Helly-Type Theorems for Approximate Covering

Let ℱ∪{U} be a collection of convex sets in ℝ ^d such that ℱ covers U. We prove that if the elements of ℱ and U have comparable size, then a small subset of ℱ suffices to cover most of the volume of U; we analyze how small this subset can be depending on the geometry of the elements of ℱ and show that smooth convex sets and axis parallel squares behave differently. We obtain similar results for surface-to-surface visibility amongst balls in three dimensions for a notion of volume related to form factor. For each of these situations, we give an algorithm that takes ℱ and U as input and computes in time O(|ℱ|*|ℋ _ε |) either a point in U not covered by ℱ or a subset ℋ _ε covering U up to a measure ε, with |ℋ _ε | meeting our combinatorial bounds. The authors acknowledge support from the French–Korean Science and Technology amicable relationship program (STAR) 11844QJ.     

Discrete maximal surfaces with singularities in Minkowski space

We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the “singular set” for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case.     

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

We prove Helly-type theorems for line transversals to disjoint unit balls in ℝ ^d . In particular, we show that a family of n≥2d disjoint unit balls in ℝ ^d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n≥4d−1 disjoint unit balls in ℝ ^d admits a line transversal if any subfamily of size 4d−1 admits a transversal. Andreas Holmsen was supported by the Research Council of Norway, prosjektnummer 166618/V30. Otfried Cheong and Xavier Goaoc acknowledge support from the French-Korean Science and Technology Amicable Relationships program (STAR).     

Variational Principles, Minimization Theorems, and Fixed-Point Theorems on Generalized Metric Spaces

In this paper, we prove a new minimization theorem by using the generalized Ekeland variational principle. We apply our minimization theorem to obtain some fixed-point theorems. Our results extend, improve, and unify many known results due to Kui, Ekeland, Takahashi, Caristi, iri, and others.     

离散随机序列随机和的一类强偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic se-quences,as a measure of “dissimilarity“ between their joint distributions and the product of theirmarginals,is introduced. Construct a. s. convergence supermartingale by means of truncation methodand under suitable restrict Chung-Teicher type conditions,some strong deviation theorems for arbi-trary discrete stochastic sequence are obtained.     

达布定理和比较定理思想在中值定理类问题中的运用

微分达布定理表明区间内的导函数具有介值性,这使得我们在考虑一些数学分析问题时,往往可以不需要最高阶导函数的连续性.而在微分方程理论中,比较定理的思想对于解的估计非常重要.本文利用比较定理的思想将中值定理类问题转化为微分方程解的估计问题,对于在数学分析的学习中提高学生的认识和兴趣很有意义.     

Intersection Theorems for Infinite Families of Convex Sets in Graphs

We prove several Helly-type theorems for infinite families of geodesically convex sets in infinite graphs. That is, we determine the least cardinal n such that any family of (particular) convex sets in some infinite graph has a nonempty intersection whenever each of its subfamilies of cardinality less than n has a nonempty intersection. We obtain some general compactness theorems, and some particular results for pseudo-modular graphs, strongly dismantlable graphs and ball-Helly graphs.     

Approximation Theorems and Fixed Point Theorems for Various Classes of 1-set-contractive Mappings in Banach Spaces

In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234–240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with intK≠0. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors. Project supported by the National Natural Science Foundation of China and Natural Science Foundation of Shandong Province of China     

Fixed Point Theorems for Non-convex Valued Multifunctions

The purpose of this paper is to establish fixed point theorems for non-convex valued multifunctions which generalize known results in the literature. We also derive coincidence theorems in the non-compact setting.AMS Subject Classification (2000): 47H, 54H     

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.     

Fixed Point Theorems and Convergence Theorems for a New Generalized Nonexpansive Mapping

Abstract

This paper introduces a new mapping that is weaker than nonexpansive mapping. The new mapping is different from the Suzuki’s generalized nonexpansive mapping. This paper introduces a new iteration process for the fixed point, and gives fixed point theorems and Convergence theorems for a new generalized nonexpansive mapping in Banach space, insteading of uniformly convex Banach space.     

离散三角子模

为了构造离散三角模,引入了离散三角子模的概念,给出了生成离散三角(子)模的方法;然后讨论了光滑离散三角子模的结构.     

On Implicit Multifunction Theorems

We give implicit multifunction results generalizing to multifunctions the Robinson’s implicit function theorem (Robinson, Math Oper Res 16(2):292–309, 1991). To this end, we use parametric error bounds estimates for a suitable function refining the one given in Azé and Corvellec (ESAIM Control Optim Calc Var 10:409–425, 2004). Sharp approximations of the implicit multifunctions are given extending the results of Nachi and Penot (Control Cybernet 35:871–901, 2005). Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

H型群上的非线性Liouville定理

对H型群上的次线性方程ΔGu up=0进行了研究,其中ΔG是对应的sublaplace算子．通过将移动平面方法推广到H型群上,证明了在一定条件下的p和函数u,次线性方程ΔGu up=0的唯一解是u≡0．     

KKM Type Theorems and Coincidence Theorems with Applications to the Existence of Equilibria

In this paper, we obtain some coincidence theorems and some KKM-type theorems. We apply these results to establish the existence of the solution to generalized vector equilibrium problems, where a bimap f : X × Y 2^Z is involved and some sufficient conditions are imposed on f.     

论邓-王定理

对王曾叙述了并证明了若干重要定理,本文给出在稍稍不同条件下的类似定理的更简单的证明,讨论这些定理的含义、作用并予以推广。     

Abelian Theorems,Farey Series and the Riemann Hypothesis

The aim of this paper is to provide unconditional estimates for the error terms associated with Farey series that are comparable to error terms in the prime number theorem, and also to provide error terms for Farey series based on implications of the RH().     

A Helly-Type Theorem for Intersections of Compact Connected Sets in the Plane

For a family C of nonempty compact sets in the plane, the following conditions are equivalent:(1) Every two (not necessarily distinct) members of C have a connected union and every three (not necessarily distinct) members of C have a simply connected union.(2) C is a family of simply connected sets such that every two (not necessarily distinct) members of C have a connected intersection and every three (not necessarily distinct) members of C have a nonempty intersection.If either set of conditions is satisfied, then { C : C in C } is nonempty, simply connected, and connected. Furthermore, if the collection C is finite, then it is also true that { C : C in C } is simply connected.     

Vanishing Theorems for Conformally Compact Manifolds

Abstract

We study L ² harmonic p-forms on conformally compact manifolds with a rather weak boundary regularity assumption. We proved that if the lower bound of the curvature operator is great than or equal to ?1 and the infimum of the L ² spectrum of the Laplacian great than p(n ? p) for some p ≤ n/2, then there is no nontrivial L ² harmonic p-form.