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...ntroduced. 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 should be well versed with not only linear algebra but also n-linear algebras of type I....

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... american research press rehoboth 2002 { φ} {a} {b} {c} {d} {a,b,c} {a,b,c,d} {a,b} {a,c} {a,d} {b,c} {b,d} {d,c} {a... ... {a,b,c} {a,b,c,d} {a,b} {a,c} {a,d} {b,c} {b,d} {d,c} {a,b,d} {a,d,c} {b,d,c} 1 Smarandache Semirings, Semifields, and Semivector spaces... ... 1 Smarandache Semirings, Semifields, and Semivector spaces W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... ...ver is a Boolean algebra constructed using the power set P(X) where X = {a, b, c} which is a finite Smarandache semiring of order 16. This book ... ...r spaces with examples 103 6.2 S-subsemivector spaces 104 6.3 Smarandache linear transformation 106 6.4 S-anti semivector spaces 109 7. Res... ...nd properties about vector spaces. DEFINITION 1.4.1: A vector space (or a linear space) consists of the following 1. a field F of scalars. 2. a... ... a vector space over the field F. Let β be a vector in V, β is said to be a linear combination of vectors α 1 , … , α n in V provided there exists s... ...NITION 1.4.5: Let V be a vector space over F. A subset S of V is said to be linearly dependent (or simply dependent) if there exist distinct vectors ... ...l of which are 0 such that α 1 c 1 + … + α n c n = 0. A set that is not linearly dependent is called linearly independent. If the set S contains ...

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...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 li...

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... spaces or n-vector spaces of type I. Several interesting properties about them are derived in this chapter. DEFINITION 2.1: A n-vector space or a n-linear space of type I (n 2)...

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two n-VECTOR SPACES OF TYPE I AND THEIR PROPERTIES 13 Chapter Three APPLICATIONS OF n-LINEAR ALGEBRA OF TYPE I 81 Chapter Four SUGGESTED PROBLEMS 103 FURTHER READING 111 INDEX 116 ABOUT THE AUTHORS 120...

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... Structures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... .... Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, University ... ...ience, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kanda... ...and neutrosophic understanding of Periyar's ideology and approach. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One ... ...Z ) ( is a group. For P = 2Z (I ) serves as the additive identity. Take a, b ∈ P I Z ) ( . If a,b ∈ Z(I ) \ P then two possibilities occur. a + b ... ...rosophic vector space over the neutrosophic field K(I ). By basis we mean a linearly independent set which spans the neutrosophic space. Now we ill... ...developed the theory of trees, in order to solve the system of simultaneous linear equations, which give the current in each branch and each circuit...

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...ur SUPER INTERVAL SEMILINEAR ALGEBRAS 137 Chapter Five SUPER FUZZY INTERVAL MATRICES 203 5.1 Super Fuzzy Interval Matrices 203 5.2 Special Fuzzy Linear Algebras Using Super Fuzzy Interval Matrices 246 Chapter Six APPLICATION OF SUPER INTERVAL MATRICES AND SET LINEAR ALGEBRAS BUILT USING SUPER INTERVAL MATRICES 255 Chapter Seven SUGGESTED PROBLEMS 257 FURT...

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...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 space...

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...Matrices provide a very powerful tool for dealing with linear models. Bimatrices which we are going to define in this chapter are still a powerful and an advanced tool which can handle over one linear model at a time. Bimatrices will be useful when time bound comparisons are ne...

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... 1 INTRODUCTION TO BIMATRICES W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... ..., Georgia State University, Atlanta, GA 30302, USA Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral La... ...EIR PROPERTIES 2.1 Introduction to Bivector Spaces 79 2.2. Systems of linear biequations 87 2.3 Bitransformation in bivector spaces 96 2.4 ... ...paces 105 2.5 Bieigen values and Bieigen vectors 117 2.6 Bidiagonalizable linear bioperator and its properties 124 4 Chapter Three ... ...zy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices ... ...odels. 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 eac... ...vention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are ... ...my for cover designing and Kama Kandasamy for formatting this book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K. ILANTHENRAL 7 Ch... ...and define some of basic operation on it. DEFINITION 1.1.1: A bimatrix A B is defined as the union of two rectangular away of numbers A 1 and A...

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...This book has six chapters. First chapter is introductory in nature. The new concept of non-associative semi-linear algebras is introduced in chapter two. This structure is built using groupoids over semi-fields. Third chapter introduces the notion of non-associative linear algebras. These algebraic structures are built using the ne...

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...ever the resultant product may not be a loop in general. For the concept of semifield refer [41]. We will be using these loops and groupoids to build linear algebra and semilinear algebras which are non associative. ...

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two NON ASSOCIATIVE SEMILINEAR ALGEBRAS 13 Chapter Three NON ASSOCIATIVE LINEAR ALGEBRAS 83 Chapter Four GROUPOID VECTOR SPACES 111 Chapter Five APPLICATION OF NON ASSOCIATIVE VECTOR SPACES / LINEAR ALGEBRAS 161 Chapter Six SUGGESTED PROBLEMS 163 FURTHER READING 225 INDEX 229 ABOUT THE ...

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...e 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...

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... REHOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...e, Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ... 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 algeb... ...sociative rings boils down to the study of properties of vector spaces or linear algebras over fields. But study of non-associative algebras usin... ...e definition of fields or rings. DEFINITION 1.1.1: A vector space or a linear space consists of the following: 1. a field K of scalars. 2. a s... ...α and β in such a way that a. addition is commutative α + β = β + α . b. addition is associative that is α + ( β + γ) = ( α + β) + γ. c. There i... ... the product of c and α in such a way That a. 1. α = α for all α ∈ V. b. (a 1 a 2 ) α = a 1 (a 2 α). c. a( α+ β) = a α + a β. d. (a 1 + a 2 ) ... ...ltiplication on V. The reader is advised to learn the concept of basis, linear transformation, linear operator etc. We say a vector space is finit... ... say a vector space is finite dimensional if there exists a finite set of linearly independent elements (v 1 , v 2 , …, v n ) in V which generate V, ...

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...This book is organized into seven chapters. Chapter one is introductory in content. The notion of neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras are introduced and their properties analysed in chapter two. Chapter three introduces the notion of neutrosophic semigroup linear algebras and neutrosophic gro...

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...Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET NEUTROSOPHIC LINEAR ALGEBRA 13 2.1 Type of Neutrosophic Sets 13 2.2 Set Neutrosophic Vector Space 16 2.3 Neutrosophic Neutrosophic Integer Set Vector Spaces 40 2.4 Mixed Set Neutrosophic Rational Vector Spaces and their Properties 52 ...

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...d group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...tion by: Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ... DEFINITION 1.2.8: Let µ be a fuzzy subgroup of a group G. Then for any a, b ∈ G a fuzzy middle coset a µ b of the group G is defined by (a µ b) (x) ... ...y middle coset a µ b of the group G is defined by (a µ b) (x) = µ (a –1 x b –1 ) for every x ∈ G. The following example from [139] is interesting... ...x ( , , ) x ( L where 0 < t i < A(x i ) for i = 1, 2, …, k is said to be linearly independent in A if and only if n 1 t t k k t 1 0 ) x ( n ) x ( k... ...let C and D be fuzzy subfields of F such that D, C ⊇ δ K . If C and D are linearly disjoint over δ K then we write CD = C ⊗ D where CD is the compo... ...p {A(x) | x ∈ V \ {(0)}}. Then ζ is fuzzy free over k if and only if x is linearly independent over F. Proof: It is a matter of routine [75]. ... ...pose {0} ⊂ A * . Then k * = f. Suppose ζ is maximal fuzzy free, then χ is linearly independent over F. Using earlier theorems we get with simple cal... ...generated over k, then A has a fuzzy basis over k. Now we define fuzzy linearly independent set. DEFINITION 1.9.7: Let A∈ A k and let ζ ⊆ {x λ...

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...e basics. The second chapter deals with different types of neutrosophic bilinear algebras and bivector spaces and proves several results analogous to linear bialgebra. In chapter three the authors introduce the notion of n-linear algebras and prove several theorems related to them. Many of the classical theorems for neutrosophic algebras are proved with appropriate modific...

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...1 Introduction to Bilinear Algebras and their Generalizations 9 1.2 Introduction to Neutrosophic Algebraic Structures 11 Chapter Two NEUTROSOPHIC LINEAR ALGEBRA 15 2.1 Neutrosophic Bivector Spaces 15 2.2 Strong Neutrosophic Bivector Spaces 32 2.3 Neutrosophic Bivector Spaces of Type II 50 2.4 Neutrosophic Biinner Product Bivector Space 198 Chapter Three NEUTROSOP...

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... in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research...

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... APPLICATIONS OF BIMATRICES TO SOME FUZZY AND NEUTROSOPHIC MODELS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of T... ...y, Anchorage, AK 99508, Alaska, USA. Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K.Ilanthenral ... ...s book to Prof. Dr. Bart Kosko, the father of Fuzzy Cognitive Maps. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K.ILANTHENRAL 7 ... ... define some of the basic operations on it. DEFINITION 1.1.1: A bimatrix A B is defined as the union of two rectangular array of numbers A 1 an... ...apply the bimatrices and their extensions to these models. 1. Abraham, R. Linear and Multilinear Algebra, W. A. Benjamin Inc., 1966. 2. Adams, ... ...g, USA. http://www.fuzzysys.com/ftaprod.html 43. Gel'fand, I.M. Lectures on linear algebra, Interscience, New York, 1961. 44. Georgopoulous, V.C.... ...Artificial Neural Networks, Espoo, Finland, 1245-1248, 1991. 46. Greub, W.H. Linear Algebra, Fourth Edition, Springer- Verlag, 1974. 47. Hadjiski, ... ...opics in Algebra, John Wiley, New York, 1975. 55. Hoffman, K. and Kunze, R. Linear algebra, Prentice Hall of India, New Delhi, 1991. 56. Hummel,... ...acob Chen and Anil Milal (eds.), 63-68, 1997. 65. Katsumi, N. Fundamentals of Linear Algebra, McGraw Hill, New York, 1966. 66. Kemeni, J. and J. Sn...

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...rm, the Consecutive Sequence is considered in an arbitrary numeration base B.” For example if we use base 3, the elements of the sequence are 1, 12, 1... ...four sequence. Problem: Is there any other case where an element of a base B 1 Consecutive Sequence is equal to an element of another base B 2 seque... ... one simple pattern of these sequences that is easy to verify. Theorem: If B > 2 is an arbitrary base and we form the sequence of numbers 1, 12, 123, ... ...ary base and we form the sequence of numbers 1, 12, 123, . . . , 123 . . .(B-1), 123 . . . (B-1)10, . . . and let m represent the nth element of the ... ...present the nth element of the sequence, we have the following pattern. If B is odd, then m is odd if n is congruent to 1 or 2 modulo 4 ... ... of the plotted graph shows a relationship that grows faster than a simple linear one. 63 However, while this gives us some hints as to the lower bou... ...ructure of the graph, it appears that the sum is generally increasing in a linear manner, although the values vary considerably from this norm. A litt... ...he values as r gets larger would be very small, hence the appearance of a linear increase. Theorem: There is no upper limit to the sums of the elemen... ...e general trend of increasing sums of the elements of the family of series linearly increasing as r increases? 9) The series is convergent to a number...

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...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 o...

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...In this section we give the definition of linear algebra and just state the few important theorems like Cayley Hamilton theorem, Cyclic Decomposition Theorem, Generalized Cayley Hamilton Theorem and give some properties about linear algebra....

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... 1 INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tech... ...ustin Priescu, Academia Technica Militaria, Bucharest, Romania, Prof. Dr. B. S. Kirangi, Department of Mathematics and Computer Science, University... ...ence, University of Mysore, Karnataka, India. Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral Lay... ... CONTENTS Preface 5 Chapter One INTRODUCTION TO LINEAR ALGEBRA AND S- LINEAR ALGEBRA 1.1 Basic properties of linear al... ...GEBRA 1.1 Basic properties of linear algebra 7 1.2 Introduction to s-linear algebra 15 1.3 Some aapplications of S-linear algebra 30 ... ...Introduction of S-bigroups and S-bivector spaces 46 Chapter Three LINEAR BIALGEBRA, S-LINEAR BIALGEBRA AND THRIR PROPERTIES 3.1 Basic ... ...EBRA, S-LINEAR BIALGEBRA AND THRIR PROPERTIES 3.1 Basic properties of linear bialgebra 51 3.2 Linear bitransformation and linear bioperators... ...on 123 3.9 Markov bichains–biprocess 129 Chapter Four NEUTROSOPHIC LINEAR BIALGEBRA AND ITS APPLICATION 4.1 Some basic neutrosophic alg... ...istance and encouragement in every manner made this book possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K. ILANTHENRAL 7 ...

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...super FAMs etc. In chapter five we define interval matrix and matrix interval using the natural class of intervals, that is intervals of the form [a, b] where a b or a and b not comparable. In the final chapter we have defined the new notion of bimatrices and n-matrices. Using these new concepts we have constructed linear algebra of type I and linear algebra of type II. U...

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...easily understood by a beginner. Binary symmetric channel is an illustration of a model for a transmission channel. Now we will proceed onto define a linear code algebraically....

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two ALGEBRAIC LINEAR CODES AND THEIR PROPERTIES 29 Chapter Three ERASURE DECODING OF MAXIMUM RANK DISTANCE CODES 49 3.1 Introduction 49 3.2 Maximum Rank Distance Codes 51 3.3 Erasure Decoding of MRD Codes 54 Chapter Four MRD CODES ...

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...hat a semigroup S is called an rpp semigroup if all its principal right ideals aS1, regarded as right S1-systems, are projective. According to J.B. Fountain[5], a semigroup S is rpp if and only if, for any a 2 S, the set Ma=fe 2 E j S1a µ Se and for all x; y 2 S1, ax = ay ) ex = eyg is a non-empty set, where E is the set of all idempotents of S. An rpp semigroup S is ...

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...X. Pan and B. Liu : On the irrational root sieve sequence 1 H. Liu : Sub-self-conformal sets 4 K. Liu : On mean values of an arithmetic function 12 L. Wang and Y. Liu : On an infinite series related to Hexagon-numbers 16 X. Ren, etc...

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...ive belief conditioning rules . . . . . 66 1.10 Conclusion . . . . . . . . . . 68 1.11 References . . . . . . . . . . . 68 Chapter 2 DSm field and linear algebra of refined labels 75 by F. Smarandache, J. Dezert and X. Li 2.1 Introduction . . . . . . . . . . 76 2.2 DSm field and linear algebra of refined labels . . . . 77 2.2.1 Qualitative operators on DSm-LARL . ....