Gleason parts separated by smooth curves

A Geometric Characterization of Gleason Parts

Smooth Self-Similar Curves

I present a technique for constructing self-similar curves from smooth base curves. The technique is similar to that used in Iterated Function Systems like the Koch curve, except that it does not require a piecewise linear path in order to induce a set of similarities. I explain the mathematical machinery behind the technique, describe a practical numerical approximation that can be implemented...

Partitions into Distinct Parts and Elliptic Curves

A positive integer D is called a ‘congruent number’ if there exists a right triangle with rational sidelengths with area D. Over the centuries there have been many investigations attempting to classify the congruent numbers, but little was known until Tunnell [T] brilliantly applied a tour de force of methods and provided a conditional solution to this problem. It turns out that a square-free i...

Unstable Products of Smooth Curves

We give examples of smooth manifolds with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kähler classes that do not admit any constant scalar curvature Kähler metrics. These manifolds also have unstable Hilbert and Chow points. We compare this to the work of Song-Weinkove on the J-flow.

The Estimation of Smooth Curves*

Smooth curves a.re often used to illustrate the relationship between two variables. They are also an important building block in many recent statistical models. A procedure to estimate such a curve is called a smoother. This paper discusses currently available smoothers and introduces the class of maximum likelihood smoothers. A variety of other statistical techniques are shown to be applicable...