Warmstarting the homogeneous and self-dual interior point method for linear and conic quadratic problems

We present two strategies for warmstarting primal-dual interior point methods for the homogeneous self-dual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and their negligible computational cost. This is a major advantage when compared to previously suggested strategies that require a pool of iterates from the solution process of the initial problem. Consequently our strategies are better suited for users who use optimization algorithms as black-box routines which usually only output the final solution. Our two strategies differ in that one assumes knowledge only of the final primal solution while the other assumes the availability of both primal and dual solutions. We analyze the strategies and deduce conditions under which they result in improved theoretical worst-case complexity. We present extensive computational results showing work reductions when warmstarting compared to coldstarting in the range 30–75% depending on the problem class and magnitude of the problem perturbation. The computational experiments thus substantiate that the warmstarting strategies are useful in practice.     

Types of von Neumann algebras associated with extremal invariant states

A globalized version of the following is proved. Let be a factor acting on a Hilbert space ,G a group of unitary operators on inducing automorphisms of ,x a vector separating and cyclic for which is up to a scalar multiple the unique vector invariant under the unitaries inG. Then either is of type III or _x is a trace of . The theorem is then applied to study the representations due to invariant factors state of asymptotically abelianC*-algebras, and to show that in quantum field theory certain regions in the Minkowski space give type III factors.     

On the completeness of eigenfunctions of some higher order operators

The completeness of diverse eigenfunction systems is of interest, e.g., in connection with their use for expansion purposes. Whereas second-order operators have been treated extensively, it is more difficult to find valid proofs even for comparatively simple fourth- and higher-order operators. A criterion is developed that allows known properties of second-order differential operators to be used as a basis for conclusions regarding the completeness of the eigenfunctions of related fourth-order operators. The completeness of the “flatclamped-plate modes” is proved as an example, and it is demonstrated that the detailed form of the “conditions of finiteness at the singular point” can be crucial for the definition of operators corresponding to differential expressions with singular points.     

Relative reactivity enhancement of (Me5Cp)2ZrCl2 in mixture with (1,2,4-Me3Cp)2ZrCl2 during ethene/1-hexene copolymerization

Dual-site ethene/1-hexene copolymerizations with MAO-activated (1,2,4-Me₃Cp)₂ZrCl₂ and (Me₅Cp)₂ZrCl₂ catalysts were performed. Copolymers with narrow molecular weight distributions and bimodal short chain branching distributions could be produced. The combined catalyst system demonstrates a number of discrepancies from an expected average behavior of the individual sites. Dual-site (1,2,4-Me₃Cp)₂ZrCl₂/(Me₅Cp)₂ZrCl₂ systems produce copolymers with lower incorporation than expected. Clear evidences for relative activity enhancement of the (Me₅Cp)₂ZrCl₂ catalyst in the mixture were observed in melting endotherms and Crystaf profiles. Molecular weights obtained by the mixture were higher than for any of the individual catalysts. A similar effect is observed for a dual-site system of the (1,2,4-Me₃Cp)₂ZrCl₂ catalyst together with the Me₄Si₂(Me₄Cp)₂ZrCl₂ catalyst as an alternative to (Me₅Cp)₂ZrCl₂.     

On the solid solution of the spinel phase in the system NiO-Al2O3

The solid solubility of Al₂O₃ in NiAl₂O₄ spinel has been investigated by powder X-ray diffraction of samples prepared by solid state synthesis. The solid solution region found was in agreement with a previous report. The cubic cell parameter of the spinel solid solution was observed to decrease with increasing alumina content. Spinel with high alumina content was shown to be close to an inverse spinel as previously reported for stoichiometric NiAl₂O₄ and the inversion parameter proved to be relatively independent of the overall composition.     

Cryogenic Correlative Single‐Particle Photoluminescence Spectroscopy and Electron Tomography for Investigation of Nanomaterials

Cryogenic single‐particle photoluminescence (PL) spectroscopy has been used with great success to directly observe the heterogeneous photophysical states present in a population of luminescent particles. Cryogenic electron tomography provides complementary nanometer scale structural information to PL spectroscopy, but the two techniques have not been correlated due to technical challenges. Here, we present a method for correlating single‐particle information from these two powerful microscopy modalities. We simultaneously observe PL brightness, emission spectrum, and in‐plane excitation dipole orientation of CdSSe/ZnS quantum dots suspended in vitreous ice. Stable and fluctuating emitters were observed, as well as a surprising splitting of the PL spectrum into two bands with an average energy separation of 80 meV. In some cases, the onset of the splitting corresponded to changes in the in‐plane excitation dipole orientation. These dynamics were assigned to structures of individual quantum dots and the excitation dipoles were visualized in the context of structural features.