**Previous months:**

2009 - 0902(1) - 0910(1) - 0911(1)

2010 - 1003(17) - 1004(1) - 1005(14) - 1006(1) - 1007(5) - 1008(4) - 1011(3) - 1012(1)

2011 - 1101(2) - 1102(1) - 1103(1) - 1105(2) - 1106(2) - 1107(1) - 1110(1) - 1111(3)

2012 - 1201(2) - 1202(1) - 1203(1) - 1204(1) - 1205(1) - 1207(2) - 1208(2) - 1209(1) - 1210(4) - 1211(3) - 1212(9)

2013 - 1301(2) - 1302(1) - 1303(2) - 1304(1) - 1305(9) - 1306(17) - 1307(2) - 1309(5) - 1311(2) - 1312(2)

2014 - 1403(1) - 1404(2) - 1406(2) - 1407(2) - 1408(3) - 1409(1) - 1410(1) - 1411(7) - 1412(2)

2015 - 1501(1) - 1503(2) - 1504(2) - 1505(1) - 1507(3) - 1508(2) - 1509(1) - 1511(3)

2016 - 1602(1) - 1604(1) - 1605(4) - 1606(1) - 1607(64) - 1608(3) - 1609(1) - 1610(2) - 1612(2)

2017 - 1702(3) - 1705(1) - 1706(1) - 1708(3) - 1709(1) - 1710(1) - 1712(2)

2018 - 1801(1) - 1802(3) - 1804(1) - 1805(2) - 1806(3) - 1807(3) - 1809(1) - 1810(2) - 1811(5) - 1812(2)

2019 - 1901(6) - 1902(1) - 1903(3) - 1904(2) - 1905(5) - 1906(8) - 1907(5) - 1908(2) - 1909(5) - 1910(4) - 1911(8) - 1912(2)

Any replacements are listed farther down

[337] **viXra:1912.0073 [pdf]**
*submitted on 2019-12-04 07:10:56*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[336] **viXra:1912.0002 [pdf]**
*submitted on 2019-12-01 07:44:01*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate the standard definition of the quaternion as four equations to produce its multiplication table. We then derive the implication table for the quaternion which is not tautologous, and hence refutes the quaternion. This result forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Algebra

[335] **viXra:1911.0479 [pdf]**
*submitted on 2019-11-28 13:22:22*

**Authors:** Christopher Goddard

**Comments:** 27 Pages.

This article takes an oblique sidestep from two previous papers, wherein an approach to reformulation of game theory in terms of information theory, topology, as well as a few other notions was indicated. In this document a description is provided as to how one might determine an approach for an agent to choose a policy concerning which actions to take in a game that constrains behaviour of subsidiary agents. It is then demonstrated how these results in algebraic information theory, together with previous investigations in geometric and topological information theory, can be unified into a single cohesive framework.

**Category:** Algebra

[334] **viXra:1911.0452 [pdf]**
*submitted on 2019-11-26 11:57:04*

**Authors:** Clumsy Foo

**Comments:** 8 Pages. Every mathematical proof is only valid for the range of the few axioms, on which its construction is based! English and German press release.

To all appearances mathematicians are not interested in mathematics, but they are only interested in constructing proofs. After having found a proof that a mathematical statement is correct or wrong, or that a mathematical construction is possible or impossible, mathematicians stop thinking about these statements or constructions. This is problematic for a science which is axiomatically founded in such a severe way.

**Category:** Algebra

[333] **viXra:1911.0345 [pdf]**
*submitted on 2019-11-20 11:53:48*

**Authors:** Clumsy Foo

**Comments:** 15 Pages. Shorter equations will result in better mathematics.

It is possible to model rotations in three-dimensional space with triplets.

**Category:** Algebra

[332] **viXra:1911.0275 [pdf]**
*submitted on 2019-11-16 02:53:50*

**Authors:** Eckhard Hitzer, Stephen J. Sangwine

**Comments:** 18 Pages. Submitted to Adv. in Appl. Cliff. Algs., 2019.

In this paper we consider general multivector elements of Clifford algebras $C(p,q)$, $p+q \leq 3$, and study multivector equivalents of polar decompositions and factorization into products of exponentials, where the exponents are frequently blades of grades zero (scalar) to $n$ (pseudoscalar).

**Category:** Algebra

[331] **viXra:1911.0258 [pdf]**
*submitted on 2019-11-15 06:50:19*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Note that Disqus comments here are not read by the author; reply by email only to: info@cec-services dot com. Include a list publications for veracity. Updated abstract at ersatz-systems.com.

We evaluate fundamental sets for algebraic subgroups by way of the seminal Theorem 6.12. It is not tautologous and refutes the conjecture. What follows is refutation of quantitative reduction theory and unlikely intersections. These results form a non tautologous fragment of the universal logic VŁ4.

**Category:** Algebra

[330] **viXra:1911.0226 [pdf]**
*submitted on 2019-11-12 10:50:35*

**Authors:** Valery Timin

**Comments:** timinva@yandex.ru, 3 pages in Russian

This work is devoted to the search, study and compilation of the multiplication table of a compound hyperbolic (Hypercomplex) number of dimension nine (11):
q = {1, i1, i2, i3, …, i10, }: in2 = 1.
On this basis, as a template, you can choose other multiplication tables with a different target or arbitrary arrangement of plus (+) and minus (-) characters in the cells of the multiplication table. The disadvantage of multiplication tables of dimension 11 is that they are neither associative nor commutative (not even anticommutative). But this is their fundamental property..
Эта работа посвящена поиску, изучению и составлению таблицы умножения составного гиперболического (гиперкомплексного) числа размерностью девять (11):
q = {1, i1, i2, i3, …, i11, }: in2 = 1.
На этой основе как на шаблоне можно подобрать другие таблицы умножения с другой целевой или произвольной расстановкой знаков "плюс" (+) и "минус" (-) в ячейках таблицы умножения.
Ассоциативных и коммутативных таблиц умножения не имеется.

**Category:** Algebra

[329] **viXra:1911.0174 [pdf]**
*submitted on 2019-11-09 04:11:04*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to field theory.

**Category:** Algebra

[328] **viXra:1911.0023 [pdf]**
*submitted on 2019-11-01 12:27:07*

**Authors:** Martin Erik Horn

**Comments:** 6 pages, bilingual version (in English and German)

Together with complex conjugation, complex numbers are misused to model anti-commutative structures.
German Abstract:
Komplexe Zahlen werden zur Modellierung anti-kommutativer Strukturen missbraucht, wenn die komplexe Konjugation genutzt wird.

**Category:** Algebra

[327] **viXra:1910.0613 [pdf]**
*submitted on 2019-10-29 00:18:26*

**Authors:** William F. Gilreath

**Comments:** 6 Pages. Published in Transmathematica 2019

The Two Couriers Problem is an algebra problem, originally stated in 1746 by the French mathematician Clairaut. For over a century, the Two Couriers Problem has been re-used in various forms as a mathemat- ical problem, in textbooks and journals, by different mathematicians and authors.
The Two Couriers Problem involves cases where division by zero arises in practice, where each has a real-world, actual result for the solution. Thus the Two Couriers Problem is a centuries old algebra problem with actual applied results that involve division by zero. It is an excellent mathematical problem to evaluate different methods for dividing by zero.
Division by zero has many different mathematical approaches. Conventional mathematics handles division by zero as an indeterminate or undefined result. Transmathematics defines division by zero as either nul- lity or explicitly positive or negative infinity. Two other approaches are by Saitoh, who defines division by zero simply as zero, and Barukčić who defines division by zero as either unity or explicitly positive or implicitly negativity infinity. The question is, which approach is best to solve the mathematical problem of division by zero?
The paramount goal of this paper is to use the Two Couriers Problem as an objective test to examine and evaluate mathematical approaches to division by zero – and find which one is best.

**Category:** Algebra

[326] **viXra:1910.0345 [pdf]**
*submitted on 2019-10-19 11:43:49*

**Authors:** J Gregory Moxness

**Comments:** 5 Pages.

We introduce a unimodular Determinant=1 8x8 rotation matrix to produce four 4 dimensional copies of H4 600-cells from the 240 vertices of the Split Real Even E8 Lie group. Unimodularity in the rotation matrix provides for the preservation of the 8 dimensional volume after rotation, which is useful in the application of the matrix in various fields, from theoretical particle physics to 3D visualization algorithm optimization.

**Category:** Algebra

[325] **viXra:1910.0293 [pdf]**
*submitted on 2019-10-17 09:08:41*

**Authors:** Timothy W. Jones

**Comments:** 7 Pages.

There are natural lead ins to abstract algebra that occur in elementary algebra. We explore function composition and permutations as such lead ins to group theory and abstract algebra.

**Category:** Algebra

[324] **viXra:1910.0037 [pdf]**
*submitted on 2019-10-05 09:50:32*

**Authors:** Valery Timin

**Comments:** timinva@yandex.ru, 4 pages in Russian

This work is devoted to the search, study and compilation of the multiplication table of a compound hyperbolic (Hypercomplex) number of dimension nine (10):
q = {1, i1, i2, i3, …, i9, }: in2 = 1.
On this basis, as a template, you can choose other multiplication tables with a different target or arbitrary arrangement of plus (+) and minus (-) characters in the cells of the multiplication table. The disadvantage of multiplication tables of dimension 10 is that they are neither associative nor commutative (not even anticommutative). But this is their fundamental property..
Эта работа посвящена поиску, изучению и составлению таблицы умножения составного гиперболического (гиперкомплексного) числа размерностью девять (10):
q = {1, i1, i2, i3, …, i9, }: in2 = 1.
На этой основе как на шаблоне можно подобрать другие таблицы умножения с другой целевой или произвольной расстановкой знаков "плюс" (+) и "минус" (-) в ячейках таблицы умножения. Только коммутативных таблиц размерности 10 может быть до 1750, а ассоциативных – до 65201 (не окончательно). Все коммутативные таблицы имеют эластичную ассоциативность (Am).

**Category:** Algebra

[323] **viXra:1909.0570 [pdf]**
*submitted on 2019-09-26 18:58:38*

**Authors:** Zeolla Gabriel Martín

**Comments:** 29 Pages.

This document develops and demonstrates the discovery of a new cube potentiation algorithm that works absolutely with all the numbers using the formula of the square of a binomial, trinomial, tetranomial and pentanomial. This presents the expansion of terms to the cube, the ideal order of the coefficients to obtain a sum that generates the results of the power.

**Category:** Algebra

[322] **viXra:1909.0429 [pdf]**
*submitted on 2019-09-20 20:45:35*

[321] **viXra:1909.0292 [pdf]**
*submitted on 2019-09-13 10:33:39*

**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.

We evaluate the first definition for a pre-orderable group which is not tautologous. This refutes subsequent conjectures, and forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Algebra

[320] **viXra:1909.0286 [pdf]**
*submitted on 2019-09-13 18:09:39*

**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.

We evaluate a Lipschitz horizontal vector field on Heisenberg group. It is not tautologous and further exemplifies that vector spaces are not bivalent. This forms a non tautologous fragment of the universal logic VŁ4.

**Category:** Algebra

[319] **viXra:1909.0255 [pdf]**
*submitted on 2019-09-11 13:40:44*

**Authors:** Valery Timin

**Comments:** 2 Pages. Russian

This work is devoted to the search, study and compilation of the multiplication table of a compound hyperbolic (Hypercomplex) number of dimension nine (9):
q = {1, i1, i2, i3, …, i8, }: in2 = 1.
On this basis, as a template, you can choose other multiplication tables with a different target or arbitrary arrangement of plus (+) and minus (-) characters in the cells of the multiplication table. The disadvantage of multiplication tables of dimension 9 is that they are neither associative nor commutative (not even anticommutative). But this is their fundamental property.

**Category:** Algebra

[318] **viXra:1908.0481 [pdf]**
*submitted on 2019-08-23 18:06:27*

**Authors:** M. D. Sheppeard

**Comments:** 2 Pages.

This two page note summarises the quantum gravity perspective on motives for a mathematician.

**Category:** Algebra

[317] **viXra:1908.0355 [pdf]**
*submitted on 2019-08-16 08:15:44*

**Authors:** Anna Knezevic, Greg Cohen

**Comments:** 75 Pages. This research was partly supported by the School of Electrical Engineering, Computing and Mathematical Sciences of the Curtin University (Australia)

The permanent’s polynomial-time computability over fields of characteristic 3 for k-semi-unitary matrices (i.e. n×n-matrices A such that rank(AA^T-I_n )=k) in the case k ≤ 1 and its #_3P-completeness for any k > 1 (Ref. 9) is a result that essentially widens our understanding of the computational complexity boundaries for the permanent modulo 3. Now we extend this result to study more closely the case k > 1 regarding the (n-k)×(n-k)-sub-permanents (or permanent-minors) of a unitary n×n-matrix and their possible relations, because an (n-k)×(n-k)-submatrix of a unitary n×n-matrix is generically a k-semi-unitary (n-k)×(n-k)-matrix.
The following paper offers a way to receive a variety of such equations of different sorts, in the meantime also extending (in its second chapter divided into subchapters) this direction of research to reviewing all the set of polynomial-time permanent-preserving reductions and equations for a generic matrix’s sub-permanents they might yield, including a number of generalizations and formulae (valid in an arbitrary prime characteristic) analogical to the classical identities relating the minors of a matrix and its inverse. Moreover, the second chapter also deals with the Hamiltonian cycle polynomial in characteristic 2 that surprisingly demonstrates quite a number of properties very similar to the corresponding ones of the permanent in characteristic 3.
Besides, the paper’s third chapter is devoted to the computational complexity issues of the permanent and some related functions on a variety of Cauchy matrices and their certain generalizations, including constructing a polynomial-time algorithm (based on them) for the permanent of an arbitrary square matrix in characteristic 5 and conjecturing the existence of a similar scheme in characteristic 3.
Throughout the paper, we investigate various matrix compressions and transformations preserving the permanent and related functions in certain finite characteristics. And, as an auxiliary algebraic tool supposed for an application when needed in all the constructions we’re going to discuss in the present article, we’ll introduce and utilize a special principle involving a field’s extension by a formal infinitesimal and allowing, provided a number of conditions are fulfilled, to reduce the computation of a polynomial over a field to solving a system of algebraic equations in polynomial time.
greg.cohen.math@gmail.com

**Category:** Algebra

[316] **viXra:1908.0043 [pdf]**
*submitted on 2019-08-02 09:58:14*

**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.

We evaluate the seminal equation for Heyting algebra of a∧b≤c⇔a≤b→c. It is not tautologous, hence refuting Heyting algebra as stated, and forming another non tautologous fragment of Heyting algebra in the universal logic V�?4.

**Category:** Algebra

[315] **viXra:1907.0443 [pdf]**
*submitted on 2019-07-23 14:38:41*

**Authors:** Valery Timin

**Comments:** language: Russian, number of pages: 6, mailto:timinva@yandex.ru, Creative Commons Attribution 3.0 License

This work is devoted to the search, study and compilation of the multiplication table of a composite hyperbolic (Hypercomplex) number of dimension five, six, seven (5,6,7):
q = {1, i, j, k, l}: i2 = 1, j2 = 1, k2 = 1, l2 = 1.
On this basis, as a template, you can pick up other multiplication tables with a different target or arbitrary arrangement of plus (+) and minus (-) signs in the cells of the multiplication table. The disadvantage of this multiplication table is its weak associativity and not commutativity (not even anti-commutativity). But this is its fundamental property.
Эта работа посвящена поиску, изучению и составлению таблицы умножения составного гиперболического (гиперкомплексного) числа размерностью пять, шесть и семь (5,6,7):

**Category:** Algebra

[314] **viXra:1907.0395 [pdf]**
*submitted on 2019-07-20 09:15:19*

**Authors:** Valery Timin

**Comments:** language: Russian, number of pages: 4, mailto:timinva@yandex.ru, Creative Commons Attribution 3.0 License

This work is devoted to the search, study and compilation of the multiplication table of a composite hyperbolic (Hypercomplex) number of dimension five (5):
q = {1, i, j, k, l}: i2 = 1, j2 = 1, k2 = 1, l2 = 1.
On this basis, as a template, you can pick up other multiplication tables with a different target or arbitrary arrangement of plus (+) and minus (-) signs in the cells of the multiplication table. The disadvantage of this multiplication table is its weak associativity and not commutativity (not even anti-commutativity). But this is its fundamental property.
Эта работа посвящена поиску, изучению и составлению таблицы умножения составного гиперболического (гиперкомплексного) числа размерностью пять (5).

**Category:** Algebra

[313] **viXra:1907.0332 [pdf]**
*submitted on 2019-07-18 01:16:19*

**Authors:** Kujonai

**Comments:** 48 Pages.

A compilation of drawings sprung during a email exchanging with T. Golden, author of the Polysigns numbers. It must be mentioned that he accepts only some of the concepts presented here. Although we do share some common ground while talking about Polysigns and/or "simplexogonal" arithmetics, we do have some differences in the approach. In anycase, it is required some understanding of the Polysign Notation to fully appreciate the drawings. A few important bits arose as a direct consequence of the
interaction and some are presented here, in a rather highly informal way.

**Category:** Algebra

[312] **viXra:1907.0162 [pdf]**
*submitted on 2019-07-09 09:12:46*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[311] **viXra:1907.0117 [pdf]**
*submitted on 2019-07-07 10:09:59*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

We propose a generalization of the derivations with help of homomorphism.

**Category:** Algebra

[310] **viXra:1906.0485 [pdf]**
*submitted on 2019-06-25 08:38:38*

**Authors:** Edgar Valdebenito

**Comments:** 2 Pages.

We give a formula for Pi.

**Category:** Algebra

[309] **viXra:1906.0475 [pdf]**
*submitted on 2019-06-26 01:58:32*

**Authors:** Han Xiao

**Comments:** 19 Pages.

Traditionally, an infinitesimal is regarded as a variable that runs toward 0. Since a differential is a kind of infinitesimal, a differential is essentially a variable running toward 0 too. As a result, differentials are form invariant but not meaning invariant. This paper proposes a Number Field of Ordered Infinitesimals and Infinities (OII Number Field) which can be seen as a kind of extension of real number field. The terminus of a variable running toward 0 is no longer 0, but a point in the OII Number Field, with an Order and a Weight. In this way, the process of running is recorded in the destination, making infinitesimals a kind of number which can be compared and operated easily. On this basis, the differential of a variable is invariant not only in form, but also in meaning. As a differential becomes a variable on another number axis parallel to the real number axis in OII Number Field, a differential can generate differential too, thus giving rise to high order differentials which are also invariant both in form and in meaning.

**Category:** Algebra

[308] **viXra:1906.0447 [pdf]**
*submitted on 2019-06-23 12:36:54*

**Authors:** Ajda Foˇsner, Mehsin Jabel Atteya

**Comments:** 24 Pages.

The main purpose of this paper is to study and investigate certain results concerning the (σ,τ)-generalized derivation D associated with the (σ,τ)-derivation d of semiprime and prime rings R, where σ and τ act as two automorphism mappings of R. We focus on the composition of (σ,τ)-generalized derivations of the Leibniz’s formula, where we introduce the general formula to compute the composition of the (σ,τ)-generalized derivation D of R.

**Category:** Algebra

[307] **viXra:1906.0343 [pdf]**
*submitted on 2019-06-18 09:09:35*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[306] **viXra:1906.0304 [pdf]**
*submitted on 2019-06-16 15:57:29*

**Authors:** Anamitra Palit

**Comments:** 4 Pages.

We endeavor to show certain contradictions in the theory of linear vector spaces.

**Category:** Algebra

[305] **viXra:1906.0145 [pdf]**
*submitted on 2019-06-09 22:32:11*

**Authors:** Henry Wong

**Comments:** 2 Pages.

An addendum to group theory.

**Category:** Algebra

[304] **viXra:1906.0120 [pdf]**
*submitted on 2019-06-07 08:34:30*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

We give some roots of the equation: 8*x*x*(cos(x))*(cos(x))-(4-2*sqrt(2))*x*x-1=0.

**Category:** Algebra

[303] **viXra:1906.0002 [pdf]**
*submitted on 2019-06-01 04:49:44*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[302] **viXra:1905.0530 [pdf]**
*submitted on 2019-05-27 10:01:35*

**Authors:** Likai Fareed

**Comments:** 1 Page.

Through my most extensive research, I have made a groundbreaking discovery. While many of us are familiar with binomial theorem and the polynomial multiplication algorithm, "First Outside Inside Last", there are also a set of rules that apply in field R. For instance, (a+b)^2 expands to a^2+b^2, and (a+b)(a-b) expands to a+b. This paper seeks to prove these rules in a simple and geometric manner.

**Category:** Algebra

[301] **viXra:1905.0430 [pdf]**
*submitted on 2019-05-23 01:50:13*

**Authors:** Kevin

**Comments:** 1 Page.

Fundamental-Methods-Of-Mathematical-Economics-by-Kevin-Wainwright-Professor-Alpha-C-Chiang

**Category:** Algebra

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

**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

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

**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

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

**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

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

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

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

**Category:** Algebra

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

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

En esta nota se muestran algunas relaciones del tipo arcotangente.

**Category:** Algebra

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

**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

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

**Authors:** Volker W. Thürey

**Comments:** 3 Pages.

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

**Category:** Algebra

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

**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

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

**Authors:** Ведерников Сергей Иванович

**Comments:** 11 Pages.

Простое доказательство инструментами элементарной алгебры.

**Category:** Algebra

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages.

A historical definition of division by zero is reconsidered.

**Category:** Algebra

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

**Authors:** Hiroshi Okumura

**Comments:** 1 Page.

A remark of the definition of division by zero is given.

**Category:** Algebra

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

**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

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

**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

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

**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

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

**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

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

**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

[56] **viXra:1911.0275 [pdf]**
*replaced on 2019-11-17 23:52:46*

**Authors:** Eckhard Hitzer, Stephen J. Sangwine

**Comments:** 18 Pages. Submitted to Adv. in Appl. Cliff. Algs., 2019.

In this paper we consider general multivector elements of Clifford algebras $Cl(p,q)$, $n=p+q \leq 3$, and study multivector equivalents of polar decompositions and factorization into products of exponentials, where the exponents are frequently blades of grades zero (scalar) to $n$ (pseudoscalar).

**Category:** Algebra

[55] **viXra:1910.0293 [pdf]**
*replaced on 2019-10-18 07:34:11*

**Authors:** Timothy W. Jones

**Comments:** 9 Pages. Additional material added with corrections.

There are natural lead-ins to abstract algebra that occur in elementary algebra. We explore function composition using linear functions and permutations on letters in misspellings of words. Groups and the central idea of abstract algebra, proving 5th degree and greater polynomials are unsolvable, are put into focus for college students.

**Category:** Algebra

[54] **viXra:1907.0162 [pdf]**
*replaced on 2019-07-21 05:56:55*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[53] **viXra:1907.0162 [pdf]**
*replaced on 2019-07-13 05:18:17*

**Authors:** Henry Wong

**Comments:** 1 Page.

An addendum to group theory.

**Category:** Algebra

[52] **viXra:1906.0447 [pdf]**
*replaced on 2019-06-25 14:27:50*

**Authors:** Ajda Foˇsner, Mehsin Jabel Atteya

**Comments:** 22 Pages.

The main purpose of this paper is to study and investigate certain results concerning the (σ,τ)-generalized derivation D associated with the (σ,τ)-derivation d of semiprime and prime rings R, where σ and τ act as two automorphism mappings of R. We focus on the composition of (σ,τ)-generalized derivations of the Leibniz’s formula, where we introduce the general formula to compute the composition of the (σ,τ)-generalized derivation D of R.

**Category:** Algebra

[51] **viXra:1906.0304 [pdf]**
*replaced on 2019-08-17 10:29:26*

**Authors:** Anamitra Palit

**Comments:** 2 Pages.

The short writing seeks to demonstrate certain lapses in the theory of the linear vector spaces.

**Category:** Algebra

[50] **viXra:1905.0361 [pdf]**
*replaced on 2019-09-20 16:51:40*

**Authors:** Zeolla Gabriel Martín

**Comments:** 28 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

[49] **viXra:1903.0560 [pdf]**
*replaced on 2019-06-24 21:35:25*

**Authors:** Anamitra Palit

**Comments:** 6 Pages.

The direct sum decomposition of a vector space has been explored to bring out a conflicting feature in the theory. It has been proved that a subspace cannot have dimension less than a third of the dimension of the parent vector space.

**Category:** Algebra

[48] **viXra:1901.0340 [pdf]**
*replaced on 2019-08-07 07:50:43*

**Authors:** Ведерников Сергей Иванович

**Comments:** 9 Pages. Стать�? �?вл�?ет�?�? доработанным вариантом одноимённой �?татьи, опубликованной в журнале "�?аука через призму времени" №23 февраль 2019 .

Доказатель�?тво о�?новано на выражении чётного чи�?ла разно�?тью квадратов двух нечётных чи�?ел и его разложени�? на множители.

**Category:** Algebra

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

**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

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

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages.

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

**Category:** Algebra

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

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages.

A historical definition of division by zero is reconsidered.

**Category:** Algebra

[44] **viXra:1809.0485 [pdf]**
*replaced on 2019-06-12 14:05:41*

**Authors:** Saulo Jorge

**Comments:** 7 Pages.

In this work I axiomatize the result of $x \cdot 0$ ($x\in\real_{\ne 0}$) as a number $\inull(x)$ that has a null real part (denoted as $\Re(\inull(x))=0$) but that is not real. This implies that $y+\Re(\inull(x)) = y$ but $y+\inull(x) = y + x\cdot 0 \not=y$, $y\in\real_{\ne 0}$. From this I define the set of null imaginary numbers $\nullset=\{\inull(x)=x\cdot 0|\forall x\in\real_{\ne 0}\}$
and present its elementary algebra taking the axiom of uniqueness as base (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$).
Under the condition of existence of $\nullset$ I show that division by zero can be defined without causing inconsistencies in elementary algebra.

**Category:** Algebra

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

**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

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

**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

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

**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