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Let X and Y be CW-complexes, and let f:X->Y be a continuous map. Then the cellular approximation theorem states that any such f is homotopic to a cellular map. In fact, if ...

A theorem which effectively describes how lengths, areas, volumes, and generalized n-dimensional volumes (contents) are distorted by differentiable functions. In particular, ...

For every positive integer n, there exists a circle which contains exactly n lattice points in its interior. H. Steinhaus proved that for every positive integer n, there ...

Elder's theorem is a generalization of Stanley's theorem which states that the total number of occurrences of an integer k among all unordered partitions of n is equal to the ...

If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the existence of ...

If a function f(x) is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. If f(x) has an extremum on an open interval (a,b), then the ...

Let the squares square ABCD and square AB^'C^'D^' share a common polygon vertex A. The midpoints Q and S of the segments B^'D and BD^' together with the centers of the ...

The fourth group isomorphism theorem, also called the lattice group isomorphism theorem, lets G be a group and let N⊴G, where N⊴G indicates that N is a normal subgroup of G. ...

The Gershgorin circle theorem (where "Gershgorin" is sometimes also spelled "Gersgorin" or "Gerschgorin") identifies a region in the complex plane that contains all the ...

The Hadwiger conjecture is a generalization of the four-color theorem which states that for any loopless graph G with h(G) the Hadwiger number and chi(G) the chromatic ...