Über die isoperimetrische Aufgabe imn-dimensionalen Raum konstanter negativer Krümmung

Ohne ZusammenfassungHerrn Oskar Perron zum 60. Geburtstag am 7. Mai 1940 gewidmet     

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.     

Two-step photoionization of Na2: dependence on alignment

Laser light is used for two-step photoionization of Na₂. the first step depends on the moleculer orientation, the second step doesnot. This yields a method for the analysis of molecular alignment.     

Kinetik und mechanismus der elektrochemischen reduktion von 2,2′-dipyridyl I. Primärprozesse: Vorgelagerte isomerisierung eines komplexes

Polarography and chronopotentiometry are used to study the primary processes of the reduction of 2,2′-bipyridine at mercury electrodes in aqueous solutions (12 < pH < 14). Bipyridine and its first reduction product are strongly adsorbed. The reduction is preceded by a complex equilibrium between bipyridine and the cation of the supporting electrolyte (Li, Na, K, Ba). Due to the great excess of the cations, the equilibrium is shifted completely to the side of the complex, which exists as cis- and trans-isomer. The cis-form is more stable, forming a chelate, the trans-form however is more easily reducible. The kinetic and thermodynamic data of the isomerization are determined. The first reduction product reacts irreversibly giving a substance which can be reoxidized to bipyridine via a radical intermediate.     

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