In this work, we aim to characterize solutions of ABA = BAB, including identifying the necessary, and where possible sufficient, conditions under which distinct matrices A and B satisfy this relation. The approach is somewhat analogous to the study of AB = BA, i.e., determining when two distinct matrices commute.

We aimed to develop a conjugate gradient method by reformulating parameters so that iterative optimization techniques perform more effectively. Instead of using a set of basic conjugate gradient formulas, the methodology introduces a new parameter that produces better convergence. The method is applied in Fortran to test how many iterations and how many evaluations of the function are needed, as listed in table 1. The behavior of convergence and the results used for comparison are created with Matplotlib on Python and Ggplot2 on R programming for the chart. We like to compare our method to the proven LS (Liu-Storey) method when checking how effective our proposed method is. We found that the technique provides better results with lower iteration numbers and better convergence speeds for many test cases, proving it can challenge conventional methods in optimizing problems without constraints.

Geometrical objects are not always static, they are subject to changes in real life problems. This work underlines the principles of the topic and gives a systematic reasoning for such problems. The initial geometry, the rules/constraints of deformation and the final deformed geometry are the major steps in treating such problems. Real life example problems are given to exploit the ideas.

Aims/ Objectives: The aim of the article is to provide the proof for use of trits in coding of the signal for Ternary processors The author substantiates the aforementioned use by formula and properties of Ternary Algebra In order to do so an analysis of the incoming signal is done and the wave is presented in the form of 𝑓(𝑥)=𝑎 sin⁡(bx+C)+D By converting this formula we obtain numeric value of the energy conveyed by the standing wave The same numeric value is the mathematical representation of a signal in the Ternary circuits (Pierce, 1973)

A new classification of functions is presented in this work in relation to ordinary differential equations. A function for which the n’th derivative of it can be expressed in terms of linear combinations of lower derivatives and the function itself is defined as the linear differential function where n is the lowest derivative for such a relation to hold. The functions that do not obey the rule are defined as nonlinear differential functions. The properties of the differential functions are discussed through theorems and examples. One elementary application is the method of undetermined coefficients where the non-homogenous function should be a linear differential function. Variable coefficient equations are also treated within the context of linear differential functions. The approximation of nonlinear differential functions in terms of linear differential functions are discussed. An application to the perturbation solutions is given. The ideas and definitions presented in this work will add to the understanding of differential equations and their solutions as functions.

This paper attempts to describe the Fibonacci numbers, their qualities, completeness, and mathematical completeness; the Fibonacci sequence may be seen in a number of stunning natural events; as a result, this paper begins by introducing the Fibonacci sequence and then goes on to discuss some of its properties; the Fibonacci sequence is mathematically complete, as well as having natural and artistic manifestations; the series itself is infinite, and any positive integer may be expressed as the sum of individual Fibonacci numbers; the Fibonacci sequence and golden ratio act as testimonies; the Fibonacci sequence, which has ancient mathematical foundations, continues to fascinate our minds and show subtle patterns in a variety of fields; as we interpret the complexities of these mathematical masterworks, we develop a better understanding of how mathematics and the world around us.

This study examine the significant contribution of Nigeria Stock Market towards the Gross Domestic Product. The main objective of this work is to assess the level of Stock Market stability in Nigeria by applying Dynamic Regression to modeling of Gross Domestic Product (GDP) using the Quarterly Gross Domestic Product at 1990 constant basic prices of Nigeria from the first quarter of 1985 to the third quarter of 2013. Quarterly all Share Index of the Nigerian Stock Exchange from the first quarter of 1985 to the fourth quarter of 2013 and finally Quarterly Market Capitalization of the Nigerian Stock Exchange for the same period. It was observed that the relationship is statistically significant, which allows the stock market to have an impact on the Nigerian Economy. It was concluded that Government and Economic planner should take more advantage of statistical tools in studying the relationship between the GDP movement and the Stock Exchange.

In this study, linear Volterra-Fredholm integral equations are approximatively solved in terms of Lucas polynomials about any point in this study using a practical matrix approach. This technique uses collocation points and Lucas polynomials to transform the aforementioned linear Volterra-Fredholm integral problem into a matrix equation. Lucas coefficients are unknown in the system of linear algebraic equations. With the use of an error estimation, some illustrated examples are also provided. The outcomes demonstrate how effective and practical the suggested methodology is. Code was created in MATLAB to acquire the matrix equations and answers for the chosen issues.

Linear programming is a mathematical method for choosing the best product combination to maximize profit or decrease cost within the limitations of available resources. In this article, we have proposed the linear programming application for profit Maximization of a Biscuit Factory in Bangladesh. For linear programming problems, the Simplex algorithm provides a powerful computational tool, able to provide fast solution to very large-scale application. The optimization principal examined unit cost of production, the selling price and the quantity of different raw materials used in production. A model for the problem was formulated and optimum results derived using software that employed simplex method. Our main goal is to emphasize the uniqueness of linear programming modeling at the business level as an optimization technique, and to encourage manufacturing companies to employ linear programming to determine their optimal profit.

The different methods used in the multiplication of numbers are long method (column method) of multiplication, grid methods and addition methods of multiplication. The utilization of the methods mentioned to solve power (index) of base number (number that multiply itself in a number of times) are horrendous, outrageous, tedious, time consuming and too long to be carried out. This study develops computation of the power of base of number using Kifilideen (matrix, combination and distributive (MCD)) approach. The Kif matrix method of multiplication is a shorter version of the long method (column method) of multiplication. The matrix method provides a straight forward, direct and systematic means of multiplication of digit number that multiply itself repetitively