Systems of functional equations

Abstract

The aim of this paper is to discuss the asymptotic properties of the coefficients of generating functions which satisfy a system of functional equations. It turns out that under certain general conditions these coefficients are related to the distribution of a multivariate random variable that is asymptotically normal. As an application it turns out that the distribution of the terminal symbols in context-free languages is typically asymptoti-Ž .cally normal.

An Unbiased Adaptive Sampling Algorithm for the Exploration of RNA Mutational Landscapes Under Evolutionary Pressure

“…and aperiodic (Words of any parity can be generated.). Applying a striking result by Drmota (Drmota, 1997), we directly conclude that the distribution of the number of occurrence of G + C is asymptotically normal of mean ln…”

Exponentially few RNA structures are designable

“…One could therefore determine, through a trivial modification of Equation (4), the bivariate generating function S(z, u) = n,k ≥0 s n,v z n u k , s n,v being an upper bound for the number of structures of size n having v ensemble defect. An application of the famous Drmota theorem [11] would then very likely provide sharper estimates, by accounting the accumulation of local defects rather than only consider the worst one, as currently done in this work.…”

Combinatorics of locally optimal RNA secondary structures

“…Limit law for the number of links. Using a theorem of Drmota [7] (closely related to the Drmota-Lalley-Wood theorem) we show that the number of links in a random secondary structure (general, saturated, or G-saturated) of length n is asymptotically a gaussian law with Θ(n) expectation and Θ( √ n) standard deviation.…”