Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2021
Volume
Vol. 41
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 41 (2021)
Projects
Pages
Articles
The achromatic number of K6 □ K7 is 18
(Wydawnictwa AGH, 2021) Horňák, Mirko
A vertex colouring $f:V(G) \to C$ of a graph $G$ is complete if for any two distinct colours $c_{1},c_{2} \in C$ there is an edge $\{v1,v2\} \in E(G)$ such that $f(v_{i})=c_{i}$, $i=1,2$. The achromatic number of $G$ is the maximum number $\text{achr}(G)$ of colours in a proper complete vertex colouring of $G$. In the paper it is proved that $\text{achr}(K_6 \square K_7)=18$. This result finalises the determination of $\text{achr}(K_6 \square K_q)$.
Influence of an Lp-perturbation on Hardy-Sobolev inequality with singularity a curve
(Wydawnictwa AGH, 2021) Ijaodoro, Idowu Esther; Thiam, El Hadji Abdoulaye
We consider a bounded domain $\Omega$ of $\mathbb{R}^{N}$, $N \geq 3$, $h$ and $b$ continuous functions on $\Omega.$ Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^{1}_{0}(\Omega)$ to the perturbed Hardy-Sobolev equation: $-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,$ where $2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $\sigma \in [0,2)$, $0\lt\delta\lt\frac{4}{N-2}$ and $\rho_{\Gamma}$ is the distance function to $\Gamma$. We show that the existence of minimizers does not depend on the local geometry of $\Gamma$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-\Delta+h$ and or on $b$. This is due to the perturbative term of order $1+\delta$.
On the gauge-natural operators similar to the twisted Dorfman-Courant bracket
(Wydawnictwa AGH, 2021) Mikulski, Włodzimierz M.
All $\mathcal{VB}_{m,n}$-gauge-natural operators $C$ sending linear $3$-forms $H \in \Gamma^{l}_E(\bigwedge^3T^*E)$ on a smooth ($\mathcal{C}^\infty$) vector bundle $E$ into $\mathbf{R}$-bilinear operators $C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)$ transforming pairs of linear sections of $TE \oplus T^*E \to E$ into linear sections of $TE \oplus T^*E \to E$ are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets $C$ (i.e. $C$ as above such that $C_0$ is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear $3$-forms $H$. An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.
Dimension of the intersection of certain Cantor sets in the plane
(Wydawnictwa AGH, 2021) Pedersen, Steen; Shaw, Vincent T.
In this paper we consider a retained digits Cantor set $T$ based on digit expansions with Gaussian integer base. Let $F$ be the set all $x$ such that the intersection of $T$ with its translate by $x$ is non-empty and let $F_{\beta}$ be the subset of $F$ consisting of all $x$ such that the dimension of the intersection of $T$ with its translate by $x$ is $\beta$ times the dimension of $T$. We find conditions on the retained digits sets under which $F_{\beta}$ is dense in $F$ for all $0\leq\beta\leq 1$. The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane.
Introduction to dominated edge chromatic number of a graph
(Wydawnictwa AGH, 2021) Piri, Mohammad R.; Alikhani, Saeid
We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$, is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $\chi_{dom}^{\prime}(G)$. We obtain some properties of $\chi_{dom}^{\prime}(G)$ and compute it for specific graphs. Also examine the effects on $\chi_{dom}^{\prime}(G)$, when $G$ is modified by operations on vertex and edge of $G$. Finally, we consider the $k$-subdivision of $G$ and study the dominated edge chromatic number of these kind of graphs.