Description Details
...Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several ...

Excerpt Details
... dealing with linear models. Bimatrices which we are going to define in this chapter are still a powerful and an advanced tool which can handle over one linear model at a time. Bimatrices will be useful when time bound comparisons are needed in the analysis of the model....

Full Text Search Details
...SBN: 1931233-95-0 HEXIS Publ. Hse., Phoenix, Arizona Standard Address Number: 297-5092 Printed in the United States of America 3 C... ...tes of America 3 CONTENTS Preface 5 Chapter One BIMATRICES AND THEIR PROPERTIES 1.1 Definition and basic opera... ... ABOUT THE AUTHORS 181 5 Preface Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic model... ... applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form... ...ave started with. Only this incompleteness led to the invention of complex number or the imaginary number “i”. (Complex number i = 1 − ). But, aft... ...ive with zero, R the reals, Q the set of rationals and C the set of complex numbers. Z n will denote the set of integers modulo n. 8 1.1 Defi... ...rtible. 169 BIBLIOGRAPHY Here we suggest the following book which my help the reader for further development of bimatrices. 1. ABRAHAM,...

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

Full Text Search Details
...ooks-otherformats.htm ISBN: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed i... ...se we also define the Smarandache semifields of level II. This study makes one relate, compare and contrasts weaker and stronger structures of the ... ... outset that the reader has a strong background in algebra that will enable one to follow and understand the book completely. This book consists ... ...mivector spaces are introduced in the third chapter. Chapter four, which is one of the major parts of this book contains a complete systematic intro... ...ache notions in semirings and its generalizations. I deeply acknowledge my children Meena and Kama whose joyful persuasion and support encourage... ...contained. The serious study of semirings is very recent and to the best of my knowledge we do not have many books on semirings or semifields or semi... ...∗ b = b ∗ a for all a, b ∈ S we say S is a commutative semigroup. If the number of elements in the semigroup S is finite we say S is a finite semi... ... G a • b = b • a. A group, which is not abelian, is called non-abelian. The number of distinct elements in G is called the order of G; denoted by ο ... ...ND ITS PROPERTIES The study of the concept of semiring is very meagre. In my opinion I have not come across a textbook that covers completely all ...

Full Text Search Details
...e/eBooks-otherformats.htm ISBN: 1-931233-59-4 Standard Address Number: 297-5092 Printed in the United States of America ... ...he concepts are completely understood by illustrating with examples and a great number of problems. Solutions to all the problems need extraordinary... ...tions. Smarandache notions are revolutionary because group theory does not make one think of the unthinkable that can naturally occur but Smarandach... ... see how the mixture of a group and a semigroup behaves. I deeply acknowledge my family members for their constant encouragement and support which... ... Abaddaba, Oradea, 78-81, (2000). 7 CHAPTER ONE One of the essential and outstanding fe... ...e world of Smarandache notions in groups and Smarandache semigroups. In Chapter one, we introduce some basic notation, Binary relations, mappings an... ...2, 3) be the set S. Let S(3) denote the set of all mappings of S to itself. The number of elements in S(3) is 27 = 3 3 . 10 Example 1.2.2: Let...

Description Details
...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 proble...

Full Text Search Details
...RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (Collected papers) Edited by ZHANG WEN... ... Hexis 200 4 RESEARCH ON SMARANDACHE PROBLEMS IN NUMBER THEORY (Collect ed papers) Edited by ... ...by Hexis (Phoenix, USA) and Zhang Wenpeng, a nd A uthors . Many books one can download from the Digit al Library : http://www.gallup.unm.edu/~sm... ...ats.htm ISBN : 1 - 931233 - 88 - 8 Standard Address Number : 297 - 5092 Printed in the United States of America This book ... ...udied in several papers of this book. Most authors of these papers are my students. After this chance, I hope they will be more interested in th... ...ical and P hysical S ciences of N. S. F. C. I would also like to thank my student s Xu Zhefeng and Zhang Xiaobeng for the ir careful typeset and ... ...u Zhefeng and Zhang Xiaobeng for the ir careful typeset and design works. My special gratitude is due to all contributors of th is book for their gre... ...ed great interests in them (See references [2], [3]). But it seems that no one knows the relationship between these two arithmetic func- tions before.... ...THEORY Corollary. For any real number , we have the asymptotic formula 2. One simple lemma To complete the proof of the theorem, we need the followin...

Description Details
... 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 se...

Excerpt Details
...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....

Table of Contents Details
...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 Involvin...

Full Text Search Details
.../~smarandache/eBooks-otherformats.htm ISBN: 1-931233-23-3 Standard Address Number: 297-5092 Printed in the United States of America 3 Preface This i... ... website http://www.gallup.unm.edu/~smarandache/. The deepest thanks go to my mother Paula Ashbacher, who encouraged me to play sports, but in the off... ...his proved to be a wise career strategy. Finally, I would like to express my deep love for Kathy Brogla, my partner/soul mate/best friend. So pretty ... ...y single day. She is a remarkable woman and I am so blessed to have her in my life. Kathy is also the creator of the image on the front cover. Dedicat... ...rs …………………. 5 Chapter 2 Sequences Formed by Concatenating Selected Natural Numbers …………………. 44 Chapter 3 Smarandache Stereograms ………………….. 70 3.1 The ... ... 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 ho... ...es 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. Pro... ...t write text to the file"); } } theCount=theCount.add(BigInteger.ONE); } System.out.println("The search was through "+UPPERLIMIT); try... ...table is that there are no primes for the base 4 and 10 sequences and only one small one for the base 8. To further pursue the matter, the program was...

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

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

Full Text Search Details
...s.htm ISBN: 1-931233-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ... the study of loop rings and above all Smarandache loop rings. Further in my opinion the deeper study of loop rings will certainly branch off into a... ...150 problems which will help any researcher. I first deeply acknowledge my indebtness to Dr. Minh Perez of American Research Press, whose constant... ...tant mental support and encouragement made me write this book. I thank my husband Dr. Kandasamy and my daughters Meena and Kama, who tirelessly w... ...) is a mapping [x, y] → (x, y) of V × V into the set of (real or complex) numbers such that (ax + bx', y) = a (x, y) + b (x', y) and (x, a... ...tor space V is a mapping [x, y] → (x, y) of V × V into the set of complex numbers such that (x, y) = ) x , y ( and (ax + bx', y) = a (x, y) + b (x',... ...oroial space as given in [43]. We further illustrate with examples. As no one has worked on the Smarandache properties of Smarandache K-vectorial sp... ...ra over Q. Thus by this example we illustrate that there can be more than one proper subset which is a K-algebra, for K = Q in this example. Now we ... ...es the notion of Smarandache loop rings. It has five sections. In section one we just recall the definition of loops and Smarandache loops and give ...

Full Text Search Details
...her of fuzzy sets, remarked that: "So, this whole thing started because of my perception at that time, that the world of classical mathematics – was ... ...oks-otherformats.htm ISBN: 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS ... ...inted in the United States of America 3 CONTENTS Preface 5 PART ONE 1. SOME RESULTS ON FUZZY ALGEBRA 1.1 Fuzzy subsets 9 1.2 Grou... ...h those concepts that clearly have relevant applications. When I informed my interest in writing this book, Dr. Minh Perez of the American Research ... ... rigour that is expected of serious research, are now a permanent part of my personal faith. Periyar's ideology and political praxis have influenced... ... to dissent and to dare. 7 PART ONE 8 PART ONE 9 Chapte... ...E 8 PART ONE 9 Chapter One SOME RESULTS ON FUZZY ALGEBRA This chapter has twelve sections. F... ...f p ) x ( 3 2 1 Clearly µ is a fuzzy subgroup of a group G and o(µ ) = number of elements of the set {x ∈ G | µ (x) = µ (e)} = number of elements... ...by an example. 25 Example 1.2.10: Let X = N be the set of all natural numbers and µ be defined by µ (x) = x 1 for every x ∈ X. Now consider th...