In this paper, we develop and analyze the Runge-Kutta discontinuous Galerkin (RKDG) method to solve weakly coupled hyperbolic multi-domain problems. Such problems involve transfer type boundary conditions with discontinuous fluxes between different domains, calling for special techniques to prove stability of the RKDG methods. We prove both stability and error estimates for our RKDG methods on simple models, and then apply them to a biological cell proliferation model \cite{EMSC}. Numerical results are provided to illustrate the good behavior of our RKDG methods.

Errors-in-variables regression is the study of the association between covariates and responses where covariates are observed with errors. In this paper, we consider the estimation of multivariate regression functions for dependent data with errors in covariates. Nonparametric deconvolution technique is used to account for errors-in-variables. The asymptotic behavior of regression estimators depends on the smoothness of the error distributions, which are characterized as either ordinarily smooth or super smooth. Asymptotic normality is established for both strongly mixing and -mixing processes, when the error distribution function is either ordinarily smooth or super smooth.

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We study harmonic maps from an admissible flat simplicial complex to a non-positively curved Riemannian manifold. Our main regularity theorem is that these maps are C1, at the interfaces of the top-dimensional simplices in addition to satisfying a balancing condition. If we assume that the domain is a 2-complex,then these maps are C1. As an application, we show that the regularity, the balancing condition and a Bochner formula lead to rigidity and vanishing theorems for harmonic maps. Furthermore, we give an explicit relationship between our techniques and those obtained via combinatorial methods.

Let M 4n be a complete quaternionic K¨ahler manifold with scalar curvature bounded below by .16n(n + 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.