Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity

This paper deals with a control-constrained linear-quadratic optimal control problem governed by the Stokes equations. It is concerned with situations where the gradient of the velocity field is not bounded. The control is discretized by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes that are not necessarily conforming. The approximate control is constructed as projection of the discrete adjoint velocity in the set of admissible controls. It is proved that under certain assumptions on the discretization of state and adjoint state this approximation is of order 2 in L ²(Ω). As first example a prismatic domain with a reentrant edge is considered where the impact of the edge singularity is counteracted by anisotropic mesh grading and where the state and the adjoint state are approximated in the lower order Crouzeix-Raviart finite element space. The second example concerns a nonconvex, plane domain, where the corner singularity is treated by isotropic mesh grading and state and adjoint state can be approximated by a couple of standard element pairs.     

Four-Dimensional Compact Projective Planes of Orbit Type (1,1)

We consider 4-dimensional flexible projective planes with the following properties: The collineation group is a 6-dimensional solvable Lie group which fixes some flag ∞ ∈ W. Furthermore, the collineation group has a 1-dimensional orbit both on W and on the pencil of lines through {∞}. We show that there are three different families of planes with these properties.     

Flying qubits and entangled photons

Efficient generation of polarized single photons or entangled photon pairs is crucial for the implementation of quantum key distribution (QKD) systems. Self organized semiconductor quantum dots (QDs) are capable of emitting on demand one polarized photon or an entangled photon pair upon current injection. Highly efficient single‐photon sources consist of a pin structure inserted into a microcavity where single electrons and holes are funneled into an InAs QD via a submicron AlOx aperture, leading to emission of single polarized photons with record purity of the spectrum and non‐classicality of the photons. A new QD site‐control technique is based on using the surface strain field of an AlO_x current aperture below the QD. GaN/AlN QD based devices are promising to operate at room temperature and reveal a fine‐structure splitting (FSS) depending inversely on the QD size. Large GaN/AlN QDs show disappearance of the FSS. Theory also suggests QDs grown on (111)‐oriented GaAs substrates as source of entangled photon pairs.     

Packet loss characteristics for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis> queueing systems

In this contribution we investigate higher-order loss characteristics for M/G/1/N queueing systems. We focus on the lengths of the loss and non-loss periods as well as on the number of arrivals during these periods. For the analysis, we extend the Markovian state of the queueing system with the time and number of admitted arrivals since the instant where the last loss occurred. By combining transform and matrix techniques, expressions for the various moments of these loss characteristics are found. The approach also yields expressions for the loss probability and the conditional loss probability. Some numerical examples then illustrate our results.     

Preparation of Cr2O3-Al2O3 Solid Solutions by Reactive Magnetron Sputtering

 (Al,Cr)₂O₃ layers were deposited on cemented carbide insert tips at a substrate temperature of 500 °C by means of reactive magnetron sputtering. An Al target was sputtered in RF mode and a Cr target in DC mode simultaneously in an oxygen/argon plasma. The influence of the Al and Cr sputter power and of the oxygen partial pressure on composition and structure of the (Al,Cr)₂O₃ layers as well as on the binding states of their components were investigated. Special attention was paid to the interpretation of the O ls and O-KLL fine structure and peak shifts. For the binary phases γ-Al₂O₃ and Cr₂O₃, a good agreement with literature values was observed in each case. In case of the ternary phases a continuous shift of the energetic position of the O1s peak, the O-KL₂₃L₂₃ transition and the modified Auger parameter α ′ of oxygen between the two binary phases γ-Al₂O₃ and Cr₂O₃ could be detected, indicating a wide range of solid solubility between Al₂O₃ and Cr₂O₃. As revealed by grazing incidence X-ray diffraction, the crystallinity of the ternary phases is less pronounced as compared to the binaries and increases with increasing oxygen flow rate.     

Sandwich-type tetranuclear lanthanide complexes with cucurbit[6]uril: from molecular compounds to coordination polymers

Sandwich-type lanthanide complexes with macrocyclic ligand cucurbit[6]uril (C 36H 36N 24O 12, CB[6]) were synthesized under hydrothermal conditions from aqueous solutions of lanthanide(III) bromides, CB[6], and 4-cyanopyridine. According to X-ray analysis (Ln = La, Pr, Dy, Ho, Er, and Yb), the compounds with different structural types of lanthanide cores have a common fragment where the tetranuclear hydroxo complex is sandwiched between two macrocycles {(IN@CB[6])Ln 4(mu 3-OH) 4(IN@CB[6])} (6+) (IN = isonicotinate). The photoluminescence (for Ln = Eu) and Fourier transform ion cyclotron resonance mass spectra (for Ln = Pr, Dy, and Er) were studied. The compounds are used for the first time as precursors for the synthesis of lanthanide-silver heterometallic coordination polymers. The chainlike crystal structure of polymers (Ln = La, Pr, and Dy) is constituted by the sandwich complexes linked via the coordination of IN nitrogen atoms to the silver atoms.