Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

Semi-classical Green kernel asymptotics for the Dirac operator

We consider a semi-classical Dirac operator in arbitrary spatial dimensions with a smooth potential whose partial derivatives of any order are bounded by suitable constants. We prove that the distribution kernel of the inverse operator evaluated at two distinct points fulfilling a certain hypothesis can be represented as the product of an exponentially decaying factor involving an associated Agmon distance and some amplitude admitting a complete asymptotic expansion in powers of the semi-classical parameter. Moreover, we find an explicit formula for the leading term in that expansion.Comment: 46 page

Semiclassical Approach to Chaotic Quantum Transport

We describe a semiclassical method to calculate universal transport properties of chaotic cavities. While the energy-averaged conductance turns out governed by pairs of entrance-to-exit trajectories, the conductance variance, shot noise and other related quantities require trajectory quadruplets; simple diagrammatic rules allow to find the contributions of these pairs and quadruplets. Both pure symmetry classes and the crossover due to an external magnetic field are considered.Comment: 33 pages, 11 figures (appendices B-D not included in journal version

Erfahrungen bei der Entwicklung eines Informationssystems auf RDBMS- und 4GL-Basis

Anläßlich der Entwicklung eines großen Informationssystems wurde die Entscheidung getroffen, ein Werkzeug der vierten Generation (4GL) und ein relationales Datenbanksystem (Oracle) zu verwenden. Der Beitrag beschreibt die Erfahrungen, die bei der Datenmodellierung auf Basis des Entity-Relationship-Ansatzes und der Überführung in ein Relationenmodell sowie mit den Oracle-spezifischen 4GL-Werkzeugen und der dadurch ermöglichten Entwicklungsmethodik (Prototyping) gesammelt wurden. Das Informationssystem selbst ist an anderer Stelle beschrieben.<br

From error bounds to the complexity of first-order descent methods for convex functions

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with H\"olderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple methodology: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form $O(q^{k})$ where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with $\ell^1$ regularization

The Dwarf Galaxy Population of the Dorado group down to Mv=-11

We present V and I CCD photometry of suspected low-surface brightness dwarf galaxies detected in a survey covering ~2.4 deg^2 around the central region of the Dorado group of galaxies. The low-surface brightness galaxies were chosen based on their sizes and magnitudes at the limiting isophote of 26.0V\mu. The selected galaxies have magnitudes brighter than V=20 (Mv=-11 for an assumed distance to the group of 17.2 Mpc), with central surface brightnesses \mu0>22.5 V mag/arcsec^2, scale lengths h>2'', and diameters > 14'' at the limiting isophote. Using these criteria, we identified 69 dwarf galaxy candidates. Four of them are large very low-surface brightness galaxies that were detected on a smoothed image, after masking high surface brightness objects. Monte Carlo simulations performed to estimate completeness, photometric uncertainties and to evaluate our ability to detect extended low-surface brightness galaxies show that the completeness fraction is, on average, > 80% for dwarf galaxies with $-17<M_{V}<-10.5$ and 22.5<\mu0<25.5 V mag/arcsec^2, for the range of sizes considered by us (D>14''). The V-I colors of the dwarf candidates vary from -0.3 to 2.3 with a peak on V-I=0.98, suggesting a range of different stellar populations in these galaxies. The projected surface density of the dwarf galaxies shows a concentration towards the group center similar in extent to that found around five X-ray groups and the elliptical galaxy NGC1132 studied by Mulchaey and Zabludoff (1999), suggesting that the dwarf galaxies in Dorado are probably physically associated with the overall potential well of the group.Comment: 32 pages, 16 postscript figures and 3 figures in GIF format, aastex v5.0. To appear in The Astronomical Journal, January 200

Systematic NLTE study of the -2.6 < [Fe/H] < 0.2 F and G dwarfs in the solar neighbourhood. I. Stellar atmosphere parameters

We present atmospheric parameters for 51 nearby FG dwarfs uniformly distributed over the -2.60 < [Fe/H] < +0.20 metallicity range that is suitable for the Galactic chemical evolution research. Lines of iron, Fe I and Fe II, were used to derive a homogeneous set of effective temperatures, surface gravities, iron abundances, and microturbulence velocities. We used high-resolution (R>60000) Shane/Hamilton and CFHT/ESPaDOnS observed spectra and non-local thermodynamic equilibrium (NLTE) line formation for Fe I and Fe II in the classical 1D model atmospheres. The spectroscopic method was tested with the 20 benchmark stars, for which there are multiple measurements of the infrared flux method (IRFM) Teff and their Hipparcos parallax error is < 10%. We found NLTE abundances from lines of Fe I and Fe II to be consistent within 0.06 dex for every benchmark star, when applying a scaling factor of S_H = 0.5 to the Drawinian rates of inelastic Fe+H collisions. The obtained atmospheric parameters were checked for each program star by comparing its position in the log g-Teff plane with the theoretical evolutionary track in the Yi et al. (2004) grid. Our final effective temperatures lie in between the T_IRFM scales of Alonso et al. (1996) and Casagrande et al. (2011), with a mean difference of +46 K and -51 K, respectively. NLTE leads to higher surface gravity compared with that for LTE. The shift in log g is smaller than 0.1 dex for stars with either [Fe/H] > -0.75, or Teff 4.20. NLTE analysis is crucial for the VMP turn-off and subgiant stars, for which the shift in log g between NLTE and LTE can be up to 0.5 dex. The obtained atmospheric parameters will be used in the forthcoming papers to determine NLTE abundances of important astrophysical elements from lithium to europium and to improve observational constraints on the chemo-dynamical models of the Galaxy evolution.Comment: 18 pages, 14 figures, accepted for publication in Ap

On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds

Asymptotic laws for mean multiplicities of lengths of closed geodesics in arithmetic hyperbolic three-orbifolds are derived. The sharpest results are obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o) and some congruence subgroups. Similar results hold for cocompact arithmetic quaternion groups, if a conjecture on the number of gaps in their length spectra is true. The results related to the groups above give asymptotic lower bounds for the mean multiplicities in length spectra of arbitrary arithmetic hyperbolic three-orbifolds. The investigation of these multiplicities is motivated by their sensitive effect on the eigenvalue spectrum of the Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as the Hamiltonian of a three-dimensional quantum system being strongly chaotic in the classical limit.Comment: 29 pages, uuencoded ps. Revised version, to appear in NONLINEARIT

Learning filter functions in regularisers by minimising quotients

Learning approaches have recently become very popular in the field of inverse problems. A large variety of methods has been established in recent years, ranging from bi-level learning to high-dimensional machine learning techniques. Most learning approaches, however, only aim at fitting parametrised models to favourable training data whilst ignoring misfit training data completely. In this paper, we follow up on the idea of learning parametrised regularisation functions by quotient minimisation as established in [3]. We extend the model therein to include higher-dimensional filter functions to be learned and allow for fit- and misfit-training data consisting of multiple functions. We first present results resembling behaviour of well-established derivative-based sparse regularisers like total variation or higher-order total variation in one-dimension. Our second and main contribution is the introduction of novel families of non-derivative-based regularisers. This is accomplished by learning favourable scales and geometric properties while at the same time avoiding unfavourable ones