596
个编辑
更改
跳到导航
跳到搜索
第11行：
第11行： − + + + 第17行：
第19行： − + − − − − In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. − − 在数学中，一个自相似的物体与它自身的某一部分完全或近似地相似(例如:。整体具有相同的形状作为一个或多个部分)。现实世界中的许多物体，例如海岸线，在统计学上是自相似的: 它们的某些部分在许多尺度上表现出相同的统计特性。自相似是分形的一个典型性质。尺度不变性是自相似的一种精确形式，在任何放大倍数下，物体中有一小块与整体相似的部分。例如，科赫雪花的一侧既对称又具有尺度不变性; 它可以连续放大3倍而不改变形状。分形中明显的非平凡的相似性是通过它们的精细结构或任意小尺度上的细节来区分的。作为一个反例，尽管直线的任何部分都可能类似于整体，但是进一步的细节没有透露。 − − A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity + − + − 一个时间发展的现象被称为自相似性，如果某个可观测量的数值+ − + − <math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.+ − 在不同时间测量的 math f (x，t) / math 是不同的，但是在给定的 math x / t ^ z / math 值下相应的无量纲量是不变的。如果数学量 f (x，t) / math 表现出动态缩放，就会发生这种情况。这个概念只是两个三角形相似性概念的延伸。请注意，如果两个三角形的边的数值不同，那么它们是相似的，但是对应的无量纲量，例如它们的角重合。
无编辑摘要
Notoc
Notoc
[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]]
[[Image:KochSnowGif16 800x500 2.gif|thumb|right|250px|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.
当无限放大科赫曲线时，它会展示出无限重复的自相似性。]]
A [[Koch curve has an infinitely repeating self-similarity when it is magnified.]]
A [[Koch curve has an infinitely repeating self-similarity when it is magnified.]]
[[[科赫曲线被放大后有无限重复的自相似性]]
[[[科赫曲线被放大后有无限重复的自相似性]]
[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref>Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>]]
[[File:Standard self-similarity.png|thumb|300px|Standard (trivial) self-similarity.<ref name=":0">Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. {{ISBN|978-0716711865}}.</ref>标准(平凡)自相似性。<ref name=":0" />]]
Standard (trivial) self-similarity.
Standard (trivial) self-similarity.
标准(平凡)自相似性。
标准(平凡)自相似性。
In [[mathematics]], a '''self-similar''' object is exactly or approximately [[similarity (geometry)|similar]] to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as [[coastline]]s, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref name="Mandelbrot_Science_1967">{{cite journal | title=How long is the coast of Britain? Statistical self-similarity and fractional dimension | journal=[[Science (journal)|Science]] | date=5 May 1967 | author=Mandelbrot, Benoit B. | pages=636–638 | volume=156 |number=3775 |doi=10.1126/science.156.3775.636 |series=New Series | pmid=17837158| bibcode=1967Sci...156..636M }} [http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf PDF]</ref> Self-similarity is a typical property of [[fractal]]s. [[Scale invariance]] is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is [[Similarity (geometry)|similar]] to the whole. For instance, a side of the [[Koch snowflake]] is both [[symmetrical]] and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a [[counterexample]], whereas any portion of a [[straight line]] may resemble the whole, further detail is not revealed.
In [[mathematics]], a '''self-similar''' object is exactly or approximately [[similarity (geometry)|similar]] to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as [[coastline]]s, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref name="Mandelbrot_Science_1967">{{cite journal | title=How long is the coast of Britain? Statistical self-similarity and fractional dimension | journal=[[Science (journal)|Science]] | date=5 May 1967 | author=Mandelbrot, Benoit B. | pages=636–638 | volume=156 |number=3775 |doi=10.1126/science.156.3775.636 |series=New Series | pmid=17837158| bibcode=1967Sci...156..636M }} [http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf PDF]</ref> Self-similarity is a typical property of [[fractal]]s. [[Scale invariance]] is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is [[Similarity (geometry)|similar]] to the whole. For instance, a side of the [[Koch snowflake]] is both [[symmetrical]] and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a [[counterexample]], whereas any portion of a [[straight line]] may resemble the whole, further detail is not revealed.
在数学中，一个自相似的物体与它自身的某一部分完全或近似地相似（例如：整体和一个或多个部分具有相同的形状）。现实世界中的许多物体，例如海岸线，在统计学上是自相似的：它们的某些部分在许多不同尺度上表现出相同的统计特性。自相似是分形的一个典型性质。标度不变性是自相似的一种精确形式：在任何放大倍数下，物体中总有更小的部分与整体相似。例如，科赫雪花的一边既对称又具有标度不变性；它可以连续放大3倍而不改变形状。分形中明显的非平凡的相似性是通过它们的精细结构或任意小尺度上的细节来区分的。作为一个反例，尽管直线的任何部分都可能类似于整体，但是进一步放大之后，却没有更多的细节显露。
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity <math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits [[dynamic scaling]]. The idea is just an extension of the idea of similarity of two triangles.<ref>{{cite journal | author = Hassan M. K., Hassan M. Z., Pavel N. I. | year = 2011 | title = Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks | url = | journal = J. Phys. A: Math. Theor. | volume = 44 | issue = 17| page = 175101 | doi=10.1088/1751-8113/44/17/175101| arxiv = 1101.4730| bibcode = 2011JPhA...44q5101K}}</ref><ref>{{cite journal | author = Hassan M. K., Hassan M. Z. | year = 2009 | title = Emergence of fractal behavior in condensation-driven aggregation | url = | journal = Phys. Rev. E | volume = 79 | issue = 2| page = 021406 | doi=10.1103/physreve.79.021406| pmid = 19391746 | arxiv = 0901.2761| bibcode = 2009PhRvE..79b1406H}}</ref><ref>{{cite journal | author = Dayeen F. R., Hassan M. K. | year = 2016 | title = Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice | url = | journal = Chaos, Solitons & Fractals | volume = 91 | issue = | page = 228 | doi=10.1016/j.chaos.2016.06.006| arxiv = 1409.7928| bibcode = 2016CSF....91..228D}}</ref> Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.
A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity <math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits dynamic scaling. The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.
<math>f(x,t)</math> measured at different times are different but the corresponding dimensionless quantity at given value of <math>x/t^z</math> remain invariant. It happens if the quantity <math>f(x,t)</math> exhibits [[dynamic scaling]]. The idea is just an extension of the idea of similarity of two triangles.<ref>{{cite journal | author = Hassan M. K., Hassan M. Z., Pavel N. I. | year = 2011 | title = Dynamic scaling, data-collapseand Self-similarity in Barabasi-Albert networks | url = | journal = J. Phys. A: Math. Theor. | volume = 44 | issue = 17| page = 175101 | doi=10.1088/1751-8113/44/17/175101| arxiv = 1101.4730| bibcode = 2011JPhA...44q5101K}}</ref><ref>{{cite journal | author = Hassan M. K., Hassan M. Z. | year = 2009 | title = Emergence of fractal behavior in condensation-driven aggregation | url = | journal = Phys. Rev. E | volume = 79 | issue = 2| page = 021406 | doi=10.1103/physreve.79.021406| pmid = 19391746 | arxiv = 0901.2761| bibcode = 2009PhRvE..79b1406H}}</ref><ref>{{cite journal | author = Dayeen F. R., Hassan M. K. | year = 2016 | title = Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice | url = | journal = Chaos, Solitons & Fractals | volume = 91 | issue = | page = 228 | doi=10.1016/j.chaos.2016.06.006| arxiv = 1409.7928| bibcode = 2016CSF....91..228D}}</ref> Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide.
对于一个依赖时间发展的现象，如果其相关观测量<math>f(x,t)</math>在不同时间所测得的数值不同，但是对应的无量纲量在给定的<math>x/t^z</math>下保持不变，则可以说该现象具有自相似性。
一个时间发展的现象被称为自相似性，如果某个可观测量的数值，在不同时间测量的 math f (x，t) / math 是不同的，但是在给定的 math x / t ^ z / math 值下相应的无量纲量是不变的。如果数学量 f (x，t) / math 表现出动态缩放，就会发生这种情况。这个概念只是两个三角形相似性概念的延伸。请注意，如果两个三角形的边的数值不同，那么它们是相似的，但是对应的无量纲量，例如它们的角重合。