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[5] viXra:2406.0171 [pdf] submitted on 2024-06-28 20:52:11
Authors: Aziz Arbai, Amina Bellekbir
Comments: 8 Pages.
A new way to teach and to solve problems in complex numbers, as how to use the Moivre Formula, Newton's Binomial and Euler formula for the linearization. We will also share new results like The "Magic Formula" of the second root of any complex number, and the true method to solve the equation of second degree in general case (with complex coefficients) with the explicit formula of the solution and an algorithm to program it.
Category: Algebra
[4] viXra:2406.0155 [pdf] submitted on 2024-06-26 15:22:15
Authors: Huanyin Chen
Comments: 24 Pages.
In this paper, we introduce weighted weak group inverse in a ring with proper involution.This is a natural generalization of weak group inverse for a complex matrix and weighted weak group inverse for a Hilbert operator.We characterize this weighted generalized by using a kind of decomposition involving weighted group inverses and nilpotents. The relations among weighted weak group inverse, weighted Drazin inverse and weighted core-EP inverse are thereby presented.
Category: Algebra
[3] viXra:2406.0137 [pdf] submitted on 2024-06-23 13:41:03
Authors: Huanyin Chen, Yueming Xiang
Comments: 22 Pages.
Recently, Gao, Zuo and Wang introduced the $W$-weighted $m$-weak group inverse for complex matrices which generalizedthe (weighted) core-EP inverse and the WC inverse. The main purpose of this paper is to extend the concept of $W$-weighted $m$-weak group inverse for complex matrices to elements in a Banach *-algebra. This extension is called $w$-weighted $m$-generalized group inverse. We present various properties, presentations of such new weighted generalized inverse. Related (weighted) $m$-generalized core inverses are investigated as well. Many properties of the $W$-weighted $m$-weak group inverse are thereby extended to wider cases.
Category: Algebra
[2] viXra:2406.0024 [pdf] submitted on 2024-06-06 05:40:01
Authors: Abhishek Kumar
Comments: 4 Pages. Keywords : Mathematics in Machine Learning, Statistics, Calculus, Linear Algebra, Probability
Machine learning (ML) is a prominent branch of artificial intelligence (AI) that has drastically transformed various fields by providing sophisticated tools for data analysis and prediction. This paper reviews the pivotal role of mathematics in the development and refinement of machine learning algorithms. The core objective is to illustrate how mathematical principles underpin the processes of training and optimizing ML models, ensuring their effectiveness in recognizing patterns and making autonomous decisions from data.
Category: Algebra
[1] viXra:2406.0017 [pdf] submitted on 2024-06-04 01:48:39
Authors: Shao-Dan Lee
Comments: 4 Pages.
We have constructed an ideal with respect to a subset of binary operations. In this paper, we construct a prime ideal with respect to a nonempty subset of binary operations in an algebra. Let P and Q be two prime ideals with respect to Φ and Ψ, respectively. Then we have that P ∪ Q is a prime ideal if some conditions hold.
Category: Algebra
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