The paper construct continuous and discrete distribution laws, used to assess risks in information systems. Generalized expressions for continuous distribution laws with maximum entropy are obtained. It is shown that, in the general case, the entropy also depends on the type of moments used to determine the numerical characteristics of the distribution law. Also, probabilistic model have been developed to analyze the sequence of independent trials with three outcomes. Expressions for their basic numerical characteristics are obtained, as well as for calculating the probabilities of occurrence of the corresponding events.

The equation \(v=v_0+\int_0^t(t-s)^{\lambda -1}v(s)ds\) is considered, \(\lambda\neq 0,-1,-2…\) and \(v_0\) is a smooth function rapidly decaying with all its derivatives. It is proved that the solution to this equation does exist, is unique and is smoother than the singular function \(t^{-\frac 5 4}\).

The Wiener index is a topological index of a molecule, defined as the sum of distances between all pairs of vertices in the chemical graph. Hexagonal chains consist of hexagonal rings connected with each other by edges. This class of chains contains molecular graphs of unbranched catacondensed benzenoid hydrocarbons. A segment of length \(\ell\) of a chain is its maximal subchain with \(\ell\) linear annelated hexagons. We consider chains in which all segments have equal lengths. Such chains can be uniquely represented by binary vectors. The Wiener index of hexagonal chains under some operations on the corresponding binary vectors are investigated. The obtained results may be useful in studying of topological indices for sets of hexagonal chains induced by algebraic constructions.

For every \(n\ge 3\), we determine the minimum number of edges of graph with \(n\) vertices such that for any non edge \(xy\) there exits a hamiltonian cycle containing \(xy\).

In this paper we characterize a bounded vector measure on Lie compact group \(G = GL(2;\mathbb{R})\). It is a question of considering a bounded vector measure \(m\) defined from \(K(G;E)\) the space of \(E\) valued functions with compact support on \(G\) and giving its integral form.

In this research work, we applied some solving techniques on Travelling salesman problem (TSP) and Capaciated Vehicle routing problem (CVRP) which are some of the variants of vehicle routing problem. For each of the considered problem, Branch-and-cut was applied on TSP and Column generation technique was used on CVRP. We obtained optimal solution and tour. Hence, these methods can be used to solve similar problem for optimal solution.

In this paper, we investigate the equivalence between the Backlund transformation of Riccati equation method and the extended tanh-function method. It is proved that the two methods are equivalent when applying them to partial differential equations and differential-difference equations. Two examples are introduced to justify our results.

In this paper, we will determine the 3-total edge product cordial (3-TEPC) labeling. We study certain classes of graphs namely tadpole, book and flower graphs in the context of 3-TEPC labeling.

In this paper, the notion of some algebraic properties of fundamental group of intuitionistic fuzzy topological spaces (IFTSs) are introduced. We give a necessary and sufficient condition for a fundamental group of IFTSs to be abelian, a necessary and sufficient conditions for a subset of fundamental group of IFTSs to be subgroup, a necessary and sufficient condition for a subgroup of fundamental group of IFTSs to be normal and a necessary and sufficient condition for an element to be in a center of fundamental group of IFTSs. We also describe the set of centralizers of an element in a fundamental group of IFTSs and the quotient fundamental group of IFTSs.

In this paper, a coupled fixed point theorem for maps satisfying rational type contractive condition in the perspective of dislocated quasi b-metric space have been formed and the existence and uniqueness of a couple fixed point have been proved. Our result improves and generalizes comparable results in the literature.