Geometric compression plays a fundamental rolein virtual reality and augmented reality (VR/AR) applications. Dense meshesare re-sampled and re-tessellatedtoreduce thecomplexity. This process is called remeshing. In this work,we propose a novel remeshing algorithm based on both angle-preservingparameterization, and measurecontrollable parameterization. The conformal parameterization iscarried out by discrete surface Ricci flow method, the measurecontrollable parameterization is obtaind by an optimal mass transportation map. The sampling is performed on the measurecontrollable parameterization domain, the triangulation is computed on the conformal parameterization domain using Delaunay refinement algorithm. This method gives the user full control of sampling distribution, and produces mesheswith curvature measure convergence. The meshing resultcan emphsizethe region of interests, is curvature sensitive. Experimental results demonstrate the efficiency and efficacy of the proposed method.

In this paper, we propose a new algorithm for solving the general approximate common divisors (GACD) problems, which is based on lattice reduction algorithms on certain special lattices and linear equation solving algorithms over integers. Through both theoretical arguments and experimental data, we show that our new algorithm works in polynomial time but under roughly the following condition: \begin{itemize} \item There is a positive integer t such that \frac{\gamma+\eta}{t} + \frac{t}{H} + \rho < \eta; \item We have more than t GACD samples. \end{itemize} or equivalently \begin{itemize} \item H(\eta-\rho)^2 - 4(\gamma+\eta) > 0; \item We have more than t = \lceil\frac{H(\eta-\rho) - \sqrt{H^2(\eta-\rho)^2 - 4H(\gamma+\eta)}{2}\rceil GACD samples. \end{itemize} Here \gamma, \eta and \rho are parameters describing a GACD problem, H = 1/\log_2 F and F is the Hermite Factor. In our experiments, H is shown to be roughly 40 when using the LLL reduction algorithm and it should be around 80 when using Deep or BKZ algorithms. % We use our algorithm to solve concrete problems that no other algorithm could solve before. We show how our algorithm can be applied to attack the fully homomorphic encryption schemes which are based on the general approximate common divisors problem and its limitations.

The paper develops an existence theory for solutions of the Abreu equation, which include extremal metrics on toric surfaces. The technique employed is a continuity method, combined with “blow-up” arguments. General existence results are obtained, assuming a hypothesis (the “M-condition”) on the solutions, which is shown to be related to the injectivity radius.

We show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective. Moreover, we give a stability estimate and develop a reconstruction procedure.

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