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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 ...

Let M be a finitely generated module over a commutative Noetherian ring R. Then there exists a finite set {N_i|1<=i<=l} of submodules of M such that 1. intersection ...

Let V and W be vector spaces over a field F, and let T:V->W be a linear transformation. Assuming the dimension of V is finite, then dim(V)=dim(Ker(T))+dim(Im(T)), where ...

The space of immersions of a manifold in another manifold is homotopically equivalent to the space of bundle injections from the tangent space of the first to the tangent ...

A special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles (also called the kissing circles problem). There are two ...

As originally stated by Gould (1972), GCD{(n-1; k),(n; k-1),(n+1; k+1)} =GCD{(n-1; k-1),(n; k+1),(n+1; k)}, (1) where GCD is the greatest common divisor and (n; k) is a ...

The general orthogonal group GO_n(q,F) is the subgroup of all elements of the projective general linear group that fix the particular nonsingular quadratic form F. The ...