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}
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: 10.1080/01621459.2026.2622104
Wenhao Cui
{"title":"The Explicative Market Microstructure Noise","authors":"Wenhao Cui","doi":"10.1080/01621459.2026.2622104","DOIUrl":"https://doi.org/10.1080/01621459.2026.2622104","url":null,"abstract":"","PeriodicalId":17227,"journal":{"name":"Journal of the American Statistical Association","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146148","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: 10.1080/01621459.2026.2626478
Brenda Betancourt
{"title":"Model to Meaning: How to Interpret Statistical Models with R and Python","authors":"Brenda Betancourt","doi":"10.1080/01621459.2026.2626478","DOIUrl":"https://doi.org/10.1080/01621459.2026.2626478","url":null,"abstract":"","PeriodicalId":17227,"journal":{"name":"Journal of the American Statistical Association","volume":"176 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146146150","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/s10444-026-10281-z
Elena Zampieri
We approximate the acoustic wave equation in two-dimensional regions using collocation and Galerkin isogeometric analysis (IGA) in space, coupled with implicit second-order Newmark schemes for time integration. We present a detailed numerical study that examines and compares the behavior of extreme eigenvalues and condition numbers of the mass and iteration IGA matrices, varying the polynomial degree p , mesh size h , regularity k , and the boundary conditions, that can be either Dirichlet or absorbing in order to simulate unbounded domains. We propose and validate numerically some conjectures related to the IGA collocation and Galerkin matrices for the wave equation with different types of boundary conditions, extending similar results that are known for the IGA Galerkin approximation, limitedly to the case of the Poisson problem with Dirichlet boundary conditions, and generalizing earlier results obtained within the framework of the collocation method. The results show that the spectral properties of the IGA collocation matrices are analogous and in most cases better than the corresponding IGA Galerkin discretization of the Poisson problem with Dirichlet or absorbing boundary conditions.
{"title":"A comparative numerical study of spectral properties in isogeometric collocation and Galerkin methods for acoustic waves","authors":"Elena Zampieri","doi":"10.1007/s10444-026-10281-z","DOIUrl":"https://doi.org/10.1007/s10444-026-10281-z","url":null,"abstract":"We approximate the acoustic wave equation in two-dimensional regions using collocation and Galerkin isogeometric analysis (IGA) in space, coupled with implicit second-order Newmark schemes for time integration. We present a detailed numerical study that examines and compares the behavior of extreme eigenvalues and condition numbers of the mass and iteration IGA matrices, varying the polynomial degree <jats:italic>p</jats:italic> , mesh size <jats:italic>h</jats:italic> , regularity <jats:italic>k</jats:italic> , and the boundary conditions, that can be either Dirichlet or absorbing in order to simulate unbounded domains. We propose and validate numerically some conjectures related to the IGA collocation and Galerkin matrices for the wave equation with different types of boundary conditions, extending similar results that are known for the IGA Galerkin approximation, limitedly to the case of the Poisson problem with Dirichlet boundary conditions, and generalizing earlier results obtained within the framework of the collocation method. The results show that the spectral properties of the IGA collocation matrices are analogous and in most cases better than the corresponding IGA Galerkin discretization of the Poisson problem with Dirichlet or absorbing boundary conditions.","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146145954","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}
Pub Date : 2026-02-09DOI: 10.1007/s10878-026-01398-4
Qi Zhang, Hao Zhong
{"title":"On positive influence dominating sets with bi-directional weighted constraint in social networks","authors":"Qi Zhang, Hao Zhong","doi":"10.1007/s10878-026-01398-4","DOIUrl":"https://doi.org/10.1007/s10878-026-01398-4","url":null,"abstract":"","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"70 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146145941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 127-149, February 2026. Abstract. We present a variational framework for studying functions learned by deep neural networks with rectified linear unit nonlinearities. We introduce a function space built from compositions of functions of second-order Radon-domain bounded variation. The compositional form of these functions captures the structure of deep neural networks. We prove a representer theorem that shows that deep neural networks with finite width solve regularized data-fitting problems over this space. The critical width is controlled by the square of the number of training data. This perspective explains the effect of weight-decay regularization in neural network training, the importance of skip connections, and the role of sparsity in neural networks. By considering the function-space perspective, we provide sharp links between deep learning and variational methods.
{"title":"Compositional Function Spaces for Deep Learning","authors":"Rahul Parhi, Robert D. Nowak","doi":"10.1137/25m1802948","DOIUrl":"https://doi.org/10.1137/25m1802948","url":null,"abstract":"SIAM Review, Volume 68, Issue 1, Page 127-149, February 2026. <br/> Abstract. We present a variational framework for studying functions learned by deep neural networks with rectified linear unit nonlinearities. We introduce a function space built from compositions of functions of second-order Radon-domain bounded variation. The compositional form of these functions captures the structure of deep neural networks. We prove a representer theorem that shows that deep neural networks with finite width solve regularized data-fitting problems over this space. The critical width is controlled by the square of the number of training data. This perspective explains the effect of weight-decay regularization in neural network training, the importance of skip connections, and the role of sparsity in neural networks. By considering the function-space perspective, we provide sharp links between deep learning and variational methods.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2026-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146138822","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}
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