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This book extends the natural operations defined on intervals, finite complex numbers and matrices. Secondly the authors introduce the new notion of finite complex modulo numbers just defined as for usual reals. Finally we introduced the notion of natural product Xn on matrices. This enables one to define product of two column matrices of same order. We can find also product of m*n matrices even if m not does equal n. This natural product Xn is nothing but the usual product performed on the row matrices. So we have extended this type of product to all types of matrices....

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In this chapter we just give a analysis of why we need the natural operations on intervals and if we have to define natural operations existing on reals to the intervals what changes should be made in the definition of intervals. Here we redefine the structure of intervals to adopt or extend to the operations on reals to these new class of intervals....

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Dedication 4 Preface 5 Chapter One INTRODUCTION 7 Chapter two EXTENSION OF NATURAL OPERATIONS TO INTERVALS 9 Chapter Three FINITE COMPLEX NUMBERS 41 Chapter Four NATURAL PRODUCT ON MATRICES 81 FURTHER READING 147 INDEX 148 ABOUT THE AUTHORS 150...

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While your child will enjoy doing Gol-Gol Games, you as the parent/ teacher/ facilitator may need to figure out how to guide your child through Gol-Gol Games, or play the game with your child. This is a useful companion book to Gol-Gol Games Volume 1, designed to help you guide the child, while both of you have fun....

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We are trying to use the “Socratic” method, as in asking questions and letting children think about the answers. We suggest you allow children to think and write the answers. Try not to draw either the objects or the circles for the child. Try to avoid suggesting techniques and methods of thinking. Children can develop their own ways of thinking. Less help in finding answers is a good thing for children! It makes them think. More time in finding answers is a very good thing for children! It makes them think without pressure. Read a book while the child draws. Relax....

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The role of the guide Suggested solutions for words and circles Suggested solutions for adding circles

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Gol-Gol Games introduces you and your parents to a game of drawing circles, a game for all ages. We try to make it interesting and fun and challenging....

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Get something to draw: A board and pieces of chalk, OR Paper, pencils and erasers In this game, you have to draw circles. You have to draw circles around things of the same type. Not a purr-fect circle that your art teacher will say “wow!” to; badly twisted circles are also okay. Let's play; it will be easy to explain....

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Circles around things Words and circles Adding things in circles

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1. O ke Anahonua ka mea e i ike ai ke ano o na mea i hoopalahaiahaia, oia na kaha, a me na ili, a me na paa. Ekolu mau ano o na mea i hoopalahaiahaia, he loa, he laula, a he manoanoa. 2. O ke kaha ; he loa wale no ko ke kaha; aole laula, aole manoanoa. O na welau o ke kaha he mau kiko ia: nolaila, o ke kiko, aole ona loa, aole laula, aole manoanoa, aka he wahi e ku wale ai no. 3. O ke kaha pololei ka loa pokole mai kekahi kiko a i kekahi kiko. 4. O ke kaha pololei ole, a i hui ole ia na kaha pololei, he kaha poai ia Penei, AE he kaha pololei ia; AIOE he kaha hakiia oia, i huiia na kaha pololei AI, IO, OE; AUE he kahapoai ia. Ina ma keia palapala, ua loaa ka olelo kaha wale no, ua manaoia he kaha pololei....

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This book has five chapters. Chapter one is introductory in nature. Fuzzy linguistic spaces are introduced in chapter two. Fuzzy linguistic vector spaces are introduced in chapter three. Chapter four introduces fuzzy linguistic models. The final chapter suggests over 100 problems and some of them are at research level....

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Preface 5 Chapter One Introduction 7 Chapter Two Fuzzy Linguistic Spaces 9 Chapter Three Fuzzy Linguistic Set Vector Spaces 23 Chapter Four Fuzzy Linguistic Models 129 4.1 Operations on Fuzzy Linguistics Matrices 129 4.2 Fuzzy Linguistic Cognitive Models 139 4.3 Fuzzy Linguistic Relational Map Model 148 Chapter Five Suggested Problems 165 Further Reading 187 Index 189 About the Authors 192...

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Florentin Smarandache a lansat un nou gen literar-artistic, adaptat cerinţelor cititorului şi scriitorului contemporan grăbit, numit FOTOJURNAL INSTANTANEU bazat pe abundenţa de imagini corespunzătoare textului, plus scrierea directă, rapidă, la prima mȃnă, pe locul vizitei, cu verbe puţine, şi exprimări scurte, neşlefuite. Smarandache launched a new literary genre, artistic, adapted to the contemporary reader and writer hurriedly called INSTANT FOTOJURNAL appropriate image based on the abundance of text, plus direct writing, fast at first mȃnă on the site visit, with fewer verbs and expressions short, unpolished....

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08.07.2010 Viaţa în Franţa e mai lejeră. În America e mai stresantă. Am scris pe foaia de lichidare că mi ar plăcea să revin în Fran-ţa ca lucrător (cercetător sau profesor la vreo Universitate). Cei de la HAL (prescurtare de la Hyper Articles en Ligne – Articole în exces online; este o arhivă deschisă gesti-onată de către Centrul pentru Comunicarea Ştiinţifică Directă, din Franţa) pretind că au o arhivă deschisă tuturor savanţilor, dar nu este deloc liberă, cum ar trebui. Articolele mele de matematică tradiţiona-lă (nu de avangardă, cum mi-e stilul) aşteaptă cu săptămânile pentru a fi validate. Cele de fizică sunt respinse (de exemplu, unul despre nematerie – unmatter, în engleză): imediat. 08.07.2010     Life in France is lighter. In America's most stressful. I wrote the winding sheet that I would like to return to Fran-ce as a worker (researcher or professor at a university).     Most of the HAL (short for Hyper Articles en Ligne - Online Articles excess, is an open archive pol-Onate by the Centre for Direct Scientific Communication, France) claim to have an archive open to all scientists, but it is not free , as it ...

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Prefaţă, de Mircea Monu ………………………………… 4 Călătorul călătorului, eseu de George Anca …………… 9 FRANŢA (la Brest) ……………………………………… 12 UNGARIA (la Pécs) …………………………………… 41 FRANŢA (la Brest, Carnac şi la Paris) ………………… 51 ANGLIA (la Londra) …………………………………… 96 SCOŢIA (la Edinburg) ………………………………… 137 În loc de postfaţă ………………………………………… 165...

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Urmărirea oficială a început pe data de 22.03.1985 (deşi am găsit în dosare şi rapoarte din ianuarie 1985) şi a continuat şi după fuga obiectivului „SAVU” pe data de 07.09.1988 în lagărul de refugiaţi politici din Istambul şi Ankara (Turcia). A fost luat în atenţie de Serviciul III din cadrul Inspectoratul Judeţean Dolj, pe linia problemei „Atlas”. Ȋn vreme ce ei „mă lucrau în urmărire informativă” cu atȃta zel … eu habar n-aveam! Ba chiar mă băgaseră în atenţia Serviciului de Contraspionaj (conform Notei RD. 5/890/1985): fiindcă aveam multe legături cu străinii. Ca-n filme poliţiste, numai că antieroul eram... EU însumi! Existau zile cȃnd mă „filau” pas cu pas (unde mergeam, cu cine mă întȃlneam, la ce oră anume), ba mă şi pozau... Eu nici nu mi-am dat seama! Cȃnd i-am istorisit chestia asta lui Ion Soare, el a adăugat rȃzȃnd: - Păăi, aveam Securitate, domnule (!) Revăzȃnd acest D.U.I. 742 „Savu”, format din 4 volume şi însumȃnd peste 1200 file! – cu menţiunea -, m-a apucat deodată o nemulţumire generală, o scȃrbă pentru această viaţă irosită, în zbatere continuă... cu o soartă mereu în cumpănă. Cȃt timp am pierdut şi cȃtă e...

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Pierderi de manuscrise Am pierdut vreo 350 pagini de manuscrise în perioada aceea. A fost şi greşeala mea că mă grăbeam să le public ori să le scot din ţară... n-am avut răbdare să aştept. Dar cine se gȃndea atunci că o să cadă dictatura, cȃnd ea părea aşa de stabilă şi tare ca betonul?! 1) Trei plicuri mari şi groase, albe, puse la poşta din Craiova, din spatele Universităţii Centrale, şi expediate către aceeaşi Chantal Signoret în primele zile din septembrie 1988 – înaintea „fugii mele”. Acestea au fost confiscate de către Securitate şi menţionate în Dosariada lor, dar neînapoiate manuscrisele (în jur de 100 pagini) - în ciuda apelului meu către CNSAS, cu toate că majoritatea erau inofensive, adică: probleme propuse şi rezolvate de matematică, rebusuri, şi poeme nepolitice. Nu înţeleg nici acum de ce nu-mi înapoiază CNSAS-ul manuscrisele? Au fost distruse? Dar... au vrut să-mi facă rău cu tot dinadinsul... Curios, că alte doua plicuri, la fel de mari, trimise din Bucureşti de la o poşta din Drumul Taberei, în aceleaşi zile, au ajuns la Chantal, care mi le-a retrimis în Turcia. Deci Securitatea din capitală nu mă avea încă în...

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Dosariada.ro (prefaţă) ................................................................. 4 Spicuiri din scrisori................................................................................... 15 Pierderi de manuscrise....................................................................... 16 Erori şi exagerări informative.................................................. 18 Scrisoare de la CNSAS........................................................................ 22 D. U. I . 742 “Savu” .................................................................. 24 Vol. 1: (Note sau Rapoarte informative, Note de relaţii, Note extrase, Note explicative, Note de analiză, Note de stadiu, Note de filare şi identificare, Radiograme, Telegrame, Serviciul Contrainformaţii, Situaţii operative, Planuri de măsuri, Date de interes operativ, Cereri de verificari la evidenţa populaţiei, Dispoziţii de verificări, Legăturile obiectivului, Fotografii camuflate)................ 25 Vol. 2: (Scrisori trimise sau primite interceptate, Relatări verbale, Manuscrise interceptate, Traduceri de scrisori şi manuscrise interceptate din engleză şi fran...

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Generally, combinatorics deals with twofold: Question 1.1. to determine or find structures or properties of configurations, such as those structure results appeared in graph theory, combinatorial maps and design theory,..., etc.. Question 1.2. to enumerate configurations, such as those appeared in the enumeration of graphs, labeled graphs, rooted maps, unrooted maps and combinatorial designs,...,etc.....

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Combinatorial Speculation and Combinatorial Conjecture for Mathematics BY LINFAN MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Structures of Cycle Bases with Some Extremal Properties BY HAN REN and YUN BAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 The Crossing Number of K1,5,n BY HANFEI MEI and YUANQIU HUANG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Pseudo-manifold Geometries with Applications BY LINFAN MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Minimum Cycle Base of Graphs Identified by Two Planar Graphs BY DENGJU MA and HAN REN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 A Combinatorially generalized Stokes Theorem on Integrations BY LINFAN MAO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A No...

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This book is organized into 5 chapters. The first chapter introduces real neutrosophic complex numbers. Chapter two introduces the notion of finite complex numbers; algebraic structures like groups, rings etc are defined using them. Matrices and polynomials are constructed using these finite complex numbers. Chapter three introduces the notion of neutrosophic complex modulo integers. Algebraic structures using neutrosophic complex modulo integers are built and around 90 examples are given. Some probable applications are suggested in chapter four and chapter five suggests around 160 problems some of which are at research level....

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In this chapter we for the first time we define the notion of integer neutrosophic complex numbers, rational neutrosophic complex numbers and real neutrosophic complex numbers and derive interesting properties related with them....

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Preface 5 Chapter One REAL NEUTROSOPHIC COMPLEX NUMBERS 7 Chapter Two FINITE COMPLEX NUMBERS 89 Chapter Three NEUTROSOPHIC COMPLEX MODULO INTEGERS 125 Chapter Four APPLICATIONS OF COMPLEX NEUTROSOPHIC NUMBERS AND ALGEBRAIC STRUCTURES USING THEM 175 Chapter Five SUGGESTED PROBLEMS 177 FURTHER READING 215 INDEX 217 ABOUT THE AUTHORS 220...

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Professor Florentin Smrandache is invited to spend a period of three months time at the Research Institute of Extension Engineering at Guangdong University of Technology in China in order to conduct research on extenics....

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Cu moralul extrem de ridicat, alerg pe autostrada I-40 din Gallup spre aeroportul din Albuquerque. Urlă muzica. Ferestrele deschise. Muzică populară românească, din Banat şi Oltenia, la casetă. Am înregistraţi şi pe cântăreţii mei din Bălceşti: Vasile Oprea (Vasilică a lu’ Tirina, vecinul meu), Mărin Covrig (coleg de şcoală primară şi generală), Gheorghe Lupu (mai mare cu doi ani ca mine, stătea peste drum de casa mea)....

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Application / 3 Invitation / 8 Extensica. Fotojurnal instantaneu din Canton / 9 Certificate / 170 Letter / 171

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This paper contains exercises and problems of algebra and grouped chapters, the upper classes of secondary schools and coli culture media general. Its purpose is to prepare students from high schools mathematics categories and will be helpful in working independently. Also work can be used for extra school work because the reader will find in it important formulas, theorems, concepts, and basic definitions which are not always included in school textbooks....

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Prezenta lucrare conţine exerciţii §i probleme de algebră, grupate pe capitole, pentru clasele superioare de licee §i §coli medii de cultură generală. Scopul ei este pregătirea matematică a elevilor din liceele de toate categoriile §i va fi utilă în lucrul de sine stătător. De asemenea, lucrarea poate fi folosită pentru lucrul extra§colar, deoarece cititorul va găsi în ea teoreme §i formule importante, noţiuni §i definiţii de bază care nu întotdeauna sunt incluse în manualele §colare....

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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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F. Smarandache has defined many sequences of integers formed by concatenating the positive integers in specific ways. A question that is often asked about most of these sequences concerns the number of primes in the sequence. As Paul Erdos pointed out, determining how many primes are in sequences of this type is a problem that the current state of mathematical knowledge is unprepared to solve. In general, the sequences grow rather fast, so after a very short time, the terms are too large for modern computers to determine if they are prime. The only hope is to find some property of the elements of the sequence that can be used to very quickly determine if it can be factored....

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Using the bases from three through nine to form new sequences, we can then ask the same question regarding the number of primes in the sequence. That appears to be a question just as difficult as the one for base ten numbers, and is summarized in the following statement....

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Preface.3 -- Table of Contents.4 -- Chapter 1 Sequences Formed by Concatenating All the Positive Integers.5 -- Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers .44 -- Chapter 3 Smarandache Stereograms .70 3.1 -- The Stereogram.70 -- 3.2 Altering the Background.87 -- 3.3 Additional Stereograms .91 -- 3.4 Possibilities for Additional Figures .92 -- Chapter 4 Constants Involving the Smarandache Function .95 -- Index.133 --...

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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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Surfing on the Ocean of Numbers - why this title? Because this little book does not attempt to give theorems and rigorous proofs in the theory of numbers. Instead it will attempt to throw light on some properties of numbers, nota bene integers, through a study of the behaviour of large numbers of integers in order to draw some reasonably certain conclusions or support already made conjectures. But no matter how far we extend our search or increase our samples in these studies we are in fact, in spite of more and more powerful technologies, merely skimming the surface of the immense sea of numbers. - Hence the title....

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Number theory is an ancient subject, but we still cannot answer many simplest and most natural questions about the integers. Some old problems have been solved, but more arise. All the research for these ancient or new problems implicated and are still promoting the development of number theory and mathematics....

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Kuai kekahi keiki i ka ohia a me ka alani i na keneta he 12, no ia mau mea. Ua oi pakolu hoi na keneta o ka alani imua o ko ka ohia. Ehia na keneta o kela a o keia? E kau iho i ka w i hoailona no na keneta o ka ohia. A o ka w ke kumukuai i ka ohia, a he pakolu ko ka alani i ko ka ohia; nolaila, he mau w ekolu ke kumukuai i ka alani. He w hookahi ko ka ohia, a he akolu mau w ko ka alani, ina e huia lakou, he mau w eha o ka huina. Aka, he 12 na keneta i lilo no ia mau mea a elua, nolaila, ua like na w eha me ia mau keneta he 12. A, ina he like na w eha me ka 12, alaila, ua like ka w kookahi me ka hapaha o ka 12, oia 3 mau keneta. O ka w ka hoailona o na keneta kuai i ka ohia, nolaila, ua make ka ohia i na keneta 3. Ina i make ka ohia i 3 keneta, ua pakolu na keneta i make ai ka alani, oia 3 ekolu keneta. Nolaila, he 9 na keneta ke kumukuai i ka alani. Ma keia hana ana, ua kauia ka w i hoailona no ka mea i ike ole ia; oia no na keneta o ka ohia, a o ka alani. E hiki pono no ka mea e ae i hoailona. Aka, ma keia helu, he mau no ke kakau i na hua hope o ka Pia-pa, i m, n, a me ka w, i hoailona no na mea i ike ole ia; a e hoa...

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Ua oi pa 4 aku na makahiki o Ioane imua o ko Iakobo; a o ka huina o ko laua mau. makahiki, he 20 ia. Ehia na makahiki o kela, o keia? E hoailona i na makahiki o Iakobo i ka w, no ka mea, he pa 4 na makahiki o Ioane i ko Iakobo, 4 mau w ka hoailona o kona mau makahiki. Nolaila, hookahi w a me 4 w, oia no 5 w ka huina o ko laua mau makahiki. Aka, he 20 ka huina o ko laua mau makahiki; nolaila, ua like 5 w me ka 20, a o ka w hookahi me ka hapa 5 o ka 20, oia na makahiki 4; oia ko Iakobo mau makahiki. Ina hoi ua like ka w hookahi me ka 4, alaila, ua like na w 4 me ka 16, oia no na makahiki o Ioane. Aole kakou e kakau hookahi w, elua w, ekolu w, aka, e kakau no i ka huahelu mamua o ka hoailona; penei, he w, 2 w, 3 w, 4 w, a pela aku. Ua kapaia na hua helu imua o ka w, he heluna. Ina aole i kauia. ka heluna imua o ka hua, hookahi wale no ia; penei, he w, a 1 w, ua like pu no laua. I ka hana ana i ko ka ui 2, ka hoailona no na makahiki o Ioane, 4 w ia. A i ka loaa ana o kona makahiki, ua hoonuiia ka w me ka 4, 4 w. Nolaila, o na hoailona a pau, e kau ana ka heluna mamua ana, ua hoonuiia ua hoailona la i kona heluna. Penei, in...

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This is an amazing little volume and a paradox all of its own. It is published in the United States, in a French translation of a compilation of essays from the original Romanian made jointly by Paul Georgelin and the man it is all about: Smarandache....

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POEMATIO LUDENS A1. En bas d’en haut Rien n’est plus beau En haut d’en bas… Quel bla bla bla : Règnent Ne et Pas !

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C.M.Popa – Le mouvement littéraire paradoxiste -------- 7 F.Smarandache – Neutrosophic degree of a paradoxicity 9 L.Kuciuk, M.Antholy – Paradoxism in geometry -------- 13 Un paradox aritmetic ----------------------------------------- 18 Lediana Paja – Paradoksizmi – rryma më e fundit… ---- 19 A.Balog – Viaţa şi opera scriitorului F. Smarandache --- 24 Paradoxurile bogăţiei ----------------------------------------- 33 Azulejo colgado en muro de Sevilla ------------------------ 34 Quantum Smarandache quasi paradoxes and quantum Smarandache sorites paradoxes --------- 34 T.Negoescu – În miezul paradoxismului ------------------- 39 Folclor românesc din America – Balada lui… ------------ 40 Taifasuri – chat: J.Nică – M. Monu – F.Smarandache --- 43 Folclor paradoxist România – Decalog -------------------- 62 Alt paradox aritmetic ----------------------------------------- 62 Mirifica Românie în imagini -------------------------------- 63 Folklore English ----------------------------------------------- 65 Auzite din popor ---------------------------------------------- 65 Constituţia României revizuită în 2010 (pamflet) ---...

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Florentin Smarandache, an American mathematician of Romanian descent has generated a vast variety of mathematical problems. Some problems are easy, others medium, but many are interesting or unsolved and this is the reason why the present book appears. Here, of course, there are problems from various types. Solving these problems is addictive like eating pumpkin seed: having once started, one cannot help doing it over and over again....

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Preface 5 Definitions, Problems, Theorems 7 References 76

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A collection of short nonfiction works in the public domain. The selections included in this collection were independently chosen by the readers, and the topics encompass history, travel, embroidery, science, mathematics, humor, philosophy, poliltics, and nature. (summary by J. M. Smallheer)...

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Essay/Short nonfiction