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...nal products 164 Chapter Two SOME APPLICATIONS OF FRE 2.1 Use of FRE in Chemical Engineering 167 2.2 New FRE to estimate the peak hours o... ... motor-drive system 197 2.10 Application of genetic algorithm to problems in chemical industries 199 2.11 Semantics of implication operators ... ... has 10 sections: we elaborately give the applications of FRE in flow rates in chemical industry problems, preference and determination of peak hou... ...m X to Y, binary relations R(X, Y) may assign to each element of X two or more elements of Y. Some basic operations on functions such as the invers... ...n that each element of Y has to an element of X is equal to the degree of that elements membership in the range of R. In addition, the height of a ... ...responding to one set is distinguished from nodes representing the other set. Elements of X × Y with non-zero membership grades in R(X, Y) are rep... ... ik ] ° [q kj ] where r ij = k max min (p ik q kj ). Observe that the same elements of P and Q are used in the calculation of R as would be use... ... give several of the applications of fuzzy relational equations in studies like chemical engineering, transportation, medicine etc. The fuzzy relati... ... applied to special problem. [81, 85, 100-101,111, 114] 2.1 Use of FRE in chemical engineering The use of fuzzy relational equations (FRE) f...

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....6: Let G(I) be a neutrosophic vector space over the field K. The number of elements in the neutrosophic basis is called the neutrosophic dimension ... ...g neutrosophic vector space over the neutrosophic field K(I). The number of elements in the strong neutrosophic basis is called the strong neutrosop... ...tinguished from the others), with point of degree at most 4 and finally the chemical problem of trees in which every point has degree 1 or 4. Jorda... ...e that Jordan did so without having any suspicion of its bearing on modern chemical doctrine. The most famous problem in graph theory and perhaps i... ...nite non empty set V = V (G) of p points. (vertex, node, junction O-simplex elements) together with a prescribed set X of q unordered pairs of disti... ...Each pair x = {u, v} of points in X is a line (edge, arc, branch, 1-simplex elements) of G and x is said to join u and v. We write x = uv and say t... ...graph is a complete graph. Thus in a tournament for every pair of distinct elements u and v either (u, v) or (v, u) but not both is an arc of D. To... ...ry, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars...

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...belian or a commutative group. If the group G has only a finite number of elements we call G a group of finite order otherwise G is said to be of in... ...G is said to be of infinite order. If a b ≠ b a, for atleast a pair of elements a, b ∈ G, then we call G a non-commutative group. Notation: Let... ...up of G then o(H) is a divisor of o(G). Note: o(G) means the number of elements in G it will also be denoted by |G|. CAUCHY THEOREM (FOR ABELIA... ...ontain any non-trivial normal subloop. DEFINITION 1.1.4: If x and y are elements of a loop L, the commutator (x, y) is defined by xy = (yx) (x, y)... ... 〉 denotes the subloop generated by A. DEFINITION 1.1.5: If x, y, z are elements of a loop L an associator (x, y, z) is defined by (xy)z = (x(yz))... ..., fuzzy groups, and applications of fuzzy theory to the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars a...

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...(B 1 + B 2 ). Thus two bimatrices are added by adding the corresponding elements only when compatibility of usual matrix addition exists. No... ...y square bimatrix then ( ) 2 mm mm II × × = . A bimatrix A B all of whose elements are zero is called a null or zero bimatrix and it is denoted by... ...This is a m th order square bimatrix. This will not have 2m 2 arbitrary elements since 11 ij ji aa = and 22 ij ji aa = (where A 1 = 1 ij (a ... ...nd A 2 = 2 ij (a )) both below and above the main diagonal. The number of elements above the main diagonal of A B = A 1 ∪ A 2 is (m 2 – m). Th... ... above the main diagonal of A B = A 1 ∪ A 2 is (m 2 – m). The diagonal elements are also arbitrary. Thus the total number of arbitrary elements... ...ry, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars...

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...ndence 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 composite structures, which may not play any independent part in theoretical physics." / Einstein then went on to say that, in spite of the foregoing comment, we should temporarily support the use of the...

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...ck do not in the conceptual edifice of physics play the part of irreducible elements, but that of composite structures, which may not play any indep... ...like the “light cone” equation 0 2 2 2 = − σ τ d d c ( 0 ≠ σ d , 0 ≠ τ d ), elements of which are world-trajectories of light-like particles. But, i... ...rvable quantities. So teleporting photons move along trajectories which are elements of the world-cone (like the light cone) in that space- time area... ...onents of the circuit like a battery or energy supply. The passive circuit elements like the capacitors and inductors and resistors can be modeled ... ...ion in the function space can be represented as a long sum whose terms have elements from the spectral set as factors. For example, the set {sin(nx)... ...echnology, proceedings of the Third International Conference of The Electro-chemical Society, Inc. 30 East 42 Street, New York, NY 10017; Editor Ro...

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...nal products 164 Chapter Two SOME APPLICATIONS OF FRE 2.1 Use of FRE in Chemical Engineering 167 2.2 New FRE to estimate the peak hours o... ... motor-drive system 197 2.10 Application of genetic algorithm to problems in chemical industries 199 2.11 Semantics of implication operators ... ... has 10 sections: we elaborately give the applications of FRE in flow rates in chemical industry problems, preference and determination of peak hou... ...m X to Y, binary relations R(X, Y) may assign to each element of X two or more elements of Y. Some basic operations on functions such as the invers... ...n that each element of Y has to an element of X is equal to the degree of that elements membership in the range of R. In addition, the height of a ... ...responding to one set is distinguished from nodes representing the other set. Elements of X × Y with non-zero membership grades in R(X, Y) are rep... ... ik ] ° [q kj ] where r ij = k max min (p ik q kj ). Observe that the same elements of P and Q are used in the calculation of R as would be use... ... give several of the applications of fuzzy relational equations in studies like chemical engineering, transportation, medicine etc. The fuzzy relati... ... applied to special problem. [81, 85, 100-101,111, 114] 2.1 Use of FRE in chemical engineering The use of fuzzy relational equations (FRE) f...

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...∪ (B 1 + B 2 ). Thus two bimatrices are added by adding the corresponding elements only when compatibility of usual matrix addition exists. ... ...dence maps that associates with each element of E i (G) an unordered pair of elements of V i (G) i = 1, 2. It is to be noted that E 1 (G 1 ) ... ...simple. A bigraph G = G 1 ∪ G 2 is bisimple if |G| = n that is number of elements in G 1 and G 2 without repetition i.e. o (G 1 ∪ G 2 ) = o(... ...ion between the health hazards the agricultural labourer suffers 94 due to chemical pollution. Let us taken the 11 attributes (P 1 , P 2 ,…, P 11... ... – Ulcer / skin ailments in legs and hands P 3 – Manuring the fields with chemical fertilizers P 4 – Vomiting P 5 – Mouth and stomach ... ...e at a fixed bipoint as the bihidden pattern. Thus manuring the field with of chemical fertilizers makes all the coordinates on except P 4 which im... ...ey may not suffer the symptom of vomiting because of manuring the field with chemical fertilizers. Now we study the effect of the attribute Head... ...ed by them due to spray of pesticides insecticides and manuring the plants by chemical fertilizers. At the first stage we take arbitrary attributes ... ...concepts. We further assume no intermediate relation exists within the domain elements or node and the range spaces elements. The number of elements...

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... loop-rings and introduction to Smarandache loop-rings 20 2.3 Smarandache elements in loop rings 30 2.4 Smarandache substructures in loop rings 40 ... ... Groupoid rings and Smarandache Groupoid rings 61 3.3 Smarandache special elements in Groupoid rings 69 3.4 Smarandache substructures in Groupoid ri... ...pace consists of the following: 1. a field K of scalars. 2. a set V of elements called vectors. 3. a rule or operation called vector addition whi... ...ly it is a vector quantity, only to mention that it is different from the elements of the field which are termed as scalars. 8 DEFINITION 1.1.2... ...s finite dimensional if there exists a finite set of linearly independent elements (v 1 , v 2 , …, v n ) in V which generate V, otherwise it is infi... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. scholars a...

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...ngs and its properties 2.1 Definition and examples 21 2.2 Special elements in rings 24 2.3 Substructures of a ring 26 2.4 Homomorph... .... and S. D.C.C. 67 3.8 Some special types of rings 68 3.9 Special elements in S-rings 71 3.10 Special properties about S-rings 78 ... ... 115 4.2 Smarandache rings of level II 119 4.3 Some new Smarandache elements and their properties 121 4.4 New Smarandache substructures and th... ... a -1 a = aa -1 = e. A group, which contains only a finite number of elements, is called a finite group, otherwise it is termed as an infinite ... ...s an infinite group. By the order of a finite group we mean the number of elements in the group. 8 It may happen that a group G consists enti... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. scholars ...

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...n P by declaring a ∗ b = c if σ (a, b) = c. DEFINITION: A non empty set of elements G is said to form a groupoid if in G is defined a binary oper... ...all a ∈ G. We call the order of the groupoid G to be the number of distinct elements in it denoted by o(G) or |G|. If the number of elements in G ... ...order of this groupoid is 5. Example 1.2.2: Let (S, ∗) be a groupoid with 3 elements given by the following table: ∗ x 1 x 2 x 3 x 1 x 1 x... ... commutative semigroup. DEFINITION: Let S be a semigroup, if the number of elements in a semigroup is finite we say the semigroup S is of finite ... ...= {0, 1}} is a semigroup under matrix multiplication. PROBLEM 9: How many elements are there in S 3×3 given in example 1.3.6.? PROBLEM 10: Ca... ...eory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. schola...

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...3 2 1 Clearly µ is a fuzzy subgroup of a group G and o(µ ) = number of elements of the set {x ∈ G | µ (x) = µ (e)} = number of elements of the se... ... [0, min{µ 1 (e 1 ), µ 2 (e 2 )}] where e 1 and e 2 denote the identity elements of the groups G 1 and G 2 respectively. Then t 1 t 2 t 1 2 G G... ...R µ , the ring of fuzzy cosets of µ in R is free from non-zero nilpotent elements. THEOREM 1.4.23: Let µ be any fuzzy ideal of a ring R such that... ... that Im µ = {t, s} with t > s. If the ring R µ has no non-zero nilpotent elements, then the fuzzy ideal µ is fuzzy semiprime. THEOREM [99]: A ri... ... If P : R → L is a prime fuzzy ideal and P P (0) ≠ 0, then P (R) has two elements. P is properly fuzzy if and only if P(R) has three elements. We s... ...Rings and Ideals, Addison-Wesley, Reading, MA, 1970. 30. CHAUVIN, Remi, Chemical Algebra I: Fuzzy subgroups, J. Math. Chemistry, 16, 245-256 (1994... ..., fuzzy groups, and applications of fuzzy theory to the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars a...

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...concepts. We further assume no intermediate relation exists within the domain elements or node and the range spaces elements. The number of elements... ...of elements in the range space need not in general be equal to the number of elements in the domain space. Thus throughout this section we assume t... ...mediate relations exist within the domain and the range spaces. The number of elements or 37 nodes in the range space need not be equal to the num... ...elements or 37 nodes in the range space need not be equal to the number of elements or nodes in the domain space. Throughout this section we assu... ... elements or nodes in the domain space. Throughout this section we assume the elements of a domain space are taken from the neutrosophic vector spac... ...eory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. schola...

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...s the level of service and the convenience factors. We have the following elements, Frequency of the service, in-vehicle travel time, the travel far... ...behaviour of a model are determined by activating the individual involved elements. This is achieved by ‘Clamping’ concepts of interest and includin... ...done by fixing the corresponding element in the input and the rest of the elements are given a value zero. Thus we have the input vector as a 1 × n ... ... effective strategic planning support with object- oriented expert systems elements and a hyperknowledge user interface. In this paper Carlsson and F... ...onversion link (black arrow, shown as a dotted line), a node (typically a chemical(s)) is converted into another node, and used up in the process. In... ... (blue arrows, shown as a thick line) represents an enzyme that enables a chemical conversion and does not get used up in the process. For figures pl... ...ustrial sectors. As an example a particular model may be suitable for the chemical industry but appropriate to service industry. 4. Models as desc... ...due to industries and in particular the problems faced by workers of some chemical industries and so on. Now we proceed on to define the notion of... ...lved in the inter-relations of concepts. 22. Can NRM be used to model a chemical plant? Justify your claim. 23. When the data is unsupervised wh...

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...by default of notation just say V a vector space over the field F and call elements of V as vectors only as matter of convenience for the vectors in... ... dimensional vector space) of k if every S- basis has only finite number of elements in it. It is interesting to note that if L is a finite dimension... ...We say the order of the bigroup G = G 1 ∪ G 2 is finite if the number of elements in them is finite; otherwise we say the bigroup G to be of infin... ...ace V = V 1 ∪ V 2 to be of dimension m + n. Thus there exists only m + n elements which are linearly independent and has the capacity to generate... ...0) if v ∈ V 2 then v = (0, 0, …, 0, y 1 , y 2 , …, y n ). We never add elements of V 1 and V 2 . We keep them separately as no operation may be... ...y, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars ...

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...n by artificial brains. Neural networks consists of many simple computing elements generally simple non linear summing junctions connected together ... ... classes say C 1 , C 2 , …, C t where C i may have different number of elements in them, where even n can be a prime number then we call such str... ...int cognitive maps here only the blocks need not contain equal number of elements. We illustrate as well as exhibit these models in the study of ... ...he attributes A 1 , 176 A 2 …, A n in to m classes with a overlap of s elements ie A 1 , A 2 ,…, A n is divided into classes C 1 ,…, C m where ... ... 2 ,…, A n is divided into classes C 1 ,…, C m where we have number of elements in each C i is s where there are common elements between the C i... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars a...