Graph connectivity and Wiener index

The graphs with a given number n of vertices and given (vertex or edge) connectivity k, having minimum Wiener index are determined. In both cases this is Kk + (K1 U Kn-k-1), the graph obtained by connecting all vertices of the complete graph Kk with all vertices of the graph whose two components are Kn-k-1 and K1. AMS Mathematics Subject Classification (2000): 05C12, 05C40 05C35.

Distribution analogue of the Tumarkin result

We give a distribution analogue of the Tumarkin result that concerns approximation of some functions by sequence of rational functions with given poles. AMS Mathematics Subject Classification (2000): 46F20, 30E25, 32A35.

On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets

Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functional and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis. AMS Mathematics Subject Classification (2000): 46F30, 46A20, 47G30.

Positivity in twisted convolution algebra and Fourier modulation spaces

Let Wp,q be the Fourier modulation space FMp,q and let *? be the twisted convolution. I? ? ? D' such that (a *? ?,?)? 0 for every ? ? C?0, and ? ? such that X(0) ? 0, then we prove that ?? ? Wp,? iff ? ? Wp,?. We also present some extensions to the case when weighted Fourier modulation spaces are used. AMS Mathematics Subject Classification (2000): 47B65, 35A21, 35S05.

Regular nonlinear generalized functions and applications

We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of G8-regularity introduced by M. Oberguggenberger. As a first application we show that these new spaces are useful in a problem of representation of linear maps by integral operators, giving an analogon to Schwartz kernel theorem in the framework of nonlinear generalized functions. Secondly, we remark that these new regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to micro local analysis of singularities of generalized functions, with respect to these regularities. AMS Mathematics Subject Classification (2000): 35A18, 35A27, 42B10, 46E10, 46F30.

Generalized solutions to singular initial-boundary hyperbolic problems with non-Lipshitz nonlinearities

We prove the existence and uniqueness of global generalized solutions in a Colombeau algebra of generalized functions to semilinear hyperbolic systems with nonlinear boundary conditions. Our analysis covers the case of non-Lipschitz nonlinearities both in the differential equations and in the boundary conditions. We admit strong singularities in the differential equations as well as in the initial and boundary conditions. AMS Mathematics Subject Classification (2000): 35L50, 35L67, 35D05.

On a model equation that reflects some of the shear flow hydrodynamic stability properties

A model equation is proposed in the paper that mimics some of the shear flow hydrodynamic stability properties. It contains the basic velocity profile which can be arbitrarily chosen, and a nonlinear term, whose form can be appropriately adjusted to any particular problem. Full linear and weakly nonlinear theories for the Bickley jet velocity profile are elaborated. The solution of the linear problem is obtained in terms of associated Legendre functions. Within the weakly nonlinear theory a Landau equation is derived that describes the evolution of the perturbations near the critical wave number. The conditions for supercritical stability and subcritical instability are revealed. AMS Mathematics Subject Classification (2000) 76E05, 76E30.

Inequalities between distance-based graph polynomials

In a recent paper [ I. Gutman, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur) 131 (2005) 1-7], the Hosoya polynomial H = H(G,?) of a graph G, and two related distance-based polynomials H1 = H1(G, ?) and H2 = H2(G, ?) were examined. We now show that max-?H1 - ?2H, ?H1 - ?2H} ? H2 ? ?H1 - ??H holds for all graphs G and for all ? ? 0, where ? and ? are the smallest and greatest vertex degree in G. The answer to the question which of the terms ?H1 -?2H and ?H1 -?2H is greater, depends on the graph G and on the value of the variable ?. We find a number of particular solutions of this problem. AMS Mathematics Subject Classification (2000): 05C12, 05C05.

Inequalities which include q-integrals

The main problem in analyzing inequalities which include q-integrals is the fact that q-integral of a function over an interval [a; b] (0 < a < b) is defined by the difference of two infinite sums. Thus defined q- integral properties must include the points outside of interval of integration. In this paper, we will signify to some directions for solving this problem and derive some inequalities which are analogues to well-known ones in standard integral calculus. AMS Mathematics Subject Classification (2000): 33D60, 26D15.

Inhomogeneous Gevrey ultradistributions and Cauchy problem

After a short survey on Gevrey functions and ultradistributions, we present the inhomogeneous Gevrey ultradistributions introduced recently by the authors in collaboration with A. Morando, cf. [7]. Their definition depends on a given weight function ?, satisfying suitable hypotheses, according to Liess-Rodino [16]. As an application, we define (s, ?)-hyperbolic partial differential operators with constant coefficients (for s > 1), and prove for them the well-posedness of the Cauchy problem in the frame of the corresponding inhomogeneous ultradistributions. This sets in the dual spaces a similar result of Calvo [4] in the inhomogeneous Gevrey classes, that in turn extends a previous result of Larsson [14] for weakly hyperbolic operators in standard homogeneous Gevrey classes. AMS Mathematics Subject Classification (2000): 46F05, 35E15, 35S05.