Pub Date : 2023-12-20DOI: 10.1134/s0081543823040119
V. V. Kozlov
Abstract
We consider linear autonomous systems of second-order differential equations that do not contain first derivatives of independent variables. Such systems are often encountered in classical mechanics. Of particular interest are cases where external forces are not potential. An important special case is given by the equations of nonholonomic mechanics linearized in the vicinity of equilibria of the second kind. We show that linear systems of this type can always be represented as Lagrange and Hamilton equations, and these equations are completely integrable: they admit complete sets of independent involutive integrals that are quadratic or linear in velocity. The linear integrals are Noetherian: they appear due to nontrivial symmetry groups.
{"title":"On Linear Equations of Dynamics","authors":"V. V. Kozlov","doi":"10.1134/s0081543823040119","DOIUrl":"https://doi.org/10.1134/s0081543823040119","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider linear autonomous systems of second-order differential equations that do not contain first derivatives of independent variables. Such systems are often encountered in classical mechanics. Of particular interest are cases where external forces are not potential. An important special case is given by the equations of nonholonomic mechanics linearized in the vicinity of equilibria of the second kind. We show that linear systems of this type can always be represented as Lagrange and Hamilton equations, and these equations are completely integrable: they admit complete sets of independent involutive integrals that are quadratic or linear in velocity. The linear integrals are Noetherian: they appear due to nontrivial symmetry groups. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"48 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817266","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}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040156
A. G. Petrova, V. V. Pukhnachev, O. A. Frolovskaya
Abstract
The second-grade fluid equations describe the motion of relaxing fluids such as aqueous solutions of polymers. The existence and uniqueness of solutions to the initial–boundary value problems for these equations were studied by D. Cioranescu, V. Girault, C. Le Roux, A. Tani, G. P. Galdi, and others. However, their studies do not contain information about the qualitative properties of solutions of these equations. Such information can be obtained by analyzing their exact solutions, which is the main goal of this work. We study layered flows and a model problem with a free boundary, construct an analog of T. Kármán’s solution, which describes the stationary motion of a second-grade fluid in a half-space induced by the rotation of the plane bounding it, and propose a generalization of V. A. Steklov’s solution of the problem on unsteady helical flows of a Newtonian fluid to the case of a second-grade fluid.
摘要 二级流体方程描述了聚合物水溶液等松弛流体的运动。D. Cioranescu、V. Girault、C. Le Roux、A. Tani、G. P. Galdi 等人研究了这些方程初界值问题解的存在性和唯一性。然而，他们的研究并不包含有关这些方程的解的定性属性的信息。这些信息可以通过分析其精确解来获得，而这正是本研究的主要目标。我们研究了层流和自由边界的模型问题，构建了 T. Kármán 的类似解，该解描述了第二级流体在其边界平面旋转引起的半空间中的静止运动，并提出了将 V. A. Steklov 的牛顿流体非稳态螺旋流动问题解推广到第二级流体的情况。
{"title":"Exact Solutions of Second-Grade Fluid Equations","authors":"A. G. Petrova, V. V. Pukhnachev, O. A. Frolovskaya","doi":"10.1134/s0081543823040156","DOIUrl":"https://doi.org/10.1134/s0081543823040156","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The second-grade fluid equations describe the motion of relaxing fluids such as aqueous solutions of polymers. The existence and uniqueness of solutions to the initial–boundary value problems for these equations were studied by D. Cioranescu, V. Girault, C. Le Roux, A. Tani, G. P. Galdi, and others. However, their studies do not contain information about the qualitative properties of solutions of these equations. Such information can be obtained by analyzing their exact solutions, which is the main goal of this work. We study layered flows and a model problem with a free boundary, construct an analog of T. Kármán’s solution, which describes the stationary motion of a second-grade fluid in a half-space induced by the rotation of the plane bounding it, and propose a generalization of V. A. Steklov’s solution of the problem on unsteady helical flows of a Newtonian fluid to the case of a second-grade fluid. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"67 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817258","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}
Pub Date : 2023-12-20DOI: 10.1134/s008154382304020x
A. P. Chugainova, R. R. Polekhina
Abstract
We study self-similar solutions of the Riemann problem in the nonuniqueness region for weakly anisotropic elastic media with a negative nonlinearity parameter. We show that all discontinuities contained in the solutions in the nonuniqueness region have a stationary structure. We also show that in the nonuniqueness region one can construct two types of self-similar solutions.
{"title":"Nonuniqueness of a Self-similar Solution to the Riemann Problem for Elastic Waves in Media with a Negative Nonlinearity Parameter","authors":"A. P. Chugainova, R. R. Polekhina","doi":"10.1134/s008154382304020x","DOIUrl":"https://doi.org/10.1134/s008154382304020x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study self-similar solutions of the Riemann problem in the nonuniqueness region for weakly anisotropic elastic media with a negative nonlinearity parameter. We show that all discontinuities contained in the solutions in the nonuniqueness region have a stationary structure. We also show that in the nonuniqueness region one can construct two types of self-similar solutions. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"82 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821592","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}
Pub Date : 2023-12-20DOI: 10.1134/s008154382304017x
D. V. Treschev
Abstract
We obtain a complete set of explicit necessary and sufficient conditions for the isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented in terms of the Taylor coefficients of the Hamiltonian function and have the form of an infinite collection of polynomial equations.
{"title":"On Isochronicity","authors":"D. V. Treschev","doi":"10.1134/s008154382304017x","DOIUrl":"https://doi.org/10.1134/s008154382304017x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We obtain a complete set of explicit necessary and sufficient conditions for the isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented in terms of the Taylor coefficients of the Hamiltonian function and have the form of an infinite collection of polynomial equations. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"40 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821545","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}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040089
I. G. Goryacheva, A. A. Yakovenko
Abstract
We study the contact interaction of a periodic system of axisymmetric rigid indenters with two height levels with an elastic half-space in the absence of friction forces. To construct a solution of the problem, we use the localization method. We obtain analytical expressions for the characteristics of the contact interaction (the radius of contact spots and the distribution of contact pressure) as well as for the components of the internal stress tensor on the symmetry axes of indenters of both levels. We analyze the effect of the shape of the contact surface of indenters, which is described by a power function (with arbitrary integer exponent), and the spatial arrangement of indenters on the contact characteristics and the stressed state of the elastic half-space.
{"title":"Internal Stresses in an Elastic Half-space under Discrete Contact Conditions","authors":"I. G. Goryacheva, A. A. Yakovenko","doi":"10.1134/s0081543823040089","DOIUrl":"https://doi.org/10.1134/s0081543823040089","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the contact interaction of a periodic system of axisymmetric rigid indenters with two height levels with an elastic half-space in the absence of friction forces. To construct a solution of the problem, we use the localization method. We obtain analytical expressions for the characteristics of the contact interaction (the radius of contact spots and the distribution of contact pressure) as well as for the components of the internal stress tensor on the symmetry axes of indenters of both levels. We analyze the effect of the shape of the contact surface of indenters, which is described by a power function (with arbitrary integer exponent), and the spatial arrangement of indenters on the contact characteristics and the stressed state of the elastic half-space. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"70 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817262","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}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823060093
Abstract
For an arbitrary partition (sigma) of the set (mathbb{P}) of all primes, a sufficient condition for the (sigma)-subnormality of a subgroup of a finite group is given. It is proved that the Kegel–Wielandt (sigma)-problem has a positive solution in the class of all finite groups all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank 1.
{"title":"On the Kegel–Wielandt $$sigma$$ -Problem","authors":"","doi":"10.1134/s0081543823060093","DOIUrl":"https://doi.org/10.1134/s0081543823060093","url":null,"abstract":"<h3>Abstract</h3> <p>For an arbitrary partition <span> <span>(sigma)</span> </span> of the set <span> <span>(mathbb{P})</span> </span> of all primes, a sufficient condition for the <span> <span>(sigma)</span> </span>-subnormality of a subgroup of a finite group is given. It is proved that the Kegel–Wielandt <span> <span>(sigma)</span> </span>-problem has a positive solution in the class of all finite groups all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank 1. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757435","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}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823050048
Abstract
We discuss various definitions of Sobolev and Besov classes on infinite-dimensional spaces, give a survey of the results on coincidence of some of these classes, and obtain a number of new results.
{"title":"Sobolev and Besov Classes on Infinite-Dimensional Spaces","authors":"","doi":"10.1134/s0081543823050048","DOIUrl":"https://doi.org/10.1134/s0081543823050048","url":null,"abstract":"<span> <h3>Abstract</h3> <p> We discuss various definitions of Sobolev and Besov classes on infinite-dimensional spaces, give a survey of the results on coincidence of some of these classes, and obtain a number of new results. </p> </span>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"88 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140057695","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}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823060019
Abstract
The paper is devoted to subgradient methods with switching between productive and nonproductive steps for problems of minimization of quasiconvex functions under functional inequality constraints. For the problem of minimizing a convex function with quasiconvex inequality constraints, a result is obtained on the convergence of the subgradient method with an adaptive stopping rule. Further, based on an analog of a sharp minimum for nonlinear problems with inequality constraints, results are obtained on the geometric convergence of restarted versions of subgradient methods. Such results are considered separately in the case of a convex objective function and quasiconvex inequality constraints, as well as in the case of a quasiconvex objective function and convex inequality constraints. The convexity may allow to additionally suggest adaptive stopping rules for auxiliary methods, which guarantee that an acceptable solution quality is achieved. The results of computational experiments are presented, showing the advantages of using such stopping rules.
{"title":"Adaptive Subgradient Methods for Mathematical Programming Problems with Quasiconvex Functions","authors":"","doi":"10.1134/s0081543823060019","DOIUrl":"https://doi.org/10.1134/s0081543823060019","url":null,"abstract":"<h3>Abstract</h3> <p>The paper is devoted to subgradient methods with switching between productive and nonproductive steps for problems of minimization of quasiconvex functions under functional inequality constraints. For the problem of minimizing a convex function with quasiconvex inequality constraints, a result is obtained on the convergence of the subgradient method with an adaptive stopping rule. Further, based on an analog of a sharp minimum for nonlinear problems with inequality constraints, results are obtained on the geometric convergence of restarted versions of subgradient methods. Such results are considered separately in the case of a convex objective function and quasiconvex inequality constraints, as well as in the case of a quasiconvex objective function and convex inequality constraints. The convexity may allow to additionally suggest adaptive stopping rules for auxiliary methods, which guarantee that an acceptable solution quality is achieved. The results of computational experiments are presented, showing the advantages of using such stopping rules. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757191","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}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823060226
Abstract
We consider the energy operator of six-electron systems in the Hubbard model and study the structure of the essential spectrum and the discrete spectrum of the system for the second singlet state of the system. In the one- and two-dimensional cases, it is shown that the essential spectrum of the six-electron second singlet state operator is the union of seven closed intervals, and the discrete spectrum of the system consists of a single eigenvalue lying below (above) the domain of the lower (upper, respectively) edge of the essential spectrum of this operator. In the three-dimensional case, there are the following situations for the essential and discrete spectra of the six-electron second singlet state operator: (a) the essential spectrum is the union of seven closed intervals, and the discrete spectrum consists of a single eigenvalue; (b) the essential spectrum is the union of four closed intervals, and the discrete spectrum is empty; (c) the essential spectrum is the union of two closed intervals, and the discrete spectrum is empty; (d) the essential spectrum is a closed interval, and the discrete spectrum is empty. Conditions are found under which each of the situations takes place.
{"title":"The Structure of the Essential Spectrum and the Discrete Spectrum of the Energy Operator for Six-Electron Systems in the Hubbard Model. The Second Singlet State","authors":"","doi":"10.1134/s0081543823060226","DOIUrl":"https://doi.org/10.1134/s0081543823060226","url":null,"abstract":"<h3>Abstract</h3> <p>We consider the energy operator of six-electron systems in the Hubbard model and study the structure of the essential spectrum and the discrete spectrum of the system for the second singlet state of the system. In the one- and two-dimensional cases, it is shown that the essential spectrum of the six-electron second singlet state operator is the union of seven closed intervals, and the discrete spectrum of the system consists of a single eigenvalue lying below (above) the domain of the lower (upper, respectively) edge of the essential spectrum of this operator. In the three-dimensional case, there are the following situations for the essential and discrete spectra of the six-electron second singlet state operator: (a) the essential spectrum is the union of seven closed intervals, and the discrete spectrum consists of a single eigenvalue; (b) the essential spectrum is the union of four closed intervals, and the discrete spectrum is empty; (c) the essential spectrum is the union of two closed intervals, and the discrete spectrum is empty; (d) the essential spectrum is a closed interval, and the discrete spectrum is empty. Conditions are found under which each of the situations takes place. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757577","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}
Pub Date : 2023-12-01DOI: 10.1134/s0081543823060044
Abstract
The triangle-free Krein graph Kre((r)) is strongly regular with parameters (((r^{2}+3r)^{2},)(r^{3}+3r^{2}+r,0,r^{2}+r)). The existence of such graphs is known only for (r=1) (the complement of the Clebsch graph) and (r=2) (the Higman–Sims graph). A.L. Gavrilyuk and A.A. Makhnev proved that the graph Kre((3)) does not exist. Later Makhnev proved that the graph Kre((4)) does not exist. The graph Kre((r)) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex Kre((r)^{prime}) is strongly regular. The graph Kre((r)^{prime}) has parameters (((r^{2}+2r-1)(r^{2}+3r+1),r^{3}+2r^{2},0,r^{2})). This work clarifies Makhnev’s result on graphs in which the neighborhoods of vertices are strongly regular graphs without (3)-cocliques. As a consequence, it is proved that the graph Kre((r)) exists if and only if the graph Kre((r)^{prime}) exists and is the complement of the block graph of a quasi-symmetric (2)-design.
{"title":"On Graphs in Which the Neighborhoods of Vertices Are Edge-Regular Graphs without 3-Claws","authors":"","doi":"10.1134/s0081543823060044","DOIUrl":"https://doi.org/10.1134/s0081543823060044","url":null,"abstract":"<h3>Abstract</h3> <p>The triangle-free Krein graph Kre<span> <span>((r))</span> </span> is strongly regular with parameters <span> <span>(((r^{2}+3r)^{2},)</span> </span><span> <span>(r^{3}+3r^{2}+r,0,r^{2}+r))</span> </span>. The existence of such graphs is known only for <span> <span>(r=1)</span> </span> (the complement of the Clebsch graph) and <span> <span>(r=2)</span> </span> (the Higman–Sims graph). A.L. Gavrilyuk and A.A. Makhnev proved that the graph Kre<span> <span>((3))</span> </span> does not exist. Later Makhnev proved that the graph Kre<span> <span>((4))</span> </span> does not exist. The graph Kre<span> <span>((r))</span> </span> is the only strongly regular triangle-free graph in which the antineighborhood of a vertex Kre<span> <span>((r)^{prime})</span> </span> is strongly regular. The graph Kre<span> <span>((r)^{prime})</span> </span> has parameters <span> <span>(((r^{2}+2r-1)(r^{2}+3r+1),r^{3}+2r^{2},0,r^{2}))</span> </span>. This work clarifies Makhnev’s result on graphs in which the neighborhoods of vertices are strongly regular graphs without <span> <span>(3)</span> </span>-cocliques. As a consequence, it is proved that the graph Kre<span> <span>((r))</span> </span> exists if and only if the graph Kre<span> <span>((r)^{prime})</span> </span> exists and is the complement of the block graph of a quasi-symmetric <span> <span>(2)</span> </span>-design. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757431","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}
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