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An Aztec diamond of order n is the region obtained from four staircase shapes of height n by gluing them together along the straight edges. It can therefore be defined as the ...

Let Gamma(z) be the gamma function and n!! denote a double factorial, then [(Gamma(m+1/2))/(Gamma(m))]^2[1/m+(1/2)^21/(m+1)+((1·3)/(2·4))^21/(m+2)+...]_()_(n) ...

If C_1, C_2, ...C_r are sets of positive integers and union _(i=1)^rC_i=Z^+, then some C_i contains arbitrarily long arithmetic progressions. The conjecture was proved by van ...

The Bernoulli inequality states (1+x)^n>1+nx, (1) where x>-1!=0 is a real number and n>1 an integer. This inequality can be proven by taking a Maclaurin series of (1+x)^n, ...

The blow-up lemma essentially says that regular pairs in Szemerédi's regularity lemma behave like complete bipartite graphs from the point of view of embedding bounded degree ...

Any continuous function G:B^n->B^n has a fixed point, where B^n={x in R^n:x_1^2+...+x_n^2<=1} is the unit n-ball.

Calabi's triangle is the unique triangle other that the equilateral triangle for which the largest inscribed square can be inscribed in three different ways (Calabi 1997). ...

The symbol intersection , used for the intersection of sets, and sometimes also for the logical connective AND instead of the symbol ^ (wedge). In fact, for any two sets A ...

According to Pólya, the Cartesian pattern is the resolution method for arithmetical or geometrical problems based on equations. The first step is to translate the question ...

If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the ...