Related papers: Statistical Regularity of Apollonian Gaskets
The Apollonian packings (APs) are fractals that result from a space-filling procedure with spheres. We discuss the finite size effects for finite intervals $s\in[s_\mathrm{min},s_\mathrm{max}]$ between the largest and the smallest sizes of…
Continued star formation over the lifetime of a galaxy suggests that low metalicity gas is steadily flowing in from the circumgalactic medium. Also, cosmological simulations of large-scale structure formation imply that gas is accreted onto…
The construction of a Laplacian on a class of fractals which includes the Sierpinski gasket ({\bf $SG$}) has given rise to an intensive research on analysis on fractals. For instance, a complete theory of polynomials and power series on…
We develop a theory of Sobolev orthogonal polynomials on the Sierpi\'nski gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of…
We study caustics of an elliptical paraboloid and the history of their various representations from 3D models in XIX century to the recent computer graphics. In the paper two ways of generating the surface, one with cartesian coordinates…
The depth function of three numbers representing curvatures of three mutually tangent circles is introduced. Its 2D plot leads to a partition of the moduli space of the triples of mutually tangent circles/disks that is unexpectedly a…
The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent…
In this paper, we study the topological behavior of elementary planes in the Apollonian orbifold $M_A$, whose limit set is the classical Apollonian gasket. The existence of these elementary planes leads to the following failure of…
The solution of Apollonius' problem on constructing a circle (line), tangent to three given circles (lines), is presented in terms of oriented circles and inversive invariants. Tangency is understood as the coincidence of tangent vectors at…
The Apollonius problem asks for a sphere tangent to $n+1$ given spheres or hyperplanes in $\mathbb{R}^n$. This problem has been widely studied for an isolated configuration of $n+1$ spheres. In this paper, we study relations among the…
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,…
We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different…
Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the…
The focus of this paper is on the study of specific circle formations known as orthogonal Pappus chains and the related incidence results that involve points of tangency between the circles in the construction. These chains give rise to new…
Four mutually tangent spheres form two gaps. In each of these, one can inscribe in a unique way four mutually tangent spheres such that each one of these spheres is tangent to exactly three of the original spheres. Repeating the process…
In an Abelian gauge symmetry, spontaneously broken at a first order phase transition, we investigate the evolution of two and three bubbles of the broken symmetry phase. The full field equations are evolved and we concentrate in particular…
The arcs of Neptune - Fraternit\'e, Egalit\'e, Libert\'e, and Courage - are four incomplete rings immersed in the Adams ring. A recent confinement model for the arcs proposes that the structures are azimuthally confined by four co-orbital…
We propose a scheme for dynamically creating orbitally entangled electron-hole pairs through a time-dependent variation of the electrical potential in a mesoscopic conductor. The time-dependent potential generates a superposition of…
We consider two seemingly unconnected problems: first, under what circumstances are arithmetic groups like SL(2,O) generated by elementary matrices; second, when do certain classes of circle/sphere packings fill up space? We show that these…
We describe how AGN-produced cosmic rays form large X-ray cavities and radio lobes in the hot diffuse gas in galaxy groups and clusters. Cosmic rays are assumed to be produced in a small shocked region near the cavity center, such as at the…