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In this paper we extend the ideas of the so-called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higherdimensional spatial domains. For that effect we present a new set of general analytic estimates. These estimates are valid for any dimension and are used, together with rigorous computations, to construct a finite number of radii polynomials. These polynomials provide a computationally efficient method to prove, via a contraction argument, the existence and local uniqueness of solutions for a rather large class of nonlinear problems. We apply this technique to prove existence and local uniqueness of equilibrium solutions for the Cahn-Hilliard and the Swift-Hohenberg equations defined on two-and three-dimensional spatial domains.

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A method is described for quantitatively analyzing the level of interconnectivity of solid-oxide fuel cell electrode phases. The method was applied to the three-dimensional microstructure of a Ni-Y2O3-stabilized ZrO2 (Ni-YSZ) anode active layer measured by focused ion beam scanning electron microscopy. Each individual contiguous network of Ni, YSZ, and porosity was identified and labeled according to whether it was contiguous with the rest of the electrode. It was determined that the YSZ phase was 100% connected, whereas at least 86% of the Ni and 96% of the pores were connected. Triple-phase boundary (TPB) segments were identified and evaluated with respect to the contiguity of each of the three phases at their locations. It was found that 11.6% of the TPB length was on one or more isolated phases and hence was not electrochemically active.

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We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conley's topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications.

We introduce a technique based on algebraic topology for quantifying spatio-temporally chaotic dynamics. The technique is illustrated using the Gray-Scott and the FitzHugh-Nagumo models.