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[Un cabinet medk:aI Inzestrat cu instalatii moderne, tactnd parte dintr-un sanatoriu. Seam!nA cu 0 anticamed avtnd patru u§i: una In strnga, douA In falA §i alta In dreapta, cu inscriptii1e: •0", "Salonul 1" , "Salonul 2", respectiv "Salonul 3". Interior negru. Oameni ImpAiali. Un Om-NOll tinut In SlielA, congelat, de reclamA - JnSOlit de un extras revelatoriu din pres!. Pc pereli: citate, slogane pc aeeea§i temA. Doctori Ii Asistenti In balate de spital, dar negre, cu priviri Slicloasc, Inumane. Pacienlii tn balate wrgate de bolnavL Figuratia inutiiA de personal medical, care §CbioapAtA up', spre deosebire de padenp care §CbioapAtA rAu. InvAluit In mister, Doctorul stA tntr-un cArudor tantasmagoric, cu rotile, avtnd spatelc a public. Nu i se va vedea pc toatA durata plesei dectt ceata, de,i eAruciorul se mai ms. De ctteva ori dA seazalia . eA ar vrea sA se IntoareA spre public, dar renuntA. Nid In antracte, nid dupA, actorul nu va iqi Ia aplauze. Glasul cam nepAmlntesc, sinal. Asistentul, In picioare, sterilizeazA 0 siringA uriap.]...

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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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In a lecture that Einstein gave to the Prussian Academy of Sciences in 1921, he said the following: "Geometry predicates nothing about relations of real things, but only geometry together with the purport of physical laws can do so... The idea of the measuring rod and the idea of the clock contained with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which may not play any independent part in theoretical physics." / Einstein then went on to say that, in spite of the foregoing comment, we should temporarily support the use of the length and time transformations as though they were physically real....

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Preface. 5 -- About the cover 9 -- Welcome from Campus President, DrRaul Ramirez. 10 -- Registrants/Attendees. 11 -- Keynotes. 12 -- The light barrier: its construction and demolition. 14 -- Jim Malmberg. 17 -- DrDavid Iadevaia. 25 -- DrLarissa BBorrisova & Dmitri DRabounski. 29 -- DrFlorentin Smarandache. 36 -- Greg Holmberg AKA BBash. 43 -- Homer BTilton. 83 -- Credits . 100 -- Photos. 101 --...

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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 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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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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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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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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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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In this book, we study the subject of Smarandache Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field founded by Florentine Smarandache, is alike to that of Fuzzy Theory: both the fields imperatively questioned the dogmas of classical mathematics....

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An associative ring is just realized or built using reals or complex; finite or infinite by defining two binary operations on it. But on the contrary when we want to define or study or even introduce a non-associative ring we need two separate algebraic structures say a commutative ring with 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algebras and Jordan algebras are mainly built using a vector space over a field satisfying special identities called the Jacobi identity and Jordan identity respectively. Study of these algebras started as early as 1940s. Hence the study of non-associative algebras or even non-associative rings boils down to the study of properties of vector spaces or linear algebras over fields....

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Near-rings are one of the generalized structures of rings. The study and research on near-rings is very systematic and continuous. Near-ring newsletters containing complete and updated bibliography on the subject are published periodically by a team of mathematicians (Editors: Yuen Fong, Alan Oswald, Gunter Pilz and K. C. Smith) with financial assistance from the National Cheng Kung University, Taiwan. These newsletters give an overall picture of the research carried out and the recent advancements and new concepts in the field. Conferences devoted solely to near-rings are held once every two years. There are about half a dozen books on near-rings apart from the conference proceedings. Above all there is a online searchable database and bibliography on near-rings. As a result the author feels it is very essential to have a book on Smarandache near-rings where the Smarandache analogues of the near-ring concepts are developed. The reader is expected to have a good background both in algebra and in near-rings; for, several results are to be proved by the reader as an exercise....

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The aim of this book is two fold. At the outset the book gives most of the available literature about Fuzzy Relational Equations (FREs) and its properties for there is no book that solely caters to FREs and its applications. Though we have a comprehensive bibliography, we do not promise to give all the possible available literature about FRE and its applications. We have given only those papers which we could access and which interested us specially. We have taken those papers which in our opinion could be transformed for neutrosophic study....

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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 essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have an arithmetic character (as 9-12) on number theoretic problems in Geometry. Chapter 2 includes various diophantine equations, some of which are treatable by elementary methods; others are partial solutions of certain unsolved problems. An important method is based on the famous Euler-Bell-Kalm ar lemma, with many applications. Article 20 may be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many papers study the famous Smarandache function, the source of inspiration of so many mathematicians or scientists working in other elds. The author has discovered various generalizations, extensions, or analogues functions. Other topics are connected to the composition of arithmetic funct...

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This is a book on empirical number theory concentrating on the analysis of number sequences. Its focus is on a small part of a very large number of integer sequences defined by Florentin Smarandache. The author has. however. when appropriate included some other of his research results which organically belongs to this area. The content is organized into chapters according to the main considerations to make when programming the analysis. They are not mutually exclusive. As in my previous book Surfing on the ocean of numbers - a few Smarandache Notions and Similar Topics an attempt has been made to present results so that they are easy to understand and not to burdensome to read. There are many tables. some of which may be used for reference. but in general they are there to show the overall results obtained at a glance. In most cases it is the way in which sequences behave. not the individual figures. which is of importance. Some graphs have been included to illustrate important findings....

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A collection ot papers concerninq Smarandache type functions, numbers, sequences, inteqer alqoritbms, paradoxes, experimental qeometries, alqebraic structures, neutrosophic probability, set, and loqic, etc. is published this year....

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Florentin Smarandache, a lliathematician from Eastern Europe, escaped fran his country because the communist authorities had prohibited the publication of his resea~ch papers and his participation in international congresses. After two years of waiting in a political ~efugee camp in Turkey, he emigrated to the united states. As research workers, receiving our co-worker, we decided to publish a selection of his papers....

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Over the past 25 years, I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings) specially to motivate both ring theorists and Smarandache algebraists to develop and study several important and innovative properties about S-rings. Writing this book essentially involved a good deal of reference work. As a researcher, I felt that it will be a great deal better if we thrust importance on results given in research papers on ring theory rather than detail the basic properties or classical results that the 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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Last autumn I received a letter from a student at Arizona State University. He sent me a response to my letter to the editor in Mathematical Spectrum, including some pages of F. Smarandache's open problems. At first, I was not interested in the enclosure, for some of the problems are not so new and creative. But reading carefully, there are also some problems which stimulate the curiosity on arithmetic functions and number sequences. Then I needed almost no time to understand his talent in mathematics. I returned a letter to the student with a copy of my publication in The Mathematical Scientist and including a response where I stated that I was willing to write additional articles....

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The study of Smarandache Algebraic Structure was initiated in the year 1998 by Raul Padilla following a paper written by Florentin Smarandache called “Special Algebraic Structures”. In his research, Padilla treated the Smarandache algebraic structures mainly with associative binary operation. Since then the subject has been pursued by a growing number of researchers and now it would be better if one gets a coherent account of the basic and main results in these algebraic structures. This book aims to give a systematic development of the basic non-associative algebraic structures viz. Smarandache groupoids. Smarandache groupoids exhibits simultaneously the properties of a semigroup and a groupoid. Such a combined study of an associative and a non associative structure has not been so far carried out. Except for the introduction of smarandacheian notions by Prof. Florentin Smarandache such types of studies would have been completely absent in the mathematical world. Thus, Smarandache groupoids, which are groupoids with a proper subset, which is a semigroup, has several interesting properties, which are defined and studied in this book...

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K.R.Narayanan was a lauded hero and a distinguished victim of his Dalit background. Even in an international platform when he was on an official visit to Paris, the media headlines blazed, ‘An Untouchable at Elysee’. He was visibly upset and it proved that a Dalit who rose up to such heights was never spared from the pangs of outcaste-ness and untouchability, which is based on birth. Thus, if the erstwhile first citizen of India faces such humiliation, what will be the plight of the last man who is a Dalit? As one of the world’s largest socio-economically oppressed, culturally subjugated and politically marginalized group of people, the 138 million Dalits in India suffer not only from the excesses of the traditional oppressor castes, but also from State Oppression— which includes, but is not limited to, authoritarianism, police brutality, economic embargo, criminalization of activists, electoral violence, repressive laws that aim to curb fundamental rights, and the non-implementation of laws that safeguard Dalit rights. The Dalits were considered untouchable for thousands of years by the Hindu society until the Constitution of India...

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A collection of papers concerning Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc. is published this year....

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This document will describe the current status on the search for factors of Smaranda.che consecutive numbers and their reverse. A complete list up to index 200 will be given, with all known factors. Smarandache numbers are the concatenation of the natural numbers from one up to the given index, and reverse Smaranda.che numbers are the concatenation of the natural numbers from the given index dO\vIJ. to 1....

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As an alternative to the existing logics we propose the Neutrosophic Logic to represent a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undefined, unknown, incompleteness, inconsistency, redundancy, contradiction. It is a non-classical logic. Eksioglu (1999) explains some of them: “Imprecision of the human systems is due to the imperfection of knowledge that humain receives (observation) from the external world. Imperfection leads to a doubt about the value of a variable, a decision to be taken or a conclusion to be drawn for the actual system. The sources of uncertainty can be stochasticity (the case of intrinsic imperfection where a typical and single value does not exist), incomplete knowledge (ignorance of the totality, limited view on a system because of its complexity) or the acquisition errors (intrinsically imperfect observations, the quantitative errors in measures).”...

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In a world of chaotic alignments, traditional logic with its strict boundaries of truth and falsity has not imbued itself with the capability of reflecting the reality. Despite various attempts to reorient logic, there has remained an essential need for an alternative system that could infuse into itself a representation of the real world. Out of this need arose the system of Neutrosophy, and its connected logic, Neutrosophic Logic. Neutrosophy is a new branch of philosophy that studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra. This was introduced by one of the authors, Florentin Smarandache. A few of the mentionable characteristics of this mode of thinking are [90-94]: It proposes new philosophical theses, principles, laws, methods, formulas and movements; it reveals that the world is full of indeterminacy; it interprets the uninterpretable; regards, from many different angles, old concepts, systems and proves that an idea which is true in a given referential system, may be false in another, and vice versa; attempts to make peace in the war of ideas, and to make w...

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Papers concerning any of the Smarandache type functions, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, conjectures, open problems, neutrosophic logic/set/probability, etc. have been selected for this volume. The...

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Papers concerning any of the Smarandache type functions, sequences, numbers, algorithms, inferior/superior f-parts, magic squares, palindromes, functional iterations, semantic paradoxes, Non-Euclidean geometries, manifolds, conjectures, open problems, algebraic structures, neutrosophy, neutrosophic logic/set/probability, hypothesis that there is no speed barrier in the universe} quantum paradoxes, etc. have been selected for this volume. Contributors are from Australia, China, England, Germany, India, Ireland, Israel, Italy, Japan, Malaysia, Morocco, Portugal, Romania, Spain, USA. Most of the papers are in English, a few of them are in Spanish, Portuguese, or Gennan....

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In these proceedings of the second international conference on smarandache type notions in mathematics and quantum physics (december 21-24, 2000, university ofcraiova, romania; organizers: v. Seleacu and m. L perez) are collected articles and notes: in mathematics: related to smarandache anti-geometry, function, f-inferior part function, k-k additive relationships, 2-2 subtractive relationships, sequences, coprime functions, double factorial function, magic squares, problems, conjectures, equations, partitions, paradoxes, series, algebraic structures, pseodo-smarandache function, erdos-smarandache moments numbers; and in physics: related to smarandache hypothesis that there is no speed barrier in the universe, srm-theory of the possibility of constructing arbitrary speeds, and quantum smarandache paradoxes....

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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 of mathematics, and hardly a day goes by where I do not use some topic from those courses. Logic is also a fundamental component of advanced computer classes. I am currently teaching advanced courses in assembly language programming and computer organization. Reference is constantly being made to how the rules of logic are incorporated into the fundamental circuits of a computer. The logic used in these classes is known as classical or Boolean logic. Neutrosophic logic is an extension of classical logic, but as you will see in the book, there are two intermediate steps between them. Neutrosophic logic is yet another idea generated by Florentin Smarandache, who seems to be a perpetual idea machine. Like classical logic, it can be used in many ways, everywhere from statistics to quantum me...