... AMERICAN RESEARCH PRESS REHOBOTH 2002 A 1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Depart... ...1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...(8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...exas 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can... ...s 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be... ...just give the basic concepts or notion like equivalence relation greatest common divisor etc. Second section is devoted to giving the definition of ... ...nd b are integers both not zero, then an integer d is called the greatest common divisor of a and b if i. d > 0 ii. d is a common divisor of a a... ... property). 2. a, b, c ∈ G implies a • (b • c) = (a • b) • c (associative law). 3. There exist an element e ∈ G such that a • e = e • a = a for all ... ...tional condition (L 4 ) x ∩ x = x, x ∪ x = x. (L 1 ) is the commutative law (L 2 ) is the associative law (L 3 ) is the absorption law and (L 4...

... 1 Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...aic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Nigeria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...geria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ...3 ∈ N; (n 1 + n 2 ) n 3 = n 1 n 3 + n 2 n 3 (right distributive law). This near-ring will be termed as right near-ring. If n 1 (n 2 ... ...ents of N are left cancelable we say that N fulfils the left cancellation law. Similarly one defines right cancelable. N is an integral domain if N ... ... a S-seminear-ring. We call a non empty subset I of R to be a Smarandache common ideal related to A (S-common ideal related to A) (A ⊂ R) and A a ne... ...quipped with associative binary operations + and ' '. Bisemigroups with a common neutral element for the two associative binary operations are calle...

...d group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology Madras... ... Indian Institute of Technology Madras Chennai – 600 036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv ... ...tion by: Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...tive biring. We say the biring R has a monounit if a unit exists which is common to both R 1 and R 2 . If R 1 and R 2 are rings which has separat... ...t (right) cancelable we say that N fulfills the left (right) cancellation law. N is a bi-integral domain if both N 1 and N 2 has no zero divisors. ... ... for all i, j. Proof: Let q i = Π{n j ; i ≠ j }> 1 since the greatest common divisor of q 1 , q 2 , …, q r is 1 then there exists a i ∈ Z suc... ...erly contained in 〈S 〉; as in 〈S 〉 we may or may not have the associative law to be true. Remark: Even (ab)c ≠ a(bc) in general for all a, b, c ∈ ...

... American Research Press Rehoboth 2002 A 1 × A 2 Z 2 Z 1 B 1 × B 2 A 1 A 2 B 1 B 2 4 0 1 2 3 1 W. B. Vasantha ... ... Z 2 Z 1 B 1 × B 2 A 1 A 2 B 1 B 2 4 0 1 2 3 1 W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tech... ... Z 1 B 1 × B 2 A 1 A 2 B 1 B 2 4 0 1 2 3 1 W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology M... ...ipiona 11550, Spain. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books... ...ona 11550, Spain. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books ca... ..., Spain. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be down... ...p > 1 is a prime if its only divisor are ± 1 and ± p. DEFINITION: The least common multiple of two positive integers a and b is defined to be the ... ...osure). 2. For all a, b, c ∈ S we have a  (b  c) = (a  b)  c (associative law). (S, ) is a semigroup. DEFINITION: If in a semigroup (S, ) w... ...s properly contained in <S>; as in <S> we may or may not have the associative law to be true. Remark: Even (ab) c ≠ a (bc) in general for all a, b...

... REHOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...HOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...M 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...e, Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... (closure axiom) 2. a. (b.c) = (a.b).c for all a, b, c ∈ L. (associative law) (S, .) is called the semigroup. The semigroup in general may not ha... ...b) . c for some a, b, c ∈ R. 3. a. (b + c) = a.b + a.c (The distributive law are true) for all a, b, c, ∈ R. Let R be a non-assoiciative ring. R ... ...ne all non- isomorphic Lie algebras. The following is a very important and commonly used example. Example. 4.1.3: Let gl (n, C) be the set of all ... ...t we are happy to state that we have introduced several notions which are common to associative algebras and analysed it in the case of Jordan algeb...

...DS With Specific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of T... ... With Specific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tech... ...pecific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, M... ...tics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...ova, Bulgarian Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Ka... ...a although rare among women, premarital and extramarital sexual behaviors are common and tolerated among men even though it is publicly disapproved... ...and differential competitive. I. The deterministic signal Hebbian learning law correlates local neuronal signals. XY ij ij i i j j m m S( x) S... ...0, inhibitory if m ij < 0. 17 II. The deterministic competitive learning law modulates the signal-synaptic differences S i (x i ) – m ij wit... ...get a male child or in some cases the man marries another woman. This is very common in rural India. Thus even to opt for a child is not in the hand...

...UZZY AND NEUTROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iit... ...Y AND NEUTROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iitm.a... ...TROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv... ...Russia. Prof. G. Tica, M. Viteazu College, Bailesti, jud. Dolj, Romania. Dr. B.S.Kirangi, University of Mysore, Mysore, Karnataka, India ... ... of Mysore, Mysore, Karnataka, India Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K. Kandasamy ... .... Combined conflicting opinions tend to cancel out and assisted by the strong law of large numbers, a consensus emerges as the sample opinion appro... ...s, in case of a model, add two or more FCMs to produce a new FCM. The strong law of large numbers ensures in some sense that, knowledge reliability... ...otion of untouchability from the time of the ancient lawgiver Manu (approx. 1 Common Era) 1 . So, we specifically study the social effects of untouc... ..., ride bicycles or wear new clothes. They are not allowed to draw water from common ponds, or graze their cattle on public grounds. In teashops in ...

...VICE 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...E 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...uzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Indian Institute of Technology, Madras Chennai – 600036, India E-mail: vasantha@iitm.ac.in Florentin Smarandache Department of Mathematics... ... Finance, Griffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...ombined conflicting opinions tend to cancel out and assisted by the strong law of large numbers, a consensus emerges as the sample opinion approximat... ...in case of a model, add two or more FCMs to produce a new FCM. The strong law of large numbers ensures in some sense that knowledge reliability incr... ...ts (Levels and Variations), several kinds of Relations (Causal, Inference Common, similarity, ill-defined, Crisp, Level-Variation, and other common r... ... effect during a single iteration, but an event which has a real delay. A common example of this kind of timing events is the effect of oil-price var...

...GNITIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...TIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ... With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarand... ... University of Hyderabad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...y. The tools used in section 1 and section 2 are made very simple for any common reader to follow. By this we have to a large extent suppressed so... ...n case of a model, add two or more FCMs to produce a new FCM. The strong law of large numbers ensures in some sense that knowledge reliability incr... ...mbined conflicting opinions tend to cancel out and assisted by the strong law of large numbers, a consensus emerges as the sample opinion approximat... ..., C m where we have number of elements in each C i is s where there are common elements between the C i ’s and C i+1 ’s. Now we using the experts ...

... Structures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... ...ructures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techno... ...and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Mad... ...cs Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smara... .... Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, University ... ...rsity of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover... ...t countries are those having a boundary line (not merely a single point) in common. The conjecture has acted as a catalyst in the branch of mathem... ...he edges or lines of G. If two distinct lines x and y are incident with a common point, then they are adjacent lines. A graph with p points and q ... ...ism Debate: Expert Reasoning, Fuzzy Logic and Complex Statutes, Vanderbilt Law Review, 52 (1999), 1243-1340. http://law.vanderbilt.edu/lawreview/vo...

...4 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of... ... 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Te... ... FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology,... ...matics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...alia. Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, Univers... ...res [1, 4, 10-12, 18, 34, 52, 75-80, 82, 108, 111]. The max-min composition is commonly used when a system requires conservative solutions in the s... ... .6 12 Consider now two binary fuzzy relations P(X, Y) and Q(Y, Z) with a common set Y. The standard composition of these relations, which is ... ... 2 ) given in Figure 2.1.3. 173 The linear equation is based on Ohm’s law, the drop in voltage V across a resistor R is given by the linea... ...in voltage V across a resistor R is given by the linear relation V = iR (Ohm’s law). The hydrodynamic analogue to the mean velocity v for laminar f...