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The n×n square matrix F_n with entries given by F_(jk)=e^(2piijk/n)=omega^(jk) (1) for j,k=0, 1, 2, ..., n-1, where i is the imaginary number i=sqrt(-1), and normalized by ...

Given a sequence of values {a_k}_(k=1)^n, the high-water marks are the values at which the running maximum increases. For example, given a sequence (3,5,7,8,8,5,7,9,2,5) with ...

Given a homogeneous linear second-order ordinary differential equation, y^('')+P(x)y^'+Q(x)y=0, (1) call the two linearly independent solutions y_1(x) and y_2(x). Then ...

The abundancy of a number n is defined as the ratio sigma(n)/n, where sigma(n) is the divisor function. For n=1, 2, ..., the first few values are 1, 3/2, 4/3, 7/4, 6/5, 2, ...

An acyclic graph is a graph having no graph cycles. Acyclic graphs are bipartite. A connected acyclic graph is known as a tree, and a possibly disconnected acyclic graph is ...

Consider the process of taking a number, adding its digits, then adding the digits of the number derived from it, etc., until the remaining number has only one digit. The ...

(dy)/(dx)+p(x)y=q(x)y^n. (1) Let v=y^(1-n) for n!=1. Then (dv)/(dx)=(1-n)y^(-n)(dy)/(dx). (2) Rewriting (1) gives y^(-n)(dy)/(dx) = q(x)-p(x)y^(1-n) (3) = q(x)-vp(x). (4) ...

The entire function B(z) = [(sin(piz))/pi]^2[2/z+sum_(n=0)^(infty)1/((z-n)^2)-sum_(n=1)^(infty)1/((z+n)^2)] (1) = 1-(2sin^2(piz))/(pi^2z^2)[z^2psi_1(z)-z-1], (2) where ...

The quartic curve given by the implicit equation (x^2-a^2)(x-a)^2+(y^2-a^2)^2=0, (1) so-named because of its resemblance to a tooth. The bicuspid curve has cusps at (a,-a) ...

A binary quadratic form is a quadratic form in two variables having the form Q(x,y)=ax^2+2bxy+cy^2, (1) commonly denoted <a,b,c>. Consider a binary quadratic form with real ...