Wavelength division (de)multiplexing based on dispersive self-imaging

We proposed and experimentally demonstrated wavelength division (de)multiplexers (WDMs) utilizing the wavelength dispersive nature of self-imaging multimode interferometers. Proof-of-principle devices fabricated on the silicon-on-insulator platform operated as 4-channel WDMs with a free spectral range of >90 nm, an averaging cross talk of <-20 dB for a 1 nm band, and an insertion loss of <2.0 dB. The potential for higher channel counts and smaller channel wavelength spacing was also predicted. This type of WDM is easy to design and fabricate. The underlying concept is applicable to all planar waveguide platforms.     

On the parameter choice in grad-div stabilization for the Stokes equations

Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be $\mathcal O(1)$ . This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^{1}(\Omega )$ error of the velocity and the $L^{2}(\Omega )$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the $H^{1}(\Omega )$ error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the $L^{2}(\Omega )$ error of the pressure requires in many cases smaller stabilization parameters than the minimization of the $H^{1}(\Omega )$ velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.     

Coupling nonlinear Stokes and Darcy flow using mortar finite elements

We study a system composed of a nonlinear Stokes flow in one subdomain coupled with a nonlinear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes–Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the algorithm and to compare two treatments of the defective boundary conditions.     

On the local behavior of certain homeomorphisms. II

In the previous paper by the authors, it is shown that a sense-preserving homeomorphism, absolutely continuous on lines (ACL), under certain integral conditions on the complex dilation is conformal at a point. This result includes the Teichmüller-Wittich-Belinski theorem. In the present paper, it is shown that under weakened conditions the local behavior of such a homeomorphism is similar to that of a quasiconformal mapping. Bibliography: 2 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 11–20.Dedicated to the 90th anniversary of G. M. Goluzin's birth.     

Time correlation functions in classical dense fluids

The velocity autocorrelation functions and memory functions of dense classical fluids may be directly obtained from the static radial distribution function g(r) in an approximate way. Following the Mori projection operator formalism, the memory functions may be related to the fluctuating force correlation. At low densities, these functions may be evaluated by following the trajectories of particle pairs in the interatomic potential. At higher densities, the force correlation functions can be evaluated approximately from particle pair trajectories via the potential of the mean force. The contributions to the memory function come mainly from particle pairs with rather specific and rather short interatomic distances. At higher temperatures, this specific distance is even shorter, hence the memory function decays quickly with time. At lower temperatures, a negative region of the memory function may develop. On the other hand, there is relatively little density dependence of the normalized memory function. The results for argon fluids at various densities and temperatures agree satisfactorily with the molecular dynamics and the Enskog values. The decrease of the diffusion coefficient with density is partly due to the nature of g(r) which reflects the stronger clustering of atoms at higher densities.     

Selective mono reduction of bis-phosphine oxides under mild conditions

Bis-phosphine oxides can be selectively reduced to bis-phosphine monoxides under exceptionally mild conditions using triflic anhydride and a thiol.     

Children's Books Reviewed

Maps and Globes Jack Knowlton Hurricane Watch Franklyn M. Branley Sunshine Makes the Seasons Franklyn M. Branley Heart Disease America's No. 1 Killer Dr. Alvin and Virginia B. Silverstein Cuts, Breaks, Bruises and Burns Joanna Cole Auks, Rocks, and the Odd Dinosaur Inside Stories from the Smithsonian Institute by Peggy Thomson The Secret Life of Cosmetics: A Science Experiment Book Vicki Cobb A Children's Zoo Tana Hoban Milk Donald Carrick Wallaby Creek Joyce Powzyk Being a Twin, Having a Twin Maxine Rosenberg     

On Some Families of Modules for the Current Algebra

Given a finite-dimensional module V for a finite-dimensional, complex semi-simple Lie algebra \(\mathcal {g}\), and a positive integer m, we construct a family of graded modules for the current algebra \(\mathcal {g}[t]\) indexed by simple C \(\mathcal {S}_{m}\)-modules. These modules are free of finite rank for the ring of symmetric polynomials and so can be localized to give finite-dimensional graded \(\mathcal {g}[t]\)-modules. We determine the graded characters of these modules and show that these graded characters admit a curious duality.     

Zur Bestimmung des Stickstoffs in organischen Substanzen

Ohne Zusammenfassung