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A Turing machine which, by appropriate programming using a finite length of input tape, can act as any Turing machine whatsoever. In his seminal paper, Turing himself gave ...

A vector is formally defined as an element of a vector space. In the commonly encountered vector space R^n (i.e., Euclidean n-space), a vector is given by n coordinates and ...

The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel set theory. In the following (Jech 1997, p. 1), exists stands for exists, forall means for all, in stands for ...

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides ...

3 is the only integer which is the sum of the preceding positive integers (1+2=3) and the only number which is the sum of the factorials of the preceding positive integers ...

According to the novel The Hitchhiker's Guide to the Galaxy (Adams 1997), 42 is the ultimate answer to life, the universe, and everything. Unfortunately, it is left as an ...

A quasigroup with an identity element e such that xe=x and ex=x for any x in the quasigroup. All groups are loops. In general, loops are considered to have very little in the ...

The mathematical study of abstract computing machines (especially Turing machines) and the analysis of algorithms used by such machines. A connection between automata theory ...

The covariant derivative of the Riemann tensor is given by (1) Permuting nu, kappa, and eta (Weinberg 1972, pp. 146-147) gives the Bianchi identities ...

A Bill picture is a sequence of nested regular polygons in which subsequent polygons are each rotated so that they begin one vertex further. The term was coined by Trott ...