对称多胞形的极值性质及其应用

凸多胞形现代理论的主要成就是被称之为Dehn-Sommerville关系的上界定理和下界定理,它们属于凸多胞形的经典组合理论．本文建立了关于对称凸多胞形的两个极值定理,它们可视为凸多胞形度量理论中的上界定理和下界定理,另外给出了两个极值定理的一个应用．     

Connectivity of the Mycielskian of a digraph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This work investigates the vertex connectivity and arc connectivity of the Mycielskian of a digraph D. This generalizes the recent results due to Balakrishnan and Raj [R. Balakrishnan, S.F. Raj, Connectivity of the Mycielskian of a graph, Discrete Math, 308 (2008), 2607–2610].     

On the Number of Birch Partitions

Birch and Tverberg partitions are closely related concepts from discrete geometry. We show two properties for the number of Birch partitions: Evenness and a lower bound. This implies the first nontrivial lower bound for the number of Tverberg partitions that holds for arbitrary q, where q is the number of partition blocks. The proofs are based on direct arguments and do not use the equivariant method from topological combinatorics.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

On a theorem of Wigner on products of positive matrices

A simple algebraic proof of a theorem due to Wigner on the product of three positive matrices is given. It is shown that the theorem holds for four matrices under an additional condition. The proofs are valid in the more general case of operators in a Hilbert space.     

On a tauberian theorem of Hardy and Littlewood

In this paper, we give a simple alternative proof of a Tauberian theorem of Hardy and Littlewood (Theorem E stated below, [3]).     

图的代数连通度的界(英文)

本文首先给出了简单图的度序列的平方和的上界,利用这些结果,求出了简单图的代数连通度的几个上下界并确定了它们的临界图。另外,文章也给出了加权图的代数连通度的一个下界。     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

A Bipartite Analogue of Dilworth’s Theorem

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is greater than every element of B or every element of A is incomparable with every element of B. We prove that . Moreover, for fixed ε ∈ (0,1) and n sufficiently large, we construct a partially ordered set P with n elements such that no element of P is comparable with other elements of P and for every two disjoint subsets A and B of P each with cardinality at least , there is an element of A that is comparable with an element of B.     

On a class of nonlinear integral equations

In this paper we consider abstract equations of the typeK _ν ν +ν =w ₀, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K _v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations of the typeν(s)+ν(s)∫ ₀ ¹ k(s,t)ν(t)dt =W ₀(s) in the spaceL ² [0,1].     

On the ingham divisor problem

The main result of this paper is the following theorem. Suppose thatτ(n) = ∑ _d|n l and the arithmetical functionF satisfies the following conditions:

On functions uniquely determined by their asymptotic expansion

We present a maximal class of analytic functions. The elements of this class are uniquely determined by their asymptotic expansions. We also discuss the possibility of recovery of a function from the coefficients of its asymptotic series. In particular, we consider the problem of recovering by using Borel summation. The last published result in this direction was obtained by Alan Sokal in 1980, but his paper well known to physicists (in quantum field theory) seems to have remained unnoticed by mathematicians.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups. It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A₃(2), A₂(4)) and(B _n (q), C _n (q)) forn ≥ 3,q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H( ) for a split semisimple algebraic groupH defined over , does not determine the groupH up to isomorphism, but it determines the field under some mild conditions. We then put a group structure on the pairs(H ₁,H ₂) of split semisimple groups defined over a fixed field such that the orders of the finite groups H₁( ) and H₂( ) are the same and the groupsH _i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.     

On the complete convergence of sums of negatively associated random variables

This paper extends results on complete convergence in the law of large numbers for subsequences to the case of negatively associated nonidentically distributed random variables. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 411–420, September, 2000.     

An illustrated theory of hyperbolic virtual polytopes

The paper gives an illustrated introduction to the theory of hyperbolic virtual polytopes and related counterexamples to A.D. Alexandrov’s conjecture.      

On the Existence of Specific Stars in Planar Graphs

Given a graph G, a (k;a,b,c)-star in G is a subgraph isomorphic to a star K_1,3 with a central vertex of degree k and three leaves of degrees a, b and c in G. The main result of the paper is: Every planar graph G of minimum degree at least 3 contains a (k;a,b,c)-star with a≤ b≤ c and (i) k = 3, a≤ 10, or (ii) k = 4, a = 4, 4≤ b≤ 10, or (iii) k = 4, a = 5, 5≤ b≤ 9, or (iv) k = 4, 6≤ a≤ 7, 6≤ b≤ 8, or (v) k = 5, 4≤ a≤ 5, 5≤ b≤ 6 and 5≤ c≤ 7, or (vi) k = 5 and a = b = c = 6.     

A Jordan–Brouwer Separation Theorem for Polyhedral Pseudomanifolds

The Jordan Curve Theorem referring to a simple closed curve in the plane has a particularly simple proof in the case that the curve is polygonal, called the “raindrop proof”. We generalize the notion of a simple closed polygon to that of a polyhedral (d−1)-pseudomanifold (d≥2) and prove a Jordan–Brouwer Separation Theorem for such a manifold embedded in ℝ ^d . As a by-product, we get bounds on the polygonal diameter of the interior and exterior of such a manifold which are almost tight. This puts the result within the frame of computational geometry. The research of Y.S. Kupitz was partially supported by the Landau Center at the Mathematics Institute of the Hebrew University of Jerusalem (supported by Minerva Foundation, Germany), and by Deutsche Forschungsgemeinschaft.     

On Morrey’s estimate of the Sobolev norms of solutions of elliptic equations

We give a complete proof of Morrey’s estimate for the W ^1,p -norm of a solution of a second-order elliptic equation on a domain in terms of the L ₁-norm of this solution. The dependence of the constant in this estimate on the coefficients of the equation is investigated.