An Exotic {T_{1}\mathbb{S}^4} with Positive Curvature

We construct a metric with positive sectional curvature on a 7-manifold which supports an isometry group with orbits of codimension 1. It is a connection metric on the total space of an orbifold 3-sphere bundle over an orbifold 4-sphere. By a result of S. Goette, the manifold is homeomorphic but not diffeomorphic to the unit tangent bundle of the 4-sphere.     

A novel monochromator for experiments with ultrashort X‐ray pulses

Aiming at advancing storage‐ring‐based ultrafast X‐ray science, over the past few years many upgrades have been undertaken to continue improving beamline performance and photon flux at the Femtoslicing facility at BESSY II. In this article the particular design upgrade of one of the key optical components, the zone‐plate monochromator (ZPM) beamline, is reported. The beamline is devoted to optical pump/soft X‐ray probe applications with 100 fs (FWHM) X‐ray pulses in the soft X‐ray range at variable polarization. A novel approach consisting of an array of nine off‐axis reflection zone plates is used for a gapless coverage of the spectral range between 410 and 1333 eV at a designed resolution of E/ΔE = 500 and a pulse elongation of only 30 fs. With the upgrade of the ZPM the following was achieved: a smaller focus, an improved spectral resolution and bandwidth as well as excellent long‐term stability. The beamline will enable a new class of ultrafast applications with variable optical excitation wavelength and variable polarization.     

A note on the stability theory of buoyancy-driven ocean currents over a sloping bottom

In a previous paper, a new model was derived describing the baroclinic dynamics of buoyancy-driven ocean currents over a sloping bottom. In particular, a normal mode stability analysis and a general stability analysis based on an appropriately constrained energy invariant were presented. However, these two sets of stability results were not identical to each other. Here, we show that the normal-mode stability results previously described may be derived from a general stability analysis based on an appropriately constrained linear momentum invariant. In addition, we establish conditions for the nonlinear stability in the sense of Liapunov of these flows. The analysis presented here eliminates the need to introduce a Poincare inequality between the perturbation energy and the enstrophy which the previous analysis was forced to assume. Relaxing this assumption means the present analysis is applicable to a much larger range of flow geometries and is therefore a substantially stronger result.     

On the cardinality of the Pareto set in bicriteria shortest path problems

Computing shortest paths with two or more conflicting optimization criteria is a fundamental problem in transportation and logistics. We study the problem of finding all Pareto-optimal solutions for the multi-criteria single-source shortest-path problem with nonnegative edge lengths. The standard approaches are generalizations of label-setting (Dijkstra) and label-correcting algorithms, in which the distance labels are multi-dimensional and more than one distance label is maintained for each node. The crucial parameter for the run time and space consumption is the total number of Pareto optima. In general, this value can be exponentially large in the input size. However, in various practical applications one can observe that the input data has certain characteristics, which may lead to a much smaller number—small enough to make the problem efficiently tractable from a practical viewpoint. For typical characteristics which occur in various applications we study in this paper whether we can bound the size of the Pareto set to a polynomial size or not. These characteristics are also evaluated (1) on a concrete application scenario (computing the set of best train connections in view of travel time, fare, and number of train changes) and (2) on a simplified randomized model. It will turn out that the number of Pareto optima on each visited node is restricted by a small constant in our concrete application, and that the size of the Pareto set is much smaller than our worst case bounds in the randomized model. A preliminary short version of this paper appeared in the Proceedings of the 5th Workshop on Algorithm Engineering (WAE 2001), LNCS 2141, Springer Verlag, pp. 185–197 (2001) under the title “Pareto shortest paths is often feasible in practice.”     

Convenient synthetic route of versatile 21-monothiatetraphenylporphyrins of the A4 and AB3 type

A novel convenient synthetic route for poly-functional 21-monothiatetraphenylporphyrins of the type A₄ und AB₃ having base labile substituents in meso position was developed. Using this method a series of symmetric and asymmetric 21-thiaporphyrins containing different functional groups at the meso position is reported. The new products were characterized by NMR, UV-Vis and mass spectroscopy.     

A convenient synthesis of long-chain 4-substituted benzyloxycarbonyl alkanethiols for the formation of self assembled monolayers on metal substrates

New 4-substituted benzyl esters of 16-mercaptohexadecanoic acid were prepared by developing a practicable synthetic procedure and using readily available staring materials. The compounds synthesized have been characterized by NMR, MS, IR spectra and elemental analysis. The mercapto derivatives are precursors for the formation of self-assembled monolayers on metal substrates. Dedicated to Professor Dr. Manfred Hesse on the occasion of his 70 ^th birthday     

Coupling of FEM and BEM in Shape Optimization

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape, especially L ²-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.This research has been carried out when the second author stayed at the Department of Mathematics, Utrecht University, The Netherlands, supported by the EU-IHP project Nonlinear Approximation and Adaptivity: Breaking Complexity in Numerical Modelling and Data Representation     

Über endlich erzeugte Fittingklassen

Ohne Zusammenfassung     

The influence of nonfundamental pump modes in erbium-doped fiber amplifiers

This work analyzes how the characteristics of an erbium-doped fiber amplifier (EDFA) depends on the mode distribution of 800-nm pump light. The analysis is performed by using a large-signal model of an optimally pumped erbium-doped fiber amplifier. Special attention is given to the gain degradation, and noise figures of the amplifier for co- and counterpropagating pump light.