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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 group linear algebras. A study of their substructures are systematically carried out in this chapter. The fuzzy analogue of neutrosophic group linear algebras, neutrosophic semigroup linear algebras and neutrosophic set linear algebras are introduced in chapter four of this book. Chapter five introduces the concept of neutrosophic group bivector spaces, neutrosophic bigroup linear algebras, neutrosophic semigroup (bisemigroup) linear algebras and neutrosophic biset bivector spaces. The fuzzy analogue of these concepts are given in chapter six. An interesting feature of this book is it contains nearly 424 examples of these new notions. The final chapter suggests over 160 problems which is another interesting feature of this book....

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Now we proceed onto define the notion of neutrosophic subgroup of a neutrosophic group. DEFINITION 1.2: Let N(G) = (GuI) be a neutrosophic group generated by G and I. A proper subset P(G) is said to be a neutrosophic subgroup if P(G) is a neutrosophic group i.e. P(G) must contain a (sub) group. Example 1.3: Let N(Z2) = 􀂢Z2 􀂉 I􀂲 be a neutrosophic group under addition. N(Z2) = {0, 1, I, 1 + I}. Now we see {0, I} is a group under + in fact a neutrosophic group {0, 1 + I} is a group under ‘+’ but we call {0, I} or {0, 1 + I} only as pseudo neutrosophic groups for they do not have a proper subset which is a group. So {0, I} and {0, 1 + I} will be only called as pseudo neutrosophic groups (subgroups). We can thus define a pseudo neutrosophic group as a neutrosophic group, which does not contain a proper subset which is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different. Now we see a neutrosophic group ...

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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 Chapter Three NEUTROSOPHIC SEMIGROUP LINEAR ALGEBRA 91 3.1 Neutrosophic Semigroup Linear Algebras 91 3.2 Neutrosophic Group Linear Algebras 113 Chapter Four NEUTROSOPHIC FUZZY SET LINEAR ALGEBRA 135 Chapter Five NEUTROSOPHIC SET BIVECTOR SPACES 155 Chapter Six NEUTROSOPHIC FUZZY GROUP BILINEAR ALGEBRA 219 Chapter Seven SUGGESTED PROBLEMS 247 FURTHER READING 277 INDEX 280 ABOUT THE AUTHORS 286 ...

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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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Now we proceed onto define the notion of neutrosophic subgroup of a neutrosophic group. DEFINITION 1.1.2: Let N(G) = 〈G ∪ I〉 be a neutrosophic group generated by G and I. A proper subset P(G) is said to be a neutrosophic subgroup if P(G) is a neutrosophic group i.e. P(G) must contain a (sub) group. Example 1.1.3: Let N(Z2) = 〈Z2 ∪ I〉 be a neutrosophic group under addition. N(Z2) = {0, 1, I, 1 + I}. Now we see {0, I} is a group under + in fact a neutrosophic group {0, 1 + I} is a group under ‘+’ but we call {0, I} or {0, 1 + I} only as pseudo neutrosophic groups for they do not have a proper subset which is a group. So {0, I} and {0, 1 + I} will be only called as pseudo neutrosophic groups (subgroups). We can thus define a pseudo neutrosophic group as a neutrosophic group, which does not contain a proper subset which is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different. Now we see a neutrosophi...

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Preface 5 Chapter One INTRODUCTION 1.1 Neutrosophic Groups and their Properties 7 1.2 Neutrosophic Semigroups 20 1.3 Neutrosophic Fields 27 Chapter Two NEUTROSOPHIC RINGS AND THEIR PROPERTIES 2.1 Neutrosophic Rings and their Substructures 29 2.2 Special Type of Neutrosophic Rings 41 Chapter Three NEUTROSOPHIC GROUP RINGS AND THEIR GENERALIZATIONS 3.1 Neutrosophic Group Rings 59 3.2 Some special properties of Neutrosophic Group Rings 73 3.3 Neutrosophic Semigroup Rings and their Generalizations 85 Chapter Four SUGGESTED PROBLEMS 107 REFERENCES 135 INDEX 149 ABOUT THE AUTHORS 154 ...

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...roups or neutrosophic biinterval semigroups. We derive results pertaining to them. The new notion of quasi bisubsemigroups and ideals are introduced. Smarandache interval neutrosophic bisemigroups are also introduced and analysed. Also notions like neutrosophic interval bigroups and their substructures are studied in section two of this chapter. Neutrosophic interval bigro...

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Now we will be using these intervals and work with our results. However by the context the reader can understand whether we are working with pure neutrosophic intervals or neutrosophic of rationals or integers or reals or modulo integers. This chapter has four sections. Section one introduces neutrosophic interval bisemigroups, neutrosophic interval bigroups are introduced in section two. Section three defines biinterval neutrosophic bigroupoids. The final section gives the notion of neutrosophic interval biloops....

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Preface 5 Chapter One BASIC CONCEPTS 7 1.1 Neutrosophic Interval Bisemigroups 8 1.2 Neutrosophic Interval Bigroups 32 1.3 Neutrosophic Biinterval Groupoids 41 1.4 Neutrosophic Interval Biloops 57 Chapter Two NEUTROSOPHIC INTERVAL BIRINGS AND NEUTROSOPHIC INTERVAL BISEMIRINGS 75 2.1 Neutrosophic Interval Birings 75 2.2 Neutrosophic Interval Bisemirings 85 2.3 Neutrosophic Interval Bivector Spaces and their Generalization 93 Chapter Three NEUTROSOPHIC n- INTERVAL STRUCTURES (NEUTROSOPHIC INTERVAL n-STRUCTURES) 127 Chapter Four APPLICATIONS OF NEUTROSOPHIC INTERVAL ALGEBRAIC STRUCTURES 159 Chapter Five SUGGESTED PROBLEMS 161 FURTHER READING 187 INDEX 189 ABOUT THE AUTHORS 195 ...

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The first chapter is introductory in nature and gives a few essential definitions and references for the reader to make use of the literature in case the reader is not thorough with the 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 modifications. Chapter four indicates the probable applications of these algebraic structures. The final chapter suggests about 80 innovative problems for the reader to solve....

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Dedication 5 Preface 7 Chapter One INTRODUCTION TO BASIC CONCEPTS 9 1.1 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 NEUTROSOPHIC n-VECTOR SPACES 205 3.1 Neutrosophic n-Vector Space 205 3.2 Neutrosophic Strong n-Vector Spaces 276 3.3 Neutrosophic n-Vector Spaces of Type II 307 Chapter Four APPLICATIONS OF NEUTROSOPHIC n-LINEAR ALGEBRAS 359 Chapter Five SUGGESTED PROBLEMS 361 FURTHER READING 391 INDEX 396 ABOUT THE AUTHORS 402...

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This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described 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 level....

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...t defined on column matrices, m * n (m ≠ n) matrices. This extension is the same in case of row matrices. We make use of the notion of semigroups and Smarandache semigroups. Also the notion of semirings, Smarandache semirings, semi vector spaces and semifields are used....

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This book has four chapters. Chapter one is introductory in nature. The reader is expected to have a good background of algebra and graph theory in order to derive maximum understanding of this research. The second chapter represents groups as graphs. The main feature of this chapter is that it contains 93 examples with diagrams and 18 theorems. In chapter three we describe commutative semigroups, loops, commutative groupoids and commutative rings as special graphs. The final chapter contains 52 problems....

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Preface 5 Chapter One INTRODUCTION TO SOME BASIC CONCEPTS 7 1.1 Properties of Rooted Trees 7 1.2 Basic Concepts 9 Chapter Two GROUPS AS GRAPHS 17 Chapter Three IDENTITY GRAPHS OF SOME ALGEBRAIC STRUCTURES 89 3.1 Identity Graphs of Semigroups 89 3.2 Special Identity Graphs of Loops 129 3.3 The Identity Graph of a Finite Commutative Ring with Unit 134 Chapter Four SUGGESTED PROBLEMS 157 FURTHER READING 162 INDEX 164 ABOUT THE AUTHORS 168...

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This volume is a collection of six papers on the use of auxiliary information and a priori values in construction of improved estimators. The work included here will be of immense application for researchers and students who employ auxiliary information in any form....

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Introduction In survey sampling, the use of auxiliary information can increase the precision of an estimator when study variable y is highly correlated with the auxiliary variable x. but in several practical situations, instead of existence of auxiliary variables there exists some auxiliary attributes, which are highly correlated with study variable y, such as (i) Amount of milk produced and a particular breed of cow. (ii) Yield of wheat crop and a particular variety of wheat etc. (see Shabbir and Gupta (2006)). In such situations, taking the advantage of point biserial correlation between the study variable and the auxiliary attribute, the estimators of parameters of interest can be constructed by using prior knowledge of the parameters of auxiliary attribute....

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Preface...............................................4 1.Ratio Estimators in Simple Random Sampling Using Information on Auxiliary Attribute...............................................7 2.Ratio-Product Type Exponential Estimator for Estimating Finite Population Mean Using Information on Auxiliary Attribute......................18 3.Improvement in Estimating the Population Mean Using Exponential Estimator in Simple Random Sampling....................................33 4.Almost Unbiased Exponential Estimator for the Finite Population Mean......................41 5.Almost Unbiased Ratio and Product Type Estimator of Finite Population Variance Using the Knowledge of Kurtosis of an Auxiliary Variable in Sample Surveys....................................54 6.A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s).....................62-73...

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In this book the authors study dual numbers in a special way. The main aim of this book is to find rich sources of new elements g such that g2 = 0. The main sources of such new elements are from Zn, n a composite number. We give algebraic structures on them. This book is organized into six chapters. The final chapter suggests several research level problems. Fifth chapter indicates the applications of dual numbers. The forth chapter introduces the concept of interval dual numbers, we also extend it to the concept of neutrosophic and 8 fuzzy dual numbers. Higher dimensional dual numbers are defined, described and developed in chapter three. Chapter two gives means and methods to construct the new element g such that g2 = 0. The authors feel Zn (n a composite positive integer) is a rich source for getting new element, the main component of the dual number x = a + bg....

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Dedication 5 Preface 7 Chapter One INTRODUCTION 9 Chapter Two DUAL NUMBERS 11 Chapter Three HIGHER DIMENSIONAL DUAL NUMBERS 55 Chapter Four DUAL INTERVAL NUMBERS AND INTERVAL DUAL NUMBERS 89 Chapter Five APPLICATION OF THESE NEW TYPES OF DUAL NUMBERS 129 Chapter Six SUGGESTED PROBLEMS 131 FURTHER READING 153 INDEX 156 ABOUT THE AUTHORS 159...

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This book extends the natural operations defined on intervals, finite complex numbers and matrices. Secondly the authors introduce the new notion of finite complex modulo numbers just defined as for usual reals. Finally we introduced the notion of natural product Xn on matrices. This enables one to define product of two column matrices of same order. We can find also product of m*n matrices even if m not does equal n. This natural product Xn is nothing but the usual product performed on the row matrices. So we have extended this type of product to all types of matrices....

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In this chapter we just give a analysis of why we need the natural operations on intervals and if we have to define natural operations existing on reals to the intervals what changes should be made in the definition of intervals. Here we redefine the structure of intervals to adopt or extend to the operations on reals to these new class of intervals....

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Dedication 4 Preface 5 Chapter One INTRODUCTION 7 Chapter two EXTENSION OF NATURAL OPERATIONS TO INTERVALS 9 Chapter Three FINITE COMPLEX NUMBERS 41 Chapter Four NATURAL PRODUCT ON MATRICES 81 FURTHER READING 147 INDEX 148 ABOUT THE AUTHORS 150...

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This book has five chapters. Chapter one is introductory in nature. Fuzzy linguistic spaces are introduced in chapter two. Fuzzy linguistic vector spaces are introduced in chapter three. Chapter four introduces fuzzy linguistic models. The final chapter suggests over 100 problems and some of them are at research level....

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Preface 5 Chapter One Introduction 7 Chapter Two Fuzzy Linguistic Spaces 9 Chapter Three Fuzzy Linguistic Set Vector Spaces 23 Chapter Four Fuzzy Linguistic Models 129 4.1 Operations on Fuzzy Linguistics Matrices 129 4.2 Fuzzy Linguistic Cognitive Models 139 4.3 Fuzzy Linguistic Relational Map Model 148 Chapter Five Suggested Problems 165 Further Reading 187 Index 189 About the Authors 192...

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This book is organized into 5 chapters. The first chapter introduces real neutrosophic complex numbers. Chapter two introduces the notion of finite complex numbers; algebraic structures like groups, rings etc are defined using them. Matrices and polynomials are constructed using these finite complex numbers. Chapter three introduces the notion of neutrosophic complex modulo integers. Algebraic structures using neutrosophic complex modulo integers are built and around 90 examples are given. Some probable applications are suggested in chapter four and chapter five suggests around 160 problems some of which are at research level....

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In this chapter we for the first time we define the notion of integer neutrosophic complex numbers, rational neutrosophic complex numbers and real neutrosophic complex numbers and derive interesting properties related with them....

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Preface 5 Chapter One REAL NEUTROSOPHIC COMPLEX NUMBERS 7 Chapter Two FINITE COMPLEX NUMBERS 89 Chapter Three NEUTROSOPHIC COMPLEX MODULO INTEGERS 125 Chapter Four APPLICATIONS OF COMPLEX NEUTROSOPHIC NUMBERS AND ALGEBRAIC STRUCTURES USING THEM 175 Chapter Five SUGGESTED PROBLEMS 177 FURTHER READING 215 INDEX 217 ABOUT THE AUTHORS 220...

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This paper contains exercises and problems of algebra and grouped chapters, the upper classes of secondary schools and coli culture media general. Its purpose is to prepare students from high schools mathematics categories and will be helpful in working independently. Also work can be used for extra school work because the reader will find in it important formulas, theorems, concepts, and basic definitions which are not always included in school textbooks....

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Prezenta lucrare conţine exerciţii §i probleme de algebră, grupate pe capitole, pentru clasele superioare de licee §i §coli medii de cultură generală. Scopul ei este pregătirea matematică a elevilor din liceele de toate categoriile §i va fi utilă în lucrul de sine stătător. De asemenea, lucrarea poate fi folosită pentru lucrul extra§colar, deoarece cititorul va găsi în ea teoreme §i formule importante, noţiuni §i definiţii de bază care nu întotdeauna sunt incluse în manualele §colare....

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...Florentin Smarandache, an American mathematician of Romanian descent has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason wh...

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Preface 5 Definitions, Problems, Theorems 7 References 76

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The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of low energy reaction (coldfusion) phenomena. While some of these discussions may be found a bit too theoretical, our view is that once these phenomena can be put into rigorous theoretical framework, thereafter more 'openminded' physicists may be more ready to consider these New Energy methods more seriously. Our basic proposition in the present book is that considering these new theoretical insights, one can expect there are new methods to generate New Energy technologies which are clearly within reach of human knowledge in the coming years....

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In the preceding article we argue that biquaternionic extension of Klein-Gordon equation has numerical solution with sinusoidal form, which differs appreciably from conventional Yukawa potential. In the present article we interpret and compare this result from the viewpoint of EQPET/TSC model described by Takahashi. Further observation is of course recommended in order to refute or verify this proposition....

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...ction and calculation of New Energy development – Fu Yuhua 111 4. Some unsolved problems in the physics of elementary particle – V. Christianto & F. Smarandache (PiP, vol. 3 no. 4, 2007) 127 5. About some unsolved problems in physics – M. Pitkanen 132 Beyond Standard Model, Unmatter and Yang-Mills Field 6. Bifurcations and pattern formation in particle physics: an intr...

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...cs, and therefore we can think that the quantization of orbit distances can be caused by superfluid helium quantization. This issue is explored by F. Smarandache and V. Christianto. Moreover, F. Smarandache also discusses possible new era of research that is pertaining to superluminal physics and instantaneous physics. Ion Patrascu and D. Rabounski discuss superluminality ...

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...RN. (I. Patrascu) Progress in Physics Vol.4, 2011...................127 Superluminal physics and Instantaneous physics as new trends in research (F. Smarandache) ...................129 The Hypergeometrical Universe: Cosmology and Standard Model. (M.A. Pereira). Jan. 2012...................134 On astrometric data and time-varying sun earth distance in light of Carmeli me...

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...This book contains 23 papers, most of which were written by participants to the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo University, China, in March, 2009. In this Conference, several professors gave a talk on Smarandache Notions and many participants lectured on them both extensively and intensively. All th...

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3. Remarks Sandor [2] has considered the problem of finding the S-perfect and completely S-perfect numbers, but his proof is not complete. He has proved that the only S-perfect of the form n = p q is n = 6 and there is no S-perfect number of the form n = 2kq where k ¸ 2 is an integer and q is an odd prime. On the other hand, Theorem 2.1 gives all the S-perfect numbers. Again, Sandor only proved that, the only completely S-perfect number of the form n = p2q is n = 28, and all completely S-perfect numbers are given by Theorem 2.2. Theorem 2.1 and Theorem 2.2 ¯nd respectively the S-perfect and completely S-perfect numbers when S(1) = 1. The situation is quite different if one adopts the convention that S(0) = 1. In the latter case, as has been proved by Gronas [3], all completely S-perfect numbers are n = p(prime), 9, 16, 24. All that is known about the S-perfect numbers is that, among the ¯rst 106 numbers, n = 12 is the only S-perfect number (see Ashbacher [4]). In exactly the same way, the Z-perfect and completely Z-perfect numbers may be defined. Thus, given an integer n, 1. ...

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...J. Wang : An equation related to the Smarandache power function 1 X. Lu and J. Hu : On the F.Smarandache 3n-digital sequence 5 B. Cheng : An equation involving the Smarandache double factorial function and Euler function 8 A. A. K. Majumdar : S-perfect and c...

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This book is organized into seven chapters. The first one is introductory in nature. In the second chapter authors give the unique and new way of analyzing the data which is time dependent. In chapter three the authors for the first time define the new concept of matrices with linguistic variables. In chapter four authors have used super matrices in super fuzzy models like super FCMs, super FRMs 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. Using these matrices we can build n-eigen values and n-eigen vectors....

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Preface 5 Chapter One INTRODUCTION 7 Chapter Two AVERAGE TIME DEPENDENT (ATD) DATA MATRIX 9 Chapter Three FUZZY LINGUISTIC MATRICES AND THEIR APPLICATIONS 43 Chapter Four SUPERMATRICES AND THEIR APPLICATIONS 65 Chapter Five INTERVAL MATRICES AND NATURAL CLASS OF INTERVALS 149 Chapter Six DSM MATRIX OF REFINED LABELS 203 Chapter Seven N-MATRICES AND THEIR APPLICATIONS 205 FURTHER READING 217 INDEX 222 ABOUT THE AUTHORS 226...

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In this book authors study and analyze the problem of school dropouts and their life after. The problems can by no means be analyzed by collecting the numerical data. For such data can only serve as information beyond that the data can be of no use, for the school dropouts suffer an environment change after becoming a school dropout. Thus the emotions of the school dropout; is technically involved....

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The basic tools used in the analysis of the problem of school dropout and their life after are described briefly in this chapter. We provide also the references for these concepts. Bart Kosko introduced the Fuzzy Cognitive Maps (FCMs) in the year 1986. Fuzzy Cognitive Maps are fuzzy structures that strongly resemble neural networks, and they have powerful and far-reaching consequences as a mathematical tool for modeling complex systems. FCM was a fuzzy extension of the cognitive map pioneered in 1976 by political scientist Robert Axelord, who used it to represent knowledge as an interconnected, directed, bilevel-logic graph....

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Preface 5 Chapter One INTRODUCTION 7 Chapter Two BASIC CONCEPTS 21 Chapter Three CAUSES OF SCHOOL DROPOUTS – A MATHEMATICAL ANALYSIS 35 Chapter Four SCHOOL DROPOUTS AS CHILD LABOURERS 59 Chapter Five SCHOOL DROPOUTS AS RAG PICKERS USING FUZZY MODELS 79 Chapter Six PERFORMANCE ASPECTS OF SCHOOL STUDENTS USING RULE BASED CONTROL SYSTEM 89 Chapter Seven MIGRATION OF PARENTS AND THE SCHOOL DROPOUTS A STUDY USING THE FUZZY RELATIONAL MAPS MODEL 99 Chapter Eight THE IMPACT OF MISSIONARY INTERVENTIONS ON THE EDUCATION AND REHABILITATION OF DEPRIVED CHILDREN – A FUZZY ANALYSIS 117 Chapter Nine CONCLUSIONS AND SUGGESTIONS 129 FURTHER READING 139 INDEX 143 ABOUT THE AUTHORS 145...

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In this book the authors study the erasure techniques in concatenated Maximum Rank Distance (MRD) codes. The authors for the first time in this book introduce the new notion of concatenation of MRD codes with binary codes, where we take the outer code as the RD code and the binary code as the inner code....

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We want to find efficient algebraic methods to improve the realiability of the transmission of messages. In this chapter we give only simple coding and decoding algorithms which can be 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 –SOME PROPERTIES AND A DECODING TECHNIQUE 63 4.1 Introduction 63 4.2 Error-erasure Decoding Techniques to MRD Codes 64 4.3 Invertible q-Cyclic RD Codes 83 4.4 Rank Distance Codes With Complementary Duals 101 4.4.1 MRD Codes With Complementary Duals 103 4.4.2 The 2-User F-Adder Channel 106 4.4.3 Coding for the 2-user F-Adder Channel 107 Chapter Five EFFECTIVE ERASURE CODES FOR RELIABLE COMPUTER COMMUNICATION PROTOCOLS USING CODES OVER ARBITRARY RINGS 111 5.1 Integer Rank Distance Codes 112 5.2 Maximum Integer Rank Distance Codes 114 Chapter Six CONCATENATION OF ALGEBRAIC CODES 129 6.1 Concatenation of Linear Block Codes 129 6.2 Concatenation of RD Codes With CR-Metric 140 FURTHER READING 153 INDEX 157 ABOUT THE AUTHORS 161...

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科学面临的难题 _ 中智学为何诞生 中智学(neutrosophy)起源于1995年美国, 它站在东西文化交融的立场上, 从对立统一的角度探索从科学技术到文学艺术的一切宏观及微观结构, 构造超越一切学科、超越自然科学与社会科学界限的统一场, 以解决当今认知科学、信息科学、系统科学、经济学、量子力学等科学技术前沿难题——非确定性问题。中智学努力通过新型开放模式改造当今各自然科 与社会科学, 实现它们的新陈代谢、改革创新和更新换代。中智学在我们中国还属空白, 故借 对学科正式命名并引入中国。...

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中智学, 新的哲学分支 _(Neutrosophy - A New Branch of Philosophy) 摘要: 本文推出了一个新的哲学分支, 中智学 _(neutrosphy), 研究中性的起源、本质和范畴以及和不同思想观念的作用。它的基本点是: 任何观念具有T%的真实性、I%的不确定性以及 的谬误性, 其中T, I, F为╟-0, 1+╢的标准或非标准子集。 _基本理论:任何观念 _ 趋于被 _ 所中和、削弱和平衡 _(不仅仅是被黑格尔主 的), 达到一种平衡状态。 中智学是中智逻辑学 _(在模糊逻辑的基础上总结出来的多值逻辑)、中智集合论 _(模糊 合论的概括总结)、中智概率论和中智统计学 _(分别是经典及非精确概率论、统计学的概括 结) 的基础。 _ 关键字与短语: 非标准分析, 超实数, 无穷小, 单子, 非标准实数单位区间, 集合运算。 _...

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译者序 _Preface by the Translator ........................................5 作者简介 _Author’s Biography ..........................................9 译者简介 _Biography of the Translator.................................12 原书前言 _查尔斯·李 _.....................................................19 Preface by Charles T. Le 0．引言: 非标准实数单位区间 _.......................................24 Introduction: The Non-Standard Real Unit Interval 1．中智学——哲学的崭新分支 _.......................................27 Neutrosophy - a new branch of philosophy 2．中智逻辑——逻辑学的统一 _........................................90 Neutrosophic Logic - a unifying field in logics 3．中智集合论——集合论的统一 _....................................109 Neutrosophic Set - a unifying field in sets 4．中智概率论 _..........................................................113 ——传统概率论和非精确概率论的概括总结 _ ——以及中智统计学 _ Neutrosophic Probability - a generalization of classical and imprecise probabilities - and Neutrosophic Statistics 附录: 中智学产生的定义 _..............................................117 Addenda: Definitions derived from Neutrosophics...