Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2011
Volume
Vol. 31
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 31 (2011)
Projects
Pages
Articles
A note on global alliances in trees
(2011) Bouzefrane, Mohamed; Chellali, Mustapha
For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V - S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive (respectively, defensive) alliance if for each vertex in $V - S$ (respectively, in $S$) at least half the vertices from the closed neighborhood of $v$ are in $S$. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set of $G$, and the global offensive alliance number $\gamma_{o}(G)$ (respectively, global defensive alliance number $\gamma_{a}(G)$) is the minimum cardinality of a global offensive alliance (respectively, global deffensive alliance) of $G$. We show that if $T$ is a tree of order $n$, then $\gamma_{o}(T)\leq 2\gamma(T)-1$ and if $n\geq 3$, then $\gamma_{o}(T)\leq \frac{3}{2}\gamma_{a}(T)-1$. Moreover, all extremal trees attaining the first bound are characterized.
Convolution algebras for topological groupoids with locally compact fibres
(2011) Buneci, Mădălina Roxana
The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on $G$, we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the »left invariance« condition in the groupoid sense.
The Hardy potential and eigenvalue problems
(2011) Chabrowski, Jan
We establish the existence of principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We consider the Dirichlet and Neumann boundary conditions.
On the Bochner subordination of exit laws
(2011) Hmissi, Mohamed; Maaouia, Wajdi
Let $\mathbb{P}=(P_t)_{t\ge 0}$ be a sub-Markovian semigroup on $L^{2}(m)$, let $\beta=(\beta_t)_{t\ge 0}$ be a Bochner subordinator and let $\mathbb{P}^{\beta}=(P_t^{\beta})_{t\ge 0}$ be the subordinated semigroup of $\mathbb{P}$ by means of $\beta$, i.e. $P^{\beta}_s:=\int_0^{\infty} P_r\,\beta_s(dr)$. Let $\varphi:=(\varphi_t)_{t\gt 0}$ be a $\mathbb{P}$-exit law, i.e. $P_t\varphi_s= \varphi_{s+t}, \qquad s,t\gt 0$ and let $\varphi^{\beta}_t:=\int_0^{\infty} \varphi_s\,\beta_t(ds)$. Then $\varphi^{\beta}:=(\varphi_t^{\beta})_{t\gt 0}$ is a $\mathbb{P}^{\beta}$-exit law whenever it lies in $L^{2}(m)$. This paper is devoted to the converse problem when $\beta$ is without drift. We prove that a $\mathbb{P}^{\beta}$-exit law $\psi:=(\psi_t)_{t\gt 0}$ is subordinated to a (unique) $\mathbb{P}$-exit law $\varphi$ (i.e. $\psi=\varphi^{\beta}$) if and only if $(P_tu)_{t\gt 0}\subset D(A^{\beta})$, where $u=\int_0^{\infty} e^{-s} \psi_s ds$ and $A^{\beta}$ is the $L^{2}(m)$-generator of $\mathbb{P}^{\beta}$.
A sampling theory for infinite weighted graphs
(2011) Jørgensen, Palle E.T.
We prove two sampling theorems for infinite (countable discrete) weighted graphs $G$, one example being »large grids of resistors« i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing $G$, and there are Hilbert spaces of functions on $X$ that allow interpolation by sampling values of the functions restricted only on the vertices in $G$. We sample functions on $X$ from their discrete values picked in the vertex-subset $G$. We prove two theorems that allow for such realistic ambient spaces $X$ for a fixed graph $G$, and for interpolation kernels in function Hilbert spaces on $X$, sampling only from points in the subset of vertices in $G.$ A continuum is often not apparent at the outset from the given graph $G$. We will solve this problem with the use of ideas from stochastic integration.