arXiv subjects

W. B. Vasantha Kandasamy

Publications and source records attributed to W. B. Vasantha Kandasamy.

50 records · Page 3Linked to original sources

arXiv2006

Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures

In this book, for the first time we introduce the notion of neutrosophic algebraic structures for groups, loops, semigroups and groupoids; and also their neutrosophic N-algebraic structures. One is fully aware of the fact that many classical theorems like Lagrange, Sylow and Cauchy have been studied only in the context of finite groups. Here we try to shift the paradigm by studying and introducing these theorems to neutrosophic semigroups, neutrosophic groupoids, and neutrosophic loops. This book has seven chapters. Chapter one provides several basic notions to make this book self-contained. Chapter two introduces neutrosophic groups and neutrosophic N-groups and gives several examples. The third chapter deals with neutrosophic semigroups and neutrosophic N-semigroups, giving several interesting results. Chapter four introduces neutrosophic loops and neutrosophic N-loops. We introduce several new, related definitions. In fact we construct a new class of neutrosophic loops using modulo integer Z_n, n > 3, where n is odd. Several properties of these structures are proved using number theoretic techniques. Chapter five just introduces the concept of neutrosophic groupoids and neutrosophic N-groupoids. Sixth chapter innovatively gives mixed neutrosophic structures and their duals. The final chapter gives problems for the interested reader to solve.

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arXiv2006

N-Algebraic Structures and S-N-Algebraic Structures

For the first time, we have introduced the concept of N-groups, N-semigroups, N-loops, and N-groupoids. We also define a mixed N-algebraic structure. The main aim of this book is to attract young mathematicians to this interesting field. It contains more than 200 new definitions. These concepts find applications in fields like finite automaton, coloring problems and coding theory.

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arXiv2005

Fuzzy and Neutrosophic Analysis of Periyar's views on Untouchability

This book is organized into four chapters. In Chapter One we just introduce the basic Fuzzy and Neutrosophic tools used in the analysis of the social evil of Untouchability. Since the notion of caste is based on themind, it is appropriate to use Fuzzy and Neutrosophic theory. In Chapter Two we use the opinion of several experts to analyze the various aspects of untouchability. Here, we use the tools that have been described in Chapter One. Fuzzy Directed Graphs and Neutrosophic Directed Graphs of these Fuzzy and Neutrosophic models happen to be very dense. We have represented 16 such graphs in this book. In Chapter Three we give a brief introduction about the life and struggle of Periyar. The fourth and concluding chapter gives observations drawn from our mathematical results based on the Fuzzy and Neutrosophic analysis.

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arXiv2005

Introduction to Linear Bialgebra

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. With the recent introduction of bimatrices (2005)we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic structures. It is important to mention here it is a matter of simple exercise to extend these to linear n-algebra for any n greater than 2; for n = 2 we get the linear bialgebra. This book has five chapters. In the first chapter we just introduce some basic notions of linear algebra and Slinear algebra and their applications. Chapter two introduces some new algebraic bistructures. In chapter three we introduce the notion of linear bialgebra and discuss several interesting properties about them. Also, application of linear bialgebra to bicodes is given. A remarkable part of our research in this book is the introduction of the notion of birepresentation of bigroups. The fourth chapter introduces several neutrosophic algebraic structures since they help in defining the new concept of neutrosophic linear bialgebra, neutrosophic bivector spaces, Smarandache neutrosophic linear bialgebra and Smarandache neutrosophic bivector spaces. Theirprobable applications to real-world models are discussed.

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arXiv2005

Fuzzy and Neutrosophic Analysis of Women with HIV/AIDS

This book has six chapters. The first chapter gives a brief introduction about HIV/AIDS among rural women in Tamil Nadu. Chapter 2 is an analysis of the situation using fuzzy theory in general and Fuzzy Relational Maps (FRM)in particular. The FRM tool is specially chosen for two reasons: It can give the hidden pattern of the dynamical system and as the socio-economic condition of women infected with HIV/AIDS is interdependent we choose this special fuzzy tool called FRM. In Chapter 3, we use Fuzzy Associative Memories (FAM) to analyze the problem because FAM is the only fuzzy tool that can give the gradations of each of the nodes/concepts. In Chapter 4 for the first time we use neutrosophic theory in general and neutrosophic relational maps (NRM) in particular to analyze this problem. Interested readers can compare and analyze the two models FRM and NRM. This chapter introduces the notion of Neutrosophic Associative Memories (NAMs) that is an analogous model of Fuzzy Associative Memories. NAMs are applied to this problem and conclusions are based on this analysis. The sixth chapter gives the translated version of the verbatim interviews of the 101 HIV/AIDS infected, rural, uneducated, and poor women of Tamil Nadu.

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arXiv2005

Introduction to Bimatrices

Generally any real-world problem is not always solvable, because in that not only a percentage of uncertainty is present, but also, a certain percentage of indeterminacy is present. The presence of uncertainty has been analyzed using fuzzy logic. In this book the amount of indeterminacy is being analyzed using neutrosophic logic. Most of these models use the concept of matrices. Matrices have certain limitation; when the models are time-dependent and any two experts' opinions are being studied simultaneously, one cannot compare both of them at each stage. The new concept of bimatrices would certainly cater to these needs. A bimatrix AB = A1 U A2, where A1 and A2 are distinct matrices of arbitrary order. This book introduces the concept of bimatrices, and studies several notions like bieigen values, bieigen vectors, characteristic bipolynomials, bitransformations, bioperators and bidiagonalization. Further, we introduce and explore the concepts like fuzzy bimatrices, neutrosophic bimatrices and fuzzy neutrosophic bimatrices, which will find its application in fuzzy and neutrosophic logic.

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arXiv2004

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

The involvement of uncertainty of varying degrees when the total of the membership degree exceeds one or less than one, then the newer mathematical paradigm shift, Fuzzy Theory proves appropriate. For the past two or more decades, Fuzzy Theory has become the potent tool to study and analyze uncertainty involved in all problems. But, many real-world problems also abound with the concept of indeterminacy. In this book, the new, powerful tool of neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic models are described. The theory of neutrosophic graphs is introduced and applied to fuzzy and neutrosophic models. This book is organized into four chapters. In Chapter One we introduce some of the basic neutrosophic algebraic structures essential for the further development of the other chapters. Chapter Two recalls basic graph theory definitions and results which has interested us and for which we give the neutrosophic analogues. In this chapter we give the application of graphs in fuzzy models. An entire section is devoted for this purpose. Chapter Three introduces many new neutrosophic concepts in graphs and applies it to the case of neutrosophic cognitive maps and neutrosophic relational maps. The last section of this chapter clearly illustrates how the neutrosophic graphs are utilized in the neutrosophic models. The final chapter gives some problems about neutrosophic graphs which will make one understand this new subject.

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arXiv2004

Fuzzy Relational Equations and Neutrosophic Relational Equations

The introduction of Fuzzy Relational Equations (FREs) has made problems that were unsolvable using algebraic linear equations into solvable ones. FREs have been applied to problemsin medicine, industry, transportation and all types of social problems where the data is an unsupervised one. Yet, FREs lack the capacity to tackle problems where an element of indeterminacy is involved. This book develops the new concept of Neutrosophic Relational Equations (NREs) that have the capacity to analyze problems with indeterminacy. Here, earlier models on FREs are analyzed and new NRE models, with practical applications, are presented.

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arXiv2004

Analysis of Social Aspects of Migrant Labourers Living With HIV/AIDS Using Fuzzy Theory and Neutrosophic Cognitive Maps: With Special Reference to Rural Tamil Nadu in India

This book has seven chapters. The first chapter is introductory in nature and it speaks about the migrant labourers. In chapter two we use Fuzzy Cognitive Maps to analyze the socio-economic problems of HIV/AIDS infected migrant labourers in rural areas of Tamil Nadu. In chapter three we analyze the role played by the government helping these migrant labourers with HIV/AIDS and factors of migration and their vulnerability in catching HIV/AIDS. For the first time Neutrosophic Cognitive Maps are used in the study of migrant labourers who have become HIV/AIDS victims. This study is done in Chapter IV. In chapter V we use Neutrosophic Relational Maps and we define some new neutrosophic tools like Combined Disjoint Block FRM, Combined Overlap NRM and linked NRM. We adopt these new techniques in the study and analysis of this problem. Chapter VI gives a very brief sketch of the life history of these 60 HIV/AIDS infected migrant labourers so that people from different social and cultural backgrounds follow our analysis. The last chapter gives suggestions and conclusions based on our study.

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arXiv2003

Smarandache Loops

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B which is embedded with a stronger structure S. By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A. These types of structures occur in our everyday's life, thats why we study them in this book. As an example: A non-empty set L is said to form a loop, if on L is defined a binary operation called product, denoted by '.' such that: for all a, b belonging to L we have a . b belonging to L (closure property); there exists an element e belonging to L such that a . e = e . a = a for all a belonging to L (e is the identity element of L); for every ordered pair (a, b) belonging to L x L there exists a unique pair (p, q) in L such that ap = b and qa = b. Whence: A Smarandache Loop (or S-Loop) is a loop L such that a proper subset M of L is a subgroup (with respect to the same induced operation).

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arXiv2003

Smarandache Near-rings

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B contained in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A Near-ring is a non-empty set N together with two binary operations '+' and '.' such that (N, +) is a group (not necessarily abelian), (N, .) is a semigroup. For all a, b, c belonging to N we have (a + b) . c = a . c + b . c A Near-field is a non-empty set P together with two binary operations '+' and '.' such that (P, +) is a group (not-necessarily abelian), {P\{0}, .) is a group. For all a, b, c belonging to P we have (a + b) . c = a . c + b . c A Smarandache Near-ring is a near-ring N which has a proper subset P contained in N, where P is a near-field (with respect to the same binary operations on N).

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arXiv2003

Smarandache Non-Associative Rings

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B contained in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A non-associative ring is a non-empty set R together with two binary operations '+' and '.' such that (R, +) is an additive abelian group and (R, .) is a groupoid. For all a, b, c belonging to R we have (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b. A Smarandache non-associative ring is a non-associative ring (R, +, .) which has a proper subset P contained in R, that is an associative ring (with respect to the same binary operations on R).

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arXiv2003

Smarandache semirings, semifields and semivector spaces

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B which is embedded with a stronger structure S. By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A. These types of structures occur in our every day's life, that's why we study them in this book. Thus, as three particular cases: 1) A Smarandache semiring is a semiring A such that a proper subset B of A is a semifield (with respect to the same induced operation). 2) A Smarandache semifield is a semifield A such that a proper subset B of A is a k-semi algebra, with respect to the same induced operations and an external operator. 3) A Smarandache semivector space is a semivector space A (over a semifield B) which is a Smarandache semigroup.

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arXiv2003

Groupoids and Smarandache Groupoids

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