Asymptotic behavior of positive solution branches of elliptic problems with linear part at resonance

We consider Dirichlet problems for semilinear elliptic equations whose linear part is at resonance. For various types of nonlinearities we provide sufficient conditions for the solvability of these problems for certain types of forcing terms. Depending upon the dimension of the independent variable and the shape of the domain, we obtain multiple existence results and show that in some cases infinitely many solutions may exist.Dedicated to Klaus Kirchgässner on the occasion of his 60th birthdayThe authors' work on this project has been supported by several institutions and thanks are due to the University of Utah for granting Schaaf a one year's visiting professorship during 1989–1990, to the DFG and the University of Stuttgart for providing Schmitt with the opportunity to visit that institution during the summer semester 1989. Final preparation was supported by a grant from NSF.     

The fuzzy supersphere

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative ₂-graded algebras tending in a suitable limit to a dense subalgebra of the ₂-graded algebra of ^∞-functions on the (2|2)-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the “fuzzy level”.     

Improved linear multi-step methods for stochastic ordinary differential equations

We consider linear multi-step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise. We present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations. In previous work, we considered Maruyama-type schemes, where only the increments of the driving Wiener process are used to discretize the diffusion part. Here, we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical-stochastic integrals. We show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile. Simulation results illustrate the theoretical findings.     

Constructive influence of noise-flatness in correlation ratchets

The influence of noise-flatness on overdamped motion of Brownian particles in a 1D periodic system with a simple sawtooth potential subjected to both unbiased thermal noise and three-level telegraph noise is considered. The exact formula for the stationary probability flux (current) is presented. The phenomenon of multiple current reversals and some topological properties of the hypersurface of zero current in the parameter space of noises are investigated and illustrated by phase diagrams. The conditions for the existence of four current reversals versus the switching rate of nonequilibrium noise are given. An alternative interpretation of the results in terms of cross-correlation between two dichotomous noises is presented.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

The Physical Tourist¶Vienna: A Random Walk in Science

I provide a guided tour of sites of scientific interest in Vienna with particular reference to the history of physics.     

Switching layer stability in a polymer bilayer by thickness variation

We use optical and scanning force microscopy to explore the possibility of switching the stability of a bilayer of poly(methyl methacrylate) (PMMA) on polystyrene by simply changing the film thickness. We show that for thin PMMA layers on thicker polystyrene films the PMMA layer is unstable to thickness fluctuations. However, polystyrene layers are unstable when they are substantially thinner than the now stable PMMA film. Dewetting morphologies are cataloged as a function of the thickness of individual polymer layers by identifying which layer is unstable by which mechanism, be it spinodal dewetting or heterogeneous thermal nucleation. Our results are in good agreement with a linear stability analysis of the influence of long-range dispersion forces, but also indicate the influence of film preparation and small variations of material properties.     

Akkommodationskoeffizienten von Edelgasen an Pt im Temperaturbereich von 80 K bis 450 K

Zusammenfassung Die Akkommodationskoeffizienten von He, Ne und Ar wurden von 80 K bis 450 K, die von Kr und Xe von 250 K bis 450 K an gasbedeckten und teilweise gereinigten Pt-Oberflächen gemessen. Zur Anwendung kam die stationäre Hitzdrahtmethode bei sehr niedrigen Drucken.Der Fehler in den Bestimmungen an der gasbedeckten Oberfläche wird auf 2–3% geschätzt. Die Messungen an der teilweise gereinigten Oberfläche wurden extrapoliert aus dem zeitlichen Anstieg des Akkommodationskoeffizienten nach einer Aufheizung des Hitzdrahtes auf 900 K. Diese Ergebnisse sind mit einem wesentlich höheren Fehler behaftet.

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Accommodation coefficients of noble gases on Pt-surfaces from 80 K up to 450 K

Accommodation coefficients of He, Ne and Ar have been measured from 80 K up to 450 K, those of Kr and Xe from 250 K up to 450 K on gas covered and partially cleaned Pt-surfaces by the stationary hot wire method at very low pressures.The error of determinations on gas covered surfaces is estimated to be about 2 or 3%. The measurements on partially cleaned surfaces were extrapolated from the increase of the accommodation coefficient with time after a heating of the Pt-wire up to 900 K. In this case the error will be considerably higher.

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Bezeichnungen A Apparaturkonstante - a_i Konstante - C_v Molwärme bei konstantem Volumen - F Oberfläche des Meßdrahtes aus Pt - f_i Druckexpansionsfaktor - g (T) Korrekturfunktion, die die thermomolekulare Druck-differenz in verschieden temperierten Teilen des Versuchsraumes berücksichtigt - I Heizstrom im Pt-Draht - l Länge des Pt-Drahtes - M Molmasse - p Druck - q_D vom Pt-Draht durch Drahtwärmeleitung abgeführte Leistung - q_E dem Pt-Draht zugeführte elektrische Leistung - q_G vom Pt-Draht durch Gaswärmeleitung abgeführte Leistung - q_s vom Pt-Draht durch Strahlung abgeführte Leistung - R allgemeine Gaskonstante - R_i Widerstand des Pt-Drahtes - r Radius des Pt-Drahtes - T Gastemperatur - t₀ Raumtemperatur - T Differenz der Temperatur des Meßdrahtes und der Gas-temperatur - t Zeit - t_A charakteristische Adsorptionszeit - V_i Volumina - z Koordinate in Längsrichtung des Pt-Drahtes - Akkommodationskoeffizient (AK) - ₀ AK der teilweise gereinigten Pt-Oberfläche - _G AK der gasbedeckten Pt-Oberfläche - mittlere freie Weglänge - _Pt Wärmeleitfähigkeit des Pt