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A function f:X->R is measurable if, for every real number a, the set {x in X:f(x)>a} is measurable. When X=R with Lebesgue measure, or more generally any Borel measure, then ...

A monoid is a set that is closed under an associative binary operation and has an identity element I in S such that for all a in S, Ia=aI=a. Note that unlike a group, its ...

Given a set S of n nonnegative integers, the number partitioning problem requires the division of S into two subsets such that the sums of number in each subset are as close ...

The set of octonions, also sometimes called Cayley numbers and denoted O, consists of the elements in a Cayley algebra. A typical octonion is of the form ...

The orthogonal complement of a subspace V of the vector space R^n is the set of vectors which are orthogonal to all elements of V. For example, the orthogonal complement of ...

Let Y^X be the set of continuous mappings f:X->Y. Then the topological space Y^X supplied with the compact-open topology is called a mapping space. If (Y,*) is a pointed ...

Let K be a number field and let O be an order in K. Then the set of equivalence classes of invertible fractional ideals of O forms a multiplicative Abelian group called the ...

Given a set of ordered quantities, a list of the fractions of values at or below each quantity is know as a plotting position. A quantile-quantile plot compares observed ...

The maximum distance between n points in three dimensions can occur no more than 2n-2 times. Also, there exists a fixed number c such that no distance determined by a set of ...

In its simplest form, the principle of permanence states that, given any analytic function f(z) defined on an open (and connected) set U of the complex numbers C, and a ...