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The Pólya enumeration theorem, also known as the Redfield–Pólya theorem and Pólya counting, is a theorem in combinatorics that both follows from and ultimately generalizes Burnside's lemma on the number of orbits of a group action on a set. The theorem was first published by J. Howard Redfield in 1927.
The book was unique at the time because of its arrangement, less by topic and more by method of solution, so arranged in order to build up the student's problem-solving abilities. The preface of the book contains some remarks on general problem solving and mathematical heuristics which anticipate Pólya's later works on that subject ...
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 =,
The Hermite or Pólya class is a set of entire functions satisfying the requirement that if E(z) is in the class, then: [1] [2]. E(z) has no zero in the upper half-plane. | (+) | | | for x and y real and y positive.
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.
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 ...
Cases of tularemia, also known as “rabbit fever," are on the rise in the U.S., according to a new report from the CDC. The report identifies symptoms and the groups most at risk.
The monodromy theorem gives a sufficient condition for the existence of a direct analytic continuation (i.e., an extension of an analytic function to an analytic function on a bigger set). Suppose D ⊂ C {\displaystyle D\subset \mathbb {C} } is an open set and f an analytic function on D .