A new proof of the Mahler conjecture in R2 is given. In order to prove the result, we introduce a new method  the vertex removal method; i.e., for any origin-symmetric polygon P, there exists a linear image P contained in the unit disk B2, and there exist three contiguous vertices of P lying on the boundary of B2. We can show that the volume product of P decreases when we remove the middle vertex of the three vertices.

Real-time crime forecasting is important. However, accurate prediction of when and where the next crime will happen is difficult. No known physical model provides a reasonable approximation to such a complex system. Historical crime data are sparse in both space and time and the signal of interests is weak. In this work, we first present a proper representation of crime data. We then adapt the spatial temporal residual network on the well represented data to predict the distribution of crime in Los Angeles at the scale of hours in neighborhood-sized parcels. These experiments as well as comparisons with several existing approaches to prediction demonstrate the superiority of the proposed model in terms of accuracy. Finally, we present a ternarization technique to address the resource consumption issue for its deployment in real world. This work is an extension of our short conference proceeding paper [Wang et al, Arxiv 1707.03340].

We present a uniformly accurate finite difference method and establish rigorously its uniform error bounds for the Zakharov system (ZS) with a dimensionless parameter 0 < ε ≤ 1, which is inversely proportional to the speed of sound. In the subsonic limit regime, i.e., 0 < ε << 1, the solution propagates highly oscillatory waves and/or rapid outgoing initial layers due to the perturbation of the wave operator in the ZS and/or the incompatibility of the initial data, which is characterized by two parameters α ≥ 0 and β ≥ −1. Specifically, the solution propagates waves with wavelength of O(ε) and O(1) in time and space, respectively, and amplitude at O(ε^{min{2,α,1+β}}). This high oscillation of the solution in time brings significant difficulties in designing numerical methods and establishing their error bounds, especially in the subsonic limit regime. A uniformly accurate finite difference method is proposed by reformulating the ZS into an asymptotic consistent formulation and adopting an integral approximation of the oscillatory term. By applying the energy method and using the limiting equation via a nonlinear Schr ̈odinger equation with an oscillatory potential, we rigorously establish two independent error bounds at O(h^2 + τ^2/ε) and O(h^2 + τ^2 + τε^{α^∗} + ε^{1+α^∗}), respectively, with h the mesh size, τ the time step and α^∗ = min{1, α, 1 + β}. Thus we obtain error bounds at O(h^2 + τ^{4/3}) and O(h^2 + τ^{1+α^∗/(2+α^∗)}) for well-prepared (α^∗ = 1) and ill-prepared (0 ≤ α^∗ < 1) initial data, respectively, which are uniform in both space and time for 0 < ε ≤ 1 and optimal at the second order in space. Other techniques in the analysis include the cut-off technique for treating the nonlinearity and inverse estimates to bound the numerical solution. Numerical results are reported to demonstrate our error bounds.

In this paper, we develop an energy stable fully discrete discontinuous Galekin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.

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