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{{Short description|Features that do not change if length or energy scales are multiplied by a common factor}}

{{Short description|Features that do not change if length or energy scales are multiplied by a common factor}}

[[File:Wiener process animated.gif|thumb|right|500px|The [[Wiener process]] is scale-invariant.]]

[[File:Wiener process animated.gif|thumb|right|500px|The [[Wiener process]] is scale-invariant.|链接=Special:FilePath/Wiener_process_animated.gif]]

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.

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.

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.

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 (mathematics)|transformation]] is a '''dilatation''' (also known as '''dilation'''), and the dilatations can also form part of a larger [[conformal symmetry]].

The technical term for this [[transformation (mathematics)|transformation]] is a '''dilatation''' (also known as '''dilation'''), and the dilatations can also form part of a larger [[conformal symmetry]].

===Fractals===

===Fractals===

[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].]]

[[File:Kochsim.gif|thumb|right|250px|A [[Koch curve]] is [[self-similar]].|链接=Special:FilePath/Kochsim.gif]]

It is sometimes said that [[fractal]]s are scale-invariant, although more precisely, one should say that they are [[self-similar]]. A fractal is equal to itself typically for only a discrete set of values {{mvar|λ}}, and even then a translation and rotation may have to be applied to match the fractal up to itself.

It is sometimes said that [[fractal]]s are scale-invariant, although more precisely, one should say that they are [[self-similar]]. A fractal is equal to itself typically for only a discrete set of values {{mvar|λ}}, and even then a translation and rotation may have to be applied to match the fractal up to itself.

:\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.

:\frac{1}{c^2} \frac{\partial^2 \varphi}{\partial t^2}-\nabla^2 \varphi+g\varphi^3=0.

给定这个尺度维数，无质量纯量场理论的非线性修正也是尺度不变的。一个例子是 = 4的无质量 φ < sup > 4 </sup > 理论。场方程是: frac {1}{ c ^ 2} frac { partial ^ 2 varphi }{ partial t ^ 2}-nabla ^ 2 varphi + g varphi ^ 3 = 0。

给定这个尺度维数，无质量纯量场理论的非线性修正也是尺度不变的。一个例子是 = 4的无质量 φ < sup > 4 理论。场方程是: frac {1}{ c ^ 2} frac { partial ^ 2 varphi }{ partial t ^ 2}-nabla ^ 2 varphi + g varphi ^ 3 = 0。

(Note that the name {{mvar|φ}}⁴ derives from the form of the [[Phi to the fourth#The Lagrangian|Lagrangian]], which contains the fourth power of {{mvar|φ}}.)

(Note that the name {{mvar|φ}}⁴ derives from the form of the [[Phi to the fourth#The Lagrangian|Lagrangian]], which contains the fourth power of {{mvar|φ}}.)