Fixed points of Feigenbaum's type for the equationf p (λx)≡λf(x)

Existence and hyperbolicity of fixed points for the mapN _p :f(x) ^–1 f ^p (x), withf ^p p-fold iteration and =f ^p (0) are given forp large. These fixed points come close to being quadratic functions, and our proof consists in controlling perturbation theory about quadratic functions.Supported in part by the Swiss National Science Foundation     

Harmonische Funktionen und Randwertaufgaben in einem Komplex

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Der Cohomologie-Ring einer beliebigen Gruppe

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Über monothetische Gruppen

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A hybrid formalism combining fluctuating hydrodynamics and generalized Langevin dynamics for the simulation of nanoparticle thermal motion in an incompressible fluid medium

A novel hybrid scheme based on Markovian fluctuating hydrodynamics of the fluid and a non-Markovian Langevin dynamics with the Ornstein-Uhlenbeck noise perturbing the translational and rotational equations of motion of the nanoparticle is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian fluid medium. A direct numerical simulation adopting an arbitrary Lagrangian-Eulerian (ALE) based finite element method (FEM) is employed in simulating the thermal motion of a particle suspended in the fluid confined in a cylindrical vessel. The results for thermal equilibrium between the particle and the fluid are validated by comparing the numerically predicted temperature of the nanoparticle with that obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation function (VACF) and mean squared displacement (MSD) with well-known analytical results. For nanoparticle motion in an incompressible fluid, the fluctuating hydrodynamics approach resolves the hydrodynamics correctly but does not impose the correct equipartition of energy based on the nanoparticle mass because of the added mass of the displaced fluid. In contrast, the Langevin approach with an appropriate memory is able to show the correct equipartition of energy, but not the correct short- and long-time hydrodynamic correlations. Using our hybrid approach presented here, we show for the first time, that we can simultaneously satisfy the equipartition theorem and the (short- and long-time) hydrodynamic correlations. In effect, this results in a thermostat that also simultaneously preserves the true hydrodynamic correlations. The significance of this result is that our new algorithm provides a robust computational approach to explore nanoparticle motion in arbitrary geometries and flow fields, while simultaneously enabling us to study carrier adhesion mediated by biological reactions (receptor-ligand interactions) at the vessel wall at a specified finite temperature.     

Systeme von Richtungsfeldern in Sphären und stetige Lösungen komplexer linearer Gleichungen

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Nonequilibrium Energy Profiles for a Class of 1-D Models

As a paradigm for heat conduction in 1 dimension, we propose a class of models represented by chains of identical cells, each one of which contains an energy storage device called a ``tank'. Energy exchange among tanks is mediated by tracer particles, which are injected at characteristic temperatures and rates from heat baths at the two ends of the chain. For stochastic and Hamiltonian models of this type, we develop a theory that allows one to derive rigorously – under physically natural assumptions – macroscopic equations for quantities related to heat transport, including mean energy profiles and tracer densities. Concrete examples are treated for illustration, and the validity of the Fourier Law in the present context is discussed. JPE is partially supported by the Fonds National Suisse. LSY is partially supported by NSF Grant #0100538.     

Social choice and topology a case of pure and applied mathematics

The existence of a social choice model on a preference space P is a topological, even homotopical problem. It has been solved 50 years ago, under different terminology, by the author and, a little later, jointly with T. Ganea and P.J. Hilton. P must be an H-space and either contractible or homotopy equivalent to a product of Eilenberg-MacLane spaces over the rationals.