Afrika Matematika最新文献

Let G be a nontrivial connected graph with an edge coloring, and let (u,v in V(G)). A (u-v) path in G is said to be a rainbow path if no color is repeated on the edges of the path. Similarly, we define a rainbow geodesic. A rainbow connected graph G is a graph with an edge coloring such that every two vertices in G are connected by a rainbow path. Further, a strong rainbow connected graph G is a graph with an edge coloring such that every two vertices in G is connected by a rainbow geodesic. The minimum number of colors needed to make a graph rainbow connected is called the rainbow connection number, denoted ({{,textrm{rc},}}(G)), and the minimum number of colors needed to make a graph strong rainbow connected is called the strong rainbow connection number, denoted ({{,textrm{src},}}(G)). In this paper we determine ({{,textrm{rc},}}(G)) and ({{,textrm{src},}}(G)) when G is a n-dimensional rectangular grid graph, triangular grid graph, hexagonal grid graph, and a (weak) Bruhat graph, respectively. We show for all these families that ({{,textrm{src},}}(G)={{,textrm{diam},}}(G)).

The energy En(G) of a graph G is defined as the sum of the absolute values of its eigenvalues. The Hosoya index Z(G) of a graph G is the number of independent edge subsets of G, including the empty set. For any given degree sequence D, we characterize the caterpillar (mathcal {S}(D)) that has the minimum Z and En. We also show that (Z(mathcal {S}(D))<Z(mathcal {S}(Y))) and (En(mathcal {S}(D))<En(mathcal {S}(Y))) for any degree sequences (Y=(y_1,dots ,y_n)) and (D=(d_1,dots ,d_n)) with

Plithogenic product intuitionistic fuzzy graph is a novel graphical model for representing complex systems characterised by multi-valued dyadic attributes. It provides four or more dyadic attributes to its elements, consisting of ({mu }) membership values and ({nu }) non-membership values. The dyadic attribute values of the edges are calculated using the (*) operator. In this paper, various properties and characterisations of Plithogenic product intuitionistic fuzzy graphs, including order, size, path, and cycle, are analysed to show the utility of the Plithogenic product intuitionistic fuzzy graphs. Additionally, weight, strength, the strength of connectedness, and subgraphs of Plithogenic product intuitionistic fuzzy graphs are newly introduced, accompanied by examples and figures, to examine the connectivity between parts and the significance of each part.

The decorated Partial Brauer algebras are finite dimensional diagram algebras contain Brauer algebras, Partial Brauer algebras and the group algebras (Rwidetilde{S_{n}}), where (widetilde{S_{n}}) is the wreath product group (mathbb {Z}_{2}wr S_{n}) of (mathbb {Z}_{2}) with (S_{n}). In this paper, we study the semisimplicity criterion of the decorated partial Brauer algebras using two functors F and G. In particular, we determine for which value of the parameters this algebra is semisimple. This result can be considered as a generalization of Hanlon–Wales conjecture on Brauer algebra.

Not many of the congruence properties of the eighth-order mock theta function (V_1(q)):

have been considered to date. We show that there are self-similarities of the coefficients of (V_1(q)). As consequences, we find congruences like the one below. For all (nge 0) and (kge 1), we have

for (0le s< 29), (sne 13).

In this paper, a numerical method based on the conservative finite difference scheme is constructed to numerically solve the strongly coupled nonlinear Schrödinger (SCNLS) equation. Conservative properties such as energy and mass of the SCNLS equation have been proven. In particular a fourth-order central difference scheme is used to discretize the the spatial derivative and a second-order Crank-Nicolson type discretization is used to discretize the temporal derivative. It has been shown that the proposed scheme preserves the discrete mass and energy. The existence of discrete solution is also investigated. Several numerical results are given to demonstrate the preservation properties of the new method. Also, the effect of the linear coupling parameters on the evolution of solitary waves is investigated.

In this work, we provide an elementary proof of an integral product formula for Jack polynomials of two variables, extending the well-known case of zonal polynomials. As an application, we derive an explicit integral representation for the Dunk- Bessel function of type (B_2).

The Linear Canonical Transform (LCT) serves as a powerful generalization of the Fourier and fractional Fourier transforms, with significant implications in signal processing, optics, and quantummechanics. This paper develops a novel representation-theoretic framework for the LCT by leveraging the unitary dual of the Heisenberg group and the metaplectic representation of the symplectic group. Beyond recovering known uncertainty principles, we present refined inequalities that explicitly depend on the LCT parameter matrix and derive new structural results for the spectral decomposition of LCT operators. In particular, we provide a distributional spectral analysis for degenerate LCT cases ((b = 0)), introduce entropic uncertainty bounds tailored to the LCT domain, and propose a group-theoretic formulation of sparsity constraints. These findings significantly extend classical results and offer a deeper understanding of the LCT in both theoretical and applied contexts. We conclude with suggestions for quantum state manipulation via LCTs and numerical illustrations that bridge abstract theory with practical computation.

We investigate the essential spectrum of linear pencils in Banach algebras, particularly their behavior under ideal perturbations. Building upon the foundational work of J. Shapiro and M. Snow in [The Fredholm spectrum of the sum and product of two operators, Transactions of the American Mathematical Society, 191 (1974), 387-393] on the Fredholm spectrum in Banach spaces, this study introduces novel characterizations of quasi-inverses and their role in spectral analysis. By leveraging these characterizations, we derive conditions ensuring that the essential spectrum of a linear pencil is confined within a specific sector of the complex plane. Our findings establish a refined connection between Fredholm theory and the algebraic structure of Banach algebras, offering both theoretical advancements and geometric insights into spectral containment. These results extend existing frameworks and open avenues for exploring spectral properties in more general algebraic settings with applications in operator theory and differential equations.

In recent years, the notions of discrete homotopy for graphs, such as A-homotopy and (times)-homotopy fundamental groupoid for graphs, have been introduced. This paper develops the notion of a looped fundamental groupoid for graphs to the isotropy group of the vertex v in the fundamental groupoid. We show that it is similar to the classical fundamental group of a topological space and the A-theory of graphs in some sense. Then, we generalize the concept of this theory to the looped fundamental group of weighted reflexive graphs and investigate some properties for the classification of weighted reflexive graphs in various situations. Moreover, we show that it satisfies the Seifert–Van Kampen property, with analogous results similar to the classical theory. Finally, we discuss the conclusion, its applications in the real world, and future works.