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A double-toroidal graph is a graph with graph genus 2 (West 2000, p. 266). Planar and toroidal graphs are therefore not double-toroidal. Some known double-toroidal graphs on ...

Let (A,<=) and (B,<=) be disjoint totally ordered sets with order types alpha and beta. Then the ordinal sum is defined at set (C=A union B,<=) where, if c_1 and c_2 are both ...

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers ...

The direct product of the rings R_gamma, for gamma some index set I, is the set product_(gamma in I)R_gamma={f:I-> union _(gamma in I)R_gamma|f(gamma) in R_gamma all gamma in ...

The tangent plane to a surface at a point p is the tangent space at p (after translating to the origin). The elements of the tangent space are called tangent vectors, and ...

Given a complex measure mu, there exists a positive measure denoted |mu| which measures the total variation of mu, also sometimes called simply "total variation." In ...

Algebra

Determined the possible values of r and n for which there is an identity of the form (x_1^2+...+x_r^2)(y_1^2+...+y_r^2)=z_1^2+...+z_n^2.

For a sequence {a_n}, if a_(n+1)-a_n>0 for n>=x, then a_n is increasing for n>=x. Conversely, if a_(n+1)-a_n<0 for n>=x, then a_n is decreasing for n>=x. If a_n>0 and ...

The Tits group is a group of order 17971200. It is implemented in the Wolfram Language as TitsGroupT[].