Features that do not change if length or energy scales are multiplied by a common factor
The Wiener process is scale-invariant
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). Dilatations can form part of a larger conformal symmetry.
In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity.
In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale.
In quantum field theory, scale invariance has an interpretation in terms of particle physics. In a scale-invariant theory, the strength of particle interactions does not depend on the energy of the particles involved.
In statistical mechanics, scale invariance is a feature of phase transitions. The key observation is that near a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena. Such theories are scale-invariant statistical field theories, and are formally very similar to scale-invariant quantum field theories.
Universality is the observation that widely different microscopic systems can display the same behaviour at a phase transition. Thus phase transitions in many different systems may be described by the same underlying scale-invariant theory.
In general, dimensionless quantities are scale-invariant. The analogous concept in statistics are standardized moments, which are scale-invariant statistics of a variable, while the unstandardized moments are not.
physics, mathematics and statistics, scaleinvariance is a feature of objects or laws that do not change if scales of length, energy, or other variables...
attribute of power laws is their scaleinvariance. Given a relation f ( x ) = a x − k {\displaystyle f(x)=ax^{-k}} , scaling the argument x {\displaystyle...
classically scale-invariant scalar field theory in D = 4 is the massless φ4 theory. Classical scaleinvariance, however, normally does not imply quantum scale invariance...
problem of time may be related to an underlying scaleinvariance of gravity-matter systems. Scaleinvariance has also been proposed to resolve the hierarchy...
include linearity, shift invariance, semi-group structure, non-enhancement of local extrema, scaleinvariance and rotational invariance. In the works, the uniqueness...
2019. Mood, A. M.; Graybill, F. A.; Boes, D. C. (1974). "VII.6.2 Scale invariance". Introduction to the theory of statistics (3rd ed.). New York: McGraw-Hill...
list (link) Labini, F. Sylos; Montuori, M. & Pietronero, L. (1998). "Scale-invariance of galaxy clustering". Physics Reports. 293 (1): 61–226. arXiv:astro-ph/9711073...
(the first note in a scale) and the tonic chord (the first note in the scale with the third and fifth note) with the rest of the scale. The tonic is the...
field theory, scaleinvariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal...
admit conformal symmetry due to it typically being implied by local scaleinvariance (see here for motivation and counterexamples). A famous example is...
an additive random variable is then E(Z) = λμ and var(Z) = λV(μ). Scaleinvariance implies that the variance function obeys the relationship V(μ) = μ...
that enlarges or diminishes objects Scaleinvariance, a feature of objects or laws that do not change if scales of length, energy, or other variables...
solution to a two-person bargaining problem that satisfies the axioms of scaleinvariance, symmetry, efficiency, and independence of irrelevant alternatives...
Rotational invariance, the property of function whose value does not change when arbitrary rotations are applied to its argument Scaleinvariance, a property...
to the notion of scaleinvariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor...
dimension. It is instructive to see how the scaleinvariance at the upper critical dimension becomes a scaleinvariance below this dimension. For small external...
breaking the Weyl (or Scale) invariance of the theory. In quantum chromodynamics in the chiral limit, the classical theory has no mass scale so there is a conformal...
Orthogonal regression Weighted geometric distance: Deming regression Scaleinvariance: Major axis regression Linear least squares Linear segmented regression...
to scaleinvariance and conformal invariance, symmetries in which a system appears the same at all scales (so-called self-similarity). As the scale varies...
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