Algebra

2507 Submissions

[3] viXra:2507.0225 [pdf] submitted on 2025-07-31 20:22:16

Relationship Between Consecutive Quadratic Products: A Generalized Algebraic Identity

Authors: Jose Acevedo Jimenez
Comments: 6 Pages. In Spanish (Note by viXra Admin: Please cite listed scientific references)

This paper presents and proves a novel algebraic identity involving quadratic expressions with two parameters, b and c. The identity equates the product of two consecutive quadratic expressions to a structured combination of squares and linear terms of a single quadratic. A brief overview of the concept and history of algebraic identities is provided, followed by a detailed step by-step verification of the inherent symmetry and algebraic elegance of parameterized expressions and highlights potentials educational and analytical applications.
Category: Algebra

[2] viXra:2507.0050 [pdf] submitted on 2025-07-06 21:05:59

The Weighted Core-ep Inverse and Its Associated Pre-Order

Authors: Huanyin Chen, Marjan Sheibani
Comments: 32 Pages.

In this paper, we introduce a new pre-order stemming from the $w$-core-EP inverse in a ring. We characterize this generalized inverse by combing the $w$-core inverse with nilpotent elements. This characterization allows us to explore a new binary relation among $w$-core-EP invertible elements, leveraging the $w$-core pre-order. Using the Pierce matrix for two idempotents as a new tool, we find equivalent conditions for the forward and reverse order laws of $w$-core-EP invertibility. In addition, we extend the *-DMP property to encompass a broader range of cases within the$w$-core-EP pre-order.
Category: Algebra

[1] viXra:2507.0046 [pdf] submitted on 2025-07-06 21:22:24

Revolutionary Inequality Using in General Quartic Equations and Proving Non Exsistence of Real Roots

Authors: Bouazad El Bachir
Comments: 6 Pages. (Note by ai.viXra.org Admin: Author name is required in the article; please cite listed sceintific references)

We present a revolutionary inequality that deterministically excludes real roots in general quartic equations Ax4+Bx3+Cx2+Dx+E=0 (A,E≠0). Our condition: If A > 0 and If E > 0 and if bd ≤ U and if bd ≥ L and if L > UIf A < 0 and If E < 0 and if bd ≥ U and if bd ≤ L and if L < U Hence , L = (4EC−D2 ) /4E U = B2/4A provably guarantees the quartic is globally positive or negative, eliminating real roots without solving the equation. Validated across 1 trillion random quartics (with zero counterexamples), this rule:u2022Outperforms classical discriminants: Computes 1,000× faster than the 256-term quartic discriminant. u2022Stronger guarantees: Ensures definiteness (not just complex roots).u2022Novel foundation: Derived from a sum-of-squares decomposition and residual quadratic analysis.Applications span real-time control systems, cryptographic key validation, and numerical optimization. This work redefines efficiency in polynomial analysis.
Category: Algebra