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.matpur.2026.103874
Shikun Cui, Lili Wang, Wendong Wang
As is well known, for the 3D Patlak-Keller-Segel system, regardless of whether they are parabolic-elliptic or parabolic-parabolic forms, finite-time blow-up may occur for arbitrarily small values of the initial mass. In this paper, it is proved for the first time that one can prevent the finite-time blow-up when the initial mass is less than a certain critical threshold via the stabilizing effect of the moving Navier-Stokes flows. In details, we investigate the nonlinear stability of the Couette flow in the Patlak-Keller-Segel-Navier-Stokes system and show that if the Couette flow is sufficiently strong (A is large enough), then the solutions for Patlak-Keller-Segel-Navier-Stokes system are global in time provided that the initial velocity is sufficiently small and the initial cell mass is less than .
{"title":"Suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow","authors":"Shikun Cui, Lili Wang, Wendong Wang","doi":"10.1016/j.matpur.2026.103874","DOIUrl":"10.1016/j.matpur.2026.103874","url":null,"abstract":"<div><div>As is well known, for the 3D Patlak-Keller-Segel system, regardless of whether they are parabolic-elliptic or parabolic-parabolic forms, finite-time blow-up may occur for arbitrarily small values of the initial mass. In this paper, it is proved for the first time that one can prevent the finite-time blow-up when the initial mass is less than a certain critical threshold via the stabilizing effect of the moving Navier-Stokes flows. In details, we investigate the nonlinear stability of the Couette flow <span><math><mo>(</mo><mi>A</mi><mi>y</mi><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math></span> in the Patlak-Keller-Segel-Navier-Stokes system and show that if the Couette flow is sufficiently strong (A is large enough), then the solutions for Patlak-Keller-Segel-Navier-Stokes system are global in time provided that the initial velocity is sufficiently small and the initial cell mass is less than <span><math><mfrac><mrow><mn>24</mn></mrow><mrow><mn>5</mn></mrow></mfrac><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"210 ","pages":"Article 103874"},"PeriodicalIF":2.3,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147428","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: 10.1080/01621459.2026.2621517
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 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}
SIAM Review, Volume 68, Issue 1, Page 3-90, February 2026. Abstract. “One of the ways to help make computer science respectable is to show that it is deeply rooted in history [math]” (Donald E. Knuth, Comm. ACM, 15 (1972), p. 671). A great many of the “respectable” modern numerical methods proceed iteratively, and we give an overview of them in the final section . Teaching and learning science from a historical perspective also leads to a “respectable” deeper understanding. The first problems requiring iterative processes were square-root calculations in Babylon, Greece, and India. More complicated problems such as sine tables in the Arabic, Indian, and medieval calculations, including Kepler’s Problem, were performed with fixed point iterations. With Newton, Raphson, and Simpson we enter the “respectable” realm of methods based on derivatives. Mourraille and Cayley contribute geometric insights in both [math] and [math], while Fourier, Cauchy, and Kantorovich provide rigorous error estimations. Surprisingly, even linear problems became interesting for very large dimensions, beginning with the work of Gauss, Seidel, Young, Richardson, and Krylov to domain decomposition and multigrid methods. We explain all of these methods and illustrate them using the “Montreal test problem.”
SIAM评论，第68卷，第1期，第3-90页，2026年2月。摘要。“使计算机科学受人尊敬的方法之一是表明它深深植根于历史[数学]”（Donald E. Knuth, Comm. ACM, 15(1972)，第671页）。许多“值得尊敬的”现代数值方法都是迭代进行的，我们将在最后一节对它们进行概述。从历史的角度来教授和学习科学也会带来“可敬的”更深层次的理解。第一个需要迭代过程的问题是巴比伦、希腊和印度的平方根计算。更复杂的问题，如阿拉伯、印度和中世纪计算中的正弦表，包括开普勒问题，都是用定点迭代来完成的。随着牛顿、拉夫森和辛普森的出现，我们进入了基于衍生方法的“体面”领域。Mourraille和Cayley在[数学]和[数学]两方面都贡献了几何见解，而Fourier、Cauchy和Kantorovich则提供了严格的误差估计。令人惊讶的是，从Gauss、Seidel、Young、Richardson和Krylov的领域分解和多重网格方法开始，即使是线性问题在非常大的维度上也变得有趣起来。我们将解释所有这些方法，并使用“蒙特利尔测试问题”来说明它们。
{"title":"Landmarks in the History of Iterative Methods","authors":"Martin J. Gander, Philippe Henry, Gerhard Wanner","doi":"10.1137/24m1680428","DOIUrl":"https://doi.org/10.1137/24m1680428","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 3-90, February 2026. <br/> Abstract. “One of the ways to help make computer science respectable is to show that it is deeply rooted in history [math]” (Donald E. Knuth, Comm. ACM, 15 (1972), p. 671). A great many of the “respectable” modern numerical methods proceed iteratively, and we give an overview of them in the final section . Teaching and learning science from a historical perspective also leads to a “respectable” deeper understanding. The first problems requiring iterative processes were square-root calculations in Babylon, Greece, and India. More complicated problems such as sine tables in the Arabic, Indian, and medieval calculations, including Kepler’s Problem, were performed with fixed point iterations. With Newton, Raphson, and Simpson we enter the “respectable” realm of methods based on derivatives. Mourraille and Cayley contribute geometric insights in both [math] and [math], while Fourier, Cauchy, and Kantorovich provide rigorous error estimations. Surprisingly, even linear problems became interesting for very large dimensions, beginning with the work of Gauss, Seidel, Young, Richardson, and Krylov to domain decomposition and multigrid methods. We explain all of these methods and illustrate them using the “Montreal test problem.”","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"182 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138553","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}
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