Laura K. Potter

Estimation of Probability Distributions for Individual Parameters Using Aggregate Population Observations

In this paper we discuss a general methodology for estimating the distribution of individual growth rates in a size-structured population using aggregate population data. The method, for which rigorous theoretical formulations have been developed, is presented in the context of an inverse problem methodology and its use is illustrated with application to mosquitofish, Gambusia affinis, populations in rice fields.

Modeling of Nonlinear Hysteresis in Elastomers under Uniaxial Tension

As a fundamental component of an overall program in modeling smart material damping devices, we consider inactive host material models for moderate to highly filled rubbers undergoing uniaxial tensile deformations. Beginning from a neo-Hookean strain energy function formulation for nonlinear extension, we develop general constitutive models for both quasi-static and dynamic deformations of a viscoelastic rod. The constitutive laws are nonlinear and contain hysteresis through a Boltzmann superposition integral term. The resulting integropartial differential equations models are shown to be equivalent to the usual Lagrangian dynamic distributed parameter models coupled with linear ordinary differential equations for internal variables (internal strains). Comprehensive well-posedness results (existence, uniqueness and continuous dependence) are summarized in a discussion of theoretical aspects of the systems. The models are validated with experiments designed and carried out explicitly for this study. In particular, quasi-static Instron experimental data are used in a least squares inverse problem formulation to estimate nonlinear elastic and nonlinear viscoelastic contributions to the general stress-strain constitutive laws proposed. It is shown that the models provide an excellent prediction of nested hysteresis loops manifested in the data. These models are then used as initial estimates in determining the nonlinear hysteretic constitutive laws for the dynamic experiments. It is shown that in cases of more highly filled rubbers, multiple internal variable models lead to best fits to the data.

Improved upper bounds for the reliability of d-dimensional consecutive-k-out-of-n : F systems

Consider a 2-dimensional consecutive-k-out-of-n : F system, as described by Salvia and Lasher [9], whose components have independent, perhaps identical, failure probabilities. In this paper, we use Jansons exponential inequalities [5]; to derive improved upper bounds on such a systems reliability, and compare our results numerically to previously determined upper bounds. In the case of equal component-failure probabilities, we determine analytically, given k and n, those component-failure probabilities for which our bound betters the upper bounds found by Fu and Koutras [4] and Koutras et al. [6]. A different kind of analytic comparison is made with the upper bound of Barbour et al. [3]. We further generalize our upper bound, given identical component-failure probabilities, to suit d-dimensional systems for d ≤ 3.

Stress-strain laws for carbon black and silicon filled elastomers

In this note we present results for nonlinear and hysteretic constitutive laws for filled elastomers. Theoretical, computational and experimental results are given.

Model predictions and comparisons for three toxicokinetic models for the systemic transport of trichloroethylene

In this paper, we present and compare three physiologically based pharmacokinetic models for the systemic transport of trichloroethylene (TCE), a common environmental toxicant. Of particular interest is the disposition of TCE in the adipose tissue, where TCE is known to accumulate. The first two systemic models utilize standard ODE-based adipose compartments that assume rapid equilibrium and uniformity. The third model includes a PDE-based axial dispersion model that is designed to capture the heterogeneous physiology of adipose tissue and the expected transport of TCE there. Using numerical methods and model simulations, we compare the predicted concentration profiles of TCE in the adipose tissue for the three systemic models. Our results suggest that the dispersion-based adipose compartmental model is best able to capture the physiological heterogeneities of adipose tissue and their expected effects on TCE adipose concentrations.

Sign-balanced covering matrices

Abstract A q × n array with entries from 0, 1,..., q − 1 is said to form a difference matrix if the vector difference (modulo q ) of each pair of columns consists of a permutation of [0, 1,... q − 1]; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate δ (-[−1, 1] t , a λq × n array will be said to form a ( t, q, λ, Δ ) sign-balanced matrix if for each choice C 1 , C 2 ,..., C t of t columns and for each choice ɛ = ( ɛ 1 ,..., ɛ t ) ∈ Δ of signs, the linear combination ∑ j = 1 t e j C j contains (mod q ) each entry of [0, 1,..., q − 1] exactly λ times. We consider the following extremal problem in this paper: How large does the number k = k ( n, t, q, λ, δ ) of rows have to be so that for each choice of t columns and for each choice ( ɛ 1 , ..., ɛ t ) of signs in δ, the linear combination ∑ j = 1 t e j C j contains each entry of [0, 1,..., q − 1] at least λ times? We use probabilistic methods, in particular the Lovasz local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k ( n, t, q, λ, δ ), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array - in a sense to be described.