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...b]. 8 DEFINITION 1.1.3: Any function whose domain is some subset of set of integers is called an arithmetic function. DEFINITION 1.1.4: An ... ...ion if f(mn) = f(m) f(n) whenever (m, n) = 1. Notation: If x ∈ R (R the set of reals), Then [x] denotes the largest integer that does not exceed x... ...fs are left as exercises for the reader. DEFINITION 1.2.1: A non-empty set of elements G is said to form a group if on G is defined a binary oper... ...n of the group G. DEFINITION 1.2.6: Let X = (a 1 , a 2 , … , a n ). The set of all one to one mappings of the set X to itself under the compositio... ...enoted by S n and S n has n! elements in it. A permutation σ of the set X is a cycle of length n if there exists a 1 , a 2 , …, a n ∈ X such ... ...{L 9 (3), ∗, 2}, {L 9 (3),∗, 5} and {L 9 (3), ∗, 8} can be calculated and test whether the S-hyperloops partition L 9 (5) × L 9 (5); on similar lines... ...nd S-A-hyperloop, S-hyperloop II and S-A hyperloop II and compare them and test whether they partition or not. PROBLEMS: 1. Find a S-hyperloop...

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...veral new neutrosophic concepts starting from the notion of neutrosophic fuzzy set, neutrosophic fuzzy matrix, neutrosophic lattices, neutrosophic ... ...f neural networks and genetic algorithm is used in solving problems. A complete set of references is given in the end of the book citing the names o... ...binary relations as well. Given a fuzzy relation R(X, Y), its domain is a fuzzy set on X, dom R, whose membership function is defined by dom (R(x... ...y dom (R(x)) = Y y∈ max R (x, y) for each x ∈ X. That is, each element of set X belongs to the domain of R to the degree equal to the strength... ...[107] for medical diagnosis, and in [60] for telecommunication equipment module test. Contrary to the traditional optimization problems [62], probl... ...orithms cannot provide measurement for the performance empirical computational testing is necessary. Test problems for computational experiments us... ...xamples. 2. Problems taken from real world applications. 3. Randomly generated test problems. 55 Since the NFRC problem is in its early rese... ...ed example is available. In this section, we propose a method for constructing test problems for NFRC. To characterize the system of fuzzy relation... ... A goal of multi-objective optimization is to obtain the Pareto optimal set. We tested some optimization problems with both linear and nonlinear obj...

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...at many other things A Smarandache notion is an element of an ill-defined set, sometimes being almost an accident oflabeling. However, that takes no... .... b) A problem posed by someone else that is an extension ofan element of set (a). See Some Notions and Questions in Number Theory, edited by C Dumi... ...s. For purposes of notation the acronym will be used to denote the entire set and the acronym with a subscript will refer to a specific element of t... ...and the acronym with a subscript will refer to a specific element of that set. For example, SPS = { 12, 1342, 135642, 13578642, 13579108642, 1357911... ...o store the current candidate as an additional term in the sequence. long test; II This integer is used to count the number of the term currently be... ...ber :=tail_ ofJist) { tmember=pmember ->pnext; while(tmember!=XLLL) { test=pmember ->value * pmember ->value+tmember ->value* tmember ->value; i... ...=pmember ->value * pmember ->value+tmember ->value* tmember ->value; if( (test>tail_ ofJist ->value )&&(test<mingood)) { mingood=test; } tmember=... ...umbers formed using the rules (a) and (b) are in CS2. Replacing the line test = pmember->value*pmember->value -l- tmember->value*tmember->value; by... ...ber->value*pmember->value -l- tmember->value*tmember->value; by the line test = pmember ->value*pmember ->value*pmember ->value + tmember->value*tm...

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...up under addition. Z(I) denote the additive abelian group generated by the set Z and I, Z(I) is called the neutrosophic abelian group under ‘+’. ... ... over the neutrosophic field K(I ). By basis we mean a linearly independent set which spans the neutrosophic space. Now we illustrate with an exampl... ...er oriented proof of [2, 29] settled the conjecture in 1976 and has stood a test of time, a theoretical proof of the four colour problem is still to... ...he diagraphs are used in the representation of Binary relations on a single set. Thus one of the forms of representation of a fuzzy relation R(X, X)... ...rties. [5, 23] DEFINITION 2.1.1: A graph G consists of a finite non empty set V = V (G) of p points. (vertex, node, junction O-simplex elements) ... ...nts. (vertex, node, junction O-simplex elements) together with a prescribed set X of q unordered pairs of distinct points of V. Each pair x = {u, ...

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... and Smarandache semigroup. 1.1 Binary Relation Let A be any non-empty set. We consider the Cartesian product of a set A with itself; A × A... ... consider the Cartesian product of a set A with itself; A × A. Note that if the set A is a finite set having n elements, then the set A × A is also ... ...he set A is a finite set having n elements, then the set A × A is also a finite set, but has n 2 elements. The set of elements (a, a) in A × A is ca... ...tant one and plays a central role in all mathematics. DEFINITION: If A is a set and if ~ is an equivalence relation on A, then the equivalence c... ...f ~ is an equivalence relation on A, then the equivalence class of a ∈ A is the set {x ∈ A/ a ~ x}. We write this set as cl(a) or [a]. THEOREM 1.... ...claim. Finally we conclude this chapter with an interesting example study and test the validity of all the results proved in this chapter. Exam... ...e S into disjoint sets such that their union is S. Clearly from examples. To test whether Smarandache double coset relation is an equivalence rela...

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...tial for our study and future reference. DEFINITION 1.1.1: A non-empty set G, is said to form a group if in G there is defined a binary operation... ... a non-commutative group. Notation: Let X = {x 1 , x 2 , … , x n }, the set of all one to one mappings of the set X to itself under the product ca... ... and several other properties about them. DEFINITION 1.1.2: A non-empty set L is said to form a loop if in L is defined a binary operation called ... ... N λ ∩ N µ ∩ N p . THE MOUFANG CENTER: The Moufang center C(L) is the set of elements of the loop L which commute with every element of L i.e. C... ... = C(L) ∩ N(L). DEFINITION 1.1.7: Let L n (m) = {e, 1, 2, 3, …, n} be a set where n > 3 , n is odd and m a positive integer such that (m, n) = 1 a... ...iding undesirable influences in the experiment. In general, if we have to test the effect of r different conditions with m possibilities for each co...

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... 0 4 . G D T P a r t i c l e F i e l d s 2 0 - 3 2 5. The SQP Set in MT & GDT Comparative Physics 32-41 Bibliography ... ...ol. 413, No. 6854] and Smarandache Hypothesis (FTL) are defined as a formal set of (anti-logic) statements inclusive in modern quantum theory. To de... ...s, together with the Smarandache (FTL) Hypothesis, [38] as a physical logic test, the "SQP Set". The set members are restated here with the intentio... ...h the Smarandache (FTL) Hypothesis, [38] as a physical logic test, the "SQP Set". The set members are restated here with the intention of enabling e... ...4) is expressed as, A given object {S}, having physical characteristics in set {A} ∋ { a 1 , a 2 , a 3 ,…a n }, is composed of elements, {S} ∋ {s 1... ...a, (1687), as a given principle and treat it instead as an hypothesis to be tested within these parameters. In isolated system S’, the mass m, has a... ...ught-experiment with unrealistically ideal conditions: (requirement that at test-instant, the arriving light-signal is from only one time-instant of ... ...test-instant, the arriving light-signal is from only one time-instant of the test- object.) [43]. The reason for SRT's non-accord with experimental ... ...ongitudinal length of an object in motion relative to an observer requires a test- window having time-length greater than zero, with signals arriving ...

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...The purpose of this book is to postulate some theories and test them numerically. Estimation is often a difficult task and it has wide application in social sciences and financial market. In order to obtain the optimum efficiency for some classes of estimators, we have devoted this b...

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... 4 Forward The purpose of this book is to postulate some theories and test them numerically. Estimation is often a difficult task and it ha... ... extensively in recent years to deal with such problems of reliability and life-testing is the Weibull distribution introduced by Weibull(1939), who ... ...Weibull distribution has been done since 1970. An experimenter involved in life-testing experiments becomes quite familiar with failure data and henc... .... For example, Singh and Bhatkulikar(1978) suggested performing a significance test of the validity of the prior value of β (which they took as 1). ... ...of β (which they took as 1). Pandey(1983) also suggested a similar preliminary test shrunken estimator for β. In the present investigation, it is ... ... the second one corresponds to ARBs of β ∧ (,) pq . The last two rows of each set of q includes the range of dominance of ∆ and Best ∆ . The AR... ...of ∆ and Best ∆ . The ARBs of β M has also been given at the end of each set of table. 14 Table 3.1 PREs of proposed estimator β ∧ (,) ...

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...veral new neutrosophic concepts starting from the notion of neutrosophic fuzzy set, neutrosophic fuzzy matrix, neutrosophic lattices, neutrosophic ... ...f neural networks and genetic algorithm is used in solving problems. A complete set of references is given in the end of the book citing the names o... ...binary relations as well. Given a fuzzy relation R(X, Y), its domain is a fuzzy set on X, dom R, whose membership function is defined by dom (R(x... ...y dom (R(x)) = Y y∈ max R (x, y) for each x ∈ X. That is, each element of set X belongs to the domain of R to the degree equal to the strength... ...[107] for medical diagnosis, and in [60] for telecommunication equipment module test. Contrary to the traditional optimization problems [62], probl... ...orithms cannot provide measurement for the performance empirical computational testing is necessary. Test problems for computational experiments us... ...xamples. 2. Problems taken from real world applications. 3. Randomly generated test problems. 55 Since the NFRC problem is in its early rese... ...ed example is available. In this section, we propose a method for constructing test problems for NFRC. To characterize the system of fuzzy relation... ... A goal of multi-objective optimization is to obtain the Pareto optimal set. We tested some optimization problems with both linear and nonlinear obj...

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...CE In 1965, Lofti A. Zadeh introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. It provoked, at first (and a... ...rmulated the fuzzy subset concept. He defined fuzzy subset of a non-empty set as a collection of objects with grade of membership in a continuum, wi... ...ct being assigned a value between 0 and 1 by a membership function. Fuzzy set theory was guided by the assumption that classical sets were not natur... ...is not a probabilistic concept. DEFINITION 1.1.1: Let X be a non-empty set. A fuzzy set (subset) µ of the set X is a function µ : X → [0, 1]. ... ...function µ : X → [0, 1]. DEFINITION 1.1.2: Let µ be a fuzzy subset of a set X. For t ∈ [0, 1], the set {} t ) x ( X x X t ≥ ∈ = µ µ is called a t-... ... = δ it δ js . and j it t ' ij e ) f ( δ φ = . Now we will just test whether we can define Smarandache fuzzy continuity of linear maps of ... ...iding undesirable influences in the experiment. In general, if we have to test the effect of r different conditions with m possibilities for each co...

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... to make them build more Smarandache structures. DEFINITION 1.1.1: A set G that is closed under a given operation '.' is called a group if the ... ...n '.' is called a group if the following axioms are satisfied. 1. The set G is non-empty. 2. If a, b, c ∈ G then a(bc) = (ab) c. 3. There are ... ... and the element a is said to generate G. Example 1.1.1: Let Q be the set of rationals. Q\{0} is a group under multiplication. This is an infini... ...te group. Example 1.1.2: Z p = {0, 1, 2, … , p – 1}, p a prime be the set of integers modulo p. Z p \{0} is a group under multiplication modulo ... ... abelian. DEFINITION 1.1.3: Let X = {1, 2, … , n}. Let S n denote the set of all one to one mappings of the set X to itself. Define operation on... ...g? 2. Find all the S-nilpotent elements of the group ring Z 4 S 3 . 3. Test whether the semigroup ring Z 6 S(4) is a reduced ring. 4. Find all S... ..., 6, 4, 8 10}, A 7 = Z 3 × Z 12 = R. Test whether this lattice satisfies the quasi distributive identity DEFI...

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...h.*; import java.io.*; public class BaseNPrime { // The upper limit to the test for primeness static final int UPPERLIMIT=500; // This is the base us... ... value of the nth term in another base? A computer program was written to test this problem for all bases 3 through 9 and no other solutions were fou... ...ne 6 2, 11, 12, 77, 207 7 493 8 144, 226 91 7 10 2, 19, 20 Once again, the testing yields few primes, one base where none were found and there is no d... ...element of this sequence to be prime, in fact, it is a modification of the set of repunit numbers. 1, 11, 111, 1111, 11111, 1... ... 1, 11, 111, 1111, 11111, 111111, 1111111, . . . A set of additional sequences can be formed by constructing the sequence usin... ...ng five will compute all elements of a base B permutation sequence up to a set limit. Listing 5 /* This program will create the elements of the Permut... ...ents in these three sequences that are congruent to 0 modulo 3? To further test this, the program was rerun to examine the first 1000 elements in the ... ...west of the counts was 37 and the highest was 53. As the last part of the test, the program was modified to count the elements of these three sequenc... ...idpoint. If the value we are looking for is less than the midpoint, we set the high value to the midpoint, otherwise we set the low value to t...

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...zy nodes. DEFINITION 1.1.3: FCMs with edge weights or causalities from the set {–1, 0, 1} are called simple FCMs. DEFINITION 1.1.4: Consider th... ...pace of dimension m (m in general need not be equal to n). We denote by R the set of nodes R 1 ,…, R m of the range space, where R = {(x 1 ,…, x m ... ...R j called relations. Every edge in the FRM is weighted with a number in the set {0, ±1}. Let e ij be the weight of the edge D i R j , e ij ∈ {0,... ...the American visa, a% have the chance to be accepted – where a varies in the set A, r% to be rejected – where r varies in the set R, and p% to be i... ...n other cases one may have the inferior sum < 0, while in the classical fuzzy set theory the superior sum should be 100 and the inferior sum µ 0. In... ...tion. If a man has shame, he has to go to the jail at least once. That is the test of integrity for a man now. If our elders backtrack by chance, o...

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...unction . . . . . . . . . . . . . . . 150 10 The Smarandache function of a set . . . . . . . . . . . . . . . . . . . . . . 152 5 11 On the Pseudo-Sma... ... with distinct parities, so the solutions (2) (with t) are reobtained. The set of solutions, however is much wider, as can be seen by other selections... ... integers. In fact, the used method permits to obtain all solutions in the set of integersZ. We shall use the following auxiliary results: Lemma 1. Le... ...seXY = 1. HenceX =Y = 1, implyingx =y =d,z = 2. This gives the (in nitely) set of solutions of the given equation. Remark. Particular cases of our equ... ...ous equation of degree 3: x 2 +y 2 =z(1 +xy): (1) We will study (1) in the set N of positive integers. First remark that for x =y from (1) we get 2x... ...(n) n ! 1 as n!1, giving by (1): lim n!1 n p a(n) =e: (2) Now, by Cauchy’s test of convergence of series of positive terms, this gives immediately tha...

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...y /\ XTYl) :::} (y = x + 1 /\ Yl = X + 1) :::} Y = Yl, ce demonstreaza ca Teste relatie funqionala. e) Pentru relatia T- l , obtinem: yT-lx ¢} xTy ... ..., pentru fie care denumire eutia sa. In ccite moduri poate fi eomandat un set de cinei eutii? " 5 (3 + 5 - I)! 7! Solu!le (vezl problema 11). C 3 ... ...Bucure§ti: Editura didactica §i pedagogica, 1994. 6. Petrica 1., Lazar 1. Teste de matematica pentru treapta I-a §i a II-a de liceu. - Bucure§ti: Al... ...i ?i literatura englezii pentru clasa a X-a Le '::- 'n<;ais. Exercices et tests de grammaire. Bacalaureat Limba romana. Teste la limba ~i literatura...

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...so on. Theorem 1.1.10 The range of Zt(n) function is N-{0} where N is the set of positive integers numbers. Proof. The theorem is a direct consequen... ...a squarefree number. 29 Theorem 1.2.15 The range of Zw function is the set of squarefree numbers. Proof. A direct consequence of the fact that ... ...er Theory, Dover Publications, Inc. New York, 1996 40 Chapter II A set of new Smarandache-type notions in Number Theory. In [1] F. Smarandache... ... in number theory. Several concepts of number theory can be applied to the set of Smarandache sequences as has already been done for example by Castil... ...p n + = 2 where p and q are two odd primes or 1. The conjecture has been tested for all odd numbers 6 10 < References [1] E. Weisstein, CRC Conci... ...is always lower or equal to 1. Then an Ubasic program has been written to test the new version of Smarandache conjecture for all primes lower than ... ...f three primes. This reduce the problem to finding this number N, and then testing all odd numbers up to N to verify that they, too, can be written as... ...ze was approximately [6]: 6846168 10 But this is a rather large number; to test all odd numbers up to this limit would take more time and computer pow... ... than the previous one, and suggests that some day soon we will be able to test all odd numbers up to this limit to see if they can be written as the ...

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...ic notion of utility. Extrinsic utility is proposed as a cardinally measurable quantity; measurable in terms of the expected information content of a set of alternative choices. The entropic notion of utility is subsequently used to model the financial behavior of individual investors based on their governing risk-return preferences involving financial structured products ...

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...le quantity; measurable in terms of the expected information content of a set of alternative choices. The entropic notion of utility is subsequently... ...ss. But there 9 will definitely be enterprising people who will happily set off in their cars without a thought! The economics of general ... ... ) from non-linear alternatives (H A ). The approach generally taken is to test the H 0 of linearity against a pre-chosen particular non-linear H A ... ...lar non-linear H A . Using the classical theory of statistical hypothesis testing, several test statistics have been developed for this purpose. The... ...oped for this purpose. They can be classified as Lagrange Multiplier (LM) tests, likelihood ratio (LR) tests and Wald (W) tests. The LR test require... ...imation of the model parameters both under H 0 and H A , whereas the LM test requires estimation only under H 0 . Hence in case of a 28 complica... ...ntaining many more parameters as compared to the model under H 0 , the LM test is far more convenient to use. On the other hand, the LM test is desi... ...wn Dempster’s rule (Zadeh, 1979). Let Θ = {θ 1 , θ 2 … θ n } stand for a set of n mutually exhaustive, elementary events that cannot be precisely d... ...ecisely defined and classified making it impossible to construct a larger set Θ ref of disjoint elementary hypotheses. The assumption of ex...

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...ble lives? The good part of life (and, therefore, the differential and meaningful test) is in the experience itself – not in the desire to experie... ...nd a sperm is even more serious philosophically: the creation of a fetus limits the set of all potentials embedded in the genetic material to the on... ... The fetus is the collapse of the wave function: it represents a much more limited set of potentials. If killing an embryo is murder because of the... ... where we have no direct observational access to the events and we have to rely on testimonials. If they do not occur (or are not reported to have ... ...Politicians cannot be trusted, regardless of the system in which they operate. No set of constraints, checks, and balances, is proved to work and ... ...ate people. Every social arrangement and institution should be put to the dual acid tests of personal autonomy and freedom and moral law. If it fail... ...enesses decide whether one deserves to live, pain-free and happy. The quantitative test includes the ability to communicate (manipulate vocal-verba... ...ee that it is - in its essence - like us. This is the subject of the famous Turing Test: there is no effective way to distinguish a machine from a ... ... necessarily compatible or mutually reinforcing in their goals and actions. If we set aside the dewy-eyed arguments of altruism, we are compelled ...

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...nodes. DEFINITION 1.1.3: FCMs with edge weights or causalities from the set {–1, 0, 1} are called simple FCMs. DEFINITION 1.1.4: Consider the n... ...d output window */ } else if (label.equals("Load Sample Matrix")) { /* set the number of values to 5 */ try { dataURL=new URL ("http://members... ...r from the model. An equilibrium in the system is attained when we have a set of repeated patterns. Repeating patterns can be fixed points or limiti... ...pattern A 1 ⇒ A 2 ⇒ A 3 ⇒ A 4 ⇒ A 4 ⇒ A 1 . A "limiting cycle" is a set of multiple repeating patterns such as A 3 ⇒ A 4 ⇒ A 5 in the patte... ...ly alert level of the network. Causal relations can be used to denote a set of fuzzy FCM rules. The FCM in Figure 1.3.3 describes two such causal ... ...rrelationships which can be used as the foundation for the development and testing of a Business Metrics theory. Our approach, is highly customizable... ... FCModeler tool is intended to capture the intuitions of biologists, help test hypotheses and provide a modeling framework for assessing the large a... ... is expected to compare these results. Several other state vectors can be tested in this model. The union leader of the same company was asked to...

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...ave to be at r 0 =0. The point-mass can be located at any point r 0 ∈ . A test particle can be located at any point r∈ . The point-mass and the test... ...tance between these 2-spheres enters into consideration. Therefore, if two test particles are located, one at any point on the 2-sphere r 0 =0 and on... ...ation of the relevant radiiR c andR p on (6). Consider (7) and introduce a test particle at each of the pointsr 0 andr. Let the particle located atr 0... ...ravitational feld (M g ,g g ) the distance between the point- mass and the test particle, and the radius of curvature of a great circle, centred atr 0... ...ent derived by Abrams [7] by a dif- ferent method. Alternatively one could set r=R c in (6), as Hilbert in his work [8] effectively did, to obtain the... ...must be obtained rigorously from the geometry of equation (17), and that a test particle is located at somer =r 0 in parameter space, wherer, r 0 ∈ .... ...hosen. To fnd the r-parameter in terms of a particular coordinate sys- tem set, χ 2π = p C n (D(r))= r−r 0 n +α n 1 n , so |r−r 0 |= h χ ... ... +sin 2 θdϕ 2 , the metric of Special Relativity when r 0 =0. One cannot set the lower bound r 0 =α=0 in (18) and simultaneously keepα =0 in the com... ...ik , hg 00 =−g, g=det g αβ . Observable inhomogeneity of the space is set up by the chr.inv.-Christoffel symbols Δ i jk =h im Δ jk,m , which are ...

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...npredictable. Either one cannot design an experiment to rigorously and validly test theorems and conjectures in economics - or something is ver... ...recisely constitutes their subject matter. Is economics about the construction and testing of models in accordance with certain basic assumptions? ... ...h like physics yields technology. Economists must develop "economic technology" - a set of tools, blueprints, rules of thumb, and mechanisms with t... ...conditions is not enough. Scientific theories must also pass the crucial hurdles of testability, verifiability, refutability, falsifiability, and re... ... Yet, many economists go as far as to argue that no experiments can be designed to test the statements of economic theories. It is difficult - pe... ...st the statements of economic theories. It is difficult - perhaps impossible - to test hypotheses in economics for four reasons. a. Ethical –... ...ell as the competitors' brand recognition and ability and inclination to collude to set prices. Victory is not guaranteed, losses loom constantly,... ...atory costs, economies of scale and of scope, excess profits due to the ability to set prices in a less competitive market - allow firms in an olig... ...tive technologies" and "destructive creation" to check the power of oligopolies to set extortionate prices, lower customer care standards, or inhib...

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...nce, and Growth, 2010, p. 16. 5. World Bank, The Crisis Hits Home – Stress-Testing Households in Europe and Central Asia, 2010, p. xiii. 6. Ibid., p. ... ...a and Brazil) in the global economy, the landscape of financial centers is set to change – both geographically and functionally. Despite the fact that... ...xports, the proceeds of privatizations or fiscal surpluses. SWFs have been set up to diversify and improve the return on a country’s foreign exchange ... ...eploy surplus forex reserves; but since SWFs 43 Fixing Global Finance are set up to diversify investment, they undertake long-term investments in ill... ... a US hedge fund considered to be “the Rolls Royce of hedge funds.” It was set up in 1994 and had returns of over 40 percent in its initial years. Wit... ...wed excessive leverage due to strict regulatory norms. In 2009, the stress tests undertaken by the Committee on Financial Sector Assessment found that... ...pt to restructure the present international financial system presupposes a set of guiding principles under which several specific steps could be initi...

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...tion to DSmT 3 by J. Dezert and F. Smarandache 1.1 Introduction . . . . . . . . . . 4 1.2 Foundations of DSmT . . . . . . . . . 4 1.2.1 The power set, hyper-power set and super-power set . . 6 1.2.2 Notion of free and hybrid DSm models . . . . 18 1.2.3 Generalized belief functions . . . . . . 20 1.2.4 The classic DSm rule of combination . . . . . 21 1.2.5 The hybri...