2,032 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
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 -Betti numbers
of closed manifolds.Comment: 8 pages, A4 pape
Following Strain-Induced Mosaicity Changes of Ferroelectric Thin Films by Ultrafast Reciprocal Space Mapping
We investigate coherent phonon propagation in a thin film of ferroelectric
PbZr0.2Ti0.8O3 (PZT) by ultrafast x-ray diffraction (UXRD) experiments, which
are analyzed as time-resolved reciprocal space mapping (RSM) in order to
observe the in- and out-of-plane structural dynamics simultaneously. The mosaic
structure of the PZT leads to a coupling of the excited out-of-plane expansion
to in-plane lattice dynamics on a picosecond timescale, which is not observed
for out-of-plane compression.Comment: 5 pages, 4 figure
Thermoelastic study of nanolayered structures using time-resolved x-ray diffraction at high repetition rate
We investigate the thermoelastic response of a nanolayered sample composed of
a metallic SrRuO3 (SRO) electrode sandwiched between a ferroelectric
Pb(Zr0.2Ti0.8)O3 (PZT) film with negative thermal expansion and a SrTiO3
substrate. SRO is rapidly heated by fs-laser pulses with 208 kHz repetition
rate. Diffraction of x-ray pulses derived from a synchrotron measures the
transient out-of-plane lattice constant c of all three materials simultaneously
from 120 ps to 5 mus with a relative accuracy up to Delta c/c = 10^-6. The
in-plane propagation of sound is essential for understanding the delayed out of
plane expansion.Comment: 5 pages, 3 figure
Liquid drops on a surface: using density functional theory to calculate the binding potential and drop profiles and comparing with results from mesoscopic modelling
The contribution to the free energy for a film of liquid of thickness on
a solid surface, due to the interactions between the solid-liquid and
liquid-gas interfaces is given by the binding potential, . The precise
form of determines whether or not the liquid wets the surface. Note that
differentiating gives the Derjaguin or disjoining pressure. We develop a
microscopic density functional theory (DFT) based method for calculating
, allowing us to relate the form of to the nature of the molecular
interactions in the system. We present results based on using a simple lattice
gas model, to demonstrate the procedure. In order to describe the static and
dynamic behaviour of non-uniform liquid films and drops on surfaces, a
mesoscopic free energy based on is often used. We calculate such
equilibrium film height profiles and also directly calculate using DFT the
corresponding density profiles for liquid drops on surfaces. Comparing
quantities such as the contact angle and also the shape of the drops, we find
good agreement between the two methods. We also study in detail the effect on
of truncating the range of the dispersion forces, both those between the
fluid molecules and those between the fluid and wall. We find that truncating
can have a significant effect on and the associated wetting behaviour of
the fluid.Comment: 16 pages, 13 fig
What do emulsification failure and Bose-Einstein condensation have in common?
Ideal bosons and classical ring polymers formed via self-assembly, are known
to have the same partition function, and so analogous phase transitions. In
ring polymers, the analogue of Bose-Einstein condensation occurs when a ring
polymer of macroscopic size appears. We show that a transition of the same
general form occurs within a whole class of systems with self-assembly, and
illustrate it with the emulsification failure of a microemulsion phase of
water, oil and surfactant. As with Bose-Einstein condensation, the transition
occurs even in the absence of interactions.Comment: 7 pages, 1 figure, typeset with EUROTeX, uses epsfi
Interfacial fluctuations near the critical filling transition
We propose a method to describe the short-distance behavior of an interface
fluctuating in the presence of the wedge-shaped substrate near the critical
filling transition. Two different length scales determined by the average
height of the interface at the wedge center can be identified. On one length
scale the one-dimensional approximation of Parry et al. \cite{Parry} which
allows to find the interfacial critical exponents is extracted from the full
description. On the other scale the short-distance fluctuations are analyzed by
the mean-field theory.Comment: 13 pages, 3 figure
Droplet shapes on structured substrates and conformal invariance
We consider the finite-size scaling of equilibrium droplet shapes for fluid
adsorption (at bulk two-phase co-existence) on heterogeneous substrates and
also in wedge geometries in which only a finite domain of the
substrate is completely wet. For three-dimensional systems with short-ranged
forces we use renormalization group ideas to establish that both the shape of
the droplet height and the height-height correlations can be understood from
the conformal invariance of an appropriate operator. This allows us to predict
the explicit scaling form of the droplet height for a number of different
domain shapes. For systems with long-ranged forces, conformal invariance is not
obeyed but the droplet shape is still shown to exhibit strong scaling
behaviour. We argue that droplet formation in heterogeneous wedge geometries
also shows a number of different scaling regimes depending on the range of the
forces. The conformal invariance of the wedge droplet shape for short-ranged
forces is shown explicitly.Comment: 20 pages, 7 figures. (Submitted to J.Phys.:Cond.Mat.
Hodge Theory on Metric Spaces
Hodge theory is a beautiful synthesis of geometry, topology, and analysis,
which has been developed in the setting of Riemannian manifolds. On the other
hand, spaces of images, which are important in the mathematical foundations of
vision and pattern recognition, do not fit this framework. This motivates us to
develop a version of Hodge theory on metric spaces with a probability measure.
We believe that this constitutes a step towards understanding the geometry of
vision.
The appendix by Anthony Baker provides a separable, compact metric space with
infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version,
to appear in Foundations of Computational Mathematics. Minor changes and
addition
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