. In this paper we establish the existence of an energy function for structurally stable diffeomorphisms of closed three-dimensional manifolds whose nonwan-dering set contains a two-dimensional expanding attractor.
{"title":"The construction of an energy function for three-dimensional cascades with a two-dimensional expanding attractor","authors":"G. Vyacheslav, Pochinka Olga, M. Nosková","doi":"10.1090/MOSC/249","DOIUrl":"https://doi.org/10.1090/MOSC/249","url":null,"abstract":". In this paper we establish the existence of an energy function for structurally stable diffeomorphisms of closed three-dimensional manifolds whose nonwan-dering set contains a two-dimensional expanding attractor.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"76 1","pages":"237-249"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/MOSC/249","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60559926","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}
Pub Date : 2014-11-06DOI: 10.1090/S0077-1554-2014-00239-9
M. Belishev, M. N. Demchenko, A. Popov
The tomography of manifolds describes a range of inverse problems in which we seek to reconstruct a Riemannian manifold from its boundary data (the “Dirichlet–Neumann” mapping, the reaction operator, and others). Different types of data correspond to physically different situations: the manifold is probed by electric currents or by acoustic or electromagnetic waves. In our paper we suggest a unified approach to these problems, using the ideas of noncommutative geometry. Within the framework of this approach, the underlying manifold for the reconstruction is obtained as the spectrum of an adequate Banach algebra determined by the boundary data.
{"title":"Noncommutative geometry and the tomography of manifolds","authors":"M. Belishev, M. N. Demchenko, A. Popov","doi":"10.1090/S0077-1554-2014-00239-9","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00239-9","url":null,"abstract":"The tomography of manifolds describes a range of inverse problems in which we seek to reconstruct a Riemannian manifold from its boundary data (the “Dirichlet–Neumann” mapping, the reaction operator, and others). Different types of data correspond to physically different situations: the manifold is probed by electric currents or by acoustic or electromagnetic waves. In our paper we suggest a unified approach to these problems, using the ideas of noncommutative geometry. Within the framework of this approach, the underlying manifold for the reconstruction is obtained as the spectrum of an adequate Banach algebra determined by the boundary data.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"133-149"},"PeriodicalIF":0.0,"publicationDate":"2014-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00239-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627425","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}
Pub Date : 2014-11-06DOI: 10.1090/S0077-1554-2014-00240-5
V. Zhikov, A. Shkalikov
{"title":"In memory of Boris Moiseevich Levitan (1914–2004)","authors":"V. Zhikov, A. Shkalikov","doi":"10.1090/S0077-1554-2014-00240-5","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00240-5","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"87-88"},"PeriodicalIF":0.0,"publicationDate":"2014-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00240-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627437","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}
Pub Date : 2014-11-06DOI: 10.1090/S0077-1554-2014-00241-7
Trudy Moskov, N. Solodovnikov
. We construct an open set of C 2 -diffeomorphisms which preserve the boundary of some manifold, and which have the following property: for each mapping, the basin of attraction of one component of the attractor is open and everywhere dense, but the basin of attraction of the second component is nowhere dense, though its measure is positive.
{"title":"Boundary-preserving mappings of a manifold with intermingling basins of components of the attractor, one of which is open","authors":"Trudy Moskov, N. Solodovnikov","doi":"10.1090/S0077-1554-2014-00241-7","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00241-7","url":null,"abstract":". We construct an open set of C 2 -diffeomorphisms which preserve the boundary of some manifold, and which have the following property: for each mapping, the basin of attraction of one component of the attractor is open and everywhere dense, but the basin of attraction of the second component is nowhere dense, though its measure is positive.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"69-76"},"PeriodicalIF":0.0,"publicationDate":"2014-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00241-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627447","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}
Pub Date : 2014-11-06DOI: 10.1090/S0077-1554-2014-00238-7
N. Valeev, É. A. Nazirova, Ya. T. Sultanaev
. A significant part of B. M. Levitan’s scientific activity dealt with questions on the distribution of the eigenvalues of differential operators [1]. To study the spectral density, he mainly used Carleman’s method, which he perfected. As a rule, he considered scalar differential operators. The purpose of this paper is to study the spectral density of differential operators in a space of vector-functions. The paper consists of two sections. In the first we study the asymptotics of a fourth-order differential operator
{"title":"Distribution of the eigenvalues of singular differential operators in a space of vector-functions","authors":"N. Valeev, É. A. Nazirova, Ya. T. Sultanaev","doi":"10.1090/S0077-1554-2014-00238-7","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00238-7","url":null,"abstract":". A significant part of B. M. Levitan’s scientific activity dealt with questions on the distribution of the eigenvalues of differential operators [1]. To study the spectral density, he mainly used Carleman’s method, which he perfected. As a rule, he considered scalar differential operators. The purpose of this paper is to study the spectral density of differential operators in a space of vector-functions. The paper consists of two sections. In the first we study the asymptotics of a fourth-order differential operator","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"89-102"},"PeriodicalIF":0.0,"publicationDate":"2014-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00238-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627391","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}
Pub Date : 2014-11-05DOI: 10.1090/S0077-1554-2014-00232-6
V. Zhikov, S. Pastukhova
Let Ω be a domain in Rd. We establish the uniform convexity of the Γ-limit of a sequence of Carathéodory integrands f(x, ξ) : Ω×Rd → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with exponents α and β, 1 < α ≤ β < ∞, and having a common modulus of convexity with respect to ξ. In particular, the Γ-limit of a sequence of power-law integrands of the form |ξ|p(x), where the variable exponent p : Ω → [α, β] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-closed extension of the class of power-law integrands is found. Applications to the homogenization theory for functionals of the calculus of variations and for monotone operators are given.
{"title":"Uniform convexity and variational convergence","authors":"V. Zhikov, S. Pastukhova","doi":"10.1090/S0077-1554-2014-00232-6","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00232-6","url":null,"abstract":"Let Ω be a domain in Rd. We establish the uniform convexity of the Γ-limit of a sequence of Carathéodory integrands f(x, ξ) : Ω×Rd → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with exponents α and β, 1 < α ≤ β < ∞, and having a common modulus of convexity with respect to ξ. In particular, the Γ-limit of a sequence of power-law integrands of the form |ξ|p(x), where the variable exponent p : Ω → [α, β] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-closed extension of the class of power-law integrands is found. Applications to the homogenization theory for functionals of the calculus of variations and for monotone operators are given.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"205-231"},"PeriodicalIF":0.0,"publicationDate":"2014-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00232-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627193","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}
Pub Date : 2014-11-05DOI: 10.1090/S0077-1554-2014-00233-8
Yu. A. Alkhutov, V. N. Denisov
We consider the first boundary value problem in a cylindrical domain for a uniformly parabolic second-order equation in nondivergence form. The solution satisfies the homogeneous Dirichlet condition on the lateral surface of the cylinder, and the initial function is bounded. We show that if the coefficients of the equation satisfy the local and global Dini conditions, then a necessary and sufficient condition for the stabilization of the solution to zero coincides with a similar condition for the heat equation.
{"title":"Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero","authors":"Yu. A. Alkhutov, V. N. Denisov","doi":"10.1090/S0077-1554-2014-00233-8","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00233-8","url":null,"abstract":"We consider the first boundary value problem in a cylindrical domain for a uniformly parabolic second-order equation in nondivergence form. The solution satisfies the homogeneous Dirichlet condition on the lateral surface of the cylinder, and the initial function is bounded. We show that if the coefficients of the equation satisfy the local and global Dini conditions, then a necessary and sufficient condition for the stabilization of the solution to zero coincides with a similar condition for the heat equation.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"233-258"},"PeriodicalIF":0.0,"publicationDate":"2014-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00233-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627203","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}
Pub Date : 2014-11-05DOI: 10.1090/s0077-1554-2014-00237-5
M. D. SURNACH¨EV, Boris Moiseevich Levitan
. We study local regularity of solutions of nonlinear parabolic equations with a double degeneracy and a weight. We impose the condition of p -admissibility on the weight; in particular this allows weights in the Muckenhoupt classes A p . We prove that solutions are locally H¨olderian without any restriction on the sign being constant. We prove a Harnack inequality for nonnegative solutions. We examine the stability of the constants as the parameters in the equation approach the linear case.
{"title":"Regularity of solutions of parabolic equations with a double nonlinearity and a weight","authors":"M. D. SURNACH¨EV, Boris Moiseevich Levitan","doi":"10.1090/s0077-1554-2014-00237-5","DOIUrl":"https://doi.org/10.1090/s0077-1554-2014-00237-5","url":null,"abstract":". We study local regularity of solutions of nonlinear parabolic equations with a double degeneracy and a weight. We impose the condition of p -admissibility on the weight; in particular this allows weights in the Muckenhoupt classes A p . We prove that solutions are locally H¨olderian without any restriction on the sign being constant. We prove a Harnack inequality for nonnegative solutions. We examine the stability of the constants as the parameters in the equation approach the linear case.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"259-280"},"PeriodicalIF":0.0,"publicationDate":"2014-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/s0077-1554-2014-00237-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627376","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}
Pub Date : 2014-11-05DOI: 10.1090/S0077-1554-2014-00231-4
V. V. Vlasov, N. Rautian
. The aim of the present paper is to study the asymptotic behavior of solutions of integro-differential equations on the basis of spectral analysis of their symbols. To this end, we obtain representations of strong solutions of these equations in the form of a sum of terms corresponding to the real and nonreal parts of the spectrum of the operator functions that are the symbols of these equations. These representations are new for the class of integro-differential equations considered in the paper.
{"title":"Properties of solutions of integro-differential equations arising in heat and mass transfer theory","authors":"V. V. Vlasov, N. Rautian","doi":"10.1090/S0077-1554-2014-00231-4","DOIUrl":"https://doi.org/10.1090/S0077-1554-2014-00231-4","url":null,"abstract":". The aim of the present paper is to study the asymptotic behavior of solutions of integro-differential equations on the basis of spectral analysis of their symbols. To this end, we obtain representations of strong solutions of these equations in the form of a sum of terms corresponding to the real and nonreal parts of the spectrum of the operator functions that are the symbols of these equations. These representations are new for the class of integro-differential equations considered in the paper.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"75 1","pages":"185-204"},"PeriodicalIF":0.0,"publicationDate":"2014-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0077-1554-2014-00231-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60627175","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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