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Critical analysis of the foundations and tenets of capitalism and of the dismal science - economics.

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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 these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theroretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences. ...

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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 completely S-perfect numbers 12 B. Zhao and S. Wang: Cyclic dualizing elements in Girard quantales 21 P. Chun and Y. Zhao : On an equation involving the Smarandache function and the Dirichlet divisor function 27 F. Li and Y. Wang : An equation involving the Euler function and the Smarandache m-th power residues function 31 H. Gunarto and A. Majumdar : On numerical values of Z(n) 34 K. Nagarajan, etc.: M-graphoidal path covers of a graph 58 H. Liu : On the cubic Gauss sums and its fourth power mean 68 A. A. K. Majumdar : On the dual functions Z¤(n) and S¤(n) 74 S. S. Gupta : Smarandache sequence of Ulam numbers 78 L. Li : A new Smarandache multiplicative function and its arithmetical properties 83 Y. Yang and X. Kang : A predator-prey epidemic model with infected predator 86 R. Fu and H. Yang : An equation involving the Lucas numbers 90 W. Yang : Two rings in IS-a...

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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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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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A collection of papers concerning Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc. is published this year....

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This document will describe the current status on the search for factors of Smaranda.che consecutive numbers and their reverse. A complete list up to index 200 will be given, with all known factors. Smarandache numbers are the concatenation of the natural numbers from one up to the given index, and reverse Smaranda.che numbers are the concatenation of the natural numbers from the given index dO\vIJ. to 1....

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Duality Theorems of Multiobjective Generalized Disjunctive Fuzzy Nonlinear Fractional Programming Abstract: This paper is concerned with the study of duality conditions to convex-concave generalized multiobjective fuzzy nonlinear fractional disjunctive programming problems for which the decision set is the union of a family of convex sets. The Lagrangian function for such problems is defined and the Kuhn-Tucker Saddle and Stationary points are characterized. In addition, some important theorems related to the Kuhn-Tucker problem for saddle and stationary points are established. Moreover, a general dual problem is formulated together with weak; strong and converse duality theorems are proved. Key Words: Generalized multiobjective fractional programming; Disjunctive programming; Convexity; Concavity; fuzzy parameters Duality. ...

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Contents Duality Theorems of Multiobjective Generalized Disjunctive Fuzzy Nonlinear Fractional Programming BY E.E.AMMAR . . . . . . . . . . . . . . . . . . . . 01 Surface Embeddability of Graphs via Joint Trees BY YANPEI LIU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Plick Graphs with Crossing Number 1 BY B.BASAVANAGOUD AND V.R.KULLI . . . . . . . . . . . . . . . . . . . . . . 21 Effects of Foldings on Free Product of Fundamental Groups BY M.El-GHOUL, A. E.El-AHMADY, H.RAFAT AND M.ABU-SALEEM. . . . . . . 29 Absolutely Harmonious Labeling of Graphs BY M.SEENIVASAN AND A.LOURDUSAMY . . . . . . . . . . . . . . . . . . . . 40 The Toroidal Crossing Number of K4n BY SHENGXIANG LV, TANG LING AND YUANGQIU HUANG . . . . . . . . . . 52 On Pathos Semitotal and Total Block Graph of a Tree BY MUDDEBIHAL M. H. AND SYED BABAJAN. . . . . . . . . . . . . . . . . . 64 Varieties of Groupoids and Quasigroups Generated by Linear-Bivariate Polynomials Over Ring Zn BY E.ILOJIDE, T.G.JAIYEOLA AND O.O.OWOJORI . . . . . 79 New Characterizations for Bertrand Curves in Minkowski 3-Space BY BAHADDIN BUKCU, MURAT K...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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x4: Radio labeling of P3n for n is less than or equal to 5 or n = 7 In this section we determine radio numbers of cube path of small order as a special case....

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Contents Tangent Space and Derivative Mapping on Time Scale BY EMIN ¨OZYILMAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01 Basic Properties Of Second Smarandache Bol Loops BY T.G.JA´IY´E O. L´A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Smarandachely Precontinuous maps and Preopen Sets in Topological Vector Spaces BY SAYED ELAGAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Path Double Covering Number of Product Graphs BY A. NAGARAJAN, V. MAHESWARI, S. NAVANEETHAKRISHNAN. . . . . . 27 Some Remarks on Fuzzy N-Normed Spaces BY SAYED ELAGAN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 σ-Coloring of the Monohedral Tiling BY M. E. BASHER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 The Forcing Domination Number of Hamiltonian Cubic Graphs BY H.ABDOLLAHZADEH AHANGAR AND PUSHPALATHA L. . . . . . . . . . 53 Permutation Polynomials modulo n, n 6= 2w and Latin Squares BY VADIRAJA BHATTA G. R. and SHANKAR B. R. . . . . . . . . . . . . . . . . 58 Graphoidal Tree d - Cover BY S.SOMASUNDARAM, A.NAGARAJAN AND G.MA...

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This book aims to assist social scientists to analyze their problems using fuzzy models. The basic and essential fuzzy matrix theory is given. The book does not promise to give the complete properties of basic fuzzy theory or basic fuzzy matrices. Instead, the authors have only tried to give those essential basically needed to develop the fuzzy model. The authors do not present elaborate mathematical theories to work with fuzzy matrices; instead they have given only the needed properties by way of examples. The authors feel that the book should mainly help social scientists who are interested in finding out ways to emancipate the society. Everything is kept at the simplest level and even difficult definitions have been omitted. Another main feature of this book is the description of each fuzzy model using examples from real-world problems. Further, this book gives lots of references so that the interested reader can make use of them. This book has two chapters. In Chapter One, basic concepts about fuzzy matrices are introduced. Basic notions of matrices are given in section one in order to make the book self contained. Section two g...

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Preface 4 Dedication 6 Chapter One BASIC MATRIX THEORY AND FUZZY MATRIX THEORY 7 1.1 Basic Matrix Theory 7 1.2 Basic Concepts on Fuzzy Matrices 33 1.3 Basic Concepts on Graphs 47 Chapter Two DESCRIPTION OF SIMPLE FUZZY MODELS AND THEIR APPLICATIONS TO REAL WORLD PROBLEMS 71 2.1 Description of Simple Fuzzy Matrix Model 72 2.2 Definition of Fuzzy Cognitive Maps with real world model representation 146 2.3 Definition and Illustration of Fuzzy Relational Maps 205 2.4 Introduction to Bidirectional Associative Memories (BAM) Model and their Application 238 2.5 Description of Fuzzy Associative Memories (FAM) model and their illustrations 264 2.6 Fuzzy Relational Equations (FRE) and their application 277 REFERENCE 317 INDEX 345 ABOUT THE AUTHORS 352...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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On (∈ vq)-Fuzzy Bigroup Abstract: In this paper, we introduce the concept of fuzzy singleton to bigroup, and uses it to define (∈ v q)- fuzzy bigroup and discuss its properties. We investigate whether or not the fuzzy point of a bigroup will belong to or quasi coincident with its fuzzy set if the constituent fuzzy points of the constituting subgroups both belong to or quasi coincident with their respective fuzzy sets, and vise versa. We also prove that a fuzzy bisubset μ is an (∈ vq)-fuzzy subbigroup of the bigroup G if its constituent fuzzy subsets are (∈ vq)-fuzzy subgroups of their respective subgroups among others. Key Words: Bigroups, fuzzy bigroups, fuzzy singleton on bigroup, (∈ vq)- fuzzy subgroups, (∈ vq)- fuzzy bigroup ...

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Contents On (∈ vq)- Fuzzy Bigroup BY AKINOLA L.S. and AGBOOLA A.A.A. . . . . . . . . . . . . . . . . . . . . . . . . . . 01 Connectivity of Smarandachely Line Splitting Graphs BY B.BASAVANAGOUD and VEENA MATHAD . . . . . . . . . . . . . . . . . . . . . . 08 Separation for Triple-Harmonic Di?erential Operator in Hilbert Space BY E.M.E.ZAYED and S.A.OMRAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Classification of Differentiable Graph BY A. El-Abed . . . . . . . . . . . . . . . . . 24 On Equitable Coloring of Helm Graph and Gear Graph BY KALIRAJ.K and VERNOLD VIVIN.J . . . . . . . . . . . . . . . . . . . . . . . . . . 32 On the Roman Edge Domination Number of a Graph BY K. EBADI, E. KHODADADI and L. PUSHPALATHA. . . . . . . . . . . . . . . . . . 38 The Upper Monophonic Number of a Graph BY J.JOHN and S.PANCHALI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Some Results on Pair Sum Labeling of Graphs BY R. PONRAJ, J.VIJAYA XAVIER PARTHIPAN and R.KALA . . . . . . . . . . . . . . 53 Weierstrass Formula for Minimal Surface in the Special Three-Dimensional Kenmotsu Manifold K with _-...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Extending Homomorphism Theorem to Multi-Systems Abstract: The multi-laterality of WORLD implies multi-systems to be its best candidate model for ones cognition on nature, which is also included in an ancient book of China, TAO TEH KING written by Lao Zi, an ancient philosopher of China. Then how it works to mathematics, not suspended in thought? This paper explains this action by mathematical logic on mathematical systems generalized to Smarandache systems, or such systems with combinatorial structures, i.e., combinatorial systems, and shows how to extend the homomorphism theorem in abstract algebra to multi-systems or combinatorial systems. All works in this paper are motivated by a combinatorial speculation of mine which is reformed on combinatorial systems and can be also applied to geometry. Key Words: Homomorphism theorem, multi-system, combinatorial system. ...

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Contents Extending Homomorphism Theorem to Multi-Systems BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01 A Double Cryptography Using the Smarandache Keedwell Cross Inverse Quasigroup BY T. G. JA´IY´EOL´A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 On the Time-like Curves of Constant Breadth in Minkowski 3-Space BY SUHA YILMAZ AND MELIH TURGUT. . . . . . . . . . . . . . . . . . . . . 34 On the Basis Number of the Strong Product of Theta Graphs with Cycles BY M.M.M. JARADAT, M.F. JANEM AND A.J. ALAWNEH. . . . . . . . . . . .40 Smarandache Curves in Minkowski Space-time BY MELIH TURGUT AND S¨UHA YILMAZ . . . . . . . . . . . . . . . . . . . . . 51 The Characterization of Symmetric Primitive Matrices with exponent n − 3 BY LICHAO, HUANGFU AND JUNLIANG CAI . . . . . . . . . . . . . . . . . . 56 The Crossing Number of the Circulant Graph C(3k − 1; {1, k}) BY JING WANG AND YUANQIU HUANG. . . . . . . . . . . . . . . . . . . .79 On the Edge Geodetic and k-Edge Geodetic Number of a Graph BY A.P. SANTHAKUMARAN AND S.V. ULLAS CHANDRAN. . . . . . . . . .85 Simple Path Cove...

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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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Aims and Scope: The International J.Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,· · · , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretic...

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Abstract: Let G be a group having a partially closed subset S such that S contains the identity element of G and each element in S has an inverse in S. Such subsets of G are called halfsubgroups of G. If a halfsubgroup S generates the group G, then S is called a halfsubgroup generating the group or hsgg in short. In this paper we prove some results on hsggs of a group. Order class of a group are special halfsubgroupoids. Elementary abelian groups are characterized as groups with maximum special halfsubgroupoids. Order class of a group with unity forms a typical halfsubgroup. Key words: halfsubgroup, hsgg, order class of an element. AMS(2000): 20Kxx, 20L05....

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Halfsubgroups BY ARUN S. MUKTIBODH. . . . . . . . . . . . . . . . . . . . . . . . . . . . .01 Flexibility of Embeddings of a Halin Graph on the Projective Plane BY HAN REN and YUN BAI. . . . . . . . . . . . . . . . . . . . . . . . . . . .06 Curvature Equations on Combinatorial Manifolds with Applications to Theoretical Physics BY LINFAN MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety BY T. G. JA´IY´EOL´A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 A Revision to G¨odel’s Incompleteness Theorem by Neutrosophy BY YUHUA FU and ANJIE FU. . . . . . . . . . . . . . . . . . . . . . . . . . . 45 On the basis Number and the Minimum Cycle Bases of the Wreath Product of Two Wheels BY M.M.M. JARADAT and M.K. AL-QEYYAM . . . . . . . . . . . . . . . . . . 52 Theory of Relativity on the Finsler Spacetime BY SHENGLIN CAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 On the Number of Graceful Trees BY GUANGXUAN WANG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 ...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Contents On the Crypto-Automorphism of the Buchsteiner Loops BY J.O.AD´EN´IRN and Y.T.OYEBO. . . . . . . . . . . . . . . . . . . . . . . . .01 Generalizations of Poly-Bernoulli Numbers and Polynomials BY HASSAN JOLANY, M.R.DARAFSHEH AND R.EIZADI ALIKELAYE . . . . 07 Open Alliance in Graphs BY N.JAFARI RAD AND H.REZAZADEH . . . . . . . . . . . . . . . . . . . . . 15 The Forcing Weak Edge Detour Number of a Graph BY A.P.SANTHAKUMARAN AND S.ATHISAYANATHAN . . . . . . . . . . . . . 22 Special Smarandache Curves in the Euclidean Space BY AHMAD T. ALI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 The H-Line Signed Graph of a Signed Graph BY R.RANGARAJAN, M. S. SUBRAMANYA AND P. SIVA KOTA REDDY. . . .37 Min-Max Dom-Saturation Number of a Tree BY S. ARUMUGAM AND S. SUDHA . . . . . . . . . . . . . . . . . . . . . . . 45 Embeddings of Circular graph C(2n + 1, 2) (n ≥ 2) on the Projective Plane BY XINQIU LIU, YUANQIU HUANG AND JING WANG. . . . . . . . . . . . .53 A Note On Jump Symmetric n-Sigraph BY H. A.MALATHI AND H. C.SAVITHRI . . . . . . . . . . . . . . . . . . . . 65 New Families of Mean Graphs ...

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Papers concerning any of the Smarandache type functions, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, conjectures, open problems, neutrosophic logic/set/probability, etc. have been selected for this volume. The...

Description Details
Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Abstract: In this paper we find the interrelations and the hidden pattern of the problems faced by the PWDs and their caretakers using Fuzzy Relational Maps (FRMs). Here we have taken the problems faced by the rural persons with disabilities in Melmalayanur and Kurinjipadi Blocks, Tamil Nadu, India. This paper is organized with the following four sections. Section one is introductory in nature giving the overall contents from the survey made about PWDs in the above said Blocks. Section two gives description of FRM models and the attributes taken for the study related with the PWDs and the caretakers, the FRM model formed using these attributes and their analysis. The third section gives the suggestions and conclusions derived from the survey as well as the FRM model. Key Words: FRM model, fixed point, hidden pattern, relational matrix, limit cycle. AMS(2000): 04A72....

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Study of the Problems of Persons with Disability (PWD) Using FRMs BY W.B.VASANTHA KANDASAMY, A.PRAVEEN PRAKASH AND K.THIRUSANGU 01 Topological Multi-groups and Multi-fields BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 08 Shortest Co-cycle Bases of Graphs BY HAN REN AND JING HAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 On Involute and Evolute Curves of Spacelike Curve with a Spacelike Principal Normal in Minkowski 3 -Space BY BAHADDIN BUKCU AND MURAT KEMAL KARACAN . . . . . . . . . . . . . . . 27 Notes on the Curves in Lorentzian Plane L2 BY S¨ UHA YILMAZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Cycle-Complete Graph Ramsey Numbers r(C4,K9), r(C5,K8) ≤ 33 BY M.M.M. JARADAT AND B.M.N. ALZALEQ . . . . . . . . . . . . . . . . . . . . 42 Smarandache Breadth Pseudo Null Curves in Minkowski Space-time BY MELIH TURGUT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Smarandachely k -Constrained labeling of Graphs BY SHREEDHARK, B. SOORYANARAYANA AND RAGHUNATHP . . . . . . . . . . .50 Equiparity Path D...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Sequences on Graphs with Symmetries Linfan Mao Chinese Academy of Mathematics and System Science, Beijing, 10080, P.R.China Beijing Institute of Civil Engineering and Architecture, Beijing, 100044, P.R.China E-mail: maolinfan@163.com Abstract: An interesting symmetry on multiplication of numbers found by Prof.Smarandache recently. By considering integers or elements in groups on graphs, we extend this symmetry on graphs and find geometrical symmetries. For extending further, Smarandache’s or combinatorial systems are also discussed in this paper, particularly, the CC conjecture presented by myself six years ago, which enables one to construct more symmetrical systems in mathematical sciences. Key Words: Smarandache sequence, labeling, Smarandache beauty, graph, group, Smarandache system, combinatorial system, CC conjecture...

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Contents Lucas Graceful Labeling for Some Graphs BY M.A.PERUMAL, S.NAVANEETHAKRISHNAN AND A.NAGARAJAN . . . . . 01 Sequences on Graphs with Symmetries BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Supermagic Coverings of Some Simple Graphs BY P.JEYANTHI AND P.SELVAGOPAL . . . . . . . . . . . . . . . . . . . . . . . 33 Elementary Abelian Regular Coverings of Cube BY FURONG WANG AND LIN ZHANG. . . . . . . . . . . . . . . . . . . . . .49 Super Fibonacci Graceful Labeling of Some Special Class of Graphs BY R.SRIDEVI, S.NAVANEETHAKRISHNAN AND K.NAGARAJAN. . . . . . . . 59 Surface Embeddability of Graphs via Tree-travels BY YANPEI LIU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 Edge Maximal C3 and C5-Edge Disjoint Free Graphs BY M.S.A.BATAINEH AND M.M.M.JARADAT . . . . . . . . . . . . . . . . . . 82 A Note on Admissible Mannheim Curves in Galilean Space G3 BY S.ERSOY, M.AKYIGIT AND M.TOSUN . . . . . . . . . . . . . . . . . . . . 88 The Number of Spanning Trees in Generalized Complete Multipartite Graphs of Fan-Type BY JUNLIANGE CAI AND XIAOLI LIU . . . . . ....

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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§1. Why is the WORLD a Combinatorial One? The multiplicity of the WORLD results in modern sciences overlap and hybrid, also implies its combinatorial structure. To see more clear, we present two meaningful proverbs following. Proverb 1. Ames Room An Ames room is a distorted room constructed so that from the front it appears to be an ordinary cubic-shaped room, with a back wall and two side walls parallel to each other and perpendicular to the horizontally level floor and ceiling. As a result of the optical illusion, a person standing in one corner appears to the observer to be a giant, while a person standing in the other corner appears to be a dwarf. The illusion is convincing enough that a person walking back and forth from the left corner to the right corner appears to grow or shrink. ...

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Contents Combinatorial Fields - An Introduction BY LINFAN MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .01 A Spacetime Geodesics of the Schwarzschild Space and Its Deformation Retract BY H. RAFAT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 On Degree Equitable Sets in a Graph BY A. ANITHA, S. ARUMUGAM AND E. SAMPATHKUMAR . . . . . . . . . . . . 32 Smarandachely k-Constrained Number of Paths and Cycles BY P. DEVADAS RAO, B. SOORYANARAYANA, M. JAYALAKSHMI . . . . . . . . . 48 On Functions Preserving Convergence of Series in Fuzzy n-Normed Spaces BY SAYED ELAGAN AND MOHAMAD RAFI SEGI RAHMAT . . . . . . . . . . . . 61 Achromatic Coloring on Double Star Graph Families BY VERNOLD VIVIN J., VENKATACHALAM M. AND AKBAR ALI M.M. . . . . . 71 Some Results on Super Mean Graphs BY R.VASUKI AND A.NAGARAJAN. . . . . . . . . . . . . . . . . . . . . . . . .82 Chromatic Polynomial of Smarandache νE-Product of Graphs BY KHALIL PARYAB AND EBRAHIM ZARE. . . . . . . . . . . . . . . . . . . .97 Open Distance-Pattern Uniform Graphs BY BIBIN K. JOSE . . . . . . . . . . . . . . . . . ....

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Papers concerning any of the Smarandache type functions, sequences, numbers, algorithms, inferior/superior f-parts, magic squares, palindromes, functional iterations, semantic paradoxes, Non-Euclidean geometries, manifolds, conjectures, open problems, algebraic structures, neutrosophy, neutrosophic logic/set/probability, hypothesis that there is no speed barrier in the universe} quantum paradoxes, etc. have been selected for this volume. Contributors are from Australia, China, England, Germany, India, Ireland, Israel, Italy, Japan, Malaysia, Morocco, Portugal, Romania, Spain, USA. Most of the papers are in English, a few of them are in Spanish, Portuguese, or Gennan....

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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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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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Smarandache-Zagreb Index on Three Graph Operators Abstract: Many researchers have studied several operators on a connected graph in which one make an attempt on subdivision of its edges. In this paper, we show how the Zagreb indices, a particular case of Smarandache-Zagreb index of a graph changes with these operators and extended these results to obtain a relation connecting the Zagreb index on operators. Key Words: Subdivision graph, ladder graph, Smarandache-Zagreb index, Zagreb index, graph operators. ...

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Contents Smarandache-Zagreb Index on Three Graph Operators BY RANJINI P.S. and V.LOKESHA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01 Total Minimal Dominating Signed Graph BY P.SIVA KOTA REDDY and S.VIJAY . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Number of Minimum Dominating Sets in Pn Å~ P2 BY H.B.WALIKAR, K.P. NARAYANKAR and S.S.SHIRAKOL . . . . . . . . . . . . . . . 17 Super Fibonacci Graceful Labeling BY R.SRIDEVI, S.NAVANEETHAKRISHNAN AND K.NAGARAJAN. . . . . . . . . . .22 A Note on Smarandachely Consistent Symmetric n-Marked Graphs BY P.SIVA KOTA REDDY, V. LOKESHA and GURUNATH RAO VAIDYA. . . . . . . . .41 Some Fixed Point Theorems in Fuzzy n-Normed Spaces BY SAYED KHALIL ELAGAN and MOHAMAD RAFI SEGI RAHMAT . . . . . . . . . . 45 A Result of Ramanujan and Brahmagupta Polynomials Described by a Matrix Identity BY R. RANGARAJAN . . . . . . . . . . . . . . . . 57 Biharmonic Slant Helices According to Bishop Frame in E3 BY ESSIN TURHAN and TALAT K¨ORPINAR . . . . . . . . . . . . . . . . . . . . . . 64 Combinatorial Optimization in VLSI Hypergraph Partitioning Using Taguchi Methods...

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During my research period in the Summer of 2012 at the Research Institute of Extenics and Innovation Methods, from Guangdong University of Technology, in Guangzhou, China, I have introduced the Linear and Non-Linear Attraction Point Principle and the Network of Attraction Curves, have generalized the 1D Extension Distance and the 1D Dependent Function to 2D, 3D, and in general to n-D Spaces, and have generalized Qiao-Xing Li’s and Xing-Sen Li’s definitions of the Location Value of a Point and the Dependent Function of a Point on a Single Finite Interval from one dimension (1D) to 2D, 3D, and in general n-D spaces. Then I used the Extenics, together with Victor Vlădăreanu, Mihai Liviu Smarandache, Tudor Păroiu, and Ştefan Vlăduţescu, in 2D and 3D spaces in technology, philosophy, and information theory....

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Contribution to Extenics (Preface by Florentin Smarandache): 4 1. Generalizations of the Distance and Dependent Function in Extenics to 2D, 3D, and n-D, by Florentin Smarandache: 22 2. Applications of Extenics to 2D-Space and 3D-Space, by Florentin Smarandache, Victor Vlădăreanu: 39 3. Generalization of the Dependent Function in Extenics for Nested Sets with Common Endpoints to 2D-Space, 3D-Space, and generally to n-D-Space, by Florentin Smarandache: 52 4. Generalizations in Extenics of the Location Value and Dependent Function from A Single Finite Interval to 2D, 3D, and n-D Spaces, by Florentin Smarandache, Mihai Liviu Smarandache: 60 5. Extention Transformation Used in I Ching, by Florentin Smarandache: 69 6. Extenics in Conventional and Nonconventional [Romanian], by Florentin Smarandache, Tudor Păroiu: 79 7. Extension Communication for Solving the Ontological Contradiction between Communication and Information, by Florentin Smarandache, Ştefan Vlăduţescu: 99-112...

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In these proceedings of the second international conference on smarandache type notions in mathematics and quantum physics (december 21-24, 2000, university ofcraiova, romania; organizers: v. Seleacu and m. L perez) are collected articles and notes: in mathematics: related to smarandache anti-geometry, function, f-inferior part function, k-k additive relationships, 2-2 subtractive relationships, sequences, coprime functions, double factorial function, magic squares, problems, conjectures, equations, partitions, paradoxes, series, algebraic structures, pseodo-smarandache function, erdos-smarandache moments numbers; and in physics: related to smarandache hypothesis that there is no speed barrier in the universe, srm-theory of the possibility of constructing arbitrary speeds, and quantum smarandache paradoxes....

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I have met the Smarandache's world for the first time about one year ago reading some articles and problems published in the Journal of Recreational Mathematics. From then on I discovered the interesting American Research Press web site dedicated to the Smarandache notions and held by Dr. Perez (address: http://www.gallup.unm.edu/~smarandache/), the Smarandache Notions Journal always published by American Research Press, and several books on conjectures, functions, unsolved problems, notions and other proposed by Professor F. Smarandache in "The Florentin Smarandache papers" special collections at: the Arizona State University (Tempe, USA), Archives of American Mathematics (University of Texas at Austin, USA), University of Craiova Library (Romania), and Archives of State (Rm. Valcea, Romania). The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting. This book, following the Smarandache spirit, presents new ...

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Abstract. The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the apparently unscientific ideas into scientific ones, and to develop their study (The Theory of Imperfections). And finally, to interconnect opposite (and not only) human fields of knowledge into a.s-heterogeneous-as-possible another fields. The author welcomes any co=ents, notes, articles on this paper and/or the 120 open questions bothering him, which will be published in a collective monograph about the paradoxist mathematics....

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The "Paradoxist Mathematics" may be understood as Experimental Mathematics, NonMathematics, or even Anti-Mathematics: not in a nihilistic way, but in positive one. The truly innovative researchers will banish the old concepts in oder of check, by heuristic processes, some new ones: their opposites. Don't simply follow the crowd, and don't accept to be manipulated by any (political, economical, social, even scientific, or artistic, cultural, etc.) media! Learn to conradict everuthing and everybody!! "Duibito, ergo cogito; cogito, ergo sum", said Rene Descartes, "I doubt, therefore I think; I think, therefore I exist" (metaphysical doubt). See what happens if you deny the classics' theory?...

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Paradoxist Mathematics [Anti-Mathematics, Multi-Structure and Multi-Space, Space of Non-Integer or Negative Dimention, Inconsistent Sysytem ofAxiomas or Contradictory Theory, Non-Euclidean Geometries: Paradoxist Geometry, Non-Geometry, CounterProjective Geometry and Counter-Axioms, Anti-Geometry and Anti-Axioms, Model of an Anti-Geometry, Discontinuous Geometries] 5 -- Logica sau iogica Matematica? .29 -- Mathematics and Alcohol and God . 30 -- Subjective Questions and Answers for a MathInstructor of Higher Education .32 -- o Geometrie Paradoxista .49 -- Geometric Conjecture .50 -- A Function in the ~umber Theory . 51 -- An Infinity of Unsolved Problems Concerning a Function in the :'-lumber Theory . 57 -- Solving Problems by Using a Function in the Number Theory .79 -- Some Linear Equations Involving a Function in the Number Theory .82 -- Contributii la Studiul unor Functii §i Conjecturi in Teoria Numerelor .85 -- "The Function that You Bear its Name" .112 -- Smarandache Type Functions Obtained by Duality .113 -- Func!;ii Analitice .129 -- Funct;ii Prime §i Coprime .137 -- Asupra unor Conjecturi §i Probleme Nerezolvate Referitoare ...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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On the Bicoset of a Bivector Space Abstract: The study of bivector spaces was first intiated by Vasantha Kandasamy. The objective of this paper is to present the concept of bicoset of a bivector space and obtain some of its elementary properties. Key Words: bigroup, bivector space, bicoset, bisum, direct bisum, inner biproduct space, biprojection. ...

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Contents On the Bicoset of a Bivector Space BY AGBOOLA A.A.A. AND AKINOLA L.S.. . . . . . . . . . . . . . . . . . . . .01 Smarandachely Bondage Number of a Graph BY KARAM EBADI AND L.PUSHPALATHA . . . . . . . . . . . . . . . . . . . . . 09 Domination Number in 4-Regular Graphs BY H.Abdollahzadeh Ahangar AND Pushpalatha L. . . . . . . . . . . . . . . . . . 20 Computing Smarandachely Scattering Number of Total Graphs BY AYSUN AYTAC AND ELGIN KILIC. . . . . . . . . . . . . . . . . . . . . . .31 On the 3ψ3 Basic Bilateral Hypergeometric Series Summation Formulas BY K. R.VASUKI AND G.SHARATH. . . . . . . . . . . . . . . . . . . . . . . .41 Minimal Retraction of Space-time and Their Foldings BY A. E. El-AHMADY aND H. RAFAT. . . . . . . . . . . . . . . . . . . . . . . .49 Efficient Domination in Bi-Cayley Graphs BY T. TAMIZH CHELVAM AND SIVAGNANAM MUTHARASU. . . . . . . . . .56 Independent Complementary Distance Pattern Uniform Graphs BY GERMINA K.A. AND BEENA KOSHY. . . . . . . . . . . . . . . . . . . . . .63 On Smarandachely Harmonic Graphs BY D.D.SOMASHEKARA AND C.R.VEENA . . . . . . . . . . . . . . . . . . . . . 75 S...

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Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, and published quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,. . . , etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitatio...

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The Characterization of Symmetric Primitive Matrices with Exponent n − 2 Junliang Cai Abstract: In this paper the symmetric primitive matrices of order n with exponent n − 2 are completely characterized by applying a combinatorial approach, i.e., mathematical combinatorics ([7]). Key words: primitive matrix, primitive exponent, graph....

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The Characterization of Symmetric Primitive Matrices with Exponent n − 2 BY JUNLIANG CAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01 Characterizations of Some Special Space-like Curves in Minkowski Space-time BY MELIH TURGUT AND SUHA YILMAZ . . . . . . . . . . . . . . . . . . . . . . . 17 Combinatorially Riemannian Submanifolds BY LINFAN MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Smarandache Half-Groups BY ARUN S.MUKTIBODH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 On Smarandache Bryant Schneider Group of a Smarandache Loop BY T. G. JA´IY´EOL´A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Some Properties of Nilpotent Lattice Matrices BY QIYI FAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 On the Crossing Number of the Join of Some 5-Vertex Graphs and Pn BY BO LI, JING WANG AND YUANQIU HUANG . . . . . . . . . . . . . . . . . . 70 Identities by L-summing Method (II) BY MEHDI HASSANI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 On the Basis Numbe...

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We focus our discussions in this book on cities, because in our opinion a city is the smallest economic entity which has ‘self-organizing’ character, in a sense that a city can grow by itself (with minimum intervention). Nonetheless, this book will not discuss the self-organization character itself, but a new concept called ‘Cultural Economy’ development. Cultural Economic here is part of leisure and tourism industry, and depends on taste, advertisement, history, and the quality of being diverse, distinctive, with a large spectrum of varieties....

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Content ................... 3 Prologue ....................... 4 Biography of authors .................. 7 Chapter 1. Historical Precursor ............. 8 Pitfalls of Industrialization Is there alternative to industrialization? A review of economics thinking Box I: Cultural Advantage and Cultural Studies Chapter 2. Cultural Advantage as Alternative ............. 20 De-Industrialization and Emergence of Experience Economy Cultural Advantage, a new proposition Chapter 3. High-Tech/High-Touch and Cultural Advantage ..... 31 To automate or not to automate? How to implement Cultural Advantage in city development Box III: Cultural Advantage and Anthropology Chapter 4. A few examples: Lessons to learn ............. 40 a. Nicosia b. Toronto c. Venice d. Huzhou e. Valencia f. Chicago g. Romanian Lesson h. Eastern Europe Lesson i. Argentina Lesson Chapter 5. Introduction to Poly-Emporium Theory ............. 49 A new proposition: Extended Tobin Tax Bibliography.........60...

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In this work the authors apply concepts of Neutrosophic Logic to the General Theory of Relativity to obtain a generalisation of Einstein’s four dimensional pseudo-Riemannian differentiable manifold in terms of Smarandache Geometry (Smarandache manifolds), by which new classes of relativistic particles and non-quantum teleportation are developed....

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1.2 The basics of neutrosophy Neutrosophy studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. It considers that every idea tends to be neutralized, balanced by ideas; as a state of equilibrium. Neutrosophy is the basis of neutrosophic logic, neutrosophic set which generalizes the fuzzy set, and of neutrosphic probability and neutrosophic statistics, which generalize the classical and imprecise probability and statistics respectively. Neutrosophic Logic is a multiple-valued logic in which each proposition is estimated to have percentages of truth, indeterminacy, and falsity in T, I, and F respectively, where T, I, F are standard or non-standard subsets included in the non-standard unit interval ] −0, 1+[. It is an extension of fuzzy, intuitionistic, paraconsistent logics. ...

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Preface of the Editor 4 Chapter 1 PROBLEM STATEMENT . THE BASICS OF NEUTROSOPHY 1.1 Problem statement 5 1.2 The basics of neutrosophy 8 1.3 Neutrosophic subjects 13 1.4 Neutrosophic logic. The origin of neutrosophy 14 1.5 Definitions of neutrosophic 16 Chapter 2 TRAJECTORIES AND PARTICLES 2.1 Einstein’s basic space-time 18 2.2 Standard set of trajectories and particles. A way to expand the set 22 2.3 Introducing trajectories of mixed isotropic/non-isotropic kind 28 2.4 Particles moving along mixed isotropic/non-isotropic trajectories 31 2.5 S-denying the signature conditions. Classification of the expanded spaces 35 2.6 More on an expanded space-time of kind IV 45 2.7 A space-time of kind IV: a home space for virtual photons 52 2.8 A space-time of kind IV: non-quantum teleportation of photons 55 2.9 Conclusions 59 Chapter 3 ENTANGLED STATES AND QUANTUM CAUSALITY THRESHOLD 3.1 Disentangled and entangled particles in General Relativity. Problem statement 61 3.2 Incorporating entangled states into General Relativity 64 3.3 Quantum Causality Threshold in General Relativity 69 3.4 Conclusions 72 Biblio...

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We have used the 2-adaptive fuzzy model having the two fuzzy models, fuzzy matrices model and BAMs viz. model to analyze the views of public about HIV/ AIDS disease, patient and the awareness program. This book has five chapters and 6 appendices. The first chapter just recalls the definition of four fuzzy models used in this book and gives illustration of some of them. Chapter two introduces the new n-adaptive fuzzy models. Chapter three uses for the first time 2 adaptive fuzzy models to study psychological and sociological problems about HIV/AIDS. Chapter four gives an outline of the interviews. Chapter five gives the suggestions and conclusion based on our study. Of the 6 appendices four of them are C-program made to make the working of the fuzzy model simple....

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In this chapter for the first time we introduce the new class of n-adaptive fuzzy models (n a positive integer and n ≥ 2) and illustrate it. These n-adaptive fuzzy models can analyze a problem using different models so one can get in one case the hidden pattern, in one case the maximum time period in some case output state vector for a given input vector and so on. So this new model has the capacity to analyze the problem in different angles, which can give multiple suggestions and solutions about the problem. This chapter has two sections. Section one defines the new model gives illustrations and section two defines some special n-adaptive models and proposes some problems....

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Chapter One SOME BASIC FUZZY MODELS WITH ILLUSTRATIONS 1.1 Fuzzy matrices and their applications 9 1.2 Definition and illustration of Fuzzy Relational Maps (FRMs) 16 1.3 Some basic concepts of BAM with illustration 20 1.3.1 Some basic concepts of BAM 22 1.3.2 Use of BAM Model to study the cause of vulnerability to HIV/AIDS and factors for migration 28 1.4 Introduction to Fuzzy Associative Memories (FAM) 35 Chapter Two ON A NEW CLASS OF N-ADAPTIVE FUZZY MODELS WITH ILLUSTRATIONS 2.1 On a new class of n-adaptive fuzzy models with illustrations 39 2.2 Some special n-adaptive models 49 Chapter Three USE OF 2-ADAPTIVE FUZZY MODEL TO ANALYZE THE PUBLIC AWARENESS OF HIV/AIDS 3.1 Study of the psychological and social problems the public have about HIV/AIDS patients using CETD matrix 52 3.2 Use of 2 adaptive fuzzy model to analyze the problem 72 Chapter Four PUBLIC ATTITUDE AND AWARENESS ABOUT HIV/AIDS 4.1 Introduction 77 4.2 Interviews 81 Chapter Five CONCLUSIONS 167 Appendix 1. Questionnaire 177 2. Table of Statistics 191 3. C-program for CETD and RTD Matrix 198 4. C-program for FRM 202 5. C-program for CFRM 2...

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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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Scientia Magna is published annually in 200-300 pages per volume and 1,000 copies on topics such as mathematics, physics, philosophy, psychology, sociology, and linguistics....

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0. In 1999, the second author of this remarks published a book over 30 of Smarandache's problems in area of elementary number theory (see [1, 2]). After this, we worked over new 20 problems that we collected in our book [28]. These books contain Smarandache's problems, described in [10, 16]. The present paper contains some of the results from [28]. In [16] Florentin Smarandache formulated 105 unsolved problems, while in [10] C.Dumitresu and V. Seleacu formulated 140 unsolved problems of his. The second book contains almost all the problems from [16], but now each problem has unique number and by this reason in [1, 28] and here the authors use the numeration of the problems from [10]. In the text below the following notations are used....

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V. Mladen and T. Krassimir : Remarks on some of the Smarandache's problem. Part 2 1 W. Kandasamy : Smarandache groupoids 27 L. Ding : On the primitive numbers of power P and its mean value properties 36 D. Torres and V. Teca : Consecutive, reversed, mirror, and symmetric Smarandache sequence of triangular numbers 39 D. Ren : On the square-free number sequence 46 T. Ramaraj and N. Kannappa : On ¯nite Smarandache near-rings 49 X. Kang : Some interesting properties of the Smarandache function 52 L. Mao : On Automorphism Groups of Maps, Surfaces and Smarandache Geometries 55 L. Ding : On the mean value of Smarandache ceil function 74 M. Le : An equation concerning the Smarandache function 78 M. Bayat, H. Teimoori and M. Hassani : An extension of ABC-theorem 81 J. Ma : An equation involving the Smarandache function 89 C. Chen : Inequalities for the polygamma functions with application 91 W. Vasantha and M. Chetry : On the number of Smarandache zero-divisors and Smarandache weak zero-divisors in loop rings 96 M. Le : The function equation S(n) = Z(n) 109 Z. Li : On the Smarandache Pseudo-number Sequences 111 D. Mehendale :...

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This book was intended to discuss some paradoxes in Quantum Mechanics from the viewpoint of Multi-Valued-logic pioneered by Lukasiewicz, and a recent concept Neutrosophic Logic. Essentially, this new concept offers new insights on the idea of ‘identity’, which too often it has been accepted as given....

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2 Lukasiewicz Multi-Valued-logic: History and Introduction to Multi- Valued Algebra 2.1 Introduction to trivalent logic and plurivalent logic We all have heard of typical binary logic, Yes or No. Or in a famous phrase by Shakespeare: “To be or not to be.” In the same way all computer hardwares from early sixties up to this year are built upon the same binary logic. It is known that the Classical Logic, also called Bivalent Logic for taking only two values {0, 1}, or Boolean Logic from British mathematician George Boole (1815-64), was named by the philosopher Quine (1981) “sweet simplicity.” [57] But this typical binary logic is not without problems. In the light of aforementioned ‘garment analogue’, we can compare this binary logic with a classic black-and-white tuxedo. It is timeless design, but of course you will not wear it for all occasions. Aristotle himself apparently knew this problem; therefore he introduced new terms ‘contingency’ and ‘possibility’ into his modal logic [5]. And then American logician Lewis first formulated these concepts of logical modality. ...

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Contents Foreword 6 1 Introduction: Paradoxes, Lukasiewicz, Multi-Valued logic 7 2 Lukasiewicz Multi-Valued Logic: History and Introduction to Multi-Valued Algebra 10 2.1. Introduction to trivalent logic and plurivalent logic 10 2.2. History of Lukasiewicz and Multi-Valued Logic 12 2.3. Introduction to Multi-Valued Algebra, Chang’s Notation 15 2.4. Linkage between Multi-Valued Logic and Quantum Mechanics 15 2.5. Exercise 17 3 Neutrosophy 25 3.1. Introduction to Neutrosophy 25 3.2. Introduction to Non-Standard Analysis 26 3.3. Definition of Neutrosophic Components 27 3.4. Formalization 28 3.5. Evolution of an Idea 30 3.6. Definition of Neutrosophic Logic 31 3.7. Differences between Neutrosophic Logic and IFL 32 3.8. Operations with Sets 33 3.9. Generalizations 34 4 Schrödinger Equation 39 4.1. Introduction 39 4.2. Quantum wave dynamics and classical dynamical system 43 4.3. A new derivation of Schrödinger-type Equation 45 5 Solution to Schrödinger’s Cat Paradox 47 5.1. Standard interpretation 47 5.2. Schrödinger’s Cat Paradox 48 5.3. Hidden-variable hypothesis 50 5.4. Hydrodynamic viewpoint and diffusion i...

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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 Generalized belief functions . . . . . . 20 1.2.4 The classic DSm rule of combination . . . . . 21 1.2.5 The hybrid DSm rule of combination . . . . . 22 1.2.6 Examples of combination rules . . . . . . 24 1.2.7 Fusion of imprecise beliefs . . . . . . 29 1.3 Proportional Conflict Redistribution rule . . . . . 33 1.3.1 PCR formulas . . . . . . . . . 34 1.3.2 Examples . . . . . . . . . 35 1.3.3 Zadeh’s example . . . . . . . . 39 1.4 Uniform and partially uniform redistribution rules . . . 41 1.5 RSC Fusion rules . . . . . . . . . 43 1.6 The generalized pignistic transformation (GPT) . . . . 45 1.6.1 The classical pignistic transformation . . . . 45 1.6.2 Notion of DSmcardinality . . . . . . 46 1.6.3 The Generalized Pignistic Transformation . . . 47 1.7 The DSmP transformation . . . . . . . 48 1.7.1 The Probabilistic In...

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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 investigations while, Part 2 presents several applications of this new theory. Some ideas in this book are still under current development or improvements, but we think it is important to propose them in order to share ideas and motivate new debates with people interested in new reasoning methods and information fusion. So, we hope that this second volume on DSmT will continue to stir up some interests to researchers and engineers working in data fusion and in artificial intelligence....

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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 rule . . . . . . . . . 6 1.2.3 The disjunctive rule . . . . . . . . . 8 1.2.4 Dempster’s rule of combination . . . . . . . 8 1.2.5 Smets’ rule of combination . . . . . . . 9 1.2.6 Yager’s rule of combination . . . . . . . 9 1.2.7 Dubois & Prade’s rule of combination . . . . . . 9 1.2.8 The hybrid DSm rule . . . . . . . . 10 1.3 The general weighted operator (WO) . . . . . . . 11 1.4 The weighted average operator (WAO) . . . . . . . 12 1.4.1 Definition . . . . . . . . . 12 1.4.2 Example for WAO . . . . . . . . . 13 1.4.3 Limitations of WAO . . . . . . . . . 13 1.5 Daniel’s minC rule of combination . . . . . . . 14 1.5.1 Principle of the minC rule . . . . . . . 14 1.5.2 Example for minC . . . . . . . . . 15 1.6 Principle of the PCR rules . . . . . . . . . 20 1.7 The...

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In The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by combinatorialization, implicated in the foreword of Chapter 5 of my book Automorphism groups of Maps, Surfaces and Smarandache Geometries (USA, 2005). This conjecture is essentially a philosophic notion for developing mathematical sciences of 21st century, which means that we can combine different fields into a union one and then determines its behavior quantitatively. It is this notion that urges me to research mathematics and physics by combinatorics, i.e., mathematical combinatorics beginning in 2004 when I was a post-doctor of Chinese Academy of Mathematics and System Science. It finally brought about me one self-contained book, the first edition of this book, published by InfoQuest Publisher in 2009. This edition is a revisited edition, also includes the development of a few topics discussed in the first edition....

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1.5 ENUMERATION TECHNIQUES 1.5.1 Enumeration Principle. The enumeration problem on a finite set is to count and find closed formula for elements in this set. A fundamental principle for solving this problem in general is on account of the enumeration principle: For finite sets X and Y , the equality |X| = |Y | holds if and only if there is a bijection f : X → Y . Certainly, if the set Y can be easily countable, then we can find a closed formula for elements in X....

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Contents Preface to the Second Edition . . . . . . . . . . . . . . . . . . . i Chapter 1. Combinatorial Principle with Graphs . . . . . . . . . . 1 1.1 Multi-sets with operations. . . . . . . . . . . . . . . . . . . . .2 1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Multi-Set . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.2 Multi-posets . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Partially ordered set . . . . . . . . . . . . . . . . . . . . .11 1.2.2 Multi-Poset . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Countable sets . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Countable set . . . . . . . . . . . . . . . . . . . . 16 1.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . .18 1.4.2 Subgraph . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.3 Labeled graph. . . . . . ...

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This book is devoted to an emerging branch of Information Fusion based on new approach for modeling the fusion problematic when 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 theoretical investigations while Part 2 presents several applications of this new theory. We hope that this first book on DSmT will stir up some interests to researchers and engineers working in data fusion and in artificial intelligence. Many simple but didactic examples are proposed throughout the book. As a young emerging theory, DSmT is probably not exempt from improvements and its development will continue to evolve over the years. We just want through this book to propose a new look at the Information Fusion problematic and open a new track to attack the combination of information....

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Preamble xi Prefaces xiii I Advances on DSmT 1 1 Presentation of DSmT 3 1.1 Introduction . . . . . 3 1.2 Short introduction to the DST . . . . 5 1.2.1 Shafer’s model and belief functions . . . 5 1.2.2 Dempster’s rule of combination . . . 5 1.2.3 Alternatives to Dempster’s rule of combination . . . 6 1.2.4 The discounting of sources of evidence . . . 10 1.3 Foundations of the DSmT . . . . 11 1.3.1 Notion of free and hybrid DSm models . . . 11 1.3.2 Notion of hyper-power set D_ . . . 13 1.3.3 Generalized belief functions . . . . 15 1.3.4 The classic DSm rule of combination . . . 16 1.3.5 The hybrid DSm rule of combination . . . 17 1.3.6 On the refinement of the frames . . . 18 1.3.7 On the combination of sources over different frames . . . 20 1.4 Comparison of different rules of combinations . . . 21 1.4.1 First example . . . . . 21 1.4.2 Second example . . . . 25 1.4.3 Third example . . . . 26 1.4.4 Fourth example . . . . 27 1.4.5 Fifth example . . . . . 27 1.5 Summary . . . . . 29 1.6 References . . . . . 31 2 The generation of hyper-power sets 37 2.1 Introduction . . . . . 37 2.2 Definition of hyper-po...