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This series is arranged in levels of reading difficulty ranging from 1 to 5. The easiest is level 1, the hardest level 5. All these books are royalty free and can be copied used printed and distributed, scrawled on walls, acted out in plays and told (very slowly) to your pet dog, if you don’t have a dog a cat will do, but hamsters do not listen as they only speak Spanish. They are designed to be easily read on a computer and can be printed on standard A4 size paper (landscape format) There is no bad language or offensive words anywhere in these books....

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Alfi has a dog called Podd. Podd was a round with pudgy legs and a small pudgy nose. Podd liked playing ball in the garden and chasing birds round the garden. ...

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This series is arranged in levels of reading difficulty ranging from 1 to 5. The easiest is level 1, the hardest level 5. All these books are royalty free and can be copied used printed and distributed, scrawled on walls, acted out in plays and told (very slowly) to your pet dog, if you don’t have a dog a cat will do, but hamsters do not listen as they only speak Spanish. They are designed to be easily read on a computer and can be printed on standard A4 size paper (landscape format) There is no bad language or offensive words anywhere in these books....

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I'm knocking on metal That comes from err ???? Cam did not know what metal came from.

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This series is arranged in levels of reading difficulty ranging from 1 to 5. The easiest is level 1, the hardest level 5. All these books are royalty free and can be copied used printed and distributed, scrawled on walls, acted out in plays and told (very slowly) to your pet dog, if you don’t have a dog a cat will do, but hamsters do not listen as they only speak Spanish. They are designed to be easily read on a computer and can be printed on standard A4 size paper (landscape format) There is no bad language or offensive words anywhere in these books....

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"Ahh..." shouted Pod "I'm stuck on something!" "Let me see." said Spike, sticking his paw in the substance.

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Relocation is often defined as a stressful and nerve-racking experience that eats up too much time and money, leaving you emotionally and physically drained. And yet, it doesn’t have to be this way. The key to successfully writing your own happy ending to the grim moving story is to be adequately prepared for Moving day by following the required packing steps. Cleverly structured by esteemed relocation experts and, the purpose of this moving book is to help you organize and execute a smooth and trouble-free move with the least effort on your part. Be well informed. Be a smart decision-maker. Make all the right moves....

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It is universally known that moving often turns out to be a stressful experience with an insatiable appetite for time and money. The psychological impact of any relocation process cannot be predicted with any certainty and its side effects may last from a couple of weeks to a few years depending on what type of person you are. Being well prepared for the upcoming move is fundamental not only for your emotional well-being, but for that of the people around you as well....

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Introduction How To Mentally Prepare For A Move “Plans are nothing; planning is everything” It’s time to say goodbye You are what matters most A change might do you good Top 10 Best Organizing Ideas For Moving Be prudent when choosing your moving service Box it in and pack it up Devise a clever sorting system Make a survival kit Guard your valuables Get rid of excess stuff Accept any help you can get Let your camera be your pal Spread the word Make it children and pet safe Bonus idea: Start early and don’t rush yourself Top 10 Moving Mistakes And How To Avoid Them 1. Not planning your move 2. Failing to ask for an in-house estimate 3. Not obtaining multiple moving quotes 4. Not researching the moving companies 5. Choosing the company with the lowest quote 6. Bypassing items during the estimate 7. Packing poorly the day before the move 8. Failing to understand your insurance options 9. Entrusting irreplaceable valuables in the hands of the movers 10. Not understanding the moving paperwork Non-Allowable Items: What To Do When Movers Say No Hazardous materials Perishable food Pets and plants Irreplaceab...

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This series is arranged in levels of reading difficulty ranging from 1 to 5. The easiest is level 1, the hardest level 5. All these books are royalty free and can be copied used printed and distributed, scrawled on walls, acted out in plays and told (very slowly) to your pet dog, if you don’t have a dog a cat will do, but hamsters do not listen as they only speak Spanish. They are designed to be easily read on a computer and can be printed on standard A4 size paper (landscape format) There is no bad language or offensive words anywhere in these books. ...

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Snozel went to the kitchen window and looked out. One of the lorries moving the museum across town had bumped over the roadworks outside the house. Something had fallen off....

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INTRODUCTION THIS IS AN INVITATION TO UNDERSTANDING. THIS IS AN INVITATION TO YOU. NOT YOU AS THE READER OF THESE WORDS ONLY, BUT TO THE (GENERIC YOU) ALL AND EVERY YOU. Millions are ready today to understand who they are and how they operate as a HUMAN BEING. There are a number of steps necessary in the learning process. 1.The individual must be ready (desire) to learn. 2.The individual must be able (potential) to learn. 3.The individual must practice (motivation) that which is being learned. 4.The individual must be (acceptance) the learned. ...

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The Read it LOUD! Foundation is a non-profit 501©3 organization started by Wally Amos, focusing on the profound positive impact that reading out loud has on children, their ability to read, their desire to imagine and their life-long love of learning. We challenge families to read out loud together at least 10 minutes a day to ensure their children have the best possible chance for a promising future....

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Our Mission: To help close the literacy gap by motivating families to read out loud every day together, for at least 10 minutes

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A National Campaign The Read it LOUD! Foundation is a non-profit, tax-exempt 501(C))(3) with the purpose of supporting all children’s literacy education. Working nationally, the foundation’s sole purpose is to have over 1,000,000 parents reading aloud to their children each day, creating a supporting, nurturing, and caring home and community environments....

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A National Campaign The Read it LOUD! Foundation is a non-profit, tax-exempt 501(C))(3) with the purpose of supporting all children’s literacy education. Working nationally, the foundation’s sole purpose is to have over 1,000,000 parents reading aloud to their children each day, creating a supporting, nurturing, and caring home and community environments....

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Subject: How argument mapping can make discussions better (and shorter). Why different perceptions of reality can cause disagreement, and how diagrams can make people aware of this. Target audience: Everyone interested. No special knowledge required. Estimated reading time: 20-25 minutes. Should you prefer watching over reading, have a look at this video: http://youtu.be/_JuDJUx7drU Page layout: Allows easy reading without scrolling, even on very small screens. License: Free for non-commercial use, Creative Commons BY-NC-SA. Commercial use requires separate agreement. Technical information: Original PDF/A file size: 685 913 Byte MD5: 3abc45069b734d7d2a60b9331d55f33b (file integrity checksum) ...

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As a Captain, Lieutenant, Master Sergeant, Sergeant, or Petty Officer, YOU have more leadership ability than most managers. This book has three purposes: First, it teaches you how your military experience has prepared you to be a leader and successfully transfer those skills to an exciting career in the corporate world. Second, it summarizes the best leadership practices of both the military and corporations, and Third, it helps you find good employer prospects and wow them with your resume and interview skills. It also helps you understand the winning ways of true leaders. Each aspect of quality leadership has its own chapter, with appropriate subheadings....

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Chapter 1–About the Author Chapter 2–Introduction Chapter 3–Leading Trust and Honesty Taking Responsibility Taking the Long-Term View Statements of Mission, Values, Good Behaviors, and Permission Lead by Example Create a Personal Vision Humility and Humor Focus Reporting The quality and direction of information Reports Meetings Chapter 4–Building Teams Improve the Quality of Life Build Your Team Listen to Team Members Listen and analyze human resources Find out what’s really happening in your unit Hiring Chapter 5–Motivating, Training, and Coaching Motivating Recognition Punishment Training Help your people become their best: How to train Who trains team members? Finding Talent Coaching Ask instead of tell Encourage collaboration and cooperation. Offer honest help Give continual and honest feedback, performance evaluations, and support. Chapter 6–Seeking Results Chapter 7–Goals Chapter 8–Change Chapter 9–Intelligence Chapter 10–Strategy Chapter 11–Taking Calculated Risks Chapter 12–Dealing with Bad News Chapter 13–Decisions Chapter 14–Crises Chapter 15–Dealing with Failure Chapter 16–Sea...

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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 :...

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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 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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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. First chapter is introductory in nature. The new concept of non-associative semi-linear algebras is introduced in chapter two. This structure is built using groupoids over semi-fields. Third chapter introduces the notion of non-associative linear algebras. These algebraic structures are built using the new class of loops. All these non-associative linear algebras are defined over the prime characteristic field Zp, p a prime. However if we take polynomial non associative, linear algebras over Zp, p a prime; they are of infinite dimension over Zp. We in chapter four introduce the notion of groupoid vector spaces of finite and infinite order and their generalizations. Only when this study becomes popular and familiar among researchers several applications will be found. The final chapter suggests around 215 problems some of which are at research level. ...

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Ln(m) is a loop of order n + 1 that is Ln(m) is always of even order greater than or equal to 6. For more about these loops please refer [37]. All properties discussed in case of groupoids can be done for the case of loops. However the resultant product may not be a loop in general. For the concept of semifield refer [41]. We will be using these loops and groupoids to build linear algebra and semilinear algebras which are non associative. ...

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Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two NON ASSOCIATIVE SEMILINEAR ALGEBRAS 13 Chapter Three NON ASSOCIATIVE LINEAR ALGEBRAS 83 Chapter Four GROUPOID VECTOR SPACES 111 Chapter Five APPLICATION OF NON ASSOCIATIVE VECTOR SPACES / LINEAR ALGEBRAS 161 Chapter Six SUGGESTED PROBLEMS 163 FURTHER READING 225 INDEX 229 ABOUT THE AUTHORS 231 ...

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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 has four chapters. Chapter one is introductory in nature, for it recalls some basic definitions essential to make the book a self-contained one. Chapter two, introduces for the first time the new notion of neutrosophic rings and some special neutrosophic rings like neutrosophic ring of matrix and neutrosophic polynomial rings. Chapter three gives some new classes of neutrosophic rings like group neutrosophic rings, neutrosophic group neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this book is that it gives 246 problems in Chapter four. Some of the problems are direct and simple, some little difficult and some can be taken up as a research problem....

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Now we proceed onto define the notion of neutrosophic subgroup of a neutrosophic group. DEFINITION 1.1.2: Let N(G) = 〈G ∪ I〉 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 1.1.3: 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 neutrosophi...

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Preface 5 Chapter One INTRODUCTION 1.1 Neutrosophic Groups and their Properties 7 1.2 Neutrosophic Semigroups 20 1.3 Neutrosophic Fields 27 Chapter Two NEUTROSOPHIC RINGS AND THEIR PROPERTIES 2.1 Neutrosophic Rings and their Substructures 29 2.2 Special Type of Neutrosophic Rings 41 Chapter Three NEUTROSOPHIC GROUP RINGS AND THEIR GENERALIZATIONS 3.1 Neutrosophic Group Rings 59 3.2 Some special properties of Neutrosophic Group Rings 73 3.3 Neutrosophic Semigroup Rings and their Generalizations 85 Chapter Four SUGGESTED PROBLEMS 107 REFERENCES 135 INDEX 149 ABOUT THE AUTHORS 154 ...