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All closed surfaces, despite their seemingly diverse forms, are topologically equivalent to spheres with some number of handles or cross-caps. The traditional proof follows ...

The grid shading problem is the problem of proving the unimodality of the sequence {a_1,a_2,...,a_(mn)}, where for fixed m and n, a_i is the number of partitions of i with at ...

If f(z) is continuous in a region D and satisfies ∮_gammafdz=0 for all closed contours gamma in D, then f(z) is analytic in D. Morera's theorem does not require simple ...

The regular polygon of 17 sides is called the heptadecagon, or sometimes the heptakaidecagon. Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is ...

Inference of the truth of an unknown result obtained by noting its similarity to a result already known to be true. In the hands of a skilled mathematician, analogy can be a ...

A symmetric block design (4n+3, 2n+1, n) which is equivalent to a Hadamard matrix of order 4n+4. It is conjectured that Hadamard designs exist for all integers n>0, but this ...

Let K be a field of field characteristic 0 (e.g., the rationals Q) and let {u_n} be a sequence of elements of K which satisfies a difference equation of the form ...

If J is a simple closed curve in R^2, the closure of one of the components of R^2-J is homeomorphic with the unit 2-ball. This theorem may be proved using the Riemann mapping ...

A Cantor set C in R^3 is said to be scrawny if for each neighborhood U of an arbitrary point p in C, there is a neighborhood V of p such that every map f:S^1->V subset C ...

A generalization of Ramsey theory to mathematical objects in which one would not normally expect structure to be found. For example, there exists a graph with very few ...