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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.
The pinch point (in this case the origin) is a limit of normal crossings singular points (the -axis in this case). These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole v {\displaystyle v} -axis and not only the pinch point.
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.
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]
Consider a smooth real-valued function of two variables, say f (x, y) where x and y are real numbers.So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target.
This means that the subset of points without a second derivative has measure zero in the curve. However, in other senses, the set of points with a second derivative can be small. In particular, for the graphs of generic non-smooth convex functions, it is a meager set, that is, a countable union of nowhere dense sets. [42]
Khetarpal points to the Huangluo village in China, whose women are famous for their long hair. "Rice water is actually part of the regimen that this town attributes to their hair to get to that ...
Repeatedly blowing up the singular points of a curve will eventually resolve the singularities. The main task with this method is to find a way to measure the complexity of a singularity and to show that blowing up improves this measure. There are many ways to do this. For example, one can use the arithmetic genus of the curve.