Stable Boundary Conditions and Discretization for $P_N$ Equations

A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic $(P_N)$ approximation, which ensure that this fundamental energy bound is satisfied by the $P_N$ approximation. Our BCs are compatible with the characteristic waves of $P_N$ equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown on abstract formulations of $P_N$ equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the $P_N$ method. We show that summation by parts (SBP) finite difference on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level.     

O2 Activation and Double CH Oxidation by a Mononuclear Manganese(II) Complex

A Mn^II complex, [Mn(dpeo)₂]²⁺ (dpeo=1,2‐di(pyridin‐2‐yl)ethanone oxime), activates O₂, with ensuing stepwise oxidation of the methylene group in the ligands providing an alkoxide and ultimately a ketone group. X‐ray crystal‐structure analysis of an intermediate homoleptic alkoxide Mn^III complex shows tridentate binding of the ligand via the two pyridyl groups and the newly installed alkoxide moiety, with the oxime group no longer coordinated. The structure of a Mn^II complex of the final ketone ligand, cis‐[MnBr₂(hidpe)₂] (hidpe=2‐(hydroxyimino)‐1,2‐di(pyridine‐2‐yl)ethanone) shows that bidentate oxime/pyridine coordination has been resumed. H₂¹⁸O and ¹⁸O₂ labeling experiments suggest that the inserted O atoms originate from two different O₂ molecules. The progress of the oxygenation was monitored through changes in the resonance‐enhanced Raman bands of the oxime unit.     

Maximal regularity of the spatially periodic stokes operator and application to nematic liquid crystal flows

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L^p-regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group \(G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z}\) to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal L^p regularity of a sectorial operator applies.     

Kaizen teaching and the learning habits of engineering students in a freshman mathematics course

Which are the teaching methods that actually contribute to the learning of mathematics? The answer to this certainly is the holy grail of didactic and pedagogy, and should be supported by large scale statistical evidence. Our article aims at providing an initial step into this direction by first illustrating a teaching paradigm that is suited for the generation of large scale data sets: based on industry best practice quality assurance standards we introduce the Kaizen teaching paradigm which enforces Kolb’s reflective learning cycle on the students’ side. Second, we present and analyze the data we obtained through our pilot implementation at a engineering freshman mathematics course in the Sultanate of Oman. These emphasize the effectiveness of Kaizen teaching and once again show the necessity of continuous learning. A practice that seems to be forgotten in traditional university engineering courses due to the mere size of the audience. In particular it seems that a Markovian estimator for students’ performance may have to be considered.     

Mechanism of Ru(II)-catalyzed olefin insertion and C-H activation from quantum chemical studies

The mechanism of catalytic hydroarylation of olefins by the homogeneous Ru(Tp)(CO)(Ph)(NCCH3) catalyst recently reported by Gunnoe et al. is characterized using quantum mechanics (density functional theory). The catalytic cycle features two key steps, 1,2-olefin insertion and C-H activation via an unusual mechanism, oxidative hydrogen migration. We find that these two key steps are competitive and that improving the rate of one step is detrimental to the rate of the other. The Ru catalyst has better balance and consequently higher activity than the previously explored Ir-based system.     

A model for π-selfadjoint operators in π1 and a special linear pencil

The eigenvalue of nonpositive type of a -selfadjoint operator A in a Pontrjagin space of index one is characterized in a model of A. Similar questions are studied for an eigen-value problem G-T in a Hilbert space with a selfadjoint operator T and a special operator G.