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Homeomorphisms of Compact Sets in Certain Hausdorff Spaces  [PDF]
Arthur D. Grainger
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/493290
Abstract: We construct a class of Hausdorff spaces (compact and noncompact) with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic. Also, it is shown that these spaces contain compact subsets that are infinite. 1. Introduction In this paper, we construct a class of Hausdorff spaces with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic (Theorem 3.7). Conditions are given for these spaces to be compact (Corollary 2.10). Also, it is shown that these spaces contain compact subsets that are infinite (Corollary 2.10). This paper uses the Zermelo-Fraenkel axioms of set theory with the axiom of choice (see [1–3]). We let denote the finite ordinals (i.e., the natural numbers) and denotes the counting numbers (i.e., ). Also, for a given set , we denote the collection of all subsets of by , and we denote the cardinality of by . In other words, is the smallest ordinal number for which a bijection of onto exists. In this paper, we will only consider compact topologies that are Hausdorff. A topology on a set is compact if and only if and imply for some and . Therefore, compact topologies need not be Hausdorff. 2. A Class of Hausdorff Spaces Let , , and be sets such that is infinite and the collection is pairwise disjoint. For example, let , , and . Unless otherwise stated, we let Recall that for set and , we have Definition 2.1. Let be an infinite set. Define We call ??the Fréchet filter on . Note that being infinite implies that is a filter (see [4, Definition??3.1, page 48]). Definition 2.2. Consider the collection defined as follows: Proposition 2.3. The collection generates a Hausdorff topology on . Proof. Clearly, is a basis for a topology (see [5, Section??13]). Let such that . If , then , , , and If , then either or . Assume that , and let . Since , and , we have Assume that . Note that . Also, , which implies and Observe that, We infer that is Hausdorff. Proposition 2.4. If , then is compact in if and only if is a finite set. Proof. Note that finite sets are compact in any topological space. So, assume that is an infinite, and let which implies Let be a nonempty, finite subcollection of . Therefore, there exists , for some , such that which implies If , then we would have , contradicting being an infinite set. Consequently, infinite subsets of are not compact in the topological space . Corollary 2.5. The set is not compact in . Corollary 2.6. The set is compact in if and only if is finite. Proposition 2.7. Let . The set is compact in if and only if is a
The Hausdorff dimension of the visible sets of connected compact sets  [PDF]
Toby C O'Neil
Mathematics , 2003,
Abstract: For a compact subset K of the plane and a point x, we define the visible part of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes the closed line segment joining x to u.) In this paper, we use energies to show that if K is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point x in the plane, the Hausdorff dimension of K_x is strictly less than the Hausdorff dimension of K. In fact, for almost every x, dim(K_x)\leq {1/2}+\sqrt{dim(K)-{3/4}}. We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than s+{1/2}+\sqrt{dim(K)-{3/4}}, for s>0.
The directed Hausdorff distance between imprecise point sets  [PDF]
Christian Knauer,Maarten L?ffler,Marc Scherfenberg,Thomas Wolle
Computer Science , 2009,
Abstract: We consider the directed Hausdorff distance between point sets in the plane, where one or both point sets consist of imprecise points. An imprecise point is modelled by a disc given by its centre and a radius. The actual position of an imprecise point may be anywhere within its disc. Due to the direction of the Hausdorff Distance and whether its tight upper or lower bound is computed there are several cases to consider. For every case we either show that the computation is NP-hard or we present an algorithm with a polynomial running time. Further we give several approximation algorithms for the hard cases and show that one of them cannot be approximated better than with factor 3, unless P=NP.
Generalized Hausdorff measure for generic compact sets  [PDF]
Richárd Balka,András Máthé
Mathematics , 2012,
Abstract: Let $X$ be a Polish space. We prove that the generic compact set $K\subseteq X$ (in the sense of Baire category) is either finite or there is a continuous gauge function $h$ such that $0<\mathcal{H}^{h}(K)<\infty$, where $\mathcal{H}^h$ denotes the $h$-Hausdorff measure. This answers a question of C. Cabrelli, U. B. Darji, and U. M. Molter. Moreover, for every weak contraction $f\colon K\to X$ we have $\mathcal{H}^{h} (K\cap f(K))=0$. This is a measure theoretic analogue of a result of M. Elekes.
On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets  [PDF]
Frédéric Bayart,Yanick Heurteaux
Mathematics , 2011,
Abstract: Let $K$ be a compact set in $\rd$ with positive Hausdorff dimension. Using a Fractional Brownian Motion, we prove that in a prevalent set of continuous functions on $K$, the Hausdorff dimension of the graph is equal to $\dim_{\mathcal H}(K)+1$. This is the largest possible value. This result generalizes a previous work due to J.M. Fraser and J.T. Hyde which was exposed in the conference {\it Fractal and Related Fields~2}. The case of $\alpha$-H\"olderian functions is also discussed.
Computation of the Hausdorff distance between sets of line segments in parallel  [PDF]
Helmut Alt,Ludmila Scharf
Computer Science , 2012,
Abstract: We show that the Hausdorff distance for two sets of non-intersecting line segments can be computed in parallel in $O(\log^2 n)$ time using O(n) processors in a CREW-PRAM computation model. We discuss how some parts of the sequential algorithm can be performed in parallel using previously known parallel algorithms; and identify the so-far unsolved part of the problem for the parallel computation, which is the following: Given two sets of $x$-monotone curve segments, red and blue, for each red segment find its extremal intersection points with the blue set, i.e. points with the minimal and maximal $x$-coordinate. Each segment set is assumed to be intersection free. For this intersection problem we describe a parallel algorithm which completes the Hausdorff distance computation within the stated time and processor bounds.
An Improved Algorithm for 2D Shape Matching Based on Hausdorff Distance

ZHANG Wen-jing,XU Xiao-ming,SU Jian-feng,

中国图象图形学报 , 2000,
Abstract: Matching between two images is often needed in automated visual inspection. Template matching, which is the most principle approach for shape match, is time consuming in case of variation in position and rotation. In this paper, an improved algorithm for 2D shape matching based on Hausdorff Distance is proposed. Hausdorff Distance is used to measure the degree of similarity between two objects to make matching more efficiently. A high dimensional, non diferentiable, and multi modal objective function can be derived based on Hausdorff Distance. Although Genetic Algorithm is a powerful and attractive procedure for function optimization, the solution generated by the procedure do not guarantee to be the global optimal. A follow up optimization scheme such as the line search method is applied, which is capable of finding the minimum value of a unimodal function over a finite search interval. Initially the non differentiable function is solved using multi point stochastic search, and the solution is further improved by executing a sequence of successive line searches that approach the optimal to a pre determined precision. The experimental results show that the proposed method is capable of matching 2D shape with higher speed and precision.
A note on Gromov-Hausdorff-Prokhorov distance between (locally) compact measure spaces  [PDF]
Romain Abraham,Jean-Francois Delmas,Patrick Hoscheit
Mathematics , 2012, DOI: 10.1214/EJP.v18-2116
Abstract: We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended Gromov-Hausdorff-Prokhorov metric is a Polish space. This generalization is needed to define L\'evy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure.
A Hausdorff dimension for finite sets  [PDF]
Juan M. Alonso
Computer Science , 2015,
Abstract: The classical Hausdorff dimension of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension which is non-trivial. It turns out that a finite bound for the finite Hausdorff dimension guarantees that every point of the set has "nearby" neighbors. This property is important for many computer algorithms of great practical value, that obtain solutions by finding nearest neighbors. We also define an analog for finite sets of the classical box-counting dimension, and compute examples. The main result of the paper is a Convergence Theorem. It gives conditions under which, if a sequence of finite sets converges to a compact set (convergence of compact subsets of Euclidean space under the Hausdorff metric), then the finite Hausdorff dimension of the finite sets will converge to the classical Hausdorff dimension of the compact set.
Berezin Quantization From Ergodic Actions of Compact Quantum Groups, and Quantum Gromov-Hausdorff Distance  [PDF]
Jeremy Sain
Mathematics , 2009,
Abstract: Over the last 25 years, the notion of "fuzzy spaces" has become ubiquitous in the high-energy physics literature. These are finite dimensional noncommutative approximations of the algebra of functions on a classical space. The most well known examples come from the Berezin quantization of coadjoint orbits of compact semisimple Lie groups. We develop a theory of Berezin quantization for certain quantum homogeneous spaces coming from ergodic actions of compact quantum groups. This allows us to construct fuzzy versions of these quantum homogeneous spaces. We show that the finite dimensional approximations converge to the homogeneous space in a continuous field of operator systems, and in the quantum Gromov-Hausdorff distance of Rieffel. We apply the theory to construct a fuzzy version of an ellipsoid which is naturally endowed with an orbifold structure, as well as a fuzzy version of the $\theta$-deformed coset spaces $C(G/H)_{\hbar \theta}$ of Varilly. In the process of the latter, we show that our Berezin quantization commutes with Rieffel's deformation quantization for actions of $\R^d$.
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