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Hence, it is technically more correct to discuss singular points of a smooth mapping here rather than a singular point of a curve. The above definitions can be extended to cover implicit curves which are defined as the zero set of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be ...
When a is a regular singular point, which by definition means that has a pole of order at most i at a, the Frobenius method also can be made to work and provide n independent solutions near a. Otherwise the point a is an irregular singularity .
Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers). [1]
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
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 ...
The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve.
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
This sort of definition extends to differentiable maps between and , a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds , as the points where the rank of the Jacobian matrix decreases.