This chapter provides an overview of the background material assumed in the remainder of the book. We state major results and provide sketches of the less technical proofs, particularly where the ideas presented are instrumental in subsequent constructions. The first topic is the theory of linear systems of ordinary differential equations (ODEs). Much of this material is standard for first-year graduate courses in ODEs, such as presented in Hartman [114], Perko [234]. This is followed by a review of the basic theory of functional analysis as applied to linear partial differential equations; further details can be found in the standard references Evans [81], Kato [162]. We finish the general overview by discussing the point spectrum in the context of the Sturm-Liouville theory for second-order operators. These operators have a one-to-one relationship between the ordering of the eigenvalues and the number of zeros for the associated eigenfunctions, which is extremely useful in applications.

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A simplified model for water management in a polymer electrolyte membrane ͑PEM͒ fuel cell operating under prescribed current with iso-potential plates is presented. The consumption of gases in the flow field channels, coupled to the electric potential and water content in the polymer membrane, is modeled in a two-dimensional slice from inlet to outlet and through the membrane. Both co-and counter-flowing air and fuel streams are considered, with attention paid to sensitivity of along-the-channel current density to inlet humidities, gas stream composition, and fuel and oxygen stoichiometries. The parameters describing the nonequilibrium kinetics of the membrane/catalyst interface are found to be fundamental to accurate fuel cell modeling. A new parameter which models nonequilibrium membrane water uptake rates is introduced. Four parameters, the exchange current, a membrane water transfer coefficient, an effective oxygen diffusivity, and an average membrane resistance, are fit to a subset of data and then held constant in subsequent runs which compare polarization curves, current density and membrane hydration distributions, water transfer, and stoichiometric sensitivity to the balance of the experimental data.

The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytic and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schrödinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.

We present a microfield approach for studying the dependence of the orientational polarization of the water in aqueous electrolyte solutions upon the salt concentration and temperature. The model takes into account the orientation of the solvent dipoles due to the electric field created by ions, and the effect of thermal fluctuations. The model predicts a dielectric functional dependence of the form ɛ(c)=ɛ_{w}-βL(3αc/β),β=ɛ_{w}-ɛ_{ms}, where L is the Langevin function, c is the salt concentration, ɛ_{w} is the dielectric of pure water, ɛ_{ms} is the dielectric of the electrolyte solution at the molten salt limit, and α is the total excess polarization of the ions. The functional form gives a remarkably accurate description of the dielectric constant for a variety of salts and a wide range of concentrations.