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A 4-sphere has positive curvature, with R^2=x^2+y^2+z^2+w^2 (1) 2x(dx)/(dw)+2y(dy)/(dw)+2z(dz)/(dw)+2w=0. (2) Since r=xx^^+yy^^+zz^^ (3) dw = -(xdx+ydy+zdz)/w (4) = ...

There are 14 families of spherical groups, which can be written in orbifold notation as *532, 532, *432, 432, *332, 3*2, 332, *22N, 2*N, 22N, *NN, N*, Nx, and NN.

A method of stochastic optimization including techniques such as gradient search or Robbins-Monro stochastic approximation.

A C^infty (infinitely differentiable) manifold is said to be a submanifold of a C^infty manifold M^' if M is a subset of M^' and the identity map of M into M^' is an ...

psi=(A_0+alpha_1A_1+alpha_2A_2+...)e^(iS/alpha).

A specific type of ultraproduct that can be used to construct nonstandard universes and obtain the transfer principle as a corollary of Łoś' theorem for ultraproducts.

A set of sample problems in unconstrained optimization is given by loading Optimization`UnconstrainedProblems` and evaluating $FindMinimumProblems.

Solving the wave equation on a disk gives a solution in terms of Bessel functions.

A function w(x) used to normalize orthogonal functions int[f_n(x)]^2w(x)dx=N_n.

A method of constructing uniform polyhedra.