Opuscula Mathematica
Loading...
ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2023
Volume
Vol. 43
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 43 (2023)
Projects
Pages
Articles
Operators induced by certain hypercomplex systems
(Wydawnictwa AGH, 2023) Alpay, Daniel; Cho, Ilwoo
In this paper, we consider a family $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$ of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations $\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}$ of the hypercomplex system $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$, and study the realizations $\pi_{t}(h)$ of hypercomplex numbers $h \in \mathbb{H}_{t}$, as $(2\times 2)$-matrices acting on $\mathbb{C}^{2}$, for an arbitrarily fixed scale $t \in \mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems
(Wydawnictwa AGH, 2023) Antczak, Tadeusz; Arana-Jimenéz, Manuel; Treanţă, Savin
In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems.
Axiomatic characterizations of Ptolemaic and chordal graphs
(Wydawnictwa AGH, 2023) Changat, Manoj; Sheela, Lekshmi Kamal K.; Narasimha-Shenoi, Prasanth G.
The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set $V$ to the power set of $V$ satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.
Existence and asymptotic stability for generalized elasticity equation with variable exponent
(Wydawnictwa AGH, 2023) Dilmi, Mohamed; Otmani, Sadok
In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor $\sigma^{p(\cdot)}$ has the form $\sigma^{p(\cdot)}(u)=(2\mu +|d(u)|^{p(\cdot)-2})d(u)+\lambda Tr(d(u)) I_{3},$ where $u$ is the displacement field, $\mu$, $\lambda$ are the given coefficients $d(\cdot)$ and $I_{3}$ are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.
On local antimagic total labeling of complete graphs amalgamation
(Wydawnictwa AGH, 2023) Lau, Gee-Choon; Shiu, Wai Chee
Let $G=(V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g:E \to \{1,2,\dots,q\}$ is called a local antimagic labeling of $ if for any two adjacent vertices $u$ and $v$, we have $g^+(u) \ne g^+(v)$, where $g^+(u) = \sum_{e\in E(u)} g(e)$, and $E(u)$ is the set of edges incident to $u$. Similarly, a bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w_{f}(u) \ne w_{f}(v)$, where $w_f(u) = f(u) + \sum_{e\in E(u)} f(e)$. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $g^{+}(v)$ (respectively, $w_{f}(u)$). The local antimagic (total) chromatic number, denoted $\chi_{la}(G)$ (respectively $\chi_{lat}(G)$), is the minimum number of induced colors taken over local antimagic (total) labeling of $G$. In this paper, we determined $\chi_{lat}(G)$ where $G$ is the amalgamation of complete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of $K_1$ and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.