关于超空间连通性的一个注记

本文回答了由E.Klein与A.C.Thompson在其著作《Thery of Correspondences》中提出的一个问题。主要结果是:若X是一度量空间,且在X中存在一条弧,则在P(X)中有一条序弧α,满足条件i)α(0)是连通子集;ii) (?)α(t)是不连通的。     

Stability of Linear Inequality Systems Measured by the Hausdorff Metric

In this paper, we propose a Hausdorff metric to measure the distance between two linear inequality systems on a real normed space X. For this topology, which comes through a pseudo-metric in the set of linear inequality systems, the closedness of the feasible set mapping is studied, and at the same time a characterization of the stability of the subset _c of consistent sytems is given.     

Best Approximations of Convex Compact Sets by Balls in the Hausdorff Metric

We deduce an upper bound for the Hausdorff distance between a nonempty bounded set and the set of all closed balls in a strictly convex straight geodesic space X of nonnegative curvature. We prove that the set $\chi \left[ {\rm M} \right]$ of centers of closed balls approximating a convex compact set in the Hausdorff metric in the best possible way is nonempty _X [M] and is contained in M. Some other properties of $\chi \left[ {\rm M} \right]$ also are investigated.     

A Remark on The Geometry of Spaces of Functions with Prime Frequencies

For any positive integer r, denote by \({\mathcal{P}_{r}}\) the set of all integers \({\gamma \in \mathbb{Z}}\) having at most r prime divisors. We show that \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) , the space of all continuous functions on the circle \({\mathbb{T}}\) whose Fourier spectrum lies in \({\mathcal{P}_{r}}\) , contains a complemented copy of \({\ell^{1}}\) . In particular, \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) is not isomorphic to \({C(\mathbb{T})}\) , nor to the disc algebra \({A(\mathbb{D})}\) . A similar result holds in the L ¹ setting.     

Missing edge coverings of bipartite graphs and the geometry of the Hausdorff metric

In this paper, we examine the problem of finding the number ${k}$ of elements at a given location on the line segment between two elements in the geometry the Hausdorff metric imposes on the set ${\mathcal{H} (\mathbb{R}^{n})}$ of all nonempty compact sets in n-dimensional real space. We demonstrate that this problem is equivalent to counting the edge coverings of simple bipartite graphs. We prove the novel results that there exist no simple bipartite graphs with exactly 19 or 37 edge coverings, and hence there do not exist any two elements of ${\mathcal{H} (\mathbb{R}^{n})}$ with exactly 19 or 37 elements at a given location lying between them—although there exist pairs of elements in ${\mathcal{H} (\mathbb{R}^{n})}$ that have k elements at any given location between them for infinitely many values of k, including k from 1 to 18 and 20 to 36. This paper extends results in the geometry of the Hausdorff metric given in J. Geom. 92: 28–59 (2009). In addition to our results about counting edge coverings, we give a brief introduction to this geometry.     

矩阵代数收敛至球面与非交换度量几何

介绍了Rieffel定义的紧致量子度量空间与量子Gromov-Hausdorff距离和近来Latrémolière定义的量子Gromov-Hausdorff邻距,分别讨论了矩阵代数如何在这两种量子距离下收敛至球面.     

Geometry of Configuration Spaces of Tensegrities

Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in ℝ ^d to admit a nonzero self-stress with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in ℝ ^d . In particular we find surgeries on graphs that give relations between different strata. Further we discuss questions related to geometric conditions defining the strata for plane tensegrities. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices.     

On Angles and The Fundamental Theorems of Metric Geometry

In this paper we investigate the abstract angle measure for affine metric spaces. Common features and differences between orthogonal angles and angles with measure ≠ 0 are examined. It turns out that an affine collineation which maps angles with a certain fixed measure α ≠ 0,4 to angles with another fixed measure β is already a metric collineation in nearly all cases (fundamental theorem). An analogous result is stated for projective metric spaces. Some applications concerning minimal conditions for metric collineations are given.     

On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound \(k\) and upper dimension bound \(N\ge 1\). In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.     

Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from to such that f(N)/ln N+ and f(N)/N0 as N goes to the laws of {x ² : d(x, C)f(N)}/N satisfy a large deviations principle with respect to the L ¹ metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.     

Geometry and Algebra of Prime Fano 3-folds of Genus 12

The connection between these Fano 3-folds and plane quartic curves is explained.     

On the Geometry of the （1, 1）-Tensor Bundle with Sasaki Type Metric

Curvature properties are studied for the Sasaki metric on the (1,1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-K(a)hler-Nordenian B-metrics are constructed on the (1,1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.     

基于Hausdorff距离的模糊数互补判断矩阵排序研究

基于Bonissone近似计算、Hausdorff距离和模糊折衷型决策方法,给出带有梯形模糊数互补判断矩阵的一种排序方法。同时给出精确值、三角模糊数的互补判断矩阵转化为梯形模糊数互补判断矩阵的方法,因此本文方法同样适合于处理精确值、三角模糊数的互补判断矩阵的排序问题。最后用算例说明了计算过程。     

关于Hausdorff距离

本文对Ｈａｕｓｄｏｒｆｆ距离的定义及有关性质作了深入讨论．     

A consistent counterexample in the theory of collectionwise Hausdorff spaces

It is shown to be consistent that there is a normal first countable locally countable space which is not collectionwise Hausdorff and in which there is a closed discrete non-G _δ set which provides the counterexample to collectionwise Hausdorffness. This answers a question of P. Nyikos. Publication 349, partially supported by the BSF.     

A Heuristic for the Prime Number Theorem

Conclusion  Might there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linear functionL(x) is increasing becausex/π(x) → ∞ asx → ∞. Moreover, using PNT, we can give a proof thatL(x) is indeed asymptotic tox/π(x). But the point of our work in this article is that for someone who wishes to understand why the growth of primes is governed by natural logarithms, a reasonable approach is to convince oneself via computation that the convex hull just mentioned satisfies the hypothesis of our theorem, and then use the relatively simple proof to show that this hypothesis rigorously implies the prime number theorem.     

A note on dyadic Hausdorff capacities

In this paper, the authors give a counterexample to show that the classical dyadic Hausdorff capacity on Rn when n?2 and 0<d?n−1 is not a capacity in the sense of Choquet. A variant of the classical dyadic Hausdorff capacity, , is then introduced and is further proved to be a capacity in the sense of Choquet.