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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 absent. This is in marked contrast with group theory where books are abundantly available for all levels: as graduate texts and as advanced research books....

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Performantele matematicii actuale,ca si descoperirile din viitor isi au,desigur, inceputul in cea mai veche si mai aproape de filozofie ramura a matematicii, in teoria numerelor. Matematicienii din toate timpurile au fost, sunt si vor fi atrasi de frumusetea si varietatea problemelor specifice acestei ramuri a matematicii. Regina a matematicii, care la randul ei este regina a stiintelor, dupa cum spunea Gauss, teoria numerelor straluceste cu lumina si atractiile ei, fascinandu-ne si usurandu-ne drumul cunoasterii legitatilor ce guverneaza macrocosmosul si microcosmosul. De la etapa antichitatii, cand teoria numerelor era cuprinsa in aritrnetica, la etapa aritrneticii superioare din perioada Renasterii, cand teoria numerelor a de venit parte de sine statatoare si ajungand la etapa moderna din secolele XIX si XXcand in teoria numerelor isi fac loc metode de cercetare din domenii vecine, ca teoria functiilor de variabila complexa si analiza matematicii drumul este lung, insa dens, cu rezultate surprinzatoare, ingenioase si culte nu numai teoriei propriuzise a numerelor, dar si altor domenii ale cunoasterii, atat teoretice cat si pra...

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The Chinese edition of Advances and Applications of DSmT for Information Fusion (Collected Works) that explains the Dezert-Smarandache Theory.

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作者在近年来 提出的似是而非和自相矛盾推理，DSmT，可以看作是 经典的 DSmT 的扩展，但是它们又存在着重要的差异。比如，DSmT 可以处理由 信度函数表示的任意类型独立信息源间的信息融合问题，但它的重点是处理不确 定、高度冲突和不精确的证据源的融合问题。DSmT 能够不受 DST 框架的限制， 处理复杂的静态或动态融合问题，特别是当信息源间的冲突非常大时，或者是所考 虑问题的框架（一般情况下用Θ表示）由于Θ中命题 之间的界限模糊、不确定、 不精确而很难细分时，DSmT 便发挥了它的优势。...

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This book has four chapters. In chapter one we just recall the notion of RD codes, MRD codes, circulant rank codes and constant rank codes and describe their properties. In chapter two we introduce few new classes of codes and study some of their properties. In this chapter we introduce the notion of fuzzy RD codes and fuzzy RD bicodes. Rank distance m-codes are introduced in chapter three and the property of m-covering radius is analysed. Chapter four indicates some applications of these new classes of codes....

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DEFINITION 1.12: The ‘norm’ of a word v _ VN is defined as the ‘rank’ of v over GF(2) (By considering it as a circulant matrix over GF(2)). We denote the ‘norm’ of v by r(v). We just prove the following theorem. THEOREM 1.2: Suppose ____GF(2N) has the polynomial representation g(x) over GF(2) such that the gcd(g(x), xN +1) has degree N – k, where 0 _ k _ N. Then the ‘norm’ of the word generated by _ is ‘k’. ...

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Preface 5 Chapter One BASIC PROPERTIES OF RANK DISTANCE CODES 7 Chapter Two RANK DISTANCE BICODES AND THEIR PROPERTIES 25 Chapter Three RANK DISTANCE m-CODES 77 Chapter Four APPLICATIONS OF RANK DISTANCE m-CODES 131 FURTHER READING 133 INDEX 144 ABOUT THE AUTHORS 150...

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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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The development of mathematics continues in a rapid rhythm, some unsolved problems are elucidated and simultaneously new open problems to be solved appear. 1. "Man is the measure of all things". Considering that mankind will last to infinite, is there a terminus point where this competition of development will end? And, if not, how far can science develop: even so to the infinite? That is . The answer, of course, can be negative, not being an end of development, but a period of stagnation or of small regression. And, if this end of development existed, would it be a (self) destruction? Do we wear the terms of selfdestruction in ourselves? (Does everything have an end, even the infinite? Of course, extremes meet.) I, with my intuitive mind, cannot imagine what this infinite space means (without a beginning, without an end), and its infinity I can explain to myself only by mAans of a special property of space, a kind of a curved line which obliges me to always come across the same point, something like Moebus Band, or Klein Bottle, which can be put up/down (!)...

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

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

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This book has three chapters. In the first chapter the notion of n-vector spaces of type II are introduced. This chapter gives over 50 theorems. Chapter two introduces the notion of n-inner product vector spaces of type II, n-bilinear forms and n-linear functionals. The final chapter suggests over a hundred problems. It is important that the reader should be well versed with not only linear algebra but also n-linear algebras of type I....

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

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

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The function named in the title of this book is originated from the e:riled Romanian mathematician Florentin Smaranda.che, who has significant contributions not only in mathematics, but also in li~ratuIe. He is the father of The Paradorut Literary Movement and is the author of many stories, novels, dramas, poems....

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In writing a book, one encounters and overcomes many obstacles. Not the least of which is the occasional case of writer’s block. This is especially true in mathematics where sometimes the answer is currently and may for all time be unknown. There is nothing worse than writing yourself into a corner where your only exit is to build a door by solving unsolved problems. In any case, it is my hope that you will read this volume and come away thinking that I have overcome enough of those obstacles to make the book worthwhile. As always, your comments and criticisms are welcome. Feel free to contact me using any of the addresses listed below, although e-mail is the preferred method....

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

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

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CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two INTERVAL SUPERMATRICES 9 Chapter Three SEMIRINGS AND SEMIVECTOR SPACES USING SUPER INTERVAL MATRICES 83 Chapter Four SUPER INTERVAL SEMILINEAR ALGEBRAS 137 Chapter Five SUPER FUZZY INTERVAL MATRICES 203 5.1 Super Fuzzy Interval Matrices 203 5.2 Special Fuzzy Linear Algebras Using Super Fuzzy Interval Matrices 246 Chapter Six APPLICATION OF SUPER INTERVAL MATRICES AND SET LINEAR ALGEBRAS BUILT USING SUPER INTERVAL MATRICES 255 Chapter Seven SUGGESTED PROBLEMS 257 FURTHER READING 281 INDEX 285 ABOUT THE AUTHORS 287...

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Prima culegere de "Problemes avec et sans …problemes!" a aparut in Maroc in 1983. Am strans aceste proleme, aparute prill diverse reviste romanel;lti sau straine (printre care: "Gazeta Matematica", revista Ia care Ill-am format ca problemisi, "American Mathematical Monihly", "Crux Mathematicorum" (Canada), "E1emcllte del' Mathematik" (Elvetia), "Gaceta Matematica" (Spania), "Nieuw voor Archief" (Olanda), etc.) ori inedite intr-un al doilea volum....

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

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

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

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The study of bialgebraic structures started very recently. Till date there are no books solely dealing with bistructures. The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published in 1998. In the year 1999, bivector spaces was introduced. In 2001, concept of free De Morgan bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the construction of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category theory or linear logic....

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This is a collection of puzzles requiring in most cases – high school math. The purpose is not necessarily to solve these problems, but to get you to think about different types of math puzzles. Many of the puzzles are introductions to different areas of math. Most of these puzzles can be done mentally. If you have to write, expect to write no more than a page. Pictures are useful....

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“There are three planes”, said Onko, “which take off simultaneously - from Bangalore, Athens, and New York.” “Each plane goes 500 km north, then 500 km east, then 500 km south, and then 500 km west.” “Which plane is closest to its starting point?”, said Onko with a smile....

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Puzzles More puzzles Some more puzzles

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Smarandache notions, which can be undoubtedly characterized as interesting mathematics, has the capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or identification as a substructure. Several 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. The study of this concept is very meagre except for a very few research papers. Now, when we study the Smarandache semirings (S-semiring), we make the richer structure of semifield to be contained in an S-semiring; and this S-semiring is of the first level. To have the second level of S-semirings, we need a still richer structure, viz. field to be a subset in a S-semiring. This is achieved by defining a new notion called the Smarandache mixed direct product. Likewise we also define the Smarandache semif...

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This book contains a group of Smarandache's theories and lessons on mathematics. It includes his famous "Smarandache Function" and how it is executed through with math....

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During the five years since publishing, we have obtained many new results related to the Smarandache problems. We are happy to have the opportunity to present them in this book for the enjoyment of a wider audience of readers. The problems in Chapter two have also been solved and published separately by the authors, but it makes sense to collate them here so that they can be better seen in perspective as a whole, particularly in relation to the problems elucidated in Chapter one. Many of the problems, and more especially the techniques employed in their solution, have wider applicability than just the Smarandache problems, and so they should be of more general interest to other mathematicians, particularly both professional and amateur number theorists....

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The main 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 background on group theory; we give some recollection about groups and some of its properties just for quick reference....