Russian Mathematical Surveys最新文献

{"title":"Vasilii Mikhailovich Babich","authors":"M. Belishev, S. Dobrokhotov, I. Ibragimov, A. P. Kiselev, S. Kislyakov, M. Lyalinov, Y. Matiyasevich, V. Romanov, V. Smyshlyaev, T. Suslina, N. Ural'tseva","doi":"10.1070/RM9987","DOIUrl":"https://doi.org/10.1070/RM9987","url":null,"abstract":"On 13 June 2020 the prominent mathematician and expert in mechanics, head of the St. Petersburg school in the theory of diffraction and wave propagation Vasilii Mikhailovich Babich observed his 90th birthday. He is the author of many now classical results on the structure of high-frequency asymptotics of solutions of various problems in mathematical physics. The pioneering works in which he developed the ray method for elastic body and surface waves are particularly notable, as are his asymptotic constructions of localized solutions of linear partial differential equations, which have found many applications, and also a series of his papers justifying formulae for high-frequency asymptotics. Babich is an Honoured Scientist of the Russian Federation (2010). His achievements have been marked by the USSR State Prize, which he received together with A. S. Alekseev, V. S. Buldyrev, I. A. and L. A. Molotkov, G. I. Petrashen, and T.B. Yanovskaya for the development of the ray method (1982), the V.A. Fock prize of the Russian Academy of Sciences for the development of asymptotic methods in diffraction theory (1998), and the prize “A Life Devoted to Mathematics” of the Dynasty Foundation (2014). In previous issues of this journal there are tributes on the occasions of his 70th and 80th birthdays1 to Babich’s research, teaching, and organizational activities in science. A. P. Kiselev and V. P. Smyshlyaev analysed his role in the development of the St. Petersburg school of the theory of diffraction and wave propagation in the paper “The 70th birthday of V. M. Babich” (Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 275 (2001), 9–16).2 Babich continues to do fruitful research in mathematical physics; in particular, he works on the theory of complex interference waves [1], [2]. In recent years he","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"193 - 194"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005337","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":"Interpolation properties of Hermite–Padé polynomials","authors":"S. Suetin","doi":"10.1070/RM10000","DOIUrl":"https://doi.org/10.1070/RM10000","url":null,"abstract":"where σ1 is a positive measure with support supp σ1 on a compact set E ⊂ R and h ∈ H (E) is a holomorphic function on E. If h(z) = σ̂2(z), where σ2 is a positive measure with support supp σ2 ⊂ F , where F ⊂ R E is a compact set, then the pair of functions f1, f2 forms a Nikishin system (see [6], and also [7], [5], [10], and the bibliography therein). Let Qn,j , j = 0, 1, 2, be the Hermite–Padé polynomials of the first type for the collection [1, f1, f2] with multi-index n = (n − 1, n, n), which means that deg Qn,j ⩽ n and (Qn,0 + Qn,1f1 + Qn,2f2)(z) = O(z−2n−2), z →∞. (2) For an arbitrary polynomial Q ∈ C[z] 0, let","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"543 - 545"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011762","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":"[IMG align=ABSMIDDLE alt=$ 3$]tex_rm_5298_img1[/IMG]-manifolds given by [IMG align=ABSMIDDLE alt=$ 4$]tex_rm_5298_img2[/IMG]-regular graphs with three Euler cycles","authors":"Andryi Valer'evich Malyutin, E. Fominykh, E. Shumakova","doi":"10.1070/rm10013","DOIUrl":"https://doi.org/10.1070/rm10013","url":null,"abstract":"","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012253","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":"Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves","authors":"V. Buchstaber, A. Mikhailov","doi":"10.1070/RM10007","DOIUrl":"https://doi.org/10.1070/RM10007","url":null,"abstract":"This survey is devoted to integrable polynomial Hamiltonian systems associated with symmetric powers of plane algebraic curves. We focus our attention on the relations (discovered by the authors) between the Stäckel systems, Novikov’s equations for the th stationary Korteweg– de Vries hierarchy, the Dubrovin–Novikov coordinates on the universal bundle of Jacobians of hyperelliptic curves, and new systems obtained by considering the symmetric powers of curves when the power is not equal to the genus of the curve. Bibliography: 52 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"587 - 652"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011815","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":"Chaos and integrability in -geometry","authors":"A. Bolsinov, A. Veselov, Y. Ye","doi":"10.1070/RM10008","DOIUrl":"https://doi.org/10.1070/RM10008","url":null,"abstract":"We review the integrability of the geodesic flow on a threefold admitting one of the three group geometries in Thurston’s sense. We focus on the case. The main examples are the quotients , where is a cofinite Fuchsian group. We show that the corresponding phase space contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively. As a concrete example we consider the case of the modular threefold with the modular group . In this case is known to be homeomorphic to the complement of a trefoil knot in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on . Bibliography: 60 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"557 - 586"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59011883","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":"Equivariant minimal model program","authors":"Yuri Prokhorov","doi":"10.1070/RM9990","DOIUrl":"https://doi.org/10.1070/RM9990","url":null,"abstract":"The purpose of the survey is to systematize a vast amount of information about the minimal model program for varieties with group actions. We discuss the basic methods of the theory and give sketches of the proofs of some principal results. Bibliography: 243 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"112 1","pages":"461 - 542"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005611","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":"Newton polytopes and tropical geometry","authors":"Boris Yakovlevich Kazarnovskii, A. Khovanskii, A. Esterov","doi":"10.1070/RM9937","DOIUrl":"https://doi.org/10.1070/RM9937","url":null,"abstract":"The practice of bringing together the concepts of ‘Newton polytopes’, ‘toric varieties’, ‘tropical geometry’, and ‘Gröbner bases’ has led to the formation of stable and mutually beneficial connections between algebraic geometry and convex geometry. This survey is devoted to the current state of the area of mathematics that describes the interaction and applications of these concepts. Bibliography: 68 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"91 - 175"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005162","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 resolution of singularities of one-dimensional foliations on three-manifolds","authors":"J. Rebelo, H. Reis","doi":"10.1070/RM9993","DOIUrl":"https://doi.org/10.1070/RM9993","url":null,"abstract":"This paper is devoted to the resolution of singularities of holomorphic vector fields and one-dimensional holomorphic foliations in dimension three, and it has two main objectives. First, within the general framework of one-dimensional foliations, we build upon and essentially complete the work of Cano, Roche, and Spivakovsky (2014). As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan–Panazzolo (2013) but proved by means of rather different methods. The other objective of this paper is to consider a special class of singularities of foliations containing, in particular, all the singularities of complete holomorphic vector fields on complex manifolds of dimension three. We then prove that a much sharper resolution theorem holds for this class of holomorphic foliations. This second result was the initial motivation for this paper. It relies on combining earlier resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations. Bibliography: 34 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"291 - 355"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005926","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":"Local groups in Delone sets: a conjecture and results","authors":"N. Dolbilin, M. Shtogrin","doi":"10.1070/RM10037","DOIUrl":"https://doi.org/10.1070/RM10037","url":null,"abstract":"In the framework of a new approach to the concept of local symmetry in arbitrary Delone sets we obtain new results for such sets without any restrictions. These results have important consequences for lattices and regular systems. A conjecture about the crystal kernel is stated, which generalises significantly the classical theorem on the non-existence of a five-fold symmetry in three-dimensional lattices. The following theorems related to the foundations of geometric crystallography are proved.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"1137 - 1139"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012763","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 compatible diagonal metrics","authors":"Alexander Mikhailovich Gagonov, Олег Иванович Мохов","doi":"10.1070/RM10031","DOIUrl":"https://doi.org/10.1070/RM10031","url":null,"abstract":"In this note the well-known important problem of a complete description of compatible diagonal metrics is solved. In 2000 (see [1] and [2]) Mokhov obtained a complete explicit description of pairs of compatible metrics for which all eigenvalues are distinct. In the case of distinct eigenvalues such a pair of metrics can be simultaneously diagonalized and, as shown in [1] and [2], it is compatible if and only if the Nijenhuis tensor of the affinor associated with this pair of metrics vanishes. This made it possible in [1] and [2] to describe all such compatible metrics explicitly. The general case of pairs of compatible diagonal metrics that have coincident eigenvalues has remained unexplored despite its importance for applications. This case is completely investigated in this work. The general case of describing all pairs of compatible metrics remains an open problem to date. Compatible metrics play an important role in the theory of integrable systems, the Hamiltonian and bi-Hamiltonian theory of systems of hydrodynamic type, integrable hierarchies, the theory of Frobenius manifolds and their generalizations, the theory of multidimensional Poisson brackets, differential geometry and mathematical physics (see [3]–[12] and the review paper [13]). The general notion of compatible metrics was introduced by Mokhov in [1] and [2], and was motivated by the study of the compatibility conditions for local and non-local Poisson structures of hydrodynamic type, the theory of which was developed by Dubrovin and Novikov ([14], local theory) and Mokhov and Ferapontov ([15] and [16], non-local theory) for the purposes of the theory of systems of hydrodynamic type. Recall that a pair of contravariant (Riemannian or pseudo-Riemannian) metrics g 1 (u) and g ij 2 (u) is called almost compatible [1], [2] if for any linear combination g λ1,λ2(u) = λ1g ij 1 (u) + λ2g ij 2 (u) of these metrics, where λ1 and λ2 are arbitrary constants, the same linear relation holds for the Christoffel symbols corresponding to these metrics (the compatibility condition for the Levi-Civita connections of these metrics): Γ λ1,λ2;k(u) = λ1Γ ij 1;k(u) + λ2Γ ij 2;k(u), where Γ λ1,λ2;k(u) = g is λ1,λ2 (u)Γjλ1,λ2;sk(u), Γ ij 1;k(u) = g is 1 (u)Γ j 1;sk(u), and Γ ij 2;k(u) = g 2 (u)Γ j 2;sk(u). A pair of almost compatible metrics g ij 1 (u) and g ij 2 (u) is called compatible [1], [2] if for any linear combination g λ1,λ2(u) = λ1g ij 1 (u) + λ2g ij 2 (u) of these metrics, where λ1 and λ2 are arbitrary constants, the same linear relation holds for the Riemann curvature tensors corresponding to these metrics (the compatibility condition for the curvatures of these metrics):","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"1140 - 1142"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59012914","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}