奇点

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In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as the lack of differentiability or analyticity.

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as the lack of differentiability or analyticity.

在数学中，奇点通常是一个给定的数学对象没有定义的点，或者是一个数学对象不再以某种特定方式表现良好的点，例如缺乏可微性或分析性。

For example, the real function

Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types:

除孤立奇点外，一个变量的复变函数还可能表现出其他奇异行为。这些称为非孤立奇点，其中有两种类型:

[math]\displaystyle{ f(x)=\frac{1}{x} }[/math]

has a singularity at [math]\displaystyle{ x = 0 }[/math], where it seems to "explode" to [math]\displaystyle{ \pm\infty }[/math] and is hence not defined. The absolute value function [math]\displaystyle{ g(x) = |x| }[/math] also has a singularity at x = 0, since it is not differentiable there.

The algebraic curve defined by [math]\displaystyle{ \{(x,y):y^3-x^2=0\} }[/math] in the [math]\displaystyle{ (x, y) }[/math] coordinate system has a singularity (called a cusp) at [math]\displaystyle{ (0, 0) }[/math]. For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.

Branch points are generally the result of a multi-valued function, such as [math]\displaystyle{ \sqrt{z} }[/math] or [math]\displaystyle{ \log(z) }[/math], which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as [math]\displaystyle{ z=0 }[/math] and [math]\displaystyle{ z=\infty }[/math] for [math]\displaystyle{ \log(z) }[/math]) which are fixed in place.

分支点通常是多值函数的结果，比如 < math > sqrt { z } </math > 或 < math > log (z) </math > ，这些分支点定义在一个特定的有限域内，因此函数可以在域内成为单值函数。切割是一条从区域中排除的直线或曲线，它在函数的不连续值之间引入了一种技术上的分离。当切割是真正需要的，功能将有明显不同的价值观在每一侧的分支削减。分支切口的形状是一个选择的问题，即使它必须连接两个不同的分支点(如 < math > z = 0 </math > 和 < math > z = infty </math > for < math > log (z) </math >) ，这两个分支点是固定的。

Real analysis

In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).

The reciprocal function, exhibiting hyperbolic growth.

[[倒数函数，显示双曲增长。[] < ! -- 一个更好的图像应该是1/(1-x)或类似的图像，显示一个正的奇点，并且随着 x 的增加而增长 -- >

To describe the way these two types of limits are being used, suppose that [math]\displaystyle{ f(x) }[/math] is a function of a real argument [math]\displaystyle{ x }[/math], and for any value of its argument, say [math]\displaystyle{ c }[/math], then the left-handed limit, [math]\displaystyle{ f(c^-) }[/math], and the right-handed limit, [math]\displaystyle{ f(c^+) }[/math], are defined by:

[math]\displaystyle{ f(c^-) = \lim_{x \to c}f(x) }[/math], constrained by [math]\displaystyle{ x \lt c }[/math] and

A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and PDEs (Partial Differential Equations) – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form [math]\displaystyle{ x^{-\alpha}, }[/math] of which the simplest is hyperbolic growth, where the exponent is (negative) 1: [math]\displaystyle{ x^{-1}. }[/math] More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses [math]\displaystyle{ (t_0-t)^{-\alpha} }[/math] (using t for time, reversing direction to [math]\displaystyle{ -t }[/math] so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time [math]\displaystyle{ t_0 }[/math]).

当一个输入变量为时间，输出变量在有限时间内向无穷大方向增长时，出现了有限时间奇异性。这些在运动学和偏微分方程(偏微分方程)中很重要——物理上不存在无穷大，但奇点附近的行为通常是有趣的。数学上，最简单的有限时间奇点是形式为 < math > x ^ {-alpha }的各种指数的幂律，其中最简单的是双曲增长，其指数是(负)1: < math > x ^ {-1}。更准确地说，为了在正时间得到一个奇点，随着时间的推移(所以输出增长到无穷大) ，取而代之的是使用 < math > (t _ 0-t) ^ {-alpha } </math > (使用 t 表示时间，将方向倒转到 < math >-t </math > ，使时间增长到无穷大，并将奇点从0向前移动到一个固定的时间 < t _ 0 </math >)。

[math]\displaystyle{ f(c^+) = \lim_{x \to c}f(x) }[/math], constrained by [math]\displaystyle{ x \gt c }[/math].

An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy).

一个例子是一个非弹性球在平面上的反弹运动。如果考虑理想运动，即每次弹跳都损失相同的动能，那么弹跳的频率就是无限的，因为球停在一个有限的时间里。有限时间奇点的其他例子包括 painlevé 悖论的各种形式(例如，当粉笔被拖过黑板时，粉笔倾向于跳过) ，以及在平面上旋转的硬币的进动速率如何加速到无限ー然后突然停止(使用欧拉圆盘玩具进行的研究)。

The value [math]\displaystyle{ f(c^-) }[/math] is the value that the function [math]\displaystyle{ f(x) }[/math] tends towards as the value [math]\displaystyle{ x }[/math] approaches [math]\displaystyle{ c }[/math] from below, and the value [math]\displaystyle{ f(c^+) }[/math] is the value that the function [math]\displaystyle{ f(x) }[/math] tends towards as the value [math]\displaystyle{ x }[/math] approaches [math]\displaystyle{ c }[/math] from above, regardless of the actual value the function has at the point where [math]\displaystyle{ x = c }[/math] .

Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time).

假设的例子包括 Heinz von Foerster 滑稽的“末日方程”(简单化的模型在有限的时间里产生无限的人口)。

There are some functions for which these limits do not exist at all. For example, the function

[math]\displaystyle{ g(x) = \sin\left(\frac{1}{x}\right) }[/math]

In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation − x = 0 }} defines a curve that has a cusp at the origin . One could define the -axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the -axis is a "double tangent."

在代数几何中，代数簇的奇点是各种各样的切线空间可能没有规则定义的一点。奇点最简单的例子就是它们自己交叉的曲线。但是还有其他类型的奇点，比如尖点。例如，方程 -x = 0}}定义了一条在原点有尖点的曲线。可以将-轴定义为这一点的切线，但这个定义不能与其他点的定义相同。实际上，在这种情况下,-轴是一个“双切线”

does not tend towards anything as [math]\displaystyle{ x }[/math] approaches [math]\displaystyle{ c = 0 }[/math]. The limits in this case are not infinite, but rather undefined: there is no value that [math]\displaystyle{ g(x) }[/math] settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.

对于仿射变种和射影变种，奇异点是指雅可比矩阵的秩低于其它变种点的点。

The possible cases at a given value [math]\displaystyle{ c }[/math] for the argument are as follows.

• A point of continuity is a value of [math]\displaystyle{ c }[/math] for which [math]\displaystyle{ f(c^-) = f(c) = f(c^+) }[/math], as one expects for a smooth function. All the values must be finite. If [math]\displaystyle{ c }[/math] is not a point of continuity, then a discontinuity occurs at [math]\displaystyle{ c }[/math].

An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring.

可以给出一个关于交换代数的等价定义，它扩展到抽象的簇和方案: 如果局部环在这一点上不是一个正则局部环，那么一个点就是奇异的。

• A type I discontinuity occurs when both [math]\displaystyle{ f(c^-) }[/math] and [math]\displaystyle{ f(c^+) }[/math] exist and are finite, but at least one of the following three conditions also applies:

• • [math]\displaystyle{ f(c^-) \neq f(c^+) }[/math];

• • [math]\displaystyle{ f(x) }[/math] is not defined for the case of [math]\displaystyle{ x = c }[/math]; or

• • [math]\displaystyle{ f(c) }[/math] has a defined value, which, however, does not match the value of the two limits.

Type I discontinuities can be further distinguished as being one of the following subtypes:

• A jump discontinuity occurs when [math]\displaystyle{ f(c^-) \neq f(c^+) }[/math], regardless of whether [math]\displaystyle{ f(c) }[/math] is defined, and regardless of its value if it is defined.

• A removable discontinuity occurs when [math]\displaystyle{ f(c^-) = f(c^+) }[/math], also regardless of whether [math]\displaystyle{ f(c) }[/math] is defined, and regardless of its value if it is defined (but which does not match that of the two limits).

• A type II discontinuity occurs when either [math]\displaystyle{ f(c^-) }[/math] or [math]\displaystyle{ f(c^+) }[/math] does not exist (possibly both). This has two subtypes, which are usually not considered separately:

• • An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote.

• • An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits [math]\displaystyle{ f(c^-) }[/math] or [math]\displaystyle{ f(c^+) }[/math] does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include [math]\displaystyle{ \pm\infty }[/math].

In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function.

Coordinate singularities

Main article: Coordinate singularity

A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an n-vector representation).

Category:Mathematical analysis

分类: 数学分析

This page was moved from wikipedia:en:Singularity (mathematics). Its edit history can be viewed at 奇点/edithistory