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Se confunda adesea paradoxul cu exercl- siul plat si plicticos al reabilitarii truismelor. Nu se poate nega fapsll ca pentru a intoarce 10- curile comune isl trebule curaj.Nimic mai rlscant decit a lucra cu banalltatea.si totuGi,paradoxul bine facut atinge pragul filosoflel.Devine 0 forrna penetranta de cllnoastere.Valoarea de excep1tie a paradoxului care di unor acte aparent insignifiante un sens Ildinc !;Ii revelator a fost intuita de Alexandru Paleologu : "Paradoxul e 0 alarmS. a inteligentel,un contact inedit cu adevarul n....

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Problemes Distrayants. 5 – Arithetique. 16 – Logique: Athekatique. 39 – Trigonometrie. 42 – Geometrie. 48 – Analyse. 68 – Algebre. 84 --

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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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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 ...