Journal of Mathematical Analysis and its Contemporary Applications
Volume:4 Issue: 4, Autumn 2022

Let $\mathcal{A},$ $\mathcal{B},$ and $\mathcal{X}$ be complex algebras, $\theta : \mathcal{B} \longrightarrow \mathcal{X}$ be an algebra homomorphism, and let $\mathcal{A}$ be an $\mathcal{X}$-bimodule. We define a product on $\mathcal{A} \times \mathcal{B}$ as $(a_1, b_1)(a_2,b_2) = (a_1 a_2 + a_1 \cdot \theta(b_2) + \theta(b_1) \circ a_2,b_1 b_2)$ for all $(a_1, b_1), (a_2,b_2) \in \mathcal{A} \times \mathcal{B}$ and write $\mathcal{A} \times \mathcal{B}$ with this product by $\mathcal{A} \times_\theta\mathcal{B}$. We shall study some basic properties of $\mathcal{A} \times_\theta \mathcal{B}$. When $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{X}$ are Banach algebras, $\mathcal{A}$ is a Banach $\mathcal{X}$-bimodule, and $\theta$ is a continuous homomorphism with the norm at most $1$, we determine the ideals of $\mathcal{A} \times_\theta \mathcal{B}$ of a certain type, the Gelfand space of this Banach algebra, and the module multipliers of this Banach algebra.

The objectives of this article are to present the notions of anti complex fuzzy subalgebras and anti complex fuzzy ideals of Lie algebras under  $S$-norms and we prove that the level subset of them are also subalgebras and ideals of Lie algebras, respectively. Next, we introduce the union and summation of them and we prove that the union and summation of them are also anti complex fuzzy subalgebras and anti complex fuzzy ideals of Lie algebras under  $S$-norms, respectively.  Finally, we investigate the homomorphic image (pre-image)  of them under Lie homomorphisms.

In this article, the concept of semi-metric spaces is extended to semi-cone metric spaces. Also, some theorems for quasi-fixed points of Banach contractions, Kannan's contractions, and Chatterjea's contractions on semi-cone metric spaces are proved. Moreover, some quasi-fixed point results for different types of contraction mappings on semi-cone metric spaces are derived.

In this paper, we propose a Taylor series expansion method for the second kind of Fredholm integral equation with smooth kernels. This method converts the integral equation into a linear equation system to reduce the amount of computation. We present convergence conditions. Finally, we show the efficiency of the method using some numerical examples.

In this paper, the behavioural analysis of a single-unit subdivision after a complete failure using the Regenerative Point Graphical Technique (RPGT) is discussed. Initially, the unit is fully operational which may have two types of malfunction, one is direct and the second is the partial failure mode. There is one server (fixer) that checks and fixes the unit for each failure. Incomplete failure of the unit cannot be restored to its original capacity. Each repair unit is subject to vandalism if the server reports that the unit cannot be repaired and replaced, following a normal distribution. The vague concept is used to determine unit failure/performance. Considering the levels of failure to be exponential, adjustment values are common and taking into account various possibilities, the system transformation diagram is developed to determine Primary, Second Circuits & Higher Education Foundations and Basic Status. The problem is constructed and resolved using RPGT to determine system parameters. System behaviour is discussed with the help of graphs.

In this work, we have brought in the notion of cyclic contraction into the intuitionistic generalized fuzzy metric spaces. We have set up some common fixed point theorems for these contractions and continuous mappings by considering their weakly commuting nature.