The discrete most degenerate principal series of irreducible Hermitian representations of the Lie algebra of an arbitrary noncompact as well as compact rotation group SO(p, q) are derived. The properties of these representations are discussed and the explicit form of the corresponding harmonic functions is given.

The eigenfunction expansions associated with the second-order invariant operator on hyperboloids and cones are derived. The global unitary irreducible representations of the SO0(p, q) groups related to hyperboloids and cones are obtained. The decomposition of the quasi-regular representations into the irreducible ones is given and the connection with the Mautner theorem and nuclear spectral theory is discussed.

Three principal continuous series of most degenerate unitary irreducible representations of an arbitrary noncompact rotation group SO(p, q) have been derived and their properties discussed in detail. The corresponding harmonic functions have been constructed.

We consider a second order differential operator A(x) = − d i,j=1, on a bounded domain D with Dirichlet boundary conditions on ∂D, under mild assumptions on the coefficients of the diffusion tensor aij . The object is to construct monotone numerical schemes to approximate the solution to the problem A(x) u(x) = µ(x), x ∈ D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness, introducing function spaces needed to prove convergence results. Then, we define non-standard stencils on grid-knots that lead to extended discretization schemes by matrices possesing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W 1 2 -convergence in detail, and mention convergence in C and L1 spaces. We conclude by a numerical example illustarting the schemes and convergence results.