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The inverse limit of a family of R-modules is the dual notion of a direct limit and is characterized by the following mapping property. For a directed set I and a family of ...

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups ...

The series for the inverse tangent, tan^(-1)x=x-1/3x^3+1/5x^5+.... Plugging in x=1 gives Gregory's formula 1/4pi=1-1/3+1/5-1/7+1/9-.... This series is intimately connected ...

Consider the Lagrange interpolating polynomial f(x)=b_0+(x-1)(b_1+(x-2)(b_3+(x-3)+...)) (1) through the points (n,p_n), where p_n is the nth prime. For the first few points, ...

The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar ...

Given a sequence {a_i}_(i=1)^N, an n-moving average is a new sequence {s_i}_(i=1)^(N-n+1) defined from the a_i by taking the arithmetic mean of subsequences of n terms, ...

Let generalized hypergeometric function _pF_q[alpha_1,alpha_2,...,alpha_p; beta_1,beta_2,...,beta_q;z] (1) have p=q+1. Then the generalized hypergeometric function is said to ...

The ratio X/Y of independent normally distributed variates with zero mean is distributed with a Cauchy distribution. This can be seen as follows. Let X and Y both have mean 0 ...

In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors v ...

A piecewise linear function is a function composed of some number of linear segments defined over an equal number of intervals, usually of equal size. For example, consider ...