Abstract By the work of P. Lévy, sample paths Brownian motion are known to satisfy a certain Hölder regularity condition almost surely. This was later improved by Ciesielski, who studied these in Besov and Besov-Orlicz spaces. We review results propose new function spaces type, strictly smaller than those Ciesielski which surely lie. In same spirit, we extend Kamont, investigated question for m...

Abstract The sample paths of white noise are proved to be elements certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse increasing space dimension. Consequently, is actually much smoother than known sharp results suggest. An application our techniques yields new for solutions Poisson and heat equation on half boundary noise. m...

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

A set D ⊆ V of a graph G = (V,E) is called an outer-connected dominating set of G if for all v ∈ V , |NG[v] ∩ D| ≥ 1, and the induced subgraph of G on V \D is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality of the input graph G. Given a positive integer k and a graph G = (V,E), the Outer-connected Domination Decision ...

Roman domination in graphs is concerned with the problem of finding a vertex labelling, with minimum weight, satisfaying certain conditions. In this work, the authors initiate the study of a generalization to labellings of both vertices and edges in a graph.

In this paper, we define a new domination-like invariant of graphs. Let $${\mathbb {R}}^{+}$$ be the set non-negative numbers. $$c\in {\mathbb {R}}^{+}-\{0\}$$ number, and let G graph. A function $$f:V(G)\rightarrow is c-self-dominating if for every $$u\in V(G)$$ , $$f(u)\ge c$$ or $$\max \{f(v):v\in N_{G}(u)\}\ge 1$$ . The c-self-domination number $$\gamma ^{c}(G)$$ defined as ^{c}(G):=\min \{...

The Roman domination problem is considered. An improvement of two existing Integer Linear Programing (ILP) formulations is proposed and a comparison between the old and new ones is given. Correctness proofs show that improved linear programing formulations are equivalent to the existing ones regardless of the variables relaxation and usage of lesser number of constraints.

We study, for a countably categorical theory T, the complexity of computing and the complexity of dominating the function specifying the number of n-types consistent with T.