Effective action in spherical domains

The effective action on an orbifolded sphere is computed for minimally coupled scalar fields. The results are presented in terms of derivatives of Barnes zeta-functions and it is shown how these may be evaluated. Numerical values are shown. An analytical, heat-kernel derivation of the Ces\`aro-Fedorov formula for the number of symmetry planes of a regular solid is also presented.Comment: 18 pages, Plain TeX (Mailer oddities possibly corrected.

Studying Wythoff and Zometool Constructions using Maple

We describe a Maple package that serves at least four purposes. First, one can use it to compute whether or not a given polyhedral structure is Zometool constructible. Second, one can use it to manipulate Zometool objects, for example to determine how to best build a given structure. Third, the package allows for an easy computation of the polytopes obtained by the kaleiodoscopic construction called the Wythoff construction. This feature provides a source of multiple examples. Fourth, the package allows the projection on Coxeter planesComment: 11 pages, 11 figure

Exploiting classical nucleation theory for reverse self-assembly

In this paper we introduce a new method to design interparticle interactions to target arbitrary crystal structures via the process of self-assembly. We show that it is possible to exploit the curvature of the crystal nucleation free-energy barrier to sample and select optimal interparticle interactions for self-assembly into a desired structure. We apply this method to find interactions to target two simple crystal structures: a crystal with simple cubic symmetry and a two-dimensional plane with square symmetry embedded in a three-dimensional space. Finally, we discuss the potential and limits of our method and propose a general model by which a functionally infinite number of different interaction geometries may be constructed and to which our reverse self-assembly method could in principle be applied.Comment: 7 pages, 6 figures. Published in the Journal of Chemical Physic

Mixing Convex Polytopes

The mixing operation for abstract polytopes gives a natural way to construct the minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope, and completely determining the structure of the mix in each case

Geometrical Frustration and Static Correlations in Hard-Sphere Glass Formers

We analytically and numerically characterize the structure of hard-sphere fluids in order to review various geometrical frustration scenarios of the glass transition. We find generalized polytetrahedral order to be correlated with increasing fluid packing fraction, but to become increasingly irrelevant with increasing dimension. We also find the growth in structural correlations to be modest in the dynamical regime accessible to computer simulations.Comment: 21 pages; part of the "Special Topic Issue on the Glass Transition

Infinitesimal rigidity of a compact hyperbolic 4-orbifold with totally geodesic boundary

Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.Comment: 9 page

Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams

This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures

Majority-vote model on hyperbolic lattices

We study the critical properties of a non-equilibrium statistical model, the majority-vote model, on heptagonal and dual heptagonal lattices. Such lattices have the special feature that they only can be embedded in negatively curved surfaces. We find, by using Monte Carlo simulations and finite-size analysis, that the critical exponents $1/\nu$, $\beta/\nu$ and $\gamma/\nu$ are different from those of the majority-vote model on regular lattices with periodic boundary condition, which belongs to the same universality class as the equilibrium Ising model. The exponents are also from those of the Ising model on a hyperbolic lattice. We argue that the disagreement is caused by the effective dimensionality of the hyperbolic lattices. By comparative studies, we find that the critical exponents of the majority-vote model on hyperbolic lattices satisfy the hyperscaling relation $2\beta/\nu+\gamma/\nu=D_{\mathrm{eff}}$, where $D_{\mathrm{eff}}$ is an effective dimension of the lattice. We also investigate the effect of boundary nodes on the ordering process of the model.Comment: 8 pages, 9 figure

On the epistemic view of quantum states

We investigate the strengths and limitations of the Spekkens toy model, which is a local hidden variable model that replicates many important properties of quantum dynamics. First, we present a set of five axioms that fully encapsulate Spekkens' toy model. We then test whether these axioms can be extended to capture more quantum phenomena, by allowing operations on epistemic as well as ontic states. We discover that the resulting group of operations is isomorphic to the projective extended Clifford Group for two qubits. This larger group of operations results in a physically unreasonable model; consequently, we claim that a relaxed definition of valid operations in Spekkens' toy model cannot produce an equivalence with the Clifford Group for two qubits. However, the new operations do serve as tests for correlation in a two toy bit model, analogous to the well known Horodecki criterion for the separability of quantum states.Comment: 16 pages, 9 figure