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A curve with a triple point at the origin: x(t) = sin(2t) + cos(t), y(t) = sin(t) + cos(2t) In general, if all the terms of degree less than k are 0, and at least one term of degree k is not 0 in f, then curve is said to have a multiple point of order k or a k-ple point.
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]
One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the x-axis is a "double tangent." 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.
The problem is to find a resolution that is an isomorphism over the set of smooth and simple normal crossing points. When X is a divisor, i.e. it can be embedded as a codimension-one subvariety in a smooth variety it is known to be true the existence of the strong resolution avoiding simple normal crossing points.
This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials.
Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points.Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory.
Point a is an ordinary point when functions p 1 (x) and p 0 (x) are analytic at x = a. Point a is a regular singular point if p 1 (x) has a pole up to order 1 at x = a and p 0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point.
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 .