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...cles 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. O...

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...l Numbers - a few elementary studies on Smarandache sequences and other number problems Henry Ibstedt American Research Press Reho... ... The surprising behaviour of a sequence Mainly Natural Numbers - a few elementary studies on Smarandache sequences and other... ...randache/eBooks-otherformats.htm ISBN: 1-931233-73-X Standard Address Number: 297-5092 Printed in the United States of America 3 ... ...n publication Only Problems, Not Solutions. All topics are independent of one another and can be read separately. Findings are illustrated with di... ... 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 ... ... 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 u... ...ber of terms equal to 1 is convergent. This study, like several others from my earlier books, has been reviewed in the Zentralblatt für Mathematik, ... ...r divisibilty properties are often brought into focus. Finally I express my sincere thanks to Dr. Minh Perez for his support for this book. Last ... ...to Dr. Minh Perez for his support for this book. Last but not least I thank my dear wife Anne-Marie for her patience with me when I am in my world o...

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...ed from a translation by the observant of the observed, and by translation one falsifies. The cars jingling on the street” was a “city poem”, “peasan... ...dictions. Why? Because we lived in that society a double life: on official one – propagated by the political system, and another one real. In mass- m... ...s an intellectual breathing, were superb springs. The “No” and “Anti” from my paradoxist manifestos had a creative character, not all nihilistic (C.... ... Soare, Ion Rotaru, Marian Barbu, Gheorghe Niculescu studied paradoxism in my literary work, Nicolae Manolescu asserted, about one of my manuscripts... ... studied paradoxism in my literary work, Nicolae Manolescu asserted, about one of my manuscripts of non-poems, that they are against-the-hair. I di... ... terms, new procedures, methodes or even algorithms of creation. In one of my manifestos, I proposed the sense of embezzlings, changes from figurati... ... 20 Engineer’s Terminologies from Bertolucci’s “The Sheltering Sky”: 1. A NUMBER OF DIFFERENT APPROACHES ARE BEING TRIED – We are still pleasing in... ... The (Non)Artist. 85 II. R E V I E W S PI O AND THE “NUMBER POEMS” I was tired, my eyes red, and I don`t know why Pi O`s “n... ...e in our unconscious. Atwinkle of mind is in every poem, no letters – only numbers, comas, and some times a small sentence in front of it. He`s tale...

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...sfies the Smarandache Lagrange criteria. But for the Smarandache concepts one wouldn't have studied the collection of subgroups of a loop. This... ...he/eBooks-otherformats.htm ISBN: 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS ... ... books on loops is the fact that it is more abstract than groups. Further one is not in a position to give a class of loops which are as concrete as... ...kground in algebra and more specifically a strong foundation in loops and number theory. 6 This book introduces over 75 Smarandache concept... ...arandache notions. This book is structured into five chapters. Chapter one which is introductory in nature covers some notions about groups, grap... ...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... ...search books on Smarandache algebraic notions. I dedicate this book to my parents, Worathur Balasubramanian and Krishnaveni for their love. ... ...x, y, z such that (x – y, n) = 1 and z = xy. Now z = xy implies. z ≡ (my – (m – 1)x) (mod n) …2 Since L n (m) is a WIP loop ...

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...I have met the Smarandache's world for the first time about one year ago reading some articles and problems published in the Journal of Recreational Mathematics. From then on I discovered the interesting American Research Press web site dedicated to the Smarandache notions and held by...

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...A SET OF NEW SMARANDACHE FUNCTIONS, SEQUENCES AND CONJECTURES IN NUMBER THEORY Felice Russo AMERICAN RESEARCH PRESS Lupton USA 2000 Wrongnes... ...0 600 700 800 900 1000 0 200 400 600 800 1000 n W(n) 2 To two stars: my doughters Gilda Aldebaran and Francesca Carlotta Antares To the memory o... ... doughters Gilda Aldebaran and Francesca Carlotta Antares To the memory of my brother Felice and my late uncles Raffaele and Angelo and my late ant Jo... ...3 INTRODUCTION I have met the Smarandache's world for the first time about one year ago reading some articles and problems published in the Journal of... ...ache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this ... ...tics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and inte... ...ty of Craiova Library, and Archivele Statului, Filiala Valcea) and another one (V) to the study of some conjectures and open questions proposed always... ...e the equation: Zt(n)+Zt(n+1)=Zt(n+2) For the first 1000 values of Zt(n), one solution has been found: Zt(6)+Zt(7)=Zt(8). Unsolved question: Is the n...

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...g of what something is must include an understanding of what it is not. In his paper, “Paradoxist Mathematics” [19], Florentin Smarandache proposed a number of ways in which we could explore “new math concepts and theories, especially if they run counter to the classical ones.” In a manner consistent with his unique point of view, he defined several types of geometry that ...

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...ublic/display/conference_info/fabz54. ISBN: 1-931233-44-6 Standard Address Number: 297-5092 Printed in the United States of America 3 Tab... .... In his paper, “Paradoxist Mathematics” [19], Florentin Smarandache proposed a number of ways in which we could explore “new math concepts and theor... ...idean geometry, the lines in hyperbolic geometry are less prone to intersecting one another. Whereas even the slightest change upsets the delicate b... ... 10]). Although the term Riemannian geometry sometimes refers specifically to one of the geometries just mentioned (elliptic or spherical), it is ... ...that the geometries of Euclid, Gauss, Bolyai, et al, are particular examples of one kind of space, and extending attention to non-uniform spaces brin... ...Marsh, and Catherine D’Ortona. Prof. Marsh was the teacher most responsible for my turning to mathematics, and Prof. Hass, my thesis advisor, introd... ...sor, introduced me to the real world of geometry and topology. Virtually all of my thinking in mathematics can be traced back in some way to these tw... ...e way to these two 8 mathematicians. Prof. D’Ortona, a valued colleague, and my wife Linda are currently the most active forces on my professiona... ... be referred to as singular. The measure of the two equal angles depends on the number of triangular disks around that vertex. Figure 7. A de...

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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. The...

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...P. Popescu, V. Seleacu, "About the behaviour of some new functions in the number theory" (to appear this year) . "Collection of Problems on Smarand... ...shbacher and Erhus University Press ISBN 1-879585-50-2 Standard Address Number 297-5092 Printed in the United States of America Preface The pre... ...ively with some "basic" consequences of the Smarandache function. In this one, the universe of discourse has been expanded to include a great many o... ...ed. 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... ...r demons. Rather than attempt to cite them in the text, I will now extend my gratitude to all those who combed the Smarandache archives for much of ... ...mond Thanks must again be extended to Brian Dalziel and Toufic Yloubarek, my supervisors who agreed to let me use Decisionmark resources in the purs... ...rvisors who agreed to let me use Decisionmark resources in the pursuit of my professional objectives Sometimes, there is simply no substitute for si... ...published[ 1], the Smarandache function SCm) = n, where n is the smallest number such that m divides n'. and associated consequences has spawned man... ...e answer is both simple and complex. A Smarandache notion is a problem in one of the following sets: a) A problem posed by Florentin Smarandache. ...

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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; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals esse... ...), where the same author may appear more times. Finally, I wish to express my warmest gratitude to a number of persons and organiza- tions from whom I... ...T oth, V.E.S. Szab o, D.M. Milo sevi c and the late D.S. Mitrinovi c. My appreciation is due also to American Research Press of Rehoboth for e ci... ...2 +AC 02 2AB 0 AC 0 cosA =a 2 cos 2 A; so C 0 B 0 =a cosA. Similarly one obtains A 0 C 0 =b cosB; A 0 B 0 =c cosC: Therefore p 0 = 1 2 X A 0 B 0... ...inB sinC (where R is the radius of the circumcircle). By a = 2R sinA, etc. one has p 0 = 2R Y a 2R = S R ; where S =area(ABC). By p = S r (r = radius ... ... sinv sinu : (1) We wish to mention here that relation (1) appears also in my book [3] on page 112, where it is used for a generalization of Steiner’s... ...AB sin AC sin(A ) (i.e. relation (1) with u = , v = ). Similarly one has BE CE = AB sin(A ) AC sin : Therefore BD BE CD CE = AB AC ...

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... of shrinkage estimators is proposed with its properties. Part2. In this section we have analyzed the two classes of estimators for population median MY of the study character Y using information on two auxiliary characters X and Z in double sampling. In this section we have shown that the suggested classes of estimators are more efficient than the one suggested by Singh e...

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..., India. ISBN: 1-931233-68-3 Standard Address Number 297-5092 Printed in the United States of America 4 F... ...ave shown that the suggested classes of estimators are more efficient than the one suggested by Singh et al (2001). Estimators based on estimated op... ...ified time until it fails, is represented by the continuous random variable X. One distribution that has been used extensively in recent years to de... ...q β β β β β β 2 2 2 1 2 1 ) , ( , (2.1) where p and q are real numbers such that p ≠ 0 and q > 0, w is a stochastic variable which m... ...tely small magnitude of bias for fixed ∆ or for fixed () β β 1 and () β β 2 , one should choose q in the vicinity of 1 − ∆ = q . It is interesting... ...sted classes of shrunken estimators have considerable gain in efficiency for a number of choices of scalars comprehend in it, particularly for heavil...

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...he standard textbooks contain. I feel that such a venture, which has consolidated several ring theoretic concepts, has made the current book a unique one from the angle of research. One of the major highlights of this book is by creating the Smarandache analogue of the various ring theoretic concepts we have succeeded in defining around 243 Smarandache concepts....

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...hingOnline.com Peer reviewers: Ion Goian, Department of Algebra and Number Theory, University of Kishinev, Moldova. Sebastian Martin Ruiz, Av... ...che/eBooks-otherformats.htm ISBN: 1-931233-64-0 Standard Address Number: 297-5092 Printed in the United States of America 3 CONT... ...idated several ring theoretic concepts, has made the current book a unique one from the angle of research. One of the major highlights of this bo... ...e properties locally makes the study easier and also gives way to greater number of newer concepts. Also localization of properties automatically co... ... product. The integral domain Z failed to be a Smarandache ring but it is one of the most natural Smarandache ring of level II. This book is orga... ...e ring of level II. This book is organized into five chapters. Chapter one is introductory in nature and introduces the basic algebraic structure... ...that has remained open for over 60 years. I firstly wish to put forth my sincere thanks and gratitude to Dr. Minh Perez. His making my books on ... ... book and other future titles. It gives me immense happiness to thank my children Meena and Kama for single- handedly helping me by spending all ... ...me in formatting and correcting this book. I dedicate this book to be my beloved mother-in-law Mrs. Salagramam Alamelu Ammal, whose only son, an...

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...y defining a new notion called the Smarandache mixed direct product. Likewise we also define the Smarandache semifields of level II. This study makes one relate, compare and contrasts weaker and stronger structures of the same set....

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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... ...se we also define the Smarandache semifields of level II. This study makes one relate, compare and contrasts weaker and stronger structures of the ... ... outset that the reader has a strong background in algebra that will enable one to follow and understand the book completely. This book consists ... ...mivector spaces are introduced in the third chapter. Chapter four, which is one of the major parts of this book contains a complete systematic intro... ...ache notions in semirings and its generalizations. I deeply acknowledge my children Meena and Kama whose joyful persuasion and support encourage... ...contained. The serious study of semirings is very recent and to the best of my knowledge we do not have many books on semirings or semifields or semi... ...∗ 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 ... ...ND ITS PROPERTIES The study of the concept of semiring is very meagre. In my opinion I have not come across a textbook that covers completely all ...

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... satisfaction and anxiety. The first situation, during an instant, is derived from the appearance of something with a convincing meaning; the second one, that comes immediately, is the perception of something concealed and absurd. Something, that is dissimulated under the level of the logical acceptance, jumps out abruptly in the main point to consider and constrains t...

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...p://www.gallup.unm.edu/~smarandache/ ISBN: 1-931233-53-5 Standard Address Number 297-5092 Printed in the United States of America 4 Foreword I... ...ase. In addition, the profession/ specialty of the sender came to support my mistrust. A mathematician in literature is a rare thing, because it sou... ... it is a very grave, serious, deep play, directed towards the face of the one who follows it. He, the poet, laughs with an eye and weeps with the oth... ...hrough all this essay on Paradoxism - since Florentin Smarandache, trying my structural weakness for unpredictable curls of his mind and my tolerant... ... came on until I was about to reach Paradoxism, almost filling a shelf of my library. Plus his letters, plus writings that have punctuated the stage... ...st concrete, of a movement put into action - to its exponential level, the one given by Smarandache. I have preferred to debate of ideas about Par... ...ay is that the paradoxist mentor is a kind of notoriety in the science of numbers, where he is also the holder of an overwhelming biography. Sensati... ...is book in 1995. Till this moment, when I have written the last page (this one of the foreword), I have not succeeded in shaking hands with its inspi... ...f the subject, this one being himself an “author” multiplied by the total number of the human beings who exist on the Earth. Of course, theoreticall...

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...As someone who works heavily in both math and computers, I can truly appreciate the role that logic plays in our modern world. One cannot understand the foundations of mathematics while lacking knowledge of the basics of logic and how proofs are constructed. Two of the first classes I took as a graduate student in mathematics were in the foundations ...

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...ington, Texas 76019, USA. ISBN: 1-931233-60-8 Standard Address Number: 297-5092 Printed in the United States of America 3 Forew... ...rs, I can truly appreciate the role that logic plays in our modern world. One cannot understand the foundations of mathematics while lacking knowled... ...he values of the variables can be any value in the continuum from zero to one. Finally, chapter four makes an introduction to neutrosophic logic, w... ...e to thank Florentin Smarandache for all of his ideas that I have used in my work. This includes three books, over twenty research papers and severa... ...ce man, I have books of his art, poetry, philosophy and mathematics among my collection. Yes, the plural can be applied to each category several tim... ...e all their help, all errors that remain in the book are of course solely my responsibility. Particular praise needs to be given to Minh Perez, who ... ...2 1 5 The order of evaluation of the connectives is marked by the numbers under the connectives. Therefore, the value of the proposition is... ...emonstrated in the truth table of table 2, where the result column is the one above the five. Table 2 p q r s p /\ q \/ (r /... ...o free variables. Example: If the universe of discourse is the set of numbers U = { 1, 2, 3, 4, 5 }, then the expression (Ex) x > 0 is tru...

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... 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 ignore...

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...nm.edu/~smarandache/a/outer-art.htm ISBN: 1-931233-32-2 Standard Address Number 297-5092 Printed in the United States of America 4 Ion Soar... ...e 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... ...compulsory anymore, everyone having- virtually- the chance to be “caliph for one day”? Every one of these hypothesis, or both at the same time, or ... ...ndache is the name that I write and utter perhaps with the deepest emotion in my mind; after Nichita Stanescu’s disappearance only Levantul of the b... ...terary movement and of Florentin Smarandache himself is followed by a limited number of literary critics which manifest a kind of ambiguity (or even... ...hird group in the middle category is formed by those who establish a hasty and one- sided relation between paradoxism and paradox. Starting from this ... ... and in literature. Returning to literature, Florin Vasiliu seems to reveal in one sentence his whole conception about Smarandache’s paradoxism: ”The... ...nded in ears the ironical/gentle verse of their brilliant fellow countryman: ”My dears, this path has been walked before!” The smarandachian “constr... ... disowned this manner of writing, when there is something to say: “Come home, my pet children,-/ I shout the eyes, the ears/ and the pavement stone ...

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...d from a translation by the observant of the observed, and by translation one falsifies. 'The cars jingling on the street' was a "city poem", 'peasa... ...ictions. Why? Because we lived in that society a double life: an official one - propagated by the political system, and another one real. In mass-med... ... an intellectual breathing, were superb springs. The "No" and "Anti" from my paradoxist manifestos had a creative character, not at all nihilistic (... ... While I. Soare, I. Rotaru, M. Barbu, Gh. Niculescu studied paradoxism in my literary work. N. Manolescu asserted, about one of my manuscripts of no... ...escu studied paradoxism in my literary work. N. Manolescu asserted, about one of my manuscripts of non-poems, that they are against-the-hair. I did... ... terms, new procedures, methods or even algorithms of creation. In one of my manifestos I proposed the sense of embezzling, changes from figurative ... ...e set, i indeterminate in the set, and f false, where t, i, and f are real numbers taken from the sets T, I, and F with no restriction on T, I, F, no... ...the next papers. Keywords and Phrases: Non-standard analysis, hyper-real number, infinitesimal, monad, non-standard real unit interval, operations ... ...s the infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all po...

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... ISBN: 1-931233-48-9 Standard Address Number 297-5092 Printed in the United States of America 4 De... ...position, should not be reason for cataloguing this method as a syncretic one, taken in its pejorative sense. In respect to such an approach, the th... ... connection with the validity of a new hypothesis: < Of all theories, the one that meets the criterion of ″internal perfection and external confirm... ... possible the Einsteinian principle, and at the same time, the Leibnizian one of sufficient reason. ... ...cluded in a future polemic edition. 6 “In another life, probably, my name was Thales” I. MODELLING OF RATIONALITY 1. A SPECULATIVE EXISTENT... ...his way if the trajectories of the movements did no bend due to the great number of interacting bodies. Furthermore, due to the movement over curved... ... transportation means for rational extraterrestrial beings. The increased number of reports on the presence of and ″contact″ with extraterrestrial b... ...al confirmations. (Compelled by objective reasons I am forced to refer to my own experience. My personal oneiric activity was marked by a number of ... ...e of the house. Furthermore, each time I have this dream I become aware of my ability to fly – a reason that makes me think that this dream represent...

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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 ... ... the study of loop rings and above all Smarandache loop rings. Further in my opinion the deeper study of loop rings will certainly branch off into a... ...150 problems which will help any researcher. I first deeply acknowledge my indebtness to Dr. Minh Perez of American Research Press, whose constant... ...tant mental support and encouragement made me write this book. I thank my husband Dr. Kandasamy and my daughters Meena and Kama, who tirelessly w... ...) 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',... ...oroial space as given in [43]. We further illustrate with examples. As no one has worked on the Smarandache properties of Smarandache K-vectorial sp... ...ra over Q. Thus by this example we illustrate that there can be more than one proper subset which is a K-algebra, for K = Q in this example. Now we ... ...es the notion of Smarandache loop rings. It has five sections. In section one we just recall the definition of loops and Smarandache loops and give ...

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.../˜ smarandache/eBooks-otherformats.htm ISBN:1-931233-92-6 Standard Address Number: 297-5092 Printed in the United States of America Preface A combina... ...tablished in this chapter. The last section determines the relation of the number of embeddings and rooted maps of a graph on genus. A general equatio... ...d a difficult problem. The unrooted complete maps, the semi-regular maps and one face maps on orientable and non-orientable surfaces are enumerated in t... ...orial maps. Although it is called open problems, in fact, any solution for one of these problems needs to establish a new mathematical system first. Bu... ...ems considered by me in the following times. The main part of this book is my post-doctor report in the Chinese Academy Preface iii of Sciences in 20... ...pears. Here I must mention some of them. On the first, I would like to give my sincerely thanks to Professor Feng Tian for his encouraging and invaluab... ...ed total map and embedding polynomial of a graph................20 4.2 The number of rooted maps underlying a graph on genus...................24 Chap... ...regular graph on surfaces ..................... 73 §4 The automorphisms of one face maps........................................76 Chapter 4 Applicati...

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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 ... ...he concepts are completely understood by illustrating with examples and a great number of problems. Solutions to all the problems need extraordinary... ...tions. Smarandache notions are revolutionary because group theory does not make one think of the unthinkable that can naturally occur but Smarandach... ... see how the mixture of a group and a semigroup behaves. I deeply acknowledge my family members for their constant encouragement and support which... ... Abaddaba, Oradea, 78-81, (2000). 7 CHAPTER ONE One of the essential and outstanding fe... ...e world of Smarandache notions in groups and Smarandache semigroups. In Chapter one, we introduce some basic notation, Binary relations, mappings an... ...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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...her of fuzzy sets, remarked that: "So, this whole thing started because of my perception at that time, that the world of classical mathematics – was ... ...oks-otherformats.htm ISBN: 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...inted in the United States of America 3 CONTENTS Preface 5 PART ONE 1. SOME RESULTS ON FUZZY ALGEBRA 1.1 Fuzzy subsets 9 1.2 Grou... ...h those concepts that clearly have relevant applications. When I informed my interest in writing this book, Dr. Minh Perez of the American Research ... ... rigour that is expected of serious research, are now a permanent part of my personal faith. Periyar's ideology and political praxis have influenced... ... to dissent and to dare. 7 PART ONE 8 PART ONE 9 Chapte... ...E 8 PART ONE 9 Chapter One SOME RESULTS ON FUZZY ALGEBRA This chapter has twelve sections. F... ...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...

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...ormats.htm ISBN: 1-931233-71-3 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...inted in the United States of America 3 CONTENTS Preface Chapter One: INTRODUCTORY CONCEPTS 1.1 Groups, Loops and S-loops 7 1.2 Semigr... ...y classify the study under four heads : i. bialgebraic structures with one binary closed associative operation : bigroups and bisemigroups ii. b... ...ve operation : bigroups and bisemigroups ii. bialgebraic structures with one binary closed non-associative operation: biloops and bigroupoids iii... ...study of the revolutionary and fascinating Smarandache Notions, and I owe my thanks to Dr. Perez for all the intellectual delight and research produ... ...rch productivity I have experienced in this span of time. I also thank my daughters Meena and Kama and my husband Dr. Kandasamy without whose com... ... have been overwhelmingly busy because of this self-sufficiency – juggling my teaching and research schedules and having my family working along with... ...y G is a abelian or a commutative group. If the group G has only a finite number of elements we call G a group of finite order otherwise G is said t... ...s a subgroup of G then o(H) is a divisor of o(G). Note: o(G) means the number of elements in G it will also be denoted by |G|. CAUCHY THEOREM (...