Random walks on inhomogeneous lattices

For lattices with two kinds of points (black and white), distributed according to a translation-invariant joint probability distribution, we study statistical properties of the sequence of consecutive colors encountered by a random walker moving through the lattice. The probability distribution for the single steps of the walk is considered to be independent of the colors of the points. Several exact results are presented which are valid in any number of dimensions and for arbitrary probability distributions for the coloring of the points and the steps of the walk. They are used to derive a few general properties of random walks on lattices containing traps.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.     

Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0, n]

An elementary majorant-minorant method to construct the most stringent Bonferroni-type inequalities is presented. These are essentially Chebyshev-type inequalities for discrete probability distributions on the set {0, 1,..., n}, where n is the number of concerned events, and polynomials with specific properties on the set lead to the inequalities. All the known results are proved easily by this method. Further, the inequalities in terms of all the lower moments are completely solved by the method. As examples, the most stringent new inequalities of degrees three and four are obtained. Simpler expressions of Mrgritescu's inequality (1987, Stud. Cerc. Mat., 39, 246–251), improving Galambos' inequality, are given.     

Facets for the cut cone I

We study facets of the cut coneC _n , i.e., the cone of dimension 1/2n(n – 1) generated by the cuts of the complete graph onn vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a lifting procedure for constructing facets ofC _n+1 from given facets of the lower dimensional coneC _n . After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented.     

A long-time tail for random walk in random scenery

Consider a simple random walk on ^d whose sites are colored black or white independently with probabilityq, resp. 1–q. Walk and coloring are independent. Letn _k be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and let _k =E(n _k )–q ^–1. Our main result is a proof that (*) lim _k k ^d/2 _k = (1 –q)q ^{d/2 – 2}(d/2) ^d/2. Since it is known thatq ^{– 1} _k =E(n ₁ n _{k + 1} B) –E(n ₁ B)E(n _{k + 1} B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations of _k (1k100) ind=1, 2, and 3 show that there is an oscillatory behavior of _k for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay of _k cannot be faster than (*).     

用带误差项的Ishikawa迭代过程逼近φ-强增生算子的零点

本文使用新的分析技巧研究了一致光滑Ｂａｎａｃｈ空间中φ 强增生算子的零点逼近问题，所得结果改进和扩展了近期许多相应的结果     

On critical cases of Sobolev's inequalities for Heisenberg groups

We prove some Trudinger-type inequalities and Brezis-Gallouet-Wainger inequality on the Heisenberg group, extending to this context the Euclidean results by T. Ozawa.     

再谈Minc-Sathre不等式的上下界

用调和平均值、均值不等式之间的关系及对数平均不等式,对Minc-Sathre不等式的上下界进行改进,使结论更精确.     

L~p空间中的Fejér和Hermite-Hadamard型不等式

给出了二阶导数属于Lp空间时Fejér和Hermite-Hadamard型不等式的推广,得到两个新结果.     

The Generalization on Inequalities of Hermite-Hadamard's Integration

Some new inequalities of Hermite-Hadamard's integration are established.As for as inequalities about the righthand side of the classical Hermite-Hadamard's integral inequality refined by S Qaisar in [3],a new upper bound is given.Under special conditions,the bound is smaller than that in [3].     

混合质点系的一类几何恒等式及其应用

首先给出了混合质点系的概念,然后又给出了关于混合质点系的一类几何恒等式,并且给出了一个有趣的应用.