We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications. (a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli ...

We provide a treatment of algebro-geometric solutions of the classical massive Thirring system in the framework of the Weierstrass–Klein theory of hyperelliptic functions. We show that the equations of this model generate the characteristic relations of hyperelliptic theory of even hyperelliptic curves, the same role that the KdV equation plays for odd hyperelliptic curves. We also consider the...

A new five-point binary subdivision scheme with high continuity and convexity preservation is proposed in this paper. It is shown that the limit curves are C (k = 0, 1, . . . , 7) continuous for the certain range of the parameter. The range of the parameter for the property of convexity preservation of the limit curves is also provided. Experimental results demonstrate the efficiency and flexib...

The flattening of spiral-galaxy rotation curves is unnatural in view the expectations from Kepler's third law and a central mass. It interesting, however, that radius-independence velocity what one expects less dimension. In our three-dimensional space, curve natural if, outside galaxy's center, gravitational potential corresponds to very prolate ellipsoid, filament, string, or otherwise cylind...

New criteria are established for upper bounds on the number of limit cycles periodic Abel differential equations having two invariant curves, one them bounded. The applied to obtain either zero or cycle planar systems.

This paper illustrates and extends an efficient framework, called the square-root-elastic (SRE) framework, for studying shapes of closed curves, that was first introduced in [2]. This framework combines the strengths of two important ideas - elastic shape metric and path-straightening methods - for finding geodesics in shape spaces of curves. The elastic metric allows for optimal matching of fe...