Approach to Equilibrium for a Class of Random Quantum Models of Infinite Range

We consider random generalizations of a quantum model of infinite range introduced by Emch and Radin. The generalizations allow a neat extension from the class l ₁ of absolutely summable lattice potentials to the optimal class l ₂ of square summable potentials first considered by Khanin and Sinai and generalised by van Enter and van Hemmen. The approach to equilibrium in the case of a Gaussian distribution is proved to be faster than for a Bernoulli distribution for both short-range and long-range lattice potentials. While exponential decay to equilibrium is excluded in the nonrandom l ₁ case, it is proved to occur for both short and long range potentials for Gaussian distributions, and for potentials of class l ₂ in the Bernoulli case. Open problems are discussed.     

Time evolution of a fractal distribution: Particle concentrations in free-surface turbulence

Steady-state turbulence is generated in a tank of water and the trajectories of particles forming a compressible system on the surface are tracked in time. The initial uniformly distributed floating particles coagulate and form a fractal structure, a rare manifestation of a strange attractor observable in real space. The surface pattern reaches a steady state in approximately 1 s. Measurements are made of the fractal dimensions D_q(t) (q=1 to 6) of the floating particles starting with the uniform distribution D_q(0)=2 for Taylor Microscale Reynolds number Re_λ?160. Focus is on the time evolution of the correlation dimension D₂(t) as the steady state is approached. This steady state is reached in several large eddy turnover times and does so at an exponential rate.     

Upper bound on the slope of steady water waves with small adverse vorticity

We consider the angle of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. There is an upper bound of 31.15° in the irrotational case [1] and an upper bound of 45° in the case of favorable vorticity [13]. On the other hand, if the vorticity is adverse, the profile can become vertical. We prove here that if the adverse vorticity is sufficiently small, then the angle still has an upper bound which is slightly larger than 45°.     

Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Hölder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.     

A Note on the Morozov Principle via Lagrange Duality

Considering a general linear ill-posed equation, we explore the duality arising from the requirement that the discrepancy should take a given value based on the estimation of the noise level, as is notably the case when using the Morozov principle. We show that, under reasonable assumptions, the dual function is smooth, and that its maximization points out the appropriate value of Tikhonov’s regularization parameter. The numerical relevance of our approach is established by means of an illustrative example from nonparametric instrumental regression, a standard problem in statistics.     

Pressure beneath a Stokes wave

We investigate the pressure within an irrotational fluid in a periodic, steady, two‐dimensional gravity wave above a flat bed. We prove that the pressure in the fluid strictly decreases horizontally away from the crest line. Furthermore, the pressure strictly increases with depth. We also investigate the effect of an underlying current on the paths of the particles. © 2009 Wiley Periodicals, Inc.     

Potential-modulated, attenuated total reflectance spectroscopy of poly(3,4-ethylenedioxythiophene) and poly(3,4-ethylenedioxythiophene methanol) copolymer films on indium-tin oxide

We report the first application of a potential-modulated spectroelectrochemical ATR (PM-ATR) instrument utilizing multiple internal reflections at an optically transparent electrode to study the charge-transfer kinetics and electrochromic response of adsorbed films. A sinusoidally modulated potential waveform was applied to an indium-tin oxide (ITO) electrode while simultaneously monitoring the optical reflectivity of thin (2-6 equivalent monolayers) copolymer films of poly(3,4-ethylenedioxythiophene) (PEDOT) and poly(3,4-ethylenedioxythiophene methanol) (PEDTM), previously characterized in our laboratory. At high modulation frequencies the measured response of the polymer film is selective toward the fastest electrochromic processes in the film, presumably those occurring within the first adsorbed monolayer. Quantitative determination of the electrochromic switching rate, derived from the frequency response of the attenuated reflectivity, shows a linear decrease in the rate, from 11 x 10(3) s(-1) to 3 x 10(3) s(-1), with increasing proportions of PEDTM in the copolymer, suggesting that interactions between the methanol substituent on EDTM and the ITO surface slow the switching process by limiting the rate of conformational change in the polymer film.     

Dependence of the zero shear-rate viscosity and the viscosity function of linear high-density polyethylenes on the mass-average molar mass and polydispersity

Linear high-density polyethylenes with molar masses M _w between 240 and 1,000,000 g/mol, obtained by metallocene catalysts, were characterized in shear using oscillatory and creep tests. The polydispersities of the molar mass distributions (MMDs) lay between 1 and 16. The resulting zero shear-rate viscosities η₀ covered a range from 2.5×10⁻³ to around 10⁸ Pas. Above a critical molar mass of M _c≈2,900 g/mol, the experimental results can be described by the relation η₀ ∼ M _w^3.6, independently of the MMD. The oscillatory data were fitted with a Carreau–Yasuda equation. The resulting parameters were correlated to molecular structure. The parameter a, being a quantity for the width of the transition between the Newtonian and the non-Newtonian regime, showed a dependence on the molar mass M _w but not on M _w/M _n. The parameter λ of the Carreau-Yasuda equation was found to be the reciprocal crossover frequency for all samples with a log-Gaussian MMD. λ depends on the molar mass M _w and also on M _w/M _n.

Equations on real intervals

A (finite or infinite) set ∑ of equations, in operation symbols F_t (t ∈T) and variables x_i, is said to be compatible with iff there exist continuous operations F_t^A on such that the algebra satisfies the equations ∑ (with the variables x_i understood as universally quantified). It is proved that there is no algorithm to decide -compatibility for all finite ∑. If the definition is restricted to C¹ idempotent operations F_t^A , then there does exist an algorithm for compatibility. Received August 9, 2005; accepted in final form February 14, 2006.