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By D M Brink; G R Satchler

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Abramowitz & I. A. Stegun (1964)]. 8. fc^M Fig. 43). 3665... corresponds to the median. The negative—w tail is exponential while the positive—w tail decreases as the super-exponential : ~ exp(— exp(w)). 44) = 0 for w < 0 38 Central Limit Theorem and Stable Laws with the parameter c larger than 1. More precisely, suppose that the distribution fx{%) is exactly 0 for x < XQ, and fx{x) ~ (a; — XQ)C~1 near the threshold. 44) when the number N tends to infinity. Moments of the Weibull distribution are all given by : {wq)we%bull =T(q/c+l) when the corresponding value if finite.

45) "& with est the static value, and e^ its high-frequency limit. In the cases of strongly disordered systems, a large number of various independent relaxation times is expected to occur. 46) Jo where 9(T) is the distribution of the relaxation times r . The decrease of v is then simply the Laplace transform of the distribution 0. In such complex materials, the relaxation typically has the stretched exponential form : v(t)~voexp(-(t/T0)>*) 42 Central Limit Theorem and Stable Laws with a positive exponent n smaller than 1 [K.

5) especially for higher orders. 5). 2b. This is the basic for a 'quick and dirty' pseudo-random generator with the normal distribution, when accuracy is not essential *. "Let us take half of the sum of four pseudo-random variables uniformly distributed over [—\/3, \/3]. The first three moments are the same as for the normal law and the fourth one is 10% off the normal value. 2. 2 13 Stable laws for sum of uncorrelated variables We return now to the additive unsealed case : MN = Y,Xi (2-6) • The quest for a statistical investigation of large objects corresponds to the characterization of the limiting distribution of such variable as M/v, with N the (large) number of subunits of the system.

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Angular momentum by D M Brink; G R Satchler


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