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In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it.
Singularity (system theory), in dynamical and social systems, a context in which a small change can cause a large effect Gravitational singularity, in general relativity, a point in which gravity is so intense that spacetime itself becomes ill-defined
Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. [ 1 ] [ 2 ] [ 3 ] The functions are notated with brackets, as x − a n {\displaystyle \langle x-a\rangle ^{n}} where n is an integer.
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity).
In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties: f is continuous on [ a , b ]. there exists a set N of measure 0 such that for all x outside of N, the derivative f ′ ( x ) exists and is zero; that is, the derivative of f vanishes almost everywhere .
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation
The Whitney umbrella x 2 = y 2 z has singular set the z axis, most of whose point are ordinary double points, but there is a more complicated pinch point singularity at the origin, so blowing up the worst singular points suggests that one should start by blowing up the origin. However blowing up the origin reproduces the same singularity on one ...