Topology for Computing (Cambridge Monographs on Applied and by Afra J. Zomorodian

By Afra J. Zomorodian

Written by way of a working laptop or computer scientist for computing device scientists, this ebook teaches topology from a computational viewpoint, and indicates how one can remedy actual difficulties that experience topological facets concerning desktops. Such difficulties come up in lots of components, equivalent to special effects, robotics, structural biology, and chemistry. the writer starts off from the fundamentals of topology, assuming no previous publicity to the topic, and strikes speedily as much as fresh advances within the sector, together with topological endurance and hierarchical Morse complexes. Algorithms and information constructions are offered whilst applicable.

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The dimension of the chart ϕ is d. The coordinate functions of the chart are xi = ui ◦ϕ : U → R, where ui : Rn → R are the standard coordinates on Rd . 20 2 Spaces and Filtrations ϕ −1 p’ U’ U p X d ϕ IR Fig. 3. A chart at p ∈ X. ϕ maps U ⊂ X containing p to U ⊆ Rd . As ϕ is a homeomorphism, ϕ−1 also exists and is continuous. 19 (Hausdorff) A topological space X is Hausdorff if, for every x, y ∈ X, x = y, there are neighborhoods U,V of x, y, respectively, such that U ∩ V = ∅. A metric space is always Hausdorff.

M} of the complex indices. A signature function is a map λ : [m] → R. 4 Alpha Shapes We have now seen the types of spaces that will be examined in this book, as well as their representation. What remains is the derivation of meaningful filtrations, encoding the geometry of the space in the ordering. 96 Angstroms 0 2 4 6 8 10 Separation (Angstroms) (a) The van der Waals force for two carbon atoms, as modeled by the Leonard-Jones potential function (b) Gramicidin A, a protein, modeled as the union of spheres with van der Waals radii Fig.

This format first describes the map that places the vertices in R3 . A vertex with location (x, y, z) ∈ R3 gets the line “v x y z” in the file. 31). The vertices are numbered according to their order in the file and numbered from 1. A triangle with vertices v1 , v2 , v3 is specified with line “f v1 v2 v3 ”. The description in an OBJ file is often called a “triangle soup,” as the topology is specified implicitly and must be extracted. 3 Subcomplexes Recall that a simplex is the power set of its simplices.

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