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This book systematically introduces the works obtained by using analytic methods on Smarandache problems, the book includes the basic knowledge of analytic number theory, mean value on some Smarandache sequences, infinite series involving some Smarandache functions, hybrid mean value of divisor function and so on. This book could open up the reader’s perspective, and inspire the reader to these fields....

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A book on number theory.

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A book on problems arising with some of Florentin Smarandache's theories.

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The main purpose of this paper is using the elementary method to study the mean value properties of the Smarandache function, and give an interesting asymptotic formula....

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x1. Introduction In reference [1], the Smarandache Sum of Composites Between Factors function SCBF(n) is defined as: The sum of composite numbers between the smallest prime factor of n and the largest prime factor of n. For example, SCBF(14)=10, since 2£7 = 14 and the sum of the composites between 2 and 7 is: 4 + 6 = 10. In reference [2]: A number n is called simple number if the product of its proper divisors is less than or equal to n. Let A denotes set of all simple numbers. That is, A = f2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 13; 14; 15; 17; 19; 21. ...

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On the Smarandache function and square complements 1 Zhang Wenpeng , Xu Zhefeng On the integer part of the k -th root of a positive integer 5 Zhang Tianping , Ma Yuankui Smarandache “Chopped” NN and N + 1N¡1 9 Jason Earls The 57 -th Smarandache’s problem II 13 Liu Huaning , Gao Jing Perfect Powers in Smarandache n - Expressions 15 Muneer Jebreel Karama On the m -th power residue of n 25 Li Junzhuang and Zhao Jian Generalization of the divisor products and proper divisor products sequences 29 Liang Fangchi The science of lucky sciences 33 Jon Perry Smarandache Sequence of Unhappy Numbers 37 Muneer Jebreel Karama On m -th power free part of an integer 39 Zhao Xiaopeng and Ren Zhibin On two new arithmetic functions and the k -power complement number sequences 43 Xu Zhefeng Smarandache Replicating Digital Function Numbers 49 Jason Earls On the m -power residues numbers sequence 53 Ma Yuankui , Zhang Tianping Smarandache Reverse Power Summation Numbers 57 Jason Earls Some Smarandache Identities 59 Muneer Jebreel Karama On the integer part of a positive integer’s k -th root 61 Yang Hai , F...

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A short collection of mathematical papers from various authors relating to Florentin Smaradache's theories.

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This dual will be studied in a separate paper (in preparation). 2. The additive analogues of the functions 5 and 5. are real variable functions, and have been defined and studied in paper [31. (See also one book [61, pp. 171-174). These functions have been recently further extended, by the use of Euler's gamma function, ill place of the factorial (see [1]). We note that in what follows, we could define also the additive analogues functions by the use of Euler's gamma function. However, we shall apply the more transparent notation of a factorial of a positive integer....

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Jozsef Sandor ON ADDITIVE ANALOGUES OF CERTAIN ARITHMETIC FUNCTIONS 29-32 M. Perez (Ed.) ON SOME SMARANDACHE PROBLEMS 33-38 M. Perez (Ed.) ON SOME PROBLEMS RELATED TO SMARANDACHE NOTIONS 39-40 G. Gregory (Ed.) GENERALIZED SMARANDACHE PALINDROME 41 M. Vassilev - lVlissana ON 15-TH SMARANDACHE'S PROBLEM 42-45 K. Atanassov ON THE SECOND SMARANDACHE'S PROBLEM 46-48...

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This book has six chapters. The first one is introductory in nature. Second chapter introduces complex modulo integer groupoids and complex modulo integer loops using C(Zn). This chapter gives 77 examples and forty theorems. Chapter three introduces the notion of nonassociative complex rings both finite and infinite using complex groupoids and complex loops. This chapter gives over 120 examples and thirty theorems. Forth chapter introduces nonassociative structures using complex modulo integer groupoids and quasi loops. This new notion is well illustrated by 140 examples. These can find applications only in due course of time, when these new concepts become familiar. The final chapter suggests over 300 problems some of which are research problems....

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THEOREM 2.1: Let G = {C(Zn), *, (t, u); t, u ∈ Zn} be a complex modulo integer groupoid. If H ⊆ G is such that H is a Smarandache modulo integer subgroupoid, then G is a Smarandache complex modulo integer groupoid. But every subgroupoid of G need not be a Smarandache complex modulo interger subgroupoid even if G is a Smarandache groupoid. Proof is direct and hence is left as an exercise to the reader. Example 2.28: Consider G = {C(Z8), *, (2, 4)}, a complex modulo integer groupoid....

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This book is organized into seven chapters. Chapter one is introductory in content. The notion of neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras are introduced and their properties analysed in chapter two. Chapter three introduces the notion of neutrosophic semigroup linear algebras and neutrosophic group linear algebras. A study of their substructures are systematically carried out in this chapter. The fuzzy analogue of neutrosophic group linear algebras, neutrosophic semigroup linear algebras and neutrosophic set linear algebras are introduced in chapter four of this book. Chapter five introduces the concept of neutrosophic group bivector spaces, neutrosophic bigroup linear algebras, neutrosophic semigroup (bisemigroup) linear algebras and neutrosophic biset bivector spaces. The fuzzy analogue of these concepts are given in chapter six. An interesting feature of this book is it contains nearly 424 examples of these new notions. The final chapter suggests over 160 problems which is another interesting feature of this book....

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Now we proceed onto define the notion of neutrosophic subgroup of a neutrosophic group. DEFINITION 1.2: Let N(G) = (GuI) be a neutrosophic group generated by G and I. A proper subset P(G) is said to be a neutrosophic subgroup if P(G) is a neutrosophic group i.e. P(G) must contain a (sub) group. Example 1.3: Let N(Z2) = 􀂢Z2 􀂉 I􀂲 be a neutrosophic group under addition. N(Z2) = {0, 1, I, 1 + I}. Now we see {0, I} is a group under + in fact a neutrosophic group {0, 1 + I} is a group under ‘+’ but we call {0, I} or {0, 1 + I} only as pseudo neutrosophic groups for they do not have a proper subset which is a group. So {0, I} and {0, 1 + I} will be only called as pseudo neutrosophic groups (subgroups). We can thus define a pseudo neutrosophic group as a neutrosophic group, which does not contain a proper subset which is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different. Now we see a neutrosophic group ...

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Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET NEUTROSOPHIC LINEAR ALGEBRA 13 2.1 Type of Neutrosophic Sets 13 2.2 Set Neutrosophic Vector Space 16 2.3 Neutrosophic Neutrosophic Integer Set Vector Spaces 40 2.4 Mixed Set Neutrosophic Rational Vector Spaces and their Properties 52 Chapter Three NEUTROSOPHIC SEMIGROUP LINEAR ALGEBRA 91 3.1 Neutrosophic Semigroup Linear Algebras 91 3.2 Neutrosophic Group Linear Algebras 113 Chapter Four NEUTROSOPHIC FUZZY SET LINEAR ALGEBRA 135 Chapter Five NEUTROSOPHIC SET BIVECTOR SPACES 155 Chapter Six NEUTROSOPHIC FUZZY GROUP BILINEAR ALGEBRA 219 Chapter Seven SUGGESTED PROBLEMS 247 FURTHER READING 277 INDEX 280 ABOUT THE AUTHORS 286 ...

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This book will mainly make part of the research results of current domestic and foreign scholars on Smarandache problems and unsolved problems into a book. Its main purpose is to introduce some of the research of Smarandache problems to readers, comprehensively and systematically, including the mean value of arithmetic functions, identities and inequalities, infinite series, the solutions of special equations, and put forward to some new interesting problems. We hope that the readers could be interested in these issues. At the same time, this book could open up the reader’s perspective, guide and inspire the readers to these fields....

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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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This volume contains basic mathematics (in Hawaiian). It teaches you the numbers in Hawaiian up to one hundred and also basio useful mathematics.

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Ehia kahi iloko o ka 10? He 10 a me na kahi ehia iloko o ka 12? He 10 a me na kahi ehia iloko o ka 13? 14? 16? 19? 15? 18? 17? 11? Ehia na umi iloko o ka 20? iloko o ke 30? 40? 60? 80? 60? 70? 50? 90? 100? Ehia na umi a me na kahi iloko o ka 21? iloko o ka 23? 28? 26? 32? 35? 37? 44? 49? 41? 53? 57? 62? 65? 68? 71? 76? 99? 85? 87? 88? 92? 94? 99? He umi a me 1, heaha ia? 10 me 3? 10 me 7? 10 me 9? 2 umi? 2 umi me 1? 2 umi me 5? 2 umi me 7? 3 umi? 3 umi me 2? 3 umi me 8? 4 umi? me 6? 5 umi? 5 umi me 3? 5 umi me 5? 6 umi? 6 umi me 4? 7 umi? 8 umi? 8 umi me 6? 9 umi? 9 umi me 2? 9 umi me 9? 10 ka umi? E kakau oe ma na huahelu i ka papa i hookahi; i elua; a pela a hiki i ka umi. E kakau ma na huahelu i ka umikumamaha, umikumamaono, a hiki i ka iwakalua, a hiki i ke kanakolu, a ke kanaha, a ke kanalima, a ke kanaono, a hiki i ka haneri....

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This volume teaches you children's basic arithmetic in Hawaiian.

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No ka hana ana i keia Helu, e ahu no ke kumu i mau hua poepoe he kanaha a keu paha i mea heluia; pela no kela keiki keia keiki e ahu no lakou i na hua like. A like me ka hana ana a ke kumu, pela hoi e hana?i kela keiki keia keiki i kana mau hua iho....

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În perioada 24-25 aprilie 2010 s-a desfăşurat A Şasea Conferinţă Internaţională în Teoria Numerelor şi Noţiuni Smarandache la Universitatea Normală din Tianshui, China. Organizatorii conferinţei au fost prof. univ. dr. Zhang Wenpeng de la Universitatea Nordvest din Xi’an şi prof. univ. dr. Wangsheng He de la Universitatea Normală din Tianshui. Lucrările conferinţei au fost publicate într-un volum special în care se găsesc şi articole matematice despre noţiuni legate de cercetările omului de ştiinţă roman, Florentin Smarandache, precum: semigrupuri, inele neasociative, funcţii aritmetice, şi conjecturi sau probleme Smarandache etc. Cartea poate fi descărcată din site-ul. During the period 24 to 25 April 2010 was conducted Sixth International Conference on Number Theory and Smarandache Notions of Tianshui Normal University, China. Conference organizers were prof. Dr. Zhang Wenpeng together from Northwest University in Xi'an and prof. Dr. Wangsheng He Tianshui Normal University. The conference proceedings were published in a special volume that are mathematics articles about concepts and research related to Romanian scientist, Flore...

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În loc de prefaţă – de Ion Pătraşcu .................................... 8 I. Pe aripile... zborului, în Dayton – Ohio! ........................ 11 . „Înaripare” spre Dayton ............................................. 12 . DWF – marele „nod aerian” ...................................... 13 . Conferinţă de... 1 Mai! ............................................... 16 . La Muzeul Naţional al Aviaţiei Militare .................... 19 . Paradoxurile Fraţilor Wright ...................................... 28 . Totuşi, cine a zburat primul? ..................................... 26 . „Nebuni” contra curentului ........................................ 28 . „Aripile de liliac” ....................................................... 30 . Bimotorul mamut ....................................................... 32 . Cu avionul priponit pe... gheaţă! ................................ 33 . Motorul cu... aer comprimat! ..................................... 34 . De la motorul cu aburi, la cel cu kerosen ................... 35 . Un inventator ciudat ................................................... 37 . Prototipul avionului uşor ........

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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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In this book the authors study dual numbers in a special way. The main aim of this book is to find rich sources of new elements g such that g2 = 0. The main sources of such new elements are from Zn, n a composite number. We give algebraic structures on them. This book is organized into six chapters. The final chapter suggests several research level problems. Fifth chapter indicates the applications of dual numbers. The forth chapter introduces the concept of interval dual numbers, we also extend it to the concept of neutrosophic and 8 fuzzy dual numbers. Higher dimensional dual numbers are defined, described and developed in chapter three. Chapter two gives means and methods to construct the new element g such that g2 = 0. The authors feel Zn (n a composite positive integer) is a rich source for getting new element, the main component of the dual number x = a + bg....

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Dedication 5 Preface 7 Chapter One INTRODUCTION 9 Chapter Two DUAL NUMBERS 11 Chapter Three HIGHER DIMENSIONAL DUAL NUMBERS 55 Chapter Four DUAL INTERVAL NUMBERS AND INTERVAL DUAL NUMBERS 89 Chapter Five APPLICATION OF THESE NEW TYPES OF DUAL NUMBERS 129 Chapter Six SUGGESTED PROBLEMS 131 FURTHER READING 153 INDEX 156 ABOUT THE AUTHORS 159...

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In this paper we defme a function L with will allowus to generalize (separately or simultaneously) some theorems from Numbers Theory obtained by Wilson, Fermat, Euler, Gauss, Lagrange, Leibnitz, Moser, Sierpinski. (Some papers not included in the volume were confiscated by the Secret Police in September 1988, when the author left Romania. He spent 19 months in a Turkish political refugee camp, and emigrated to the United Stales in March 1990. Despite efforts by his fric; is, the papers were not rcncvcred ...)...

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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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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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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 line of thought explored herein is the correspondence between cosmological phenomena and condensed matter physics, and therefore we can think that the quantization of orbit distances can be caused by superfluid helium quantization. This issue is explored by F. Smarandache and V. Christianto. Moreover, F. Smarandache also discusses possible new era of research that is pertaining to superluminal physics and instantaneous physics. Ion Patrascu and D. Rabounski discuss superluminality from their perspectives. And M. Pereira discusses his Hypergeometrical Universe model....

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Preface III Contents IV Planetary orbits in Solar and Extrasolar systems (A. Rubcic & J. Rubcic) Fizika A, 19 no.3, 2010...................1 Areal velocities of planets and their comparison (P. Pintr, V. Per\inova% , A. Luks\) unpubl. 2011...................15 Distribution of distances in solar system (P. Pintr, V. Perinova , A. Luks) Chaos , Soliton, Fractals 2007...................27 New cosmological model of universe and possible quantization of the Hubble parameter (P.Pintr) unpublished paper, Dec. 2011...................39 Do we really understand the solar system? (M. Pitkanen) Nov. 27th 2011...................50 Inflation and TGD (M. Pitkanen) Dec. 10th 2011...................70 QCD and TGD (M. Pitkanen) Dec. 19th 2011...................78 Quantum arithmetics and the relationship between real and p-adic physics (M. Pitkanen) Dec. 12th 2011...................94 A blind pilot: who is super-luminal observer? (D. Rabounski) Progress in Physics Vol.2, 2008................126 Scientist deduced the existence of particles with faster-than-light speeds recently discovered by CERN. (I. Patrascu) Progress in Physics Vol.4, 20...

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This book contains 23 papers, most of which were written by participants to the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo University, China, in March, 2009. In this Conference, several professors gave a talk on Smarandache Notions and many participants lectured on them both extensively and intensively. All these papers are original and have been refereed. The themes of these papers range from the mean value or hybrid mean value of Smarandache type functions, the mean value of some famous number theroretic functions acting on the Smarandache sequences, to the convergence property of some infinite series involving the Smarandache type sequences. ...

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3. Remarks Sandor [2] has considered the problem of finding the S-perfect and completely S-perfect numbers, but his proof is not complete. He has proved that the only S-perfect of the form n = p q is n = 6 and there is no S-perfect number of the form n = 2kq where k ¸ 2 is an integer and q is an odd prime. On the other hand, Theorem 2.1 gives all the S-perfect numbers. Again, Sandor only proved that, the only completely S-perfect number of the form n = p2q is n = 28, and all completely S-perfect numbers are given by Theorem 2.2. Theorem 2.1 and Theorem 2.2 ¯nd respectively the S-perfect and completely S-perfect numbers when S(1) = 1. The situation is quite different if one adopts the convention that S(0) = 1. In the latter case, as has been proved by Gronas [3], all completely S-perfect numbers are n = p(prime), 9, 16, 24. All that is known about the S-perfect numbers is that, among the ¯rst 106 numbers, n = 12 is the only S-perfect number (see Ashbacher [4]). In exactly the same way, the Z-perfect and completely Z-perfect numbers may be defined. Thus, given an integer n, 1. ...

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J. Wang : An equation related to the Smarandache power function 1 X. Lu and J. Hu : On the F.Smarandache 3n-digital sequence 5 B. Cheng : An equation involving the Smarandache double factorial function and Euler function 8 A. A. K. Majumdar : S-perfect and completely S-perfect numbers 12 B. Zhao and S. Wang: Cyclic dualizing elements in Girard quantales 21 P. Chun and Y. Zhao : On an equation involving the Smarandache function and the Dirichlet divisor function 27 F. Li and Y. Wang : An equation involving the Euler function and the Smarandache m-th power residues function 31 H. Gunarto and A. Majumdar : On numerical values of Z(n) 34 K. Nagarajan, etc.: M-graphoidal path covers of a graph 58 H. Liu : On the cubic Gauss sums and its fourth power mean 68 A. A. K. Majumdar : On the dual functions Z¤(n) and S¤(n) 74 S. S. Gupta : Smarandache sequence of Ulam numbers 78 L. Li : A new Smarandache multiplicative function and its arithmetical properties 83 Y. Yang and X. Kang : A predator-prey epidemic model with infected predator 86 R. Fu and H. Yang : An equation involving the Lucas numbers 90 W. Yang : Two rings in IS-a...

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In this book the authors study the erasure techniques in concatenated Maximum Rank Distance (MRD) codes. The authors for the first time in this book introduce the new notion of concatenation of MRD codes with binary codes, where we take the outer code as the RD code and the binary code as the inner code....

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We want to find efficient algebraic methods to improve the realiability of the transmission of messages. In this chapter we give only simple coding and decoding algorithms which can be easily understood by a beginner. Binary symmetric channel is an illustration of a model for a transmission channel. Now we will proceed onto define a linear code algebraically....

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Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two ALGEBRAIC LINEAR CODES AND THEIR PROPERTIES 29 Chapter Three ERASURE DECODING OF MAXIMUM RANK DISTANCE CODES 49 3.1 Introduction 49 3.2 Maximum Rank Distance Codes 51 3.3 Erasure Decoding of MRD Codes 54 Chapter Four MRD CODES –SOME PROPERTIES AND A DECODING TECHNIQUE 63 4.1 Introduction 63 4.2 Error-erasure Decoding Techniques to MRD Codes 64 4.3 Invertible q-Cyclic RD Codes 83 4.4 Rank Distance Codes With Complementary Duals 101 4.4.1 MRD Codes With Complementary Duals 103 4.4.2 The 2-User F-Adder Channel 106 4.4.3 Coding for the 2-user F-Adder Channel 107 Chapter Five EFFECTIVE ERASURE CODES FOR RELIABLE COMPUTER COMMUNICATION PROTOCOLS USING CODES OVER ARBITRARY RINGS 111 5.1 Integer Rank Distance Codes 112 5.2 Maximum Integer Rank Distance Codes 114 Chapter Six CONCATENATION OF ALGEBRAIC CODES 129 6.1 Concatenation of Linear Block Codes 129 6.2 Concatenation of RD Codes With CR-Metric 140 FURTHER READING 153 INDEX 157 ABOUT THE AUTHORS 161...

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前言 数论这门学科最初是从研究整数开始的, 所以叫做整数论. 后来整数 论又进一步发展, 就叫做数论了. 确切的说, 数论就是一门研究整数性质 的学科. 它是最古老的数学分支. 按照研究方法来说, 数论可以分成初等 数论, 解析数论, 代数数论, 超越数论, 计算数论, 组合数论等. Foreword Number theory, this discipline was originally started from the study integer, so called Number Theory. Later integer on further development of number theory called it. Rather, number theory is an integer nature of disciplines and it is the oldest branch of mathematics concerned by the study methods, can be divided into elementary number theory, number theory, analytic number theory, algebraic number theory, transcendental number theory, computational number theory, combinatorics number theory and so on....

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第一章Smarandache函数. . . . . . . . . . . . 1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . 1 1.2 关于F.Smarandache可乘数函数的一类均值. . . . . . 1 1.3 Smarandache函数值的分布. . . . . . . . . . . . . 5 1.3.1 几个引理. . . . . . . . . . . . . . . . . . . 6 1.3.2 证明. . . . . . . . . . . . . . . . . . . . . 7 1.4 Smarandache函数df (n) 的均值. . . . . . . . . . . . 9 1.5 关于F.Smarandache LCM 函数以及它的主值. . . . . . 12 1.6 Smarandache Pierced 链. . . . . . . . . . . . . . . 16 1.7 Smarandache 函数的几个相关结论. . . . . . . . . . 18 1.7.1 关于Smarandache 函数的一个等式. . . . . . . . 18 1.7.2 关于文章\一个新的算术函数的主值"的一些注释. . 20 1.7.3 Smarandache 函数的一个推广. . . . . . . . . . 23 1.7.4 关于F.Smarandache函数及其k次补数. . . . . . . 27 1.7.5 关于F.Smarandache函数的奇偶性. . . . . . . . 32 第二章伪Smarandache 函数. . . . . . . . . . . . 36 2.1 伪Smarandache 函数的定义及性质. . . . . . . . . . 36 2.2 关于伪Smarandache函数的几个定理. . . . . . . . . 38 2.3 关于伪Smarandache 函数的几个方程. . . . . . . . . 40 2.3.1 一个与Smarandache函数有关的函数方程及其正整 数解. . . . . . . . . . . . . . . . . . . . . 41 2.3.2 一个包含伪Smarandache函数及其对偶函数的方程. 42 2.3.3 一个包含伪Smarandache 函数及Smarandache 可乘 函数的方程. . . . . . . . . . . . . . . . . . 45 2.4 伪Smarandache函数的...

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前言 数论这门学科最初是从研究整数开始的, 所以叫做整数论. 后来整数 论又进一步发展, 就叫做数论了. 确切的说, 数论就是一门研究整数性质 的学科. 在我国, 数论也是发展最早的数学分支之一. 许多著名的数学著 作中都有关于数论内容的论述, 比如求最大公约数、勾股数组、某些不 定方程整数解的问题等等... Foreword Number theory, this discipline was originally started from the study integer, so called Number Theory. Later integer on further development of number theory called it. Rather, number theory is an integer nature of Discipline in our country, the development of number theory is one of the oldest branches of mathematics and many well-known mathematical forward work on number theory in both the content of discourse, such as the common denominator, Pythagorean, some do not Equation given integer solution problems, and so…....

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第一章Smarandache函数的问题及其新进展1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Smarandache非构造序列. . . . . . . . . . . . . . 1 1.3 Smarandache数字和. . . . . . . . . . . . . . . . 2 1.4 Smarandache数字乘积. . . . . . . . . . . . . . . 2 1.5 Smarandache Pierced链. . . . . . . . . . . . . . . 3 1.6 Smarandache因子乘积. . . . . . . . . . . . . . . 4 1.7 Smarandache真因子乘积. . . . . . . . . . . . . . 5 1.8 Smarandache平方补数. . . . . . . . . . . . . . . 6 1.9 Smarandache立方补数. . . . . . . . . . . . . . . 7 1.10 Smarandache广义剩余序列. . . . . . . . . . . . . 7 1.11 Smarandache素数列. . . . . . . . . . . . . . . . 8 1.12 Smarandache平方列. . . . . . . . . . . . . . . . 13 1.13 Smarandache素数可加补数. . . . . . . . . . . . . 15 1.14 Smarandache函数S(n) . . . . . . . . . . . . . . . 19 1.15 Smarandache双阶乘函数. . . . . . . . . . . . . . 31 1.16 Smarandache商函数. . . . . . . . . . . . . . . . 42 1.17 Smarandache p次幂原函数. . . . . . . . . . . . . . 43 1.18 第一类伪Smarandache素数. . . . . . . . . . . . . 43 1.19 第一类伪Smarandache平方数. . . . . . . . . . . . 44 1.20 Goldbach-Smarandache序列. . . . . . . . . . . . . 46 1.21 Vinogradov-Smarandache序列. . . . . . . . . . . . 46 ...

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This book includes part of the research results about the Smarandache problems written by Chinese scholars at present, and its main purpose is to introduce various results about the Smarandache problems, such as Smarandache function and its asymptotic properties, series convergence, solutions about special equations. At the same time, we put forward to some new interesting problems either in order to research further. We hope this booklet will guide and inspire readers to these fields....

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前言 数论这门学科最初是从研究整数开始的, 所以叫整数数论. 后来整数 数论又进一步发展, 就叫做数论了. 确切地说, 数论就是一门研究整数性 质的学科. 数论和几何学一样, 是古老的数学分支. 数论在数学中的地位是特殊的, 高斯曾经说过:“数学是科学的皇后, 数论是数学中的皇冠”. 虽然数论中的许多问题在很早就开始了研究, 并得到了丰硕的成果, 但是至今仍有许多被数学家称之为“皇冠上的明 珠”的悬而未解的问题等待人们去解决. 正因如此, 数论才能不断地充 实和发展, 才能既古老又年轻, 才能始终活跃在数学领域的前沿. Foreword Number theory, this discipline was originally started from the study integer, so called integer number theory. Later integer further development of number theory, number theory called up. Rather, number theory is an integer of study qualitative disciplines. Number theory and geometry, is an ancient branch of mathematics. Number theory in mathematics position is special, Gauss once said: "Mathematics is the queen of sciences, number theory is the mathematics of the crown. "Although many of the problems in number theory began very early in the research, And has been fruitful, but there are still many of the mathematicians call "crown Ming Pearl "of unsolved problems waiting to be solved for this reason, number theory can continue to charge Real and development in order to both old and young, can always active in the forefront of the field of mathematics....

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目录 第一章Smarandache 函数1 1.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 S(n) 函数和d(n) 函数的混合均值. . . . . . . . . . . . 4 1.3 关于F.Smarandache 函数S(mn) 的渐近性质. . . . . . . . 6 1.4 复合函数S(Z(n)) 的均值. . . . . . . . . . . . . . . . 7 1.5 是否为整数的问题. . . . . . . . . . . . . . . 10 1.6 关于函数S(n) 的一个方程. . . . . . . . . . . . . . . 13 1.7 关于函数S(nk) 的一个方程. . . . . . . . . . . . . . . 15 1.8 关于Smarandache 函数值的分布. . . . . . . . . . . . . 17 1.9 S(ak(n)) 函数的值分布. . . . . . . . . . . . . . . . . 21 1.10 两个包含Smarandache 函数的方程. . . . . . . . . . . . 25 1.11 S(n) 函数及其均值. . . . . . . . . . . . . . . . . . 27 第二章Smarandache 对偶函数 . . . . . . . . . . . . . . . . .30 2.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Smarandache 对偶函数的渐近公式. . . . . . . . . . . . 30 2.3 关于Smarandache 对偶函数的一个方程. . . . . . . . . . 33 2.4 关于Smarandache 对偶函数S¤¤(n) . . . . . . . . . . . 37 2.5 一个包含SM(n) 函数的方程. . . . . . . . . . . . . . 40 2.6 一个包含Smarandache 对偶函数的方程. . . . . . . . . . 44 第三章关于SL(n) 函数及其对偶函数的性质 . . . . . . . . . . . . . . . . .48 3.1 引言. . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 SL(n) 函数的渐近公式. . . . . . . . . . . . . . ....

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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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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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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 Dr. Perez (address: http://www.gallup.unm.edu/~smarandache/), the Smarandache Notions Journal always published by American Research Press, and several books on conjectures, functions, unsolved problems, notions and other proposed by Professor F. Smarandache in "The Florentin Smarandache papers" special collections at: the Arizona State University (Tempe, USA), Archives of American Mathematics (University of Texas at Austin, USA), University of Craiova Library (Romania), and Archives of State (Rm. Valcea, Romania). The Smarandache's universe is undoubtedly very fascinating and is halfway between the number theory and the recreational mathematics. Even though sometime this universe has a very simple structure from number theory standpoint, it doesn't cease to be deeply mysterious and interesting. This book, following the Smarandache spirit, presents new ...

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Abstract. The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the apparently unscientific ideas into scientific ones, and to develop their study (The Theory of Imperfections). And finally, to interconnect opposite (and not only) human fields of knowledge into a.s-heterogeneous-as-possible another fields. The author welcomes any co=ents, notes, articles on this paper and/or the 120 open questions bothering him, which will be published in a collective monograph about the paradoxist mathematics....

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The "Paradoxist Mathematics" may be understood as Experimental Mathematics, NonMathematics, or even Anti-Mathematics: not in a nihilistic way, but in positive one. The truly innovative researchers will banish the old concepts in oder of check, by heuristic processes, some new ones: their opposites. Don't simply follow the crowd, and don't accept to be manipulated by any (political, economical, social, even scientific, or artistic, cultural, etc.) media! Learn to conradict everuthing and everybody!! "Duibito, ergo cogito; cogito, ergo sum", said Rene Descartes, "I doubt, therefore I think; I think, therefore I exist" (metaphysical doubt). See what happens if you deny the classics' theory?...

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Paradoxist Mathematics [Anti-Mathematics, Multi-Structure and Multi-Space, Space of Non-Integer or Negative Dimention, Inconsistent Sysytem ofAxiomas or Contradictory Theory, Non-Euclidean Geometries: Paradoxist Geometry, Non-Geometry, CounterProjective Geometry and Counter-Axioms, Anti-Geometry and Anti-Axioms, Model of an Anti-Geometry, Discontinuous Geometries] 5 -- Logica sau iogica Matematica? .29 -- Mathematics and Alcohol and God . 30 -- Subjective Questions and Answers for a MathInstructor of Higher Education .32 -- o Geometrie Paradoxista .49 -- Geometric Conjecture .50 -- A Function in the ~umber Theory . 51 -- An Infinity of Unsolved Problems Concerning a Function in the :'-lumber Theory . 57 -- Solving Problems by Using a Function in the Number Theory .79 -- Some Linear Equations Involving a Function in the Number Theory .82 -- Contributii la Studiul unor Functii §i Conjecturi in Teoria Numerelor .85 -- "The Function that You Bear its Name" .112 -- Smarandache Type Functions Obtained by Duality .113 -- Func!;ii Analitice .129 -- Funct;ii Prime §i Coprime .137 -- Asupra unor Conjecturi §i Probleme Nerezolvate Referitoare ...

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We have used the 2-adaptive fuzzy model having the two fuzzy models, fuzzy matrices model and BAMs viz. model to analyze the views of public about HIV/ AIDS disease, patient and the awareness program. This book has five chapters and 6 appendices. The first chapter just recalls the definition of four fuzzy models used in this book and gives illustration of some of them. Chapter two introduces the new n-adaptive fuzzy models. Chapter three uses for the first time 2 adaptive fuzzy models to study psychological and sociological problems about HIV/AIDS. Chapter four gives an outline of the interviews. Chapter five gives the suggestions and conclusion based on our study. Of the 6 appendices four of them are C-program made to make the working of the fuzzy model simple....

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In this chapter for the first time we introduce the new class of n-adaptive fuzzy models (n a positive integer and n ≥ 2) and illustrate it. These n-adaptive fuzzy models can analyze a problem using different models so one can get in one case the hidden pattern, in one case the maximum time period in some case output state vector for a given input vector and so on. So this new model has the capacity to analyze the problem in different angles, which can give multiple suggestions and solutions about the problem. This chapter has two sections. Section one defines the new model gives illustrations and section two defines some special n-adaptive models and proposes some problems....

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Chapter One SOME BASIC FUZZY MODELS WITH ILLUSTRATIONS 1.1 Fuzzy matrices and their applications 9 1.2 Definition and illustration of Fuzzy Relational Maps (FRMs) 16 1.3 Some basic concepts of BAM with illustration 20 1.3.1 Some basic concepts of BAM 22 1.3.2 Use of BAM Model to study the cause of vulnerability to HIV/AIDS and factors for migration 28 1.4 Introduction to Fuzzy Associative Memories (FAM) 35 Chapter Two ON A NEW CLASS OF N-ADAPTIVE FUZZY MODELS WITH ILLUSTRATIONS 2.1 On a new class of n-adaptive fuzzy models with illustrations 39 2.2 Some special n-adaptive models 49 Chapter Three USE OF 2-ADAPTIVE FUZZY MODEL TO ANALYZE THE PUBLIC AWARENESS OF HIV/AIDS 3.1 Study of the psychological and social problems the public have about HIV/AIDS patients using CETD matrix 52 3.2 Use of 2 adaptive fuzzy model to analyze the problem 72 Chapter Four PUBLIC ATTITUDE AND AWARENESS ABOUT HIV/AIDS 4.1 Introduction 77 4.2 Interviews 81 Chapter Five CONCLUSIONS 167 Appendix 1. Questionnaire 177 2. Table of Statistics 191 3. C-program for CETD and RTD Matrix 198 4. C-program for FRM 202 5. C-program for CFRM 2...

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Scientia Magna is published annually in 200-300 pages per volume and 1,000 copies on topics such as mathematics, physics, philosophy, psychology, sociology, and linguistics....

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0. In 1999, the second author of this remarks published a book over 30 of Smarandache's problems in area of elementary number theory (see [1, 2]). After this, we worked over new 20 problems that we collected in our book [28]. These books contain Smarandache's problems, described in [10, 16]. The present paper contains some of the results from [28]. In [16] Florentin Smarandache formulated 105 unsolved problems, while in [10] C.Dumitresu and V. Seleacu formulated 140 unsolved problems of his. The second book contains almost all the problems from [16], but now each problem has unique number and by this reason in [1, 28] and here the authors use the numeration of the problems from [10]. In the text below the following notations are used....

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V. Mladen and T. Krassimir : Remarks on some of the Smarandache's problem. Part 2 1 W. Kandasamy : Smarandache groupoids 27 L. Ding : On the primitive numbers of power P and its mean value properties 36 D. Torres and V. Teca : Consecutive, reversed, mirror, and symmetric Smarandache sequence of triangular numbers 39 D. Ren : On the square-free number sequence 46 T. Ramaraj and N. Kannappa : On ¯nite Smarandache near-rings 49 X. Kang : Some interesting properties of the Smarandache function 52 L. Mao : On Automorphism Groups of Maps, Surfaces and Smarandache Geometries 55 L. Ding : On the mean value of Smarandache ceil function 74 M. Le : An equation concerning the Smarandache function 78 M. Bayat, H. Teimoori and M. Hassani : An extension of ABC-theorem 81 J. Ma : An equation involving the Smarandache function 89 C. Chen : Inequalities for the polygamma functions with application 91 W. Vasantha and M. Chetry : On the number of Smarandache zero-divisors and Smarandache weak zero-divisors in loop rings 96 M. Le : The function equation S(n) = Z(n) 109 Z. Li : On the Smarandache Pseudo-number Sequences 111 D. Mehendale :...