999 research outputs found
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
Analysis of a four-mirror cavity enhanced Michelson interferometer
We investigate the shot noise limited sensitivity of a four-mirror cavity
enhanced Michelson interferometer. The intention of this interferometer
topology is the reduction of thermal lensing and the impact of the
interferometers contrast although transmissive optics are used with high
circulating powers. The analytical expressions describing the light fields and
the frequency response are derived. Although the parameter space has 11
dimensions, a detailed analysis of the resonance feature gives boundary
conditions allowing systematic parameter studies
Optimal time-domain combination of the two calibrated output quadratures of GEO 600
GEO 600 is an interferometric gravitational wave detector with a 600 m arm-length and which uses a dual-recycled optical configuration to give enhanced sensitivity over certain frequencies in the detection band. Due to the dual-recycling, GEO 600 has two main output signals, both of which potentially contain gravitational wave signals. These two outputs are calibrated to strain using a time-domain method. In order to simplify the analysis of the GEO 600 data set, it is desirable to combine these two calibrated outputs to form a single strain signal that has optimal signal-to-noise ratio across the detection band. This paper describes a time-domain method for doing this combination. The method presented is similar to one developed for optimally combining the outputs of two colocated gravitational wave detectors. In the scheme presented in this paper, some simplifications are made to allow its implementation using time-domain methods
Superconductor-Ferromagnet Bi-Layers: a Comparison of s-Wave and d-Wave Order Parameters
We study superconductor-ferromagnet bi-layers, not only for s-wave but also
for d-wave superconductors. We observe oscillations of the critical temperature
when varying the thickness of the ferromagnetic layer for both s-wave and
d-wave superconductors. However, for a rotated d-wave order parameter the
critical temperature differs considerably from that for the unrotated case. In
addition we calculate the density of states for different thicknesses of the
ferromagnetic layer; the results reflect the oscillatory behaviour of the
superconducting correlations.Comment: 11 pages, 5 figures, accepted for publication in J. Phys.: Condens.
Matte
The universal functorial equivariant Lefschetz invariant
We introduce the universal functorial equivariant Lefschetz invariant for
endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We
use K_0 of the category of "phi-endomorphisms of finitely generated free
RPi(G,X)-modules". We derive results about fixed points of equivariant
endomorphisms of cocompact proper smooth G-manifolds.Comment: 33 pages; shortened version of the author's PhD thesis, supervised by
Wolfgang Lueck, Westfaelische Wilhelms-Universitaet Muenster, 200
Thermal noise of folding mirrors
Current gravitational wave detectors rely on the use of Michelson interferometers. One crucial limitation of their sensitivity is the thermal noise of their optical components. Thus, for example fluctuational deformations of the mirror surface are probed by a laser beam being reflected from the mirrors at normal incidence. Thermal noise models are well evolved for that case but mainly restricted to single reflections. In this work we present the effect of two consecutive reflections under a non-normal incidence onto mirror thermal noise. This situation is inherent to detectors using a geometrical folding scheme such as GEO\,600. We revise in detail the conventional direct noise analysis scheme to the situation of non-normal incidence allowing for a modified weighting funtion of mirror fluctuations. An application of these results to the GEO\,600 folding mirror for Brownian, thermoelastic and thermorefractive noise yields an increase of displacement noise amplitude by 20\% for most noise processes. The amplitude of thermoelastic substrate noise is increased by a factor 4 due to the modified weighting function. Thus the consideration of the correct weighting scheme can drastically alter the noise predictions and demands special care in any thermal noise design process
Surgery groups of the fundamental groups of hyperplane arrangement complements
Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that
the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any
group which contains a finite index strongly poly-free normal subgroup, in
particular, for the Artin full braid groups. As a consequence we explicitly
compute the surgery groups of the Artin pure braid groups. This is obtained as
a corollary to a computation of the surgery groups of a more general class of
groups, namely for the fundamental group of the complement of any fiber-type
hyperplane arrangement in the complex n-space.Comment: 11 pages, AMSLATEX file, revised following referee's comments and
suggestions, to appear in Archiv der Mathemati
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
Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory
We study D-branes and Ramond-Ramond fields on global orbifolds of Type II
string theory with vanishing H-flux using methods of equivariant K-theory and
K-homology. We illustrate how Bredon equivariant cohomology naturally realizes
stringy orbifold cohomology. We emphasize its role as the correct cohomological
tool which captures known features of the low-energy effective field theory,
and which provides new consistency conditions for fractional D-branes and
Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from
equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings
of D-branes which generalize previous examples. We propose a definition for
groups of differential characters associated to equivariant K-theory. We derive
a Dirac quantization rule for Ramond-Ramond fluxes, and study flat
Ramond-Ramond potentials on orbifolds.Comment: 46 pages; v2: typos correcte
Analytic and Reidemeister torsion for representations in finite type Hilbert modules
For a closed Riemannian manifold we extend the definition of analytic and
Reidemeister torsion associated to an orthogonal representation of fundamental
group on a Hilbert module of finite type over a finite von Neumann algebra. If
the representation is of determinant class we prove, generalizing the
Cheeger-M\"uller theorem, that the analytic and Reidemeister torsion are equal.
In particular, this proves the conjecture that for closed Riemannian manifolds
with positive Novikov-Shubin invariants, the L2 analytic and Reidemeister
torsions are equal.Comment: 78 pages, AMSTe
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