The Ore condition, affiliated operators, and the lamplighter group

Let G be the wreath product of Z and Z/2, the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1-x is invertible, with x a generator of Z considered as subset of G. Then R is not flat over kG. If k is the field of complex numbers, this applies in particular to the algebra UG of unbounded operators affiliated to the group von Neumann algebra of G. We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.Comment: LaTex2e, 7 pages. Added a new proof of the main result (due to Warren Dicks) which is shorter, easier and more elementary, and at the same time yields a slightly more general result. Additionally: misprints removed. to appear in Proceedings of "Higher dimensional manifold theory", Conference at ICTP Trieste 200

The anomalous accretion disk of the Cataclysmic Variable RW Sextantis

Synthetic spectra covering the wavelength range 900\AA~to 3000\AA~provide an accurate fit, established by a ${\chi}_{\nu}^2$ analysis, to a combined observed spectrum of RW Sextantis. Two separately calibrated distances to the system establish the synthetic spectrum comparison on an absolute flux basis but with two alternative scaling factors, requiring alternative values of $\dot{M}$ for final models. Based on comparisons for a range of $\dot{M}$ values, the observed spectrum does not follow the standard model. Rather than the exponent 0.25 in the expression for the radial temperature profile, a value close to 0.125 produces a synthetic spectrum with an accurate fit to the combined spectrum. A study of time-series $FUSE$ spectra shows that a proposed warped or tilted disk is not supported by the data; an alternative proposal is that an observed non-axisymmetric wind results from an interaction with the mass transfer stream debris.Comment: 56 pages, 15 figures, 11 tables. Accepted for The Astrophysical Journa

On a conjecture of Atiyah

In this note we explain how the computation of the spectrum of the lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a strong version of the Atiyah conjectures about the range of $L^2$-Betti numbers of closed manifolds.Comment: 8 pages, A4 pape

The anomalous accretion disk of the Cataclysmic Variable RW Sextantis

Synthetic spectra covering the wavelength range 900\AA~to 3000\AA~provide an accurate fit, established by a ${\chi}_{\nu}^2$ analysis, to a combined observed spectrum of RW Sextantis. Two separately calibrated distances to the system establish the synthetic spectrum comparison on an absolute flux basis but with two alternative scaling factors, requiring alternative values of $\dot{M}$ for final models. Based on comparisons for a range of $\dot{M}$ values, the observed spectrum does not follow the standard model. Rather than the exponent 0.25 in the expression for the radial temperature profile, a value close to 0.125 produces a synthetic spectrum with an accurate fit to the combined spectrum. A study of time-series $FUSE$ spectra shows that a proposed warped or tilted disk is not supported by the data; an alternative proposal is that an observed non-axisymmetric wind results from an interaction with the mass transfer stream debris.Comment: 56 pages, 15 figures, 11 tables. Accepted for The Astrophysical Journa

On Turing dynamical systems and the Atiyah problem

Main theorems of the article concern the problem of M. Atiyah on possible values of l^2-Betti numbers. It is shown that all non-negative real numbers are l^2-Betti numbers, and that "many" (for example all non-negative algebraic) real numbers are l^2-Betti numbers of simply connected manifolds with respect to a free cocompact action. Also an explicit example is constructed which leads to a simply connected manifold with a transcendental l^2-Betti number with respect to an action of the threefold direct product of the lamplighter group Z/2 wr Z. The main new idea is embedding Turing machines into integral group rings. The main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio

GALEX and Optical Light Curves of WX LMi, SDSSJ103100.5+202832.2 and SDSSJ121209.31+013627.7

{\it GALEX} near ultraviolet (NUV) and far-ultraviolet (FUV) light curves of three extremely low accretion rate polars show distinct modulations in their UV light curves. While these three systems have a range of magnetic fields from 13 to 70 MG, and of late type secondaries (including a likely brown dwarf in SDSSJ121209.31+013627.7), the accretion rates are similar, and the UV observations imply some mechanism is operating to create enhanced emission zones on the white dwarf. The UV variations match in phase to the two magnetic poles viewed in the optical in WX LMi and to the single poles evident in the optical in SDSSJ1212109.31+013627.7 and SDSSJ103100.55+202832.2. Simple spot models of the UV light curves show that if hot spots are responsible for the UV variations, the temperatures are on the order of 10,000-14,000K. For the single pole systems, the size of the FUV spot must be smaller than the NUV and in all cases, the geometry is likely more complicated than a simple circular spot.Comment: 29 pages, 4 tables, 10 figures, Astrophysical Journal, accepte

MR Cygni revisited

New analysis tools and additional unanalyzed observations justify a reanalysis of MR Cygni. The reanalysis applied successively more restrictive physical models, each with an optimization program. The final model assigned separate first and second order limb darkening coefficients, from model atmospheres, to individual grid points. Proper operation of the optimization procedure was tested on simulated observational data, produced by light synthesis with assigned system parameters, and modulated by simulated observational error. The iterative solution converged to a weakly-determined mass ratio of 0.75. Assuming the B3 primary component is on the main sequence, the HR diagram location of the secondary from the light ratio (ordinate) and adjusted T sub eff (abscissa) was calculated. The derived mass ratio, together with a main-sequence mass for the B3 component, implies a main-sequence secondary spectral type of B4. The photometrically-determined secondary radii agree with this spectral type, in marginal disagreement with the B7 type from the HR diagram analysis. The individual masses, derived from the radial velocity curve of the primary component, the photometrically-determined i, and alternative values of derived mass ratio are seriously discrepant with main sequence objects. The imputed physical status of the system is in disagreement with representations that have appeared in the literature