Mathematical singularity
Encyclopedia
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 y2 = x2 in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent
there.
singularities are also called discontinuities
. There are three kinds: 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, suppose that
is a function of a real argument 
, and for any value of its argument, say 
, the symbols 
and 
are defined by:
, constrained by 
and
, constrained by 
.
The limit
is called the left-handed limit, and 
is called the right-handed limit. 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.
.
there are four classes of singularities, described below. Suppose U is an open subset
of the complex number
s 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}.
A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinite at a finite time. These are important in kinematics and PDEs – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically the simplest finite-time singularities are power law
s 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 (asd 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).
and commutative algebra
, a singularity is a prime ideal
whose localization
is not a regular local ring
(alternately a point of a scheme
whose stalk
is not a regular local ring
). For example,
defines an isolated singular point (at the cusp) 
. The ring in question is given by
The maximal ideal
of the localization at
is a height one local ring generated by two elements and thus not regular.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Well-behaved
Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...
in some particular way, such as differentiability
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
. See Singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
for general discussion of the geometric theory, which only covers some aspects.
For example, the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
has a singularity at x = 0, where it seems to "explode" to ±∞ and is not defined. The function g(x) = |x| (see absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
) 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 y2 = x2 in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
there.
Real analysis
In real analysisReal analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
singularities are also called discontinuities
Classification of discontinuities
Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there...
. There are three kinds: 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, suppose that
is a function of a real argument , and for any value of its argument, say , the symbols and are defined by:, constrained by and, constrained by .The limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
is called the left-handed limit, and is called the right-handed limit. 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.- A point of continuity, which is not a singularity, is a value of
for which 
, as one usually expects. All the values must be finite. - A type I discontinuity occurs when both
and 
exist and are finite, but one of three conditions also apply: 
; 
does not exist for that value of 
; or 
does not match the value that the two limits tend towards. Two subtypes occur:
- A jump discontinuity occurs when
, regardless of whether 
exists, and regardless of what value it might have if it does exist. - A removable discontinuity occurs when
, but either the value of 
does not match the limits, or the function does not exist at the point 
.
- A jump discontinuity occurs when
- A type II discontinuity occurs when either
or 
does not exist (possibly both). This has two subtypes, which are usually not considered separately:
- An infinite discontinuity is the special case when either the left hand or right hand limit does not exist specifically because it is infinite, and the other limit is either also infinite or is some well defined finite number.
- An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits
or 
does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if legal answers are extended to include 
.
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-vectorN-vector
n-vector is a three parameter non-singular horizontal position representation well-suited for replacing latitude and longitude in mathematical calculations and computer algorithms. Geometrically, it is a unit vector that is normal to the reference ellipsoid. The vector is decomposed in an Earth...
.
Complex analysis
In complex analysisComplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
there are four classes of singularities, described below. Suppose U is an open subset
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s C, and the point a is an element of U, and f is a complex differentiable function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
defined on some neighborhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
around a, excluding a: U \ {a}.
- Isolated singularities: Suppose the function f is not defined at a, although it does have values 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 function g is a continuous replacement for the function f.
- The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U and a natural numberNatural numberIn mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n such that f(z) = g(z) / (z − a)n for all z in U \ {a}. The derivative at a non-essential singularity may or may not exist. If g(a) is nonzero, then we say that a is a pole of order n. - The point a is an essential singularityEssential singularityIn complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...
of f if is neither a removable singularity nor a pole. The point a is an essential singularity if and only ifIFFIFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
the Laurent seriesLaurent seriesIn mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
has infinitely many powers of negative degree.
- Branch pointBranch pointIn the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...
s are generally the result of a multi-valued function, such as
or 
being defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The location and shape of most of the branch cut is usually a matter of choice, with perhaps only one point (like 
for 
) which is fixed in place.
Finite-time singularity
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
s for various exponents,
of which the simplest is hyperbolic growthHyperbolic growth
When a quantity grows towards a singularity under a finite variation it is said to undergo hyperbolic growth.More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as x \to 0 is infinity: any similar graph is said to exhibit...
, 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
Kinetic energy
The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
is lost on each bounce, the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of 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
Painlevé paradox
The Painlevé paradox is a well known example by Paul Painlevé in rigid-body dynamics that showed that rigid-body dynamics with both contact friction and Coulomb friction is inconsistent. This is due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in...
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
Coin
A coin is a piece of hard material that is standardized in weight, is produced in large quantities in order to facilitate trade, and primarily can be used as a legal tender token for commerce in the designated country, region, or territory....
spun on a flat surface accelerates towards infinite, before abruptly stopping (asd studied using the Euler's Disk
Euler's disk
Euler's Disk is a scientific educational toy, used to illustrate and study the dynamic system of a spinning disk on a flat surface , and has been the subject of a number of scientific papers. This phenomenon has been studied since Leonard Euler in the 18th century, hence the name.It is manufactured...
toy).
Hypothetical examples include Heinz von Foerster
Heinz von Foerster
Heinz von Foerster was an Austrian American scientist combining physics and philosophy. Together with Warren McCulloch, Norbert Wiener, John von Neumann, Lawrence J. Fogel, and others, Heinz von Foerster was an architect of cybernetics.-Biography:Von Foerster was born in 1911 in Vienna, Austria,...
's facetious "Doomsday's Equation" (simplistic models yield infinite human population in finite time).
Algebraic geometry and commutative algebra
In algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, a singularity is a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
whose localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
is not a regular local ring
Regular local 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 equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
(alternately a point of a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
whose stalk
Stalk (sheaf)
The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.-Motivation and definition:Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a...
is not a regular local ring
Regular local 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 equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
). For example,
defines an isolated singular point (at the cusp) . The ring in question is given byThe maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
of the localization at
is a height one local ring generated by two elements and thus not regular.See also
- AsymptoteAsymptoteIn analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
- Catastrophe theoryCatastrophe theoryIn mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry....
- Defined and undefined
- Division by zeroDivision by zeroIn mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
- Hyperbolic growthHyperbolic growthWhen a quantity grows towards a singularity under a finite variation it is said to undergo hyperbolic growth.More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as x \to 0 is infinity: any similar graph is said to exhibit...
- Singular solutionSingular solutionA singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...