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A contraction is a transformation T that reduces the distance between every pair of points. That is, there is a number r < 1 with. dist (T (x, y), T (x', y')) ≤ r⋅dist ( (x, y), (x', y')) for all pairs of points (x, y) and (x', y'). Here dist denotes the Euclidean distance between points:
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The answer is 1/2, the larger of the individual factors....
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First, we need a way to measure distances between compact...
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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.
X is called a contraction if there exists k < 1 such that for any x;y 2 X, kd(x;y) ‚ d(f(x);f(y)). Example 1. Consider the metric space (R;d) where d is the Euclidean distance metric, i.e. d(x;y) = jx¡yj. The function f: R! Rwhere f(x) = x a + b is a contraction if a > 1. In this speciflc case we can flnd a flxed point. Since a flxed ...
28 Οκτ 2024 · Geometric Contraction, Edge Contraction, Ideal Contraction, Tensor Contraction, Vertex Contraction
The contraction mapping theorem concerns maps f: X!X, (X;d) a metric space, and their xed points. A point xis a xed point of fif f(x) = x, i.e. f xes x. A contraction mapping is a map f: X!Xsuch that there is 2(0;1) such that d(f(x);f(y)) d(x;y) for all x;y2X. Theorem 1 Suppose Xis a complete metric space, and f: X!Xis a contraction mapping. Then
19 Οκτ 2015 · A function is a contraction mapping if, after you apply the function to the two points, they get closer together no matter which two points you started with. What a contraction mapping does, in other words, is to squeeze all the points closer together.
Theorem 2 (Uniform Contraction Principle I). Let X;Y be two metric spaces with Xbeing complete. Assume f : X Y !Xis continuous and uniformly contractive with a contraction constant 0 < <1. Then the unique fixed point x (y) is continuous and d( x(z); x(y)) 1 1 d(f( x(y);z);f( x(y);y)): Proof. Let 0 < <1 be the uniform contraction constant. Then ...
