Related papers: Euler's explorations of extremal ellipses

We give an overview of results on irregular complex surfaces of general type, discussing in particular the distribution of the numerical invariants self-intersection of a canonical divisor and holomorphic Euler characteristic for the…

Inspired by Frehse's [1] 1973 work, we show that his elliptic system $\Delta u = F(u, \nabla u)$ in the plane has bounded weak solutions $u$ with arbitrarily prescribed singular sets.

In this article we consider the Euler-$\alpha$ system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary…

We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of…

Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far…

We show that if the eccentricity of an ellipse is sufficiently small then up to isometries it is spectrally unique among all smooth domains. We do not assume any symmetry, convexity, or closeness to the ellipse, on the class of domains. In…

We survey and study some aspects of the distribution of primes in very short intervals.

Suppose that we are given two distinct points, $P_1$ and $P_2$, in the interior of a triangle, $T$. Is there always an ellipse inscribed in $T$ which also passes through $P_1$ and $P_2$ ? If yes, how many such ellipses ? We answer those…

We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…

We prove existence of strong solutions to a family of some semilinear parabolic free boundary problems by means of elliptic regularization. Existence of solutions is obtained in two steps: we first show some uniform energy estimates and…

Inspired by the concept of evolutoids of planar curves, we present the concept of evolutoids for regular surfaces as an envelope of a two-parameter family of lines in Euclidean 3-space. We give an explicit parametrization for such…

This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply…

This paper examines an averaging technique in which the nonlinear flux term is expanded and the convective velocities are passed through a low-pass filter. It is the intent that this modification to the nonlinear flux terms will result in…

We investigate the existence of elliptic islands for a special family of periodic orbits of a two-parameter family of maps corresponding to the billiard problem on the elliptical stadium. The hyperbolic or elliptical character of these…

On the perimeter length determination of the eight-centered oval. Several studies have shown that an eight-centered oval coincides almost perfectly with the ellipse constructed on the same axes and can be considered as a representation of…

We give a quick tour through many of the classical results in the field of minimal submanifolds, starting at the definition. The field of minimal submanifolds remains extremely active and has very recently seen major developments that have…

We determine when a quasi-isometry between discrete spaces is at bounded distance from a bilipschitz map. From this we prove a geometric version of the Von Neumann conjecture on amenability. We also get some examples in geometric groups…

Leonhard Euler, the most prolific mathematician in history, contributed to advance a wide spectrum of topics in celestial mechanics. At the Saint Petersburg Observatory, Euler observed sunspots and tracked the movements of the Moon.…

During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th century). In the words of Andre Weil (1989) "as…

A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally…