# Algebra

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## Recent submissions

Any replacements are listed farther down

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

### None Complex Numbers

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

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

[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

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

Authors: James A. Smith

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

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

Authors: Jesús Álvarez Lobo

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

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

Authors: Jesús Álvarez Lobo

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

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

Authors: A.Polorovskii

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

### Approximate A Slice of Pi Essay

Authors: Cres Huang

A simple way of approximating &pi; by slice.
Category: Algebra

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

### On Fermat's Last Theorem. Revised.

Authors: Richard Wayte

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

### Mathematical Closure

Authors: Paris Samuel Miles-Brenden

Mathematical Closure.
Category: Algebra

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

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

Authors: Charanjeet Singh Bansrao

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

### The Quintic Equation: X^5+10*x^3+20*x-1=0

Authors: Edgar Valdebenito

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

### Discussion Sur Les Structures Algébriques Des Infinis Réels et L’irrationalité de la Constante D’Euler-Mascheroni

Authors: F.L.B.Périat

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

### Real Roots of the Equation: X^6-3x^4-2x^3+9x^2+3x-26=0

Authors: Edgar Valdebenito

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

### 3D Matrix Ring with a “Common” Multiplication

Authors: Orgest ZAKA

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

### Double Conformal Geometric Algebra (long CGI2016/GACSE2016 paper in SI of AACA)

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

### On the K-Macga Mother Algebras of Conformal Geometric Algebras and the K-Cga Algebras

Authors: Robert B. Easter

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

### Levi-Civita Rhymes with Lolita

Authors: William O. Straub

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

### Solving Inconsistent Systems of Linear Equations (Part I)

Authors: Martin Erik Horn

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

### A Two-Dimensional Vector Space Algebra with Identity 2x2 Matrix Basis Matrix Multiplication Homomorphism

Authors: Claude Michael Cassano

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

### Conic and Cyclidic Sections in Double Conformal Geometric Algebra G_{8,2}

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

### Alternative Representations of X/2

Authors: Carauleanu Marc

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

### Duplex Fraction Method To Compute The Determinant Of A 4 × 4 Matrix

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

### MOD Natural Neutrosophic Semirings

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

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), _I, _I, _I and _I. Several interesting properties about this structure is derived. Using these MOD natural neutrosophic semirings MOD natural neutrosophic matrix semirings and MOD natural neutrosophic polynomial semirings and defined and described. Special elements of these structures are analysed. When MOD intervals [0, n) and MOD natural neutrosophic intervals [0, n) are used we see the MOD semirings do not in general satisfy the associative laws and the distributive laws leading to the definition of pseudo semirings of infinite order. These are also introduced in this book. We also define and develop MOD subset pseudo semiring and MOD subset natural neutrosophic pseudo semirings. This study is innovative and interesting by providing a large class of MOD pseudo semirings. Special elements in them are analysed. Using these MOD subset matrix pseudo semirings and MOD subset polynomial pseudo semirings and developed. They enjoy very many special features. Several problems are suggested and these notions will certainly attract semiring theorists.
Category: Algebra

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### 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

[244] 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

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

### Unit Graph of Some Finite Group Zn, Cn and Dn

Authors: A. D. Godase, M. B. Dhakne

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

### Solving Boolean Equation

Authors: Oh Jung Uk

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

### On Neutrosophic Quadruple Algebraic Structures

Authors: S.A. Akinleye, F. Smarandache, A.A.A. Agboola

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

### Aﬃrmative Resolve of Kothe Conjecture

Authors: T.Nakashima

Aﬃrmative resolve of Kothe conjecture
Category: Algebra

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

### The Counter Example of Jacobson Conjecture

Authors: T.Nakashima

The counter example of Jacobson conjecture
Category: Algebra

## Replacements of recent Submissions

[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

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

Authors: Charanjeet Singh Bansrao

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

### The Difference of Any Real Transcendental Number and Complex Number is Always a Complex Transcendental Number.

Authors: Charanjeet Singh Bansrao

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

### Duplex Fraction Method To Compute The Determinant Of A 4 × 4 Matrix

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

### Duplex Fraction Method To Compute The Determinant Of A 4 × 4 Matrix

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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 2017-12-31 12:26:50

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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 2017-04-24 20:06:06

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

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

### Symmetric Normal Matrices and the Marcus-de Oliveira Determinantal Conjecture

Authors: Ameet Sharma

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

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

### Structure Theorem for Abelian Groups (General Case, Finite or Infinite)

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

### Structure Theorem for Abelian Groups (General Case, Finite or Infinite)

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