A Criterion for the Existence of Transversals of Set Systems

Nash-Williams [6] formulated a condition that is necessary andsufficient for a countable family A=(A_i)_iI of sets to have atransversal. In [7] he proved that his criterion applies alsowhen we allow the set I to be arbitrary and require only that_iJA_i=Ø for any uncountable JI. In this paper, we formulateanother criterion of a similar nature, and prove that it isequivalent to the criterion of Nash-Williams for any familyu. We also present a self-contained proof that if _iJA_i=Øfor any uncountable JI, then our condition is necessary andsufficient for the family u to have a transversal.     

关于定向极小集

本文对定向极小集作了进一步的研究，得到一系列重要性质，文章最后给出连续格为完全分配格的一个充分条件．     

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ?M. These minimal surfaces are either disjoint from ?M or meet ?M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ?M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.     

On the Existence of Min-Max Minimal Torus

In this paper, we will study the existence problem of min-max minimal torus. We use classical conformal invariant geometric variational methods. We prove a theorem about the existence of min-max minimal torus in Theorem 5.1. First we prove a strong uniformization result (Proposition 3.1) using the method of Ahlfors and Bers (Ann. Math. 72(2):385–404, 1960). Then we use this proposition to choose good parameterization for our min-max sequences. We prove a compactification result (Lemma 4.1) similar to that of Colding and Minicozzi (Width and finite extinction time of Ricci flow, [math.DG], 2007), and then give bubbling convergence results similar to that of Ding et al. (Invent. math. 165:225–242, 2006). In fact, we get an approximating result similar to the classical deformation lemma (Theorem 1.1).     

A Frequency Function and Singular Set Bounds for Branched Minimal Immersions

We establish a frequency function monotonicity formula for two‐valued C^1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C^{1, 1/2}, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.     

拟序定向集最小共尾部分的结构

It is shown that for any directed quasi ordered set (Q,≤),there is a minimal ordinal number λ such that every cofinal subset of Q contains a cofinal subset which is the o-th class original set of a λ-th class chain of Q. A special case of our results gives necessary and sufficient conditions for a directed set to contain a cofinal chain.     

向量组的极小生成组

分析了线性相关概念的教学难点.引入了向量组的生成组和极小生成组的概念.由此给出了线性相关和极大线性无关组的刻画.     

平面凸集存在最小凸生成集的充要条件

本文在平面上解决了StevenRLay在 [1 ]中提出的开放性问题“什么样的凸集存在唯一的最小凸生成子集” ,给出并证明了“平面上的凸集存在唯一的最小凸生成子集”的一个充要条件 .同时证明了En 中的开集一定不存在最小凸生成集 .     

Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

Journal of Optimization Theory and Applications - In this paper, we give some properties concerned with weak minimal solutions of nonconvex set optimization problems. We also give some properties...     

Existence and Lyapunov Pairs for the Perturbed Sweeping Process Governed by a Fixed Set

Set-Valued and Variational Analysis - The aim of this paper is to prove existence results for a class of sweeping processes in Hilbert spaces by using the catching-up algorithm. These processes are...     

一类极小本原对称图的指数集

运用有向图方法完全确定出顶点带环的n阶极小本原对称有向图的本原指数集,所得的结论是:1)顶点全部自带环的n阶极小本原对称有向图所成的子图类之本原指数集E1={2,3,…,n-1};2)顶点不全带环的n阶极小本原对称有向图所成的子图类之本原指数集E2={2,3,…,2n-2}\S,其中S是{n,n+1,…,2n-2}中的所有奇数之集;3)顶点带环的n阶极小本原对称有向图所成的特殊图类之本原指数集En=E1∪E2={2,3,…,2n-2}\S.     

一类拟线性椭圆障碍问题极小正解的存在性

该文使用Ｌｉｏｎｓ的集中紧性原理和变分方法,证明了一类非齐次拟线性椭圆型方程对应的障碍问题中极小正解的存在性．     

On the Action of the Restricted Weyl Group on the Set of Orbits a Minimal Parabolic Subgroup

Doklady Mathematics - We construct the action of the restricted Weyl group on the set of principal families of orbits of a minimal parabolic subgroup over an algebraically nonclosed field....     

脉冲集值微分方程解的存在性

利用Leray-Schauder不动点定理,证明一类一阶脉冲集值微分方程解的存在性.     

Set Intersection Theorems and Existence of Optimal Solutions

The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We consider asymptotic directions of a sequence of closed sets, and introduce associated notions of retractive, horizon, and critical directions, based on which we provide new conditions that guarantee the nonemptiness of the corresponding intersection. We show how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions, and to derive some new results, including a new extension of the Frank–Wolfe Theorem for (nonconvex) quadratic programming.     

On The Existence of a Countable Set of Positive Solutions for a Nonlocal Boundary-Value Problem with Vector-Valued Response

ABSTRACT

The existence of a countable set of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated by some variational methods based on the idea of the Fenchel conjugate. As a consequence of a duality developed here, we obtain the existence of a countable set of solutions for our problem that are minimizers to a certain integral functional. We derive (also in the superlinear case) a measure of a duality gap between primal and dual functional for approximate solutions.     

On the Existence of Real Entire Functions with a Prescribed Ordered Set of Stationary Values

To each ordered set of real stationary values (with multiplicities) is shown to correspond a real entire function ƒ taking on the stationary values in the given order and with the given multiplicities along the real axis. This function has a derivative ƒ′, whose roots are all real, and which is of the Pólya-Laguerre class; the function ƒ is, apart from an arbitrary real affine transformation (conserving order) of the independent variable, the only function satisfying all of the conditions above.     

Existence Theorems of Random Nash Equilibrium Points for Random Set-valued Maps

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欧氏空间闭集最佳逼近元的唯一存在性

使用点集拓扑和凸泛函的部分基本知识与技巧,讨论有限维欧氏空间闭集最佳逼近元的存在性和唯一性,并对类似结论在无穷维空间中的情形给出反例.