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The prime number theorem gives an asymptotic form for the prime counting function pi(n), which counts the number of primes less than some integer n. Legendre (1808) suggested ...

A run is a sequence of more than one consecutive identical outcomes, also known as a clump. Let R_p(r,n) be the probability that a run of r or more consecutive heads appears ...

A graph corresponding to the skeleton of one of the Archimedean solids. There are 13 Archimedean graphs, all of which are regular, planar, polyhedral, and Hamiltonian. The ...

Continuum percolation can be thought of as a continuous, uncountable version of percolation theory-a theory which, in its most studied form, takes place on a discrete, ...

Taking a connected graph or network with a high graph diameter and adding a very small number of edges randomly, the diameter tends to drop drastically. This is known as the ...

The blossom algorithm (Edmonds 1965) finds a maximum independent edge set in a (possibly weighted) graph. While a maximum independent edge set can be found fairly easily for ...

The Buffon-Laplace needle problem asks to find the probability P(l,a,b) that a needle of length l will land on at least one line, given a floor with a grid of equally spaced ...

A formula for the Bell polynomial and Bell numbers. The general formula states that B_n(x)=e^(-x)sum_(k=0)^infty(k^n)/(k!)x^k, (1) where B_n(x) is a Bell polynomial (Roman ...

An extended rooted binary tree satisfying the following conditions: 1. Every node has two children, each colored either red or black. 2. Every tree leaf node is colored ...

A leaf of an unrooted tree is a node of vertex degree 1. Note that for a rooted or planted tree, the root vertex is generally not considered a leaf node, whereas all other ...