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Encyclopedia > Mathematical singularity

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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is well-behaved or not. ... In mathematics, a derivative is the rate of change of a quantity. ... For non-mathematical singularity theories, see singularity. ...

For example, the function Partial plot of a function f. ...

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

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. In mathematics, the real line is simply the set of real numbers. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

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. In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

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}. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

• 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) / (za)n for all z in U − {a}.

These three types of singularities are isolated. The fourth type is branch points; they require a more verbose definition, see branch point. In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In complex analysis, an essential singularity of a function is a severe singularity near which the function exhibits extreme behavior. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... A Laurent series is defined with respect to a particular point c and a path of integration Î³. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... In complex analysis, a branch of mathematics, an isolated singularity is a singularity which contains no other singularities close to it. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ...

## 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. Kinetic energy is the energy by virtue of the motion of an object. ... Sine waves of various frequencies; the bottom waves have higher frequencies than those above. ... Eulers disk, named after Leonhard Euler, is a circular disk that spins, without slipping, on a surface. ...

## Algebraic geometry and commutative algebra

See main article singular point In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...

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, y2x3 = 0 defines an isolated singular point (at the cusp) x = y = 0. The ring in question is given by Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules based on such rings; and of fields and their algebras. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension. ...

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

The maximal ideal of the localization at (t2,t3) is a height one local ring generated by two elements and thus not regular.

## 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. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

## 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 Km and v in Kn such that In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

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

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

$M = USigma V^* ,!$

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. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Linear least squares is a mathematical optimization technique to find an approximate solution for a system of linear equations that has no exact solution. ... In statistics, principal components analysis (PCA) is a technique for simplifying a dataset. ...

Results from FactBites:

 Mathematical singularity - Wikipedia, the free encyclopedia (599 words) 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. 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). In linear algebra a square matrix is said to be singular when it is not invertible, that is when its determinant is zero.
 Singularity - Wikipedia, the free encyclopedia (178 words) mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved. gravitational singularity (physics) - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum, namely fl holes, white holes and worm holes. technological singularity - a theoretical point in the development of a scientific civilization at which technological progress accelerates into infinity or beyond prediction.
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