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Define O = lim_(->)O(n),F=R (1) U = lim_(->)U(n),F=C (2) Sp = lim_(->)Sp(n),F=H. (3) Then Omega^2BU = BU×Z (4) Omega^4BO = BSp×Z (5) Omega^4BSp = BO×Z. (6)

If X is any space, then there is a CW-complex Y and a map f:Y->X inducing isomorphisms on all homotopy, homology, and cohomology groups.

The radius of convergence of the Taylor series a_0+a_1z+a_2z^2+... is r=1/(lim_(n->infty)^_(|a_n|)^(1/n)).

The general displacement of a rigid body (or coordinate frame) with one point fixed is a rotation about some axis. Furthermore, a rotation may be described in any basis using ...

If it is possible to transform a coordinate system to a form where the metric elements g_(munu) are constants independent of x^mu, then the space is flat.

The curvature and torsion functions along a space curve determine it up to an orientation-preserving isometry.

If sets E and F are independent, then so are E and F^', where F^' is the complement of F (i.e., the set of all possible outcomes not contained in F). Let union denote "or" ...

Any finite semigroup is a divisor for an alternating wreath product of finite groups and semigroups.

In space, the only conformal mappings are inversions, similarity transformations, and congruence transformations. Or, restated, every angle-preserving transformation is a ...

The lengths of the tangents from a point P to a conic C are proportional to the cube roots of the radii of curvature of C at the corresponding points of contact.