Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2019
Volume
Vol. 39
Number
No. 6
Description
Journal Volume
Opuscula Mathematica
Vol. 39 (2019)
Projects
Pages
Articles
Oscillatory criteria for second order differential equations with several sublinear neutral terms
(Wydawnictwa AGH, 2019) Baculíková, Blanka
In this paper, sufficient conditions for oscillation of the second order differential equations with several sublinear neutral terms are established. The results obtained generalize and extend those reported in the literature. Several examples are included to illustrate the importance and novelty of the presented results.
Vertices with the second neighborhood property in Eulerian digraphs
(Wydawnictwa AGH, 2019) Cary, Michael
The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.
Deformation of semicircular and circular laws via p-adic number fields and sampling of primes
(Wydawnictwa AGH, 2019) Cho, Ilwoo; Jørgensen, Palle E.T.
In this paper, we study semicircular elements and circular elements in a certain Banach $∗$-probability space $(\mathfrak{LS},\tau ^{0})$ induced by analysis on the $p$-adic number fields $\mathbb{Q}_{p}$ over primes $p$. In particular, by truncating the set $\mathcal{P}$ of all primes for given suitable real numbers $t\lt s$ in $\mathbb{R}$, two different types of truncated linear functionals $\tau_{t_{1}\lt t_{2}}$, and $\tau_{t_{1}\lt t_{2}}^{+}$ are constructed on the Banach $∗$-algebra $(\mathfrak{LS}$. We show how original free distributional data (with respect to $\tau ^{0}$) are distorted by the truncations on $\mathcal{p}$ (with respect to $\tau_{t\lt s}$, and $\tau_{t\lt s}^{+}$). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.
Graphs with equal domination and certified domination numbers
(Wydawnictwa AGH, 2019) Dettlaff, Magda; Lemańska, Magdalena; Miotk, Mateusz; Topp, Jerzy; Ziemann, Radosław; Żyliński, Paweł
A set $D$ of vertices of a graph $G=(V_{G},E_{G})$ is a dominating set of $G$ if every vertex in $V_{G}-D$ is adjacent to at least one vertex in $D$. The domination number (upper domination number, respectively) of $G$, denoted by $\gamma(G)$ ($\Gamma(G)$, respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of $G$. A subset $D\subseteq V_G$ is called a certified dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ has either zero or at least two neighbors in $V_{G}-D$. The cardinality of a smallest (largest minimal, respectively) certified dominating set of $G$ is called the certified (upper certified, respectively) domination number of $G$ and is denoted by $\gamma_{\rm cer}(G)$ ($\Gamma_{\rm cer}(G)$, respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.
Lightweight paths in graphs
(Wydawnictwa AGH, 2019) Harant, Jochen; Jendroľ, Stanislav
Let $k$ be a positive integer, $G$ be a graph on $V(G)$ containing a path on $k$ vertices, and w be a weight function assigning each vertex $v \in V(G)$ a real weight $w(v)$. Upper bounds on the weight $w(P)=\sum_{v\in V(P)}w(v)$ of $P$ are presented, where $P$ is chosen among all paths of $G$ on $k$ vertices with smallest weight.