相关论文: Scaling Ratios and Triangles in Siegel Disks

It is known that solutions of the parabolic elliptic Keller-Segel equations in the two dimensional plane decay, as time goes to infinity, provided the initial data admits sub-critical mass and finite second moments, while such solution…

The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is…

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group $\mathbb{F}_2^n$. A triangle in $\mathbb{F}_2^n$ is a triple $(x,y,z)$ such that $x+y+z = 0$. The triangle removal lemma for…

A simple proof of the classical subconvexity bound $\zeta(1/2+it) \ll_\epsilon t^{1/6+\epsilon}$ for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor…

In this paper, we study a particular conservative standard map in complex dimension 2. In this example, Siegel disks can be visualized and analyzed numerically as to the smoothness of their boundaries. We formulate and numerically support…

For an infinite word $\mathbf{x}$, Bugeaud and Kim introduced a new complexity function $\text{rep}(\mathbf{x})$ which is called the exponent of repetition of $\mathbf{x}$. They showed $1\le \text{rep}(\mathbf{x}) \le \sqrt{10}-\frac{3}{2}$…

We consider the eigenvalue problem $\Delta^{\mathbb{S}^2} \xi + 2 \xi=0 $ in $ \Omega $ and $\xi = 0 $ along $ \partial \Omega $, being $\Omega$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in…

For $\Gamma={\hbox{PSL}_2( {\mathbb Z})}$ the hyperbolic circle problem aims to estimate the number of elements of the orbit $\Gamma z$ inside the hyperbolic disc centered at $z$ with radius $\cosh^{-1}(X/2)$. We show that, by averaging…

In [B.Gruenbaum, G.C. Shephard, Spherical tilings with transitivity properties, in: The geometric vein, Springer, New York, 1981, pp. 65-98], they proved "for every spherical normal tiling by congruent tiles, if it is isohedral, then the…

Under the assumption of infinitely many Siegel zeroes $s$ with $Re(s)>1-\frac{1}{(\log q)^{R}}$ for a sufficiently large value of $R$, we prove that there exist infinitely many $m$-tuples of primes that are $\ll e^{1.9828m}$ apart. This…

The Einstein tile is a novel type of non-periodic tile that can cover the plane without repeating itself. It has a simple shape that resembles a fedora. This research paper unveils the aperiodicity of the newly discovered Einstein tile…

A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…

A measure for the description of the chirality of triangles is introduced. The measure $X\delta$ is zero for triangles with at least one mirror axe, i.e. equilateral or isosceles triangles, and positive or negative for scalane, i.e. left or…

In a previous work [Scientific Reports 4, 6193(2014)] we proved the existence of scale symmetry in square and triangular (thus honeycomb) lattices by investigating the functiony=\arcsin(\sin(2\pinx)), where the parameter is either the…

We prove that for typical rotation numbers polynomial Siegel disks are Jordan domains with boundaries containing at least one critical point.

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…

In this paper we prove that as N goes to infinity, the scaling limit of the correlation between critical points z1 and z2 of random holomorphic sections of the N-th power of a positive line bundle over a compact Riemann surface tends to…

Let $\Omega$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha…

We define and study when a polynomial mapping has a local or global time average. We conjecture that a polynomial f in the complex plane has a time average near a point z if and only if z is eventually mapped into a Siegel-disc of f. We…