Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

Untersuchungen zur 3-Phononenabsorption in partiell kristallinen Polymeren

Zusammenfassung In dieser Arbeit wird die Dreiphononenabsorption in Polymeren, die den Hauptanteil an den dielektrischen Verlusten im Millimeterwellengebiet stellt, theoretisch und experimentell untersucht. Aus der Störungsrechnung bis zur 2. Ordnung wird die Temperaturabhängigkeit der Absorption abgeschätzt. Damit ist eine befriedigende Deutung des gesamten Temperaturverlaufs der dielektrischen Verlust-Kurven möglich. Im Hochtemperaturbereich läßt sich die Mehrphononenabsorption eindeutig gegenüber restlichen Relaxationsprozessen abgrenzen. Im Tieftemperaturgebiet konnte erstmals auch für das Millimeterwellengebiet ein temperaturunabhängiger Bestandteil fehlordnungsinduzierter Einphononenabsorption nachgewiesen werden. Im Übergangsbereich durchlaufen die Absorptionskurven einen Wendepunkt bei einer charakteristischen TemperaturT ₀, die dem Schwerpunkt der am Absorptionsprozeß beteiligten Phononenfrequenzen entspricht. Daraus konnten in Übereinstimmung mit dem Experiment die Frequenzen der niedrigsten optisch aktiven Phononen für die verschiedenen Polymeren abgeschätzt werden.

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Summary The 3-phonon absorption in polymers, which gives the main contribution to their dielectric loss in the microwave region, is theoretically and experimentally investigated. Perturbation theory up to the second order is used for the evaluation of its temperature dependence. The theoretical results allow an interpretation of the total absorption-temperature-curves: In the high temperature region 3-phonon absorption can be distinguished from residual relaxation losses (above the glass temperature). In the low temperature region the existence of a low frequency tail of temperature independent disorderinduced one-phonon absorption became evident. At intermediate temperatures (50–150°K) the curves show an inflexion point or plateau at a characteristic temperatureT ₀, which according toT ₀= ⁰, corresponds to the frequencies of those phonons, that mainly contribute to the multiphonon absorption. The plateau gives, in good agreement with the experiment, the frequencies of the lowest optically active phonons in the various polymers.

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Mit 6 Abbildungen in 8 Einzeldarstellungen

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Vorgetragen auf der Arbeitssitzung des Fachausschusses Physik der Hochpolymeren in der Frühjahrstagung 1970 des Regionalverbandes Hessen-Mittelrhein-Saar der Deutschen Physikalischen Gesellschaft in Darmstadt, 10.–13. März 1970.     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995     

Bestimmung der Beladung von Aktivkohle mit organischen Stoffen über die Geschwindigkeit der Wasserstoffperoxidzersetzung

A simple method is described for the rapid determination of the amount of organic substances adsorbed on activated carbon. The technique is based on the catalytic decomposition of H₂O₂ by free active centres on the carbon; occupied adsorption sites are not catalytically effective. The data yield additional information as to the type of binding between adsorbed substance and adsorbent as well as to the specific space requirement of the adsorbed molecules. A number of possible applications are discussed.     

Phosphinerhodium complexes as homogeneous catalysts : VIII. Enantioselective hydrogenation of ketones: evidence for several mechanisms with catalysts containing diop

Optical yields obtained in the hydrogenation of acetophenone with cationic and in situ rhodium complex catalysts depend on the P/Rh ratio and on the ionic or non-ionic character of the active species. The enantioselectivity of the in situ catalyst containing (+)-DIOP is reversed by addition of achiral tri-n-alkyl-phosphines. On the basis of these observations and the amount of H₂ consumed in preforming the catalysts, several different mechanisms are suggested: for example: cycles involving cationic rhodium complexes containing two (or three) phosphorus ligands and cycles involving non-ionic rhodium complexes with two phosphorus ligands in cis or trans positions. In the in situ catalyst with a Rh/(+)-DIOP/P-n-Bu₃  1/1/1 ratio (+)-DIOP functions as a monodentate ligand.     

Measuring methotrexate polyglutamates in red blood cells: a new LC-MS/MS-based method

The folate antagonist methotrexate (MTX) is the anchor drug in the treatment of rheumatoid arthritis. The therapeutic effects of MTX are attributed to the intracellular levels of MTX, present in the cell as polyglutamates (MTXPGn). We developed a new liquid chromatography-electrospray ionization-tandem mass spectrometry (LC-ESI-MS/MS)-based assay to separately quantitate MTXPGn in red blood cells using stable-isotope-labelled internal standards. Samples were analyzed by LC-ESI-MS/MS using a Waters Acquity UPLC BEH C18 column with a 5–100% organic gradient of 10 mM ammonium bicarbonate (pH 10) and methanol. The analysis consisted of simple sample preparation and a 6-min run time. Detection was done using a Waters Acquity UPLC coupled to a Waters Quattro Premier XE with electrospray ionization operating in the positive ionization mode. Assay validation was performed following recent Food and Drug Administration guidelines. The method was linear from 1–1,000 nM for all MTXPGn (R ²?>?0.99). The coefficient of variation ranged from 1–4% for intraday precision and 6–15% for interday precision. Samples were stable for at least 1 month at ?80 °C. Recovery ranged from 98–100%, and the relative matrix-effect varied from 95–99%. The lower limit of quantitation was 1 nM for each MTXPGn. Fifty patient samples from the tREACH study were analyzed. The MTXPGn concentration and distribution of these samples were comparable with values reported in literature. The developed LC-ESI-MS/MS method for the quantitative measurement of MTXPGn in red blood cells is both sensitive and precise within the clinically relevant range. The method can be easily applied in clinical laboratories due to the combination of simple pre-treatment with robust LC-ESI-MS/MS.

Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario

We demonstrate the influence of vectorial coupling on the synchronization behavior of complex systems. We study two semiconductor lasers subject to delayed optical feedback which are unidirectionally coherently coupled via their optical fields. Our experimental and numerical results demonstrate a characteristic synchronization scenario in dependence on the relative feedback phase leading cyclically from chaos synchronization to almost uncorrelated states, and back to chaos synchronization. Finally, we reveal the influence of the feedback phase on the dynamics of the solitary delay system.