In the paper, the Cauchy problem for the wave equation with variable (symmetric) velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional sphere is considered. It is assumed that the initial conditions are localized on the ray and the velocity on the sphere depends only on the geodesic distance to the gluing point. The asymptotic series of the solution of the problem as parameter characterizing the initial perturbation tends to zero is given. Since the sphere is compact, then the wave propagating over the sphere is reflected at the pole opposite to the gluing point and returns to the ray. Thus, the question of the distribution of wave energy at every moment of time is also interested and discussed in this work.
{"title":"The wave equation with symmetric velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional sphere","authors":"A. Shafarevich, A. Tsvetkova","doi":"10.1090/mosc/326","DOIUrl":"https://doi.org/10.1090/mosc/326","url":null,"abstract":"In the paper, the Cauchy problem for the wave equation with variable (symmetric) velocity on the hybrid manifold obtained by gluing a ray to a three-dimensional sphere is considered. It is assumed that the initial conditions are localized on the ray and the velocity on the sphere depends only on the geodesic distance to the gluing point. The asymptotic series of the solution of the problem as parameter characterizing the initial perturbation tends to zero is given. Since the sphere is compact, then the wave propagating over the sphere is reflected at the pole opposite to the gluing point and returns to the ray. Thus, the question of the distribution of wave energy at every moment of time is also interested and discussed in this work.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48557321","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}
This paper is devoted to studying a class of nonlinear two-dimensional convolution-type integral equations on R 2 mathbb {R}^2 . This class of equations has applications in the theory of p p -adic open-closed strings and in the mathematical theory of the spread of epidemics in space and time. The existence of an alternating bounded solution is proved. The asymptotic behaviour of the constructed solution is also studied in a particular case. At the end of the paper, specific applied examples of these equations are given to illustrate the results. UDK 517.968.4.
{"title":"Alternating bounded solutions of a class of nonlinear two-dimensional convolution-type integral equations","authors":"K. Khachatryan, A. Petrosyan","doi":"10.1090/mosc/329","DOIUrl":"https://doi.org/10.1090/mosc/329","url":null,"abstract":"This paper is devoted to studying a class of nonlinear two-dimensional convolution-type integral equations on \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 2\u0000 \u0000 mathbb {R}^2\u0000 \u0000\u0000. This class of equations has applications in the theory of \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic open-closed strings and in the mathematical theory of the spread of epidemics in space and time. The existence of an alternating bounded solution is proved. The asymptotic behaviour of the constructed solution is also studied in a particular case. At the end of the paper, specific applied examples of these equations are given to illustrate the results. UDK 517.968.4.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46197107","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}
{"title":"Letter to the Editors","authors":"A. Pirkovskii","doi":"10.1090/mosc/325","DOIUrl":"https://doi.org/10.1090/mosc/325","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45644839","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}
Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of f f -graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graphically by an example where the invariant of the well-known Joukowsky system (the Euler case with a gyrostat) is realized for a certain energy range. It turns out that the entire Liouville foliation, rather than just the class of its base, is realized there; that is, the billiard and mechanical systems turn out to be Liouville equivalent. Results due to Vedyushkina and Kibkalo on constructing billiards with arbitrary values of numerical invariants are also presented. For billiard books without potential that possess a certain property, the existence of a Fomenko–Zieschang invariant is shown; it is also proved that they belong to the class of topologically stable systems. Finally, an example is presented when the addition of a Hooke potential to a planar billiard produces a splitting nondegenerate 4-singularity of rank 1.
{"title":"Realizing integrable Hamiltonian systems by means of billiard books","authors":"V. Kibkalo, A. Fomenko, I. Kharcheva","doi":"10.1090/mosc/324","DOIUrl":"https://doi.org/10.1090/mosc/324","url":null,"abstract":"Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of \u0000\u0000 \u0000 f\u0000 f\u0000 \u0000\u0000-graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graphically by an example where the invariant of the well-known Joukowsky system (the Euler case with a gyrostat) is realized for a certain energy range. It turns out that the entire Liouville foliation, rather than just the class of its base, is realized there; that is, the billiard and mechanical systems turn out to be Liouville equivalent. Results due to Vedyushkina and Kibkalo on constructing billiards with arbitrary values of numerical invariants are also presented. For billiard books without potential that possess a certain property, the existence of a Fomenko–Zieschang invariant is shown; it is also proved that they belong to the class of topologically stable systems. Finally, an example is presented when the addition of a Hooke potential to a planar billiard produces a splitting nondegenerate 4-singularity of rank 1.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43516976","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}
In 1968, V. I. Oseledets formulated the question of the convergence in Birkhoff’s theorem and in the multiplicative ergodic theorem for measurable cocycles over flows, under the condition of integrability at any fixed time. In 2016, A. M. Stepin and the author of this paper established convergence along subsets of density 1 on the time axis. Here we show that, moreover, convergence takes place modulo subsets of finite measure of the time axis.
1968年，V. I. Oseledets在任意固定时间可积条件下，给出了流上可测环的Birkhoff定理和乘法遍历定理的收敛性问题。2016年，A. M. Stepin和本文作者在时间轴上沿密度1的子集建立了收敛性。这里我们进一步证明，收敛发生于时间轴的有限测度的模子集。
{"title":"The asymptotic behaviour of cocycles over flows","authors":"M. Lipatov","doi":"10.1090/mosc/320","DOIUrl":"https://doi.org/10.1090/mosc/320","url":null,"abstract":"In 1968, V. I. Oseledets formulated the question of the convergence in Birkhoff’s theorem and in the multiplicative ergodic theorem for measurable cocycles over flows, under the condition of integrability at any fixed time. In 2016, A. M. Stepin and the author of this paper established convergence along subsets of density 1 on the time axis. Here we show that, moreover, convergence takes place modulo subsets of finite measure of the time axis.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43232217","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}
Given a mixing action LL of a group GG and a set AA of half measure we consider the possible limits of the measures μ(A∩LmiA∩LniA)mu (Acap L^{m_{i}}Acap L^{n_{i}}A) as i→∞ito infty and
给定群G G和半测度集合a a的混合作用L L，我们考虑测度μ (a∩L mi a∩L n i a) mu (a cap L^{m＿iA{}}cap)的可能极限L^{n＿iA{)}}当i→∞i toinfty和m i,n i,m i-n i→∞m i,n i,m i-n i {}{}{}{}toinfty。如果动作是3混合，那么这些限制总是等于1/8 1/8。在Ledrappier的例子中，这个极限对于某些序列是零。研究了以下问题:如果这些限制中的一个是正的但很小，那么对行动可以说什么?在本文中，我们对这个话题做了一些观察。参考书目:11篇。
{"title":"A violation of multiple mixing close to an extremal","authors":"S. Tikhonov","doi":"10.1090/mosc/322","DOIUrl":"https://doi.org/10.1090/mosc/322","url":null,"abstract":"<p>Given a mixing action <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of a group <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and a set <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of half measure we consider the possible limits of the measures <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu left-parenthesis upper A intersection upper L Superscript m Super Subscript i Superscript Baseline upper A intersection upper L Superscript n Super Subscript i Superscript Baseline upper A right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo>∩<!-- ∩ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:msub>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>i</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo>∩<!-- ∩ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:msub>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>i</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mu (Acap L^{m_{i}}Acap L^{n_{i}}A)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> as <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i right-arrow normal infinity\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>i</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ito infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m Subscript i Baseline comma n Subscript i Baseline comma m Subscript i Baseline minus n Subscript ","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47591073","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}
G. Kazaryan, A. Karapetyants, V. Margaryan, G. Mkrtchyan, A. Sergeev
This article is dedicated to the memory of Garnik Al’bertovich Karapetyan and it contains a review of results obtained by G. A. Karapetyan and his colleagues within the joint Russian–Armenian project of RFBR. In the first section, we look at multi-anisotropic spaces which were intensively studied by Karapetyan and his students. The second section is devoted to a new class of singular Hausdorff and Hausdorff–Berezin operators. In the third section, we study the connection between real function spaces and operator algebras in a Hilbert space, established by means of a quantization procedure. UDK: 517.518.
{"title":"New classes of function spaces and singular operators","authors":"G. Kazaryan, A. Karapetyants, V. Margaryan, G. Mkrtchyan, A. Sergeev","doi":"10.1090/mosc/327","DOIUrl":"https://doi.org/10.1090/mosc/327","url":null,"abstract":"This article is dedicated to the memory of Garnik Al’bertovich Karapetyan and it contains a review of results obtained by G. A. Karapetyan and his colleagues within the joint Russian–Armenian project of RFBR. In the first section, we look at multi-anisotropic spaces which were intensively studied by Karapetyan and his students. The second section is devoted to a new class of singular Hausdorff and Hausdorff–Berezin operators. In the third section, we study the connection between real function spaces and operator algebras in a Hilbert space, established by means of a quantization procedure. UDK: 517.518.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46351850","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}
We present a holomorphic version of the inverse scattering method for soliton equations of parabolic type in two-dimensional space-time. It enables one to construct examples of solutions holomorphic in both variables and study the properties of all such solutions. We show that every local holomorphic solution of any of these equations admits an analytic continuation to a globally meromorphic function of the spatial variable. We also discuss the role of the Riemann problem in the theory of integrable systems, solubility conditions for the Cauchy problem, the property of trivial monodromy for all solutions of the auxiliary linear system, and the Painlevé property for soliton equations.
{"title":"Holomorphic solutions of soliton equations","authors":"A. Domrin","doi":"10.1090/mosc/323","DOIUrl":"https://doi.org/10.1090/mosc/323","url":null,"abstract":"We present a holomorphic version of the inverse scattering method for soliton equations of parabolic type in two-dimensional space-time. It enables one to construct examples of solutions holomorphic in both variables and study the properties of all such solutions. We show that every local holomorphic solution of any of these equations admits an analytic continuation to a globally meromorphic function of the spatial variable. We also discuss the role of the Riemann problem in the theory of integrable systems, solubility conditions for the Cauchy problem, the property of trivial monodromy for all solutions of the auxiliary linear system, and the Painlevé property for soliton equations.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43409409","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}
We formulate a new condition, weaker than any already known, for the existence of a sequence of finite subgraphs of an infinite loaded linear graph along which the sequence of equilibrium measures converges to the equilibrium measure of the original infinite graph.
{"title":"Conditions for the existence of a regular sequence of finite subgraphs of an infinite loaded linear graph","authors":"B. Gurevich","doi":"10.1090/mosc/319","DOIUrl":"https://doi.org/10.1090/mosc/319","url":null,"abstract":"We formulate a new condition, weaker than any already known, for the existence of a sequence of finite subgraphs of an infinite loaded linear graph along which the sequence of equilibrium measures converges to the equilibrium measure of the original infinite graph.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49170729","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}
An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.
{"title":"CR-manifolds of infinite type in the sense of Bloom and Graham","authors":"M. Stepanova","doi":"10.1090/mosc/330","DOIUrl":"https://doi.org/10.1090/mosc/330","url":null,"abstract":"An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45970361","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}
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