Related papers: Circle Packings in the Unit Disc

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is…

We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analysing the rules for phyllotactic indices of related structures and the variation…

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

S. Blank solved the question of classifying immersed circles in $\mathbb{R}^{2}$ that extend to immersed disks, and how many topologically inequivalent disks can be extended. The quetions of various cases in $2$-dimension have already been…

We prove uniqueness, up to diffeomorphism, of symplectically aspherical fillings of certain unit cotangent bundles, including those of higher-dimensional tori.

We provide some new sharp embeddings for p-Carleson and related measures in the unit disk of the complex plane

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature,…

For each k >= 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce m(k) = max[(k-1)!/ 2, 1] packings of h(k) equal disks inside a circle which we call "the curved hexagonal packings". The curved hexagonal packing of 7 disks…

We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a…

In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to…

We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result…

Studies of random close packing of spheres have advanced our knowledge about the structure of systems such as liquids, glasses, emulsions, granular media, and amorphous solids. When these systems are confined their structural properties…

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over…

Given N points in the plane $P_1 P_2...P_N$ and a location $\Omega$, the union of discs with diameters $[\Omega P_i], i = 1, 2,...N$ covers the convex hull of the points. The location $\Omega_s$ minimizing the area covered by the union of…

We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This…

Let $L \subset {\Bbb R}^3$ be the union of unit balls, whose centres lie on the $z$-axis, and are equidistant with distance $2d \in [2, 2\sqrt{2}]$. Then a packing of unit balls in ${\Bbb R}^3$ consisting of translates of $L$ has a density…

Let $\mathbb{D}$ be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: $$B(z)=e ^{is}\prod_{k=1}^d \frac{z-a_k}{1-z \overline{a_k}}.$$ The Lebesgue measure of the…

The multiskyrmions with large baryon number B given by rational map (RM) ansaetze can be described reasonably well within the domain wall approximation, or as spherical bubbles with energy and baryon number density concentrated at their…

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.

Consider a smooth complex surface $X$ which is a double cover of the projective plane $\mathbb{P}^2$ branched along a smooth curve of degree $2s$. In this article, we study the geometric conditions which are equivalent to the existence of…