The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a two-manifold M, based on a geodesic metric. We present a formal definition of the skeleton S(\Omega) for a shape V in M and show several properties that make S(\Omega) distinct from its Euclidean counterpart in R^2. We further prove that for a shape sequence {\Omega_i} that converge to a shape V in M, the mapping \Omega\rightarrow S(\Omega_i) is lower semi-continuous. A direct application of this result is that we can use a set P of sample points to approximate the boundary of a 2D shape V in M, and the Voronoi diagram of P inside \Omega\subset M gives a good approximation to the skeleton S(\Omega). Examples of skeleton computation in topography and brain morphometry are illustrated.

High order path-conservative schemes have been developed for solving nonconservative hyperbolic systems. Recently, it has been observed that this approach may have some computational issues and shortcomings. In this paper, a modification to the high order path-conservative scheme in is proposed to improve its computational performance and to overcome some of the shortcomings. This modification is based on the high order finite volume WENO scheme with subcell resolution and it uses an exact Riemann solver to catch the right paths at the discontinuities. An application to one-dimensional compressible two-medium flows of nonconservative or primitive Euler equations is carried out to show the effectiveness of this new approach.

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The prolongation g^{(k)} of a linear Lie algebra g \subset gl(V) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras g \subset gl(V) with non-zero prolongations. If g is the Lie algebra aut(\hat{S}) of infinitesimal linear automorphisms of a projective variety S \subset \BP V, its prolongation g^{(k)} is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation aut(\hat{S})^{(k)} is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety with non zero prolongations, which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when S is linearly normal and Sec(S) \neq P(V), the blow-up of P(V) along S has the target rigidity property, i.e., any deformation of a surjective morphism Y \to Bl_S(PV) comes from the automorphisms of Bl_S(PV).