226 research outputs found
Limit laws for distorted return time processes for infinite measure preserving transformations
We consider conservative ergodic measure preserving transformations on
infinite measure spaces and investigate the asymptotic behaviour of distorted
return time processes with respect to sets satisfying a type of Darling-Kac
condition. We identify two critical cases for which we prove uniform
distribution laws. For this we introduce the notion of uniformly returning sets
and discuss some of their properties.Comment: 18 pages, 2 figure
The interval ordering problem
For a given set of intervals on the real line, we consider the problem of
ordering the intervals with the goal of minimizing an objective function that
depends on the exposed interval pieces (that is, the pieces that are not
covered by earlier intervals in the ordering). This problem is motivated by an
application in molecular biology that concerns the determination of the
structure of the backbone of a protein.
We present polynomial-time algorithms for several natural special cases of
the problem that cover the situation where the interval boundaries are
agreeably ordered and the situation where the interval set is laminar. Also the
bottleneck variant of the problem is shown to be solvable in polynomial time.
Finally we prove that the general problem is NP-hard, and that the existence of
a constant-factor-approximation algorithm is unlikely
Operator renewal theory and mixing rates for dynamical systems with infinite measure
We develop a theory of operator renewal sequences in the context of infinite
ergodic theory. For large classes of dynamical systems preserving an infinite
measure, we determine the asymptotic behaviour of iterates of the
transfer operator. This was previously an intractable problem.
Examples of systems covered by our results include (i) parabolic rational
maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly
expanding interval maps with indifferent fixed points.
In addition, we give a particularly simple proof of pointwise dual ergodicity
(asymptotic behaviour of ) for the class of systems under
consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for and rates of mixing. Also, we obtain
error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a
minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated
version includes minor corrections in Sections 10 and 11, and corresponding
modifications of certain statements in Section 1. All main results are
unaffected. In particular, Sections 2-9 are unchanged from the published
versio
A new proof of the Vorono\"i summation formula
We present a short alternative proof of the Vorono\"i summation formula which
plays an important role in Dirichlet's divisor problem and has recently found
an application in physics as a trace formula for a Schr\"odinger operator on a
non-compact quantum graph \mathfrak{G} [S. Egger n\'e Endres and F. Steiner, J.
Phys. A: Math. Theor. 44 (2011) 185202 (44pp)]. As a byproduct we give a new
proof of a non-trivial identity for a particular Lambert series which involves
the divisor function d(n) and is identical with the trace of the Euclidean wave
group of the Laplacian on the infinite graph \mathfrak{G}.Comment: Enlarged version of the published article J. Phys. A: Math. Theor. 44
(2011) 225302 (11pp
Distribution of Eigenvalues for the Modular Group
The two-point correlation function of energy levels for free motion on the
modular domain, both with periodic and Dirichlet boundary conditions, are
explicitly computed using a generalization of the Hardy-Littlewood method. It
is shown that ion the limit of small separations they show an uncorrelated
behaviour and agree with the Poisson distribution but they have prominent
number-theoretical oscillations at larger scale. The results agree well with
numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital,
but can be obtained upon request from the first Autho
Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble
The paper studies the spectral properties of large Wigner, band and sample
covariance random matrices with heavy tails of the marginal distributions of
matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at
Giens, France on September 13-17, 200
Axiomatic relation between thermodynamic and information-theoretic entropies
Thermodynamic entropy, as defined by Clausius, characterizes macroscopic observations of a system based on phenomenological quantities such as temperature and heat. In contrast, information-theoretic entropy, introduced by Shannon, is a measure of uncertainty. In this Letter, we connect these two notions of entropy, using an axiomatic framework for thermodynamics [Lieb, Yngvason, Proc. Roy. Soc.(2013)]. In particular, we obtain a direct relation between the Clausius entropy and the Shannon entropy, or its generalisation to quantum systems, the von Neumann entropy. More generally, we find that entropy measures relevant in non-equilibrium thermodynamics correspond to entropies used in one-shot information theory
Speckle-free laser imaging
Many imaging applications require increasingly bright illumination sources,
motivating the replacement of conventional thermal light sources with light
emitting diodes (LEDs), superluminescent diodes (SLDs) and lasers. Despite
their brightness, lasers and SLDs are poorly suited for full-field imaging
applications because their high spatial coherence leads to coherent artifacts
known as speckle that corrupt image formation. We recently demonstrated that
random lasers can be engineered to provide low spatial coherence. Here, we
exploit the low spatial coherence of specifically-designed random lasers to
perform speckle-free full-field imaging in the setting of significant optical
scattering. We quantitatively demonstrate that images generated with random
laser illumination exhibit higher resolution than images generated with
spatially coherent illumination. By providing intense laser illumination
without the drawback of coherent artifacts, random lasers are well suited for a
host of full-field imaging applications from full-field microscopy to digital
light projector systems.Comment: 5 pages, 4 figure
On a relation between Schur, Hardy-Littlewood-Pólya and Karamata’s theorem and an inequality of some products of x p − 1 derived from the Furuta inequality
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