We prove that there are finitely many simple closed geodesics on regular tetrahedra in spherical space. Also, for any pair of coprime positive integers , we find constants and depending on and and satisfying the inequality , such that a regular spherical tetrahedron with planar angle has a unique simple closed geodesic of type , up to tetrahedron isometry, whilst a regular spherical tetrahedron with planar angle has no such geodesic. Bibliography: 19 titles.
{"title":"Simple closed geodesics on regular tetrahedra in spherical space","authors":"A. Borisenko, D. Sukhorebska","doi":"10.1070/SM9433","DOIUrl":"https://doi.org/10.1070/SM9433","url":null,"abstract":"We prove that there are finitely many simple closed geodesics on regular tetrahedra in spherical space. Also, for any pair of coprime positive integers , we find constants and depending on and and satisfying the inequality , such that a regular spherical tetrahedron with planar angle has a unique simple closed geodesic of type , up to tetrahedron isometry, whilst a regular spherical tetrahedron with planar angle has no such geodesic. Bibliography: 19 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75319218","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}
Hua Loo-Keng’s problem is solved for primes, four of which have binary expansions of a special form, whilst the fifth satisfies the inequality , where . Bibliography: 13 titles.
{"title":"Hua Loo-Keng’s problem for primes of a special form","authors":"K. M. Éminyan","doi":"10.1070/SM9394","DOIUrl":"https://doi.org/10.1070/SM9394","url":null,"abstract":"Hua Loo-Keng’s problem is solved for primes, four of which have binary expansions of a special form, whilst the fifth satisfies the inequality , where . Bibliography: 13 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74849147","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}
A nonlinear conflict control system in a finite-dimensional Euclidean space on a finite time interval is considered. Two interrelated game-theoretic problems of making a system approach a compact set at a fixed moment of time are studied. A method for constructing approximate solutions to game problems of approach is presented. Most attention is paid to problems related to constructing approximations of the solvability sets of game problems in the phase space. Bibliography: 35 titles.
{"title":"Two game-theoretic problems of approach","authors":"A. Ershov, A. V. Ushakov, V. Ushakov","doi":"10.1070/SM9496","DOIUrl":"https://doi.org/10.1070/SM9496","url":null,"abstract":"A nonlinear conflict control system in a finite-dimensional Euclidean space on a finite time interval is considered. Two interrelated game-theoretic problems of making a system approach a compact set at a fixed moment of time are studied. A method for constructing approximate solutions to game problems of approach is presented. Most attention is paid to problems related to constructing approximations of the solvability sets of game problems in the phase space. Bibliography: 35 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77517313","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}
Consider the set of irreducible denominators of the rational numbers representable by finite continued fractions all of whose partial quotients belong to some finite alphabet . Let the set of infinite continued fractions with partial quotients in this alphabet have Hausdorff dimension satisfying . Then contains a positive share of positive integers. A previous similar result of the author of 2017 was related to the inequality 0.7807dots$?> ; in the original 2011 Bourgain-Kontorovich paper, 0.9839dots$?> . Bibliography: 28 titles.
{"title":"A strengthening of the Bourgain-Kontorovich method: three new theorems","authors":"I. D. Kan","doi":"10.1070/SM9437","DOIUrl":"https://doi.org/10.1070/SM9437","url":null,"abstract":"Consider the set of irreducible denominators of the rational numbers representable by finite continued fractions all of whose partial quotients belong to some finite alphabet . Let the set of infinite continued fractions with partial quotients in this alphabet have Hausdorff dimension satisfying . Then contains a positive share of positive integers. A previous similar result of the author of 2017 was related to the inequality 0.7807dots$?> ; in the original 2011 Bourgain-Kontorovich paper, 0.9839dots$?> . Bibliography: 28 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75529742","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}
Criteria for the uniform approximation of functions by solutions of second-order strongly elliptic equations on compact subsets of are obtained using the method of reduction to similar problems in , which were previously investigated by Mazalov. A number of metric properties of the capacities used are established. Bibliography: 16 titles.
{"title":"Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of","authors":"P. V. Paramonov","doi":"10.1070/SM9503","DOIUrl":"https://doi.org/10.1070/SM9503","url":null,"abstract":"Criteria for the uniform approximation of functions by solutions of second-order strongly elliptic equations on compact subsets of are obtained using the method of reduction to similar problems in , which were previously investigated by Mazalov. A number of metric properties of the capacities used are established. Bibliography: 16 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77792816","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}
Threshold resonance arises on the lower bound of the continuous spectrum of a quantum waveguide (the Dirichlet problem for the Laplace operator), provided that for this value of the spectral parameter a nontrivial bounded solution exists which is either a trapped wave decaying at infinity or an almost standing wave stabilizing at infinity. In many problems in asymptotic analysis, it is important to be able to distinguish which of the waves initiates the threshold resonance; in this work we discuss several ways to clarify its properties. In addition, we demonstrate how the threshold resonance can be preserved by fine tuning the profile of the waveguide wall, and we obtain asymptotic expressions for the near-threshold eigenvalues appearing in the discrete or continuous spectrum when the threshold resonance is destroyed. Bibliography: 60 titles.
{"title":"The preservation of threshold resonances and the splitting off of eigenvalues from the threshold of the continuous spectrum of quantum waveguides","authors":"S. A. Nazarov","doi":"10.1070/SM9426","DOIUrl":"https://doi.org/10.1070/SM9426","url":null,"abstract":"Threshold resonance arises on the lower bound of the continuous spectrum of a quantum waveguide (the Dirichlet problem for the Laplace operator), provided that for this value of the spectral parameter a nontrivial bounded solution exists which is either a trapped wave decaying at infinity or an almost standing wave stabilizing at infinity. In many problems in asymptotic analysis, it is important to be able to distinguish which of the waves initiates the threshold resonance; in this work we discuss several ways to clarify its properties. In addition, we demonstrate how the threshold resonance can be preserved by fine tuning the profile of the waveguide wall, and we obtain asymptotic expressions for the near-threshold eigenvalues appearing in the discrete or continuous spectrum when the threshold resonance is destroyed. Bibliography: 60 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88106226","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}
An asymptotic solution of the linear Cauchy problem in the presence of a ‘weak’ turning point of the limit operator is built using Lomov’s regularization method. The major singularities of the problem are written out in an explicit form. Estimates are given with respect to , which characterise the behaviour of the singularities as . The asymptotic convergence of the regularized series is proved. The results of the work are illustrated by an example. Bibliography: 8 titles.
{"title":"The regularized asymptotics of a solution of the Cauchy problem in the presence of a weak turning point of the limit operator","authors":"A. Eliseev","doi":"10.1070/SM9444","DOIUrl":"https://doi.org/10.1070/SM9444","url":null,"abstract":"An asymptotic solution of the linear Cauchy problem in the presence of a ‘weak’ turning point of the limit operator is built using Lomov’s regularization method. The major singularities of the problem are written out in an explicit form. Estimates are given with respect to , which characterise the behaviour of the singularities as . The asymptotic convergence of the regularized series is proved. The results of the work are illustrated by an example. Bibliography: 8 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84998835","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}
It is proved that in each homotopy class of continuous mappings of the two-dimensional torus to itself that induce a hyperbolic action on the fundamental group, as long as it is free of expanding mappings, there exists an -endomorphism whose nonwandering set consists of an attracting hyperbolic sink and a nontrivial one-dimensional collapsing repeller, which is a one-dimensional orientable lamination, locally homeomorphic to the direct product of a Cantor set and a line segment. Moreover, the unstable -invariant subbundle of the tangent space to the repeller has the property of uniqueness. Bibliography: 23 titles.
{"title":"On -endomorphisms of the two-dimensional torus","authors":"V. Grines, E. Zhuzhoma, E. D. Kurenkov","doi":"10.1070/SM9372","DOIUrl":"https://doi.org/10.1070/SM9372","url":null,"abstract":"It is proved that in each homotopy class of continuous mappings of the two-dimensional torus to itself that induce a hyperbolic action on the fundamental group, as long as it is free of expanding mappings, there exists an -endomorphism whose nonwandering set consists of an attracting hyperbolic sink and a nontrivial one-dimensional collapsing repeller, which is a one-dimensional orientable lamination, locally homeomorphic to the direct product of a Cantor set and a line segment. Moreover, the unstable -invariant subbundle of the tangent space to the repeller has the property of uniqueness. Bibliography: 23 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87117880","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}
We obtain a first-order trace formula for a higher order differential operator on a closed interval in the case where the perturbation operator is the operator of multiplication by a finite complex-valued charge. For operators of even orders , the result contains a term of new type, previously unknown. Bibliography: 15 titles.
{"title":"A trace formula for higher order ordinary differential operators","authors":"E. D. Gal’kovskii, A. Nazarov","doi":"10.1070/SM9449","DOIUrl":"https://doi.org/10.1070/SM9449","url":null,"abstract":"We obtain a first-order trace formula for a higher order differential operator on a closed interval in the case where the perturbation operator is the operator of multiplication by a finite complex-valued charge. For operators of even orders , the result contains a term of new type, previously unknown. Bibliography: 15 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76199274","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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