Algebra

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Recent submissions

Any replacements are listed further down

[147] viXra:1407.0167 [pdf] submitted on 2014-07-21 16:17:56

Distribution of Prime Numbers

Authors: Ouannas Moussa
Comments: 08 Pages.

In this paper I present the distribution of prime numbers which was treated in many researches by studying the function of Riemann; because it has a remarkable property; its non trivial zeros are prime numbers; but in this work I will show that we can find the distribution of prime numbers on remaining in natural numbers only.
Category: Algebra

[146] viXra:1406.0189 [pdf] submitted on 2014-06-30 20:48:05

The Generalized Continuum Hypothesis

Authors: Daniel Cordero Grau
Comments: 2 Pages.

In this paper we prove the generalized continuum hypothesis by categorical logic which is as the theory of categories a first-order theory and so is a logically complete theory. To prove that the theory of initial ordinals and the theory of transfinite cardinals are isomorphic, that is, to prove that the theorems of the theory of transfinite cardinals are also theorems of the theory of initial ordinals and conversely, the theorems of the theory of initial ordinals are also theorems of the theory of transfinite cardinals, since the theories of isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic and the theories of isomorphic categories are isomorphic theories which is the fundamental theorem of categorical logic for isomorphic categories are isomorphic structures, we prove these structures are isomorphic categories. We use the definition of a theory, the definition of an isomorphism of structures in its equivalent form the definition of an isomorphism of categories, the definition of a functor, the axioms of the theory of categories and the axioms of mathematical logic, so as to apply both the theorem on the comparablity of initial ordinals to the theory of transfinite cardinals and the fundamental theorem of transfinite cardinal arithmetic to the theory of initial ordinals.
Category: Algebra

[145] viXra:1406.0183 [pdf] submitted on 2014-06-30 08:46:28

Plots of Cycle Graphs of the Finite Groups up to Order 36.

Authors: Richard J. Mathar
Comments: 182 Pages. 131 Figures and 129 Tables.

The cycle graphs for all finite abstract groups up to order 37 are plotted, except the trivial ring graphs of the cyclic groups themselves.
Category: Algebra

[144] viXra:1406.0098 [pdf] submitted on 2014-06-15 18:28:07

Soft Neutrosophic Algebraic Structures and Their Generalization

Authors: Florentin Smarandache, Mumtaz Ali, Muhammad Shabir
Comments: 264 Pages.

Study of soft sets was first proposed by Molodtsov in 1999 to deal with uncertainty in a non-parametric manner. The researchers did not pay attention to soft set theory at that time but now the soft set theory has been developed in many areas of mathematics. Algebraic structures using soft set theory are very rapidly developed. In this book we developed soft neutrosophic algebraic structures by using soft sets and neutrosophic algebraic structures. In this book we study soft neutrosophic groups, soft neutrosophic semigroups, soft neutrosophic loops, soft neutrosophic LA-semigroups, and their generalizations respectively.
Category: Algebra

[143] viXra:1404.0120 [pdf] submitted on 2014-04-14 11:09:26

Algebraic Structures on Fuzzy Unit Square and Neutrosophic Unit Square

Authors: W. B. vasantha Kandasamy, Florentin Smarandache
Comments: 221 Pages.

In this book authors build algebraic structures on fuzzy unit semi-open square UF = {(a,b), with a, b in [0, 1)} and on neutrosophic unit semi-open square UN = {a+bI, with a, b in [0, 1)}. As distributive laws are not true, we are not in a position to develop several properties of rings, semirigs and linear algebras. Seven open conjectures are proposed.
Category: Algebra

[142] viXra:1404.0119 [pdf] submitted on 2014-04-14 11:10:37

Groupoids of Type I and II Using [0, n)

Authors: W. B. vasantha Kandasamy, Florentin Smarandache
Comments: 178 Pages.

Study of algebraic structures built using [0, n) looks to be one of interesting and innovative research. Here we define two types of groupoids using [0, n), both of them are of infinite order. It is an open conjecture to find whether this new class of groupoids satisfy any of the special identities like Moufang identity or Bol identity and so on.
Category: Algebra

[141] viXra:1403.0958 [pdf] submitted on 2014-03-28 09:56:58

Generalized Determinant

Authors: Nikolay Dementev
Comments: 7 Pages.

The report suggests an approach to extend a concept of determinant to the systems of any order.
Category: Algebra

[140] viXra:1312.0213 [pdf] submitted on 2013-12-26 19:15:38

Algebraic Structues Using [0,n)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 216 Pages.

The algebraic structures built using [0, n) interval are new and innovative. They happen to have different properties. The interval [0, n) can be realized as the real algebraic closure of the modulo ring Zn. The algebraic behavior of [0, n) is different from the ring Zn.
Category: Algebra

[139] viXra:1312.0212 [pdf] submitted on 2013-12-26 19:16:40

Subset Non Associative Topological Spaces

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 211 Pages.

The concept of non associative topological space is very new and interesting. In this book we have built non associative topological spaces using subsets of non associative algebraic structures like loops, groupoids, non associative rings and non associative semirings. We also find conditions on these non associative subset topological spaces to satisfy special identities like Bol, Moufang, right alternative, etc. The new notion will find several applications.
Category: Algebra

[138] viXra:1312.0064 [pdf] submitted on 2013-12-10 05:54:25

Teorema de Wedderburn e o Teorema de Jacobson

Authors: Heitor Baldo
Comments: 10 Pages.

O objetivo principal desse texto e enunciar é provar o Teorema de Jacobson para anéis, por vezes chamado Teorema de Comutatividade de Jacobson. Também enunciamos e provamos outros teoremas, como o Teorema de Wedderburn, além de apresentar uma concisa introdução de alguns conceitos muito utilizados em Teoria dos Anéis (principalmente para anéis não-comutativos) como anéis semisimples e o radical de Jacobson.
Category: Algebra

[137] viXra:1312.0063 [pdf] submitted on 2013-12-10 05:57:09

Teorema de Lagrange

Authors: Heitor Baldo
Comments: 3 Pages.

Neste texto enunciamos e provamos o Teorema de Lagrange para grupos finitos.
Category: Algebra

[136] viXra:1312.0062 [pdf] submitted on 2013-12-10 06:11:14

Um Anel Finito Não-Associativo

Authors: Heitor Baldo
Comments: 6 Pages.

Nesse texto construímos um anel finito não-associativo, não-comutativo e sem elemento unidade.
Category: Algebra

[135] viXra:1312.0061 [pdf] submitted on 2013-12-09 17:35:44

Bases de Espaços Vetoriais

Authors: Heitor Baldo
Comments: 3 Pages.

Neste texto introduzimos o conceito de base de um Espaco Vetorial e sua conexão com a dimensão do espaço. Também expomos alguns exemplos.
Category: Algebra

[134] viXra:1312.0057 [pdf] submitted on 2013-12-09 05:26:22

Poblema Inverso de Galois

Authors: Heitor Baldo
Comments: 11 Pages.

A Teoria de Galois, o estudo da estrutura e da simetria de uma extensão de corpos polinomiais ou associados, e uma ferramenta padrão para mostrar a insolubilidade de uma equação de quinto grau por radicais. Por outro lado, o Problema Inverso de Galois, i.e. o problema de encontrar uma extensão nita do corpo dos racionais Q cujo grupo de Galois seja G, onde G e um grupo nito dado, e ainda um problema em aberto. Nesse texto damos uma introdução bastante resumida do Problema Inverso de Galois.
Category: Algebra

[133] viXra:1311.0040 [pdf] submitted on 2013-11-06 00:52:42

Special Type of Subset Topological Spaces

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 241 Pages.

Special type of subset topological spaces introduced by the authors pave way for the topological spaces, which basically inherit the algebraic structures from which the subsets are taken. This study is new and happens to be a mixture of algebra and topology.
Category: Algebra

[132] viXra:1311.0039 [pdf] submitted on 2013-11-06 00:54:17

Subset Semilinear Algebras

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 254 Pages.

The authors use the subset semigroups over the semifields to build the semilinear algebras of both finite order and infinite order. The concept of subset linear independence and subset linear dependence which leads to the dimension and basis of subset semilinear algebras is analyzed in this book.
Category: Algebra

[131] viXra:1309.0120 [pdf] submitted on 2013-09-17 10:05:33

Smarandache BE-Algebras

Authors: Arsham Borumand Saeid
Comments: 63 Pages.

There are three types of Smarandache Algebraic Structures: 1.A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure. 2.A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker structure. 3.A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure. By proper subset of a set S, one understands a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any. Having two structures {u} and {v} defined by the same operations, one says that structure {u} is stronger than structure {v}, i.e. {u} > {v}, if the operations of {u} satisfy more axioms than the operations of {v}. Each one of the first two structure types is then generalized from a 2-level (the sets P ⊂ S and their corresponding strong structure {w1}>{w0}, respectively their weak structure {w1}<{w0}) to an n-level (the sets Pn-1 ⊂ Pn-2 ⊂ … ⊂ P2 ⊂ P1 ⊂ S and their corresponding strong structure {wn-1} > {wn-2} > … > {w2} > {w1} > {w0}, or respectively their weak structure {wn-1} < {wn-2} < … < {w2} < {w1} < {w0}). Similarly for the third structure type, whose generalization is a combination of the previous two structures at the n-level. A Smarandache Weak BE-Algebra X is a BE-algebra in which there exists a proper subset Q such that 1 belongs to Q, |Q| ≥ 2, and Q is a CI-algebra. And a Smarandache Strong CI-Algebra X is a CI-algebra X in which there exists a proper subset Q such that 1 belongs to Q, |Q| ≥ 2, and Q is a BE-algebra. The book elaborates a recollection of the BE/CI-algebras, then introduces these last two particular structures and studies their properties.
Category: Algebra

[130] viXra:1309.0107 [pdf] submitted on 2013-09-17 09:01:32

Subset Semirings

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 260 Pages.

The authors have constructed subset semirings using rings of both finite and infinite order. Thus, using finite rings we construct infinite number of finite semirings, both commutative as well as non-commutative, which is the main advantage of using this algebraic structure. For finite distribute lattices alone contribute for finite semirings.
Category: Algebra

[129] viXra:1309.0027 [pdf] submitted on 2013-09-05 21:38:31

Subset Interval Groupoids

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 246 Pages.

When the subsets of a loop are taken they also form only a subset groupoid and not a subset loop. Thus the concept of subset interval loop is not there, and they only form a subset interval groupoid. Subset matrix interval groupoid S using the loops Ln(m) has no S-Cauchy elements.
Category: Algebra

[128] viXra:1309.0026 [pdf] submitted on 2013-09-05 21:41:06

Subset Non Associative Semirings

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 207 Pages.

In this book the authors introduce the notion of subset non associative semirings. It is pertinent to keep on record that study of non associative semirings is meager and books on this topic are still rare. Some open problems are suggested in this book.
Category: Algebra

[127] viXra:1309.0019 [pdf] submitted on 2013-09-04 21:27:50

Vector Field Computations in Clifford's Geometric Algebra

Authors: Eckhard Hitzer, Roxana Bujack, Gerik Scheuermann
Comments: 5 Pages. Proc. of the Third SICE Symposium on Computational Intelligence, August 30, 2013, Osaka University, Osaka, pp. 91-95.

Exactly 125 years ago G. Peano introduced the modern concept of vectors in his 1888 book "Geometric Calculus - According to the Ausdehnungslehre (Theory of Extension) of H. Grassmann". Unknown to Peano, the young British mathematician W. K. Clifford (1846-1879) in his 1878 work "Applications of Grassmann's Extensive Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern concept of geometric algebras, including the measurement of lengths (areas and volumes) and angles (between arbitrary subspaces). This leads currently to new ideal methods for vector field computations in geometric algebra, of which several recent exemplary results will be introduced.
Category: Algebra

[126] viXra:1307.0017 [pdf] submitted on 2013-07-03 09:19:40

Subset Groupoids

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 149 Pages.

The authors in this book construct a large class of finite subset groupoids as well as a large class of infinite subset groupoids. Here the conditions under which these subset groupoids satisfy special identities like Bol Identity, Moufang Identity, right alternative identity and so on are found. In fact, it is an open problem to find subset groupoids to satisfy special identities, even if the groupoid over which they are defined do not satisfy the special identities.
Category: Algebra

[125] viXra:1307.0016 [pdf] submitted on 2013-07-03 09:21:52

Subset Polynomial Semirings and Subset Matrix Semirings

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 267 Pages.

In this book the authors introduce the new notions of subset polynomial semirings and subset matrix semirings. Solving subset polynomial equations is an interesting exercise. Open problems about the solution set of subset polynomials are proposed.
Category: Algebra

[124] viXra:1306.0234 [pdf] submitted on 2013-06-29 19:34:04

Exceptional Isomorphisms in the Qi Men Dun Jia Cosmic Board Model

Authors: John Frederick Sweeney
Comments: 6 Pages.

In math, especially within the Lie Algebra groups, there exist a small group of "exceptional isomorphisms" or "accidental isomorphisms." Following the lead of Swiss psychologist Carl Jung, the author does not accept the existence of "exceptional" or "accidental;" instead, these are merely phenomena which heretofore have not been explained satisfactorily by mathematical theory. However, articulation of the Qi Men Dun Jia Cosmic Board Model in a recent and forthcoming paper has helped to explain several of the exceptional isomorphisms.
Category: Algebra

[123] viXra:1306.0178 [pdf] submitted on 2013-06-21 04:43:19

Axioms of Geometric Algebra

Authors: Eckhard Hitzer
Comments: 8 Pages. 3 figures, 1 table. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Axioms for Geometric Algebra R_{p,q} - Definitions using quadratic form, basic multiplication rules. Grade r subspaces, geometric algebra R_2, complex numbers, reflections and rotations, 2-dim. point groups, geometric algebra R_3 and quaternions.
Category: Algebra

[122] viXra:1306.0177 [pdf] submitted on 2013-06-21 04:46:56

Determinants in Geometric Algebra

Authors: Eckhard Hitzer
Comments: 3 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Definition, Adjoint and inverse linear mappings, References.
Category: Algebra

[121] viXra:1306.0176 [pdf] submitted on 2013-06-21 04:49:53

Gram-Schmidt Orthogonalization in Geometric Algebra

Authors: Eckhard Hitzer
Comments: 2 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

We describe Gram-Schmidt orthogonalization in W.K. Clifford's geometric algebra.
Category: Algebra

[120] viXra:1306.0175 [pdf] submitted on 2013-06-21 04:54:51

The Geometric Product and Derived Products

Authors: Eckhard Hitzer
Comments: 11 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

The aim of this work is to show how the geometric product of multivectors is defined in general, extending the basic geometric product of vectors given by Clifford. An alternative definition of Clifford geometric algebra, that guarantees existence as quotient algebra of the tensor algebra was given by Chevalley in 1954.[2] We further treat the scalar product, the outer product, the cross product in three dimensions, linear dependence and independence, as well as right- and left contractions. !"#$%
Category: Algebra

[119] viXra:1306.0174 [pdf] submitted on 2013-06-21 04:59:20

The Use of Quadratic Forms in Geometric Algebra

Authors: Eckhard Hitzer
Comments: 6 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Definition of geometric algebra with quadratic form - examples of quadratic forms and associated geometric algebras - geometric algebras with degenerate quadratic forms - new interpretation of the geometric algebra of the Minkowski plane - generalizing to the geometric mother algebra with p=q=n.
Category: Algebra

[118] viXra:1306.0173 [pdf] submitted on 2013-06-21 05:03:08

What is an Imaginary Number?

Authors: Eckhard Hitzer
Comments: 4 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

The previous Japanese emperor is said to have asked this question. Today many students and scientists still ask it, but the traditional canon of mathematics at school and university needs to be widened for the answer.
Category: Algebra

[117] viXra:1306.0150 [pdf] submitted on 2013-06-19 03:03:40

Three Vector Generation of Crystal Space Groups in Geometric Algebra

Authors: Eckhard Hitzer, Christian Perwass
Comments: 2 Pages. 1 figure, 1 table. Bulletin of the Society for Science on Form, 21(1), pp. 55,56 (2006).

This paper focuses on the symmetries of crystal space lattices. All two dimensional (2D) and three dimensional (3D) point groups of 2D and 3D crystal cells are exclusively described by vectors (two in 2D, three in 3D for one particular cell) taken from the physical cells. Geometric multiplication of these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary-reflections and rotary-inversions. We then extend this treatment to 3D space groups by including translations, glide reflections and screw rotations. We focus on the monoclinic case as an example. A software demonstration shows the spacegroup visualizer. Keywords: Crystal lattice, space group, geometric algebra, OpenGL, interactive software.
Category: Algebra

[116] viXra:1306.0125 [pdf] submitted on 2013-06-17 02:20:49

Geometric Algebra Illustrated by Cinderella

Authors: Eckhard Hitzer, Luca Redaelli
Comments: 22 Pages. 27 figures. Advances in Applied Clifford Algebras, 13(2), pp. 157-181 (2003). DOI: 10.1007/s00006-003-0013-x .

Conventional illustrations of the rich elementary relations and physical applications of geometric algebra are helpful, but restricted in communicating full generality and time dependence. The main restrictions are one special perspective in each graph and the static character of such illustrations. Several attempts have been made to overcome such restrictions. But up till now very little animated and fully interactive, free, instant access, online material is available. This report presents therefore a set of over 90 newly developed (freely online accessible [1]) JAVA applets. These applets range from the elementary concepts of vector, bivector, outer product and rotations to triangle relationships, oscillations and polarized waves. A special group of 21 applets illustrates three geometrically different approaches to the representation of conics; and even more ways to describe ellipses. Next Clifford's famous circle chain theorem is illustrated. Finally geometric applications important for crystallography and structural mechanics give a glimpse of the vast potential for applied mathematics. The interactive geometry software Cinderella [2] was used for creating these applets. The interactive features of many of the applets invite the user to freely explore by a few mouse clicks as many different special cases and perspectives as he likes. This is of great help in "visualizing" geometry encoded by geometric algebra.
Category: Algebra

[115] viXra:1306.0113 [pdf] submitted on 2013-06-17 04:25:16

A Real Explanation for Imaginary Eigenvalues and Complex Eigenvectors

Authors: Eckhard Hitzer
Comments: 21 Pages. in T. M. Karade (ed.), Proc. of the Nat. Symp. on Math. Sc., 1-5 March, 2001, Nagpur, India, Einst. Foundation Int. 1, pp. 1-26 (2001).

This paper first reviews how anti-symmetric matrices in two dimensions yield imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to rotations by means of the Cayley transformation. Then the necessary tools from real geometric algebra are introduced and a real geometric interpretation is given to the eigenvalues and eigenvectors. The latter are seen to be two component eigenspinors which can be further reduced to underlying vector duplets. The eigenvalues are interpreted as rotors, which rotate the underlying vector duplets. The second part of this paper extends and generalizes the treatment to three dimensions. The final part shows how all entities and relations can be obtained in a constructive way, purely assuming the geometric algebras of 2-space and 3-space.
Category: Algebra

[114] viXra:1306.0112 [pdf] submitted on 2013-06-17 04:30:00

Antisymmetric Matrices are Real Bivectors

Authors: Eckhard Hitzer
Comments: 16 Pages. Mem. Fac. Eng. Fukui Univ. 49(2), pp. 283-298 (2001).

This paper briefly reviews the conventional method of obtaining the canonical form of an antisymmetric (skew-symmetric, alternating) matrix. Conventionally a vector space over the complex field has to be introduced. After a short introduction to the universal mathematical "language" Geometric Calculus, its fundamentals, i.e. its "grammar" Geometric Algebra (Clifford Algebra) is explained. This lays the groundwork for its real geometric and coordinate free application in order to obtain the canonical form of an antisymmetric matrix in terms of a bivector, which is isomorphic to the conventional canonical form. Then concrete applications to two, three and four dimensional antisymmetric square matrices follow. Because of the physical importance of the Minkowski metric, the canonical form of an antisymmetric matrix with respect to the Minkowski metric is derived as well. A final application to electromagnetic fields concludes the work. Keywords: Geometric Calculus, Geometric Algebra, Clifford Algebra, antisymmetric (alternating, skewsymmetric) matrix, Real Geometry
Category: Algebra

[113] viXra:1306.0038 [pdf] submitted on 2013-06-06 10:12:21

Some Results on Smarandache Groupoids

Authors: H. J. Siamwalla, A.S.Muktibodh
Comments: 7 Pages.

In this paper we prove some results towards classifying Smarandache groupoids which are in Z*(n) and not in Z(n) when n is even and n is odd.
Category: Algebra

[112] viXra:1306.0036 [pdf] submitted on 2013-06-06 10:15:44

Smarandache Mukti-Squares

Authors: Arun S. Muktibodh
Comments: 7 Pages.

In [4] we have introduced Smarandache quasigroups which are Smarandache non-associative structures. A quasigroup is a groupoid whose composition table is a Latin square. There are squares in the Latin squares which seem to be of importance to study the structure of Latin Squares. We consider the particular type of squares properly contained in the Latin squares which themselves contain a Latin square. Such Latin squares are termed as Smarandache Mukti-Squares or SMS. Extension of some SMS to Latin squares is also considered.
Category: Algebra

[111] viXra:1306.0035 [pdf] submitted on 2013-06-06 10:17:09

Smarandache Quasigroup Rings

Authors: Arun S. Muktibodh
Comments: 6 Pages.

In this paper, we have introduced Smarandache quasigroups which are Smarandache non- associative structures. W.B.Kandasamy [2] has studied groupoid ring and loop ring. We have de¯ned Smarandache quasigroup rings which are again non-associative structures having two binary operations. Substructures of quasigroup rings are also studied.
Category: Algebra

[110] viXra:1306.0034 [pdf] submitted on 2013-06-06 10:18:29

Smarandache Quasigroups

Authors: Arun S. Muktibodh
Comments: 7 Pages.

In this paper, we have introduced Smarandache quasigroups which are Smaran- dache non-associative structures. W.B.Kandasamy [2] has studied Smarandache groupoids and Smarandache semigroups etc. Substructure of Smarandache quasigroups are also studied.
Category: Algebra

[109] viXra:1306.0033 [pdf] submitted on 2013-06-06 10:20:01

Smarandache Semiquasi Near-Rings

Authors: Arun S. Muktibodh
Comments: 3 Pages.

G. Pilz [1] has dened near-rings and semi-near-rings. In this paper we introduce the concepts of quasi-near ring and semiquasi-near ring. We have also dened Smarandache semiquasi-near-ring. Some examples are constructed. We have posed some open problems.
Category: Algebra

[108] viXra:1306.0029 [pdf] submitted on 2013-06-06 10:29:40

Chains of Smarandache Semifields

Authors: A. S. Muktibodh, V. M. Wagh
Comments: 3 Pages.

In this paper we have constructed two chains of semifields. All the semifields in the chains are Smarandache semifields. Every member of the chain is an extension semifield of Ordered equilateral Integral triangles with Zero triangle such that it is a semivector space over R^I_e\delta.
Category: Algebra

[107] viXra:1306.0019 [pdf] submitted on 2013-06-05 00:29:16

A Phenomenon in Matrix Analysis

Authors: S. Kalimuthu
Comments: 4 Pages. NA

The history of matrices goes back to ancient times! But the term "matrix" was not applied to the concept until 1850."Matrix" is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced. The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of the use of matrix methods to solve simultaneous equations. Since their first appearance in ancient China, matrices have remained important mathematical tools. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps! The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory. In this work, by applying the fundamental concepts of matrices, the author attempts to study the fifth Euclidean postulate problem and Godel’s incompleteness theorems.
Category: Algebra

[106] viXra:1305.0183 [pdf] submitted on 2013-05-30 00:59:35

Pretty Algebra

Authors: S.Kalimuthu
Comments: 6 Pages. This is a new attempt in set theory 7 geometry

The sum of the interior angles of a number triangles were transformed into linear algebraic equations. The analysis of these equations without assuming the fifth Euclidean postulate established the following theorem: There exists a triangle whose interior angle sum is equal to two right angles
Category: Algebra

[105] viXra:1305.0182 [pdf] submitted on 2013-05-30 01:05:03

On Applied Algebra

Authors: S.Kalimuthu
Comments: 4 Pages. This is an entirely new approach in algebra & geometry

By the application of abstract algebra the famous unsolved classical problems trisection of an angle,squaring the circle , duplicating the cube and drawing a regular septagon were shown not possible to solve.. On the other hand,in this work, the application of classical algebra proposed a proposition for the origin of a new field of geometry.
Category: Algebra

[104] viXra:1305.0181 [pdf] submitted on 2013-05-30 01:11:48

Pretty Set Theory

Authors: S.Kalimuthu
Comments: 3 Pages. Quo Vadis

By applying the basic operations of number theory and set theory to geometry, a new challenging finding has been obtained
Category: Algebra

[103] viXra:1305.0152 [pdf] submitted on 2013-05-24 09:24:45

A New Concept in Spherical Geometry

Authors: S. Kalimuthu
Comments: 8 Pages. This is an extra-ordinary math. article.

After establishing the fundamental physics prizes, Yuri Milner said: “Unlike the Nobel in physics, the Fundamental Physics Prize can be awarded to scientists whose ideas have not yet been verified by experiments, which often occurs decades later. Sometimes a radical new idea “really deserves recognition right away because it expands our understanding of at least what is possible.”. Keeping this mind the author formulated two spherical geometrical theorems which may applied for the studies and probes of fundamental particles, quantum gravity, gravitational waves ,dark matter and dark energy.
Category: Algebra

[102] viXra:1305.0151 [pdf] submitted on 2013-05-24 09:27:04

On Gödel and Miles Mathis

Authors: S. Kalimuthu
Comments: 4 Pages. This is an extra-ordinary math. article.

Miles Mathis has shown the following equivalent propositions to Gödel’s incompleteness theorems: [ http://milesmathis.com/godel.html ] Theorem 1: In any logical system one can construct statements that are neither true nor false (mathematical variations of the liar’s paradox). Theorem 2: Therefore no consistent system can be used to prove its own consistency. No proof can be proof of itself. In this work , we attempt to prove the first theorem mention above. Key words: Gödel’s incompleteness theorems ; Miles Mathis’s proposition
Category: Algebra

[101] viXra:1305.0150 [pdf] submitted on 2013-05-24 09:29:12

A Phenomenon in Hyperbolic Math.

Authors: S. Kalimuthu
Comments: 2 Pages. This is an extra-ordinary math. article.

The classical Euclidean geometry is widely used in all fields of science, engineering and other art fields. The hyperbolic and elliptic non Euclidean geometries are applied in theoretical physics and cosmology. In this short paper, we have come across an interesting non geometrical phenomenon.
Category: Algebra

[100] viXra:1305.0058 [pdf] submitted on 2013-05-08 14:47:25

Algebraic Structures Using Subsets

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 197 Pages.

Study of algebraic structures using subsets was started by George Boole. After the invention of Boolean algebra, subsets were not used in any algebraic structures. In this book we develop algebraic structures using subsets of a set, or of a group, or semigroup, or a ring and we get algebraic structures. Using ring or semiring we get only subset semirings. BY the method we get an infinite number of non-commutative semirings of finite order. We build subset semivector spaces, and describe and develop interesting properties over them.
Category: Algebra

[99] viXra:1305.0057 [pdf] submitted on 2013-05-08 14:50:08

Fuzzy Neutrosophic Models for Social Scientists

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 167 Pages.

Use of fuzzy neutrosophic models in the study of social problem or technical problem has become a demand for the day due to the involvement of indeterminacy. For always one is not in the position to say true or false, or assign a fuzzy value to any problem. At times, on is not in the position to commit to any of the three statements. In that situation, neutrosophic lmodels play a vital role. In this book we introduce several fuzzy neutrosophic models to study social problems.
Category: Algebra

[98] viXra:1305.0045 [pdf] submitted on 2013-05-07 21:51:26

Smarandache Seminormal Subgroupoids

Authors: H. J. Siamwalla, A.S.Muktibodh
Comments: 15 Pages.

In this paper, we defined the Smarandache seminormal subgroupoids. We have proved some results for finding the Smarandache seminormal subgroupoids in Z(n) when n is even and n is odd.
Category: Algebra

[97] viXra:1304.0021 [pdf] submitted on 2013-04-04 12:05:34

Set Theoretic Approach to Algebraic Structures in Mathematics

Authors: Vasantha Kandasamy, Florentin Smarandache
Comments: 166 Pages.

This book brings out how sets in algebraic structures can be used to construct the most generalized algebraic structures, like set linear algebra / vector space, set ideals in groups and rings and semigroups, and topological set vector spaces. This sort of study is innovative and will find applications in data handling.
Category: Algebra

[96] viXra:1303.0016 [pdf] submitted on 2013-03-03 12:42:55

The Transit Over Ultimate Problem

Authors: Nasir ermain
Comments: 2 Pages.

Nasir Germain newest problem
Category: Algebra

[95] viXra:1303.0014 [pdf] submitted on 2013-03-03 11:09:03

Innovative Uses of Matrices

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, Indra Venkatbabu
Comments: 227 Pages.

In this book authors have given the innovative ways in which they have used matrices, starting from ATD, RTD & CETD matrices which can be applied to any collected data depending on time. A new type of matrices, called fuzzy linguistic matrices, which play a major role in the construction of fuzzy linguistic models, is defined. Then, the super matrices are used in the construction of super linear algebra and super fuzzy models. Further, interval matrices using natural class of intervals is introduced. The ingenious way of defining DSm matrices of refined labels is an interesting feature. Finally the construction of n-matrices and its use in the building of n-codes is described.
Category: Algebra

[94] viXra:1302.0088 [pdf] submitted on 2013-02-13 11:27:51

Indefinite Summation

Authors: Joerg Siegler
Comments: 39 Pages.

This paper about indefinite summation describes a natural approach to discrete calculus. Two natural operators for discrete difference and summation are defined. They preserve symmetry and show a duality in contrast to the classical operators. Several summation and differentiation algorithms will be presented.
Category: Algebra

[93] viXra:1301.0175 [pdf] submitted on 2013-01-28 04:43:38

Some New Wedge Products

Authors: Martin Erik Horn
Comments: 25 Pages. 6 Figures.

Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric Algebra it is helpful to restate Geometric Algebra in the mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. Special emphasis will be given to the definition of some wedge products which fifit better to this algebra of (3 x 3) matrices than the usual Geometric Algebra wedge product. And as S3 permutation symmetry is flavour symmetry a unifified flavour picture of Geometric Algebra will emerge.
Category: Algebra

[92] viXra:1301.0049 [pdf] submitted on 2013-01-10 09:26:05

Quasi Set Topological Vector Subspaces

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 152 Pages.

In this book, the authors introduce the notion of quasi set topological vector subspaces. The advantage of such study is that given a vector subspace we can have only one topological space associated with the collection of all subspaces. However, we can have several quasi set topological vector subspaces of a given vector space. Further, we have defined topological spaces for set vector spaces, semigroup vector spaces and group vector spaces.
Category: Algebra

[91] viXra:1212.0157 [pdf] submitted on 2012-12-27 16:40:31

Countability Properties in Interior Algebras

Authors: Colin Naturman
Comments: 20 Pages.

Certain topological countability properties are generalized to interior algebras and basic reaults concerning these properies are investigated. The preservation of these properties under the formation of principal quotients and under a new construction called a join of interior algebras, is investigated.
Category: Algebra

[90] viXra:1212.0151 [pdf] submitted on 2012-12-27 03:05:04

Convergence and Accumulation in Interior Algebras

Authors: Colin Naturman
Comments: 26 Pages. Results published in "Naturman, C.A., 1991, Interior Algebras and Topology, PhD Thesis, University of Cape Town Department of Mathematics" and this paper presented as "Research Reports, Department of Mathematics. The University of Cape Town, Volume 139

Interior algebras are Boolean algebras enriched with an interior operator and corresponding closure operator. Alternative descriptions of interior algebras in terms of generalized topologies in Boolean algebras and neighbourhood functions on Boolean algebras are found. The topological concepts of convergence and accumulation of systems and nets are generalized to interior algebras. Relationships between different forms of convergence and accunulation are found.
Category: Algebra

[89] viXra:1212.0148 [pdf] submitted on 2012-12-26 04:38:39

Interval Algebras of Interior Algebras

Authors: Colin Naturman
Comments: 17 Pages. Results published in "Naturman, C.A., 1991, Interior Algebras and Topology, PhD Thesis, University of Cape Town Department of Mathematics" and this paper presented as "Research Reports, Department of Mathematics. The University of Cape Town, Volume 144

The intervals in an interior algebra A can be turned into interior algebras called interval algebras. Generalizations of homomorphisms, called topomorphisms, are introduced and certain quotient structures of A in the category of interior algebras and topomorphisms (the principal quotients) are shown to be (up to isomorphism) precisely the interval algebras of A.
Category: Algebra

[88] viXra:1212.0146 [pdf] submitted on 2012-12-25 11:02:30

Amalgamation in the Pentagon Variety

Authors: Peter Bruyns, Colin Naturman, Henry Rose
Comments: 17 Pages.

The amalgamation class Amal (N) of a lattice variety generated by a pentagon is considered. It is shown that Amal (N) is closed under reduced products and therefore is an elementary class determined by Horn sentences. The above result is based on a new characterization of Amal (N). The lattice varieties whose amalgamation classes contain Amal (N) as a subclass are considered.
Category: Algebra

[87] viXra:1212.0138 [pdf] submitted on 2012-12-23 08:11:59

Elementary Equivalent Pairs of Algebras Associated with Sets

Authors: Colin Naturman, Henry Rose
Comments: 26 Pages.

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.
Category: Algebra

[86] viXra:1212.0133 [pdf] submitted on 2012-12-21 12:55:50

Interior Algebras: Some Universal Algebraic Aspects

Authors: Colin Naturman, Henry Rose
Comments: 25 Pages. Published in Journal of the Korean Mathematical Society, 30(1), 1993, pp.1–23 content is free for download but PDFs distributed by the publisher are missing the diagrams and/or abstract and errata.

An interior algebra is a Boolean algebra enriched with an interior operator. Congruences on interior algebras are investigated. Simple, subdirectly irreducible, finitely subdirectly irreducible and directly indecomposable interior algebras are characterized and the classes of these are shown to be finitely axiomatizable elementary classes. Quotients by open elements, dissectable and openly decomposable interior algebras are investigated. Basic results concerning interior algebras and their connection to topology are discussed.
Category: Algebra

[85] viXra:1212.0131 [pdf] submitted on 2012-12-21 03:39:27

Ideal Algebras of Interior Algebras

Authors: Colin Naturman, Henry Rose
Comments: 6 Pages. Published in Ordered Set and Lattices, 11, 1995 pp. 39-44, publisher does not provide offprints

An interior algebra is a Boolean algebra enriched with an interior operator. Given an interior algebra there is a natural way of forming interior algebras from its principal ideals. Basic results concerning these ideal algebras, Stone spaces of ideal algebras and preservation properties of ideal algebras are investigated.
Category: Algebra

[84] viXra:1212.0124 [pdf] submitted on 2012-12-20 04:53:25

NP-Hardness of Optimizing the Sum of Rational Linear Functions Over an Asymptotic-Linear-Program

Authors: Deepak Ponvel Chermakani
Comments: 6 Pages, 6 Theorems, 5 Figures

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real-variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.
Category: Algebra

[83] viXra:1212.0018 [pdf] submitted on 2012-12-03 12:30:39

Neutrosophic Super Matrices and Quasi Super Matrices

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 200 Pages.

The authors introduce the concept of neutrosophic super matrices and the new notion of quasi super matrices. This new notion of quasi super matrices contains the class of super matrices. The larger class contains more partitions of the usual simple matrices. Studies in this direction are interesting and find more applications in fuzzy models. The authors also suggest in this book some open problems.
Category: Algebra

[82] viXra:1211.0044 [pdf] submitted on 2012-11-08 12:33:29

Supermodular Lattices

Authors: Iqbal Unnisa, W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 132 Pages.

In this book the authors introduce a new class of lattices called Supermodular Lattices, which is an equational class of lattices lying between the equational class of distributive lattices and modular lattices. Several other new properties related with these lattices are introduced, described and developed.
Category: Algebra

[81] viXra:1211.0038 [pdf] submitted on 2012-11-07 13:25:27

Set Ideal Topological Spaces

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 114 Pages.

In this book the authors use set ideal of rings (or semigroups) to build topological spaces. These spaces are dependent on the set over which the set ideals are defined. It is left as an open problem whether this newly constructed topological space of finite order increases the existing number of finite topological spaces.
Category: Algebra

[80] viXra:1211.0029 [pdf] submitted on 2012-11-06 06:11:24

A Rewriting System Applied to the Simplest Algebraic Identities

Authors: J. S. Markovitch
Comments: 2 Pages.

A rewriting system applied to the simplest algebraic identities is shown to yield second- and third-order equations that share a property associated with 137.036.
Category: Algebra

[79] viXra:1210.0142 [pdf] submitted on 2012-10-25 16:03:56

Geometric Algebra of Quarks

Authors: Martin Erik Horn
Comments: 14 Pages. Poster presentation at AGACSE 2012 in La Rochelle

Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric algebra it is helpful to restate Geometric algebra in the mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. And as S3 permutation symmetry is flavour symmetry a unified flavour picture of Geometric algebra will emerge.
Category: Algebra

[78] viXra:1210.0086 [pdf] submitted on 2012-10-17 11:01:47

Special Quasi Dual Numbers and Groupoids

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 193 Pages.

A new notion of special quasi dual numbers is introduced. If a+bg is the special quasi dual number with a, b reals, g the new element is such that g^2 = - g. The rich source of getting new elements of the form g^2 = - g is from Z_n, the ring of modulo integers. For the first time we construct non associative structures using them. We have proposed some research problems.
Category: Algebra

[77] viXra:1210.0085 [pdf] submitted on 2012-10-17 11:06:59

Special Dual Like Numbers and Lattices, by W. B. Vasantha Kandasamy, Florentin Smarandache

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 246 Pages.

In this book we define x = a+bg to be a special dual like number, where a, b are reals and g is a new element such that g^2 = g. The new element which is idempotent can be got from Z_n or from lattices or from linear operators. Mixed dual numbers are constructed using dual numbers and special dual like numbers. Neutrosophic numbers are a natural source of special dual like numbers, since they have the form a+bI, where I = indeterminate and I^2 = I.
Category: Algebra

[76] viXra:1210.0084 [pdf] submitted on 2012-10-17 11:08:46

Semigroups as Graphs, by W. B. Vasantha Kandasamy, Florentin Smarandache

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 153 Pages.

The zero divisor graph of semigroups of finite modulo integers n under product is studied and characterized. If n is a non-prime, the zero divisor graph is not a tree. We introduce the new notion of tree covering a pseudo lattice. When n is an even integer of the form 2p, p a prime, then the modulo integer zero divisor graph is a tree-covering pseudo lattice.
Category: Algebra

[75] viXra:1209.0014 [pdf] submitted on 2012-09-05 00:31:28

Dual Numbers

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 159 Pages.

The concept of dual numbers was introduced in 1873 by W.K.Clifford. In this book the authors build higher dimensional dual numbers, interval dual numbers and impose some algebraic structures on them. The S-vector space of dual numbers built over a Smarandache dual ring can have eigen values and eigen vectors to be dual numbers. Complex modulo integer dual numbers and neutrosophic dual numbers are also introduced. The notion of fuzzy dual numbers can play a vital role in fuzzy models. Some research level problems are also proposed here.
Category: Algebra

[74] viXra:1208.0200 [pdf] submitted on 2012-08-20 03:38:01

A Mild Generalization of Zariski's Lemma

Authors: Pierre-Yves Gaillard
Comments: 2 Pages.

We give a mild generalization of Zariski's Lemma.
Category: Algebra

[73] viXra:1208.0020 [pdf] submitted on 2012-08-07 13:16:43

Smarandache Weak BE-Algebras

Authors: Arsham Borumand Saeid
Comments: Pages.

In this paper, we introduce the notions of Smarandache weak BE-algebra, Q-Smarandache filters and Q-Smarandache ideals. We show that a nonempty subset F of a BE-algebra X is a Q-Smarandache filter if and only if A(x, y) is included in or equal to F, which A(x, y) is a Q-Smarandache upper set. The relationship between these notions are stated and proved.
Category: Algebra

[72] viXra:1207.0076 [pdf] submitted on 2012-07-19 20:57:11

Exploring the Extension of Natural Operations on Intervals, Matrices and Complex Numbers

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 150 Pages.

In this book we explore the possibility of extending the natural operations on reals to intervals and matrices. The extension to intervals makes us define a natural class of intervals in which we accept [a, b], a > b. Further, we introduce a complex modulo integer in Zn (n, a positive integer) and denote it by i_F with i_F^2 = n-1.
Category: Algebra

[71] viXra:1207.0051 [pdf] submitted on 2012-07-13 00:51:32

The Fundamental Theorem of Galois Theory

Authors: Pierre-Yves Gaillard
Comments: 2 Pages.

We give a short and self-contained proof of the Fundamental Theorem of Galois Theory for finite degree extensions.
Category: Algebra

[70] viXra:1206.0009 [pdf] submitted on 2012-06-02 04:13:28

A Review on Classification of Complex Semi-Simple Lie Algebras

Authors: Ren Shiquan
Comments: 24 Pages. this is the bachelor thesis of Mr. Ren Shiquan in 2010.

We give a review on the classification of complex semi-simple Lie algebras, according to our study process.
Category: Algebra

[69] viXra:1204.0005 [pdf] submitted on 2012-04-02 14:11:51

Non Associative Linear Algebras

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

In this book we introduce the notion of Non Associative Linear Algebras. We mainly use the concepts of loops and groupoids to build these structures. We have also introduced the concept of Non Associative Semi-Linear Algebras. In the future, non-associative linear algebras will find applications in mathematical models that do not in general need to be associative.
Category: Algebra

[68] viXra:1203.0008 [pdf] submitted on 2012-03-02 14:01:07

Fuzzy Linguistic Topological Spaces

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Amal
Comments: 193 Pages.

In this book the authors introduce the notion of fuzzy linguistic topological spaces. These linguistic topological spaces enjoy many properties depending on the problem and the linguistic variables associated with them. These structures find applications in mathematical modeling and fuzzy linguistic neural networks.
Category: Algebra

[67] viXra:1202.0019 [pdf] submitted on 2012-02-07 20:30:57

Natural Product Xn on Matrices

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 342 Pages.

In this book the authors introduce a new type of product on matrices called the natural product Xn. This is an extension of product carried out in the case or row matrices of the same order. Further, when two column matrices of same order can be added, nothing prevents one from multiplying them. This sort of multiplication which is natural is defined as natural product Xn on matrices. We suggest 100 problems and some of them are at the research level.
Category: Algebra

[66] viXra:1201.0098 [pdf] submitted on 2012-01-25 14:05:51

Non Associative Algebraic Structures Using Finite Complex Numbers

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 213 Pages.

The authors have used the concept of finite complex modulo integers to construct non associative algebraic structures like groupoids, loops and quasi-loops. Using these structures we built non associative complex matrix groupoids and complex polynomial groupoids. The authors suggest over 300 problems and some are at the research level.
Category: Algebra

[65] viXra:1201.0066 [pdf] submitted on 2012-01-16 10:44:27

Study of Natural Class of Intervals Using (-∞, ∞) and (∞, ∞)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, D. Datta, H. S. Kushwaha, P. A. Jadhav
Comments: 181 Pages.

In this book the authors introduce and study the properties of natural class of intervals built using (-∞, ∞) and (∞, -∞). The operations on these matrices with entries from natural class of intervals behave like usual reals. So working with these interval matrices takes the same time as usual matrices. Hence, when applying them to fuzzy finite element methods or finite element methods the determination of solution is simple and time saving.
Category: Algebra

[64] viXra:1111.0078 [pdf] submitted on 22 Nov 2011

Finite Neutrosophic Complex Numbers

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 222 pages

In this book for the first time the authors introduce the notion of real neutrosophic complex numbers.
Category: Algebra

[63] viXra:1111.0077 [pdf] submitted on 22 Nov 2011

DSm Super Vector Space of Refined Labels

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 299 pages

In this book authors for the first time introduce the notion of supermatrices of refined labels. Authors prove super row matrix of refined labels form a group under addition. However super row matrix of refined labels do not form a group under product; it only forms a semigroup under multiplication. In this book super column matrix of refined labels and m x n matrix of refined labels are introduced and studied.
Category: Algebra

[62] viXra:1111.0076 [pdf] submitted on 22 Nov 2011

Neutrosophic Interval Bialgebraic Structures

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 197 pages

In this book the authors for the first time introduce the notion of neutrosophic intervals and study the algebraic structures using them. Concepts like groups and fields using neutrosophic intervals are not possible. Pure neutrosophic intervals and mixed neutrosophic intervals are introduced and by the very structure of the interval one can understand the category to which it belongs.
Category: Algebra

[61] viXra:1110.0038 [pdf] submitted on 12 Oct 2011

DSm Vector Spaces of Refined Labels

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 216 pages

The study of DSm linear algebra of refined labels have been done by Florentin Smarandache, Jean Dezert, and Xinde Li. In this book the authors introduce the notion of DSm vector spaces of refined labels. The reader is requested to refer the paper as the basic concepts are taken from that paper
Category: Algebra

[60] viXra:1107.0041 [pdf] submitted on 21 Jul 2011

Algebraic Structures Using Super Interval Matrices

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 289 pages

In this book, super interval matrices using the special type of intervals of the form [0, a] are introduced. Several algebraic structures like semigroups, groups, semirings, rings, semivector spaces and vector spaces are introduced. Special fuzzy linear algebras are introduced using the concept of super fuzzy interval matrices.
Category: Algebra

[59] viXra:1106.0050 [pdf] submitted on 23 Jun 2011

Interval Algebraic Bistructures

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 210 pages.

Authors in this book construct interval bistructures using only interval groups, interval loops, interval semigroups and interval groupoids.
Category: Algebra

[58] viXra:1106.0019 [pdf] submitted on 11 Jun 2011

Algebraic Structures Using Natural Class of Intervals

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 172 pages

In this book the authors introduce a new class of intervals called the natural class of intervals. Using these intervals, algebraic structures are defined. Over 200 problems are given, of which some of them are at the research level.
Category: Algebra

[57] viXra:1105.0009 [pdf] submitted on 6 May 2011

Why the Colombeau Algebras Cannot Handle Arbitrary Lie Groups ?

Authors: Elemér E Rosinger
Comments: 18 pages.

It is briefly shown that, due to the growth conditions in their definition, the Colombeau algebras cannot handle arbitrary Lie groups, and in particular, cannot allow the formulation, let alone, solution of Hilbert's Fifth Problem.
Category: Algebra

[56] viXra:1105.0007 [pdf] submitted on 4 May 2011

Why the Colombeau Algebras Cannot Formulate, Let Alone Prove the Global Cauchy-Kovalevskaia Theorem ?

Authors: Elemér E Rosinger
Comments: 9 pages.

It is briefly shown that, due to the growth conditions in their definition, the Colombeau algebras cannot handle arbitrary analytic nonlinear PDEs, and in particular, cannot allow the formulation, let alone, give the proof of the global Cauchy-Kovalevskaia theorem.
Category: Algebra

[55] viXra:1103.0031 [pdf] submitted on 10 Mar 2011

On the Relation Between Double Summations and Tetrahedral Numbers

Authors: Marco Ripà
Comments: 3 pages

In this paper we provide an inverse proof of the relation between a particular class of double sums and tetrahedral numbers. Thus, we present a compact formula to reduce the number of calculations necessary to solve such a kind of problems. The initial identity is confirmed "a posteriori" using the formula mentioned above.
Category: Algebra

[54] viXra:1102.0045 [pdf] submitted on 23 Feb 2011

Multi-Matrices and Arithmetical Operations with Multi-Matrices

Authors: Constantin Scheau
Comments: 8 pages.

The multi-space structure has been defined by Fl Smarandache as a union spaces with some additional conditions hold. The mathematician L. Mao wrote a series of works in which he introduces the concepts of multi-group, multi-ring, multivector - space etc. In [1] (Smarandache Multi-Space Theory (I)), at open problems section, he suggests the introduction of a theory of matrices and applications defined on the multi-linear spaces. This paper will give an example of a multi-ring structure, introduces the notion of multi-matrix and defines the multi-matrix addition and multiplication.
Category: Algebra

[53] viXra:1101.0072 [pdf] submitted on 22 Jan 2011

Smarandache GT-Algebras

Authors: Jaedoek Kim, Youngmi Kim, Eun Hwan Roh
Comments: 7 pages

We introduce the notion of Smarandache GT-algebras, and the notion of Smarandache GT-Filters of the Smarandache GT-algebra related to the Tarski algebra, and related some properties are investigated.
Category: Algebra

[52] viXra:1101.0063 [pdf] submitted on 21 Jan 2011

Interval Semigroups

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 167 pages

In this book we introduce the notion of interval semigroups using intervals of the form [0, a], a is real. Several types of interval semigroups like fuzzy interval semigroups, interval symmetric semigroups, special symmetric interval semigroups, interval matrix semigroups and interval polynomial semigroups are defined and discussed. This book has eight chapters. The main feature of this book is that we suggest 241 problems in the eighth chapter. In this book the authors have defined 29 new concepts and illustrates them with 231 examples. Certainly this will find several applications. The authors deeply acknowledge Dr. Kandasamy for the proof reading and Meena and Kama for the formatting and designing of the book.
Category: Algebra

[51] viXra:1012.0028 [pdf] submitted on 12 Dec 2010

Intuitionistic Fuzzy Γ-Ideals of Γ-la-Semigroups.

Authors: Muhammad Aslam, Saleem Abdullah
Comments: 14 pages

We consider the intuitionistic fuzzi?cation of the concept of several Γ-ideals in Γ-LA-semigroup S, and investigate some related properties of such Γ-ideals. We also prove in this paper the set of all intuitionistic fuzzy left(right) Γ-ideal of S is become LA-semigroup. We prove In Γ-LA band intuitionistic fuzzy right and left Γ-ideals are coincide..
Category: Algebra

[50] viXra:1011.0038 [pdf] submitted on 17 Nov 2010

Interval Linear Algebra

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 249 pages

This Interval arithmetic or interval mathematics developed in 1950's and 1960's by mathematicians as an approach to putting bounds on rounding errors and measurement error in mathematical computations. However no proper interval algebraic structures have been defined or studies. In this book we for the first time introduce several types of interval linear algebras and study them.
Category: Algebra

[49] viXra:1011.0037 [pdf] submitted on 14 Nov 2010

Lattice Rationals

Authors: Nathaniel S. K. Hellerstein
Comments: 17 pages

This paper redefines the addition of rational numbers, in a way that allows division by zero. This requires defining a "compensator" on the integers, plus extending least-common-multiple (LCM) to zero and negative numbers. "Compensated addition" defines ordinary addition on all ratios, including the 'infinities' n/0, and also 'zeroids' 0/n. The infinities and the zeroids form two 'double ringlets'. The lattice rationals modulo the zeroids yields the infinities plus the 'wheel numbers'. Due to the presence of the 'alternator' @ = 0/-1, double-distribution does not apply, but triple-distribution still does.
Category: Algebra

[48] viXra:1011.0019 [pdf] submitted on 11 Nov 2010

On Reduction

Authors: Nathaniel S. K. Hellerstein
Comments: 33 pages

In this paper I discuss "reduction", a.k.a. "reciprocal addition"; addition conjugated by reciprocal. I discuss reduction's definition, its laws, its graphs, its geometry, its algebra, its calculus, and its practical applications. This paper contains a problem set with answer key.
Category: Algebra

[47] viXra:1010.0021 [pdf] submitted on 10 Oct 2010

On Smarandache Rings II

Authors: A. K. S. Chandra Sekhar Rao
Comments: 12 pages

In this paper we show that a commutative semisimple ring is always a Smarandache ring. We will also give a necessary and sufficient condition for group algebra to be a Smarandache ring. Examples are provided for justification.
Category: Algebra

[46] viXra:1008.0090 [pdf] submitted on 31 Aug 2010

Interval Groupoids

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, Moon K. Chetry
Comments: 242 pages

This book introduces several new classes of groupoid, like polynomial groupoids, matrix groupoids, interval groupoids, polynomial interval groupoids, matrix interval groupoids and their neutrosophic analogues.
Category: Algebra

[45] viXra:1008.0040 [pdf] submitted on 13 Aug 2010

Smarandache Hyper (∩, ∈)-Idealson Smarandache Hyper K-Algebras

Authors: Kyung Ho Kim, Young Bae Jun, Eun Hwan Roh, Habib Harizavi
Comments: 6 Pages.

We introduce the notion of a Smarandache hyper (∩, ∈)-ideal and Ω-reflexive in hyper K-algebra, and some related properties are given.
Category: Algebra

[44] viXra:1008.0039 [pdf] submitted on 13 Aug 2010

Divisibility Tests for Smarandache Semigroups

Authors: A.K.S. Chandra Sekhar Rao.
Comments: 12 Pages.

Two Divisibility Tests for Smarandache semigroups are given . Further, the notion of divisibility of elements in a semigroup is applied to characterize the Smarandache semigroups. Examples are provided for justification.
Category: Algebra

[43] viXra:1008.0014 [pdf] submitted on 6 Aug 2010

A New Proof Viorel Vîjîitu Inequality

Authors: Marian Dincă
Comments: 2 pages.

In the paper given new proof the inequality using convex function
Category: Algebra

[42] viXra:1008.0013 [pdf] submitted on 6 Aug 2010

Generalisation of the Inequalities Proposed I.m.o. Madrid 2008 and India-International Mathematical Olympiad Training Camp2010

Authors: Marian Dincă
Comments: 3 pages.

In the paper given generalisation inequalities using Lagrange identity.
Category: Algebra

[41] viXra:1007.0029 [pdf] submitted on 13 Mar 2010

Super Linear Algebra

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 294 pages

In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra.
Category: Algebra

[40] viXra:1007.0027 [pdf] submitted on 13 Mar 2010

Superbimatrices and Their Generalizations

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 410 pages

The systematic study of supermatrices and super linear algebra has been carried out in 2008. These new algebraic structures find their applications in fuzzy models, Leontief economic models and data-storage in computers.
Category: Algebra

[39] viXra:1007.0014 [pdf] submitted on 13 Mar 2010

Special Set Linear Algebra and Special Set Fuzzy Linear Algebra

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K Ilanthenral
Comments: 469 pages

This book for the first time introduces the notion of special set linear algebra and special set fuzzy linear algebra. This is an extension of the book set linear algebra and set fuzzy linear algebra. These algebraic structures basically exploit only the set theoretic property, hence in applications one can include a finite number of elements without affecting the systems property. These new structures are not only the most generalized structures but they can perform multi task simultaneously; hence they would be of immense use to computer scientists.
Category: Algebra

[38] viXra:1007.0009 [pdf] submitted on 7 Jul 2010

Neutrosophic Bilinear Algebras and Their Generalizations

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 404 pages.

This book introduces the concept of neutrosophic bilinear algebras and their generalizations to n-linear algebras, n>2.
Category: Algebra

[37] viXra:1007.0004 [pdf] submitted on 5 Jul 2010

On Smarandache Semigroups

Authors: A.K.S.Chandra Sekhar Rao
Comments: 4 pages.

The notion of completely regular element of a semigroup is applied to characterize Smarandache Semigroups. Examples are provided for justification.
Category: Algebra

[36] viXra:1006.0013 [pdf] submitted on 11 Mar 2010

Smarandache Idempotents in Loop Rings ZtLn(m) of the Loops Ln(m):

Authors: W.B.Vasantha, Moon K. Chetry
Comments: 9 pages

In this paper we establish the existance of S-idempotents in case of loop rings ZtLn(m) for a special class of loops Ln(m); over the ring of modulo integers Zt for a specific value of t. These loops satisfy the conditions gi2 = 1 for every gi ε Ln(m). We prove ZtLn(m) has an S-idempotent when t is a perfect number or when t is of the form 2ip or 3ip (where p is an odd prime) or in general when t = p1ip2 (p1 and p2 are distinct odd primes). It is important to note that we are able to prove only the existance of a single S-idempotent; however we leave it as an open problem wheather such loop rings have more than one S-idempotent. This paper has three sections. In section one, we give the basic notions about the loops Ln(m) and recall the definition of S-idempotents in rings. In section two, we establish the existance of S-idempotents in the loop ring ZtLn(m). In the final section, we suggest some interesting problems based on our study.
Category: Algebra

[35] viXra:1005.0110 [pdf] submitted on 11 Mar 2010

Smarandache Zero Divisors

Authors: W.B.Vasantha Kandasamy
Comments: 5 pages

In this paper, we study the notion of Smarandache zero divisor in semigroups and rings. We illustrate them with examples and prove some interesting results about them.
Category: Algebra

[34] viXra:1005.0104 [pdf] submitted on 11 Mar 2010

Factors and Primes in Two Smarandache Sequences

Authors: Ralf W. Stephan
Comments: 7 pages

Using a personal computer and freely available software, the author factored some members of the Smarandache consecutive sequence and the reverse Smarandache sequence. Nearly complete factorizations are given up to Sm(80) and RSm(80). Both sequences were excessively searched for prime members, with only one prime found up to Sm(840) and RSm(750): RSm(82) = 828180...10987654321.
Category: Algebra

[33] viXra:1005.0103 [pdf] submitted on 11 Mar 2010

Smarandache Neutrosophic Algebraic Structures

Authors: W. B. Vasantha Kandasamy
Comments: 203 pages

In this book for the first time we introduce the notion of Smarandache neutrosophic algebraic structures. Smarandache algebraic structures had been introduced in a series of 10 books. The study of Smarandache algebraic structures has caused a shift of paradigm in the study of algebraic structures.
Category: Algebra

[32] viXra:1005.0082 [pdf] submitted on 21 May 2010

Infinite Smarandache Groupoids

Authors: A.K.S. Chandra Sekhar Rao
Comments: 6 pages

It is proved that there are infinitely many infinite Smarandache Groupoids.
Category: Algebra

[31] viXra:1005.0070 [pdf] submitted on 11 Mar 2010

Set Linear Algebra and Set Fuzzy Linear Algebra

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K Ilanthenral
Comments: 345 pages.

In this book, the authors define the new notion of set vector spaces which is the most generalized form of vector spaces. Set vector spaces make use of the least number of algebraic operations, therefore, even a non-mathematician is comfortable working with it. It is with the passage of time, that we can think of set linear algebras as a paradigm shift from linear algebras. Here, the authors have also given the fuzzy parallels of these new classes of set linear algebras. This book abounds with examples to enable the reader to understand these new concepts easily. Laborious theorems and proofs are avoided to make this book approachable for nonmathematicians. The concepts introduced in this book can be easily put to use by coding theorists, cryptologists, computer scientists, and socio-scientists. Another special feature of this book is the final chapter containing 304 problems. The authors have suggested so many problems to make the students and researchers obtain a better grasp of the subject. This book is divided into seven chapters. The first chapter briefly recalls some of the basic concepts in order to make this book self-contained. Chapter two introduces the notion of set vector spaces which is the most generalized concept of vector spaces. Set vector spaces lends itself to define new classes of vector spaces like semigroup vector spaces and group vector 6 spaces. These are also generalization of vector spaces. The fuzzy analogue of these concepts are given in Chapter three. In Chapter four, set vector spaces are generalized to biset bivector spaces and not set vector spaces. This is done taking into account the advanced information technology age in which we live. As mathematicians, we have to realize that our computer-dominated world needs special types of sets and algebraic structures. Set n-vector spaces and their generalizations are carried out in Chapter five. Fuzzy n-set vector spaces are introduced in the sixth chapter. The seventh chapter suggests more than three hundred problems. When a researcher sets forth to solve them, she/he will certainly gain a deeper understanding of these new notions.
Category: Algebra

[30] viXra:1005.0069 [pdf] submitted on 11 Mar 2010

Smarandache Semirings and Semifields

Authors: W. B. Vasantha Kandasamy
Comments: 4 pages.

In this paper we study the notion of Smarandache semirings and semifields and obtain some interesting results about them. We show that not every semiring is a Smarandache semiring. We similarly prove that not every semifield is a Smarandache semifield. We give several examples to make the concept lucid. Further, we propose an open problem about the existence of Smarandache semiring S of finite order.
Category: Algebra

[29] viXra:1005.0065 [pdf] submitted on 11 Mar 2010

Smarandache Pseudo-Ideals

Authors: W. B. Vasantha Kandasamy
Comments: 5 pages

In this paper we study the Smarandache pseudo-ideals of a Smarandache ring. We prove every ideal is a Smarandache pseudo-ideal in a Smarandache ring but every Smarandache pseudo-ideal in general is not an ideal. Further we show that every polynomial ring over a field and group rings FG of the group G over any field are Smarandache rings. We pose some interesting problems about them.
Category: Algebra

[28] viXra:1005.0046 [pdf] submitted on 11 Mar 2010

N-Linear Algebra of Type II

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

This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of type II which is introduced in this book. In case of n-linear algebra of type II, we are in a position to define linear functionals which is one of the marked difference between the n-vector spaces of type I and II. However all the applications mentioned in n-linear algebras of type I can be appropriately extended to n-linear algebras of type II. Another use of n-linear algebra (n-vector spaces) of type II is that when this structure is used in coding theory we can have different types of codes built over different finite fields whereas this is not possible in the case of n-vector spaces of type I. Finally in the case of n-vector spaces of type II we can obtain neigen values from distinct fields; hence, the n-characteristic polynomials formed in them are in distinct different fields. An attractive feature of this book is that the authors have suggested 120 problems for the reader to pursue in order to understand this new notion. This book has three chapters. In the first chapter the notion of n-vector spaces of type II are introduced. This chapter gives over 50 theorems. Chapter two introduces the notion of n-inner product vector spaces of type II, n-bilinear forms and n-linear functionals. The final chapter 6 suggests over a hundred problems. It is important that the reader should be well versed with not only linear algebra but also nlinear algebras of type I.
Category: Algebra

[27] viXra:1005.0045 [pdf] submitted on 11 Mar 2010

N-Linear Algebra of Type I and Its Applications

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 120 pages

With the advent of computers one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure namely n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of nvector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several related results which are analogues of the classical linear algebra theorems. In case of n-vector spaces we can define several types of linear transformations. The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in finite element analysis of mechanical structures with uncertain parameters. Further it is suggested that the concept of nmatrices can be used in real world problems which adopts fuzzy models like Fuzzy Cognitive Maps, Fuzzy Relational Equations and Bidirectional Associative Memories. The applications of 6 these algebraic structures are given in Chapter 3. Chapter four gives some problem to make the subject easily understandable. The authors deeply acknowledge the unflinching support of Dr.K.Kandasamy, Meena and Kama.
Category: Algebra

[26] viXra:1005.0021 [pdf] submitted on 11 Mar 2010

Neutrosophic Rings

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 154 pages

In this book we define the new notion of neutrosophic rings. The motivation for this study is two-fold. Firstly, the classes of neutrosophic rings defined in this book are generalization of the two well-known classes of rings: group rings and semigroup rings. The study of these generalized neutrosophic rings will give more results for researchers interested in group rings and semigroup rings. Secondly, the notion of neutrosophic polynomial rings will cause a paradigm shift in the general polynomial rings. This study has to make several changes in case of neutrosophic polynomial rings. This would give solutions to polynomial equations for which the roots can be indeterminates. Further, the notion of neutrosophic matrix rings is defined in this book. Already these neutrosophic matrixes have been applied and used in the neutrosophic models like neutrosophic cognitive maps (NCMs), neutrosophic relational maps (NRMs) and so on.
Category: Algebra

[25] viXra:1005.0007 [pdf] submitted on 10 Mar 2010

Smarandache Near-Rings and Their Generalizations

Authors: W. B. Vasantha Kandasamy
Comments: 5 pages

In this paper we study the Smarandache semi-near-ring and nearring, homomorphism, also the Anti-Smarandache semi-near-ring. We obtain some interesting results about them, give many examples, and pose some problems. We also define Smarandache semi-near-ring homomorphism.
Category: Algebra

[24] viXra:1005.0005 [pdf] submitted on 10 Mar 2010

Basic Neutrosophic Algebraic Structures and Their Application to Fuzzy and Neutrosophic Models

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 149 pages

Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs or bipartite graphs. Thus the use of graphs has become inevitable in fuzzy models. The neutrosophic models are fuzzy models that permit the factor of indeterminacy. It also plays a significant role, and utilizes the concept of neutrosophic graphs. Thus neutrosophic graphs and neutrosophic bipartite graphs plays the role of representing the neutrosophic models. Thus to construct the neutrosophic graphs one needs some of the neutrosophic algebraic structures viz. neutrosophic fields, neutrosophic vector spaces and neutrosophic matrices. So we for the first time introduce and study these concepts. As our analysis in this book is application of neutrosophic algebraic structure we found it deem fit to first introduce and study neutrosophic graphs and their applications to neutrosophic models.
Category: Algebra

[23] viXra:1005.0004 [pdf] submitted on 10 Mar 2010

Smarandache Non-Associative (SNA-) rings

Authors: W. B. Vasantha Kandasamy
Comments: 13 pages

In this paper we introduce the concept of Smarandache non-associative rings, which we shortly denote as SNA-rings as derived from the general definition of a Smarandache Structure (i.e., a set A embedded with a week structure W such that a proper subset B in A is embedded with a stronger structure S). Till date the concept of SNA-rings are not studied or introduced in the Smarandache algebraic literature. The only non-associative structures found in Smarandache algebraic notions so far are Smarandache groupoids and Smarandache loops introduced in 2001 and 2002. But they are algebraic structures with only a single binary operation defined on them that is nonassociative. But SNA-rings are non-associative structures on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from the right and left. Further to understand the concept of SNA-rings one should be well versed with the concept of group rings, semigroup rings, loop rings and groupoid rings. The notion of groupoid rings is new and has been introduced in this paper. This concept of groupoid rings can alone provide examples of SNA-rings without unit since all other rings happens to be either associative or nonassociative rings with unit. We define SNA subrings, SNA ideals, SNA Moufang rings, SNA Bol rings, SNA commutative rings, SNA non-commutative rings and SNA alternative rings. Examples are given of each of these structures and some open problems are suggested at the end.
Category: Algebra

[22] viXra:1005.0002 [pdf] submitted on 1 May 2010

Almost Unbiased Estimator for Estimating Population Mean Using Known Value of Some Population Parameter(s)

Authors: Rajesh Singh, Mukesh Kumar, Florentin Smarandache
Comments: 14 pages

In this paper we have proposed an almost unbiased estimator using known value of some population parameter(s). Various existing estimators are shown particular members of the proposed estimator. Under simple random sampling without replacement (SRSWOR) scheme the expressions for bias and mean square error (MSE) are derived. The study is extended to the two phase sampling. Empirical study is carried out to demonstrate the superiority of the proposed estimator.
Category: Algebra

[21] viXra:1004.0084 [pdf] submitted on 9 Mar 2010

New Classes of Neutrosophic Linear Algebras

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 288 pages.

In this book we introduce mainly three new classes of linear algebras; neutrosophic group linear algebras, neutrosophic semigroup linear algebras and neutrosophic set linear algebras. The authors also define the fuzzy analogue of these three structures.
Category: Algebra

[20] viXra:1003.0231 [pdf] submitted on 7 Mar 2010

Smarandache Special Definite Algebraic Structures

Authors: W. B. Vasantha Kandasamy
Comments: 141 pages

In this book we introduce the notion of Smarandache special definite algebraic structures. We can also call them equivalently as Smarandache definite special algebraic structures. These new structures are defined as those strong algebraic structures which have in them a proper subset which is a weak algebraic structure. For instance, the existence of a semigroup in a group or a semifield in a field or a semiring in a ring. It is interesting to note that these concepts cannot be defined when the algebraic structure has finite cardinality i.e., when the algebraic structure has finite number of elements in it.
Category: Algebra

[19] viXra:1003.0168 [pdf] submitted on 6 Mar 2010

K-Nomial Coefficients

Authors: Florentin Smarandache
Comments: 4 pages

In this article we will widen the concepts of "binomial coefficients" and "trinomial coefficients" to the concept of "k-nomial coefficients", and one obtains some general properties of these. As an application, we will generalize the" triangle of Pascal".
Category: Algebra

[18] viXra:1003.0115 [pdf] submitted on 6 Mar 2010

Special Algebraic Structures

Authors: Florentin Smarandache
Comments: 4 pages

New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> make an important such contribution.
Category: Algebra

[17] viXra:1003.0098 [pdf] submitted on 6 Mar 2010

Applications of Bimatrices to Some Fuzzy and Neutrosophic Models

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 273 pages

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. These concepts are also extended neutrosophically in this book.
Category: Algebra

[16] viXra:1003.0097 [pdf] submitted on 6 Mar 2010

Introduction to Bimatrices

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 181 pages

Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other.
Category: Algebra

[15] viXra:1003.0096 [pdf] submitted on 6 Mar 2010

Introduction to Linear Bialgebra

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 238 pages

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications.
Category: Algebra

[14] viXra:1003.0079 [pdf] submitted on 7 Mar 2010

Linear Algebra and Smarandache Linear Algebra

Authors: W. B. Vasantha Kandasamy
Comments: 175 pages

While I began researching for this book on linear algebra, I was a little startled. Though, it is an accepted phenomenon, that mathematicians are rarely the ones to react surprised, this serious search left me that way for a variety of reasons. First, several of the linear algebra books that my institute library stocked (and it is a really good library) were old and crumbly and dated as far back as 1913 with the most 'new' books only being the ones published in the 1960s.
Category: Algebra

[13] viXra:1003.0078 [pdf] submitted on 7 Mar 2010

Smarandache Fuzzy Algebra

Authors: W. B. Vasantha Kandasamy
Comments: 455 pages

In 1965, Lofti A. Zadeh introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. It provoked, at first (and as expected), a strong negative reaction from some influential scientists and mathematicians - many of whom turned openly hostile. However, despite the controversy, the subject also attracted the attention of other mathematicians and in the following years, the field grew enormously, finding applications in areas as diverse as washing machines to handwriting recognition. In its trajectory of stupendous growth, it has also come to include the theory of fuzzy algebra and for the past five decades, several researchers have been working on concepts like fuzzy semigroup, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings and so on.
Category: Algebra

[12] viXra:1003.0077 [pdf] submitted on 7 Mar 2010

Bialgebraic Structures and Smarandache Bialgebraic Structures

Authors: W. B. Vasantha Kandasamy
Comments: 272 pages

The study of bialgebraic structures started very recently. Till date there are no books solely dealing with bistructures. The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published in 1998. In the year 1999, bivector spaces was introduced. In 2001, concept of free De Morgan bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the construction of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category theory or linear logic.
Category: Algebra

[11] viXra:1003.0076 [pdf] submitted on 7 Mar 2010

Smarandache Non-Associative Rings

Authors: W. B. Vasantha Kandasamy
Comments: 201 pages

An associative ring is just realized or built using reals or complex; finite or infinite by defining two binary operations on it. But on the contrary when we want to define or study or even introduce a non-associative ring we need two separate algebraic structures say a commutative ring with 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algebras and Jordan algebras are mainly built using a vector space over a field satisfying special identities called the Jacobi identity and Jordan identity respectively. Study of these algebras started as early as 1940s. Hence the study of non-associative algebras or even non-associative rings boils down to the study of properties of vector spaces or linear algebras over fields.
Category: Algebra

[10] viXra:1003.0075 [pdf] submitted on 7 Mar 2010

Smarandache Near-Rings

Authors: W. B. Vasantha Kandasamy
Comments: 201 pages

Near-rings are one of the generalized structures of rings. The study and research on near-rings is very systematic and continuous. Near-ring newsletters containing complete and updated bibliography on the subject are published periodically by a team of mathematicians (Editors: Yuen Fong, Alan Oswald, Gunter Pilz and K. C. Smith) with financial assistance from the National Cheng Kung University, Taiwan. These newsletters give an overall picture of the research carried out and the recent advancements and new concepts in the field. Conferences devoted solely to near-rings are held once every two years. There are about half a dozen books on near-rings apart from the conference proceedings. Above all there is a online searchable database and bibliography on near-rings. As a result the author feels it is very essential to have a book on Smarandache near-rings where the Smarandache analogues of the near-ring concepts are developed. The reader is expected to have a good background both in algebra and in near-rings; for, several results are to be proved by the reader as an exercise.
Category: Algebra

[9] viXra:1003.0074 [pdf] submitted on 7 Mar 2010

Smarandache Rings

Authors: W. B. Vasantha Kandasamy
Comments: 222 pages

Over the past 25 years, I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings) specially to motivate both ring theorists and Smarandache algebraists to develop and study several important and innovative properties about S-rings.
Category: Algebra

[8] viXra:1003.0073 [pdf] submitted on 7 Mar 2010

Smarandache Loops

Authors: W. B. Vasantha Kandasamy
Comments: 129 pages

The theory of loops (groups without associativity), though researched by several mathematicians has not found a sound expression, for books, be it research level or otherwise, solely dealing with the properties of loops are absent. This is in marked contrast with group theory where books are abundantly available for all levels: as graduate texts and as advanced research books.
Category: Algebra

[7] viXra:1003.0072 [pdf] submitted on 7 Mar 2010

Smarandache Semirings, Semifields, and Semivector Spaces

Authors: W. B. Vasantha Kandasamy
Comments: 122 pages

Smarandache notions, which can be undoubtedly characterized as interesting mathematics, has the capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or identification as a substructure. Several researchers around the world working on Smarandache notions have systematically carried out this study. This is the first book on the Smarandache algebraic structures that have two binary operations.
Category: Algebra

[6] viXra:1003.0071 [pdf] submitted on 7 Mar 2010

Groupoids and Smarandache Groupoids

Authors: W. B. Vasantha Kandasamy
Comments: 115 pages

The study of Smarandache Algebraic Structure was initiated in the year 1998 by Raul Padilla following a paper written by Florentin Smarandache called "Special Algebraic Structures". In his research, Padilla treated the Smarandache algebraic structures mainly with associative binary operation. Since then the subject has been pursued by a growing number of researchers and now it would be better if one gets a coherent account of the basic and main results in these algebraic structures. This book aims to give a systematic development of the basic non-associative algebraic structures viz. Smarandache groupoids. Smarandache groupoids exhibits simultaneously the properties of a semigroup and a groupoid. Such a combined study of an associative and a non associative structure has not been so far carried out. Except for the introduction of smarandacheian notions by Prof. Florentin Smarandache such types of studies would have been completely absent in the mathematical world.
Category: Algebra

[5] viXra:1003.0070 [pdf] submitted on 7 Mar 2010

Smarandache Semigroups

Authors: W. B. Vasantha Kandasamy
Comments: 95 pages

The main motivation and desire for writing this book, is the direct appreciation and attraction towards the Smarandache notions in general and Smarandache algebraic structures in particular. The Smarandache semigroups exhibit properties of both a group and a semigroup simultaneously. This book is a piece of work on Smarandache semigroups and assumes the reader to have a good background on group theory; we give some recollection about groups and some of its properties just for quick reference.
Category: Algebra

[4] viXra:1003.0066 [pdf] submitted on 5 Mar 2010

Theory and Problems on Algebraic Structures.

Authors: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache
Comments: 119 pages, In Romanian language.

Theory and problems on algebraic structures.
Category: Algebra

[3] viXra:0911.0034 [pdf] submitted on 13 Nov 2009

Opuestos, Grafos Y Arithmeticas

Authors: Por Kujonai
Comments: 82 pages, In Spanish

A continuación, pretendo relacionar varios conceptos como modulo, opuestos (o signos), aritmética, el cuarto nivel de hypernumeros de Musean, politopos, especialmente el triangulo, matrices y determinantes, complejos, raices, ..., ya que de esta sopa de conceptos nace mi trabajo, aunque a un nivel mas profundo nace por darle un sentido matemático simple al concepto de opuesto, especialmente a una aritmética de 3 signos, y lo demás fue saliendo a medida de que avanzaba en esto, mientras iba adquiriendo sentido y fuerza.
Category: Algebra

[2] viXra:0910.0026 [pdf] submitted on 16 Oct 2009

Proof Without Words: the Expansion of (1 + X + X2 + ... + Xn)3

Authors: Hideyuki Ohtsuka
Comments: 2 Pages

In this paper, we show a geometry approach to the expansion of (1 + x + x2 + ... + xn)3. This proof is a "Proof Without Words"
Category: Algebra

[1] viXra:0902.0006 [pdf] submitted on 14 Feb 2009

A Study Of New Concepts In Smarandache Quasigroups And Loops

Authors: Jaiyeola Temitope Gbolahan
Comments: recovered from sciprint.org

A Study Of New Concepts In Smarandache Quasigroups And Loops
Category: Algebra

Replacements of recent Submissions

[9] viXra:1404.0466 [pdf] replaced on 2014-06-21 19:29:49

The Idea of the Arithmetica

Authors: Hajime Mashima
Comments: 8 Pages.

During the 360 years of Fermat's last theorem is to be proved, this proposition was the presence appear full-length novel in "The Lord of the Rings", such as the "One Ring". And finally in 1994, it was proved completely by Andrew Wiles. However interesting proof is Fermat has been is still unknown. This will be assumed in the category of algebra probably.
Category: Algebra

[8] viXra:1403.0958 [pdf] replaced on 2014-03-29 00:04:16

Generalized Determinant

Authors: Nikolay Dementev
Comments: 7 Pages.

The report suggests an approach to extend a concept of determinant to the systems of any order.
Category: Algebra

[7] viXra:1403.0037 [pdf] replaced on 2014-03-23 12:35:47

Algebraic Combinatorics and Combinatorial Analysis

Authors: Giuseppe Rauti
Comments: 2 Pages.

Algebraic Combinatorics, Combinatorial Analysis, Additive Combinatorics.
Category: Algebra

[6] viXra:1312.0213 [pdf] replaced on 2013-12-28 19:47:58

Algebraic Structues Using [0,n)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 216 Pages.

The algebraic structures built using [0, n) interval are new and innovative. They happen to have different properties. The interval [0, n) can be realized as the real algebraic closure of the modulo ring Zn. The algebraic behavior of [0, n) is different from the ring Zn.
Category: Algebra

[5] viXra:1212.0124 [pdf] replaced on 2012-12-21 04:13:24

NP-Hardness of Optimizing the Sum of Rational Linear Functions Over an Asymptotic-Linear-Program

Authors: Deepak Ponvel Chermakani
Comments: There are 6 Pages, 6 Theorems, 7 Figures. I also made a small correction that in Theorem-1, the correct word is "NP-Hard" and not "NP-Complete".

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.
Category: Algebra

[4] viXra:1211.0029 [pdf] replaced on 2012-11-27 13:25:20

A Rewriting System Applied to the Simplest Algebraic Identities

Authors: J. S. Markovitch
Comments: 2 Pages.

A rewriting system applied to the simplest algebraic identities is shown to yield second- and third-degree equations that share a property associated with the constant 137.036, which is a minimal case.
Category: Algebra

[3] viXra:1205.0093 [pdf] replaced on 2012-06-08 19:15:06

Mathematical Analysis of the Problems faced by the People With Disabilities (PWDs) (With Specific Reference to Tamil Nadu in India)

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, A. Praveen Prakash
Comments: 167 Pages.

The authors in this book have analyzed the socio-economic and psychological problems faced by People with Disabilities (PWDs) and their families. The study was made by collecting data using both fuzzy linguistic questionnaire / by interviews in case they are not literates from 2,15,811 lakhs people. This data was collected using the five Non Government Organizations (NGOs) from northern Tamil Nadu.
Category: Algebra

[2] viXra:1005.0104 [pdf] replaced on 25 Aug 2011

Factors and Primes in Two Smarandache Sequences

Authors: Ralf W. Stephan
Comments: 10 Pages

Using a personal computer and freely available software, the author factored some members of the Smarandache consecutive sequence and the reverse Smarandache sequence. Nearly complete factorizations are given up to Sm(80) and RSm(80). Both sequences were excessively searched for prime members, with only one prime found up to Sm(840) and RSm(750): RSm(82) = 828180 ... 10987654321.
Category: Algebra

[1] viXra:1003.0066 [pdf] replaced on 6 Mar 2010

Theory and Problems on Algebraic Structures.

Authors: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache
Comments: 119 pages, v1 in Romanian language, v2 in Russian language.

Theory and problems on algebraic structures.
Category: Algebra