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We propose several statistics to test the Markov hypothesis for -mixing stationary processes sampled at discrete time intervals. Our tests are based on the ChapmanKolmogorov equation. We establish the asymptotic null distributions of the proposed test statistics, showing that Wilkss phenomenon holds. We compute the power of the test and provide simulations to investigate the finite sample performance of the test statistics when the null model is a diffusion process, with alternatives consisting of models with a stochastic mean reversion level, stochastic volatility and jumps.

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.

The SimpleMatrix encryption scheme as proposed by Tao et al. \cite{TD13} is one of the very few existing approaches to create a secure and efficient encryption scheme on the basis of multivariate polynomials. However, in its basic version, decryption failures occur with non-negligible probability. Although this problem has been addressed in several papers \cite{DP14,TX15}, a general solution to it is still missing. In this paper we propose an improved version of the SimpleMatrix scheme, which eliminates decryption failures completely and therefore solves the biggest problem of the SimpleMatrix encryption scheme. Additionally, we propose a second version of the scheme, which reduces the blow-up factor between plain and ciphertext size to a value arbitrary close to 1.

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