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This paper is concerned with an interior penalty discontinuous Galerkin (IPDG) method based on a flexible type of non-polynomial local approximation space for the Helmholtz equation with varying wavenumber. The local approximation space consists of multiple polynomial-modulated phase functions which can be chosen according to the phase information of the solution. We obtain some {approximation} properties for this space and \textit{a prior} $L^2$ error estimates for the \textit{h}-convergence of the IPDG method using duality argument. We also provide ample numerical examples to show that, building phase information into the local spaces often gives more accurate results comparing to using the standard polynomial spaces.

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm is quite robust and converges in much fewer iterations and is more efficient than many existing methods.

Short integer solution (SIS) and learning with errors (LWE) are two hard lattice problems. These two problems are believed having huge potential in application of cryptography. In 2012, Ding et al. [5] introduced the first provably secure key exchange based on LWE problem. On the other hand, we believe that it is very difficult to do key exchange on SIS problem only. In 2014, Wang et al. [6] did an attempt, but it was not successful. Mao et al. [7] broke the protocol by an attack based on CBi-SIS problem in 2016. However, their attack is not efficient. In this paper, we present a extremely straightforward and simple attack to Wang’s key exchange and then we will construct a key exchange based on SIS and LWE problems.

We present an algorithm for shape reconstruction from incomplete 3D scans by fusing together two acquisition modes: 2D photographs and 3D scans. The two modes exhibit complementary characteristics: scans have depth information, but are often sparse and incomplete; photographs, on the other hand, are dense and have high resolution, but lack important depth information. In this work we fuse the two modes, taking advantage of their complementary information, to enhance 3D shape reconstruction from an incomplete scan with a 2D photograph. We compute geometrical and topological shape properties in 2D photographs and use them to reconstruct a shape from an incomplete 3D scan in a principled manner. Our key observation is that shape properties such as boundaries, smooth patches and local connectivity, can be inferred with high confidence from 2D photographs. Thus, we register the 3D scan with the 2D photograph and use scanned points as 3D depth cues for lifting 2D shape structures into 3D. Our contribution is an algorithm which significantly regularizes and enhances the problem of 3D reconstruction from partial scans by lifting 2D shape structures into 3D. We evaluate our algorithm on various shapes which are loosely scanned and photographed from different views, and compare them with state-of-the-art reconstruction methods.