Algebra

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(3) - 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(7) - 1606(1) - 1607(64) - 1608(3) - 1609(1)

Recent submissions

Any replacements are listed further down

[249] viXra:1609.0262 [pdf] submitted on 2016-09-17 15:49:01

Structure Theorem for Abelian Groups (General Case, Finite or Not)

Authors: Eli Halylaurin
Comments: 4 Pages. This document is french written.

This document is an attempt to demonstrate a general structure theorem for abelian groups (finite or not). Such a theorem already exists in the finite case, but the infinite case does not seem to have been deeply studied. This is what it is proposed to do in this document. To achieve this task, Zorn Lemma will be used. We will try to prove each abelian group can be written, modulo isomorphism, as a direct product of groups that we will called elementary, because they can be represented upon a circle or a line. This work may be very valuable for every mathematicians who like to better understand the structure of groups.
Category: Algebra

[248] viXra:1608.0308 [pdf] submitted on 2016-08-24 06:17:36

DSm Field and Linear Algebra of Refined Labels

Authors: Florentin Smarandache, Jean Dezert, Xinde Li
Comments: 11 Pages.

This chapter presents the DSm Field and Linear Algebra of Refined Labels (FLARL) in DSmT framework in order to work precisely with qualitative labels for information fusion. We present and justify the basic operators on qualitative labels (addition, subtraction, multiplication, division, root, power, etc).
Category: Algebra

[247] viXra:1608.0136 [pdf] submitted on 2016-08-12 21:45:01

Unit Graph of Some Finite Group Zn, Cn and Dn

Authors: A. D. Godase, M. B. Dhakne
Comments: 10 Pages.

We represent finite group in the form of a graph, these graphs are called unit graph. Since the main role in obtaining the graph is played by the unit element of the group, this study is innovative. Also study of different properties like the subgroups of a group, normal subgroups of a group are carried out using the unit graph of the group.
Category: Algebra

[246] viXra:1608.0039 [pdf] submitted on 2016-08-03 18:36:06

Solving Boolean Equation

Authors: Oh Jung Uk
Comments: 19 Pages.

If ∀P:proposition, B(P) is the truth value(0 or 1) of P then we can solve a boolean equation by using these below. B(p_1∨p_2∨…∨p_n )≡1+∏_(k=1)^n▒(1+p_k ) (mod 2) { (x_1,x_2,…,x_n ) | ∏_(i=1)^n▒B(x_i ) ≡0(mod 2)}=(⋂_(i=1)^n▒{ (x_i ) | B(x_i )≡1(mod 2)} )^c={(x_1,x_2,…,x_n ) |(1,1,1,…,1)}^c
Category: Algebra

[245] viXra:1607.0508 [pdf] submitted on 2016-07-27 01:56:49

On Neutrosophic Quadruple Algebraic Structures

Authors: S.A. Akinleye, F. Smarandache, A.A.A. Agboola
Comments: 5 Pages.

In this paper we present the concept of neutrosophic quadruple algebraic structures. Specially, we study neutrosophic quadruple rings and we present their elementary properties.
Category: Algebra

[244] viXra:1607.0499 [pdf] submitted on 2016-07-27 03:00:47

Affirmative Resolve of Kothe Conjecture

Authors: T.Nakashima
Comments: 1 Page.

Affirmative resolve of Kothe conjecture
Category: Algebra

[243] viXra:1607.0498 [pdf] submitted on 2016-07-27 03:02:03

The Counter Example of Jacobson Conjecture

Authors: T.Nakashima
Comments: 1 Page.

The counter example of Jacobson conjecture
Category: Algebra

[242] viXra:1607.0350 [pdf] submitted on 2016-07-18 07:16:53

Multi-Structure and Multi-Space

Authors: Florentin Smarandache
Comments: 3 Pages.

I consider that life and practice do not deal with pure spaces, but with a group of many spaces, with a mixture of structures, a 'mongrel', a heterogeneity - the ardently preoccupation is to reunite them! to constitute a multi-structure.
Category: Algebra

[241] viXra:1607.0345 [pdf] submitted on 2016-07-18 07:22:25

Notes On Using T.I.-82 Calculators To Find Sums

Authors: Florentin Smarandache
Comments: 4 Pages.

A sequence is the range of a function whose domain is a subset of Z.
Category: Algebra

[240] viXra:1607.0340 [pdf] submitted on 2016-07-18 07:28:27

On an Erdös’ Open Problems

Authors: Florentin Smarandache
Comments: 3 Pages.

In one of his books (“Analysis…”) Mr. Paul Erdös proposed the following problem.
Category: Algebra

[239] viXra:1607.0339 [pdf] submitted on 2016-07-18 07:29:13

On Another Erdös’ Open Problem

Authors: Florentin Smarandache
Comments: 2 Pages.

Paul Erdös has proposed the following problem.
Category: Algebra

[238] viXra:1607.0337 [pdf] submitted on 2016-07-18 07:31:31

On Carmichaël’s Conjecture

Authors: Florentin Smarandache
Comments: 4 Pages.

In this article we prove that the equation ϕ (x) = n admits a finite number of solutions, we find the general form of these solutions
Category: Algebra

[237] viXra:1607.0336 [pdf] submitted on 2016-07-18 07:32:21

On A Diophantine Equation

Authors: Florentin Smarandache
Comments: 3 Pages.

In this note we present a method of solving this Diophantine equation, method which is different from Ljunggren’s, Mordell’s, and R.K.Guy’s.
Category: Algebra

[236] viXra:1607.0335 [pdf] submitted on 2016-07-18 07:33:54

One Application of Wallis Theorem

Authors: Bencze MihÁly, Florentin Smarandache
Comments: 4 Pages.

By multiplication we obtain the statement. We prove in the same way for cos x.
Category: Algebra

[235] viXra:1607.0332 [pdf] submitted on 2016-07-18 07:37:01

Où se Trouve la Faute ? (Equations Diophantiennes)

Authors: Florentin Smarandache
Comments: 3 Pages.

Où se trouve la faute ? (equations diophantiennes).
Category: Algebra

[234] viXra:1607.0328 [pdf] submitted on 2016-07-18 07:49:06

P-Q Relationships and Sequences

Authors: Florentin Smarandache
Comments: 3 Pages.

This article is on P-Q Relationships and Sequences.
Category: Algebra

[233] viXra:1607.0327 [pdf] submitted on 2016-07-18 07:50:34

Prime Conjecture

Authors: Florentin Smarandache
Comments: 3 Pages.

Any odd number can be expressed as a sum of two primes.
Category: Algebra

[232] viXra:1607.0326 [pdf] submitted on 2016-07-18 07:51:57

Quelques Propriétés Des Medianes

Authors: Florentin Smarandache
Comments: 2 Pages.

Cet article généralise cerrtains résultats sur les nédiannes.
Category: Algebra

[231] viXra:1607.0321 [pdf] submitted on 2016-07-18 07:57:53

Sequences of Sub-Sequences

Authors: Florentin Smarandache
Comments: 4 Pages.

This article is on Sequences of Sub-Sequences.
Category: Algebra

[230] viXra:1607.0320 [pdf] submitted on 2016-07-18 07:59:18

C. Dumirescu, N. VÎRLAN, ȘT. Zamfir, e. RĂDESCU, N. RĂDESCU, Florentin Smarandache

Authors: Smarandache Type Function Obtained by Duality
Comments: 17 Pages.

In this paper we extended the Smarandache function from the set N' of positive integers to the set Q of rationals.
Category: Algebra

[229] viXra:1607.0318 [pdf] submitted on 2016-07-18 08:00:58

Solving Problems by Using a Function in The Number Theory

Authors: Florentin Smarandache
Comments: 4 Pages.

liVe have constructed a function n which associates to each non-null integer m the smallest positive n such that n! is a multiple of m.
Category: Algebra

[228] viXra:1607.0317 [pdf] submitted on 2016-07-18 08:01:42

Some Periodical Sequences

Authors: Florentin Smarandache
Comments: 4 Pages.

Let N be a positive integer with not all digits the same, and N' its digital reverse.
Category: Algebra

[227] viXra:1607.0312 [pdf] submitted on 2016-07-18 08:06:10

Subjective Questions and Anwers for A Math Instructor of Higher Education

Authors: Florentin Smarandache
Comments: 16 Pages.

What are the instructor's general responsabilities ?
Category: Algebra

[226] viXra:1607.0311 [pdf] submitted on 2016-07-18 08:07:19

Sur la Resolution Dans L’ensemble Des Naturels Des Equationes Linéaires

Authors: Florentin Smarandache
Comments: 4 Pages.

L'utilité du cet article est qu'il établit si le nombre des solutions naturelles d'une équation linéaire est limité ou non.
Category: Algebra

[225] viXra:1607.0310 [pdf] submitted on 2016-07-18 08:08:59

Sur la Resolution D’équations du Second Degré a Deux Inconnues Dans Z

Authors: Florentin Smarandache
Comments: 3 Pages.

Cet article nous presente la resolution l’équations du second degré a deux inconnues dans Z.
Category: Algebra

[224] viXra:1607.0309 [pdf] submitted on 2016-07-18 08:10:57

Sur Quelques Progressions

Authors: Florentin Smarandache
Comments: 3 Pages.

Dans cet article on construit des ensembles qui ont la proprieté suivente: quel que soit leur partage en (Leux sous-ensembles, au moins l'un de ces sous-ensembles contient au moins trois éléments en progression arithmétique (ou bien géométrique).
Category: Algebra

[223] viXra:1607.0308 [pdf] submitted on 2016-07-18 08:11:45

The A.m.a.t.y.c. Competition

Authors: Florentin Smarandache
Comments: 3 Pages.

The American Mathematical Association of Two-Year Colleges organizes each year a mathematical competition.
Category: Algebra

[222] viXra:1607.0299 [pdf] submitted on 2016-07-18 08:28:30

Une Application de la Generalisation du Théorème du Ceva

Authors: Florentin Smarandache
Comments: 4 Pages.

Cet article presente une application de la generalisation du théorème du Ceva.
Category: Algebra

[221] viXra:1607.0298 [pdf] submitted on 2016-07-18 08:29:56

Une Classe D’ensembles Recursifs

Authors: Florentin Smarandache
Comments: 8 Pages.

Dans cet article on construit une classe d' ensembles récursifs, on établit des propriétés de ces ensembles et on propose des applications.
Category: Algebra

[220] viXra:1607.0297 [pdf] submitted on 2016-07-18 08:30:57

Une Généralisation de L’inegalité Cauchy Bouniakovski-Schwartz

Authors: Florentin Smarandache
Comments: 2 Pages.

Cet article presente une généralisation de l’inegalité Cauchy-Bouniakovski-Schwartz.
Category: Algebra

[219] viXra:1607.0296 [pdf] submitted on 2016-07-18 08:32:37

Une Generalisation du Theoreme D’Euler

Authors: Florentin Smarandache
Comments: 6 Pages.

Dans les paragraphes qui suivent nous alions démontrer un resultat qui remplace le teorème d' Euler.
Category: Algebra

[218] viXra:1607.0295 [pdf] submitted on 2016-07-18 08:33:58

Une Généralisation de L’inegalité de Hölder

Authors: Florentin Smarandache
Comments: 3 Pages.

On géneralise l'inégalité de Holder grâce à un raisonement par récurrence.
Category: Algebra

[217] viXra:1607.0294 [pdf] submitted on 2016-07-18 08:34:54

Une Généralisation de L’inégalité de Minkowski

Authors: Florentin Smarandache
Comments: 2 Pages.

Cet article presente une généralisation de l’inégalité de Minkowski.
Category: Algebra

[216] viXra:1607.0293 [pdf] submitted on 2016-07-18 08:35:50

Une Généralisation de L’inegalité de Tcebychev

Authors: Florentin Smarandache
Comments: 2 Pages.

Cet article presente une généralisation de l’inegalité de Tcebychev.
Category: Algebra

[215] viXra:1607.0292 [pdf] submitted on 2016-07-18 08:36:49

Une Généralisation D’une Théorème de Carnot

Authors: Florentin Smarandache
Comments: 3 Pages.

Cet article presente une généralisation d’une théorème de Carnot.
Category: Algebra

[214] viXra:1607.0287 [pdf] submitted on 2016-07-18 04:49:33

About Bernoulli's Numbers

Authors: Bencze MihÁly, Florentin Smarandache
Comments: 4 Pages.

Many methods to compute the sum ofthe same powers of the first n natural numbers are well-known. In this paper we present a simple proof of the method.
Category: Algebra

[213] viXra:1607.0285 [pdf] submitted on 2016-07-18 04:52:25

About the Characteristic Function of the Set

Authors: Bencze MihÁly, Florentin Smarandache
Comments: 12 Pages.

In our paper we give a method, based on characteristic function of the set, of resolving some difficult problem of set theory found in high school study.
Category: Algebra

[212] viXra:1607.0284 [pdf] submitted on 2016-07-18 04:55:44

About Very Perfect Numbers

Authors: Bencze MihÁly, Florin Popovici, Florentin Smarandache
Comments: 3 Pages.

the square of an odd prime number can't be very perfect number.
Category: Algebra

[211] viXra:1607.0282 [pdf] submitted on 2016-07-18 04:58:02

A Function in the Number Theory

Authors: Florentin Smarandache
Comments: 7 Pages.

In this paper I shall construct a function n having the following properties.
Category: Algebra

[210] viXra:1607.0277 [pdf] submitted on 2016-07-18 05:10:32

A Generalization of Euler’s Theorem

Authors: Florentin Smarandache
Comments: 7 Pages.

In the paragraphs which follow we will prove a result which replaces the theorem of Euler.
Category: Algebra

[209] viXra:1607.0276 [pdf] submitted on 2016-07-18 05:11:16

A Generalization of the Inequality of Hölder

Authors: Florentin Smarandache
Comments: 3 Pages.

One generalizes the inequality of Hödler thanks to a reasoning by recurrence. As particular cases, one obtains a generalization of the inequality of Cauchy-Buniakovski-Scwartz, and some interesting applications.
Category: Algebra

[208] viXra:1607.0275 [pdf] submitted on 2016-07-18 05:12:13

A Generalization of the Leibniz Theorem

Authors: Bencze MihÁly, Florin Popovici, Florentin Smarandache
Comments: 6 Pages.

In this paper we show a generalization of Leibniz's theorem and an application of this.
Category: Algebra

[207] viXra:1607.0263 [pdf] submitted on 2016-07-18 05:28:21

An Infinity Of Unsolved Problems Concerning A Function In The Number Theory

Authors: Florentin Smarandache
Comments: 23 Pages.

W.Sierpinski has asserted to an international conference that if mankind lasted for ever and numbered the unsolved problems, then in the long run all these unsolved problems would be solved.
Category: Algebra

[206] viXra:1607.0262 [pdf] submitted on 2016-07-18 05:29:13

An Integer Number Algorithm To Solve Linear Equations

Authors: Florentin Smarandache
Comments: 6 Pages.

An algorithm is given that ascertains whether a linear equation has integer number solutions or not; if it does, the general integer solution is determined.
Category: Algebra

[205] viXra:1607.0261 [pdf] submitted on 2016-07-18 05:30:26

Another Integer Algorithm To Solve Linear Equations (Using Congruences)

Authors: Florentin Smarandache
Comments: 8 Pages.

In this section is presented a new integer number algorithm for linear equation.This algorithm is more “rapid” than W. Sierpinski’s presented in the sense that it reaches the general solution after a smaller number of iterations. Its correctness will be thoroughly demonstrated.
Category: Algebra

[204] viXra:1607.0259 [pdf] submitted on 2016-07-18 05:33:51

A Numerical Function in Congruence Theory

Authors: Florentin Smarandache
Comments: 5 Pages.

In this article we define a function L which will allow us to generalize (separately or simultaneously) some theorems from Numbers Theory obtained by Wilson, Fermat, Euler, Gauss, Lagrange, Leibnitz, Moser, Sierpinski.
Category: Algebra

[203] viXra:1607.0255 [pdf] submitted on 2016-07-18 05:37:55

A Property For A Counterexample To Carmichaël’s Conjec

Authors: Florentin Smarandache
Comments: 3 Pages.

There are many papers on this subject, but the author cites the papers which have influenced him, especially Klee’s papers.
Category: Algebra

[202] viXra:1607.0252 [pdf] submitted on 2016-07-18 05:41:53

Asupra Unei Metode a Lui W. Sierpinski de Rezolvare în Numere întregi a Ecuațiilor Liniare

Authors: Florentin Smarandache
Comments: 3 Pages.

In nota urmatoare se fac cateva remarci privind metoda expusa de Sierpinski, remarci ce au ca scop sirnplificarea si extinderea acestei metode.
Category: Algebra

[201] viXra:1607.0251 [pdf] submitted on 2016-07-18 05:43:32

Asupra Unor Conjecturi si Probleme Nerezolvate Referitoare la O Functie in Teoria Numerelor

Authors: Florentin Smarandache
Comments: 15 Pages.

Am construit o functie care asociaza fiecarui intreg nenul n cel mai mic intreg pozitiv m astfel incat m! este multiplu de n.
Category: Algebra

[200] viXra:1607.0247 [pdf] submitted on 2016-07-18 05:58:29

Coefficients K-Nomiaux

Authors: Florentin Smarandache
Comments: 4 Pages.

Dans cet article on élargit les notions de "coefficients binomiaux" et de "coefficients trinomiaux" à la notion de"coefficients k-nomiaux~ et on obtient quelques propriétés générales de ceux-ci. Comme application, on généralisera le "triangle de Pascal".
Category: Algebra

[199] viXra:1607.0246 [pdf] submitted on 2016-07-18 06:00:10

Conjectures On Primes' Summation

Authors: Florentin Smarandache
Comments: 4 Pages.

Any odd integer n can be expressed as a combination of three primes.
Category: Algebra

[198] viXra:1607.0245 [pdf] submitted on 2016-07-18 06:01:23

Conjectures Which Generalize Andrica's Conjecture

Authors: Florentin Smarandache
Comments: 5 Pages.

Five conjectures on paires of consecutive primes are listed below with examples in each case.
Category: Algebra

[197] viXra:1607.0243 [pdf] submitted on 2016-07-18 06:03:33

Contributii la Studiul Unor Functii si Conjecturi in Teoria Numerelor

Authors: Florentin Smarandache
Comments: 28 Pages.

Teoria Numerelor reprezinta pentru mine o pasiune. Rezultatele expuse mai departe constituie rodul catorva ani buni de cercetari si cautari.
Category: Algebra

[196] viXra:1607.0242 [pdf] submitted on 2016-07-18 06:04:48

Convergence D’une Famille de Series in

Authors: Florentin Smarandache
Comments: 4 Pages.

Dans cet article, on construit une famille d'expressions E (n).
Category: Algebra

[195] viXra:1607.0237 [pdf] submitted on 2016-07-18 06:12:59

Des Fantaisies Mathématiques

Authors: Florentin Smarandache
Comments: 2 Pages.

Ces fantaisies mathematiques sont des divertissements, des problèmes amusants : elles font abstraction de la logique communne, mais elles ont quand meme leur "logique", une logique fantaisiste.
Category: Algebra

[194] viXra:1607.0236 [pdf] submitted on 2016-07-18 06:14:14

Digital Subsequences

Authors: Florentin Smarandache
Comments: 4 Pages.

We screen this sequence, selecting only the terms whose digits also satisfy the property or relationship.
Category: Algebra

[193] viXra:1607.0235 [pdf] submitted on 2016-07-18 06:24:48

Enseignement et Recherche Scientifique (Dans Les Mathematiques

Authors: Florentin Smarandache
Comments: 4 Pages.

De nos jours on met un accent puissant sur la correlation de l'enseignement avec la recherche et la production.
Category: Algebra

[192] viXra:1607.0232 [pdf] submitted on 2016-07-18 06:27:47

Functii Aritmetice

Authors: Florentin Smarandache
Comments: 9 Pages.

Este bine cunoscuta importanta functiilor aritmetice in teoria numerelor, importanta datorata pe de-a parte bogatiei rezultatelor ce se obtin cu ajutorul acestor functii, si pe de alta parte frumusetii acestor rezultate.
Category: Algebra

[191] viXra:1607.0231 [pdf] submitted on 2016-07-18 06:28:46

Functii Prime si Coprime

Authors: Florentin Smarandache
Comments: 2 Pages.

Vom construi urmatoare1e functii pe care le numim prime.
Category: Algebra

[190] viXra:1607.0224 [pdf] submitted on 2016-07-18 06:43:25

Happy New Mathematical Years!

Authors: Florentin Smarandache
Comments: 2 Pages.

Due to professor Gane Policarp’s kindness, I have several issues of “Caietul de informare matematică” (“The Notebook of Mathematical Information”), which has beenput together with attention to detail and skill, and which attracted and persuaded me, fromthe very beginning, to collaborate with small materials.
Category: Algebra

[189] viXra:1607.0220 [pdf] submitted on 2016-07-18 06:48:58

Inconsistent Systems of Axioms and Contradictory Theory

Authors: Florentin Smarandache
Comments: 3 Pages.

We construct the system of n+2 axioms.
Category: Algebra

[188] viXra:1607.0219 [pdf] submitted on 2016-07-18 06:49:51

Inequalities for The Integer Part Function In

Authors: Bencze MihÁly, Florin Popovici, Florentin Smarandache
Comments: 4 Pages.

In this paper we prove some inequalities for the integer part function and we give some applications in the number theory.
Category: Algebra

[187] viXra:1607.0218 [pdf] submitted on 2016-07-18 06:51:02

In Legatura cu O Problema de la Concursul de Matematica, Faza Locala, Ramnicu Valcea

Authors: Florentin Smarandache
Comments: 2 Pages.

Se prezinta in aceasta nota o extindere a unei probleme data la Olimpiada de matematica, faza locala, la Ramnicul VaIcea, clasa a VI-a, 1980.
Category: Algebra

[186] viXra:1607.0216 [pdf] submitted on 2016-07-18 06:53:18

Integer Solutions of Linear Equations

Authors: Florentin Smarandache
Comments: 9 Pages.

In this article are presented Definitions and properties of the integer solutions of linear equations.
Category: Algebra

[185] viXra:1607.0212 [pdf] submitted on 2016-07-18 06:58:28

Où se Trouve la Faute Dans ce Raisonnement Par Reccurence?

Authors: Florentin Smarandache
Comments: 2 Pages.

A un concours d'entré en faculté on pose le problème suivant.
Category: Algebra

[184] viXra:1607.0211 [pdf] submitted on 2016-07-18 06:59:23

Où se Trouve la Faute Sur Les Integrales?

Authors: Florentin Smarandache
Comments: 2 Pages.

Où se trouve la faute sur les integrales?
Category: Algebra

[183] viXra:1607.0160 [pdf] submitted on 2016-07-13 11:02:35

General Integration Theory Defined from Extended Cohomology.

Authors: Johan Noldus
Comments: 5 Pages.

We engage in an approach towards integration theory divorced from measure theory concentrating on the dierentiable functions instead of the measurable ones. In a sense, we do for \measure theory" what dierential geometry does for topology; the nal goal of this paper being the rigorous denition of a generalization of the Feynman path integral. The approach taken is an axiomatic one in which it is more important to understand relationships between certain quantities rather than to calculate them exactly. In a sense, this is how the eld of algebraic geometry is developed in opposition to the study of partial dierential equations where in the latter case, the stress is unfortunately still too much on the construction of explicit solutions rather than on structural properties of and between solutions.
Category: Algebra

[182] viXra:1607.0075 [pdf] submitted on 2016-07-06 23:42:10

Schanuel’s Conjecture’s Partial Resolve

Authors: T.Nakashima
Comments: 1 Page.

Schanuel’s conjecture’s partial resolve
Category: Algebra

[181] viXra:1606.0209 [pdf] submitted on 2016-06-20 10:16:01

The Complex Form of the Law of Cosines

Authors: Louai Hassan Elzein Basheir
Comments: 5 Pages.

This paper is prepared to show the mathematical derivation of the complex form of the law of cosines and show how it can help in the vector algebra.
Category: Algebra

[180] viXra:1605.0306 [pdf] submitted on 2016-05-30 15:53:54

Algebraic Poincare Duality 1

Authors: Bin Wang
Comments: 25 Pages. This is the second of three papers, all of which are posted on this site.

This paper includes two main chapters, \S 2 and \S3. Each deals with one type of algebraic Poincar\'e duality (APD) on linear spaces originated from algebraic cycles. Two types of APD confirm the following conjectures: (1) the Griffiths' conjecture on the incidence equivalence versus Abel-Jacobi equivalence. (2) the standard conjectures including the ``D" conjecture over $\mathbb C$.
Category: Algebra

[179] viXra:1605.0305 [pdf] submitted on 2016-05-30 15:57:17

Algebraic Poincare Duality 2

Authors: Bin Wang
Comments: 27 Pages. This is the last of three papers, all of which are posted on this site.

This is the sequel of three papers. In this paper we apply algebraic Poincar\'e duality to the maximal sub Hodge structures to show (1) Generalized Hodge conjecture of level 1 is correct. (2) The generalized Hodge conjecture of level 0, i.e. the usual Hodge conjecture, is correct.
Category: Algebra

[178] viXra:1605.0302 [pdf] submitted on 2016-05-30 15:49:46

Algebraic Poincare Duality 0

Authors: Bin Wang
Comments: 20 Pages. This is the first of three papers, all of which are posted on this site.

We discuss a structure that exists in many problems on smooth projective varieties over the field of complex numbers, and name it as ``Algebraic Poincar\'e duality" or ``APD" for abbreviation. In particular, over the complex numbers with singular cohomology, it is a solution to (1) Griffiths' conjecture on the incidence equivalence versus Abel-Jacobi equivalence, (2) Generalized Hodge conjecture of level 1, (3) Generalized Hodge conjecture of level 0, i.e. the usual Hodge conjecture, (4) The standard conjectures, (5) Grothendieck's ``D" conjecture. However it is not the goal of this paper to show APD implies these conjectures. In this paper we'll build the foundation for the structure by introducing the APD in its simplest form over the complex numbers.
Category: Algebra

[177] viXra:1605.0139 [pdf] submitted on 2016-05-13 12:00:15

Root of a Sum as the Sum of Roots

Authors: José de Jesús Camacho Medina
Comments: 13 Pages.

In the following document shows a particular form of simplify the root of a sum as the sum of roots, through an algebraic expression entitled: "Camacho Identity".
Category: Algebra

[176] viXra:1605.0022 [pdf] submitted on 2016-05-03 01:16:32

Special Type of Fixed Point Pairs using MOD Rectangular Matrix Operators

Authors: W. B. Vasantha Kandasamy, Ilanthenral K, Florentin Smarandache
Comments: 257 Pages.

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling.
Category: Algebra

[175] viXra:1605.0021 [pdf] submitted on 2016-05-03 01:17:45

Special Type of Fixed Points of MOD Matrix Operators

Authors: W. B. Vasantha Kandasamy, Ilanthenral K, Florentin Smarandache
Comments: 202 Pages.

In this book authors for the first time introduce a special type of fixed points using MOD square matrix operators. These special type of fixed points are different from the usual classical fixed points.
Category: Algebra

[174] viXra:1605.0013 [pdf] submitted on 2016-05-02 05:34:51

A Consistent Descriptive Logographic Onomatology of Algebraic Systems

Authors: Yakov A. Iosilevskii
Comments: 34 Pages. An additional category is "Mathematics: Set Theory and Logic"

There are two presently common onamastic (onomatological) methods of logographically naming and thus concisely describing an algebraic system; both methods are often used simultaneously. According to one method, an algebraic system is equivocally denoted by an atomic logographic symbol that originally denotes a certain underlying set of elements, which is regarded as the principal one, while all other objects of the algebraic system, properly named, are kept in mind and are regarded as implicit properties of that set or of its separate elements. That is to say, according to this method, an algebraic system is its principal underlying set of elements together with all its properties, which are implied and are not mentioned explicitly. According to the other method, an algebraic system is regarded as an ordered multiple, whose coordinates properly denote the defining objects of the algebraic system, and consequently the ordered multiple name is equivocally used as a proper name of the algebraic system. Thus, in this case, the togetherness of all constituents of the algebraic system is expressed by the pertinent ordered multiple name in terms of its coordinate names. In my recent article available at http://viXra.org/abs/1604.0124¸ I have demonstrated that both above onomastic methods are inconsistent. Therefore, in that article and also in my earlier article appearing at http://arxiv.org/abs/1510.00328, I suggested and used another onomastic method of logographically naming the pertinent algebraic systems, namely that employing, as a name of an algebraic system, a complex logographic name the union of all explicit constituent sets of the system, namely, the underlying sets of elements, the surjective binary composition functions, and the bijective singulary inversion functions; a function is a set (class) of ordered pairs. In the present article, the latter onomastic method is substantiated and generalized in two respects. First, the set of explicit constituent sets of an algebraic system is now extended to include the injective choice, or selection, functions of all additive and multiplicative identity elements of the algebraic system, belonging to its underlying sets, so that all those elements are now mentioned by the logographic name of the system. A general definition of an algebraic system is elaborated in such a way so as to make the new onomastic method universally applicable to any algebraic system.
Category: Algebra

[173] viXra:1604.0124 [pdf] submitted on 2016-04-06 07:25:43

An Axiomatic Algebraico-Functional Theory of an N-Dimensional Euclidean Affine Space

Authors: Yakov A. Iosilevskii
Comments: 68 Pages.

A concise rigorous axiomatic algebraico-functional theory of a real affine Euclidean space of any given dimension n>=1 (nDRAfES), which is an underlying discipline of differential and integral calculus and particularly of my recent theory of nD wave fields, presented in arXiv:1510.00328, is developed from an algebraic system, called an affine additive group (AAG). The latter consists of a certain underlying set of points, called an affine additive group manifold (AAGM), and of a certain commutative [abstract] additive group (CAG), called the adjoint group of the AAG, whose elements, called vectors, are related to pairs of points of AAGM by a binary surjection, satisfying the appropriate version of the Chasle, or triangle, law. An AAG is illustrated by an nD primitive (Bravais) affine lattice. When the CAG is successively supplemented by the appropriate additional attributes to become ultimately an nD real abstract vector Euclidean space (nDRAbVES), the AAG is automatically self-adjusted to all current metamorphoses of its adjoint CAG to become ultimately an nDRAfES, of which the above nDRAbVES is adjoint. Relative to its any orthonormal basis, the nDRAbVES, adjoint of the nDRAfES, is isomorphic to the nD real arithmetical vector Euclidean space (nDRArVES), whose underlying set consists of ordered n-tuples of real numbers, being coordinates of the respective abstract vectors of the underlying vector set of the nDRAbVES. A time continuum (TC) is a special interpretation of 1DRAfES. A real-valued functional form (FF) that is initially defined on a certain region of the direct product (DP) of a 1DRAfES and a nDRAfES can rigorously be mapped onto a certain real-valued FF defined on a certain region of the DP of the TC and nDRArVES and vice versa.
Category: Algebra

[172] viXra:1602.0067 [pdf] submitted on 2016-02-06 02:37:55

About "Categories and Sheaves"

Authors: Pierre-Yves Gaillard
Comments: 191 Pages.

The purpose of this text is to make a few comments about the book "Categories and Sheaves" by Kashiwara and Schapira, Springer 2006.
Category: Algebra

[171] viXra:1511.0267 [pdf] submitted on 2015-11-27 16:20:45

Algebra of Paravectors

Authors: Józef Radomański
Comments: 17 Pages.

Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually similar to vector counterparts, known from the Euclidean geometry. Paravectors meet the idea of parallelogram law, Pythagorean theorem and many other properties well-known to everyone from school.
Category: Algebra

[170] viXra:1511.0034 [pdf] submitted on 2015-11-03 11:48:25

Algebraic Structures on MOD Planes

Authors: W. B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache
Comments: 215 Pages.

Study of MOD planes happens to a very recent one. In this book, systematically algebraic structures on MOD planes like, MOD semigroups, MOD groups and MOD rings of different types are defined and studied. Such study is innovative for a large four quadrant planes are made into a small MOD planes. Several distinct features enjoyed by these MOD planes are defined, developed and described.
Category: Algebra

[169] viXra:1511.0032 [pdf] submitted on 2015-11-03 11:52:39

MOD Functions: A New Approach to Function Theory

Authors: W. B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandache
Comments: 205 Pages.

In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration.
Category: Algebra

[168] viXra:1509.0023 [pdf] submitted on 2015-09-01 04:55:38

Smarandache Fuzzy Strong Ideal and Smarandache Fuzzy N-Fold Strong Ideal of a BH-Algebra

Authors: Shahrezad Jasim Mohammed
Comments: 6 Pages.

In this paper, we define the concepts of a Q-Smarandache n-fold strong ideal and a Q-Smarandache fuzzy (strong, n-fold strong) ideal of a BH- algebra .Also, we study some properties of these fuzzy ideals.
Category: Algebra

[167] viXra:1508.0331 [pdf] submitted on 2015-08-31 06:20:44

Smarandache’s Conjecture on Consecutive Primes

Authors: Octavian Cira
Comments: 23 Pages.

Let p and q two consecutive prime numbers, where q > p.
Category: Algebra

[166] viXra:1508.0181 [pdf] submitted on 2015-08-21 21:30:47

Methods of Solving Calculus Problems

Authors: Constantin Dumitrescu, Florentin Smarandache
Comments: 241 Pages.

In this book, we discuss a succession of methods encountered in the study of high school mathematical analysis to students and teachers, to higher education entry examination candidates, to all those interested, in order to allow them to reduce as many diverse problems as possible to already own work schemes. We tried to present in a methodical manner, advantageous for the reader, the most frequent calculation methods encountered in the study of the mathematical analysis at this level. { First imprint: Metode de calcul în analiza matematică (in Romanian), INFOMED Publishing House, Craiova, Romania, 1995.}
Category: Algebra

[165] viXra:1507.0148 [pdf] submitted on 2015-07-19 17:54:56

Algebraic Problems and Exercises for High School (Sets, Sets Operations, Relations, Functions, Aspects of Combinatorics)

Authors: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache
Comments: 144 Pages. Translation from Romanian to English by Ana Maria Buzoianu

In this book, you will find algebra exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories, but can be equally helpful in a standalone workout. The book can also be used as an extracurricular source, as the reader shall find enclosed important theorems and formulas, standard definitions and notions that are not always included in school textbooks.
Category: Algebra

[164] viXra:1507.0122 [pdf] submitted on 2015-07-16 09:28:54

Theory of Abel Grassmann's Groupoids

Authors: Madad Khan, Florentin Smarandache, Saima Anis
Comments: 208 Pages.

An AG-groupoid is an algebraic structure that lies in between a groupoid and a commutative semigroup. It has many characteristics similar to that of a commutative semigroup. If we consider x^2y^2= y^2x^2, which holds for all x, y in a commutative semigroup, on the other hand one can easily see that it holds in an AG-groupoid with left identity e and in AG**-groupoids. This simply gives that how an AG-groupoid has closed connections with commutative algebras. We extend now for the first time the AG-Groupoid to the Neutrosophic AG-Groupoid. A neutrosophic AG-groupoid is a neutrosophic algebraic structure that lies between a neutrosophic groupoid and a neutrosophic commutative semigroup.
Category: Algebra

[163] viXra:1507.0089 [pdf] submitted on 2015-07-13 11:51:55

Irreducible Representations of Small Abstract Groups Computed with GAP

Authors: Richard J. Mathar
Comments: 102 Pages.

This is a table of all irreducible matrix representations of all 181 groups up to order 40, generated with the GAP software.
Category: Algebra

[162] viXra:1505.0165 [pdf] submitted on 2015-05-23 11:23:16

(T, i, f)-Neutrosophic Structures & I-Neutrosophic Structures (Revisited)

Authors: Florentin Smarandache
Comments: 7 Pages.

This paper is an improvement of our paper “(t, i, f)-Neutrosophic Structures” [1], where we introduced for the first time a new type of structures, called (t, i, f)-Neutrosophic Structures, presented from a neutrosophic logic perspective, and we showed particular cases of such structures in geometry and in algebra. In any field of knowledge, each structure is com-posed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy of the form (t, i, f) ≠ (1, 0, 0), that structure is a (t, i, f)-Neutrosophic Structure. The (t, i, f)-Neutrosophic Structures [based on the components t = truth, i = numerical indeterminacy, f = falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a + bI, where I = literal indeterminacy and In = I], that we rename as I-Neutrosophic Algebraic Structures (meaning algebraic structures based on indeterminacy “I” only). But we can combine both and obtain the (t, i, f)-I-Neutrosophic Algebraic Structures, i.e. algebraic struc-tures based on neutrosophic numbers of the form a+bI, but also having indeterminacy of the form (t, i, f) ≠ (1, 0, 0) related to the structure space (elements which only partially belong to the space, or elements we know noth-ing if they belong to the space or not) or indeterminacy of the form (t, i, f) ≠ (1, 0, 0) related to at least one axiom (or law) acting on the structure space. Then we extend them to Refined (t, i, f)- Refined I-Neutrosophic Algebra-ic Structures.
Category: Algebra

[161] viXra:1504.0192 [pdf] submitted on 2015-04-23 20:54:28

Fuzzy Abel Grassmann Groupoids

Authors: Madad Khan, Florentin Smarandache, Tariq Aziz
Comments: 82 Pages.

In this book, we introduce the concept of (/in, /in /or q_k)-fuzzy ideals and (/in_gamma, in_gamma /or q_delta)-fuzzy ideals in a non-associative algebraic structure called Abel Grassmann’s groupoid, discuss several important features of a regular AG-groupoid, investigate some characterizations of regular and intra-regular AG-groupoids and so on.
Category: Algebra

[160] viXra:1504.0032 [pdf] submitted on 2015-04-03 18:57:03

Smallest Symmetric Supergroups Of the Abstract Groups Up to Order 37

Authors: Richard J. Mathar
Comments: 18 Pages.

Each finite group is a subgroup of some symmetric group, known as the Cayley theorem. We find the symmetric group of smallest order which hosts the finite groups in that sense for most groups of order less than 37. For each of these small groups this is made concrete by providing a permutation group with a minimum number of moved elements in terms of a list of generators of the permutation group in reduced cycle notation.
Category: Algebra

[159] viXra:1503.0146 [pdf] submitted on 2015-03-18 11:30:24

Infinite Quaternion Pseudo Rings Using [0, n)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 231 Pages.

Here finite real quaternion rings, and finite complex modulo integer quaternion rings, plus neutrosophic finite quaternion rings, and complex neutrosophic quaternion rings for the first time are introduced and analyzed. All these rings behave in a very unique way.
Category: Algebra

[158] viXra:1503.0072 [pdf] submitted on 2015-03-10 19:26:40

Neutrosophic Crisp Set Theory

Authors: A. A. Salama, Florentin Smarandache
Comments: 182 Pages.

In this book the authors introduce and study the following notions: Neutrosophic Crisp Points, Neutrosophic Crisp Relations, Neutrosophic Crisp Sets, Neutrosophic Set Generated by (Characteristic Function), α-cut Level for Neutrosophic Sets, Neutrosophic Crisp Continuous Function, Neutrosophic Crisp Compact Spaces, Neutrosophic Crisp Nearly Open Sets, Neutrosophic Crisp Ideals, Neutrosophic Crisp Filter, Neutrosophic Crisp Local Functions, Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals, Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity, Neutrosophic Topological Region, Neutrosophic Closed Set and Neutrosophic Continuous Function, etc. They compute the distances between neutrosophic sets and extend it to Neutrosophic Hesitancy Degree. The authors also generalize the Crisp Topological Space and Intuitionistic Topological Space to the notion of Neutrosophic Crisp Topological Space. At the end, they present applications to Neutrosophic Database, and show a security scheme based on Public Key Infrastructure (PKI) using Neutrosophic Logic Manipulation. The authors utilize neutrosophic sets in order to analyze social networks data conducted through learning activities, and for the Geographical Information Systems (GIS) they employ fundamental concepts and properties of a Neutrosophic Spatial Region.
Category: Algebra

[157] viXra:1501.0240 [pdf] submitted on 2015-01-27 20:54:40

Distance in Matrices and Their Applications to Fuzzy Models and Neutrosophic Models

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K.
Comments: 169 Pages.

The distance between row matrices was defined by Hamming. However the distance between column matrices or for that matter and any two matrices of same order are not defined systematically. Here in this book this concept is defined and it can help in the analysis of storage of data, deviation in eigen values and in difference in opinions of experts using some fuzzy models. Some open problems are suggested.
Category: Algebra

[156] viXra:1412.0253 [pdf] submitted on 2014-12-27 09:42:49

A Nonstandard Cubic Equation

Authors: J. S. Markovitch
Comments: 5 Pages.

A nonstandard cubic equation is shown to have an unusually economical solution, where this solution incorporates an angle that serves as the equation's discriminant.
Category: Algebra

[155] viXra:1412.0252 [pdf] submitted on 2014-12-27 10:24:17

A Substitution Map Applied to the Simplest Algebraic Identities

Authors: J. S. Markovitch
Comments: 3 Pages.

A substitution map applied to the simplest algebraic identities is shown to yield second- and third-order equations that share an interesting property at the minimum 137.036.
Category: Algebra

[154] viXra:1411.0159 [pdf] submitted on 2014-11-15 09:45:29

Algebraic Structures on the Fuzzy Interval [0, 1)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 252 Pages.

In this book we introduce several algebraic structures on the special fuzzy interval [0, 1). This study is different from that of the algebraic structures using the interval [0, n) n ≠ 1, as these structures on [0, 1) has no idempotents or zero divisors under ×. Further [0, 1) under product × is only a semigroup. However by defining min(or max) operation in [0, 1); [0, 1) is made into a semigroup.
Category: Algebra

[153] viXra:1411.0158 [pdf] submitted on 2014-11-15 09:47:23

Algebraic Structures Using [0,n)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 218 Pages.

In this book authors for the first time introduce a new method of building algebraic structures on the interval [0, n). This study is interesting and innovative. However, [0, n) is a semigroup under product, × modulo n and a semigroup under min or max operation. Further [0, n) is a group under addition modulo n.
Category: Algebra

[152] viXra:1411.0154 [pdf] submitted on 2014-11-15 10:03:06

Two Hundred Conjectures and One Hundred and Fifty Open Problems on Fermat Pseudoprimes (Collected Papers)

Authors: Marius Coman
Comments: 132 Pages.

Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me.
Category: Algebra

[151] viXra:1411.0153 [pdf] submitted on 2014-11-15 10:04:38

Conjectures on Primes and Fermat Pseudoprimes, Many Based on Smarandache Function (Collected Papers)

Authors: Marius Coman
Comments: 84 Pages.

The present collected papers aims to show new applications of Smarandache function in the study of some well known classes of numbers, like prime numbers, Poulet numbers, Carmichael numbers, Sophie Germain primes etc.
Category: Algebra

[150] viXra:1411.0149 [pdf] submitted on 2014-11-15 10:11:33

Conjectures on Primes and Pseudoprimes Based on Smarandache Function

Authors: Marius Coman
Comments: 27 Pages.

It is always difficult to talk about arithmetic, because those who do not know what is about, nor do they understand in few sentences, no matter how inspired these might be, and those who know what is about, do no need to be told what is about.
Category: Algebra

[149] viXra:1411.0148 [pdf] submitted on 2014-11-15 10:12:32

Conjectures on Some Well Known and Less Known Types of Primes (Collected Papers)

Authors: Marius Coman
Comments: 43 Pages.

This collection of papers brings together several articles regarding primes, submitted by the author to the preprint scientific database Vixra.
Category: Algebra

[148] viXra:1411.0147 [pdf] submitted on 2014-11-15 10:13:43

Conjectures on Types of Primes and Fermat Pseudoprimes, Many Based on Smarandache Function (Collected Papers)

Authors: Marius Coman
Comments: 81 Pages.

This collection of articles seeks to expand the knowledge on some well known classes of primes, like for instance Sophie Germain primes, but also to define new classes of primes, like for instance “ACPOW chains of primes”, or classes of integers directly related to primes, like for instance “chameleonic numbers”.
Category: Algebra

[147] viXra:1410.0169 [pdf] submitted on 2014-10-26 14:52:01

New Research on Neutrosophic Algebraic Structures

Authors: Mumtaz Ali, Florentin Smarandache, Muhammad Shabir
Comments: 333 Pages.

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures in a broader sense. The indeterminacy element “I“ gives rise to a bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three categories such as: neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical algebraic structures respectively. This reveals the fact that a classic algebraic structure is a part of the neutrosophic algebraic structures. This opens a new way for the researcher to think in a broader way to visualize these vast neutrosophic algebraic structures.
Category: Algebra

[146] viXra:1409.0054 [pdf] submitted on 2014-09-08 08:51:46

Generalization of Several Soft Neutrosophic Algebraic Structures

Authors: Mumtaz Ali, Florentin Smarandache, Muhammad Shabir
Comments: 288 Pages.

In this book, the authors define several new types of soft neutrosophic algebraic structures over neutrosophic algebraic structures and we study their generalizations. These soft neutrosophic algebraic structures are basically parameterized collections of neutrosophic sub-algebraic structures of the neutrosophic algebraic structure. An important feature of this book is that the authors introduce the soft neutrosophic group ring, soft neutrosophic semigroup ring with its generalization, and soft mixed neutrosophic N-algebraic structure over neutrosophic group ring, then the neutrosophic semigroup ring and mixed neutrosophic N-algebraic structure respectively.
Category: Algebra

[145] viXra:1408.0182 [pdf] submitted on 2014-08-27 06:03:34

Towards a Closed-Form Formula for Roots of Polynomials

Authors: Andrzej Okninski
Comments: 4 Pages.

In the present work we make a small step towards inding explicit formula for polynomial roots. xplicit formulae for simple real roots, based on the Frobenius companion matrix, are derived. After some modifications the method can be used to compute multiple or complex roots.
Category: Algebra

[144] viXra:1408.0124 [pdf] submitted on 2014-08-18 12:19:02

Algebraic Structures on Real and Neutrosophic Semi Open Squares

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 206 Pages.

Here for the first time we introduce the semi open square using modulo integers. This square under addition modulo ‘n’ is a group, and however under product ‘x’ this semi open square is only a semigroup as under ‘x’ the square has infinite number of zero divisors. We define the new type of rings called pseudo rings. Several open problems are suggested.
Category: Algebra

[143] viXra:1408.0118 [pdf] submitted on 2014-08-18 12:16:59

Algebraic Structures on Finite Complex Modulo Integer Interval C([0, n))

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 235 Pages.

In this book the authors introduce the notion of finite complex modulo integer intervals. Finite complex modulo integers were introduced by the authors in 2011. Now, using this finite complex modulo integer intervals several algebraic structures are built. Open problems are suggested.
Category: Algebra

Replacements of recent Submissions

[17] viXra:1604.0124 [pdf] replaced on 2016-05-02 05:47:55

An Axiomatic Algebraico-Functional Theory of an N-Dimensional Euclidean Affine Space

Authors: Yakov A. Iosilevskii
Comments: 68 Pages.

A concise rigorous axiomatic algebraico-functional theory of a real affine Euclidean space of any given dimension n>=1 (nDRAfES), which is an underlying discipline of differential and integral calculus and particularly of my recent theory of nD wave fields, presented in arXiv:1510.00328, is developed from an algebraic system, called an affine additive group (AAG). The latter consists of a certain underlying set of points, called an affine additive group manifold (AAGM), and of a certain commutative [abstract] additive group (CAG), called the adjoint group of the AAG, whose elements, called vectors, are related to pairs of points of AAGM by a binary surjection, satisfying the appropriate version of the Chasle, or triangle, law. An AAG is illustrated by an nD primitive (Bravais) affine lattice. When the CAG is successively supplemented by the appropriate additional attributes to become ultimately an nD real abstract vector Euclidean space (nDRAbVES), the AAG is automatically self-adjusted to all current metamorphoses of its adjoint CAG to become ultimately an nDRAfES, of which the above nDRAbVES is adjoint. Relative to its any orthonormal basis, the nDRAbVES, adjoint of the nDRAfES, is isomorphic to the nD real arithmetical vector Euclidean space (nDRArVES), whose underlying set consists of ordered n-tuples of real numbers, being coordinates of the respective abstract vectors of the underlying vector set of the nDRAbVES. A time continuum (TC) is a special interpretation of 1DRAfES. A real-valued functional form (FF) that is initially defined on a certain region of the direct product (DP) of a 1DRAfES and a nDRAfES can rigorously be mapped onto a certain real-valued FF defined on a certain region of the DP of the TC and nDRArVES and vice versa.
Category: Algebra

[16] viXra:1506.0145 [pdf] replaced on 2015-07-08 13:53:53

The Solution of Some Very Old Transcendence Conjectures Over Field Q by Using Algebra of Dedekind Hyperreals

Authors: Jaykov Foukzon
Comments: 52 Pages.

In this paper the important applications of the Dedekind completion *R_d in transcendental number theory is considered. Given any analytic function of one complex variable f ∈Q[[z]],we investigate the arithmetic nature of the values of f(z) at transcendental points e^n. Main results are: (i) the both numbers e+pi and e-pi are irrational, (ii) number e^e are transcendental
Category: Algebra

[15] viXra:1503.0072 [pdf] replaced on 2015-04-01 11:30:50

Neutrosophic Crisp Set Theory

Authors: A. A. Salama, Florentin Smarandache
Comments: 182 Pages.

In this book the authors introduce and study the following notions: Neutrosophic Crisp Points, Neutrosophic Crisp Relations, Neutrosophic Crisp Sets, Neutrosophic Set Generated by (Characteristic Function), α-cut Level for Neutrosophic Sets, Neutrosophic Crisp Continuous Function, Neutrosophic Crisp Compact Spaces, Neutrosophic Crisp Nearly Open Sets, Neutrosophic Crisp Ideals, Neutrosophic Crisp Filter, Neutrosophic Crisp Local Functions, Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals, Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity, Neutrosophic Topological Region, Neutrosophic Closed Set and Neutrosophic Continuous Function, etc. They compute the distances between neutrosophic sets and extend it to Neutrosophic Hesitancy Degree. The authors also generalize the Crisp Topological Space and Intuitionistic Topological Space to the notion of Neutrosophic Crisp Topological Space. At the end, they present applications to Neutrosophic Database, and show a security scheme based on Public Key Infrastructure (PKI) using Neutrosophic Logic Manipulation. The authors utilize neutrosophic sets in order to analyze social networks data conducted through learning activities, and for the Geographical Information Systems (GIS) they employ fundamental concepts and properties of a Neutrosophic Spatial Region.
Category: Algebra

[14] viXra:1412.0253 [pdf] replaced on 2015-06-14 16:16:42

A Nonstandard Cubic Equation

Authors: J. S. Markovitch
Comments: 5 Pages.

A nonstandard cubic equation is shown to have an unusually economical solution, where this solution incorporates an angle that serves as the equation's discriminant.
Category: Algebra

[13] viXra:1412.0253 [pdf] replaced on 2015-01-12 08:33:03

A Nonstandard Cubic Equation

Authors: J. S. Markovitch
Comments: 5 Pages.

A nonstandard cubic equation is shown to have an unusually economical solution, where this solution incorporates an angle that serves as the equation's discriminant.
Category: Algebra

[12] viXra:1409.0054 [pdf] replaced on 2014-09-08 12:19:39

Soft Neutrosophic Algebraic Structures and Their Generalization, Vol. 2

Authors: Mumtaz Ali, Florentin Smarandache, Muhammad Shabir
Comments: 288 Pages.

In this book, the authors define several new types of soft neutrosophic algebraic structures over neutrosophic algebraic structures and we study their generalizations. These soft neutrosophic algebraic structures are basically parameterized collections of neutrosophic sub-algebraic structures of the neutrosophic algebraic structure. An important feature of this book is that the authors introduce the soft neutrosophic group ring, soft neutrosophic semigroup ring with its generalization, and soft mixed neutrosophic N-algebraic structure over neutrosophic group ring, then the neutrosophic semigroup ring and mixed neutrosophic N-algebraic structure respectively.
Category: Algebra