The numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth. This problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces. Convergent difference approximations can be found that do not require the explicit specification of the boundary conditions at interfaces in the medium and hence are simple to program. Although such difference approximations typically can only be expected to be first-order accurate, the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth. This is proved for the one-dimensional case and demonstrated numerically for an example in two space dimensions in which the interface is not aligned with the computational mesh.

Controlling physically based characters is an ongoing

problem in animation. Motions that involve a large of

changing contacts with the environment are especially

problematic. In this thesis we present techniques for

controlling contact rich motions, specifically rolling and

falling motions. For rolling we present a method for

controlling the full body orientation of a multibody

character. We use this to describe a roll for the character

to perform in terms of the character's goal orientation,

linear velocity and angular velocity. We then use

a quadratic programming controller to compute the torques to

best achieve these goals. To deal with the many and

constantly varying contacts we introduce a heuristic that

selects only the most important contact points to be active

in the quadratic program. Falling is similar except it

requires time varying orientation and position goals for the

character. We automatically compute these goals by defining

a function to evaluate the quality of a fall and use

sampling based optimization to find the best way to control

the character for a given fall. We demonstrate our rolling

controller for several different rolls in different

environments and our falling controller on a variety of

different falls.

We extend le Stum's construction of the overconvergent site to algebraic stacks. We prove that etale morphisms are morphisms of cohomological descent for finitely presnted crystals on the overconvergent site. Finally, using the notion of an open subtopos of SGA4, we define a notion of overconvergent cohomology supported in a closed substack and show that it agrees with the classical notion of rigid cohomology supported in a closed subscheme.