An algorithm is given for computing the Hausdorff dimension of the set(s) Λ = Λ(β,D) of real numbers with representations x = ∑∞ n=1 dnβ −n, where each dn ∈ D, a finite set of “digits”, and β > 0 is a Pisot number. The Hausdorff dimension is shown to be log λ/ log β, where λ is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data β,D is given.

In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of a non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of unifor...

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R ′ . With Es(A, N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A, N) for each fixed s ≥ d. Moreover, if A has positi...

In this paper, a new class of abstract impulsive Riemann-Liouville fractional partial neutral functional differential equations with infinite delay is introduced. We apply the suitable fixed point theorem together with the Hausdorff measure of noncompactness to investigate the existence of mild solutions for these equations in Banach spaces. Finally, two examples to illustrate the applications ...

We give several partial positive answers to a question of Juhász and Szentmiklóssy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.

Let G be an abelian group, and F a downward directed family of subsets of G. In [6], I. Protasov and E. Zelenyuk describe the finest group topology T on G under which F converges to 0; in particular, their description yields a criterion for T to be Hausdorff. They then show that if F is the filter of cofinite subsets of a countable subset X ⊆ G (the Fréchet filter on X), there is a simpler crit...

We consider sets of real numbers in [0, 1) with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by 1/2, are given by a modified variational principle.

We consider a multi-criterion Boolean programming problem with partial criteria of the kind MIN MODUL of linear functions. We investigate such type of stability which can be understood as a discrete analogue of the Hausdorff lower semi-continuity. A formula of the quasi-stability radius is obtained. Mathematics Subject Classification 2000: 90C09, 90C29, 90C31.

A continuum is a compact connected Hausdorff space. If X is a continuum, then CEX is a C-set if (i) C^X, (ii) for each subcontinuum A EX that meets C we have either A EC or CEA. Thus a composant of a solenoid is a C-set. A ?«0& is a Hausdorff space together with a continuous associative multiplication. A clan is a compact connected mob with (two-sided) unit, denoted by u. In what follows we ass...

For a manifold equipped with vector field there exists the universal completion consisting of a (possibly non-Hausdorff) manifold with a complete vector field on it. We describe the universal completion of the partial differential equations ut + F (u)ux = 0 viewed as vector fields on infinite dimensional manifolds.