Counterexamples to the B-spline Conjecture for Gabor Frames

The frame set conjecture for B-splines \(B_n\), \(n \ge 2\), states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form \(ab=r\), where r is a rational number smaller than one and a and b denote the sampling and modulation rates, respectively, has infinitely many pieces, located around \(b=2,3,\dots \), not belonging to the frame set of the nth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines \(B_n\), \(n \ge 2\).     

System bandwidth and the existence of generalized shift-invariant frames

We consider the question whether, given a countable family of lattices ${({\mathrm{\Gamma }}_j)}_{j\in J}$ in a locally compact abelian group G, there exist functions ${(g_j)}_{j\in J}$ such that the resulting generalized shift-invariant system ${(g_j(??\gamma ))}_{j\in J,\gamma \in {\mathrm{\Gamma }}_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of generalized shift-invariant system via almost periodic functions, based on a novel unconditional convergence property. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the system bandwidth as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. Without the unconditional convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system.     

Polar sets for anisotropic Gaussian random fields

This paper studies polar sets for anisotropic Gaussian random fields, i.e. sets which a Gaussian random field does not hit almost surely. The main assumptions are that the eigenvalues of the covariance matrix are bounded from below and that the canonical metric associated with the Gaussian random field is dominated by an anisotropic metric. We deduce an upper bound for the hitting probabilities and conclude that sets with small Hausdorff dimension are polar. Moreover, the results allow for a translation of the Gaussian random field by a random field, that is independent of the Gaussian random field and whose sample functions are of bounded Hölder norm.     

On the use of Helmholtz resonators as sound attenuators

Summary In the present paper a nonlinear acoustic theory is proposed, to accurately describe the properties of a generalized type of Helmholtz resonators. The theory may be used as a layout tool to design sound attenuators for combustion facilities. The present investigation shows that, in addition to the nonlinear effects, the effects due to thermoacoustic boundary layers and the first frequency correction to the Helmholtz approximation should be taken into account, in order to predict the frequency-dependent attenuation properties of Helmholtz attenuators with the accuracy required by typical technical applications. A series of experiments is presented which is used to validate the theoretical predictions.

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Zusammenfassung In der vorliegenden Abhandlung wird eine nichtlineare akustische Theorie vorgeschlagen, mit der das Verhalten einer verallgemeinerten Art von Helmholtzresonatoren beschrieben werden kann. Die Theorie kann als Werkzeug zur Auslegung von Schalldämpfern für Verbrennungsanlagen benutzt werden. Die vorliegende Arbeit zeigt, dass neben den nichtlinearen Effekten auch die Einflüsse der thermoakustischen Grenzschichten und der ersten Frequenzkorrektur zur Helmholtzapproximation berücksichtigt werden müssen, damit die frequenzabhängigen Eigenschaften von Helmholtzdämpfern mit technisch ausreichender Genauigkeit beschrieben werden können. Mit einer Reihe von Experimenten werden die theoretischen Ergebnisse untermauert.

     

NMR-microscopy with TrueFISP at 11.75T

The purpose of this paper is to demonstrate that a fully balanced gradient echo technique (TrueFISP) can be used for microscopic experiments at high static magnetic field strengths. TrueFISP experiments were successfully performed on homogeneous and inhomogeneous objects at 11.75T. High-resolution TrueFISP images were obtained from phantoms, plants, formalin-fixed samples, and from an isolated beating rat heart with an in-plane resolution of 78 micro m and a slice thickness of 500 micro m. The signal-to-noise ratio (SNR) gain of TrueFISP compared to conventional gradient echo or spin echo sequences will allow faster acquisition times or an improvement in spatial resolution for microscopic experiments.     

Quantitative Imaging by Optical Diffraction Tomography

Optical diffraction tomography (ODT) is applied to reconstruct the cross-sectional, complex refractive index distribution of cylindrical objects. Experimental reconstructions showing noticeable image contrast for refractive index variations of about 0.001, and having a spatial resolution of about 2 μm, are obtained for objects with cross-sectional diameters of about 100 μm. The results demonstrate that ODT can be used for quantitative imaging of semitransparent fibers.Presented at the International Commission of Optics Topical Meeting, Kyoto, 1994.     

Generalized parton distributions from nucleon form factor data

We present a simple empirical parameterization of the x- and t-dependence of generalized parton distributions at zero skewness, using forward parton distributions as input. A fit to experimental data for the Dirac, Pauli and axial form factors of the nucleon allows us to discuss quantitatively the interplay between longitudinal and transverse partonic degrees of freedom in the nucleon (nucleon tomography). In particular we obtain the transverse distribution of valence quarks at given momentum fraction x. We calculate various moments of the distributions, including the form factors that appear in the handbag approximation to wide-angle Compton scattering. This allows us to estimate the minimal momentum transfer required for reliable predictions in that approach to be around . We also evaluate the valence contributions to the energy-momentum form factors entering Jis sum rule.Received: 16 September 2004, Revised: 15 October 2004, Published online: 17 December 2004     

Synthetic protocols and building blocks for molecular electronics

Simple and readily accessible aryl bromides are useful building blocks for thiol end-capped molecular wires. Thus, 4-bromophenyl tert-butyl sulfide and 1-bromo-4-(methoxymethyl)benzene serve as precursors for a variety of oligo(phenylenevinylene) and oligo(phenyleneethynylene) wires via efficient synthetic transformations as presented in this paper.     

Splitless on-line coupling of capillary high-performance liquid chromatography,capillary electrochromatography and pressurized capillary electrochromatography with nuclear magnetic resonance spectroscopy

A mixture of unsaturated fatty acid methyl esters was separated with a new splitless capillary set-up. With the employed apparatus configuration different capillary separation techniques such as capillary high-performance liquid chromatography (cHPLC), capillary electrochromatography (CEC) and pressurized capillary electrochromatography (pCEC) could be applied. The detection and identification of the sample compounds were accomplished by hyphenating these capillary separation techniques with nuclear magnetic resonance (NMR) spectroscopy using a novel configuration of the detection capillary set-up. Using modified electrokinetically driven separation techniques, the electric field was applied solely across the separation column. With this improved interface for capillary liquid chromatography-NMR on-line coupling, the stereochemical assignment of the cis and trans configuration of unsaturated fatty acids could be easily accomplished. Finally, the results of cHPLC-NMR, CEC-NMR and pCEC-NMR coupling experiments were compared.Dedicated to Professor Günter Häfelinger on the occasion of his 65th birthday