We describe a method to improve both the accuracy and computational
efficiency of a given finite difference scheme used to simulate a
geophysical flow. The resulting modified scheme is at least as
accurate as the original, has the same time step, and often uses the
same spatial stencil. However, in certain parameter regimes it is
higher order. As examples we apply the method to the shallow water
equations, the Navier-Stokes equations, and to a sea breeze model.

Best constants are found for a class of multiplicative inequalities
that give an estimate of the C-norm of a function in terms of the product
of the L_2-norms of the powers of the Laplace operator.
Special attention is given to functions defined on the sphere S^n.

We discuss the problem of extending a recent
result due to G. Gunatillake concerning fixed points of
analytic self-maps of the disk and the spectrum (relative to
weighted Hardy spaces) of a compact weighted composition
operator induced by such a map and a weight function that is
bounded away from zero to a general class of Hilbert spaces
over bounded convex domains in n-dimensional complex
Euclidean space.

In this joint work with Pierre Dolbeault and Giuseppe Tomassini
we consider the problem of characterizing compact real submanifolds of C^n
that bound Levi flat hypersurfaces. The problem is well studied in C^2 but
surprisingly little is known in higher dimension. In this talk I will, in
particular, explain the fundamental difference between n=2 and higher dimension
showing why the known methods in C^2 do not apply.

We prove the continuity of solutions with respect to parameter for
a semilinear elliptic eigenvalue problem with constraint by using
variational methods.
and then show the bifurcating solutions to a semilinear elliptic eigenvalue
problem
without constraint