Related papers: Apollonian circle packings: Dynamics and Number th…
We use the planar circular restricted three-body problem in order to numerically investigate the orbital dynamics of orbits of a spacecraft, or a comet, or an asteroid in the Saturn-Titan system in a scattering region around the Titan. The…
We introduce a new class of fractal circle packings in the plane, generalizing the polyhedral packings defined by Kontorovich and Nakamura. The existence and uniqueness of these packings are guaranteed by infinite versions of the…
We study Apollonian circle packings in relation to a certain rank 4 indefinite Kac-Moody root system $\Phi$. We introduce the generating function $Z(\mathbf{s})$ of a packing, an exponential series in four variables with an Apollonian…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
This paper concerns the number of lattice points in a circle.
We overview our recent studies of cosmological models with expansion and global rotation. Problems of the early rotating models are discussed, and the class of new viable cosmologies is described in detail. Particular attention is paid to…
These are notes for a mini-course of 3 lectures given at the St. Petersburg School in Probability and Statistical Physics (June 2012). My aim was to explain, on the example of a particular model, how ideas from the representation theory of…
The different dynamical processes (relaxation, dynamical friction, tides and mergers) operating in groups and clusters are reviewed. The small-scale substructure observed in clusters is argued to be the remnants of the cores of rich…
Motivated by a recently identified severe discrepancy between a static and a dynamic theory of glasses, we numerically investigate the behavior of dense hard spheres in spatial dimensions 3 to 12. Our results are consistent with the static…
This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the…
A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so…
During the last few years several new results on packing problems were obtained using a blend of tools from semidefinite optimization, polynomial optimization, and harmonic analysis. We survey some of these results and the techniques…
Table of contents (abridged): COLD FRONTS Origin and evolution of merger cold fronts Cold fronts in cluster cool cores . . . Simulations of gas sloshing. Origin of density discontinuity. . . . Effect of sloshing on cluster mass estimates…
Efforts to understand and map the possible explanations for the late time acceleration of the universe have led to a broad range of suggestions, ranging from the cosmological constant and straightforward dark energy, to exotically coupled…
Identification of first and second turnaround radii in groups of galaxies provide interesting scaling relations and may constrain cosmological parameters.
How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of…
In this chapter we review concepts and theories of polymer dynamics. We think of it as an introduction to the topic for scientists specializing in other subfields of statistical mechanics and condensed matter theory, so, for the readers…
This paper deals with the problem of circle packing, in which the largest radii circle is to be fit in a confined space filled with arbitrary circles of different radii and centers. A circle packing problem is one of a variety of cutting…
The article presents the mathematical sequences describing circle packing densities in four different geometric configurations involving a hexagonal lattice based equal circle packing in the Euclidian plane. The calculated sequences take…
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…