# partition of unity

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## partition of unity

[pär′tish·ən əv ′yü·nəd·ē]
(mathematics)
On a topological space X, this is a covering by open sets Uαwith continuous functions ƒαfrom X to [0,1], where each ƒαis zero on all but a finite number of the Uα, and the sum of all these ƒαat any point equals 1.
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In addition, those blocks satisfy the partition of unity for a given F, E, and V, respectively.
Whether it has explicitly defined a partition of unity function or not, most of the mesh-free methods can be categorized as a partition of unity method.
These five indexes verify the condition of partition of unity, namely:
There are many meshless methods, such as smoothed particle hydrodynamics method (SPH) [9, 10], the element-free Galerkin (EFG) method [11, 12], the collocation point method [13], the finite point method (FPM) [14, 15], reproducing kernel particle method (RKPM) [16], multiscale reproducing kernel particle method [17], the hp cloud method [18], the partition of unity finite element method [19, 20], generalized finite element method [21], meshless local Petrov-Galerkin method [22], least-square collocation meshless method [23], meshless weighted least-square method [24], complex variable meshless method (CVMM) [25, 26], and complex variable meshless manifold method (CVMMM) [27].
Buay Keke, a deputy principal for Kampala rebel based representative, rubbished partition of Unity state, referring to the commissioner's claim as lacking basis.
(a) Partition of unity: [[summation].sup.3.sub.i=0][B.sub.i](t; [alpha], [beta]) = 1.
Once the multiscale implicit functions for each subdomain are constructed, function values at arbitrary position in overall domain can be evaluated by partition of unity method that calculates a reconstructed function value by weighted sum of local functions for neighbor subdomains.
In the third step we use the family of cubes [{[K.sub.i]}.sub.i[member of]I] to construct a partition of unity and to define a near-minimizer [f.sub.t].
We recall the partition of unity method in Section 2 and the primal-dual active set algorithm in Section 3.
Some methods coupling with the partition of unity method (PUM) [36, 37] have also been investigated in mechanical problems in recent years.
Shape functions for meshless techniques need to satisfy certain conditions such as adherence to partition of unity ([[summation].sup.n.sub.i = 1] [[PHI].sub.I](x) = 1), compact domain of influence, and adapting to randomness of nodes, to name a few.
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