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Vasilii Mikhailovich Babich Vasilii Mikhailovich Babich报道

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM9987

M. Belishev, S. Dobrokhotov, I. Ibragimov, A. P. Kiselev, S. Kislyakov, M. Lyalinov, Y. Matiyasevich, V. Romanov, V. Smyshlyaev, T. Suslina, N. Ural'tseva

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

2020年6月13日，著名数学家和力学专家、圣彼得堡衍射和波传播理论学院院长瓦西里·米哈伊洛维奇·巴比奇庆祝了他的90岁生日。他是许多关于数学物理中各种问题解的高频渐近结构的经典结果的作者。在他的开创性工作中，他发展了弹性体和表面波的射线方法，他对线性偏微分方程的局部解的渐近构造，发现了许多应用，以及他的一系列证明高频渐近公式的论文，都特别值得注意。巴比奇是俄罗斯联邦荣誉科学家(2010年)。他的成就被苏联国家奖所标志，他与A. S. Alekseev, V. S. Buldyrev, I. A. Molotkov和L. A. Molotkov, G. I. Petrashen和T.B. Yanovskaya一起获得了射线方法的发展(1982年)，俄罗斯科学院的V.A. Fock奖用于发展衍射理论的渐近方法(1998年)，以及王朝基金会的“一生致力于数学”奖(2014年)。在本杂志的前几期中，在巴比奇70岁和80岁生日的时候，有对他在科学领域的研究、教学和组织活动的致敬。A. P. Kiselev和V. P. Smyshlyaev在论文“V. M. Babich的70岁生日”(Zap)中分析了他在圣彼得堡衍射和波传播理论学派发展中的作用。Nauchn。扫描电镜。S.-Peterburg。Otdel。斯特克洛夫博士。(pomi) 275 (2001)， 9-16).2巴比奇继续在数学物理方面进行卓有成效的研究;特别是，他研究的是复杂干涉波[1]b[2]的理论。近年来，他

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引用次数: 0

Interpolation properties of Hermite–Padé polynomials hermite - pad<s:1>多项式的插值性质

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM10000

S. Suetin

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

其中σ1是紧集合E∧R上支持σ1的正测度，h∈h(E)是E上的全纯函数。若h(z) = σ σ2 (z)，其中σ2是支持σ2∧F的正测度，其中F∧R E是紧集合，则函数f1、f2对形成尼基辛系统(见[6]，也见[7]、[5]、[10]及其参考文献)。设Qn,j,j = 0,1,2，是多指标n = (n−1,n, n)集合[1,f1, f2]的第一类hermite - pad多项式，即deg Qn,j≤n和(Qn,0 + Qn,1f1 + Qn,2f2)(z) = O(z−2n−2)，z→∞。(2)对于任意多项式Q∈C[z] 0，令

引用次数: 1

[IMG align=ABSMIDDLE alt=$ 3$]tex_rm_5298_img1[/IMG]-manifolds given by [IMG align=ABSMIDDLE alt=$ 4$]tex_rm_5298_img2[/IMG]-regular graphs with three Euler cycles [IMG align=ABSMIDDLE alt=$ 3$]tex_rm_5298_img1[/IMG]-由[IMG align=ABSMIDDLE alt=$ 4$]tex_rm_5298_img2[/IMG]给出的流形-具有三个欧拉循环的正则图

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/rm10013

Andryi Valer'evich Malyutin, E. Fominykh, E. Shumakova

引用次数: 0

Integrable polynomial Hamiltonian systems and symmetric powers of plane algebraic curves 可积多项式哈密顿系统与平面代数曲线的对称幂

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM10007

V. Buchstaber, A. Mikhailov

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.

本文研究与平面代数曲线对称幂相关的可积多项式哈密顿系统。我们的重点是(作者发现的)Stäckel系统、平稳Korteweg - de Vries层次的Novikov方程、超椭圆曲线jacobian泛束上的Dubrovin-Novikov坐标，以及考虑曲线的对称幂不等于曲线的格时得到的新系统之间的关系。参考书目:52篇。

引用次数: 3

Chaos and integrability in -geometry 几何中的混沌与可积性

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM10008

A. Bolsinov, A. Veselov, Y. Ye

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.

在瑟斯顿意义上，我们讨论了三种群几何中的一种的三重曲面上测地线流的可积性。我们专注于案件。主要的例子是商，它是一个有限的Fuchsian群。我们证明了相应的相空间包含两个具有可积和混沌行为的开放区域，分别具有零和正拓扑熵。作为一个具体的例子，我们考虑具有模群的模三重。在这种情况下，已知是同胚的补三叶结在一个3球。Ghys证明了一个值得注意的事实，即周期测地线在模表面上的抬升会产生与著名的洛伦兹系统的混沌版本中出现的相同的同位素类型的结，并且由Birman和Williams详细研究过。在测地线系统的可积极限上，我们证明了这些结被三叶结索所取代。参考书目:60篇。

引用次数: 0

Equivariant minimal model program 等变最小模型程序

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM9990

Yuri Prokhorov

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.

调查的目的是将关于具有群体行动的品种的最小模型程序的大量信息系统化。我们讨论了该理论的基本方法，并给出了一些主要结果的证明。参考书目:243种。

引用次数: 17

Newton polytopes and tropical geometry 牛顿多面体和热带几何

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM9937

Boris Yakovlevich Kazarnovskii, A. Khovanskii, A. Esterov

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.

将“牛顿多面体”、“环面变异”、“热带几何”和“Gröbner基”等概念结合在一起的实践，导致代数几何和凸几何之间形成了稳定和互利的联系。本调查致力于描述这些概念的相互作用和应用的数学领域的现状。参考书目:68种。

引用次数: 7

On the resolution of singularities of one-dimensional foliations on three-manifolds 三维流形上一维叶形奇异性的解析

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM9993

J. Rebelo, H. Reis

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.

本文研究了三维全纯向量场和一维全纯叶的奇异性的解析，主要有两个目的。首先，在一维叶理的一般框架内，我们建立并基本上完成了Cano, Roche和Spivakovsky(2014)的工作。因此，我们得到了一个与McQuillan-Panazzolo(2013)的分解定理相当的一般分解定理，但通过不同的方法证明。本文的另一个目的是考虑一类特殊的叶形奇点，特别是包含三维复流形上完全全纯向量场的所有奇点。在此基础上，我们证明了该类全纯叶的一个更清晰的分辨定理。这第二个结果是本文的最初动机。它依赖于将早期的(一般)叶分的分解定理与微分方程解的渐近展开的一些经典材料相结合。参考书目:34篇。

引用次数: 0

Local groups in Delone sets: a conjecture and results 德龙集合中的局部群:一个猜想与结果

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM10037

N. Dolbilin, M. Shtogrin

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.

在对任意Delone集合局部对称概念的一种新方法的框架下，我们得到了不受任何限制的任意Delone集合的新结果。这些结果对晶格和规则系统具有重要意义。提出了一个关于晶体核的猜想，它极大地推广了三维晶格中五重对称不存在的经典定理。证明了下列与几何晶体学基础有关的定理。

引用次数: 1

On compatible diagonal metrics 关于相容对角度量

IF 0.9 4区数学 Q1 MATHEMATICS

Russian Mathematical Surveys

Pub Date : 2021-01-01 DOI: 10.1070/RM10031

Alexander Mikhailovich Gagonov, Олег Иванович Мохов

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):

本文解决了相容对角度量的完备描述这一众所周知的重要问题。在2000年(见[1]和[2])，Mokhov获得了所有特征值不同的兼容度量对的完整显式描述。在特征值不同的情况下，这样一对度量可以同时对角化，如[1]和[2]所示，当且仅当与这对度量相关的仿射的Nijenhuis张量消失时，它是相容的。这使得在[1]和[2]中可以显式地描述所有这些兼容指标。具有相同特征值的兼容对角度量对的一般情况尽管在应用中具有重要意义，但仍未被探索。在这项工作中，对这个案件进行了彻底的调查。迄今为止，描述所有兼容度量对的一般情况仍然是一个开放的问题。相容度量在可积系统理论、流体动力型系统的哈密顿和双哈密顿理论、可积层次、Frobenius流形理论及其推广、多维泊松括号理论、微分几何和数学物理中起着重要的作用(参见[3]-[12]和综述论文[13])。相容度规的一般概念是由Mokhov在[1]和[2]中提出的，其动机是对水动力型局部和非局部泊松结构的相容条件的研究，其理论是由Dubrovin和Novikov([14]，局部理论)和Mokhov和Ferapontov([15]和[16]，非局部理论)为水动力型系统的理论而发展起来的。回想一下，一对逆变(黎曼或伪黎曼)度量g 1 (u)和g ij 2(u)被称为几乎相容的[1]，[2]，如果对于这些度量的任何线性组合g λ1，λ2(u) = λ1g ij 1 (u) + λ2g ij 2(u)，其中λ1和λ2是任意常数，对应于这些度量的Christoffel符号也成立相同的线性关系(这些度量的Levi-Civita连接的相容条件):Γλ1,λ2;k (u) =λ1Γij 1; k (u) +λ2Γij 2; k (u),在Γλ1λ2;k (u) = g是λ1λ2 (u)Γjλ1,λ2;sk (u),Γij 1; k (u) = g 1 (u)Γj 1; sk (u)和Γij 2; k (u) = g 2 (u)Γj 2; sk (u)。一对几乎相容的度量g ij 1 (u)和g ij 2(u)被称为相容的[1]，[2]如果对于这些度量的任何线性组合g λ1，λ2(u) = λ1g ij 1 (u) + λ2g ij 2(u)，其中λ1和λ2是任意常数，对应于这些度量的黎曼曲率张量的线性关系成立(这些度量的曲率的相容条件):

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引用次数: 0

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期刊

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