Towards a unified bifurcation theory

Bifurcation theories for the instability of slowly evolving systems have been developed in various disciplines, and a first step is here taken towards some desirable unification. A modern account of the authors' general branching theory for discrete systems is first presented, some new features being the introduction of principal imperfections and the delineation of the important semi-symmetric points of bifurcation. This theory, embedded in a perturbation approach ideal for quantitative analysis, is complementary to the far-reaching qualitative catastrophe theory of René Thom which offers a profound topological classification of instability phenomena. For this reason, we present here a detailed correlation of the two theories. Also presented in the paper is a survey of some fields of application ranging from classical fields such as hydrodynamics, through thermodynamics, crystallography and cosmology, to the newer domains of biology and psychology.     

A computer program for fuzzy reasoning

An interactive computer program is described which implements the procedure proposed in “A Formal System for Fuzzy Reasoning” [1]. The problem in question is that of deciding what conclusions may be drawn in the presence of (posibly conflicting) evidence provided, generally with associated partial degrees of belief, by several sources of differing reliability. In using the program, each piece of evidence is entered as a sentence (using the terms NOT, AND, OR, IMPLIES as necessary), with an associated ‘degree of belief’ and ‘weight’; followed by a tentative conclusion. The system returns the degree(s) of belief and weight(s) which may rationally be attached to the conclusion. Copies of the program, written in FORTRAN IV (870 lines) have been lodged with the program libraries CUBE, DECUS, and SHARE, or may be obtained by writing to the author.     

Pion production from deuterium bombarded with polarized protons of 277 and 500 MeV

Analyzing power measurements of the $\text{p}\text{→ tτ}{}^{\mathrm{+}}$ reaction are reported at incident proton energies of 277 and 500 MeV. The 277 MeV results span the angular range from 70° to 130° in the centre of mass while the two 500 MeV measurements at large angles were taken as a check of published results. With the angular distribution of the analyzing power at 277 MeV now known, it is possible to show that the energy dependence of the analyzing power exhibits characteristics closely resembling the shape and magnitude of the analyzing power distribution observed for 1p shell nuclei of similar excitation energies.     

Influence of the carrier-envelope phase of few-cycle pulses on ponderomotive surface-plasmon electron acceleration

Control over basic processes through the electric field of a light wave can lead to new knowledge of fundamental light-matter interaction phenomena. We demonstrate, for the first time, that surface-plasmon (SP) electron acceleration can be coherently controlled through the carrier-envelope phase (CEP) of an excitation optical pulse. Analysis indicates that the physical origin of the CEP sensitivity arises from the electron's ponderomotive interaction with the oscillating electromagnetic field of the SP wave. The ponderomotive electron acceleration mechanism provides sensitive (nJ energies), high-contrast, single-shot CEP measurement capability of few-cycle laser pulses.     

Three-stage multilayer formation kinetics during adsorption of an anionic fluorinated surfactant onto germanium. 1. Concentration effect

The adsorption of tetraethylammonium perfluorooctylsulfonate (TEA-FOS) from aqueous solution onto hydroxylated germanium is studied using in situ polarized attenuated total reflection Fourier transform infrared spectroscopy (ATR-FTIR). The adsorption is monitored at a series of bulk solution concentrations spanning from well below to above the critical micelle concentration (CMC; 1.0 mM). The kinetics of adsorption is followed by monitoring the intensity of the fluorocarbon bands. The orientation of the fluorocarbon director with respect to the germanium surface is determined by circular dichroism measurements of CF2 stretching bands. At bulk concentrations ranging from 10% of the CMC to at least 500% of the CMC, the adsorption occurs in an unusual sequence of three stages. Initially, rapid adsorption occurs within 200 min, leading to coverage of a monolayer or less. A long period of slow adsorption follows, during which we hypothesize that surfactant molecules form clusters, some of which serve as nuclei for multilayer growth. This stage concludes suddenly with an acceleration in the rate of adsorption, which eventually leads to multilayer formation. Because this is an anionic surfactant adsorbing onto a negatively charged surface at pH 6, the tetraethylammonium ions must mediate the interactions between the surfactant headgroups and the surface. The dichroism measurements show that TEA-FOS is initially oriented randomly or somewhat parallel to the surface, but over time adopts an orientation somewhat normal to the surface. This behavior is consistent with initial adsorption at isolated sites, followed by aggregation into isotropic admicelles, and finally growth into flattened admicelles. The sudden onset of accelerated adsorption can be explained either by autoaccelerating adsorption or nucleation and growth of a hydrophobic multilayer structure.     

Methane oxidation at redox stable fuel cell electrode La0.75Sr0.25Cr0.5Mn0.5O(3-delta)

Because of its widespread availability, natural gas is the most important fuel for early application of stationary fuel cells, and furthermore, methane containing biogases are one of the most promising renewable energy alternatives; thus, it is very important to be able to efficiently utilize methane in fuel cells. Typically, external steam reforming is applied to allow methane utilization in high temperature fuel cells; however, direct oxidation will provide a much better solution. Recently, we reported good electrochemical performance for an oxide anode La0.75Sr0.25Cr0.5Mn0.5O3 (LSCM) in low moisture (3% H2O) H2 and CH4 fuels without significant coking in CH4. Here, we investigate the catalytic activity of this oxide with respect to its ability to utilize methane. This oxide is found to exhibit fairly low reforming activity for both H2O and CO2 reforming but is active for methane oxidation. LSCM is found to be a full oxidation catalyst rather than a partial oxidation catalyst as CO2 production dominates CO production even in CH4-rich CH4/O2 mixtures. X-ray adsorption spectroscopy was utilized to confirm that Mn was the redox active species, clearly demonstrating that this material has the oxidation catalytic behavior that might be expected from a Mn perovskite and that the Cr ion is only present to ensure stability under fuel atmospheres.     

Coulomb blockade anisotropic magnetoresistance effect in a (Ga,Mn)As single-electron transistor

We observe low-field hysteretic magnetoresistance in a (Ga,Mn)As single-electron transistor which can exceed 3 orders of magnitude. The sign and size of the magnetoresistance signal are controlled by the gate voltage. Experimental data are interpreted in terms of electrochemical shifts associated with magnetization rotations. This Coulomb blockade anisotropic magnetoresistance is distinct from previously observed anisotropic magnetoresistance effects as it occurs when the anisotropy in a band structure derived parameter is comparable to an independent scale, the single-electron charging energy. Effective kinetic-exchange model calculations in (Ga,Mn)As show chemical potential anisotropies consistent with experiment and ab initio calculations in transition metal systems suggest that this generic effect persists to high temperatures in metal ferromagnets with strong spin-orbit coupling.     

High finesse opto-mechanical cavity with a movable thirty-micron-size mirror

We report on the demonstration of a high finesse micro-optomechanical system and identify potential applications ranging from optical cooling to weak force detection to massive quantum superpositions. The system consists of a high quality diameter flat dielectric mirror cut from a larger substrate with a focused ion beam and attached to an atomic force microscope cantilever. Cavity ring-down measurements performed on a 25 mm long Fabry-Pérot cavity with the 30 microm mirror at one end show an optical finesse of 2100. Numerical calculations show that the finesse is not diffraction limited and that orders of magnitude higher finesse should be possible. A mechanical quality factor of more than 10(5) at pressures below 10(-3) mbar is demonstrated for the cantilever with a mirror attached.     

Finite energy solutions of self‐adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H¹‐solutions of second order, self‐adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H¹(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H¹(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.     

Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out in Charrier et al. (SIAM J Numer Anal, 2013), to more difficult problems, posed on non-smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fréchet differentiable non-linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen–Loève expansion of the random coefficient. Numerical results complete the paper.