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We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

We present a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.

In this paper, high order central finite difference schemes in a finite interval are analyzed for the diffusion equation. Boundary conditions of the initial-boundary value problem (IBVP) are treated by the simplified inverse Lax-Wendroff (SILW) procedure. For the fully discrete case, a third order explicit Runge-Kutta method is used as an example for the analysis. Stability is analyzed by both the GKS (Gustafsson, Kreiss and Sundstr\"om) theory and the eigenvalue visualization method on both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate and validate the analysis results.

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