We discuss differences between the exact<i>S</i>-matrix for scattering on serial structures and a known factorized expression constructed of single-element<i>S</i>-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities.

We develop a specification test for the transition density of a discretely-sampled continuous-time diffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts assumed by the null hypothesis. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are critically investigated via simulation studies and are compared with other tests. Our approaches are illustrated by applications to implied volatility data and Treasury bill data.

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Anderson mixing (AM) is a powerful acceleration method for fixed-point iterations, but its computation requires storing many historical iterations. The extra memory footprint can be prohibitive when solving high-dimensional problems in a resource-limited machine. To reduce the memory overhead, we propose a novel class of short-term recurrence AM methods (ST-AM). The ST-AM methods only store two previous iterations with cheap corrections. We prove that the basic version of ST-AM is equivalent to the full-memory AM in strongly convex quadratic optimization, and with minor changes it has local linear convergence for solving general nonlinear fixed-point problems. We further analyze the convergence properties of the regularized ST-AM for nonconvex (stochastic) optimization. Finally, we apply ST-AM to several applications including solving root-finding problems and training neural networks. Experimental results show that ST-AM is competitive with the long-memory AM and outperforms many existing optimizers.

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