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...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... ... > 0 ii. d is a common divisor of a and b and iii. if any integer f is a common divisor of both a and b then f is also a divisor of d. The grea... ...n divisor of both a and b then f is also a divisor of d. The greatest common divisor of a and b is denoted by g.c.d (a, b) or simply (g.c.d). If... ...and b are relatively prime then (a, b) = 1. DEFINITION 1.1.2: The least common multiple of two positive integers a and b is defined to be the smal... ... 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... ...sociative law (L 3 ) is the absorption law and (L 4 ) is the idempotent law. DEFINITION 1.3.7: A partial order relation ≤ on A is called a total... ...Orin Chein et al [11]). DEFINITION [41]: Let L be a loop. The flexible law FLEX: x • yx = xy • x for all x, y ∈ L. If a loop L satisfies left alt...

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...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... ...enerated by A 1 and A 2 so A denotes the free bimonoid as both have the common identity, Λ the empty sequence . . z ) , z ( = Λ δ ) a , z ( ) a... ...L 1 ∪ L 2 where each L i is an even order loop should we paste or have common paths between the two K 2n graphs. This question is an interesting ... ...tive biring. We say the biring R has a monounit if a unit exists which is common to both R 1 and R 2 we call the common unit as the 138 monouni... ...t) cancellable we say that N fulfils the left (right) 196 cancellation law. N is a bi-integral domain if both N 1 and N 1 has no zero-divisors. ...

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...d in the array 16 3 2 13 5 10 II 8 9 6 7 12 ,4 15 14 I such that the common sum is 34. It is well-known that magic squares can be constructed u...

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...adox, specifically defines the statement as being false. Nevertheless it is common practice in both scientific and pure logic systems to measure the ... ... as beta ( β), by Einstein, i.e., β = (1-(v/c) 2 ) -1/2 , [14] γ is now the common symbol for this factor.] With (3.1), we see that the secant of an... ...d thus has conjunctive motion with the body. We now suspend Newton's first law of motion, Principia, (1687), as a given principle and treat it inst... ... o = 0: - ϒ = mvsec(θ) /t 2 . (15) From Newton's second law of motion, the resultant acceleration (A), of the body is then, A ... ...s in that quanta. This definition is in complete accord with Newton's first law of motion and moreover provides a causal definition for inertial moti... ... expansion in the motion vector body. In accordance with the inverse square law then, longitudinal length in the gravity vector S' body is defined by... ...Periodic Table), should possess perfect nuclear cubic symmetry, i.e. have a common characteristic of completely filled η potential shells. Analysis ... ...nts should be available as well ―as deduced from the physical properties of common isotopes and their tendencies for noble-like structures ―within t... ...rine, we find exact n-p pairing with 6 C-12, 7 N-14, and 8 O-16, but the common isotopes of boron, 5 B-11 and fluorine, 9 F-19 have an addition...

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.... 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... ...Combined conflicting opinions tend to cancel out and assisted with the strong law of large numbers, a consensus emerges as the sample opinion approx... ...s to construct FCMs with iterative interviews or questionnaire mailings. The law of large numbers require that the random samples be independent, i... ...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 ... ...terminate in his opinion. Now we proceed on to take a few of the attributes, common in the study of all the five aspects of untouchability viz., re... ... and investigate the effect. Thus we take these five as major nodes and their common attributes of as the other nodes: N 1 - Religious untoucha... ... use all public roads by Dalits R 10 - Fight against denial of right to use common wells and lakes by Dalits R 11 - Rights denied to Dalits to ...

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...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... ...t A be a S-T-fuzzy subsemigroup of P ⊂ G, G a S-semigroup. Then the least common multiple of the fuzzy order of the elements of P in G with respect ... ...: Let A be a S-T-fuzzy sub-bisemigroup of the S-bigroup G. Then the least common multiple of the S-fuzzy order of the elements of G with respect to A... ...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 ∈ ... ...comparison of one BIBD with the another and also give the analysis of the common features. Thus the S-planar near-ring has more advantages than t...

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...d to prime numbers . . . . . . . . . . . . . . . . . . 247 11 On the least common multiple of the rst n positive integers . . . . . . . . 255 Chapter... ...lt a +b +c 3 p 3R is another source for inequalities. For example, by the law of sinus, a = 2R sinA, etc., we get the classical trigonometric inequal... ...) d (24) where d;m;n;k are positive integers; m > n; and d is an arbitrary common divisor of (m n)(k 2 +mn), m(k 2 +n 2 ), n(k 2 +m 2 ). For a comple... ...iangles, and where r is integer are given by formulae (24), where d is any common divisor of the following expressions: (m n)(k 2 +mn); m(k 2 +n 2 );... ...s of (28) can be obtained from (30), by multiplying each term of (30) by a common factor d. These give all H-triangles with the required conditions. R... ...::b x i :::x n jx 1 x 2 :::x n (3) Let d = (x 1 ;x 2 ;:::;x n ) = greatest common divisor of x 1 ;:::;x n . Then x =dy 1 ;:::x n =dy n ; where (y 1 ;:...

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...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 ... ...t is reflexive, anti symmetric and transitive is called a partial ordering. The common symbol ≤ is suggestive of the properties of this class of rel... ... procedures [1, 11, 16, 24, 30, 82]. 33 The “max-min” composition [117] is commonly used when a system requires conservative solutions in the s... ... 2 ) In this way, Ω 1 contains sets of I j ’s which have some element(s) in common and we can decompose the original problem into k sub-problems... ... 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... ...ting), from accepted / rejected / pending, etc. and guided by the fact that the law of excluded middle did not work any longer in the modern logics,...

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...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... ... Alternative, 17, 20, 40-44, 56, 62, 65, 91, 92, 106-108 Associative Law, 11, 93 Automaton, 88, 93-96 Binary operation, 8, 11, 16, 25, 3...

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... (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 ... ...is said to be associative if its multiplication satisfies the associative law that is (xy) z = x (yz). 92 A non-associative algebra L is a Lie ... ...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...

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...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... ... Combined conflicting opinions tend to cancel out and assisted the strong law of large numbers a consensus emerges as the sample opinion approximate... ...bined (and normalized) FCM entry f ij tends to be in {–1, 1}. The strong law of large numbers ensures that f ij provides a rational approximation ... ...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... ... might result in assigning grades of membership in a way unsupportable by common sense reasoning. In order to facilitate a more reasonable represe... ...firm to issue guarantees. The nested FCMs are shown below linked to their common nodes, N3, “Insiders Waive Claim to Firm’s Assets” and N4, “Firm Is... ...on, I translate these variables into singleton values. This approach is a common practice. The resulting FCM is still fuzzy, because both edge and c...

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...nce XXXVII. The Lessons of Transition XXXVIII. The Morality of Child Labor XXXIX. Law and Technology XL. The Author Economics - Psychology's Neg... ...t to disguise its nature as a social science by applying complex mathematics where common sense and direct experimentation would have yielded far b... ...dministered by some supreme, transcendental principle is restored. This sense of "law and order" is further enhanced when the theory yields predic... ...ons which come true, either because they are self-fulfilling or because some real "law", or pattern, has emerged. Alas, this happens rarely. As "Th... ...wer. Influential antitrust thinkers, such as Robert Bork, recommended to revise the law to focus solely on consumer welfare. These - and other ins... ...ofit and loss centers. Competition used to be extraneous to the firm - now it is commonly an internal affair among autonomous units within a loos... ...r assets as well. They end up charging borrowers the wrong interest rates or, more common, financing risky projects. Badly managed banks pay highe... ... Governments would like to maintain the stability of their financial systems. The common interest is overwhelming - and moral hazard seems to be a... ...cording to an August 2000 article written by Albert Beer of America Re. ART is also common in the public and not for profit sectors. Captive.com co...

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... In some countries, the obligation to save life is legally codified. But while the law of the land may create a LEGAL right and corresponding LEGAL... ...d, if so (b) in which circumstances – can be easily settled using modern contract law. Yes, a contract can be annulled and voided if signed under ... ...s to provide goods and services (always at a personal cost to the provider) are the commonest of contracts. We open a business. We sell a software ... ...poses numerous and insurmountable philosophical problems. No one disputes the now common view that the main crime committed in aborting a pregnanc... ... Why is there a debate regarding this question? "Thou shalt not kill" is a natural law, it appears in virtually every legal system. It is easily an... ..., the quantity, quality, or functions are wrong, lacking, altogether missing". The common error - that we criticize the absent - is the outcome of ... ...ive self-definitions I share with someone (i.e., the more affiliations we have in common) , the larger and more overriding is my moral obligation ... ...y do shareholders cooperate with such corporate brigandage? In an important Chicago Law Review article whose preprint was posted to the Web a few we... ...akunin), trade unions (anarcho- syndicalists), ideal communism (Kropotkin). What is common to this smorgasbord is the affirmation of freedom as the ...

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... Credit Default Swap CECA Comprehensive Economic Cooperation Agreement CIS Commonwealth of Independent States CRAs Credit Rating Agencies ECA Europe a... ... Actis (which was spun off as a management buyout from the UK government’s Commonwealth Development Corporation [CDC] in 2004, with CDC remaining its ... ...eform the present financial system. If there is a strong political will, a common international framework could be evolved which national regulatory a... ...r called the European System of Financial Supervision, which could enforce common standards among European regulators. While supporting the need for n... ...n update on the Basel Accords. The new rules propose strict definitions of common equity held by banks, introduction of a leverage ratio and provision... ...nstraints Imposed by the EEA Agreement,” Society of International Economic Law (SIEL), Second Biennial Global Conference, University of Barcelona, Jul... ...ding Crisis: Causes and Effects of the Mort- gage Meltdown, Wolters Kluwer Law & Business, 2008. Kavaljit Singh, Globalization of Finance: A Citizen’s...