奇点理论


In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U".This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the "U" away from the "underline".

In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the "U" away from the "underline".

在数学中,奇点理论研究的空间几乎都是流形,但不完全是流形。如果忽略弦的厚度,弦可以作为一维流形的例子。一个奇点的形成可以通过把它团起来,扔在地板上,然后把它压扁。完成以上步骤即可得到一个奇点。在某些地方,扁平的字符串会以近似“ x”的形状交叉自身。这些扔在地板上的点是一种奇点。也许字符串也会在没有交叉的情况下接触自己,就像u的下划线那样。这是另一种奇点。与双点不同,它是不稳定的,在某种意义上说,一个小的推动将提升底部的“ u”远离“下划线”。



Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These situations are called perestroika (模板:Lang-ru), bifurcations or catastrophes. Classifying the types of changes and characterizing sets of parameters which give rise to these changes are some of the main mathematical goals. Singularities can occur in a wide range of mathematical objects, from matrices depending on parameters to wavefronts.[1]


Vladimir Arnold 将奇点理论的主要目标定义为描述物体如何依赖于参数,尤其强调在参数发生微小变化的情况下,参数的性质会发生突然变化。这些情况被称为 perestroika ,意为分叉或灾难。分类变化的类型并确定引起这些变化的参数集是研究奇点时的重要数学目标。奇点可以出现在很多数学对象中,从依赖于参数的矩阵到波前。


How singularities may arise 奇点理论是怎样产生的

In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.

在奇点理论中,常规状态下的点和奇点的集合都是研究的对象,也会涉及到对没有奇点存在的空间的研究。

Singularities in algebraic 代数学下的奇点

Historically, singularities were first noticed in the study of algebraic curves. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point), in accounting for intersections of curves. It was then a short step to define the general notion of a singular point of an algebraic variety.In addition, singularities allow higher dimensions.

在奇点理论的发展史中,它被第一次注意到是在对函数图象的研究中。牛顿对三次方的函数图象进行了详细的研究,这些图象跟奇点理论也是相关的。在裴蜀定理中,奇点在黎曼曲面投影到的射影平面上的曲线最多只含有通常二重点作为奇点,且涉及到两曲线函数图象相交的情况。裴蜀定理定义了奇点在代数学图象的一般状态。此外,奇点能存在于高维空间中。


This page was moved from wikipedia:en:Singularity theory. Its edit history can be viewed at 奇点理论/edithistory

  1. Arnold, V. I. (2000). "Singularity Theory". www.newton.ac.uk. Isaac Newton Institute for Mathematical Sciences. Retrieved 31 May 2016.