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... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS HEXI... ...OSOPHIC RELATIONAL MAPS HEXIS Church Rock 2004 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasan... ... i c a t i o n 6 P r e f a c e 7 Chapter One FUZZY RELATIONAL EQUATIONS: BASIC CONCEPTS AND PROPERTIES 1.1 Fuzzy Rel... ... Fuzzy Relational Equations and their properties 10 1.2 Properties of Fuzzy Relations 16 1.3 Fuzzy compatibility relations and composition o... ...In this section we just recollect the properties of fuzzy relations like, fuzzy equivalence relation, fuzzy compatibility relations, fuzzy ordering ... ...more about these concepts please refer [43]. Now we proceed on to define fuzzy equivalence relation. A crisp binary relation R(X, X) that is reflex... ...inary relation R(X, X) that is reflexive, symmetric and transitive is called an equivalence relation. For each element x in X, we can define a crisp... ...sp set A x , which contains all the elements of X that are related to x, by the equivalence relation. A x = {y ⏐(x, y) ∈ R (X, X)} A x is cl... ...ed to any element of X not included in A x . This set A x is referred to an as equivalence class 17 of R (X, X) with respect to x. The members...

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... Preface 5 1. Preliminary notions 1.1 Binary Relation 7 1.2 Mappings 9 1.3 Semigroup and Smarandache Semigroup... ...marandache semigroups. In Chapter one, we introduce some basic notation, Binary relations, mappings and the concept of semigroup and Smarandache sem... ...appings and the concept of semigroup and Smarandache semigroup. 1.1 Binary Relation Let A be any non-empty set. We consider the Cartesian pr... ...s called the 8 diagonal of A × A. A subset S of A × A is said to define an equivalence relation on A if (a, a) ∈ S for all a ∈ A (a, b) ∈ S ... ... 8 diagonal of A × A. A subset S of A × A is said to define an equivalence relation on A if (a, a) ∈ S for all a ∈ A (a, b) ∈ S implies (b, ... ...ollowing definition. DEFINITION: The binary relation ~ on A is said to be an equivalence relation on A if for all a, b, c in A i. a ~ a ii. a... ...reflexivity, the second, symmetry and the third transitivity. The concept of an equivalence relation is an extremely important one and plays a centr... ... a central role in all mathematics. DEFINITION: If A is a set and if ~ is an equivalence relation on A, then the equivalence class of a ∈ A is th... ... ∈ A/ a ~ x}. We write this set as cl(a) or [a]. THEOREM 1.1.1: The distinct equivalence classes of an equivalence relation on A provide us with ...

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... and applies it to the case of neutrosophic cognitive maps and neutrosophic relational maps. The last section of this chapter clearly illustrates ho... ...out this book by ‘I ’ we denote the indeterminacy of any notion / concept / relation. That is when we are not in a position to associate a relation ... ...f the neutrosophic vector space V over R. A study of these basis and its relations happens to be an interesting form of research. DEFINITION 1... ... Cognitive Maps and use rectangular neutrosophic matrices for Neutrosophic Relational Maps. 13 Chapter Two SOME BASIC RESULTS... ...ion of a graph in which people are represented by points and interpersonal relations by lines. Such relations include love, hate, communication and... ...is reachable from the other. It is easily verified that disconnection is an equivalence relation on the vertex set of D and if the equivalence class... ...onigsberg bridge problem has no solution. The following theorem which gives equivalence of three conditions is left for the reader as an exercise to...

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...velopment and dissemination of G-dimensional theory (GDT). K. Toshihara, from Japan, proposed the concept of a paper defining G-Dimensional theory in relation to the Smarandache quantum paradoxes. After reviewing a draft, Dr. M. L. Perez, Editor of the SNJ, suggested the inclusion of the Smarandache (FTL) Hypothesis as well. So it is hoped that a determination of parametri...

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...rom Japan, proposed the concept of a paper defining G-Dimensional theory in relation to the Smarandache quantum paradoxes. After reviewing a draft, ... ...(FTL) Hypothesis as well. So it is hoped that a determination of parametric relationships between the quantum paradoxes and GDT proves helpful in adv... ...e greater the similarity between set B and set A, the more logical is their relationship. We see then, (somewhat obviously), that these statements a... ...roportional focal length" (f), defines the general kinetic vector geometric relationship with elliptic eccentricity established in terms of (a/b); f... ...ongitudinal lengths {a 1 , a 2 }, as in accordance with the principle of equivalence, Einstein (1907), are as yet undetermined. That is, we have n... ... σ e (1 + vu) . (14) 3.5 Inertial Causality & Gravitational Equivalence We find a simple premise reveals causality of inertial motio... ...r, and thus is not contradicted by (24). 3.10 Particle Gravity Identity & Equivalence It is proposed that matter particles are extrinsic and int... ... longitudinal length contraction, and therefore does not determine relative equivalence. Given the "intrusion principle" (gravity identities 1-3) abo... ...intrusion principle" (gravity identities 1-3) above, we must infer "reverse equivalence" i.e. longitudinal length contraction in the gravitational ve...

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...The aim of this book is two fold. At the outset the book gives most of the available literature about Fuzzy Relational Equations (FREs) and its properties for there is no book that solely caters to FREs and its applications. Though we have a comprehensive bibliography, we do not promise to give all the possible available literature...

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... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS HEXI... ...OSOPHIC RELATIONAL MAPS HEXIS Church Rock 2004 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasan... ... i c a t i o n 6 P r e f a c e 7 Chapter One FUZZY RELATIONAL EQUATIONS: BASIC CONCEPTS AND PROPERTIES 1.1 Fuzzy Rel... ... Fuzzy Relational Equations and their properties 10 1.2 Properties of Fuzzy Relations 16 1.3 Fuzzy compatibility relations and composition o... ...In this section we just recollect the properties of fuzzy relations like, fuzzy equivalence relation, fuzzy compatibility relations, fuzzy ordering ... ...more about these concepts please refer [43]. Now we proceed on to define fuzzy equivalence relation. A crisp binary relation R(X, X) that is reflex... ...inary relation R(X, X) that is reflexive, symmetric and transitive is called an equivalence relation. For each element x in X, we can define a crisp... ...sp set A x , which contains all the elements of X that are related to x, by the equivalence relation. A x = {y ⏐(x, y) ∈ R (X, X)} A x is cl... ...ed to any element of X not included in A x . This set A x is referred to an as equivalence class 17 of R (X, X) with respect to x. The members...

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...s-manifolds are a very restricted subclass of the 7 polyhedral surfaces. The relationship between polyhedral surfaces and Riemannian manifolds is... ... The non-Euclidean vertices introduce a sort of curvature, and this affects the relationships between s-lines. The topology of an s-manifold can all... ...lines being parallel or not parallel at different points along them, since this relationship between s-lines changes as 21 we move from point to ... ...y, to change from region to region, this does not necessarily carry over to the relationships between lines, or 26 geodesics. This is, in fact, o... ...e to S-deny all of Hilbert’s axioms, and our choice to have congruence to be an equivalence relation further reduces our ability to S-deny Hilbert’s... ...all of Hilbert’s axioms, and our choice to have congruence to be an equivalence relation further reduces our ability to S-deny Hilbert’s axioms. Mos... ...nsitive. In fact, our definition of s- congruence satisfies the properties of an equivalence relation, i.e., it is reflexive, symmetric, and transiti... ...lexive, symmetric, and transitive. Using a notion of congruence that was not an equivalence relation would complicate the study of all related issue...

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... Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathemati...

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... how the Zagreb indices, a particular case of Smarandache-Zagreb index of a graph changes with these operators and extended these results to obtain a relation connecting the Zagreb index on operators. Key Words: Subdivision graph, ladder graph, Smarandache-Zagreb index, Zagreb index, graph operators. ...

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...SI Hypergraph Partitioning Using Taguchi Methods BY P.SUBBARAJ, S.SARAVANASANKAR and S.ANAND . . . . . . . . . . . . . . . . 69 Negation Switching Equivalence in Signed Graphs BY P.SIVA KOTA REDDY, K.SHIVASHANKARA and K.V.MADHUSUDHAN. . . . . . 85 Weak and Strong Reinforcement Number For a Graph BY PINAR DUNDAR, TUFAN TURACI and DERYA DOGAN. . . . . . . . . . . . . ....

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... Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combinatorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathemati...

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.... . . . . . . . 70 Entire Semitotal-Point Domination in Graphs BY B.BASAVANAGOUD, S.M.HOSAMANI and S.H.MALGHAN. . . . . . . . . . . . . . .79 On k-Equivalence Domination in Graphs BY S.ARUMUGAM and M.SUNDARAKANNAN. . . . . . . . . . . . . . . . . . . . . . . 86 On Near Mean Graphs BY A.NAGARAJAN, A.NELLAI MURUGAN and S.NAVANEETHA KRISHNAN. . . . . . 94 On Pathos Lic...

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... manifolds with higher dimension are inspired by the Riemann surfaces. The relation of maps with Riemann surfaces has been known in 80s in the last ce... ...etries, maps and the semi-arc automorphism group of a graph. Preface ii A relation for maps and Smarandache manifolds (abbreviated s-manifolds) and a... ...oted maps are established in this chapter. The last section determines the relation of the number of embeddings and rooted maps of a graph on genus. A... ...me for enumerating maps underlying a graph.........................17 §4 A relation among the total embeddings and rooted maps of a graph on genus20 4... ...pty boundary and to be analytic functions. Whence, we have the following relation: {Riemann Sufaces}⊂{Klein surfaces}. The upper half planeH ={z∈C|I... ...re exists an isomorphism ζ between them induced by an element ξ. Call ζ an equivalence between M 1 ,M 2 . Certainly, on an orientable surface, an equi... ...t1 2 Γ such that P ζ 1 = P 2 or P ζ 1 = P −1 2 . Proof By the definition of equivalence between maps, if κ is an equivalence between M 1 and M 2 , then... ...ducedtheconformalmappingbetween maps. Then, how can we define the conformal equivalence for maps enabling us to get the uniformiza- tion theorem of map...

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...el each of which assumes some economic parameter which describe the inter relations between the industries in the economy 32 under considerations... ...he collection of all polynomials of degree less than or equal to 4. So no relation among elements of V 1 and V 2 is possible. Thus we also show t... ...gebra and the linear bialgebra. Clearly the strong linear bialgebra has no relation with the linear bialgebra or a linear bialgebra has no relation ... ...d where as the strong linear bialgebra is defined over a bifield, hence no relation can ever be derived. In the similar means one cannot derive any... ...n can ever be derived. In the similar means one cannot derive any form of relation between the weak linear bialgebra and linear bialgebra. All th... ...d to define the notion of Jordan biform. The notion of Jordan form in each equivalence class of matrices under similarity and so it has been proved ...

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...gy and the second revolves around the role of technological progress in re-defining relationships between stakeholders. Both issues are related to ... ...mous centrifugal shift is underway. Power percolates back to the people. Thus, the relationships between user and supplier, customer and company, s... ...o discrepancies in accrued wealth. The doctrine of equal access is founded on the equivalence of Men. That all men are created equal and deserve t... ...f the increase in the share price and the resulting capital gain. But whatever the relationship, there is no doubt that earnings are a good proxy t... ...eigh the physical or monetary kinds. This led to an ongoing revolution in economic relations. Ironically, dehumanizing totalitarian regimes, such a... ...the exact nature and extent of the rot - but they witnessed approvingly the public relations antics, insider trading, stock option resetting , unwi... ...raction between different definitions of definition gives rise to numerous forms of equivalence, all called "definitions". Real definitions, nominal...

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...the exact nature and extent of the rot - but they witnessed approvingly the public relations antics, insider trading, stock option resetting , unwi... ...nkenscientist, in due course, science will even be able to demonstrate a monovalent relationship between a pattern of brain activity in situ and the... ...l communication between humans would have ceased to exist. It would appear that the relationship between this universal language and the idiosyncrat... ...tween this universal language and the idiosyncratic, individualistic languages is a relation of correlation. Pain is correlated to brain activity, o... ... Whatever the disparity between the two, they must be related to one another. This relation is by virtue of two facts: (1) Both the brain and the c... ...raction between different definitions of definition gives rise to numerous forms of equivalence, all called "definitions". Real definitions, nominal... ...o discrepancies in accrued wealth. The doctrine of equal access is founded on the equivalence of Men. That all men are created equal and deserve t... ...rinciple. The general Relativity Theory (GRT) was founded on the principle of the equivalence (i.e., indistinguishability) of gravity and inertia.... ... desirable and two identically-preferable courses of action. Where there is no such equivalence – there is no choice, merely the pre-ordained (given...