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Piecewise Linear (other)
Piecewise linear may refer to: * Piecewise linear curve, a connected sequence of line segments * Piecewise linear function, a function whose domain can be decomposed into pieces on which the function is linear * Piecewise linear manifold, a topological space formed by gluing together flat spaces * Piecewise linear homeomorphism, a topological equivalence between two piecewise linear manifolds * Piecewise linear cobordism, a cohomology theory * Piecewise linear continuation Simplicial continuation, or piecewise linear continuation (Allgower and Georg),Eugene L. Allgower, K. Georg, "Introduction to Numerical Continuation Methods", ''SIAM Classics in Applied Mathematics'' 45, 2003.E. L. Allgower, K. Georg, "Simplicial an ..., a method for approximating functions by piecewise linear functions {{mathdab [Baidu] |
Piecewise Linear Curve
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. Name A polygonal chain may also be called a polygonal curve, polygonal path, polyline,. piecewise linear curve, broken line or, in geographic information systems, a linestring or linear ring. Variations A simple polygonal chain is one in which only consecutive (or the first and the last) segments intersect and only at their endpoints. A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Manifold
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Homeomorphism
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Cobordism
This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at the end of this article. Notation *''S'' = π = ''S''''0'' is the sphere spectrum. *''S''''n'' is the spectrum of the ''n''-dimensional sphere *''S''''n''''Y'' = ''S''''n''∧''Y'' is the ''n''th suspension of a spectrum ''Y''. * 'X'',''Y''is the abelian group of morphisms from the spectrum ''X'' to the spectrum ''Y'', given (roughly) as homotopy classes of maps. * 'X'',''Y''sub>''n'' = 'S''''n''''X'',''Y''* 'X'',''Y''sub>''*'' is the graded abelian group given as the sum of the groups 'X'',''Y''sub>''n''. *π''n''(''X'') = 'S''''n'', ''X''= 'S'', ''X''sub>''n'' is the ''n''th stable homotopy group of ''X''. *π''*''(''X'') is the sum of the groups π''n''(''X''), and is called the coefficient ring of ''X'' when ''X'' is a rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
