Mathematical singularity

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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 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

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.

1 Complex analysis
2 From the point of view of dynamics
3 Algebraic geometry and commutative algebra
4 Singular matrices
5 Singular value decomposition
6 See also

[edit] Complex analysis

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.

[edit] From the point of view of dynamics

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.

[edit] Algebraic geometry and commutative algebra

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, <math>y^2 - x^3 = 0</math> defines an isolated singular point (at the cusp) <math>x = y = 0</math>. The ring in question is given by

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

[edit] Singular matrices

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

[edit] Singular value decomposition

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

<math>Mv = \sigma u \,\mbox{ and } M^*u = \sigma v. \,\!</math>

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

<math>M = U\Sigma V^* \,\!</math>

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.