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...introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use f...

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... HEXIS Church Rock 2004 2 This book can be ordered in a paper bound reprint from: Books on Demand ProQ... ...eutrosophic Vector spaces 8 Chapter Two SOME BASIC RESULTS ON GRAPH THEORY AND THEIR APPLICATION TO FUZZY MODELS 2.1 Some basics on Gr... ...ophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to i... ...n almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Mo... ... father of graph theory. In 1847 Kirchoff developed the theory of trees, in order to solve the system of simultaneous linear equations, which give t... ...y to consider every cycle in the graph of an electric network separately in order to solve the system of equations. T G N FIGURE: 2A 14 Inste... ...loiting these properties can be had from [3, 25, 58]. Just for the sake of completeness we recall the notions of perfect matchings and the Tutte ma... ...used. For more about these please refer [32-4, 64-6]. Just for the sake of completeness we at each stage illustrate each of these by giving a brief...

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...roup and a semigroup simultaneously. This book is a piece of work on Smarandache semigroups and assumes the reader to have a good background on group theory; we give some recollection about groups and some of its properties just for quick reference....

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... American Research Press Rehoboth 2002 2 This book can be ordered in a paper bound reprint from: Books on Demand Pro... ...liminary results 14 2.4 Subgroups 15 3. Some Classical Theorems in Group Theory 3.1 Lagrange's Theorem 21 3.2 Cauchy's Theorems 22 3.3 ... ...marandache semigroups and assumes the reader to have a good background on group theory; we give some recollection about groups and some of its proper... ...e used in this book. Chapters 2 and 3 gives most of the basic concepts on group theory and results in group theory which have been used in this text... ...otions in groups or Smarandache semigroups. This text does not in any way claim completeness in giving the properties of groups. Chapter 4 starts ... ...bset M of S × T such that for every s ∈ S there is a unique t ∈ T such that the ordered pair (s, t) is in M. This definition serves to make the ... ... illustrations. PROBLEM 1: Given an example of a Smarandache semigroup of order 42. PROBLEM 2: Find all subgroups in the Smarandache semigro...

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...ower set P(X) where X = {a, b, c} which is a finite Smarandache semiring of order 16. This book can be ordered in a paper bound reprint from: ... ...o the Smarandache semirings, semifields and semivector spaces. Secondly, in order to make an organized study possible, we have also included all the ... ...migroup S is finite we say S is a finite semigroup or a semigroup of finite order, otherwise S is of infinite order. If the semigroup S contains an ... ... to prove the result). Supplementary Reading 1. Birkhoff, G. Lattice Theory. American Mathematical Society, Providence, R.I., 1967. 2. Bi... ...n Applied Algebra. Mc-Graw Hill, New York, 1970. 3. Gratzer, G. Lattice Theory. Freeman, San Francisco, 1971. 4. Hall, Marshall. Theory of Gr... ..., 1975. 6. Lang, S. Algebra. Addison-Wesley, 1967. 7. Nagata, M. Field Theory. Marcel Dekker, NewYork-Basel, 1977. 8. Passman, D.S. Infinite ... ...rties about it, analogous to rings. This chapter also gives for the sake of completeness the definition of several types of special semirings like ∗-...

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...edu HEXIS Church Rock 2004 2 This book can be ordered in a paper bound reprint from: Books on Demand ... ...S patients 211 Chapter Three SOME NEW AND BASIC DEFINITIONS ON NEUTROSOPHIC THEORY 3.1 Neutrosophic sets and neutrosophic logic 222 3.2 Fuzzy... ...nalysis of Social Aspects of Migrant Labourers Living With HIV/AIDS Using Fuzzy Theory and Neutrosophic Cognitive Maps: With Specific Reference to R... ...lations like, fuzzy equivalence relation, fuzzy compatibility relations, fuzzy ordering relations, fuzzy morphisms and sup-i-compositions of fuzzy ... ...p i ° Q = r i we need only methods for solving equations of the later form in order to arrive at a solution. We may therefore restrict our further... ... of the natural evolution of genetics. GAs emulate the biological evolutionary theory to solve optimization problems. In general, 50 GAs start... ... and a ∨ b = b for a ≤ b. The existence of least upper bound is assured by the completeness. Here a fuzzy set is defined as a mapping from a nonemp...

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...This book has four chapters. Chapter one is introductory in nature. The reader is expected to have a good background of algebra and graph theory in order to derive maximum understanding of this research. The second chapter represents groups as graphs. The main feature of this chapter is that it contains 93 examples with diagrams and 18 theorems. In chapter thre...

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...In this book, we study the subject of Smarandache Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that challenged the status quo of existing mathematics. The genesis of Smarandache Notions, a field founded by Florentine Smarandache, is alike to that of Fuzzy ...

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...is book, which is an amalgamation of alternatives. This book can be ordered in a paper bound reprint from: Books on Demand ProQuest ... .... In its trajectory of stupendous growth, it has also come to include the theory of fuzzy algebra and for the past five decades, several researchers... ...y the subject of Smarandache Fuzzy Algebra. Originally, the revolutionary theory of Smarandache notions was born as a paradoxist movement that chall... ...ns, a field founded by Florentine Smarandache, is alike to that of Fuzzy Theory: both the fields imperatively questioned the dogmas of classical ma... ... x t G µ = t x G µ for every x ∈ G and t ∈ [0, 1]. We now define the order of the fuzzy subgroup µ of a group G. DEFINITION 1.2.7: Let µ be a... ... a fuzzy subgroup of a group G and H = {x ∈ G/ µ (x) = µ (e)} then o(µ ), order of µ is defined as o(µ ) = o(H). 13 THEOREM 1.2.6: Any subgroup ... ...field is defined for the first time in this book. We just for the sake of completeness recall the definition of bifields. DEFINITION 1.6.2: A bir... ...y absent. This concept is defined only in this book. Just for the sake of completeness we start to give the definition of fuzzy semirings and proceed... ...andache primary fuzzy subsemigroup of a S- semigroup G. We for the sake of completeness recall some of the definitions. DEFINITION [ 14]: Let A be ...

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...uch real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. These concepts are also exten...

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...nthenral@gmail.com HEXIS Phoenix, Arizona 2005 2 This book can be ordered in a paper bound reprint from: Books ... ...an give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathem... ...atrix A = A 1 ∪ A 2 both A 1 and A 2 are square matrices but of different order say A 1 is a n × n matrix and A 2 a s × s matrix then we call ... ...in [154, 155]. Here we only recall just the definition merely for the sake of completeness. 3. Kirchhoff bimatrix will always to be a square or a m... ...bigraph will prove a major role in the applications of bimodels both in fuzzy theory and neutrosophic theory. Finally we give the matrix represen... ...that gives the hidden pattern of the situation. As we have a very well known theory, which states that the strength of the data 76 depends on, th... ... 3.2 Fuzzy Cognitive Bimaps and their Applications Just for the sake of completeness we have given the basic concepts of FCM in the earlier se... ...ps will also be directed neutrosophic graphs. It is no coincidence that graph theory has been independently discovered many times since it may quite...

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...Over the past 25 years, I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings) specially to motivate both ring theorists and Smarandache algebraists to develop and study several important and innovative properties about S-rings. Writi...

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...resentation of S-ideal need not in general be modular. This book can be ordered in a paper bound reprint from: Books on Demand ProQuest ... ...ine.com Peer reviewers: Ion Goian, Department of Algebra and Number Theory, University of Kishinev, Moldova. Sebastian Martin Ruiz, Avda. De ... ...I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (S- rings) spe... ...etter if we thrust importance on results given in research papers on ring theory rather than detail the basic properties or classical results that t... ...alled a finite group, otherwise it is termed as an infinite group. By the order of a finite group we mean the number of elements in the group. ... ...s a group under multiplication modulo p. This is a finite cyclic group of order p-1. DEFINITION 1.1.2: Let G be a group. If a . b = b . a for all... ...itor group. Proof: Follows by the definition. Just for the sake of completeness we give the definition of semiring as it used to define semio...

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...nly geometry together with the purport of physical laws can do so... The idea of the measuring rod and the idea of the clock contained with it in the theory of relativity do not find their exact correspondence in the real world. It is also clear that the solid body and the clock do not in the conceptual edifice of physics play the part of irreducible elements, but that of ...

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...Smarandache Pima Community College, East Campus 2 This book can be ordered in a paper bound reprint from: Books on Demand ProQues... ...dea of the measuring rod and the idea of the clock contained with it in the theory of relativity do not find their exact correspondence in the real ... ...on of Einstein's as reported by Rosser, p.398:[3] 7 The theory of relativity is often criticized for giving, without justificati... ...of propagation of light. The situation is, however, somewhat as follows. In order to give physical significance to the concept of time, processes of... ...chooses for such a definition of time. It is advantageous, however, for the theory to choose only those processes concerning which we know something... ... the light wave vibration; - if one uses the entanglement phenomenon, in order to transfer the polarization between two photons, then: whatever h... ...erved, the language of the description must be continually altered. Godel's completeness theorem plainly states that a fixed language can only descr... ...lar state, this state is called a metric state. Godel's consistency and completeness theorems - A mathematical theory which says that there are ...

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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 ...

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... American Research Press Rehoboth 2002 This book can be ordered in microfilm format from: Books on Demand ... ...s or articles, as well as studies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric the- orems; Di... .... . . . . . . . . . . . . 241 8 On certain new conjectures in prime number theory . . . . . . . . . . . . . 243 9 On certain conjectures by Russo . . ... ...ve made it clear that there exists a very intimate correlation between the Theory of numbers and other departments of Mathematics, not excluding geome... ... 1 (4) 16 We shall use in what follows this relation, but for the sake of completeness, we give a sketch of proof: it is known that m a l a p(p ... ...ying (xy 4)(z +t) 2 8(x +y)(z +t) (xy + 4)(x +y) 2 0 (4) This second order inequality gives at once z +t 4(x +y) + p 16(x +y) 2 + (x +y) 2 (xy ... ...n positive integers occurs many times for particular values of a;b;c;d. In order to obtain all solutions of (1), remark that (1) can be written as bdx... ...known inequality d(n) 2 p n (3) (see W. Sierpinski [12]). For the sake of completeness, we give the short proof of (3). If d is a divisor of n, the s...

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...n 1980’s of a new literary and artistic avant-garde movement that he called “paradoxism”, because I received some books and papers dealing with it in order to review them for the German journal “Zentralblatt fár Mathematik”. It was an inspired connection he made between literature/arts and science, philosophy....

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...thus opening new ways of research in four fields: philosophy, logics, set theory, and probability/statistics. It was known to me his setting up in 1... ...“paradoxism”, because I received some books and papers dealing with it in order to review them for the German journal “Zentralblatt f ár Mathematik”... ...so, to experiment." 2.2. Etymology: Paradoxism = paradox+ism, means the theory and school of using paradoxes in literary, artistic, philosophical... ... with different ideational spectra. Neutrosophy considers a proposition, theory, event, concept, or entity, "A" in relation to its opposite, "Anti- ... ...unctured neighborhoods (balls) of c. Of course, – a < a and b + > b. No order between – c + and c. Addition of non-standard finite numbers with ... ...ut-A>, and <Anti-A> are disjoint two by two. <Non-A> is the completeness of <A> with respect to the universal set. ... ...er. c) Law of Complementarity: <A> feels like completing with <Non-A> in order to form a whole. Examples: Persons, who are different, ...

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...rt articles are preferable. All that has been accepted for the online issue of Progress in Physics is printed in the paper version of the journal. To order printed issues, contact Chief Editor. This journal is non-commercial, academic edition. It is printed from private donations....

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...lectronic version of this journal: http://www.geocities.com/ptep_online To order printed issues of this journal, con- tact the Editor in Chief. Chief ... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 D. Rabounski A Theory of Gravity Like Electrodynamics. . . . . . . . . . . . . . . . . . .... ...sova Gravitational Waves and Gravitational Inertial Waves in the Gen- eral Theory of Relativity: A Theory and Experiments. . . . . . . . . . . . . . .... ...i, L. Borissova, F. Smarandache Quantum Causality Threshold in the General Theory of Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . .... ... of Progress in Physics is printed in the paper version of the journal. To order printed issues, contact Chief Editor. This journal is non-commercial,... ...arth’s surface near the equator the value isΛ 2.8 × 10 −20 μ, so it is in order of 10 42 less than in atoms. Therefore, because the Lagrangian of a s... ...esent Situation in Quantum Mechanics. He turns to EPR paradox and analyses completeness of the description by the wave function for the entangled part...

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... Xiquan (second printed edition) 2002 This book can be ordered in microfilm format from: Books on Demand ProQuest I... ...n punctured neighborhoods (balls) of c. Of course, – a < a and b + > b. No order between – c + and c. Addition of non-standard finite numbers ... ... and instability of stable systems. Let's note by <A> an idea, or proposition, theory, event, concept, entity, by <Non-A> what is not <A>, and by <A... ... to measure the truth, indeterminacy, and falsehood. My hypothesis is that no theory is exempted from paradoxes, because of the language imprecisio... ...rsively (Dubois, Prade), changing/adjusting the definitions after each step in order to improve the next result. This might be comparable to approxi... ...eria and the complementarity of various sources. For example, the Possibility Theory (Zadeh 1978, Dubois, Prade) gives a better approach than the F... ...mes an instinct. It is even severe that we develop a fallacy instinct. 5. Completeness and incompleteness: knowledge and practice There alway... ...incompleteness: knowledge and practice There always is contradiction between completeness and incompleteness of knowledge. In various papers prese... ...is Tractatus Logicus Philosophicus), because we can incompletely represent its completeness... As Wittgenstein himself (but not most of his follower...

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...American Research Press Rehoboth 2002 2 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest I... ...of Inference Section 7 Formal Reasoning Section 8 Quantification Theory Section 9 Uses of Logic In Computer Programming 7 13 14 1... ...osophic Connectives in Computer Programs Section 6 Ordering The Elements of Neutrosophic Logic Section 7 Rules Of Inferenc... ...ives are at the same level, absent parentheses, they are evaluated in the order encountered when you move from left to right. Example: Given the ... ...e many other binary connectives that can be defined. Of course, given the completeness of the set CS = { /\, \/, ¬ } all of them can be constructed u... ...o use truth tables to perform the computations. For that we need a formal theory. Definition 1.5.1: A formal theory is a system S constructed from... ...heorem in the system S. Example: The following is a formal axiomatic theory L a) The set of symbols in L are { ¬, →, (, ), T, F, p 1 , p 2 , ...

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...y weariness, which would reconcile self-despair with the rapturous consciousness of life beyond self. She found her epos in the reform of a religious order....

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...sness of life beyond self. She found her epos in the reform of a religious order. That Spanish woman who lived three hundred years ago, was certainly ... ... for these later born Theresas were helped by no coherent social faith and order which could perform the function of knowledge for the ardently willin... ...lled up in time. But not too hard. I have always been in favor of a little theory: we must have Thought; else we shall be landed back in the dark ages... ...r of my life and purposes: a tenor unsuited, I am aware, to the com moner order of minds. But I have discerned in you an elevation of thought and a c... ...whose plodding application, rows of note books, and small taper of learned theory exploring the tossed ruins of the world, seemed to enforce a moral e... ... physicist and interpreter of hieroglyphs write detestable verses? Has the theory of the solar system been advanced by graceful manners and conversati... ...r was the chief outcome of her existence— the divinity passing into higher completeness and all but exhausted in the act of covering your bit of canva... ... would have become firm again. Naumann’s apparatus was at hand in wonderful completeness, and the sketch went on at once as well as the conversation. D... ...efs and indefinable joys, yearnings of one life towards another, visions of completeness, indefinite trust. And Lydgate fell to spinning that web from h...