d. ebrahimibagha

In this paper, bipolar inverse fuzzy graphs are introduced. Several operations are made on such graphs and whether these operations result in a bipolar inverse fuzzy graph is investigated. A type of bipolar inverse fuzzy graphs called strong bipolar inverse fuzzy graphs have been studied. The complement graph and dependent threshold of a bipolar inverse fuzzy graph are introduced and it is concluded that the complement of a bipolar inverse fuzzy graph is a strong bipolar inverse fuzzy graph. An inclusion condition regarding threshold graphs corresponding to two bipolar inverse fuzzy graphs with special conditions will result.

In this paper we study the relation between module amenability of $theta$-Lau product $A×_theta B$ and that of Banach algebras $A, B$. We also discuss module biprojectivity of $A×theta B$. As a consequent we will see that for an inverse semigroup $S$, $l^1(S)×_theta l^1(S)$ is module amenable if and only if $S$ is amenable.

In this paper, we present the numerical solution of ordinary differential equations (or SDEs), from each order especially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multiplicative noises). Making stochastic differential equations system from this equation, it could be approximated or solved numerically by different numerical methods. In the case of linear stochastic differential equations system by Computing fundamental matrix of this system, it could be calculated based on the exact solution of this system. Finally, this stochastic equation is solved by numerically method like Euler-Maruyama and Milstein. Also its Asymptotic stability and statistical concepts like expectation and variance of solutions are discussed.

Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with the set of idempotents $E_S$, the module extension Banach algebra $S=l^1(E_S)oplus l^1(S)$ is $triangle$-amenable as a $l^1(E_S)$-module if and only if $l^1(E_S)$ is amenable as Banach algebra.