Anmerkungen zu einem lemma von hoàng tuy

In [1], [2] HOÀNG TUY gave an approach to the main theorems of the convex analysis and the convex optimization being based on a lemma; he has proved it by means o induction. In [1] the equivalence of the main theorems of convex optimization given in [1], [2] does not use a separation theorem or equivalent statements. In this note the author has proved that the lemma of HOÀNG TUY can be characterized as a special separation theorem and be obtained from a separation theorem of Eidelheit. That means that the lemma is equivalent to the theorem of Hahn-Banach.     

Sperner’s lemma and zero point theorems on a discrete simplex and a discrete simplotope

We show two discrete zero point theorems that are derived from Sperner’s lemma and a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] [5]; Freund [Math. Oper. Res. (1986)]) [3]. Applications to economic and game models are also presented.     

Elements of the KKM theory for generalized convex spaces

In the present paper, we introduce fundamental results in the KKM theory for G-convex spaces which are equivalent to the Brouwer theorem, the Sperner lemma, and the KKM theorem. Those results are all abstract versions of known corresponding ones for convex subsets of topological vector spaces. Some earlier applications of those results are indicated. Finally, we give a new proof of the Himmelberg fixed point theorem andG-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.     

A rainbow blow‐up lemma

We prove a rainbow version of the blow‐up lemma of Komlós, Sárközy, and Szemerédi for μn‐bounded edge colorings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow‐up lemma can be used to transfer the bandwidth theorem of Böttcher, Schacht, and Taraz to the rainbow setting. It can also be employed as a tool beyond the setting of μn‐bounded edge colorings. Kim, Kühn, Kupavskii, and Osthus exploit this to prove several rainbow decomposition results. Our proof methods include the strategy of an alternative proof of the blow‐up lemma given by Rödl and Ruciński, the switching method, and the partial resampling algorithm developed by Harris and Srinivasan.     

Sharp Dirac's theorem for DP‐critical graphs

Correspondence coloring, or DP‐coloring, is a generalization of list coloring introduced recently by Dvořák and Postle [11]. In this article, we establish a version of Dirac's theorem on the minimum number of edges in critical graphs [9] in the framework of DP‐colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz [15] on classifying list‐critical graphs that satisfy Dirac's bound with equality.     

关于解析算子值函数的几个结果

关于复值解析函数Riesz—Dunford积分的Ky Fan定理由[1]推广到算子值解析函数,由此函数论中的很多定理得到了推广.本文的目的在于改进[1]中的结果,得到了较弱条件下的Pick定理,从而推广了[2]中的Julia引理,并简化了其证明过程.     

Tverberg’s Theorem and Graph Coloring

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Δ, where Δ is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>Δ ² then the topological Tverberg theorem still works. It is conjectured that q>KΔ is also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.     

A variant of the hypergraph removal lemma

Recent work of Gowers [T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001) 465-588] and Nagle, Rödl, Schacht, and Skokan [B. Nagle, V. Rödl, M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Applications of the regularity lemma for uniform hypergraphs, preprint] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 299-345], and Furstenberg and Katznelson [H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 34 (1978) 275-291] concerning one-dimensional and multidimensional arithmetic progressions, respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [T. Tao, The Gaussian primes contain arbitrarily shaped constellations, preprint] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes.     

AZ-Identities and Strict 2-Part Sperner Properties of Product Posets

One of central issues in extremal set theory is Sperner’s theorem and its generalizations. Among such generalizations is the best-known LYM (also known as BLYM) inequality and the Ahlswede–Zhang (AZ) identity which surprisingly generalizes the BLYM into an identity. Sperner’s theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.     

Generalized Sperner lemma and subdivisions into simplices of equal volume

A generalization of the well-known Sperner lemma is suggested, which covers the case of arbitrary subdivisions of (convex) polyhedra into (convex) polyhedra. It is used for giving a new proof of the Thomas-Monsky-Mead theorem saying that the n-cube can be subdivided into N simplices of equal volume if and only if N is divisible by n!. Some new related results are announced. Bibliography: 6 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 245–254.     

基于非协调元的加法型Schwarz交替法──弱重迭情形

基于非协调元的加法型Ｓｃｈｗａｒｚ交替法──弱重迭情形黄建国（上海交通大学应用数学系）ＡＤＤＩＴＩＶＥＳＣＨＷＡＲＺＡＬＴＥＲＮＡＴＩＮＧＭＥＴＨＯＤＦＯＲＮＯＮＣＯＮＦＯＲＭＩＮＧＦＩＮＩＴＥＥＬＥＭＥＮＴ──ＣＡＳＥＯＦＷＥＡＫＯＶＥＲＬＡＰ￥Ｈ...     

无限凸规划的约束规范条件和Farkas引理

假设X是局部凸Hausdorff拓扑向量空间,C是X的闭凸子集,T是一下标集(可以是无限集),假设{ft:t∈T}是一X到R ∪{+∞}的真凸下半连续函数簇,而f:x→R∪{+∞}是一真凸下半连续函数.     

Topological degree and the Sperner lemma

In this paper, we give a new proof of Sperner's lemma, in its superstrong from, using the topological degree. Thus, we point out a relation between several methods for fixed-point theorems using either the topological degree, or the KKM lemma, or the Sperner lemma.The author would like to thank Dr. G. Leitmann for his remarks and suggestions.     

A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper,we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma" which is quite different from the method in [12] where "the generalized Banach lemma" was used. By the method of the perturbation analysis of bounded linear operators,we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.     

Fuzzy Integrals of Set-vatued Function

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Finsler流形上的Laplace比较定理和有界估计(英文)

本文研究了Finsler流形上的距离函数的Laplacian.利用指标引理和文献[4]中主要方法,获得了Ricci曲率有函数下界的Laplacian比较定理,改进了文献[6]和文献[7]的相关结果.     

Triangulated Spheres and Colored Cliques

   Abstract. In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S ^d-1 can be extended to a triangulation of the ball B ^d , by adding one point at a time, having degree at most 2d to its predecessors.     

Triangulated Spheres and Colored Cliques

Abstract. In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S ^d-1 can be extended to a triangulation of the ball B ^d , by adding one point at a time, having degree at most 2d to its predecessors.     

Other Proofs of Old Results

We transform the proof of the second incompleteness theorem given in [3] to a proof-theoretic version, avoiding the use of the arithmetized completeness theorem. We give also new proofs of old results: The Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth; the proofs in which the construction of a sentence by means of diagonalization lemma is not needed.     

Decomposition theorems

The countable-decomposition theorem for linear functionals has become a useful tool in the theory of representing measures (see [4–7]). The original proof of this theorem was based on a rather involved study of extreme points in the state space of a convex cone. Recently M. Neumann [9] gave an independent proof using a refined form of Simons convergence lemma and Choquet's theorem. In this paper a (relatively) short proof of an extension (to a more abstract situation) of the countable-decomposition theorem is given. Furthermore a decomposition criterion is obtained which even works in the case when not all states are decomposable. All the work is based on a complete characterization of those states which are partially decomposable with respect to a given sequence of sublinear functionals.