Supercompact spaces

We prove the following theorem: Let Y be a Hausdorff space which is the continuous image of a supercompact Hausdorff space, and let K be a countably infinite subset of Y. Then (a) at least one cluster point of K is the limit of a nontrivial convergent sequence in Y (not necessarily in K), and (b) at most countably many cluster points of K are not the limit of some nontrivial sequence in Y. This theorem implies that spaces like β$\text{N}$ and β$\text{N}$?$\text{N}$ are not supercompact. Moreover we will give an example of a separable first countable compact Hausdorff space which is not supercompact.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution RK(X) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then RK(X) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X×P. Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X^′, such that X is strong shape dominated by X^′ and the Cartesian product X^′×P is a direct product in Sh(Top), then also X×P is a direct product in the shape category Sh(Top).     

CH, a problem of Rolewicz and bidiscrete systems

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable ‘nice’ semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum.We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K² which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that for every infinite compact Hausdorff space K, the space C(K) has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron II

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In a subsequent paper, published in 2007, the author proved that R(X,K) is a covariant functor in the first variable. In the present paper it is proved that R(X,K) is a covariant functor also in the second variable.     

On the fractional calculus of Besicovitch function

Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann–Liouville fractional calculus and Weyl–Marchaud fractional derivative of Besicovitch function have been discussed.     

A remark on CD 0(K)-spaces

A representation of the CD ₀(K)-space is given in [1, 2] for a compact Hausdorff space K without isolated points. We generalize this to an arbitrary countably compact space K without any assumption on isolated points.     

Fractal Measures for Parabolic IFS

Let h be the Hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d?2. In case the system of maps is finite, we provide necessary and sufficient conditions for the h-dimensional Hausdorff measure to be positive and finite and also, assuming the strong open set condition holds, characterize when the h-dimensional packing measure of the limit set is positive and finite. We also prove that the upper ball (box)-counting dimension and the Hausdorff dimension of this limit set coincide. As a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3.     

A combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

In a paper from 1954 Marstrand proved that if K⊂R² has a Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^1+α, α>0, for which the sum of their Hausdorff dimension is greater than 1.     

Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the g-closureg_K(H) of H in K as the subgroup consisting of χ∈K such that χ(a_n)?0 in T=R/Z for every sequence {a_n} in (the Pontryagin dual of K) that converges to 0 in the topology that H induces on . We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the G_δ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.     

True trees are dense

We show that any compact, connected set K in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, K can be approximated in the Hausdorff metric by a true tree in the sense of Grothendieck’s dessins d’enfants.     

Compactness by the Hausdorff measure of noncompactness

In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary BK-space into the sequence spaces c₀, c, ?_∞ and ?₁, and into the matrix domains of triangles in these spaces. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on the BK-spaces.     

Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.     

Vietoris topology on partial maps with compact domains

The space PK of partial maps with compact domains (identified with their graphs) forms a subspace of the hyperspace of nonempty compact subsets of a product space endowed with the Vietoris topology. Various completeness properties of PK, including ?ech-completeness, sieve completeness, strong Choquetness, and (hereditary) Baireness, are investigated. Some new results on the hyperspace K(X) of compact subsets of a Hausdorff X with the Vietoris topology are obtained; in particular, it is shown that there is a strongly Choquet X, with 1st category K(X).     

Travelling wave solutions for a class of generalized KdV equation

In this work, the K^∗(l,p) equation is investigated. The sine-cosine method, the tanh method and the extended tanh method are efficiently used for analytic study of this equation. New solitary patterns solutions and compactons solutions are formally derived. The proposed schemes are reliable and manageable.     

Extreme operator-valued continuous maps

Let ?(H) denote the space of operators on a Hilbert spaceH. We show that the extreme points of the unit ball of the space of continuous functionsC(K, ?(H)) (K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dimH<∞ and cardK<∞, (b) exposed if and only ifH is separable andK carries a strictly positive measure.     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.