用拼图法研究Ramsey数下界的一些注记

指出论文《关于Ramsey数下界的部分结果》中的一些错误,评注用拼图法研究Ramsey数下界的一些困难问题,并提出两个猜想.     

Some Open Questions

In this paper, several longstanding problems that the author has tried to solve, are described. An exposition of these questions was given in Luminy in January 2002, and now three years later the author is pleased to report some progress on a couple of them.2000 Mathematics Subject Classification: Primary—11A25, 11N37Research partially supported by CNRS, Institut Camille Jordan, UMR 5208     

B值拟鞅差序列加权和的收敛性与大数律

{Xn,Fn,n≥1}为B值p拟鞅，{anj}为实值常数阵列，在{‖djX‖^r}关于{｜anj｜^r}一致可积的条件下，建立了^kn∑j=1anjdjX的收敛性与Banach空间p光滑性的关系，并在p光滑Banach空间中，给出^kn∑j=1anjdjX的强大数定律及完全收敛定理。     

General theorems on rates of convergence in distribution of random variables II. Applications to the stable limit laws and weak law of large numbers

This part is concerned with the applications of the general limit theorems with rates of Part I, achieved by specializing the limiting r.v. X. This leads to new convergence theorems with higher order rates in the one- and multi-dimensional case for the stable limit law, for the central limit theorem, and the weak law of large numbers.     

Heegner Divisors and Nonholomorphic Modular Forms

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.     

关于错排数与Bell数的一些循环公式

In this paper, a further investigation for the number of Derangements and Bell numbers is performed, and some new recursion formulae for the number of Derangements and Bell numbers are established by applying the generating function methods and Padé approximation techniques. Illustrative special cases of the main results are also presented.     

The Development of the Theory of Strong Electrolyte Solutions

The expression for Gibbs energy of electrolyte solutions has been derived taking into account the contribution of interactions of the hydrated shells of ions. It is in good agreement with data for the dependence of the hydration number on concentration. The contribution of different interactions to the activity coefficient depending on concentration has been analyzed. The possibility of determining parameters which characterize the composition of hydrated shells of individual atoms from experimental data has been considered.     

New identities in the Catalan triangle

In this paper we prove new identities in the Catalan triangle whose (n,p) entry is defined by

     

A bijective proof of Jackson's formula for the number of factorizations of a cycle

Factorizations of the cyclic permutation into two permutations with respectively n and m cycles, or, equivalently, unicellular bicolored maps with N edges and n white and m black vertices, have been enumerated independantly by Jackson and Adrianov using evaluations of characters of the symmetric group. In this paper we present a bijection between unicellular partitioned bicolored maps and couples made of an ordered bicolored tree and a partial permutation, that allows for a combinatorial derivation of these results.Our work is closely related to a recent construction of Goulden and Nica for the celebrated Harer-Zagier formula, and indeed we provide a unified presentation of both bijections in terms of Eulerian tours in graphs.