Partition of unity

To coarse-grain the Fukui function, consider an atomic partition of unity,... 

An alternative construction, termed fragment of molecular response [24], inserts the atomic partition of unity outside the derivative,... 

J. Fish, Z. Multiscale enrichment based on partition of unity for nonperiodic fields and nonlinear problem. Comp. Mech. 40, 249-259 (2007)... 

This equation is the algebraic analog of the partitions of unity which are so useful in differential geometry. The fact that one always has such an equation whenever one is given a covering of Spec (R) by distinguished open subsets is the reason for the cohomological triviality of affine schemes, (the so-called Theorems A and B - cf. Chapter 6). 

The expression for the overlap densities (88) has the same structure as in the non-relativistic FPLO approach. Hence, a redistribution between the two lattice sites involved can be obtained with the same technique using a partition of unity as discussed in Ref [1]. 

Note that the shape functions for any k in the MLS are partitions of unity. The view point used in partition of unity inspired researchers to develop several meshless methods such as the hp-clouds method and the partition of unity finite element method. 

If ll is not assumed to be locally finite, then we define a partition of unity subordinate to U to be a partition of unity for some locally finite refinement ofU. 

If X is paracompact, and lA is an open covering of X, then there exists a partition of unity that is subordinate to lA. Moreover, iflA is locally finite, then there exists a partition of unity for lA. 

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