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We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homol-ogy computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that… (More)
Two implementations of a homology algorithm based on the Forman's discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms.
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,… (More)
We present an efficient algorithm for constructing piecewise constant Lyapunov functions for dynamics generated by a continuous nonlinear map defined on a compact metric space. We provide a memory efficient data structure for storing nonuniform grids on which the Lyapunov function is defined and give bounds on the complexity of the algorithm for both time… (More)
We show how a graph algorithm for finding matching labeled paths in pairs of labeled directed graphs can be used to perform model validation for a class of dynamical systems including regulatory network models of relevance to systems biology. In particular, we extract a partial order of events describing local minima and local maxima of observed quantities… (More)