The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

These notes introduce the basic concepts of integral equations and their application in global illumination. Much of the discussion is expressed in the language of linear operators to simplify the notation and to emphasize the algebraic properties of the integral equations. We start by reviewing some facts about linear operators and examining some of the operators that occur in global illumination. Six general methods of solving operator and integral equations are then discussed: the Neumann series, successive approximations, the Nystrom method, collocation, least squares, and the Galerkin method. Finally, we look at some of the steps involved in applying these techniques in the context of global illumination. 1 Introduction The transfer of energy by radiation has a character fundamentally different from the processes of conduction and convection. One reason for this difference is that the radiant energy passing through a point in space cannot be completely described by a single scala...

We analyze asymptotically a differential-difference equation, that arises in a Markovmodulated fluid model. Here there are N identical sources that turn on and off, and when on they generate fluid at unit rate into a buffer, which process the fluid at a rate c < N. In the steady state limit, the joint probability distribution of the buffer content and the number of active sources satisfies a system of N + 1 ODEs, that can also be viewed as a differential-difference equation analogous to a backward/forward parabolic PDE. We use singular perturbation methods to analyze the problem for N → ∞, with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used. 1

We solve exactly the problem of a finite slab receiving an isotropic radiation on one side and no radiation on the other side. This problem- to be more precise the calculation of the source function within the slab- was first formulated by K. Schwarzschild in 1914. We first solve it for unspecified albedos and optical thicknesses of the atmosphere, in particular for an albedo very close to 1 and a very large optical thickness in view of some astrophysical applications. Then we focus on the conservative case (albedo = 1), which is of great interest for the modeling of grey atmospheres in radiative equilibrium. Ten-figure tables of the conservative source function are given. From the analytical expression of this function, we deduce 1) a simple relation between the effective temperature of a grey atmosphere in radiative equilibrium and the temperature of the black body that irradiates it, 2) the temperature at any point of the atmosphere when it is in local thermodynamical equilibrium. This temperature distribution is the counterpart, for a finite slab, of Hopf’s distribution in a half-space. Its graphical representation is given for various optical thicknesses of the atmosphere.

We consider the analysis and numerical solution of a forward-backward boundary value problem. We provide some motivation, prove existence and uniqueness in a function class especially geared to the problem at hand, provide various energy estimates, prove a priori error estimates for the Galerkin method, and show the results of some numerical computations. Key Words: forward-backward, heat equation, degenerate parabolic problem, Brownian motion, electron and neutron scattering, separated flow boundary layers AMS(MOS) Subject Classification: 65M60, 65N30, 65N15, 76D15, 76M10 1 Introduction: We study a class of forward-backward heat equations in this report. Problems such as these arise in a remarkable variety of physical applications which we will describe in the next section. It seems that this problem-type has been avoided to some degree due to the nontrivial task of finding a proper formulation. Let\Omega ae IR 2 be a rectangle (0; L) \Theta (0; H). Let oe(x; y) be a smooth func...

Abstract. We give a description of the classical Wiener-Hopf factorization from the point of view of excursion theory concentrating mainly on the case of random walks as opposed to Lévy processes. The exposition relies primarily on the ideas of Greenwood and Pitman (1979, 1980). Key words: Lévy process, Wiener-Hopf factorization, infinite divisibility. 1

An exact model is proposed for a gray, isotropically scattering planetary atmosphere in radiative equilibrium. The slab is illuminated on one side by a collimated beam and is bounded on the other side by an emitting and partially reflecting ground. We provide expressions for the incident and reflected fluxes on both boundary surfaces, as well as the temperature of the ground and the temperature distribution in the atmosphere, assuming the latter to be in local thermodynamic equilibrium. Tables and curves of the temperature distribution are included for various values of the optical thickness. Finally, semi-infinite atmospheres illuminated from the outside or by sources at infinity will be dealt with.