We consider a simple model of a one-locus, two-allele population inhibiting a two-patch system and experiencing spatially heterogeneous viability selection. The populaton size is finite. We use a diffusion approximation and singular perturbation techniques to find the probability of fixation of a mutant allele. We focus on situations in which each allele is advantageous in one patch and deleterious in the other patch. Our theoretical results support the previous conclusions that, under certain conditions, small populations respond faster to selection than do large populations. We emphasize that knowledge of the dependence of migration rates on population size is crucial in evaluating the effects of population size on the rate of evolution.

Abstract. We consider electromagnetic interrogation problems for complex materials involving distributions of polarization mechanisms and also distributions for the parameters in these mechanisms. A theoretical and computational framework for such problems is given. Computational results for specific problems with multiple Debye mechanisms are given in the case of discrete, uniform, log-normal, and log-bi-Gaussian distributions. Introduction.For at least the past century [52,53], scientific investigators have sought to understand what happens to electromagnetic fields (and how to mathematically model the associated phenomena) when they are introduced into complex materials such as biotissue and more general dielectrics, conductors and magnetics. More specifically, a fundamental question is how to model dispersion and dissipation of the fields in these complex materials. This has most often led to the use of Maxwell's equations in a nonvacuum environment which entails constitutive relationships for polarization (in dielectrics), magnetization (in magnetic materials) and conductivity. We focus here on modeling polarization in dielectric materials for which we develop a new modeling framework. Even though we treat only polarization as our dispersive mechanism in our formulation (adopting Ohm's law for conductivity and considering nonmagnetic materials), the approach is sufficiently general so as to be readily extended to treat magnetization and conductivity in materials (each in some type of convolution representation involving susceptibility kernels, e.g., see [2,3]). We develop a framework that allows not only uncertainty (through

We present existence, uniqueness and continuous dependence (with respect to probability distributions on polarization parameters) of solutions in Maxwell systems. This provides a theoretical and computational foundation for associated inverse problems.

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In this paper we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the cases of circular and square microstructures in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic field is much larger than the relevant dimensions of the microstructure. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTESThe original document contains color images. ABSTRACTIn this paper we employ the periodic unfolding method for simulating the electromagnetic eld in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the e ective parameters for a Debye dielectric medium in the cases of circular and square microstructures in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic eld is much larger than the relevant dimensions of the microstructure.