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In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.
Contraction mapping. In mathematics, a contraction mapping, or contraction or contractor, on a metric space ( M , d) is a function f from M to itself, with the property that there is some real number such that for all x and y in M , The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called ...
The Common Core State Standards Initiative, also known as simply Common Core, was a multi-state educational initiative begun in 2010 with the goal of increasing consistency across state standards, or what K–12 students throughout the United States should know in English language arts and mathematics at the conclusion of each school grade.
Tensor contraction. In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor (s) caused by applying the summation convention to a pair of dummy indices that are bound to ...
e. Mathematics education in the United States varies considerably from one state to the next, and even within a single state. However, with the adoption of the Common Core Standards in most states and the District of Columbia beginning in 2010, mathematics content across the country has moved into closer agreement for each grade level.
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
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