arXiv subjects

W. B. Vasantha Kandasamy

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

At least 37 records · Page 2Linked to original sources

arXiv2009

Special set linear algebra and special set fuzzy linear algebra

The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analogue is introduced in chapter two. In chapter three the notion of special set semigroup linear algebra is introduced. The concept of special set n-vector spaces, n greater than or equal to three is defined and their fuzzy analogue is their fuzzy analogue is given in chapter four. The probable applications are also mentioned. The final chapter suggests 66 problems.

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arXiv2009

Reservation for Other Backward Classes in Indian Central Government Institutions like IITs, IIMs and AIIMs- A Study of the Role of Media using Fuzzy Super FRM models

In this book the authors introduce three new types of fuzzy model called the super column Fuzzy Relational Model using super column matrices, super row fuzzy relational model using super row matrices and super mixed fuzzy relational model using supermatrices. These new models are used to study the role of media on 27 percent reservation for the other backward classes in the educational institutions run by the Indian central Government. This book has four chapters. Chapter one introduces the new notion of super fuzzy relational models using supermatrices. In chapter two these new models are used to study the problem. Conclusions and opinions are given in chapters four and three respectively.

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arXiv2009

Superbimatrices and their generalizations

In this book, the authors introduce the new notion of superbimatrices and generalize it to supertrimatrices and super n-matrices. Study of these structures is not only interesting and innovative but is also best suited for the computerize world. The main difference between simple bimatrices and super bimatrices is that in case of simple bimatrices we have only one type of product defined on them, whereas in case of superbimatrices we have different types of products called minor and major defined using them. This book has four chapters. Chapter one describes the basic concepts to make this book a selfcontained one. Superbimatrices, semi superbimatrices, symmetric superbimatrices are introduced in chapter two. Chapter three introduces the notion of super trimatrices and the products defined using them. Chapter four gives the most generalized form of the superbimatrix, viz. super n-matrix

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arXiv2009

Groups as Graphs

For the first time we represent every finite group in the form of a graph in this book. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties, like the subgroups of a group, normal subgroups of a group, p-sylow subgroups of a group and conjugate elements of a group are carried out using the identity graph of the group in this book. This book has four chapters. The first chapter is introductory. The second chapter represents groups as graphs. In the third chapter, we have defined similar types of graphs for algebraic structures like commutative semigroups, loops, commutative groupoids and commutative rings. The final chapter poses 52 problems.

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arXiv2009

Smarandache Special Definite Algebraic Structures

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. This book has four chapters. Chapter one is introductory in nature. In chapter two, the notion of Smarandache special definite groups and Smarandache special definite fields are introduced and several interesting properties are derived. The notion of Smarandache definite special rings, vector spaces and linear algebras are introduced and analyzed in chapter three. The final chapter suggests over 200 problems.

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arXiv2009

n-Linear Algebra of type II

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 n-eigen 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 suggests over a hundred problems. It is important that the reader is well-versed not only with linear algebra but also n-linear algebra of type I.

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arXiv2008

n-Linear Algebra of type I and its applications

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 n-vector 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 n-matrices 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 these algebraic structures are given in the third chapter. The fourth chapter suggests problems to further a reader's understanding of the subject.

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arXiv2008

Super Linear Algebra

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. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized world. Some other structures ought to replace linear algebras which are over a century old. Super linear algebras that use super matrices can store data not only in a block but in multiple blocks so it is certainly more powerful than the usual matrices. This book has 3 chapters. Chapter one introduces the notion of super vector spaces and enumerates a number of properties. Chapter two defines the notion of super linear algebra, super inner product spaces and super bilinear forms. Several interesting properties are derived. The main application of these new structures in Markov chains and Leontief economic models are also given in this chapter. The final chapter suggests 161 problems mainly to make the reader understand this new concept and apply them.

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arXiv2008

Methods in Industrial Biotechnology for Chemical Engineers

In keeping with the definition that biotechnology is really no more than a name given to a set of techniques and processes, the authors apply some set of fuzzy techniques to chemical industry problems such as finding the proper proportion of raw mix to control pollution, to study flow rates, to find out the better quality of products. We use fuzzy control theory, fuzzy neural networks, fuzzy relational equations, genetic algorithms to these problems for solutions. When the solution to the problem can have certain concepts or attributes as indeterminate, the only model that can tackle such a situation is the neutrosophic model. The authors have also used these models in this book to study the use of biotechnology in chemical industries. This book has six chapters. First chapter gives a brief description of biotechnology. Second chapter deals will proper proportion of mix of raw materials in cement industries to minimize pollution using fuzzy control theory. Chapter three gives the method of determination of temperature set point for crude oil in oil refineries. Chapter four studies the flow rates in chemical industries using fuzzy neutral networks. Chapter five gives the method of minimization of waste gas flow in chemical industries using fuzzy linear programming. The final chapter suggests when in these studies indeterminancy is an attribute or concept involved, the notion of neutrosophic methods can be adopted.

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arXiv2008

Set Linear Algebra and Set Fuzzy Linear Algebra

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 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 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.

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arXiv2008

Super Fuzzy Matrices and Super Fuzzy Models for Social Scientists

This book introduces the concept of fuzzy super matrices and operations on them. This book will be highly useful to social scientists who wish to work with multi-expert models. Super fuzzy models using Fuzzy Cognitive Maps, Fuzzy Relational Maps, Bidirectional Associative Memories and Fuzzy Associative Memories are defined here. The authors introduce 13 multi-expert models using the notion of fuzzy supermatrices. These models are described with illustrative examples. This book has three chapters. In the first chaper, the basic concepts about super matrices and fuzzy super matrices are recalled. Chapter two introduces the notion of fuzzy super matrices adn their properties. The final chapter introduces many super fuzzy multi expert models.

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arXiv2008

New Classes of Codes for Cryptologists and Computer Scientists

In this book, we have introduced several new classes of codes to aid cryptologists and computer scientists. We have explained these codes very non-technically so that a strong mathematical foundation is not needed to understand them. This book also provides an easy method to detect and correct errors that occur during transmission. Further, some of the codes are constructed so as to mislead an intruder/ hacker. False n-codes, whole n-codes can serve this pupose. These codes can be used to ensure security in networks and safe transmission of identity. We have named a few new classes of codes after Periyar, the south-Indian social leader, to mark his services to humanity. This book is divided into three chapters. Chapter one is introductory in nature. The notion of bicodes and their generalization, and n-codes are introduced in the second chapter. Periyar linear codes are introduced in the third chapter. We have used two methods, viz. pseudo best n-approximations and n-coset leader properties to detect and correct errors.

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arXiv2007

Special Fuzzy Matrices for Social Scientists

This book introduces special classes of Fuzzy and Neutrosophic matrices. These special classes of matrices are used in the construction of multi-expert special fuzzy models using FCM, FRM and FRE and their Neutorosophic analogues (simultaneous or otherwise, according to ones need). Using the six basic models, we have constructed a multi-expert multi-model called the Super Special Hexagonal Fuzzy and Neutrosophic model. Given any special input vector, these models can give the resultant using special operations. When coupled with computer programming, these operations can give the solution within a reasonable time period. Such multi-expert multi-model systems are not only a boon to social scientists, but also to anyone who wants to use Fuzzy or Neutrosophic models.

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arXiv2007

Elementary fuzzy matrix theory and fuzzy models for social scientists

This book gives the basic notions of fuzzy matrix theory and its applications to simple fuzzy models. The approach is non-traditional in order to attract many students to use this methodology in their research. The traditional approach of mathematicians has conditioned students of sociology in such a manner that they are averse to using mathematical tools. Six simple types of fuzzy models that make use of fuzzy matrices are given. These models are distinct because they are time-dependent and can even be used for statistical data. The Fuzzy Cognitive Maps models gives the hidden pattern. Fuzzy Relational Maps model not only gives the hidden pattern but also gives the inter-relations between two sets of disjoint attributes. The Bidirectional Associative Memories model analyzes data depending on the time-period, while the Fuzzy Associative Memories model can give the gradation of importance of each attribute. Finally, the Fuzzy Relational Equation model is capable of giving a solution closer to the predicted solution. All the models are illustrated through elaborate examples of particular social problems.

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arXiv2006

Vedic Mathematics: 'Vedic' or 'Mathematics' -- A Fuzzy and Neutrosophic Analysis

In this book the authors probe into Vedic Mathematics (a concept that gained renown in the period of the religious fanatic and revivalist Hindutva rule in India): and explore whether it is really 'Vedic' in origin or 'Mathematics' in content. We analyzed this problem using fuzzy models like Fuzzy Cognitive Maps (FCM), Fuzzy Relational Maps (FRM) and the newly constructed fuzzy dynamical system (and its Neutrosophic analogue) that can analyze multi-experts opinion at a time using a single model. The issue of Vedic Mathematics involves religious politics, caste supremacy, apart from elementary arithmetic -- so we cannot use simple statistics for our analysis.

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arXiv2006

Fuzzy Interval Matrices, Neutrosophic Interval Matrices and their Applications

The new concept of fuzzy interval matrices has been introduced in this book for the first time. The authors have not only introduced the notion of fuzzy interval matrices, interval neutrosophic matrices and fuzzy neutrosophic interval matrices but have also demonstrated some of its applications when the data under study is an unsupervised one and when several experts analyze the problem. Further, the authors have introduced in this book multiexpert models using these three new types of interval matrices. The new multi expert models dealt in this book are FCIMs, FRIMs, FCInMs, FRInMs, IBAMs, IBBAMs, nIBAMs, FAIMs, FAnIMS, etc. Illustrative examples are given so that the reader can follow these concepts easily. This book has three chapters. The first chapter is introductory in nature and makes the book a self-contained one. Chapter two introduces the concept of fuzzy interval matrices. Also the notion of fuzzy interval matrices, neutrosophic interval matrices and fuzzy neutrosophic interval matrices, can find applications to Markov chains and Leontief economic models. Chapter three gives the application of fuzzy interval matrices and neutrosophic interval matrices to real-world problems by constructing the models already mentioned. Further these models are mainly useful when the data is an unsupervised one and when one needs a multi-expert model. The new concept of fuzzy interval matrices and neutrosophic interval matrices will find their applications in engineering, medical, industrial, social and psychological problems. We have given a long list of references to help the interested reader.

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arXiv2006

Neutrosophic Rings

This book has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings,neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this book is that it gives 246 problems in Chapter four. Some of the problems are direct and simple, some little difficult and some can be taken up as a research problem.

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arXiv2006

Smarandache Neutrosophic Algebraic Structures

This book has seven chapters. In Chapter one, an elaborate recollection of Smarandache structures like S-semigroups, S-loops, and S-groupoids is given. It also gives notions about N-ary algebraic stuctures and their Smarandache analogue, Neutrosophic structures viz. groups, semigroups, groupoids and loops are given in Chapter one to make the book a self-contained one. For the first time, S-neutrosophic groups and S-neutrosophic N-groups are introduced in Chapter two and their properties are given. S-neutrosophic semigroups and S-neutrosophic N-semigroups are defined and discussed in Chapter three. Chapter four defines S-neutrosophic S-loops and S-N-neutrosophic groupoids and their generalizations are given in Chapter five. Chapter six gives S-neutrosophic mixed N-structures and their duals. Chapter seven gives 68 problems for any interested reader.

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