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The Cayley–Hamilton Theorem and Resolvent Representation 卡莱-汉密尔顿定理和残差表示法
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S001626632304010X
I. D. Kostrub
{"title":"The Cayley–Hamilton Theorem and Resolvent Representation","authors":"I. D. Kostrub","doi":"10.1134/S001626632304010X","DOIUrl":"10.1134/S001626632304010X","url":null,"abstract":"<p> For the Frobenius matrix accompanying an algebraic (differential) equation in a complex Banach algebra, the Cayley–Hamilton theorem is proved, which is used to obtain a representation of the resolvent. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"371 - 373"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590225","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}
引用次数: 0
Free Topological Algebra with Separately Continuous Mal’tsev Operation 具有独立连续马尔采夫操作的自由拓扑代数
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S001626632304007X
O. V. Sipacheva, A. A. Solonkov
{"title":"Free Topological Algebra with Separately Continuous Mal’tsev Operation","authors":"O. V. Sipacheva,&nbsp;A. A. Solonkov","doi":"10.1134/S001626632304007X","DOIUrl":"10.1134/S001626632304007X","url":null,"abstract":"<p> Topological spaces with separately continuous Mal’tsev operation, called quasi-Mal’tsev spaces, are considered. The existence of the free quasi-Mal’tsev space generated by an arbitrary completely regular Hausdorff space is proved. It is shown that any quasi-Mal’tsev space is a quotient of a free quasi-Mal’tsev space. It is also shown that the topology of a free quasi-Mal’tsev space has a simple and natural description in terms of the generating space. Finally, it is proved that any completely regular Hausdorff quasi-Mal’tsev space is a retract of a quasi-topological group. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"337 - 345"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590224","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}
引用次数: 0
Full Symmetric Toda System: Solution via QR-Decomposition 全对称户田系统：通过 QR 分解求解
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040081
D. V. Talalaev, Yu. B. Chernyakov, G. I. Sharygin
{"title":"Full Symmetric Toda System: Solution via QR-Decomposition","authors":"D. V. Talalaev,&nbsp;Yu. B. Chernyakov,&nbsp;G. I. Sharygin","doi":"10.1134/S0016266323040081","DOIUrl":"10.1134/S0016266323040081","url":null,"abstract":"<p> The full symmetric Toda system is a generalization of the open Toda chain, for which the Lax operator is a symmetric matrix of general form. This system is Liouville integrable and even superintegrable. Deift, Lee, Nando, and Tomei (DLNT) proposed the chopping method for constructing integrals of such a system. In the paper, a solution of Hamiltonian equations for the entire family of DLNT integrals is constructed by using the generalized QR factorization method. For this purpose, certain tensor operations on the space of Lax operators and special differential operators on the Lie algebra are introduced. Both tools can be interpreted in terms of the representation theory of the Lie algebra <span>(mathfrak{sl}_n)</span> and are expected to generalize to arbitrary real semisimple Lie algebras. As is known, the full Toda system can be interpreted in terms of a compact Lie group and a flag space. Hopefully, the results on the trajectories of this system obtained in the paper will be useful in studying the geometry of flag spaces. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"346 - 363"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590306","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}
引用次数: 0
Remarks on Yangian-Type Algebras and Double Poisson Brackets 关于扬基类型代数和双泊松括弧的评论
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040068
G. I. Olshanski, N. A. Safonkin
{"title":"Remarks on Yangian-Type Algebras and Double Poisson Brackets","authors":"G. I. Olshanski,&nbsp;N. A. Safonkin","doi":"10.1134/S0016266323040068","DOIUrl":"10.1134/S0016266323040068","url":null,"abstract":"<p> In a recent paper, one of the authors proposed a construction of associative algebras which share a number of properties of the Yangians of series A but are more massive. We show that this construction admits a generalization which reveals a direct connection with a large family of double Poisson brackets on free associative algebras, which was described by Pichereau and Van de Weyer (in 2008). </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"326 - 336"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590151","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}
引用次数: 0
The Mumford Dynamical System and Hyperelliptic Kleinian Functions 芒福德动力系统和超椭圆克莱因函数
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040032
V. M. Buchstaber
{"title":"The Mumford Dynamical System and Hyperelliptic Kleinian Functions","authors":"V. M. Buchstaber","doi":"10.1134/S0016266323040032","DOIUrl":"10.1134/S0016266323040032","url":null,"abstract":"<p> We develop a differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the <span>((P,Q))</span>-recursion, which defines a sequence of functions <span>(P_1,P_2,ldots)</span> given the first function <span>(P_1)</span> of this sequence and a sequence of parameters <span>(h_1,h_2,dots)</span>. The general solution of the <span>((P,Q))</span>-recursion is shown to give a solution for the parametric graded Korteweg–de Vries hierarchy. We prove that all solutions of the Mumford dynamical <span>(g)</span>-system are determined by the <span>((P,Q))</span>-recursion under the condition <span>(P_{g+1} = 0)</span>, which is equivalent to an ordinary nonlinear differential equation of order <span>(2g)</span> for the function <span>(P_1)</span>. Reduction of the <span>(g)</span>-system of Mumford to the Buchstaber–Enolskii–Leykin dynamical system is described explicitly, and its explicit <span>(2g)</span>-parameter solution in hyperelliptic Klein functions is presented. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"288 - 302"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590222","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}
引用次数: 0
Reconstructions of the Asymptotics of an Integral Determined by a Hyperbolic Unimodal Singularity 双曲单峰奇点确定的积分渐近线的重构
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040056
S. V. Zakharov
{"title":"Reconstructions of the Asymptotics of an Integral Determined by a Hyperbolic Unimodal Singularity","authors":"S. V. Zakharov","doi":"10.1134/S0016266323040056","DOIUrl":"10.1134/S0016266323040056","url":null,"abstract":"<p> The asymptotic behavior of an exponential integral is studied in which the phase function has the form of a special deformation of the germ of a hyperbolic unimodal singularity of type <span>(T_{4,4,4})</span>. The integral under examination satisfies the heat equation, its Cole–Hopf transformation gives a solution of the vector Burgers equation in four-dimensional space-time, and its principal asymptotic approximations are expressed in terms of real solutions of systems of third-degree algebraic equations. The obtained analytical results make it possible to trace the bifurcations of an asymptotic structure depending on the parameter of the modulus of the singularity. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"314 - 325"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590245","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}
引用次数: 0
The Nonlinear Kantorovich Transportation Problem with Nonconvex Costs 成本非凸的非线性康托洛维奇运输问题
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040019
K. A. Afonin
{"title":"The Nonlinear Kantorovich Transportation Problem with Nonconvex Costs","authors":"K. A. Afonin","doi":"10.1134/S0016266323040019","DOIUrl":"10.1134/S0016266323040019","url":null,"abstract":"<p> The paper is devoted to the study of the Kantorovich optimal transportation problem with nonlinear cost functional generated by a cost function depending on the conditional measures of the transport plan. The case of a cost function nonconvex in the second argument is considered. It is proved that this nonlinear Kantorovich problem with general cost function on a Souslin space can be reduced to the same problem with a convex cost function. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"267 - 278"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590309","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}
引用次数: 0
The Mumford Dynamical System and the Gelfand–Dikii Recursion 芒福德动力系统和格尔芬-迪基递归
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040020
P. G. Baron
{"title":"The Mumford Dynamical System and the Gelfand–Dikii Recursion","authors":"P. G. Baron","doi":"10.1134/S0016266323040020","DOIUrl":"10.1134/S0016266323040020","url":null,"abstract":"<p> In his paper “The Mumford dynamical system and hyperelliptic Kleinian functions” [Funkts. Anal. Prilozhen. <b>57</b> (4), 27–45 (2023)] Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the <span>((P,Q))</span>-recursion introduced in his paper. </p><p> In the present paper, we further develop the theory of the <span>((P,Q))</span>-recursion and describe its connections to the Korteweg–de Vries hierarchy, the Lenard operator, and the Gelfand–Dikii recursion. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"279 - 287"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590307","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}
引用次数: 0
Classification of Measurable Functions of Several Variables and Matrix Distributions 多变量可测量函数和矩阵分布的分类
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-04-01 DOI: 10.1134/S0016266323040044
A. M. Vershik
{"title":"Classification of Measurable Functions of Several Variables and Matrix Distributions","authors":"A. M. Vershik","doi":"10.1134/S0016266323040044","DOIUrl":"10.1134/S0016266323040044","url":null,"abstract":"<p> We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, this is an invariant of this function with respect to a certain group of transformations of variables; on the other hand, this is a special probability measure in the space of matrices (tensors) that is invariant under actions of natural infinite permutation groups. The intricate interplay of both interpretations of matrix (tensor) distributions makes them an important subject of modern functional analysis. We formulate and prove a theorem that, under certain conditions on a measurable function of two variables, its matrix distribution is a complete invariant. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 4","pages":"303 - 313"},"PeriodicalIF":0.6,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590230","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}
引用次数: 0
Self-Joinings and Generic Extensions of Ergodic Systems Ergodic 系统的自连接和通用扩展
IF 0.6 4区 数学
Functional Analysis and Its Applications Pub Date : 2024-03-12 DOI: 10.1134/S0016266323030048
V. V. Ryzhikov
{"title":"Self-Joinings and Generic Extensions of Ergodic Systems","authors":"V. V. Ryzhikov","doi":"10.1134/S0016266323030048","DOIUrl":"10.1134/S0016266323030048","url":null,"abstract":"<p> It is proved that the generic extensions of a dynamical system inherit the triviality of pairwise independent self-joinings. This property is related to well-known problems of joining theory and to Rokhlin’s famous multiple mixing problem. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"57 3","pages":"236 - 247"},"PeriodicalIF":0.6,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128282","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}
引用次数: 0
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