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Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds. Before stating Thurston's geometrization conjecture in ...

A permutation cycle is a subset of a permutation whose elements trade places with one another. Permutations cycles are called "orbits" by Comtet (1974, p. 256). For example, ...

First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of ...

For any prime number p and any positive integer n, the p^n-rank r_(p^n)(G) of a finitely generated Abelian group G is the number of copies of the cyclic group Z_(p^n) ...

Let L_n be the n×n matrix whose (i,j)th entry is 1 if j divides i and 0 otherwise, let Phi_n be the n×n diagonal matrix diag(phi(1),phi(2),...,phi(n)), where phi(n) is the ...

Tarski's theorem says that the first-order theory of reals with +, *, =, and > allows quantifier elimination. Algorithmic quantifier elimination implies decidability assuming ...

A distribution with zero kurtosis excess, i.e., gamma_2=0.

A unigraphic graph (or simply a "unigraph") is a graph that is isomorphic to every graph having that degree sequence. All graphs on four are fewer vertices are unigraphic. ...

The ideal generated by a set in a vector space.

An impossible hexnut modeled after the impossible torus. The figure above shows three impossible figures: the ambihelical hexnut in the lower left-hand corner, tribox in the ...