Many fundamental cellular processes involve small numbers of molecules. When numbers are small, fluctuations dominate, and stochastic models, which account for these fluctuations, are required. In this chapter, we describe minimal stochastic models of three fundamental cellular processes: gene expression, protein modification, and polymerization. We introduce key analytic tools for solving each...

We study the scattering of acoustic waves by an obstacle embedded in a uniform waveguide with planar boundaries, which is a fundamental model in shallow ocean acoustics. Under certain geometric assumptions on the shape of the obstacle, we prove a Rellichtype uniqueness theorem in the case of soft obstacles. The solvability of the scattering problem is proved via boundary integral equations and ...

The Debye-Wolf electromagnetic diffraction integral is now routinely used to describe focusing by high numerical (NA) lenses. We obtain an eigenfunction expansion of the electric vector field in the focal region in terms of Bessel and generalized prolate spheroidal functions. Our representation has many optimal and desirable properties which offer considerable simplification to the evaluation a...

GLOSSARY dispersive: Characteristic of waves whose velocity of propagation depends on wave frequency. The shape of a dispersive wave packet changes as it moves along. eigenfunction: Functional shape of the horizontal and vertical components of wave motion versus depth in the ocean for a specific wave frequency. geometrical spreading: Process of amplitude reduction resulting from the progressive...

An exact analytical solution is obtained for the problem of three-dimensional transient heat conduction in the multilayered sphere. The sphere has multiple layers in the radial direction and, in each layer, time-dependent and spatially nonuniform volumetric internal heat sources are considered. To obtain the temperature distribution, the eigenfunction expansion method is used. An arbitrary comb...

The goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains. In particular, we will locate the place where u achieves its maximum to within a distance comparable to the inradius, uniformly for arbitrarily large diameter. In addition, we will estimate the location of other level sets of u by showing that u is well-approximated by the first eigenfu...

Motivated by practical applications in heat conduction and contaminant transport, we consider mass diffusion across a perturbed interface separating two finite regions of distinct diffusivity. Under the assumption continuity solution diffusive flux at interface, use perturbation theory to develop an asymptotic expansion valid for small perturbations. Each term satisfies initial-boundary value p...