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The function f(x) = x 2 − 4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2 − √ 17 /2.. In order theory, a branch of mathematics, the least fixed point (lfp or LFP, sometimes also smallest fixed point) of a function from a partially ordered set ("poset" for short) to itself is the fixed point which is less than each other fixed point, according ...
A function need not have a least fixed point, but if it does then the least fixed point is unique. One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixed point that coincides with its least prefixpoint (and similarly its greatest fixed point coincides with its greatest ...
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
In mathematics, the Lefschetz fixed-point theorem [1] is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mappings on the homology groups of .
Since complete lattices cannot be empty (they must contain a supremum and infimum of the empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point (or greatest fixed point). In many practical cases, this is the most important implication of the theorem.
Computation of the least fixpoint of f(x) = 1 / 10 x 2 +atan(x)+1 using Kleene's theorem in the real interval [0,7] with the usual order. In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem.
The notation . (and its dual) are inspired from the lambda calculus; the intent is to denote the least (and respectively greatest) fixed point of the expression where the "minimization" (and respectively "maximization") are in the variable , much like in lambda calculus . is a function with formula in bound variable; [6] see the denotational ...
Several variations of fixed point logics have been studied. In least fixed point logic, the right hand side of the operator in the defining formula must use the predicate only positively (that is, each appearance should be nested within an even number of negations) in order to make the least fixed point well defined. In another variant with ...