Russian Mathematical Surveys最新文献

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Viktor Stepanovich Kulikov (on his seventieth birthday) 维克托·斯捷潘诺维奇·库利科夫(庆祝70岁生日)

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/rm10060

F. Bogomolov, S. Gorchinskiy, A. Zheglov, V. Nikulin, Dmitri Orlov, D. Osipov, A. N. Parshin, V. Popov, V. Przyjalkowski, Yuri Prokhorov, M. Reid, A. Sergeev, D. Treschev, A. K. Tsikh, I. Cheltsov, E. Chirka

引用次数: 0

Structures of non-classical discontinuities in solutions of hyperbolic systems of equations 双曲型方程组解中的非经典不连续结构

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/RM10033

A. Kulikovskii, A. P. Chugainova

Discontinuity structures in solutions of a hyperbolic system of equations are considered. The system of equations has a rather general form and, in particular, can describe the longitudinal and torsional non-linear waves in elastic rods in the simplest setting and also one-dimensional waves in unbounded elastic media. The properties of discontinuities in solutions of these equations have been investigated earlier under the assumption that only the relations following from the conservation laws for the longitudinal momentum and angular momentum about the axis of the rod and the displacement continuity condition hold on the discontinuities. The shock adiabat has been studied. This paper deals with stationary discontinuity structures under the assumption that viscosity is the main governing mechanism inside the structure. Some segments of the shock adiabat are shown to correspond to evolutionary discontinuities without structure. It is also shown that there are special discontinuities on which an additional relation must hold, which arises from the condition that a discontinuity structure exists. The additional relation depends on the processes in the structure. Special discontinuities satisfy evolutionary conditions that differ from the well-known Lax conditions. Conclusions are discussed, which can also be of interest in the case of other systems of hyperbolic equations. Bibliography: 58 titles.

研究了一类双曲型方程组解中的不连续结构。该方程组具有相当一般的形式，尤其可以描述最简单情况下弹性杆中的纵向和扭转非线性波，也可以描述无界弹性介质中的一维波。这些方程的解的不连续性质已经在前面的假设下进行了研究，假设不连续只适用于沿杆轴的纵动量和角动量的守恒律和位移连续条件。对激波绝热进行了研究。本文在假定黏性是结构内部的主要控制机制的前提下，研究了平稳不连续结构。激波绝热区的某些部分显示出与无结构的演化不连续相对应。还证明了在一些特殊的不连续点上必须有附加关系，这是由不连续结构存在的条件引起的。附加关系取决于结构中的过程。特殊不连续满足不同于众所周知的松弛条件的演化条件。讨论了一些结论，这些结论对其他双曲型方程组也有意义。参考书目:58种。

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引用次数: 0

A number-theoretic part of control theory 控制论的数论部分

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/RM10050

Aleksandr Iosifovich Ovseevich

The importance of Q is primarily due to the fact that it defines a common Lyapunov function for the two stable matrices M = −diag(1, 2, . . . , N) and A + BC, where Aei = −iei+1 for i = 1, . . . , N , B = e1, and C = −(1/2)B∗Q. Here the ei, i = 1, . . . , N , form the standard basis of R and eN+1 = 0. In [6] it was shown that Q is an even integer matrix, that is, Qij ∈ 2Z, and it was conjectured that all elements of the matrix are divisible by N(N + 1). The proofs in [6] were based on considering orthogonal polynomials. Here we prove this conjecture using similar methods.

Q的重要性主要是由于它为两个稳定矩阵M = - diag(1,2，…)定义了一个公共Lyapunov函数。， N)和A + BC，其中对于i = 1, Aei =−iei+1，…， N, B = e1，且C =−(1/2)B * Q。这里是ei, i = 1，…， N，构成R和eN+1 = 0的标准基。[6]中证明了Q是一个偶数矩阵，即Qij∈2Z，并推测了矩阵的所有元素都可以被N(N + 1)整除。[6]中的证明是基于考虑正交多项式的。这里我们用类似的方法证明了这个猜想。

引用次数: 0

Spectrum of a convolution operator with potential 具有势的卷积算子的谱

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/rm10055

A. Kupavskii, A. Sagdeev, Nóra Frankl

引用次数: 2

Effective results in the theory of birational rigidity 有效的结果在理论的birational刚性

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/RM10039

A. Pukhlikov

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.

本文综述了近年来关于高维Fano品种和Fano - mori纤维空间的双胞刚度理论的有效结果。参考书目:59种。

引用次数: 1

Sergei Viktorovich Bochkarev 谢尔盖·维克托罗维奇·博奇卡列夫

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/RM10051

B. S. K. A. S. V. Konyagin

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

著名的苏联和俄罗斯数学家、泛函理论和泛函分析专家谢尔盖·维克托罗维奇·博奇卡列夫于2021年6月8日去世。再过不到一个月，他就要庆祝80岁生日了。在这篇文章中，我们阐述了他的主要研究成果，并回顾了他一生中重要的里程碑。他于1941年7月24日出生在古比雪夫(现萨马拉)。他的父亲维克多·亚历克山德罗维奇·博奇卡廖夫出生于萨拉托夫，是一位著名的文学专家，教授，理学博士(语言学)，长期担任古比雪夫国立教育学院俄罗斯和外国文学系主任。他的母亲tatyana Georgievna Demina是一名工程师，在古比雪夫建筑学校任教。1958年，他从北京第一中学毕业。1994年在古比雪夫获得金牌后，博奇卡列夫考入莫斯科物理技术学院。1964年至1967年，他在该研究所高等数学系攻读研究生课程，p.l. Ul 'yanov是他的研究顾问。1969年，他在苏联科学院斯特克洛夫数学研究所为自己的博士论文辩护。他的第一份工作是在苏联科学院中央经济和数学研究所，在那里呆了三年。从1971年4月到他生命的最后一段时间，大约有50多年的时间，谢尔盖·博奇卡列夫在斯特克洛夫数学研究所的函数理论系工作，专注于研究。1974年，他为自己的博士论文进行答辩，题目是关于完全标准正交系统的函数类和傅立叶系数。Bochkarev的出版物列表包含70多个标题。我们只提到他在我们看来最重要的成就，这些成就使他在分析专家中赢得了“解决基本问题的人”的无可争议的声誉。Bochkarev的第一个大周期的论文，构成了他的博士论文，致力于在不同的函数空间中关于经典或一般标准正交系统的函数的傅里叶系数的性质。由

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引用次数: 0

On estimating the local error of a numerical solution of the parametrized Cauchy problem 参数化柯西问题数值解的局部误差估计

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/rm10056

E. Kuznetsov, Sergei Sergeevich Leonov

引用次数: 0

Roots of the characteristic equation for symplectic groupoid 辛群拟的特征方程的根

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/rm9999

L. Chekhov, Michael Zalmanovich Shapiro, H. Shibo

引用次数: 0

On the Dirichlet problem for not strongly elliptic second-order equations 非强椭圆型二阶方程的Dirichlet问题

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/RM10011

A. Bagapsh, M. Mazalov, K. Fedorovskiy

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.)

1. 设L为C中复系数常的二阶椭圆偏微分算子，即Lf = af ' xx + bf ' xy + cf ' yy，其中a, b, C∈C。L的椭圆性意味着对应的特征方程λ + bλ + C = 0的根λ1和λ2不实数。如果L满足λ1和λ2相对于实直线属于复平面的不同半平面，则称L为强椭圆。强椭圆算子的经典例子是拉普拉斯算子∆，其中∆f = f“xx + f”yy，而Bitsadze算子∂，其中∂f = (f ' x + if ' y)/2是柯西-黎曼算子，作为一个非强椭圆算子的例子。我们用C(E)表示集合E∧C上所有连续复值函数的空间，并令∥f∥E = supz∈E |f(z)|对于f∈C(E)。有界单连通域G⊂C是定期对L的狄利克雷问题(或短暂,L-regular)如果为每一个函数h C∈(∂G),有一个f∈C (G),这样如果= 0 G和f∣∣∂G = h。经典的定理由于勒贝格[1]指出,任何有界单连通域G⊂C∆规律,也就是说,它是定期对经典的狄利克雷问题调和函数。对于所考虑的形式的一般强椭圆算子L，存在一个关于任何有界单连通域都是L正则的猜想(见[2]，问题4.2)。这一猜想仅在∂G边界正则性的某些附加的相当严格的条件下得到了证明，即在[3]中对于具有分段c光滑边界的Jordan域得到了相应的结果，并且在过去的20年里，∂G上的这一条件并没有明显减弱。(例如，关于任意Jordan域G的l -正则性的问题，即使在∂G可校正的情况下仍然是开放的。)

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引用次数: 0

Inverse function theorem on the class of holomorphic self-maps of a disc with two fixed points 具有两个不动点的圆盘的全纯自映射的反函数定理

IF 0.9 4区数学 Q1 Mathematics

Russian Mathematical Surveys

Pub Date : 2022-01-01 DOI: 10.1070/RM10042

O. Kudryavtseva, A. Solodov

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

本文解决了具有两个(内不动点和边界不动点)的单位圆盘的全纯自映射类上的尖锐可逆性定义域问题和边界不动点处角导数的约束问题。对这类极值问题的兴趣主要源于Bloch的著名定理[1]，即在单位圆盘D = {z∈C: |z| < 1}中任何全纯函数在半径为R|的圆盘f '(0)|中是可逆的，其中R是一个绝对常数。寻找这样的R的明显上界B(称为布洛赫常数)是几何函数理论中最重要(但仍未解决)的问题之一。接近最佳的较低估计B小于√3/4是由于Ahlfors bb0。相当之后，Bonk[3]借助他在Bloch类上的畸变定理证明了Ahlfors的估计是不尖锐的，即B >√3/4。不久之后，Chen和Gauthier通过稍微改进Bonk证明的技术细节，证明了B >√3/4 + 2·10−4。在我们看来，朗道的方法可能能够为布洛赫常数提供新的下界。考虑圆盘D的内不动点z = 0且f '(0) = 1的有界全纯映射f的类，Landau[5]证明了该类上存在一个一价公共圆盘，并求出了它的精确半径。此外，他还发现存在一个圆盘，其中这类函数的所有函数都是可逆的，他还确定了这个圆盘的精确半径。利用这些结果，朗道给出了布洛赫常数的最早估计之一。同时，Landau所研究的类中每个函数的可逆性定义域比一般的可逆性圆盘要宽得多。这就提出了在这个类的子类上寻找唯一性和可逆性的尖锐定义域的自然问题。在某种意义上，作为这样的子类，研究具有几个不动点的单位圆盘D的全纯自映射的类(见[6])是很自然的，它们具有重要的应用。我们让B表示D的全纯自映射的类。令B[0] = {f∈B: f(0) = 0}，我们可以将Landau的结果写成如下:如果f∈B[0]和|f '(0)|与M[0]的小于1/M，那么f在Z = {Z∈D: | Z | < M−√M2−1}中是一元的，并且在W = {W∈D: | W | < (M−√M2−1)}中是可逆的。此外，我们既不能取半径更大的圆盘，也不能取更宽的域来代替Z和W。这个结果的证明是基于下面的不等式(见[5]):如果f∈B[0]，如果a, B∈D与a α = B使得f(a) = f(B) = c，则|c|≤|a| | B |。在B类上{1}= {f∈B:∠limz→1 f(z) = 1}， Becker和Pommerenke[7]得到了一个在一定意义上类似于Landau不等式的不等式。他们证明了

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