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...The primal natm:e of the effort to establish relation is already to be seen in the earliest and most confined stage. Before anything isolated can be perceived, timid glances mOve out into indistinct space, towards something indefinite; and in times when there seems to ...

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... to other men and his attitude to things. The attitude to other men is a relation between persons, to things it is a connexi~n with objects. In the... ...etween persons, to things it is a connexi~n with objects. In the personal relation one subject - I - confronts another subject-Thou 1; in the conne... ...the. "world of Thou", and the latter the " world of It" 'The content and relation of these two worlds is the theme of I and Thou. The other person,... ...ncing subject and experienced object, is possible. So long as I remain in relation with ;my Tlunt, I cannot experience it, but can only know it in ... ...tion with ;my Tlunt, I cannot experience it, but can only know it in the relation itself. U In the act of experience Thou is far away." The world ... ...and. in it. * The life of human beings is not passed in the sphere : of transitive verbs alone. It does not exist in virtue ,,: of a.otivities alo...

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...e first section we just give the basic concepts or notion like equivalence relation greatest common divisor etc. Second section is devoted to giving ... ...ain aim of this section is to introduce the basic concepts of equivalence relation, equivalence class and introduce some number theoretic results use... ...= c -1 ac. We denote a conjugate to b by a ~ b and we shall refer to this relation as conjugacy relation on G. 9 DEFINITION 1.2.4: Let G be a... ...properties of them. DEFINITION 1.3.1: Let A and B be non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to B are cal... ...R is antisymmetric if for all a and b in A, aRb and bRa imply a = b. R is transitive if for a, b, c in A, aRb and bRc imply aRc. A relation R on A... ...lation R on A is an equivalence relation if R is reflexive, symmetric and transitive. In this case [a] = {b ∈ A / aRb} is called the equivalence clas... ...s called a partial order (relation) if R is reflexive, anti-symmetric and transitive. In this case (A, R) is called a partial ordered set or a poset... ... the representation of a loop (L, •) if and only if a. I ∈ π b. π is transitive on L and c. for α, β ∈ π, if αβ -1 fixes one element of L, th... ...sults of Albert [1]. a. Clearly π has identity. b. To show that π is transitive on L. Now these n non-identity distinct permutations have (n(n+...

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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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...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... ... x 5 x 1 .6 x 5 x 2 .1 x 5 x 5 .5 A crisp relation R (X, Y) is called transitive if and only if (x, z) ∈ R, whenever both (x, y) ∈ R and (... ...other words the relation of x to y and of y to z imply the relation x to z is a transitive relation. A relation that does not satisfy this property ... ...ansitive relation. A relation that does not satisfy this property is called non-transitive. If (x, z) ∉ R whenever both (x, y) ∈ R and (y, z) ∈ R, t... ... ∉ R whenever both (x, y) ∈ R and (y, z) ∈ R, then the relation is called anti-transitive. The reflexivity, symmetry and transitivity is described ... ...x, y ∈ X the relation R is called anti symmetric. A fuzzy relation R (X, X) is transitive if R (x, z) ≥ Y y∈ max min [R (x, y), R (y, z)] is satis...

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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... ... x 5 x 1 .6 x 5 x 2 .1 x 5 x 5 .5 A crisp relation R (X, Y) is called transitive if and only if (x, z) ∈ R, whenever both (x, y) ∈ R and (... ...other words the relation of x to y and of y to z imply the relation x to z is a transitive relation. A relation that does not satisfy this property ... ...ansitive relation. A relation that does not satisfy this property is called non-transitive. If (x, z) ∉ R whenever both (x, y) ∈ R and (y, z) ∈ R, t... ... ∉ R whenever both (x, y) ∈ R and (y, z) ∈ R, then the relation is called anti-transitive. The reflexivity, symmetry and transitivity is described ... ...x, y ∈ X the relation R is called anti symmetric. A fuzzy relation R (X, X) is transitive if R (x, z) ≥ Y y∈ max min [R (x, y), R (y, z)] is satis...

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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... ... a fuzzy partial ordering if and only if it is reflexive antisymmetric and transitive under some form of fuzzy transitivity. We represent this b... ...3 .2 1 .6 .3 .2 .2 a b c d e .3 96 A fuzzy relation R N (X, X) is transitive (or more specifically max-min transitive) if R N (x, z) ... ...ailing to satisfy the above inequality for some members of X is called non- transitive and if R N (x, x) < Y y ∈ max min [R N (x, y), R N (y, z)] f... ...(x, y), R N (y, z)] for all (x, x) ∈ X 2 , then the relation is called anti-transitive. Given a relation R N (X, X) its transitive closure R NT (X,... ... transitive closure R NT (X, X) can be analyzed in the following way. The transitive closure on a crisp relation R N (X, X) is defined as the rel...

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... n . (D 2n is called the dihedral group of order 2n given by the following relation, D 2n = {a, b/ a 2 = b n = 1; bab = a}. 13. Give an examp... ...ces used in this book. DEFINITION 1.2.1: Let A and B be non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to B are c... ...lation R from A to B is a subset of A × B. Relations from A to B are called relations on A, for short, if (a, b) ∈ R then we write aRb and say that ... ...ions on A, for short, if (a, b) ∈ R then we write aRb and say that 'a is in relation R to b’. Also if a is not in relation R to b, we write b R a / ... ... R is anti symmetric if for all a and b in A; aRb and bRa imply a = b. R is transitive if for a, b, c in A; aRb and bRc imply aRc. A relation R on... ... relation R on A is an equivalence relation if R is reflexive, symmetric and transitive. In the case [a] = {b ∈ A| aRb}, is called the equivalence cl... ... is called a partial order (relation) if R is reflexive, anti symmetric and transitive. In this case (A, R) is a partially ordered set or poset. ... ...tric equivalence, see Equivalence Reflexive, 11-12 Symmetric, 11-12 Transitive, 11-12 Right ideal, see Ideal, right Ring characteristic 0... ...up, symmetric T Totally ordered set, 12 Transitivity, see Relation, transitive U UDO HEBISCH, 29, 51, 93 Unit, 8, 19, 29-30, 32-33,...

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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... ...ollowing two conditions: (Ci) αP =P −1 α; (Cii) the group Ψ J =<α,β,P > is transitive on X α,β . For a given map M = (X α,β ,P), it can be shown that ... ...t, the following condition is needed. (Ciii) If the group Ψ I =< αβ,P > is transitive on X α,β , then M is non- orientable. Otherwise, orientable. It ... ...results. Corollary 2.2.3 If a finite groupG,G AutM is fixed-free onV(M) and transitive onF(M), for example,M is regular andG =AutM, thenM/G is an one f... ...(G,g). ♮ Corollary 3.1.1 For any positive integer n, there exists a vertex transitive map M underlying a circultant such that Z n is an orientation-pr... ...diner et al proved in [20] that if add an additional condition , i.e, G is transitive on the vertices in Γ, then there is a regular map underlying the...

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...× µ ) (x, y) = min {λ(x), µ (y)} for every (x, y) ∈ X × Y. A fuzzy binary relation R λ on a set X is defined as a fuzzy subset of X × X. The co... ...X is defined as a fuzzy subset of X × X. The composition of two fuzzy relations R λ and R µ is defined by (R λ o R µ )(x, y) = sup X t∈ {min R... ... y)}, for every x, y ∈ X. DEFINITION 1.1.12: Let R λ be a fuzzy binary relation on a set X. A fuzzy subset µ of the set X is said to be a pre cla... ... {µ (x), µ (y) } ≤ R λ (x, y) for every x, y ∈ X. 11 A fuzzy binary relation R λ on a set X is said to be a similarity relation on the set X i... ...o be a similarity relation on the set X if it is reflexive, symmetric and transitive that is, for every x, y, z ∈ X. R λ (x, x) = 1 R λ (x, y) ... ...s said to be a similarity relation on X if it is reflexive, symmetric and transitive i.e. S (x, x) = 1. S (x, y) = S (y, x). S (x, y) ∧ S (y, z)... ... , z ( ), z , x ( min ) y , x ( sup M z α α α ∈ ≥ for all x, y in M (fuzzy transitive). DEFINITION 1.8.5: A fuzzy equivalence relation α on an R-mo... .... 133 DEFINITION [54]: Let S be a non-empty set completely ordered by transitive, irreflexive relation '<'. For all x, y ∈ S, exactly one relati... ... X; λ is symmetric on X if λ (a, b) = λ (b, a) for all a, b ∈ X; and λ is transitive on X if λ o λ ⊆ λ . A reflexive, symmetric and transitive fu...

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...dular lattices. DEFINITION 1.3.1: Let A and B be two non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to A are ca... ...ion R from A to B is a subset of A × B. Relations from A to A are called relation on A, for short. If (a, b) ∈ R then we write aRb and say that a i... ...ion on A, for short. If (a, b) ∈ R then we write aRb and say that a is in relation R to b. Also if a is not in relation R to b we write a R / b. A r... ... is anti symmetric if for all a,b in A, aRb and bRa imply a = b. R is transitive if for all a,b,c in A aRb and bRc imply aRc. A relation R on A i... ... a = b. R is transitive if for all a,b,c in A aRb and bRc imply aRc. A relation R on A is an equivalence relation, if R is reflexive, symmetric a... ...ation R on A is an equivalence relation, if R is reflexive, symmetric and transitive. In this case, [a] = {b ∈ A / aRb} is called the equivalence... ...s called a partial order (relation) if R is reflexive, anti symmetric and transitive. In this case (A, R) is called a partially ordered set or poset... ... DEFINITION [5]: A ring R is called filial if the relation ideal in R is transitive, that is if a subring J is an ideal in a subring I, and I is an... ...a Smarandache filial ring (S- filial ring) if the relation S-ideal in R is transitive, that is if a S-subring, J is an S- ideal in a S-subring I and I...

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...lly, chapter four makes an introduction to neutrosophic logic, where the relationship between it and the other logics will be clear. 4 As alway... ...to be logically equivalent if p ↔ q is a tautology. The notation for this relationship is p q. It is possible to show that two expressions... ...pressions are taken to be true by assumption. d) There is a finite set of relations R 1 , R 2 , . . . , R n between sets of the wffs in S, called t... ...ent. p→ q, ¬q ¬p • Resolution Implication is transitive p → q, q → r p → r 19 Section 7 Formal... ... truth value of the expression is dependent on internal variables and the relationships between the features. For example, in the first expression, ... ...s defined in NL1 is 121 (i) irreflexive, (ii) asymmetric, (iii) transitive, (iv) connected. b) The order relation (<) as defined in N... ...n (<) as defined in NL2 is (v) irreflexive, (vi) asymmetric, (vii) transitive, (viii) connected. c) The order relation (<) as defined in ... ...n (<) as defined in INL1 is (ix) irreflexive, (x) asymmetric, (xi) transitive, (xii) connected. d) The order relation (<) as defined in I... ...) as defined in INL2 is (xiii) irreflexive, (xiv) asymmetric, (xv) transitive, (xvi) connected. e) The order relation (<) as defined in P...

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...hereA,B,C are analytic functions. I emphatically remark that the geometric relations between the components of the metric tensor of (2a) are precisely... ...lthough the absolute value is suppressed in (1) and (7a)). The geometrical relations between the components of the metric tensor are inviolable. There... ...ue arbitrary nature of the origin r 0 . Therefore, the correct geometrical relations have gone unrecognized by the orthodox analysis. I note, for inst... ...located at somer =r 0 in parameter space, wherer, r 0 ∈ . The geometrical relationships between the components of the metric tensor of (1) must be pr... ... q p C(D)+ q p C(D)−α K , (20) D=|r−r 0 |, K = const. The relationship betweenr andR p is, asr→r ± 0 , R p (r)→0 + , or equivalently,... ...ing that the action of the isotropy groupK of the origin is not necesarily transitive on the ordinary topolo- gical boundary. Shilov boundaries are th...

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...p (S-strongly commutative loop). The author is requested to obtain the relations between these concepts. THEOREM [72]: Let L be a power associ... ...easily obtained. Hence left for the reader to prove. We now give some relation between the quasi regular elements and idempotents, how the exist... ...dempotent element cannot be right quasi regular. 25 Now we give the relation between the J(FL) and W(FL) in a loop ring. THEOREM 2.2.10: Le... ...rongly ideal ring (S-strongly ideal ring). We do not have any form of relations existing between them but we have several problems in this direc... ...ould be if L is a loop having P (L) to be the pseudo commutator; find the relation between P (RL) and R (P (L)). DEFINITION 2.4.20: Let RL be a l... ...e filial Lie ring (S-filial Lie ring) if the relation S-Lie ideal in L is transitive that is a S-subring J is a S-Lie ideal in S-subring I and I a S...

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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... ...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... ...ne correspondence is preserved under composition, s-congruence of s-segments is transitive. In fact, our definition of s- congruence satisfies the pr... ...he properties of an equivalence relation, i.e., it is reflexive, symmetric, and transitive. Using a notion of congruence that was not an equivalence ...

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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... ...rnal data is weak in comparison (it is never absent) – we are likely to experience Transitive Emotions. The latter are emotions, which involve obse...

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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... ...rnal data is weak in comparison (it is never absent) – we are likely to experience Transitive Emotions. The latter are emotions, which involve obse...

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...R be a ring. We say R is a Smarandache filial ring (S-filial ring) if the relation S-ideal in R is transitive, that is if a S-subring J is an S-ideal... ...a Smarandache filial ring (S-filial ring) if the relation S-ideal in R is transitive, that is if a S-subring J is an S-ideal in a S-subring I and I ... ... a ring, a pair x, y ∈ R is said to have a Smarandache subring right link relation (S-subring right link relation) if there exists a S-subring P in ... ... \ {x, y} such that x ∈ Py and y ∈ P x. Similarly Smarandache subring link relation (S-subring link relation) if x ∈ y P and y ∈ x P. If it has both ... ... if x ∈ y P and y ∈ x P. If it has both a Smarandache left and right link relation for the same S-subring P, then we say x and y have a Smarandache ... ... using groupoids over rings. These non-associative rings enjoy a varied relation only when they satisfy the special types of identities like Moufa...

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...s by those of the characters in their scripture, so was time (its tattered relationships, its solitary and jejune rides in coffins on board trains) a ... ...(he was going to consider himself 39 for as long as he could), arm broken, relationship with a wife broken, and girlfriend deceased most horrifically,... ... he were painting over the background that tainted him? Perversion had the transitive root 'to pervert', and it was rooted deep with the years: it was... ...s long as they had good health, the necessities of food and shelter, scant relationships at the very least, and some occupation to direct time and tho... ..., such as this obsession with an aging self and his ineptness at long-term relationships, were quickly, dizzying, and incessantly being repeated in hi... ...ng to himself, "As if she had not urged it on--as if I would have had this relationship with Kimberly, as much as I may have wanted it, without this b...

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... under this rule, may properly be run into the text, if it bears an organic relation to the argument pre- sented. On the other hand, a quotation of ... .... If among several adjectives preceding a noun the last bears a more direct relation to the noun than the others, it should not be preceded by a com... ... rule, nouns formed by the combina- tion of two nouns standing in objective relation to each other-that is, one of whose components is de- rived fr... ...elation to each other-that is, one of whose components is de- rived from a transitive verb: mind-reader, story-teller, fool-killer, office-holder, ...

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...companied on these occasions; and, in the tried security of their personal relations, had been driven to conclude that his book was bothering him, and... ...n the gauntlet, to flaunt his private code in the face of society; and the relation 93 Wharton of the sexes being a topic always sure of an audience,... .... It was a room with which she had never been able to establish any closer relation than that between a traveller and a railway station; and now, as s... ... the old porcelains, seemed to typify a superficial refinement that had no relation to the deeper significances of life. Suddenly she heard her husban... ...means of self-defence?” “Some preliminary knowledge of human nature in its relation to the marriage tie.” She made an impatient gesture. “How should y... ... for: the readjustment of personal relations. As soon as their necessarily transitive nature was recognized they would gain in dignity as well as in h...

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...ne man, it is all to be explained from indi- vidual experience. There is a relation between the hours of our life and the centuries of time. As the ai... ...and other accidents of appearance; others by intrinsic likeness, or by the relation of cause and effect. The progress of the intellect is to the clear... ...ct in nature, to reduce it under the dominion of man. A man is a bundle of relations, a knot of roots, whose flower and fruitage is the world. His fac... ...en it before me. For my perception of it is as much a fact as the sun. The relations of the soul to the divine spirit are so pure that it is profane t... ...sorbs past and future into the present hour. All things are made sacred by relation to it,—one as much as another. All things are dissolved to their c... ...ld and false. For all symbols are fluxional; all language is vehicular and transitive, and is good, as ferries and horses are, for conveyance, not as ...

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...case; _255 And if I should perceive in your relation That either has the right to satisfaction From the other, I give y... ...he intercourse would be promiscuous: on the contrary, it appears, from the relation of parent to child, that this union is generally of long duration,... ...and rejected the delusions of free-will. Religion is the perception of the relation in which we stand to the principle of the universe. But if the pri... ... the universe be not an organic being, the model and prototype of man, the relation between it and human beings is abso- lutely none. Without some ins... ...iverse, otherwise than as the events to which we apply these epithets have relation to our own peculiar mode of being. Still less than with the hypoth... ... as adjectival clauses qualifying whirlwinds (3), and taking bore (3) as a transitive verb governing All who approached their sphere (5). This, which ... ...d and fed his everliving woe. Wake, accordingly, is to be con- strued as a transitive (Locock). 3. 3. 3. 3. 3. Lines 130-169. This entire passage is d...