In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. See singularity theory for general discussion of the geometric theory, which only covers some aspects.

For example, the function

$f(x)=\backslash frac\{1\}\{x\}$

on the real line has a singularity at x = 0, where it seems to "explode" to ±∞ and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it isn't differentiable there. Similarly, the graph defined by y² = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point.

The algebraic set defined by y² = x² in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.

Complex analysis

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In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, a is an element of U, and f is a holomorphic function defined on U \ {a}.

• The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − {a}.

• The point a is a pole of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (z − a)ⁿ for all z in U − {a}.

• The point a is an essential singularity of f if is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.

These three types of singularities are isolated. The fourth type is branch points; they require a more verbose definition, see branch point.

From the point of view of dynamics

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A finite-time singularity occurs when a kinematic variable increases towards infinity at a finite time. 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 Euler's disk and the Painlevé paradox.

Algebraic geometry and commutative algebra

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See main article singular point

In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a scheme (mathematics) with a stalk that is not a regular local ring). For example, $y^2\; -\; x^3\; =\; 0$ defines an isolated singular point (at the cusp) $x\; =\; y\; =\; 0$. The ring in question is given by

$C/\; (y^2\; -\; x^3)\; \backslash cong\; C[t^2,\; t^3.$

The maximal ideal of the localization at $(t^2,\; t^3)$ is a height one local ring generated by two elements and thus not regular.

Singular matrices

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In linear algebra a square matrix is said to be singular when it is not invertible, that is when its determinant is zero.

Singular value decomposition

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Singular value decomposition (SVD) is a method of factorizing matrices. A non-negative real number σ is a singular value for M if and only if there exist normalized vectors u in K^m and v in Kⁿ such that

$Mv\; =\; \backslash sigma\; u\; \backslash ,\backslash mbox\{\; and\; \}\; M^*u\; =\; \backslash sigma\; v.\; \backslash ,\backslash !$

The vectors u and v are called left-singular and right-singular vectors for σ, respectively. The factorisation is

$M\; =\; U\backslash Sigma\; V^*\; \backslash ,\backslash !$

where diagonal entries of Σ are equal to the singular values of M. The columns of U and V are left- resp. right-singular vectors for the corresponding singular values. It is widely used in statistics where it is used as a technique for solving linear least squares problems and is related to principal components analysis.

See also

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• Singular solution of a differential equation
• Catastrophe theory

Mathematical analysis

Singularität (Mathematik) | 特異点 | Singularidade matemática | Особенность | Matematická singularita | matematična singularnost