Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


osu1060976912.pdf (978.64 KB)

ETD Abstract Container

Abstract Header

Asymptotic enumeration via singularity analysis

Lladser, Manuel Eugenio
http://rave.ohiolink.edu/etdc/view?acc_num=osu1060976912

Abstract Details

2003, Doctor of Philosophy, Ohio State University, Mathematics.
Asymptotic formulae for two-dimensional arrays (f r,s ) r,s≥0 where the associated generating function F(z,w):=Σ r,s≥0 f r,s z r w s is meromorphic are provided. Our approach is geometrical. To a big extent it generalizes and completes the asymptotic description of the coefficients f r,s along a compact set of directions specified by smooth points of the singular variety of the denominator of F(z,w) . The scheme we develop can lead to a high level of complexity. However, it provides the leading asymptotic order of f r,s if some unusual and pathological behavior is ruled out. It relies on the asymptotic analysis of a certain type of stationary phase integral of the form ∫ e -s·P(d,θ) A(d,θ) dθ , which describes up to an exponential factor the asymptotic behavior of the coefficients f r,s along the direction d=r⁄s in the (r,s) -lattice. The cases of interest are when either the phase term P(d,θ) or the amplitude term A(d,θ) exhibits a change of degree as d approaches a degenerate direction. These are handled by a generalized version of the stationary phase and the coalescing saddle point method which we propose as part of this dissertation. The occurrence of two special functions related to the Airy function is established when two simple saddles of the phase term coalesce. A scheme to study the asymptotic behavior of big powers of generating functions is proposed as an additional application of these generalized methods.
Robin Pemantle (Advisor)
227 p.
Airy phenomena; amplitude; analytic combinatorics; asymptotic enumeration; asymptotic expansion; bandwidth; big powers of generating functions; bivariate generating function; Cauchy integral formula; central limit theorem; coalescing saddle point method

Recommended Citations

Citations

  • Lladser, M. E. (2003). Asymptotic enumeration via singularity analysis [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1060976912

    APA Style (7th edition)

  • Lladser, Manuel. Asymptotic enumeration via singularity analysis. 2003. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1060976912.

    MLA Style (8th edition)

  • Lladser, Manuel. "Asymptotic enumeration via singularity analysis." Doctoral dissertation, Ohio State University, 2003. http://rave.ohiolink.edu/etdc/view?acc_num=osu1060976912

    Chicago Manual of Style (17th edition)

osu1060976912
889
© 2003, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.