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In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with$d$-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.

Delaunay meshes (DM) are a special type of manifold triangle meshes --- where the local Delaunay condition holds everywhere --- and find important applications in digital geometry processing. This paper addresses the \yongjin{general} DM simplification problem: given an arbitrary manifold triangle mesh M with n vertices and the user-specified resolution m(< n), compute a Delaunay mesh M* with m vertices that has the least Hausdorff distance to M. To solve the problem, we abstract the simplification process using a 2D Cartesian grid model, in which each grid point corresponds to triangle meshes with a certain number of vertices and a simplification process is a monotonic path on the grid. We develop a novel differential-evolution-based method to compute a low-cost path, which leads to a high quality Delaunay mesh. Extensive evaluation shows that our method consistently outperforms the existing methods in terms of approximation error. In particular, our method is highly effective for small-scale CAD models and man-made objects with sharp features but less details. Moreover, our method is fully automatic and can preserve sharp features well and deal with models with multiple components, whereas the existing methods often fail.

Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function . We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., <i>f</i>(<i>t</i>) = <i>t</i> ² <i>t</i> ³ 1/<i>t</i> <i>t</i> ^1/2, |<i>t</i>|, and [<i>t</i>]₊. Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.

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