Quaestiones Mathematicae

Analytical solutions of virus propagation model in blockchain networks

Y. Chatibi — 2024-11-29

The main goal of this paper is to find analytical solutions of a system of nonlinear ordinary differential equations arising in the virus propagation in blockchain networks. The presented method reduces the problem to an Abel differential equation of the first kind and solve it directly.

On (k, ℓ)-locating colorings of graphs

Michael A. Henning — 2024-11-29

Let c : V (G) → {1, . . . , ℓ} = [ℓ] be a proper vertex coloring of G and C(i) = {u ∈ V (G): c(u) = i} for i ∈ [ℓ]. The k-color code rk(v|c) of vertex v is the ordered ℓ-tuple (aG(v, C(1)), . . . , aG(v, C(ℓ))) where aG(v, C(i)) = min{k, min{dG(v, x) : x ∈ C(i)}}. If every two vertices have different color codes, then c is a (k, ℓ)-locating coloring of G. The k-locating chromatic number of graph G, denoted by χLk (G), is the smallest integer ℓ such that G has a (k, ℓ)-locating coloring. In this paper, we propose this concept as an extension of diam(G)-locating chromatic number and 2-locating chromatic number which are known as the locating chromatic number, denoted χL(G), and neighbor-locating chromatic number, denoted χLN (G), respectively. In this paper, we give sharp bounds for χLk (G ◦ H) and χL(G ⋄ H) where G ◦ H and G ⋄ H are the corona and edge corona of G and H, respectively. We formulate an integer linear programming model to determine χL2 (G), noting that almost all graphs have diameter 2 and χLk (G) = χL2 (G) for every graph G of diameter 2.

Atomicity of positive monoids

Scott T. Chapman — 2024-11-29

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey many of the recent advances regarding positive monoids, and we provide numerous examples to illustrate the complexity of their atomic and arithmetic structures.

Measure pseudo s-asymptotically ω-periodic solution in distribution for some stochastic differential equations with Stepanov pseudo s-asymptotically ω-periodic coefficients

Mondher Damak — 2024-11-29

In this research paper, a mathematical concept known as the ω-periodic process is discussed, and a new type of processes called the doubly measure pseudo S-asymptotically ω-periodic in distribution process is set forward. Additionally, the properties of these processes are identified and invested to tackle the solution of a stochastic differential equation guided by Brownian motion. The basic objective of the current work resides in corroborating the existence and uniqueness of the solution which is doubly measure pseudo S-asymptotically ω-periodic in distribution.

On the general position number of the k-th power graphs¹

Jing Tian — 2024-11-29

No Abstract

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

M.K. Dauylbayev — 2024-11-29

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Geometric polynomials via a differential operator

Said Taharbouchet — 2024-11-29

In this paper, we use differential operator to present new identities and provide alternative proofs for certain established identities related to geometric polynomials. Additionally, by using the differential operator associated with geometric polynomials, Rolle’s theorem and a theorem of Wang and Yeh, we present some results on the real rootedness of polynomials.

Characterizing some finite groups by the average order

Z. Akhlaghi — 2024-11-29

The average order of a finite group G is denoted by o(G). In this note, we classify groups whose average orders are less than o(S4), where S4 is the symmetric group on four elements. Moreover, we prove that G ∼= S4 if and only if o(G) = o(S4). As a consequence of our results we give a characterization for some finite groups by the average order. In [9, Theorem 1.2], the groups whose average orders are less than o(A4) are classified. It is worth mentioning that to get our results we avoid using the main theorems of [9] and our results leads to reprove those theorems.

Norm and relative uniform convergence

Zili Chen — 2024-11-29

In this paper, we show that if H is a Dedekind complete normed Riesz space with the local Egoroff property, then H is lattice isomorphic to an ideal of an AM-space if and only if xn o−→ 0 for any order bounded norm null sequence in H. Based on this, several characterizations of discrete Banach lattices with order continuous norms are provided. As an application, we find the special regulator for every relative uniform convergent sequence in discrete Banach lattices with order continuous norms.

Total coalitions in graphs

Saeid Alikhani — 2024-11-29

We define a total coalition in a graph G as a pair of disjoint subsets A1, A2 ⊆ A that satisfy the following conditions: (a) neither A1 nor A2 constitutes a total dominating set of G, and (b) A1 ∪ A2 constitutes a total dominating set of G. A total coalition partition of a graph G is a partition Υ = {A1, A2, . . . , Ak} of its vertex set such that no subset of Υ acts as a total dominating set of G, but for every set Ai ∈ Υ, there exists a set Aj ∈ Υ such that Ai and Aj combine to form a total coalition. We define the total coalition number of G as the maximum cardinality of a total coalition partition of G, and we denote it by Ct(G). The purpose of this paper is to begin an investigation into the characteristics of total coalition in graphs.

On positive solutions of the Cauchy problem for doubly nonlocal equations

Sanping Rao — 2024-11-29

No Abstract

Some characterizations of ergodicity in Riesz spaces

Youssef Azouzi — 2024-11-29

No Abstract

Quaestiones Mathematicae

Analytical solutions of virus propagation model in blockchain networks

On (k, ℓ)-locating colorings of graphs

Atomicity of positive monoids

Measure pseudo s-asymptotically ω-periodic solution in distribution for some stochastic differential equations with Stepanov pseudo s-asymptotically ω-periodic coefficients

On the general position number of the k-th power graphs1

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

Geometric polynomials via a differential operator

Characterizing some finite groups by the average order

Norm and relative uniform convergence

Total coalitions in graphs

On positive solutions of the Cauchy problem for doubly nonlocal equations

Some characterizations of ergodicity in Riesz spaces

On the general position number of the k-th power graphs¹