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A module M is Noetherian if it obeys the ascending chain condition with respect to inclusion, i.e., if every set of increasing sequences of submodules eventually becomes ...

A null function delta^0(x) satisfies int_a^bdelta^0(x)dx=0 (1) for all a,b, so int_(-infty)^infty|delta^0(x)|dx=0. (2) Like a delta function, they satisfy delta^0(x)={0 x!=0; ...

Let K be a number field of extension degree d over Q. Then an order O of K is a subring of the ring of integers of K with d generators over Z, including 1. The ring of ...

There are a number of attractive polyhedron compounds consisting of ten octahedra. These compounds will be implemented in a future version of the Wolfram Language as ...

There are a number of attractive polyhedron compounds consisting of 25 octahedra. These compounds will be implemented in a future version of the Wolfram Language as ...

There are a number of attractive polyhedron compounds consisting of six octahedra. The two illustrated above can be constructed as the duals of cube 6-compounds. These ...

An n-dimensional open ball of radius r is the collection of points of distance less than r from a fixed point in Euclidean n-space. Explicitly, the open ball with center x ...

A pair of functions phi_i(x) and phi_j(x) are orthonormal if they are orthogonal and each normalized so that int_a^b[phi_i(x)]^2w(x)dx = 1 (1) int_a^b[phi_j(x)]^2w(x)dx = 1. ...

The Pappus spiral is the name given to the conical spiral with parametric equations x(t) = asin(alphat)cost (1) y(t) = asin(alphat)sint (2) x(t) = acos(alphat) (3) by ...

Each subsequent row of Pascal's triangle is obtained by adding the two entries diagonally above. This follows immediately from the binomial coefficient identity (n; r) = ...