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f(x)=1-2x^2 for x in [-1,1]. Fixed points occur at x=-1, 1/2, and order 2 fixed points at x=(1+/-sqrt(5))/4. The natural invariant of the map is rho(y)=1/(pisqrt(1-y^2)).

Two open manifolds M and M^' are cobordant if there exists a manifold with boundary W^(n+1) such that an acceptable restrictive relationship holds.

The geometry of Finsler space.

If the average number of envelopes per pigeonhole is a, then some pigeonhole will have at least a envelopes. Similarly, there must be a pigeonhole with at most a envelopes.

A polynomial function of the elements of a vector x can be uniquely decomposed into a sum of harmonic polynomials times powers of |x|.

S_n=sum_(i)eta_imu(E_i), where mu(E_i) is the measure of the set E_i of points on the x-axis for which f(x) approx eta_i.

An 80-sided polygon.

A parameterization of a surface x(u,v) in u and v is regular if the tangent vectors (partialx)/(partialu) and (partialx)/(partialv) are always linearly independent.

If A is true unless B, then not-B implies A, but B does not necessarily imply not-A.

A finite division algebra is a field.