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Aim of this book is to present basic mathematics that is needed by computer scientists The reader is not expected to be a mathematician and we hope you will find what follows to be useful....

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Teoria Analitics a Numerelor reprezents pentru mine 0 pasiune. Rezultatele expuse mai departe constituie rodul catorva ani buni de cercetsri si csutsri. Lucrarea de fass se compune din 9 articole, publicate toate prin reviste de matematics romanesti sau strsine, iar unele prezentate chiar la congrese si conferinse nasionale cat si internasionale [vezi "Lista publicasiilor autorului pe tema tezei"). Ea se structureazs in patru capitole: - in primele trei capi tole se introduc noi funsii in teoria numerelor, se studiazs proprietssile lor, probleme nerezol vate legate de ele, implicasii in lumea stiinsific! internasional! (ce alsi matematicieni au abordat nosiunile acestea), conexisi cu alte funcsii bine stiute, importansa rezultatelor obsinute: - in ul timul capitol se aduc contribusii la studierea unei- funsii cunoscute in teoria numerelor (totient sau phi a lui Euler), in principal referitoare la conjectura lui carmichael. [Exist! referinse particulare dup! fiecare paraqraf (articol), iar referinse generale in finalul tezei.)...

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Ne bucuram exprimAnd mult-umirile noast.re f'at-a de t.ot-i acei.cunescut- i sau mai put-in cunoscut-i.care de-a Iungul aniler ne-au ajut.at. sa ajungem la aceast.a cart.e.Sunt. mult-i.sunt. f'oart.e mult-i cei care ne-au ajut.at. ... Unii ne-au dat. sugest.ii.alsii ne-au oferit. idei. uneori ne-am st.raduit. impreunA sa descif'ram un amanunt. nelamurit.. alleori Invst-am din Int.rebarile mest.esugit.e sau poale chiar naive ale inlerloculorilor nost.rii:...

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Mi-a sosit, de curand, cu po~ta aeriana, 0 carte din Amcrica. Era expcdiata de Florentin Smarandache, din Phoenix. Arizona. un valcean de-al nostru din Balce~ti, fugit Inainte de revolutic prin Turcia ~i "aclimatizat" In Statele Unite. Cinc e Florentin Smarandache: profesor de matematica, aut or a peste cincisprczece volume. poet. initiator al curentului paradoxise acum cercetator la Corporatia de computere Honeywell. Arizona. L-am cunoscut bine pe acest domn In perioada studentiei, cand incerca sa publice la revista studentcasca pe care 0 conduceam, tot felul de jocuri matematice. pe care nici nu Ie Intelegeam, nici nu Ie gustam, dar pe care Ie-am publicae in final. de dragul omului....

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In 1996 the author wrote reviews for "Zentralblatt fUr Mathematik" for books [11 and [21 and this was him first contact of with the Smarandache's problems. In [1] Florentin Smarandache formulated 105 unsolved problems, while in [21 C. Dumitrescu and V. Seleacu formulated 140 unsolved problems. The second book contains almost all problems from [11, but now everyone problem has unique number and by this reason the author will use the numeration of the problems from [2]. Also, in [2] there are some problems, which are not included in [1]. On the other hane, there are problems from [1], which are not included in [2]. One of them is Problem 62 from [1], which is included here under the same number. In the summer of 1998 the author found the books in his library and for a first time tried to solve a problem from them. After some attempts one of the problems was solved and this was a power impulse for the next research. In the present book are collected the 27 problems solved by the middle of February 1999....

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Preface 5 -- 1. On The 4-Th Smarandache's Problem 7 -- 2. On The 16-Th Smarandache's Problem 12 -- 3. On The 22-Nd, The 23-Rd, And The 24th -- Smarandache's Problems 16 -- 4. On The 37-Th And The 38th -- Smarandache's Problems 22 -- 5. On The 39-Th, The 40-Th, The 41st, And -- The 42-Nd Smarandache's Problems 27 -- 6. On The 43-Rd And 44-Th Smarandache's -- Problems 33 -- 7. On The 61-St, The 62-Nd, And The 63red -- Smarandache's Problems 38 -- 8. On The 97-Th, The 98-Th, And The 99th -- Smarandache's Problems 50 -- 9. On The 100-Th, The 101-St, And The 102nd -- Smarandache's Problems 57 -- 10. On The Ll7-Th Smarandache's Problem 62 -- Ll. On The Ll8-Th Smarandache's Problem 64 -- 12. On The 125-Th Smarandache's Problem 66 -- 13. On The I26-Th Smarandache's Problem 68 -- 14. On The 62-Nd Smarandache's Problem 71 -- Is. Conclusion 74 -- 16. Appendix 76 -- References 83 -- Curriculum Vitae Of K. Atanassov 86 --...

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As if it wanted to be in accordance with the paradoxism and even to confirm it, the way the mathematician writer Florentin Smarandache is received is ... paradoxical. Tens of books, studies and articles have already been written about both, scientist and smarandachism - the name given to the movement by Ion Rotaru and Titu Popescu. However the paradoxist number 1 in the world is less known in his native country. At the same time some researchers ( Jacques Sarthou, Dan Tarchila, Doru Motoc and even... Ion Rotaru) declared him a man of genius (or close by!), while others ignored him or they haven’t yet heard about him. Without being disturbed by any of the two contradictory tendencies, the Oltenian- Romanian settled in New Mexico creates further on important works in the two large fieldsliteratures and mathematics (but also in other art and scientific fields)....

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A cycle based approach to stock and options trading

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The main purpose of this paper is using the elementary method to study the mean value properties of the Smarandache function, and give an interesting asymptotic formula....

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x1. Introduction In reference [1], the Smarandache Sum of Composites Between Factors function SCBF(n) is defined as: The sum of composite numbers between the smallest prime factor of n and the largest prime factor of n. For example, SCBF(14)=10, since 2£7 = 14 and the sum of the composites between 2 and 7 is: 4 + 6 = 10. In reference [2]: A number n is called simple number if the product of its proper divisors is less than or equal to n. Let A denotes set of all simple numbers. That is, A = f2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 13; 14; 15; 17; 19; 21. ...

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On the Smarandache function and square complements 1 Zhang Wenpeng , Xu Zhefeng On the integer part of the k -th root of a positive integer 5 Zhang Tianping , Ma Yuankui Smarandache “Chopped” NN and N + 1N¡1 9 Jason Earls The 57 -th Smarandache’s problem II 13 Liu Huaning , Gao Jing Perfect Powers in Smarandache n - Expressions 15 Muneer Jebreel Karama On the m -th power residue of n 25 Li Junzhuang and Zhao Jian Generalization of the divisor products and proper divisor products sequences 29 Liang Fangchi The science of lucky sciences 33 Jon Perry Smarandache Sequence of Unhappy Numbers 37 Muneer Jebreel Karama On m -th power free part of an integer 39 Zhao Xiaopeng and Ren Zhibin On two new arithmetic functions and the k -power complement number sequences 43 Xu Zhefeng Smarandache Replicating Digital Function Numbers 49 Jason Earls On the m -power residues numbers sequence 53 Ma Yuankui , Zhang Tianping Smarandache Reverse Power Summation Numbers 57 Jason Earls Some Smarandache Identities 59 Muneer Jebreel Karama On the integer part of a positive integer’s k -th root 61 Yang Hai , F...

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Volumul de faţă adună o mână de studii, articole şi consemnări din presa românească despre scriitorul şi omul de ştiinţă Florentin Smarandache, mişcarea lite-rar-artistică pe care a iniţiat-o (Paradoxismul) şi una dintre teoriile pe care le-a dezvoltat (Viteza Supra-luminală); câteva mesaje adresate acestuia şi o addenda ilustrată vin să contureze peisajul ştiinţific, artistic şi uman în care se mişcă unul dintre cei mai prolifici, mai interesanţi şi mai apreciaţi români ai momentului. This volume gathers a handful of studies, articles and records of Romanian press about the writer and scientist Smarandache rare literary and artistic movement that initiated it (Paradoxism) and one of the theories he developed (Over-luminal speed) address and a few messages come to shape the landscape addenda illustrated scientific, artistic and human that moves one of the most prolific, most interesting and Romanian appreciate the moment....

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Florentin Smarandache este unul dintre cei mai cunoscuți scriitor români în afara ţării natale. Activităţile sale literare şi ştiinţifice sunt impresionante. Peste 3.000 de pagini de jurnal nepublicate (datorate călătoriei şi traiului în jurul lumii, dornic să ştie şi să întâlnească oameni şi să studieze diferite culturi). La Arizona State University, Hayden Library, în Tempe, Arizona, există o mare colecţie specială numită “The Florentin Smarandache Papers” (care are mai mult de 30 de picioare liniare / linear feet) cu cărţi, reviste, manuscrise, documente, CD-uri, DVD-uri, benzi video realizate de el sau despre opera sa. Altă colecţie specială “The Florentin Smarandache Papers” se află la The University of Texas, la Austin, Archives of American Mathematics (în cadrul Centrului American de Istorie). ...

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NORMA SMARANDACHE ..................................... 7 Florentin Smarandache: Fişă de dicţionar ..................................... 9 Nominalizarea lui Florentin Smarandache pentru Premiul Nobel pentru literatură (Geo Stroe)..... 18 Fizicianul care l-a contrazis pe Einstein (Florin Grieraşu) ..................................... 25 FAŢETELE PARADOXULUI ..................................... 37 De la paradox la paradoxism (Titu Popescu) ............................... 38 De la multistructură şi multispaţiu la “recepţionarea multidimensională estetică şi paradoxistă Smarandache” (Ştefan Vlăduţescu) ..................................... 51 Teoria S-negării (Olga Popescu) ..................................... 54 Antiscrisori paradoxiste (Ion Segărceanu) ..................................... 57 Antologia a şasea paradoxistă în cotidianul internaţional (Marinela Preoteasa) ..................................... 61 MATEMATICA LITERELOR ..................................... 65 Libertatea s-a născut la mahala (Tudor Negoescu) ..................................... 66 Florentin Smarandache: poetul matematician (Ion M...

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A short collection of mathematical papers from various authors relating to Florentin Smaradache's theories.

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This dual will be studied in a separate paper (in preparation). 2. The additive analogues of the functions 5 and 5. are real variable functions, and have been defined and studied in paper [31. (See also one book [61, pp. 171-174). These functions have been recently further extended, by the use of Euler's gamma function, ill place of the factorial (see [1]). We note that in what follows, we could define also the additive analogues functions by the use of Euler's gamma function. However, we shall apply the more transparent notation of a factorial of a positive integer....

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Jozsef Sandor ON ADDITIVE ANALOGUES OF CERTAIN ARITHMETIC FUNCTIONS 29-32 M. Perez (Ed.) ON SOME SMARANDACHE PROBLEMS 33-38 M. Perez (Ed.) ON SOME PROBLEMS RELATED TO SMARANDACHE NOTIONS 39-40 G. Gregory (Ed.) GENERALIZED SMARANDACHE PALINDROME 41 M. Vassilev - lVlissana ON 15-TH SMARANDACHE'S PROBLEM 42-45 K. Atanassov ON THE SECOND SMARANDACHE'S PROBLEM 46-48...

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This book has six chapters. First chapter is introductory in nature. The new concept of non-associative semi-linear algebras is introduced in chapter two. This structure is built using groupoids over semi-fields. Third chapter introduces the notion of non-associative linear algebras. These algebraic structures are built using the new class of loops. All these non-associative linear algebras are defined over the prime characteristic field Zp, p a prime. However if we take polynomial non associative, linear algebras over Zp, p a prime; they are of infinite dimension over Zp. We in chapter four introduce the notion of groupoid vector spaces of finite and infinite order and their generalizations. Only when this study becomes popular and familiar among researchers several applications will be found. The final chapter suggests around 215 problems some of which are at research level. ...

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Ln(m) is a loop of order n + 1 that is Ln(m) is always of even order greater than or equal to 6. For more about these loops please refer [37]. All properties discussed in case of groupoids can be done for the case of loops. However the resultant product may not be a loop in general. For the concept of semifield refer [41]. We will be using these loops and groupoids to build linear algebra and semilinear algebras which are non associative. ...

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Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two NON ASSOCIATIVE SEMILINEAR ALGEBRAS 13 Chapter Three NON ASSOCIATIVE LINEAR ALGEBRAS 83 Chapter Four GROUPOID VECTOR SPACES 111 Chapter Five APPLICATION OF NON ASSOCIATIVE VECTOR SPACES / LINEAR ALGEBRAS 161 Chapter Six SUGGESTED PROBLEMS 163 FURTHER READING 225 INDEX 229 ABOUT THE AUTHORS 231 ...

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This book has six chapters. The first one is introductory in nature. Second chapter introduces complex modulo integer groupoids and complex modulo integer loops using C(Zn). This chapter gives 77 examples and forty theorems. Chapter three introduces the notion of nonassociative complex rings both finite and infinite using complex groupoids and complex loops. This chapter gives over 120 examples and thirty theorems. Forth chapter introduces nonassociative structures using complex modulo integer groupoids and quasi loops. This new notion is well illustrated by 140 examples. These can find applications only in due course of time, when these new concepts become familiar. The final chapter suggests over 300 problems some of which are research problems....

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THEOREM 2.1: Let G = {C(Zn), *, (t, u); t, u ∈ Zn} be a complex modulo integer groupoid. If H ⊆ G is such that H is a Smarandache modulo integer subgroupoid, then G is a Smarandache complex modulo integer groupoid. But every subgroupoid of G need not be a Smarandache complex modulo interger subgroupoid even if G is a Smarandache groupoid. Proof is direct and hence is left as an exercise to the reader. Example 2.28: Consider G = {C(Z8), *, (2, 4)}, a complex modulo integer groupoid....

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

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

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Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET NEUTROSOPHIC LINEAR ALGEBRA 13 2.1 Type of Neutrosophic Sets 13 2.2 Set Neutrosophic Vector Space 16 2.3 Neutrosophic Neutrosophic Integer Set Vector Spaces 40 2.4 Mixed Set Neutrosophic Rational Vector Spaces and their Properties 52 Chapter Three NEUTROSOPHIC SEMIGROUP LINEAR ALGEBRA 91 3.1 Neutrosophic Semigroup Linear Algebras 91 3.2 Neutrosophic Group Linear Algebras 113 Chapter Four NEUTROSOPHIC FUZZY SET LINEAR ALGEBRA 135 Chapter Five NEUTROSOPHIC SET BIVECTOR SPACES 155 Chapter Six NEUTROSOPHIC FUZZY GROUP BILINEAR ALGEBRA 219 Chapter Seven SUGGESTED PROBLEMS 247 FURTHER READING 277 INDEX 280 ABOUT THE AUTHORS 286 ...

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This book has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings, neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this book is that it gives 246 problems in Chapter four. Some of the problems are direct and simple, some little difficult and some can be taken up as a research problem....

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

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

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We in this book introduce the notion of pure (mixed) neutrosophic interval bisemigroups or neutrosophic biinterval semigroups. We derive results pertaining to them. The new notion of quasi bisubsemigroups and ideals are introduced. Smarandache interval neutrosophic bisemigroups are also introduced and analysed. Also notions like neutrosophic interval bigroups and their substructures are studied in section two of this chapter. Neutrosophic interval bigroupoids and the identities satisfied by them are studied in section three of this chapter. The final section of chapter one introduces the notion of neutrosophic interval biloops and studies them. Chapter two of this book introduces the notion of neutrosophic interval birings and bisemirings. Several results in this direction are derived and described. Even new bistructures like neutrosophic interval ring-semiring or neutrosophic interval semiring-ring are introduced and analyzed. Further in this chapter the concept of neutrosophic biinterval vector spaces or neutrosophic interval bivector spaces are introduced and their properties are described. In the third chapter we introduce the n...

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

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

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The first chapter is introductory in nature and gives a few essential definitions and references for the reader to make use of the literature in case the reader is not thorough with the basics. The second chapter deals with different types of neutrosophic bilinear algebras and bivector spaces and proves several results analogous to linear bialgebra. In chapter three the authors introduce the notion of n-linear algebras and prove several theorems related to them. Many of the classical theorems for neutrosophic algebras are proved with appropriate modifications. Chapter four indicates the probable applications of these algebraic structures. The final chapter suggests about 80 innovative problems for the reader to solve....

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Dedication 5 Preface 7 Chapter One INTRODUCTION TO BASIC CONCEPTS 9 1.1 Introduction to Bilinear Algebras and their Generalizations 9 1.2 Introduction to Neutrosophic Algebraic Structures 11 Chapter Two NEUTROSOPHIC LINEAR ALGEBRA 15 2.1 Neutrosophic Bivector Spaces 15 2.2 Strong Neutrosophic Bivector Spaces 32 2.3 Neutrosophic Bivector Spaces of Type II 50 2.4 Neutrosophic Biinner Product Bivector Space 198 Chapter Three NEUTROSOPHIC n-VECTOR SPACES 205 3.1 Neutrosophic n-Vector Space 205 3.2 Neutrosophic Strong n-Vector Spaces 276 3.3 Neutrosophic n-Vector Spaces of Type II 307 Chapter Four APPLICATIONS OF NEUTROSOPHIC n-LINEAR ALGEBRAS 359 Chapter Five SUGGESTED PROBLEMS 361 FURTHER READING 391 INDEX 396 ABOUT THE AUTHORS 402...

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This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research level....

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In this chapter we only indicate as reference of those the concepts we are using in this book. However the interested reader should refer them for a complete understanding of this book. In this book we define the notion of natural product in matrices so that we have a nice natural product defined on column matrices, m * n (m ≠ n) matrices. This extension is the same in case of row matrices. We make use of the notion of semigroups and Smarandache semigroups. Also the notion of semirings, Smarandache semirings, semi vector spaces and semifields are used....

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

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

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

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

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This book will mainly make part of the research results of current domestic and foreign scholars on Smarandache problems and unsolved problems into a book. Its main purpose is to introduce some of the research of Smarandache problems to readers, comprehensively and systematically, including the mean value of arithmetic functions, identities and inequalities, infinite series, the solutions of special equations, and put forward to some new interesting problems. We hope that the readers could be interested in these issues. At the same time, this book could open up the reader’s perspective, guide and inspire the readers to these fields....