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

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

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

A Far Ultraviolet Archival Study of Cataclysmic Variables: I. FUSE and HST/STIS Spectra of the Exposed White Dwarf in Dwarf Nova Systems

We present a synthetic spectral analysis of Far Ultraviolet Spectroscopic Explorer (FUSE) and Hubble Space Telescope/Space Telescope Imaging Spectrograph (HST/STIS) spectra of 5 dwarf novae above and below the period gap during quiescence. We use our synthetic spectral code, including options for the treatment of the hydrogen quasi-molecular satellite lines (for low temperature stellar atmospheres), NLTE approximation (for high temperature stellar atmospheres), and for one system (RU Peg) we model the interstellar medium (ISM) molecular and atomic hydrogen lines. In all the systems presented here the FUV flux continuum is due to the WD. These spectra also exhibit some broad emission lines. In this work we confirm some of the previous FUV analysis results but we also present new results. For 4 systems we combine the FUSE and STIS spectra to cover a larger wavelength range and to improve the spectral fit. This work is part of our broader HST archival research program, in which we aim to provide accurate system parameters for cataclysmic variables above and below the period gap by combining FUSE and HST FUV spectra.Comment: 49 pages (preprint format), 11 figures, 6 tables, ApJ in pres

GALEX, Optical and IR Light Curves of MQ Dra: UV Excesses at Low Accretion Rates

Ultraviolet light curves constructed from NUV and FUV detectors on GALEX reveal large amplitude variations during the orbital period of the Low Accretion Rate Polar MQ Dra (SDSSJ1553+55). This unexpected variation from a UV source is similar to that seen and discussed in the Polar EF Eri during its low state of accretion, even though the accretion rate in MQ Dra is an order of magnitude lower than even the low state of EF Eri. The similarity in phasing of the UV and optical light curves in MQ Dra imply a similar location for the source of light. We explore the possibilities of hot spots and cyclotron emission with simple models fit to the UV, optical and IR light curves of MQ Dra. To match the GALEX light curves with a single temperature circular hot spot requires different sizes of spots for the NUV and FUV, while a cyclotron model that can produce the optical harmonics with a magnetic field near 60 MG requires multipoles with fields > 200 MG to match the UV fluxes.Comment: accepted for ApJ; 15 pages, 7 tables, 8 fig

Contact Discontinuities in Models of Contact Binaries Undergoing Thermal Relaxation Oscillations

In this paper we pursue the suggestion by Shu, Lubow & Anderson (1979) and Wang (1995) that contact discontinuity (DSC) may exist in the secondary in the expansion TRO (thermal relaxation oscillation) state. It is demonstrated that there is a mass exchange instability in some range of mass ratio for the two components. We show that the assumption of {\it constant} volume of the secondary should be relaxed in DSC model. For {\it all} mass ratio the secondary alway satisfies the condition that no mass flow returns to the primary through the inner Lagrangian point. The secondary will expand in order to equilibrate the interaction between the common convective envelope and the secondary. The contact discontinuity in contact binary undergoing thermal relaxation does not violate the second law of thermodynamics. The maintaining condition of contact discontinuity is derived in the time-dependent model. It is desired to improve the TRO model with the advanced contact discontinuity layer in future detailed calculations.Comment: 5 pages in emulateapj, 1 figur

Complementary network-based approaches for exploring genetic structure and functional connectivity in two vulnerable, endemic ground squirrels

The persistence of small populations is influenced by genetic structure and functional connectivity. We used two network-based approaches to understand the persistence of the northern Idaho ground squirrel (Urocitellus brunneus) and the southern Idaho ground squirrel (U. endemicus), two congeners of conservation concern. These graph theoretic approaches are conventionally applied to social or transportation networks, but here are used to study population persistence and connectivity. Population graph analyses revealed that local extinction rapidly reduced connectivity for the southern species, while connectivity for the northern species could be maintained following local extinction. Results from gravity models complemented those of population graph analyses, and indicated that potential vegetation productivity and topography drove connectivity in the northern species. For the southern species, development (roads) and small-scale topography reduced connectivity, while greater potential vegetation productivity increased connectivity. Taken together, the results of the two network-based methods (population graph analyses and gravity models) suggest the need for increased conservation action for the southern species, and that management efforts have been effective at maintaining habitat quality throughout the current range of the northern species. To prevent further declines, we encourage the continuation of management efforts for the northern species, whereas conservation of the southern species requires active management and additional measures to curtail habitat fragmentation. Our combination of population graph analyses and gravity models can inform conservation strategies of other species exhibiting patchy distributions