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For n>=1, let u and v be integers with u>v>0 such that the Euclidean algorithm applied to u and v requires exactly n division steps and such that u is as small as possible ...

The space join of a topological space X and a pair of points S^0, Sigma(X)=X*S^0.

The space of immersions of a manifold in another manifold is homotopically equivalent to the space of bundle injections from the tangent space of the first to the tangent ...

A topology induced by the metric g defined on a metric space X. The open sets are all subsets that can be realized as the unions of open balls B(x_0,r)={x in X|g(x_0,x)<r}, ...

The first practical algorithm for determining if there exist integers a_i for given real numbers x_i such that a_1x_1+a_2x_2+...+a_nx_n=0, or else establish bounds within ...

In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let A, B, and C be the basis vectors defining a three-dimensional parallelepiped. Then ...

A point is a 0-dimensional mathematical object which can be specified in n-dimensional space using an n-tuple (x_1, x_2, ..., x_n) consisting of n coordinates. In dimensions ...

A Borel set is an element of a Borel sigma-algebra. Roughly speaking, Borel sets are the sets that can be constructed from open or closed sets by repeatedly taking countable ...

The roots of a semisimple Lie algebra g are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely ...

Given a subspace A of a space X and a map from A to a space Y, is it possible to extend that map to a map from X to Y?