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2018 - 1801(1) - 1802(3) - 1804(2)

Any replacements are listed farther down

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

**Authors:** Said Amharech

**Comments:** 5 Pages. the work is written in french

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

**Category:** Algebra

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

**Authors:** Antoine Balan

**Comments:** 4 pages, written in french

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

**Category:** Algebra

[267] **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

[266] **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

[265] **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

[264] **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

[263] **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

[262] **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

[261] **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

[260] **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

[259] **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

[258] **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

[257] **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

[256] **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

[255] **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

[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

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

**Authors:** Antoine Balan

**Comments:** 4 pages, written in french

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

**Category:** Algebra

[36] **viXra:1709.0131 [pdf]**
*replaced on 2018-03-09 07:22:47*

**Authors:** Charanjeet Singh Bansrao

**Comments:** 8 Pages.

The Difference of Any Real Transcendental Number and Complex Number E^i is Always a Complex Transcendental Number.

**Category:** Algebra

[35] **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

[34] **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

[33] **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

[32] **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

[31] **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

[30] **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

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

**Authors:** Ameet Sharma

**Comments:** 17 Pages.

**Category:** Algebra

[28] **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

[27] **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

[26] **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

[25] **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

[24] **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

[23] **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

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

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

**Category:** Algebra

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

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

**Category:** Algebra

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

**Authors:** Ameet Sharma

**Comments:** 15 Pages.

**Category:** Algebra

[19] **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

[18] **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