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[8] viXra:2202.0180 [pdf] submitted on 2022-02-28 07:24:50
Authors: Theophilus Agama
Comments: 5 Pages.
In this paper, we study the notion of dominating number of expansions.
Category: Algebra
[7] viXra:2202.0149 [pdf] submitted on 2022-02-23 20:10:52
Authors: Nikos Mantzakouras
Comments: 31 Pages.
The difference between the Beal equation and the Fermat equation is the different exponents of the variables and the method of solving it. As we will show, for the proof of the Beal equation to be complete, Fermat's theorem will be must hold. There are only 10 known solutions and all of them appear with exponent 2. This very fact is proved here using a uniform method. Therefore, Beal's conjecture is true under the above conditions because it accepts that there is no solution if the condition that all exponent values are greater than 2 occurs, the truth of which is proved in Theorem 6, based on the results of Theorem 5. The primary purpose for solving the equation is to see what happens for solving the equation ax + by = cz i.e. for Pythagorean triples of degree 1. This is the generator of the theorems and programs that follow.
Category: Algebra
[6] viXra:2202.0120 [pdf] submitted on 2022-02-18 09:03:26
Authors: Theophilus Agama
Comments: 6 Pages.
In this paper, we study the notion of an index of sub-expansions in an expansion. We prove the index inequality as an application.
Category: Algebra
[5] viXra:2202.0109 [pdf] submitted on 2022-02-16 07:20:31
Authors: Eckhard Hitzer
Comments: 24 Pages. 6 figures, 7 tables.
We study the embedding of octonions in the Clifford geometric algebra for spacetime STA Cl(1,3), as suggested by Anthony Lasenby at AGACSE 2021. As far as possible, we extend the approach to similar octonion embeddings for all three- and four dimensional Clifford geometric algebras Cl(p,q), n = p + q = 3,4. Noticeably, the lack of a quaternionic subalgebra in Cl(2,1), seems to prevent the construction of an octonion embedding in this case, and necessitates a special approach in Cl(2,2). As examples, we present for Cl(3,0) the non-associativity of the octonionic product in terms of multivector grade parts with cyclic symmetry, show how octonion products
and involutions can be combined to make the opposite transition from octonions to the Clifford geometric algebra Cl(3,0), and how octonionic multiplication can be represented with (complex) biquaternions or Pauli matrix algebra.
Category: Algebra
[4] viXra:2202.0107 [pdf] submitted on 2022-02-15 07:38:12
Authors: Theophilus Agama
Comments: 6 Pages.
In this paper, we continue the development of multivariate expansivity theory. We introduce and study the notion of an exact expansion and exploit some applications.
Category: Algebra
[3] viXra:2202.0099 [pdf] submitted on 2022-02-14 09:20:00
Authors: Stephane Breuils, Kanta Tachibana, Eckhard Hitzer
Comments: 43 Pages. Revised (expanded) version of journal article https://doi.org/10.1007/s00006-021-01196-7.
The new applications of Clifford's geometric algebra surveyed
in this paper include kinematics and robotics, computer graphics and
animation, neural networks and pattern recognition, signal and image
processing, applications of versors and orthogonal transformations,
spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.
Category: Algebra
[2] viXra:2202.0045 [pdf] submitted on 2022-02-08 20:05:49
Authors: Philippe Gibone
Comments: 12 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]
We will show that it is possible to find a few parameters that describes several "interesting"
R-Algebras of dimension 4 and 8 (we will recall the very easy and well known 1 and 2 cases).
Category: Algebra
[1] viXra:2202.0023 [pdf] replaced on 2024-02-12 16:18:06
Authors: Richard Wayte
Comments: 7 Pages.
The received Theorem is transformed into a new symmetrical expression, for it to be compared with a parallel all-integer expression. Differences in the configurations of the expressions prove that components of the Theorem cannot all be integers.
Category: Algebra
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