In this paper we will give the definition of almost Multiplicative linear functionals on commutative Banach algebras and investigate the closeness of these linear functionals with Multiplicative linear functionals (AMNM property). َSeveral Banach algebra has this property, but not all of them. Due to relationship between the spectrum of an element of a complex Banach algebra and Multiplicative linear functionals. Thus this investigation about almost Multiplicative linear functionals give us some information about the spectrum of an element of a Banach algebra. In this paper We will give a very helpful theorem, for this investigation. Also we will show that, if in two commutative Banach algebras A and B, near any almost Multiplicative linear functional, there is a Multiplicative linear functional, then A×B also has this property and vice versa. We investigate this property for finite product of commutative Banach algebras. Also we will refer to some of the sequential spaces.

We generalize a theorem due to Jarosz, by proving that every almost $n$-Multiplicative linear functional on Banach algebra $A$ is automatically continuous. The relation between almost Multiplicative and almost $n$-Multiplicative linear functional on Banach algebra $A$ is also investigated. Additionally, for commutative Banach algebra $A$, we prove that every almost Jordan homomorphism $\varphi:A\longrightarrow \mathbb{C}$ is an almost $n$-Jordan homomorphism.

I In this paper, we propose an algorithm base on decomposition technique for solving the mixed integer linear Multiplicative-linear bilevel problems. In fact, this algorithm is an application of the algorithm given by G. K. Saharidis et al for the case in which the first level objective function is linear Multiplicative. We use properties of quasi-concave of bilevel programming problems and decompose the initial problem into two subproblems named RMP and SP. The lower and upper bound provided from the RMP and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is less than an arbitrary tolerance. In conclusion, some numerical examples are presented in order to show the efficiency of algorithm.

Our main purpose of this paper is to study the relationship between Multiplicative maps and almost Multiplicative maps between probabilistic normed algebras. We first derive some properties of invertible elements and their relation with Multiplicative maps. Then we show that every complex homomorphism on elements whose probabilistic norm is equal to 1, is bounded. In the following, we give an open problem about the functionally continuous of unital commutative probabilistic Banach algebra. Finally, we prove that every almost Multiplicative map that is not a Multiplicative map is continuous.

In a Multiplicative hyperring, the multiplication is a hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on Multiplicative hyperrings is introduced and some related properties are investigated.

For Banach algebras $\mathcal{A}$ and $\mathcal{B}$, we show that if $\mathfrak{A}=\mathcal{A}\times \mathcal{B}$ is unital, then each bi-Multiplicative mapping from $\mathfrak{A}$ into a semisimple commutative Banach algebra $\mathcal{D}$ is jointly continuous. This conclusion generalizes  a famous result due to$\check{\text{S}}$ilov, concerning the automatic continuity of homomorphisms between Banach algebras. We also prove that every $n$-bi-Multiplicative functionals on $\mathfrak{A}$ is continuous if and only if it is continuous for the case $n=2$.

We extend the notion of approximately Multiplicative to approximately n{Multiplicative maps be-tween locally Multiplicatively convex algebras and study some properties of these maps. We prove that every approximately n{Multiplicative linear functional on a functionally continuous locally mul-tiplicatively convex algebra is continuous. We also study the relationship between approximately Multiplicative linear functionals and approximately n{Multiplicative linear functionals.

functional analysis was an  important development in the viewpoint of mathematicians in examining the behavior of functions as a set (space of functions). The important concept of single space and the distance of its points, invented by ‎Fréchet‎ in the early 20th century, and the fusion of algebraic and topological concepts eventually led to the establishment of this branch of mathematics around 1933. The first Iranian mathematician to make a mark in this field was Heydar Radjavi, who contributed to the field in Iran from 1963 to 1968. Another branch of this field is called nonlinear functional analysis, and Behzad Djafari-Rouhani was the first specialist in this branch who worked in the country from 19‎87‎ onwards. Another key figure in the development and expansion of this branch in the country is Jafar Zafarani. This article, after outlining the history of functional analysis, will provide a brief overview of his contributions to this field. Through a conversation with him, we will delve into his experiences and insights from various aspects of his professional life.