Algebra

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Recent submissions

Any replacements are listed farther down

[301] viXra:1905.0379 [pdf] submitted on 2019-05-19 06:10:10

Refutation of Lattice Effect Algebra

Authors: Colin James III
Comments: 1 Page. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com.

A seminal definition of lattice effect algebra is not tautologous. This refutes lattice effect and lattice pseudoeffect algebras along with the chain effect of quasiresiduation. The conjectures form a non tautologous fragment of the universal logic VŁ4.
Category: Algebra

[300] viXra:1905.0361 [pdf] submitted on 2019-05-19 16:23:26

Expansion of Terms Squared, Square of a Binomial, Trinomial, Tetranomial and Pentanomial.

Authors: Zeolla Gabriel Martín
Comments: 12 Pages.

This document develops and demonstrates the discovery of a new square potentiation algorithm that works absolutely with all the numbers using the formula of the square of a binomial, trinomial, tetranomial and pentanomial.
Category: Algebra

[299] viXra:1905.0122 [pdf] submitted on 2019-05-07 09:45:28

The Burnside Q-Algebras of a Monoid

Authors: Pierre-Yves Gaillard
Comments: 4 Pages.

To each monoid M we attach an inclusion A --> B of Q-algebras, and ask: Is B flat over A? If our monoid M is a group, A is von Neumann regular, and the answer is trivially Yes in this case.
Category: Algebra

[298] viXra:1904.0299 [pdf] submitted on 2019-04-15 08:25:32

Sobre Una Ecuación Polinomial de Grado Nueve

Authors: Edgar Valdebenito
Comments: 4 Pages.

En esta nota se muestra una raíz real de una ecuación polinomial de grado nueve.
Category: Algebra

[297] viXra:1904.0024 [pdf] submitted on 2019-04-01 07:27:29

Algunas Relaciones Del Tipo Arcotangente

Authors: Edgar Valdebenito
Comments: 3 Pages.

En esta nota se muestran algunas relaciones del tipo arcotangente.
Category: Algebra

[296] viXra:1903.0560 [pdf] submitted on 2019-03-31 22:49:35

Direct Sum Decomposition of a Linear Vector Space

Authors: Anamitra Palit
Comments: 5 Pages.

The direct sum decomposition of a vector space has been explored to bring out a conflicting feature in the theory. We decompose a vector space using two subspaces. Keeping one subspace fixed we endeavor to replace the other by one which is not equal to the replaced subspace. Proceeding from such an effort we bring out the conflict. From certain considerations it is not possible to work out the replacement with an unequal subspace. From alternative considerations an unequal replacement is possible.
Category: Algebra

[295] viXra:1903.0367 [pdf] submitted on 2019-03-21 04:32:57

The Klein Four-Group

Authors: Volker W. Thürey
Comments: 3 Pages.

We describe alternative ways to present the famous Klein four-group
Category: Algebra

[294] viXra:1903.0099 [pdf] submitted on 2019-03-05 06:31:22

Finite Faithful G-Sets Are Asymptotically Free

Authors: Pierre-Yves Gaillard
Comments: 3 Pages.

Let G be a finite nontrivial group, let X be a finite faithful G-set, let P^i(X) be the i-th power set of X, let n(i) be the number of points of P^i(X), let m(i) be the number of points of P^i(X) with non-trivial stabilizer, let k be the number of prime order subgroups of G, and set E(j):=2^j for any integer j. We prove that n(i)/m(i) is at least E(n(i-1)/4)/k for i>1.
Category: Algebra

[293] viXra:1902.0116 [pdf] submitted on 2019-02-07 01:05:38

Полное доказательство великой теоремы Ферма методом деления

Authors: Ведерников Сергей Иванович
Comments: 11 Pages.

Простое доказательство инструментами элементарной алгебры.
Category: Algebra

[292] viXra:1901.0377 [pdf] submitted on 2019-01-26 00:54:05

A Simple Introduction & Suggestion to Using Nested Relational Algebra Theory,Data Processing & Related Concepts in the Context of Protein Folding Mechanisms/Metabolomics/Other Bio-informatics Applications.

Authors: Nirmal Tej Kumar
Comments: 2 Pages. Short Communication & Technical Notes

A Simple Introduction & Suggestion to Using Nested Relational Algebra Theory,Data Processing & Related Concepts in the Context of Protein Folding Mechanisms/Metabolomics/Other Bio-informatics Applications.
Category: Algebra

[291] viXra:1901.0340 [pdf] submitted on 2019-01-23 00:37:15

Доказательство гипотезы Эндрю Била

Authors: Ведерников Сергей Иванович
Comments: 7 Pages. Научный журнал "Интернаука" №48(82), 2018.

Доказательство гипотезы Била в контексте "Полного доказательства великой теоремы Ферма методом деления".
Category: Algebra

[290] viXra:1901.0306 [pdf] submitted on 2019-01-20 22:03:16

Refutation of Heyting Algebra

Authors: Colin James III
Comments: 4 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only at: info@ersatz-systems dot com. See website ersatz-systems.com . (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

Using the Lindenbaum method, we show pseudo-complementation is not tautologous along with its eight properties. This refutes Heyting algebra. Based thereon, what follows is the Gödel n-valued matrix logic is refuted and the derivative intuitionistic propositional logic.
Category: Algebra

[289] viXra:1901.0246 [pdf] submitted on 2019-01-16 21:58:12

Construction of Multivector Inverse for Clif Ford Algebras Over 2m+1-Dimensional Vector Spaces from Multivector Inverse for Clifford Algebras Over 2m-Dimensional Vector Spaces

Authors: Eckhard Hitzer, Stephen J. Sangwine
Comments: Advances of Applied Clifford Algebras, Vol. 29, article #29, 22 pages, First Online: 19 February 2019. DOI: 10.1007/s00006-019-0942-7.

Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space R^{p',q'), n' = p' +q' = 2m, we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over R^{p,q}, n = p+q = p'+q'+1 = 2m+1. Explicit examples are provided for dimensions n' = 2,4,6, and the resulting inverses for n = n' +1 = 3,5,7. The general result for n = 7 appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p,q), n = p + q = 7, only involving a single addition of multivector products in forming the determinant.
Category: Algebra

[288] viXra:1901.0142 [pdf] submitted on 2019-01-10 08:52:27

Classification Des Formes Quadratiques

Authors: BOUCHOUAT El Mehdi
Comments: 23 Pages.

La théorie des formes quadratiques est très vaste et dans ce mémoire je n'en mentionne qu'une très petite partie. Dans la première section, j'introduis les formes quadratiques et leurs formes bilinéaires associées et je présente quelques résultats généraux. La deuxième section est consacrée à la classification des formes quadratiques sur $\mathbb{C}$, $\mathbb{R}$, et sur les corps finis $\mathbb{F}_{q}$, et dans la troisième section je me concentre sur l'une des applications de la classification des formes quadratiques : La loi de réciprocité quadratique.
Category: Algebra

[287] viXra:1901.0128 [pdf] submitted on 2019-01-09 09:14:26

The Modified Clifford Algebra

Authors: Antoine Balan
Comments: 1 page, written in english

We propose here a modification of the Clifford algebra with relations using three vectors instead of two.
Category: Algebra

[286] viXra:1812.0203 [pdf] submitted on 2018-12-12 00:03:23

Review on Rationality Problems of Algebraic K-Tori

Authors: Youngjin Bae
Comments: 12 Pages.

Rationality problems of algebraic $k-tori$ is closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic $k-tori$ can be identified as an invariant field under a group action and that a $k-tori$ is rational if and only if its function field is rational over $k$. We also introduce character group of $k-tori$ and numerical approach to determine rationality of $k-tori$.
Category: Algebra

[285] viXra:1812.0039 [pdf] submitted on 2018-12-02 06:25:19

Refutation of Kent Algebras on Rough Set Concept Analysis

Authors: Colin James III
Comments: 1 Page. © Copyright 2018 by Colin James III All rights reserved. Respond to the author by email at: info@ersatz-systems dot com.

We use modal logic to evaluate definitions for Kent algebras, as presented for rough set concept analysis. Some definitions are not tautologous, hence refuting Kent algebras on rough sets.
Category: Algebra

[284] viXra:1811.0441 [pdf] submitted on 2018-11-27 21:36:42

An Insight into Higher Order Logic (HOL) based Ontology NeuroInformatics Framework by Considering Ontology Oriented Concepts & Language/s based on HOL/Grobner Bases/Scala/Jikes RVM/JVMTechnologies/IoT Computing Environments.

Authors: Nirmal Tej Kumar
Comments: 7 Pages. Short Communication & Technical Notes

“Gröbner Bases Theory” & Ontology could be implemented as explained - An Insight into Higher Order Logic (HOL) based Ontology Neuroinformatics Framework by Considering Ontology Oriented Concepts & Language/s based on HOL/Grobner Bases/Scala/Jikes RVM/JVM Technologies/IoT Computing Environments.
Category: Algebra

[283] viXra:1811.0310 [pdf] submitted on 2018-11-20 23:28:02

Generalized Definition of Division in Any Field

Authors: Hiroshi Okumura
Comments: 2 Pages.

A historical definition of division by zero is reconsidered.
Category: Algebra

[282] viXra:1811.0283 [pdf] submitted on 2018-11-18 20:46:54

To Divide by Zero is to Multiply by Zero

Authors: Hiroshi Okumura
Comments: 1 Page.

A remark of the definition of division by zero is given.
Category: Algebra

[281] viXra:1811.0236 [pdf] submitted on 2018-11-15 19:52:46

Splitting of Quasi-Definite Linear System Maintains Inertia

Authors: Martin Neuenhofen
Comments: 4 Pages.

We show that there is a variety of Schur complements that yield a decoupling of a quasi-definite linear system into two quasi-definite linear systems of half the size each. Splitting of linear systems of equations via Schur complements is widely used to reduce the size of a linear system of equations. Quasi-definite linear systems arise in a variety of computational engineering applications.
Category: Algebra

[280] viXra:1811.0144 [pdf] submitted on 2018-11-10 00:56:54

The Importance of Quaternions & Rotational Systems in the Context of Cryo-EM Image Processing – A Simple Suggestion On Using HOL/JVM/JikesRVM/Image J Based Computing Environments.

Authors: Nirmal Tej Kumar
Comments: 3 Pages. Short Communication & Technical Notes

As the title suggests it is our sincere desire to explore “Quaternions & Rotational Systems” in the highly promising domains of Cryo-EM Image Processing to probe the frontiers of Nano-Bio Systems.
Category: Algebra

[279] viXra:1810.0381 [pdf] submitted on 2018-10-24 01:07:02

Deterministic Method for Special Exponential Equations

Authors: Obiwulu Solomon
Comments: 26 Pages.

In this note, some mathematical equations where solved using a modified approach that introduces logarithm with its rules as well as presented as a certain determinant. While some ideas and theories presented in this note could generate issues of dispute, yet the progressive orderliness and agreement in the method cannot easily be set aside. Diverse equations were developed and conveniently solved by the proposed model. And this modification is called Determinant Method
Category: Algebra

[278] viXra:1810.0009 [pdf] submitted on 2018-10-01 09:11:15

The Complex Clifford Algebra

Authors: Antoine Balan
Comments: 2 pages, written in english

We define here a complex Clifford algebra by help of two intertwined (by a Heisenberg algebra) usual Clifford algebras. We deduce two Dirac operators.
Category: Algebra

[277] viXra:1809.0485 [pdf] submitted on 2018-09-23 23:17:37

On The Non-Real Nature of x.0 (x \in R_{\ne 0}): The Set of Null Imaginary Numbers $\nullset$

Authors: Saulo Queiroz
Comments: 6 Pages.

In this letter we discuss the inconsistencies of $0/0\cdot x=y$, $x,y\in\real_{\ne 0}$ from the perspective of the zero property multiplication (ZPM) $x\cdot 0 = y\cdot 0$ on $\real$. We axiomatize $x\cdot 0$ as a number $\inull(x)$ that has a real part $\Re(\inull(x))=0$ but indeed is not real. From this we define the set of null imaginary numbers $\nullset$ as $\{\inull(x)|\forall x\in\real_{\ne 0}\} \cup \{0\}$. Based on the definition of uniqueness (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$ and the null division (i.e., $\inull(x)/0=x$) we show the elementar algebra of $\nullset$. Hence, under the condition of existence of $\nullset$, we show that $0/0=1$ does cause the logic trivialism of mathematic.
Category: Algebra

[276] viXra:1807.0240 [pdf] submitted on 2018-07-12 08:50:24

Some Finite Series and Their Application

Authors: Saikat sarkar
Comments: 4 Pages.

This is only for maths students
Category: Algebra

[275] viXra:1807.0131 [pdf] submitted on 2018-07-05 07:07:06

The Upper Bound of Composition Series

Authors: Abhijit Bhattacharjee
Comments: 9 Pages. The paper was submitted to journal of combinatorial theory a, after referee 's review they told me to submit it algebra related journal.

The upper bound of composition series for finite group is obtained .
Category: Algebra

[274] viXra:1807.0091 [pdf] submitted on 2018-07-03 11:10:16

Triple Conformal Geometric Algebra for Cubic Plane Curves (long CGI2017/ENGAGE2017 paper in SI of MMA)

Authors: Robert B. Easter, Eckhard Hitzer
Comments: 20 pages. Revision, 3 July 2018, with corrections and improvements to the published version, 18 Sep 2017 DOI:10.1002/mma.4597, in MMA 41(11)4088-4105, 30 July 2018, Special Issue: ENGAGE. 9 tables, 4 figures, 28 references.

The Triple Conformal Geometric Algebra (TCGA) for the Euclidean R^2-plane extends CGA as the product of three orthogonal CGAs, and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3-vectors that linearize the representation of non-linear curves, and the entities are inner product null spaces (IPNS) with respect to all points on the represented curves. Each IPNS entity also has a dual geometric outer product null space (OPNS) form. Orthogonal or conformal (angle-preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and, by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA, also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line or circle are possible. TCGA defines commutator-based differential operators in the coordinate directions that can be combined to yield a general n-directional derivative.
Category: Algebra

[273] viXra:1806.0467 [pdf] submitted on 2018-06-30 09:35:47

Clifford Algebras :New Results

Authors: Jean Claude Dutailly
Comments: 28 Pages.

The purpose of the paper is to present new results (exponential, real structure, Cartan algebra,...) but, as the definitions are sill varying with the authors, the paper covers all the domain, and can be read as a comprehensive presentation of Clifford algebras.
Category: Algebra

[272] viXra:1806.0430 [pdf] submitted on 2018-06-29 03:36:32

Note on Mathematical Inequality.

Authors: Saikat sarkar
Comments: 3 Pages.

This artical has been prepared for basic inequality concept.
Category: Algebra

[271] viXra:1806.0250 [pdf] submitted on 2018-06-16 20:42:30

The Pagerank Algorithm: Theory & Implementation in Scilab

Authors: Ayoub ABRACH, El Mehdi BOUCHOUAT
Comments: 27 Pages.

Search engines are huge power factors on the Web, guiding people to information and services. Google is the most successful search engine in recent years,his research results are very complete and precise. When Google was an early research project at Stanford, several articles have been written describing the underlying algorithms. The dominant algorithm has been called PageRank and is still the key to providing accurate rankings for search results. A key feature of web search engines is sorting results associated with a query in order of importance or relevance. We present a model allowing to define a quantification of this concept (Pagerank) a priori fuzzy and elements of formalization for the numerical resolution of the problem. We begin with a natural first approach unsatisfactory in some cases. A refinement of the algorithm is introduced to improve the results.
Category: Algebra

[270] viXra:1805.0528 [pdf] submitted on 2018-05-31 00:49:48

The Matricial Clifford Algebras

Authors: Antoine Balan
Comments: 1 page, written in french

We introduce here the notion of matricial Clifford algebras with help of the product of matrices and the tensor product.
Category: Algebra

[269] viXra:1805.0355 [pdf] submitted on 2018-05-20 05:12:52

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 8 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 3 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[268] viXra:1804.0093 [pdf] submitted on 2018-04-06 07:41:06

None Complex Numbers

Authors: Said Amharech
Comments: 5 Pages. the work is written in french

This work is dealing with something that we’re not sure it exists, but it’s just an attempt to solve some problems in algebra, it gives a general idea about the complexe numbers we got from the great mathematicians of all times, I believe that these complexe numbers we notice in algebra, calculus or even quantum physics are not enough, in general, i think there is another infinite dimension of numbers, and real or complexe numbers are just a simple projection of the complexity of that world. As a method of work instead of adding a function of rotation pi over two as we did with the real line to expand the complexe plan, i’ve thought to make a function of translation… But the importance in here is that the function we need is not a linear function to solve some kind of problems like for example dividing over zero, and as you may notice if the function isn’t linear so that the neutral element of the complexe plan for the additive law which it the trivial zero we know is completely different of the neutral element of the new mother group. at the end we can find some roots easily of riemann’s zeta function but it is not a complexe roots.
Category: Algebra

[267] viXra:1804.0003 [pdf] submitted on 2018-04-01 04:40:39

About the Hamilton Numbers

Authors: Antoine Balan
Comments: 4 pages, written in french

We introduce here some algebraic theory about the Hamilton numbers and develop a quaternionic geometry of fiber bundles.
Category: Algebra

[266] viXra:1802.0294 [pdf] submitted on 2018-02-21 10:54:53

Solution of a High-School Algebra Problem to Illustrate the Use of Elementary Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 5 Pages.

This document is the first in what is intended to be a collection of solutions of high-school-level problems via Geometric Algebra (GA). GA is very much "overpowered" for such problems, but students at that level who plan to go into more-advanced math and science courses will benefit from seeing how to "translate" basic problems into GA terms, and to then solve them using GA identities and common techniques.
Category: Algebra

[265] viXra:1802.0096 [pdf] submitted on 2018-02-08 06:48:35

Solution to the Problem Pmo33.5. Problema Del Duelo Matemático 08 (Olomouc – Chorzow Graz).

Authors: Jesús Álvarez Lobo
Comments: 3 Pages. Spanish.

Solution to the problem PMO33.5. Problema del Duelo Matemático 08 (Olomouc – Chorzow - Graz). Let a, b, c in ℝ. Prove that V = 4(a² + b² + c² ) - (a + b)² - (b + c)² - (c + a)² >= 0, and determine all values of a, b, c for which V = 0.
Category: Algebra

[264] viXra:1802.0022 [pdf] submitted on 2018-02-02 16:54:13

Discarding Algorithm for Rational Roots of Integer Polynomials (DARRIP).

Authors: Jesús Álvarez Lobo
Comments: 20 Pages.

The algorithm presented here is to be applied to polynomials whose independent term has many divisors. This type of polynomials can be hostile to the search for their integer roots, either because they do not have them, or because the first tests performed have not been fortunate. This algorithm was first published in Revista Escolar de la Olimpíada Iberoamericana de Matemática, Number 19 (July - August 2005). ISSN – 1698-277X, in Spanish, with the title ALGORITMO DE DESCARTE DE RAÍCES ENTERAS DE POLINOMIOS. When making this English translation 12 years later, some erratum has been corrected and when observing from the perspective of time that some passages were somewhat obscure, they have been rewritten trying to make them more intelligible. The algorithm is based on three properties of divisibility of integer polynomials, which, astutely implemented, define a very compact systematic that can simplify significantly the exhaustive search of integer roots and rational roots. Although there are many other methods for discarding roots, for example, those based on bounding rules, which sometimes drastically reduce the search interval, for the sake of simplicity, they will not be considered here. The study presented here could be useful to almost all the young people of the planet, since at some stage of their academic training they will have to solve polynomial equations with integer coefficients, looking for rational solutions, integer or fractional. The author thinks that DARRIP's algorithm should be incorporated into the curricula of all the elementary study centers over the world.
Category: Algebra

[263] viXra:1801.0106 [pdf] submitted on 2018-01-09 08:48:03

A Simpler Classification Paradigm for Finite Simple Groups and an Application to the Riemann Hypothesis

Authors: A.Polorovskii
Comments: 2 Pages.

In this paper we propose a new system of classification that greatly simplifies the task of classifying (or setifying) all finite simple groups (Hereafter referred to as FSGs.) We propose classification of FSGs by identifying each group with the equivalence class of certain groups up to isomorphism. Furthermore, it is shown that every FSG belongs to at least one of the equivalence classes herein. Using our new classification, the Generalized Riemann Hypothesis is proven.
Category: Algebra

[262] viXra:1712.0575 [pdf] submitted on 2017-12-24 00:18:53

Approximate A Slice of Pi Essay

Authors: Cres Huang
Comments: Pages.

A simple way of approximating π by slice.
Category: Algebra

Replacements of recent Submissions

[44] viXra:1812.0203 [pdf] replaced on 2018-12-12 10:43:42

Review on Rationality Problems of Algebraic K-Tori

Authors: Youngjin Bae
Comments: 12 Pages.

Rationality problems of algebraic k-tori are closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic k-tori can be identified as an invariant field under a group action and that a k-tori is rational if and only if its function field is rational over k. We also introduce character group of k-tori and numerical approach to determine rationality of k-tori.
Category: Algebra

[43] viXra:1811.0310 [pdf] replaced on 2018-11-25 06:15:59

A General Definition of Division in a Field

Authors: Hiroshi Okumura
Comments: 2 Pages.

The historical definition of division by zero given by Brahmagupta is correct.
Category: Algebra

[42] viXra:1811.0310 [pdf] replaced on 2018-11-21 06:24:15

A General Definition of Division in a Field

Authors: Hiroshi Okumura
Comments: 2 Pages.

A historical definition of division by zero is reconsidered.
Category: Algebra

[41] viXra:1809.0485 [pdf] replaced on 2018-10-23 08:04:46

On The Non-Real Nature of x.0 (x in R*): The Set of Null Imaginary Numbers

Authors: Saulo Queiroz
Comments: 7 Pages.

In this letter we discuss the inconsistencies of $0/0\cdot x=y$, $x,y\in\real_{\ne 0}$ from the perspective of the zero property multiplication (ZPM) $x\cdot 0 = y\cdot 0$ on $\real$. We axiomatize $x\cdot 0$ as a number $\inull(x)$ that has a real part $\Re(\inull(x))=0$ but indeed is not real. From this we define the set of null imaginary numbers $\nullset$ as $\{\inull(x)|\forall x\in\real_{\ne 0}\} \cup \{0\}$. We present the elementary algebra on $\nullset$ based on the definitions of uniqueness (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$) and the null division (i.e., $\inull(x)/0=x\ne 0$). Also, \emph{under the condition of existence of $\nullset$}, we show that $0/0=\inull(0)/\inull(0)=1$ does not cause the logic trivialism of the real mathematic.
Category: Algebra

[40] viXra:1809.0485 [pdf] replaced on 2018-10-01 16:23:37

On The Non-Real Nature of x.0 (x in R*): The Set of Null Imaginary Numbers

Authors: Saulo Queiroz
Comments: 6 Pages.

In this letter we discuss the inconsistencies of $0/0\cdot x=y$, $x,y\in\real_{\ne 0}$ from the perspective of the zero property multiplication (ZPM) $x\cdot 0 = y\cdot 0$ on $\real$. We axiomatize $x\cdot 0$ as a number $\inull(x)$ that has a real part $\Re(\inull(x))=0$ but indeed is not real. From this we define the set of null imaginary numbers $\nullset$ as $\{\inull(x)|\forall x\in\real_{\ne 0}\} \cup \{0\}$. We present the elementary algebra on $\nullset$ based on the definitions of uniqueness (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$) and the null division (i.e., $\inull(x)/0=x\ne 0$). Also, \emph{under the condition of existence of $\nullset$}, we show that $0/0=\inull(0)/\inull(0)=1$ does not cause the logic trivialism of the real mathematic.
Category: Algebra

[39] viXra:1809.0485 [pdf] replaced on 2018-09-24 10:18:41

On The Non-Real Nature of x.0 (x in R*): The Set of Null Imaginary Numbers

Authors: Saulo Queiroz
Comments: 6 Pages.

In this letter we discuss the inconsistencies of $0/0\cdot x=y$, $x,y\in\real_{\ne 0}$ from the perspective of the zero property multiplication (ZPM) $x\cdot 0 = y\cdot 0$ on $\real$. We axiomatize $x\cdot 0$ as a number $\inull(x)$ that has a real part $\Re(\inull(x))=0$ but indeed is not real. From this we define the set of null imaginary numbers $\nullset$ as $\{\inull(x)|\forall x\in\real_{\ne 0}\} \cup \{0\}$. We present the elementary algebra on $\nullset$ based on the definitions of uniqueness (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$) and the null division (i.e., $\inull(x)/0=x\ne 0$). Also, \emph{under the condition of existence of $\nullset$}, we show that $0/0=\inull(0)/\inull(0)=1$ does not cause the logic trivialism of the real mathematic.
Category: Algebra

[38] viXra:1805.0355 [pdf] replaced on 2019-05-12 04:34:52

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 12 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[37] viXra:1805.0355 [pdf] replaced on 2019-05-09 11:22:56

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 12 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[36] viXra:1805.0355 [pdf] replaced on 2019-04-22 09:15:16

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 10 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[35] viXra:1805.0355 [pdf] replaced on 2019-04-21 19:48:35

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 10 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[34] viXra:1805.0355 [pdf] replaced on 2019-03-19 21:37:25

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 10 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[33] viXra:1805.0355 [pdf] replaced on 2019-02-16 01:58:22

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 10 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[32] viXra:1805.0355 [pdf] replaced on 2019-02-09 08:45:32

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 9 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.
Category: Algebra

[31] viXra:1805.0355 [pdf] replaced on 2018-10-04 19:34:38

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 11 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 3 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[30] viXra:1805.0355 [pdf] replaced on 2018-06-12 05:12:19

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 10 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 4 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[29] viXra:1805.0355 [pdf] replaced on 2018-06-06 15:20:33

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 10 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 4 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[28] viXra:1805.0355 [pdf] replaced on 2018-06-02 05:19:02

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 8 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 3 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[27] viXra:1805.0355 [pdf] replaced on 2018-06-01 17:03:00

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 8 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 3 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[26] viXra:1805.0355 [pdf] replaced on 2018-05-24 17:28:18

Boundary Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma
Comments: 8 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 3 theorems regarding these boundary matrices. We propose 2 conjectures related to the Marcus-de Oliveira conjecture.
Category: Algebra

[25] viXra:1804.0003 [pdf] replaced on 2018-04-07 11:38:11

About the Hamilton Numbers

Authors: Antoine Balan
Comments: 4 pages, written in french

We introduce here some algebraic theory about the Hamilton numbers and develop a quaternionic geometry of fiber bundles.
Category: Algebra