Pub Date : 2026-02-09DOI: 10.1016/j.amc.2026.129986
Wuquan Li, Yanzhu Jiang, Meiqiao Wang, Hui Wang
This paper studies the distributed tracking for the nonlinear multi-agent systems perturbed by second-order moment processes. What makes the case here unique from the prior results is that on the one hand, the system is more practical than those with white noise since it is more in line with practical engineering applications, and on the other hand, the system has inherently nonlinear distinguishing from those with second-order moment processes. In this paper, a distributed tracking controller is designed by employing the algebraic graph theorem and the distributed backstepping method, in which, extensive computational skills are taken to offset the difficulties from the system’s high-order. Furthermore, by using stochastic control theory, the tracking errors can be adjusted to an arbitrarily small value while all the states of the closed-loop system composed by the multi-agent systems are bounded in probability. At last, a simulation example with three agents demonstrates the effectiveness of the distributed tracking controller.
{"title":"Distributed tracking of high-order nonlinear multi-agent systems perturbed by second-order moment processes","authors":"Wuquan Li, Yanzhu Jiang, Meiqiao Wang, Hui Wang","doi":"10.1016/j.amc.2026.129986","DOIUrl":"https://doi.org/10.1016/j.amc.2026.129986","url":null,"abstract":"This paper studies the distributed tracking for the nonlinear multi-agent systems perturbed by second-order moment processes. What makes the case here unique from the prior results is that on the one hand, the system is more practical than those with white noise since it is more in line with practical engineering applications, and on the other hand, the system has inherently nonlinear distinguishing from those with second-order moment processes. In this paper, a distributed tracking controller is designed by employing the algebraic graph theorem and the distributed backstepping method, in which, extensive computational skills are taken to offset the difficulties from the system’s high-order. Furthermore, by using stochastic control theory, the tracking errors can be adjusted to an arbitrarily small value while all the states of the closed-loop system composed by the multi-agent systems are bounded in probability. At last, a simulation example with three agents demonstrates the effectiveness of the distributed tracking controller.","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"69 1","pages":""},"PeriodicalIF":4.0,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-09DOI: 10.1080/01621459.2026.2621517
Elena Bortolato, Antonio Canale
{"title":"Adaptive Partition Factor Analysis","authors":"Elena Bortolato, Antonio Canale","doi":"10.1080/01621459.2026.2621517","DOIUrl":"https://doi.org/10.1080/01621459.2026.2621517","url":null,"abstract":"","PeriodicalId":17227,"journal":{"name":"Journal of the American Statistical Association","volume":"35 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 68, Issue 1, Page 208-210, February 2026. Linear algebra is often viewed as one of the most foundational courses in a mathematics or computer science curriculum, yet it is also one that can intimidate students with its abstract formalism and steep learning curve. In Linear Algebra: A Problem-Centered Approach, Róbert Freud reimagines the subject by presenting it not as a procession of theorems and proofs, but as an unfolding narrative of problems, motivations, and applications. Published as part of the AMS’s Pure and Applied Undergraduate Texts series, this book brings together the rigor of traditional mathematics with the accessibility and playfulness of the Hungarian problem-solving tradition.
{"title":"Book Review:; Linear Algebra: A Problem-Centered Approach","authors":"Anita T. Layton","doi":"10.1137/25m1799192","DOIUrl":"https://doi.org/10.1137/25m1799192","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 208-210, February 2026. <br/> Linear algebra is often viewed as one of the most foundational courses in a mathematics or computer science curriculum, yet it is also one that can intimidate students with its abstract formalism and steep learning curve. In Linear Algebra: A Problem-Centered Approach, Róbert Freud reimagines the subject by presenting it not as a procession of theorems and proofs, but as an unfolding narrative of problems, motivations, and applications. Published as part of the AMS’s Pure and Applied Undergraduate Texts series, this book brings together the rigor of traditional mathematics with the accessibility and playfulness of the Hungarian problem-solving tradition.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"9 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 68, Issue 1, Page 210-211, February 2026. Control in Finite and Infinite Dimension is an excellent textbook based on many years of in-depth teaching experience, as well as on the author’s expertise in the field. It provides a concise and (mostly) self-contained introduction to mathematical control theory, for both finite- and infinite-dimensional systems. It is written at the level of a Master’s/Ph.D. program, but for more experienced researchers it can also serve as a good overview of the basic results and the main tools used in the field, as well as material for lectures in specialised courses on the topic. It covers the most important parts of the classical control theory: controllability, observability and their duality, optimal controls, and stabilization (the latter two only in the finite-dimensional case).
{"title":"Book Review:; Control in Finite and Infinite Dimension","authors":"Martin Lazar","doi":"10.1137/25m1732787","DOIUrl":"https://doi.org/10.1137/25m1732787","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 210-211, February 2026. <br/> Control in Finite and Infinite Dimension is an excellent textbook based on many years of in-depth teaching experience, as well as on the author’s expertise in the field. It provides a concise and (mostly) self-contained introduction to mathematical control theory, for both finite- and infinite-dimensional systems. It is written at the level of a Master’s/Ph.D. program, but for more experienced researchers it can also serve as a good overview of the basic results and the main tools used in the field, as well as material for lectures in specialised courses on the topic. It covers the most important parts of the classical control theory: controllability, observability and their duality, optimal controls, and stabilization (the latter two only in the finite-dimensional case).","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"1 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-09DOI: 10.1016/j.chaos.2026.118035
Hadi Susanto
We introduce a new class of nonlinear Schrödinger (NLS) equations with a logarithmic–power nonlinearity that admits exact localized solutions of super-Gaussian form. The resulting stationary states possess flat-top profiles with sharp edges and are referred to as super-Gaussons, in analogy with the Gaussian Gaussons of the classical logarithmic NLS (log-NLS). The model, which we call the logarithmic-power NLS (logp-NLS), is parameterized by an exponent p≥1 that controls the degree of flatness of the soliton core and the sharpness of its decay. Mathematically, p interpolates between the standard log-NLS (p=1) and increasingly flat-top profiles as p increases, while physically it governs the stiffness of an underlying logarithmic–power compressibility law. The proposed equation is constructed so as to admit super-Gaussian stationary states and can be interpreted within a generalized pressure-law framework, thereby extending the log-NLS. We investigate the dynamics of super-Gaussons in one spatial dimension through numerical simulations for various values of p, demonstrating how this parameter affects the internal structure of the soliton and its collision dynamics. The logp-NLS thus generalizes the standard log-NLS by admitting a broader family of localized states with distinctive structural and dynamical properties, suggesting its relevance for flat-top solitons in nonlinear optics, Bose–Einstein condensates, and related nonlinear media.
{"title":"New logarithmic power nonlinear Schrödinger equations with super-Gaussons","authors":"Hadi Susanto","doi":"10.1016/j.chaos.2026.118035","DOIUrl":"https://doi.org/10.1016/j.chaos.2026.118035","url":null,"abstract":"We introduce a new class of nonlinear Schrödinger (NLS) equations with a logarithmic–power nonlinearity that admits exact localized solutions of super-Gaussian form. The resulting stationary states possess flat-top profiles with sharp edges and are referred to as super-Gaussons, in analogy with the Gaussian Gaussons of the classical logarithmic NLS (log-NLS). The model, which we call the logarithmic-power NLS (logp-NLS), is parameterized by an exponent <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:mrow><mml:mi>p</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> that controls the degree of flatness of the soliton core and the sharpness of its decay. Mathematically, <mml:math altimg=\"si97.svg\" display=\"inline\"><mml:mi>p</mml:mi></mml:math> interpolates between the standard log-NLS (<mml:math altimg=\"si98.svg\" display=\"inline\"><mml:mrow><mml:mi>p</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>) and increasingly flat-top profiles as <mml:math altimg=\"si97.svg\" display=\"inline\"><mml:mi>p</mml:mi></mml:math> increases, while physically it governs the stiffness of an underlying logarithmic–power compressibility law. The proposed equation is constructed so as to admit super-Gaussian stationary states and can be interpreted within a generalized pressure-law framework, thereby extending the log-NLS. We investigate the dynamics of super-Gaussons in one spatial dimension through numerical simulations for various values of <mml:math altimg=\"si97.svg\" display=\"inline\"><mml:mi>p</mml:mi></mml:math>, demonstrating how this parameter affects the internal structure of the soliton and its collision dynamics. The logp-NLS thus generalizes the standard log-NLS by admitting a broader family of localized states with distinctive structural and dynamical properties, suggesting its relevance for flat-top solitons in nonlinear optics, Bose–Einstein condensates, and related nonlinear media.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"32 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-09DOI: 10.1007/s10444-025-10273-5
Christian Kuehn, Sara-Viola Kuntz
Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input–output dynamics of finite and infinite depth neural networks with scalar output. In the finite-depth case, the input is a state associated with a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps an affine linear transformation of the input to an affine linear transformation of its time- T map. We show that, depending on the specific structure of the network, the input–output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions where every critical point is non-degenerate. We prove that critical points cannot exist if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller than or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that, except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite-depth neural networks, the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs. Our dynamical systems viewpoint on the geometric structure of the input–output map provides a fundamental understanding of why certain architectures perform better than others.
{"title":"Analysis of the geometric structure of neural networks and neural ODEs via morse functions","authors":"Christian Kuehn, Sara-Viola Kuntz","doi":"10.1007/s10444-025-10273-5","DOIUrl":"https://doi.org/10.1007/s10444-025-10273-5","url":null,"abstract":"Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input–output dynamics of finite and infinite depth neural networks with scalar output. In the finite-depth case, the input is a state associated with a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps an affine linear transformation of the input to an affine linear transformation of its time- <jats:italic>T</jats:italic> map. We show that, depending on the specific structure of the network, the input–output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions where every critical point is non-degenerate. We prove that critical points cannot exist if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller than or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that, except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite-depth neural networks, the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs. Our dynamical systems viewpoint on the geometric structure of the input–output map provides a fundamental understanding of why certain architectures perform better than others.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146145956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 68, Issue 1, Page 211-212, February 2026. This valuable and unique book delivers a comprehensive lecture on a wide range of control theory issues in relation to matrix computing. Individual problems are illustrated with examples of sufficient dimensionality to ensure they can be manually recalculated, while still illustrating all the intricacies of the relevant calculations and algorithms. The book also contains numerous drawings and diagrams that clarify the various issues.
{"title":"Book Review:; Time-Variant and Quasi-Separable Systems","authors":"Jerzy S. Respondek","doi":"10.1137/25m1758283","DOIUrl":"https://doi.org/10.1137/25m1758283","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 211-212, February 2026. <br/> This valuable and unique book delivers a comprehensive lecture on a wide range of control theory issues in relation to matrix computing. Individual problems are illustrated with examples of sufficient dimensionality to ensure they can be manually recalculated, while still illustrating all the intricacies of the relevant calculations and algorithms. The book also contains numerous drawings and diagrams that clarify the various issues.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"121 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：