Related papers: A note on unbounded Apollonian disk packings
A parametrization of integral Descartes configurations (and effectively Apollonian disk packings) by pairs of two-dimensional integral vectors is presented. The vectors, called here tangency spinors defined for pairs of tangent disks, are…
We give a necessary and sufficient condition so that a pair of disjoint Jordan regions in the sphere can be quasiconformally mapped to a pair of disks. As a consequence, we obtain a simple characterization that involves Lipschitz functions…
We use results of [6] to enlarge our knowledge of the approximate fixed point property (AFPP) for digital images in $\mathbb{Z}^2$. In particular, we study conditions under which the union of two convex digital disks has the AFPP.
It is shown how to embed the polydisk algebras (finite and infinite ones) into the disk algebra $A(\overline{\mathbb D})$. As a consequence, one obtains uniform closed subalgebras of $A(\overline{\mathbb D})$ which have arbitrarily…
The current knowledge of the abundance structure in the Milky Way is reviewed. Special emphasis is put on recent results for stars with kinematics typical of the thin and the thick disks, respectively, and how these results can, apart from…
A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices are familiarly linked to Clifford algebra and quaternions. Somewhat more loosely, we develop connections between the su(4) algebra of two spins and of its…
An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal…
In this paper we present a new formula for the number of unrestricted partitions of $n$. We do this by introducing a correspondence between the number of unrestrited partitions of $n$ and the number of non-negative solutions of systems of…
We aim to quantify the chemical and kinematical properties of the Galactic disks with a sample of 119,558 giant stars having abundances and 3D velocities taken or derived from the APOGEE DR17 and Gaia EDR3 catalogs. The Gaussian Mixture…
Positively graded algebras are fairly natural objects which are arduous to be studied. In this article we query quotients of non-standard graded polynomial rings with combinatorial and commutative algebra methods.
Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of…
The plane partition polynomial $Q_n(x)$ is the polynomial of degree $n$ whose coefficients count the number of plane partitions of $n$ indexed by their trace. Extending classical work of E.M. Wright, we develop the asymptotics of these…
The Golden Ratio is fascinating topic that continually generated news ideas. A Riemannian manifold endowed with a Golden Structure will be called a Golden Riemannian manifold. The main purpose of the present paper is to study the geometry…
We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$…
This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each…
We investigate accreting disk systems with polytropic gas in Keplerian motion. Numerical data and partial analytic results show that the self-gravitation of the disk speeds up its rotation -- its rotational frequency is larger than that…
In a previous paper we introduced examples of Hamiltonian mappings with phase space structures resembling circle packings. It was shown that a vast number of periodic orbits can be found using special properties. We now use this information…
We call a packing of hyperspheres in $n$ dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the $n$-dimensional space; and every sphere in the packing is a member of a cluster of $n+2$…
We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…
We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere, with nonnegative normal bundles.