Journal of Prime Research in Mathematics

Abstract: A Monad graph is a graph Γ in which each of its vertices belongs to a finite group G and connects with its image under the action of a linear map f. This kind of graph was introduced by V. Arnold in 2003. In this paper, we compute the Monad graphs in which G is isomorphic to a cyclic group C_n of order n and f the fifth power function, i.e. f(g) = g⁵. Furthermore, some algebraic and dynamical properties of the studied Monad graphs are obtained. The proofs of our results are based on various tools and results with regard to the fields of number theory, algebra and graph theory.

Abstract: In this paper, we introduce the concept of morphisms between two grey sets and defined a new category, namely, GSet, of grey sets and grey morphisms. We investigate some categorical notions such product, coproduct, pullback, and pushout. Additionally, we study equalizer and coequalizer for pairs of grey morphisms and show that any grey set has an injective hull.

Abstract: In this paper, we propose a new type of module by focusing on the second cosingular submodule of a module. We define a module M as weak T-dual Rickart if, for any homomorphism φ ∈ EndR(M), the submodule φ(z̄²(M)) lies above a direct summand of M. We prove that this property is inherited by direct summands of M. We also introduce weak T-dual Baer modules and provide a complete characterization of such modules where the second cosingular submodule is a direct summand. Furthermore, we present a characterization of (semi)perfect rings in which every (finitely generated) module is weak T-dual Rickart.

Abstract: Consider the following general three dimensional system of difference equations
\begin{cases}
x_{n+1}=f^{-1}\left(\frac{g(y_{n} )g(y_{n-1} )(f(x_{n-1} ))^{p}}{f(x_{n} )[a_{n} (g(y_{n-2} ))^{q} +b_{n} g(y_{n} )g(y_{n-1} )]}\right),\\
y_{n+1}=g^{-1} \ \left(\frac{h(z_{n} )h(z_{n−1} )(g(y_{n−1} ))^{q}}{g(y_{n} )[c_{n} (h(z_{n−2} ))^{r} +d_{n} h(z_{n} )h(z_{n−1} )]}\right),\\
\ z_{n+1} =h^{−1} \ \ \left(\frac{f(x_{n} )f(x_{n−1} )(h(z_{n−1} ))^{r}}{h(z_{n} )[s_{n} (f(x_{n−2} ))^{p} +t_{n} f(x_{n} )f(x_{n−1} )]}\right),\end{cases}
where⁢ 𝑛∈ℕ0,𝑝,𝑞,𝑟∈ℕ,𝑓,𝑔,ℎ :D→ℝ⁢ are continuous one-to-one functions on D⊆ℝ, the coefficients ⁡(𝑎𝑛)⁢𝑛∈ℕ0,(𝑏𝑛)⁢𝑛∈ℕ0⁡(𝑐𝑛)⁢𝑛∈ℕ0,(𝑑𝑛)⁢𝑛∈ℕ0,(𝑠𝑛)⁢𝑛∈ℕ0,(𝑡𝑛)⁢𝑛∈ℕ0⁢ are non-zero real numbers and the initial values 𝑥−𝑖,𝑦−𝑖,𝑧−𝑖,𝑖=0,1,2, are real numbers. We will give explicit formulas for well-defined solutions of the aforementioned system in both variable and constant cases of the coefficients. As an application, we will deduce the formulas of the solutions of the particular system obtained from the general one by taking 𝑓⁡(𝑥)=𝑔⁡(𝑥)=ℎ⁡(𝑥)=𝑥.

Abstract: The resolving set for connected graphs has become one of the most important concept due to its applicability in networking, robotics and computer sciences. Let G be a simple and connected graph, an ordered-subset B of V (G) is called resolving set of G, if every distinct vertex of G have different metric code w.r.t B. Smallest resolving set of G is known as basis of G and size of basis set is called as metric dimension(MD) of graph G. A resolving set B′ of G is known as fault-tolerant resolving set(FTRS), ifB′\{v} is also resolving set, ∀ v ϵ B′. Such set B′ with smallest size is termed as fault-tolerant metric basis and the cardinality of this set is called fault-tolerant metric dimension(FTMD) of graph G. A FTMD set B′ for which the system failure at vertex location v of any station still provide us a resolving set. In this article, we have provided the MD and FTMD for triangular benzenoid system and hourglass benzenoid system.

Abstract: The forced SIR system’s synchronization is the main subject of this study. Through the use of several control techniques, including active control (AC), active backstepping control (ABC), adaptive control (AdC), and sliding mode control (SMC). We redesigned the three-dimensional system to be autonomous and made it four-dimensional to ease numerical computations. The system’s phase portrait, Lyapunov exponent graph, and bifurcation diagram are used to analyze its dynamic properties through numerical simulation. The effectiveness of the AC, SMC, ABC, and AdC approaches is examined using dynamical error and necessary control inputs. By contrasting the integral square error with the required control energy measurements, the best control for synchronization is found.