Related papers: A note on unbounded Apollonian disk packings
We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either…
We consider the Bergman space on the complex plane. We prove an analogue of Schwarz's reflection principle for unbounded quasidisks.
This review summarises the invited presentation I gave on the Milky Way disc. The idea underneath was to touch those topics that can be considered hot nowadays in the Galactic disk research: the reality of the thick disk, the spiral…
It is known that the hyperbolic plane admits a countable infinity of compactifications into a closed disk such that the isometric action of SL(2;R) acts analytically on the compactified space. We prove that among those compactifications,…
Inspired by a question of Sarnak, we introduce the notion of a prime component in an Apollonian circle packing: a maximal tangency-connected subset having all prime curvatures. We also consider thickened prime components, which are…
The half-liberated orthogonal group $O_n^*$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^+$. We discuss here its basic algebraic properties, and we classify its irreducible…
In this paper we describe the homology and cohomology of some natural bimodules over the little discs operad, whose components are configurations of non-$k$-overlapping discs. At the end we briefly explain how this algebraic structure…
In this work we revisit classical systems of integers inside the real normed division algebras from the point of view of finite norm shells and root systems. Building on the icosian framework of Moody--Patera and on the integral root-system…
This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a…
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…
Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which…
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence…
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…
In this work, almost product and almost golden structures are studied. Conditions for those structures being Integrable and parallel are investigated. Also harmonicity of a map between almost pruduct or almost golden manifolds with pure or…
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…
We study the orbit of $\widehat{\mathbb{R}}$ under the M\"obius action of the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$ on $\widehat{\mathbb{C}}$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$.…
We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a…
In spherical symmetry with radial coordinate $r$, classical Newtonian gravitation supports circular orbits and, for $-1/r$ and $r^2$ potentials only, closed elliptical orbits [1]. Various families of elliptical orbits can be thought of as…
The connection between some features of the metallicity gradient in the Galactic disk, best revealed by Open Clusters and Cepheids, and the spiral structure, is explored. The step-like abrupt decrease in metallicity at 8.5 kpc (with R_0=…
A long-term research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings…