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A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us in this study. On the other hand, some very general theories have been developed for the study of smooth manifolds. One of the most powerful, and useful, of(More)
x0. Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices x 0 , x 1 ; : : :; x n , and M a subcomplex of S, known to both the hider and the seeker. Let be a simplex of S, known only to the hider. The seeker is permitted to ask questions of the sort \Is vertex x i in ?" The(More)
A variety of questions in combinatorics lead one to the task of analyzing the topology of a simplicial complex, or a more general cell complex. However, there are few general techniques to aid in this investigation. On the other hand, the subjects of differential topology and geometry are devoted to precisely this sort of problem, except that the(More)
In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which(More)
During anterior-posterior axis specification in the Drosophila embryo, the Hunchback (Hb) protein forms a sharp boundary at the mid-point of the embryo with great positional precision. While Bicoid (Bcd) is a known upstream regulator for hb expression, there is evidence to suggest that Hb effectively filters out "noisy" data received from varied Bcd(More)
In classical Morse theory 23], one begins with a smooth manifold M and a smooth function on M: The main point is a very precise relationship between the critical points of the function and the topology of M: In 24, 25], Novikov introduced a generalization of this theory. Novikov's theory begins with a closed 1-form ! on M: Locally we can write ! = df for(More)
A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated order complex. However, there are few general techniques to aid in this investigation. On(More)
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