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The polyhedral formula generalized to a surface of genus g, V-E+F=chi(g) where V is the number of polyhedron vertices, E is the number of polyhedron edges, F is the number of ...

A nonsimply connected 3-manifold, also called a dodecahedral space.

Let {y^k} be a set of orthonormal vectors with k=1, 2, ..., K, such that the inner product (y^k,y^k)=1. Then set x=sum_(k=1)^Ku_ky^k (1) so that for any square matrix A for ...

Solutions to holomorphic differential equations are themselves holomorphic functions of time, initial conditions, and parameters.

Poinsot's spirals are the two polar curves with equations r = acsch(ntheta) (1) r = asech(ntheta). (2)

To find the minimum distance between a point in the plane (x_0,y_0) and a quadratic plane curve y=a_0+a_1x+a_2x^2, (1) note that the square of the distance is r^2 = ...

The point-slope form of a line through the point (x_1,y_1) with slope m in the Cartesian plane is given by y-y_1=m(x-x_1).

Members of a coaxal system satisfy x^2+y^2+2lambdax+c=(x+lambda)^2+y^2+c-lambda^2=0 for values of lambda. Picking lambda^2=c then gives the two circles (x+/-sqrt(c))^2+y^2=0 ...

If (X,x) and (Y,y) are pointed spaces, a pointed map is a continuous map F:X->Y with the additional requirement that F(x)=y.

D_P(x)=lim_(epsilon->0)(lnmu(B_epsilon(x)))/(lnepsilon), where B_epsilon(x) is an n-dimensional ball of radius epsilon centered at x and mu is the probability measure.