Capacity Density of Subanalytic Sets in Higher Dimension

We show that capacity density exists for subanalytic sets in any dimension. Moreover, we state connections between capacity density and volume density for subanalytic sets.      

Local Flattening in Rigid Analytic Geometry

We prove a rigid analytic version of Hironaka's Local FlatteningTheorem, and establish a framework in which to study rigid subanalyticsets in the style of Denef and van den Dries. 1991 MathematicsSubject Classification: 32P05, 32B20, 12J25, 03C10.     

关于空间解析几何的学术形态和教育形态

空间解析几何知识需要形式化的表述,而学生掌握空间解析几何知识需要经过思考,需要追根求源．作为教师,必须善于将空间解析几何的学术形态转化为教育形态,激发学生的学习兴趣和主动性．结合自己在空间解析几何教学实践中的体会,给出了一些实例．     

高等院校数学专业解析几何课程改革研究

本文从高校几何学课程改革的一般现状、解析几何课程改革的必要性、课程内容改革和课程安排设想以及相关课程内容及设置时间的处理建议等角度对高等院校数学专业解析几何课程改革做了一定的探讨,并提出了解析几何教材改革的具体内容和结构安排设想.     

线性代数与空间解析几何教学内容改革实践

科学技术的发展深刻地影响着大学教育体系和教育模式 ,大学教育的改革势在必行 ,面向二十一世纪高等数学教学改革是高等教育改革中一个非常重要的组成部分 ,线性代数与空间解析几何是高等数学的三大模块之一 ,本文主要介绍面向二十一世纪高等数学教学改革教材《线性代数与空间解析几何》教学内容的改革实践 .     

Constructive Axiomatizations of Plane Absolute,Euclidean and Hyperbolic Geometry

In this paper we provide quantifier‐free, constructive axiomatizations for 2‐dimensional absolute, Euclidean, and hyperbolic geometry. The main novelty consists in the first‐order languages in which the axiom systems are formulated.     

线性代数与空间解析几何课程全面改革的思考

首先阐述为加强大学数学教育过程中的实践性教学环节,注重大学生对数学知识应用能力的培养,对线性代数与空间解析几何课程进行全方位改革的必要性;其次主要介绍我校在以学习教育过程为中心教育理论指导下,对线性代数与空间解析几何课程正在进行的教学方法、教学内容和课程考核等方面的改革.     

Rigid sets and nonexpansive mappings

We introduce a new class of normed spaces (not necessarily finite dimensional), which contains the finite dimensional normed spaces with polyhedral norm. We study the properties of rigid sets of the spaces of this class and we apply the results to limit sets of the sequences of iterates of nonexpansive maps.

     

Remarks on the Analytic Centers of Convex Sets

In this paper we analyze properties of the analytic centers for perturbed convex problems. In particular we study the continuity of convex regions with respect to such characteristics as boundedness, full dimensionality and the existence of an analytic centre. Furthermore, we state the necessary and sufficient conditions for the existence of the analytic centre for possibly unbounded regions and we show that the analytic centre is identical for any minimal representation of the convex set. We are also concerned with the limiting behaviour of the central path as it approaches the set of optimal solutions of the problem.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

Rigid Analytic Uniformization of Curves and the Study of Isogenies

Complex uniformization of curves is a popular tool in Number Theory. There are, however, some arithmetic and computational advantages in the use of p-adic uniformization. This paper compares the two theories and discusses how they can be used to study isogenies, with explicit examples of p-adic uniformization of hyperelliptic curves.      

$G$核开集,$G$核邻域与$G$ 核导集

在拓扑空间中, 在$G$方法意义下以$G$壳与$G$核为基础, 引入$G$壳闭集,$G$核开集,$G$核邻域与$G$核导集的概念, 讨论其相应的一些性质. 特别的, 定义了点式$G$方法, 提供了在此方法下$G$闭集与$G$壳闭集, $G$开集与$G$核开集, $G$邻域与$G$核邻域, $G$导集与$G$核导集的一致性, 丰富了拓扑空间中关于$G$闭集, $G$开集, $G$内部, $G$邻域和$G$导集的一些结果. 同时, 提出一些问题以供进一步研究.     

随机模糊集与随机集

本文研究了三个方面的工作：一是定义了一种模糊集上的可测结构，从而定义了随机模糊集，这些定义都与论域Ｘ上的拓扑结构无关。将通常意义下的集合看成特殊模糊集得到的通常集合上的超可测结构与文（３）中的定义一致；二是给出了随机模糊集、随机集的一些等价条件；三是研究了随机模糊集、随机集的分布与其有限维落影族的关系。     

谈线性代数与空间解析几何的整合

线性代数与空间解析几何的产生、发展,意味着它们是不可分割的两部分,本文分析和论述了线性代数与空间解析几何整合的必要性和可行性.     

型和余型的一个解析鞅特征

应用解析鞅的不等式及其收敛性给出了Ｂａｎａｃｈ空间的型和余型的刻划．     

自反Banach空间中锥线性优化问题初始——对偶目标函数水平集的几何性质

讨论自反Banach空间中的原——对偶锥线性优化问题的目标函数水平集的几何性质．在自反Banach空间中,证明了原目标函数水平集的最大模与对偶目标函数水平集的最大内切球半径几乎是成反比例的．     

Join-Independent and Meet-Independent Sets in Complete Lattices

In this paper join-independent and meet-independent sets in a complete lattice are defined. The relations between them are presented and substructures of a complete lattice induced by such sets are investigated.     

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called directed sets is constructed, in which the convex cone of all nonempty convex compact sets in R ⁿ is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in R ⁿ are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a support function and directed supporting faces of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the support function and recursively on the directed supporting faces. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.     

Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

Nodal Sets and Doubling Conditions in Elliptic Homogenization

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients. We show that the(d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L_ε(u_ε) = 0 in a ball in Rdare bounded uniformly in ε 0. The proof relies on a uniform doubling condition and approximation of u_ε by solutions of the homogenized equation.