Fern Y. Hunt

On the Approximation of Invariant Measures

Given a discrete dynamical system defined by the map τ:X →X, the density of the absolutely continuous (a.c.) invariant measure (if it exists) is the fixed point of the Frobenius-Perron operator defined on L1(X). Ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator. T. Y. Li proved the convergence of the scheme for expanding maps of the interval. G. Keller and M. Blank extended this result to piecewise expanding maps of the cube in ℝn. We show convergence of a variation of Ulams scheme for maps of the cube for which the Frobenius-Perron operator is quasicompact. We also give sufficient conditions onτ for the existence of a unique fixed point of the matrix approximation, and if the fixed point of the operator is a function of bounded variation, we estimate the convergence rate.

Microstructure of weathered paint and its relation to gloss loss: Computer simulation and modeling

The role of pigment particle size, pigment volume concentration, and dispersion in gloss loss of paint films on weathered surfaces is unclear. Because reproducible and cost effective data are difficult to obtain, an approach based on computer simulation and modeling is a promising supplementary tool. We describe the simulation of a painted surface consisting of pigment particles of known size, pigment volume concentration and dispersion, situated in a binder that erodes over time due to exposure to ultraviolet radiation. Pigment particle size, geometry, pigment volume concentration, and pigment particle dispersion are parameters of the model. Our purpose is to illustrate how simulation can be used to aid the development of formulation strategies for the design of coatings with desirable gloss characteristics.

Efficient Algorithms for Computing Fractal Dimensions

Our purpose is to describe a new class of methods for computing the “capacity dimension” and related quantities for point-sets. The techniques presented here build on existing work which has been described in the literature. The novelty of our methods lies first in the approach taken to the definition of computation of dimension (namely, via Monte Carlo calculation of the volume of an e-cover of the point-set), and second in the use of data structures which result in extremely efficient codes for vector computers such as the Cyber 205 (the computation is reduced to the sorting and searching of one-dimensional arrays so that a calculation employing one million points requires less than 2 minutes).

Probabilistic computation of Poiseuille flow velocity fields

Velocity fields for Poiseuille flow through tubes having general cross section are calculated using a path integral method involving the first‐passage times of random walks in the interior of the cross sectional domain D of the pipe. This method is applied to a number of examples where exact results are available and to more complicated geometries of practical interest. These examples include a tube with ‘‘fractal’’ cross section and open channel flows. The calculations demonstrate the feasibility of the probabilistic method for pipe flow and other applications having an equivalent mathematical description (e.g., torsional rigidity of rods, membrane deflection). The example of flow through a fractal pipe shows an extended region of diminished flow velocity near the rough boundary which is similar to the suppressed vibration observed near the boundaries of fractal drums.

How to estimate capacity dimension

Abstract We describe a class of robust, computationally efficient algorithms for estimating capacity dimension and related quantities for compact subsets of R n . Our algorithms are based on Monte Carlo quadrature, data compression, and generalized distance functions.

Regulation of population cycles by genetic feedback: Existence of periodic solutions of a mathematical model

Populations of voles, and lemmings of the Northern hemisphere exhibit cyclic fluctuations with a cycle of three to four years. Krebs et al. presented evidence that the cycles are driven by changes in the genotypic structure of the population [9]. Incorporating some of their hypotheses we present a mathematical model of a one locus two allele population with density dependent selection and assuming a slow selection hypothesis, the existence of periodic solutions is proved. These solutions arise by Hopf bifurcation in δ1/¦β1¦, the ratio of the residual death and birth rates of the density sensitive homozygote.

On the persistence of spatially homogeneous solutions of a population genetics model with slow selection.

Abstract We consider a model of a nonconstant population with a genetic trait determined by one locus and two alleles. Natural selection as reflected in small differences among the genotype birth and death rates acts slowly and depends on the density. This model was first considered by F. Hoppensteadt for the density independent case. The assumption of slow selection allows the construction of solutions of the model using matched asymptotic expansions. It is found that to leading order the allele frequency is the solution of a logistic type differential equation, so that its ultimate value can be determined once the birth rates, death rates, and population size to leading order are given. In Sec. 2 solutions of the corresponding diffusion model and a related system which leads to Fishers equation are constructed under two assumptions: first that dispersal is fast relative to selection, and then that it is comparable. With the first assumption, and to leading order, the genotype frequencies are slowly varying functions of time t for large t . In the second case the allele frequency is the solution of a generalization of Fishers equation.

A linear programming based algorithm for multiple sequence alignments

In this brief paper we discuss an approach to multiple sequence alignment based on treating the alignment process as a stochastic control problem. The use of a model based on a Markov decision process leads to a linear programming problem whose solution is linked to a suggested alignment. Our goal is to avoid the expense in time and computation encountered when dynamic programming based algorithms are used to align a large number of sequences. The dual linear programming problem can also be defined. We implemented the method on a set of cytochrome p450 sequences and compared our suggested alignments of 3 sequences with that obtained by CLUSTALW.

Finite Precision Representation of the Conley Decomposition

We present a theoretical basis for a novel way of studying and representing the long-time behavior of finite-dimensional maps. It is based on a finite representation of ε-pseudo orbits of a map by the sample paths of a suitable Markov chain based on a finite partition of phase space. The use of stationary states of the chain and the corresponding partition elements in approximating the attractors of maps and differential equations was demonstrated in Refs. 7 and 3 and proved for a class of stable attracting sets in Ref. 8. Here we explore the relationship between the communication classes of the Markov chain and basic sets of the Conley decomposition of a dynamical system. We give sufficient conditions for the existence of a chain transitive set and show that basic sets are isolated from each other by neighborhoods associated with closed communication classes of the chain. A partition element approximation of an isolating block is introduced that is easy to express in terms of sample paths. Finally, we show that when the map supports an SBR measure there is a unique closed communication class and the Markov chain restricted to those states is irreducible.

Patterns of LPS synthesis in Gram negative bacteria

Lipopolysaccharide (LPS), a lipid based carbohydrate polymer, is found in the outer membrane of Gram negative bacteria where it plays a vital role in its structure and function. It is chiefly responsible for the toxic effects of the bacterial diseases caused by these organisms and plays a role in the organisms defense against host immune attack. In recent experiments using high resolution gel techniques Goldman & Leive (1980) have revealed an unexpected heterogeneity in the distribution of polymer lengths found in the membrane with lengths ranging from 0 (lipid-A-core) to nearly 40 sugar units. Monomer units are an essential element of the synthesis process. Working with mutant strains of E. coli and Salmonella typhimurium, they have also shown that these bacteria will continue to synthesize LPS molecules with very long chains even though monomer unit production is severely reduced. The steps involved in the synthesis of LPS are known and in this paper it is shown that the results of Goldman & Leive cannot be obtained assuming the synthesis process is length independent as has been suggested. Moreover, the paradoxical persistence of long chains in spite of monomer suppression has a simple explanation once length dependence is assumed. These conclusions result from the analysis of a Markov chain model of synthesis.