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第7行： − − 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), and the dilatations can also form part of a larger conformal symmetry. 第24行：
第20行： − − *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. 第42行：
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第50行： − − An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates , the spiral can be written as 第69行：
第56行： − − Allowing for rotations of the curve, it is invariant under all rescalings ; that is, is identical to a rotated version of . 第76行：
第61行： − − The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a homogeneous polynomial, and more generally to a homogeneous function. Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of schemes, it has connections to various topics in string theory.
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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.
在物理学、数学和统计学中，'''Scale Invariance 标度不变性'''是物体或者物理定律的一种特征，如果长度、能量或者其他变量的标度与一个公因子相乘，而不发生改变，因此也就代表某种普遍性。
在物理学、数学和统计学中，'''Scale Invariance 标度不变性'''是物体或者物理定律的一种特征，如果长度、能量或者其他变量的标度与一个公因子相乘，而不发生改变，因此也就代表某种普遍性。
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]].
这种变换的专业名称是'''Dilatation 膨胀'''，膨胀也可以形成一个更大'''Conformal Symmetry 共形对称'''的一部分。
这种变换的专业名称是'''Dilatation 膨胀'''，膨胀也可以形成一个更大'''Conformal Symmetry 共形对称'''的一部分。
*[[universality (dynamical systems)|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.
*[[universality (dynamical systems)|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 moment]]s, which are scale invariant statistics of a variable, while the unstandardized moments are not.
*In general, [[dimensionless quantities]] are scale invariant. The analogous concept in [[statistics]] are [[standardized moment]]s, which are scale invariant statistics of a variable, while the unstandardized moments are not.
* 在数学中，标度不变性通常指单个函数或曲线的不变性。与此密切相关的概念是'''Self-similarity 自相似性'''，其中函数或曲线在膨胀的离散子集下是不变的。随机过程的概率分布也可能表现出这种标度不变性或自相似性。
* 在数学中，标度不变性通常指单个函数或曲线的不变性。与此密切相关的概念是'''Self-similarity 自相似性'''，其中函数或曲线在膨胀的离散子集下是不变的。随机过程的概率分布也可能表现出这种标度不变性或自相似性。
In mathematics, one can consider the scaling properties of a [[function (mathematics)|function]] or [[curve]] {{math|''f'' (''x'')}} under rescalings of the variable {{mvar|x}}. That is, one is interested in the shape of {{math|''f'' (''λx'')}} for some scale factor {{mvar|λ}}, which can be taken to be a length or size rescaling. The requirement for {{math|''f'' (''x'')}} to be invariant under all rescalings is usually taken to be
In mathematics, one can consider the scaling properties of a [[function (mathematics)|function]] or [[curve]] {{math|''f'' (''x'')}} under rescalings of the variable {{mvar|x}}. That is, one is interested in the shape of {{math|''f'' (''λx'')}} for some scale factor {{mvar|λ}}, which can be taken to be a length or size rescaling. The requirement for {{math|''f'' (''x'')}} to be invariant under all rescalings is usually taken to be
:<math>f(\lambda x)=\lambda^{\Delta}f(x)</math>
:<math>f(\lambda x)=\lambda^{\Delta}f(x)</math>
for some choice of exponent {{mvar|Δ}}, and for all dilations {{mvar|λ}}. This is equivalent to {{mvar|f}} being a [[homogeneous function]] of degree {{mvar|Δ}}.
for some choice of exponent {{mvar|Δ}}, and for all dilations {{mvar|λ}}. This is equivalent to {{mvar|f}} being a [[homogeneous function]] of degree {{mvar|Δ}}.
在数学中，我们会考虑函数或曲线在变量{{mvar|x}}重新标度下的标度性质。也就是说，人们对某些标度因子{{mvar|λ}} 对应下{{math|''f'' (''λx'')}}的形状感兴趣，这些标度因子可以被视为长度或大小的重新标度。对于某些选择的指数{{mvar|Δ}}和所有的膨胀{{mvar|λ}}，要求{{math|''f'' (''x'')}} 在所有重新标度下保持不变需要满足：
在数学中，我们会考虑函数或曲线在变量{{mvar|x}}重新标度下的标度性质。也就是说，人们对某些标度因子{{mvar|λ}} 对应下{{math|''f'' (''λx'')}}的形状感兴趣，这些标度因子可以被视为长度或大小的重新标度。对于某些选择的指数{{mvar|Δ}}和所有的膨胀{{mvar|λ}}，要求{{math|''f'' (''x'')}} 在所有重新标度下保持不变需要满足：
An example of a scale-invariant curve is the [[logarithmic spiral]], a kind of curve that often appears in nature. In [[polar coordinates]] {{math|(''r'', ''θ'')}}, the spiral can be written as
An example of a scale-invariant curve is the [[logarithmic spiral]], a kind of curve that often appears in nature. In [[polar coordinates]] {{math|(''r'', ''θ'')}}, the spiral can be written as
一个标度不变曲线的例子是'''Logarithmic Spiral 对数螺线（等角螺线）'''，这是一种在自然界中经常出现的曲线。在极坐标系中，螺旋线可以写成
一个标度不变曲线的例子是'''Logarithmic Spiral 对数螺线（等角螺线）'''，这是一种在自然界中经常出现的曲线。在极坐标系中，螺旋线可以写成
Allowing for rotations of the curve, it is invariant under all rescalings {{mvar|λ}}; that is, {{math|''θ''(''λr'')}} is identical to a rotated version of {{math|''θ''(''r'')}}.
Allowing for rotations of the curve, it is invariant under all rescalings {{mvar|λ}}; that is, {{math|''θ''(''λr'')}} is identical to a rotated version of {{math|''θ''(''r'')}}.
在任意重新标度{{mvar|λ}}下，标度不变也允许曲线进行旋转；换句话说，{{math|''θ''(''λr'')}}与其旋转后的{{math|''θ''(''r'')}}一模一样。
在任意重新标度{{mvar|λ}}下，标度不变也允许曲线进行旋转；换句话说，{{math|''θ''(''λr'')}}与其旋转后的{{math|''θ''(''r'')}}一模一样。
===Projective geometry 射影几何===
===Projective geometry 射影几何===
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a [[homogeneous polynomial]], and more generally to a [[homogeneous function]]. Homogeneous functions are the natural denizens of [[projective space]], and homogeneous polynomials are studied as [[projective varieties]] in [[projective geometry]]. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of [[scheme (mathematics)|schemes]], it has connections to various topics in [[string theory]].
The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a [[homogeneous polynomial]], and more generally to a [[homogeneous function]]. Homogeneous functions are the natural denizens of [[projective space]], and homogeneous polynomials are studied as [[projective varieties]] in [[projective geometry]]. Projective geometry is a particularly rich field of mathematics; in its most abstract forms, the geometry of [[scheme (mathematics)|schemes]], it has connections to various topics in [[string theory]].
单项式标度不变性的概念在高维时推广到'''Homogeneous Polynomial 齐次多项式'''，更一般地推广到'''Homogeneous Function 齐次函数'''。齐次函数是射影空间的“土著”，齐次多项式在射影几何中作为'''Projective Varieties 射影簇'''进行研究。射影几何是数学中一个内容特别丰富的领域；在其最抽象的形式——'''Schemes 概型'''的几何学中，它与'''String Theory 弦理论'''中的各种主题都有联系。
单项式标度不变性的概念在高维时推广到'''Homogeneous Polynomial 齐次多项式'''，更一般地推广到'''Homogeneous Function 齐次函数'''。齐次函数是射影空间的“土著”，齐次多项式在射影几何中作为'''Projective Varieties 射影簇'''进行研究。射影几何是数学中一个内容特别丰富的领域；在其最抽象的形式——'''Schemes 概型'''的几何学中，它与'''String Theory 弦理论'''中的各种主题都有联系。