Dirichlet's approximation theoremIn number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers and , with , there exist integers and such that and Here represents the integer part of . This is a fundamental result in Diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality is satisfied by infinitely many integers p and q.
Champernowne constantIn mathematics, the Champernowne constant C10 is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. For base 10, the number is defined by concatenating representations of successive integers: C10 = 0.12345678910111213141516... . Champernowne constants can also be constructed in other bases, similarly, for example: C2 = 0.11011100101110111... 2 C3 = 0.12101112202122..
Liouville numberIn number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.
Mess dress uniformMess dress uniform is the most formal (or semi-formal, depending on the country) type of evening-wear uniform used by military personnel, police personnel, and other uniformed services members. It frequently consists of a mess jacket, trousers, white dress shirt and a black bow tie, along with orders and medals insignia. Design may depend on regiment or service branch, e.g. army, navy, air force, marines, etc. In modern Western dress codes, mess dress uniform is the supplementary alternative equivalent to the civilian black tie for evening wear.
Induction of regular languagesIn computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction.
