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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 ...
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 ...
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness.
An example where it does not is given by the isolated singularity of x 2 + y 3 z + z 3 = 0 at the origin. Blowing it up gives the singularity x 2 + y 2 z + yz 3 = 0. It is not immediately obvious that this new singularity is better, as both singularities have multiplicity 2 and are given by the sum of monomials of degrees 2, 3, and 4.
The singular values are the absolute values of the eigenvalues of a normal matrix A, because the spectral theorem can be applied to obtain unitary diagonalization of as . Therefore, . Most norms on Hilbert space operators studied are defined using singular values. For example, the Ky Fan - k -norm is the sum of first k singular values, the ...
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