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A sequence of random variates X_0, X_1, ... is called absolutely fair if for n=1, 2, ..., <X_1>=0 and <X_(n+1)|X_1,...,X_n>=0 (Feller 1971, p. 210).

An abstract manifold is a manifold in the context of an abstract space with no particular embedding, or representation in mind. It is a topological space with an atlas of ...

The identity element of an additive group G, usually denoted 0. In the additive group of vectors, the additive identity is the zero vector 0, in the additive group of ...

A curve which has at least multiplicity r_i-1 at each point where a given curve (having only ordinary singular points and cusps) has a multiplicity r_i is called the adjoint ...

An algebraic identity is a mathematical identity involving algebraic functions. Examples include the Euler four-square identity, Fibonacci identity, Lebesgue identity, and ...

A number of the form p^a·A is said to be an antisquare if it fails to be a square number for the two reasons that a is odd and A is a nonsquare (modulo p). The first few ...

A theorem from information theory that is a simple consequence of the weak law of large numbers. It states that if a set of values X_1, X_2, ..., X_n is drawn independently ...

The Bolyai expansion of a real number x is a nested root of the form x=a_0-1+RadicalBox[{{a, _, 1}, +, RadicalBox[{{a, _, 2}, +, RadicalBox[{{a, _, 3}, +, ...}, m]}, m]}, m], ...

Every continuous map f:S^n->R^n must identify a pair of antipodal points.

A metric space X is boundedly compact if all closed bounded subsets of X are compact. Every boundedly compact metric space is complete. (This is a generalization of the ...