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Contemporary Mathematics. Fundamental Directions最新文献

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On Representations of Groups and Algebras in Spaces with Indefinite Metric 不定度量空间中群与代数的表示
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-2-295-315
E. Kissin, V. Shulman
{"title":"On Representations of Groups and Algebras in Spaces with Indefinite Metric","authors":"E. Kissin, V. Shulman","doi":"10.22363/2413-3639-2021-67-2-295-315","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-2-295-315","url":null,"abstract":"The paper contains a survey of known results on the structure of J-symmetric operator algebras in Pontryagin and Krein spaces, as well as on representations of groups and *-algebras in these spaces.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128171978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Asymptotics of the Spectrum of Variational Problems Arising in the Theory of Fluid Oscillations 流体振动理论中变分问题谱的渐近性
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-2-363-407
T. Suslina
{"title":"Asymptotics of the Spectrum of Variational Problems Arising in the Theory of Fluid Oscillations","authors":"T. Suslina","doi":"10.22363/2413-3639-2021-67-2-363-407","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-2-363-407","url":null,"abstract":"This work is a survey of results on the asymptotics of the spectrum of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. The problems were posed by N. D. Kopachevsky in the late 1970s and cover various fluid models. The formulations of problems are given both in the form of boundary-value problems for eigenvalues in the domain R3, which is occupied by the fluid in the equilibrium state, and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an elliptic constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface . The spectrum in the considered problems is discrete; the spectrum distribution functions have power-law asymptotics.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130677716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Bi-Variationality, Symmetries and Approximate Solutions 双变分性、对称性与近似解
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-3-596-608
V. Filippov, V. Savchin, S. Budochkina
{"title":"Bi-Variationality, Symmetries and Approximate Solutions","authors":"V. Filippov, V. Savchin, S. Budochkina","doi":"10.22363/2413-3639-2021-67-3-596-608","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-3-596-608","url":null,"abstract":"By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator S to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130070029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Injectivity and Nuclearity Properties for Real C*-Algebras 实C*-代数的注入性和核性
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-4-755-765
A. Rakhimov, M. Nurillaev, Kh. Kh. Boltaev
{"title":"Injectivity and Nuclearity Properties for Real C*-Algebras","authors":"A. Rakhimov, M. Nurillaev, Kh. Kh. Boltaev","doi":"10.22363/2413-3639-2021-67-4-755-765","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-4-755-765","url":null,"abstract":"In this paper, we study injective and nuclear real W*- and C*-algebras. The connection of these concepts with similar concepts of enveloping W*- and C*-algebras is considered. The equivalence of the concepts of injectivity and nuclearity for real C*-algebras is shown. As a consequence, nuclear real factors of types II1, II, III1, III0 and III (0 1) are completely described.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127034315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Algorithm for the Numerical Solution of the Stefan Problem and Its Application to Calculations of the Temperature of Tungsten under Impulse Action Stefan问题的数值解算法及其在脉冲作用下钨的温度计算中的应用
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-3-442-454
D. Apushkinskaya, G. Lazareva
{"title":"Algorithm for the Numerical Solution of the Stefan Problem and Its Application to Calculations of the Temperature of Tungsten under Impulse Action","authors":"D. Apushkinskaya, G. Lazareva","doi":"10.22363/2413-3639-2021-67-3-442-454","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-3-442-454","url":null,"abstract":"In this paper, we present the numerical solution of the Stefan problem to calculate the temperature of the tungsten sample heated by the laser pulse. Mathematical modeling is carried out to analyze field experiments, where an instantaneous heating of the plate to 9000 K is observed due to the effect of a heat flow on its surface and subsequent cooling. The problem is characterized by nonlinear coefficients and boundary conditions. An important role is played by the evaporation of the metal from the heated surface. Basing on Samarskiis approach, we choose to implement the method of continuous counting considering the heat conductivity equation in a uniform form in the entire domain using the Dirac delta function. The numerical method has the second order of approximation with respect to space, the interval of smoothing of the coefficients is 5 K. As a result, we obtain the temperature distributions on the surface and in the cross section of the sample during cooling.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124657352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions 边界条件频繁交替型拉普拉斯算子边值问题的渐近分析
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-1-14-129
D. Borisov
{"title":"Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions","authors":"D. Borisov","doi":"10.22363/2413-3639-2021-67-1-14-129","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-1-14-129","url":null,"abstract":"This work, which can be considered as a small monograph, is devoted to the study of twoand three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"53 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117351311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Right-Sided Invertibility of Binomial Functional Operators and Graded Dichotomy 二项式泛函算子的右侧可逆性与分级二分
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-2-208-236
A. Antonevich
{"title":"Right-Sided Invertibility of Binomial Functional Operators and Graded Dichotomy","authors":"A. Antonevich","doi":"10.22363/2413-3639-2021-67-2-208-236","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-2-208-236","url":null,"abstract":"In this paper, we consider the right-sided invertibility problem for binomial functional operators. It is known that such operators are invertible iff there exists dichotomy of solutions of the homogeneous equation. New property of solutions of the homogeneous equation named graded dichotomy is introduced and it is proved that right-sided invertibility of binomial functional operators is equivalent to existence of graded dichotomy.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127751424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stochastic Lagrange Approach to Viscous Hydrodynamics 粘性流体力学的随机拉格朗日方法
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-2-285-294
Y. Gliklikh
{"title":"Stochastic Lagrange Approach to Viscous Hydrodynamics","authors":"Y. Gliklikh","doi":"10.22363/2413-3639-2021-67-2-285-294","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-2-285-294","url":null,"abstract":"The work is a survey of the authors results with modifications and preliminary information on the use of stochastic analysis on Sobolev groups of diffeomorphisms of a flat n-dimensional torus to describe the motion of viscous fluids (nonrandom ones). The main idea is to replace the covariant derivatives on the groups of diffeomorphisms in the equations introduced by D. Ebin and J. Marsden to describe ideal fluids by the so-called mean derivatives of random processes.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134362863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures 无限测度对称空间的统计遍历定理
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-4-654-667
A. Veksler, V. Chilin
{"title":"Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures","authors":"A. Veksler, V. Chilin","doi":"10.22363/2413-3639-2021-67-4-654-667","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-4-654-667","url":null,"abstract":"Let (,) be a measurable space with -finite continuous measure, ()=. A linear operator T:L1()+L()L1()+L() is called the Dunford-Schwartz operator if ||T(f)||1||f||1 (respectively, ||T(f)||||f||) for all fL1() (respectively, fL()). {Tt}t0is a strongly continuous in L1() semigroup of Dunford-Schwartz operators, then each operator At(f)=1t∫0tTs(f)ds∈L1(Ω){{{A_t(f)} ={frac{1}{t}} {int_0^t} {T_s(f)} ds in L_1(Omega)}} has a unique extension to the Dunford-Schwartz operator, which is also denoted by At, t0. It is proved that in the completely symmetric space of measurable functions on (,) the means At converge strongly as t+ for each strongly continuous in L1() semigroup {Tt}t0 of Dunford-Schwartz operators if and only if the norm ||.||E() is order continuous.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130684545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations 一类微分-差分算子方程变分原理的构造
Contemporary Mathematics. Fundamental Directions Pub Date : 2021-12-15 DOI: 10.22363/2413-3639-2021-67-2-316-323
I. Kolesnikova
{"title":"On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations","authors":"I. Kolesnikova","doi":"10.22363/2413-3639-2021-67-2-316-323","DOIUrl":"https://doi.org/10.22363/2413-3639-2021-67-2-316-323","url":null,"abstract":"In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator P(t) depending on t and the nonlinear operator Q. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differentialdifference operators.","PeriodicalId":280824,"journal":{"name":"Contemporary Mathematics. Fundamental Directions","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134467267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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