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Singularity (mathematics) 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. [1][2][3] For example, the reciprocal function has a singularity at , where the ...
An example is given by the map from the affine plane A 2 to the conical singularity x 2 + y 2 = z 2 taking (X,Y) to (2XY, X 2 − Y 2, X 2 + Y 2). The XY -plane is already nonsingular so should not be changed by resolution, and any resolution of the conical singularity factorizes through the minimal resolution given by blowing up the singular ...
But it is important to note that a real variety may be a manifold and have singular points. For example the equation y 3 + 2x 2 y − x 4 = 0 defines a real analytic manifold but has a singular point at the origin. [2] This may be explained by saying that the curve has two complex conjugate branches that cut the real branch at the origin.
Singularity theory. 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. In some places the flat string will cross itself ...
Milnor number. In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor number of f, denoted μ (f), is either a nonnegative integer, or is infinite. It can be considered both a geometric invariant and ...
Point a is an ordinary point when functions p1(x) and p0(x) are analytic at x = a. Point a is a regular singular point if p1(x) has a pole up to order 1 at x = a and p0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point. We can check whether there is an irregular singular point at infinity by using the ...
Monodromy. The imaginary part of the complex logarithm. Trying to define the complex logarithm on C \ {0} gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of C \ {0} by a helicoid (an example of a Riemann surface). In mathematics, monodromy is the study of how objects from ...
Singular solution. A singular solution ys (x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) 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 ...
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