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A point p on a regular surface M in R^3 is said to be planar if the Gaussian curvature K(p)=0 and S(p)=0 (where S is the shape operator), or equivalently, both of the ...

A number r is an nth root of unity if r^n=1 and a primitive nth root of unity if, in addition, n is the smallest integer of k=1, ..., n for which r^k=1.

The semimajor axis (also called the major semi-axis, major semiaxis, or major radius) of an ellipse (or related figure) is half its extent along the longer of the two ...

The semiminor axis (also called the minor semi-axis, minor semiaxis, or minor radius) of an ellipse (or related figure) is half its extent along the shorter of the two ...

A simple pole of an analytic function f is a pole of order one. That is, (z-z_0)f(z) is an analytic function at the pole z=z_0. Alternatively, its principal part is c/(z-z_0) ...

On an algebraic curve, the sum of the number of coincidences at a noncuspidal point C is the sum of the orders of the infinitesimal distances from a nearby point P to the ...

Differential Geometry

Take K a number field and m a divisor of K. A congruence subgroup H is defined as a subgroup of the group of all fractional ideals relative prime to m (I_K^m) that contains ...

Given a number z, the cube root of z, denoted RadicalBox[z, 3] or z^(1/3) (z to the 1/3 power), is a number a such that a^3=z. The cube root is therefore an nth root with ...

The first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic. Its discovery was a shock to the ...