Analytically Continued Hypergeometric Expression of the Incomplete Beta Function

The Incomplete Beta Function is rewritten as a Hypergeometric Function that is the analytic continuation of the conventional form, a generalization of the finite series, which simpifies the Stieltjes transform of powers of a monomial divided by powers of a binomial.     

Modelling thermal front dynamics in microwave heating

The formation and propagation of thermal fronts in a cylindricalmedium that is undergoing microwave heating is studied in detail.The model consists of Maxwell's wave equation coupled to a temperaturediffusion equation containing a bistable nonlinear term. When the thermal diffusivity is sufficiently small the leading-ordertemperature solution of a singular perturbation analysis isused to reduce the system to a free boundary problem. This approximationis then used to derive predictions for the steady-state penetrationand profiles of the temperature and electric fields. These solutionsare valid for arbitrary values of the electric conductivity,and thus extend the previous (small conductivity) results foundin the literature. A quasi-static approximation for the electric field is thenused to obtain an ordinary differential equation for the relaxationdynamics to the steady state. This equation appears to accuratelydescribe the time scale of the electric field's evolution bothwith and without the presence of a strongly coupled temperaturefront, and may be of wider interest than the model for microwaveheating studied here.     

Kinetics of nonequilibrium methylsiloxane polymerization and rearrangement

Nonrandom reactions were demonstrated experimentally during the polymerization of (CH₃)₃SiOSi(CH₃)₃ plus [(CH₃)₂SiO]₄ and the rearrangement of (CH₃)₃SiO[(CH₃)₂SiO]₅Si(CH₃)₃ or (CH₃)₃SiO[(CH₃)₂SiO]₈Si(CH₃)₃, using sulfuric acid-treated Fuller's earth as catalyst. Cyclic and linear reaction products were analyzed by gas–liquid chromatography. A four-step polymerization mechanism was proposed to account for the approach to equilibrium from either the forward or reverse direction. Reaction rate expressions for this mechanism were reduced to a finite set of nonlinear ordinary differential equations. These were solved by using a fourth-order Runge-Kutta numerical integration on a Burroughs B5500 computer. The calculated molecular weight distributions thus obtained were found to agree at all times with the distributions from polymerization and rearrangement experiments.