We investigate the SL2(R) invariant geodesic curves with the associated invariant distance function in parabolic geometry. Parabolic geometry naturally occurs in the study of SL2(R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2dimensional Minkowski space-time) geometries. Initially we attempt to use standard methods of finding geodesics...

The null controllability of parabolic operators in bounded domains, with both boundary or locally distributed controls, is a well-established property, see, e.g., (Bensoussan et al., 1993) and (Fattorini, 1998). Such a property brakes down, however, for degenerate parabolic operators even when degeneracy occurs on ”small” subsets of the space domain, such as subsets of the boundary. This talk w...

This work focuses on the development of computationally efficient predictive control algorithms for nonlinear parabolic and hyperbolic PDEs with state and control constraints arising in the context of transport-reaction processes. We first consider a diffusion-reaction process described by a nonlinear parabolic PDE and address the problem of stabilization of an unstable steady-state subject to ...

We prove that D0, the set of all parameters λ ∈ C \ {0} for which the cubic polynomial fλ is parabolic and has no other parabolic or finite attracting periodic cycles, contains a deleted neighborhood of the origin 0. Our main result is that the function D0 3 λ 7→ HD(J(fλ)) ∈ R is real-analytic. This function ascribes to the polynomial fλ the Hausdorff dimension of its Julia set J(fλ). The theor...

In this paper, we study the relation between the cohomology of Coxeter groups and their parabolic subgroups of nite order. Let W be a Coxeter group and k a commutative ring with identity. We investigate the natural map : H (W; k) ! lim:inv: H (W F ; k), where W F runs all parabolic subgroups of nite order, and prove that the kernel and the cokernel of consist of nilpotent elements. This general...

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g. a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundle over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally ...

Recent improvements in the parabolic equation method are combined to extend this approach to a larger class of seismo-acoustics problems. The variable rotated parabolic equation [J. Acoust. Soc. Am. 120, 3534-3538 (2006)] handles a sloping fluid-solid interface at the ocean bottom. The single-scattering solution [J. Acoust. Soc. Am. 121, 808-813 (2007)] handles range dependence within elastic s...

Talk Abstract We describe the application of arbitrary-order Hermite methods to hyperbolic-parabolic systems, including parabolic systems as a special case, emphasizing situations where the parabolic terms are small (e.g. large Reynolds or Peclet numbers). Using Fourier methods, we study the dispersion relation for the semidiscretized problem, leading to an understanding of time-step stability ...