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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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This book is 4 and 5 volumes based on Smarandache's math theories.

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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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This book consists of a selection of papers most of which were produced during the period 1999-2002. They have been inspired by questions raised in recent articles in current Mathematics journals and in Florentin Smarandache’s wellknown publication Only Problems, Not Solutions. All topics are independent of one another and can be read separately. Findings are illustrated with diagrams and tables. The latter have been kept to a minimum as it is often not the numbers but the general behaviour and pattern of numbers that matters. One of the facinations with number problems is that they are often easy to formulate but hard to solve – if ever, and if one finds a solution, new questions present themselves and one may end up having more new questions than questions answered....

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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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A Smarandache notion is an element of an ill-defined set, sometimes being almost an accident oflabeling. However, that takes nothing away from the interest and excitement that can be generated by exploring the consequences of such a problem It is a well-known cliche among writers that the best novels are those where the author does not know what is going to happen until that point in the story is actually reached. That statement also holds for some of these problems. In mathematics, one often does not know what the consequences of a statement are. Cnlike a novel however, there are no complete plot resolutions in mathematics as there are no villains to rub out. As the French emphatically say in another context, "Vive la difference'"...

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The methodology of how mathematics is untaught.

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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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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 fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs or bipartite graphs. Thus the use of graphs has become inevitable in fuzzy models. The neutrosophic models are fuzzy models that permit the factor of indeterminacy. It also plays a significant role, and utilizes the concept of neutrosophic graphs. Thus neutrosophic graphs and neutrosophic bipartite graphs plays the role of representing the neutrosophic models. Thus to construct the neutrosophic graphs one needs some of the neutrosophic algebraic structures viz. neutrosophic fields, neutrosophic vector spaces and neutrosophic matrices. So we for the first time introduce and study these concepts. As our analysis in this book is application of neutrosophic algebraic structure we found it deem fit t...

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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 applications in several 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 component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in the construction of biautomatons. So we felt it is important to construct nonabstract bistructures which can give itself for more and more lucid applications....

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Matrices provide a very powerful tool for dealing with linear models. Bimatrices which we are going to define in this chapter are still a powerful and an advanced tool which can handle over one linear model at a time. Bimatrices will be useful when time bound comparisons are needed in the analysis of the model....

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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 collection of poems compiled by the author Florentin Smarandache.

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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....