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{{short description|Point where a function, a curve or another mathematical object does not behave regularly}}

{{short description|Point where a function, a curve or another mathematical object does not behave regularly}}

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>x^{-\alpha},</math> of which the simplest is hyperbolic growth, where the exponent is (negative) 1: <math>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>(t_0-t)^{-\alpha}</math> (using t for time, reversing direction to <math>-t</math> so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).

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>x^{-\alpha},</math> of which the simplest is hyperbolic growth, where the exponent is (negative) 1: <math>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>(t_0-t)^{-\alpha}</math> (using t for time, reversing direction to <math>-t</math> so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time <math>t_0</math>).

当一个输入变量为时间时，而一个输出变量在有限时间趋于无穷大时，就会出现<font color=“#ff8000”>有限时间奇点 finite-time singularity </font>，。这些在运动学和偏微分方程中很重要——无穷大在物理上并不存在，但在<font color=“#ff8000”>奇点</font>附近的行为通常是令人感兴趣的。在数学上，最简单的<font color=“#ff8000”>有限时间奇点</font>是x-α形式的各种指数的幂律，其中最简单的是双曲增长，其中指数为（负）1:x−1。更准确地说，为了随着时间的推移在正时间处获得<font color=“#ff8000”>奇点</font>（因此输出增长到无穷大），可以使用（t0−t）−α（使用t表示时间，将方向反转为−t，以便时间增加到无穷大，并将<font color=“#ff8000”>奇点</font>从0向前移动到固定时间t0）。

当一个输入变量为时间时，而一个输出变量在有限时间趋于无穷大时，就会出现<font color="#ff8000">有限时间奇点 finite-time singularity</font>。这些在运动学和偏微分方程中很重要——无穷大在物理上并不存在，但在<font color="#ff8000">奇点</font>附近的行为通常是令人感兴趣的。在数学上，最简单的<font color="#ff8000">有限时间奇点</font>是x-α形式的各种指数的幂律，其中最简单的是双曲增长，其中指数为（负）1:x−1。更准确地说，为了随着时间的推移在正时间处获得<font color="#ff8000">奇点</font>（因此输出增长到无穷大），可以使用（t0−t）−α（使用t表示时间，将方向反转为−t，以便时间增加到无穷大，并将<font color="#ff8000">奇点</font>从0向前移动到固定时间t0）。

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).

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。如果考虑理想化的运动，即每次弹跳动能损失的比例相同，反弹的频率就变得无限大，因为球在有限时间内静止。<font color=“#ff8000”>有限时间奇点</font>的其他例子包括潘列夫悖论的各种形式（例如，在黑板上拖动粉笔时，粉笔会跳跃的趋势），以及在平面上旋转的硬币的进动率如何在突然停止之前加速到无限大（正如使用欧拉圆盘玩具所研究的那样）。

一个例子是一个非弹性球在平面上的反弹运动。如果考虑理想化的运动，即每次弹跳动能损失的比例相同，反弹的频率就变得无限大，因为球在有限时间内静止。<font color="#ff8000">有限时间奇点</font>的其他例子包括潘列夫悖论的各种形式（例如，在黑板上拖动粉笔时，粉笔会跳跃的趋势），以及在平面上旋转的硬币的进动率如何在突然停止之前加速到无限大（正如使用欧拉圆盘玩具所研究的那样）。

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."

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<font color=“#ff8000”>奇点</font>是簇中切线空间可能没有规则定义的一点。<font color=“#ff8000”>奇点</font>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<font color=“#ff8000”>奇点</font>，比如尖点。例如，方程 -x = 0定义了一条在原点有一个尖点的曲线。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”。

在代数几何中，代数簇的<font color="#ff8000">奇点</font>是簇中切线空间可能没有规则定义的一点。<font color="#ff8000">奇点</font>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<font color="#ff8000">奇点</font>，比如尖点。例如，方程 -x = 0定义了一条在原点有一个尖点的曲线。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”。

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.

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.

对于仿射簇和射影簇，<font color=“#ff8000”>奇点</font>是指<font color="#ff8000"> 雅可比矩阵Jacobian matrix</font>的秩低于簇中其他点的秩的点。

对于仿射簇和射影簇，<font color="#ff8000">奇点</font>是指<font color="#ff8000"> 雅可比矩阵Jacobian matrix</font>的秩低于簇中其他点的秩的点。

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.

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，那么该点为<font color=“#ff8000”>奇点</font>。

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，那么该点为<font color="#ff8000">奇点</font>。