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

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 standpoint, and exhibits the best way to clear up actual difficulties that experience topological features related to desktops. Such difficulties come up in lots of parts, resembling 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 quickly as much as fresh advances within the zone, together with topological endurance and hierarchical Morse complexes. Algorithms and knowledge constructions are provided whilst acceptable.

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If our original manifold X is compact, nothing “nasty” can happen, and an immersion F : X → Y is simply a local embedding. In other words, for any point p ∈ X, there exists a neighborhood U containing p such that F|U is an embedding. However, F need not be an embedding within the neighborhood of F(p) in Y. That is, immersed compact spaces may self-intersect. 3 Simplicial Complexes 23 1 1 2π π 0 0 0 -1 -2 0 -40 2 (a) F(t) -20 0 20 40 (b) g(t) -1 -2 0 2 ˆ = F(g(t)) (c) F(t) Fig. 6. Mapping of R into R2 with topological consequences.

We also assume the following: (i) Set S is made up of elements a ∈ S. (ii) There is only one empty set ∅. (iii) We may describe a set by characterizing it ({x | P(x)}) or by enumerating elements ({1, 2, 3}). Here P is a predicate. (iv) A set S is well defined if, for each object a, either a ∈ S or a ∈ S. Note that “well defined” really refers to the definition of a set, rather than to the set itself. |S| or card S is the size of the set. We may multiply sets in order to get larger sets. 1 (Cartesian) The Cartesian product of sets S1 , S2 , .

3 Group Theory Having examined the structure of the input to our computations in the last chapter, we now turn to developing the machinery we need for characterizing the topology of spaces. Recall that we are interested in classification systems. Group theory provides us with powerful tools to define equivalence relations using homomorphisms and factor groups. In the next chapter, we shall utilize these tools to define homology, a topological classification system. Unlike homeomorphy and homotopy, homology is discrete by nature.

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