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In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval [0,1] such that for every point : there is a neighbourhood of x {\displaystyle x} where all but a finite number of the functions of R {\displaystyle R} are 0, and
Partitions of unity are useful because they often allow one to extend local constructions to the whole space. For instance, the integral of differential forms on paracompact manifolds is first defined locally (where the manifold looks like Euclidean space and the integral is well known), and this definition is then extended to the whole space ...
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set, partition of a graph, partition of an integer, partition of an interval, partition of unity, partition of a matrix; see block matrix, and
The extended finite element method (XFEM), is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method (FEM) approach by enriching the solution space for solutions to differential equations with discontinuous functions.
Partition of unity A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1. Path
Smooth functions with given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence.
Then a partition of unity subordinate to the cover {U α} is a collection of real-valued C k functions φ i on M satisfying the following conditions: The supports of the φ i are compact and locally finite; The support of φ i is completely contained in U α for some α; The φ i sum to one at each point of M: () =
If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U. This shows the relationship of normal spaces to paracompactness. In fact, any space that satisfies any one of these three conditions must be normal. A product of normal spaces is not necessarily normal.