Fibered orbifolds and crystallographic groups

In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurability class of normal subgroups of G. In particular, we prove that E^n/G is a fiber bundle, with totally geodesic fibers, over a b-dimensional torus, where b is the first Betti number of G. Let N be a normal subgroup of G which is maximal in its commensurability class. We study the relationship between the exact sequence 1 -> N -> G -> G/N -> 1 splitting and the corresponding fibration projection having an affine section. If N is torsion-free, we prove that the exact sequence splits if and only if the fibration projection has an affine section. If the generic fiber F = Span(N)/N has an ordinary point that is fixed by every isometry of F, we prove that the exact sequence always splits. Finally, we describe all the geometric fibrations of the orbit spaces of all 2- and 3-dimensional crystallographic groups building on the work of Conway and Thurston.Comment: 26 pages, 1 Table. Some new theorems have been added to v

Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function

A new method is presented for Fourier decomposition of the Helmholtz Green Function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Helmholtz Green function are split into their half advanced+half retarded and half advanced-half retarded components. Closed form solutions are given for these components in terms of a Horn function and a Kampe de Feriet function, respectively. The systems of partial differential equations associated with these two-dimensional hypergeometric functions are used to construct a fourth-order ordinary differential equation which both components satisfy. A second fourth-order ordinary differential equation for the general Fourier coefficent is derived from an integral representation of the coefficient, and both differential equations are shown to be equivalent. Series solutions for the various Fourier coefficients are also given, mostly in terms of Legendre functions and Bessel/Hankel functions. These are derived from the closed form hypergeometric solutions or an integral representation, or both. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented

Ammonia Mono Hydrate IV: An Attempted Structure Solution

The mixed homonuclear and heteronuclear hydrogen bonds in ammonia hydrates have been of interest for several decades. In this manuscript, a neutron powder diffraction study is presented to investigate the structure of ammonia monohydrate IV at 170 K at an elevated pressure of 3–5 GPa. The most plausible structure that accounts for all features in the experimental pattern was found in the P21/c space group and has the lattice parameters a=5.487(3) Å, b=19.068(4) Å, c=5.989(3) Å, and β=99.537(16) deg. While the data quality limits discussion to a proton-ordered structure, the structure presented here sheds light on an important part of the ammonia–water phase diagram

The Free Will Theorem

On the basis of three physical axioms, we prove that if the choice of a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. We show that this result is robust, and deduce that neither hidden variable theories nor mechanisms of the GRW type for wave function collapse can be made relativistic. We also establish the consistency of our axioms and discuss the philosophical implications.Comment: 31 pages, 6figure

The Apparent Host Galaxy of PKS 1413+135: HST, ASCA and VLBA Observations

PKS 1413+135 (z=0.24671) is one of very few radio-loud AGN with an apparent spiral host galaxy. Previous authors have attributed its nearly exponential infrared cutoff to heavy absorption but have been unable to place tight limits on the absorber or its location in the optical galaxy. In addition, doubts remain about the relationship of the AGN to the optical galaxy given the observed lack of re-emitted radiation. We present new HST, ASCA and VLBA observations which throw significant new light on these issues. The HST observations reveal an extrremely red color (V-H = 6.9 mag) for the active nucleus of PKS 1413+135, requiring both a spectral turnover at a few microns due to synchrotron aging and a GMC-sized absorber. We derive an intrinsic column N_H = 4.6^{+2.1}_{-1.6} times 10^{22}cm^{-2} and covering fraction f = 0.12^{+0.07}_{-0.05}. As the GMC is likely in the disk of the optical galaxy, our sightline is rather unlikely (P ~ 2 times 10^{-4}). The properties of the GMC typical of GMCs in our own galaxy. The HI absorber appears centered 25 milliarcseconds away from the nucleus, while the X-ray and nearly all of the molecular absorbers must cover the nucleus, implying a complicated geometry and cloud structure, with a molecular core along our line of sight to the nucleus. Interestingly, the HST/NICMOS data require the AGN to be decentered relative to the optical galaxy by 13 +/- 4 milliarcseconds. This could be interpreted as suggestive of an AGN location far in the background compared to the optical galaxy, but it can also be explained by obscuration and/or nuclear structure, which is more consistent with the observed lack of multiple images.Comment: 27 pages, 8 figures; accepted to A

Criticality for the Gehring link problem

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality. Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehring's problem and our natural extension.Comment: This is the version published by Geometry & Topology on 14 November 200

Mean Area of Self-Avoiding Loops

The mean area of two-dimensional unpressurised vesicles, or self-avoiding loops of fixed length $N$, behaves for large $N$ as $A_0 N^{3/2}$, while their mean square radius of gyration behaves as $R^2_0 N^{3/2}$. The amplitude ratio $A_0/R_0^2$ is computed exactly and found to equal $4\pi/5$. The physics of the pressurised case, both in the inflated and collapsed phases, may be usefully related to that of a complex O(n) field theory coupled to a U(1) gauge field, in the limit $n \to 0$.Comment: 12 pages, plain TeX, (one TeX macro omission corrected