Singularity (mathematics)

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 is not 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 in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.

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 sub-types, and type II, which also can be divided into two subtypes, but normally is not.

To describe these types two limits are used. Suppose that is a function of a real argument , and for any value of its argument, say , then the left-handed limit, , and the right-handed limit, , are defined by:

, constrained by and
, constrained by .

The value is the value that the function tends towards as the value approaches from below, and the value is the value that the function tends towards as the value approaches from above, regardless of the actual value the function has at the point where  .

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

does not tend towards anything as approaches . The limits in this case are not infinite, but rather undefined: there is no value that settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.

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

A coordinate singularity (or coördinate singularity) occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example 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 latitude/longitude with n-vector.

Complex analysis

In complex analysis there are several classes of singularities, described below.

Isolated singularities

Suppose U is an open subset of the complex numbers C, and the point a is an element of U, and f is a complex differentiable function defined on some neighborhood around a, excluding a: U \ {a}.

Nonisolated singularities

Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. Namely, two kinds of nonisolated singularities exist:

Branch points

Finite-time singularity

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 which the simplest is hyperbolic growth, where the exponent is (negative) 1: More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses (using t for time, reversing direction to so time increases to infinity, and shifting the singularity forward from 0 to a fixed 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 the Painlevé paradox in various forms (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).

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

Algebraic geometry and commutative algebra

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 y2 x3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-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 x-axis is a "double tangent."

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.

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.

See also

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