“fixed-point theorem”的概念、定义、翻译、参考文献-科学参考

fixed-point theorem

Mathematics

Any theorem which gives conditions under which a function must have a fixed point (see Brouwer’s fixed-point theorem, contraction mapping theorem). Such theorems also have importance in economics.

Computer

A theorem concerning the existence and nature of fixed points used to give solutions to equations. A fixed point of a function f
$f : X \to X$
is an element x such that f(x) = x. A least fixed point is one that, among all the fixed points of f, is lowest in some partial ordering that has been imposed on the elements of X. Specifically, if ≤ is a partial ordering of X then x is a least fixed point if for fixed point y we have x ≤ y.
The most often-cited form of fixed-point theorem to do with computing is due originally to S. C. Kleene, and originated in recursive function theory. It states that, subject to certain assumptions, notably that f is continuous, f has a least fixed point, x_f, which moreover can be characterized as the limit of a sequence x₀,x₁,x₂,… of approximations. This abstract fact is of great relevance to the semantics of programming languages, in particular in specifying the precise meaning of constructs like iteration, recursion, and recursive types using equations.

单词	fixed-point theorem
释义	fixed-point theorem Mathematics Any theorem which gives conditions under which a function must have a fixed point (see Brouwer’s fixed-point theorem, contraction mapping theorem). Such theorems also have importance in economics. Computer A theorem concerning the existence and nature of fixed points used to give solutions to equations. A fixed point of a function f $f : X \to X$ is an element x such that f(x) = x. A least fixed point is one that, among all the fixed points of f, is lowest in some partial ordering that has been imposed on the elements of X. Specifically, if ≤ is a partial ordering of X then x is a least fixed point if for fixed point y we have x ≤ y. The most often-cited form of fixed-point theorem to do with computing is due originally to S. C. Kleene, and originated in recursive function theory. It states that, subject to certain assumptions, notably that f is continuous, f has a least fixed point, x_f, which moreover can be characterized as the limit of a sequence x₀,x₁,x₂,… of approximations. This abstract fact is of great relevance to the semantics of programming languages, in particular in specifying the precise meaning of constructs like iteration, recursion, and recursive types using equations.
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