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The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory. The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups .
Manifold learning can draw a decision boundary between the natural classes of the unlabeled data, under the assumption that close-together points probably belong to the same class, and so the decision boundary should avoid areas with many unlabeled points. This is one version of semi-supervised learning.
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right.
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k) th homology group of M, for all integers k
In a sense, this means that there are k-dimensional holes in the complex. For example, consider the complex S obtained by gluing two triangles (with no interior) along one edge, shown in the image. The edges of each triangle can be oriented so as to form a cycle. These two cycles are by construction not boundaries (since every 2-chain is zero).
A Guide to the Classification Theorem for Compact Surfaces is a textbook in topology, on the classification of two-dimensional surfaces. It was written by Jean Gallier and Dianna Xu , and published in 2013 by Springer-Verlag as volume 9 of their Geometry and Computing series ( doi : 10.1007/978-3-642-34364-3 , ISBN 978-3-642-34363-6 ).
Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object. The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes:
The entity–control–boundary approach finds its origin in Ivar Jacobson's use-case–driven object-oriented software engineering (OOSE) method published in 1992. [1] [2] It was originally called entity–interface–control (EIC) but very quickly the term "boundary" replaced "interface" in order to avoid the potential confusion with object-oriented programming language terminology.