By Sylvie Corteel, Guy Louchard, Robin Pemantle (auth.), Michael Drmota, Philippe Flajolet, Danièle Gardy, Bernhard Gittenberger (eds.)
This ebook includes invited and contributed papers on combinatorics, random graphs and networks, algorithms research and bushes, branching procedures, constituting the lawsuits of the third overseas Colloquium on arithmetic and machine technological know-how that would be held in Vienna in September 2004. It addresses a wide public in utilized arithmetic, discrete arithmetic and machine technology, together with researchers, academics, graduate scholars and engineers. they are going to locate the following present questions in computing device technological know-how and the similar smooth and robust mathematical tools. the variety of purposes is especially broad and is going past computing device Science.
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Extra resources for Mathematics and Computer Science III: Algorithms, Trees, Combinatorics and Probabilities
Sample text
For tl:= (logl)jlogm and t E [tl,tl+d, we set Gm(a ; t) = + (Sm(a; tl+ 1 ) - Sm(a; tL)) Sm(a; tt) t - tl tl+ 1 - tl , where 1 ::; l ::; m - 1. Our problem now is to describe the set of limit curves of {Sm(a; t)} and {Gm(a; t)} "for almost all a E Sn" in the sense of Section 2. Again, to overcome some obstacles appearing in the sequence of probability spaces, we have to introduce more general notions. Let (U, d) be a separable metric space and Y, Y1 , .. , Yn be U-valued random elements all defined on the probability space {nn,:7n ,Pn }, 'J> = {Pn }, where n = 1,2, ..
Then H(q) = ( (q) ~ 2 [0] ) ( -1 jqoo-z ) ( -1 jqoo ) Z (-zqjqoo-z - q, q) 00 ( ) ~[zO] ( - q)~ (q)oo f L zn qn(n+1)/2 L nEZ L ( - q)~ n=O (q)n (q)oo -n n(n-1)/2 _z----=q-:-:-__ n2:0 (q)n 20 Sylvie Corteel, Jeremy Lovejoy, and Ae Ja Yee by the first Rogers-Ramanujan identity, 00 qn 2 1 L -() n==O q n = ( q, q4; q5) 00 (25) . 3. L PD,oD(m, n)ymqn = m,n2:0 [ZO] (-zq; q)oo( -z-\ q)oo( - yz-\ q)oo (Z-l; q)oo L L 00 (n+l ) 00 [zO] ( -~; q)oo Znq. 4. "" ~ R,m,n2:0 = = 00 of (24). 0 ( ) R m n [OJ(-zq,-l/z,-l/az)oo PO,02D f,m,n a b q = z (azq ) 00 (b 2/ z, 2.
The main results are analogues to Feller's and Strassen's theorems proved for partial sums of independent random variables. 1. Weak convergence of distributions Let 0' E Sn be an arbitrary permutation and W = w(O') , (1) be its unique up to the order expression by the product of the independent cycles K,. Denote by IIn (. )-ll{a E Sn : ... }I the uniform probability measure on Sn. In what follows we assume that n ---4 00 . Despite to the long list of asymptotic results (see [1], [2], [7-10], [12- 15], and other publications), we examine the ordered statistics and 8 = 8(0'), consisting of the cycle lengths and the different cycle lengths appearing in decomposition (1).