Motivated by recent work on studying massive imaging data in various neuroimaging studies, we propose a novel spatially varying coefficient model (SVCM) to capture the varying association between imaging measures in a three-dimensional volume (or two-dimensional surface) with a set of covariates. Two stylized features of neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations. To specifically account for these two features, SVCM includes a measurement model with multiple varying coefficient functions, a jumping surface model for each varying coefficient function, and a functional principal component model. We develop a three-stage estimation procedure to simultaneously estimate the varying coefficient functions and the spatial correlations. The estimation procedure includes a fast multiscale adaptive estimation and

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be Hlder or Dini continuous in the time variable and all but one spatial variable. This type of system arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanatos approach in a novel way. Non-divergence type equations under similar conditions are also discussed.

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold M may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram
filling a word that represents the identity in a finitely presented group 􀀀 can either be measured intrinsically in the 1–skeleton of
or extrinsically in the Cayley graph of 􀀀. We construct the first examples of closed manifolds M and finitely presented groups 􀀀 = π1M for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min–diameter filling functions.

In this paper, we provide a priori $L^2$ error estimates for the semi-discrete discontinuous Galerkin method and the local discontinuous Galerkin method for one- and two-dimensional nonlinear Hamilton-Jacobi equations with smooth solutions. With a special Gauss-Radau projection, the optimal error estimates on rectangular meshes are obtained.

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