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It follows that the class of Randers metrics induced by trans-Sasakian manifolds (M, η, ξ, ϕ, g) are (2n + 1)-dimensional generalized Berwald metrics on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Here, we give a two-dimensional Randers metric F = α +β with vanishing S-curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' We show that if F is a generalized Berwald metric then it reduces to a Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let y = u∂/∂x + v∂/∂y ∈ T(x,y)R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Consider the Randers metric F = α + β, where α = α(y) and β = β(y) are given by α := �� 1 + (1 − ǫ2)(x2 + y2) � (u2 + v2) + � 1 + ǫ2 + x2 + y2 � (xv − yu)2 � 1 + (1 − ǫ2)(x2 + y2) �� 1 + x2 + y2 , β := − ǫ(xv − yu) 1 + (1 − ǫ2)(x2 + y2), On a Class of Generalized Berwald Manifolds 17 and ǫ is a real constant (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' F is defined on the whole sphere for |ǫ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It is remarkable that, one can rewrite F in a polar coordinate system, x = r cos(θ), y = r sin(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Express α = � a11µ2 + a12µν + a21νµ + a22ν2, β = b1µ + b2ν, where a11 = 1 (1 + r2) � 1 + (1 − ǫ2)r2�, a12 = a21 = 0, a22 = r2(1 + r2) � 1 + (1 − ǫ2)r2�2, b1 = 0, b2 = − ǫr2 1 + (1 − ǫ2)r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let us put A := det(aij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, we get a11 = r2(1 + r2) A � 1 + (1 − ǫ2)r2�2, a22 = 1 A(1 + r2) � 1 + (1 − ǫ2)r2�, a12 = a21 = 0, b1 = 0, b2 = − ǫr2 A(1 + r2) � 1 + (1 − ǫ2)r2�2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Therefore, we obtain ∥β∥2 α = aijbibj = bibi = ǫ2r4 A(1 + r2) � 1 + (1 − ǫ2)r2�3 ̸= constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2) means that F is not a generalized Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On the other hand, a direct computa- tion yields r11 = r22 = 0, r12 = r21 = ǫ3r3 (1 + r2) � 1 + (1 − ǫ2)r2�2, s11 = s22 = 0, s12 = ǫr � 1 + (1 − ǫ2)r2�2 = −s21 s1 = ǫ2r (1 + r2) � 1 + (1 − ǫ2)r2�, s2 = 0, r1 = − ǫ4r5 A(1 + r2)2� 1 + (1 − ǫ2)r2�4, r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' It is easy to find that rij + bisj + bjsi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='1 in [5], we get S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, one can see that the following holds ri + si = ǫ2r � A(1 + r2)(1 + (1 − ǫ2)r2)3 − ǫ2r4� A(1 + r2)2� 1 + (1 − ǫ2)r2�4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3) According to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='3), F is a generalized Berwald metric (equivalently, ri + si = 0) if and only if ǫ = 0 or the following holds � 1 + (1 − ǫ2)r2�4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='4) contradicts with |ǫ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Therefore, F is a generalized Berwald metric if and only if ǫ = 0 or equivalently β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In this case, F reduces to the standard Riemannian metric F = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' On a Class of Generalized Berwald Manifolds 18 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (Xu) It is proved that a Finsler metric F = F(x, y) is of Randers type F = α+β if and only if it is a solution of the navigation problem on a Riemannian manifold (M, h) (see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Zermelo navigation is an efficient method to study of Randers metrics with certain Riemannian and non-Riemannian curvature properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' More precisely, any Randers metric F = α + β on a manifold M is a solution of the following Zermelo navigation problem h � x, y F − Wx � = 1, where h = � hij(x)yiyj is a Riemannian metric and W = Wi(x)∂/∂xi is a vector field such that h(x, −Wx) = � hij(x)Wi(x)Wj(x) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' In fact, α and β are given by α = √λh2 + W0 λ , β = −W0 λ , respectively and moreover, λ := 1 − ∥W∥2 h, W0 := hijWiyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' For more details, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, F can be written as follows F = � λh2 + W2 0 λ − W0 λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='5) In this case, the pair (h, W) is called the navigation data of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Now, let G/H be any homogeneous manifold and g = h + m is its reductive decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Suppose m = m0 + m1 be an Ad(H)-invariant decomposition, in which m0 is 1-dimensional and the Ad(H)-action on m0 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let h be a G-invariant Riemannian metric on G/H, such that m0 and m1 are h-orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Let W be a G-invairant vector field on G/H, such that W(o) ∈ m0\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, the navigation process with the data (h, W) provides a G-invariant generalized Berwald Randers metric with S = 0 (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' (Xu) As we mentioned in Introduction, the Bao-Shen’s Randers metrics on S3 are concrete generalized Berwald metrics, namely they are not Berwaldian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Any non-Riemannian homogeneous Randers sphere S3 = SU(3)/SU(2) (including Bao-Shen’s Randers metrics) sat- isfies S = 0 with constant pointwise ∥β∥α-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Then, every non-Riemannian homogeneous Randers sphere is a generalized Berwald metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' An S3 × S1, in which S3 = SU(3)/SU(2) and the navigation field is tangent to the S3-factor, is a 4-dimensional generalized Berwald Randers metric (see [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Acknowledgments: The authors are so grateful to Ming Xu for his valuable comments on this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Likewise, we thank him for providing us with examples 5 and 6 which improve the quality of our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Also, we are thankful to Behzad Najafi, Mansoor Barzegari and Libing Huang for their reading of this manuscript and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Debrecen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' 90(2017), 507-523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Akbar Tayebi and Faezeh Eslami Department of Mathematics, Faculty of Science University of Qom Qom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content=' Iran Email: akbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='tayebi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='com Email: faezeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='eslami70@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tAzT4oBgHgl3EQfE_oc/content/2301.01001v1.pdf'} diff --git a/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf b/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f821f7b31c83468ce1b24f7add1f12d079e5ffa8 --- /dev/null +++ b/7tE1T4oBgHgl3EQfBwI8/content/2301.02855v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:63f1d74c2d7cbf7129daea6e275e70b483848df463f2ab2fdaf3044bdea45372 +size 725928 diff --git a/7tE1T4oBgHgl3EQfBwI8/vector_store/index.faiss b/7tE1T4oBgHgl3EQfBwI8/vector_store/index.faiss new file mode 100644 index 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A security and social policy approach. +Andrea Russo*,1,2 and Davide Coco3 +1University of Catania, Italy +2University College Dublin, Ireland +3University of Rome, Sapienza University, Italy +*Corresponding author. Email: Andrea.russo@phd.unict.it +(Received –; revised –; accepted –; first published online –) +Abstract +We present a computational method to quantify the geopolitical impact of a space mission, based on the +national budget and data logs of previous mission, and evidencing how even if some missions succeed, +they can bring negative effect to the sponsored country. The objective of this research is to study how the +success (or failure) of a space mission can bring an economical and political benefit (or loss) to a country. +By retrieving various data, including sentiment from #hashtags related to the considered space missions, +national budgets for space exploration, and the reliability of space launch systems, from social networks, +public institutions, and online repositories, we propose an equation to evaluate the geopolitical importance +of a space mission for a particular country or space agency. The geopolitical equation can be used by public +institutions or private companies to estimate the potential impact of a space mission on the public opinion +and international relationships, which can be either positive or negative, as even successful missions may +negatively affect the international relationships and negotiation with some countries and their partners. +Also we combine the ideology of classic social policy with a security and space mission point of view, +to enlighten cultural, institutional, and political limits in public spending decisions. +Keywords: Geopolitical dynamics, Space Mission, Social policy, Political dynamics and Computational methods +1. +Introduction +———————————- +Since the end of World War II, the proposal of ambitious space programs by governments and +national space agencies has always been a means not only to push forward space exploration and +research but also to alter the prestige and geopolitical influence in the international context. For +instance, the social policy action by the 35th United States president John Fitzgerald Kennedy to +invest $25 billion (1961 US dollar value) in the Apollo mission [Bozzo 2018] was not only a social +investment to the increase public work with high qualification skills, but also a geopolitical plan +action against the URSS space expansionism. +Geopolitical value appears since the first space missions, for instance, after Russian first space satellite +"Sputnik 1" successfully orbited the Earth. The Space Race that characterized the 20th century, was +actually a geopolitical and propaganda race to determine which country would have finally had +© Cambridge University Press 2022. This is an Open Access article, distributed under the terms of the Creative Commons Attribu- +tion licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, +provided the original work is properly cited. +arXiv:2301.03538v1 [physics.soc-ph] 9 Jan 2023 + +2 +Andrea Russo et al. +access (and conquered) the “new and endless world above us”. This geopolitical space race has been +sustained by a huge effort from a social policy prospective. The $25 billion USD for the Apollo +mission on 12th September 1962 (same day of the “Address at Rice University on the Nation’s Space +Effort” speech by United States President John F. Kennedy to further inform the public about his plan +to land a man on the Moon before 1970), are the equivalent of $231 billion USD on 7th February +2022 [Appendix 1]. +Nowadays, superpowers like the United States, China, or the Russian federation, have increased +the frequency of space missions to show their presence and value in a geopolitical and international +perspective. The surge of space missions’ proposals in the last decades was due also to the cost of +access to space, which significantly decreased thanks to the development of reusable launch systems, +performing hardware, and IT and IoT improvement. +Space missions have attracted huge money investments by public and private actors, with a social +and business impact, due to the their potential economic return and their socioeconomic impact, as +the design of a space mission encourages public high quality work and many public services originate +from space activity (GPS, global mapping, high speed connection, global communication, and many +others) [ASI 2020]. +Given the aforementioned result, thus the spin-offs of space services to the population, can we +define space missions (both research and security missions) as social policy, given the socio-economic +effects and the infrastructure-related services? +The security policy of the US government in early 2000s launched by the “Pentagon” (i.e., the +headquarters building of the United States Department of Defense) highlights how the difference +between social policy and security policy became less evident. Indeed, the financial investment was +aimed at developing a security program to defend social and public infrastructures and resulted in +the X37-B program, a small shuttle that could defend other satellites, which provide social services, +from Russian and Chinese physical and cyber-attacks, thus interlacing the security and social aspects +[Spagnulo 2020]. Today, the relationship among social policy, security, space missions, and geopoli- +tics is more intricate and complex than we could have imagined years ago. +Designing a space mission is an extremely difficult task, with a high probability of failure due +to the complexity of aerospace systems and the harsh conditions under which they are supposed to +operate. The launch system plays an essential role in a space mission, as rockets must be “perfect” +systems that respond seamlessly to all the perturbations that they experience during the atmospheric +ascent and the exoatmospheric flight [Benedikter et al. 2021, 2022]. Every phase of the ascent +trajectory must be carefully studied and planned before flight, as the margin of error is extremely +small, even for the apparently simple scenarios. A first distinction by difficulty in space launches can +be made by considering suborbital flights and orbital flights. In the former, the greatest difficulty +is reaching a high altitude and then re-enter the atmosphere before completing a full revolution +around the Earth. Suborbital flights generally cross what is called the Karman line, an imaginary line +at an altitude of 100 km (330’000 ft) above sea level that conventionally marks the boundary between +the Earth’s atmosphere and outer space. Although it is not a necessary requirement, the apogee (i.e., +the maximum altitude) is a benchmark for more or less complex suborbital flights, as it indirectly +determines the speed that the vehicle will reach during re-entry and therefore the thermal and +mechanical stresses that it will undergo. For orbital flights, instead, the inherent difficulty depends +on the target orbit that the payload must be released into. The range of possible scenarios increases +enormously when it comes to orbital flights from one planet (or celestial body) to another. For +instance, a flyby, that is, the short passage of a high-speed probe near a celestial body, is seen as +an extremely critical moment compared to the simple time spent cruising. On the other hand, the +orbit insertion of a probe around a celestial body is a more critical moment than the flyby because it + +British Journal of Social Policy +3 +requires several active maneuvers that involve multiple simultaneously operating systems. In a similar +way, landing on a planet or asteroid is even a more critical accomplishment, as it involves much more +complex operations. +This paper is inspired by the growing amount of usable data from social and space science. +During recent years, social science has acquired methods and skills to collect data and use it like hard +science to highlight and identify social patterns or social dynamics. Information technology has also +increased the quality of social research, thanks to huge advancements in algorithms and data analysis. +This social-informatics improvement allows social science to connect with others disciplines, such as +space science, engineering, and complex systems. Social science can finally prove or evidence social +complex interactions in political science and others social disciplines. For example, in this work, we +try to evidence a complex interaction between space missions and international cooperation, based +on risk-cost-benefit analysis and political affinity. +Due to the great inherent complexity of aerospace projects, international cooperation allows for +mitigating the risks and costs (both financial and time-related) of space missions. There are several +examples that show that the cooperation among national space agencies or research institutes has +brought benefits to all the parties involved, not only relative to the economic return of scientific +discoveries and patented technologies, but also to a positive outcome in terms of reputation and +geopolitical prestige associated with these missions. +Besides the technical aspects, the organization and management of a space mission are also +quite challenging because every political interaction or action during the mission has a series of +emerging behaviors in international affairs, and nonlinear interactions affect also reactions on political, +economical, and security layers, which go beyond the space system [Dittmer 2014]. Complexity +science tries to explain the dynamics of a system (e.g., physical, biological, social, or economical) that +bring a different result than the expected one. A simple way to define a complex system is when the +whole system is bigger than the sum of its parts. Since space missions are related not only to social +and security policies but also to international relationships and geopolitical dynamics, then they can +be considered as complex systems. +Social policy, social network activity, and space exploration are just a slice of the whole part, +but sufficient to understand how space missions influence the geopolitical strategy. In complexity +sciences, the online social network activity patterns consist of successive, spike-like perturbations, +generated by consecutive shocks bearing conceptual similarities to the tremors preceding or following +an earthquake [Lymperopoulos 2017]. Indeed, the earthquake’s dynamics follow the Self-Organized +Critically (SOC) model [Bak and Chen 1991], present not only in nature but also in social systems +[Dmitriev, Dmitriev, and Balybin 2019]. +These dynamics could have been studied only with an hard science methodology and high quality +data, which, in the pasts years, could have been very hard to collect. +In this paper, we used computational method to collect high quality data with a rigorous +methodology to understand geopolitical and public policy effect from space mission. More precisely, +we acquired data from social networks like Twitter, to evaluate the sentiment of space missions, which +evidences the social reaction about the related space event. The sentiment analysis has been used by +many others scientist over different research areas, and, if combined with a high quality computational +method, it could highlight important patterns related to social events. This methodology has been +called Computational Social Science (CSS) [Cioffi-Revilla 2014]. +1.1 +Related Work +Computational Social Science, since its introduction, provided a valuable approach to asses and +explain social dynamical events. In this short section, we present some noteworthy works, relative to + +4 +Andrea Russo et al. +computational social complexity and sentiment analysis, helpful to understand our project. +Joseph Downing and Richard Dron [Downing and Dron 2020] have contributed to the under- +standing of constructivist security by analysing social media outputs to understand who is influential +in the security debate. Jie Yin et al. [Yin et al. 2015] have focused on analyzing Twitter messages +generated during humanitarian crises and disasters. They presented relevant methods for burst +detection, tweet filtering and classification, online clustering, and geotagging to classify whether or +not a tweet is talking about a disastrous event. A disaster type classifier groups tweets according to +representative disaster types: earth-quake, flooding, fire, storm, other disasters (e.g., traffic accident +and civil disorders), and non-disaster. +Sancheng Peng et al. [Peng et al. 2017] have worked on a framework that quantifies social +influence in mobile social networks (Phones). The social influence of users was measured by analyzing +the SMS/MMS-based communication behaviors among individuals. They have also revealed and +characterized the social relations among mobile users through the analysis of the entropy of friend +nodes and the entropy of interaction frequency. The extensive analytical results demonstrate that +the influence spread of their proposed method outperforms a random method and a degree-based +method. +Kolli et al. [Kolli, Balakrishnan, and Ramakrishnan 2017] have quantified the predictability of +cascade volumes in online social media communications, and tried to understand to what degree +are future cascade trajectories predictable and to what degree are they random. The predictability +analysis in their work reveals that for methods that combine information on frequency with temporal +correlation of the trajectory, provides the theoretical limit on predictability. Hence, their methods +such as AR and ARMA models that rely on the time order of data have the potential to achieve +prediction accuracy as high as 83% in the MemeTracker dataset and 87% in the Twitter Hashtag +dataset. Correspondingly, these methods have the potential to achieve prediction accuracy as high as +94% in the MemeTracker dataset. +Cihon and Yasseri [Cihon and Yasseri 2016], have written a short review that considers a small +number of selected papers on computational social science related to sociology and social science; they +analyze their theoretical approaches, review their methodological innovations, and offer suggestions +given the relevance of their results to political scientists and sociologists. They evidence how the +sentiment analysis content using semantic and sentiment analytic algorithms on individual users +opinions from the 15M movement in Spain ( analyzing up to 200 tweets on the topic per user) is a +promising technique for future studies of political activity, and, indeed, any activity, on Twitter. +Vidgen and Yasseri [Vidgen and Yasseri 2020] built a multi-class classifier that distinguishes +between non-Islamophobic, weak Islamophobic, and strong Islamophobic content. Accuracy is +77.6% and balanced accuracy is 83%. They applied the classifier to a dataset of 109 488 tweets +produced by far right Twitter accounts during 2017. While most tweets resulted not Islamophobic, +weak Islamophobia was considerably more prevalent (36 963 tweets) than strong (14 895 tweets). +1.2 +Geopolitical Efects of Space Missions +Sentiment analysis has been used in different research areas and with different goals. In this paper, we +apply it to assess the geopolitical effect of space missions by collecting data from latest and legacy space +mission that have or have not been successful. Improving the success rate of space missions implies, +from a geopolitical standpoint, an improvement of the international status. However, on the other +hand, a failure can damage the relationship among international partners. The success of the Apollo +11 mission by NASA made the United States the winner of the space race and raised its geopolitical +value even though many milestones were reached earlier by the URSS (first orbiting satellite and first +human in space, to name a couple). Recently, numerous space missions were successfully launched and +fulfilled their planned goals or even performed beyond expectations, receiving positive reception from +the public opinion and altering (or consolidating) the international status of the involved countries. + +British Journal of Social Policy +5 +Noteworthy examples of recent successful missions are the Ingenuity helicopter, the first helicopter +to fly outside the planet Earth, which was sent to Mars together with the Perseverance rover as part +of NASA’s Mars2020 mission, the Rosetta mission, which carried Philae, the first spacecraft ever to +accomplish a soft landing on the surface of a comet (67P/Churyumov-Gerasimenko), launched in +2014 by ESA (European Space Agency), the latest James Webb Space Telescope, placed in a very +challenging orbit (Sun–Earth Lagrangian point L2) in 2021 by NASA and ESA, and the Chang’e 4 +mission, featuring the first soft landing rover on the far side of the Moon, lunched in 2018 by CNSA +(China National Space Administration). +Even when a mission succeeds, there may be criticism from the society or even consequences +and repercussions from others international actors, affecting the geopolitical status and international +relationships of the sponsoring country. A computational social science methodology [Lazer et +al. 2020] [Lazer et al. 2009] [Cihon and Yasseri 2016] could assess how much space missions could +increase or decrease (in case of failures or partially successful missions) the geopolitical status of each +country. In particular, the reaction of people on social networks generally gives a "feel strong" status +to each country during international negotiations. Also, a statistical qualitative evaluation of the +success-to-failure ratio of space missions and rocket launches indicates the reliability of space launch +systems and mission design and management of every country. The geopolitical value depends on +other people’s perception of strength. Therefore, if many people make negative comments (or have a +negative sentiment) toward something that has been accomplished by some entity, implicitly it will +come to change other people’s perception of it, and thus also its geopolitical value. +Over the past decades, the goal of numerous space missions was not only to meet the social +policy agenda of each country or related to scientific investigations, but also to reinforce the strength +of international relationships between countries. For instance, the ExoMars mission has helped +reinforcing the relationship between ESA and Roscosmos, the Russian space agency, before the +Russian-Ukrainian war. Likewise, the DART mission is a cooperation between ASI (Italian Space +Agency) and NASA, which strengthened the bond between European countries and the United States +in space operations. The number of cooperative space missions holds the promise to significantly +increase in the near future, not only to strengthen and seal international relationships, but also to +reduce the cost and time needed to design and accomplish a space mission. +The United States was the first country to understand this new political frontier [Bozzo 2018], +and its actions have paved the way (from the industrial side) for the design of new engines and +the world’s first reusable rocket, giving a huge advantage from the other space competitors. The +objective of this paper is to quantify the geopolitical score of each state, and evaluate how much it +may fluctuate depending on space missions. +1.3 +Paper Outline +This paper is organized as follows. Section 2 introduces the research hypotheses on space missions +and geopolitical dynamics. Section 3 describes the methodological approach. In Section 4, we present +the considered data sets and discuss the results. Section 5 explores the opportunity for a Space Mission +Proposal by ESA, which can improve its geopolitical score. A conclusion section ends the manuscript. +2. +Research hypotheses +Our research hypothesis, is to establish whether it is possible to create an equation that gives a +geopolitical scores inherent to space missions, since, we think that as a ’social policy’, are included all +those services deriving from space missions that also increase the security and geopolitical prestige +of the state. To equate the outcome of space missions, as a public policy, is not so wrong given the +increases in public services and highly skilled labour that are initiated at the beginning and end of +each space mission. Defence missions can also be part of an increase in work and services to citizens. +Indeed, the US, Russian Federation, China, and EU countries use defense budgets to organize space + +6 +Andrea Russo et al. +missions for security and communication operation. Space missions can provide useful information +to the state, companies and citizens, such as weather alerts to prevent catastrophes, prediction and +prevention of potentially dangerous events, increased communication areas (internet and telephones) +and also precise information on possible antagonists of the state to which they belong. In fact, over +the past decades, many administrations have financed security space mission to obtain data about +their competitors (using for example spy satellites) or just to amplify the satellite communication +network. Anyway, the political tension about the dynamical and unpredictable security situation +between US and China, did not touch only the military and economic sector, but have reached also +the space and intelligence system [Giannangeli 2022]. As a matter of fact, Admiral Rob Bauer, the +Dutchman who since June 2021 chairs the NATO Military Committee, the highest military body +of the Atlantic Alliance, did not turn around when it comes to describing the framework of the +challenges pressing on the Euro-Atlantic area. He said that "Russia and China are increasing their +respective military capabilities, conventional and nuclear, space and cyber: it’s a fact, not an opinion." +[Pioppi 2021] +However, space missions can also be used to improve conditions between states. Indeed, inter- +national events, such as space missions, are often used as a time to reconcile or strengthen relations +between states. In fact, projects between the US and ESA are not only tools for providing jobs and +services to citizens, thus moving the economy, but are also areas of cultural inclusion and knowledge, +which is jealously guarded, especially the last one since it can be copied by foreign competitors. +There can therefore be real international as well as economic strategies to get a space project off the +ground. +In order to calculate, how space missions have a geopolitical impact, hence on security and citizen +services, we took a large part of the space missions inherent to the various countries. +2.1 +Considered Space Missions +We searched which kind of recent (i.e., after social networks existence) space missions had had +repercussions in the geopolitical and security domains, and we collected data for the missions +reported in Table 1. The list includes only missions sponsored by countries that develop launch +vehicles, as launch success and failure data are used as a way to estimate the country’s experience in +space missions (see Section Methodology or Table 4). +Table 1. Space mission data collected +Mission +Country +Result +Tianwen-1 +China +Succeeded +Tianhe +China +Succeeded +ASAT +Russia +Succeeded +Mars 2020 +US +Succeeded +James Webb +US/EU +Succeeded +Rosetta +EU +Semi-Succeeded +The Tianwen-1 is a Chinese interplanetary mission to send a robotic spacecraft on Mars. It was +launched July 23, 2020 [CNSA.gov 2020], and it landed on Mars on May 14, 2021 [CNSA 2021]. +The mission consisted of an orbiter, a lander, and a rover called "Zhurong". The mission succeeded, +and NASA’s associate administrator Thomas Zurbuchen and the director general of Roscosmos +Dmitry Rogozin congratulated the China National Space Administration (CNSA). +China has also lunched the Tianhe core module, which is the first module of the Tiangong space +station, on April 29, 2021 atop a Long March 5B rocket [S. CNSA 2021]. The core stage of the + +British Journal of Social Policy +7 +LM-5B crashed back to Earth on Saturday, May 8, 2021, after 10 controversial days that captured +the world’s attention and started a wider conversation about orbital debris and the responsibility for +the return of spent stages. +On November 15, 2021, Russia conducted a direct-ascent anti-satellite test (ASAT), destroying +one of its own space objects, a defunct satellite, in low-earth orbit [BBC 2021]. The test captured +international attention and was quickly and widely condemned as threatening and irresponsible +action, not least for the cloud of uncontrollable debris it created, which will endanger both space assets +and human spaceflight for years to come. Other countries in the past have already organised ASAT +missions, like the Mission Shakti (India), the ASM-135 ASAT (US) and the 2007 Chinese anti-satellite +missile test (China). Others to object in the wake of the test included Australia, the European Union, +Japan, NATO, and South Korea. China and India – the two countries other than Russia and the +US that have previously conducted destructive ASAT tests —- are yet to comment publicly. Also, +following the Russian ASAT mission, the International Space Station started emergency procedures +due to the debris, closing its security hatches while the crew sheltered. +Mars 2020 is a Mars rover mission [NASA.gov 2021] that includes the rover Perseverance and +a small helicopter called Ingenuity. It was launched from Earth on July 30, 2020 and landed in +Martian crater Jezero on the 18th of February of the following year. The Mars 2020 mission is +forming part of NASA’s Mars Exploration Program, which will continue with a sample return from +Mars. Ingenuity is a robotic helicopter that demonstrated the technology for rotor-craft flight in the +extremely thin atmosphere of Mars, becoming the first controlled helicopter on another planet. The +budget for the Perseverance rover was US$2.8 billion in 2020 and was cheaper than its predecessor, +the Curiosity rover, which costed $3.2 billion. Wikipedia 2022 +The James Webb Space Telescope [J. NASA 2020] is a space telescope designed primarily to conduct +infrared astronomy. NASA led the development of the telescope in collaboration with ESA and +CSA (Canadian Space Agency). The mission duration is expected to be about 20 years and, the time +planning for all the missions was 20 years (10 for planning and 10 for realization). It was Launched +on December 25, 2021 by the contractor Arianespace from the Centre Spatial Guyanais, with an +Ariane 5 rocket. The James Webb space telescope had a total budget of USD ∼ 9.70 billion (2002 +to 2021) and has several scientific goals, including the search for light coming from the first stars +and galaxies that formed in the universe after the Big Bang, the study of the galaxy, star, and planet +formation and evolution, and studying planetary systems and the origins of life. +The Rosetta mission [ESA 2014] was a space probe built by the ESA (European Space Agency) +launched on March 2, 2004. The mission’s goal was to perform a detailed and comprehensive study +of comet 67P/Churyumov–Gerasimenko (67P). On August 2014, the spacecraft reached the comet +and performed a series of manoeuvers to eventually orbit the comet at distances of 30 to 10 kilometres. +The probe also housed a lander called Philae, which unfortunately was unable to last long on the +comet’s surface after a less-than-perfect landing. This was, indeed, the first mission landing on a +comet. Yet, despite the problems, the probe’s instruments obtained the first images from a comet’s +surface, and several instruments made the first direct analysis of a comet, sending back data that +would be analysed to determine the composition of the surface. The mission cost was about ∼ 1.3 +billion € (US ∼ $ 1.8 billion). +It is worth noting that even when the mission succeeds, it may happen that the mission goal has +external or side effects that provoke a social or political reaction. Such reactions is spread, when there +is something that does not belong to the common day-order, or that may provoke instability in + +8 +Andrea Russo et al. +some country-system. We collect data from Twitter to see the social reaction to the space missions in +Table 1. In all those cases, the missions succeeded, but, in some cases, the mission can cause a social +reaction due to side effects. +3. +Methodology +Online social networks have evolved into valuable sources of information and pervasive commu- +nication platforms where people, businesses, and organizations generate and share content, build +relationships, and join public conversations. In this online ecosystem, social networks are where +information propagation is affected by external sources of influence, such as mass media, socioe- +conomic circumstances, advertising, or events, giving rise to collective intelligence or collective +behaviour patterns. +We have collected social reactions from Twitter, for every mission in Table 1 and compared them +with the difficulty and quality of the national space organization and the country that launched the +space mission to quantify the "Authority and the status of power" status in geopolitical interaction +dynamics. To simplify this operation, we create an equation called GSS, to quantify how space +mission can influence the geopolitical feelings during international affairs dynamics. +The geopolitical space score (GSS) index is the geopolitical score value from each country, +depending on the result of the spaceflight mission, related to statistical and social events [Equation (1)]. +GSS = +S +G ∗ B ∗ (F + Q) +(1) +S is the Sentiment value from the Twitter event; B is the amount of money invested (budget); G is a +difficulty rate associated with the country that launched the space mission, "not because they are +easy, but because they are hard" that imply Geopolitical effects; F is related to the success or Failure +of the mission; and Q stands for the statistical Quality and difficulty of the country in spaceflight +launch organization. +The S and Q parameters are evaluated by data scientists through statistical methods. The G score +is the only parameter that needs a subjective value because it depends on personal evaluation and +hypotheses to evaluate the difficulty rate of reaching that goal. +3.1 +S Factor +To collect data for the selected topic, we use the public API by Twitter and Tweepy. Both services +use permission from Twitter to obtain and gather data. We collected a total of ∼ 7000 tweets, but +any downloaded topic needs revisions and a cleaning process to increase the quality of the research. +This has significantly reduced the volume of the tweets. We used the same methodology for each +topic to obtain standard and quality data. In addition, to obtain the correct amount of tweets for +each day we use getdaytrends.com, a specific site where it is possible to monitor every topic in real +time as well as aged topics. +To calculate the sentiment for the selected topic, we used the VADER sentiment analysis tools +provided by MIT. The VADER sentiment quantifies each selected post or sentence’s negative, neutral, +or positive sentiment, giving in the end of the analysis, a compound score, i.e. the average between +all sentence. +However, to evaluate the sentiment for every mission, it is necessary that each mission be very +important to the general public or, at least, sufficiently viral among the space community. + +British Journal of Social Policy +9 +3.2 +G Factor +As mentioned above, the G factor is the only parameter without computational or statistical data. +This parameter evaluates the country that launches the space mission, corresponding to a difficulty +rate implying geopolitical effects. For example, if a small state succeeds in a mission with the same +budget and other factors in comparison to a big state such as the US etc., the small state will get a +bigger bonus. +Unfortunately, there is no universal value factor that gives a score to quantify the difficulty of space +missions. +Geopolitics is a social evaluation, therefore deriving from the perceptual fluctuations of people. +Since people make up a complex system, they cannot give a univocal value to the G factor, because it +always depends on the value of the individual and on the oscillations (provoked by news, newspapers, +friends, etc.) that modify the perception and evaluation of individuals and society on certain topics. +Due to this problem, we decided to self-evaluate the difficulty of a space mission, even if it is +hard to make an assessment that takes into account everything. There are many pros and cons, and, +in each case, we know that people cannot evaluate sufficiently well the difficulty of a space mission. +Yet, we believe that people can understand whether it is a surprise if a very small country alone (like +Ireland or Pakistan) can accomplish, for example, to build a Martian base, and the US could not do it. +We estimate a self-score from 0.1 to 1, where 0.1 is the maximum and 1 is the minimum possible +score. For example, a sub-orbital lunch mission could have a 1 score, an orbital mission could be a 0.8 +score. The Tianwen-1 could be evaluated as 0.3, like the Martian Ingenuity helicopter on Mars. A +full Moon base could be evaluated as 0.2 (or 0.1 if it is on Mars). All other space missions beyond the +state-of-the-art technology level cannot be evaluated, because, we do not have sufficient information +to be able to carry out the mission successfully, like a human base on the surface of Mercury, Titan +or Europa, for example. +3.3 +B Factor +In the equation, we thought it was important to evaluate, then quantify, the level of resources +invested versus the expected outcome (successful or failed mission). We thought this based on the +logic "why should a mission cost a lot of money, while it is possible doing the same things while +spending fewer resources?". To evaluate and quantify this factor, we collected data from the most +important space agencies of each country. Since the experience gained in the design and management +of past missions can be exploited in newer missions, we chose not to use a specific budget for each +mission in the equation, but, rather, the annual funds dedicated to space missions. In this logic, it +is hard to quantify how much money the oldest mission have helped (with knowledge, moneys, +competence, technology) the newest space mission. Also, huge space missions like the JWST, or the +Rosetta mission’s budget, grows up during years. The budget for the mission is spread over years, +and we cannot only take a single space mission budget, because there are also many other funded +missions different from the JWST in the same year. Therefore, we thought that the Budget factor, +imply also public opinion logic. Public opinion is usually skeptic about spending money for space +mission, because they did not (unfortunately) see the huge policy investment on research, security +and works employments, "Rockets don’t run on cash". And nowadays, is still difficult to quantify +the investment return (knowledge, moneys, competence, technology) form the policy investment +from space mission. In addition, the budget factor imply also a economic strength from the country +that invest on space mission, For example, in the equation, if a small country achieve a successful +space mission with a low budget, the GSS score will be higher than a the same country (or a bigger +country) will achieve the same mission with a higher budget. +However, to give an insight into space investment, in 2020, the policy plan for the major gov- +ernment in space mission amount of a $73.98 billion, and it is the ∼ 0,927 % as a share of gross + +10 +Andrea Russo et al. +domestic product (GDP), with a medium of ∼0,115875 % for each country. The Organisation for +Economic Co-operation and Development (OECD) as show the total value of space budgets from +the G20 country [Table 2], and we add in comparison, the years military budget for each country. +Table 2. G20 government space and military budgets (2020) +Country +2020 Space Budget in Billions +∼ National Space Budget % +∼ National Military Budget % +US +22.62 +0.480% +3.74 % +Russia +3.58 +0.210 % +4.26 % +France +4.04 +0.122% +2.07 % +Japan +3.32 +0.076 % +1 % +Saudi Arabia +2.1 +0.076 % +8.45 % +China +8.85 +0.075% +1.75 % +Italy +2.0 +0.069 % +1.56 % +Germany +2.40 +0.049 % +1.4 % +% in billion U.S. dollars [Appendix 2] +3.4 +F Factor +The factor F, was needed to the equation to weigh/ponder the space mission, and it shows if the +mission succeeded or failed. The F factor is equal to 1 if the mission succeeded, or 0 if it failed. +3.5 +Q Factor +The factor Q, evidence the statistical risk factor and the reliability for each Country about spaceflight +missions. Usually, every space mission have a risk factor, like the Apollo mission had the 95% of failure +[NASA 1965], but this information (if they still made it) is not available to the public. Therefore, since +we cannot obtain data for every single mission, we hypothesize a different evaluation method, so we +rely as a risk factor on collecting data on the success/failure of each country’s space launch. Those +data give a specific statistical risk and reliability factor to each country, since year 2010. [Appendix 3] +It can happen that some space mission fail. In this scenario, we had imagined a failure factor that +influence as a feature on GSS equation. The failures factor arises when you make mistakes over and +over again, and in this case you always lose trust from others. The "Success/Authority improvement" +comes from maintaining your own (high) standard for as long as possible. We chose to not put the +Failures Factor in the equation, because factor Q evidence all the space mission (satellite mission, +supplying mission, scientific mission etc.), and not only the scientific space mission, like those we +have chosen as subject of this paper (Table 1). +A well know example for a Failures Factor, could be the Tianhe mission from China, that had +unfortunately an uncontrolled stage reentry, and the vector crashed back to Earth without having +the possibility to calculate the final crash site, due the amount of variables on the descent stage. Sadly, +this inconvenience will arise often by the China, any time when they decide to add a core module +on his space station, because the Long March 5B (Y2) cannot claim to get the core module in orbit +(hooked to the Tianhe core stage), without losing control of the rocket on re-entry. [Jones 2021] +The Long March 5B re-entry had provoke concern about the security for some city, because +the rocket had an orbital inclination of 41.5 degrees, means the rocket body passes a little farther +north than New York, Madrid and Beijing and as far south as southern Chile and Wellington, New +Zealand, and could make it is reentry at any point within this area. With obvious concerns for those +country. + +British Journal of Social Policy +11 +4. +Data & Result +According to our model/equation, the most important and determinant score are the sentiment +score, which refers as the reaction and evaluation about the space mission’s result. The resulting +online activity give us a social input valuable to quantify and qualify the international social reaction +about space mission. Through the analysis of online activity topics, we identified the sentiment that +describing the dynamic between space activity mission and geopolitical dynamics. In table 3 we +show the sentiment results (S) from the most recently and most know space mission of the last decade. +Table 3. S factor - Sentiment analysis +Mission +Hashtag +Sentiment +Result +Tianwen-1 +Tianwen-1 +0,46447 +Succeeded +Tianhe* +ChineseRocket* +-0,05151 +Succeeded (Sides efects) +ASAT +ASAT +-0,16607 +Succeeded (Sides efects) +Mars 2020* +Perseverance* +0,428263 +Succeeded +Mars 2020 +Mars2020* +0,487525 +Succeeded +James Webb +JWST* +0,480994 +Succeeded +Rosetta +Rosetta +0,429542 +Semi-Succeeded +* Actually, some hashtags are derived not from the mission itself, but from the ef- +fect achieved (losing control of the Chinese rocket on re-entry) or the main sub- +ject of the mission (perseverance). +The results clearly evidence that an increase of side effect during mission, increases the negative +score sentiments from the space mission community. +Regarding the economical resources invested (B) we collect data from different source, and +arranged it on USD Billions dollars. Table 4 was the difficult one to make, because it is hard to obtain +data from not-direct-democratic state, like China, and also because the inclusion of both civilian and +military space budget for security missions. +Table 4. B factor - Space budgets +Country +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +Japan +1.67 +1.59 +1.68 +1.6 +1.76 +1.56 +1.33 +1.32 +3.06 +1.34 +3.32 +4.14 +Russia +2.4 +3.8 +- +5.6 +4.39 +2.42 +3.18 +- +4.17 +3.58 +3.58 +1.92 +EU +4,19 +4,52 +4,56 +4,85 +4,85 +4,65 +5,95 +6,52 +6,35 +6,49 +5,52 +5,16 +China +- +- +- +- +2.66 +- +4.91 +- +5.83 +8.00 +8.85 +10.28 +US +18.72 +18.44 +17.77 +16.86 +17.64 +18.01 +19.3 +19.50 +20.73 +21.5 +22.629 +23.27 +In billion U.S. dollars [Appendix 2] [Appendix 3] +Therefore, the data from the statistical failures presence in our equation, that mark the quality of +space launch system for each country (Q) are shows in Table 5. The data are collected since the year +2010 to 2021 (the 2022 is not finished yet), show the total number of Core Stage Manufacture send +in space, between overall launch log outside brackets and failures inside brackets. In the end, we +shows also the total launch and failures between 2010 and 2021 and the failures percentage. The +failures percentage is the Q value in our equation, evidencing the failures probability to each space +launch system-country for each launch. +It is possible to see that the Chinese government have many more launch than the US and Europe, +this is due to the low presence of space satellite (for communication and security mission) orbiting +the Earth by the Chinese government. The Chinese government had invested a huge amount of + +12 +Andrea Russo et al. +Table 5. Q factor - Statistical failures +Country +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +TOT +Failures % +China +55(3) +39(4) +34(2) +39(1) +18(2) +22(2) +19(0) +16(0) +15(1) +19(0) +19(1) +15(0) +310 (16) +5,2 +Russia +25(2) +17(0) +25(0) +20(1) +20(1) +19(1) +27(3) +35(3) +32(1) +26(2) +32(4) +31(1) +309(19) +6,1 +US +43(2) +35(3) +19(0) +29(0) +28(0) +21(0) +20(2) +20(0) +17(0) +13(1) +15(1) +15(0) +275(9) +3,3 +Europe +6(0) +5(1) +6(1) +8(1) +9(0) +9(0) +8(0) +7(0) +5(0) +8(0) +7(0) +6(0) +84(3) +3,6 +India +2(1) +2(0) +6(0) +7(0) +5(1) +7(0) +5(0) +4(0) +3(0) +2(0) +3(0) +3(2) +49(4) +8,2 +Japan +3(0) +4(0) +2(0) +6(0) +7(1) +4(0) +4(0) +4(0) +3(0) +2(0) +3(0) +2(0) +42(1) +2,4 +New Zeal. +6(1) +7(1) +6(0) +3(0) +1(1) +- +- +- +- +- +- +- +23(3) +13,6 +Iran +1(1) +2(1) +2(2) +- +- +- +1(0) +- +- +3(2) +1(0) +- +10(6) +60 +Total launch by year (total launch failures by year) [Appendix 4] +money to get enough satellites to make public infrastructures work. Also, Russian launch operation +have been increased since the 2011, due to the retirement of the space shuttle (last mission 21th July +2011), and as result, the US astronauts had to traveling by Russian Soyuz spacecraft to get to the +international space station. We have compared the data from the equation (S, T, R and Q parameters), +and in Figure 1 and Table 6 we show the score after the mission succeeded or failed. +Figure 1. GSS’s info-graphic of mains space mission +As is easiest to see, there is a negative score (we have highlight it with red on Figure 1), due +to the risk of the ASAT and the space debris rocket had on population on heart and on the ISS. +Tianwen-1 and Rosetta mission have a high GSS score also because it was a first huge milestone +for the respectively space agency (ESA and CNSA). These data evidence the difference between a +successful mission, achieving its goal; wile a successful mission, still achieving is goal but with side +effect. +4.1 +GSS trend in time +We have tried to quantify a GSS value for space Research Mission only between 2010 and 2021, for +each country, due the fact that ASAT is a military space mission. So we have gather data from all + +Geopolitical Space Score +Rosetta +JWST +Mars 2020 +Perseverance +ASAT +ChineseRock +Tianwen-1 +-2 +0 +2 +4 +6 +8 +10British Journal of Social Policy +13 +Table 6. Geopolitical Space Score +Mission +Geopolitical Space Score +Tianwen-1 +9,547438889 +ChineseRock +-0,208492857 +ASAT +-0,659007937 +Perseverance +2,198416733 +Mars 2020 +2,502628333 +JWST +2,469102533 +Rosetta +7,588575333 +factor for the equation, but unfortunately we did not get all the Sentiment Factor (S), because usually +unknown space mission does not have much reaction on social network; As said before to evaluate +the sentiment for each missions, those missions should be very important for the Humankind, or at +least "virality" for the space community, and many mission unfortunately become known to major +society only from newspaper and occasionally from TV news. To solve this problem, we estimate a +medium 0,425 S factor from successful mission, and a medium -0,125 for failed missions. For the +other well know mission (Table 1) we have used the original sentiment. Regarding the economical +resources invested (B) toward the data accumulated, we did not get the full budget planning over +years for China and Russia, so we estimate a progressive linear regression between the missing data +on table 4. +Appendix 5 - 6 +Figure 2. GSS - Research missions +As Figure 2 shows, it is possible to see the fluctuations characterised by the missions. In addition, +it is also possible to see the type of strategy and activity for each country/agencies. +For example, ESA, which is characterised by the limitation of not having a launch station on the +east coast, has specialised in international collaboration, and it is possible to see how ESA manages +missions with fluctuations of about one year (given the many collaborations, especially with NASA). + +6 +5 +4 +. +3 +GSS +2 +1 +0 +-1 +2009 +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +China +Russia +·USA +ESA14 +Andrea Russo et al. +Moreover, it should be noted that the Rosetta mission, like the Tianwen-1 mission, could be designed +as a "baptism of independence", derived from "not because they are easy, but because they are hard" +philosophy; Thy was indeed, the first to do that, showing his competence and skill upon the others +superpowers space country like US and Russia. China, on the other hand, is concentrating on a few +space research missions, since it is a new superpower and has yet to stabilise its infrastructure and +network telecommunication on space. Instead, the US have/had many active missions for a long +time, in fact the score is relatively lower than the others precisely because of their resilience. People +expect them to fail less than the others, so the astonishing/sensational felling can only be found in +very difficult missions, or when they failed badly. +Appendix 5 - 6 +Figure 3. GSS - Research missions cumulative +5. +Analysis +The analyses support the expectation that: 1) that it is possible to numerically assess geopolitical +factors; 2) that even in the case of a successful mission, there is the possibility of a negative geopolitical +feedback; 3) that public spending on space missions can also positively improve the geopolitical level +of the investing state; 4) that the geopolitical level can vary over time, depending on the success of +the missions, proportional to the investment. +An increase in the social policy inherent in space missions, in addition to providing public +employment with low-high qualification (and therefore removing people from the poverty line), +would improve the risk factor (Q Factor) decreasing it to each future space mission, thus improving +the geopolitical sentiment and also the geopolitical weight of the state. +In fact, as already mentioned, the most resilient state is the US. They are not first on the geopolitical +level despite having a high number of missions both quantitatively and qualitatively, in indeed, this +score is given by the fact that they rarely miss, thus showing a high quality work, given the quality +of the workers. +6. +Conclusions +Our study shows that can be possible using computational social method to quantify geopolitical +dynamics for every country. In our case we have shown how space mission influence geopolitical + +20 +15 +10 +GSS +5 +0 +-5 +2009 +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +China +Russia +·USA +ESABritish Journal of Social Policy +15 +dynamics, supported by a security and social policy approach. In a more specific way, we have +demonstrated (1) A socio-physical method for assessing the geopolitical level of any country, based +on its goal and its system-organization; (2) That space missions, even if successful, can bring negative +sentiment, which goes to reflect on the geopolitical value to the state itself; (3) That social policy is +also an investment in the defense and security system, and that increased investment of government +spending on space missions, also has a spillover effect on geopolitical value. From a sociological and +socio-physical point of view, through the contribution of a Computational social science model, the +GSS equation can be used to evaluate the geopolitical level not only in the spatial domain, but also in +other "competitions", such as sports competitions (Olympics, world sports championships) or music +competitions or international wars, where there are enough social reactions (S Factor) and enough +statistical data (Q - B Factor) to evaluate both the event and the organization/country participating +in the event. Therefore, our work would emphasise the most general Political policy, and in the +most specific way, the Social policy, as not only a economical financial aids, but also a more complex +systems, related to complex dynamics. As we demonstrated that Social policy is also funding the +defense, security, and geopolitical systems. It has become not only an aid to the population, but much +more. Moreover, an increase in the spending policy for space missions (which we define as part of +social policy), also has a geopolitical impact on the state that promotes them, and by reflection, to its +society. +7. +Declaration of interest statement +The authors declare no conflict of interest. +8. +Authors details +8.1 +Andrea Russo +Is a PhD candidate in Complex Systems at the University of Catania. He is currently working at the +Department of Physics and Astronomy. He collaborated with CNR Ibam, he also has worked purely +on projects involving technology and society. His main research field and interests are focused on +the study and the development of Computational social method to explain social complexity, in +particular field like Politics - Economics - Business and Defense-Security sector applications. ORCID: +0000-0003-3816-0539 +Corresponding author. Email: Andrea.russo@phd.unict.it +8.2 +Davide Coco +M.Sc. in Space and Astronautical Engineering at University of Rome "Sapienza", with several years +of experience in mission design and preliminary studies of space missions. +His main goal is to further specialize and be involved in conceptual studies, system requirements +definition, mission proposal writing, space system modeling and simulation, launcher or mission +design and operations. +He is currently leading the design and development of Cubesat’s subsystems in a small Italian startup, +Human4Research. +ORCID: 0000-0001-8010-9468 +Email: davide.coco@outlook.it +Acknowledgement +We thank Taha Yasseri for for his crucial help in the equation. + +16 +Andrea Russo et al. +References +ASI. 2020. Documento di visione strategica per lo spazio (dvss). ASI Archive 1. +Bak, Per, and Kan Chen. 1991. Self-organized criticality. Scientific American 264 (1): 46–53. +BBC, News. 2021. 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Computational social science: obstacles and opportunities. +Science 369 (6507): 1060–1062. +Lymperopoulos, Ilias N. 2017. Dynamic response and transfer function of social systems: a neuro-inspired model of collective +human activity patterns. Neural Networks 94:125–140. +NASA. 1965. Apollo reliability and quality assurance program quarterly status report, second quarter 1965. NasaArchive 1:83. +NASA, JWST. 2020. James webb space telescope. nasa.gov. +NASA.gov. 2021. Mars 2020 perseverance rover. nasa.gov. +Peng, Sancheng, Aimin Yang, Lihong Cao, Shui Yu, and Dongqing Xie. 2017. Social influence modeling using information +theory in mobile social networks. Information Sciences 379:146–159. +Pioppi, Stefano. 2021. Ipersonica e cyber, russia e cina sfidano la nato. parla l’ammiraglio bauer. formiche.net 1 (1): 1. +Salas, Erick Burgueño. 2021. Government space program spending of the leading countries in the world 2014-2020. statista.com. +Spagnulo, Marcello. 2020. Ecco lo spazioplano del pentagono, tra misteri e tecnologie futuribili. formiche.net 1 (1): 2. + +British Journal of Social Policy +17 +Vidgen, Bertie, and Taha Yasseri. 2020. Detecting weak and strong islamophobic hate speech on social media. Journal of +Information Technology & Politics 17 (1): 66–78. +Wikipedia. 2022. Curiosity (rover, https://en.wikipedia.org/wiki/Curiosity_(rover)#Cost. +Yin, Jie, Sarvnaz Karimi, Andrew Lampert, Mark Cameron, Bella Robinson, and Robert Power. 2015. Using social media to +enhance emergency situation awareness. In Twenty-fourth international joint conference on artificial intelligence. +Appendix 1. +Equivalence of the 1962 USD to 2022 USD +More specifically: 231 408 697 113.64 USD $ . +Inflation over the period: 825.63 % +Index used: USCPI31011913 (Bureau of Labor Statistics). +Initial Index: 309.12, Final Index: 2 861.33 +Link=https://fxtop.com/it/conversione-valute-passato.php +Appendix 2. +Space annual budget (∼ approximately) over years and percentage of the na- +tional budget +It is very hard to collect data for the most important Space Agency, this is due to the different money +value (Yen - EURO - USD), and also the difficult to obtain data from not-democratic state, like +China, due to information security and also due to the inclusion of both civilian and military space +spending global security. +Source: +1. https://www.statista.com/statistics/745717/global-governmental-spending-on-space-programs- +leading-countries/ +2. https://www.euroconsult-ec.com/press-release/government-space-budgets-driven-by-space-exploration- +and-militarization-hit-record-92-billion-investment-in-2021-despite-covid-with-1-trillion-forecast- +over-the-decade/ +3. https://spacenews.com/op-ed-global-government-space-budgets-continues-multiyear-rebound/ +4. https://stacker.com/stories/2524/countries-spend-most-space-exploration +5. https://www.oecd.org/sti/inno/space-forum/space-economy-for-people-planet-and-prosperity.pdf +6. https://global,jaxa,jp/about/transition/index,html +7. esa.int +8. https://en.wikipedia.org/wiki/Budget_of_NASA +9. https://global.jaxa.jp/ +[Salas 2021] +Military budget: +1. https://databank.worldbank.org/reports.aspx?source=2&type=metadata&series=MS.MIL.XPND.GD.ZS# +2. https://www.defensenews.com/global/2021/04/26/the-world-spent-almost-2-trillion-on-defense- +in-2020/ +Appendix 3. +Billion U.S. dollars +Exchange rate 1.1269 at 27/02/2022 02:50 28 Feb 2020 - 25 Feb 2022 +Appendix 4. +Statistical failures +Annual space reports: +From 2010 to 2021 Launch Log +Source = https://www.spacelaunchreport.com/index.html + +18 +Andrea Russo et al. +Appendix 5. +GSS - Research Missions +Table 7. GSS related to space research missions +Country +2009 +2010 +2011 +2012 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +2020 +2021 +China +- +- +-0,541666667 +4,969230769 +5,383333333 +- +- +- +- +- +1,615 +1,490577956 +- +Russia +- +- +-0,334429825 +- +- +- +- +2,389937107 +- +- +1,75719055 +- +- +USA +- +0,808621288 +0,464438894 +1,026104163 +-0,040508492 +0,492783694 +0,512710274 +0,976252159 +- +0,776134433 +0,876356589 +0,011896792 +0,942333613 +ESA +- +4,647391567 +- +- +2,648339061 +- +2,762246117 +1,128012708 +- +2,654855643 +- +1,903820817 +2,11289354 +Japan +- +4,542027057 +- +-2,976190476 +- +4,277597403 +- +- +- +5,740740741 +-7,462686567 +- +- +TOT +- +3,332679971 +-0,137219199 +1,006381485 +2,663721301 +2,385190548 +1,637478196 +1,498067325 +- +3,057243606 +-0,803534857 +1,135431855 +1,527613577 +GSS - Research missions +Appendix 6. +Country Succeeded and Failed Space Research Mission +Table 8. Country Succeeded(Failed) Space Research Mission +Year +China +Russia +USA +ESA +Japan +India +2010 +- +- +Deep Impact +Rosetta +Akatsuki +- +- +- +Stardust +- +IKAROS (Shin’en) +- +2011 +(Yinghuo-1) +(Fobos-Grunt) +Dawn +- +- +- +2012 +Chang’e 2 +- +MSL Curiosity +- +(PROCYON) +- +2013 +Chang’e 3 +- +(Deep Impact) +Gaia +- +- +2014 +- +- +MAVEN +- +Shin’en 2 +Mangalyaan +2015 +- +- +DSCOVR +LISA Pathfinder +- +- +- +- +New Horizons +- +- +- +- +- +Dawn +- +- +- +2016 +- +ExoMars 2016 (Schiaparelli EDM lander) +Juno +ExoMars 2016 (Schiaparelli EDM lander) +- +- +2017 +- +- +- +- +- +- +2018 +- +- +Parker Solar Probe +MASCOT +Hayabusa2 +- +- +- +MarCO A "WALL-E" +BepiColombo +BepiColombo +- +- +- +MarCO B "EVE" +- +- +- +- +- +OSIRIS-REx +- +- +- +- +- +InSight +- +- +- +2019 +Chang’e 4 +Spektr-RG +New Horizons +- +(Minerva II-2) +- +- +- +Spektr-RG +- +- +- +2020 +Chang’e 5 +- +Mars 2020 +Solar Orbiter +- +- +Tianwen-1 +- +- +- +- +- +Beidou +- +- +- +- +- +2021 +- +- +James Webb +James Webb +- +- +GSS - Research missions + diff --git a/89E1T4oBgHgl3EQf7wWX/content/tmp_files/load_file.txt b/89E1T4oBgHgl3EQf7wWX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df833620d7f413a2be37b6e87188398b0807abf3 --- /dev/null +++ b/89E1T4oBgHgl3EQf7wWX/content/tmp_files/load_file.txt @@ -0,0 +1,767 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf,len=766 +page_content='British Journal of Social Policy (2022), 2, 1–18 doi:– British Journal of Social Policy ARTICLE Quantify how space mission influence geopolitical dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A security and social policy approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Andrea Russo*,1,2 and Davide Coco3 1University of Catania, Italy 2University College Dublin, Ireland 3University of Rome, Sapienza University, Italy Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Email: Andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='russo@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='it (Received –;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' revised –;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' accepted –;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' first published online –) Abstract We present a computational method to quantify the geopolitical impact of a space mission, based on the national budget and data logs of previous mission, and evidencing how even if some missions succeed, they can bring negative effect to the sponsored country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The objective of this research is to study how the success (or failure) of a space mission can bring an economical and political benefit (or loss) to a country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' By retrieving various data, including sentiment from #hashtags related to the considered space missions, national budgets for space exploration, and the reliability of space launch systems, from social networks, public institutions, and online repositories, we propose an equation to evaluate the geopolitical importance of a space mission for a particular country or space agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The geopolitical equation can be used by public institutions or private companies to estimate the potential impact of a space mission on the public opinion and international relationships, which can be either positive or negative, as even successful missions may negatively affect the international relationships and negotiation with some countries and their partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Also we combine the ideology of classic social policy with a security and space mission point of view, to enlighten cultural, institutional, and political limits in public spending decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Keywords: Geopolitical dynamics, Space Mission, Social policy, Political dynamics and Computational methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Introduction ———————————- Since the end of World War II, the proposal of ambitious space programs by governments and national space agencies has always been a means not only to push forward space exploration and research but also to alter the prestige and geopolitical influence in the international context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For instance, the social policy action by the 35th United States president John Fitzgerald Kennedy to invest $25 billion (1961 US dollar value) in the Apollo mission [Bozzo 2018] was not only a social investment to the increase public work with high qualification skills, but also a geopolitical plan action against the URSS space expansionism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Geopolitical value appears since the first space missions, for instance, after Russian first space satellite "Sputnik 1" successfully orbited the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The Space Race that characterized the 20th century, was actually a geopolitical and propaganda race to determine which country would have finally had © Cambridge University Press 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This is an Open Access article, distributed under the terms of the Creative Commons Attribu- tion licence (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='03538v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='soc-ph] 9 Jan 2023 2 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' access (and conquered) the “new and endless world above us”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This geopolitical space race has been sustained by a huge effort from a social policy prospective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The $25 billion USD for the Apollo mission on 12th September 1962 (same day of the “Address at Rice University on the Nation’s Space Effort” speech by United States President John F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Kennedy to further inform the public about his plan to land a man on the Moon before 1970), are the equivalent of $231 billion USD on 7th February 2022 [Appendix 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Nowadays, superpowers like the United States, China, or the Russian federation, have increased the frequency of space missions to show their presence and value in a geopolitical and international perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The surge of space missions’ proposals in the last decades was due also to the cost of access to space, which significantly decreased thanks to the development of reusable launch systems, performing hardware, and IT and IoT improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Space missions have attracted huge money investments by public and private actors, with a social and business impact, due to the their potential economic return and their socioeconomic impact, as the design of a space mission encourages public high quality work and many public services originate from space activity (GPS, global mapping, high speed connection, global communication, and many others) [ASI 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Given the aforementioned result, thus the spin-offs of space services to the population, can we define space missions (both research and security missions) as social policy, given the socio-economic effects and the infrastructure-related services?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The security policy of the US government in early 2000s launched by the “Pentagon” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=', the headquarters building of the United States Department of Defense) highlights how the difference between social policy and security policy became less evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Indeed, the financial investment was aimed at developing a security program to defend social and public infrastructures and resulted in the X37-B program, a small shuttle that could defend other satellites, which provide social services, from Russian and Chinese physical and cyber-attacks, thus interlacing the security and social aspects [Spagnulo 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Today, the relationship among social policy, security, space missions, and geopoli- tics is more intricate and complex than we could have imagined years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Designing a space mission is an extremely difficult task, with a high probability of failure due to the complexity of aerospace systems and the harsh conditions under which they are supposed to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The launch system plays an essential role in a space mission, as rockets must be “perfect” systems that respond seamlessly to all the perturbations that they experience during the atmospheric ascent and the exoatmospheric flight [Benedikter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2021, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Every phase of the ascent trajectory must be carefully studied and planned before flight, as the margin of error is extremely small, even for the apparently simple scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A first distinction by difficulty in space launches can be made by considering suborbital flights and orbital flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In the former, the greatest difficulty is reaching a high altitude and then re-enter the atmosphere before completing a full revolution around the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Suborbital flights generally cross what is called the Karman line, an imaginary line at an altitude of 100 km (330’000 ft) above sea level that conventionally marks the boundary between the Earth’s atmosphere and outer space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Although it is not a necessary requirement, the apogee (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=', the maximum altitude) is a benchmark for more or less complex suborbital flights, as it indirectly determines the speed that the vehicle will reach during re-entry and therefore the thermal and mechanical stresses that it will undergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For orbital flights, instead, the inherent difficulty depends on the target orbit that the payload must be released into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The range of possible scenarios increases enormously when it comes to orbital flights from one planet (or celestial body) to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For instance, a flyby, that is, the short passage of a high-speed probe near a celestial body, is seen as an extremely critical moment compared to the simple time spent cruising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' On the other hand, the orbit insertion of a probe around a celestial body is a more critical moment than the flyby because it British Journal of Social Policy 3 requires several active maneuvers that involve multiple simultaneously operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In a similar way, landing on a planet or asteroid is even a more critical accomplishment, as it involves much more complex operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This paper is inspired by the growing amount of usable data from social and space science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' During recent years, social science has acquired methods and skills to collect data and use it like hard science to highlight and identify social patterns or social dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Information technology has also increased the quality of social research, thanks to huge advancements in algorithms and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This social-informatics improvement allows social science to connect with others disciplines, such as space science, engineering, and complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Social science can finally prove or evidence social complex interactions in political science and others social disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For example, in this work, we try to evidence a complex interaction between space missions and international cooperation, based on risk-cost-benefit analysis and political affinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Due to the great inherent complexity of aerospace projects, international cooperation allows for mitigating the risks and costs (both financial and time-related) of space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' There are several examples that show that the cooperation among national space agencies or research institutes has brought benefits to all the parties involved, not only relative to the economic return of scientific discoveries and patented technologies, but also to a positive outcome in terms of reputation and geopolitical prestige associated with these missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Besides the technical aspects, the organization and management of a space mission are also quite challenging because every political interaction or action during the mission has a series of emerging behaviors in international affairs, and nonlinear interactions affect also reactions on political, economical, and security layers, which go beyond the space system [Dittmer 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Complexity science tries to explain the dynamics of a system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=', physical, biological, social, or economical) that bring a different result than the expected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A simple way to define a complex system is when the whole system is bigger than the sum of its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Since space missions are related not only to social and security policies but also to international relationships and geopolitical dynamics, then they can be considered as complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Social policy, social network activity, and space exploration are just a slice of the whole part, but sufficient to understand how space missions influence the geopolitical strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In complexity sciences, the online social network activity patterns consist of successive, spike-like perturbations, generated by consecutive shocks bearing conceptual similarities to the tremors preceding or following an earthquake [Lymperopoulos 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Indeed, the earthquake’s dynamics follow the Self-Organized Critically (SOC) model [Bak and Chen 1991], present not only in nature but also in social systems [Dmitriev, Dmitriev, and Balybin 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' These dynamics could have been studied only with an hard science methodology and high quality data, which, in the pasts years, could have been very hard to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In this paper, we used computational method to collect high quality data with a rigorous methodology to understand geopolitical and public policy effect from space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' More precisely, we acquired data from social networks like Twitter, to evaluate the sentiment of space missions, which evidences the social reaction about the related space event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The sentiment analysis has been used by many others scientist over different research areas, and, if combined with a high quality computational method, it could highlight important patterns related to social events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This methodology has been called Computational Social Science (CSS) [Cioffi-Revilla 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 Related Work Computational Social Science, since its introduction, provided a valuable approach to asses and explain social dynamical events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In this short section, we present some noteworthy works, relative to 4 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' computational social complexity and sentiment analysis, helpful to understand our project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Joseph Downing and Richard Dron [Downing and Dron 2020] have contributed to the under- standing of constructivist security by analysing social media outputs to understand who is influential in the security debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Jie Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' [Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2015] have focused on analyzing Twitter messages generated during humanitarian crises and disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' They presented relevant methods for burst detection, tweet filtering and classification, online clustering, and geotagging to classify whether or not a tweet is talking about a disastrous event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A disaster type classifier groups tweets according to representative disaster types: earth-quake, flooding, fire, storm, other disasters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=', traffic accident and civil disorders), and non-disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Sancheng Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' [Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2017] have worked on a framework that quantifies social influence in mobile social networks (Phones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The social influence of users was measured by analyzing the SMS/MMS-based communication behaviors among individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' They have also revealed and characterized the social relations among mobile users through the analysis of the entropy of friend nodes and the entropy of interaction frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The extensive analytical results demonstrate that the influence spread of their proposed method outperforms a random method and a degree-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Kolli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' [Kolli, Balakrishnan, and Ramakrishnan 2017] have quantified the predictability of cascade volumes in online social media communications, and tried to understand to what degree are future cascade trajectories predictable and to what degree are they random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The predictability analysis in their work reveals that for methods that combine information on frequency with temporal correlation of the trajectory, provides the theoretical limit on predictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Hence, their methods such as AR and ARMA models that rely on the time order of data have the potential to achieve prediction accuracy as high as 83% in the MemeTracker dataset and 87% in the Twitter Hashtag dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Correspondingly, these methods have the potential to achieve prediction accuracy as high as 94% in the MemeTracker dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Cihon and Yasseri [Cihon and Yasseri 2016], have written a short review that considers a small number of selected papers on computational social science related to sociology and social science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' they analyze their theoretical approaches, review their methodological innovations, and offer suggestions given the relevance of their results to political scientists and sociologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' They evidence how the sentiment analysis content using semantic and sentiment analytic algorithms on individual users opinions from the 15M movement in Spain ( analyzing up to 200 tweets on the topic per user) is a promising technique for future studies of political activity, and, indeed, any activity, on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Vidgen and Yasseri [Vidgen and Yasseri 2020] built a multi-class classifier that distinguishes between non-Islamophobic, weak Islamophobic, and strong Islamophobic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Accuracy is 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='6% and balanced accuracy is 83%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' They applied the classifier to a dataset of 109 488 tweets produced by far right Twitter accounts during 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' While most tweets resulted not Islamophobic, weak Islamophobia was considerably more prevalent (36 963 tweets) than strong (14 895 tweets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 Geopolitical Efects of Space Missions Sentiment analysis has been used in different research areas and with different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In this paper, we apply it to assess the geopolitical effect of space missions by collecting data from latest and legacy space mission that have or have not been successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Improving the success rate of space missions implies, from a geopolitical standpoint, an improvement of the international status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' However, on the other hand, a failure can damage the relationship among international partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The success of the Apollo 11 mission by NASA made the United States the winner of the space race and raised its geopolitical value even though many milestones were reached earlier by the URSS (first orbiting satellite and first human in space, to name a couple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Recently, numerous space missions were successfully launched and fulfilled their planned goals or even performed beyond expectations, receiving positive reception from the public opinion and altering (or consolidating) the international status of the involved countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' British Journal of Social Policy 5 Noteworthy examples of recent successful missions are the Ingenuity helicopter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' the first helicopter to fly outside the planet Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' which was sent to Mars together with the Perseverance rover as part of NASA’s Mars2020 mission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' the Rosetta mission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' which carried Philae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' the first spacecraft ever to accomplish a soft landing on the surface of a comet (67P/Churyumov-Gerasimenko),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' launched in 2014 by ESA (European Space Agency),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' the latest James Webb Space Telescope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' placed in a very challenging orbit (Sun–Earth Lagrangian point L2) in 2021 by NASA and ESA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' and the Chang’e 4 mission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' featuring the first soft landing rover on the far side of the Moon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' lunched in 2018 by CNSA (China National Space Administration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Even when a mission succeeds, there may be criticism from the society or even consequences and repercussions from others international actors, affecting the geopolitical status and international relationships of the sponsoring country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A computational social science methodology [Lazer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2020] [Lazer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2009] [Cihon and Yasseri 2016] could assess how much space missions could increase or decrease (in case of failures or partially successful missions) the geopolitical status of each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In particular, the reaction of people on social networks generally gives a "feel strong" status to each country during international negotiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Also, a statistical qualitative evaluation of the success-to-failure ratio of space missions and rocket launches indicates the reliability of space launch systems and mission design and management of every country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The geopolitical value depends on other people’s perception of strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Therefore, if many people make negative comments (or have a negative sentiment) toward something that has been accomplished by some entity, implicitly it will come to change other people’s perception of it, and thus also its geopolitical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Over the past decades, the goal of numerous space missions was not only to meet the social policy agenda of each country or related to scientific investigations, but also to reinforce the strength of international relationships between countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For instance, the ExoMars mission has helped reinforcing the relationship between ESA and Roscosmos, the Russian space agency, before the Russian-Ukrainian war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Likewise, the DART mission is a cooperation between ASI (Italian Space Agency) and NASA, which strengthened the bond between European countries and the United States in space operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The number of cooperative space missions holds the promise to significantly increase in the near future, not only to strengthen and seal international relationships, but also to reduce the cost and time needed to design and accomplish a space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The United States was the first country to understand this new political frontier [Bozzo 2018], and its actions have paved the way (from the industrial side) for the design of new engines and the world’s first reusable rocket, giving a huge advantage from the other space competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The objective of this paper is to quantify the geopolitical score of each state, and evaluate how much it may fluctuate depending on space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='3 Paper Outline This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Section 2 introduces the research hypotheses on space missions and geopolitical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Section 3 describes the methodological approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In Section 4, we present the considered data sets and discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Section 5 explores the opportunity for a Space Mission Proposal by ESA, which can improve its geopolitical score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A conclusion section ends the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Research hypotheses Our research hypothesis, is to establish whether it is possible to create an equation that gives a geopolitical scores inherent to space missions, since, we think that as a ’social policy’, are included all those services deriving from space missions that also increase the security and geopolitical prestige of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' To equate the outcome of space missions, as a public policy, is not so wrong given the increases in public services and highly skilled labour that are initiated at the beginning and end of each space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Defence missions can also be part of an increase in work and services to citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Indeed, the US, Russian Federation, China, and EU countries use defense budgets to organize space 6 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' missions for security and communication operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Space missions can provide useful information to the state, companies and citizens, such as weather alerts to prevent catastrophes, prediction and prevention of potentially dangerous events, increased communication areas (internet and telephones) and also precise information on possible antagonists of the state to which they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In fact, over the past decades, many administrations have financed security space mission to obtain data about their competitors (using for example spy satellites) or just to amplify the satellite communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Anyway, the political tension about the dynamical and unpredictable security situation between US and China, did not touch only the military and economic sector, but have reached also the space and intelligence system [Giannangeli 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' As a matter of fact, Admiral Rob Bauer, the Dutchman who since June 2021 chairs the NATO Military Committee, the highest military body of the Atlantic Alliance, did not turn around when it comes to describing the framework of the challenges pressing on the Euro-Atlantic area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' He said that "Russia and China are increasing their respective military capabilities, conventional and nuclear, space and cyber: it’s a fact, not an opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='" [Pioppi 2021] However, space missions can also be used to improve conditions between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Indeed, inter- national events, such as space missions, are often used as a time to reconcile or strengthen relations between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In fact, projects between the US and ESA are not only tools for providing jobs and services to citizens, thus moving the economy, but are also areas of cultural inclusion and knowledge, which is jealously guarded, especially the last one since it can be copied by foreign competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' There can therefore be real international as well as economic strategies to get a space project off the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In order to calculate, how space missions have a geopolitical impact, hence on security and citizen services, we took a large part of the space missions inherent to the various countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 Considered Space Missions We searched which kind of recent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=', after social networks existence) space missions had had repercussions in the geopolitical and security domains, and we collected data for the missions reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The list includes only missions sponsored by countries that develop launch vehicles, as launch success and failure data are used as a way to estimate the country’s experience in space missions (see Section Methodology or Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Space mission data collected Mission Country Result Tianwen-1 China Succeeded Tianhe China Succeeded ASAT Russia Succeeded Mars 2020 US Succeeded James Webb US/EU Succeeded Rosetta EU Semi-Succeeded The Tianwen-1 is a Chinese interplanetary mission to send a robotic spacecraft on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' It was launched July 23, 2020 [CNSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='gov 2020], and it landed on Mars on May 14, 2021 [CNSA 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The mission consisted of an orbiter, a lander, and a rover called "Zhurong".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The mission succeeded, and NASA’s associate administrator Thomas Zurbuchen and the director general of Roscosmos Dmitry Rogozin congratulated the China National Space Administration (CNSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' China has also lunched the Tianhe core module, which is the first module of the Tiangong space station, on April 29, 2021 atop a Long March 5B rocket [S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' CNSA 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The core stage of the British Journal of Social Policy 7 LM-5B crashed back to Earth on Saturday, May 8, 2021, after 10 controversial days that captured the world’s attention and started a wider conversation about orbital debris and the responsibility for the return of spent stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' On November 15, 2021, Russia conducted a direct-ascent anti-satellite test (ASAT), destroying one of its own space objects, a defunct satellite, in low-earth orbit [BBC 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The test captured international attention and was quickly and widely condemned as threatening and irresponsible action, not least for the cloud of uncontrollable debris it created, which will endanger both space assets and human spaceflight for years to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Other countries in the past have already organised ASAT missions, like the Mission Shakti (India), the ASM-135 ASAT (US) and the 2007 Chinese anti-satellite missile test (China).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Others to object in the wake of the test included Australia, the European Union, Japan, NATO, and South Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' China and India – the two countries other than Russia and the US that have previously conducted destructive ASAT tests —- are yet to comment publicly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Also, following the Russian ASAT mission, the International Space Station started emergency procedures due to the debris, closing its security hatches while the crew sheltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Mars 2020 is a Mars rover mission [NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='gov 2021] that includes the rover Perseverance and a small helicopter called Ingenuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' It was launched from Earth on July 30, 2020 and landed in Martian crater Jezero on the 18th of February of the following year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The Mars 2020 mission is forming part of NASA’s Mars Exploration Program, which will continue with a sample return from Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Ingenuity is a robotic helicopter that demonstrated the technology for rotor-craft flight in the extremely thin atmosphere of Mars, becoming the first controlled helicopter on another planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The budget for the Perseverance rover was US$2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='8 billion in 2020 and was cheaper than its predecessor, the Curiosity rover, which costed $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 billion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Wikipedia 2022 The James Webb Space Telescope [J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' NASA 2020] is a space telescope designed primarily to conduct infrared astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' NASA led the development of the telescope in collaboration with ESA and CSA (Canadian Space Agency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The mission duration is expected to be about 20 years and, the time planning for all the missions was 20 years (10 for planning and 10 for realization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' It was Launched on December 25, 2021 by the contractor Arianespace from the Centre Spatial Guyanais, with an Ariane 5 rocket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The James Webb space telescope had a total budget of USD ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='70 billion (2002 to 2021) and has several scientific goals, including the search for light coming from the first stars and galaxies that formed in the universe after the Big Bang, the study of the galaxy, star, and planet formation and evolution, and studying planetary systems and the origins of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The Rosetta mission [ESA 2014] was a space probe built by the ESA (European Space Agency) launched on March 2, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The mission’s goal was to perform a detailed and comprehensive study of comet 67P/Churyumov–Gerasimenko (67P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' On August 2014, the spacecraft reached the comet and performed a series of manoeuvers to eventually orbit the comet at distances of 30 to 10 kilometres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The probe also housed a lander called Philae, which unfortunately was unable to last long on the comet’s surface after a less-than-perfect landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This was, indeed, the first mission landing on a comet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Yet, despite the problems, the probe’s instruments obtained the first images from a comet’s surface, and several instruments made the first direct analysis of a comet, sending back data that would be analysed to determine the composition of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The mission cost was about ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='3 billion € (US ∼ $ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='8 billion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' It is worth noting that even when the mission succeeds, it may happen that the mission goal has external or side effects that provoke a social or political reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Such reactions is spread, when there is something that does not belong to the common day-order, or that may provoke instability in 8 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' some country-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We collect data from Twitter to see the social reaction to the space missions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In all those cases, the missions succeeded, but, in some cases, the mission can cause a social reaction due to side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Methodology Online social networks have evolved into valuable sources of information and pervasive commu- nication platforms where people, businesses, and organizations generate and share content, build relationships, and join public conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In this online ecosystem, social networks are where information propagation is affected by external sources of influence, such as mass media, socioe- conomic circumstances, advertising, or events, giving rise to collective intelligence or collective behaviour patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We have collected social reactions from Twitter, for every mission in Table 1 and compared them with the difficulty and quality of the national space organization and the country that launched the space mission to quantify the "Authority and the status of power" status in geopolitical interaction dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' To simplify this operation, we create an equation called GSS, to quantify how space mission can influence the geopolitical feelings during international affairs dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The geopolitical space score (GSS) index is the geopolitical score value from each country, depending on the result of the spaceflight mission, related to statistical and social events [Equation (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' GSS = S G ∗ B ∗ (F + Q) (1) S is the Sentiment value from the Twitter event;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' B is the amount of money invested (budget);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' G is a difficulty rate associated with the country that launched the space mission, "not because they are easy, but because they are hard" that imply Geopolitical effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' F is related to the success or Failure of the mission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' and Q stands for the statistical Quality and difficulty of the country in spaceflight launch organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The S and Q parameters are evaluated by data scientists through statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The G score is the only parameter that needs a subjective value because it depends on personal evaluation and hypotheses to evaluate the difficulty rate of reaching that goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 S Factor To collect data for the selected topic, we use the public API by Twitter and Tweepy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Both services use permission from Twitter to obtain and gather data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We collected a total of ∼ 7000 tweets, but any downloaded topic needs revisions and a cleaning process to increase the quality of the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This has significantly reduced the volume of the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We used the same methodology for each topic to obtain standard and quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In addition, to obtain the correct amount of tweets for each day we use getdaytrends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='com, a specific site where it is possible to monitor every topic in real time as well as aged topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' To calculate the sentiment for the selected topic, we used the VADER sentiment analysis tools provided by MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The VADER sentiment quantifies each selected post or sentence’s negative, neutral, or positive sentiment, giving in the end of the analysis, a compound score, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' the average between all sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' However, to evaluate the sentiment for every mission, it is necessary that each mission be very important to the general public or, at least, sufficiently viral among the space community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' British Journal of Social Policy 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 G Factor As mentioned above, the G factor is the only parameter without computational or statistical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' This parameter evaluates the country that launches the space mission, corresponding to a difficulty rate implying geopolitical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For example, if a small state succeeds in a mission with the same budget and other factors in comparison to a big state such as the US etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=', the small state will get a bigger bonus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Unfortunately, there is no universal value factor that gives a score to quantify the difficulty of space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Geopolitics is a social evaluation, therefore deriving from the perceptual fluctuations of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Since people make up a complex system, they cannot give a univocal value to the G factor, because it always depends on the value of the individual and on the oscillations (provoked by news, newspapers, friends, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=') that modify the perception and evaluation of individuals and society on certain topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Due to this problem, we decided to self-evaluate the difficulty of a space mission, even if it is hard to make an assessment that takes into account everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' There are many pros and cons, and, in each case, we know that people cannot evaluate sufficiently well the difficulty of a space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Yet, we believe that people can understand whether it is a surprise if a very small country alone (like Ireland or Pakistan) can accomplish, for example, to build a Martian base, and the US could not do it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We estimate a self-score from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 to 1, where 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 is the maximum and 1 is the minimum possible score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For example, a sub-orbital lunch mission could have a 1 score, an orbital mission could be a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='8 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The Tianwen-1 could be evaluated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='3, like the Martian Ingenuity helicopter on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A full Moon base could be evaluated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 if it is on Mars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' All other space missions beyond the state-of-the-art technology level cannot be evaluated, because, we do not have sufficient information to be able to carry out the mission successfully, like a human base on the surface of Mercury, Titan or Europa, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='3 B Factor In the equation, we thought it was important to evaluate, then quantify, the level of resources invested versus the expected outcome (successful or failed mission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We thought this based on the logic "why should a mission cost a lot of money, while it is possible doing the same things while spending fewer resources?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' To evaluate and quantify this factor, we collected data from the most important space agencies of each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Since the experience gained in the design and management of past missions can be exploited in newer missions, we chose not to use a specific budget for each mission in the equation, but, rather, the annual funds dedicated to space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In this logic, it is hard to quantify how much money the oldest mission have helped (with knowledge, moneys, competence, technology) the newest space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Also, huge space missions like the JWST, or the Rosetta mission’s budget, grows up during years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The budget for the mission is spread over years, and we cannot only take a single space mission budget, because there are also many other funded missions different from the JWST in the same year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Therefore, we thought that the Budget factor, imply also public opinion logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Public opinion is usually skeptic about spending money for space mission, because they did not (unfortunately) see the huge policy investment on research, security and works employments, "Rockets don’t run on cash".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' And nowadays, is still difficult to quantify the investment return (knowledge, moneys, competence, technology) form the policy investment from space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In addition, the budget factor imply also a economic strength from the country that invest on space mission, For example, in the equation, if a small country achieve a successful space mission with a low budget, the GSS score will be higher than a the same country (or a bigger country) will achieve the same mission with a higher budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' However, to give an insight into space investment, in 2020, the policy plan for the major gov- ernment in space mission amount of a $73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='98 billion, and it is the ∼ 0,927 % as a share of gross 10 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' domestic product (GDP), with a medium of ∼0,115875 % for each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The Organisation for Economic Co-operation and Development (OECD) as show the total value of space budgets from the G20 country [Table 2], and we add in comparison, the years military budget for each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' G20 government space and military budgets (2020) Country 2020 Space Budget in Billions ∼ National Space Budget % ∼ National Military Budget % US 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='480% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='74 % Russia 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='210 % 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='26 % France 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='122% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='07 % Japan 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='076 % 1 % Saudi Arabia 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='076 % 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='45 % China 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='075% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='75 % Italy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='069 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='56 % Germany 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='049 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='4 % % in billion U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' dollars [Appendix 2] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='4 F Factor The factor F, was needed to the equation to weigh/ponder the space mission, and it shows if the mission succeeded or failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The F factor is equal to 1 if the mission succeeded, or 0 if it failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='5 Q Factor The factor Q, evidence the statistical risk factor and the reliability for each Country about spaceflight missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Usually, every space mission have a risk factor, like the Apollo mission had the 95% of failure [NASA 1965], but this information (if they still made it) is not available to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Therefore, since we cannot obtain data for every single mission, we hypothesize a different evaluation method, so we rely as a risk factor on collecting data on the success/failure of each country’s space launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Those data give a specific statistical risk and reliability factor to each country, since year 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' [Appendix 3] It can happen that some space mission fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In this scenario, we had imagined a failure factor that influence as a feature on GSS equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The failures factor arises when you make mistakes over and over again, and in this case you always lose trust from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The "Success/Authority improvement" comes from maintaining your own (high) standard for as long as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We chose to not put the Failures Factor in the equation, because factor Q evidence all the space mission (satellite mission, supplying mission, scientific mission etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' ), and not only the scientific space mission, like those we have chosen as subject of this paper (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' A well know example for a Failures Factor, could be the Tianhe mission from China, that had unfortunately an uncontrolled stage reentry, and the vector crashed back to Earth without having the possibility to calculate the final crash site, due the amount of variables on the descent stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Sadly, this inconvenience will arise often by the China, any time when they decide to add a core module on his space station, because the Long March 5B (Y2) cannot claim to get the core module in orbit (hooked to the Tianhe core stage), without losing control of the rocket on re-entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' [Jones 2021] The Long March 5B re-entry had provoke concern about the security for some city, because the rocket had an orbital inclination of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='5 degrees, means the rocket body passes a little farther north than New York, Madrid and Beijing and as far south as southern Chile and Wellington, New Zealand, and could make it is reentry at any point within this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' With obvious concerns for those country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' British Journal of Social Policy 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Data & Result According to our model/equation, the most important and determinant score are the sentiment score, which refers as the reaction and evaluation about the space mission’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The resulting online activity give us a social input valuable to quantify and qualify the international social reaction about space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Through the analysis of online activity topics, we identified the sentiment that describing the dynamic between space activity mission and geopolitical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In table 3 we show the sentiment results (S) from the most recently and most know space mission of the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' S factor - Sentiment analysis Mission Hashtag Sentiment Result Tianwen-1 Tianwen-1 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='46447 Succeeded Tianhe* ChineseRocket* 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='05151 Succeeded (Sides efects) ASAT ASAT 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='16607 Succeeded (Sides efects) Mars 2020* Perseverance* 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='428263 Succeeded Mars 2020 Mars2020* 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='487525 Succeeded James Webb JWST* 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='480994 Succeeded Rosetta Rosetta 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='429542 Semi-Succeeded Actually,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' some hashtags are derived not from the mission itself,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' but from the ef- fect achieved (losing control of the Chinese rocket on re-entry) or the main sub- ject of the mission (perseverance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The results clearly evidence that an increase of side effect during mission, increases the negative score sentiments from the space mission community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Regarding the economical resources invested (B) we collect data from different source, and arranged it on USD Billions dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Table 4 was the difficult one to make, because it is hard to obtain data from not-direct-democratic state, like China, and also because the inclusion of both civilian and military space budget for security missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' B factor - Space budgets Country 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 Japan 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='14 Russia 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='92 EU 4,19 4,52 4,56 4,85 4,85 4,65 5,95 6,52 6,35 6,49 5,52 5,16 China 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='83 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='85 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='28 US 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='72 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='44 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='77 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='86 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='64 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='01 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='50 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='73 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='629 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='27 In billion U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' dollars [Appendix 2] [Appendix 3] Therefore, the data from the statistical failures presence in our equation, that mark the quality of space launch system for each country (Q) are shows in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The data are collected since the year 2010 to 2021 (the 2022 is not finished yet), show the total number of Core Stage Manufacture send in space, between overall launch log outside brackets and failures inside brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In the end, we shows also the total launch and failures between 2010 and 2021 and the failures percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The failures percentage is the Q value in our equation, evidencing the failures probability to each space launch system-country for each launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' It is possible to see that the Chinese government have many more launch than the US and Europe, this is due to the low presence of space satellite (for communication and security mission) orbiting the Earth by the Chinese government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' The Chinese government had invested a huge amount of 12 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Q factor - Statistical failures Country 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 TOT Failures % China 55(3) 39(4) 34(2) 39(1) 18(2) 22(2) 19(0) 16(0) 15(1) 19(0) 19(1) 15(0) 310 (16) 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 Russia 25(2) 17(0) 25(0) 20(1) 20(1) 19(1) 27(3) 35(3) 32(1) 26(2) 32(4) 31(1) 309(19) 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 US 43(2) 35(3) 19(0) 29(0) 28(0) 21(0) 20(2) 20(0) 17(0) 13(1) 15(1) 15(0) 275(9) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='3 Europe 6(0) 5(1) 6(1) 8(1) 9(0) 9(0) 8(0) 7(0) 5(0) 8(0) 7(0) 6(0) 84(3) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='6 India 2(1) 2(0) 6(0) 7(0) 5(1) 7(0) 5(0) 4(0) 3(0) 2(0) 3(0) 3(2) 49(4) 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 Japan 3(0) 4(0) 2(0) 6(0) 7(1) 4(0) 4(0) 4(0) 3(0) 2(0) 3(0) 2(0) 42(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='4 New Zeal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 6(1) 7(1) 6(0) 3(0) 1(1) 23(3) 13,6 Iran 1(1) 2(1) 2(2) 1(0) 3(2) 1(0) 10(6) 60 Total launch by year (total launch failures by year) [Appendix 4] money to get enough satellites to make public infrastructures work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Also, Russian launch operation have been increased since the 2011, due to the retirement of the space shuttle (last mission 21th July 2011), and as result, the US astronauts had to traveling by Russian Soyuz spacecraft to get to the international space station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' We have compared the data from the equation (S, T, R and Q parameters), and in Figure 1 and Table 6 we show the score after the mission succeeded or failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' GSS’s info-graphic of mains space mission As is easiest to see, there is a negative score (we have highlight it with red on Figure 1), due to the risk of the ASAT and the space debris rocket had on population on heart and on the ISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Tianwen-1 and Rosetta mission have a high GSS score also because it was a first huge milestone for the respectively space agency (ESA and CNSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' These data evidence the difference between a successful mission, achieving its goal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' wile a successful mission, still achieving is goal but with side effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 GSS trend in time We have tried to quantify a GSS value for space Research Mission only between 2010 and 2021, for each country, due the fact that ASAT is a military space mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' So we have gather data from all Geopolitical Space Score Rosetta JWST Mars 2020 Perseverance ASAT ChineseRock Tianwen-1 2 0 2 4 6 8 10British Journal of Social Policy 13 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Geopolitical Space Score Mission Geopolitical Space Score Tianwen-1 9,547438889 ChineseRock 0,208492857 ASAT 0,659007937 Perseverance 2,198416733 Mars 2020 2,502628333 JWST 2,469102533 Rosetta 7,588575333 factor for the equation, but unfortunately we did not get all the Sentiment Factor (S), because usually unknown space mission does not have much reaction on social network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' As said before to evaluate the sentiment for each missions, those missions should be very important for the Humankind, or at least "virality" for the space community, and many mission unfortunately become known to major society only from newspaper and occasionally from TV news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' To solve this problem, we estimate a medium 0,425 S factor from successful mission, and a medium -0,125 for failed missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For the other well know mission (Table 1) we have used the original sentiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Regarding the economical resources invested (B) toward the data accumulated, we did not get the full budget planning over years for China and Russia, so we estimate a progressive linear regression between the missing data on table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Appendix 5 - 6 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' GSS - Research missions As Figure 2 shows, it is possible to see the fluctuations characterised by the missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In addition, it is also possible to see the type of strategy and activity for each country/agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' For example, ESA, which is characterised by the limitation of not having a launch station on the east coast, has specialised in international collaboration, and it is possible to see how ESA manages missions with fluctuations of about one year (given the many collaborations, especially with NASA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 6 5 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 3 GSS 2 1 0 1 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 China Russia USA ESA14 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Moreover, it should be noted that the Rosetta mission, like the Tianwen-1 mission, could be designed as a "baptism of independence", derived from "not because they are easy, but because they are hard" philosophy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Thy was indeed, the first to do that, showing his competence and skill upon the others superpowers space country like US and Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' China, on the other hand, is concentrating on a few space research missions, since it is a new superpower and has yet to stabilise its infrastructure and network telecommunication on space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Instead, the US have/had many active missions for a long time, in fact the score is relatively lower than the others precisely because of their resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' People expect them to fail less than the others, so the astonishing/sensational felling can only be found in very difficult missions, or when they failed badly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Appendix 5 - 6 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' GSS - Research missions cumulative 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Analysis The analyses support the expectation that: 1) that it is possible to numerically assess geopolitical factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2) that even in the case of a successful mission, there is the possibility of a negative geopolitical feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 3) that public spending on space missions can also positively improve the geopolitical level of the investing state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 4) that the geopolitical level can vary over time, depending on the success of the missions, proportional to the investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' An increase in the social policy inherent in space missions, in addition to providing public employment with low-high qualification (and therefore removing people from the poverty line), would improve the risk factor (Q Factor) decreasing it to each future space mission, thus improving the geopolitical sentiment and also the geopolitical weight of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In fact, as already mentioned, the most resilient state is the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' They are not first on the geopolitical level despite having a high number of missions both quantitatively and qualitatively, in indeed, this score is given by the fact that they rarely miss, thus showing a high quality work, given the quality of the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Conclusions Our study shows that can be possible using computational social method to quantify geopolitical dynamics for every country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In our case we have shown how space mission influence geopolitical 20 15 10 GSS 5 0 5 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 China Russia USA ESABritish Journal of Social Policy 15 dynamics, supported by a security and social policy approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In a more specific way, we have demonstrated (1) A socio-physical method for assessing the geopolitical level of any country, based on its goal and its system-organization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' (2) That space missions, even if successful, can bring negative sentiment, which goes to reflect on the geopolitical value to the state itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' (3) That social policy is also an investment in the defense and security system, and that increased investment of government spending on space missions, also has a spillover effect on geopolitical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' From a sociological and socio-physical point of view,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' through the contribution of a Computational social science model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' the GSS equation can be used to evaluate the geopolitical level not only in the spatial domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' but also in other "competitions",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' such as sports competitions (Olympics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' world sports championships) or music competitions or international wars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' where there are enough social reactions (S Factor) and enough statistical data (Q - B Factor) to evaluate both the event and the organization/country participating in the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Therefore, our work would emphasise the most general Political policy, and in the most specific way, the Social policy, as not only a economical financial aids, but also a more complex systems, related to complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' As we demonstrated that Social policy is also funding the defense, security, and geopolitical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' It has become not only an aid to the population, but much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Moreover, an increase in the spending policy for space missions (which we define as part of social policy), also has a geopolitical impact on the state that promotes them, and by reflection, to its society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Declaration of interest statement The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Authors details 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1 Andrea Russo Is a PhD candidate in Complex Systems at the University of Catania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' He is currently working at the Department of Physics and Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' He collaborated with CNR Ibam, he also has worked purely on projects involving technology and society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' His main research field and interests are focused on the study and the development of Computational social method to explain social complexity, in particular field like Politics - Economics - Business and Defense-Security sector applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' ORCID: 0000-0003-3816-0539 Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Email: Andrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='russo@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='it 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='2 Davide Coco M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' in Space and Astronautical Engineering at University of Rome "Sapienza", with several years of experience in mission design and preliminary studies of space missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' His main goal is to further specialize and be involved in conceptual studies, system requirements definition, mission proposal writing, space system modeling and simulation, launcher or mission design and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' He is currently leading the design and development of Cubesat’s subsystems in a small Italian startup, Human4Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' ORCID: 0000-0001-8010-9468 Email: davide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='coco@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='it Acknowledgement We thank Taha Yasseri for for his crucial help in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 16 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' References ASI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Documento di visione strategica per lo spazio (dvss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' ASI Archive 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' British Journal of Social Policy 17 Vidgen, Bertie, and Taha Yasseri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Detecting weak and strong islamophobic hate speech on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Journal of Information Technology & Politics 17 (1): 66–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Curiosity (rover, https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='org/wiki/Curiosity_(rover)#Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Yin, Jie, Sarvnaz Karimi, Andrew Lampert, Mark Cameron, Bella Robinson, and Robert Power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Using social media to enhance emergency situation awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' In Twenty-fourth international joint conference on artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Equivalence of the 1962 USD to 2022 USD More specifically: 231 408 697 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='64 USD $ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Inflation over the period: 825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='63 % Index used: USCPI31011913 (Bureau of Labor Statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Initial Index: 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='12, Final Index: 2 861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='33 Link=https://fxtop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='com/it/conversione-valute-passato.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='php Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Space annual budget (∼ approximately) over years and percentage of the na- tional budget It is very hard to collect data for the most important Space Agency, this is due to the different money value (Yen - EURO - USD), and also the difficult to obtain data from not-democratic state, like China, due to information security and also due to the inclusion of both civilian and military space spending global security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Source: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='statista.' metadata={'source': 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+page_content='org/sti/inno/space-forum/space-economy-for-people-planet-and-prosperity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='pdf 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' https://global,jaxa,jp/about/transition/index,html 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='int 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='org/wiki/Budget_of_NASA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' https://global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='jaxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='jp/ [Salas 2021] Military budget: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' https://databank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='worldbank.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='1269 at 27/02/2022 02:50 28 Feb 2020 - 25 Feb 2022 Appendix 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Statistical failures Annual space reports: From 2010 to 2021 Launch Log Source = https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='spacelaunchreport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='com/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='html 18 Andrea Russo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' Appendix 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' GSS - Research Missions Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content=' GSS related to space research missions Country 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 China 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='541666667 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='969230769 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} +page_content='383333333 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQf7wWX/content/2301.03538v1.pdf'} 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b/BdAzT4oBgHgl3EQfv_7e/content/tmp_files/2301.01717v1.pdf.txt @@ -0,0 +1,10056 @@ +Prepared for submission to JHEP +JLAB-THY-23-3741 +Rapidity-only TMD factorization at one loop +Ian Balitsky +Physics Dept., Old Dominion University, Norfolk, VA 23529 & Theory Group, JLAB, 12000 +Jefferson Ave, Newport News, VA 23606 +E-mail: balitsky@jlab.org +Abstract: Typically, a production of a particle with a small transverse momentum in +hadron-hadron collisions is described by CSS-based TMD factorization at moderate Bjorken +xB ∼ 1 and by kT -factorization at small xB. A uniform description valid for all xB is +provided by rapidity-only TMD factorization developed in a series of recent papers at the +tree level. In this paper the rapidity-only TMD factorization for particle production by +gluon fusion is extended to the one-loop level. +arXiv:2301.01717v1 [hep-ph] 4 Jan 2023 + +Contents +1 +Introduction +1 +2 +TMD factorization for particle production by gluon fusion +2 +3 +TMD factorization from functional integral +4 +4 +Coefficient function from background-field diagrams +7 +5 +Virtual contributions +13 +6 +“Production” diagrams +18 +6.1 +Power counting for production terms +18 +6.2 +Calculation of leading production terms +20 +6.3 +Handbag diagrams +26 +7 +Result for the sum of diagrams in Figs. 5,6,7,8 minus TMD matrix ele- +ments in Figs. 11,12 +28 +8 +Subtraction of soft/Glauber contributions +29 +8.1 +sG-contributions to virtual diagrams +30 +8.2 +sG-contributions to production diagrams +32 +8.3 +The sum of sG-terms +33 +8.4 +sG-contributions to TMD matrix elements +34 +9 +Result for the coefficient function +36 +9.1 +Factorization of integral over A ∪ B fields +39 +9.1.1 +Cancellation of soft and Glauber gluons +39 +9.1.2 +Factorization in terms of generalized TMDs +40 +10 Conclusions and outlook. +41 +11 Appendix +43 +11.1 Gluon “cut” propagator in the background field A +43 +11.2 TMD matrix elements +43 +11.3 Soft factor with rapidity-only cutoffs +48 +11.4 Approximation x∥ +12 = 0 for the calculation of coefficient function +50 +11.5 Diagrams with correction field ¯C +52 +11.6 Necessary integrals +55 +11.6.1 Integrals for virtual diagrams +55 +11.6.2 Integrals for “production” diagrams +57 +11.7 Coefficient function from the calculation with background gluons on the mass +shell +58 +– i – + +11.7.1 Virtual contributions +58 +11.7.2 Production terms minus TMD matrix elements +60 +11.7.3 J(i>1) +1 +are power corrections +62 +11.7.4 sG-contribution +64 +1 +Introduction +Rapidity factorization and rapidity evolution are main tools for study of QCD processes at +small x [1]. On the other hand, at moderate x conventional methods are based on CSS equa- +tion [2] and closely related SCET approach (see Refs. [3] and [4] reviews). However, with +the advent of EIC accelerator the region of energies intermediate between low and moderate +x needs to be investigated. One of the ideas is to extend rapidity factorization methods +beyond the “pure” small-x region. In a series of recent papers A. Tarasov, G.A. Chirilli +and the author applied the method of rapidity-only factorization to processes of particle +production in hadron-hadron collisions in the so-called Sudakov region where transverse +momentum of produced particle(s) is much smaller than their invariant mass. The typical +examples of such processes are the Drell-Yan process or Higgs production by gluon fusion. +At moderate x such processes are studied by CSS-based TMD factorization [3, 5] +dσ +dηd2q⊥ += +� +f +� +d2b⊥ei(q,b)⊥Df/A(xA, b⊥, ηa)Df/B(xB, b⊥, η2) +× σff→X(η, ηa, ηb) + power corrections + Y − terms +(1.1) +where η = +1 +2 ln q+ +q− is the rapidity, Df/h(x, z⊥, ηi) is the TMD density of a parton f in +hadron h with rapidity cutoff ηi, and σff→X(η, ηa, ηb) is the cross section of production of +particle(s) X of invariant mass m2 +X = q2 ≡ Q2 ≫ q2 +⊥ in the scattering of two partons. The +TMD parton densities are regularized with a combination of UV and rapidity cutoffs and +the relevant Sudakov logarithms are obtain by solving double evolution with respect to µUV +and the rapidity cutoffs ηi [3]. +It should be emphasized that the CSS approach and hence the formula (1.1) are valid +at xA ∼ xB ∼ 1. At small xA and/or xB one should resort to other factorization methods. +As I mentioned above, a perspective approach is to apply methods based on rapidity-only +factorization used in small-x/BFKL physics. In a series of papers [6–9] A. Tarasov and +the author applied rapidity-only factorization approach to get for the first time power +corrections ∼ q2 +⊥ +Q2 restoring EM gauge invariance of DY hadronic tensor both at moderate +and small x. Also, in recent papers [9, 10] G.A. Chirilli and the author calculated the +rapidity-only evolution of TMD operators, again both at moderate and small x. In the +present paper I calculate coefficient function multiplying two TMD distributions at the +one-loop level. This completes the task of performing the rapidity-only factorization at the +one-loop accuracy. +– 1 – + +Apart from requirement Q2 = xAxBs ≫ q2 +⊥, in this paper it is assumed that +Q2 +q2 +⊥ +≫ +q2 +⊥ +m2 +N +(1.2) +The region (1.2) can be understood in terms of rescaling s → ζs0, ζ → ∞ with q2 +⊥ fixed: +s ∼ ζs0, +q⊥ ∼ O(ζ0) +(1.3) +It should be emphasized that we will not use the small-x approximation s ≫ Q2 so our +formulas are correct both at x ≪ 1 and x ∼ 1 provided that the condition (1.2) is satisfied. +Thus, at x ∼ 1 our rapidity-evolution formulas should be equivalent to usual CSS approach, +although the exact relation between our rapidity evolution and CSS double evolution in +rapidity and UV cutoff remains to be established, see the discussion in the Conclusions +section. +The rapidity evolution of TMDs Df/A(xA, b⊥, ηa), Df/B(xB, b⊥, ηb) should match the +one-loop rapidity evolution of the coefficient function σff→H(η, ηa, ηb) so that the cutoffs +ηa and ηb disappear from physical amplitude. I will demonstrate that the result for the +coefficient function in Eq. (1.1) for rapidity-only gluon TMD factorization is proportional +to (γ ≡ γE ≃ 0.577) +exp +�αsNc +2π +� +(ln b2 +⊥s +4 +− ηa − ηb)2 − 2(ln xA − ηa − γ)(ln xB − ηb − γ) + π2�� +(1.4) +and check that ηa, ηb dependence matches the rapidity-only TMD evolution obtained in +Refs. [9, 10]. +The paper is organized as follows. In Sect. 2 I define hadronic tensor and TMD oper- +ators for particle production by gluon fusion. In Sect. 3 I discuss separation of functional +integral for particle production in three integrals according to the rapidity of the fields +involved. In Sect. 4 I set up the calculation of the coefficient function in front of TMD op- +erators by computing diagrams in two background fields. Sections 5, 6, 7, and 8 are devoted +to calculation of these diagrams at the one-loop level. The result of the calculation and +check of matching to evolution of TMD operators are presented in Sect. 9. The necessary +technical and sidelined results are presented in the Appendix. +2 +TMD factorization for particle production by gluon fusion +Let us consider production of an (imaginary) scalar particle Φ by gluon fusion in proton- +proton scattering. The particle is connected to gluons by the vertex +LΦ = gΦ +� +d4x Φ(x)g2F 2(x), +F 2(x) ≡ F a +µν(x)F aµν(x) +(2.1) +This is a mH +mt ≪ 1 approximation for Higgs production via gluon fusion at the LHC. The +differential cross section of Φ production is defined by the “hadronic tensor” W(pA, pB, q) +W(pA, pB, q) +def += +N2 +c − 1 +16 +� +X +� +d4x e−iqx⟨pA, pB|g2F 2(x)|X⟩⟨X|g2F 2(0)|pA, pB⟩ += +N2 +c − 1 +16 +� +d4x e−iqx⟨pA, pB|g4F 2(x)F 2(0)|pA, pB⟩ +(2.2) +– 2 – + +p +p +q +A +B +ACS3icbZBLS8NAFIUn9VXrq+rSzWARXEhJiqg +bQXTjRmjBVqFpw2R6mw7OJOnMRCil+X1u3LjzT7hxoYgLJ32ArwMDJ+e7l8kcP+ZMadt+tnJz8wuLS/nlwsrq2vpGcXOroaJEUqjTiEfy1icKOAuhrpnmcBtLIMLncOPfXWT85h6kYlF4rQcxtAQJQtZlGgTeUVfuYE2WOBau5KmbhCkac1 +zY5BxuzJDIkNGrlvomxT6CbvP5k+vPLfaY+3KQfp9a8L7s2+vWL9lj4r3GmpoSmqnrFJ7cT0URAqCknSjUdO9atIZGaUQ6jgpsoiAm9IwE0jQ2JANUajrsY4T2TdHA3kuaEGo/T7xtDIpQaCN9MCqJ76jfLwv9YM9Hdk9aQhXGiIaSTi7oJ +xzrCWbG4wyRQzQfGECqZ+VdMe0QSqk39BVOC8/vJf02jUnaOyk7tsHR2Pq0j3bQLtpHDjpGZ+gSVEdUfSAXtAbercerVfrw/qcjOas6c42+qHcwhd+27Q4 +s & Q2 +� +Q2 +? & m2 +q2 ⌘ Q2 = M 2 +�, +Q2 +? ⌘ q2 +? +AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cI9gPaUDbTbN0dx +N2N0IJ/QtePCji1T/kzX/jps1BWx8MPN6bYWZemHKmjet+O5W19Y3Nrep2bWd3b/+gfnjU0Um +mCG2ThCeqF2JNOZO0bZjhtJcqikXIaTec3BV+94kqzRL5aKYpDQeSxYxgk0hDfyYDesNt+n +OgVaJV5IGlPCH9a/BKCGZoNIQjrXue25qghwrwins9og0zTFZILHtG+pxILqIJ/fOkNnVhmh +KFG2pEFz9fdEjoXWUxHaToFNrJe9QvzP62cmuglyJtPMUEkWi6KMI5Og4nE0YoSw6eWYKYv +RWRGCtMjI2nZkPwl9eJZ2LpnfV9B4uG63bMo4qnMApnIMH19Ce/ChDQRieIZXeHOE8+K8Ox ++L1opTzhzDHzifP+SxjiU=� +Figure 1. Particle production by gluon-gluon fusion +where the factor N2 +c −1 +16 +is added to simplify factorization formulas. As usual, � +X denotes +the sum over full set of “out” states. +We will study the hadronic tensor (2.2) with non-zero momentum transfer in t-channel +defined as a matrix element of the operator +ˆW(x1, x2) ≡ N2 +c − 1 +16 +g4F a +µν(x2)F µν;a(x2)F b +λρ(x1)F λρ;b(x1) +(2.3) +between initial and final states with slightly non-equal momenta +W(pA, pB, p′ +A, p′ +B; x1, x2) = ⟨p′ +A, p′ +B| ˆW(x1, x2)|pA, pB⟩ +(2.4) +where +pA = p1 + m2 + l2 +⊥ +s +p2 − l⊥ +2 , +p′ +A = p1 + m2 + l2 +⊥ +s +p2 + l⊥ +2 , +pB = p2 + m2 + l2 +⊥ +s +p1 + l⊥ +2 , +p′ +B = p2 + m2 + l2 +⊥ +s +p1 − l⊥ +2 +(2.5) +Here p1 and p2 are light-like vectors close to pA and pB, respectively. 1 We will use light-cone +coordinates with respect to the frame where p1 = +� √s +2 , 0, 0, +√s +2 +� +and p2 = +� √s +2 , 0, 0, − +√s +2 +� +so that p+ +1 = p− +2 = � s +2, p+ +2 = p− +1 = 0 and p1⊥ = p2⊥ = 0. +The kinematical region (1.2) in the coordinate space translates to +x2 +∥ ≪ x4 +12⊥m2 +N +(2.6) +where x2 +∥ ≡ 2x+ +12x− +12. Also, we must assume x2 +12⊥ ≤ m−2 +N +so that the coupling constant +αs(x⊥) is a valid perturbative parameter. +In the coordinate space, TMD factorization (1.1) for hadronic tensor in Eq. (2.4) should +look like +g4 +16(N2 +c − 1)⟨p′ +A, p′ +B|F a +µνF aµν(x2)F b +λρF bλρ(x1)|pA, pB⟩ += +� +dz− +2 dz2⊥dz− +1 dz1⊥dw+ +1 dw1⊥dw+ +2 dw2⊥C(x2, x1; z− +i , zi⊥, w+ +i , wi⊥; σp, σt) +× ⟨p′ +A| ˆOσp +ij (z− +2 , z2⊥; z− +1 , z1⊥)|pA⟩⟨p′ +B| ˆOij;σt(w+ +2 , w2⊥; w+ +1 , w1⊥)|pB⟩ + ... +(2.7) +1We assume that t = −l2 +⊥ ∼ m2 +N. If there is a longitudinal component of momentum transfer, one can +redefine p1 and p2 in such a way that with respect to new p′ +1 and p′ +2 the formulas are those of Eq. (2.5). +– 3 – + +where the dots stand for power corrections ∼ q2 +⊥ +Q2 and +ˆOij(z+ +2 , z2⊥; z+ +1 , z1⊥) = F a +i (z2)[z2 − ∞+, z1 − ∞+]abF b +j (z1) +��� +z− +2 =z− +1 =0 , +(2.8) +ˆOij(z− +2 , z2⊥; z− +1 , z1⊥) = F a +i (z2)[z2 − ∞−, z1 − ∞−]abF b +j (z1) +��� +z+ +2 =z+ +1 =0 , +F i,a(z⊥, z+) ≡ gF −i,m(z)[z+, −∞+]ma +z +��� +z−=0, +F i,a(z⊥, z−) ≡ gF +i,m(z)[z−, −∞−]ma��� +z+=0 +are gluon TMD operators (the precise definitions of rapidity-only cutoffs σa = eηa and +σb = eηb for gluon TMDs will be given later). Hereafter, we use the notation +[x, y] ≡ Peig +� 1 +0 du (x−y)µAµ(ux+y−uy) +(2.9) +for the straight-line ordered gauge link between points x and y, and space-saving notations +[x+, y+]z ≡ [x+ + z⊥, y+ + z⊥], +[x−, y−]z ≡ [x− + z⊥, y− + z⊥] +(2.10) +The coefficient function C represents a Fourier transform of σff→H(η, η1, η2) in Eq. +(1.1) with ηi = ln σi. The normalization in the l.h.s. of Eq. (2.7) is chosen in such a way +that C = 1 + αsNc +2π C1 + O(α2 +s). The goal of this paper is to find the one-loop coefficient +function C1(x2, x1; z− +i , zi⊥, w+ +i , wi⊥; σa, σb) and check that the evolution of this coefficient +function matches the evolutions of TMD operators. +3 +TMD factorization from functional integral +The hadronic tensor (2.2) can be represented by double functional integral +W(pA, pB, p′ +A, p′ +B; x2, x1) = +� +X +⟨p′ +A, p′ +B|g2F 2(x2)|X⟩⟨X|g2F 2(x1)|pA, pB⟩ +(3.1) += +tf→∞ +lim +ti→−∞ g4 +� +˜ +A(tf)=A(tf) +D ˜AµDAµ +� +˜ψ(tf)=ψ(tf) +D ˜¯ψD ˜ψD ¯ψDψ e−iSQCD( ˜ +A, ˜ψ)eiSQCD(A,ψ) +× Ψ∗ +p′ +A( ⃗˜A(ti), ˜ψ(ti))Ψ∗ +p′ +B( ⃗˜A(ti), ˜ψ(ti)) ˜F 2(x2)F 2(x1)ΨpA( ⃗A(ti), ψ(ti))ΨpB( ⃗A(ti), ψ(ti)) +Here the fields A, ψ correspond to the amplitude ⟨X|F 2(x1)|pA, pB⟩, the fields ˜A, ˜ψ corre- +spond to complex conjugate amplitude ⟨p′ +A, p′ +B|F 2(x2)|X⟩ and Ψp( ⃗A(ti), ψ(ti)) denote the +proton wave function at the initial time ti. The boundary conditions ˜A(tf) = A(tf) and +˜ψ(tf) = ψ(tf) reflect the sum over all states X, cf. Refs. [11–13]. We will also use the +notation +{x, y} ≡ Peig +� 1 +0 du (x−y)µ ˜ +Aµ(ux+y−uy) +(3.2) +and similar notations like Eq. (2.10) for gauge links in the left sector. +For calculations in the momentum space we will use Sudakov variables related to light- +cone components p+, p−, p⊥ by α ≡ p+/ϱ and β ≡ p−/ϱ where ϱ ≡ +� +s/2. In terms of +Sudakov variables p·q = (αpβq +αqβp) s +2 −(p, q)⊥ where (p, q)⊥ ≡ −piqi. Throughout the +– 4 – + +paper, the sum over the Latin indices i, j... runs over the two transverse components while +the sum over Greek indices runs over the four components as usual. Also, since we use +Sudakov variables it is convenient to change the notations of gluon momentum fractions to +αa ≡ xA, +βb ≡ xB +(3.3) +to avoid confusion with coordinates. +Following Refs. +[6, 7], to derive the factorization formula we separate gluon (and +quark) fields in the functional integral (3.1) into three sectors: “projectile” fields Aµ, ψa +with |β| < σp ≡ σa, “ target” fields Bµ, ψb with |α| < σt ≡ σb and “central rapidity” fields +Cµ, ψ with |α| > σt and |β| > σp. Let us specify the values of the TMD cutoffs σp and σt +Figure 2. Rapidity factorization for particle production +in our factorization. Needless to say, we should take σt ≪ αa and σp ≪ βb. Moreover, as +discussed in Ref. [9], power corrections to rapidity evolution of TMDs are ∼ +Q2 +⊥ +βbσts so we +need to assume σtβbs ≫ Q2 +⊥, and similarly σpαas ≫ Q2 +⊥ for the projectile. Next, as we +shall see below, it is convenient to calculate coefficient function (4.19) at m2 +N ≫ µ2 +σ ≡ σpσts +so finally we take the region of σp and σt as follows +αa ≫ σt ≫ Q2 +⊥ +βbs, +β′ +b ≫ σp ≫ Q2 +⊥ +α′as, +m2 +N ≫ µ2 +σ ≡ σpσts ≫ Q4 +⊥ +Q2 +(3.4) +Note that due to Eq. (1.2) we can choose µ2 +σ between m2 +N and parametrically small Q4 +⊥ +Q2 . In +terms of rescaling (1.3) this means that we can choose σp, σt ∼ ζ− 3 +4 ∼ +� Q⊥ +Q +�3/2 so that +µ2 +σ ∼ ζ−1/2 +⇔ +1 ≫ µ2 +σ +Q2 +⊥ +≫ Q2 +⊥ +Q2 ∼ ζ−1 +(3.5) +and both conditions in Eq. (3.4) are satisfied. +In this paper we are calculating logarithmical corrections so the power corrections +due to the small parameters +O2 +⊥ +σpα′as, +O2 +⊥ +σtβ′ +bs will be systematically neglected. The convenient +– 5 – + +A +"Proiectile""fields: +3 +AB7XicbZC7SgNBFIbPxluMt3jpbAaDYBV2tdDOgIWEcwFkiWcnUySMb +Ozy8ysEJe8g42FIrZ2Vj6JnaVv4uRSaOIPAx/fw5zgliwbVx3S8ns7C4tLySXc2trW9sbuW +3d6o6ShRlFRqJSNUD1ExwySqG8HqsWIYBoLVgv7FK/dMaV5JG/MIGZ+iF3JO5yisVa1iSLu +YStfcIvuWGQevCkUzj/uvy/f9JyK/ZbEc0CZk0VKDWDc+NjZ+iMpwKNsw1E81ipH3soZF +iSHTfjqedkgOrdMmnUjZJw0Zu787Ugy1HoSBrQzR9PRsNjL/yxqJ6Zz5KZdxYpik486iSAmI +qPVSZsrRo0YWECquJ2V0B4qpMYeKGeP4M2uPA/V46J3Ujy+dgslFybKwj4cwBF4cAoluIyVI +DCLTzAEzw7kfPovDivk9KM+3ZhT9y3n4Aq0aS2A= +� +AB7HicbVA9SwNBEN3zM4lfUubwyBYSLiLhZYBG8sIXhJIjrC3N5cs2d +s7dueEcKSxsbSxUMTWzj9j56/RzUehiQ8GHu/NMDMvSAX6Dhf1srq2vrGZqFY2tre2d0r7x8 +0dZIpBh5LRKLaAdUguAQPOQpopwpoHAhoBcOrid+6A6V5Im9xlIf07kEWcUjeR1A0DaK1ec +qjOFvUzcOanUiw/hx/f9WaNX/uyGCctikMgE1brjOin6OVXImYBxqZtpSCkb0j50DJU0Bu3n +02PH9olRQjtKlCmJ9lT9PZHTWOtRHJjOmOJAL3oT8T+vk2F06edcphmCZLNFUSZsTOzJ53bIF +TAUI0MoU9zcarMBVZShyadkQnAX14mzVrVPa/WbkwaDpmhQI7IMTklLrkgdXJNGsQjHDySJ +7JiyWtJ+vVepu1rljzmUPyB9b7D4WbkiA= +↵� = Q2 +? +s +A 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+AB6HicdZDLSgMxFIYz9VbrV52boJ +FcFWSCrZdWehCly3YC7RDyaRpG5vJDElGqEOfwI0LRdz6AK58EncufR +MzrYK/hD4+P9zyDnHCwXBqE3J7WwuLS8kl7NrK1vbG5lt3eaOogUZ +Q0aiEC1PaKZ4JI1DeCtUPFiO8J1vLG1SRvXTGleSAvzCRkrk+Gkg84 +JcZa9Wovm0N5hBDGCaAiyfIQrlcKuASxElklTt9uX4/e96La73sa7c +f0Mhn0lBtO5gFBo3JspwKtg040CwkdkyHrWJTEZ9qNZ4NO4aF1+n +AQKPukgTP3e0dMfK0nvmcrfWJG+neWmH9lncgMSm7MZRgZJun8o0Eko +AlgsjXsc8WoERMLhCpuZ4V0RBShxt4mY4/wtSn8H5qFPD7OF+oV0Fg +rjTYBwfgCGBQBVwDmqgAShg4AbcgXvn0rl1HpzHeWnK+ezZBT/kPH0 +Aq+QvQ= +|↵| < �t +A +AB+nicdVBLSgNBEO2Jvxh/E93pjEIrsIko +0kWLgJuXLhIwHwgGYaTidp0vOhu0cJSU7g +Gdy4UMStN/AG7ryFR7DzEVT0QcHjvSq6nkR +Z1JZ1ruRWFpeWV1Lrqc2Nre2d8z0bl2GsSC +0RkIeiqYHknIW0JpitNmJCj4HqcNb3A+9Rv +XVEgWBldqGFHh17AuoyA0pJrpsdt4FEfxm +dtyXo+uMo1M1a2VCgWTwrYylozaJI/tfO2j +XMLJVM2qx+vl/u3Fd8a3dCEvs0UISDlK2cF +SlnBEIxwuk1Y4ljYAMoEdbmgbgU+mMZqdP +8JFWOrgbCl2BwjP1+8QIfCmHvqc7fVB9+dub +in95rVh1S86IBVGsaEDmi7oxyrE0xwhwl +KFB9qAkQwfSsmfRBAlE4rpUP4+hT/T+r5bM +7O5qs6DQvNkUQH6BAdoxwqojK6QBVUQwTdoD +v0gB6NsXFvPBnP89aEsZjZQz9gvHwCPZuXl +g= +��t +AB8HicdV +DLSgMxFM20Pmp9VcWVm2AR3DjMTNW2u4IblxXsQ9qh +ZNK0DU0yQ5IRytCvcONCEbf+hb/gQnDlp2j6EFT0w +IXDOfdy7z1BxKjSjvNmpdILi0vLmZXs6tr6xmZua7 +uwlhiUsMhC2UzQIowKkhNU81IM5IE8YCRjA8m/i +NayIVDcWlHkXE56gvaI9ipI10dRWtM9R3dyecun +pQcz4WO7UxhSMErH5dPoTtX8pX068fz7jupdnIv7W +6IY06Exgwp1XKdSPsJkpiRsbZdqxIhPAQ9UnLUIE +4UX4yPXgMD4zShb1QmhIaTtXvEwniSo14YDo50gP1 +25uIf3mtWPdKfkJFGsi8GxRL2ZQh3DyPexSbBmI +0MQltTcCvEASYS1yShrQvj6FP5P6p7tFmzvwqThgBk +yYA/sg0PgiKogHNQBTWAQc34A7cW9K6tR6sx1lr +yprP7IAfsJ4+AYWIlJk= +��p +A +B8HicdVA9SwNBEN3zM8avGLGyWQyCjcde4hnT +BWwsI5gPSY6wt9lLluzeHbt7QjSWtnZpFAkr +T/Hzn/jJlFQ0QcDj/dmJnx5wpjdC7tbS8s +rq2ntnIbm5t7+zm9vINFSWS0DqJeCRbPlaUs5 +DWNdOctmJsfA5bfrDy5nfvKNSsSi80aOYegL +3QxYwgrWRbk87ivUF7sbdXAHZLnIqbgUuSAk +ZgpyzsnsOHRvNUajmp/cT9+Ch1s29dXoRSQN +NeFYqbaDYu2lWGpGOB1nO4miMSZD3KdtQ0Msq +PLS+cFjeGyUHgwiaSrUcK5+n0ixUGokfNMps +B6o395M/MtrJzq48FIWxomIVksChIOdQRn38 +Mek5RoPjIE8nMrZAMsMREm4yJoSvT+H/pFG +0nZJdvDZpILBABhyCI3ACHFAGVXAFaqAOCBD +gETyBZ0taE+vFmi5al6zPmX3wA9brB9d+k1g= + +Figure 3. Regions of factorization in the momentum space +Glauber gluons (if p⊥ ≫ p+, p− ∼ ζ− 1 +4 ) or soft gluons (if p⊥ ∼ p+, p−). We will denote +both of them by notation C = A ∩ B and call them soft/Glauber (sG) gluons. We will see +later that Glauber gluons do not contribute to factorization (1.1) and soft gluons form a +soft factor which is a power correction with our rapidity cutoffs. +To discuss interaction of central gluons with either A, B, or C fields it is convenient do +denote all the latter by notation A = A ∪ B which means all fields with |α| < σa and/or +|β| < σb. +We get +W(pA, pB, p′ +A, p′ +B; x1, x2) = +lim +ti→−∞ g4 +� +DΦA DΦC Ψ∗ +p′ +A(ti)ΨpA(ti) +× Ψ∗ +p′ +B(ti)ΨpB(ti)( ˜F A + ˜F C)2(x2)(F A + F C)2(x1) +(3.8) +– 6 – + +where F A is the usual field tensor for A field and F C +µν ≡ Fµν(A + C) − Fµν(A ). Also, +we use shorthand notations +ΨpA(ti) ≡ ΨpA( ⃗A(ti), ψ(ti)), +Ψ∗ +pA(ti) ≡ Ψ∗ +pA( ⃗˜A(ti), ˜ψa(ti)) +ΨpB(ti) ≡ ΨpB( ⃗B(ti), ψb(ti)), +Ψ∗ +pB(ti) ≡ Ψ∗ +pB( ⃗˜B(ti), ˜ψb(ti)), +(3.9) +for projectile and target protons’ wave functions, and +� +DΦA +≡ +� +˜ +A (tf)=A (tf) +D ˜ +AµDAµ +× +� +˜ψA (tf)=ψA (tf) +D ˜ψA DψA e−iSQCD( ˜ +A , ˜ψA )+iSQCD(A ,ψA ), +� +DΦC ≡ +� +˜C(tf)=C(tf) +D ˜CµDCµ +� +˜ψC(tf)=ψC(tf) +DψCD ˜ψC e−i ˜SC+iSC +(3.10) +for functional integrals. In the last line SC ≡ SQCD(A + C) − SQCD(A ). +Our goal is to integrate over central fields and get the amplitude in the factorized form, +as a (sum of) products of functional integrals over A fields representing projectile matrix +elements (TMDs) and functional integrals over B fields representing target matrix elements. +In the spirit of background-field method, we “freeze” projectile and target fields and get a +sum of diagrams in these external fields. Since |β| < σp in the projectile fields and |α| < σt +in the target fields, at the tree-level one can set with power accuracy β = 0 for the projectile +fields and α = 0 for the target fields - the corrections will be O +� m2 +σps +� +and O +� m2 +σts +� +. 2 Beyond +the tree level, the integration over C fields will produce the logarithms of the cutoffs σp +and σt which cancel with the corresponding logs in gluon TMDs of the projectile and the +target, see the discussion in Sect. 9. +4 +Coefficient function from background-field diagrams +We will calculate the coefficient function C in the first non-trivial order in perturbation +theory: C = 1 + αsNc +2π C1. The desired formula looks like that +1 +16(N2 +c − 1)⟨p′ +A, p′ +B|g2F a +µνF aµν(x2)g2F b +λρF bλρ(x1)|pA, pB⟩ += +� +DΦA Ψ∗ +p′ +A(ti)ΨpA(ti)Ψ∗ +p′ +B(ti)ΨpB(ti) +� +ˆOσp +ij (x− +2 , x2⊥; z− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥) ++ +� +dz− +1 dz1⊥dz− +2 dz2⊥dw+ +1 dw1⊥dw+ +2 dw2⊥ +αsNc +2π C1(x1, x2; z− +i , zi⊥, w+ +i , wi⊥; σp, σt) +× ˆOσp +ij (z− +2 , z2⊥; z− +1 , z1⊥) ˆOij;σt(z+ +2 , z2⊥; z+ +1 , z1⊥) + ... +� +(4.1) +where dots stand for higher orders in perturbation theory and/or power corrections. +2Indeed, suppose an “A” gluon with momentum ka interacts with a “C” gluon with momentum kC. The +resulting propagator is [(α′ +a + αc)(βa + βc)s − (ka + kc)2 +⊥]−1 and one can neglect βa with βa/βc ≤ m2/s +σp +accuracy. Similarly, for the target fields one gets the accuracy αb/αc ≤ m2/s +σt . +– 7 – + +The standard way to obtain a coefficient function is to rewrite Eq. (4.1) as an operator +formula +� +dz− +2 dz2⊥dz− +1 dz1⊥dw+ +1 dw1⊥dw+ +2 dw2⊥ +αsNc +2π C1(x1, x2; z− +i , zi⊥, w+ +i , wi⊥; σp, σt) +× ˆOσp +ij (z− +2 , z2⊥; z− +1 , z1⊥) ˆOij;σt(z+ +2 , z2⊥; z+ +1 , z1⊥) + ... +(4.2) += g4 +16(N2 +c − 1)F a +µνF aµν(x2)F b +λρF bλρ(x1) − +ˆOσp +ij (x− +2 , x2⊥; x− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥), +calculate the l.h.s. and r.h.s. between two initial and two final gluon states and compare. +However, the amplitudes with real gluons have infrared divergencies so we will consider +amplitudes with virtual gluon tails instead. As is well known, a gauge-invariant way to +write down matrix elements “between virtual gluons” is to consider l.h.s. and r.h.s. of Eq. +(4.2) in a suitable background field. +Following the analysis of rapidity factorization (3.8) in Refs. +[6, 7], we choose the +background field as a result of interaction of “projectile” field ¯A and “target” field ¯B where +the “projectile” field ¯A(z) depends only on z⊥, z− (corresponding to βa=0) and “target” +field ¯B(z) depends only on z⊥, z+ (corresponding to αb=0). 3 As demonstrated in Ref. [6], +in this case one can always choose the gauge where ¯A− = ¯B+ = 0. Moreover, since we are +after logarithmical corrections to coefficient in front of the operators (2.8), it is convenient +to take ¯A+ = ¯B− = 0. 4 Thus , we choose “projectile” and “target” fields in the form +g ¯Ai = Ui(x−, x⊥), +A+ = A− = 0, +g ¯Bi = Vi(x+, x⊥), +B+ = B− = 0 +g ¯F +i(A) = ∂+Ui ≡ U +i(x−, x⊥), +g ¯F +i(B) = ∂−Vi ≡ V −i(x+, x⊥) +(4.3) +We assume that the “projectile” and “target” fields A(z−, z⊥) and B(z+, z⊥) satisfy standard +YM equations +(∂i − i[Ui, )U −i = g2 ¯ψAγ−ψA, +(∂i − i[Vi, )V +i = g2 ¯ψBγγ+ψB +(4.4) +and only “good” components of background quark fields γ−ψA(z−, z⊥) and γ+ψB(z+, z⊥) +do exist. +The “interaction” field A is defined as is a classical field solving classical YM equations +DνFa +µν = +� +f +g ¯ΨftaγµΨf, +(̸P + mf)Ψf = 0 +(4.5) +with boundary conditions 5 +Aµ(x) x+→−∞ += +¯Aµ(x−, x⊥), +Ψ(x) x+→−∞ += +ψA(x−, x⊥) +Aµ(x) x−→−∞ += +¯Bµ(x+, x⊥), +Ψ(x) x−→−∞ += +ψB(x+, x⊥) +(4.6) +3See Ref. [14] for similar approach +4The general case with background fields ¯A−, ¯B+ ̸= 0 is relevant for obtaining power corrections to Eq. +(2.7), see the discussion in Refs. [6, 7] +5It is convenient to fix redundant gauge transformations by requirements ¯Ai(−∞•, x⊥) = 0 for the +projectile and ¯Bi(−∞∗, x⊥) = 0 for the target, see the discussion in Ref. [15]. +– 8 – + +reflecting the fact that at t → −∞ we have only incoming hadrons with “A” and “B” fields. +An important property of the functional integral (3.8) is that since the projectile fields ¯˜A +and ¯A should coincide at t → ∞ (see Eq. (3.10)) and since they do not depend on x+, they +coincide everywhere. Similar property is valid for the target fields so we have the condition +¯˜A = +¯A, +¯˜B = +¯B +(4.7) +As proved in Refs. [6, 7] the solution of classical equations (4.5) has the same property: +˜A = A. In terms of perturbative diagrams the solution of Eq. (4.5) is given by the sum +¯A + ¯B + ¯C where the “correction” field ¯C is given by the sum diagrams of the type shown +in Fig. 4 with retarded propagators. +Figure 4. Typical diagram for the classical field with projectile/target sources. The Green func- +tions of the central fields are given by retarded propagators. +The solution of YM equations (4.5) in general case is yet unsolved problem, especially +important for description of scattering of two heavy nuclei in semiclassical approximation. +Fortunately, for our case of particle production with q⊥ +Q ≪ 1 one can construct the “cor- +rection” field ¯C as a series in this small parameter. The explicit form of the expansion of +correction field ¯C in powers of this small parameter is presented in Refs. [6, 7]. We will +need only one term in this expansion shown in Eq. (11.38) below. +First, let us present the estimates of relative strength of different components of pro- +jectile and target fields in our Q2 ≫ q2 +⊥ kinematics from Ref. [6] +U i, V i ∼ Q⊥, +U +i = ∂+U i ∼ Q⊥ +√s, +V −i = ∂−V i ∼ Q⊥ +√s +(4.8) +U ij = ∂iU j − ∂jUi − i[U i, Uj] ∼ Q2 +⊥, +V ij = ∂iV j − ∂jVi − i[V i, V j] ∼ Q2 +⊥ +Note that Eq. (4.8) means that characteristic scales of projectile fields are such that extra +∂+ brings √s and extra ¯Di is ∼ q⊥ so ∂+ ≫ ¯Di for the projectile fields. Similarly, for +the target fields ∂− ≫ ¯Di. The characteristic longitudinal distances are z+ ∼ 1/√s for +the projectile fields and z− ∼ 1/√s for the target ones while the characteristic transverse +distances are ∼ 1/Q⊥ for both of them. Also, in sorting out power corrections according +to rescaling parameter ζ in Eq. (1.3), we do not distinguish between m2 +N +Q2 , q2 +⊥ +Q2 , m2 +N +s , and q2 +⊥ +s +and use the common notation +O +�m2 +⊥ +s +� +∼ O +�m2 +N +Q2 , q2 +⊥ +Q2 , m2 +N +s , q2 +⊥ +s +� +– 9 – + +9000000 +0000 +00000000 +QQQQQQQQQQQQQ +0000 +00000000000000000000000000000000000000000000000000000 +80000000 +00000000 +80080800 +000000000Dand similarly for other ratios. +The relevant terms in the expansion of correction fields ¯C are [6] +¯C−(x) = +1 +2ϱ +� +dz(x| +1 +α + iϵ|z)[Uk(z−, z⊥), V k(z+, z⊥)] += +− i +2g +� x+ +−∞ +dx′+ +� x− +−∞ +dx′−(x − x′)−[U + +k(x′−, x⊥), V −k(x′+, x⊥)] ∼ m2 +⊥ +√s , +¯C+(x) = +− 1 +2ϱ +� +dz(x| +1 +β + iϵ|z)[Uk(z−, z⊥), V k(z+, z⊥)] += +i +2g +� x− +−∞ +dx′− +� x+ +−∞ +dx′+(x − x′)+[U + +k(x′−, x⊥), V −k(x′+, x⊥)] ∼ m2 +⊥ +√s +¯Ci(x) = −i +2g +� x− +−∞ +dx′− +� x+ +−∞ +dx′+� +[U j(x′−, x⊥), Vij(x′+, x⊥)] + ∂j[Ui(x′−, x⊥), Vj(x′+, x⊥)] ++ [V j(x′+, x⊥), Uij(x′−, x⊥)] + ∂j[Vi(x′+, x⊥), Uj(x′−, x⊥)] +� +∼ m3 +⊥ +s +(4.9) +where we used +U i(x−, x⊥) = +� x− +−∞ +dx′−U +i(x′−, x⊥), +V i(x+, x⊥) = +� x+ +−∞ +dx′+V +i(x′+, x⊥) +(4.10) +It should be noted that in expressions (4.9), (11.38) we neglected terms ∼ UiUjVk and +UiVjVk since they are proportional to F 3 +ξη and hence cannot contribute to our coefficient +function. +Thus, we need to calculate l.h.s. and r.h.s. of Eq. (4.2) in the background of the field +A ≃ +¯A + ¯B + ¯C +(4.11) +given by Eqs. (4.3) and (11.38) +� +dz− +2 dz2⊥dz− +1 dz1⊥dw+ +1 dw1⊥dw+ +2 dw2⊥ +αsNc +2π C1(x1, x2; z− +i , zi⊥, w+ +i , wi⊥; σp, σt) +× ⟨ ˆOσp +ij (z− +2 , z2⊥; z− +1 , z1⊥) ˆOij;σt(z+ +2 , z2⊥; z+ +1 , z1⊥)⟩A + ... += N2 +c − 1 +16 +g4⟨ ˜F a +µν ˜F aµν(x2)F b +λρF bλρ(x1)⟩A +− ⟨ ˆOσp +ij (x− +2 , x2⊥; x− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥)⟩A +(4.12) +Since we are after first order of perturbation theory, operators in the l.h.s. of this +equation can be replaced in the leading order by corresponding classical fields +⟨ ˆOij,σp(z− +2 , z2⊥; z− +1 , z1⊥)⟩ ¯ +A = U +i,a(z− +2 , z2⊥)U +j,a(z− +1 , z1⊥) + O(αs) +⟨ ˆOij;σt(z+ +2 , z2⊥; z+ +1 , z1⊥)⟩ ¯B = V −i,a(z+ +2 , z2⊥)V −j,a(z+ +1 , z1⊥) + O(αs) +(4.13) +– 10 – + +so the master formula (4.12) takes the form +� +dz− +2 dz2⊥dz− +1 dz1⊥dw+ +1 dw1⊥dw+ +2 dw2⊥ +αsNc +2π C1(x1, x2; z− +i , zi⊥, w+ +i , wi⊥; σp, σt) +× U −i,a(z+ +2 , z2⊥)U −j,a(z+ +1 , z1⊥)V +i,a(z− +2 , z2⊥)V +j,a(z− +1 , z1⊥) += N2 +c − 1 +16 +g4⟨ ˜F a +µν ˜F aµν(x2)F b +λρF bλρ(x1)⟩A +− ⟨ ˆOij,σp(x− +2 , x2⊥; x− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥)⟩A +(4.14) +In what follows we will calculate the r.h.s. of this equation in the background field (4.3), +(11.38) and get the coefficient function. +The double functional integral for the r.h.s. of Eq. (4.14) has the form +N2 +c − 1 +16 +g4⟨ ˜F a +µν ˜F aµν(x2)F b +λρF bλρ(x1) − ˆOij,σp(x− +2 , x2⊥; x− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥)⟩A += +� +D ˜ADAD ˜ψDψ e−iSA,Ψ( ˜ +A, ˜ψ)+iSA,Ψ(A,ψ) +(4.15) +× +�N2 +c − 1 +16 +g4 ˜F a +µν ˜F aµν(x2)F b +λρF bλρ(x1) − ˆOσp +ij (x− +2 , x2⊥; x− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥) +� +where SA,Ψ(A, ψ) is a standard QCD action in a background field A in the background- +Feynman gauge: +SA,Ψ(A, ψ) +(4.16) += Scl(A, Ψ) + +� +dz 2tr +� +Aµ(D2gµν + 2iGµν)Aν + igDµAν[Aµ, Aν] + g2 +2 [Aµ, Aν]2� ++ ... +where dots stand for quark terms which are not relevant for our calculation. Hereafter “Tr” +means color trace in the adjoint representation. Note that the term Scl(A, Ψ) cancels in +the exponent in Eq. (4.15) so we will ignore it in what follows. The propagators in the +background field A can be obtained as an expansion in “correction field” ¯C since it is down +by one power of m2 +⊥ +s +in comparison to ¯A and ¯B. As to propagator in ¯A + ¯B background, +it can be in principle obtained as a cluster expansion, but fortunately we will need only a +couple of terms ∼ U −i and ∼ V +i which will be easily identified. +Most frequently we will perform this calculation in the momentum space, so we intro- +duce Fourier transforms of projectile and target fields +V −i(z+, z⊥) = +� +d−βbd−kb⊥V −i(βb, kb⊥)e−iϱβbz++i(ka,z)⊥, +U +i(z−, z⊥) = +� +d−αad−ka⊥U +i(αa, ka⊥)e−iϱαaz−+i(ka,z)⊥, +V −i(βb, kb⊥) = ϱ +� +dz+dz⊥ U −i(z+, z⊥)eiϱβbz+−i(ka,z)⊥, +U +i(αa, ka⊥) = ϱ +� +dz−dz⊥ U +i(z−, z⊥)eiϱαaz−−i(ka,z)⊥ +(4.17) +To avoid cluttering of our formulas, throughout the paper we use the ℏ-inspired notation +� +d−np ≡ +� +dnp +(2π)n +(4.18) +– 11 – + +where n is the dimension of corresponding momentum space. +Thus, the object of our calculations is the Fourier transform of Eq. (4.14) +C1(x1, x2; α′ +a, αa, ka⊥, k′a⊥, β′ +b, βb, kb⊥, k′ +b⊥; σp, σt) += +� +dz− +1 dz− +2 dw+ +1 dw+ +2 dz1⊥dz2⊥dw1⊥dw2⊥e−iϱα′ +az− +2 +i(k′ +a,z2)⊥e−iϱαaz− +1 +i(ka,z1)⊥ +× e−iϱβ′ +bz+ +2 +i(k′ +b,z2)⊥e−iϱβbz+ +1 +i(kb,z1)⊥C1(x1, x2; z− +i , zi⊥, w+ +i , wi⊥; σp, σt) +(4.19) +For one-loop calculations in the background field A it is convenient to multiply the hadronic +tensor (2.2) by additional factor N2 +c −1 +2g2Nc π2. We define +W (x1, x2) = N2 +c − 1 +2g2Nc +π2⟨W one−loop(x1, x2)⟩A += N2 +c − 1 +Nc +8π2g2⟨F −i,a(x2)F +a +i (x2)F −j,b(x1)F +b +j (x1)⟩one−loop +A +(4.20) +The contributions to W (x1, x2) will be parametrized as follows +W (x1, x2) − W σp,σt +eik +(x1, x2) = +� +d−α′ +ad−k′a⊥d−β′ +bd−k′ +b⊥d−αad−ka⊥d−βbd−kb⊥ +× e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 e−i(k′ +a+k′ +a,x2)⊥−i(ka+kb,x1)⊥ +× U +,b +i (α′ +a, k′a⊥)V −i,a(β′ +b, k′ +b⊥)U +,b +j (αa, ka⊥)V −j,a(βb, kb⊥) +× [I − Iσp,σt +eik +](α′ +a, k′a⊥, αa, ka⊥, β′ +b, k′ +b⊥, βb, kb⊥, x2, x1) +(4.21) +where W σp,σt +eik +(x1, x2) is the contribution of eikonal TMD operators ⟨ ˆOij,σp ˆOij;σt⟩A which +has to be subtracted according to Eq. (4.15). The coefficient function C1 is then a Fourier +transform of [I − Iσp,σt +eik +]. +Recall that the kinematical region for hadronic tensor (1.2) translates to Eq. (2.6) in +the non-forward case so in our approximation longitudinal distances are smaller than the +transverse ones. 6 +In order to have parameters of the Fourier transformations with ”natural” scales re- +sembling those of forward case, in the formulas (4.17) we should take the origin somewhere +between x2 and x1, so the kinematical region where we calculate Eq. (4.19) is +α′ +a ∼ αa, β′ +b ∼ βb, ka⊥ ∼ k′ +a⊥ ∼ kb⊥ ∼k′ +b⊥ ∼ Q⊥, +αaβb, α′ +aβb, αβ′ +b, α′ +aβ′ +b ∼ Q2 +s ≫ Q2 +⊥ +s +(4.22) +which corresponds to +x+ +2 ∼ x+ +1 , x− +2 ∼ x− +1 , x2⊥ ∼ x1⊥, +x+ +2 x− +2 ∼ x+ +1 x− +1 ∼ O +�1 +ζ +� +≪ x2 +2⊥ ∼ x2 +1⊥ ∼ O(ζ0) (4.23) +in the coordinate space because x+ +i ∼ +1 +βbϱ and x− +i ∼ +1 +αaϱ and ρ ∼ +� 1 +√ζ +� +. +6It is worth noting that for tree-level calculations, the parameter +q2 +⊥ +Q2 is sufficient. A more restrictive +parameter (1.2) is necessary for our calculation of logarithmical corrections. If +q2 +⊥ +Q2 ≪ 1 but Eq. (1.2) is not +satisfied, the calculation of power corrections in Refs. [6, 7] is still valid but the logarithmical corrections +will probably be more complicated than our result (9.2). +– 12 – + +5 +Virtual contributions +It is convenient to start calculation of functional integral (4.15) from the so-called “virtual” +contribution to the first term given by diagrams in Fig. 5 a-c. (The reason for “production” +diagram (d) appearing on this Figure is explained in the end of this Section). Let us start +with the diagram in Fig. 5a. +(b) +(c) +(d) +(a) +x 1 +x 2 +k b +k a +Figure 5. Diagrams (a)-(c): virtual diagrams in the right sector. Diagram (d): related diagram +with two-gluon production. +In an arbitrary background field A we get +Fµν(A + A) = Fµν(A) + (DµAν − DνAµ) − ig[Aµ, Aν] +The diagram in Fig. 5a corresponds to +⟨(DµAν − DνAµ)(x1)(DµAν − DνAµ)(x1)⟩ +(5.1) +expanded up to two Fµν(A). Using the background-field propagator in Schwinger’s nota- +tions +i⟨T{Aµ(x)Aν(y)}⟩A = (x| +1 +P2gµν + 2iFµν + iϵ|y) +(5.2) +and identities +Pµ 1 +P2 Pµ = 1 − g2 1 +P 2 FµνFµν 1 +P 2 +(5.3) +− g2 1 +P2 PηDξFξη +1 +P2 + g2 1 +P2 {Pα, Fαξ} 1 +P2 {Pβ, Fβξ} 1 +P2 +Pµ 1 +P2 Fµν +1 +P2 Pν = ig2 +2 +1 +P2 FµνFµν 1 +P2 + ig2 1 +P2 PνDµFµν +1 +P2 +−ig2 1 +P2 {Pα, Fαξ} 1 +P2 FβξPβ 1 +P2 − ig2 1 +P2 PαFαξ 1 +P2 {Fβξ, Pβ} 1 +P2 + O(F3 +ξη) +– 13 – + +one obtains after some algebra +(Pµδξ +ν − Pνδξ +µ) +1 +P2gξη + 2iFξη + iϵ(Pµδη +ν − Pνδη +µ) = 6 − 4 g2 +P2 FµνFµν 1 +P2 +(5.4) ++ 4Fµν +g2 +P2 Fµν 1 +P2 + 4 g2 +P2 Fµν 1 +P2 Fµν − 2 g2 +P2 PηDξFξη +1 +P2 + 8 g2 +P2 DµFλρ +1 +P2 DµFλρ 1 +P2 +− 2i g2 +P2 DαFβξ +1 +P2 {Pα, Fβξ} 1 +P2 + 2i g2 +P2 {Fαξ, Pβ} 1 +P2 DβFαξ 1 +P2 + O +� +F3 +ξη, (DξFξη)2� += 6 + 2Fµν +g2 +P2 Fµν 1 +P2 + 2 g2 +P2 Fµν 1 +P2 Fµν − 2 g2 +P2 PηDξFξη +1 +P2 + 4 g2 +P2 DµFλρ +1 +P2 DµFλρ 1 +P2 ++ O +� +F3 +ξη, (DξFξη)2� +(here all P2 are P2 + iϵ). +Next, it is convenient to add to Eq. (5.4) the contribution of diagrams in Fig. 5b,c +which has the form +2⟨igfabcAb +µAc +ν(x)Fcµν(x)⟩ = +− 4ig2fabc(x| 1 +P2 Gµν +1 +P2 |x)abGcµν(x) + O +� +F3 +ξη, (DξFξη)2� += (x| − 4 g2 +P2 Fµν +1 +P2 Fµν|x)aa + O +� +F3 +ξη, (DξFξη)2� +(5.5) +We get +2g2⟨i(DµAa +ν − DνAa +µ)(x1)DµAa,ν(x1) + igfabcAb +µAc +ν(x1)Fcµν(x1)⟩ += g4(x1| − 2 1 +P2 PηDξFξη +1 +P2 + 4 1 +P2 DµFλρ +1 +P2 DµFλρ 1 +P2 |x1)aa + O +� +F3 +ξη, (DξFξη)2� +(5.6) +Note that the absence of UV divergent terms follows from one-loop renorm-invariance of +local operator g2F a +µνF aµν. +Next, the expansion of propagators (p2 + 2{p, A} + A2)−1 in powers of external field A +will bring more powers of Fξη in the numerators leading to power corrections to Eq. (4.1) +rather than the logarithmical ones. Thus, one can replace all (P2 + iϵ)−1 in the r.h.s. of +Eq. (5.4) by usual propagators (p2 + iϵ)−1. Also, for our background field A +Fµν(A) = Fµν( ¯A) + Fµν( ¯B) + Fµν( ¯C) − ig([ ¯Aµ, ¯Bν] + [ ¯Aµ + ¯Bµ, ¯Cν] − µ ↔ ν) +(5.7) +Since the correction field ¯C is proportional to commutators of U −i and V +j we can neglect +it in the r.h.s. of Eq. (5.4) - it will lead to the terms with two U −i and one V +j, or vice +versa. By the same token, one can disregard [ ¯Aµ, ¯Bν] since its Lorentz components are +either zero or made from [U −i, V +j]. Finally, we are interested only in terms with one U −i +and one V +j in the r.h.s. of Eq. (5.4) corresponding to diagrams in Fig. 5 7 +Thus, in our approximation +2g2⟨(DµAa +ν − DνAa +µ)(x1)DµAa,ν(x1) + igfabcAb +µAc +ν(x1)Fcµν(x1)⟩A +(5.8) += +− 16i(x1| +1 +p2 + iϵ +¯DµU +i +1 +p2 + iϵ +¯DµV − +i +1 +p2 + iϵ|x1)aa +7The terms with zero U −i’s and two V +j’s will lead to zero color trace after integration over projectile +fields in the integral (3.8), and similarly for terms with U +j’s +– 14 – + +This term is proportional to ¯DµU +i ⊗ ¯DµV − +i = ∂+U +i ⊗ ∂−V − +i + ¯DkU −i ⊗ ¯DkV + +i . As we +discussed after Eq. (4.8), ∂+ ⊗ ∂− brings extra factor of s and ¯Dk ⊗ ¯Dk only q2 +⊥ so we can +disregard it. We get +−16i(x1| +1 +p2 + iϵ∂−U +i +1 +p2 + iϵ∂+V − +i +1 +p2 + iϵ|x1)aa += +− 8Nc +� +d−αad−ka⊥ +� +d−βbd−kb⊥U +i,a(αa, ka⊥)V −,a +i +(βb, kb⊥)e−iϱαax− +1 −iϱβbx+ +1 +i(ka+ka,x1)⊥ +× +� d−4p +i +sαaβb +[(p + ka)2 + iϵ](p2 + iϵ)[(p − ka)2 + iϵ] += +− g2Nc +2π2 +� +d−αad−ka⊥ +� +d−βbd−kb⊥U +i,a(αa, ka⊥)V −,a +i +(βb, kb⊥)e−iϱαax− +1 −iϱβbx+ +1 +i(ka+ka,x1)⊥ +× +� +ln −αaβbs − iϵ +k2a⊥ +ln −αaβbs − iϵ +k2 +b⊥ ++ π2 +3 +� +(5.9) +where we used Eq. (11.52) in the last line. Finally, we obtain +g2⟨T{F a +µν(x1)F µν,a(x1)}⟩Fig.5a+b+c += +− +� +d−αad−ka⊥ +� +d−βbd−kb⊥U +i,a(αa, ka⊥)V −,a +i +(βb, kb⊥)e−iϱαax− +1 −iϱβbx+ +1 +i(ka+ka,x1)⊥ +× g2Nc +2π2 +� +ln −αaβbs − iϵ +k2a⊥ +ln −αaβbs − iϵ +k2 +b⊥ ++ π2 +3 +� +(5.10) +Next, let us consider the diagram in Fig. 5d. This diagram and its left-right permu- +tation are the only diagrams with two-gluon cut since gluon with ka or kb cannot solely +produce two gluons. As we will see below, the diagram in Fig. 5d will change Feynman-type +singularities in logarithms in Eq. (5.10) to causal-type singularities. +The structure of the diagram in Fig. 5d is the same as in Fig. 5a with two Feynman +propagators replaced by cut propagators and the left-sector propagator between U −i and +V +i being +i +p2−iϵ instead of +−i +p2+iϵ. Note also that the first three terms in the r.h.s. of Eq. +(5.4) do not contribute since our background fields cannot produce particles. We get +g2⟨T{F a +µν(x1)F a,µν(x1)}⟩Fig.5d +(5.11) += +− 16i(x1|˜δ−(p)∂−U +i +1 +p2 − iϵ∂+V − +i ˜δ+(p)|x1)aa +Hereafter we introduce space-saving notations +˜δ+(p) ≡ 2πδ(p2)θ(p0), +˜δ−(p) ≡ 2πδ(p2)θ(−p0) +(5.12) +Using integral (11.51) from Appendix 11.6 one easily obtains +g2⟨T{F a +µν(x1)F µν,a(x1)}⟩Fig.5d += +� +d−αad−ka⊥ +� +d−βbd−kb⊥U +i(αa, ka⊥)V − +i (βb, kb⊥)e−iϱαax− +1 −iϱβbx+ +1 +i(ka+ka,x1)⊥ +× θ(−αa)θ(−βb)g2Nc +π +(−i) ln αa2βb2s2 +k2a⊥k2 +b⊥ +(5.13) +– 15 – + +Next, using the identity +ln −αaβbs − iϵ +k2a⊥ +ln −αaβbs − iϵ +k2 +b⊥ ++ 2πiθ(−αa)θ(−βb) ln αa2βb2s2 +k2a⊥k2 +b⊥ += ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ +(5.14) +where 8 +Q2 +ab ≡ (αa + iϵ)(βb + iϵ)s +(5.15) +we get the contribution of diagrams in Fig. 5 in the form +g2⟨T{F a +µν(x1)F µν,a(x1)}⟩Fig.5 = g2Nc +2π2 +� +d−αad−ka⊥ +� +d−βbd−kb⊥ +(5.16) +× U +i,a(αa, ka⊥)V −,a +i +(βb, kb⊥)e−iϱαax− +1 −iϱβbx+ +1 +i(ka+kb,x1)⊥Ivirt +Fig.5(αa, ka⊥, βb, kb⊥) +where +Ivirt +Fig.5(αa, ka⊥, βb, kb⊥) = +− 16π2 +� d−4p +i +� +sαaβb +[(p + ka)2 + iϵ](p2 + iϵ)[(p − kb)2 + iϵ] ++ ˜δ−(p + ka) sαaβb +p2 − iϵ +˜δ+(p − kb) +� += +− ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ +− π2 +3 ++ O(λ) +(5.17) +and λ is defined in Eq. (3.6). +In coordinate space, the singularity (5.15) means that +� +d−α′ +a e−iϱα′ +ax− +1 f(−α′ +a − iϵ)U(α′ +a) = +� x− +1 +−∞ +dz− +1 +˜f(x− +1 − z− +1 )U(z− +1 ) +� +d−β′ +b e−iϱβ′ +bx+ +1 f(−β′ +b − iϵ)V (β′ +b) = +� x+ +1 +−∞ +dz+ +1 +˜f(x+ +1 − z+ +1 )V (z− +1 ) +(5.18) +Thus, after summation of the diagrams Fig. 5a,b,c and Fig. 5d we get the result that the +emission of the background-field gluon always preceeds the original point x1. +Actually, it can be seen before the calculation of integrals. To this end, consider the +identity +(x| +1 +p2 + iϵA +1 +p2 + iϵB +1 +p2 + iϵ + ˜δ−(p)A +1 +p2 − iϵB˜δ+(p)|y) + ˜δ−(p)A˜δ+(p)B +1 +p2 + iϵ ++ +1 +p2 + iϵA˜δ−(p)B˜δ+(p) = (x| +1 +p2 + iϵp0 +A +1 +p2 + iϵB +1 +p2 − iϵp0 +−i +1 +p2 + iϵp0 +A +1 +p2 + iϵp0 +B˜δ+(p) − i˜δ−(p)A +1 +p2 − iϵp0 +B +1 +p2 − iϵp0 +|y) (5.19) +valid for any operators A and B. Using this formula, the sum of ∂−U +i ⊗ ∂+V − +i terms in +Eqs. (5.8) and (5.11) can be rewritten as +1 +p2 + iϵ∂−U +i +1 +p2 + iϵ∂+V − +i +1 +p2 + iϵ + ˜δ−(p)∂−U +i +1 +p2 − iϵ∂+V − +i ˜δ+(p) += +1 +p2 + iϵp0 +∂−U +i +1 +p2 + iϵ∂+V − +i +1 +p2 − iϵp0 +(5.20) +−i˜δ−(p)∂−U +i +1 +p2 − iϵp0 +∂+V − +i +1 +p2 − iϵp0 +− i +1 +p2 + iϵp0 +∂−U +i +1 +p2 + iϵp0 +∂+V − +i ˜δ+(p) +8Later we will use similar notations Q2 +ab′ ≡ (αa + iϵ)(β′ +b + iϵ)s, Q2 +a′b ≡ (α′ +a + iϵ)(βb + iϵ)s, and Q2 +a′b′ ≡ +(α′ +a + iϵ)(β′ +b + iϵ)s +– 16 – + +from where the causal structure of the result of summation is evident. Note that we used +˜δ−(p)∂−U +i˜δ+(p) = ˜δ−(p)∂−V −i˜δ+(p) = 0 following from the fact that background fields +U +i(x−, x⊥) and V −i(x−, x⊥) cannot produce two real particles. This is similar to the case +of tree-level diagrams where the summation of the emissions from both sides of the cut +leads to the diagrams with retarded propagators, see Ref. [6]. +One can calculate diagrams in Fig. 6 in a similar way. The result for the diagrams in +(a) +(b) +(c) +(d) +x 1 +x 2 +k’a +k’b +Figure 6. Diagrams (a)-(c): virtual diagrams in the left sector. Diagram (d): related diagram +with two-gulon production +Fig. 6 a,b,c is obtained by complex conjugation of Eq. (5.10) +g2⟨ ˜T{F a +µν(x2)F µν,a(x2)}⟩Fig.6a+b+c += +− +� +d−α′ +ad−k′a⊥ +� +d−β′ +bd−k′ +b⊥U +i,a(α′ +a, k′a⊥)V −,a +i +(β′ +b, k′ +b⊥)e−iϱα′ +ax− +2 −iϱβ′ +bx+ +2 +i(k′ +a+k′ +b,x2)⊥ +× g2Nc +2π2 ln −α′ +aβ′ +bs + iϵ +k′2 +a⊥ +ln −α′ +aβ′ +bs + iϵ +k2 +b⊥ +(5.21) +Next, the diagram in Fig. 6d +g2⟨ ˜T{F a +µν(x2)F a,µν(x2)}⟩Fig.6d = +− 16i(x2|˜δ+(p)∂−U +i +1 +p2 + iϵ∂+V − +i ˜δ−(p)|x2) +(5.22) +can be obtained from the integrals in Eq. (11.52) by the replacements ka → −k′ +a, kb → −k′ +b. +The result is +g2⟨ ˜T{F a +µν(x2)F µν,a(x2)}⟩Fig.6d += +� +d−α′ +ad−k′a⊥ +� +d−β′ +bd−k′ +b⊥U +i(α′ +a, k′a⊥)V − +i (β′ +b, k′ +b⊥)e−iϱα′ +ax− +2 −iϱβ′ +bx+ +2 +i(k′ +a+k′ +b,x2)⊥ +× θ(α′ +a)θ(β′ +b)g2Nc +π +i ln α′ +a +2β′ +b +2s2 +k′2 +a⊥k2 +b⊥ +(5.23) +Using now +ln −α′ +aβ′ +bs + iϵ +k′2 +a⊥ +ln −α′ +aβ′ +bs + iϵ +k′2 +b⊥ +−2πiθ(α′ +a)θ(β′ +b) ln α′ +a +2β′ +b +2s2 +k′2 +a⊥k′2 +b⊥ += ln −Q2 +a′b′ +k′2 +a⊥ +ln −Q2 +a′b′ +k′2 +b⊥ +(5.24) +– 17 – + +we get +g2⟨ ˜T{F a +µν(x2)F µν,a(x2)}⟩Fig.6 = g2Nc +2π2 +� +d−α′ +ad−k′a⊥ +� +d−β′ +bd−k′ +b⊥ +U +i,a(α′ +a, k′a⊥)V −,a +i +(β′ +b, k′ +b⊥)e−iϱα′ +ax− +2 −iϱβ′ +bx+ +2 +i(k′ +a+k′ +b,x2)⊥Ivirt +Fig.6 +(5.25) +where +Ivirt +Fig.6 = 16π2 +� d−4p +i +� +sα′ +aβ′ +b +[(p + k′a)2 − iϵ](p2 − iϵ)[(p − k′ +b)2 − iϵ] ++ ˜δ+(p + k′ +a) sα′ +aβ′ +b +p2 + iϵ +˜δ−(p − k′ +b) +� += +− ln −Q2 +a′b′ +k′2 +a⊥ +ln −Q2 +a′b′ +k′2 +b⊥ +− π2 +3 +(5.26) +and Q2 +ab is defined in Eq. (5.15). Again, the sum of all diagrams in Fig. 6 reveals causal +structure in the coordinate space. +The final result for the virtual contributions can be presented as follows +W virt(x1, x2) = +N2 +c − 1 +Nc +8π2� +V −i,a(x2)U +a +i (x2)⟨F −j,b(x1)F +b +j (x1)⟩Fig.5 +A ++ ⟨F −i,b(x2)F +b +i (x2)⟩Fig.6 +A +V −j,a(x1)U +a +j (x1) +� += +� +d−α′ +ad−k′a⊥d−β′ +bd−k′ +b⊥d−αad−ka⊥d−βbd−kb⊥e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 +× e−i(ka+ka,x1)⊥−i(k′ +a+k′ +b,x2)⊥ U +,b +i (α′ +a, k′a⊥)V −i,a(β′ +b, k′ +b⊥)U +,b +j (αa, ka⊥)V −j,a(βb, kb⊥) +× Ivirt(α′ +a, k′a⊥, β′ +b, k′ +b⊥, αa, ka⊥, βb, kb⊥) + O(λ) +(5.27) +where +Ivirt(α′ +a, k′ +a⊥, β′ +b, kb⊥, αa, k′ +a⊥, β′ +b, k′ +b⊥) = Ivirt +Fig.5(αa, ka⊥, βb, kb⊥) +(5.28) ++ Ivirt +Fig.6(α′ +a, k′ +a⊥, β′ +b, k′ +b⊥) = +− Id.log(α′ +a, k′ +a⊥, β′ +b, k′ +b⊥) − Id.log(αa, k′ +a⊥, βb, k′ +b⊥) +For future use, we introduced the notation +Id.log(αa, ka⊥, βb, kb⊥) = ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ ++ π2 +3 +(5.29) +for the double-log contributions. The expression for Id.log(α′ +a, k′ +a⊥, β′ +b, k′ +b⊥) is similar. +6 +“Production” diagrams +6.1 +Power counting for production terms +From power counting (4.8) it is easy to see that the leading contribution to hadronic tensor +(4.20) with one-gluon production comes from the following terms +W (x1, x2) ≡ N2 +c − 1 +2Nc +π2g2Fa +µν(x2)⟨F µν,a(x2)F αβ,b(x1)⟩AFb +αβ(x1) = N2 +c − 1 +Nc +8π2 +(6.1) +× +� +V −i,a(x2)⟨F +,a +i +(x2)F −,b +j (x1)⟩AU +j,b(x1) + U +i,a(x2)⟨F −,a +i +(x2)F +,b +j (x1)⟩AV −j,b(x1) ++ U +i,a(x2)⟨F −,a +i +(x2)F −,b +j (x1)⟩AU +j,b(x1) + V −i,a(x2)⟨F +,a +i +(x2)F +,b +j (x1)⟩AV −j,b(x1) +� +– 18 – + +In this Section we calculate the first term in this equation. The second term is obtained by +trivial replacements while the third and the fourth terms correspond to “handbag” diagrams +considered in next Section. +The gluon propagator in the background field A is given by Eq. (11.5) from Appendix +11.1. +Also, in Appendix 11.5 it was proved that the contributions due to background +“correction field” ¯C can be neglected so we need to compute +V a,−i(x2)⟨(D+Ai − DiA+)(x2)(D−Aj − DjA−)(x1)⟩abU b,+j(x1) = += +− V a,−ii(x2|(P+δα +i − Pig+α) +1 +P2gαξ + 2iFαξ − iϵp2 +× ˜δ+(p)p2 +1 +P2δξ +β + 2iFξ +β + iϵ +(P−δβ +j − Pjgβ−)|x1)abU b,+j += +− V a,−i(x2|(p+δα +i − Pig+α) +� +gαβ +1 +P2 − iϵp2˜δ+(p)p2 +1 +P2 + iϵ +− 2i +1 +P2 − iϵ +� +p2˜δ+(p)p2 +1 +P2 + iϵFαβ + Fαβ +1 +P2 − iϵp2˜δ+(p)p2� +1 +P2 + iϵ|y) ++ 4 +1 +P2 − iϵ +� +Fαξ +1 +P2 − iϵF ξ +β +1 +P2 − iϵp2˜δ+(p)p2 + Fαξ +1 +P2 − iϵp2˜δ+(p)p2 +1 +P2 + iϵF ξ +β ++ p2˜δ+(p)p2 +1 +P2 + iϵFαξ +1 +P2 + iϵF ξ +β +� +1 +P2 + iϵ +� +(p−δβ +j − Pjgβ−)|x1)abU b,+j(x1) +(6.2) +The leading contribution, shown in Fig. 7, +−4V −a,i(x2)(x2| +p+ +p2 − iϵU +i +1 +p2 − iϵV −jp−˜δ+(p) + +p+ +p2 − iϵU +i˜δ+(p)V −j +p− +p2 − iϵ ++ p+˜δ+(p)U +i +1 +p2 + iϵV −j +p− +p2 − iϵ|x1)abU b,+j(x1) +(6.3) +comes from the last term in the r.h.s. of Eq. (6.2). As we will see in the next Section, this +(a) +x 1 +x 2 +z 1 +z 2 +(b) +(c) +k’a +k a +k’b +k b +Figure 7. First set of leading diagrams with gluon production. Projectile fields U +i(z2) are denoted +by green tails while target fields V −i(z1) by red tails. +contribution is logarithmic similarly to Eq. (5.9) for virtual diagram. +Next, using power counting (4.8), we demonstrate that all other contributions to the +r.h.s. of Eq. (6.2) are power corrections with respect to Eq. (6.3). To this end, we note that +– 19 – + +p+p− ∼ α′ +aβbs = Q2 +a′b and U +iV −j ∼ Q2 +⊥s so the numerator p+p−U +iV −j in the integral +of Eq. (6.3) is of order Q2 +abQ2 +⊥s. Now, if we take ∼ PiPjU +kV − +k contribution to the last +term in Eq. (6.2), it is ∼ Q4 +⊥s which is down by Q2 +⊥ +Q2 +ab factor in comparison to p+p−U +iV −j. +As to the term ∼ p+PjFikF−k ∼ α′ +asQ4 +⊥, it is O +� +α′ +a +2Q2 +⊥/Q2 +ab +� +in comparison to Eq. (6.3). +9 +Let us now consider the terms in the second line in the r.h.s. of Eq. (6.2). To get the +contribution having four gluon tails, we expand +1 +P 2 once and get terms like +V a,−i(x2|(p+δα +i − Pig+α) +1 +p2 − iϵ{pk, Ak}˜δ+(p)Fαβ +1 +p2 + iϵ(p−δβ +j − Pjgβ−)|x1)abU b,+j(x1) +(6.4) +where Ak is either Uk or Vk. The numerator in the integral of this equation is again ∼ Q4 +⊥s, +so, as we mentioned above, it is a power correction in comparison to the leading term (6.3). +Finally, the term in the first line in the r.h.s. of Eq. (6.2) should be expanded twice to get +four gluons, so we have +V a,−i(x2|(p+δα +i − Pig+α) +1 +p2 − iϵ{pk, Ak}˜δ+(p){pl, Al} +1 +p2 + iϵ(p−δβ +j − Pjgβ−)|y)abU b,+j(x1) +(6.5) +The numerator in this equation is at best ∼ Q4 +⊥s which is O(Q2 +⊥/Q2 +ab) in comparison to +the leading term. +Thus, all terms in the r.h.s. of Eq. (6.2) are power corrections ∼ O +� m2 +⊥ +s +∼ ζ−1� +to the +logarithmical leading term (6.3) which we calculate in the next Section. +6.2 +Calculation of leading production terms +In this Section we will calculate the first term in the r.h.s. of term in Eq. (6.1) given by +Eq. (6.3), see Fig. 7. First, it is convenient to use the identity +(x2| +1 +p2 − iϵA˜δ+(p)B +1 +p2 + iϵ + ˜δ+(p)A +1 +p2 + iϵB +1 +p2 + iϵ + +1 +p2 − iϵA +1 +p2 − iϵB˜δ+(p) ++ ˜δ+(p)A˜δ−(p)B˜δ+(p)|x1) = (x2| +1 +p2 + iϵp0 +A˜δ+(p)B +1 +p2 − iϵp0 ++ ˜δ+(p)A +1 +p2 − iϵp0 +B +1 +p2 − iϵp0 ++ +1 +p2 + iϵp0 +A +1 +p2 + iϵp0 +B˜δ+(p)|x1) +(6.6) +and rewrite Eq. (6.3) changing singularities of propagators accordingly +−4V −a,i(x2)(x2| +p+ +p2 + iϵp0 +U +i +1 +p2 + iϵp0 +V −jp−˜δ+(p) + +p+ +p2 + iϵp0 +U +i˜δ+(p)V −j +p− +p2 − iϵp0 ++ p+˜δ+(p)U +i +1 +p2 − iϵp0 +V −j +p− +p2 − iϵp0 +|x1)abU b,+j(x1) += +Nc +8π2(N2c − 1) +� +d−α′ +ad−k′a⊥d−β′ +bd−k′ +b⊥d−αad−ka⊥d−βbd−kb⊥ +× e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 e−i(k′ +a+k′ +b,x2)⊥−i(ka+kb,x1)⊥ +(6.7) +× U +,b +i (α′ +a, k′ +a⊥)V −i,a(β′ +b, k′ +b⊥)U +,b +j (αa, ka⊥)V −j,a(βb, kb⊥)I1(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) +9Recall that α′ +a (and βb) are either ∼ 1 or ≪ 1 depending on moderate-x or small-x kinematics +– 20 – + +where +I1(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) = 8π2s2 +� +d−αd−βd−p⊥ eiαϱx− +12+iβϱx+ +12−i(p,x12)⊥ +(6.8) +× +� +α′ +a + α +(α′a + α)βs − (p + k′a)2 +⊥ + iϵ +˜δ(αβs − p2 +⊥)θ(α) +β − βb +α(β − βb)s − (p − kb)2 +⊥ − iϵ ++ ˜δ[(α′ +a + α)βs − (p + k′ +a)2 +⊥]θ(β) +1 +αβs − p2 +⊥ − iϵ +(α + α′ +a)(β − βb) +α(β − βb)s − (p − kb)2 +⊥ − iϵα ++ +(α′ +a + α)(β − βb) +(α′a + α)βs − (p + k′a)2 +⊥ + iϵ(α′a + α) +1 +αβs − p2 +⊥ + iϵ +˜δ[α(β − βb)s − (p − kb)2 +⊥]θ(α) +� +Here we used the fact that after taking matrix elements between nucleon states only the +colorless operators survive. +Note that singularities in denominators in there expressions correspond to α′ +a + iϵ and +βb + iϵ so it is sufficient to perform calculations at, say, positive α′ +a and βb. +To calculate the integral (6.8) it is convenient to split it in two parts using identity +δ(αβs − p2 +⊥) = δ(αβs − p2 +⊥) +� +p2 +⊥ +α2sξ + p2 +⊥ ++ +p2 +⊥ +β2sξ−1 + p2 +⊥ +� +(6.9) +where ξ is an arbitrary positive number of order of 1. We get +I1 = I1a + I1b, +(6.10) +where +I1a(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) = 8π2s2 +� +d−αd−βd−p⊥ θ(α) eiαϱx− +12+iβϱx+ +12−i(p,x12)⊥ +× +� +θ(α)(α′ +a + α)(β − βb)˜δ(αβs − p2 +⊥) +[(α′a + α)βs − (p + k′a)2 + iϵ][α(β − βb)s − (p − kb)2 +⊥ − iϵ] +p2 +⊥ +α2sξ + p2 +⊥ +(6.11) ++ +(α′ +a + α)(β − βb) +[(α′a + α)βs − (p + k′a)2 +⊥ + iϵ(α′a + α)](αβs − p2 +⊥ + iϵ) +˜δ[α(β − βb)s − (p − kb)2 +⊥] +� +and +I1b(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) = 8π2s2 +� +d−αd−βd−p⊥ θ(β)eiαϱx− +12+iβϱx+ +12−i(p,x12)⊥ +× +� +(α′ +a + α)(β − βb)˜δ(αβs − p2 +⊥) +[(α′a + α)βs − (p + k′a)2 + iϵ][α(β − βb)s − (p − kb)2 +⊥ − iϵ] +p2 +⊥ +β2sξ−1 + p2 +⊥ ++ ˜δ[(α′ +a + α)βs − (p + k′ +a)2] +1 +αβs − p2 +⊥ − iϵ +(α + α′ +a)(β − βb) +α(β − βb) − (p − kb)2 +⊥ − iϵα +� +(6.12) +As we discussed above, the Eq. +(6.7) is not yet the contribution to the coefficient +function. According to Eq. (4.2), one needs to subtract relevant matrix elements of gluon +TMDs from the result of calculation in the background fields A and B. The “projectile” +matrix elements of operator ˆOij,σp(x− +2 , x2⊥; x− +1 , x1⊥) are given by the diagrams shown in +Fig. 12 and “target” matrix elements of operator ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥) are given by the +– 21 – + +diagrams shown in Fig. 11. Consequently, to get the contribution of Eq. (6.8) to the coef- +ficient function one should subtract from the integral (6.8) two eikonal-type contributions +of TMD matrix elements coming from diagrams shown in Fig. 11a,b and Fig. 12d,e. +The first contribution, coming from the “projectile” eikonals in Fig. 11a,b, corresponds +to the α ≪ α′ +a asymptotics in Eq. (6.8) cut from above according to “smooth” cutoff e−i α +σt +discussed in Ref. [10] (see also Appendix 11.2) +Ieik +Fig.11a,b(βb, kb⊥, x+ +1 , x1⊥, x+ +2 , x2⊥) +(6.13) += 8π2s +� ∞ +0 +d−α e−i α +σt +d−β +β + iϵd−p⊥ eiβϱx+ +12−i(p,x12)⊥ +× +� +˜δ(αβs − p2 +⊥) +β − βb +α(β − βb)s − (p − kb)2 +⊥ − iϵ + +(β − βb) +αβs − p2 +⊥ + iϵ +˜δ[α(β − βb)s − (p − kb)2 +⊥] +� +The second contribution coming from Fig. 12e,f corresponds to β ≪ βb asymptotics of the +integrand in Eq. (6.8) integrated with the upper “smooth cutoff” factor e +i β +σp +Ieik +Fig.12e,f(α′ +a, k′ +a⊥, x− +1 , x1⊥, x− +2 , x2⊥) +(6.14) += 8π2s +� ∞ +0 +d−β e +i β +σp +d−α +α − iϵd−p⊥ eiαϱx− +12−i(p,x12)⊥ +× +� +α′ +a + α +(α′a + α)βs − (p + k′a)2 + iϵ +˜δ(αβs − p2 +⊥) + ˜δ[(α′ +a + α)βs − (p + k′ +a)2] +α + α′ +a +αβs − p2 +⊥ − iϵ +� +To get the contribution to the coefficient function we need to calculate +J1(α′ +a, k′a⊥, βb, kb⊥, x2, x1) +(6.15) += I1(α′ +a, k′a⊥, βb, kb⊥, z2, z1) − Ieik +Fig.11a,b(βb, kb⊥, x2, x1) − Ieik +Fig.12e,f(α′ +a, k′ +a⊥, x2x1) +As demonstrated in the Appendix 11.4, one can neglect x+ +12 and x− +12 in the difference +I1a(α′ +a, k′a⊥, βb, kb⊥, x1, x2) − Ieik +Fig.11a,b(βb, kb⊥, x1, x2) +(6.16) +and similarly in +I1b(α′ +a, k′a⊥, βb, kb⊥, x1, x2) − Ieik +Fig.12e,f(α′ +a, k′ +a⊥, x1, x2) +(6.17) +Qualitatively, the argument is as follows. Consider Eq. (6.12). In order for eiαϱx− +12 to be +essential, α should be of order of α′ +a due to Eq. (4.23). Due to δ-functions in Eq. (6.12), +this means that β should be small, of order of p2 +⊥ +α′as ≪ 1 since p⊥ ∼ +1 +x12⊥ ∼ Q⊥. However, the +contribution of small β to Eq. (6.8) is subtracted by small-β eikonal (6.14) so the resulting +difference I1b−Ieik +Fig.11e,f is small and the factor eiαϱx− +12 can be neglected. Similarly, the factor +eiβϱx+ +12 in the difference I1 − Ieik +Fig.12a,b at small α can be replaced by 1. In Appendix 11.4 +it is demonstrated that the corrections due to these approximations are ∼ +Q2 +⊥ +σtβbs ∼ λt and +∼ +Q2 +⊥ +σpα′as ∼ λp, respectively. As we discussed in Sect. 2, we neglect such power corrections. +– 22 – + +We get +I1a(α′ +a, k′a⊥, βb, kb⊥, x1⊥, x2⊥) = 4π +� ∞ +0 +dα +α +� +d−p⊥ +e−i(p,x12)⊥ +αβbs + (p − kb)2 +⊥ − p2 +⊥ + iϵ +× +� +p2 +⊥ +α2sξ + p2 +⊥ +(α + α′ +a)(αβbs − p2 +⊥) +� +(α + α′a)p2 +⊥ − α(p + k′a)2 + iϵ +� ++ +(p − kb)2 +⊥(α + α′ +a) +α(α′a + α)βbs + (α′a + α)(p − kb)2 +⊥ − α(p + k′a)2 +⊥ + iϵ(α′a + α) +� +(6.18) +and +I1b(α′ +a, k′a⊥, βb, k′b⊥, x1⊥, x2⊥) = 4π +� ∞ +0 +dβ +β +� +d−p⊥ +e−i(p,x12)⊥ +α′aβs − (p + k′a)2 + p2 +⊥ + iϵ +× +�� ∞ +0 +d−β +β +p2 +⊥ξ +β2s + p2 +⊥ξ +(βb − β)(α′ +aβs + p2 +⊥) +� +(βb − β)p2 +⊥ + β(p − kb)2 + iϵ +� ++ +(βb − β)(p + k′ +a)2 +⊥ +(α′a + iϵ)(βb − β)βs − (βb − β)(p + k′a)2 +⊥ − β(p − kb)2 +⊥ +� +(6.19) +To calculate the sum of Eqs. +(6.18) and (6.19), it is convenient to perform change of +variables α = p2 +⊥ +βs in the first term in square brackets in Eq. (6.18) and change α = (p−k′ +b)2 +⊥ +(β−βb)s +in the second term. Since this change affects cancellation of logarithmic divergences at +α → 0, before the change we replace +� ∞ +0 +dα +α by +� ∞ +ε +dα +α . After some algebra one obtains +I1(α′ +a, k′a⊥, βb, kb⊥, x1⊥, x2⊥) = 4π +� +d−p⊥ e−i(p,x12)⊥ +� ∞ +0 +dβ +� +θ(β − βb) − θ(β) +� +× (β − βb)[(p − kb)2 +⊥ − α′ +a(βb − β)s] +� +β(p − kb)2 +⊥ + p2 +⊥(βb − β)s + iϵ +�−1 +� +− (α′a + iϵ)β(βb − β)s + (p − kb)2 +⊥β + (p + k′a)2 +⊥(βb − β) +� ++ lim +ε→0 +1 +β[(p − kb)2 +⊥ − p2 +⊥ + iϵ] +� +θ +� +β − p2 +⊥ +εs +� +− θ +� +β − βb − (p − kb)2 +⊥ +εs +��� +(6.20) +Let us take βb > 0, then +I1(α′ +a, k′a⊥, βb, kb⊥, x1⊥, x2⊥) = 4π +� +d−p⊥ e−i(p,x12)⊥ +(6.21) +× +�� 1 +0 +du +¯u[(p − kb)2 +⊥ − Q2 +a′b¯u] +[¯u(p + k′a)2 +⊥ + u(p − kb)2 +⊥ − Q2 +a′b¯uu][p2 +⊥¯u + u(p − kb)2 +⊥] + ln(p − kb)2 +⊥/p2 +⊥ +(p − kb)2 +⊥ − p2 +⊥ +� += +− 4π +� +d−p⊥ +� 1 +0 +du +e−i(p,x12)⊥ ¯uQ2 +ab′ +[¯u(p + k′a)2 +⊥ + u(p − kb)2 +⊥ − Q2 +a′b¯uu][p2 +⊥¯u + u(p − kb)2 +⊥] + O(λ) += +− 4π +� +d−p⊥ +� 1 +0 +du +e−i(p,x12)⊥Q2 +ab′ +[¯u(p + k′a)2 +⊥ − Q2 +a′bu][p2 +⊥¯u + u(p − kb)2 +⊥] + O(λ) += 4π +� +d−p⊥ e−i(p,x12)⊥ +Q2 +a′b +Q2 +a′bp2 +⊥ + (p + k′a)2 +⊥(p − kb)2 +⊥ +ln +−Q2 +a′bp2 +⊥ +(p + k′a)2 +⊥(p − kb)2 +⊥ ++ O(λ) +where ¯u ≡ 1 − u and Q2 +a′b ≡ (α′ +a + iϵ)(βb + iϵ)s (recall that the analytical properties of +integrals over α′ +a and βb are determined by the integral (6.8) to be α′ +a + iϵ and βb + iϵ). +– 23 – + +This integral is calculated in the Appendix 11.6.2, see Eq. (11.57): +4π +� +d−p⊥ +e−i(p,x)⊥Q2 +ab +Q2 +abp2 +⊥ + (p + ka)2 +⊥(p − kb)2 +⊥ +ln +−Q2 +abp2 +⊥ +(p + ka)2 +⊥(p − kb)2 +⊥ +(6.22) += ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ +− 1 +2 +� +ln −Q2 +abx2 +⊥ +4 ++ 2γ +�2 ++ π2 +3 ++ +� 1 +0 +du +u +� +ln k2 +a⊥x2 +⊥¯uu +4 ++ 2γ + 2eiu(k,x)⊥K0( +� +k2a⊥x2 +⊥¯uu) + ka⊥ ↔ −kb⊥ +� ++ O(λ) +It is convenient to represent it as a sum of the double-log contribution similar to virtual +term, and the remainder. We get +I1 = Id.log(α′ +a, k′a⊥, βb, kb⊥) + Irem +1 +(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) +(6.23) +where Id.log was defined in Eq. (5.29) +Id.log(α′ +a, k′a⊥, βb, kb⊥) = ln −Q2 +a′b +k′2 +a⊥ +ln −Q2 +a′b +k2 +b⊥ ++ π2 +3 ++ O(λ), +(6.24) +and +Irem +1 +(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) +(6.25) += +− 1 +2 +� +ln −Q2 +a′bx2 +12⊥ +4 ++ 2γ +�2 ++ +� 1 +0 +du +u +�� +ln k′2 +a⊥x2 +12⊥ ¯uu +4 ++ 2γ +� ++ 2e−iu(k′ +a,x12)⊥K0( +� +k′2 +a⊥x2 +12⊥ ¯uu) + k′ +a⊥ ↔ −kb⊥ +� ++ O(λ) += +− 1 +2 +� +ln −Q2 +a′bx2 +12⊥ +4 ++ 2γ +�2 ++ IK(k′ +a⊥, x12⊥) + IK(−kb⊥, x12⊥) + O(λ) +where +IK(k⊥, x⊥) ≡ +� 1 +0 +du +u +� +ln k2 +⊥x2 +⊥¯uu +4 ++ 2γ + 2eiu(k,x)⊥K0( +� +k2 +⊥x2 +⊥¯uu) +� +(6.26) +and K0 is the Macdonald function. +The second leading contribution to hadronic tensor (4.20) comes from the second term +in the r.h.s. of Eq. (6.1) +U +i,a(x2)⟨g2F −,a +i +(x2)F +,b +j (x1)⟩AV −j,b(x1) += +− 4g2U +i,a(x2)(x2| +p− +p2 + iϵp0 +V −i +1 +p2 + iϵp0 +U +jp+˜δ+(p) ++ +p− +p2 + iϵp0 +V −i˜δ+(p)U +j +p+ +p2 − iϵp0 ++ p−˜δ+(p)V −i +1 +p2 − iϵp0 +U +j +p+ +p2 − iϵp0 +|x1)abV −j,b(x1) += +g2Nc +8π2(N2c − 1) +� +d−αad−ka⊥d−βbd−kb⊥d−α′ +ad−k′ +a⊥d−β′ +bd−k′ +b⊥ +× e−iαaϱx− +1 −iα′ +aϱx− +2 e−iβbϱx+ +1 −iβ′ +bϱx+ +2 e−i(ka+kb,x1)⊥−i(k′ +a+k′ +b,x2)⊥ +(6.27) +× U +,b +i (αa, ka⊥)V −i,a(βb, kb⊥)U +,b +j (α′ +a, k′ +a⊥)V −j,a(β′ +b, k′ +b⊥)I2(αa, ka⊥, β′ +b, k′ +b⊥, x1, x2) +– 24 – + +x 1 +x 2 +k’b +k b +k’a +k a +Figure 8. Second set of leading diagrams with gluon production. Projectile fields U −i(z2) are +denoted by green tails while target fields V +i(z1) by red tails. +The corresponding diagrams are shown in Fig. 8. It is clear that they differ from the +diagrams in Fig. 7 by trivial projectile↔target replacements +x+ ↔ x−, +αa ↔ βb, +α′ +a ↔ β′ +b, +ka⊥ ↔ kb⊥, +k′ +a⊥ ↔ k′ +b⊥ +(6.28) +so we get +I2(αa, ka⊥, β′ +b, k′ +b⊥, x1, x2) = 8π2s2 +� +d−αd−βd−p⊥ eiαϱx− +12+iβϱx+ +12−i(p,x12)⊥ +(6.29) +× +� +β′ +b + β +(β′ +b + β)βs − (p + k′ +b)2 + iϵ +˜δ(αβs − p2 +⊥)θ(α) +α − αa +(α − αa)βs − (p − ka)2 +⊥ − iϵ ++ ˜δ[α(β′ +b + β)s − (p + k′ +b)2]θ(α) +1 +αβs − p2 +⊥ − iϵ +(α − αa)(β + β′ +b) +(α − αa)βs − (p − ka)2 +⊥ − iϵβ ++ +(α − αa)(β′ +b + β) +α(β′ +b + β)s − (p + k′ +b)2 +⊥ + iϵ(β′ +b + β) +1 +αβs − p2 +⊥ + iϵ +˜δ[β(α − αa)s − (p − ka)2 +⊥]θ(α) +� +Similarly to the previous case, after subtraction of the corresponding “projectile” eikonals +in Fig. 12a,b and “projectile” eikonals in Fig. 11e,f one can set x∥ +12 = 0 and get +I2(αa, ka⊥, β′ +b, k′ +b⊥, x1, x2) +x∥ +12=0 += +Id.log(αa, ka⊥, β′ +b, k′ +b⊥) + Irem +2 +(αa, ka⊥, β′ +b, k′ +b⊥, x12⊥) +(6.30) +from Eqs. (6.24) and (6.25) projectile↔target replacements (6.28) so that +Id.log(αa, ka⊥, β′ +b, k′ +b⊥) = ln −Q2 +ab′ +k2a⊥ +ln −Q2 +ab′ +k′2 +b⊥ ++ π2 +3 ++ O(λ) +(6.31) +(cf. Eq. (6.24)) and +Irem +2 +(αa, ka⊥, β′ +b, k′ +b⊥, x2⊥, x1⊥) +(6.32) += +− 1 +2 +� +ln −Q2 +ab′x2 +⊥ +4 ++ 2γ +�2 ++ IK(k′ +b⊥, x21⊥) + IK(−ka⊥, x21⊥) + O(λ) +where IK is given by Eq. (6.26). +– 25 – + +The final result for “production” contributions (at x+ +12 = x− +12 = 0) can be presented as +follows +W prod(x1, x2) = +N2 +c − 1 +Nc +8π2� +V −i,a(x2)⟨F +,a +i +(x2)F −,b +j (x1)⟩Fig.7 +A +U +j,b(x1) ++ U +i,a(x2)⟨F −,a +i +(x2)F +,b +j (x1)⟩Fig.8 +A +V −j,b(x1) +� +x∥ +12=0 += +� +d−αad−ka⊥d−βbd−kb⊥d−α′ +ad−k′ +a⊥d−βbd−k′ +b⊥e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 +× e−i(ka+ka,x1)⊥−i(k′ +a+k′ +b,x2)⊥ U +,b +i (α′ +a, k′a⊥)V −i,a(β′ +b, kb⊥)U +,b +j (αa, ka⊥)V −j,a(βb, kb⊥) +× Iprod(αa, ka⊥, βb, kb⊥, α′ +a, k′ +a⊥, β′ +b, k′ +b⊥) + O(λ) +(6.33) +where +Iprod(αa, ka⊥, βb, kb⊥, α′ +a, k′ +a⊥, β′ +b, k′ +b⊥) +(6.34) += Id.log(α′ +a, k′ +a⊥, βb, kb⊥) + Irem +1 +(α′ +a, k′a⊥, βb, kb⊥, x21⊥) ++ Id.log(αa, ka⊥, β′ +b, k′ +b⊥) + Irem +2 +(αa, ka⊥, β′ +b, k′ +b⊥, x21⊥) +where Irem +1 +and Irem +2 +are given by Eqs. (6.25) and (6.32), respectively. +6.3 +Handbag diagrams +Let us start with the third term in the r.h.s. of Eq. (6.1) given by “target handbag” diagrams +in Fig. 9. We need to subtract from these diagrams the corresponding diagrams coming +(a) +(b) +(c) +x 2 +x 1 +Figure 9. “Target” handbag diagrams. +from “target” TMD eikonals in Fig. 9. As was mentioned above (see Appendix 11.2 for +details), we use “point-splitting” regularization of integrals over α in these contributions. +The subtracted “eikonal” diagrams then look the same as those in Fig. 9 with the only +difference in points where gluon fields V −i,a, F +,a +i +, F −,b +j , and U +j,b are located. +– 26 – + +(a) +(b) +(c) +x 1 +x2 +x 2 +x 1 +t +t +Figure 10. “Target eikonal” handbag diagrams. Here xt +1 = x1⊥ + x+ +1 , x′ +1 = xt +1 + δ+ and similarly +for x2. +Using Wightman “cut” propagator (11.5) in the background field one easily obtains +⟨g2 ˜F −i,a(x2)F −j,b(x1)⟩ = +− 4(x2| +pi +p2 + iϵp0 +V − +ξ ˜δ+(p)V −ξ +pj +p2 − iϵp0 +− pi˜δ+(p)V − +ξ +1 +p2 − iϵp0 +V −ξ +pj +p2 − iϵp0 +− +pi +p2 + iϵp0 +V − +ξ +1 +p2 + iϵp0 +V −ξpj˜δ+(p)|x1)abg2 += +− 2g2 +� +d−βbd−β′ +bd−kb⊥d−k′ +b⊥e−ikax1−ik′ +bx2V −,ac +k +(β′ +b, k′ +b⊥)V −k,cb(βb, kb⊥) +� +d−p⊥ +× +� ∞ +0 +d−α +α +� (p + k′ +b)i(p − kb)j +� +e−iβ′ +bϱx+ +12+i +(p+k′ +b)2 +⊥ +αs +ϱx+ +12 − ei +p2 +⊥ +αs ϱx+ +12 +� +eiαϱx− +12−i(p,x12)⊥ +[αβ′ +bs − (p + k′ +b)2 +⊥ + p2 +⊥ + iϵ][α(β′ +b + βb)s − (p + k′ +b)2 +⊥ + (p − kb)2 +⊥ + iϵ] ++ +(p + k′ +b)i(p − kb)j +� +eiβbϱx+ +12+i +(p−kb)2 +⊥ +αs +ϱx+ +12 − ei +p2 +⊥ +αs ϱx+ +12 +� +eiαϱx− +12−i(p,x12)⊥ +[α(β′ +b + βb)s + (p − kb)2 +⊥ − (p + k′ +b)2 +⊥ + iϵ][αβbs + (p − kb)2 +⊥ − p2 +⊥ + iϵ] +� +(6.35) +so we get (see the definition (4.20)) +W handbag(x1, x2) − W handbag +eik +(x1, x2) = 8π2 +� +d−α′ +ad−k′a⊥d−β′ +bd−kb⊥d−αad−k′a⊥d−βbd−k′b⊥ +× e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 e−i(ka+ka,x2)⊥−i(k′ +a+k′ +b,x1)⊥U +,b +i (α′ +a, k′a⊥) +(6.36) +× V −i,a(β′ +b, kb⊥)U +,b +j (αa, p′ +A⊥)V −j,a(βb, k′ +b⊥)Ih1(βb, β′ +b, kb⊥, k′ +b⊥, x1, x2) +where +Ih1(β′ +b, kb⊥, βb, k′ +b⊥, x1, x2) = +− 2 +� +d−p⊥ e−i(p,x12)⊥ +� ∞ +0 +d−α +α +� +eiαϱx− +12 − 1 +� +× +� +(p + k′ +b)i(p − kb)j +� +e−iβ′ +bϱx+ +12+i +(p+k′ +b)2 +⊥ +αs +ϱx+ +12 − ei +p2 +⊥ +αs ϱx+ +12 +� +[αβ′ +bs − (p + k′ +b)2 +⊥ + p2 +⊥ + iϵ][α(β′ +b + βb)s − (p + k′ +b)2 +⊥ + (p − kb)2 +⊥ + iϵ] ++ +(p + k′ +b)i(p − kb)j +� +eiβbϱx+ +12+i +(p−kb)2 +⊥ +αs +ϱx+ +12 − ei +p2 +⊥ +αs ϱx+ +12 +� +[α(β′ +b + βb)s + (p − kb)2 +⊥ − (p + k′ +b)2 +⊥ + iϵ][αβbs + (p − kb)2 +⊥ − p2 +⊥ + iϵ] +� +(6.37) +– 27 – + +The term “-1” in the parentheses in the first line comes from subtraction of “eikonal” di- +agrams in Fig. 10. Note that there is no need for the additional cutoff for α integrals in +those diagrams since the integral (6.37) is convergent. +Now, in is easy to see that the integral over p⊥ is convergent so the characteristic +p⊥ ∼ Q⊥ ∼ x−1 +12⊥. Moreover, the integral over α is convergent at α ∼ +1 +ϱx− +12 ∼ α′ +a which +means that even at β′ +b + βb = 0 the integral in the Eq. (6.37) is ∼ Q2 +⊥ +Q2 which is a power +correction. Similarly, the fourth term in Eq. (6.1) is given by the same set of diagrams with +projectile↔target reflection so after subtractions of “projectile eikonal” handbag diagrams +of Fig. 12 g-i it becomes a power correction. +7 +Result for the sum of diagrams in Figs. 5,6,7,8 minus TMD matrix +elements in Figs. 11,12 +Assembling Eqs. (5.27), (5.28), (6.33), (6.34) and subtracting “eikonal” TMD matrix ele- +ments given by Eq. (11.16), we get +W (x1, x2) − W eik(x1, x2) = +N2 +c − 1 +Nc +8π2g2 +× +� +V −i,a(x2)⟨F +a +i (x2)F −j,b(x1)⟩Fig.7 +A +U +b +j (x1) + U +i,a(x2)⟨F −,a +i +(x2)F +,b +j (x1)⟩Fig.8 +A +V −j,b(x1) ++ V −i,a(x2)U +a +i (x2)⟨F −j,b(x1)F +b +j (x1)⟩Fig.5 +A ++ ⟨F −i,a(x2)F +a +i (x2)⟩Fig.6 +A +V −j,b(x1)U +b +j (x1) +− U +a +i (x2)V −i,n(x2)⟨[x+ +2 , −∞]na +x2⊥+δ−[−∞, x+ +1 ]bc +x1⊥+δ−F −j,c(x+ +1 , x1⊥)⟩Fig. 11a−c +A +U +j,b(x1) +− U +a +i (x2)⟨F −i,n(x+ +2 , x2⊥)[x+ +2 , −∞]na +x2⊥+δ−[−∞, x+ +1 ]bc +x1⊥+δ−⟩Fig. 11d−f +A +V −j,c(x1)U +j,b(x1) +− U +n +i (x2)V −i,a(x2)⟨[x− +2 , −∞]na +x2⊥+δ+[−∞, x− +1 ]bc +x1⊥+δ+F +j,c(x− +1 , x1⊥)⟩Fig. 12a−c +A +V −j,c(x1) +− V −i,a(x2)⟨F +i,n(x− +2 , x2⊥)[x− +2 , −∞]na +x2⊥+δ+[−∞, x− +1 ]bc +x1⊥+δ+⟩Fig. 12d−f +A +V −j,c(x1)U +b +j (x1) +� += +� +d−α′ +ad−k′a⊥d−β′ +bd−kb⊥d−αad−k′a⊥d−βbd−k′b⊥e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 +× e−i(ka+ka,x2)⊥−i(k′ +a+k′ +b,x1)⊥U +,b +i (α′ +a, k′a⊥)V −i,a(β′ +b, k′ +b⊥)U +,b +j (αa, ka⊥)V −j,a(βb, kb⊥) +× g2[I − Iσp,σt +eik +](αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥, x1, x2) +(7.1) +– 28 – + +with +[I − Iσp,σt +eik +](αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥, x1, x2) += +− Id.log(αa, βb, ka⊥, kb⊥) − Id.log(α′ +a, β′ +b, k′ +a⊥, k′ +b⊥) + Id.log(α′ +a, βb, k′ +a⊥, kb⊥) ++ Id.log(αa, β′ +b, ka⊥, k′ +b⊥) + Irem +1 +(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) + Irem +2 +(αa, ka⊥, β′ +b, k′ +b⊥, x12⊥) +− Iσp,σt +eik +(αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥, x12⊥) += +− ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ +− ln −Q2 +a′b′ +k′2 +a⊥ +ln −Q2 +a′b′ +k′2 +b⊥ ++ ln −Q2 +a′b +k′2 +a⊥ +ln −Q2 +a′b +k2 +b⊥ ++ ln −Q2 +ab′ +k2a⊥ +ln −Q2 +ab′ +k′2 +b⊥ +− 1 +2 +� +ln −Q2 +a′bx2 +⊥ +4 ++ 2γ +�2 +− 1 +2 +� +ln −Q2 +ab′x2 +⊥ +4 ++ 2γ +�2 ++ 1 +2 ln2 � +− i +4(α′ +a + iϵ)σpsx2 +⊥eγ� ++ 1 +2 ln2 � +− i +4(αa + iϵ)σpsx2 +⊥eγ� ++ 1 +2 ln2 � +− i +4(βb + iϵ)σtsx2 +⊥eγ� ++ 1 +2 ln2 � +− i +4(βb + iϵ)σtsx2 +⊥eγ� ++ π2 +(7.2) +Note that the contribution proportional to integral (6.26) canceled. After some algebra this +result can be represented as +[I − Iσp,σt +eik +](α′ +a, αa, β′ +b, βb, k′a⊥, k′a⊥, kb⊥, k′ +b⊥, x2, x1) += +− ln (−iα′ +a)k′2 +a⊥ +(−iαa)k′2 +a⊥ +ln (−iβ′ +b)k′2 +b⊥ +(−iβb)k2 +b⊥ ++ ln2 x2 +12⊥sσpσt +4 +− ln (−iα′ +a)eγ +σt +ln (−iβ′ +b)eγ +σp +− ln (−iαa)eγ +σt +ln (−iβb)eγ +σp ++ π2 +(7.3) +However, this formula is not the final result for the coefficient function (4.2) since the +integrals Ivirt get contributions form soft/Glauber gluons (sG-gluons) which need to be +subtracted. Indeed, the coefficient function (4.2) was defined as a result of integration over +C-fields with α > σt and β > σp. Since we did not impose these restrictions while calculating +the loop integrals like Eq. +(5.9) and Eq. +(6.8), we need to subtract α < σt, β < σp +contributions to these integrals. This will be done in the next Section. +8 +Subtraction of soft/Glauber contributions +As we mentioned above, the coefficient function C1 in Eq. (4.1) was defined as an integral +over large |α| > σt and |β| > σp so the contributions to our background-field diagrams +with |α| < σt and/or |β| < σp should be subtracted from the result (7.3). +We have +already subtracted the TMD matrix elements: “target eikonals” with |α| < σt and “projectile +eikonals” with |β| < σp. I. Sect. 8.4 below we prove that sG-contributions to TMD matrix +elements are power corrections so there is no double counting. Still, we need to subtract +sG-contributions from W (x1, x2) itself. +As for the case of subtractions of eikonal TMD matrix elements, we use smooth α and +β cutoffs which do not change the analytical properties of the diagrams. Let us again start +with virtual contributions. +– 29 – + +8.1 +sG-contributions to virtual diagrams +The virtual contribution of diagrams in Fig. 5 is given by Eqs. (5.16) and (5.17). Let us +now calculate contribution of sG-gluons to virtual diagram. Integral (5.17) with “smooth” +α and β restrictions has the form +Ivirt sG +Fig.5 += +− 16π2 +� d−4p +i +� +sαaβb +[(p + ka)2 + iϵ](p2 + iϵ)[(p − kb)2 + iϵ] ++ ˜δ−(p + ka) sαaβb +p2 − iϵ +˜δ+(p − kb) +� +e +−i α +σt +i β +σp +(8.1) +As we will see below, the above choice of signs of the exponential cutoffs agrees with +analytical properties of the original uncut diagram, namely that the background fields are +emitted before the point x1, see Eq. (5.18). +Let us perform the calculation for the most complicated case αa, βb < 0 where we need +both terms in the r.h.s. of Eq. (8.1). +Ivirt sG +Fig.5 += +− 16π2 +� d−4p +i e +−i α +σt +i β +σp +� +αaβbs[α(β − βb)s − (p − kb)2 +⊥ + iϵ]−1 +[(α + αa)βs − (p + ka)2 +⊥ + iϵ](αβs − p2 +⊥ + iϵ) ++ ˜δ[(αa + α)βs − (p + ka)2]θ(−β) +αaβbs +αβs − p2 +⊥ − iϵ +˜δ[α(β − βb)s − (p − kb)2 +⊥]θ(α) +� +≃ 16π2 +� d−p +i +� +αa +αaβs − (p + ka)2 +⊥ + iϵ +s +αβs − p2 +⊥ + iϵ +βb +αβbs + (p − kb)2 +⊥ − iϵ +− ˜δ[αaβs − (p + ka)2 +⊥]θ(−β) +αaβbs +αβs − p2 +⊥ − iϵ +˜δ[αβbs + (p − kb)2 +⊥]θ(α) +� +e +−i α +σt +i β +σp +(8.2) +Here we neglected α ∼ σt in comparison to αa and β ∼ σp in comparison to βb. +Next, we take residue over α and obtain +Eq. (8.2) +βb<0 += +− 8π2 +� +d−2p +� +d−β +� +αaβbsθ(β) +p2 +⊥βb + (p − kb)2 +⊥β − iϵ +e +−i +p2 +⊥ +βσts +i β +σp +αaβs − (p + ka)2 +⊥ + iϵ +− +αaβbs +p2 +⊥βb + (p − kb)2 +⊥β − iϵ +θ(αa) +αaβs − (p + ka)2 +⊥ + iϵe +i +(p−kb)2 +⊥ +βbσts +i β +σp +−2πiθ(−αa)|αaβb|s ˜δ +� +(p − kb)2 +⊥β − p2 +⊥|βb| +� +e +−i +p2 +⊥ +βσts +i β +σp +αaβs − (p + ka)2 +⊥ + iϵ +� += +− 8π2 +� +d−2p +� +d−β +−αa|βb|sθ(β) +−p2 +⊥|βb| + (p − kb)2 +⊥β + iϵ +1 +αaβs − (p + ka)2 +⊥ + iϵe +−i +p2 +⊥ +βσts +i β +σp +β=v2|βb| += +4π +� +d−2p +� ∞ +0 +dv2 +αa|βb|s +p2 +⊥ − (p − kb)2 +⊥v2 − iϵ +e +−i +p2 +⊥ +v2|βb|σts +iv2 |βb| +σp +(p + ka)2 +⊥ − αa|βb|sv2 − iϵ +p⊥=k⊥v += +4π +� +d−2k +� ∞ +0 +dv2 +αa|βb|s +k2 +⊥ − (kb − kv)2 +⊥ − iϵ +e +−i +k2 +⊥ +|βb|σts +iv2 |βb| +σp +(ka + kv)2 +⊥ − αa|βb|sv2 − iϵ +(8.3) +– 30 – + +This integral can be rescaled by change l2 +⊥ = +k2 +⊥ +|βb|σts and t2 = v2 |βb| +σp as follows +−4π +� +d−2l⊥ +� ∞ +0 +dt2 +e−il2 +l2 +⊥ − (kb−ltµσ)2 +⊥ +|βb|σts +− iϵ +eit2 +t2 − (ka+ltµσ)2 +⊥ +αaσps ++ iϵ +(8.4) +where µσ ≡ √σpσts. As we assumed, µσ ≪ q⊥ (see Eq. (3.5)) so one can neglect ltµσ in +the denominators and get +Eq. (8.2) +βb<0 += +− +� +ln −i(αa + iϵ)σps +k2a⊥ +− γ +�� +ln i|βb|σts +k2 +b⊥ +− γ +� +(8.5) +where we used integral +� ∞ +0 +dx +e−x +x + a = ln 1 +a − γ + O(a) +(8.6) +Performing similar calculation at βb > 0 we get sG-contribution to the virtual diagram +in the form +Ivirt sG +Fig.5 += +− +� +ln −i(αa + iϵ)σps +k2a⊥ +− γ +�� +ln −i(βb + iϵ)σts +k2 +b⊥ +− γ +� +(8.7) +Note that this double-log contribution comes from the region 1 ≫ l2 ≫ +k2 +b⊥ +σt|β′ +b|s and 1 ≫ +t2 ≫ +k′2 +a⊥ +σp|α′a|s in the integral (8.4) which corresponds to the region +σp ≫ β ≫ k′2 +a⊥ +|α′a|s, +σt ≫ α ≫ +k2 +b⊥ +|β′ +b|s, +σpσts ≫ p2 +⊥ ≫ +k′2 +a⊥k2 +b⊥ +|α′aβ′ +b|s +(8.8) +in the original integral (8.3). +The result for Fig. 6 integral (5.26) is obtained from Fig. 5 result (5.17) by complex +conjugation and ka ↔ −k′ +a, kb ↔ −k′ +b replacement so the sG-contribution can be obtained +in a similar way +Ivirt sG +Fig.6 += 16π2 +� d−4p +i +� +sα′ +aβ′ +b +[(p + k′a)2 − iϵ](p2 − iϵ)[(p − k′ +b)2 − iϵ] ++ ˜δ+(p + k′ +a) sα′ +aβ′ +b +p2 + iϵ +˜δ−(p − k′ +b) +� +e +−i α′ +σt +i β′ +σp += +− 16π2 +� d−p +i +� +α′ +a +α′aβs − (p + k′a)2 +⊥ − iϵ +s +αβs − p2 +⊥ − iϵ +β′ +b +αβ′ +bs + (p − k′ +b)2 +⊥ + iϵ +− ˜δ[α′ +aβs − (p + k′ +a)2 +⊥]θ(β) +α′ +aβ′ +bs +αβs − p2 +⊥ + iϵ +˜δ[αβ′ +bs + (p − k′ +b)2 +⊥]θ(−α) +� +e +−i α +σt +i β +σp += +− +� +ln −i(α′ +a + iϵ)σps +k′2 +a⊥ +− γ +�� +ln −i(β′ +b + iϵ)σts +k2 +b⊥ +− γ +� +(8.9) +To get the last line, we performed complex conjugation of Eq. +(8.2), replaced ka → +−k′ +a, kb → −k′ +b, and changed the sign of p. +– 31 – + +8.2 +sG-contributions to production diagrams +The sG-contribution to the integral (6.8) has the form +IsG +1 (α′ +a, k′ +a⊥, βb, kb⊥, x12) = 8π2s2 +� +d−αd−βd−p⊥ e +−i α +σt +i β +σp −i(p,x12)⊥ +(8.10) +× +� +α′ +a +α′aβs − (p + k′a)2 + iϵ +˜δ(αβs − p2 +⊥)θ(α) +βb +αβbs + (p − kb)2 +⊥ + iϵ ++ ˜δ[α′ +aβs − (p + k′ +a)2]θ(β) +1 +αβs − p2 +⊥ − iϵ +α′ +aβb +αβbs + (p − kb)2 +⊥ − iϵβb ++ +−α′ +aβb +α′aβs − (p + k′a)2 +⊥ + iϵα′a +1 +αβs − p2 +⊥ + iϵ +˜δ[αβbs + (p − kb)2 +⊥]θ(α) +� +To get the above equation, we neglected α ∼ σt in comparison to α′ +a and β ∼ σp in +comparison to βb in the denominators in Eq. (6.8). Also, in the exponent in Eq. (6.8) we +neglected αϱx− +12 in comparison to α +σt = αϱδ− and βϱx+ +12 in comparison to +β +σp = βϱδ+. It is +easy to see that the integral over α in the second term and over β in the last term in the +above equation vanish so we get +IsG +1 (α′ +a, k′a⊥, βb, kb⊥, x2, x1) = 8π2s +� ∞ +0 +d−βd−p⊥ e +−i +p2 +⊥ +βσts +i β +σp −i(p,x12)⊥ +(8.11) +× +α′ +aβbs +[p2 +⊥βb + (p − kb)β + iϵ][α′aβs − (p + k′a)2 + iϵ] +Let us again consider βb < 0, then change of variables β = v2|βb| yields +Eq. (8.11) +βb<0 += +− 4π +� ∞ +0 +dv2 +� +d−p⊥ e +−i +p2 +⊥ +v2|βb|σts +iv2 |βb| +σp −i(p,x12)⊥ +× +α′ +a|βb|s +[p2 +⊥ − (p − kb)v2 − iϵ][(p + k′a)2 − α′a|βb|sv2 − iϵ] +(8.12) +This integral differs from the fifth line in Eq. (8.3) by extra factor e−i(p,x12)⊥. Doing the +same rescaling, we obtain +Eq. (8.11) +βb<0 += +4π +� +d−2l⊥ +� ∞ +0 +dt2 +e−it(l,x12⊥)µσ +l2 +⊥ − (k′a−ltµσ)2 +⊥ +|βb|σts +− iϵ +e−il2+it2 +t2 − (k′a+ltµσ)2 +⊥ +α′aσps ++ iϵ +(8.13) +Again, since µσ ≪ q⊥, we can neglect all ltµσ terms and get +IsG +Fig.7 = +� +ln −i(α′ +a + iϵ)σps +k′2 +a⊥ +− γ +�� +ln −i(βb + iϵ)σts +k2 +b⊥ +− γ +� +(8.14) +Similarly to the integral (6.29) itself, the sG-contribution to the integral (6.29) can be +obtained from the result for the integral (8.14) by trivial projectile↔target replacements +IsG +Fig.8 = +� +ln −i(αa + iϵ)σps +k′2 +a⊥ +− γ +�� +ln −i(β′ +b + iϵ)σts +k2 +b⊥ +− γ +� +(8.15) +– 32 – + +8.3 +The sum of sG-terms +Assembling Eqs. (8.7), (8.1), (8.14), and (8.15) we get +W (x1, x2)sG = +N2 +c − 1 +Nc +16π2g2� +V −i,a(x2)⟨F +a +i (x2)F −j,b(x1)⟩Fig.7 +A +U +b +j (x1) ++ U +i,a(x2)⟨F −,a +i +(x2)F +,b +j (x1)⟩Fig.8 +A +V −j,b(x1) ++ V −i,a(x2)U +a +i (x2)⟨F −j,b(x1)F +b +j (x1)⟩Fig.5 +A ++ ⟨F −i,a(x2)F +a +i (x2)⟩Fig.6 +A +V −j,b(x1)U +b +j (x1) +�sG += +� +d−αad−ka⊥d−βbd−kb⊥d−α′ +ad−ka⊥d−β′ +bd−k′ +b⊥e−iα′ +aϱx− +2 −iαaϱx− +1 −iβ′ +bϱx+ +2 −iβbϱx+ +1 +× e−i(ka+ka,x1)⊥−i(k′ +a+k′ +b,x2)⊥U +,b +i (α′ +a, k′a⊥)V −i,a(β′ +b, k′ +b⊥)U +,b +j (αa, ka⊥)V −j,a(βb, kb⊥) +× g2IsG(αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥) +(8.16) +where +Iσp,σt +sG +(αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥) = Ivirt sG +Fig.5 ++ Ivirt sG +Fig.6 ++ IsG +Fig.7 + IsG +Fig.8 += +− ln (−iα′ +a)k′2 +a⊥ +(−iαa)k2a⊥ +ln (−iβ′ +b)k′2 +b⊥ +(−iβb)k2 +b⊥ ++ O +� µ2 +σ +Q2 +⊥ +� +(8.17) +It is worth noting that the soft-Glauber contribution (8.17) is actually a soft contribution +with characteristic transverse momenta p2 +⊥ ∼ +k2 +a⊥k2 +b⊥ +|αaβb|s . Indeed, while the momenta in the +individual integrals for IsG are given by Eq. (8.8), the characteristic transverse momenta +in their sum (8.17) are of the order of low limit in Eq. (8.8). To see that, we rewrite the +sum (8.17) as follows +Ivirt sG +Fig.5 ++ Ivirt sG +Fig.6 ++ IsG +Fig.7 + IsG +Fig.8 = +− +� ∞ +0 +dl2 e−il2� +1 +l2 +⊥ − +k2 +b⊥ +|βb|σts − iϵ +− +1 +l2 +⊥ − +k′2 +b⊥ +|β′ +b|σts − iϵ +� +× +� ∞ +0 +dt2 eit2� +1 +t2 − +k2a⊥ +αaσps + iϵ +− +1 +t2 − +k′2 +a⊥ +α′aσps + iϵ +� ++ O +� µ2 +σ +Q2 +⊥ +� +(8.18) +It is easy to see that the integral over l2 is determined by l2 ∼ +k2 +b⊥ +|βb|σts and the integral over +t2 by t2 ∼ +k2 +a⊥ +|αa|σps which translates to +β ∼ k2 +a⊥ +|αa|s, +α ∼ +k2 +b⊥ +|βb|s, +p2 +⊥ ∼ +k2 +a⊥k2 +b⊥ +|αaβb|s +(8.19) +which is a soft contribution since Eq. (8.19) means that +p+ ∼ p− ∼ p⊥ ∼ O +� 1 +λ +� +(8.20) +in terms of rescaling (1.3). Thus, the soft-Glauber contribution (8.17) is actually a soft +contribution in accordance with general statement that contributions from Glauber gluons +cancel. +– 33 – + +Actually, the statement that the soft-Glauber contribution (8.17) is a soft contribution +can be checked independently. Let us calculate the contribution of small p⊥ ≪ ki⊥ to a +non-restricted integrals of Fig. 5. Neglecting p⊥ in comparison to ki⊥ and using dimensional +regularization for UV integrals obtained as a result of this approximation, we get instead +of Eq. (8.2) +Ivirt soft +Fig.5 += 16π2 +� d−2+2εp +i +� +αa +αaβs − k2a⊥ + iϵ +s +αβs − p2 +⊥ + iϵ +βb +αβbs + k2 +b⊥ − iϵ +− ˜δ[αaβs − k2 +a⊥]θ(−β) +αaβbs +αβs − p2 +⊥ − iϵ +˜δ[αβbs + k2 +b⊥]θ(α) +� +(8.21) +Repeating all the steps in derivation of Eq. (8.3) we get +Eq. (8.21) +βb<0 += +− 4π +� +d−2+2εk +1 +k2 +⊥ − k2 +b⊥ + iϵ +� ∞ +0 +dv2 +v2ε +v2 − +k2a⊥ +(αa+iϵ)|βb|s += +− Γ2(−ε)Γ(1 + ε) +(4π)ε +(−k2 +b⊥ + iϵ)ε� +− +k2 +a⊥ +(αa + iϵ)|βb|s +�ε +(8.22) += +− 1 +ε2 − π2 +4 − γ2 − +�1 +ε + γ +�� +ln k2 +a⊥ ln k2 +b⊥ − ln(αa + iϵ)|βb|s +� +− 1 +2 ln2 (αa + iϵ)|βbs +k2a⊥k2 +b⊥ +The similar contribution of diagrams in Fig. +6 is obtained by replacement αa ↔ α′ +a, +βb ↔ β′ +b and ka,b⊥ ↔ k′ +a,b⊥. Moreover, for soft contributions one can neglect e−ipx12 in the +“production” terms and obtain +Isoft = Eq.(8.22) − +� +αa → α′ +a, ka⊥ → k′ +a⊥ +� +− +� +βb → β′ +b, kb⊥ → k′ +b⊥ +� +(8.23) ++ +� +αa → α′ +a, βb → β′ +b, ka⊥ → k′ +a⊥, kb⊥ → k′ +b⊥ +� += +− ln (iα′ +a)k2 +a⊥ +(iαa)k′2 +a⊥ +ln (iβ′ +b)k′2 +b⊥ +(iβb)k′2 +b⊥ +which coincides with Eq. (8.17). +As we mentioned above, in Appendix 11.3 it is demonstrated that in the first perturba- +tive order such soft contributions cancel in the sum of all diagrams. Non-perturbatively, soft +contributions form wave functions of hadrons and also presumably lead to non-perturbative +power corrections to scattering amplitude ∼ x2 +12⊥Λ2 +QCD. +8.4 +sG-contributions to TMD matrix elements +In this section we demonstrate that sG-contributions to TMD matrix elements are power +corrections. Let us start with the “target eikonals” of Fig. 11 given by Eq. (11.7). A +– 34 – + +“smooth” cutoff |β| < σp is obtained by inserting an extra e +i β +σp = eiβϱδ+ in the integrand +Ieik,β<σp +Fig.11a−d(βb, kb⊥, x+ +1 , x1⊥, x+ +2 , x2⊥) = 8π2s +� ∞ +0 +d−α e +−i α +σt +i β +σp +d−β +β + iϵd−p⊥ eiβϱx+ +12−i(p,x12)⊥ +× +� +˜δ(αβs − p2 +⊥) +β − βb +α(β − βb)s − (p − kb)2 +⊥ − iϵ + +(β − βb) +αβs − p2 +⊥ + iϵ +˜δ[α(β − βb)s − (p − kb)2 +⊥] +� +− 8π2i +� +d−αd−βd−p⊥ +e +−i α +σt +i β +σp s(β − βb) +(β + iϵ)(αβs − p2 +⊥ + iϵ)[α(β − βb)s − (p − kb)2 +⊥ + iϵ] += 8π2 +� ∞ +0 +d−α e−i α +σt +� +d−p⊥ +�βbs +p2 +⊥ +ei +p2 +⊥ +αs ϱx+ +12−i(p,x12)⊥ − 1 +αβbs + (p − kb)2 +⊥ + iϵei +p2 +⊥ +αs ϱδ+ ++ (p − kb)2 +⊥ei(p,x12)⊥� +ei(βb+ +(p−kb)2 +⊥ +αs +)ϱ(x+ +12+δ+) − ei +p2 +⊥ +αs ϱ(x+ +12+δ+)� +α[αβbs + (p − kb)2 +⊥ + iϵ][αβbs + (p − kb)2 +⊥ − p2 +⊥] +� +(8.24) +Since x+ +12 ≪ δ+ we can neglect x+ +12 and get +Ieik +Fig. 11a−d(βb, kb⊥, x12⊥) = 8π2 +� ∞ +0 +d−α +� d−p⊥ +p2 +⊥ +βbse +−i α +σt +i +p2 +⊥ +ασps � +e−i(p,x12)⊥ − 1 +� +αβbs + (p − kb)2 +⊥ + iϵ +α=v2αa += +4π +� ∞ +0 +dv2 +� d−p⊥ +p2 +⊥ +αaβbse +−i αa +σt v2+i +p2 +⊥ +v2αaσps � +e−i(p,x12)⊥ − 1 +� +v2αaβbs + (p − kb)2 +⊥ + iϵ +p⊥=k⊥v += +4π +� ∞ +0 +dv2 +� d−2k +k2 +⊥ +� +e−iv(k,x12)⊥ − 1 +� +αaβbse +−i αa +σt v2+i +k2 +⊥ +αaσps +v2αaβbs + (k⊥v − kb)2 +⊥ + iϵ +(8.25) +Performing the same rescaling as in Eq. (8.4) we get +Ieik +Fig. 11a−d(βb, kb⊥, x12⊥) = 4π +� ∞ +0 +dt2 +� d2l⊥ +l2 +⊥ +� +e−itµσ(l,x12)⊥ − 1 +� +e−it2+il2 +t2 + (kb−µσtl)2 +⊥ +σtβbs +≃ 4π +� ∞ +0 +dt2 +e−it2 +t2 + +k2 +b⊥ +σtβbs +� d2l⊥ +l2 +⊥ +� +e−itµσ(l,x12)⊥ − 1 +� +eil2 +⊥ +(8.26) +where we again used µσ ≪ q⊥ ∼ kb⊥. The first integral in the r.h.s. is ln −iβbσts +k2 +b⊥ +−γ (see Eq. +(8.6) but the second is obviously O(µ2 +σx2 +12⊥) ∼ O +� µ2 +σ +q2 +⊥ +� +so the sG-contribution to “target” +eikonal TMD matrix elements is a power correction. Similarly, one can demonstrate that +the sG-contribution to “proijectile” eikonal TMD matrix elements of Fig. 11 is a power +correction O +� µ2 +σ +q2 +⊥ +� +. Thus, with power accuracy there is no double counting and we should +subtract from the amplitude (7.3) only the sG-contributions (8.17). +– 35 – + +9 +Result for the coefficient function +According to Eq. (4.2), the coefficient function is given by the functional integral over +central fields with α > σt, β > σp minus the eikonal contributions. It is determined by +W (x1, x2) − W sG(x1, x2) − W eik(x1, x2) +(9.1) += +� +d−αad−ka⊥d−βbd−kb⊥d−α′ +ad−k′a⊥d−β′ +bd−k′b⊥ +× e−iαaϱx− +1 −iα′ +aϱx− +2 e−iβbϱx+ +1 −iβ′ +bϱx+ +2 e−i(ka+kb,x1)⊥−i(k′ +a+k′ +b,x2)⊥U +,b +i (αa, ka⊥) +× V −i,a(βb, kb⊥)U +,b +j (α′ +a, k′ +a⊥)V −j,a(β′ +b, k′ +b⊥)C1(αa, α′ +a, βb, β′ +b, x12⊥; σp, σt) += +� +d−αad−βbd−α′ +ad−β′ +bd−k′b⊥ e−iαaϱx− +1 −iα′ +aϱx− +2 e−iβbϱx+ +1 −iβ′ +bϱx+ +2 U +,b +i (αa, x1⊥) +× V −i,a(βb, x1⊥)U +,b +j (α′ +a, x2⊥)V −j,a(β′ +b, x2⊥)C1(αa, α′ +a, βb, β′ +b, x12⊥; σp, σt) +where the coefficient function in the momentum representation is +C1(αa, α′ +a, βb, β′ +b, x12⊥; σp, σt) = I − Iσp,σt +eik +− Iσp,σt +sG +The explicit form of C1 is easily found from Eq. (7.3) and Eq. (8.17) +C1(αa, α′ +a, βb, β′ +b, x12⊥; σp, σt) = ln2 x2 +12⊥sσpσt +4 +(9.2) +− ln (−iαa + ϵ)eγ +σt +ln (−iβb + ϵ)eγ +σp +− ln (−iα′ +a + ϵ)eγ +σt +ln (−iβ′ +b + ϵ)eγ +σp ++ π2 + O(λp, λt) +This formula is the main technical result of the paper. +The very important property of the coefficient function C1 is that the r.h.s. +(9.2) +does not actually depend on transverse momenta so all the dynamics at the one-loop level +proceeds in the longitudinal direction. +This fact can be used to check the algebra and +approximations leading to the result (9.2). In the Appendix 11.7 the coefficient function is +calculated using the on-shell background fields with zero transverse momenta +U +i(z−) = +� +d−αa U +i(αa) e−iϱαaz− +⇔ +U +i(αa) = ϱ +� +dz−dz⊥ U +i(z−) eiϱαaz−, +V −i(z+) = +� +d−βb V −i(βb) e−iϱβbz+ +⇔ +V −i(βb) = ϱ +� +dz+dz⊥ V −i(z+) eiϱβbz+ +(9.3) +and the result (9.2) is confirmed. +In the coordinate space our result reads +⟨ ˆW(x1, x2)⟩A = ⟨ ˆOσp +ij (x− +2 , x2⊥; x− +1 , x1⊥) ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥)⟩A ++ +� +dz− +2 dz− +1 dw+ +1 dw+ +2 +αsNc +2π C1(x2⊥, x1⊥; z− +2 , z− +1 , z+ +2 , z+ +1 ; σp, σt) +× U +,b +i (z− +2 , x2⊥)V −i,a(z+ +2 , x2⊥)U +,b +j (z− +1 , x1⊥)V −j,a(z+ +1 , x1⊥) + ... +(9.4) +– 36 – + +where +C1(x2⊥, x1⊥; z− +2 , z− +1 , z+ +2 , z+ +1 ; σp, σt) +(9.5) += +� +ln2 2δ+δ− +x2 +12⊥ ++ π2� +δ(x2 − z2)−δ(x1 − z1)−δ(x2 − z2)+δ(x1 − z1)+ +− δ(x1 − z1)−δ(x1 − z1)+ +× +�θ(x2 − z2)− +(x2 − z2)− − δ(x2 − z2)− +� δ− +0 +dz− +2 +z− +2 +��θ(x2 − z2)+ +(x2 − z2)+ − δ(x2 − z2)+ +� δ+ +0 +dz+ +2 +z+ +2 +� +− δ(x2 − z2)−δ(x2 − z2)+ +× +�θ(x1 − z1)− +(x1 − z1)− − δ(x1 − z1)− +� δ− +0 +dz− +1 +z− +1 +��θ(x1 − z1)+ +(x1 − w2)+ − δ(x1 − z1)+ +� δ+ +0 +dz+ +1 +z+ +1 +� +Let us check matching of the cutoffs, namely that the r.h.s. of Eq. (9.4) does not +depend on σp and σt. We start with σt d +dσt = −δ− d +δ− . Since +−δ− d +δ− C1(x2⊥, x1⊥; z− +i , zi⊥, z+ +i , zi⊥; σp, σt) +(9.6) += +− δ(x2 − z2)−δ(x1 − z1)−� +2 ln 2δ+δ− +x2 +12⊥ +δ(x2 − z2)+δ(x1 − z1)+ ++ δ(x1 − z1)+�θ(x2 − z2)+ +(x2 − z2)+ − δ(x2 − z2)+ +� δ+ +0 +dz+ +2 +z+ +2 +� ++ δ(x2 − z2)+�θ(x1 − z1)+ +(x1 − z1)+ − δ(x1 − z1)+ +� δ+ +0 +dz+ +1 +z+ +1 +�� += δ(x2 − z2)−δ(x1 − z1)−� +2 ln sx2 +12⊥ +4 +δ(x2 − z2)+δ(x1 − z1)+ +− δ(x1 − z1)+�θ(x2 − z2)+ +(x2 − z2)+ − δ(x2 − z2)+ +� +2 +sδ− +0 +dz+ +2 +z+ +2 +� +− δ(x2 − z2)+�θ(x1 − z1)+ +(x1 − z1)+ − δ(x1 − z1)+ +� +2 +sδ− +0 +dz+ +1 +z+ +1 +�� +we get +σt +d +dσt +� +r.h.s. of Eq. (9.4) +� += U +,b +i (x− +2 , x2⊥)U +,b +j (x− +1 , x1⊥) +(9.7) +× +� +σt +d +dσt +⟨ ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥)⟩B + αsNc +2π +� +2 ln sx2 +12⊥ +4 +V −i,a(x+ +2 , x2⊥)V −j,a(x+ +1 , x1⊥) +− +� +dz+ +2 +�θ(x2 − z2)+ +(x2 − z2)+ − δ(x2 − z2)+ +� σt/ϱ +0 +dz+ +2 +z+ +2 +� +V −,b +i (z+ +2 , x2⊥)V −,b +j (x+ +1 , x1⊥) +− +� +dz+ +1 +�θ(x1 − z1)+ +(x1 − z1)+ − δ(x1 − z1)+ +� σt/ϱ +0 +dz+ +1 +z+ +1 +� +V −,b +i (x+ +2 , x2⊥)V −,b +j (z+ +1 , x1⊥) +�� += 0 +due to Eq. (11.21). Similarly, the r.h.s. of Eq. (9.4) does not depend on σp cutoff. +It is possible to represent our result for the coefficient function as evolution equations +with respect to σt and σp. Since the differentiation over σt is represented by the integration +– 37 – + +operator in the coordinate space and by simple multiplication in the momentum space, we +will use the latter. We define the Fourier transform of the operator ˆW(x1, x2) in Eq. (2.3) +as follows +ˆW(α′ +a, αa, β′ +b, βb, x1⊥, x2⊥) +(9.8) += ρ4 +� +dx− +2 dx− +1 dx+ +2 dx+ +1 eiα′ +aϱx− +2 +iα′ +aϱx− +2 +iβ′ +bϱx+ +2 +iβ′ +bϱx+ +2 ˆW(pA, p′ +A, pB, p′ +B; x1⊥, x2⊥) +The general TMD factorization formula (2.7) for ˆW(α′ +a, αa, β′ +b, βb, x1⊥, x2⊥) can be written +as +ˆW(α′ +a, αa, β′ +b, βb, x1⊥, x2⊥) = +� +d−α′ +ad−αad−β′ +bd−βb C(x1⊥, x2⊥; α′ +a, αa, β′ +b, βb; σp, σt) +× ˆOσp +ij (α′ +a, αa, x2⊥, x1⊥) ˆOij;σt(β′ +b, βb, x2⊥, x1⊥) + ... +(9.9) +where +ˆOσp +ij (α′ +a, αa, x2⊥, x1⊥) ≡ ρ2 +� +dx− +2 dx− +1 eiα′ +aϱx− +2 +iαaϱx− +1 ˆOσp +ij (x− +2 , x2⊥; x− +1 , x1⊥) +ˆOσt +ij (β′ +b, βb, x2⊥, x1⊥) ≡ ρ2 +� +dx+ +2 dx+ +1 eiβ′ +bϱx+ +2 +iβbϱx+ +1 ˆOσt +ij (x+ +2 , x2⊥; x+ +1 , x1⊥) +(9.10) +Here we took into account the absence of dynamics in the transverse space (and tacitly +assumed that such property survives in higher orders of perturbation theory). Since the +evolution equations for TMD operators in the momentum space are given by Eqs. (11.19) +and (11.20), the coefficient function should satisfy matching evolution equations +σt +d +dσt +C(x1⊥, x2⊥; α′ +a, αa, β′ +b, βb; σp, σt) = αsNc +2π +� +2 ln sx2 +12⊥ +4 +(9.11) ++ ln(−iβ′ +bσt + ϵ) + ln(−iβbσt + ϵ) + 2γ +� +C(x1, x2; α′ +a, αa, β′ +b, βb; σp, σt) +σp +d +dσp +C(x1⊥, x2⊥; α′ +a, αa, β′ +b, βb; σp, σt) = αsNc +2π +� +2 ln sx2 +12⊥ +4 ++ ln(−iα′ +aσp + ϵ) + ln(−iαaσp + ϵ) + 2γ +� +C(x1⊥, x2⊥; α′ +a, αa, β′ +b, βb; σp, σt) +The solution of this equations compatible with first-order result (9.2) is +C(x1⊥, x2⊥; α′ +a, αa, β′ +b, βb; σp, σt) = e +αsNc +2π +C1(x12⊥,α′ +a,αa,β′ +b,βb;σp,σt) ++ O +� +α2 +s × +� +ln α′ +a +σt +ln β′ +b +σp +, ln α′ +a +σt +, ln β′ +b +σp +, const +�� +(9.12) +and the final form of “double operator expansion” reads +ˆW(α′ +a, αa, β′ +b, βb, x1⊥, x2⊥) = +� +d−α′ +ad−αad−β′ +bd−βb e +αsNc +2π +C1(x12⊥,α′ +a,αa,β′ +b,βb;σp,σt) +× ˆOσp +ij (α′ +a, αa, x2⊥, x1⊥) ˆOij;σt(β′ +b, βb, x2⊥, x1⊥) + ... +(9.13) +In the next Section we will consider matrix elements of the operator equation (9.13) between +initial and final protons’ states and demonstrate that +⟨p′ +A, p′ +B| ˆOσp +ij ˆOij;σt|pA, pB⟩ = ⟨p′ +A| ˆOσp +ij |pA⟩⟨p′ +B| ˆOij;σt|pB⟩ +(9.14) +To prove the above equation, we need to check that the contribution of sG-gluons cancel +up to power corrections terms. +– 38 – + +9.1 +Factorization of integral over A ∪ B fields +9.1.1 +Cancellation of soft and Glauber gluons +The functional integral form of our result for hadronic tensor (2.4) reads +1 +16(N2 +c − 1)ρ4 +� +dx− +1 dx− +2 dx+ +1 dx+ +2 eiαaϱx− +1 +iα′ +aϱx− +2 +iβbϱx+ +1 +iβ′ +bϱx+ +2 +(9.15) +× ⟨p′ +A, p′ +B|g2F a +µνF aµν(x2)g2F b +λρF bλρ(x1)|pA, pB⟩ = e +αsNc +2π +C1(x12⊥,αa,α′ +a,βb,β′ +b;σp,σt) +× +� +DΦA Ψ∗ +p′ +A(ti)ΨpA(ti)Ψ∗ +p′ +B(ti)ΨpB(ti) ˆOσp +ij (α′ +a, αa, x12⊥) ˆOij;σt(β′ +b, βb, x12⊥) + ... +where the TMD operators ˆOσp +ij (α′ +a, αa, x12⊥) and ˆOσt +ij (β′ +b, βb, x12⊥) are made of A and B +fields, respectively. However, as we mentioned above (see Fig. 3), there are C = A ∩ B +fields with both α < σt and β < σp so to get the desired factorization (2.7) we need +to discuss the interactions between A and B fields. +The integrals over A and B fields +give matrix elements of TMD operators (2.8) between projectile and target fields while +the integral over C fields cancels due to unitarity with power corrections accuracy. To +understand this, let us discuss the C fields which are defined as gluons (and, in principle, +quarks) with both |α| < σt and |β| < σp, see Fig. 3. As we mentioned above, depending on +the scale of characteristic transverse momenta they may be Glauber gluons with p⊥ ∼ q⊥ +or soft gluons with p⊥ ≪ q⊥). +Let us first consider Glauber gluons. As discussed in Refs. [6, 7], the integration over +Glauber gluons leads to the high-energy effective action built from Wilson lines made of A +and B fields: +� +DΦA += +� +DΦADΦB e−iSeff( ˜U,˜V)+iSeff(U,V), +Uσt(z⊥) = [−∞−, ∞−]σt +z , +˜Uσt(z⊥) = {−∞−, ∞−}σt +z +Vσp(z⊥) = [−∞+, ∞+]σp +z , +˜Vσp(z⊥) = {−∞+, ∞+}σp +z +(9.16) +where +� +DΦA ≡ +� +˜ +A(tf)=A(tf) +D ˜AµDAµ +� +˜ψA(tf)=ψA(tf) +DψAD ˜ψA e−iSQCD( ˜ +A, ˜ψA)+iSQCD(A,ψA) +(9.17) +and similarly for +� +DΦB. In Ref. [6] it is demonstrated that the two terms in the exponent +in the first line of Eq. (9.16) cancel if we have a sum over full set of intermediate states. 10 +As to soft gluons, it is argued in Refs. [3, 19, 20] that they form the correlation function +of four Wilson lines going form points x1 and x2 in the light-like directions so the result of +10The explicit form of the effective action is known only in the first two orders of expansion in powers of +g2 ln σpσt [16–18], but for the cancellation we need only the fact that the effective action is made of Wilson +lines (9.16), see the discussion in Ref. [6]. +– 39 – + +the integration over soft gluons in Eq. (9.15) is +� +DΦA Ψ∗ +p′ +A(ti)ΨpA(ti)Ψ∗ +p′ +B(ti)ΨpB(ti) ˆOA +ij (x+ +2 , x2⊥; x+ +1 , x1⊥) ˆOA +ij (x− +2 , x2⊥; x− +1 , x1⊥) += S(x2, x1; σp, σt) +� +DΦA Ψ∗ +p′ +A(ti)ΨpA(ti) ˆOA +ij(x+ +2 , x2⊥; x+ +1 , x1⊥) +(9.18) +× +� +DΦBΨ∗ +p′ +B(ti)ΨpB(ti) ˆOB +ij(x− +2 , x2⊥; x− +1 , x1⊥) += S(x2, x1; σp, σt)⟨p′ +A| ˆOij(x+ +2 , x2⊥; x+ +1 , x1⊥)|pA⟩⟨p′ +B| ˆOij(x− +2 , x2⊥; x− +1 , x1⊥)|pB⟩ +where +S(x2, x1; σp, σt) = +1 +N2c − 1Tr +� +DC {−∞+, x+ +2 }x2{x+ +2 , −∞+}x2[−∞+, z+ +1 ]z1[z+ +1 , −∞+]z1 +(9.19) +is the soft factor. Here DC is defined as in Eq. (9.17) with A → C replacement. As we +will see in Sect. 11.3, with the rapidity-only cutoff the dependence of the soft factor on σp +and σt gives power corrections ∼ Q2 +⊥ +σps and/or Q2 +⊥ +σts hence, contrary to CSS approach based +on double UV+rapidity cutoff, there is no logarithmic dependence on the cutoffs in the soft +factor. Of course, there may be the non-perturbative power corrections ∼ Λ2 +QCDx2 +12⊥ which +should be studied by some non-perturbative methods, but the claim is that the soft factor +with rapidity-only regularization does not have perturbative contributions which can mix +with the TMD evolution. +Thus, we can neglect the integration over C fields in Eq. (9.15) and get the factorized +result +ρ4 +� +dx− +2 dx− +1 dx+ +2 dx+ +1 eiα′ +aϱx− +2 +iαaϱx− +1 +iβ′ +bϱx+ +2 +iβbϱx+ +1 W(pA, pB, p′ +A, p′ +B; x1, x2) +(9.20) += e +αsNc +2π +C1(x12⊥,α′ +a,αa,β′ +b,βb;σp,σt)⟨p′ +A| ˆOσp +ij (α′ +a, αa, x12⊥)|pA⟩⟨p′ +B| ˆOij;σt(β′ +b, βb, x12⊥)|pB⟩ +9.1.2 +Factorization in terms of generalized TMDs +Let us rewrite our result in terms of generalized TMDs (gTMDs). They can be defined as +follows [21] +Gσp +ij (xA, b⊥; p′ +A, pA) = +− g−2 +πϱxA +� +dz−eixAϱz−⟨p′ +A| ˆOσp +ij (−z− +2 − b⊥ +2 , z− +2 + b⊥ +2 +� +|pA⟩, +Gσt +ij (xB, b⊥; p′ +B, pB) = +− g−2 +πϱxA +� +dz−eixBϱz−⟨p′ +B| ˆOσp +ij (−z− +2 − b⊥ +2 , z− +2 + b⊥ +2 +� +|pB⟩(9.21) +The above choice of normalization reproduces gluon TMDs for unpolarized hadrons defined +in Ref. [22] at p′ +A = pA. +⟨pA| ˆOσp +ij (z−, 0−, b⊥)|pA⟩ = +− g2ϱ2 +� 1 +0 +du uGσp +ij (u, b⊥) cos uϱz−, +⟨pA| ˆOσp +ij (αq, b⊥)|pA⟩ ≡ ρ +� +dz−eiαqϱz−⟨pA| ˆOσp +ij (z−, 0−, b⊥)|pA⟩ += +− πg2ϱ2|αq|Gσp +ij (|αq|, b⊥, pA), +Gσp +ij (u, b⊥) = gijDg(u, b⊥; σp) + +1 +2m2 +N +(2∂i∂j + gij∂2 +⊥)H(u, b⊥; σp) +(9.22) +– 40 – + +and similarly for ⟨pB| ˆOσt +ij (βq, b⊥)|pB⟩. Note that at b⊥ = 0 the TMD Dg(xB, σ) is the +gluon PDF with the rapidity-only cutoff discussed in Ref. +[23]. +At the leading order, +this is equivalent to usual UV regularization of light-ray operator ˆOσ +ij(z±) and reproduces +LO DGLAP equation [23]. At the NLO level, the two-loop DGLAP equation should be +reproduced by the combination of rapidity-only evolution of light-ray operator ˆOσ +ij(z±) and +usual µ2 evolution for self-energy and vertex Z-factors. +With the normalization (9.21) we get +⟨p′ +A| ˆOσp +ij (α′ +a, αa, x2⊥, x1⊥)|pA⟩ += +− 2π2δ(αa + α′ +a)g2ρ2|αa|e− i +2 (l,x1+x2)⊥Gσp +ij (|αa|, x12⊥; pA, p′ +A) +⟨p′ +B| ˆOσt +ij (β′ +b, βb, x2⊥, x1⊥)|pB⟩ += +− 2π2δ(βb + β′ +b)g2ρ2|βb|e +i +2 (l,x1+x2)⊥Gσt +ij (|βb|, x12⊥; pB, p′ +B) +(9.23) +The δ-functions in the above expressions for ⟨p′ +A| ˆOσp +ij |pA⟩ and ⟨p′ +B| ˆOσtp +ij |pB⟩ are present also +in the l.h.s. of Eq. (9.20) because p′ +A + p′ +B = pA + pB. Canceling them, one obtains +� +dx− +12dx+ +12 eiαaϱx− +12+iβbϱx+ +12 N2 +c − 1 +16 +⟨p′ +A, p′ +B|F 2� +− x12 +2 +� +F 2�x12 +2 +� +|pA, pB⟩ +(9.24) += π2 +2 Q2e +αsNc +2π +� +ln2 b2 +⊥sσpσt +4 +−2 ln αaeγ +σt +ln βbeγ +σp + π2 +2 +� +Gσp +ij (αa, x12⊥; pA, p′ +A)Gij;σt(βb, x12⊥; pB, p′ +B) +This is the final formula for rapidity-only TMD factorization of hadronic tensor. +10 +Conclusions and outlook. +In conclusion let us present our final formula (9.24) for the practical case of hadronic tensor +(2.2) which corresponds to “forward” matrix element with p′ +A = pA and p′ +B = pB. It reads +W(pA, pB; q) = +� +db⊥ ei(q,b)⊥W(pA, pB; αq, βq, b⊥), +W(pA, pB; αq, βq, b⊥) = π2 +2 Q2Gσp +ij (αq, b⊥; pA)Gij;σt(βq, b⊥; pB) +× exp +�αsNc +2π +� +ln2 b2 +⊥sσpσt +4 +− 2 +� +ln αq +σt ++ γ +�� +ln βq +σp ++ γ +� ++ π2 +2 +�� ++ NLO terms ∼ O +� +α2 +s) + power corrections +(10.1) +where gluon TMDs Gσp +ij (αq, b⊥) and Gij +σt(βq, b⊥) are defined in Eq. (9.22) above. Note that +this formula is actually our goal - TMD factorization (1.1) with the coefficient function +(1.4) at ηa = ln σp and ηb = ln σt. +Let us discuss the region of applicability of Eq. (10.1). The r.h.s. of the evolution +formula (10.1) does not depend on cutoffs σp and σt as long as σp ≥ ˜σp = +4b−2 +⊥ +αqs and +σt ≥ ˜σt ≡ 4b−2 +⊥ +βqs , see Eq. (3.4). Thus, the result of double-log Sudakov evolution reads +W(pA, pB; αq, βq, b⊥) = π2 +2 Q2G ˜σp +ij (αq, b⊥; pA)Gij;˜σt(βq, b⊥; pB) +(10.2) +× exp +� +− αsNc +2π +�� +ln Q2b2 +⊥ +4 ++ 2γ +�2 − 2γ2 − π2 +2 +�� ++ O +� +α2 +s) terms + power corrections +– 41 – + +This result is universal for moderate x and small-x hadronic tensor. The difference lies in +the continuation of the evolution beyond Sudakov region. This is discussed in Appendix G +of Ref. [9] and here I briefly sum up the main points of that discussion. First, if xB ∼ 1 +and q2 +⊥ ≥ m2 +N, there is no room for any evolution and one should turn to phenomenological +models of TMDs like the replacement of b by b∗ in Refs. [2, 24]. If xB ∼ 1 and q2 +⊥ ≥ m2 +N, +there is a room for DGLAP-type evolution summing logs +� +αs ln q2 +⊥/m2 +N +�n. Similarly, if +xB = βb ≪ 1, then even at βbσs = q2 +⊥ there can be the BFKL-type evolution from σ = q2 +⊥ +β′ +bs +to σ = q2 +⊥ +s +which sums up logs (αs ln xB)n. The matching between double-log Sudakov +evolution (10.1) and single-log DGLAP or BFKL evolutions can in principle be performed +by solving general rapidity evolution equations discussed in Ref. [23]. +There is another issue that should be addressed before matching to BFKL and es- +pecially to DGLAP evolutions. As usually for rapidity-only factorization, the argument of +coupling constant in Eq. (10.1) is undetermined in the leading order and should be obtained +from higher orders of perturbative expansion. Typically, argument of coupling constant in +the small-x evolution equations is fixed using the BLM/renormalon approach [25], see for +example Ref. [26] for the BFKL equation and Refs. [27, 28] for the BK equation [29–31]. +In recent paper [9] G.A. Chirilli and the author used this BLM optimal scale setting [25] +to fix the argument of coupling constant in the rapidity-only TMD evolution (11.19). The +result is that the effective argument of a coupling constant is halfway in the logarithmical +scale between the transverse momentum and energy of TMD distribution. One of the fu- +ture directions of this research is to use BLM prescription to fix the argument of coupling +constant in the coefficient function C(x12⊥; αa, βb; σp, σt) and obtain the running-coupling +generalization of Sudakov-type formula (10.1). +Another outlook is to connect to usual CSS/SCET-type evolution of TMDs at moderate +x where the two and three-loop results are available [32–35]. +It should be noted that +the “double operator expansion” method recently used in Ref. [14] is very similar to the +approach of this paper and also uses calculation of Feynman diagrams in two background +fields. However, the UV+rapidity cutoff of TMD operators in Ref. [14] is very different from +the rapidity-only cutoff used here so the hope is to fix the argument of coupling constant +in Eq. (10.1) and compare the final results for the evolution. +Also, it would be very interesting to obtain similar Sudakov-type formula (10.1) for the +Drell-Yan process. The study is in progress. +Acknowledgments +The author is grateful to G.A. Chirilli, A. Radyushkin, T. Rogers, and A. Vladimirov for +valuable discussions. This work is supported by DOE contract DE-AC05-06OR23177 and +by the grant DE-FG02-97ER41028. +– 42 – + +11 +Appendix +11.1 +Gluon “cut” propagator in the background field A +In general, the “cut” gluon propagator from left to right sector in the background-Feynman +gauge is given by the double functional integral +⟨Aα(x)Aβ(y)⟩A +(11.1) += +� +˜ +A(tf)=A(tf) +D ˜AµDAµ ˜Aα(x)Aβ(y) ei +� +dz +1 +2 +� +˜ +Aµ,a(˜D2gµν−2i˜Fµν)ab ˜ +Aν,b+Aa +µ(D2gµν−2iFµν)Ab +ν +� +(recall that in our case background field A is the same for both left and right sector). In +Schwinger’s notations, in can be written down as +⟨Aα(x)Aβ(y)⟩A = +− (x| +� +1 +P2gαξ + 2iFαξ − iϵp2� +˜δ+(p) +� +p2 +1 +P2δξ +β + 2iFξ +β + iϵ +� +|y) +(11.2) +where expressions in parenthesis in the r.h.s. can be understood as a series +p2 +1 +(p + A)2δξ +β + 2iFξ +β + iϵ += 1 − Oξ +β +1 +p2 + iϵ + Oξ +η +1 +p2 + iϵOη +β +1 +p2 + iϵ + ... +1 +(p + A)2gαξ + 2iFαξ − iϵp2 = 1 − +1 +p2 + iϵOαξ + +1 +p2 − iϵOαη +1 +p2 − iϵOη +β + ... (11.3) +with +Oαβ ≡ ({pλ, Aλ} + A2)gαβ + 2iFαβ. +(11.4) +By rearranging the series, it is possible to prove that +⟨Aa +α(x)Ab +β(y)⟩A = (x| +1 +p2 + iϵp0 +O˜δ+(p2) + ˜δ+(p2)O +1 +p2 − iϵp0 ++ +1 +p2 + iϵp0 +O˜δ+(p)O +1 +p2 − iϵp0 ++ ˜δ+(p)O +1 +p2 − iϵp0 +O +1 +p2 − iϵp0 ++ +1 +p2 + iϵp0 +O +1 +p2 + iϵp0 +O˜δ+(p) − ˜δ+(p)O˜δ−(p)O˜δ+(p) + O(O3)|y)ab +αβ +(11.5) +Thus, the only term which spoils the “retardiness” property is the last term, but it can +be proportional only to correction field ¯C since neither projectile no target field can solely +produce a pair of gluons. However, two fields ¯C involve four Fξη (two U +i and two V −j, +see Eq. (11.38)) which exceeds our accuracy. Thus, we use formula (11.5) without the last +term. +11.2 +TMD matrix elements +In this Section we list the necessary results for the “eikonal” contributions - one-loop TMD +matrix elements calculated in Ref. [9]. As discussed there, the cutoffs in α and β respect- +ing analytical properties of Feynman diagrams (and hence IR real-virtual cancellations) +are obtained by “smooth” cuts e±i α +σt and e±i β +σt . These cutoffs are visualized with “point- +splitting” regularization as shown in Figs. 11 and 12. 11 As demonstrated in Ref. [9], δ+ +11As demonstrated in Ref. [9], the violations of gauge invariance due to this point-splitting are power +corrections ∼ λp or λt. +– 43 – + +(a) +(b) +(e) +(i) +(j) +(f) +(k) +(g) +(c) +(h) +(d) +x’1 +x’2 +x1 +t +x 2 +t +Figure 11. “Target” TMD matrix elements. The e−i α +σt regularization is depicted by point splitting: +F −k shown by dots stand at xt +1 = x1⊥ + x+ +1 and xt +2 = x2⊥ + x+ +2 while Wilson lines start from +x′ +1 = xt +1 + δ− and x′ +2 = xt +2 + δ− where δ− = +1 +ϱσt . +and δ− should be positive which follows from the requirement that the distances between +the “splitted” operators should be space-like. +The results of calculation of diagrams in Fig. 11 are: +⟨[x+ +2 , −∞+]ab +x2⊥+δ−[−∞+, x+ +1 ]bc +x1⊥+δ−F −j,c(x+ +1 , x1⊥)⟩Fig. 11a−d +A +(11.6) += g2Nc +8π2 +� +d−βbd−kb⊥V −j,a(βb, kb⊥)e−iβbϱx+ +1 +i(k′ +b,x1)⊥Ieik +Fig. 11a−d(βb, kb⊥, x12) +where +Ieik +Fig.11a−d(βb, kb⊥, x+ +1 , x1⊥, x+ +2 , x2⊥) = 8π2s +� ∞ +0 +d−α e−i α +σt +d−β +β + iϵd−p⊥ eiβϱx+ +12−i(p,x12)⊥ +× +� +˜δ(αβs − p2 +⊥) +β − βb +α(β − βb)s − (p − kb)2 +⊥ − iϵ + +(β − βb) +αβs − p2 +⊥ + iϵ +˜δ[α(β − βb)s − (p − kb)2 +⊥] +� +− 8π2i +� +d−αd−βd−p⊥ +e−i α +σt s(β − βb) +(β + iϵ)(αβs − p2 +⊥ + iϵ)[α(β − βb)s − (p − kb)2 +⊥ + iϵ] += 8π2 +� ∞ +0 +d−α e−i α +σt +� +d−p⊥ +�βbs +p2 +⊥ +ei +p2 +⊥ +αs ϱx+ +12−i(p,x12)⊥ − 1 +αβbs + (p − kb)2 +⊥ + iϵ ++ (p − kb)2 +⊥ei(p,x12)⊥� +ei(βb+ +(p−kb)2 +⊥ +αs +)ϱx+ +12 − ei +p2 +⊥ +αs ϱx+ +12� +α[αβbs + (p − kb)2 +⊥ + iϵ][αβbs + (p − kb)2 +⊥ − p2 +⊥] +� +(11.7) +– 44 – + +At x+ +12 = 0 this integral is simplified to +Ieik +Fig. 11a−d(βb, kb⊥, x12⊥) = 8π2 +� ∞ +0 +d−α +� d−p⊥ +p2 +⊥ +βbse−i α +σt � +e−i(p,x12)⊥ − 1 +� +αβbs + (p − kb)2 +⊥ + iϵ += 8π2 +� ∞ +0 +d−α e−i α +σt +� d−p⊥ +p2 +⊥ +βbs +� +e−i(p,x12)⊥ − 1 +� +αβbs + p2 +⊥ + iϵ +� +1 − +p2 +⊥ − (p − kb)2 +⊥ +αβbs + (p − kb)2 +⊥ + iϵ +� += 8π2 +� ∞ +0 +d−α e−i α +σt +� d−p⊥ +p2 +⊥ +βbs +� +e−i(p,x12)⊥ − 1 +� +αβbs + p2 +⊥ + iϵ ++ 4π +� d−p⊥ +p2 +⊥ +� +e−i(p,x12)⊥ − 1 +� +ln +p2 +⊥ +(p − kb)2 +⊥ += +− 1 +2 ln2 � +− i +4(βb + iϵ)σtsx2 +12eγ� +− π2 +4 + IK(−kb⊥, x12⊥) +(11.8) +where IK is defined in Eq. (6.26). Here we neglected the e−i α +σ cutoff in the second integral +in the third line since it converges at α ∼ Q2 +⊥ +|β′ +b|s so α +σt ∼ λt. The first term in the last line is +given by Eq. (C4) from Ref. [9] and the second by Eq. (11.59) from Appendix 11.6.2. +Similarly, the result for diagrams in Fig. 11 e-h reads +⟨F −i,a(x+ +2 , x2⊥)[x+ +2 , −∞]ab +x2⊥+δ−[−∞, x+ +1 ]bc +x1⊥+δ−⟩Fig. 11e−h +A +(11.9) += g2Nc +8π2 +� +d−β′ +bd−k′ +b⊥e−iβ′ +bx+ +12+i(k′ +b,x12⊥)V −i,c(β′ +b, k′ +b⊥)Ieik +Fig. 11e−h(β′ +b, k′ +b⊥x12⊥), +Ieik +Fig. 11e−h(β′ +b, k′ +b⊥, x12) = 8π2 +� ∞ +0 +d−α +� +d−p⊥ ei α +σt +�β′ +bs +p2 +⊥ +� +e−i(p,x12)⊥+i +p2 +⊥ +αs ϱx+ +12 − 1 +� +αβ′ +bs − (p + k′ +b)2 +⊥ + iϵ ++ (p + k′ +b)2 +⊥e−i(p,x12)⊥� +ei +(p−ka)2 +⊥ +αs +ϱx+ +12−iβbϱx+ +12 − ei +p2 +⊥ +αs ϱx+ +12� +α[αβ′ +bs + p2 +⊥ − (p + k′ +b)2 +⊥ + iϵ][αβ′ +bs − (p + k′ +b)2 +⊥ + iϵ] +� +which simplifies to +Ieik +Fig. 11e−h(β′ +b, k′ +b⊥, x12) = 8π2 +� ∞ +0 +d−α +� +d−p⊥ ei α +σt β′ +bs +p2 +⊥ +� +e−i(p,x12)⊥ − 1 +� +αβ′ +bs − (p + k′ +b)2 +⊥ + iϵ(11.10) +at x+ +2 = x+ +1 . This integral can be obtained from Eq. (11.8) by complex conjugation and +replacement kb → −k′ +b so one obtains +Ieik +Fig. 11e−h(β′ +b, k′ +b⊥, x12⊥) += +− 1 +2 ln2 � +− i +4(β′ +b + iϵ)σtsx2 +12⊥eγ� +− π2 +4 + IK(k′ +b⊥, x12⊥) + O(λt) +(11.11) +As to ‘handbag” diagrams in Fig. 11i-k, they were already discussed in Sect. 6.3. +The diagrams in Fig. 12 are obtained by simple target↔projectile replacements (6.28) +in Eqs. (11.8) and (11.9). We get +– 45 – + +(a) +(b) +(e) +(f) +(i) +(j) +(k) +(d) +(h) +(g) +(c) +x’1 +x’2 +x1 +p +x2 +p +Figure 12. “Projectile” TMD matrix elements. The e−i β +σp regularization is depicted by point +splitting: F +k shown by dots stand at xp +1 = x1⊥ + x− +1 and xp +2 = x2⊥ + x− +2 while Wilson lines start +from x′ +1 = x2 + δ+ and x′ +2 = x1 + δ+ where δ+ = +1 +ϱσp . +⟨[x− +2 , −∞]ab +x2⊥+δ+[−∞, x− +1 ]bc +x1⊥+δ+F +j,c(x− +1 , x1⊥)⟩Fig. 12a−d +A +(11.12) += g2Nc +8π2 +� +d−αad−ka⊥U +j,b(αa, ka⊥)e−iαaϱx− +1 +i(ka,x1)⊥Ieik +Fig. 12a−d(αa, ka⊥, x12), +Ieik +Fig. 12a−d(αa, ka⊥, x12) = 8π2 +� ∞ +0 +d−β e +−i β +σp +� +d−p⊥ +�αas +p2 +⊥ +ei +p2 +⊥ +βs ϱx− +12−i(p,x12)⊥ − 1 +αaβs + (p − ka)2 +⊥ + iϵ ++ (p − ka)2 +⊥e−i(p,x12)⊥� +ei(αa+ +(p−ka)2 +⊥ +βs +)ϱx− +12 − ei +p2 +⊥ +βs ϱx− +12� +α[αaβs + (p − ka)2 +⊥ + iϵ][αaβs + (p − ka)2 +⊥ − p2 +⊥] +� +simplified to +Ieik +Fig. 12a−d(αa, ka⊥, x12⊥) = +− 1 +2 ln2 � +− i +4(αa + iϵ)σpsx2 +12⊥eγ� +− π2 +4 + IK(−ka⊥, x12⊥) +(11.13) +at x− +12 = 0, and +⟨F +i,a(x− +2 , x2⊥)[x− +2 , −∞]ab +x2⊥+δ+[−∞, x− +1 ]bc +x1⊥+δ+⟩Fig. 12e−h +A +(11.14) += g2Nc +� +d−αad−k′a⊥e−iαax+ +12+i(k′ +a,x12⊥)U +i,c(αa, k′a⊥)Ieik +Fig. 12e−h(αa, k′a⊥, x2, x1), +Ieik +Fig. 12e−h(α′ +a, k′ +a⊥, x12) = 8π2 +� ∞ +0 +d−β +� +d−p⊥ e +i β +σp +�α′ +as +p2 +⊥ +� +e−i(p,x12)⊥+i +p2 +⊥ +βs ϱx− +12 − 1 +� +α′aβs − (p + k′a)2 +⊥ + iϵ ++ (p + k′ +a)2 +⊥e−i(p,x12)⊥� +ei +(p+k′a)2 +⊥ +αs +ϱx− +12−iα′ +aϱx− +12 − ei +p2 +⊥ +βs ϱx− +12� +β[α′aβs + p2 +⊥ − (p + k′a)2 +⊥ + iϵ][α′aβs − (p + k′a)2 +⊥ + iϵ] +� +– 46 – + +which similarly simplifies to +Ieik +Fig. 12e−h(αa, k′a⊥, x12⊥) += +− 1 +2 ln2 � +− i +4(α′ +a + iϵ)σpsx2 +12eγ� +− π2 +4 + IK(k′ +a⊥, x12⊥) + O(λp) +(11.15) +at x+ +12 = 0. Again, the “handbag” diagrams in Fig. 12i-k were already accounted for in +Sect. 6.3. +Assembling Eqs. (11.8), (11.11), (11.13), and (11.15), we get the matrix elements of +eikonal TMD operators which have to be subtracted from W according to Eq. (4.2): +W (x1, x2)eik = +N2 +c − 1 +Nc +8π2 +× +� +U +a +i (x2)V −i,n(x2)⟨[x+ +2 , −∞]na +x2⊥+δ−[−∞, x+]bc +x1⊥+δ−F −j,c(x+, x1⊥)⟩Fig. 11a−c +A +U +j,b(x1) ++ U +a +i (x2)⟨F −i,n(x+ +2 , x2⊥)[x+ +2 , −∞]na +x2⊥+δ−[−∞, x+ +1 ]bc +x1⊥+δ−⟩Fig. 11d−f +A +V −j,c(x1)U +j,b(x1) ++ U +n +i (x2)V −i,a(x2)⟨[x−, −∞]na +x2⊥+δ+[−∞, x− +1 ]bc +x1⊥+δ+F +j,c(x−, x1⊥)⟩Fig. 12a−c +A +V −j,c(x1) ++ V −i,a(x2)⟨F +i,n(x− +2 , x2⊥)[x− +2 , −∞]na +x2⊥+δ+[−∞, x− +1 ]bc +x1⊥+δ+⟩Fig. 12d−f +A +V −j,c(x1)U +b +j (x1) +� += +� +d−α′ +ad−k′a⊥d−β′ +bd−kb⊥d−αad−k′a⊥d−βbd−k′b⊥ e−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 +× e−i(ka+ka,x2)⊥−i(k′ +a+k′ +b,x1)⊥U +,b +i (α′ +a, k′a⊥)V −i,a(β′ +b, kb⊥)U +,b +j (αa, p′ +A⊥)V −j,a(βb, k′ +b⊥) +× +� +Iσp,σt +eik +(αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥, x12⊥) + O(λp) + O(λt) +� +(11.16) +with +Iσp,σt +eik +(αa, α′ +a, βb, β′ +b, ka⊥, k′ +a⊥, kb⊥, k′ +b⊥, x12⊥) +(11.17) += +− 1 +2 ln2 � +− i +4(α′ +a + iϵ)σpsx2 +12⊥eγ� +− 1 +2 ln2 � +− i +4(αa + iϵ)σpsx2 +12⊥eγ� +− 1 +2 ln2 � +− i +4(β′ +b + iϵ)σtsx2 +12⊥eγ� +− 1 +2 ln2 � +− i +4(βb + iϵ)σtsx2 +12⊥eγ� +− π2 ++ IK(−ka⊥, x12⊥) + IK(k′ +a⊥, x12⊥) + IK(−kb⊥, x12⊥) + IK(k′ +b⊥, x12⊥) +where IK is defined in Eq. (6.26). +Let us present also the derivative 12 +σt +d +dσt +Iσp,σt +eik +(α′ +a, αa, β′ +b, βb, k′ +a⊥, ka⊥, kb⊥, k′ +b⊥, x12⊥) +(11.18) += +− ln +� +− i +4(β′ +b + iϵ)σtsx2 +12⊥eγ� +− ln +� +− i +4(βb + iϵ)σtsx2 +12⊥eγ� +which translates into evolution equation [9, 10] +σt +d +dσt +ˆOij;σt(β′ +b, βb, x2⊥, x1⊥) +(11.19) += +− αsNc +2π +� +2 ln sx2 +12⊥ +4 ++ ln(−iβ′ +bσt + ϵ) + ln(−iβbσt + ϵ) + 2γ +� +ˆOij;σt(β′ +b, βb, x2⊥, x1⊥) +12It is worth noting that handbag diagrams in Figs 11i-k and 12i-k do not contribute to evolution equation +since they do not need a rapidity cutoff, see the discussion in in Ref. [9] and in Sect. 6.3. +– 47 – + +Similarly, one obtains +σp +d +dσp +ˆOij;σt(α′ +a, αa, x2⊥, x1⊥) +(11.20) += +− αsNc +2π +� +2 ln sx2 +12⊥ +4 ++ ln(−iα′ +aσp + ϵ) + ln(−iαaσp + ϵ) + 2γ +� +ˆOij;σt(α′ +a, αa, x2⊥, x1⊥) +In the coordinate space, the evolution equation (11.19) takes the form +σt +d +dσt +ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥) +(11.21) += +− αsNc +2π +� +2 ln sx2 +12⊥ +4 +ˆOij;σt(x+ +2 , x2⊥; x+ +1 , x1⊥) +− +� +dz+ +2 +�θ(x2 − z2)+ +(x2 − z2)+ − δ(x2 − z2)+ +� σt/ϱ +0 +dz+ +2 +z+ +2 +� +ˆOσp +ij (z+ +2 , x2⊥; x+ +1 , x1⊥) +− +� +dz+ +1 +�θ(x1 − z1)+ +(x1 − z1)+ − δ(x1 − z1)+ +� σt/ϱ +0 +dz+ +1 +z+ +1 +� +ˆOσp +ij (x+ +2 , x2⊥; z+ +1 , x1⊥) +� +where we used formula +� +d−β e−iβz� +ln +� +− iβσ + ϵ +� ++ γ +� += +− θ(z) +z ++ δ(z) +� σ +0 +dz′ +z′ +(11.22) +The evolution equation of ˆOij;σp(x− +2 , x2⊥; x− +1 , x1⊥) looks like Eq. (11.21) with trivial re- +placements x+ → x− and σt → σp. +11.3 +Soft factor with rapidity-only cutoffs +In this Section we demonstrate that the soft factor with rapidity-only cutoffs is a power +correction. The soft factor is given by the correlation function of four Wilson lines +1 +N2c − 1Tr⟨{x− +2 , −∞−}x2⊥{−∞+, x+ +2 }x2⊥[x+ +1 , −∞+]x1⊥[−∞−, x− +1 ]x1⊥⟩C +(11.23) +The diagrams for the one-loop soft factor with rapidity-only regularization by “point split- +ting” is shown in Fig. 13 +– 48 – + +x’’1 +x’’2 +x’1 +x’2 +x 1 +x 2 +Figure 13. First-order perturbative diagrams for the soft factor. The rapidity regularization is +depicted by point splitting: “projectile” Wilson lines start from x′ +1 = x1 + δ+ and x′ +2 = x2 + δ+ +while “target” Wilson lines from x′′ +1 = x1 + δ− and x′′ +2 = x2 + δ− +Tr +Nc(N2c − 1)⟨{x− +2 , −∞−}x2⊥+δ+{−∞+, x+ +2 }x2⊥+δ−[x+ +1 , −∞+]x1⊥+δ−[−∞−, x− +1 ]x1⊥+δ+⟩ += +1 +Nc(N2c − 1)Tr +�� x− +2 +−∞ +dx′ +2 +− +� x+ +2 +−∞ +dx′ +2 ++ ˜A+(x′ +2 +− + δ+, x2⊥) ˜A−(x′ +2 ++ + δ−, x2⊥) +− +� x− +2 +−∞ +dx′ +2 +− +� x+ +1 +−∞ +dx′ +1 ++ ˜A+(x′ +2 +− + δ+, x2⊥)A−(x′ +1 ++ + δ−, x1⊥) +− +� x+ +2 +−∞ +dx′ +2 ++ +� x− +1 +−∞ +dx′ +1 +− ˜A−(x′ +2 ++ + δ−, x2⊥)A+(x′ +1 +− + δ+, x1⊥) ++ +� x− +1 +−∞ +dx′ +1 +− +� x+ +1 +−∞ +dx′ +1 ++ A+(x′ +1 +− + δ+, x1⊥)A−(x′ +1 ++ + δ−, x1⊥) +� += +� +d−4p +�� x− +2 +−∞ +dx′ +2 +− +� x+ +2 +−∞ +dx′ +2 ++ e−iαϱ(x′ +2 +−−δ−)+iβϱ(x′ +2 ++−δ+) +i +αβs − p2 +⊥ − iϵ ++ +� x− +2 +−∞ +dx′ +2 +− +� x+ +1 +−∞ +dx′ +1 ++ e−iαϱ(x′ +2 +−−δ−)+iβϱ(x′ +1 ++−δ+)−i(p,x12)⊥δ+(αβs − p2 +⊥) ++ +� x+ +2 +−∞ +dx′ +2 ++ +� x− +1 +−∞ +dx′ +1 +− eiαϱ(x′ +1 +−−δ−)−iβϱ(x′ +2 ++−δ+)−i(p,x12)⊥δ+(αβs − p2 +⊥) +− +� x− +1 +−∞ +dx′ +1 +− +� x+ +1 +−∞ +dx′ +1 ++ e−iαϱ(x′ +1 +−−δ−)+iβϱ(x′ +1 ++−δ+) +i +αβs − p2 +⊥ + iϵ +� +(11.24) += +� +d−αd−βd−p⊥ +� +i +(α + iϵ)(β − iϵ) +�eiαϱ(δ−−x− +2 )−iβϱ(δ+−x+ +2 ) +αβs − p2 +⊥ − iϵ +− eiαϱ(δ−−x− +1 )−iβϱ(δ+−x+ +1 ) +αβs − p2 +⊥ + iϵ +� ++ s +p2 +⊥ +e−i(p,x12)⊥ +� +eiαϱ(δ−−x− +2 )−iβϱ(δ+−x+ +1 ) + e−iαϱ(δ−−x− +1 )+iβϱ(δ+−x+ +2 )� +δ+(αβs − p2 +⊥) +� +– 49 – + +Since the x− +2 , x− +1 ∼ +1 +ϱα′a ≪ δ− and x+ +2 , x+ +1 ∼ +1 +ϱβ′ +b ≪ δ+ we can neglect x− +2 , x− +1 , x+ +2 , x+ +1 in the +above integrals and get += s +� +d−αd−β d−p⊥ +p2 +⊥ +� +− eiαϱδ−−iβϱδ+˜δ(αβs − p2 +⊥) ++ e−i(p,x12)⊥� +eiαϱδ−−iβϱδ+ + e−iαϱδ−+iβϱδ+� +δ+(αβs − p2 +⊥) +� += s +� +d−αd−β d−p⊥ +p2 +⊥ +� +eiαϱδ−−iβϱδ+ + e−iαϱδ−+iβϱδ+� +δ+(αβs − p2 +⊥) +� +e−i(p,x12)⊥ − 1 +� += +1 +2π2 +� dp2 +⊥ +p2 +⊥ +[J0(p⊥∆⊥) − 1]K0 +� +p⊥ +√ +2δ+δ− +� += +1 +4π2 Li2 +� +− x2 +12⊥ +2δ+δ− +� +(11.25) +Thus, we get the perturbative contribution to rapidity-regularized soft factor in the form +⟨{x− +2 , −∞−}x2⊥+δ+{−∞+, x+ +2 }x2⊥+δ−[x+ +1 , −∞+]x1⊥+δ−[−∞−, x− +1 ]x1⊥+δ+⟩ += +1 +4π2 Li2 +� +− x2 +12⊥ +2δ+δ− +� +∼ O +� ∆2 +⊥ +2δ+δ− +� +∼ O +�σpσts +Q2 +⊥ +� += O +� µ2 +σ +Q2 +⊥ +� +∼ ζ−1/2 +(11.26) +which is a parametrically small power correction. Of course, there are non-perturbative +contributions to the soft factor - power corrections presumably of order of Λ2 +QCDx2 +12⊥, but, +as we mentioned above, the lesson is that the soft factor with rapidity-only regularization +does not have perturbative contributions which can mix with the TMD evolution, quite +unlike the usual regularization of the soft factor with “UV+rapidity” cutoff. +11.4 +Approximation x∥ +12 = 0 for the calculation of coefficient function +In this Section we prove that for the calculation of the coefficient function C1 in Eq. (4.1) +one can set x∥ +12 = 0 with power accuracy. Let us start with the difference I1a − Ieik +1a in Eq. +(6.16). Since vitual eikonals do not depend on x, it is sufficient to consider +I1a(α′ +a, k′ +a⊥, β′ +b, kb⊥, x2, x1) − Ieik +Fig. 11a,b(α′ +a, k′a⊥, β′ +b, kb⊥, x2, x1) − (x∥ → 0) +(11.27) +The expression for I1a is given by Eq. (6.11) +I1a(α′ +a, k′a⊥, βb, kb⊥, x2⊥, x1⊥) = 8π2 +� ∞ +0 +d−α +α +� +d−p⊥ +e−i(p,x12)⊥ +αβbs + (p − k′ +b)2 +⊥ − p2 +⊥ + iϵ +× eiαϱx− +12 +� +p2 +⊥ +α2sξ + p2 +⊥ +(α + α′ +a)(αβbs − p2 +⊥)ei +p2 +⊥ +αs ϱx+ +12 +� +(α + α′a)p2 +⊥ − α(p + k′a)2 + iϵ +� ++ +(p − k′ +b)2 +⊥(α + α′ +a)eiβbϱx+ +12+i +(p−k′ +b)2 +⊥ +αs +ϱx+ +12 +α(α′a + α)βbs + (α′a + α)(p − kb)2 +⊥ − α(p + k′a)2 +⊥ + iϵ(α′a + α) +� +(11.28) +(cf. Eq.(6.18) at x∥ = 0), and the expression for Ieik +Fig. 11a,b can be taken as Eq. (11.7) +without virtual term +Ieik +Fig. 11a,b(βb, kb⊥, x2, x1) = 8π2 +� ∞ +0 +d−α +α +e−i α +σt +� +d−p⊥ +�αβbs − p2 +⊥ +p2 +⊥ +e−i +p2 +⊥ +αs ϱx+ +12 ++ (p − kb)2 +⊥ei(βb+ +(p−kb)2 +⊥ +αs +)ϱx+ +12 +[αβbs + (p − kb)2 +⊥ + iϵ] +� +e−i(p,x12)⊥ +αβbs − p2 +⊥ + (p − kb)2 +⊥ +(11.29) +– 50 – + +For definiteness, let us take βb > 0 (the case of βb < 0 is similar). The difference (11.27) +can be represented as a sum of two contributions. The first one is the difference between +first terms in Eqs. (11.28) and (11.29) minus same difference at x∥ +12 = 0. +8π2 +� ∞ +0 +d−α +α +� +d−p⊥ +� +p2 +⊥ +α2sξ+p2 +⊥ (α + α′ +a)(αβbs − p2 +⊥) +� +eiαϱx− +12+i +p2 +⊥ +αs ϱx+ +12 − 1 +� +� +(α + α′a)p2 +⊥ − α(p + k′a)2 + iϵ +�� +αβbs + (p − kb)2 +⊥ − p2 +⊥ + iϵ +� +− +� ∞ +0 +d−α +α +e−i α +σt +� +ei +p2 +⊥ +αs ϱx+ +12 − 1 +� +(αβbs − p2 +⊥) +p2 +⊥[αβbs − p2 +⊥ + (p − kb)2 +⊥ + iϵ] +� +e−i(p,x12)⊥ += 4π +� ∞ +0 +dt +t +� +d−p⊥ +� +p2 +⊥ +t2 +α′aβbs| α′a +βb |ξ + p2 +⊥ +(t + α′ +aβbs) +� +e +i +tα′aϱx− +12 +α′aβbs − 1 +� +ei +p2 +⊥ +t β′ +bϱx+ +12 +(t + α′aβbs)p2 +⊥ − t(p + k′a)2 + iϵ ++ +� +p2 +⊥ +t2 +α′aβbs| α′a +βb |ξ + p2 +⊥ +(t + α′ +aβbs) +(t + α′aβbs)p2 +⊥ − t(p + ka)2 + iϵ − e +−i +t +σtβbs +p2 +⊥ +� +× +� +e−i +p2 +⊥ +t βbϱx+ +12 − 1 +�� +t − p2 +⊥ +t + (p − kb)2 +⊥ − p2 +⊥ + iϵ = O +� Q2 +⊥ +σtβbs +� +(11.30) +Indeed, integrals over p⊥ and t converge at p⊥ ∼ x−1 +⊥ ∼ Q⊥ and t between Q2 +⊥ and α′ +aβbs. +At t ∼ α′ +aβbs the first term is ∼ Q2 +⊥ +Q2 = λ and the second ∼ Q2 +⊥ +Q2 βbϱx+ +12 ∼ λ. At t ∼ Q2 +⊥ the +first term is ∼ Q2 +⊥ +Q2 α′ +aϱx− +12 ∼ Q2 +⊥ +Q2λ while the second can be rewritten as +4π +� ∞ +0 +dt +t +� d−p⊥ +p2 +⊥ +e−i(p,x12)⊥ +� +ei +p2 +⊥ +t βbϱx+ +12 − 1 +� +� +1 − e +−i +t +σtβbs +� +(t − p2 +⊥) +[t − p2 +⊥ + (p − kb)2 +⊥ + iϵ] ∼ O(λt) +(11.31) +The second contribution to the Eq. (11.27) is the difference between second terms in +Eqs. (6.18) and (11.29) +8π2 +� ∞ +0 +d−α +α +� +d−p⊥ +� +(α + α′ +a) +� +eiαϱx− +12+iβ′ +bϱx+ +12+i +(p−kb)2 +⊥ +αs +ϱx+ +12 − 1 +� +α(α′a + α)βbs + (α′a + α)(p − kb)2 +⊥ − α(p + k′a)2 +⊥ + iϵ(α′a + α) +− e−i α +σt ei(β′ +b+ +(p−k′ +b)2 +⊥ +αs +)ϱx+ +12 − 1 +[αβbs + (p − kb)2 +⊥ + iϵ] +� +(p − k′ +b)2 +⊥e−i(p,x12)⊥ +αβbs − p2 +⊥ + (p − kb)2 +⊥ + iϵ +(11.32) +Again, at α ∼ α′ +a this integral is of order of λ whereas at small α ≪ α′ +a it turns to +8π2 +� ∞ +0 +d−α +α +� +d−p⊥ +� +eiαϱx− +12+iβbϱx+ +12+i +(p−kb)2 +⊥ +αs +ϱx+ +12 − 1 − e−i α +σt +� +ei(βb+ +(p−kb)2 +⊥ +αs +)ϱx+ +12 − 1 +�� +× +(p − kb)2 +⊥e−i(p,x12)⊥ +[αβbs + (p − kb)2 +⊥ + iϵ][αβbs − p2 +⊥ + (p − kb)2 +⊥ + iϵ] += 4π +� ∞ +0 +dt +t +� +d−p⊥ +� +ei(1+ +(p−kb)2 +⊥ +t +)βbϱx+ +12� +e +i +t +βbs ϱx− +12 − e +i +t +σtβbs � ++ e +−i +t +σtβbs − 1 +� +× +(p − kb)2 +⊥ei(p,x12)⊥ +[t + (p − kb)2 +⊥ + iϵ][t − p2 +⊥ + (p − kb)2 +⊥ + iϵ] = O(λt) +(11.33) +– 51 – + +since the integral over t converges at t ∼ Q2 +⊥. +Thus, +I1a(α′ +a, k′ +a⊥, βb, kb⊥, x2, x1) − Ieik +Fig. 11a,b(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) +(11.34) += I1a(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) − Ieik +Fig. 11a,b(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) + O(σt) +Similarly, one can demonstrate that +I1b(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) − Ieik +Fig.12e,f(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) +(11.35) += I1b(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) − Ieik +Fig.12e,f(α′ +a, k′ +a⊥, βb, kb⊥, x12⊥) + O(σp) +so we get +I1(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) − [Ieik +Fig. 11a,b + Ieik +Fig.12e,f](α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) − (x∥ +12 → 0) += O +� Q2 +⊥ +σtβbs +� ++ O +� Q2 +⊥ +σpα′as +� +∼ O(λp) + O(λt) +(11.36) +By projectile↔target replacement we get +I2(αa, ka⊥, β′ +b, k′ +b⊥, x1, x2) − [Ieik +Fig. 11e,f + Ieik +Fig.12a,b](αa, ka⊥, β′ +b, k′ +b⊥, x1, x2)] − (x∥ +12 → 0) += O +� Q2 +⊥ +σtβ′ +bs +� ++ O +� Q2 +⊥ +σpαas +� +∼ O(λp) + O(λt) +(11.37) +This justifies the calculation of the coefficient function in Eq. (6.15) at x∥ +12 = 0. +11.5 +Diagrams with correction field ¯C +In this Section we demonstrate that diagrams in the the background field ¯C lead to power +corrections. +The correction fields ¯Cµ are given by Eq. +(4.9) and we will also use the +expressions for field strengths 13 from Ref. [6] +¯Cµν ≡ Fµν(A) − Fµν( ¯A) − Fµν( ¯B), +(11.38) +g ¯C−i(x+, x−) = i +2 +� x+ +−∞ +dx′+ +� x− +−∞ +dx′−(x − x′)− +× +� +∂i[U + +k(x′−), V −k(x′+)] − ∂k[U +k(x′−), V −i(x′+)] + ∂k[U +i(x′−), V −k(x′+)] +� +∼ Q3 +⊥ +√s , +g ¯C+i(x+, x−) = +− i +2 +� x− +−∞ +dx′− +� x+ +−∞ +dx′+(x − x′)+ +× +� +∂i[U + +k(x′−), V −k(x′+)] + ∂k[U +k(x′−), V −i(x′+)] − ∂k[U +i(x′−), V −k(x′+)] +� +∼ Q3 +⊥ +√s , +g ¯C+−(x) = +− i[Uj(x−, x⊥), V j(x+, x⊥)] ∼ Q2 +⊥, +g ¯Cik(x) = Uik(x−, x⊥) + Vik(x+, x⊥) − i +� +Ui(x−, x⊥)Vk(x+, x⊥) − i ↔ k +� +∼ Q2 +⊥ +There are two types of diagrams with background field ¯C shown in Fig. 14. +13Note that ¯Cµν ̸= ∂µ ¯Cν − ∂ν ¯Cµ − ig[ ¯Cµ, ¯Cν] +– 52 – + +(c) +(a) +x 1 +x 2 +(b) +z +x 1 +x 2 +z +x 1 +x 2 +Figure 14. Typical diagrams in the background correction field ¯C. +Let us start with the first one. We get +¯C−i(x2)⟨F + +i (x2)F − +j(x1)⟩ ¯C+j(x1) = x2 +12⊥gij + 2xi +12xj +12 +π2(−x2 +12 − iϵx0 +12)3 ¯C− +i(x2) ¯C+ +j(x1) +(11.39) +≃ +� +gij + 2xi +12xj +12 +x2 +12⊥ +� +1 +x4 +12⊥ +¯C− +i(x2) ¯C+ +j(x1) ∼ O +�Q6 +⊥ +s3 +� +× U −iV +i(x2)U −jV +j(x1) +since ¯C−i, ¯C+j ∼ Q3 +⊥ +√s , see Eq. (11.38). +Next we consider diagram in Fig. 14b where we replaced Feynman propagator by the +retarded one according to Eq. (11.5). +V −i(x2)⟨(D+Ai − DiA+)(x2)(D−Aj − DjA−)(x1)⟩U +j(x1) = += +− V −i(x2)(x2|(p+δα +i − pig+α) +1 +p2 + iϵp0 +� +[{pλ, ¯Cλ} + ¯C2]gαβ + 2i ¯Cαβ� +× ˜δ+(p)(p−δβ +j − pjgβ−)|x1)U +j(x1) +(11.40) +Looking at power counting for the correction fields (4.9) and (11.38) we see that the largest +contribution to the r.h.s. of Eq. (11.40) comes from the term +V −i(x2)⟨(D+Ai − DiA+)(x2)(D−Aj − DjA−)(x1)⟩U +j(x1) = += +− V −i(x2|p+ +1 +p2 + iϵp0 +� +{pλ, ¯Cλ}gij + ¯C2gij + 2i ¯Cij�˜δ+(p)p−|x1)U +j(x1)(11.41) +Let us estimate the term with ¯Cij. It is similar to Eq. (6.3), only instead of U +i 1 +p2 V −j ≥ m2 +⊥ +we have here ¯Cij ∼ m4 +⊥ +s +or ¯C+ ¯C− ∼ m4 +⊥ +s , see Eq. (4.9). Next, let us consider term with +{pλ, ¯Cλ} = {p+, C−} + {p−, C+} + {pi, Ci}. From Eq. (4.9) we see that the last term is +O +� m2 +⊥ +s +� +with respect to the first two terms so we get 14 +(x2|p+ +1 +p2 + iϵp0 +({p+, C−} + {p−, C+})˜δ+(p)p−|x1)ab +(11.42) += (x2| 1 +p2 +� +{p+, i∂+C−} + {p−, i∂+C+})p− + ({p+, C−} + {p−, C+})p2 +⊥ +2 +� +˜δ+(p)|x1)ab +14For the estimate of power corrections the exact form of singularity in the gluon propagator is not +important +– 53 – + +Since ∂+, ∂− ∼ √s the second term is O +� p2 +⊥ +s +∼ m2 +⊥ +s +� +in comparison to the first one, the +r.h.s. of Eq. (11.42) reduces to +(x2| 1 +p2 +� +{p+, i∂+C−} + {p−, i∂+C+})p−˜δ+(p)|x1)ab +(11.43) += 1 +2 +� +d−α′ +ad−k′ +a⊥d−βbd−kb⊥ +e−ik′ +ax2−ikbx1 +α′aβ′ +b +[U + +i (α′ +a, k′ +a⊥), V −i(βb, kb⊥)]ab +� +d−αd−βd−p⊥ +× eiαϱx− +12+iβϱx+ +12−i(p,x12)⊥ +θ(β − βb)(−α′ +a) +(α + α′a)βs − (p + ka)2 +⊥ +� +1 + α +α′a +− β +βb +� +˜δ +� +α − (p − k′ +a)⊥ +(β − βb)s +� += +− +� +d−α′ +ad−k′ +a⊥d−βbd−kb⊥ +e−ik′ +ax2−ikbx1 +2βb +[U + +i (α′ +a, k′ +a⊥), V −i(βb, kb⊥)]ab +� +d−βd−p⊥ +× +θ(β)βei +(p−kb)2 +⊥ +βs +ϱx− +12+i(β+βb)ϱx+ +12−i(p,x12)⊥ +α′aβ(β + βb)s + (p − kb)2 +⊥(β + βb) − (p + k′a)2 +⊥β +� β +βb +− (p − kb)2 +⊥ +α′aβs +� += +� +d−α′ +ad−k′ +a⊥d−βbd−kb⊥[U + +i (α′ +a, k′ +a⊥), V −i(βb, kb⊥)]abe−iα′ +aϱx− +2 +i(k′ +a,x2)⊥−iβbϱx− +1 +i(kb,x1)⊥ +× 1 +4π +� +d−p⊥ +� ∞ +0 +dt[t2 − Q2 +ab(p − kb)2 +⊥]eit−1(p−kb)2 +⊥α′ +aϱx− +12+i +� +Q−2 +a′bt+1 +� +βbϱx+ +12−i(p,x12)⊥ +Q4 +a′b[t(t + Q2 +a′b) − (p − kb)2 +⊥(t + Q2 +a′b) + (p + k′a)2 +⊥t] +This should be compared to the leading order contribution (6.7) +∼ U + +i (α′ +a, k′ +a⊥)V −j(βb, kb⊥)×logs. It is clear that contribution from t ≲ Q2 +a′b to the last line +in Eq. (11.43) is O +� Q2 +⊥ +Q2 +a′b +� +, and if t ≫ Q2 +a′b the last line in the above equation reduces to +1 +4πQ4 +a′b +� +d−p⊥ +� ∞ +0 +dt eit−1(p−kb)2 +⊥α′ +aϱx− +12+iQ−2 +a′btβbϱx+ +12−i(p,x12)⊥ +(11.44) += +− ie−i(kb,x12)⊥ +16π2Q4 +a′b +� ∞ +0 +dt +t +α′aϱx− +12 +e +iQ−2 +a′btβbϱx+ +12−it +x2 +12⊥ +4α′aϱx− +12 += +− +iα′ +aϱx− +12e−i(kb,x12)⊥ +π2Q4 +a′b(x2 +12⊥ − x2 +12∥)2 +which is a power correction ∼ Q4 +⊥ +Q4 +ab since α′ +aϱx− +12 ∼ 1. Thus, the contribution of the diagram +in Fig. 14b is a power correction. +Finally, let us consider diagram in Fig. 14c. We get +¯C−i(x2)⟨(D+Ai − DiA+)(x2)(D−Aj − DjA−)(x1)⟩ ¯BU +j(x1) = += +− ¯C−i(x2)(x2|(p+δα +i − pig+α) +1 +p2 + iϵp0 +¯Bαβ2i˜δ+(p)(p−δβ +j − pjgβ−)|x1)U +j(x1) += +− 2i ¯C−i(x2)(x2| +p+ +p2 + iϵp0 +V−i˜δ+(p)pj|x1)U + +j(x1) += +− i ¯C−;a +i (x2)U +;b +j (x1) +� +d−βbd−kb⊥V −i;ab(βb, kb⊥)eiβbϱx+ +1 −i(kb,x1)⊥ +× +� +d−αd−βd−p⊥ eiαϱx− +12+i(β+βb)ϱx+ +12−i(p+kb,x12)⊥ +ϱ(β + βb)(p + kb)j +α(β + βb)s − (p + kb)2 +⊥ +θ(β) +β +˜δ +� +α − p2 +⊥ +βs +� += +− i +2πϱ ¯C−;a +i (x2)U +j;b(x1) +� +d−βbd−kb⊥V −i;ab(βb, kb⊥)eiβbϱx+ +1 −i(kb,x1)⊥ +× +� ∞ +0 +dβ +� +d−p⊥ ei +p2 +⊥ +βs ϱx− +12+i(β+βb)ϱx+ +12−i(p+kb,x12)⊥ +(β + βb)(p + kb)j +p2 +⊥(β + βb) − (p + kb)2 +⊥β +(11.45) +– 54 – + +Since ϱ ¯C−i ∼ m3 +⊥ in comparison to U +iV −k ∼ sm2 +⊥ in the leading term, we need an extra +s/m⊥ from the last line. At finite β ∼ β′ +b the integral in the last line is ∼ kj +a ∼ m⊥ so we +need to check this integral at very small or very large β. It is convenient to make change +of variables p⊥ = k⊥ +√ +b: +� ∞ +0 +dβ +� +d−p⊥ ei +p2 +⊥ +βs ϱx− +12+i(β+βb)ϱx+ +12−i(p+kb,x12)⊥ +(β + βb)(p + kb)j +p2 +⊥(β + βb) − (p + kb)2 +⊥β +(11.46) += +� ∞ +0 +dβ +� +d−k⊥ ei +k2 +⊥ +s ϱx− +12+i(β+βb)ϱx+ +12−i(k√β+kb,x12)⊥ +(β + βb)(k√β + kb)j +k2 +⊥(β + βb) − (k√β + kb)2 +⊥ +As β → 0 we get +eiβbϱx+ +12−i(kb,x12)⊥ +� ∞ +0 +dβ eiβϱx− +12 +� +d−k⊥ +kj +bei +k2 +⊥ +s ϱx− +12 +k2 +⊥ − +k2 +b⊥ +βb +∼ +βbkj +b +βbϱx+ +12 +ln +Q2 +ab +k2 +b⊥α′aϱx− +12 +∼ βbm⊥ +(11.47) +since βbϱx+ +12 ∼ αaϱx− +12 ∼ 1. Conversely, as β → ∞ one obtains +� ∞ +0 +dβ +� +d−p⊥ ei +p2 +⊥ +βs ϱx− +12+iβϱx+ +12−i(p+kb,x12)⊥ +(p + kb)j +k2 +b⊥ + 2(p, kb)⊥ +(11.48) += 2e−i(kb,x12)⊥ +� +d−p⊥ +� +p2 +⊥x− +12 +sx+ +12 +K1 +� +p⊥ +� +−x2 +12∥ +�(p + kb)je−i(p,x12)⊥ +k2 +b⊥ + 2(p, kb)⊥ +∼ +m⊥ +ρx+ +12 +∼ βbm⊥ +Thus, we got m⊥ instead of an extra s/m⊥ needed to compensate the smallness in Eq. +(11.45) so the contribution of the diagram in Fig. 14c is a power correction O +� m2 +⊥ +s +� +in +comparison to the leading term (6.7). +Summarizing, we demonstrated that the diagrams with correction fields (4.9) are power +corrections ∼ O +� 1 +ζ +� +. +11.6 +Necessary integrals +11.6.1 +Integrals for virtual diagrams +Master integral for virtual diagrams can be taken from integrals (11) - (18) of Ref. [36]. At +k2 +1, k2 +2 < 0 and (k1 + k2)2 > 0 it reads +� d−4p +i +1 +[(p + k1)2 + iϵ][(p − k2)2 + iϵ](p2 + iϵ) = +(11.49) += +1 +16π2κ +� +Li2( +−k2 +1 +(k1, k2) + κ) + Li2( +−k2 +2 +(k1, k2) + κ) + 1 +2 ln k2 +2 +k2 +1 +ln k2 +2 + k1 · k2 + κ +k2 +1 + k1 · k2 + κ ++ 1 +2 ln +� +k2 +1 +(k1, k2) + κ + iϵ +� +ln +� +k2 +2 +(k1, k2) + κ + iϵ +� ++ π2 +6 +� +where κ ≡ +� +(k1 · k2)2 − k2 +1k2 +2. In our kinematics k1 · k2 ≫ k2 +1, k2 +2 so +� d−4p +i +1 +[(p + k1)2 + iϵ][(p − k2)2 + iϵ](p2 + iϵ) +(11.50) += +1 +32π2(k1 · k2) ln −2(k1 · k2) − iϵ +−k2 +1 +ln −2(k1 · k2) − iϵ +−k2 +2 ++ O +� k2 +1, k2 +2 +(k1 · k2) +� +– 55 – + +We will also need similar integral with cut propagators. The standard calculation yields +� +d−4p ˜δ(p + k1)2(p + k1)0 +1 +p2 ˜δ(p − k2)2θ(k2 − p)0 +(11.51) += +− θ(k1 + k2)2θ(k1 + k2)0 +32πκ +ln k1 · k2 + κ +k1 · k2 − κ ≃ +− θ(k1 + k2)2θ(k1 + k2)0 +32πk1 · k2 +ln 4(k1 · k2)2 +k2 +1k2 +2 +in agreement with Eq. (11.50). Note that at k2 +1, k2 +2 < 0 the denominator 1/p2 in the l.h.s. +of this equation is not singular. +Using Eqs. (11.50) and (11.51) it is easy to obtain +� +d−4p +αa +(p + ka)2 + iϵ +s +p2 + iϵ +βb +(p − kb)2 + iϵ = +i +16π2 +� +ln −αaβbs − iϵ +k2a⊥ +ln −αaβbs − iϵ +k2 +b⊥ ++ π2 +3 +� +� +d−4p ˜δ(p + ka)2θ(−β) αaβbs +p2 − iϵ +˜δ(p − kb)2θ(α) = +− θ(−αa)θ(−βb) +8π +ln (αaβbs)2 +k2a⊥k2 +b⊥ +(11.52) +To compare to the calculation of the “production” diagrams in Sect. 6, we need also an +explicit calculation of the sum of the “virtual” integrals in Eq. +(5.20). +Performing the +integration over α, we get +16π2 +� d−4p +i +� +αaβbs +[(α + αa)βs − (p + ka)2 +⊥ + iϵ](αβs − p2 +⊥ + iϵ)[α(β − βb)s − (p − kb)2 +⊥ + iϵ] ++ ˜δ[(αa + α)βs − (p + ka)2]θ(−β) +αaβbs +αβs − p2 +⊥ − iϵ +˜δ[α(β − βb)s − (p − kb)2 +⊥]θ(α) +� += +− 4π +� +d−p⊥ +� 1 +0 +du +¯uQ2 +ab +[¯up2 +⊥ + u(p − kb)2][¯u(p + ka)2 +⊥ + u(p − kb)2 +⊥ − Q2 +ab¯uu] += ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ ++ π2 +3 ++ O(λ) +(11.53) +where Q2 +ab is defined in Eq. (5.15). To get the last line in the above equation, we performed +the integration over p⊥ +Eq. (11.53) = +� 1 +0 +dudv +−Q2 +ab +k2a⊥v(1 − ¯uv) + k2 +b⊥u − (Q2 +ab − 2(ka, kb)⊥)uv += +� 1 +0 +dudv +1 +av(1 − ¯uv) + bu + uv + O(a, b) += +� 1 +0 +dudv +� +1 +av + bu + uv + +a¯uv2 +[av + bu + uv][av(1 − ¯uv) + bu + uv] +� += +� 1 +0 +dudv +� +1 +av + bu + uv + +a¯uv2 +[av + bu + uv][av(1 − ¯uv) + bu + uv] +� ++ O(a, b) += +� 1 +0 +dudv +� +1 +av + bu + uv + +a +(a + u)(a¯v + u) +� ++ O(a, b) = ln a ln b + π2 +3 ++ O(a, b) +where a = −k2 +a⊥/Q2 +ab and b = −k2 +b⊥/Q2 +ab. +– 56 – + +11.6.2 +Integrals for “production” diagrams +To calculate the integral (6.21) we will represent it as follows +4π +� +d−p⊥ +e−i(p,x)⊥Q2 +ab +Q2 +abp2 +⊥ + (p + ka)2 +⊥(p − kb)2 +⊥ +ln +−Q2 +abp2 +⊥ +(p + ka)2 +⊥(p − kb)2 +⊥ += 4π +� +d−p⊥ +e−i(p,x)⊥Q2 +ab +Q2 +abp2 +⊥ + k2a⊥k2 +b⊥ +ln +−Q2 +abp2 +⊥ +(p + ka)2 +⊥(p − kb)2 +⊥ ++ 4π +Q2 +ab +� +d−p⊥ +e−i(p,x)⊥[k2 +a⊥k2 +b⊥ − (p + ka)2 +⊥(p − kb)2 +⊥] +� +p2 +⊥ + +k2a⊥k2 +b⊥ +Q2 +ab +�� +p2 +⊥ + (p+ka)2 +⊥(p−kb)2 +⊥ +Q2 +ab +� ln +−Q2 +abp2 +⊥ +(p + ka)2 +⊥(p − kb)2 +⊥ += 4π +� +d−p⊥ +e−i(p,x)⊥Q2 +ab +Q2 +abp2 +⊥ + k2a⊥k2 +b⊥ +ln +−Q2 +abp2 +⊥ +(p + ka)2 +⊥(p − kb)2 +⊥ ++ O(λ) +(11.54) +To get the last line we note that the integral in the third line is at best logarithmic at small +p⊥. Next, we split the integral in the last line in three parts +8π +� +d−p⊥ +e−i(p,x)⊥ +p2 +⊥ + +k2a⊥k2 +b⊥ +Q2 +ab +� +ln −Q2 +abp2 +⊥ +k2a⊥k2 +b⊥ +− ln (p + ka)2 +⊥ +k2a⊥ +− ln (p − kb)2 +⊥ +k2 +b⊥ +� +, +(11.55) +use integrals (m2 = − +k2 +a⊥k2 +b⊥ +Q2 +ab +) +4π +� +d−2p e−i(p,x) +p2 − m2 ln p2 +m2 = 1 +2 +� +ln m2x2 +4 ++ 2γ +�2 ++ π2 +3 ++ O(m2x2), +4π +� +d−2pe−i(p,x) +p2 +ln (p + k)2 +k2 += 1 +2 +� +ln k2x2 +4 ++ 2γ +�2 − IK(k, x) +(11.56) +where IK is defined in Eq. (6.26), and get +4π +� +d−p⊥ +e−i(p,x)⊥Q2 +ab +Q2 +abp2 +⊥ + (p + ka)2 +⊥(p − kb)2 +⊥ +ln +−Q2 +abp2 +⊥ +(p + ka)2 +⊥(p − kb)2 +⊥ += ln −Q2 +ab +k2a⊥ +ln −Q2 +ab +k2 +b⊥ +− 1 +2 +� +ln −Q2 +abx2 +⊥ +4 ++ 2γ +�2 ++ π2 +3 + IK(ka⊥, x⊥) + IK(−kb⊥, x⊥) + O(λ) +(11.57) +For the calculation of the integral (11.8) we need also +4π +� d−2p +p2 +� +e−ipx − 1 +� +ln (p + q)2 +⊥ +p2 += +� +4π +� +d−2p +� +e−ipx − 1 +� +1 +(p2)1+δ[(q + p)2]−δ +� +(δ) +(11.58) +where +� +.. +� +(δ) denotes the first term of the expansion in power of δ. After some algebra, one +– 57 – + +obtains +4π +� d−2p +p2 +� +e−ipx − 1 +� +ln (p + q)2 +⊥ +p2 += +� +4π +� +d−p +� +e−ipx − 1 +� +1 +(p2)1+δ[(q + p)2]−δ +� +(δ) += 2 +� 1 +0 +du +u +� +1 − ei(qx)u� +K0(qx +√ +¯uu) ++ +� +δ +Γ(1 − δ)Γ(1 + δ) +� 1 +0 +du u−δ−1¯uδ� +− (ln q2x2¯uu − ln 4 + 2γ) − 2K0(qx +√ +¯uu) +�� +δ += 2 +� 1 +0 +du +u +� +1 − ei(qx)u� +K0(qx +√ +¯uu) − +� 1 +0 +du +u +� +2K0(qx +√ +¯uu) − +� +ln q2x2¯uu +4 ++ 2γ +�� += +− +� 1 +0 +du +u +� +ln q2x2¯uu +4 ++ 2γ + 2ei(q,x)uK0(qx +√ +¯uu) +� += +− IK(q⊥, x⊥) +(11.59) +where IK is defined in Eq. (6.26). +11.7 +Coefficient function from the calculation with background gluons on the +mass shell +In this Section we double-check the calculation of the coefficient function (9.2) using back- +ground fields on the mass shell (9.3). +For the background fields on the mass shell the +hadronic tensor (4.20) is parametrized as follows +W (x2, x1) − W σp,σt +eik +(x2, x1) = +� +d−α′ +ad−β′ +bd−αad−βbe−iα′ +aϱx− +2 −iαaϱx− +1 e−iβ′ +bϱx+ +2 −iβbϱx+ +1 +× U +,b +i (α′ +a)V −i,a(β′ +b)U +,b +j (αa)V −j,a(βb)[I − Iσp,σt +eik +](α′ +a, αa, β′ +b, βb, x2, x1) +(11.60) +As demonstrated below (see Sect. 11.7.4), for the massless background fields there are no +soft contributions, and Glauber gluons cancel as usual. +11.7.1 +Virtual contributions +Let us again start from the virtual diagram and take βb < 0. From Eq. (8.1) we get +Ivirt msh +Fig.5 += +− 16π2 +� d−p +i +� +αaβbs(αβs − p2 +⊥ + iϵ)−1 +[(α + αa)βs − p2 +⊥ + iϵ][α(β − βb)s − p2 +⊥ + iϵ] ++ ˜δ[(αa + α)βs − p2]θ(−β) +αaβbs +αβs − p2 +⊥ − iϵ +˜δ[α(β − βb)s − p2 +⊥]θ(α) +� += +− 4π +� 1 +0 +du +� d−p⊥ +p2 +⊥ +¯uαa|βb|s +¯uuαa|βb|s + p2 +⊥ + iϵ = +− 1 +ε2 +�(αa + iϵ)|βb|s +4π +�ε Γ(1 − ε)Γ2(1 + ε) +Γ(1 + 2ε) += +− 1 +ε2 − 1 +ε +� +ln (αa + iϵ)|βb|s +4π ++ γ +� +− 1 +2 +� +ln (αa + iϵ)|βb|s +4π ++ γ +�2 ++ π2 +12 + π2 +6 ++ O(ε) +(11.61) +– 58 – + +where ε = d⊥ +2 − 1 = d +2 − 2. It is easy to see that for βb > 0 the singularity is changed to +ln −(αa+iϵ)βbs +4π +. Adding the similar contribution of Fig. 6 diagram, we obtain +Ivirt +mass shell = +(11.62) += +− 2 +ε2 − 1 +ε +� +ln −(αa + iϵ)(βb + iϵ)s +4π ++ γ +� +− 1 +2 +� +ln −(αa + iϵ)(βb + iϵ)s +4π ++ γ +�2 +− 1 +ε +� +ln −(α′ +a + iϵ)(β′ +b + iϵ)s +4π ++ γ +� +− 1 +2 +� +ln −(α′ +a + iϵ)(β′ +b + iϵ)s +4π ++ γ +�2 ++ O(ε) +Let us consider now virtual eikonals. From Eq. (11.8) we get +Ieik +Fig. 11c,d(βb, kb) +��� +kb⊥=0 += +− 8π2 +� ∞ +0 +d−α e−i α +σt +� d−p⊥ +p2 +⊥ +βbs +α(βb + iϵ)s + p2 +⊥ += +− 2 +ε2 +�−iβbσts +4π +�ε +Γ(1 − ε)Γ(1 + ε) += +− 1 +ε2 − 1 +ε[ln(−iβbσts) − ln 4π] − 1 +2[ln(−iβbσts) − ln 4π]2 − π2 +6 ++ O(ε) +(11.63) +Similarly, +Ieik +Fig. 11g,h(β′ +b, k′ +b) +��� +kb⊥=0 = +− 8π2 +� ∞ +0 +d−α +� d−p⊥ +p2 +⊥ +ei α +σt +β′ +bs +α(β′ +b + iϵ)s − p2 +⊥ += +− 1 +ε2 − 1 +ε[ln(−iβ′ +bσts) − ln 4π] − 1 +2[ln(−iβ′ +bσts) − ln 4π]2 − π2 +6 ++ O(ε) +where −iβ′ ≡ −iβ′ + ϵ etc. +Next, we can get contributions of virtual diagrams in Fig. diagrams in Fig. 11c,d and +Fig. 11g,h by usual projectile↔target replacements (6.28) so the final result for virtual +eikonal TMD contributions on the mass shell reads +Ivirt eik +mass shell = Ieik +Fig. 11c,d(βb) + Ieik +Fig. 11g,h(β′ +b) + Ieik +Fig. 12c,d(αa) + Ieik +Fig. 12g,h(α′ +a) += +− 4 +ε2 − 1 +ε +� +ln −iα′ +aσps +4π ++ ln −iαaσps +4π ++ ln −iβ′ +bσts +4π ++ ln −iβbσts +4π +� +(11.64) +− 1 +2 +� +ln2 −iα′ +aσps +4π ++ ln2 −iαaσps +4π ++ ln2 −iβ′ +bσts +4π ++ ln2 −iβbσts +4π +� +− 2π2 +3 ++ O(ε) +– 59 – + +11.7.2 +Production terms minus TMD matrix elements +Next we will calculate the difference J1(α′ +a, k′ +a⊥, βb, kb⊥ = 0, x1, x2) from Eq. +(6.15) at +k′ +a⊥ = 0 and kb⊥ = 0. To avoid confusing singularities, we keep x∥ +12 ̸= 0 in this calculation. +J1(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) +��� +k′a⊥=kb⊥=0 = +� +I1(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) +− Ieik +Fig.12e,f(α′ +a, k′ +a⊥, x1, x2) − Ieik +Fig.11a,b(βb, kb⊥, x1, x2) +� +k′a⊥=k′b⊥=0 += 8π2 +� +d−p⊥ e−i(p,x12)⊥ +�� +d−β +�θ(β) +β2s +(βb − β)(α′ +aβs + p2 +⊥) +(α′a + iϵ)(βb + iϵ)p2 +⊥ +eiβϱx+ +12+i +p2 +⊥ +βs ϱx− +12 ++ θ(β) +β2s +(βb − β)p2 +⊥ +(α′a + iϵ) +� +(α′a + iϵ)(βb − β)βs − βbp2 +⊥ +�eiβϱx+ +12−iα′ +aϱx− +12+i +p2 +⊥ +βs ϱx− +12 ++ +θ(β − βb)(β − βb)[α′ +a(β − βb)s + p2 +⊥] +p2 +⊥ +� +(α′a + iϵ)β(β − βb)s + βbp2 +⊥ +� +(βb + iϵ)e +iβϱx+ +12+i +p2 +⊥ +(β−βb)s ϱx− +12 +� +− 1 +p2 +⊥ +� ∞ +0 +d−β e +−i β +σp +ei +p2 +⊥ +βs ϱx− +12 +β2(α′a + iϵ)s +� +α′ +aβs + p2 +⊥ + +p4 +⊥e−iα′ +aϱx− +12 +α′aβs − p2 +⊥ + iϵ +� +− 1 +p2 +⊥ +� ∞ +0 +d−α ei α +σt +ei +p2 +⊥ +αs ϱx+ +12 +α2(βb + iϵ)s +� +αβbs − p2 +⊥ + +p4 +⊥eiβbϱx+ +12 +αβbs + p2 +⊥ + iϵ +�� +(11.65) +It is convenient to rearrange terms in the r.h.s. as follows +J1(α′ +a, k′ +a⊥, βb, kb⊥, x1, x2) +��� +k′a⊥=kb⊥=0 = J(1) +1 ++ J(2) +1 ++ J(3) +1 ++ J(4) +1 ++ J(5) +1 +(11.66) +where +J(1) +1 (α′ +a, βb, x1, x2) +(11.67) += +8π2 +� d−p⊥ +p2 +⊥ +e−i(p,x12)⊥ +�� ∞ +0 +d−β +β ei +p2 +⊥ +βs ϱx− +12� +eiβϱx+ +12 − e +i β +σp � +− +� ∞ +0 +d−α +α ei α +σt ei +p2 +⊥ +αs ϱx+ +12 +� +, +J(2) +1 (α′ +a, βb, x1, x2) +(11.68) += 8π2 +� +d−p⊥ e−i(p,x12)⊥ +�� ∞ +0 +d−α ei α +σt +ei +p2 +⊥ +αs ϱx+ +12 +α2s(βb + iϵ) − +� ∞ +0 +d−β eiβϱx+ +12+i +p2 +⊥ +βs ϱx− +12 +(βb + iϵ)p2 +⊥ +� +, +J(3) +1 (α′ +a, βb, x1, x2) +(11.69) += 8π2 +� +d−p⊥ e−i(p,x12)⊥ +� ∞ +0 +d−β +� +ei +p2 +⊥ +βs ϱx− +12 +(α′a + iϵ)β2s +� +eiβϱx+ +12 − e +−i β +σp � +− +eiβϱx+ +12+i +p2 +⊥ +βs ϱx− +12 +βs(βb + iϵ)(α′a + iϵ) +� +, +J(4) +1 (α′ +a, βb, x1, x2) = 8π2 +� +d−p⊥ e−i(p,x12)⊥ +�� +d−β e +iβϱx+ +12+i +p2 +⊥ +(β−βb)s ϱx− +12 θ(β − βb) +p2 +⊥(βb + iϵ) +× (β − βb)[α′ +a(β − βb)s + p2 +⊥] +(α′a + iϵ)β(β − βb)s + βbp2 +⊥ +− +� ∞ +0 +d−α +α +e−i α +σt +p2 +⊥ei +p2 +⊥ +αs ϱx+ +12+iβbϱx+ +12 +(αβbs + iϵ)(αβbs + p2 +⊥ + iϵ) +� +(11.70) +– 60 – + +and +J(5) +1 (α′ +a, βb, x1, x2) = +8π2 +α′a + iϵ +� +d−p⊥ +� +d−β +�� +eiβϱx+ +12 − e +i β +σp � p2 +⊥ei +p2 +⊥ +βs ϱx− +12−iα′ +aϱx− +12 +β2s[α′aβs − p2 +⊥ + iϵ] ++ +p4 +⊥eiβϱx+ +12−iα′ +aϱx− +12+i +p2 +⊥ +βs ϱx− +12 +βs +� +(α′a + iϵ)(βb − β)βs − βbp2 +⊥ +�� +(α′a + iϵ)βs − p2 +⊥ +� +� +e−i(p,x12)⊥ +(11.71) +Let us start with J(1) +1 +term. After changing α = p2 +⊥ +βs in the last term it takes the form +J(1) +1 (α′ +a, βb, x2, x1) +(11.72) += 4π +� d−p⊥ +p2 +⊥ +e−i(p,x12)⊥ +� ∞ +0 +dβ +β +� +ei +p2 +⊥ +βs ϱx− +12+iβϱx+ +12 − ei +p2 +⊥ +βs ϱx− +12−iβϱδ+ − ei +p2 +⊥ +βs ϱδ−+iβϱx+ +12 +� +Using integral +4π +� d−p⊥ +p2 +⊥ +e−i(p,x12)⊥ +� ∞ +0 +dβ +β ei +p2 +⊥ +βs ϱx− +12+iβϱx+ +12 = (πx2 +12⊥)−ϵ +� 1 +0 +du +Γ(ϵ)uϵ−1 +� +u + 2(ix+ +12−ϵ)(ix− +12−ϵ) +x2 +12⊥ +�ϵ += +1 +ϵ2 − ln[2π(ix+ +12 + ϵ)(ix− +12 + ϵ)] + γ +ϵ ++ 1 +2 +� +ln(πx2 +12⊥) + γ +� +× +� +ln(πx2 +12⊥) + γ + 2 ln 2π(ix+ +12 − ϵ)(ix− +12 − ϵ) +x2 +12⊥ +� +− π2 +12 + O +�x+ +12x− +12 +x2 +12⊥ +� ++ O(ϵ) (11.73) +we get +J(1) +1 (α′ +a, βb, x2, x1) = +− 1 +ϵ2 + 1 +ϵ +� +ln 2πδ+δ− + γ +� +(11.74) +− 1 +2 +� +ln(πx2 +12⊥) + γ +�� +ln(πx2 +12⊥) + γ + 2 ln 2πδ+δ− +x2 +12⊥ +� ++ π2 +12 + O(ϵ) + O(λp) + O(λt) +In the next Section we will prove now that all J(i) +1 +except J(1) +1 +are power corrections so +the result for “production - eikonal” terms is +J1(α′ +a, βb, x2, x1) + J2(αa, β′ +b, x2, x1) = +− 2 +ϵ2 + 2 +ϵ +� +ln 2πδ+δ− + γ +� +(11.75) +− +� +ln(πx2 +12⊥) + γ +�� +ln(πx2 +12⊥) + γ + 2 ln 2πδ+δ− +x2 +12⊥ +� ++ π2 +6 ++ O(ϵ) + O(λp) + O(λt) +where we have added second term obtained by projectile↔target replacements. +Now we are in a position to assemble the result for the coefficient function obtained by +– 61 – + +on-the-mass-shell calculation. We get +J1(α′ +a, βb, x2, x1) + J2(αa, β′ +b, x2, x1) + Ivirt +mass shell − Ivirt eik +mass shell +(11.76) += +− 2 +ϵ2 + 2 +ϵ +� +ln 2πδ+δ− + γ +� ++ +� +ln(πx2 +12⊥) + γ +�� +ln x2 +12⊥ + 2 ln σpσts +4 +− 3 ln π − γ +� ++ π2 +6 +− 2 +ε2 − 1 +ε +� +ln −(α′ +a + iϵ)(β′ +b + iϵ)s +4π ++ γ +� +− 1 +2 +� +ln −(α′ +a + iϵ)(β′ +b + iϵ)s +4π ++ γ +�2 +− 1 +ε +� +ln −(αa + iϵ)(βb + iϵ)s +4π ++ γ +� +− 1 +2 +� +ln −(αa + iϵ)(βb + iϵ)s +4π ++ γ +�2 ++ π2 +6 ++ 4 +ε2 + 1 +ε +� +ln −iα′ +aσps +4π ++ ln −iαaσps +4π ++ ln −iβ′ +bσts +4π ++ ln −iβbσts +4π +� ++ 1 +2 +� +ln2 −iα′ +aσps +4π ++ ln2 −iαaσps +4π ++ ln2 −iβ′ +bσts +4π ++ ln2 −iβbσts +4π +� ++ 2π2 +3 ++ O(ε) += ln2 x2 +12⊥sσpσt +4 +− ln (−iα′ +a)eγ +σt +ln (−iβ′ +b)eγ +σp +− ln (−iαa)eγ +σt +ln (−iβb)eγ +σp ++ π2 +which agrees with Eq. (9.2). +11.7.3 +J(i>1) +1 +are power corrections +Here we prove that all terms in the r.h.s. of Eq. (11.66) except for J(1) +1 +are power corrections. +Let us start from J(2) +1 +which can be rewritten as +J(2) +1 += 4π +βb +� +d−p⊥ e−i(p,x12)⊥ +� ∞ +0 +dα +α2sei +p2 +⊥ +αs ϱx+ +12� +ei α +σt − eiαϱx− +12� +(11.77) += +− +i +ϱx+ +12 +� ∞ +0 +dα +α e +−i +x2 +12⊥ +4ϱx+ +12 +αs� +eiαϱδ− − eiαϱx− +12� += +i +βbϱx+ +12 +ln x2 +12⊥ − 2x− +12x+ +12 +x2 +12⊥ + 2δ−x+ +12 += 2iα′ +a(ϱδ− − ϱx− +12) +Q2 +a′bx2 +12⊥ +≃ O(λt) +Next, after the integration over p⊥ the term J(3) +1 +turns to +J(3) +1 += +i +ϱx− +12 +� ∞ +0 +dβ +� 1 +α′aβ e +−i +x2 +12⊥ +4ϱx− +12 +βs� +eiβϱx+ +12 − e +−i β +σp � +− +1 +α′aβb +e +iβϱx+ +12−i +x2 +12⊥ +4ϱx− +12 +βs +� += +i +α′aϱx− +12 +ln x2 +12⊥ − 2x+ +12x− +12 +x2 +12⊥ + 2δ+x− +12 ++ +4 +Q2 +ab′ +1 +x2 +12⊥ − x2 +12∥ +≃ O(λp) +(11.78) +To calculate J(4) +1 +term, it is convenient to change variable β = βb + p2 +⊥ +αs in the first +– 62 – + +integral. For simplicity, we take βb > 0 and get +J(4) +1 += 8π2 +βb +� +d−p⊥ e−i(p,x12)⊥ +� ∞ +0 +d−α +� p2 +⊥ +α2s +� +eiαϱx− +12 − ei α +σt �ei +p2 +⊥ +αs ϱx+ +12+iβbϱx+ +12 +αβbs + p2 +⊥ + iϵ ++ p4 +⊥ +αs +eiαϱx− +12+i +p2 +⊥ +αs ϱx+ +12+iβbϱx+ +12 +[αβbs + p2 +⊥ + iϵ][(α′a + α)αβbs + α′ap2 +⊥] +� += +� ∞ +0 +dp2 +⊥ J0(px12⊥) +� ∞ +0 +dt +� +t +t2(t + p2 +⊥)ei +t+p2 +⊥ +t +βbϱx+ +12 +× +� +e +i +t +Q2 +a′b +α′ +aϱx− +12 − e +i +t +βbσts +� ++ p4 +⊥ +t +e +i +t +Q2 +a′b +α′ +aϱx− +12+i +p2 +⊥ +t βbϱx+ +12+iβbϱx+ +12 +(t + p2 +⊥)[(Q2 +a′b + t)t + Q2 +a′bp2 +⊥] +� +(11.79) +The first term in the r.h.s. can be represented as +� ∞ +0 +dp2 +⊥ J0(px12⊥) +� ∞ +0 +dt +p2 +⊥ +t2(t + p2 +⊥)ei +t+p2 +⊥ +t +βbϱx+ +12 +� +e +i +t +Q2 +a′b +α′ +aϱx− +12 − e +i +t +Q2 +a′b +α′ +aϱδ−� +(11.80) += 2 +� ∞ +0 +dv +v J0(v) +� ∞ +0 +dt +1 +t2(t + 1)ei t+1 +t βbϱx+ +12 +� +e +i +t +Q2 +a′bx2 +12⊥ +α′ +aϱx− +12v2 +− (x− +12 → δ−) +� +We need to check two regions: t ≪ 1 and t ≫ 1. Assuming essential t′s are small, we get +2 +� ∞ +0 +dv +v J0(v) +� ∞ +0 +dt 1 +t2 ei t+1 +t βbϱx+ +12 +� +e +i +t +Q2 +a′bx2 +12⊥ +α′ +aϱx− +12v2 +− (x− +12 → δ−) +� += 2ieiβbϱx+ +12 +βbϱx+ +12 +� ∞ +0 +dv J0(v) +�� +−2x+ +12x− +12 +x2 +12⊥ +K1 +� +v +� +−2x+ +12x− +12/x2 +12⊥ +� +− (x− +12 → δ−) +� += ieiβbϱx+ +12 +βbϱx+ +12 +ln x2 +12⊥ − x+ +12δ− +x2 +12⊥ − x+ +12x− +12 +≃ +iδ− +βbϱx2 +12⊥ += O(λt) +(11.81) +which is a power correction. However, quick look at the integral over t shows that t ∼ +vQa′bx12⊥ +� +α′aδ− +βbx+ +12 ∼ v σtβbs +Q2 +⊥ +so t ≪ 1 corresponds to v ≪ λt which gives negligible contribu- +tion to the integral (11.81). Thus, characteristic t’s in the integral (11.80) are not small. +Let us assume they are large and check it a posteriori. At t ≫ 1 we get +Eq. (11.80) = 2 +� ∞ +0 +dv +v J0(v) +� ∞ +0 +dt 1 +t3 ei t+1 +t βbϱx+ +12 +� +e +i +t +Q2 +ab′ x2 +12⊥ +α′ +aϱx− +12v2 +− (x− +12 → δ−) +� += +− 2 eiβbϱx+ +12 +(βbϱx+ +12)2 +� ∞ +0 +dv vJ0(v) +�� +− 2x+ +12x− +12 +x2 +12⊥ +� +K2 +� +v +� +−2x+ +12x− +12/x2 +12⊥ +� +− (x− +12 → δ−) +� += +− 2 eiβbϱx+ +12 +(βbϱx+ +12)2 +� +ln x2 +12⊥ − x+ +12δ− +x2 +12⊥ − x+ +12x− +12 ++ +x2 +12⊥ +x2 +12⊥ − x+ +12x− +12 +− +x2 +12⊥ +x2 +12⊥ − x+ +12δ− +� +≃ +− 2eiβbϱx+ +12 +sβb2 +(δ−)2 − (x− +12)2 +x4 +12⊥ += O(λ2 +t ) +(11.82) +which is smaller than a typical power correction. Also, in this integral v ∼ 1 and t ∼ +vQa′bx12⊥ +� +α′aδ− +βbx+ +12 ∼ v σtβbs +Q2 +⊥ ∼ 1 +λt so we justified large-t approximation. +– 63 – + +The last term in the r.h.s. of Eq. (11.79) is even smaller +� ∞ +0 +dp2 +⊥ J0(px12⊥) +� ∞ +0 +dv e +iv +p2 +⊥ +Q2 +a′b +α′ +aϱx− +12+i 1 +v βbϱx+ +12+iβbϱx+ +12 +v(v + 1)[Q2 +a′b(v + 1) + v2p2 +⊥] +p⊥x12⊥∼1 +≃ +� ∞ +0 +dp2 +⊥ J0(px12⊥) +� ∞ +0 +dv +ei 1 +v βbϱx+ +12+iβbϱx+ +12 +v(v + 1)[Q2 +a′b(v + 1) + v2p2 +⊥] += 2 +� ∞ +0 +dv +v2 +v + 1ei(v+1)βbϱx+ +12K0 +� +Qa′bx12⊥ +� +v(v + 1) +� +∼ Q6 +⊥ +Q6 +(11.83) +Thus, J(4) +1 += O(λ2 +t ) and can be neglected. +Next, it is easy to see that J(5) +1 +of Eq. (11.71) differs from the first two lines in Eq. +(11.79) for J(4) +1 +by projectile↔ target replacements α′ +a ↔ βb, σp ↔ σt, x+ +12 ↔ x− +12 so we have +J(5) +1 += O(λ2 +p). +11.7.4 +sG-contribution +The soft-Glauber contribution to virtual diagram in Fig. (5) can be easily obtained from +Eq. (8.4) by setting k′a⊥ = kb⊥ = 0 +Ivirt sG ms +Fig.5 += +− µ2ε +σ +(4π)ε +� ∞ +0 +dl2 +l2 l2ε +e−il2 +1 − l2σt +α′a + iϵ +� ∞ +0 +dt2 +t2 t2ε +eit2 +1 − t2σp +|β′ +b| − iϵ += +− 1 +ε2 + 1 +ε +� +2γ − ln µ2 +σ +4πi +� +− 1 +2 ln2 µ2 +σ +4πi − 2γ ln µ2 +σ +4πi − γ2 + O(λt, λp) +(11.84) +where we used integral +� ∞ +0 +dv vε−1 +e−v +1 + λv = 1 +ε − γ − λ + ε +� +γλ + γ2 +2 + π2 +12 +� ++ O(λ2) + O(ε2) +(11.85) +The expression for Ivirt sG ms +Fig.6 +will be the same as seen from Eq. (8.9). Moreover, it is easy +to see from Eqs. (8.14)-(8.15) that the sG-contribution to “production” diagrams differs in +sign from virtual contribution (11.84) +IsG ms +Fig.7 += IsG ms +Fig.8 += +− Ivirt sG ms +Fig.5 += +− Ivirt sG ms +Fig.6 +(11.86) +and therefore the total sG-contribution to the sum of the diagrams with background fields +on the mass shell is a power correction. +References +[1] Yuri V. Kovchegov and Eugene Levin. 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B, 305:136–143, 1993. +– 66 – + diff --git a/BdAzT4oBgHgl3EQfv_7e/content/tmp_files/load_file.txt b/BdAzT4oBgHgl3EQfv_7e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f0cf6250313af24f2e62c4e76ca1982867d565c --- /dev/null +++ b/BdAzT4oBgHgl3EQfv_7e/content/tmp_files/load_file.txt @@ -0,0 +1,2935 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf,len=2934 +page_content='Prepared for submission to JHEP JLAB-THY-23-3741 Rapidity-only TMD factorization at one loop Ian Balitsky Physics Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=', Old Dominion University, Norfolk, VA 23529 & Theory Group, JLAB, 12000 Jefferson Ave, Newport News, VA 23606 E-mail: balitsky@jlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='org Abstract: Typically, a production of a particle with a small transverse momentum in hadron-hadron collisions is described by CSS-based TMD factorization at moderate Bjorken xB ∼ 1 and by kT -factorization at small xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' A uniform description valid for all xB is provided by rapidity-only TMD factorization developed in a series of recent papers at the tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In this paper the rapidity-only TMD factorization for particle production by gluon fusion is extended to the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='01717v1 [hep-ph] 4 Jan 2023 Contents 1 Introduction 1 2 TMD factorization for particle production by gluon fusion 2 3 TMD factorization from functional integral 4 4 Coefficient function from background-field diagrams 7 5 Virtual contributions 13 6 “Production” diagrams 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 Power counting for production terms 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2 Calculation of leading production terms 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3 Handbag diagrams 26 7 Result for the sum of diagrams in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 5,6,7,8 minus TMD matrix ele- ments in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 11,12 28 8 Subtraction of soft/Glauber contributions 29 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 sG-contributions to virtual diagrams 30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2 sG-contributions to production diagrams 32 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3 The sum of sG-terms 33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4 sG-contributions to TMD matrix elements 34 9 Result for the coefficient function 36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 Factorization of integral over A ∪ B fields 39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 Cancellation of soft and Glauber gluons 39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2 Factorization in terms of generalized TMDs 40 10 Conclusions and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 41 11 Appendix 43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 Gluon “cut” propagator in the background field A 43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2 TMD matrix elements 43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3 Soft factor with rapidity-only cutoffs 48 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4 Approximation x∥ 12 = 0 for the calculation of coefficient function 50 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='5 Diagrams with correction field ¯C 52 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='6 Necessary integrals 55 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 Integrals for virtual diagrams 55 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2 Integrals for “production” diagrams 57 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7 Coefficient function from the calculation with background gluons on the mass shell 58 – i – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1 Virtual contributions 58 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2 Production terms minus TMD matrix elements 60 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3 J(i>1) 1 are power corrections 62 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4 sG-contribution 64 1 Introduction Rapidity factorization and rapidity evolution are main tools for study of QCD processes at small x [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' On the other hand, at moderate x conventional methods are based on CSS equa- tion [2] and closely related SCET approach (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' [3] and [4] reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' However, with the advent of EIC accelerator the region of energies intermediate between low and moderate x needs to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' One of the ideas is to extend rapidity factorization methods beyond the “pure” small-x region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In a series of recent papers A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Tarasov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Chirilli and the author applied the method of rapidity-only factorization to processes of particle production in hadron-hadron collisions in the so-called Sudakov region where transverse momentum of produced particle(s) is much smaller than their invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The typical examples of such processes are the Drell-Yan process or Higgs production by gluon fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' At moderate x such processes are studied by CSS-based TMD factorization [3, 5] dσ dηd2q⊥ = � f � d2b⊥ei(q,b)⊥Df/A(xA, b⊥, ηa)Df/B(xB, b⊥, η2) × σff→X(η, ηa, ηb) + power corrections + Y − terms (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) where η = 1 2 ln q+ q− is the rapidity, Df/h(x, z⊥, ηi) is the TMD density of a parton f in hadron h with rapidity cutoff ηi, and σff→X(η, ηa, ηb) is the cross section of production of particle(s) X of invariant mass m2 X = q2 ≡ Q2 ≫ q2 ⊥ in the scattering of two partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The TMD parton densities are regularized with a combination of UV and rapidity cutoffs and the relevant Sudakov logarithms are obtain by solving double evolution with respect to µUV and the rapidity cutoffs ηi [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' It should be emphasized that the CSS approach and hence the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) are valid at xA ∼ xB ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' At small xA and/or xB one should resort to other factorization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' As I mentioned above, a perspective approach is to apply methods based on rapidity-only factorization used in small-x/BFKL physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In a series of papers [6–9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Tarasov and the author applied rapidity-only factorization approach to get for the first time power corrections ∼ q2 ⊥ Q2 restoring EM gauge invariance of DY hadronic tensor both at moderate and small x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Also, in recent papers [9, 10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Chirilli and the author calculated the rapidity-only evolution of TMD operators, again both at moderate and small x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In the present paper I calculate coefficient function multiplying two TMD distributions at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' This completes the task of performing the rapidity-only factorization at the one-loop accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' – 1 – Apart from requirement Q2 = xAxBs ≫ q2 ⊥, in this paper it is assumed that Q2 q2 ⊥ ≫ q2 ⊥ m2 N (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) The region (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) can be understood in terms of rescaling s → ζs0, ζ → ∞ with q2 ⊥ fixed: s ∼ ζs0, q⊥ ∼ O(ζ0) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3) It should be emphasized that we will not use the small-x approximation s ≫ Q2 so our formulas are correct both at x ≪ 1 and x ∼ 1 provided that the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Thus, at x ∼ 1 our rapidity-evolution formulas should be equivalent to usual CSS approach, although the exact relation between our rapidity evolution and CSS double evolution in rapidity and UV cutoff remains to be established, see the discussion in the Conclusions section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The rapidity evolution of TMDs Df/A(xA, b⊥, ηa), Df/B(xB, b⊥, ηb) should match the one-loop rapidity evolution of the coefficient function σff→H(η, ηa, ηb) so that the cutoffs ηa and ηb disappear from physical amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' I will demonstrate that the result for the coefficient function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) for rapidity-only gluon TMD factorization is proportional to (γ ≡ γE ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='577) exp �αsNc 2π � (ln b2 ⊥s 4 − ηa − ηb)2 − 2(ln xA − ηa − γ)(ln xB − ηb − γ) + π2�� (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4) and check that ηa, ηb dependence matches the rapidity-only TMD evolution obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 2 I define hadronic tensor and TMD oper- ators for particle production by gluon fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 3 I discuss separation of functional integral for particle production in three integrals according to the rapidity of the fields involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 4 I set up the calculation of the coefficient function in front of TMD op- erators by computing diagrams in two background fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Sections 5, 6, 7, and 8 are devoted to calculation of these diagrams at the one-loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The result of the calculation and check of matching to evolution of TMD operators are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The necessary technical and sidelined results are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 2 TMD factorization for particle production by gluon fusion Let us consider production of an (imaginary) scalar particle Φ by gluon fusion in proton- proton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The particle is connected to gluons by the vertex LΦ = gΦ � d4x Φ(x)g2F 2(x), F 2(x) ≡ F a µν(x)F aµν(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) This is a mH mt ≪ 1 approximation for Higgs production via gluon fusion at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The differential cross section of Φ production is defined by the “hadronic tensor” W(pA, pB, q) W(pA, pB, q) def = N2 c − 1 16 � X � d4x e−iqx⟨pA, pB|g2F 2(x)|X⟩⟨X|g2F 2(0)|pA, pB⟩ = N2 c − 1 16 � d4x e−iqx⟨pA, pB|g4F 2(x)F 2(0)|pA, pB⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='– 2 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='p ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='Q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' & m2 q2 ⌘ Q2 = M 2 �, Q2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' ⌘ q2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cI9gPaUDbTbN0dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='N2N0IJ/QtePCji1T/kzX/jps1BWx8MPN6bYWZemHKmjet+O5W19Y3Nrep2bWd3b/+gfnjU0Um ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='mCG2ThCeqF2JNOZO0bZjhtJcqikXIaTec3BV+94kqzRL5aKYpDQeSxYxgk0hDfyYDesNt+n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='OgVaJV5IGlPCH9a/BKCGZoNIQjrXue25qghwrwins9og0zTFZILHtG+pxILqIJ/fOkNnVhmh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='KFG2pEFz9fdEjoXWUxHaToFNrJe9QvzP62cmuglyJtPMUEkWi6KMI5Og4nE0YoSw6eWYKYv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='RWRGCtMjI2nZkPwl9eJZ2LpnfV9B4uG63bMo4qnMApnIMH19Ce/ChDQRieIZXeHOE8+K8Ox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='+L1opTzhzDHzifP+SxjiU=� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Particle production by gluon-gluon fusion where the factor N2 c −1 16 is added to simplify factorization formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' As usual, � X denotes the sum over full set of “out” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' We will study the hadronic tensor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) with non-zero momentum transfer in t-channel defined as a matrix element of the operator ˆW(x1, x2) ≡ N2 c − 1 16 g4F a µν(x2)F µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='a(x2)F b λρ(x1)F λρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='b(x1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3) between initial and final states with slightly non-equal momenta W(pA, pB, p′ A, p′ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' x1, x2) = ⟨p′ A, p′ B| ˆW(x1, x2)|pA, pB⟩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4) where pA = p1 + m2 + l2 ⊥ s p2 − l⊥ 2 , p′ A = p1 + m2 + l2 ⊥ s p2 + l⊥ 2 , pB = p2 + m2 + l2 ⊥ s p1 + l⊥ 2 , p′ B = p2 + m2 + l2 ⊥ s p1 − l⊥ 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='5) Here p1 and p2 are light-like vectors close to pA and pB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 1 We will use light-cone coordinates with respect to the frame where p1 = � √s 2 , 0, 0, √s 2 � and p2 = � √s 2 , 0, 0, − √s 2 � so that p+ 1 = p− 2 = � s 2, p+ 2 = p− 1 = 0 and p1⊥ = p2⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The kinematical region (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) in the coordinate space translates to x2 ∥ ≪ x4 12⊥m2 N (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='6) where x2 ∥ ≡ 2x+ 12x− 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Also, we must assume x2 12⊥ ≤ m−2 N so that the coupling constant αs(x⊥) is a valid perturbative parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In the coordinate space, TMD factorization (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) for hadronic tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4) should look like g4 16(N2 c − 1)⟨p′ A, p′ B|F a µνF aµν(x2)F b λρF bλρ(x1)|pA, pB⟩ = � dz− 2 dz2⊥dz− 1 dz1⊥dw+ 1 dw1⊥dw+ 2 dw2⊥C(x2, x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' z− i , zi⊥, w+ i , wi⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' σp, σt) × ⟨p′ A| ˆOσp ij (z− 2 , z2⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' z− 1 , z1⊥)|pA⟩⟨p′ B| ˆOij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='σt(w+ 2 , w2⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' w+ 1 , w1⊥)|pB⟩ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7) 1We assume that t = −l2 ⊥ ∼ m2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' If there is a longitudinal component of momentum transfer, one can redefine p1 and p2 in such a way that with respect to new p′ 1 and p′ 2 the formulas are those of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' – 3 – where the dots stand for power corrections ∼ q2 ⊥ Q2 and ˆOij(z+ 2 , z2⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' z+ 1 , z1⊥) = F a i (z2)[z2 − ∞+, z1 − ∞+]abF b j (z1) ��� z− 2 =z− 1 =0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='8) ˆOij(z− 2 , z2⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' z− 1 , z1⊥) = F a i (z2)[z2 − ∞−, z1 − ∞−]abF b j (z1) ��� z+ 2 =z+ 1 =0 , F i,a(z⊥, z+) ≡ gF −i,m(z)[z+, −∞+]ma z ��� z−=0, F i,a(z⊥, z−) ≡ gF +i,m(z)[z−, −∞−]ma��� z+=0 are gluon TMD operators (the precise definitions of rapidity-only cutoffs σa = eηa and σb = eηb for gluon TMDs will be given later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Hereafter, we use the notation [x, y] ≡ Peig � 1 0 du (x−y)µAµ(ux+y−uy) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='9) for the straight-line ordered gauge link between points x and y, and space-saving notations [x+, y+]z ≡ [x+ + z⊥, y+ + z⊥], [x−, y−]z ≡ [x− + z⊥, y− + z⊥] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='10) The coefficient function C represents a Fourier transform of σff→H(η, η1, η2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) with ηi = ln σi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The normalization in the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='7) is chosen in such a way that C = 1 + αsNc 2π C1 + O(α2 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The goal of this paper is to find the one-loop coefficient function C1(x2, x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' z− i , zi⊥, w+ i , wi⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' σa, σb) and check that the evolution of this coefficient function matches the evolutions of TMD operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' 3 TMD factorization from functional integral The hadronic tensor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) can be represented by double functional integral W(pA, pB, p′ A, p′ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' x2, x1) = � X ⟨p′ A, p′ B|g2F 2(x2)|X⟩⟨X|g2F 2(x1)|pA, pB⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) = tf→∞ lim ti→−∞ g4 � ˜ A(tf)=A(tf) D ˜AµDAµ � ˜ψ(tf)=ψ(tf) D ˜¯ψD ˜ψD ¯ψDψ e−iSQCD( ˜ A, ˜ψ)eiSQCD(A,ψ) × Ψ∗ p′ A( ⃗˜A(ti), ˜ψ(ti))Ψ∗ p′ B( ⃗˜A(ti), ˜ψ(ti)) ˜F 2(x2)F 2(x1)ΨpA( ⃗A(ti), ψ(ti))ΨpB( ⃗A(ti), ψ(ti)) Here the fields A, ψ correspond to the amplitude ⟨X|F 2(x1)|pA, pB⟩, the fields ˜A, ˜ψ corre- spond to complex conjugate amplitude ⟨p′ A, p′ B|F 2(x2)|X⟩ and Ψp( ⃗A(ti), ψ(ti)) denote the proton wave function at the initial time ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The boundary conditions ˜A(tf) = A(tf) and ˜ψ(tf) = ψ(tf) reflect the sum over all states X, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' We will also use the notation {x, y} ≡ Peig � 1 0 du (x−y)µ ˜ Aµ(ux+y−uy) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) and similar notations like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='10) for gauge links in the left sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' For calculations in the momentum space we will use Sudakov variables related to light- cone components p+, p−, p⊥ by α ≡ p+/ϱ and β ≡ p−/ϱ where ϱ ≡ � s/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In terms of Sudakov variables p·q = (αpβq +αqβp) s 2 −(p, q)⊥ where (p, q)⊥ ≡ −piqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Throughout the – 4 – paper, the sum over the Latin indices i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' runs over the two transverse components while the sum over Greek indices runs over the four components as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Also, since we use Sudakov variables it is convenient to change the notations of gluon momentum fractions to αa ≡ xA, βb ≡ xB (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3) to avoid confusion with coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' [6, 7], to derive the factorization formula we separate gluon (and quark) fields in the functional integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='1) into three sectors: “projectile” fields Aµ, ψa with |β| < σp ≡ σa, “ target” fields Bµ, ψb with |α| < σt ≡ σb and “central rapidity” fields Cµ, ψ with |α| > σt and |β| > σp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Let us specify the values of the TMD cutoffs σp and σt Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Rapidity factorization for particle production in our factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Needless to say, we should take σt ≪ αa and σp ≪ βb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Moreover, as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' [9], power corrections to rapidity evolution of TMDs are ∼ Q2 ⊥ βbσts so we need to assume σtβbs ≫ Q2 ⊥, and similarly σpαas ≫ Q2 ⊥ for the projectile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' Next, as we shall see below, it is convenient to calculate coefficient function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='19) at m2 N ≫ µ2 σ ≡ σpσts so finally we take the region of σp and σt as follows αa ≫ σt ≫ Q2 ⊥ βbs, β′ b ≫ σp ≫ Q2 ⊥ α′as, m2 N ≫ µ2 σ ≡ σpσts ≫ Q4 ⊥ Q2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4) Note that due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='2) we can choose µ2 σ between m2 N and parametrically small Q4 ⊥ Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In terms of rescaling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='3) this means that we can choose σp, σt ∼ ζ− 3 4 ∼ � Q⊥ Q �3/2 so that µ2 σ ∼ ζ−1/2 ⇔ 1 ≫ µ2 σ Q2 ⊥ ≫ Q2 ⊥ Q2 ∼ ζ−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='5) and both conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content='4) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' In this paper we are calculating logarithmical corrections so the power corrections due to the small parameters O2 ⊥ σpα′as, O2 ⊥ σtβ′ bs will be systematically neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdAzT4oBgHgl3EQfv_7e/content/2301.01717v1.pdf'} +page_content=' The convenient – 5 – A "Proiectile""fields: 3