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The Pólya enumeration theorem can be used to calculate the number of graphs up to isomorphism with a fixed number of vertices, or the generating function of these graphs according to the number of edges they have. For the latter purpose, we can say that a black or present edge has weight 1, while an absent or white edge has weight 0.
[4]: 23–24 The specific topics treated bear witness to the special interests of Pólya (Descartes' rule of signs, Pólya's enumeration theorem), Szegö (polynomials, trigonometric polynomials, and his own work in orthogonal polynomials) and sometimes both (the zeros of polynomials and analytic functions, complex analysis in general).
Pólya’s theorem can be used to construct an example of two random variables whose characteristic functions coincide over a finite interval but are different elsewhere. Pólya’s theorem. If is a real-valued, even, continuous function which satisfies the conditions =,
Bondy's theorem (graph theory, combinatorics) Bondy–Chvátal theorem (graph theory) Bonnet theorem (differential geometry) Boolean prime ideal theorem (mathematical logic) Borel–Bott–Weil theorem (representation theory) Borel–Carathéodory theorem (complex analysis) Borel–Weil theorem (representation theory) Borel determinacy theorem
For example, one can tell from looking at the graph that the point at −0.1 should have been at about −0.28. The way to do this in the algorithm is to use a single round of Newton's method . Since one knows the first and second derivatives of P ( x ) − f ( x ) , one can calculate approximately how far a test point has to be moved so that ...
Marko Riedel, Pólya's enumeration theorem and the symbolic method; Marko Riedel, Cycle indices of the set / multiset operator and the exponential formula; Harald Fripertinger (1997). "Cycle indices of linear, affine and projective groups". Linear Algebra and Its Applications. 263: 133– 156. doi: 10.1016/S0024-3795(96)00530-7. Harald ...
Burnside's lemma can compute the number of rotationally distinct colourings of the faces of a cube using three colours.. Let X be the set of 3 6 possible face color combinations that can be applied to a fixed cube, and let the rotation group G of the cube act on X by moving the colored faces: two colorings in X belong to the same orbit precisely when one is a rotation of the other.
Pakistan Institute of Nuclear Science & Technology-- Pakistan Mathematical Society-- Pakistan Statistical Society-- Palais–Smale compactness condition-- Palais theorem-- Palatini identity-- Paley construction-- Paley graph-- Paley–Wiener integral-- Paley–Wiener theorem-- Paley–Zygmund inequality-- Palindromic number-- Palindromic prime-- Palm calculus-- Palm–Khintchine theorem-- Pan ...