Related papers: Statistical Regularity of Apollonian Gaskets
The initial conditions describing seed fluctuations for the formation of structure in standard cosmological models, i.e.the Harrison-Zeldovich distribution, have very characteristic ``super-homogeneous'' properties: they are statistically…
Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms.…
A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of…
We use polynomial method techniques to bound the number of tangent pairs in a collection of $N$ spheres in $\mathbb{R}^n$ subject to a non-degeneracy condition, for any $n \geq 3$. The condition, inspired by work of Zahl for $n=3$, asserts…
We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find…
The current article continues our project on representation theory, Euler elements, causal homogeneous spaces and Algebraic Quantum Field Theory (AQFT). We call a pair (h,k) of Euler elements orthogonal if $e^{\pi i \ad h} k = -k$. We show…
Galactic orbits have been constructed over long time intervals for ten globular clusters located near the Galactic center. A model with an axially symmetric gravitational potential for the Galaxy was initially applied, after which a…
The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity…
The formation of a single bubble from an orifice in a solid surface, submerged in an in- compressible, viscous Newtonian liquid, is simulated. The finite element method is used to capture the multiscale physics associated with the problem…
The shape of galaxies depends on their orbital populations. These populations change through capture into and escape from resonance. Capture problems fall into distinct cases depending upon the shape of the potential well. To visualise the…
The Orion star-forming region is the nearest active high-mass star-forming region and has created a large superbubble, the Orion-Eridanus superbubble. Recent work by Ochsendorf et al. has extended the accepted boundary of the superbubble.…
Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at…
We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…
We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces, under the assumption that the dimension of the group is at least as large as the…
We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially…
Mass generation of gauge fields can be universally described by topological couplings in gapped systems, such as the Abelian Higgs model in $(3+1)$ dimensions and the Maxwell-Chern-Simons theory in $(2+1)$ dimensions. These systems also…
We provide a derivation of the tight-binding model that emerges from a full consideration of a particle bound in a periodic one-dimensional array of square well potentials, separated by barriers of height $V_0$ and width $b$. We derive the…
Quantum field theory predicts that dynamical curved spacetimes can spontaneously excite particle pairs from the quantum vacuum, a phenomenon extensively studied in expanding universes and in scenarios involving gravitational collapse. In…
In an earlier work, we proposed a generalization for the Apollonian packing in arbitrary dimensions and showed that the resulting object in four, five, and six dimensions have properties consistent with the Apollonian circle and sphere…
This paper uses the assumptions of ergodicity and a microcanonical distribution to compute estimates of the largest Lyapunov exponents in lower-dimensional Hamiltonian systems. That the resulting estimates are in reasonable agreement with…