A. M. Odlyzko, B. Poonen, H. Widom, and H. S. Wilf, manuscript in preparation. Home/Search Document Not in Database Summary Related Articles Check |
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Asymptotic Enumeration Methods - Odlyzko (1996) (64 citations) Self-citation (Odlyzko) (Correct)
....(10.35) and (10.36) we see that u 4 (n) n ) 2 [z 2n ]U 4 (z) # (n ) 2 1536# 5 2 n 2n 16 n e 2n n 17 2 # 1536# 3 2 n 15 2 16 n as n ## . 10.37) Xi Other methods can be applied to Gessel s generating function to obtain asymptotics of u k (n) for wider ranges of k ([306]) The above example obtains a good estimate because the remainder term in (10.30) is smaller than the main term by a factor of z 1 . Had it been smaller only by a factor of z 1 2 , the resulting estimate would have been worthless, and it would have been necessary to obtain a fuller ....
A. M. Odlyzko, B. Poonen, H. Widom, and H. S. Wilf, manuscript in preparation.
Asymptotic Enumeration Methods - Odlyzko (1996) (64 citations) Self-citation (Odlyzko) (Correct)
....large singularities (such as the saddle point technique) that will be presented later. However, on circles z = n 1 2) e, n # Z , f(z) does not vary much, so there are technical problems in applying these analytic methods. On the other hand, combinatorial estimates can be used to show [233] that the b n behave in a regularly irregular way, so that, for example, b m(m 1) 2 1 # b m(m 1) 2 as m ## , 10.14) b m(m 1) 2 # mb m(m 1) 2 1 as m ## . 10.15) These estimates are obtained by expanding the product in Eq. 10.13) and noting that b n = # r 1#k 1 kr # k i =n 1 ....
A. Knopfmacher, A. Odlyzko, B. Richmond, G. Szekeres, and N. Wormald, manuscript in preparation.
Discrete Logarithms in Finite Fields and Their Cryptographic.. - Odlyzko (1984) (41 citations) Self-citation (Odlyzko) (Correct)
....up with a dramatic improvement on the GF(2 n ) version of the algorithm (and more generally on the GF(p n ) version with p fixed and n ) which is much faster and even has different asymptotic behavior. More recently, a whole series of improvements on the basic algorithm have been discovered [20]. They do not approach the Coppersmith algorithm in asymptotic performance, but they do apply to fields GF(p) as well as GF(2 n ) and they can be used to motivate Coppersmith s algorithm (although they did not perform this function, having come afterwards) so we briefly sketch them as well. The ....
....to compute logarithms rapidly in one representation of a field enables one to compute logarithms in any other representation just about as fast. The first algorithm we discuss is one of several that have the same asymptotic performance. The other 19 algorithms in this group are described in [20], at least in the form applicable to fields GF(p) It is basically an adaptation of the Schroeppel factorization algorithm [20,53] We assume that f (x) is of the form (4.18) with deg f (x) n 2, say. This time we let S = S 1 S 2 , where S 1 consists of the irreducible polynomials of degrees m, ....
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D. Coppersmith and A. M. Odlyzko, manuscript in preparation.
Some New Methods and Results in Tree Enumeration - Odlyzko (1984) (1 citation) Self-citation (Odlyzko) (Correct)
....are studied in regions where they diverge to infinity, in other cases they are investigated in regions where they converge to zero. In still other cases that could be cited, such as the work on probabilities of very large or very small heights that was done in [38] and is further developed in [30], the corresponding sequences of generating functions are studied practically right on the boundary between convergence and divergence. What this seemingly great variety of different approaches conceals is the basic similarity of the techniques used, all of which depend on very intensive and ....
A. M. Odlyzko and L. B. Richmond, manuscript in preparation.
Limit Distributions for Coefficients of Iterates of.. - Flajolet, Odlyzko (1984) (1 citation) Self-citation (Odlyzko) (Correct)
....[3,4,8] Most of the papers in that area are concerned with questions of convergence of iteration. In this paper, on the other hand, we are operating almost exclusively in the region of divergence, and we concentrate on the rate and nature of divergence. In other situations, such as those of [5,10,11,13], it is advantageous to study the iteration either within the convergence region or else right on the boundary between convergence and divergence. Methods similar to some of those used in those papers could also be used to obtain more information than is provided by Theorem 2 when r 1. 2. Proofs ....
A. M. Odlyzko and L. B. Richmond, manuscript in preparation. R-2
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