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A transition function describes the difference in the way an object is described in two separate, overlapping coordinate charts, where the description of the same set may ...

Two partially ordered sets are said to be isomorphic if their "structures" are entirely analogous. Formally, partially ordered sets P=(X,<=) and Q=(X^',<=^') are isomorphic ...

Define T as the set of all points t with probabilities P(x) such that a>t=>P(a<=x<=a+da)<P_0 or a<t=>P(a<=x<=a+da)<P_0, where P_0 is a point probability (often, the ...

Let X be an alphabet (i.e., a finite and nonempty set), and call its member letters. A word on X is a finite sequence of letters a_1...a_n, where a_1,...,a_n in X. Denote the ...

A set function mu possesses countable additivity if, given any countable disjoint collection of sets {E_k}_(k=1)^n on which mu is defined, mu( union ...

For a connected graph G of graph diameter d, the distance-k graph G_k for k=1, ..., d is a graph with the same vertex set and having edge set consisting of the pairs of ...

A knot invariant is a function from the set of all knots to any other set such that the function does not change as the knot is changed (up to isotopy). In other words, a ...

A perfect ruler also called a complete ruler, is type of ruler considered by Guy (1994) which has k distinct marks spaced such that the distances between marks can be used to ...

Let phi(x_1,...,x_m) be an L_(exp) formula, where L_(exp)=L union {e^x} and L is the language of ordered rings L={+,-,·,<,0,1}. Then there exist n>=m and f_1,...,f_s in ...

A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i!=j. That is, the vectors are mutually perpendicular. ...