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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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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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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 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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During my research period in the Summer of 2012 at the Research Institute of Extenics and Innovation Methods, from Guangdong University of Technology, in Guangzhou, China, I have introduced the Linear and Non-Linear Attraction Point Principle and the Network of Attraction Curves, have generalized the 1D Extension Distance and the 1D Dependent Function to 2D, 3D, and in general to n-D Spaces, and have generalized Qiao-Xing Li’s and Xing-Sen Li’s definitions of the Location Value of a Point and the Dependent Function of a Point on a Single Finite Interval from one dimension (1D) to 2D, 3D, and in general n-D spaces. Then I used the Extenics, together with Victor Vlădăreanu, Mihai Liviu Smarandache, Tudor Păroiu, and Ştefan Vlăduţescu, in 2D and 3D spaces in technology, philosophy, and information theory....

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Contribution to Extenics (Preface by Florentin Smarandache): 4 1. Generalizations of the Distance and Dependent Function in Extenics to 2D, 3D, and n-D, by Florentin Smarandache: 22 2. Applications of Extenics to 2D-Space and 3D-Space, by Florentin Smarandache, Victor Vlădăreanu: 39 3. Generalization of the Dependent Function in Extenics for Nested Sets with Common Endpoints to 2D-Space, 3D-Space, and generally to n-D-Space, by Florentin Smarandache: 52 4. Generalizations in Extenics of the Location Value and Dependent Function from A Single Finite Interval to 2D, 3D, and n-D Spaces, by Florentin Smarandache, Mihai Liviu Smarandache: 60 5. Extention Transformation Used in I Ching, by Florentin Smarandache: 69 6. Extenics in Conventional and Nonconventional [Romanian], by Florentin Smarandache, Tudor Păroiu: 79 7. Extension Communication for Solving the Ontological Contradiction between Communication and Information, by Florentin Smarandache, Ştefan Vlăduţescu: 99-112...

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In The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by combinatorialization, implicated in the foreword of Chapter 5 of my book Automorphism groups of Maps, Surfaces and Smarandache Geometries (USA, 2005). This conjecture is essentially a philosophic notion for developing mathematical sciences of 21st century, which means that we can combine different fields into a union one and then determines its behavior quantitatively. It is this notion that urges me to research mathematics and physics by combinatorics, i.e., mathematical combinatorics beginning in 2004 when I was a post-doctor of Chinese Academy of Mathematics and System Science. It finally brought about me one self-contained book, the first edition of this book, published by InfoQuest Publisher in 2009. This edition is a revisited edition, also includes the development of a few topics discussed in the first edition....

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1.5 ENUMERATION TECHNIQUES 1.5.1 Enumeration Principle. The enumeration problem on a finite set is to count and find closed formula for elements in this set. A fundamental principle for solving this problem in general is on account of the enumeration principle: For finite sets X and Y , the equality |X| = |Y | holds if and only if there is a bijection f : X → Y . Certainly, if the set Y can be easily countable, then we can find a closed formula for elements in X....

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Contents Preface to the Second Edition . . . . . . . . . . . . . . . . . . . i Chapter 1. Combinatorial Principle with Graphs . . . . . . . . . . 1 1.1 Multi-sets with operations. . . . . . . . . . . . . . . . . . . . .2 1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Multi-Set . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.2 Multi-posets . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Partially ordered set . . . . . . . . . . . . . . . . . . . . .11 1.2.2 Multi-Poset . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Countable sets . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Countable set . . . . . . . . . . . . . . . . . . . . 16 1.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . .18 1.4.2 Subgraph . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.3 Labeled graph. . . . . . ...