Russian Mathematical Surveys最新文献

{"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":"76 1","pages":"726 - 728"},"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}

{"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":"76 1","pages":"1146 - 1148"},"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}

{"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":"76 1","pages":"95-190"},"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}

{"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":"76 1","pages":"366 - 368"},"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}

{"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":"76 1","pages":"360 - 362"},"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}

{"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":"48 1","pages":"369 - 371"},"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}

{"title":"Dynamical Bethe algebra and functions on pairs of quasi-polynomials","authors":"A. Varchenko, A. Slinkin, D. Thompson","doi":"10.1070/RM10010","DOIUrl":"https://doi.org/10.1070/RM10010","url":null,"abstract":"We consider the space of functions on the Cartan subalgebra of with values in the zero weight subspace of a tensor product of irreducible finite-dimensional -modules. We consider the algebra of commuting differential operators on , constructed by Rubtsov, Silantyev, and Talalaev in 2009. We describe the relations between the action of on and spaces of pairs of quasi- polynomials. Bibliography: 25 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"653 - 684"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012498","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":"Ualbai Utmakhanbetovich Umirbaev","authors":"V. Artamonov, V. Drenski, Y. Ershov, M. Zaitsev, E. Zelmanov, T. Kal’menov, L. Makar-Limanov, A. A. Mikhalëv, A. Mikhalëv, V. Remeslennikov, N. Romanovskii, V. Roman’kov, I. Shestakov","doi":"10.1070/RM9985","DOIUrl":"https://doi.org/10.1070/RM9985","url":null,"abstract":"Ualbai Utmakhanbetovich Umirbaev, doctor of the physical and mathematical sciences, professor, academician of the National Academy of Sciences of the Republic of Kazakhstan, laureate of the Moore Prize of the American Mathematical Society, laureate of the State Prize of the Republic of Kazakhstan, was born on 9 May 1960 in the village of Tortkul’ in the South-Kazakhstan Oblast. His father Utmakhanbet, a veteran of World War II, worked for a long time as the editor of the newspaper of the Shayan District of the South-Kazakhstan Oblast, then was the director of a secondary school in Tortkul’ and taught mathematics to senior school students. His mother Bibizukhra was a team-leader at the harvests, including during the difficult war years, and as a reward for her work she was invited to Moscow in 1940 as a participant of the USSR Agricultural Exhibition. Everyone in the Umirbaev family was enthusiastic about mathematics and chess. After Ualbai’s sixth year at school, his father took him to a summer camp of the republic’s Physics-Mathematics School in Alma-Ata (now Almaty), the capital of the Kazakh Soviet Socialist Republic, where he successfully passed examinations and enrolled in the best school in the Kazakh Republic. Mathematics was taught there by such excellent pedagogues as D. Zh. Erzhanov and K.E. Tolymbekova, who fascinated their students by interesting and at the same time difficult problems from various sources, including the journal Kvant. In 1977 Ualbai enrolled in the Faculty of Mechanics and Mathematics at Novosibirsk State University. Novosibirsk Akademgorodok made a strong impression on him. All the conditions for life, leisure, and scientific research work had been created here for lecturers and students. Extensive woodlands, numerous parklands, proximity to the Ob Sea reservoir — all this made Akademgorodok even more attractive. There were always many interesting activities being conducted in the House of Scientists, in the Culture House “Akademiya”, and in the University itself. Lectures were given by well-known scientists from various research institutes of the Siberian","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"373 - 378"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005305","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":"Quantization of linear systems of differential equations with a quadratic invariant in a Hilbert space","authors":"V. Kozlov","doi":"10.1070/RM9992","DOIUrl":"https://doi.org/10.1070/RM9992","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"357 - 359"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59006213","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":"Functions with general monotone Fourier coefficients","authors":"Aleksandr Sergeevich Belov, M. Dyachenko, Sergei Yur'evich Tikhonov","doi":"10.1070/RM10003","DOIUrl":"https://doi.org/10.1070/RM10003","url":null,"abstract":"This paper is a study of trigonometric series with general monotone coefficients in the class with . Sharp estimates are proved for the Fourier coefficients of integrable and continuous functions. Also obtained are optimal results in terms of coefficients for various types of convergence of Fourier series. For two-sided estimates are obtained for the -moduli of smoothness of sums of series with -coefficients, as well as for the (quasi-)norms of such sums in Lebesgue, Lorentz, Besov, and Sobolev spaces in terms of Fourier coefficients. Bibliography: 99 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"36 1","pages":"951 - 1017"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011805","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}