Russian Mathematical Surveys最新文献

{"title":"Effective results in the theory of birational rigidity","authors":"A. Pukhlikov","doi":"10.1070/RM10039","DOIUrl":"https://doi.org/10.1070/RM10039","url":null,"abstract":"This paper is a survey of recent effective results in the theory of birational rigidity of higher-dimensional Fano varieties and Fano–Mori fibre spaces. Bibliography: 59 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"77 1","pages":"301 - 354"},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59013317","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":"Sergei Viktorovich Bochkarev","authors":"B. S. K. A. S. V. Konyagin","doi":"10.1070/RM10051","DOIUrl":"https://doi.org/10.1070/RM10051","url":null,"abstract":"The prominent Soviet and Russian mathematician, expert in the theory of functions and functional analysis Sergei Viktorovich Bochkarev passed away on 8 June 2021. In less than a month he would have celebrated his 80th birthday. In this article we expound on the main results of his research and recall the key milestones of his journey through life. He was born in Kuibyshev (now Samara) on 24 July 1941. His father Viktor Aleksanrdrovich Bochkarev, born in Saratov, was a well-known expert in literature, professor, Doctor of Science (philology), who had long been Head of the Department of Russian and Foreign Literature at Kuibyshev State Pedagogical Institute. His mother Tat’yana Georgievna Demina was an engineer and taught at the Kuibyshev Construction School. In 1958, having graduated from High School no. 94 in Kuibyshev with a gold medal, Bochkarev enrolled at the Moscow Institute of Physics and Technology. In 1964–1967 he pursued postgraduate studies at the Department of Higher Mathematics of the Institute, where P. L. Ul’yanov was his research advisor. In 1969 he defended his Ph.D. thesis at the Steklov Mathematical Institute of the USSR Academy of Sciences. His first job was at the Central Institute for Economics and Mathematics of the USSR Academy of Sciences, where he stayed for three years. From April 1971 until the end of his life, for slightly more than 50 years, Sergei Bochkarev worked at the Department of the Theory of Functions of the Steklov Mathematical Institute and focused exclusively on research. In 1974 he defended his D.Sc. thesis on Function classes and Fourier coefficients with respect to complete orthonormal systems. Bochkarev’s list of publications1 contains more than 70 titles. We only mention his achievements that are the most important ones in our opinion and have earned him the unchallenged reputation as ‘solver of fundamental problems’ among experts in analysis. The first large cycle of Bochkarev’s papers, which made up his doctoral dissertation, was devoted to the properties of the Fourier coefficients of functions in various function spaces with respect to classical or general orthonormal systems. By the","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"77 1","pages":"355 - 360"},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59013595","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 estimating the local error of a numerical solution of the parametrized Cauchy problem","authors":"E. Kuznetsov, Sergei Sergeevich Leonov","doi":"10.1070/rm10056","DOIUrl":"https://doi.org/10.1070/rm10056","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59013778","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 the Dirichlet problem for not strongly elliptic second-order equations","authors":"A. Bagapsh, M. Mazalov, K. Fedorovskiy","doi":"10.1070/RM10011","DOIUrl":"https://doi.org/10.1070/RM10011","url":null,"abstract":"1. Let L be a second-order elliptic partial differential operator in C with constant complex coefficients, that is, Lf = af ′′ xx + bf ′′ xy + cf ′′ yy, where a, b, c ∈ C. The ellipticity of L means that the roots λ1 and λ2 of the corresponding characteristic equation aλ + bλ + c = 0 are not real. If L is such that λ1 and λ2 belong to different half-planes of the complex plane with respect to the real line, then L is called strongly elliptic. The classical example of a strongly elliptic operator is the Laplace operator ∆, where ∆f = f ′′ xx + f ′′ yy, while the Bitsadze operator ∂, where ∂f = (f ′ x + if ′ y)/2 is the Cauchy–Riemann operator, serves as an example of a not strongly elliptic one. We denote by C(E) the space of all continuous complex-valued functions on a set E ⊂ C, and put ∥f∥E = supz∈E |f(z)| for f ∈ C(E). A bounded simply connected domain G ⊂ C is said to be regular with respect to the Dirichlet problem for L (or, briefly, L-regular) if for every function h ∈ C(∂G), there is an f ∈ C(G) such that Lf = 0 in G and f ∣∣ ∂G = h. The classical theorem due to Lebesgue [1] states that any bounded simply connected domain G ⊂ C is ∆-regular, that is, it is regular with respect to the classical Dirichlet problem for harmonic functions. For a general strongly elliptic operator L of the form under consideration, there is a conjecture that any bounded simply connected domain is L-regular (see [2], Problem 4.2). This conjecture has been proved only under certain additional fairly restrictive conditions on the regularity of the boundary of G. Namely, the corresponding result was obtained in [3] for Jordan domains with piecewise C-smooth boundary, and this condition on ∂G has not been significantly weakened during the last 20 years. (For instance, the question concerning the L-regularity of an arbitrary Jordan domain G is still open even in the case when ∂G is rectifiable.)","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"77 1","pages":"372 - 374"},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012589","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":"Inverse function theorem on the class of holomorphic self-maps of a disc with two fixed points","authors":"O. Kudryavtseva, A. Solodov","doi":"10.1070/RM10042","DOIUrl":"https://doi.org/10.1070/RM10042","url":null,"abstract":"In the present paper we solve the problem on the sharp domain of invertibility on the class of holomorphic self-maps of the unit disc with two (interior and boundary) fixed points and a constraint on the angular derivative at the boundary fixed point. The interest in such extremal problems stems primarily from Bloch’s famous theorem [1] to the effect that any function f holomorphic in the unit disc D = {z ∈ C : |z| < 1} is invertible in some disc of radius R|f ′(0)|, where R is an absolute constant. The search for the sharp upper bound B of such R’s (known as the Bloch constant) is one of the most important (and still unsolved) problems of geometric function theory. The near-best lower estimate B ⩾ √ 3/4 is due to Ahlfors [2]. Considerably later, Bonk [3] proved with the help of his distortion theorem on the Bloch class that Ahlfors’ estimate is not sharp, that is, B > √ 3/4. A little later, Chen and Gauthier [4] showed that B > √ 3/4 + 2 · 10−4 by slightly improving the technical details of Bonk’s proof. In our opinion, Landau’s approach might be capable of delivering new lower bounds for the Bloch constant. Considering the class of bounded holomorphic maps f of the disc D with interior fixed point z = 0 and such that f ′(0) = 1, Landau [5] proved the existence of a common disc of univalence on this class and found its precise radius. Moreover, he discovered that there is a disc in which all functions from this class are invertible, and he also determined the precise radius of this disc. Using these results, Landau gave one of the first estimates of the Bloch constant. At the same time, the domain of invertibility of each function in the class studied by Landau is much broader than the common disc of invertibility. This suggest the natural problem of finding sharp domains of univalence and invertibility on subclasses of this class. In a certain sense, as such subclasses it is natural to study the classes of holomorphic self-maps of the unit disc D with several fixed points (see [6]), which have important applications. We let B denote the class of holomorphic self-maps of D. Putting B[0] = {f ∈ B : f(0) = 0}, we can write Landau’s results as follows: if f ∈ B[0] and |f ′(0)| ⩾ 1/M with M > 1, then f is univalent in Z = {z ∈ D : |z| < M− √ M2 − 1 } and invertible in W = {w ∈ D : |w| < (M − √ M2 − 1 )}. Moreover, in place of Z and W one can take neither discs of larger radius nor any broader domains. The proof of this result is based on the following inequality (see [5]): if f ∈ B[0] and if a, b ∈ D with a ̸= b are such that f(a) = f(b) = c, then |c| ⩽ |a| |b|. On the class B{1} = {f ∈ B : ∠ limz→1 f(z) = 1} Becker and Pommerenke [7] obtained an inequality analogous to Landau’s in a certain sense. They showed that","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"77 1","pages":"177 - 179"},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59013392","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":"Roots of the characteristic equation for symplectic groupoid","authors":"L. Chekhov, Michael Zalmanovich Shapiro, H. Shibo","doi":"10.1070/rm9999","DOIUrl":"https://doi.org/10.1070/rm9999","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59006725","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":"What do Abelian categories form?","authors":"D. Kaledin","doi":"10.1070/RM10044","DOIUrl":"https://doi.org/10.1070/RM10044","url":null,"abstract":"Given two finitely presentable Abelian categories and , we outline a construction of an Abelian category of functors from to , which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of and . The construction is absolute, so it makes it possible to recover not only Hochschild cohomology but also Mac Lane cohomology. Bibliography: 29 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"77 1","pages":"1 - 45"},"PeriodicalIF":0.9,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43392131","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":"Solutions of a Hamiltonian system with two-dimensional control in a neighbourhood of a singular second-order extremal","authors":"M. Ronzhina, L. Manita, L. Lokutsievskiy","doi":"10.1070/RM10018","DOIUrl":"https://doi.org/10.1070/RM10018","url":null,"abstract":"was considered. If the domain U in (1) is a triangle, then the optimal synthesis can be constructed completely (see [1]). Partial synthesis, including synthesis for the problem with a triangle, has also been constructed for a Hamiltonian system of general form with U having the shape of a convex polygon [1]. In the case when U has a smooth boundary, the question of complete optimal synthesis is still open. For the problem (1) where U is a disc, trajectories with chattering and logarithmic spirals have explicitly been found for certain classes of initial conditions (see [2] and [3]). We consider the Hamiltonian system of Pontryagin’s maximum principle","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"936 - 938"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44250583","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":"Viktor Abramovich Zalgaller","authors":"D. Burago, Yu. D. Burago, A. Verner, A. Vershik, M. L. Gromov, I. Ibragimov, S. Ivanov, S. Kislyakov, Semen Samsonovich Kutateladze, A. Lodkin, Y. Matiyasevich, N. Mnev, A. Nazarov, G. Panina, Y. Reshetnyak, V. A. Ryzhik, N. Uraltseva, Y. Eliashberg","doi":"10.1070/RM10022","DOIUrl":"https://doi.org/10.1070/RM10022","url":null,"abstract":"On 2 October, 2020, just over two months before his centenary, the outstanding representative of the St Petersburg School of Geometry, honourary member of the St Petersburg Mathematical Society, Professor Viktor Abramovich Zalgaller passed away. He was born on 25 December 1920 in the village of Parfino, Novgorod Province, in the family of engineer Abram Leont’evich Zalgaller and attorney Tat’yana Markovna Shabad-Zalgaller. In 1922 the family moved to Petrograd. In 1931 his father was convicted under Article 581 and spent 16 years in Ukhtpechlag2, in prison, and then in exile. After graduating from School no. 103 in the Smol’ninskii district of Leningrad in 1937, Viktor enrolled in the Faculty of Mathematics and Mechanics of Leningrad University. His abilities were noticed by L. V. Kantorovich, his lecturer on mathematical analysis, who asked the third-year student to prepare a textbook based on his notes. In 1940, as part of a mobilisation announced by the Young Communist League, Zalgaller was transferred to the Aviation Institute. There he received an engineering education, which played a significant role in his further research career. In the first days of the war he volunteered for the People’s Militia (Second Division) and was sent to the front almost immediately. During almost the entire war he was a signaller. The combat path of Viktor Zalgaller was not easy: the defence of Leningrad, the Oranienbaum Bridgehead, an injury during the lifting of the Leningrad blockade, the storming of Vyborg, battles in the Baltic states, the storming of Danzig, reaching the Elbe. . . Zalgaller was awarded the Order of the Red Star, the medal “For Courage”, three medals “For Battle Merit”, the medal “For the Defence of Leningrad”, and others. He met the end of the war as","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"927 - 931"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44452122","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":"Lower bounds for -term approximations in the metric of the discrete space","authors":"B. Kashin","doi":"10.1070/RM10026","DOIUrl":"https://doi.org/10.1070/RM10026","url":null,"abstract":"In applied problems m-term approximations are used extensively. The approximating m-term polynomials are usually constructed by means of various ‘greedy’ algorithms (see [1] for details). As concerns lower bounds for the quantities (1), in the case when X = L(Ω) and Φ is an orthonormal basis in X such bounds are usually obtained by using the incompressibility property of an N -dimensional cube, which was established in [2]. With the aid of this geometric result, for a fixed m the problem can be reduced to finding a cube of dimension K ·m (where K is an absolute constant) with possibly large edge length such that all its vertices lie in F . This approach cannot be used for X = L(Ω) (see [3]). On the other hand, [3] contains rather general conditions on the set F ⊂ L(Ω) and the system Φ, which make it possible to find lower bounds for the quantities (1). One reason for taking the metric (2) lies in its connections with discrete mathematics and, in particular,","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"933 - 935"},"PeriodicalIF":0.9,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47769172","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}