Hypergeometric Type Functions and Their Symmetries

The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F ₁ equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed.     

Two-Sided Optimal Bounds for Green Functions of Half-Spaces for Relativistic <Emphasis Type=Italic>α</Emphasis>-Stable Process

The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α -stable process with parameter m on ℝ ^d space. This process has an infinitesimal generator of the form mI–(m ^2/α I–Δ) ^α/2, where 0<α<2, m>0, and reduces to the isotropic α-stable process for m=0. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets. Our approach combines the recent results obtained in Byczkowski et al. (Bessel Potentials, Hitting Distributions and Green Functions (2006) (preprint). ), where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process. For example, for d≥3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough. Research supported by KBN Grants.     

New lacunar-type and pendant groups containing derivatives of β-unsubstituted dibenzotetraaza[14]annulenes—syntheses and crystal structures

Three new lacunar-type derivatives of β-unsubstituted dibenzotetraaza[14]annulene have been synthesized along with a range of their open-chain pendant group containing counterparts. The crystal structures of two representative products have been determined and the noncovalent interactions explored. A number of C-H?π and π?π interactions involving aliphatic chains and aromatic regions of the macrocycle have been evidenced. The relative ease and high yield of lacunization are suggested to be largely due to self-assembly processes, driven by favorable noncovalent interactions between reacting units.     

Faces of simplexes of invariant measures

This paper is a direct continuation of [D2]. The main result proved here, combined with Theorem 1 of [D2] widens the variety of known possibilities of what the simplex of invariant measures in a minimal topological dynamical system can be. We show that it can be equivalent, modulo affine homeomorphism of Choquet simplexes and modulo isomorphisms of the ergodic measures playing the role of the extreme points, to any face of the simplex of invariant measures of any zero-dimensional topological dynamical system, as long as this face contains no periodic measures. In particular, this implies that any formal simplex spanned by any choice of countably many nonperiodic ergodic measures can be realized in a minimal system. The first draft of this paper was written during the author’s the author’s visit at Université Paris 6. Research supported from resources for science in years 2005–2008 as research project (grant MENII 1 P03A 021 29, Poland)     

Symbolic extensions and smooth dynamical systems

Let f:XX be a homeomorphism of the compact metric space X. A symbolic extension of (f,X) is a subshift on a finite alphabet (g,Y) which has f as a topological factor. We show that a generic C¹ non-hyperbolic (i.e., non-Anosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r>1, we exhibit a residual subset of an open set of C^r diffeomorphisms of a compact surface such that if , then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C diffeomorphism has symbolic extensions with the same entropy. We also show that C^r generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two.     

On some three color Ramsey numbers for paths and cycles

For given graphs G₁,G₂,…,G_k, k≥2, the multicolor Ramsey number, denoted by R(G₁,G₂,…,G_k), is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of G_i colored with i, for some 1≤i≤k. Let P_k (resp. C_k) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R(P_i,P_k,C_m) for odd m, k≥m≥3 and when i is odd, and when i is even. In addition, we provide the exact values for Ramsey numbers R(P₃,P_k,C₄) for all integers k≥3.