更改

In [[mathematics]], '''singularity theory''' studies spaces that are almost [[manifold]]s, 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, [[projection (mathematics)|dropping]] it on the floor, and flattening it. In some places the flat [[Jordan curve|string]] will cross itself in an approximate "X" shape. The points on the [[plane (geometry)|floor]] where it does this are one kind of singularity, the double point: one [[neighbourhood (topology)|bit]] of the floor corresponds to [[multimap|more than one]] bit of string. Perhaps the string will also touch itself without crossing, like an underlined "<u>U</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 [[manifold]]s, 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, [[projection (mathematics)|dropping]] it on the floor, and flattening it. In some places the flat [[Jordan curve|string]] will cross itself in an approximate "X" shape. The points on the [[plane (geometry)|floor]] where it does this are one kind of singularity, the double point: one [[neighbourhood (topology)|bit]] of the floor corresponds to [[multimap|more than one]] bit of string. Perhaps the string will also touch itself without crossing, like an underlined "<u>U</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, 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>U</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>U</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是流形，但不完全是。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，然后把它压扁来形成。在某些地方，扁平的字符串会以近似“ x”的形状交叉自身。地板上的这些点是一种奇点，双点: 地板的一个位相当于一个以上的字符串。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”。

在数学中，奇点理论研究的空间几乎是流形，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”。

将奇点理论的主要目标定义为描述物体如何依赖于参数，特别是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。

[[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。