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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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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....

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1.2 Definition of Fuzzy Cognitive Maps In this section we recall the notion of Fuzzy Cognitive Maps (FCMs), which was introduced by Bart Kosko in the year 1986. We also give several of its interrelated definitions. FCMs have a major role to play mainly when the data concerned is an unsupervised one. Further this method is most simple and an effective one as it can analyse the data by directed graphs and connection matrices. DEFINITION 1.2.1: An FCM is a directed graph with concepts like policies, events etc. as nodes and causalities as edges. It represents causal relationship between concepts. Example 1.2.1: In Tamil Nadu (a southern state in India) in the last decade several new engineering colleges have been approved and started. The resultant increase in the production of engineering graduates in these years is disproportionate with the need of engineering graduates. ...

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Dedication 5 Preface 6 Chapter One BASIC CONCEPTS 1.1 Definition of Interval Matrices and Examples 8 1.2 Definition of Fuzzy Cognitive Maps 9 1.3 An Introduction to Neutrosophy 13 1.4 Some Basic Neutrosophic Structures 16 1.5 Some Basic Notions about Neutrosophic Graphs 22 1.6 On Neutrosophic Cognitive Maps with Examples 28 1.7 Definition and Illustration of Fuzzy Relational Maps (FRMs) 33 1.8 Introduction to Fuzzy Associative Memories 40 1.9 Some Basic Concepts of BAM 43 1.10 Properties of Fuzzy Relations and FREs 49 1.11 Binary Neutrosophic Relations and their Properties 56 Chapter Two INTRODUCTION TO FUZZY INTERVAL MATRICES AND NEUTROSOPHIC INTERVAL MATRICES AND THEIR GENERALIZATIONS 2.1 Fuzzy Interval Matrices 68 2.2 Interval Bimatrices and their Generalizations 76 2.3 Neutrosophic Interval Matrices and their Generalizations 92 Chapter Three FUZZY MODELS AND NEUTROSOPHIC MODELS USING FUZZY INTERVAL MATRICES AND NEUTROSOPHIC INTERVAL MATRICES 3.1 Description of FCIMs Model 118 3.2 Description and Illustration of FRIM Model 129 3.3 Description of FCIBM model and its Generalization 139 3.4 FRIBM model and i...

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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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This book has five chapters. First chapter is introductory in nature. Here we just study chemical pollution caused by garment industries in chapter two of this book using fuzzy associative memories. In chapter three we give ways to control pollution by improving the system performance using hierarchical genetic fuzzy control algorithm. This study is carried out using the past data reported by Shimada et al (1995). Health hazards suffered by the agriculture labourers; caused by the chemicals used as fertilizers and pesticides is analysed in chapter four using RTD matrices. Chapter five gives a method of minimization of SOx and NOx using fuzzy control theory in cement industries to reduce pollution....

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In this chapter, we have used the fuzzy control method to find the correct measure of pressure and speed of kiln. Using data from Alathur cement industry, we have used mean of maximum method of fuzzy control with the data and we found the pressure and speed of kiln. Here the speed and pressure of kiln are important parameters in the kiln, because the SOx and N0X concentration is dependent on the kiln feed, coal feed and the kiln feed, coal feed are dependent on speed and pressure of kiln. That means if we increase the kiln feed and coal feed we must increase the kiln pressure and speed. By this way the mixing of SOx and NOx with kiln feed is maintained properly or otherwise the SOx and NOx will not mix fully with the kiln feed. The remaining unmixed gases of SOx and NOx are the wastage polluting the environment. For this, we convert all this kiln feed and coal feed level and pressure of kiln and speed of kiln into fuzzy control. By using fuzzy control we identify a correct pressure and speed of the kiln for a given kiln feed and coal feed. When we choose a correct pressure of kiln and speed of kiln through fuzzy control, the SO...

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Preface 5 Chapter One INTRODUCTION 7 Chapter Two USE OF FAM MODELS TO ANALYSE THE POLLUTION CAUSED BY THE GARMENT INDUSTRIES 11 Chapter Three HIERARCHICAL GENETIC ALGORITHM TO IMPROVE THE SYSTEM PERFORMANCE AND REDUCE POLLUTION BY CHEMICAL PLANTS 51 3.1 Introduction 51 3.2 Description of the Problem 53 3.3 Application of Hierarchical Genetic Fuzzy Control Algorithm to Control Process Variables of Decision Tables in Chemical Plants 56 3.4 Use of Hierarchical Genetic Fuzzy Control Algorithm in the Real Data 58 3.5 Conclusions 73 Chapter Four IDENTIFICATION OF THE MAXIMUM AGE GROUP IN WHICH THE AGRICULTURAL LABOURERS SUFFER HEALTH HAZARDS DUE TO CHEMICAL POLLUTION USING RTD MATRIX 75 4.1 Introduction 75 4.2 Estimation of the Maximum Age Group of the Agricultural Labourers having Cardio Vascular Problem due to Chemical Pollution using RTD Matrices 77 4.3 Conclusion 93 Chapter Five MINIMIZATION OF SOX AND NOX USING FUZZY CONTROL THEORY IN CEMENT INDUSTRIES TO REDUCE POLLUTION 95 5.1 Introduction 95 5.2 Description of the Problem 97 5.3 Process Fates of Sulfur in Cement Kiln 98 5.4 Process Fates of Ni...

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Applications demonstrate the power of the DSmT framework. In this third Volume, DSmT is applied to the entire spectrum of the Information Fusion that would interest any reader in data, sensor, information, and mathematical fusion topics. Highlighted in Figure 1 are the contemporary issues that include the links between (1) data conditioning and information management, (2) combined situation and impact assessment, and (2) knowledge representation between machine processing and user coordination. Various applications leverage DSmT “Advances” listed above along with DSmH (hybrid), DSmP (Probabilistic), and DSmT theoretical insights. The third volume attacks these application issues of coordination between the “levels” of information fusion....

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...Part I Advances on DSmT 1 Chapter 1 An introduction to DSmT 3 by J. Dezert and F. Smarandache 1.1 Introduction . . . . . . . . . . 4 1.2 Foundations of DSmT . . . . . . . . . 4 1.2.1 The power set, hyper-power set and super-power set . . 6 1.2.2 Notion of free and hybrid DSm models . . . . 18 1.2.3 Ge...

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...This second book devoted on advances and applications of Dezert-Smarandache Theory (DSmT) for information fusion collects recent papers from different researchers working in engineering and mathematics. Part 1 of this book presents the current state-of-the-art on theoretical investigation...

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...Preamble iii Prefaces v Part I Advances on DSmT 1 Chapter 1 Proportional Conflict Redistribution Rules for Information Fusion 3 by Florentin Smarandache and Jean Dezert 1.1 Introduction . . . . . . . . . . . 3 1.2 The principal rules of combination . . . . . . . 6 1.2.1 Notion of total and partial conflicting masses . . . . . 6 1.2.2 The conjunctive ...

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This book consists of 6 papers focusing on social and economic issues. The topics covered include graph distance and optimal communication, migration in Jaipur, urbanization, clan capitalism, world population growth rate, and scientific inquiry. These papers were written in the period between 2009-2010. Hopefully the readers will find some new insights in this collection of papers....

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In recent years, there has been a rapid increase in the literature which discusses new phenomenon associated to social network. One of the well-known phenomenon in this regards is known as ‘six degrees of separation’ [1], which implies that one can always keep a communication with each other anywhere within a six-step. A number of experiments has verified this hypothesis, either in the context of offline communication (postal mail), or online communication (email, etc.). In this article, we argue that by introducing this known ‘six degrees of separation’ into the context of group instability problem, one can find a new type of wisdom in organization. Therefore, we offer a new conjecture, which may be called ‘Group stability conjectures based on Graph/Network distance.”...

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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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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...

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1. 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 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 新型开放模式改造当今 各自然科学与社会科学, 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 空白, 故借此对学科正式 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 F%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张的), 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 (模糊集合论的概括总结)、中智概 率论和中智统计学 (分别是经典及非精确概率论、统计学的概括总结) 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 ............................................18 Preface by Charles T. Le 0．引言: 非标准实数单位区间 ..............................23 Introduction: The Non-Standard Real Unit Interval 1．中智学——哲学的崭新分支 ..............................26 Neutrosophy - a new branch of philosophy 2．中智逻辑——逻辑学的统一 ...............................83 Neutrosophic Logic - a unifying field in logics 3．中智集合论——集合论的统一 ..............................100 Neutrosophic Set - a unifying field in sets 4．中智概率论 ...........................................103 ——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 ..................................106 Addenda: Definitions derived from Neutrosophics...

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In part 1, we have defined a fuzzy utility system, with different financial goals, different levels of risk tolerance and different personal preferences, liquid assets, etc. In part 2, we have defined a computational model for a simple portfolio insurance strategy using a protective put and computationally derive the investor’s governing utility structures underlying such a strategy under alternative market scenarios. In Part 3, it is proposed an artificial classification scheme to isolate truly benign tumors from those that initially start off as benign but subsequently show metastases. In part 4, an alternative methodological approach has been proposed for quantifying utility in terms of expected information content of the decision-maker’s choice set....

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In this paper we have designed our fuzzy system so that customers are classified to belong to any one of the following three categories: *Conservative and security-oriented (risk shy) *Growth-oriented and dynamic (risk neutral) *Chance-oriented and progressive (risk happy) A neutrosophic system has three components – that’s why it may be considered as just a generalization of a fuzzy system which has only two components. Besides being useful for clients, investor classification has benefits for the professional investment consultants as well. Most brokerage houses would value this information as it gives them a way of targeting clients with a range of financial products more effectively - including insurance, saving schemes, mutual funds, and so forth. Overall, many responsible brokerage houses realize that if they provide an effective service that is tailored to individual needs, in the long-term there is far more chance that they will retain their clients no matter whether the market is up or down....

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

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In this chapter we for the first time introduce the notion of n-vector space of type II. These n-vector spaces of type II are different from the n-vector spaces of type I because the n-vector spaces of type I are defined over a field F where as the n-vector spaces of type II are defined over n-fields. Some properties enjoyed by n-vector spaces of type II cannot be enjoyed by n-vector spaces of type I. To this; we for the sake of completeness just recall the definition of n-fields in section one and n-vector spaces of type II are defined in section two and some important properties are enumerated....

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Preface 5 Chapter One n-VECTOR SPACES OF TYPE II AND THEIR PROPERTIES 7 1.1 n-fields 7 1.2 n-vector Spaces of Type II 10 Chapter Two n-INNER PRODUCT SPACES OF TYPE II 161 Chapter Three SUGGESTED PROBLEMS 195 FURTHER READING 221 INDEX 225 ABOUT THE AUTHORS 229...

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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. In Chapter I, we give a brief description of the sixteen sutras invented by the Swamiji. Chapter II gives the text of select articles about Vedic Mathematics that appeared in the media. Chapter III recalls some basic notions of some Fuzzy and Neutrosophic models used in this book. This chapter also introduces a fuzzy model to study the problem when we have to handle the opinion of multiexperts. Chapter IV analyses the problem using these models. The final chapter gives the observations made from our study....

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We then remove the common factor if any from each and we find x + 1 staring us in the face i.e. x + 1 is the HCF. Two things are to be noted importantly. (1) We see that often the subsutras are not used under the main sutra for which it is the subsutra or the corollary. This is the main deviation from the usual mathematical principles of theorem (sutra) and corollaries (subsutra). (2) It cannot be easily compromised that a single sutra (a Sanskrit word) can be mathematically interpreted in this manner even by a stalwart in Sanskrit except the Jagadguru Puri Sankaracharya. We wind up the material from the book of Vedic Mathematics and proceed on to give the opinion/views of great personalities on Vedic Mathematics given by Jagadguru. Since the notion of integral and differential calculus was not in vogue in Vedic times, here we do not discuss about the authenticated inventor, further we have not given the adaptation of certain sutras in these fields. Further as most of the educated experts felt that since the Jagadguru had obtained his degree with mathematics as one of the subjects, most of the results given in book on Ve...

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Preface 5 Chapter One INTRODUCTION TO VEDIC MATHEMATICS 9 Chapter Two ANALYSIS OF VEDIC MATHEMATICS BY MATHEMATICIANS AND OTHERS 31 2.1 Views of Prof. S.G.Dani about Vedic Mathematics from Frontline 33 2.2 Neither Vedic Nor Mathematics 50 2.3 Views about the Book in Favour and Against 55 2.4 Vedas: Repositories of Ancient Indian Lore 58 2.5 A Rational Approach to Study Ancient Literature 59 2.6 Shanghai Rankings and Indian Universities 60 2.7 Conclusions derived on Vedic Mathematics and the Calculations of Guru Tirthaji - Secrets of Ancient Maths 61 Chapter Three INTRODUCTION TO BASIC CONCEPTS AND A NEW FUZZY MODEL 65 3.1 Introduction to FCM and the Working of this Model 65 3.2 Definition and Illustration of Fuzzy Relational Maps (FRMS) 72 3.3 Definition of the New Fuzzy Dynamical System 77 3.4 Neutrosophic Cognitive Maps with Examples 78 3.5 Description of Neutrosophic Relational Maps 87 3.6 Description of the new Fuzzy Neutrosophic model 92 Chapter Four MATHEMATICAL ANALYSIS OF THE VIEWS ABOUT VEDIC MATHEMATICS USING FUZZY MODELS 95 4.1 Views of students about the use of Vedic Mathematics in their c...

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...n the information provided by the sources is both uncertain and (highly) conflicting. This approach, known in literature as DSmT (standing for Dezert-Smarandache Theory), proposes new useful rules of combinations. We gathered in this volume a presentation of DSmT from the beginning to the latest development. Part 1 of this book presents the current state-of-the-art on theo...

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...57 8.2.1 Preliminary: about probability . . . 157 8.2.2 Dempster Shafer Theory . . . . 161 8.2.3 Transferable Belief Model . . . . 162 8.3 Dezert Smarandache Theory (DSmT) . . . 163 8.3.1 Dezert Smarandache model . . . . 164 8.3.2 Fusion rule . . . . . 166 8.4 Probability over logical propositions . . . . 167 8.4.1 Definition . . . . . 167 8.4.2 Property . . . . ....

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This book has four chapters. In chapter one the authors introduce the three types of super FRM models. Chapter two uses these three new super fuzzy models to study the role of media which feverishly argued against 27 percent reservation for OBCs in Central Government-run institutions in India. The experts we consulted were divided into 19 groups depending on their profession. These groups of experts gave their opinion and comments on the news-items that appeared about reservations in dailies and weekly magazines, and the gist of these lengthy discussions form the third chapter of this book. The fourth chapter gives the conclusions based on our study....

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Dedication 5 Preface 6 Chapter One INTRODUCTION TO NEW SUPER FUZZY MODELS 7 1.1 Supermatrices and Fuzzy Supermatrices 7 1.2 Super Fuzzy Relational Maps 45 Chapter Two ANALYSIS OF THE ROLE OF MEDIA ON RESERVATION FOR OBC USING SUPER FUZZY MODELS 57 2.1 Brief Description of the Attributes Given by the Experts 58 2.2 Super Row FRM Model to Study the Role of Media on OBC Reservation 77 2.3 Super Fuzzy Mixed FRM Model to Study the Role of Media in Falsely Blaming the Government and Supporting Dr. Venugopal 88 2.4 Use of Super Column FRM Model to Study the Interrelation between the Government and Public 102 2.5 Analysis of Role of Media on Reservation for OBC using Fuzzy Cognitive Map 117 2.6 Observations based on this Analysis by Students and Experts through Seminars and Discussions 121 Chapter Three EXCERPT OF NEWS FROM PRINT MEDIA AND SUGGESTIONS AND COMMENTS BY THE EXPERTS 145 Chapter Four ANALYSIS SUGGESTIONS AND CONCLUSIONS BASED ON DISCUSSIONS, QUESTIONNAIRE, INTERVIEWS AND MATHEMATICAL MODELS 357 4.1 A View and Analysis by Group of Educationalists about Role of Media on OBC Reservation 357 4.1.1 Students Pro...

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This book has seven chapters. Chapter one provides several basic notions to make this book self-contained. Chapter two introduces neutrosophic groups and neutrosophic N-groups and gives several examples. The third chapter deals with neutrosophic semigroups and neutrosophic N-semigroups, giving several interesting results. Chapter four introduces neutrosophic loops and neutrosophic N-loops. Chapter five just introduces the concept of neutrosophic groupoids and neutrosophic Ngroupoids. Sixth chapter innovatively gives mixed neutrosophic structures and their duals. The final chapter gives problems for the interested reader to solve. Our main motivation is to attract more researchers towards algebra and its various applications....

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1.1 Groups, N-group and their basic Properties It is a well-known fact that groups are the only algebraic structures with a single binary operation that is mathematically so perfect that an introduction of a richer structure within it is impossible. Now we proceed on to define a group. DEFINITION 1.1.1: A non empty set of elements G is said to form a group if in G there is defined a binary operation, called the product and denoted by '•' such that ...

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Preface 5 Chapter One INTRODUCTION 1.1 Groups, N-group and their basic Properties 7 1.2 Semigroups and N-semigroups 11 1.3 Loops and N-loops 12 1.4 Groupoids and N-groupoids 25 1.5 Mixed N-algebraic Structures 32 Chapter Two NEUTROSOPHIC GROUPS AND NEUTROSOPHIC N-GROUPS 2.1 Neutrosophic Groups and their Properties 40 2.2 Neutrosophic Bigroups and their Properties 52 2.3 Neutrosophic N-groups and their Properties 68 Chapter Three NEUTROSOPHIC SEMIGROUPS AND THEIR GENERALIZATIONS 3.1 Neutrosophic Semigroups 81 3.2 Neutrosophic Bisemigroups and their Properties 88 3.3 Neutrosophic N-Semigroup 98 Chapter Four NEUTROSOPHIC LOOPS AND THEIR GENERALIZATIONS 4.1 Neutrosophic loops and their Properties 113 4.2 Neutrosophic Biloops 133 4.3 Neutrosophic N-loop 152 Chapter five NEUTROSOPHIC GROUPOIDS AND THEIR GENERALIZATIONS 5.1 Neutrosophic Groupoids 171 5.2 Neutrosophic Bigroupoids and their generalizations 182 Chapter Six MIXED NEUTROSOPHIC STRUCTURES 187 Chapter Seven PROBLEMS 195 REFERENCE 201 INDEX 207 ABOUT THE AUTHORS 219 ...

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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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In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p -adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert ’s style, sequent, and hypersequent. Recall that hypersequents are a natural generalization of Gentzen ’s style sequents that was introduced independently by Avron and Pottinger . In particular, we examine Hilbert ’s style, sequent, and hypersequent calculi for infinite-valued logics based on the three fundamental continuous t-norms: Lukasiewicz ’s, Godel ’s, and Product logics....

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1.1 Neutrality concept in logic Every point of view A tends to be neutralized, diminished, balanced by Non-A. At the same time, in between A and Non-A there are infinitely many points of view Neut-A. Let’s note by A an idea, or proposition, theory, event, concept, entity, by Non-A what is not A, and by Anti-A the opposite of A. Neut-A means what is neither A nor Anti-A, i.e. neutrality is also in between the two extremes. ...

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1 Introduction. . . . . . . . . . . . . . . 9 1.1 Neutrality concept in logic . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Neutrality and non-Archimedean logical multiple-validity . . . . 10 1.3 Neutrality and neutrosophic logic . . . . . . . . . . . . . . . . . . 13 2 First-order logical language. . . . . . . . . . . . . . . 17 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Hilbert’s type calculus for classical logic . . . . . . . . . . . . . . 20 2.3 Sequent calculus for classical logic . . . . . . . . . . . . . . . . . 22 3 n -valued Lukasiewicz’s logics. . . . . . . . . . . . . . . 27 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Originality of (p + 1)-valued Lukasiewicz’s logics . . . . . . . . . 30 3.3 n-valued Lukasiewicz’s calculi of Hilbert’s type . . . . . . . . . . 32 3.4 Sequent calculi for n-valued Lukasiewicz’s logics . . . . . . . . . . 33 3.5 Hypersequent calculus for 3-valued Lukasiewicz’s propositional logic . . . . . . . . . . . .37 4 Infinite valued Lukasiewicz’s logics. . . . . . . . . . . . . . . 39 4.1 Prelimi...

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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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The present book discusses, among other things, various quantization phenomena found in Astrophysics and some related issues including Brownian Motion. With recent discoveries of exoplanets in our galaxy and beyond, this Astrophysics quantization issue has attracted numerous discussions in the past few years. Most chapters in this book come from published papers in various peer-reviewed journals, and they cover different methods to describe quantization, including Weyl geometry, Supersymmetry, generalized Schrödinger, and Cartan torsion method. In some chapters Navier-Stokes equations are also discussed, because it is likely that this theory will remain relevant in Astrophysics and Cosmology While much of the arguments presented in this book are theoretical, nonetheless we recommend further observation in order to verify or refute the propositions described herein. It is of our hope that this volume could open a new chapter in our knowledge on the formation and structure of Astrophysical systems. ...

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Abstract We explore Yang’s Noncommutative space-time algebra (involving two length scales) within the context of QM defined in Noncommutative spacetimes and the holographic area-coordinates algebra in Clifford spaces. Casimir invariant wave equations corresponding to Noncommutative coordinates and momenta in d-dimensions can be recast in terms of ordinary QM wave equations in d + 2-dimensions. It is conjectured that QM over Noncommutative spacetimes (Noncommutative QM) may be described by ordinary QMin higher dimensions. Novel Moyal-Yang-Fedosov-Kontsevich star products deformations of the Noncommutative Poisson Brackets are employed to construct star product deformations of scalar field theories. Finally, generalizations of the Dirac-Konstant and Klein-Gordonlike equations relevant to the physics of D-branes and Matrix Models are presented. ...

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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 linear transformations. The applications of these algebraic structures are given in Chapter 3. Chapter four gives some problem to make the subject easily understandable....

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n-VECTOR SPACES OF TYPE I AND THEIR PROPERTIES In this chapter we introduce the notion of n-vector spaces and describe some of their important properties. Here we define the concept of n-vector spaces over a field which will be known as the type I n-vector 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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...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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It is the purpose of this book to bring these cultural advantages into more focus, in order to bring into light some ‘human’ aspects of the country, and how these can be integrated into the broader context of economics development. At the end of the day, their achievements cannot be measured by economic progress alone, but also how the people can have the proper sense of meaning (i.e. ‘feel’ at home) in their own homeland, instead of being just another ‘bolt’ in the obsolete industrial engine of economics....

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Jingshan Park: For an extraordinary and unforgettable view of Beijing and the Forbidden City, casually climb the steps to the top of the hill to the Wanchunting (Wanchun Pavilion), which overlooks this small park, directly opposite the North Gate of the Forbidden City. From this playground of Ming and Qing dynasty emperors for almost 500 years, you will be able to get a grasp on the great city that is China’s capital. On Sunday’s choral groups gather in the park to perform amateur Peking Operas, play musical instruments or to sing songs once beloved by the masses....

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...Abstract……………3 Foreword……………4 Contents……………6 Chapter 1. Cultural Advantage as an Alternative Framework: An Introduction ‐ by F. Smarandache & V. Christianto……………7 Chapter 2. Beijing: A Cultural Metropolitan – compiled, edited, augmented by Ke Haiying & F. Smarandache…………18 Chapter 3. The City of Hengshui – by Li Zhanbing……………38 Chapter 4. Ningbo...

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This book has six chapters. The first one is introductory in nature just giving only the needed concepts to make this book a self contained one. Chapter two introduces the notion of refined plane of labels, the three dimensional space of refined labels DSm vector spaces. Clearly any n-dimensional space of refined labels can be easily studied as a matter of routine....

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Preface 5 Chapter One INTRODUCTION 7 Chapter Two DSm VECTOR SPACES 17 Chapter Three SPECIAL DSm VECTOR SPACES 83 Chapter Four DSm SEMIVECTOR SPACE OF REFINED LABELS 133 Chapter Five APPLICATIONS OF DSm SEMIVECTOR SPACES OF ORDINARY LABELS AND REFINED LABELS 173 Chapter Six SUGGESTED PROBLEMS 175 FURTHER READING 207 INDEX 211 ABOUT THE AUTHORS 214...

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In this book authors for the first time introduce the notion of supermatrices of refined labels. Authors prove super row matrix of refined labels form a group under addition. However super row matrix of refined labels do not form a group under product; it only forms a semigroup under multiplication. In this book super column matrix of refined labels and m Å~ n matrix of refined labels are introduced and studied. We mainly study this to introduce to super vector space of refined labels using matrices....

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THEOREM 1.1.1: Let S = {(a1 a2 a3 | a4 a5 | a6 a7 a8 a9 | … | an-1, an) | ai ∈ R; 1 ≤ i ≤ n} be the collection of all super row vectors with same type of partition, S is a group under addition. Infact S is an abelian group of infinite order under addition. The proof is direct and hence left as an exercise to the reader. If the field of reals R in Theorem 1.1.1 is replaced by Q the field of rationals or Z the integers or by the modulo integers Zn, n < ∞ still the conclusion of the theorem 1.1.1 is true. Further the same conclusion holds good if the partitions are changed. S contains only same type of partition. However in case of Zn, S becomes a finite commutative group....

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In this book authors for the first time introduce the notion of super interval matrices using the special intervals of the form [0, a], a belongs to Z+ ∪ {0} or Zn or Q+ ∪ {0} or R+ ∪ {0}....

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SUPER INTERVAL SEMILINEAR ALGEBRAS In this chapter we for the first time introduce the notion of semilinear algebra of super interval matrices over semifields of type I (super semilinear algebra of type I) and semilinear algebra of super interval matrices over interval semifields of type II (super semilinear algebra of type II) and study their properties and illustrate them with examples. DEFINITION 4.1: Let V be a semivector space of super interval matrices defined over the semifield S of type I. If on V we for every pair of elements x, y ∈ V; x . y is in V where ‘.’ is the product defined on V, then we call V a semilinear algebra of super interval matrices over the semifield S of type I. We will illustrate this situation by some examples. Example 4.1: Let V = {([0, a1] [0, a2] | [0, a3] [0, a4] | [0, a5] [0, a6] | [0,a7]) | ai ∪ Z+ ∪ {0}; 1 ≤ i ≤ 7} be a semivector space of super interval matrices defined over the semifield S = Z+ ∪ {0} of type I. Consider x = ([0, 5] [0, 3] | [0, 9] [0, 1] [0, 2] [0, 8] | [0, 6]) and y =([0, 1] [0, 2] | [0, 3] [0, 5] [0, 3] [0, 1] | [0, 5]) in V. We define the product ‘.’ on V as x.y =...

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CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two INTERVAL SUPERMATRICES 9 Chapter Three SEMIRINGS AND SEMIVECTOR SPACES USING SUPER INTERVAL MATRICES 83 Chapter Four 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 FURTHER READING 281 INDEX 285 ABOUT THE AUTHORS 287...