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Polygon decomposition is applied in several areas: [1] Pattern recognition techniques extract information from an object in order to describe, identify or classify it. An established strategy for recognising a general polygonal object is to decompose it into simpler components, then identify the components and their interrelationships and use this information to determine the shape of the object.
This is an important technique for all types of time series analysis, especially for seasonal adjustment. [2] It seeks to construct, from an observed time series, a number of component series (that could be used to reconstruct the original by additions or multiplications) where each of these has a certain characteristic or type of behavior.
The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials. Joseph Ritt proved that m = n {\displaystyle m=n} , and the degrees of the components are the same up to linear transformations, but possibly in different order; this is Ritt's polynomial ...
Contrary to the connected components, the modules of a graph are the same as the modules of its complement, and modules can be "nested": one module can be a proper subset of another. Note that the set of vertices of a graph is a module, as are its one-element subsets and the empty set; these are called the trivial modules. A graph may or may ...
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If two matrices of order n can be multiplied in time M(n), where M(n) ≥ n a for some a > 2, then an LU decomposition can be computed in time O(M(n)). [16] This means, for example, that an O(n 2.376) algorithm exists based on the Coppersmith–Winograd algorithm.
One can always write = where V is a real orthogonal matrix, is the transpose of V, and S is a block upper triangular matrix called the real Schur form. The blocks on the diagonal of S are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from complex conjugate eigenvalue pairs).
Decomposition of time series, a statistical task that deconstructs a time series into several components; Doob decomposition theorem of an integrable, discrete-time stochastic process; Doob–Meyer decomposition theorem of a continuous-time sub- or supermartingale