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...门学科最初是从研究整数开始的, 所以叫做整数论. 后来整数 论又进一步发展, 就叫做数论了. 确切的说, 数论就是一门研究整数性质 的学科. 它是最古老的数学分支. 按照研究方法来说, 数论可以分成初等 数论, 解析数论, 代数数论, 超越数论, 计算数论, 组合数论等. Foreword Number theory, this discipline was originally started from the study integer, so called Number Theory. Later integer on further development of number theory called it. Rather, number theory is an integer nature of disciplines...

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...Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indete...

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...eBooks-otherformats.htm ISBN: 1931233-96-9 Standard Address Number: 297-5092 Printed in the United States of America 3 ... ...NCES 290 INDEX 312 ABOUT THE AUTHORS 316 5 Preface Fuzzy theory is one of the best tools to analyze data, when the data under s... ...pervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy.... ...l women in Tamil Nadu. Chapter 2 is an analysis of the situation using fuzzy theory in general and Fuzzy Relational Maps (FRM) in particular. The F... ...terns. Neural network models represent prototype patterns as vectors of real numbers. So we can alternatively view learning as a form of adaptive v... ...d range spaces. The FRMs can take edge values from the set {–1, 0, 1}. If the number of 19 concepts in the domain space is n and that of the range... ...asamy and daughters Meena and Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In t... ...ded eight Ph.D. scholars in the different fields of non- associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and appl... ...d a Book Series, consisting of ten research books on the topic of Smarandache Algebraic Structures which were published by the American Research Pr...

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...The author's motivating interest the past few years has been the development and dissemination of G-dimensional theory (GDT). K. Toshihara, from Japan, proposed the concept of a paper defining G-Dimensional theory in relation to the Smarandache quantum paradoxes. After reviewing a draft, Dr. M. L. Perez, Editor of the SNJ, suggested th...

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... L. Stephen Young G-DIMENSIONAL THEORY & THE SMARANDACHE QUANTUM PARADOXES: Comparative Logic and Mod... ...& THE SMARANDACHE QUANTUM PARADOXES: Comparative Logic and Modern Quantum Theory Table 1A. Elliptic Pa... ...Research Press Rehoboth 2001 2 L. Stephen Young G-DIMENSIONAL THEORY & THE SMARANDACHE QUANTUM PARADOXES: Comparative Logic and Mod... ...unm.edu/~smarandache/physics.htm ISBN: 1-931233-46-2 Standard Address Number 297-5092 Printed in the United States of America 4 G-... ...ht seem due solely to semantic vagueness in defining the maximum or minimum number of wheat grains constituting a "heap", however, as noted in the St... ...le quantum states (on/off); yet the computer as a whole has a near-infinite number of possible states. This then serves an example of apparent non- p... ...+ v 2 a 2 ) 1/2 . (9.1) The (x 2 ) terms cancel; and through algebraic substitution the equation is simplified; σ e = b -1 (1 + v... ...dx, -cot (Φ) = b(-x)(a 2 – x 2 ) -1/2 . (10.1) Through simple algebraic and trigonometric identities we find x, x = bsec (θ) cos (Φ)...

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...The theory of loops (groups without associativity), though researched by several mathematicians has not found a sound expression, for books, be it research level or otherwise, solely dealing with the properties of loops are absen...

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...he/eBooks-otherformats.htm ISBN: 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS ... ... n c e s 113 I n d e x 119 5 PREFACE The theory of loops (groups without associativity), though researched by severa... ...the properties of loops are absent. This is in marked contrast with group theory where books are abundantly available for all levels: as graduate te... ...d as advanced research books. The only three books available where the theory of loops are dealt with are: R. H. Bruck, A Survey of Binary System... ...n and examination of the Smarandache loops based on a paper about Special Algebraic Structures by Florentin Smarandache. As a result, this book does... ...kground in algebra and more specifically a strong foundation in loops and number theory. 6 This book introduces over 75 Smarandache concept... ...the introduction of a new class of loops of even order. We prove that the number of different representations of right alternative loop of even orde... ... I got started in this endeavour of writing research books on Smarandache algebraic notions. I dedicate this book to my parents, Worathur Balasub... ... elements of L the sup(x, y)and inf(x, y) exist. DEFINITION 1.3.6: An (algebraic) lattice (L, ∩, ∪) is a non empty set L with two binary operatio...

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...introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use f...

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... 1 Basic Neutrosophic Algebraic Structures and Their Application to Fuzzy and Neutrosophic Mo... ...Books-otherformats.htm ISBN: 1-931233-87-X Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENT... ... f a c e 5 Chapter One INTRODUCTION TO SOME NEUTROSOPHIC ALGEBRAIC STRUCTURES 1.1 Neutrosophic Fields 7 1.2 Neutrosophic Vect... ...eutrosophic Vector spaces 8 Chapter Two SOME BASIC RESULTS ON GRAPH THEORY AND THEIR APPLICATION TO FUZZY MODELS 2.1 Some basics on Gr... ... ABOUT THE AUTHORS 149 5 Preface Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic th... ...ophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to i... ...n almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Mo... ...NITION 1.2.6: Let G(I) be a neutrosophic vector space over the field K. The number of elements in the neutrosophic basis is called the neutrosophic ... ...be a strong neutrosophic vector space over the neutrosophic field K(I). The number of elements in the strong neutrosophic basis is called the strong...

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...Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applicat...

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...SBN: 1931233-95-0 HEXIS Publ. Hse., Phoenix, Arizona Standard Address Number: 297-5092 Printed in the United States of America 3 C... ...UCTION TO NEUTROSOPHIC BIMATRICES 3.1. Introduction to some neutrosophic algebraic structures 139 3.2. Neutrosophic bimatrices 148 3.3. Neutro... ... INDEX 175 ABOUT THE AUTHORS 181 5 Preface Matrix theory has been one of the most utilised concepts in fuzzy models and ne... ...eral real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable compo... ...r bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in the constructio... ...d the notion of bimatrices. These bimatrices in general cannot be given an algebraic operation like bimatrix addition and bimatrix multiplication, ... ...ave started with. Only this incompleteness led to the invention of complex number or the imaginary number “i”. (Complex number i = 1 − ). But, aft... ...ive with zero, R the reals, Q the set of rationals and C the set of complex numbers. Z n will denote the set of integers modulo n. 8 1.1 Defi... ...the study of bimatrices for a while in order to develop some aspects of the theory of determinants which will be needed in our further investigatio...

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...Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences Editor’s Note This ... ...Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences Editor’s Note This book ar... ...PIDS, IOWA 42402 USA GENERALIZED PARTITIONS AND SOME NEW IDEAS ON NUMBER THEORY AND SMARANDACHE SEQUENCES H... ...OWA 42402 USA GENERALIZED PARTITIONS AND SOME NEW IDEAS ON NUMBER THEORY AND SMARANDACHE SEQUENCES Hexis P... ...la and Charles Ashbacher ISBN: 1-931233-34-9 Standard Address Number: 297-5092 Printed in the United States of America 3 Editor... ...e On Maohua Le’s Proof of Murthy’s Conjecture On Reciprocal Partition Theory ………………. 18 1.4 Generalization of Partition Functions, Introducti... ...+1) < log(1 + 1/r) < 1/r. Summing up for r = 1 to n+1 and applying some algebraic rearrangement yields (1). Applying (1), we get the following ... ...---- (B) This wrong lucky method is applicable in proving the following algebraic identity. a 3 + b 3 + c 3 - 3 abc = ( a + b + c ) ( a...

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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 thre...

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...ME 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 READIN...

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...OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING WITH HIV/AIDS USING FUZZY THEORY AND NEUTROSOPHIC COGNITIVE MAPS With Specific Reference To Rura... ... OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING WITH HIV/AIDS USING FUZZY THEORY AND NEUTROSOPHIC COGNITIVE MAPS With Specific Reference To Rura... ...oks-otherformats.htm ISBN: 1-931233-83-7 Standard Address Number: 297-5092 Printed in the United States of America Contents Pre... ... individuals/surviving members with AIDS. We were also taken aback by the number of cases where the migrant workers (who had visited CSWs) did not a... ...he people with HIV/AIDS towards religion. We found out that a substantial number, of men had converted from their native religion to Christianity. T... ... are used in the study of the problem. In chapter III yet another fuzzy theory technique namely the bi-directional associative memories is used in... ... are Maximum Infected, Using Fuzzy Matrices In this section we give an algebraic approach to the problem of studying the psychological aspects an... ...y and daughters Meena and Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In the p... ...eight Ph.D. scholars in the different fields of non- associative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applicat...

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...The study of Smarandache Algebraic Structure was initiated in the year 1998 by Raul Padilla following a paper written by Florentin Smarandache called “Special Algebraic Structures”. In his research, Padilla treated the Smarandache algebraic structure...

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...eBooks-otherformats.htm ISBN: 1-931233-61-6 Standard Address Number: 297-5092 Printed in the United States of America 3 ... ... 5 The study of Smarandache Algebraic Structure was initiated in the year 1998 by Raul Padilla fol... ...ul Padilla following a paper written by Florentin Smarandache called “Special Algebraic Structures”. In his research, Padilla treated the Smarandach... ...ciative binary operation. Since then the subject has been pursued by a growing number of researchers and now it would be better if one gets a coheren... ... better if one gets a coherent account of the basic and main results in these algebraic structures. This book aims to give a systematic development o... ...d of algebraic structures like semigroup, group etc. and a good foundation in number theory. In Chapter 1 we just recall the basic notations and s... ...lgebraic structures like semigroup, group etc. and a good foundation in number theory. In Chapter 1 we just recall the basic notations and some imp... ...nger Verlag, (1958). 2. Ivan Nivan and H.S.Zukerman, Introduction to number theory, Wiley Eastern Limited, (1989). 3. W.B. Vasantha Kandasamy, ... ...d Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In the past decade she has guide...

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...The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief eco...

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...oks-otherformats.htm ISBN: 1931233-97-7 Standard Address Number: 297-5092 Printed in the United States of America 3 CON... ...PHIC LINEAR BIALGEBRA AND ITS APPLICATION 4.1 Some basic neutrosophic algebraic structures 131 4.2 Smarandache neutrosophic linear bialgebr... ... INDEX 229 ABOUT THE AUTHORS 238 5 Preface The algebraic structure, linear algebra happens to be one of the subjects whi... ...elds itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief econ... ...omic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerfu... ...if v 1 , v 2 ∈ V, v 1 .v 2 ∈ V. The same set of vectors may be part of a number of distinct vectors. 9 We simply by default of notation just sa... ...e reader to prove. Here we take vector spaces only over reals i.e., real numbers. We are not interested in the study of these properties in case o... ...in this direction. 1.3.2 Smarandache Leontief economic models Matrix theory has been very successful in describing the interrelations between ... ...ween the industries in the economy 32 under considerations. Using matrix theory we evaluate certain parameters. The basic equations of the input-...

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...ain motivation and desire for writing this book, is the direct appreciation and attraction towards the Smarandache notions in general and Smarandache algebraic structures in particular. The Smarandache semigroups exhibit properties of both a group and a semigroup simultaneously. This book is a piece of work on Smarandache semigroups and assumes the reader to have a good ba...

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...e/eBooks-otherformats.htm ISBN: 1-931233-59-4 Standard Address Number: 297-5092 Printed in the United States of America ... ...liminary results 14 2.4 Subgroups 15 3. Some Classical Theorems in Group Theory 3.1 Lagrange's Theorem 21 3.2 Cauchy's Theorems 22 3.3 ... ...tion and attraction towards the Smarandache notions in general and Smarandache algebraic structures in particular. The Smarandache semigroups exhibi... ...marandache semigroups and assumes the reader to have a good background on group theory; we give some recollection about groups and some of its proper... ...he concepts are completely understood by illustrating with examples and a great number of problems. Solutions to all the problems need extraordinary... ...e used in this book. Chapters 2 and 3 gives most of the basic concepts on group theory and results in group theory which have been used in this text... ... Universidade do Porto, Portugal, 9-11 July 1998. 2. R. Padilla, Smarandache Algebraic Structures, Smarandache Notions Journal, USA, Vol.9, No. 1... ...ions Journal, USA, Vol.9, No. 1-2, 36-38, (1998). 3. R. Padilla. Smarandache Algebraic Structures, Bulletin of Pure and Applied Sciences, Delhi, ... ...2, 3) be the set S. Let S(3) denote the set of all mappings of S to itself. The number of elements in S(3) is 27 = 3 3 . 10 Example 1.2.2: Let...

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...This book contains short notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essen...

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... Hungary. ISBN: 1-931233-51-9 Standard Address Number 297-5092 Printed in the United States of America "...It is just... ...rt notes or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric the- or... ...s or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric the- orems; Di... ...s; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals esse... .... . . . . . . . . . . . . 241 8 On certain new conjectures in prime number theory . . . . . . . . . . . . . 243 9 On certain conjectures by Russo . . ... ...ve made it clear that there exists a very intimate correlation between the Theory of numbers and other departments of Mathematics, not excluding geome... ...2r R gives relation (3). For the proof of (2) we shall apply the standard algebraic inequalities 3(xy +xz +yz) (x +y +z) 2 3(x 2 +y 2 +z 2 ): Now,... ...e, by using a = 2R sinA, etc. that 14 (v) X a A > 12 R. By (ii) and the algebraic inequality (x +y +z) 1 x + 1 y + 1 z 9, clearly, one can ob... ... the segmentBC. By the triangle inequality in triangle 27 APA 1 , and the algebraic inequality (x +y) 2 3 2 x 2 + 3y 2 (equivalent with (2y z) 2 ...

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...ral researchers around the world working on Smarandache notions have systematically carried out this study. This is the first book on the Smarandache algebraic structures that have two binary operations. Semirings are algebraic structures with two binary operations enjoying several properties and it is the most generalized structure — for all rings and fields are semirings...

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...ooks-otherformats.htm ISBN: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed i... ...matically carried out this study. This is the first book on the Smarandache algebraic structures that have two binary operations. Semirings are al... ...emifields and semivector spaces based on some paper by F. Smarandache about algebraic structures and anti-structures. We assume at the outset that t... ...versidade do Porto, Portugal, 9-11 July 1998. 2. R. Padilla, Smarandache Algebraic Structures, Smarandache Notions Journal, USA, Vol.9, No. 1-2, ... ...∗ b = b ∗ a for all a, b ∈ S we say S is a commutative semigroup. If the number of elements in the semigroup S is finite we say S is a finite semi... ... G a • b = b • a. A group, which is not abelian, is called non-abelian. The number of distinct elements in G is called the order of G; denoted by ο ... ... to prove the result). Supplementary Reading 1. Birkhoff, G. Lattice Theory. American Mathematical Society, Providence, R.I., 1967. 2. Bi... ...n Applied Algebra. Mc-Graw Hill, New York, 1970. 3. Gratzer, G. Lattice Theory. Freeman, San Francisco, 1971. 4. Hall, Marshall. Theory of Gr... ..., 1975. 6. Lang, S. Algebra. Addison-Wesley, 1967. 7. Nagata, M. Field Theory. Marcel Dekker, NewYork-Basel, 1977. 8. Passman, D.S. Infinite ...

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... integers formed by concatenating the positive integers in specific ways. A question that is often asked about most of these sequences concerns the number of primes in the sequence. As Paul Erdos pointed out, determining how many primes are in sequences of this type is a problem that the current state of mathematical knowledge is unprepared to solve. In general, the se...

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...Using the bases from three through nine to form new sequences, we can then ask the same question regarding the number of primes in the sequence. That appears to be a question just as difficult as the one for base ten numbers, and is summarized in the following statement....

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...Contents.4 -- Chapter 1 Sequences Formed by Concatenating All the Positive Integers.5 -- Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers .44 -- Chapter 3 Smarandache Stereograms .70 3.1 -- The Stereogram.70 -- 3.2 Altering the Background.87 -- 3.3 Additional Stereograms .91 -- 3.4 Possibilities for Additional Figures .92 -- Chapter 4 Constants Involvin...

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.../~smarandache/eBooks-otherformats.htm ISBN: 1-931233-23-3 Standard Address Number: 297-5092 Printed in the United States of America 3 Preface This i... ...rs …………………. 5 Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers …………………. 44 Chapter 3 Smarandache Stereograms ………………….. 70 3.1 The ... ... A question that is often asked about most of these sequences concerns the number of primes in the sequence. As Paul Erdos pointed out, determining ho... ...the first four terms are 1, 12, 123, 1234, 12 which, when expressed as an algebraic formula , are 1, B + 2, B 2 + 2*B + 3, B 3 + 2*B 2 + 3*B+ 4. T... ...3 For B > 4, the first four terms are 1, 12, 123, 1234 which, expressed as algebraic formulas , are 1, B + 2, B 2 + 2*B + 3, B 3 + 2*B 2 + 3*B+ 4. ... ...e Fibonacci and Lucas sequences. References 1. Unsolved Problems in Number Theory, edited by Richard K. Guy, Springer, 2004, ISBN 0387208607. 70 Chap... ...hat kind of stereograms can be created by iterating combinations of number theory functions and plotting the number of iterations? 3.2 Altering the Ba...

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...not directly related to but should fall within the scope of Smarandache type problems. They are 1. L. Liu and W. Zhou, On conjectures about the class number of binary quadratic forms; 2. W. Liang, An identity for Stirling numbers of the second kind; 3. Y. Wang and Z. Sheng, Two formulas for x^n in terms of Chebyshev polynomials. Other papers are concerned with the number-t...

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...On Algebraic Multi-Vector Spaces Abstract A Smarandache multi-space is a union of n spaces A1;A2;An with some additional conditions hold. Combining these Smarandache multi-spaces with linear vector spaces in classical linear al...

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...L. Mao : On Algebraic Multi-Vector Spaces 1 L. Liu and W. Zhou : On conjectures concerning class number of binary quadratic forms 7 W. Zhai and H. Liu : On square-free primitive roots mod p 15 X. Pan : A new limit theorem involving t...

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...fining two binary operations on it. But on the contrary when we want to define or study or even introduce a non-associative ring we need two separate algebraic structures say a commutative ring with 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algebras and Jordan algebras are m...

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...s.htm ISBN: 1-931233-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...r forms 7 1.2 Smarandache vector spaces 8 1.3 Basic definitions of other algebraic structures 11 2. LOOP-RINGS AND SMARANDACHE LOOP-RINGS 2... ...ne or study or even introduce a non-associative ring we need two separate algebraic structures say a commutative ring with 1 (or a field) together w... ...se two algebras heavily exploits the concepts of vector spaces and matrix theory we do not venture to approach Smarandache Lie algebras or Smarandac... ... way. This book is distinct and different from other non-associative ring theory books as the properties of associative rings are incorporated and s... ...act structures rings or fields is a composite structure consisting of two algebraic structures called fields and vectors, which forms an additive ab... ...) is a mapping [x, y] → (x, y) of V × V into the set of (real or complex) numbers such that (ax + bx', y) = a (x, y) + b (x', y) and (x, a... ...tor space V is a mapping [x, y] → (x, y) of V × V into the set of complex numbers such that (x, y) = ) x , y ( and (ax + bx', y) = a (x, y) + b (x',... ...of orthogonal ideals. This notion will find its place in algebraic coding theory. Now we proceed on to define the concept of Smarandache orthogonal ...

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...r 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 enumera...

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...rphism 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 N...

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...r 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 enumera...

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...roperties 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 . . . . . . . . . . . . . . . . . . . . ....

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...In this book, we study the subject of Smarandache Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field founded by Florentine Smarandache, is alike to that of Fuzzy ...

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...ture on the cover is a simple graphic illustration depicting the classical algebraic structures with single binary operations and their Smarandache a... ...ft, composed of concentric circles, depicts the traditional conception of algebraic structures, and the pictures of the right, with their liberal in... ...ures of the right, with their liberal intersections, describe Smarandache algebraic structures. In fact, Smarandache Algebra, like its predecessor, ... ...oks-otherformats.htm ISBN: 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... .... In its trajectory of stupendous growth, it has also come to include the theory of fuzzy algebra and for the past five decades, several researchers... ...y the subject of Smarandache Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that chall... ...ns, a field founded by Florentine Smarandache, is alike to that of Fuzzy Theory: both the fields imperatively questioned the dogmas of classical ma... ...f p ) x ( 3 2 1 Clearly µ is a fuzzy subgroup of a group G and o(µ ) = number of elements of the set {x ∈ G | µ (x) = µ (e)} = number of elements... ...by an example. 25 Example 1.2.10: Let X = N be the set of all natural numbers and µ be defined by µ (x) = x 1 for every x ∈ X. Now consider th...