Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Let Cld _AW (X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space Cld _AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld _AW (X) is homeomorphic to a Hilbert space with weight 2 ^w(X).     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

Reconstruction of the radial dependence of the structural characteristic of the refractive index in a supersonic gas flow from laser beam intensity fluctuations

A method for reconstructing the radial dependence of the structural characteristic of the refractive index of air in an axisymmetric supersonic gas flow from the variance of intensity fluctuations of a laser beam that is transmitted through the flow at different distances from the jet symmetry axis is proposed. An expression that relates the relative intensity variance on the beam axis as a function of the aiming distance and the radial dependence of the structural characteristic is obtained. To invert this expression, the Abelian transformation is used. The parameters of the jet module of an AT-326 wind tunnel at the Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences are modeled.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

Characterization of convexZ-sets in linear metric spaces

L'vov State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 4, pp. 77–79, October–December, 1994.     

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup exp(G)\exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F(G) contains a G-invariant element (which is an analogue of the Haar measure on G).     

Approximation by light maps and parametric Lelek maps

The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈AP(n,0) if for every ε>0 and a map g:In→M there exists a 0-dimensional map g^′:In→M which is ε-homotopic to g. It is shown that this class has very nice properties. For example, if Mi∈AP(ni,0), i=1,2, then M₁×M₂∈AP(n₁+n₂,0). Moreover, M∈AP(n,0) if and only if each point of M has a local base of neighborhoods U with U∈AP(n,0). Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of n-dimensional Lelek maps.     

Propagation of Elastic Waves in Dynamically Self-Similar Structures (Dynamic Fractals)

Acoustical Physics - The concept of a dynamically self-similar structure (dynamic fractal) is introduced, consisting in the similarity of the dynamic parameters of the cell generatrices. Elastic...