The temporal dynamics of games have been studied widely in evolutionary spatial game theory using simulation. Each player is usually represented by a vertex of a graph and plays a particular game against every adjacent player independently. These games result in payoffs to the players which affect their relative fitness. The fitness of a player, in turn, affects its ability to reproduce. In this paper, we analyse the temporal dynamics of the evolutionary 2-person, 2-strategy snowdrift game in which players are arranged along a cycle of arbitrary length. In this game, each player has the option of adopting one of two strategies, namely cooperation or defection, during each game round. We compute the probability of retaining persistent cooperation over time from a random initial assignment of strategies to players. We also establish bounds on the probability that a small number of players of a particular mutant strategy introduced randomly into a cycle of players which have established the opposite strategy leads to the situation where all players eventually adopt the mutant strategy. We adopt an analytic approach throughout as opposed to a simulation approach clarifying the underlying dynamics intrinsic to the entire class of evolutionary spatial snowdrift games.

Let \(G\) be a simple graph with vertex set \(V=\{v_1,v_2,\ldots,v_n \}\), and let \(d_i\) be the degree of the vertex \(v_i\). The Sombor matrix of \(G\) is the square matrix \(\mathbf A_{SO}\) of order \(n\), whose \((i,j)\)-element is \(\sqrt{d_i^2+d_j^2}\) if \(v_i\) and \(v_j\) are adjacent, and zero otherwise. We study the characteristic polynomial, spectrum, and energy of \(\mathbf A_{SO}\). A few results for the coefficients of the characteristic polynomial, and bounds for the energy of \(\mathbf A_{SO}\) are established.

This note presents the characterization of the families of star, helm, flower and complete graphs by total vertex stress. The note does not present results for many families of graphs but, it highlights important philosophical (math. phil.) aspects for further research. In particular the novelty concepts of forgiven contradictions denoted by, iff\(_f\) as well as iffness and \(f\)-statements are introduced. The author suggests that the characterization of other families of graphs by total vertex stress is possible.

In this paper, improved and generalized version of Ostrowski’s type inequalities is established. The parameters used in the peano kernels help us to obtain previous results. The obtained bounds are then applied to numerical integration.

We show new integral representations for dilogarithm and trilogarithm functions on the unit interval. As a consequence, we also prove (1) new integral representations for Apéry, Catalan constants and Legendre \(\chi\) functions of order 2, 3, (2) a lower bound for the dilogarithm function on the unit interval, (3) new Euler sums.

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

In this paper, we derive the explicit expressions for single and product moments of generalized order statistics from Pareto-Rayleigh distribution using hypergeometric functions. Also, some interesting remarks are presented.

Local convergence of a family of sixth order methods for solving Banach space valued equations is considered in this article. The local convergence analysis is provided using only the first derivative in contrast to earlier works on the real line using the seventh derivative. This way the applicability is expanded for these methods. Numerical examples complete the article.

In this paper, we define some new operators \([(A \$ B),(A \# B),(A\ast B),(A \rightarrow B) ]\) of Fermatean fuzzy matrices and investigate their algebraic properties. Further, the necessity and possibility operators of Fermatean fuzzy matrices are proved. Finally, we have identified and proved several of these properties, particularly those involving the operator \(A\rightarrow B\) defined as Fermatean fuzzy implication with other operators.

In this paper, by using \(S\)-norms, we defined anti fuzzy subgroups and anti fuzzy normal subgroups which are new notions and considered their fundamental properties and also made an attempt to study the characterizations of them. Next we investigated image and pre image of them under group homomorphisms. Finally, we introduced the direct sum of them and proved that direct sum of any family of them is also anti fuzzy subgroups and anti fuzzy normal subgroups under \(S\)-norms, respectively.