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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Modern applications [1] provide motivation to consider the tridiagonal Jacobi matrix (or the so-called discrete Schrödinger operator), a classical object of spectral theory, on graphs [2]. On homogeneous trees one method to implement such operators is based on Hermite–Padé interpolation problems (see [3]). Let μ⃗ = (μ1, . . . , μd) be a collection of positive Borel measures with compact supports on R. We denote by μ̂j(z) := ∫ (z−x)−1 dμj(x) their Cauchy transforms. For an arbitrary multi-index n⃗ ∈ Z+, we need to find polynomials qn⃗,0, qn⃗,1, . . . , qn⃗,d and pn⃗, pn⃗,1, . . . , pn⃗,d with deg pn⃗ = |n⃗| := n1 + · · · + nd such that the following interpolation conditions are satisfied as z →∞ for j = 1, . . . , d:
{"title":"Multilevel interpolation for Nikishin systems and boundedness of Jacobi matrices on binary trees","authors":"A. Aptekarev, V. Lysov","doi":"10.1070/RM10017","DOIUrl":"https://doi.org/10.1070/RM10017","url":null,"abstract":"Modern applications [1] provide motivation to consider the tridiagonal Jacobi matrix (or the so-called discrete Schrödinger operator), a classical object of spectral theory, on graphs [2]. On homogeneous trees one method to implement such operators is based on Hermite–Padé interpolation problems (see [3]). Let μ⃗ = (μ1, . . . , μd) be a collection of positive Borel measures with compact supports on R. We denote by μ̂j(z) := ∫ (z−x)−1 dμj(x) their Cauchy transforms. For an arbitrary multi-index n⃗ ∈ Z+, we need to find polynomials qn⃗,0, qn⃗,1, . . . , qn⃗,d and pn⃗, pn⃗,1, . . . , pn⃗,d with deg pn⃗ = |n⃗| := n1 + · · · + nd such that the following interpolation conditions are satisfied as z →∞ for j = 1, . . . , d:","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012629","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}
Let A = (A, d) be a differential graded algebra (DGA) over a field k, that is, a Z-graded algebra A = ⊕ q∈Z A q with a k-linear map d : A → A, d = 0, of degree one that satisfies the graded Leibniz rule. Denote by D(A ) the derived category of right A -modules and by perf -A ⊂ D(A ) the triangulated subcategory of perfect modules generated by A , which is equivalent to the subcategory of compact objects D(A ) ⊂ D(A ) [5]. Suppose that A is finite dimensional. We denote by J ⊂ A the (Jacobson) radical of the k-algebra A. The ideal J is graded. Let S be the graded quotient algebra A/J , and let ε : S → A be the canonical homomorphism of algebras. We assume that d ◦ ε = 0 and d(J) ⊆ J , and consider S as a DGA with the trivial differential. In this case there are morphisms ε : S → A and π : A → S of DGAs, and the DGA A will be said to be S-split. Let e ∈ A be an idempotent, and let Pe = eA and Qe = Ae be the right and left projective A-modules. Since d(e) = 0, the A-modules Pe and Qe have the natural structure of DG A -modules. We denote by Pe = (Pe, d) and Qe = (Qe, d) the corresponding right and left DG A -modules. A right (left) DG module Φ will be called semiprojective if there is a filtration 0 = Φ0 ⊂ Φ1 ⊂ · · · = Φ such that every quotient Φi+1/Φi is a direct sum of projective DG-modules Pe (respectively, Qe). The simple right A-modules Se = Pe/eJ with d = 0 become right DG A -modules Se. We consider S as a right DG A -module and denote it by S. For any S-split DGA A , every finite-dimensional DG A -module M has a filtration 0 = Ψ0 ⊂ Ψ1 ⊂ · · · ⊂ Ψk = M such that every quotient Ψi+1/Ψi is isomorphic to some Se. Recall that a DGA A is called smooth if it is perfect as a DG bimodule.
设A = (A, d)是域k上的微分分级代数(DGA)，即Z级分级代数A =⊕q∈Z A q，其k-线性映射d: A→A, d = 0，满足分级莱布尼茨规则。用D(A)表示右A -模的派生范畴，用perf -A∧D(A)表示由A生成的完美模的三角化子范畴，它等价于紧化对象的子范畴D(A)∧D(A)[5]。假设A是有限维的。我们用J∧A表示k代数A的(Jacobson)根。理想J是分级的。设S为阶商代数A/J，设ε: S→A为代数的正则同态。设d◦ε = 0, d(J)任任，S为具有平凡微分的DGA。在这种情况下，DGA存在ε: S→A和π: A→S的态射，我们称DGA A为S分裂。设e∈A是幂等的，设Pe = eA, Qe = Ae为左右投影A模。由于d(e) = 0, A模Pe和Qe具有DG A模的自然结构。我们用Pe = (Pe, d)和Qe = (Qe, d)表示对应的左、右dga -模。如果存在过滤0 = Φ0∧Φ1∧···= Φ，使得每个商Φi+1/Φi是投影DG模块Pe(分别为Qe)的直接和，则右(左)DG模块Φ称为半投影的。简单的右A模Se = Pe/eJ, d = 0变成右DG -模Se。我们认为S是一个右DG a -模并用S表示，对于任何S分裂的DG a -模M，每个有限维DG a -模M都有一个过滤0 = Ψ0∧Ψ1∧···∧Ψk = M，使得每个商Ψi+1/Ψi与某个Se同构。回想一下，如果一个DGA a作为DG双模是完美的，那么它就是光滑的。
{"title":"Twisted tensor products of DG algebras","authors":"Dmitri Orlov","doi":"10.1070/RM10027","DOIUrl":"https://doi.org/10.1070/RM10027","url":null,"abstract":"Let A = (A, d) be a differential graded algebra (DGA) over a field k, that is, a Z-graded algebra A = ⊕ q∈Z A q with a k-linear map d : A → A, d = 0, of degree one that satisfies the graded Leibniz rule. Denote by D(A ) the derived category of right A -modules and by perf -A ⊂ D(A ) the triangulated subcategory of perfect modules generated by A , which is equivalent to the subcategory of compact objects D(A ) ⊂ D(A ) [5]. Suppose that A is finite dimensional. We denote by J ⊂ A the (Jacobson) radical of the k-algebra A. The ideal J is graded. Let S be the graded quotient algebra A/J , and let ε : S → A be the canonical homomorphism of algebras. We assume that d ◦ ε = 0 and d(J) ⊆ J , and consider S as a DGA with the trivial differential. In this case there are morphisms ε : S → A and π : A → S of DGAs, and the DGA A will be said to be S-split. Let e ∈ A be an idempotent, and let Pe = eA and Qe = Ae be the right and left projective A-modules. Since d(e) = 0, the A-modules Pe and Qe have the natural structure of DG A -modules. We denote by Pe = (Pe, d) and Qe = (Qe, d) the corresponding right and left DG A -modules. A right (left) DG module Φ will be called semiprojective if there is a filtration 0 = Φ0 ⊂ Φ1 ⊂ · · · = Φ such that every quotient Φi+1/Φi is a direct sum of projective DG-modules Pe (respectively, Qe). The simple right A-modules Se = Pe/eJ with d = 0 become right DG A -modules Se. We consider S as a right DG A -module and denote it by S. For any S-split DGA A , every finite-dimensional DG A -module M has a filtration 0 = Ψ0 ⊂ Ψ1 ⊂ · · · ⊂ Ψk = M such that every quotient Ψi+1/Ψi is isomorphic to some Se. Recall that a DGA A is called smooth if it is perfect as a DG bimodule.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012734","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 many problems in analysis and geometry there is a need to investigate vector fields with singular points that are not isolated but rather form a submanifold of the phase space, which most often has codimension 2. Of primary interest are the local orbital normal forms of such fields. ‘Orbital’ means that we may multiply vector fields by scalar functions with constant sign. In what follows, all vector fields and functions are assumed without mention to be smooth (of class C∞) unless otherwise stated. Roussarie [1] investigated vector fields of a certain special type which satisfy the following conditions at all singular points: 1) the components of the field lie in the ideal (of the space of smooth functions) generated by two of the components; 2) the divergence of the vector field (the trace of its linear part) is zero. We call such fields R-fields after Roussarie. In local coordinates the germ of an R-field has the following form at its singular point:
{"title":"Hyperbolic Roussarie fields with degenerate quadratic part","authors":"N. G. Pavlova, A. O. Remizov","doi":"10.1070/RM9893","DOIUrl":"https://doi.org/10.1070/RM9893","url":null,"abstract":"In many problems in analysis and geometry there is a need to investigate vector fields with singular points that are not isolated but rather form a submanifold of the phase space, which most often has codimension 2. Of primary interest are the local orbital normal forms of such fields. ‘Orbital’ means that we may multiply vector fields by scalar functions with constant sign. In what follows, all vector fields and functions are assumed without mention to be smooth (of class C∞) unless otherwise stated. Roussarie [1] investigated vector fields of a certain special type which satisfy the following conditions at all singular points: 1) the components of the field lie in the ideal (of the space of smooth functions) generated by two of the components; 2) the divergence of the vector field (the trace of its linear part) is zero. We call such fields R-fields after Roussarie. In local coordinates the germ of an R-field has the following form at its singular point:","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59004338","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 tunnelling effect predicted by Josephson [8] in 1962 (Nobel Prize in Physics, 1973) relates to a system of two superconductors separated by a thin dielectric layer. This phenomenon is as follows: if the dielectric is sufficiently thin, then there is a superconducting current through the system (called a Josephson junction) which is described by Josephson’s equations. In this note we investigate a model of an overdamped Josephson junction (see [3] and the bibliography there), which is described by the family of equations
{"title":"On families of constrictions in the model of an overdamped Josephson junction","authors":"Yulia P Bibilo, A. Glutsyuk","doi":"10.1070/RM9982","DOIUrl":"https://doi.org/10.1070/RM9982","url":null,"abstract":"The tunnelling effect predicted by Josephson [8] in 1962 (Nobel Prize in Physics, 1973) relates to a system of two superconductors separated by a thin dielectric layer. This phenomenon is as follows: if the dielectric is sufficiently thin, then there is a superconducting current through the system (called a Josephson junction) which is described by Josephson’s equations. In this note we investigate a model of an overdamped Josephson junction (see [3] and the bibliography there), which is described by the family of equations","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005514","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}
P. Borodin, M. Dyachenko, B. Kashin, T. P. Lukashenko, I. Mel'nikov, V. A. Sadovnichii, B. Simonov, V. Skvortsov, A. P. Solodov, V. Temlyakov, S. Tikhonov, V. M. Fedorov
The well-known mathematician in the theory of functions of a real variable and a leading expert in mathematical education Mikhail Konstantinovich Potapov observed his 90th birthday on 29 January 2021. Potapov was born in Pyatigorsk and graduated from Pyatigorsk Pedagogical Institute in 1952 as a teacher of mathematics and physics in secondary school. Subsequently, after he developed into a prominent figure in mathematics, he remained always mindful of the teaching of mathematics in school and he wrote innovative textbooks. He completed his postgraduate studies in the Faculty of Mechanics and Mathematics at Moscow State University (MSU), with S. M. Nikol’skii as his scientific advisor. Since then Potapov’s research and teaching activities have been connected with MSU, where he has been one of the leading professors in the Faculty of Mechanics and Mathematics for decades. He is the author of more than 250 research papers, and the total number of his publications exceeds 800. The main topics of his investigations are the theory of approximations of functions, embedding theorems, and trigonometric series. He was one of the first authors to study approximations of functions by algebraic polynomials in an integral metric. In the 1950s he proved Jackson’s theorem for Lipschitz classes in the spaces Lp, 1 ⩽ p < ∞. He described various structural characteristics of classes of continuous functions on a closed interval or a half-line that have one or another order of best approximation by algebraic polynomials, and he answered the question of the stability of these characteristics in the classical cases of Jacobi and Laguerre weights. He proved Jackson’s theorem and its converse for best approximation by algebraic polynomials and the moduli of smoothness defined in terms of symmetric
{"title":"Mikhail Konstantinovich Potapov","authors":"P. Borodin, M. Dyachenko, B. Kashin, T. P. Lukashenko, I. Mel'nikov, V. A. Sadovnichii, B. Simonov, V. Skvortsov, A. P. Solodov, V. Temlyakov, S. Tikhonov, V. M. Fedorov","doi":"10.1070/RM9995","DOIUrl":"https://doi.org/10.1070/RM9995","url":null,"abstract":"The well-known mathematician in the theory of functions of a real variable and a leading expert in mathematical education Mikhail Konstantinovich Potapov observed his 90th birthday on 29 January 2021. Potapov was born in Pyatigorsk and graduated from Pyatigorsk Pedagogical Institute in 1952 as a teacher of mathematics and physics in secondary school. Subsequently, after he developed into a prominent figure in mathematics, he remained always mindful of the teaching of mathematics in school and he wrote innovative textbooks. He completed his postgraduate studies in the Faculty of Mechanics and Mathematics at Moscow State University (MSU), with S. M. Nikol’skii as his scientific advisor. Since then Potapov’s research and teaching activities have been connected with MSU, where he has been one of the leading professors in the Faculty of Mechanics and Mathematics for decades. He is the author of more than 250 research papers, and the total number of his publications exceeds 800. The main topics of his investigations are the theory of approximations of functions, embedding theorems, and trigonometric series. He was one of the first authors to study approximations of functions by algebraic polynomials in an integral metric. In the 1950s he proved Jackson’s theorem for Lipschitz classes in the spaces Lp, 1 ⩽ p < ∞. He described various structural characteristics of classes of continuous functions on a closed interval or a half-line that have one or another order of best approximation by algebraic polynomials, and he answered the question of the stability of these characteristics in the classical cases of Jacobi and Laguerre weights. He proved Jackson’s theorem and its converse for best approximation by algebraic polynomials and the moduli of smoothness defined in terms of symmetric","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59006312","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}
Борис Яковлевич Казарновский, Boris Yakovlevich Kazarnovskii, Аскольд Георгиевич Хованский, A. Khovanskii, Александр Исаакович Эстеров, A. Esterov
Практика совместного использования понятий "многогранники Ньютона", "торические многообразия", "тропическая геометрия", "базисы Грeбнера" привела к формированию устойчивых взаимно полезных связей между алгебраической и выпуклой геометриями. Обзор посвящен современному состоянию области математики, описывающей взаимодействие и применение перечисленных выше понятий. Библиография: 68 названий.
{"title":"Многогранники Ньютона и тропическая геометрия","authors":"Борис Яковлевич Казарновский, Boris Yakovlevich Kazarnovskii, Аскольд Георгиевич Хованский, A. Khovanskii, Александр Исаакович Эстеров, A. Esterov","doi":"10.4213/RM9937","DOIUrl":"https://doi.org/10.4213/RM9937","url":null,"abstract":"Практика совместного использования понятий \"многогранники Ньютона\", \"торические многообразия\", \"тропическая геометрия\", \"базисы Грeбнера\" привела к формированию устойчивых взаимно полезных связей между алгебраической и выпуклой геометриями. Обзор посвящен современному состоянию области математики, описывающей взаимодействие и применение перечисленных выше понятий. Библиография: 68 названий.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70334246","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：