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[4] viXra:2602.0140 [pdf] submitted on 2026-02-23 07:19:50
Authors: Sanjeev Saxena
Comments: 4 Pages.
In this note, it is shown that the Row Echelon Normal Form (RENF), also called the Reduced Row Echelon Form (RREF) of a matrix, is unique. The result is proved from first principles.
Category: Algebra
[3] viXra:2602.0133 [pdf] submitted on 2026-02-21 21:53:57
Authors: Habeeb Mohammed
Comments: 14 Pages.
We investigate how the sequence $f^{(n)}(0)$ acts for a specific class of functions: exponentiated polynomials, $e^{p(x)}$, of which we first look at $e{-x^2}$. This leads us into an textit{infinite dimensional matrix}, which can be analysed via tools from Lie algebra. To generalise this to all polynomials $p(x)$, we define a correspondence between the space of derivatives of $e^{p(x)}$, $cal{F}$, and a general vector space of polynomials, $cal{T}$; we find that the derivative in $cal{F}$ also corresponds to an operator in $cal{T}$. We then utilise Zassenhaus' formula to find how this operator iterates, hence giving us a general formula for the $n$th derivative of $e^{p(x)}$.
Category: Algebra
[2] viXra:2602.0083 [pdf] submitted on 2026-02-15 18:04:00
Authors: Ahcene Ait Saadi
Comments: 4 Pages. (Note by viXra Admin: Please cite listed scientific references!)
This paper explores the properties of para-complex numbers ({R}[j], j^2=1, j ) as the fundamental framework for hyperbolic physical systems. By introducing the zero-divisor identity (1-j)(1+j)=0, we demonstrate a major simplification of Lorentz transformations and a formal unification between the relativistic light cone and the Mach cone in fluid mechanics. This approach aligns with recent research archived on viXra.org, aiming to renew the study of applied hyperbolic algebra.
Category: Algebra
[1] viXra:2602.0065 [pdf] submitted on 2026-02-09 21:28:11
Authors: Lamarr Widmer
Comments: 14 Pages. (Note by viXra Admin: Please cite and list scientific references)
If * is a binary operation on a set S, an element a is an idempotent for * if a*a=a . In this paper, we provide an alternative equivalent definition for idempotents in a ring with unity. This definition facilitates the calculations in several theorems characterizing the idempotents in rings of the form Z_n . This material was developed by the author while teaching a one semester undergraduate class in modern algebra. Some of this material was presented to students in that class and we believe all of it is suitable for students after a basic introduction to rings. We include numerous theorems determining the number of idempotents in Z_n for various factorizations of n . These theorems, along with specific examples and calculations, lead to the eventual general theorem which shows how the number of idempotents in Z_n is determined by the number of prime divisors of n.
Category: Algebra
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