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A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, ..., n^2 arranged such that the sum of the n numbers in any horizontal, ...

Percolation, the fundamental notion at the heart of percolation theory, is a difficult idea to define precisely though it is quite easy to describe qualitatively. From the ...

An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r_1 and r_2 from two fixed points F_1 and F_2 (the foci) separated by a distance ...

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 ...

Cantellation, also known as (polyhedron) expansion (Stott 1910, not to be confused with general geometric expansion) is the process of radially displacing the edges or faces ...

The clique polynomial C_G(x) for the graph G is defined as the polynomial C_G(x)=1+sum_(k=1)^(omega(G))c_kx^k, (1) where omega(G) is the clique number of G, the coefficient ...

Let the divisor function d(n) be the number of divisors of n (including n itself). For a prime p, d(p)=2. In general, sum_(k=1)^nd(k)=nlnn+(2gamma-1)n+O(n^theta), where gamma ...

A function built up of a finite combination of constant functions, field operations (addition, multiplication, division, and root extractions--the elementary operations)--and ...

The study of the probabilities involved in geometric problems, e.g., the distributions of length, area, volume, etc. for geometric objects under stated conditions. The ...

Gram-Schmidt orthogonalization, also called the Gram-Schmidt process, is a procedure which takes a nonorthogonal set of linearly independent functions and constructs an ...