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"Light" is the 3rd of the haiga eBook series. This eBook includes related themes like fire, heat, and shadow

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Subject: How decision making, problems, politics and quality of life are connected. What can go wrong, and how to do it better. Target audience: Everyone interested. No special knowledge required. Estimated reading time: 1 hour for all chapters. 2 extra hours for all appendices. Page layout: Allows easy reading without scrolling, even on very small screens. Entertainment value: None intended. And that's no joke. License: Free for non-commercial use, Creative Commons BY-NC-SA. Commercial use requires separate agreement. Technical information: Original PDF/A file size: 5 887 981 Byte MD5: 5f01ed34c1f835afaa5202ba7d706bfc (file integrity checksum)...

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A cycle based approach to stock and options trading

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If passing through youth was like crossing the mirage of life for Chandra and Nithya, it proved to be chasing the mirage of love for Sathya and Prema though for plain Vasavi, Chandra's pitiable sibling, it was the end of the road. As life brings Chandra, who suffers from an inferiority complex for his perceived ugliness, and Nithya, who was bogged down being jilted by Vasu, together, they script their fate of fulfillment. And as poetic justice would have it, Sathya, who caused Prema's heart burn, himself was led down the garden path by Kala, doing a "Sathya on Sathya". Just not that, life has in store just deserts for Vasu owing to Nithya's retribution as he tries to stalk her. Besides, after many a fictional twist and turn, the way the story ends, challenges the perception that fact is stranger than fiction. ...

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As if all this was not enough for his tender psyche to cope up with, he had to contend with the sternness of the paternal strictness. Thus, it was only time before the seeds of alienation towards his father were sown in his impressionable mind. But the support he got from his sister and the solace he felt in his mother’s lap helped soothe his ruffled feelings a little. In time, he reached the threshold of youth, but couldn’t cross the despair of adolescence. Oblivious of the possibilities of life, man goes through his journey of disarray, in the itinerary of the past, chasing the mirages of malady even amidst the sands of hope. And that despairs him forever. Into his puberty, as his biology induced in him sexual curiosity, owing to his ungainliness, his youthful urge for reciprocity remained unfulfilled. Being naïve to the feminine nuances, his eyes couldn’t comprehend the emanations of their indifference. When in dismay, as he turned to the mirror for a clue, the reflections of his self-doubts stared him in his face. Yet, goaded by desire, he ogled women but to no avail. And as he went back to the mirror to reassess h...

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Shackles on Psyche End of the Tether Burden of Freedom Onto the Turf Respite by Death Lessons of Life Naivety of Love Dilemma of Disclosure Perils of Youth Absurd Proposal Crossing the Mirage Setting the Pace Oasis of Bliss Busy bees in Honeycomb Twist in the Tale Love in the Bind Turn for the Worse Shadows to the Fore Spurring to Err Tempting the Fate Stooping to Conquer Fouling the Soul Poetic Justice Agony of Penitence Embrace of Love Life of a Kind Just deserts?...

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Why are there so many schools of Buddhism? Are the differences just cultural, or do they have fundamentally different visions of Dhamma? This work assesses the claims of the traditions, and takes into account to findings of modern scholarship. It pays special attention to the origins of the monastic orders. If we are to understand the differences, and sometimes tensions, between the schools of Buddhism today, we must examine more closely the forces that spurred their formation....

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Foreword Abstract 0.1 The Early Period (BCE) 0.2 The Middle Period (CE) 0.3 Comparing pre-CE & post-CE evidence 0.4 The Mahāvihāravāsins 0.5 The Dharmaguptakas 0.6 The Mūlasarvāstivādins Chapter 1 The ‘Unity Edicts’ 1.1 Schism & unity 1.2 Aśoka & unity 1.3 The Third Council 1.4 What were the heretics teaching? 1.5 Aśoka in the Mahāsaṅghika Vinaya? Chapter 2 The Saints of Vedisa 2.1 Gotiputa 2.2 Mogaliputa 2.3 Vāchiputa Chapter 3 The Dīpavaṁsa 3.1 The heresy of grammar 3.2 The Sri Lankan context 3.3 Was Buddhaghosa a Theravādin? Chapter 4 Monster or Saint? 4.1 Vasumitra’s Samayabhedoparacanacakra 4.2 Bhavya’s Nikāyabhedavibhaṅgavyakhyāna 4.3 Śāripūtraparipṛcchā 4.4 Xuan-zang’s Records of the Western Lands 4.5 Kuei Chi Chapter 5 Three Sins & Five Theses 5.1 Which Mahādeva? 5.2 The five heresies 5.3 ‘Outflows’ 5.4 Dhamma or Vinaya? Chapter 6 More on the Vibhajjavādins 6.1 The Kathāvatthu 6.2 Later Mahāvihāravāsin sources 6.3 What does ‘Vibhajjavāda’ mean? Chapter 7 Vibhajjavāda vs. Sarvāstivāda? 7.1 The early controversies 7.2 What schism? Chapter 8 Dharmagupta: the...

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Serenity and insight are the two great wings of Buddhist meditation. They each have a special role to play in the path to Awakening. While some modern approaches seek to marginalize serenity in favor of ‘dry’ insight, the Buddha’s own discourses place serenity right at the center of the path. This book collects virtually all the significant passages on this topic that are found in the early discourses, carefully elucidated for the modern reader....

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Introduction 1 A Handful of Leaves 2 Samatha & Vipassanā 3 The Wings to Enlightenment 4 Dependent Liberation 5 The Gradual Training 6 The Benefits of Samādhi 7 Jhāna & the Noble Ones 8 Counterexamples A On the Translations B Contemporary Teachers on Samādhi C Instant Enlightenment...

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A view on substitute teachers.

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Preface Dedication 1 Where's Madame? 2 Yippee- a Sub 3. Getting the Call 4. Sink the Sub 5. What was your name again? 6. The most creative minds on the planet 7. I forgot to tell you 8. Embarrassingly human 9. Revelations 10. More burdens that it is fair to bear 11. Technically a challenge 12. Back-up in a crisis 13. The Detective in Me 14. Where Education Shines 15. Is this the way I want to serve the world? 16. I am Way Old 17. Mistakes, I've made a few 18. Feedback from the Peanut Gallery 19. Humor on a Blah Day 20. Privilege 21. Familiar Face 22. Food is a Bonding Experience 23. Pretty thick molasses and hard to get through 24. Reassurances 25. Surprise in the Exam room - Compassio 26. Why some people hate school, understandably 27. The Second Mile- hey, may as well 28. What Still Needs Perfecting 29. What's Right in the schools 30. The Randomness of it all - 1 31. The Randomness of it all - 2...

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Today I decided to write a book, a concise part about my cooperation with The White House and The Central Intelligence Agency and many years of cooperation with Human Rights Watch (HRW) and IFRC (International Federation of Red Cross and Red Crescent Societies); a few political leaders of England, France, agents of NSA (The National Security Agency of the United States, who didn’t let me to submit any top secret information, before.) and CIA agents and Mossad and Shin Bet (Shabak) and USSS (The United States Secret Service) and Microsoft communicated with me about what am I started to write now? My life is full of espionage and political memories. When I started my political activities and when I started to fight for freedom of people, I was only 14, one of the teenager Iranian boys. I got tortured extremely hard. All around the hell, staying in Russia where your bones would sense it’s very cold weather, inside of it’s deep underground city where you ask your self “How is it possible for some satellites with their warheads to track me here?”. During travel to Philippines with it’s girls inside of the bars and clubs. In middle east, ...

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The book is meant for resident doctors who are starting their training. Obstetrics & Gynecology is a subject that involves a lot of operative procedures. Just knowledge of which instruments to use and which steps to perform is not enough to perform an operation successfully. A lot depends on manual dexterity. A basic knowledge of the technique of holding different instruments correctly and passing sutures, ligatures, and tying knots is essential for performance of all types of operations. This aspect has not been dealt with by any book so far. I have written this book to bridge that gap in the educational resources available to students....

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For resident doctors who are starting their training in Obstetrics & Gynecology.

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Section 1 Holding Surgical Instruments Section 2 Preparing Self For Operating Section 3 Suturing Section 4 Ligatures Section 5 Surgical Knots

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Some 2500 years ago in ancient Northeast India, a young, brilliant and courageous man discovered the sublime answer to how the mind works and how to realise unconditioned happiness and bliss. For the next 45 years, he devoted his life to teaching the doctrine that helped bring unparalleled happiness, peace and relief to numerous people long after his death. In time stories began to emerge proclaiming his greatness and his near-divinity status. Over generations and into distant lands, those stories took on a life of their own. Eventually the real man is buried under a mountain of beautiful fantasies and magical tales. Scholars in Buddhist studies have never lost sight of the historical Buddha. There is a beautiful story to tell about the real man but the materials never quite made it to the mass market because academic books are generally hard to read and appreciate. This book seeks to bring the historical Buddha back into the popular psyche. That story is as intriguing, and is perhaps even more inspiring than the best fantasy story that had ever been woven....

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“Go now and wander for the welfare and happiness of many, out of compassion for the world, for the benefit, welfare and happiness of gods and men. Teach the Dhamma that is good in the beginning, good in the middle and good in the end, with the meaning and the letter. Explain a holy life that is utterly perfect and pure. There are beings with little dust in their eyes who will be lost through not hearing the Dhamma. Some will understand the Dhamma.”...

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Abbreviations for Pali Canon texts | i Map | ii Author’s request | iii Foreword | ix Life as a Layman | 1 From Home to Homelessness | 29 Enlightenment: Nibbana! Release! | 55 Legend of the Seven Weeks | 89 Unveiling Dhamma | 119 Spreading Dhamma |135 Reaching Out to the Lay Community | 159 Buddha's Foremost Lieutenants | 175 Return to Kapilavatthu |193 Glossary |229...

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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 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. 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 ...

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We in this book introduce the notion of pure (mixed) neutrosophic interval bisemigroups or neutrosophic biinterval semigroups. We derive results pertaining to them. The new notion of quasi bisubsemigroups and ideals are introduced. Smarandache interval neutrosophic bisemigroups are also introduced and analysed. Also notions like neutrosophic interval bigroups and their substructures are studied in section two of this chapter. Neutrosophic interval bigroupoids and the identities satisfied by them are studied in section three of this chapter. The final section of chapter one introduces the notion of neutrosophic interval biloops and studies them. Chapter two of this book introduces the notion of neutrosophic interval birings and bisemirings. Several results in this direction are derived and described. Even new bistructures like neutrosophic interval ring-semiring or neutrosophic interval semiring-ring are introduced and analyzed. Further in this chapter the concept of neutrosophic biinterval vector spaces or neutrosophic interval bivector spaces are introduced and their properties are described. In the third chapter we introduce the n...

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Now we will be using these intervals and work with our results. However by the context the reader can understand whether we are working with pure neutrosophic intervals or neutrosophic of rationals or integers or reals or modulo integers. This chapter has four sections. Section one introduces neutrosophic interval bisemigroups, neutrosophic interval bigroups are introduced in section two. Section three defines biinterval neutrosophic bigroupoids. The final section gives the notion of neutrosophic interval biloops....

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Preface 5 Chapter One BASIC CONCEPTS 7 1.1 Neutrosophic Interval Bisemigroups 8 1.2 Neutrosophic Interval Bigroups 32 1.3 Neutrosophic Biinterval Groupoids 41 1.4 Neutrosophic Interval Biloops 57 Chapter Two NEUTROSOPHIC INTERVAL BIRINGS AND NEUTROSOPHIC INTERVAL BISEMIRINGS 75 2.1 Neutrosophic Interval Birings 75 2.2 Neutrosophic Interval Bisemirings 85 2.3 Neutrosophic Interval Bivector Spaces and their Generalization 93 Chapter Three NEUTROSOPHIC n- INTERVAL STRUCTURES (NEUTROSOPHIC INTERVAL n-STRUCTURES) 127 Chapter Four APPLICATIONS OF NEUTROSOPHIC INTERVAL ALGEBRAIC STRUCTURES 159 Chapter Five SUGGESTED PROBLEMS 161 FURTHER READING 187 INDEX 189 ABOUT THE AUTHORS 195 ...

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The first chapter is introductory in nature and gives a few essential definitions and references for the reader to make use of the literature in case the reader is not thorough with the basics. The second chapter deals with different types of neutrosophic bilinear algebras and bivector spaces and proves several results analogous to linear bialgebra. In chapter three the authors introduce the notion of n-linear algebras and prove several theorems related to them. Many of the classical theorems for neutrosophic algebras are proved with appropriate modifications. Chapter four indicates the probable applications of these algebraic structures. The final chapter suggests about 80 innovative problems for the reader to solve....

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Dedication 5 Preface 7 Chapter One INTRODUCTION TO BASIC CONCEPTS 9 1.1 Introduction to Bilinear Algebras and their Generalizations 9 1.2 Introduction to Neutrosophic Algebraic Structures 11 Chapter Two NEUTROSOPHIC LINEAR ALGEBRA 15 2.1 Neutrosophic Bivector Spaces 15 2.2 Strong Neutrosophic Bivector Spaces 32 2.3 Neutrosophic Bivector Spaces of Type II 50 2.4 Neutrosophic Biinner Product Bivector Space 198 Chapter Three NEUTROSOPHIC n-VECTOR SPACES 205 3.1 Neutrosophic n-Vector Space 205 3.2 Neutrosophic Strong n-Vector Spaces 276 3.3 Neutrosophic n-Vector Spaces of Type II 307 Chapter Four APPLICATIONS OF NEUTROSOPHIC n-LINEAR ALGEBRAS 359 Chapter Five SUGGESTED PROBLEMS 361 FURTHER READING 391 INDEX 396 ABOUT THE AUTHORS 402...

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This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research level....

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In this chapter we only indicate as reference of those the concepts we are using in this book. However the interested reader should refer them for a complete understanding of this book. In this book we define the notion of natural product in matrices so that we have a nice natural product defined on column matrices, m * n (m ≠ n) matrices. This extension is the same in case of row matrices. We make use of the notion of semigroups and Smarandache semigroups. Also the notion of semirings, Smarandache semirings, semi vector spaces and semifields are used....