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Any replacements are listed further down

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

**Authors:** James A. Smith

**Comments:** 5 Pages.

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

**Category:** Algebra

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

**Authors:** Jesús Álvarez Lobo

**Comments:** 3 Pages. Spanish.

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

**Category:** Algebra

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

**Authors:** Jesús Álvarez Lobo

**Comments:** 20 Pages.

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

**Category:** Algebra

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

**Authors:** A.Polorovskii

**Comments:** 2 Pages.

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

**Category:** Algebra

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

**Authors:** Cres Huang

**Comments:** Pages.

A simple way of approximating π by slice.

**Category:** Algebra

[263] **viXra:1712.0140 [pdf]**
*submitted on 2017-12-06 10:51:29*

**Authors:** Richard Wayte

**Comments:** 8 Pages.

A solution of Fermat’s Last Theorem is given, using elementary function arithmetic and inference from worked examples.

**Category:** Algebra

[262] **viXra:1710.0247 [pdf]**
*submitted on 2017-10-22 16:35:07*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 1 Page. None.

Mathematical Closure.

**Category:** Algebra

[261] **viXra:1709.0131 [pdf]**
*submitted on 2017-09-11 11:21:53*

**Authors:** Charanjeet Singh Bansrao

**Comments:** 4 Pages.

The difference of any real transcendental number and complex number e^i is always a complex transcendental number.

**Category:** Algebra

[260] **viXra:1708.0417 [pdf]**
*submitted on 2017-08-28 08:38:14*

**Authors:** Edgar Valdebenito

**Comments:** 11 Pages.

This note presents the roots (in radicals) of the equations:x^5+10*x^3+20*x-1=0 , x^5-20*x^4-10*x^2-1=0 and related fractals.

**Category:** Algebra

[259] **viXra:1708.0256 [pdf]**
*submitted on 2017-08-21 18:38:34*

**Authors:** F.L.B.Périat

**Comments:** 3 Pages.

Proposition sur l'infini imaginé comme un espace vectoriel, permettant par distribution des vecteurs de démontrer l'irrationalité de certaines valeurs.

**Category:** Algebra

[258] **viXra:1708.0188 [pdf]**
*submitted on 2017-08-16 12:49:22*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

This note presents the real roots (in radicals)of the equation:x^6-3x^4-2x^3+9x^2+3x-26=0.

**Category:** Algebra

[257] **viXra:1706.0508 [pdf]**
*submitted on 2017-06-27 07:33:39*

**Authors:** Orgest ZAKA

**Comments:** 11 Pages.

In this article, starting from geometrical considerations, he was born with the idea of 3D matrices, which have developed in this article. A problem here was the definition of multiplication, which we have given in analogy with the usual 2D matrices. The goal here is 3D matrices to be a generalization of 2D matrices. Work initially we started with 3×3×3 matrix, and then we extended to m×n×p matrices. In this article, we give the meaning of 3D matrices. We also defined two actions in this set. As a result, in this article, we have reached to present 3-dimensional unitary ring matrices with elements from a field F.

**Category:** Algebra

[256] **viXra:1705.0019 [pdf]**
*submitted on 2017-05-02 04:07:01*

**Authors:** Robert B. Easter, Eckhard Hitzer

**Comments:** 25 Pages. Published online First in AACA, 20th April 2017. DOI: 10.1007/s00006-017-0784-0. 2 tables, 26 references.

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the $\mathcal{G}_{8, 2}$ Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general (quartic) Darboux cyclide surfaces in Euclidean 3D space, including circular tori and all quadrics, and all surfaces formed by their inversions in spheres. Dupin cyclides are quartic surfaces formed by inversions in spheres of torus, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversions in spheres that are centered on points of other surfaces. All DCGA entities can be transformed by versors, and reflected in spheres and planes.
Keywords: Conformal geometric algebra, Darboux Dupin cyclide, Quadric
surface
Math. Subj. Class.: 15A66, 53A30, 14J26, 53A05, 51N20, 51K05

**Category:** Algebra

[255] **viXra:1704.0273 [pdf]**
*submitted on 2017-04-21 12:51:06*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper I set down the Quantum Geometric Algebra Algorithm standing for the Theory of Quantum Computational Complexity

**Category:** Algebra

[254] **viXra:1702.0234 [pdf]**
*submitted on 2017-02-18 21:44:17*

**Authors:** Robert B. Easter

**Comments:** 8 Pages.

This note very briefly describes or sketches the general ideas of some applications of the G(p,q) Geometric Algebra (GA) of a complex vector space C^(p,q) of signature (p,q), which is also known as the Clifford algebra Cl(p,q). Complex number scalars are only used for the anisotropic dilation (directed scaling) operation and to represent infinite distances, but otherwise only real number scalars are used. The anisotropic dilation operation is implemented in Minkowski spacetime as hyperbolic rotation (boost) by an imaginary rapidity (+/-)f = atanh(sqrt(1-d^2)) for dilation factor d>1, using +f in the Minkowski spacetime of signature (1,n) and -f in the signature (n,1).
The G(k(p+q+2),k(q+p+2)) Mother Algebra of CGA (k-MACGA) is a generalization of G(p+1,q+1) Conformal Geometric Algebra (CGA) having k orthogonal G(p+1,q+1):p>q Euclidean CGA (ECGA) subalgebras and k orthogonal G(q+1,p+1) anti-Euclidean CGA (ACGA) subalgebras with opposite signature. Any k-MACGA has an even 2k total count of orthogonal subalgebras and cannot have an odd 2k+1 total count of orthogonal subalgebras.
The more generalized G(l(p+1)+m(q+1),l (q+1)+m(p+1)):p>q k-CGA algebra, for even or odd k=l+m, has any l orthogonal G(p+1,q+1) ECGA subalgebras and any m orthogonal G(q+1,p+1) ACGA subalgebras with opposite signature. Any 2k-CGA with even 2k orthogonal subalgebras can be represented as a k-MACGA with different signature, requiring some sign changes.
All of the orthogonal CGA subalgebras are corresponding by representing the same vectors, geometric entities, and transformation versors in each CGA subalgebra, which may differ only by some sign changes.
A k-MACGA or a 2k-CGA has even-grade 2k-vector geometric inner product null space (GIPNS) entities representing general even-degree 2k polynomial implicit hypersurface functions F for even-degree 2k hypersurfaces, usually in a p-dimensional space or (p+1)-spacetime. Only a k-CGA with odd k has odd-grade k-vector GIPNS entities representing general odd-degree k polynomial implicit hypersurface functions F for odd-degree k hypersurfaces, usually in a p-dimensional space or (p+1)-spacetime. In any k-CGA, there are k-blade GIPNS entities representing the usual G(p+1,q+1) CGA GIPNS 1-blade entities, but which are representing an implicit hypersurface function F^k with multiplicity k and the k-CGA null point entity is a k-point entity. In the conformal Minkowski spacetime algebras G(p+1,2) and G(2,p+1), the null 1-blade point embedding is a GOPNS null 1-blade point entity but is a GIPNS null 1-blade hypercone entity.

**Category:** Algebra

[253] **viXra:1702.0057 [pdf]**
*submitted on 2017-02-03 16:53:05*

**Authors:** William O. Straub

**Comments:** 6 Pages.

Elementary overview of the Levi-Civita symbol, emphasizing its dependence on the Kronecker delta

**Category:** Algebra

[252] **viXra:1702.0038 [pdf]**
*submitted on 2017-02-02 16:32:16*

**Authors:** Martin Erik Horn

**Comments:** 12 Pages.

Using Geometric Algebra consistent solutions of inconsistent systems of linear equations can be found.

**Category:** Algebra

[251] **viXra:1612.0259 [pdf]**
*submitted on 2016-12-16 07:05:01*

**Authors:** Claude Michael Cassano

**Comments:** 3 Pages.

A two-dimensional vector space algebra with identity 2x2 matrix basis matrix multiplication homomorphism
There exists a homomorphism between any two-dimensional vector space algebra with identity and a 2x2 matrix basis under ordinary matrix multiplication.
This is a statement of constructive existence of an algebra.
Given that the vector space of the algebra is known to be 2-dimensional, the algebra product determines the constants: A,B,b ; determining the basis of the algebra.
And showing that the basis of a two-dimensional vector space unitary algebra is a cyclic group of order 2

**Category:** Algebra

[250] **viXra:1612.0221 [pdf]**
*submitted on 2016-12-12 03:18:52*

**Authors:** Robert Benjamin Easter, Eckhard Hitzer

**Comments:** 6 Pages. Proceedings of SSI 2016, Session SS11, pp. 866-871, 6-8 Dec. 2016, Ohtsu, Shiga, Japan, 10 color figures.

The G_{8,2} Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them. Operations include projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced that include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects.

**Category:** Algebra

[249] **viXra:1611.0078 [pdf]**
*submitted on 2016-11-05 17:24:07*

**Authors:** Carauleanu Marc

**Comments:** 3 Pages.

In this paper, we prove interesting alternative representations of the simple fraction x/2 where x is a real number using complex numbers.

**Category:** Algebra

[248] **viXra:1610.0353 [pdf]**
*submitted on 2016-10-29 08:05:38*

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper, We present a new method to compute the determinant of a 4 × 4 matrix, that is very simplest than previous methods in this subject. This method is obtained by a new definition of fraction and also by using the Dodgson’s condensation method and Salihu’s method.

**Category:** Algebra

[247] **viXra:1610.0178 [pdf]**
*submitted on 2016-10-16 13:16:55*

**Authors:** W. B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandach

**Comments:** 262 Pages.

In this book for the first time authors describe and develop the new notion of MOD natural neutrosophic semirings using Z^I_n, C_I(Zn),

**Category:** Algebra

[246] **viXra:1610.0118 [pdf]**
*submitted on 2016-10-11 13:57:41*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆U . Let ∆O be the restriction of ∆U to determinants of sums of symmetric normal matrices. In this paper, we conjecture that ∆O has the same boundary as ∆U. We prove the conjecture for the cases: 1) at least one of the two matrices has just one eigenvalue, 2) at least one of the two matrices has distinct eigenvalues. The implication of this theorem is that proving the Marcus-de Oliveira conjecture for symmetric normal matrices would prove it for the general case. This paper builds on work in [1].

**Category:** Algebra

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

**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

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

**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

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

**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

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

**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

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

**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

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

**Authors:** T.Nakashima

**Comments:** 1 Page.

Aﬃrmative resolve of Kothe conjecture

**Category:** Algebra

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

**Authors:** T.Nakashima

**Comments:** 1 Page.

The counter example of Jacobson conjecture

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Paul Erdös has proposed the following problem.

**Category:** Algebra

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

**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

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

**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

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

Où se trouve la faute ? (equations diophantiennes).

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

This article is on P-Q Relationships and Sequences.

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Cet article généralise cerrtains résultats sur les nédiannes.

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

This article is on Sequences of Sub-Sequences.

**Category:** Algebra

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

**Authors:** C. Dumirescu, N. VÎrlan, Șt. Zamfir, E. RĂdescu, N. RĂdescu, Florentin Smarandache

**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

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

**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

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 16 Pages.

What are the instructor's general responsabilities ?

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

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

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

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

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 6 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Cet article presente une généralisation de l’inégalité de Minkowski.

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Cet article presente une généralisation de
l’inegalité de Tcebychev.

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

Cet article presente une généralisation d’une théorème de Carnot.

**Category:** Algebra

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

**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

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

**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

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 7 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 7 Pages.

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

**Category:** Algebra

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 5 Pages.

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

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

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

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

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

**Category:** Algebra

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

**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

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Vom construi urmatoare1e functii pe care le numim prime.

**Category:** Algebra

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

We construct the system of n+2 axioms.

**Category:** Algebra

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

**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

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

**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

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

**Authors:** Florentin Smarandache

**Comments:** 9 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

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

**Category:** Algebra

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

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Où se trouve la faute sur les integrales?

**Category:** Algebra

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

**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

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

**Authors:** T.Nakashima

**Comments:** 1 Page.

Schanuel’s conjecture’s partial resolve

**Category:** Algebra

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

**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

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

**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

[34] **viXra:1709.0131 [pdf]**
*replaced on 2018-01-09 00:03:03*

**Authors:** Charanjeet Singh Bansrao

**Comments:** 4 Pages.

The difference of any real transcendental number and complex number is always a complex transcendental number.

**Category:** Algebra

[33] **viXra:1610.0353 [pdf]**
*replaced on 2018-01-09 07:48:52*

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper, we will present a new method to compute the determinant of a square matrix of order 4.

**Category:** Algebra

[32] **viXra:1610.0353 [pdf]**
*replaced on 2017-08-12 12:40:38*

**Authors:** Reza Farhadian

**Comments:** 4 Pages.

In this paper, we present a new method to compute the determinant of a real matrix of order 4.

**Category:** Algebra

[31] **viXra:1610.0118 [pdf]**
*replaced on 2018-02-02 02:36:20*

**Authors:** Ameet Sharma

**Comments:** 18 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆ and ∆S. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

**Category:** Algebra

[30] **viXra:1610.0118 [pdf]**
*replaced on 2018-01-06 07:21:47*

**Authors:** Ameet Sharma

**Comments:** 18 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆ and ∆S. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

**Category:** Algebra

[29] **viXra:1610.0118 [pdf]**
*replaced on 2018-01-05 07:10:19*

**Authors:** Ameet Sharma

**Comments:** 18 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆ and ∆S. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

**Category:** Algebra

[28] **viXra:1610.0118 [pdf]**
*replaced on 2018-01-04 18:54:29*

**Authors:** Ameet Sharma

**Comments:** 17 Pages.

**Category:** Algebra

[27] **viXra:1610.0118 [pdf]**
*replaced on 2017-12-31 12:26:50*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [2].

**Category:** Algebra

[26] **viXra:1610.0118 [pdf]**
*replaced on 2017-12-29 04:35:01*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [2].

**Category:** Algebra

[25] **viXra:1610.0118 [pdf]**
*replaced on 2017-12-25 12:29:29*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [2].

**Category:** Algebra

[24] **viXra:1610.0118 [pdf]**
*replaced on 2017-12-23 04:52:17*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

**Category:** Algebra

[23] **viXra:1610.0118 [pdf]**
*replaced on 2017-12-22 23:48:18*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

**Category:** Algebra

[22] **viXra:1610.0118 [pdf]**
*replaced on 2017-12-21 01:01:40*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. Let ∆S be the restriction of ∆ to determinants of sums of symmetric normal matrices. This paper focuses on boundary matrices of ∆. We derive some properties of boundary matrices and boundary points. We conjecture that ∂∆ ⊆ ∂∆S. Speculations on how to prove this conjecture are given. We also present a second conjecture with regards to the form of normal matrices with magnitude symmetry. This paper builds on work in [1].

**Category:** Algebra

[21] **viXra:1610.0118 [pdf]**
*replaced on 2017-05-02 16:15:55*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

**Category:** Algebra

[20] **viXra:1610.0118 [pdf]**
*replaced on 2017-04-24 20:06:06*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

**Category:** Algebra

[19] **viXra:1610.0118 [pdf]**
*replaced on 2017-04-13 15:18:44*

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

**Category:** Algebra

[18] **viXra:1609.0262 [pdf]**
*replaced on 2016-11-06 01:57:56*

**Authors:** Eli Halylaurin

**Comments:** 4 Pages. This document is french written.

You will find here an attempt to demonstrate a general structure theorem for abelian groups (finite or infinite). 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 seen as included, modulo isomorphism, in a direct product of groups that can be represented upon a circle or a line.

**Category:** Algebra

[17] **viXra:1609.0262 [pdf]**
*replaced on 2016-10-21 13:25:57*

**Authors:** Eli Halylaurin

**Comments:** 4 Pages. This document is french written.

You will find here an attempt to demonstrate a general structure theorem for abelian groups (finite or infinite). 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 seen as included, modulo isomorphism, in a direct product of groups that can be represented upon a circle or a line.

**Category:** Algebra