We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either , or , or . We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.
{"title":"Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space","authors":"G. V. Belozerov","doi":"10.1070/SM9588","DOIUrl":"https://doi.org/10.1070/SM9588","url":null,"abstract":"We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either , or , or . We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89748243","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}
Conditions for the absence of global solutions of a system of nonlinear ordinary differential equations are found. Examples showing that these conditions are sharp are given. Bibliography: 12 titles.
{"title":"On the absence of global solutions of a system of ordinary differential equations","authors":"A. A. Kon’kov","doi":"10.1070/SM9571","DOIUrl":"https://doi.org/10.1070/SM9571","url":null,"abstract":"Conditions for the absence of global solutions of a system of nonlinear ordinary differential equations are found. Examples showing that these conditions are sharp are given. Bibliography: 12 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76972572","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 refine a bound for a short Kloosterman sum with a prime modulus using the so-called Vinogradov sieve. The number of terms in the sum can be less than an arbitrarily small fixed power of . Bibliography: 26 titles.
{"title":"Vinogradov’s sieve and an estimate for an incomplete Kloosterman sum","authors":"M. Korolev","doi":"10.1070/SM9572","DOIUrl":"https://doi.org/10.1070/SM9572","url":null,"abstract":"We refine a bound for a short Kloosterman sum with a prime modulus using the so-called Vinogradov sieve. The number of terms in the sum can be less than an arbitrarily small fixed power of . Bibliography: 26 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73895262","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}
Stanislav Igorevich Borzov, A. Ivanov, A. Tuzhilin
{"title":"Geormtry of the Gromov-Hausdorff Distance on the Class of all Metric Spaces","authors":"Stanislav Igorevich Borzov, A. Ivanov, A. Tuzhilin","doi":"10.1070/sm9651","DOIUrl":"https://doi.org/10.1070/sm9651","url":null,"abstract":"","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79510748","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}
{"title":"Canonical geometrization of orientable [IMG align=ABSMIDDLE alt=$ 3$]tex_sm_5016_img1[/IMG]-manifolds defined by vector-colourings of [IMG align=ABSMIDDLE alt=$ 3$]tex_sm_5016_img1[/IMG]-polytopes","authors":"Nikolai Yur'evich Erokhovets","doi":"10.1070/sm9665","DOIUrl":"https://doi.org/10.1070/sm9665","url":null,"abstract":"","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78731314","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}
Generalized -piecewise functions constructed from given monotone path-connected boundedly compact subsets of the space are studied. They are shown to be monotone path-connected suns. In finite-dimensional polyhedral spaces, luminosity points of sets admitting a lower semicontinuous selection of the metric projection operator are investigated. An example of a non- -connected sun in a four-dimensional polyhedral normed space is constructed. Bibliography: 14 titles.
{"title":"Solarity and connectedness of sets in the space and in finite-dimensional polyhedral spaces","authors":"I. G. Tsar’kov","doi":"10.1070/SM9554","DOIUrl":"https://doi.org/10.1070/SM9554","url":null,"abstract":"Generalized -piecewise functions constructed from given monotone path-connected boundedly compact subsets of the space are studied. They are shown to be monotone path-connected suns. In finite-dimensional polyhedral spaces, luminosity points of sets admitting a lower semicontinuous selection of the metric projection operator are investigated. An example of a non- -connected sun in a four-dimensional polyhedral normed space is constructed. Bibliography: 14 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89910424","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}
Our subject is the geometry of planar hinged mechanisms. The article contains a formalization of basic concepts of the theory of hinged-lever constructions, as well as some information from real algebraic geometry needed for their study. We consider mechanisms with variable number of degrees of freedom and mechanisms that have more than one degree of freedom but each hinge of which moves with one degree of freedom. For the last type we find the dimension of the configuration space. We give a number of examples of mechanisms with unusual geometric properties and formulate open questions. Bibliography: 17 titles.
{"title":"Configuration spaces of hinged mechanisms, and their projections","authors":"Mikhail Dmitrievich Kovalev","doi":"10.1070/SM9542","DOIUrl":"https://doi.org/10.1070/SM9542","url":null,"abstract":"Our subject is the geometry of planar hinged mechanisms. The article contains a formalization of basic concepts of the theory of hinged-lever constructions, as well as some information from real algebraic geometry needed for their study. We consider mechanisms with variable number of degrees of freedom and mechanisms that have more than one degree of freedom but each hinge of which moves with one degree of freedom. For the last type we find the dimension of the configuration space. We give a number of examples of mechanisms with unusual geometric properties and formulate open questions. Bibliography: 17 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90806570","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}
{"title":"Strong convexity of reachable sets of linear systems","authors":"M. Balashov","doi":"10.1070/sm9627","DOIUrl":"https://doi.org/10.1070/sm9627","url":null,"abstract":"","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89968883","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 parametric family of nonlocal balance equations in the space of signed measures is studied. Under assumptions that cover a number of known conceptual models we establish the existence of the solution, its uniqueness and continuous dependence on the parameter and the initial distribution. Several corollaries of this theorem, which are useful for control theory, are discussed. In particular, this theorem yields the limit in the mean field of a system of ordinary differential equations, the existence of the optimal control for an assembly of trajectories, Trotter’s formula for the product of semigroups of the corresponding operators, and the existence of a solution to a differential inclusion in the space of signed measures. Bibliography: 33 titles.
{"title":"Nonlocal balance equations with parameters in the space of signed measures","authors":"N. Pogodaev, M. Staritsyn","doi":"10.1070/SM9516","DOIUrl":"https://doi.org/10.1070/SM9516","url":null,"abstract":"A parametric family of nonlocal balance equations in the space of signed measures is studied. Under assumptions that cover a number of known conceptual models we establish the existence of the solution, its uniqueness and continuous dependence on the parameter and the initial distribution. Several corollaries of this theorem, which are useful for control theory, are discussed. In particular, this theorem yields the limit in the mean field of a system of ordinary differential equations, the existence of the optimal control for an assembly of trajectories, Trotter’s formula for the product of semigroups of the corresponding operators, and the existence of a solution to a differential inclusion in the space of signed measures. Bibliography: 33 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90240588","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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