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...th 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 fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use...

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...ainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use fuzzy theory and... ...ed we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs ... ...trices. So we for the first time introduce and study these concepts. As our analysis in this book is application of neutrosophic algebraic structure... ...es. 1.1 Neutrosophic fields In this book we assume all fields to be real fields of characteristic 0 all vector spaces are taken as real spac... ... fields of characteristic 0 all vector spaces are taken as real spaces over reals and we denote the indeterminacy by ‘I ’ as i will make a confusion... ...I is such that I . I = I 2 = I. DEFINITION 1.1.1: Let K be the field of reals. We call the field generated by K ∪ I to be the neutrosophic field... ...tes the field generated by K and I. Example 1.1.1: Let R be the field of reals. The neutrosophic field of reals is generated by 〈R ∪ I 〉 i.e. R(... ...similar representation of a directed graph arises in that part of numerical analysis involving matrix inversion and the calculation of eigen values.... ...lly in the field of applications of fuzzy theory is a grand one for most of analysis of unsupervised data are very successfully carried out by the u...

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...lem using FRE 196 2.6 Data compression with FRE 196 2.7 Applying FRE to Threat Analysis 196 2.8 FRE application to medical diagnosis 197 2.9 A fuz... ...4.10 Some properties of minimal solution for NRE 270 4.11 Application of NRE to Real-world problems 272 Chapter Five SUGGESTED PROBLEMS 279 Bi... ...sophic Relational Equations have a role to play for we see that in most of the real-world problems, the concept of indeterminacy certainly has its ... ...on 4.11 is completely devoted to suggest how one can apply NREs in the study of real world problems. We suggest many problems in chapter five for th... ... Neutrosophic Cognitive Maps (http://gallup.unm.edu/~smarandache/NCMs.pdf) and Analysis of Social Aspects of Migrant Labourers Living With HIV/AIDS... ...goodness of one value cannot compensate the badness of another value [117]. In reality there are situations that allow compensatability among the v... ... (=N 5 ). This clearly implies that all entries in the matrices P, Q, and R are real numbers from the unit interval [0, 1]. Assume now that the thre... ... we can only provide the decision maker with the Pareto optimal set for further analysis. For a problem with multiple linear objective functions, th... ...he transform function f and the threshold value θ, which is reasonable (see the analysis by [6]) in fact for our fuzzy neural network discussed late...

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...Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by u...

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... 5 Preface Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are ba... ...hs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuz... ...vented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematica... ...and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can... ...lp in easily understanding the definition and examples. We have also given 25 real world problems as applications of Bimatrices to Fuzzy and Neutros... ...y study several models that are at times time dependent or when a comparative analysis is needed can use these models. On a personal note, we th... ...FCM and overlap FCM when some of the attributes overlap i.e. in the study or analysis of a model when we have some common attributes given by exper... ...s in case of graphs in bigraphs also the following. Since a bigraph can be realized as the ‘union’ of two graphs here the ‘union’ is distinctly d... ...ol just denotes only connection or union as subsets. Thus a bigraph G can be realized as G = (V (G 1 ), E (G 1 ), 1 G I) ∪ (V (G 2 ), E (G 2 ), 2 ...

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...stake as being Euclidean, elliptic, or hyperbolic. Great insight comes from the realization that the geometries of Euclid, Gauss, Bolyai, et al, are ... ...turning to mathematics, and Prof. Hass, my thesis advisor, introduced me to the real world of geometry and topology. Virtually all of my thinking in... ...ther Objects in an s-Manifold Segments Our s-lines correspond naturally to the real line, since they extend indefinitely and are continuous. Given ... ...nd by traveling in the other direction. In terms of this distance function, the real numbers cover the s-line. In the case of a closed s-line, R cov... ...egment to be any part of an s-line that corresponds to a closed interval on the real line. If we were to choose this definition, a closed s-line, lik... ...ht angles at G and H, we have that ∠HAB + ∠ABG + ∠HCG = 120º. This angle sum analysis can, of course, be extended to polygons with four or more si... ...hought, each theorem of Euclidean geometry is an object ready for an s-manifold analysis. Here, we will consider one example. 59 The alternate... ...ll never intersect. 69 For the non-collinear triple G, H, and I, a similar analysis shows that the s-lines e and f must always intersect. It is... ...t us finish with a discussion of future research in this area. There is much analysis that can be done on the theorems of Euclidean, hyperbolic, a...

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...Gray for her encouragement and patience as I struggled to make this book a reality. The material cited in this book can be found at the website http:/... ...simultaneously, it can become overloaded and “see” structure that does not really exist. 71 F. Smarandache defined several ASCII images that have th... ...milar to that for series number 10, although more slowly, so no additional analysis will be done. 12) Let f: N → R be a function defined by ... ...mate value for the sum of the series. 131 16) The series where p > 1 is a real number, converges to a number s16. (For 0 ≤ p ≤ 2, the series diverge...

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...om 9 2 = 81, and the sum of the digits of 81 = 9. For purposes of this analysis, we will consider one to be a trivial solution and ignore it. We ... ... r=0 Smarandache (Inferior) Prime Part Sequence For any positive real number n, one can define p p (n) as the largest prime number less than...

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...s, solved/unsolved problems, new Smarandache type sequences and new Smarandache Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theory by F. Smarandache ("The Florentin Smarandache papers" special collection, University of Craiova Library, and Archivele Statului, Filiala Valcea) an...

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...che Notions in number theory. Moreover a chapter (IV) is dedicated to the analysis of Smarandache Double factorial function introduced in number theo... ...euto-Smarandache-Totient harmonic series: ¥ = > 1 0 " " ) ( 1 n a number real a is a where n Zt Problem 47. Study the convergence of the series: ... ... can be made arbitrarily close to zero. Theorem 1.2.13 For any arbitrary real number 0 > e , there is some number 1 ‡ n such that e < n n Zw ) (... ...ct of distinct primes k p p p q K 2 1 = such that e < q 1 where e is any real number greater than zero. Now take a number n such that: 2 , , 2 1 2 1... ...ibuted uniformly in the interval ]0,1]. Conjecture 4.4 For any arbitrary real number 0 > e , there is some number 1 ‡ n such that e < n n Sdf ) ( L... ...W(n)=n-1 if n is repunit (that is all the digits are 1). In the following analysis we will exclude this two species of integers. With the Ubasic pro...

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...um amount is a fair reflection of the expected loss potential. However, in reality the situation is not all that simple and not always explicable usi... ... rigorous mathematical exposition or numerical simulation and sensitivity analysis. The major part of the work attempts to build on existing, establ... ...hapter. However an alternative scenario could be that instead of buying a real put option in the options market, the payoff from a long put and long... ...l exposed to worst-case drawdown scenarios or risks of ruin even when, in reality, the actual statistical expectation of such ultra-negative scenari... ...ing as t → T. That this theorization does not satisfactorily reflect 23 reality is attributable to some kind of combined effect of overreaction of ... ...r divergence becomes unsustainable due to the extreme pressure exerted by real economic forces ‘gone out of sync’ and the gap between perceived and ... ...ropean type only but methodological extensions are possible to extend our analysis to also include American or even Bermudan options. In our ... ... to also include American or even Bermudan options. In our present analysis we will be restricted exclusively to portfolio insurance strategy ... ...d concepts from mathematical information theory can be applied in utility analysis in general and to analyzing investor utility preferences in parti...

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...cial theory of relativity are a kinematical perspective rather than being real; but "reality" is a slippery concept, and it is expected that the read... ...nition will unfold as the reader progresses through the book. The test of reality to be ultimately applied is whether or not it is really impossible... ...observers is seen from experiment. Being a theorist, I cannot undertake a real experiment; however, the grand experiment that Homer proposes is only... ...y a theoretical one and I support his right to consider it. Insofar as "reality" is concerned I claim not to know it. I only accept that all is il... ...ed I claim not to know it. I only accept that all is illusion, and that "reality" is whatever one chooses to accept. Hence, I cannot state that STR... ...r component" is the kinematical construct we call "magnetic field.") This analysis appears in Speculations in Science and Technolgy,1993, Vol.16, No.... ...c particles which are electro-dynamically accelerated. Then in the third analysis, that of a conventional rocket or Bussard jetship, no light barri... ... know of today that can enable interstellar flight." We felt a comparative analysis was in order of the acceleration to be expected from such a plan.... ...is was in order of the acceleration to be expected from such a plan. Analysis: In the theory of special relativity, it is chosen to define for...