10,148 research outputs found
Classifying -algebras with both finite and infinite subquotients
We give a classification result for a certain class of -algebras
over a finite topological space in which there exists an
open set of such that separates the finite and infinite
subquotients of . We will apply our results to -algebras
arising from graphs.Comment: Version III: No changes to the text. We only report that Lemma 4.5 is
not correct as stated. See arXiv:1505.05951 for the corrected version of
Lemma 4.5. As noted in arXiv:1505.05951, the main results of this paper are
true verbatim. Version II: Improved some results in Section 3 and loosened
the assumptions in Definition 4.
Finite-temperature linear conductance from the Matsubara Green function without analytic continuation to the real axis
We illustrate how to calculate the finite-temperature linear-response
conductance of quantum impurity models from the Matsubara Green function. A
continued fraction expansion of the Fermi distribution is employed which was
recently introduced by Ozaki [Phys. Rev. B 75, 035123 (2007)] and converges
much faster than the usual Matsubara representation. We give a simplified
derivation of Ozaki's idea using concepts from many-body condensed matter
theory and present results for the rate of convergence. In case that the Green
function of some model of interest is only known numerically, interpolating
between Matsubara frequencies is much more stable than carrying out an analytic
continuation to the real axis. We demonstrate this explicitly by considering an
infinite tight-binding chain with a single site impurity as an exactly-solvable
test system, showing that it is advantageous to calculate transport properties
directly on the imaginary axis. The formalism is applied to the single impurity
Anderson model, and the linear conductance at finite temperatures is calculated
reliably at small to intermediate Coulomb interactions by virtue of the
Matsubara functional renormalization group. Thus, this quantum many-body method
combined with the continued fraction expansion of the Fermi function
constitutes a promising tool to address more complex quantum dot geometries at
finite temperatures.Comment: version accepted by Phys. Rev.
On the coarse classification of tight contact structures
We present a sketch of the proof of the following theorems: (1) Every
3-manifold has only finitely many homotopy classes of 2-plane fields which
carry tight contact structures. (2) Every closed atoroidal 3-manifold carries
finitely many isotopy classes of tight contact structures.Comment: 12 pages, to appear in the 2001 Georgia International Topology
Conference proceeding
Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
Maximum likelihood estimation problems are, in general, intractable
optimization problems. As a result, it is common to approximate the maximum
likelihood estimator (MLE) using convex relaxations. In some cases, the
relaxation is tight: it recovers the true MLE. Most tightness proofs only apply
to situations where the MLE exactly recovers a planted solution (known to the
analyst). It is then sufficient to establish that the optimality conditions
hold at the planted signal. In this paper, we study an estimation problem
(angular synchronization) for which the MLE is not a simple function of the
planted solution, yet for which the convex relaxation is tight. To establish
tightness in this context, the proof is less direct because the point at which
to verify optimality conditions is not known explicitly.
Angular synchronization consists in estimating a collection of phases,
given noisy measurements of the pairwise relative phases. The MLE for angular
synchronization is the solution of a (hard) non-bipartite Grothendieck problem
over the complex numbers. We consider a stochastic model for the data: a
planted signal (that is, a ground truth set of phases) is corrupted with
non-adversarial random noise. Even though the MLE does not coincide with the
planted signal, we show that the classical semidefinite relaxation for it is
tight, with high probability. This holds even for high levels of noise.Comment: 2 figure
- …