Estimates of covering numbers of convex sets with slowly decaying orthogonal subsets

Covering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation.     

Four matroidal structures of covering and their relationships with rough sets

Covering is a common form of data representation, and covering-based rough sets serve as an efficient technique to process this type of data. However, many important problems such as covering reduction in covering-based rough sets are NP-hard so that most algorithms to solve them are greedy. Matroids provide well-established platforms for greedy algorithm foundation and implementation. Therefore, it is necessary to integrate covering-based rough set with matroid. In this paper, we propose four matroidal structures of coverings and establish their relationships with rough sets. First, four different viewpoints are presented to construct these four matroidal structures of coverings, including 1-rank matroids, bigraphs, upper approximation numbers and transversals. The respective advantages of these four matroidal structures to rough sets are explored. Second, the connections among these four matroidal structures are studied. It is interesting to find that they coincide with each other. Third, a converse view is provided to induce a covering by a matroid. We study the relationship between this induction and the one from a covering to a matroid. Finally, some important concepts of covering-based rough sets, such as approximation operators, are equivalently formulated by these matroidal structures. These interesting results demonstrate the potential to combine covering-based rough sets with matroids.     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

On covering of linear operators on polyhedral sets

We consider a problem of finding a covering constant for a restriction of a linear operator to a polyhedral set. The obtained result allows to reduce the initial problem to the problem of finding a covering constant for some linear bijective operators.     

On some types of neighborhood-related covering rough sets

Covering rough sets are natural extensions of the classical rough sets by relaxing the partitions to coverings. Recently, the concept of neighborhood has been applied to define different types of covering rough sets. In this paper, by introducing a new notion of complementary neighborhood, we consider some types of neighborhood-related covering rough sets, two of which are firstly defined. We first show some basic properties of the complementary neighborhood. We then explore the relationships between the considered covering rough sets and investigate the properties of them. It is interesting that the set of all the lower and upper approximations belonging to the considered types of covering rough sets, equipped with the binary relation of inclusion ?, constructs a lattice. Finally, we also discuss the topological importance of the complementary neighborhood and investigate the topological properties of the lower and upper approximation operators.     

基于模糊乘积有限自动机的覆盖关系

对Mealy-型模糊有限自动机乘积结构作了进一步的研究,并且对覆盖关系作了细致的刻画,推广了原有的覆盖概念.针对Mealy-型这类模糊有限自动机,通过性质考察了此覆盖概念的合理有效性,新的覆盖概念在乘积自动机间建立了更多的联系.特别证明了直积、级联积、圈积三种乘积之间的覆盖关系.得到了一些乘积自动机覆盖关系的传递性质.     

Quotient and product sets of thin subsets of the positive integers

We study the cardinalities of A/A and AA for thin subsets A of the set of the first n positive integers. In particular, we consider the typical size of these quantities for random sets A of zero density and compare them with the sizes of A/A and AA for subsets of the shifted primes and the set of sums of two integral squares.     

Rectangularity of products and completions of their subsets

The approach to the problem of the distribution of the functors of the Stone-?ech compactification, the Hewitt realcompactification or the Dieudonné completion with the operation of taking products is discussed using uniform structures on products. In particular, the role of different rectangular conditions is shown. Relative analogues of this question and new examples of (strongly) rectangular products are presented. Characterizations of bounded rectangular subsets of the product are given.     

On the product sets of rational numbers

A new lower bound on the size of product sets of rational numbers is obtained. An upper estimate for the multiplicative energy of two sets of rational numbers is also found.     

On multiple sum and product sets of finite sets of integers

Let $\text{A?}\text{Z}$ be a finite set of integers of cardinality |A|=N?2. Given a positive integer k, denote kA (resp. A^(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A^(k)|)>N^b. This answers affirmably questions raised in Erd?s and Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On a triangulation consistent with covering

If to every point x of a convex polyhedron M Eⁿ there corresponds an open ball with center at x and radius f(x), where f is any positive function, then M can be split into simplexes so that every simplex is covered by those balls whose centers lie at the vertices of the simplexes.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 110–114, 1992.     

On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trindX?α. W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trIndX?λ. T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dimWX?α where dimW is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dimC of the covering dimension.As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dimCX?α belonging to some absolute Borel class.     

On the topology of the effective subsets of level sets

In studying the problem of constructing selections of the distance function to subsets of a Euclidean space the concept of the effective subset of a level set arises In this paper we establish the connectivity of effective subsets, which is important for applications.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 5–9.