P. Flajolet and R. Sedgewick, The average case analysis of algorithms: complex asymptotics and generating functions, INRIA research report  Home/Search Document Details and Download
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On a Statistical Filter for RNA Secondary Structures - Nebel (Correct)
....of coecients of standard functions which occur in such singular expansions, so that the coecients of the main terms can be extracted. Second, suitable theorems which allow to extract the asymptotic order of error terms involved. Both will just be presented without proof. We refer the reader to [7, 9] for details. For our applications, only algebraic singularities can occur. Thus, the only standard functions together with the asymptotic forms of their coecients that we will use are contained in the following table. A similar table with a lot more entries can be found in [9] Function Coecient ....
....the reader to [7, 9] for details. For our applications, only algebraic singularities can occur. Thus, the only standard functions together with the asymptotic forms of their coecients that we will use are contained in the following table. A similar table with a lot more entries can be found in [9]. Function Coecient at z 3=2 1 45 1155 25 2 2 4n The basic requirement for the method is that the asymptotic expansion of the function should be valid in an area of the complex plane which extends beyond the disk of convergence of the ....
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P. Flajolet and R. Sedgewick, The average case analysis of algorithms: complex asymptotics and generating functions, INRIA rapport de recherche
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....ceases to be complex differentiable. A dominant singularity is a singularity of minimal modulus. Throughout the paper, given a series S 2 N[ x] we set S = n (Sjn)x . When applicable, we denote the modulus of the dominant singularities of S (viewed as a function) by ae S . Classically, see [1, 13, 32], the asymptotic growth rate of (Sjn) is linked to the value of the dominant singularities. Lemma 4.4. We have ae L = 1 or ae H = 1 if and only if M ( Sigma; D) is the free commutative monoid over Sigma. Proof. We have lim sup n (Ljn) 1=ae L , and lim sup n (H jn) 1=ae H (the ....
....or using the results from section 6.1, we get (Ljn) n and (H jn) n . It implies that ae L = 1 and ae H = 1. Proposition 4.5. Let ( Sigma; D) be a strongly connected dependence graph. Then L and H have a unique dominant singularity which is positive real and of order 1. It follows (see [1, 13, 32]) that when ( Sigma; D) is strongly connected, we have (Ljn) ff L ae L and (H jn) ff H ae H , with ff L = ae L Delta [L(y) ae L Gamma y) jy=ae L and ff H = ae Delta [H(x) ae H Gamma x) jx=ae H . The proof of Proposition 4.5 is based on the representation given in Proposition ....
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P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Complex asymptotics and generating functions. Reseach Report RR2026.
Investigation of the Bernoulli-Model for RNA Secondary Structures - Nebel (2001) (Correct)
....of coecients of standard functions which occur in such singular expansions so that the coecients of the main terms can be extracted. Second, suitable theorems which allow to extract the asymptotic order of error terms involved. Both will just be presented without proof. We refer the reader to [3, 4] for details. The following table of commonly encountered functions together with the asymptotic forms of their coecients contains all estimates which are used within this paper. A similar table with much more entries can be found in [4] Function Coecient at z n (1 z) 1=2 1 p n 3 1 2 ....
....just be presented without proof. We refer the reader to [3, 4] for details. The following table of commonly encountered functions together with the asymptotic forms of their coecients contains all estimates which are used within this paper. A similar table with much more entries can be found in [4]. Function Coecient at z n (1 z) 1=2 1 p n 3 1 2 3 16n 25 256n 2 O(n 3 ) 1 z) 1=2 log(1 z) 1 p n 3 1 2 log(n) 2 log(2) 2 2 O(log(n) n) 1 z) 1=2 1 p n 1 1 8n 1 128n 2 5 1024n 3 O(n 4 ) 1 z) 62 N 0 n 1 = ....
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P. Flajolet and R. Sedgewick,The Average case analysis of algorithms: complex asymptotics and generating functions, INRIA rapport de recherche
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P. Flajolet and R. Sedgewick, The average case analysis of algorithms: complex asymptotics and generating functions, INRIA research report
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P. Flajolet and R. Sedgewick, The average case analysis of algorithms: complex asymptotics and generating functions, INRIA research report
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P. Flajolet and R. Sedgewick. The average case analysis of algorithms: Complex asymptotics and generating functions. Rapport de recherche 2026.
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