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When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction or reduction. In the most general sense, a scaling includes the case in which the directions of scaling are not ...
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term for this transformation is a dilatation (also known as dilation).
Here is the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude.
The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58. [1] These Hausdorff dimensions are related to the "critical exponent" of the Master theorem for solving recurrence relations in the analysis of algorithms.
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In a scale invariant quantum field theory, by definition each operator acquires under a dilation a factor , where is a number called the scaling dimension of . This implies in particular that the two point correlation function O ( x ) O ( 0 ) {\displaystyle \langle O(x)O(0)\rangle } depends on the distance as ( x 2 ) − Δ {\displaystyle (x^{2 ...
Dilation (operator theory), a dilation of an operator on a Hilbert space; Dilation (morphology), an operation in mathematical morphology; Scaling (geometry), including: Homogeneous dilation , the scalar multiplication operator on a vector space or affine space; Inhomogeneous dilation, where scale factors may differ in different directions
Sz.-Nagy's dilation theorem, proved in 1953, states that for any contraction T on a Hilbert space H, there is a unitary operator U on a larger Hilbert space K ⊇ H such that if P is the orthogonal projection of K onto H then T n = P U n P for all n > 0.