Minimum Riesz Energy Problems for a Condenser with Touching Plates

We consider minimum energy problems in the presence of an external field for a condenser with touching plates A₁ and A₂ in \(\mathbb {R}^{n}\), \(n\geqslant 3\), relative to the α-Riesz kernel |x?y|^α?n, \(0<\alpha \leqslant 2\). An intimate relationship between such problems and minimal α-Green energy problems for positive measures on A₁ is shown. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The approach applied is mainly based on the establishment of a perfectness-type property for the α-Green kernel with \(0<\alpha \leqslant 2\) which enables us, in particular, to analyze the existence of the α-Green equilibrium measure of a set. The results obtained are illustrated by several examples.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

On a Mathematical Problem of Cusped Double-Layered Plates

The present paper is devoted to the system of degenerate partial differential equations arise from the investigation of elastic two layered prismatic shells. Harmonic vibration of cusped double-layered plates is consider. The weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Number of Points of Elliptic Curves over a Finite Field and a Problem of B. Segre

Several theorems are studied concerning the number of points of an elliptic curve with a Legendre form on a finite field, in order to analyse the distribution of regular and pseudoregular points in relation to a hyperbola in a finite affine plane.     

On the Green Function for a Complement of a Finite Number of Real Intervals

We give a new proof (and a generalization) of recent remarkable result by V. Totik concerning the smoothness properties of the equilibrium measure for compact subsets of the real line and the Green function for complement of these subsets taken with respect to the complex plane.     

On the Problem of Finite Signature

We show that not each model can be replaced by a model of finite signature so that its automorphism group remains the same.     

On the Inverse Problem for Differential Operators on a Finite Interval with Complex Weights

Mathematical Notes - Inverse problems of spectral analysis for second-order differential operators on a finite interval with complex-valued weights and with an arbitrary number of discontinuity...     

On the Number of Distinct Values of a Class of Functions with Finite Domain

By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. These bounds are then applied in several contexts. In particular, we obtain the first non-trivial upper bound for the image set of a planar function over a finite field.     

On the On-line Number of Snacks Problem

In the number of snacks problem (NSP), which was originally proposed by our team, an on-line player is given the task of deciding how many shares of snacks his noshery should prepare each day. The on-line player must make his decision and then finish the preparation before the customers come to his noshery for the snacks; in other words, he must make decision in an on-line fashion. His goal is to minimize the competitive ratio, defined as _: C_A()/C_OPT(), where denotes a sequence of numbers of customers, C _OPT() is the cost of satisfying by an optimal off-line algorithm, and C _A() is the cost of satisfying by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem P1, the Extreme Numbers Harmonic Algorithm(ENHA), with competitive ratio 1+p(M-m)/(M+m), where M and m are two extreme numbers of customers over the total period of the game, and p is a ratio concerning the cost of the two types of situations, and then prove that this competitive ratio is the best one if an on-line player chooses a fixed number of shares of snacks for any sequence of numbers of customers. We also discuss several variants of the NSP and give some results for it. Finally, we propose a conjecture for the on-line NSP.     

关于缺失有限个特征值的逆Sturm-Liouville问题

本文考虑由三组谱确定势函数和边值条件的问题,即证明Sturm-Liouville问题的势函数可以由[0,1]区间上的一组整谱和[0,α],[α,1](0<α<1)区间上的两组部分谱唯一确定.特别地,在这两个区间内,分别可以缺失任意-个特征值,势函数仍可以被唯一确定.     

The Problem of V. N. Dubinin for Symmetric Multiconnected Domains

Ukrainian Mathematical Journal - We consider a quite general problem from the geometric theory of functions, namely, the problem of finding the maximal value of the product of inner radii of n...     

On an Approach to the Solution of the Bending Problem for Laminated Plates

The paper presents a method for determining the three-dimensional state of stresses and strains in a moderately thick rectangular plate composed of isotropic layers unlimited in number and arranged symmetrically about the midplane of the plate. The method is based on a variant of Timoshenko theory [1, 2] and allows one to calculate the stresses and displacements in a shear-compliant plate for various types of boundary conditions and various external loads. According to it, the displacement field in the plate is represented as a sum of products of two unknown functions one of which is defined on the midplane and the second one depends on the thickness coordinate and satisfies the layer contact conditions. As an example, a rectangular laminated plate clamped at all its edges and loaded on the upper surface by a sinusoidal load is considered.     

关于有限Abel群直接因子的平均数

对于有限Ａｂｅｌ群Ｇ，ｔ（Ｇ）定义为其直接因子的个数，在［５］中，Ｅ．Ｋｒａｔｚｅｌ对于和式得到一渐近公式，对应余项为△＿１（ｘ）．本文利用Ｅ．Ｆｏｕｖｒｙ和Ｈ　．Ｉｗａｎｉｅｃ的方法，得到了关于△＿１（ｘ）的一个新的估计，改进了Ｈ．Ｍｅｎｚｅｒ等人的结果．     

On the Average Number of Direct Factors of a Finite Abelian Group

 Let be the number of direct factors of a finite abelian group ?. We prove that where the are absolute constants and ɛ is an arbitrarily small positive number. This improves previous results. (Received 1 December 1999; in revised form 27 March 2000)     

On the Number of Solutions to Carrier's Problem

Let N _ɛ denote the maximum number of spikes that a solution to Carrier's problem can have, where ɛ is a small positive parameter. We show that N _ɛ is asymptotically equal to [ K /ɛ], where   K = 0.4725⋯  , and the square brackets represent the greatest integer less than or equal to the quantity inside. If n (ɛ) stands for the number of solutions to this problem, then it is also shown that 4 N _ɛ− 3 ≤ n (ɛ) ≤ 4 N _ɛ. Our approach is based on the shooting method used by Ou and Wong ( Stud. Appl. Math. 111 (2003)) and on the construction of an envelope function for the minimum values of the solutions as ɛ approaches zero.     

On the Number of Solutions of an Equation Over a Finite Field

The Stöhr–Voloch approach is used to obtain a newbound for the number of solutions in (F_q)² of an equation f(X,Y) = 0, where f(X, Y) is an absolutely irreducible polynomialwith coefficients in a finite field F_q.     

具有有限多个理想的李代数

本文给出了复数域C上的具有有限多个理想的有限维李代数L的结构:L=S+ Cr+ N,其中 S为 Levi因子,Cr+N为根基,N为幂零根基.当 N为交换时,文中还给出具有这种结构的李代数L具有有限多个理想的充要条件,并且将N为Heisenberg代数或亚交换李代数的惰形的讨论归结为N是交换的情形.     

中厚板特征值问题的杂交/混合有限元分析

本文采用杂交/混合有限元对中厚板的屈曲问题和自由振动问题进行了分析,首先推导了一个修正的Reissner变分原理,它仅要求构造C0类场变量,避免了闭锁现象的产生。所有的场变量皆采用线性插值。最后得到一个矩阵型的位移广义本征值方程,刚度矩阵对称、正定。计算结果表明本文采用的方法简单、可靠,较为令人满意。