Similarly, using a smooth partition of unity and a local change of variables, the result for a half space implies the existence of an analogous extending map
22.
In fact, a T 1 space is Hausdorff and paracompact if and only if it admits partitions of unity subordinate to any open cover ( see below ).
23.
These facts are particularly important because shrinking of open covers is a common technique in the theory of differential manifolds and while constructing functions using a partition of unity.
24.
Is an open cover of, then the calculation of the integral asymptotic is reduced to the case of a singe saddle point by employing the partition of unity.
25.
Linkwitz Riley filter is an example of practical implementation of partition of unity to separate input signal into two output signals containing only high-or low-frequency components.
26.
For a cohomology class represented by closed form in, express as a difference of forms via a partition of unity subordinate to the open cover } }, for example.
27.
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 ).
28.
In a paracompact Hausdorff space X \,, if \ mathcal { O } \, is an open cover, then there exists a partition of unity subordinate to it.
29.
More generally, the integral of over is defined as follows : Let be a partition of unity associated with a cover of ( consistently oriented ) coordinate charts, then define the integral
30.
From what has just been said, partitions of unity don't apply to holomorphic functions; their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory.
