Topological multiple recurrence for polynomial configurations in nilpotent groups

We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales-Jewett theorem (Ann. Math. 150 (1999) 33) and contains nilpotent extensions of many known “abelian” results as special cases.     

A nilpotent IP polynomial multiple recurrence theorem

We generalize the IP-polynomial Szemerédi theorem due to Bergelson and McCutcheon and the nilpotent Szemerédi theorem due to Leibman. Important tools in our proof include a generalization of Leibman’s result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.     

An infinitary polynomial Hales-Jewett theorem

A joint extension of H. Furstenberg’s central sets theorem, the Hales-Jewett coloring theorem and the polynomial van der Waerden theorem of V. Bergelson and A. Leibman is obtained by an elaboration on Furstenberg and Y. Katznelson’s approach to infinitary Ramsey theory via the enveloping semigroup.     

Some applications of the Hales-Jewett theorem to field arithmetic

Let K be a field whose absolute Galois group is finitely generated. If K neither finite nor of characteristic 2, then every hyperelliptic curve over K with all of its Weierstrass points defined over K has infinitely many K-points. If, in addition, K is not an algebraic extension of a finite field, then every elliptic curve over K with all of its 2-torsion rational has infinite rank over K. These and similar results are deduced from the Hales-Jewett theorem.     

Some Ramsey-type results for the n-cube

In this note we establish a Ramsey-type result for certain subsets of the n-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sizes 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, “corners” in the integer lattice in the plane, and 3-term integer geometric progressions.     

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

We provide an elementary proff of Fulkerson's theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkerson's theorem as a corollary.     

The partition problem for finite abelian groups

By the aid of a slight generalization of the Hales-Jewett theorem [Trans. Amer. Math. Soc.106 (1963), 222–229] we investigate the partition problem for finite Abelian groups. In particular the partition problem for the class of finitely generated free modules over Z_q is solved. By the results of Deuber and Rothschild [“Coll. Math. Soc. János Bolyai 18,” 1976] this yields a complete characterization of those finite Abelian groups with respect to which the class FAB of all finite Abelian groups has the partition property. Especially it turns out that FAB has the partition property with respect to the cyclic group Z_m, m > 1.     

Vertices of Specht modules and blocks of the symmetric group

This paper studies the vertices, in the sense defined by J.A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. The main theorem is also used to find the Green correspondents of a particular family of simple Specht modules; as a corollary, this gives a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, as developed by M. Broué, together with combinatorial arguments using Young tableaux.     

马尔可夫骨架过程的极限分布

本文运用基本更新定理和Smith关键更新定理等理论和方法,对马尔可夫骨架过程的极限分布进行深入研究,得到主要结果如下:去掉了原有结果中要求的绝对连续的条件,给出了马尔可夫骨架过程极限分布存在的充分条件;得到了马尔可夫骨架过程极限分布的具体公式,并证明了该极限分布为概率分布.     

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories.This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.     

建立在L-凸空间上的截口定理及其应用

In this paper,the author gives a new section theorem in L-convex spaces.And as its applications,the author proves a coincident theorem and a two-functional minimax theorem established in L-convex spaces.     

On pairs of multidimensional matrices

Recently Bapat applied a topological theorem of Kronecker and generalized a theorem of Sinkhorn on positive matrices. Here we give an alternative proof of a slightly stronger version of his generalization. This proof combines Kakutani's fixed point theorem and the duality theorem of linear programming and gives yet another proof of a theorem of Bacharach and Menon on pairs of nonnegative matrices.     

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.     

Hurwitz' theorem implies Rouché's theorem

It is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem.     

Very Colorful Theorems

We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger’s theorem gives a generalization of the theorem of Tverberg on non-separated partitions.     

Interpolation by Generalized Polynomials with Restricted Ranges

In 1990, A. Horwitz proved a theorem about interpolation by restricted range polynomials and asked some ralted questions. This paper gives aflirmative answers to Horwitz′ questions and generalizes his theorem.     

齐次函数的一个仿射球定理

本文研究了相关齐次函数的仿射球定理.利用Hopf极大值原理,对任意给定的带凹性条件的初等对称曲率问题,获得了此类仿射球定理.特别地,这也给出了Deicke齐次函数定理的一个新证明.     

The essence of the generalized Newton binomial theorem

Under the frame of the homotopy analysis method, Liao gives a generalized Newton binomial theorem and thinks it as a rational base of his theory. In the paper, we prove that the generalized Newton binomial theorem is essentially the usual Newton binomial expansion at another point. Our result uncovers the essence of generalized Newton binomial theorem as a key of the homotopy analysis method.