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Objective of this writing is to present excellent applications of the theory of matrices in various fields. We have given some illustrative examples of uses of matrix algebra in Cryptography, Vector mechanics, Linear dependence of functions and Chemical Sciences. It is fair to say that almost all branches of STEM(Science-Technology-Engineering-Mathematics) are widely using the matrix algebra tools. Hence in this article we(the authors) have made an attempt to bring them to the notice of young researchers. [ABSTRACT FROM AUTHOR]

Extension field is the most fundamental concept in Algebraic Number Theory. It plays a vital role while examining the roots of a polynomial equations through Galois Theory. Moreover field extensions are widely used in Algebraic Geometry. This article depicts FIELD EXTENSION THEORY and ALGEBRAIC EXTENSIONS in a recreational way. As this article is written in the current year 2023 and might be published in the year 2024 we have chosen those year's numbers 2023 and 2024 and made an attempt to demonstrate the central ideas and fundamental concepts of Galois Theory in an elegant apprach. [ABSTRACT FROM AUTHOR]

The research article explores on applications of Linear Algebra in various branches of science namely Space Geometry, Chemical Sciences, Optimization and Cryptography. Though one is familiar with the concepts of Linear Algebra, real-world applications of it should be understood. In this view the authors are believing that they are successful in presenting some wonderful applications of Linear Algebra. [ABSTRACT FROM AUTHOR]

Fixed power sum of positive integers is a well-known problem in number theory. Here we have given a brief history about it by mentioning the contributions of some eminent mathematicians. An internal relation between Σλ and Σλ3 has been traced in an geometric approach and the authors are strongly believing that this geometric approach can give a generalized result for Σλn. Pascal's recurrence relation has been proposed with illustrative examples. Moreover a recurrence relationship is presented by the authors using binomial theorem and they are concluding that any fixed power sum can be easily carried out with this relationship. More importantly by applying binomial theorem we have proved that any fixed power sum is a unique polynomial over rational numbers. Finally an explicit formula has been proposed and this can give any power sum formula. [ABSTRACT FROM AUTHOR]

In this paper we have presented various generalized results on power summation with proofs and examples. All these results are given is recurrence relations so that one can get any fixed power sum starting with σ n Backward Difference operator is employed in order to get the power sum. A lemma, we proposed here, is a significant one as it can give σ n,σ n2, σ n3, .... Moreover we have discussed the applications of Binomial Coefficients in computing the desired result. [ABSTRACT FROM AUTHOR]

This research article explores on the summation of fixed positive integral powers of first m-positive integers. Authors are strongly believing that the following discussion depicts a method fromwhich one can easily compute the summation of fixed positive integral powers of first m-natural numbers. Furthermore they have given answers to two interesting questions in the research field of analytical number theory namely: Is the sum of fixed positive integral powers of first m-positive integers coincide with a polynomial?and Is such polynomial unique?. Some principles of linear algebra have been incorporated in this article to extract very interesting results regarding this summation. The uniqueness of the polynomial has been proved using Crammer's rule also. Moreover a system of linear non homogeneous equations proposed here which help us to derive the formulas for the summations Σ n, Σ n2 , Σ n3....... [ABSTRACT FROM AUTHOR]

This research article mainly focuses on Kaprekar constants named after Indian recreational mathematician Dattatreya Ramchandra Kaprekar. Kaprekar invented a mysterious property of the number 6174 in 1949. He depicted that the number 6174 is reached in the limit as one continuously subtract the smallest number from the largest number where these numbers are taken from a set of four digited numbers in which all four digits are not identical. The main task of this paper is to propose an analytical method for searching Kaprekar constants. In this research the 3-digited Kaprekar constant 495 is traced. The mystery behind the magic of the numbers 495and 6174 respectively known as three and four digit Kaprekar constants has been chased here and the authors proposed a path by which one can hunt the Kaprekar constants of any order. [ABSTRACT FROM AUTHOR]

The purpose of this discourse is to describe the estimation methods of COBB-DOUGLAS production functional model thrash cost function and with multiplicative and additive errors. Nonlinear regression models have been a subject of an intensive investigation in select years. It is no exaggeration to say that in Econometric analysis, the COBB-DOUGLAS and constant Elasticity of substitution (CES) functions are the most frequently used non-linear type of special functions. Mohammad Zakir Hussain et al, in 2010, in their research article discussed the use of Cobb-Douglas production model on some selected manufacturing industries. Md.Moyazzem Hossain et al in 2015, in their research paper estimated the parameters of Cobb-Douglas production function with additive errors and multiplicative errors for some selected manufacturing industries and suggested the most suitable Cobb-Douglas production function to forecast the production process for some selected manufacturing industries for developing countries. Besides they investigated the efficiency of both capital and labor elasticity of the two mentioned form of Cobb-Douglas production function. In 1975, DE-Min Vu in his research paper derived the exact distribution of the indirect least squares estimators of the coefficients of the Cobb-Douglas production function within the content of a stochastic production model of Marshak-Andrews type. In 2017, RVSSN Rao et al, in their research paper, presented various estimation methods for estimating parameters of Cobb-Douglas production functional model. [ABSTRACT FROM AUTHOR]

The main objective of this study is to propose the generalized least squares estimation of stochastic linear regression model with non-spherical errors. In addition to this work the problems of stochastic linear regression model have been presented here. Generalized least squares (GLS) estimators have a wide number of applications in dealing with the inferential problems of stochastic linear regression models especially in the case of the problems of Heteroscedastic errors and Auto correlated errors. NimetOzbayetal [1], in 2018, in their paper proposed the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. In 1980 P.A.V.BSwamy etc.al [2] proposed a linear regression model where the coefficient vector is a weakly stationary multivariate stochastic process and that model provider a convenient representation of a general class of non-stationary processes. They proposed prediction and estimation methods which are linear and easy to compute. Danjiang He et.al [3] in 2014, in their paper proposed a new estimator to combat the multicollinearity in the linear model when there are stochastic linear restrictions on the regression coefficients and the new estimator arcs constructed by combine’s the ordinary mixed estimator (OME) and the principal components regression (PCR) estimator which is called the stochastic restricted principal components (SRPC) regression estimator. In 2014, Shuling Wang et al [9], in their paper proposed some diagnostic methods in stochastic restricted linear regression models. Gil Ganjalez et al [10] in 2007, in their paper derived the least squares estimators for the simple estimators for the simple linear regression model and examined them from a theoretical perspective. [ABSTRACT FROM AUTHOR]