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In mathematics, a polynomial decomposition expresses a polynomial f as the functional composition of polynomials g and h, where g and h have degree greater than 1; it is an algebraic functional decomposition. Algorithms are known for decomposing univariate polynomials in polynomial time.
Processes related to functional decomposition are prevalent throughout the fields of knowledge representation and machine learning.Hierarchical model induction techniques such as Logic circuit minimization, decision trees, grammatical inference, hierarchical clustering, and quadtree decomposition are all examples of function decomposition.
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.
In mathematics, more precisely in measure theory, the Lebesgue decomposition theorem [1] ... That is, the function is a Radon ...
The spectrum has a standard decomposition into three parts: a point spectrum , consisting of the eigenvalues of T {\displaystyle T} ; a continuous spectrum , consisting of the scalars that are not eigenvalues but make the range of T − λ {\displaystyle T-\lambda } a proper dense subset of the space;
Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex k-form can be decomposed uniquely into a sum of so-called (p, q)-forms: roughly, wedges of p differentials of the holomorphic coordinates with q differentials of their complex conjugates.
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.