This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.
{"title":"Dynamical phenomena connected with stability loss of equilibria and periodic trajectories","authors":"A. Neishtadt, D. Treschev","doi":"10.1070/RM10023","DOIUrl":"https://doi.org/10.1070/RM10023","url":null,"abstract":"This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"883 - 926"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49301815","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}
S. Y. Dobrokhotov, V. Nazaikinskii, A. Shafarevich
We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.
{"title":"Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations","authors":"S. Y. Dobrokhotov, V. Nazaikinskii, A. Shafarevich","doi":"10.1070/RM9973","DOIUrl":"https://doi.org/10.1070/RM9973","url":null,"abstract":"We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"745 - 819"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46611190","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. Aizenman, B. Vainberg, I. Goldsheid, S. Jitomirskaya, L. Pastur, A. Klein, V. Konakov, M. Cranston, B. Simon, V. Jacšić
Stanislav Molchanov was born on 21 December 1940 in the village of Snetinovo in the Ivanovo Oblast. His mother, Nina Grigorievna Molchanova, was an elementary school teacher. His father, Alexei Pavlovich Molchanov, was an accountant at a collective farm. As a son of a priest who had been persecuted after the Russian revolution, Alexei Molchanov was able to enroll at University (the Pedagogical Institute in the city of Ivanovo) only at the end of the 1940s, due to his status as a wounded veteran of WWII. After he graduated from the university, Alexei Molchanov started teaching physics at School no. 54 in the settlement of Nerl’ in the Teikovo District of the Ivanovo Oblast. Nina Molchanova moved to work at the same school as a teacher of German. After spending five years at a local school in Snetinovo, Molchanov moved to Nerl’ with his parents, where he graduated from School no. 54 with a gold medal. While in high school, he studied on his own some parts of elementary mathematics and introductory calculus not included in the high school program, using brochures from the school mathematical library and a book by A. Ya. Khinchin. In 1958 Molchanov enrolled in the Faculty of Mechanics and Mathematics at the Lomonosov Moscow State University. He received straight A grades, and, starting from his third year, he was a recipient of the Lenin Scholarship. He participated actively in the seminars of Prof. E. B. Dynkin, his scientific advisor. As a participant of the seminar, he carried out his first research work. After graduating from the university, Molchanov stayed there as a graduate student from 1963 till 1966. He was a member of Dynkin’s school, which exemplified a high level of general mathematical education: not only probability theory and stochastic processes, but also partial differential equations, Riemannian geometry, Lie groups and algebras, and so on.
{"title":"Stanislav Alekseevich Molchanov","authors":"A. Aizenman, B. Vainberg, I. Goldsheid, S. Jitomirskaya, L. Pastur, A. Klein, V. Konakov, M. Cranston, B. Simon, V. Jacšić","doi":"10.1070/RM10024","DOIUrl":"https://doi.org/10.1070/RM10024","url":null,"abstract":"Stanislav Molchanov was born on 21 December 1940 in the village of Snetinovo in the Ivanovo Oblast. His mother, Nina Grigorievna Molchanova, was an elementary school teacher. His father, Alexei Pavlovich Molchanov, was an accountant at a collective farm. As a son of a priest who had been persecuted after the Russian revolution, Alexei Molchanov was able to enroll at University (the Pedagogical Institute in the city of Ivanovo) only at the end of the 1940s, due to his status as a wounded veteran of WWII. After he graduated from the university, Alexei Molchanov started teaching physics at School no. 54 in the settlement of Nerl’ in the Teikovo District of the Ivanovo Oblast. Nina Molchanova moved to work at the same school as a teacher of German. After spending five years at a local school in Snetinovo, Molchanov moved to Nerl’ with his parents, where he graduated from School no. 54 with a gold medal. While in high school, he studied on his own some parts of elementary mathematics and introductory calculus not included in the high school program, using brochures from the school mathematical library and a book by A. Ya. Khinchin. In 1958 Molchanov enrolled in the Faculty of Mechanics and Mathematics at the Lomonosov Moscow State University. He received straight A grades, and, starting from his third year, he was a recipient of the Lenin Scholarship. He participated actively in the seminars of Prof. E. B. Dynkin, his scientific advisor. As a participant of the seminar, he carried out his first research work. After graduating from the university, Molchanov stayed there as a graduate student from 1963 till 1966. He was a member of Dynkin’s school, which exemplified a high level of general mathematical education: not only probability theory and stochastic processes, but also partial differential equations, Riemannian geometry, Lie groups and algebras, and so on.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"943 - 949"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49175122","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}
The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.
{"title":"One-dimensional dynamical systems","authors":"L. Efremova, E. Makhrova","doi":"10.1070/RM9998","DOIUrl":"https://doi.org/10.1070/RM9998","url":null,"abstract":"The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"821 - 881"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48893575","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}
Here y(x) = (y1(x), . . . , yn(x)) and A(x) = {ajk(x)}j,k=1, where the ajk are complex functions in L1[0, 1], B = diag{b1, . . . , bn} with 0 ̸= bj ∈ C, ρ is a uniformly positive bounded measurable function, U0 and U1 are n × n number matrices, and λ is the spectral parameter. Special cases of such problems include spectral problems for the Dirac operator (which corresponds to n = 2, b1 = −b2 = i, and ρ(x) ≡ 1) and for the systems of telegrapher’s equations (n = 2, b1 = −b2 = 1, and ρ ∈ L1[0, 1]). We consider the case when all the bj are different. The straight lines Re(bkλ) = Re(bjλ), k ̸= j = 1, . . . , n, partition the complex plane C into (at most n − n) sectors Γl with vertex at the origin. In each Γl, 1 ⩽ l ⩽ n − n, we have a fundamental matrix of solutions of (1), which has the form Y(x, λ) = E(x, λ)(M(x) + R(x, λ)), where
{"title":"Regular spectral problems for systems of ordinary differential equations of the first order","authors":"A. Shkalikov","doi":"10.1070/RM10021","DOIUrl":"https://doi.org/10.1070/RM10021","url":null,"abstract":"Here y(x) = (y1(x), . . . , yn(x)) and A(x) = {ajk(x)}j,k=1, where the ajk are complex functions in L1[0, 1], B = diag{b1, . . . , bn} with 0 ̸= bj ∈ C, ρ is a uniformly positive bounded measurable function, U0 and U1 are n × n number matrices, and λ is the spectral parameter. Special cases of such problems include spectral problems for the Dirac operator (which corresponds to n = 2, b1 = −b2 = i, and ρ(x) ≡ 1) and for the systems of telegrapher’s equations (n = 2, b1 = −b2 = 1, and ρ ∈ L1[0, 1]). We consider the case when all the bj are different. The straight lines Re(bkλ) = Re(bjλ), k ̸= j = 1, . . . , n, partition the complex plane C into (at most n − n) sectors Γl with vertex at the origin. In each Γl, 1 ⩽ l ⩽ n − n, we have a fundamental matrix of solutions of (1), which has the form Y(x, λ) = E(x, λ)(M(x) + R(x, λ)), where","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"939 - 941"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44115165","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}
D. Alekseevskii, M. V. Belolipetskiî, S. Gindikin, V. Kac, D. Panyushev, D. A. Timashev, O. Shvartsman, A. Elashvili, O. Yakimova
The prominent mathematician Èrnest Borisovich Vinberg passed away on 12 May 2020. He was born in Moscow on 26 July 1937. His father Boris Georgievich Vinberg worked as an electrical engineer at the Dynamo plant of electrical equipment, and his mother Vera Evgen’evna Pokhval’nova was a teacher of mathematics and physics, and then worked as a calculation engineer. During the war the Vinberg family was evacuated to the Penza Oblast, and returned to Moscow in 1943. Vinberg became acquainted with mathematics and interested in it at a very young age. Already in high school, he made a firm decision to become a mathematician. In his sixth school year he started attending mathematics study groups at Moscow University on Mokhovaya street, participated successfully in mathematical olympiads and, in 1954, enrolled in the Faculty of Mechanics and Mathematics at Moscow State University. Evgeny Borisovich Dynkin was Vinberg’s scientific advisor during his years as an undergraduate student, as well as throughout his postgraduate studies, which began immediately upon Vinberg’s graduation from the University in 1959. For Vinberg this choice was extremely important, and the two mathematicians continued to maintain a close relationship until Dynkin’s death. Dynkin was not only an outstanding mathematician, but also an extremely original teacher. Undoubtedly, this had a strong influence on Vinberg’s development as a mathematician and a teacher. For him, scientific research and work with students were equally important. The main seminar attended by Vinberg was Dynkin’s seminar on Lie groups, which started in the 1956/57 academic year as a purely student-oriented seminar. However, F. A. Berezin and F. I. Karpelevich started frequenting the seminar already in the following year, and were joined by I. I. Piatetski-Shapiro a bit later. These were established brilliant young mathematicians, who participated in the previous version of Dynkin’s seminar in their student years. They willingly chatted with junior participants. In fact, Piatetski-Shapiro became Vinberg’s second supervisor. Among students, A. A. Kirillov and S. G. Gindikin, and also A. L. Onishchik (as a postgraduate student) were permanent participants of the seminar. By that
{"title":"Èrnest Borisovich Vinberg","authors":"D. Alekseevskii, M. V. Belolipetskiî, S. Gindikin, V. Kac, D. Panyushev, D. A. Timashev, O. Shvartsman, A. Elashvili, O. Yakimova","doi":"10.1070/RM10030","DOIUrl":"https://doi.org/10.1070/RM10030","url":null,"abstract":"The prominent mathematician Èrnest Borisovich Vinberg passed away on 12 May 2020. He was born in Moscow on 26 July 1937. His father Boris Georgievich Vinberg worked as an electrical engineer at the Dynamo plant of electrical equipment, and his mother Vera Evgen’evna Pokhval’nova was a teacher of mathematics and physics, and then worked as a calculation engineer. During the war the Vinberg family was evacuated to the Penza Oblast, and returned to Moscow in 1943. Vinberg became acquainted with mathematics and interested in it at a very young age. Already in high school, he made a firm decision to become a mathematician. In his sixth school year he started attending mathematics study groups at Moscow University on Mokhovaya street, participated successfully in mathematical olympiads and, in 1954, enrolled in the Faculty of Mechanics and Mathematics at Moscow State University. Evgeny Borisovich Dynkin was Vinberg’s scientific advisor during his years as an undergraduate student, as well as throughout his postgraduate studies, which began immediately upon Vinberg’s graduation from the University in 1959. For Vinberg this choice was extremely important, and the two mathematicians continued to maintain a close relationship until Dynkin’s death. Dynkin was not only an outstanding mathematician, but also an extremely original teacher. Undoubtedly, this had a strong influence on Vinberg’s development as a mathematician and a teacher. For him, scientific research and work with students were equally important. The main seminar attended by Vinberg was Dynkin’s seminar on Lie groups, which started in the 1956/57 academic year as a purely student-oriented seminar. However, F. A. Berezin and F. I. Karpelevich started frequenting the seminar already in the following year, and were joined by I. I. Piatetski-Shapiro a bit later. These were established brilliant young mathematicians, who participated in the previous version of Dynkin’s seminar in their student years. They willingly chatted with junior participants. In fact, Piatetski-Shapiro became Vinberg’s second supervisor. Among students, A. A. Kirillov and S. G. Gindikin, and also A. L. Onishchik (as a postgraduate student) were permanent participants of the seminar. By that","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"1123 - 1135"},"PeriodicalIF":0.9,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41454842","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}
1. Definitions. An infinite countable ordered set {P,≻, ∅} with minimal element ∅ and no maximal elements is called a locally finite poset if all its principal ideals are finite. A monotone numbering of P (or a part of P ) is an injective map φ : N → P from the set of positive integers to P satisfying the following conditions: if φ(n) ≻ φ(m), then n > m, with φ(0) = ∅. The distributive lattice ΓP of all finite ideals of a locally finite poset {P,≻} forms an N-graded graph (the Hasse diagram of the lattice). A monotone numbering of P can be identified in a natural way with a maximal path in the lattice ΓP . The set TP of all monotone numberings of P , that is, the space of infinite paths in the graph ΓP can be endowed with the natural structure of a Borel and topological space. In the terminology related to the Young graph, the poset P is the set of Z+-finite ideals, that is, Young diagrams, and monotone numberings are Young tableaux. Let P be a finite (|P | < n ∈ N) or locally finite (n = ∞) poset. For each i < n, we define an involution σi acting correctly on the space of numberings TP = {φ} of P :
{"title":"Groups generated by involutions, numberings of posets, and central measures","authors":"A. Vershik","doi":"10.1070/RM10016","DOIUrl":"https://doi.org/10.1070/RM10016","url":null,"abstract":"1. Definitions. An infinite countable ordered set {P,≻, ∅} with minimal element ∅ and no maximal elements is called a locally finite poset if all its principal ideals are finite. A monotone numbering of P (or a part of P ) is an injective map φ : N → P from the set of positive integers to P satisfying the following conditions: if φ(n) ≻ φ(m), then n > m, with φ(0) = ∅. The distributive lattice ΓP of all finite ideals of a locally finite poset {P,≻} forms an N-graded graph (the Hasse diagram of the lattice). A monotone numbering of P can be identified in a natural way with a maximal path in the lattice ΓP . The set TP of all monotone numberings of P , that is, the space of infinite paths in the graph ΓP can be endowed with the natural structure of a Borel and topological space. In the terminology related to the Young graph, the poset P is the set of Z+-finite ideals, that is, Young diagrams, and monotone numberings are Young tableaux. Let P be a finite (|P | < n ∈ N) or locally finite (n = ∞) poset. For each i < n, we define an involution σi acting correctly on the space of numberings TP = {φ} of P :","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"729 - 731"},"PeriodicalIF":0.9,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48925977","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}
In [1], [2] a general approach to constructing integrable non-commutative generalizations of a given integrable system with polynomial right-hand side was proposed. We apply it to finding non-commutative analogues of the Euler top. Consider the system of ODEs u′ = z1 vw, v′ = z2 uw, w′ = z3 uv, zi ∈ C, zi ̸= 0, (1) where ′ means the derivative with respect to t. The system (1) possesses the first integrals I1 = z3u − z1w and I2 = z3v − x2w. For any i and j, the system uτ = z1 vwI 1I j 2 , vτ = z2 uwI i 1I j 2 , wτ = z3 uvI i 1I j 2 (2)
在[1]，[2]中，提出了构造具有多项式右手边的给定可积系统的可积非交换推广的一般方法。我们将它应用于寻找欧拉顶的非交换类似物。考虑常微分方程组u′=z1-vw，v′=z2-uw，w′=z3-uv，zi∈C，zi̸=0，（1）其中′表示关于t的导数。系统（1）具有第一积分I1=z3u−z1w和I2=z3v−x2w。对于任何i和j，系统uτ=z1 vwI 1I j2，vτ=z2 uwI i 1I k2，wτ=z3 uvI i 1I j2（2）
{"title":"Non-Abelian Euler top","authors":"Vladimir V. Sokolov","doi":"10.1070/RM9988","DOIUrl":"https://doi.org/10.1070/RM9988","url":null,"abstract":"In [1], [2] a general approach to constructing integrable non-commutative generalizations of a given integrable system with polynomial right-hand side was proposed. We apply it to finding non-commutative analogues of the Euler top. Consider the system of ODEs u′ = z1 vw, v′ = z2 uw, w′ = z3 uv, zi ∈ C, zi ̸= 0, (1) where ′ means the derivative with respect to t. The system (1) possesses the first integrals I1 = z3u − z1w and I2 = z3v − x2w. For any i and j, the system uτ = z1 vwI 1I j 2 , vτ = z2 uwI i 1I j 2 , wτ = z3 uvI i 1I j 2 (2)","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"183 - 185"},"PeriodicalIF":0.9,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49626576","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}
The survey is devoted to applications of growth in non- Abelian groups to a number of problems in number theory and additive combinatorics. We discuss Zaremba’s conjecture, sum-product theory, incidence geometry, the affine sieve, and some other questions. Bibliography: 149 titles.
{"title":"Non-commutative methods in additive combinatorics and number theory","authors":"I. Shkredov","doi":"10.1070/RM10029","DOIUrl":"https://doi.org/10.1070/RM10029","url":null,"abstract":"The survey is devoted to applications of growth in non- Abelian groups to a number of problems in number theory and additive combinatorics. We discuss Zaremba’s conjecture, sum-product theory, incidence geometry, the affine sieve, and some other questions. Bibliography: 149 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"28 1","pages":"1065 - 1122"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012889","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}
V. Buchstaber, A. Varchenko, A. Veselov, P. Grinevich, S. Grushevsky, S. Y. Dobrokhotov, A. Zabrodin, A. Marshakov, A. E. Mironov, N. Nekrasov, S. Novikov, A. Okounkov, M. Olshanetsky, A. Pogrebkov, I. Taimanov, M. Tsfasman, L. Chekhov, O. Sheinman, S. Shlosman
Igor’ Moiseevich Krichever was born on 8 October 1950, in a military family in Kuibyshev (now Samara). His parents, Moisei Solomonovich Krichever and Mariya Leizerovna Arlievskaya, were aviation engineers. They came from Mogilev and Polotsk in Belorussia (now Belarus), and both were talented persons, who had overcome a lot of trouble in their lives. Igor’ Krichever must have inherited his mathematical abilities and his resilience from them. His school mathematics teacher Taisiya Mitrofanovna Mishchenko, who taught him in Taganrog in 1963–1965, played an important role in Krichever’s life. Many of her students enrolled in the famous Moscow Physical and Mathematical School no. 18 (only just then founded by A. N. Kolmogorov), better known under the name of the ‘Kolmogorov Boarding School’. And so did Igor’ Krichever: he was invited to enroll it in 1965, after the 8th grade, in recognition of his successful participation in the All-Union Mathematical Olympiad. In 1967, when he was a 10th grade student,
Igor ' Moiseevich krichhever于1950年10月8日出生在古比雪夫(现在的萨马拉)的一个军人家庭。他的父母Moisei Solomonovich Krichever和Mariya Leizerovna Arlievskaya都是航空工程师。他们来自白俄罗斯的莫吉廖夫和波洛茨克，都是很有才华的人，他们克服了生活中的许多困难。伊戈尔·克里切弗的数学能力和适应力一定是遗传自他们。1963年至1965年在塔甘罗格教他的数学老师塔西娅·米特罗法诺夫娜·米什琴科在克里切弗的一生中发挥了重要作用。她的许多学生就读于著名的莫斯科物理和数学学校。18(刚刚由A. N. Kolmogorov建立)，以“Kolmogorov寄宿学校”的名字而闻名。Igor ' Krichever也是如此:他在1965年8年级后被邀请加入，以表彰他成功参加了全联盟数学奥林匹克竞赛。1967年，当他还是一名10年级学生时，
{"title":"Igor’ Moiseevich Krichever","authors":"V. Buchstaber, A. Varchenko, A. Veselov, P. Grinevich, S. Grushevsky, S. Y. Dobrokhotov, A. Zabrodin, A. Marshakov, A. E. Mironov, N. Nekrasov, S. Novikov, A. Okounkov, M. Olshanetsky, A. Pogrebkov, I. Taimanov, M. Tsfasman, L. Chekhov, O. Sheinman, S. Shlosman","doi":"10.1070/RM10015","DOIUrl":"https://doi.org/10.1070/RM10015","url":null,"abstract":"Igor’ Moiseevich Krichever was born on 8 October 1950, in a military family in Kuibyshev (now Samara). His parents, Moisei Solomonovich Krichever and Mariya Leizerovna Arlievskaya, were aviation engineers. They came from Mogilev and Polotsk in Belorussia (now Belarus), and both were talented persons, who had overcome a lot of trouble in their lives. Igor’ Krichever must have inherited his mathematical abilities and his resilience from them. His school mathematics teacher Taisiya Mitrofanovna Mishchenko, who taught him in Taganrog in 1963–1965, played an important role in Krichever’s life. Many of her students enrolled in the famous Moscow Physical and Mathematical School no. 18 (only just then founded by A. N. Kolmogorov), better known under the name of the ‘Kolmogorov Boarding School’. And so did Igor’ Krichever: he was invited to enroll it in 1965, after the 8th grade, in recognition of his successful participation in the All-Union Mathematical Olympiad. In 1967, when he was a 10th grade student,","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"733 - 743"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012612","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：