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... AMERICAN RESEARCH PRESS REHOBOTH 2002 A 1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Depart... ...exas 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can... ....O., Phillips J.D., Robinson D. A., Solarin A. R. T., Tim Hsu, Wright C.R.B. and by others who have worked on Moufang loops and other loops like Bol ... ...n very abstract are illustrated by examples. DEFINITION 1.1.1: If a and b are integers both not zero, then an integer d is called the greatest com... ...ot zero, then an integer d is called the greatest common divisor of a and b if i. d > 0 ii. d is a common divisor of a and b and iii. if any in... ...NITION 1.3.3: A partial order relation ≤ on A is called a total order (or linear order) if for each a, b ∈ A either a ≤ b or b ≤ a. (A, ≤) is then c... ...NITION 1.3.7: A partial order relation ≤ on A is called a total order (or linear order) if for each a, b ∈ A either a ≤ b or b ≤ a. (A, ≤) is then ca... ...ns, Elsevier Pub., (1976). 4. Bruck.R.H, Some results in the theory of linear non-associative algebra, Trans. Amer. Math. Soc., Vol.56, 141-199 (...

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...DS With Specific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of T... ...ova, Bulgarian Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Ka... ...ok to women around the world who are bravely battling the AIDS epidemic. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One ... ...l function S j (y j ). However the competitive learning law can be reduced to linear competitive learning law ij m = S j (y j ) (x – m j ), ... ...nd it denotes the on-off position of the nodes at any instant. Similarly let B = (b 1 ,…, b m ) b i ∈ {0, 1}. B is called the instantaneous state ... ... i = 0 if a i is off and a i = 1 if a i is on for i=1, 2,…, n. Similarly, b i =0 if b i is off and b i = 1 if b i is on, for i= 1, 2,…, m. ...

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... Elliptic Parameters of S'. {θ θ θ θ = arcsin(v)} S' a b f 1 v a / b σ σ σ σ e a 1 cos(θ θ θ θ) (4) cos 2 (θ θ θ θ) (4) s... ...ARADOXES: Comparative Logic and Modern Quantum Theory T a b l e o f C o n t e n t s Page Abstract ... ...dox. a.) unresolved paradox: further classification not yet determined. b.) resolved paradox: seemingly paradoxical behavior in physical systems ... ...sible object, but there is apparently no definite point where this occurs. b) A similar paradox is developed in an opposite direction. It is always ... ... ∋ {s 1 + s 2 + s 3 +...+ s n }, having physical characteristics in set {B} ∋ {b 1 , b 2, b 3 ,…b n }; so that, B → S . ( 1 ) S... ... however, the eight ionizations of the 2p and 2s orbitals proceed in a near linear progression, so that there is no clear way to differentiate betwe... ... into the nucleus. In fact, particle accelerator experiments, e.g. Stanford Linear Accelerator (1968), indicate elastic, near-elastic or destructive ...

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...dache – Râmnicu Vâlcea: Offsetcolor, 2005. 170 p.; cm. 20 X 14 I.S.B.N. 973-7743-19-9 5 Via ţa sub pământ 11 februarie 1995 Viz... ...ional Park (dresaje de foci şi p ăs ări, banale/comune)... Sidu (Silviu, b ăiatul cel mic) e responsabil cu aparatul de fotografiat: pozeaz ă toate... ...ntâlnit pe Grigore Vieru, academician şi mare poet patriot basarabean. Am b ăut împreun ă vinul d ăruit de Sandu Banto ş, redactorul revistei „Limb... ...c ă N. Botgros continu ă cu forma ţia „Barbu L ăutaru” din Bucure şti. Am b ăut „samahoanc ă” (b ăutur ă alcoolic ă din sfecl ă, îndulcit ă) la Rais... ...de ½ or ă în Teoria jocurilor. L-am întâlnit, am fost la restaurant şi-am b ăut un vin alb. L-am c ăutat pe Alvin White (editor la „Humanistic Math... ... asupra cuno ştin ţelor viitoare, ale c ăror evenimente deviaz ă cursul linear al istoriei. Tot în expozi ţia de "science fiction" din Muzeul "N... ...icolului: nu mai g ăseam intrarea la Pima College, unde predasem algebr ă linear ă, ecua ţii diferen ţiale şi clase preg ătitoare (developmental) în...

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...able to imagine. The other philosophers are capable of reasoning only in a linear fashion, whereas I, instead, have learned to reason both in a circ... ...shion, whereas I, instead, have learned to reason both in a circular and a linear fashion, and then make a synthesis of the two, thus creating spira... ...t from the line and the circle taken separately. * A concrete example of linear thinking can be found in Saint Augustine’s linear concept of time ... ...t note that the spiritual growth of the ascetics and the saints is rigidly linear and not dialectical. We can also note the normal growth process of ... ... over others and the power to kill other people. Said in a simpler, more linear way: every time human beings are faced with the fear of death, the... ...of perfection is demolished through a process of implosion and explosion; b) the group is a place where one can experience the body, the psyche, the... ...akes sense in that it allows us to: a) proceed in a dialectical manner; b) blend together the contributions from other points of view into a unifi... ...ristotle affirms: a) that the principle of causality cannot be put aside; b) that it is impossible to proceed infinitely from one cause to another b... ...r way of thinking that integrates both linear and dialectical thinking. b) blend together the contributions from other points of view into a unifi...

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...we are from, or have been, or will be. It's a matter of getting out of the linear expressions of time, and going to fractal time. Now, not as enough ... ...e thing here that I was there, but it's the same proportion: as X is to Y, B is to C. It's the same general idea. All of the belief paths, and the '... ... repeat itself, one can learn a lot from the past... whether that past is linear, parallel, or running in any other direction, really. And no one ca...

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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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...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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... 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....

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... 1 Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... Nigeria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...fferent from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category... ...c. We do not approach the bialgebraic structures using category theory or linear logic. We can broadly classify the study under four heads : ... ...inary operation, called the product and denoted by ' ' such that i. a, b ∈ G implies a b ∈ G. ii. a, b, c ∈ G implies (a b) c = a (b c). i... ...re exists an element a -1 ∈ G such that a a -1 = a -1 a = e. If a b = b a for all a, b ∈ G we say G is a abelian or a commutative group. If ... ..., v 2 , …, v n ) in a semivector space V over a semifield S is said to be linearly dependent if there exists a non-trivial relation among them; othe... ...exists a non-trivial relation among them; otherwise the set is said to be linearly independent. (The main difference between vector spaces and semiv... ...mivector space over the semifield S. For any subset A of V the set of all linear combinations of vectors in A is called the set spanned by A and we ...

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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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...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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...4 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of... ...alia. Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, Univers... ...r Science, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama K... ... resolution based on Archimedean triangular norms 40 1.6 Solving non-linear optimization problem with FRE constraints 48 1.7 Method... ...8 4 1.19 A FRE in dynamic fuzzy systems 113 1.20 Solving FRE with a linear objective function 117 1.21 Some properties of minimal sol... ...h deneutrofication algorithm for largest solution 268 4.9 Solving NRE with a linear objective function 269 4.10 Some properties of minimal soluti... ...ing of the book. Without his help this book would not have been possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 9 Chapter One... ...ments of X are comparable by the ordering such an ordering is usually called a linear ordering, some alternative names used in the literature are t... ...A fuzzy weak ordering R is an ordering satisfying all the properties of a fuzzy linear ordering except anti-symmetry. Alternatively it can be though...

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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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...VICE 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... Finance, Griffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...the book was intact and Meena who collected the existing literature. W.B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One BASIC CO... ...1 , … , a' n ) after thresholding and updating the vector suppose we get (b 1 , … , b n ) we denote that by (a' 1 , a' 2 , … , a' n ) → (b 1 , b... ...ors C repeatedly through the FCM connection matrix E, thresholding or non-linearly transforming the result after each pass. Independent of the FCMs ... ...by Hagiwara [41]. The features of the E-FCMs are: a. Weights having non-linear membership functions. b. Conditional weights. c. Time-delay weight... ...d vectors, which are inversely related. We shall restrict our examples to linear dependencies only. If vectors V 1 and V 2 are directly relate... ... represent an approach to model the behaviour and operation of complex non-linear systems. The implementation of FCMs for modeling the mobile robot m... ...judicial decisions affect business behavior. The trustee’s argument is linear and based on a literal reading of the Bankruptcy Code. As a simple ...

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...Rocket Plane & San Francisco Cable Car by the Danbury Mint Edited by Homer B. Tilton and Florentin Smarandache Pima Community College, East Campu... ...ats.htm Production Editor: Gloria Valles © Copyright 2005 by Homer B. Tilton / Tucson, Arizona USA All rights reserved. Keywords: Phy... ...ent September 2005 PIMA COLLEGE PRESS Edited by: Homer B. Tilton and Florentin Smarandache Production Editor: Gloria Valles T... ...…………………………... 17 Dr. David Iadevaia ……………………………….………... 25 Dr. Larissa B. Borrisova & Dmitri D. Rabounski ……….... 29 Dr. Florentin Smarandach... ...) 00 2 1 g c w − = is gravitational potential, 00 0 g g c v i i − = is the linear velocity of the space rotation, k i ik dx dx h d = 2 σ is the squ... ... turn destroys local geometry. Furthermore, quantum field theory (QFT) is a linear theory which is being used to describe non- linear data. Namely, ... ...racks from verifying picture to verifying picture. This is a sign of a non- linear system. There are many physical systems whose descriptions are ... ...on. In particular it casts no new light on solar system stability. 8. Non-linear theory is not understood, and complexity theory has pretty suspic... ...dary conditions that have a lot of unjustified order to them. Again, in non-linear theory, equations and boundary conditions are adjusted until solu...

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...GNITIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... University of Hyderabad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ... men and women living with HIV/AIDS in the Third- World countries. W.B. Vasantha Kandasamy Florentin Smarandache Chapter One INTRO... ...ors C repeatedly through the FCM connection matrix E, thresholding or non-linearly transforming the result after each pass. Independent of the FCMs ... ... , a’ n ) after 41 thresholding and updating the vector suppose we get (b 1 , … , b n ) we denote that by (a’ 1 , a’ 2 , … , a’ n ) ↪ (b 1 , b 2 ,... ... … , b n ) we denote that by (a’ 1 , a’ 2 , … , a’ n ) ↪ (b 1 , b 2 , … , b n ). Thus the symbol ‘↪’ means the resultant vector has been threshold... ... networks consists of many simple computing elements generally simple non linear summing junctions connected together by connections of varying stre...