We study a system of semilinear hyperbolic equations passively advected by smooth white noise in time random velocity fields. Such a system arises in modelling non-premixed isothermal turbulent flames under single-step kinetics of fuel and oxidizer. We derive closed equations for one-point and multi-point probability distribution functions (PDFs) and closed-form analytical formulae for the one-point PDF function, as well as the two-point PDF function under homogeneity and isotropy. Exact solution formulae allow us to analyse the ensemble-averaged fuel/oxidizer concentrations and the motion of their level curves. We recover the empirical formulae of combustion in the thin reaction zone limit and show that these approximate formulae can either underestimate or overestimate average concentrations when the reaction zone is not tending to zero. We show that the averaged reaction rate slows down locally in

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.

This paper discusses two encryption schemes to fix the Square scheme. Square+ uses the Plus modification of appending randomly chosen polynomials. Double-Layer Square uses a construction similar to some signature schemes, splitting the variables into two layers, one of which depends on the other.

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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.