Contraction Principle (other)
In mathematics, contraction principle may refer to: * Contraction principle (large deviations theory), a theorem that states how a large deviation principle on one space "pushes forward" to another space * Banach contraction principle In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Contraction Principle (large Deviations Theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a probability measure) to a large deviation principle on another space ''via'' a continuous function. Statement Let ''X'' and ''Y'' be Hausdorff topological spaces and let (''μ''''ε'')''ε''>0 be a family of probability measures on ''X'' that satisfies the large deviation principle with rate function ''I'' : ''X'' →  , +∞ Let ''T'' : ''X'' → ''Y'' be a continuous function, and let ''ν''''ε'' = ''T''∗(''μ''''ε'') be the push-forward measure of ''μ''''ε'' by ''T'', i.e., for each measurable set/event ''E'' ⊆ ''Y'', ''ν''''ε''(''E'') = ''μ''''ε''(''T''−1(''E'')). Let :J(y) := \inf \, with the convention that the i ...
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