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  2. Partial fraction decomposition - Wikipedia

    en.wikipedia.org/wiki/Partial_fraction_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.

  3. Fredholm integral equation - Wikipedia

    en.wikipedia.org/wiki/Fredholm_integral_equation

    and the problem is, given the continuous kernel function and the function , to find the function .. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely (,) = (), and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions and and therefore ...

  4. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Decomposition: This is a version of Schur decomposition where and only contain real numbers. One can always write A = V S V T {\displaystyle A=VSV^{\mathsf {T}}} where V is a real orthogonal matrix , V T {\displaystyle V^{\mathsf {T}}} is the transpose of V , and S is a block upper triangular matrix called the real Schur form .

  5. Rank factorization - Wikipedia

    en.wikipedia.org/wiki/Rank_factorization

    Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .

  6. Bruhat decomposition - Wikipedia

    en.wikipedia.org/wiki/Bruhat_decomposition

    In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) = of certain algebraic groups = into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases.

  7. Lebesgue's decomposition theorem - Wikipedia

    en.wikipedia.org/wiki/Lebesgue's_decomposition...

    Lebesgue's decomposition theorem can be refined in a number of ways. First, as the Lebesgue-Radon-Nikodym theorem . That is, let ( Ω , Σ ) {\displaystyle (\Omega ,\Sigma )} be a measure space, μ {\displaystyle \mu } a σ-finite positive measure on Σ {\displaystyle \Sigma } and λ {\displaystyle \lambda } a complex measure on Σ ...

  8. Primary decomposition - Wikipedia

    en.wikipedia.org/wiki/Primary_decomposition

    In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).

  9. Banach–Tarski paradox - Wikipedia

    en.wikipedia.org/wiki/Banach–Tarski_paradox

    "Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?" The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different ...

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