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For any ordinal number alpha, the successor of alpha is alpha union {alpha} (Ciesielski 1997, p. 46). The successor of an ordinal number alpha is therefore the next ordinal, ...

An arc-transitive graph, sometimes also called a flag-transitive graph, is a graph whose graph automorphism group acts transitively on its graph arcs (Godsil and Royle 2001, ...

Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. The Weierstrass elliptic ...

The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not ...

In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice ...

A derangement is a permutation in which none of the objects appear in their "natural" (i.e., ordered) place. For example, the only derangements of {1,2,3} are {2,3,1} and ...

Let Pi be a permutation of n elements, and let alpha_i be the number of permutation cycles of length i in this permutation. Picking Pi at random, it turns out that ...

The Laguerre polynomials are solutions L_n(x) to the Laguerre differential equation with nu=0. They are illustrated above for x in [0,1] and n=1, 2, ..., 5, and implemented ...

A rooted tree is a tree in which a special ("labeled") node is singled out. This node is called the "root" or (less commonly) "eve" of the tree. Rooted trees are equivalent ...

Square triangle picking is the selection of triples of points (corresponding to endpoints of a triangle) randomly placed inside a square. n random triangles can be picked in ...