An optimal inequality between scalar curvature and spectrum of the Laplacian

For a Riemannian closed spin manifold and under some topological assumption (non-zero A-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the odd-dimensional case. On the other hand, we study the equality case for the closed spin Riemannian manifolds with non-zero A-genus. This work improves an inequality which was first proved by K. Ono in 1988. Mathematics Subject Classification (2000):58J50, 35P15, 46L10, 58G11.     

An inequality relating the order,maximum degree,diameter and connectivity of a strongly connected digraph

We prove that if there is a strongly connected digraph of ordern, maximum degreed, diameterk and connectivityc, thennc d ^k–d /d–1+d+1. It improves the previous known results, and it, in fact, is the best possible for several interesting cases. A similar result for arc connectivity is also established.This project is supported by the National Natural Science Foundation of China.     

(d,n,r)-码的构作和它的性质

利用含有Baer子平面的m~2阶射影平面的性质构作了(d,n,r)-码并计算了它的参数,给出了它的检纠错性质.     

（n1,n2）型二重（r1,r2）—循环矩阵逆矩阵的插值求法

本文用插值法给出n1n2阶（n1，n2）型二重（r1，r2）－循环矩阵逆矩阵计算公式．     

(d,1)‐total labeling of graphs with a given maximum average degree

The (d,1)‐total number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same color, no two incident edges have the same color, and the distance between the color of a vertex and its incident edges is at least d. In this paper, we prove that for connected graphs with a given maximum average degree. © 2005 Wiley Periodicals, Inc. J Graph Theory     

A Krylov-Schur-like method for computing the best rank-(r1,r2,r3) approximation of large and sparse tensors

Numerical Algorithms - The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this...     

On the equationx(x +d 1)…(x + (k - 1)d 1) =y(y +d 2)…(y + (mk - 1)d 2)

For given positive integersm ≥ 2,d ₁ andd ₂, we consider the equation of the title in positive integersx, y andk ≥ 2. We show that the equation implies thatk is bounded. For a fixedk, we give conditions under which the equation implies that max(x, y) is bounded. Dedicated to the memory of Professor K G Ramanathan     

二元(p,r,d)-码的构作及其应用

令n是一个正整数,[n]={1,2,…,n}.利用集合[叫上的s-子集族((ns))构作了二元(p,r,d)-叠加码,研究了它的容错和析取性质并介绍了它在非适应性群测(Nonadaptive Group Testing)方面的应用.     

二重(r1,r2)-循环矩阵求逆的一种新算法

本文给出了判定任意数域上二重（r1,r2)－循环矩阵非异性的一个充要条件，并提供了求这类矩阵逆的一种新方法。     

(d,r,k)-析取矩阵的卡氏积

(d,r,κ)-析取矩阵是分组测试理论中的一个Inhibitor模型.利用两个已知的(d,r,k)-析取矩阵定义了它们的卡氏积,并计算了这个新(d,r,κ)-析取矩阵的参数.     

(p,r,d)-SC码的卡氏积

(p,r,d)-superimposed codes简称为(p,r,d)-SC码,利用两个已知的(p,r,d)-SC码定义了它们的行卡氏积,并计算了这个新(p,r,d)-SC码的参数和它的汉明距离.     

A Hardy type inequality for W m,1(0, 1) functions

In this paper, we consider functions ${u\in W^{m,1}(0,1)}$ where m ≥ 2 and u(0) = Du(0) = · · · = D ^m-1 u(0) = 0. Although it is not true in general that ${\frac{D^ju(x)}{x^{m-j}} \in L^1(0,1)}$ for ${j\in \{0,1,\ldots,m-1\}}$ , we prove that ${\frac{D^ju(x)}{x^{m-j-k}} \in W^{k,1}(0,1)}$ if k ≥ 1 and 1 ≤ j + k ≤ m, with j, k integers. Furthermore, we have the following Hardy type inequality, $$\left\|{D^k\left({\frac{D^ju(x)}{x^{m-j-k}}}\right)}\right\|_{L^1(0,1)} \leq \frac {(k-1)!}{(m-j-1)!} \|{D^mu}\|_{L^1(0,1)},$$ where the constant is optimal.     

On the number of (r,r+1)‐ factors in an (r,r+1)‐factorization of a simple graph

For integers d≥0, s≥0, a (d, d+s)‐graph is a graph in which the degrees of all the vertices lie in the set {d, d+1, …, d+s}. For an integer r≥0, an (r, r+1)‐factor of a graph G is a spanning (r, r+1)‐subgraph of G. An (r, r+1)‐factorization of a graph G is the expression of G as the edge‐disjoint union of (r, r+1)‐factors. For integers r, s≥0, t≥1, let f(r, s, t) be the smallest integer such that, for each integer d≥f(r, s, t), each simple (d, d+s) ‐graph has an (r, r+1) ‐factorization with x (r, r+1) ‐factors for at least t different values of x. In this note we evaluate f(r, s, t). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 257‐268, 2009     

n维有限射影几何上(d,r)-析取矩阵的性质及其界

介绍了n维有限射影几何上子空间的性质,利用这些性质研究了非适应性群测模型(d,r)-析取矩阵,然后计算了(d,r)-析取矩阵的相关参数,给出了它的行界.     

An inequality between thep- and (p, 1)-summing norm of finite rank operators fromC(K)-spaces

We show, for any operatorT from aC(K)-space into a Banach space with rank (T)≤n, the inequality , whereC≤4.671 is a numerical constant. The factor (1+logn)^1−1/p is asymptotically correct. This inequality extends a result of Jameson top ≠ 2. Several applications are given — one is a positive solution of a conjecture of Rosenthal and Szarek: For 1≤p<q<2,