Pub Date : 2024-01-19DOI: 10.1142/s0219530524500039
Tegegne Getachew, Birilew Belayneh
{"title":"New asymptotic lower bound for the radius of analyticity of solutions to nonlinear Schrodinger equation","authors":"Tegegne Getachew, Birilew Belayneh","doi":"10.1142/s0219530524500039","DOIUrl":"https://doi.org/10.1142/s0219530524500039","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139525270","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 : 2023-12-15DOI: 10.1142/s0219530523500331
Xiaopeng Zhao, Yong Zhou
{"title":"On the strong solution for a diffuse interface model of non-Newtonian two-phase flows","authors":"Xiaopeng Zhao, Yong Zhou","doi":"10.1142/s0219530523500331","DOIUrl":"https://doi.org/10.1142/s0219530523500331","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139000519","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 : 2023-12-01DOI: 10.1142/s021953052350032x
Mike Nguyen, Charly Kirst, Nicole Mucke
{"title":"Distributed SGD in Overparameterized Linear Regression","authors":"Mike Nguyen, Charly Kirst, Nicole Mucke","doi":"10.1142/s021953052350032x","DOIUrl":"https://doi.org/10.1142/s021953052350032x","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138626975","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 : 2023-11-11DOI: 10.1142/s0219530523500306
Mourad E. H. Ismail, Plamen Simeonov, Dennis Stanton
We develop Taylor type series expansions for entire functions of order zero using general interpolation sequences that diverge to infinity sufficiently fast. We also derive two-point and multi-point Lidstone type series expansions for entire functions of order zero. The coefficients of these Taylor and Lidstone type series are expressed in terms of values of a certain integral operator. Our results imply as special cases several recent results on [Formula: see text]-Taylor and [Formula: see text]-Lidstone expansions of order zero entire functions, and provide a general technique for deriving such series.
{"title":"Interpolatory Taylor and Lidstone series","authors":"Mourad E. H. Ismail, Plamen Simeonov, Dennis Stanton","doi":"10.1142/s0219530523500306","DOIUrl":"https://doi.org/10.1142/s0219530523500306","url":null,"abstract":"We develop Taylor type series expansions for entire functions of order zero using general interpolation sequences that diverge to infinity sufficiently fast. We also derive two-point and multi-point Lidstone type series expansions for entire functions of order zero. The coefficients of these Taylor and Lidstone type series are expressed in terms of values of a certain integral operator. Our results imply as special cases several recent results on [Formula: see text]-Taylor and [Formula: see text]-Lidstone expansions of order zero entire functions, and provide a general technique for deriving such series.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135041588","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 : 2023-10-31DOI: 10.1142/s0219530523990014
{"title":"Author index Volume 21 (2023)","authors":"","doi":"10.1142/s0219530523990014","DOIUrl":"https://doi.org/10.1142/s0219530523990014","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135814064","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 : 2023-10-19DOI: 10.1142/s021953052350029x
Wentao Huang, Haizhang Zhang
Various powerful deep neural network architectures have made great contributions to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets are the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at studying the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix–vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notion of activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. We also conduct experiments on benchmark machine learning data to illustrate the potential usefulness of the results.
{"title":"Convergence Analysis of Deep Residual Networks","authors":"Wentao Huang, Haizhang Zhang","doi":"10.1142/s021953052350029x","DOIUrl":"https://doi.org/10.1142/s021953052350029x","url":null,"abstract":"Various powerful deep neural network architectures have made great contributions to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets are the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at studying the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix–vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notion of activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. We also conduct experiments on benchmark machine learning data to illustrate the potential usefulness of the results.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667014","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 : 2023-10-17DOI: 10.1142/s0219530523500276
Yarong Liu, Yejuan Wang, Peter E. Kloeden
In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in [Formula: see text] under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms.
{"title":"A class of stochastic nonlocal evolution equations with nonlocal initial conditions","authors":"Yarong Liu, Yejuan Wang, Peter E. Kloeden","doi":"10.1142/s0219530523500276","DOIUrl":"https://doi.org/10.1142/s0219530523500276","url":null,"abstract":"In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in [Formula: see text] under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944962","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 : 2023-10-06DOI: 10.1142/s0219530523500264
Jorg-Uwe Lobus
The paper is concerned with the change of probability measures [Formula: see text] along non-random probability measure-valued trajectories [Formula: see text], [Formula: see text]. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map [Formula: see text] does not exit the state space, for [Formula: see text] or for [Formula: see text], the Radon–Nikodym derivative [Formula: see text] is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula.
{"title":"Quasi-Invariance under Flows Generated by Non-Linear PDEs","authors":"Jorg-Uwe Lobus","doi":"10.1142/s0219530523500264","DOIUrl":"https://doi.org/10.1142/s0219530523500264","url":null,"abstract":"The paper is concerned with the change of probability measures [Formula: see text] along non-random probability measure-valued trajectories [Formula: see text], [Formula: see text]. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map [Formula: see text] does not exit the state space, for [Formula: see text] or for [Formula: see text], the Radon–Nikodym derivative [Formula: see text] is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302740","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 : 2023-10-06DOI: 10.1142/s0219530523500240
Claudianor O. Alves, Ismael S. da Silva
This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form [Formula: see text] where [Formula: see text] is a continuous function that satisfies some technical conditions and [Formula: see text] is a positive parameter. We will establish the multiplicity of solution for [Formula: see text] by using the notion of Lusternik–Schnirelmann category, by introducing a new function space where the energy functional is [Formula: see text].
{"title":"Existence of multiple solutions for a Schrödinger logarithmic equation via Lusternik–Schnirelmann category","authors":"Claudianor O. Alves, Ismael S. da Silva","doi":"10.1142/s0219530523500240","DOIUrl":"https://doi.org/10.1142/s0219530523500240","url":null,"abstract":"This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form [Formula: see text] where [Formula: see text] is a continuous function that satisfies some technical conditions and [Formula: see text] is a positive parameter. We will establish the multiplicity of solution for [Formula: see text] by using the notion of Lusternik–Schnirelmann category, by introducing a new function space where the energy functional is [Formula: see text].","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302894","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 : 2023-10-04DOI: 10.1142/s0219530523500239
Mendez, A. J.
In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the [Formula: see text]-dimensional setting with [Formula: see text] and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data.
{"title":"On the propagation of Regularity for Solutions of the Zakharov-Kuznetsov Equation","authors":"Mendez, A. J.","doi":"10.1142/s0219530523500239","DOIUrl":"https://doi.org/10.1142/s0219530523500239","url":null,"abstract":"In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the [Formula: see text]-dimensional setting with [Formula: see text] and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547522","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}
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