A general Farkas lemma

An abstract version of the classical Farkas lemma for locally convex spaces is given. The abstract Farkas lemma is shown to imply Farkas-type results which have been obtained by Shimizu-Aiyoshi-Katayama, Schecter, Eisenberg, Zalinescu, and Smiley.     

Book Ramsey numbers. I

A book B_p is a graph consisting of p triangles sharing a common edge. In this paper we prove that if p ≤ q/6 ?o(q) and q is large, then the Ramsey number r (B_p,B_q) is given by r (B_p,B_q) = 2q+3, and the constant 1/6 is essentially best possible. Our proof is based on Szemerédi's uniformity lemma and a stability result for books. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Boundary point lemmas and overdetermined problems

Two elliptic boundary value problems are considered: a problem of mixed type in a cylindrical domain, and a Dirichlet problem in an annular domain. Under some overdetermined conditions on the boundary gradient, symmetry results for domain and solution are proved. The method of proof involves the classical boundary point lemma by Hopf, as well as a suitable adaptation of it that works well at certain corners.     

Random iterations, fixed points and invariant CRF-horospheres in complex Banach spaces

For holomorphic noncontractive maps on (not necessarily bounded) domains in complex Banach spaces, we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations. In particular, we show that the problem, concerning the existence of the horospheres determined by Carathéodory-Reiffen-Finsler pseudometrics defined on unbounded domains, has the solution and we prove new results of type of Julia's lemma and Wolff's theorem.     

A note on bootstrapping the variance of sample quantile

Summary Letm _{n, p} denote thep-th quantile based onn observations and let λ _p denote the population quantile. In this paper consistency of the bootstrap estimate of variance of is established.     

Star-critical Ramsey numbers for large generalized fans and books

In this paper, we show that for any fixed integers $m\ge 2$ and $t\ge 2$, the star-critical Ramsey number $r_1(K_1+n{K}_t,K_{m+1})=(m?1)tn+t$ for all sufficiently large $n$. Furthermore, for any fixed integers $p\ge 2$ and $m\ge 2$, $r_1(K_p+n{K}_1,K_{m+1})=(m?1+o\left(1\right))n$ as $n\to \infty$.     

非紧非单调算子的不动点定理

本文利用和选择公理等价的Zorn引理,得到了一般非单调算子不动点存在性的一些结果.     

A Higher Level Bailey Lemma: Proof and Application

In a recent letter, new representations were proposed for the pair of sequences (,), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs (,) labelled by the Lie algebra AN – 1, two nonnegative integers and k and a partition , whose parts do not exceed N – 1. Our results give rise to what we call a higher level Bailey lemma. As an application it is shown how this lemma can be applied to yield general q-series identities, which generalize some well-known results of Andrews and Bressoud.     

A rotation method which gives linear estimates for powers of the Ahlfors–Beurling operator

In [O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2) (2003) 415–435] the Ahlfors–Beurling operator T was represented as an average of two-dimensional martingale transforms. The same result can be proven for powers Tⁿ. Motivated by [T. Iwaniec, G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996) 25–57], we deduce from here that Tⁿ_p are bounded from above by Cnp^*, . We further improve this estimate to obtain the optimal behaviour of the L^p norms in question.     

Uniqueness for Boundary Value Problems for Second Order Finite Difference Equations

We establish maximum principles for second order difference equations and apply them to obtain uniqueness for solutions of some boundary value problems.