Entropy,Approximation Quantities and the Asymptotics of the Modulus of Continuity

The paper deals with the approximation of bounded real functions f on a compact metric space (X, d) by so-called controllable step functions in continuation of [Ri/Ste]. These step functions are connected with controllable coverings, that are finite coverings of compact metric spaces by subsets whose sizes fulfil a uniformity condition depending on the entropy numbers ε_n(X) of the space X. We show that a strong form of local finiteness holds for these coverings on compact metric subspaces of IR^m and S^m. This leads to a Bernstein type theorem if the space is of finite convex information. In this case the corresponding approximation numbers ε_n(f) have the same asymptotics its ω(f, ε_n(X)) for f ε C(X). Finally, the results concerning functions f ε M(X) and f ε C(X) are transferred to operators with values in M(X) and C(X), respectively.     

Saturated packings and reduced coverings obtained by perturbing tilings

Let a normed space X possess a tiling T consisting of unit balls. We show that any packing P of X obtained by a small perturbation of T is completely translatively saturated; that is, one cannot replace finitely many elements of P by a larger number of unit balls such that the resulting arrangement is still a packing.In contrast with that, given a tiling T of Rⁿ with images of a convex body C under Euclidean isometries, there may exist packings P consisting of isometric images of C obtained from T by arbitrarily small perturbations which are no longer completely saturated. This means that there exists some positive integer k such that one can replace k−1 members of P by k isometric copies of C without violating the packing property. However, we quantify a tradeoff between the size of the perturbation and the minimal k such that the above phenomenon occurs.Analogous results are obtained for coverings.     

Adjoining an Order Unit to a Matrix Ordered Space

We prove that an order unit can be adjoined to every L ^∞-matricially Riesz normed space. We introduce a notion of strong subspaces. The matrix order unit space obtained by adjoining an order unit to an L ^∞-matrically Riesz normed space is unique in the sense that the former is a strong L ^∞-matricially Riesz normed ideal of the later with codimension one. As an application of this result we extend Arveson’s extension theorem to L ^∞-matircially Riesz normed spaces. As another application of the above adjoining we generalize Wittstock’s decomposition of completely bounded maps into completely positive maps on C ^*-algebras to L ^∞-matricially Riesz normed spaces. We obtain sharper results in the case of approximate matrix order unit spaces. Mathematics Subject Classification (2000). Primary 46L07     

On Countable Products of Finite Hausdorff Spaces

We investigate in ZF (i.e., Zermelo‐Fraenke set theory without the axiom of choice) conditions that are necessary and sufficient for countable products ∏_m∈ℕX_m of (a) finite Hausdorff spaces X_m resp. (b) Hausdorff spaces X_m with at most n points to be compact resp. Baire. Typica results: (i) Countable products of finite Hausdorff spaces are compact (resp. Baire) if and only if countable products of non‐empty finite sets are non‐empty. (ii) Countable products of discrete spaces with at most n + 1 points are compact (resp. Baire) if and only if countable products of non‐empty sets with at most n points are non‐empty.     

Exposed and denting points in duals of operator spaces

We study the extremal structure of the dual unit balls of various operator spaces. Mainly, we show that the classes of [w*-] strongly exposed, [w*-] exposed, and denting points in the dual unit balls of spaces of compact operators between Banach spacesX andY are completely — and in a canonical way — determined by the corresponding classes of points in the unit balls of the (bi-)duals of the factor spacesX andY. Applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.     

On the Existence of Completely Saturated Packings and Completely Reduced Coverings

We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W.Kuperberg: every body K in either n-dimensional Euclidean or n-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every >0, there is a body K in hyperbolic n-space which admits a completely saturated packing with density less than but which also admits a tiling.     

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non–1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn–1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onKE ^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofE ^d with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles.     

Multilinear Operators on C(K, X) Spaces

Given Banach spaces X, Yand a compact Hausdorff space K, we use polymeasures to give necessary conditions for a multilinear operator from C(K, X) into Yto be completely continuous (resp. unconditionally converging). We deduce necessary and sufficient conditions for Xto have the Schur property (resp. to contain no copy of c ₀), and for Kto be scattered. This extends results concerning linear operators.     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Simultaneous approximation and interpolation in weighted spaces

In this paper we present a result about simultaneous approximation and interpolation in weighted spaces. It generalizes a result of Prolla in the space of continuous functionsC(X;E) whereX is a compact Hausdorff space andE is a normed space. As a consequence, we prove that simultaneous approximation and interpolation is possible from certain vector subspaces.     

Typical Frobenius coverings

A covering p from a Cayley graph Cay（G, X） onto another Cay（H, Y） is called typical Frobenius if G is a Frobenius group with H as a Frobenius complement and the map p ： G →H is a group epimorphism. In this paper, we emphasize on the typical Frobenius coverings of Cay（H, Y）. We show that any typical Frobenius covering Cay（G, X） of Cay（H, Y） can be derived from an epimorphism /from G to H which is determined by an automorphism f of H. If Cay（G, X1） and Cay（G, X2） are two isomorphic typical Frobenius coverings under a graph isomorphism Ф, some properties satisfied by Фare given.     

Hilbert-generated spaces

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c₀(Γ) spaces in their weak topology. The existence of such embeddings is characterized by the existence of certain uniformly Gâteaux smooth norms.     

Covering an ellipsoid with equal balls

The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, our goal is to find the minimum number of unit balls needed to cover this ellipsoid. A tight asymptotic bound on the logarithm of this number is obtained.     

Circulant Double Coverings of a Circulant Graph of Valency Four

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices. The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com²MaC-KOSEF (R11-1999-054) in Korea.     

Circulant Double Coverings of a Circulant Graph of Valency Five

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory. A covering of G is called circulant if its covering graph is circulant. Recently, the authors [Discrete Math., 277, 73-85 （2004）1 enumerated the isomorphism classes of circulant double coverings of a certain type, called a typical covering, and showed that no double covering of a circulant graph of valency three is circulant. Also, in [Graphs and Combinatorics, 21,386 400 （2005）], the isomorphism classes of circulant double coverings of a circulant graph of valency four are enumerated. In this paper, the isomorphism classes of circulant double coverings of a circulant graph of valency five are enumerated.     

Finitely locally finite coverings of Banach spaces

A well-known result due to H. Corson states that, for any covering τ by closed bounded convex subsets of any Banach space X containing an infinite-dimensional reflexive subspace, there exists a compact subset C of X that meets infinitely many members of τ. We strengthen this result proving that, even under the weaker assumption that X contains an infinite-dimensional separable dual space, an (algebraically) finite-dimensional compact set C with that property can always be found.     

Kirkman packing and covering designs with spanned holes of size 2

A Kirkman holey packing (resp. covering) design, denoted by KHPD(g^u) (resp. KHCD(g^u)), is a resolvable (gu, 3, 1) packing (resp. covering) design of pairs with u disjoint holes of size g, which has the maximum (resp. minimum) possible number of parallel classes. Each parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3, and 4 when gu ≡ 2, 3, and 4 (mod 3) in turn. In this paper, the existence problem of a KHPD(2^u) and a KHCD(2^u) is solved with one possible exception of a KHPD(2⁸). © 2004 Wiley Periodicals, Inc.     

Essential extensions of Hausdorff spaces

In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).Dedicated to my friend Nico Pumplün on his sixtieth birthday     

Equitable resolvable coverings

In an earlier article, Willem H. Haemers has determined the minimum number of parallel classes in a resolvable 2‐(qk,k,1) covering for all k ≥ 2 and q = 2 or 3. Here, we complete the case q = 4, by construction of the desired coverings using the method of simulated annealing. Secondly, we look at equitable resolvable 2‐(qk,k,1) coverings. These are resolvable coverings which have the additional property that every pair of points is covered at most twice. We show that these coverings satisfy k < 2q ? , and we give several examples. In one of these examples, k > q. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 113–123, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10024     

Web‐compact spaces,Fréchet‐Urysohn groups and a Suslin closed graph theorem

A topological space X is strongly web‐compact if X admits a family {A_α: α ∈ ?^?} of relatively countably compact sets covering X and such that A_α ? A_β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet‐Urysohn and Baire and Y is strongly web‐compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web‐compact is not productive. This applies to show that there are quasi‐Suslin spaces X whose product X × X is not quasi‐Suslin (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)