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Given a set P with |P|=p elements consisting of c_1 numbers 1, c_2 numbers 2, ..., and c_n numbers n and c_1+c_2+...+c_n=p, find the number of permutations with k-1 rises ...

A formal extension of the hypergeometric function to two variables, resulting in four kinds of functions (Appell 1925; Picard 1880ab, 1881; Goursat 1882; Whittaker and Watson ...

The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The ...

If P be a point in the plane of an equilateral triangle DeltaABC, then the lengths of line segments AP, BP, and CP correspond the sides of a triangle, which is degenerate ...

Let z be defined as a function of w in terms of a parameter alpha by z=w+alphaphi(z). (1) Then Lagrange's inversion theorem, also called a Lagrange expansion, states that any ...

The arithmetic mean of a set of values is the quantity commonly called "the" mean or the average. Given a set of samples {x_i}, the arithmetic mean is x^_=1/Nsum_(i=1)^Nx_i. ...

Bowling, known as "ten pins" throughout most of the world, is a game played by rolling a heavy ball down a long narrow track and attempting to knock down ten pins arranged in ...

How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular ...

If f(z) is regular and of the form O(e^(k|z|)) where k<pi, for R[z]>=0, and if f(z)=0 for z=0, 1, ..., then f(z) is identically zero.

A discrete group is a topological group with the discrete topology. Often in practice, discrete groups arise as discrete subgroups of continuous Lie groups acting on a ...