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Define the correlation integral as C(epsilon)=lim_(N->infty)1/(N^2)sum_(i,j=1; i!=j)^inftyH(epsilon-|x_i-x_j|), (1) where H is the Heaviside step function. When the below ...

A standard basis, also called a natural basis, is a special orthonormal vector basis in which each basis vector has a single nonzero entry with value 1. In n-dimensional ...

A sequence of n-tuples that fills n-space more uniformly than uncorrelated random points, sometimes also called a low-discrepancy sequence. Although the ordinary uniform ...

Watson (1939) considered the following three triple integrals, I_1 = 1/(pi^3)int_0^piint_0^piint_0^pi(dudvdw)/(1-cosucosvcosw) (1) = (4[K(1/2sqrt(2))]^2)/(pi^2) (2) = ...

Let a set of random variates X_1, X_2, ..., X_n have a probability function P(X_1=x_1,...,X_n=x_n)=(N!)/(product_(i=1)^(n)x_i!)product_(i=1)^ntheta_i^(x_i) (1) where x_i are ...

In random number computation, a seed is an initial number used as the starting point in a random number generating algorithm. In a dynamical system, a seed is a starting ...

F_k[P_N(k)](x)=F_k[exp(-N|k|^beta)](x), where F is the Fourier transform of the probability P_N(k) for N-step addition of random variables. Lévy showed that beta in (0,2) for ...

The adjective "affine" indicates everything that is related to the geometry of affine spaces. A coordinate system for the n-dimensional affine space R^n is determined by any ...

The average distance between two points chosen at random inside a unit cube (the n=3 case of hypercube line picking), sometimes known as the Robbins constant, is Delta(3) = ...

A symmetric polynomial on n variables x_1, ..., x_n (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other ...