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In this paper, we present a measure of distance in a second-order cone based on a class of continuously differentiable strictly convex functions on ₊₊. Since the distance function has some favorable properties similar to those of the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]), we refer to it as a quasi D-function. Then, a proximal-like algorithm using the quasi D-function is proposed and applied to the second-cone programming problem, which is to minimize a closed proper convex function with general second-order cone constraints. Like the proximal point algorithm using the D-function (Censor and Zenios in J. Optim. Theory Appl. 73:451464 [1992]; Chen and Teboulle in SIAM J. Optim. 3:538543 [1993]), under some mild assumptions we establish the global convergence of the algorithm expressed in terms of function values; we show that the sequence generated by the

Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with non-triangular faces as holes, by either using the nodes' location or using a landmark-based scheme without node location. The conformal map is computed with Ricci flow such that all the non-triangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossip-style computation. The method applies to both unit disk graph models and quasi-unit disk graph

We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.