Packing Superballs from Codes and Algebraic Curves

In the present paper, we make use of codes with good parameters and algebraic curves over finite fields with many rational points to construct dense packings of superballs. It turns out that our packing density is quite reasonable. In particular, we improve some values for the best-known lower bounds on packing density.     

A hypersingular integral equation for the floating body problem

A new boundary integral equation formulation for the floating body problem, which is defined on a bounded domain and includes a hypersingular and a nonlocal part, is proved to satisfy a Gärding inequality. Under the assunlption of uniqueness it is shown, that an approximating sequence of related semidiscrete integral operators is uniformly bounded invertible.     

Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves

It is proved that for a simple, closed, extreme polygon  Γ⊂R³$\Gamma \subset {\mathbb{R}}^3$ every immersed, stable minimal surface spanning Γ is an isolated point of the set of all minimal surfaces spanning Γ   w.r.t. the C⁰$C^0$-topology. Since the subset of immersed, stable minimal surfaces spanning Γ is shown to be closed in the compact set of all minimal surfaces spanning Γ, this proves in particular that Γ can bound only finitely many immersed, stable minimal surfaces.     

Bubbling,riddling, blowout and critical curves

This paper provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaoti1c trajectories with the related phenomena of bubbling, on–off intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly influence the effects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.     

A free boundary and an optimal shape problem of a floating body

The present article deals with a variational formulation of a floating body in a perfect fluid. Since the considered energy functional depends on both stream function and floating body, these two minimizations are realized in two separated steps. By the use of free boundary methods, existence and Lipschitz continuity of the minimizing stream function are proved. In addition, it is shown that the corresponding free surface has Hölder boundary. The dependence on the floating body leads to an optimal shape problem which is treated by the use of Hausdorff convergence. Under the postulate of a bounded density parameter of the domain, the existence of an optimal floating body is shown.     

Hypergeometric functions and elliptic curves

We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively. 2000 Mathematics Subject Classification Primary—11G05, 33C05 This research was supported by K. Ono’s NSF grant     

Algebraic curves and maximal arcs

A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set of the Desarguesian plane PG(2,q) is obtained. The case where is a maximal (k,n)-arc is considered in greater depth. Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni.     

On the supersingular reduction of elliptic curves

Let a∈Q and denote byE _a the curvey ² = (x ² + l)(x + a). We prove thatE _a(F_p) is cyclic for infinitely many primesp. This fact was known previously only under the assumption of the generalized Riemann hypothesis. Research partially supported by NSERC grant A9418.     

Covers of elliptic curves with unique,totally ramified branch points

A well‐known and difficult problem in computational number theory and algebraic geometry is to write down equations for branched covers of algebraic curves with specified monodromy type. In this article, we present a technique for computing such covers in the case of covers of elliptic curves with unique, totally ramified branch points.     

Entire curves,integral sets and principal bundles

We compare the behaviour of entire curves and integral sets,in particular in relation to locally trivial fiber bundles,algebraic groups and finite ramified covers over semi-abelian varieties.     

On the proportion of locally soluble superelliptic curves

We investigate the proportion of superelliptic curves that have a ${\mathbb{Q}}_p$ point for every place p of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3=f(x,z)$ for an integral binary form f of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in p for the proportion of such curves over ${\mathbb{Z}}_p$ having a ${\mathbb{Q}}_p$-point.     

RATIONALITY OF THE OFFSETS TO ALGEBRAIC CURVES AND SURFACES

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Square-tiled surfaces and rigid curves on moduli spaces

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 space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

The floating body and the hyperplane conjecture

We study the relationship between a convex body ${K\subset \mathbb{R}^{n}}$ and the convex floating body K _δ inside K. Our results complement recent work of Grigoris Paouris and Elisabeth Werner.