Szegő's First Limit Theorem provides the limiting statistical distribution (LSD) of the eigenvalues of large Toeplitz matrices. Szegő's Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the first and strong limit theorems to Kac-Murdock-Szegő (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature.
We present a Hermite interpolation based partial differential equation solver for Hamilton-Jacobi equations. Many Hamilton-Jacobi equations have a nonlinear dependency on the gradient of the solution, which gives rise to discontinuities in the gradient of the solution, resulting in kinks in the solution itself. We built our solver with two goals in mind: 1) high order accuracy in smooth regions and 2) sharp resolution of kinks. To achieve this, we use Hermite interpolation with a smoothness sensor. The degrees-of-freedom of Hermite methods are tensorproduct Taylor polynomials of degree m in each coordinate direction. The method uses (m + 1) d degrees of freedom per node in d-dimensions and achieves an order of accuracy (2m + 1) when the solution is smooth. To obtain sharp resolution of kinks, we sense the smoothness of the solution on each cell at each timestep. If the solution is smooth, we march the interpolant forward in time with no modifications. When our method encounters a cell over which the solution is not smooth, it introduces artificial viscosity locally while proceeding normally in smooth regions. We show through numerical experiments that the solver sharply captures kinks once the solution losses continuity in the derivative while achieving 2m + 1 order accuracy in smooth regions.
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