Convergence Theory for Preconditioned Eigenvalue Solvers in a Nutshell

Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using multilevel preconditioning for elliptic differential operators or graph Laplacian eigenvalue problems, exhibit almost optimal complexity in practice; i.e., their computational costs to calculate a fixed number of eigenvalues and eigenvectors grow linearly with the matrix problem size. Theoretical justification of their optimality requires convergence rate bounds that do not deteriorate with the increase of the problem size. Such bounds were pioneered by E. D’yakonov over three decades ago, but to date only a handful have been derived, mostly for symmetric eigenvalue problems. Just a few of known bounds are sharp. One of them is proved in doi: 10.1016/S0024-3795(01)00461-X for the simplest preconditioned eigensolver with a fixed step size. The original proof has been greatly simplified and shortened in doi: 10.1137/080727567 by using a gradient flow integration approach. In the present work, we give an even more succinct proof, using novel ideas based on Karush–Kuhn–Tucker theory and nonlinear programming.     

Beyond Stereoselectivity,Switchable Catalysis: Some of the Last Frontier Challenges in Ring‐Opening Polymerization of Cyclic Esters

Metal‐based catalysts and initiators have played a pivotal role in the ring‐opening polymerization (ROP) of cyclic esters, thanks to their high activity and remarkable ability to control precisely the architectures of the resulting polyesters in terms of molar mass, dispersity, microstructure, or tacticity. Today, after two decades of extensive research, the field is slowly reaching maturity. However, several challenges remain, while original concepts have emerged around new types or new applications of catalysis. This Review is not intended to comprehensively cover all of these aspects. Rather, it provides a personal overview of the very recent progress achieved in some selected, important aspects of ROP catalysis—stereocontrol and switchable catalysis. Hence, the first part addresses the development of new metal‐based catalysts for the isoselective ROP of racemic lactide towards stereoblock copolymers, and the use of syndioselective ROP metal catalysts to control the monomer sequence in copolymers. A second part covers the development of ROP catalysts—primarily metal‐based catalysts, but also organocatalysts—that can be externally regulated by the use of chemical or photo stimuli to switch them between two states with different catalytic abilities. Current challenges and opportunities are highlighted.     

Hyperbolic BMOA classes

Let φ be a holomorphic mapping between complex unit balls. We obtain several characterizations of those φ for which the Möbius-invariant BMOA condition holds with respect to the Bergman metric.     

On lovely pairs of geometric structures

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil-Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.     

Molecular dynamics simulation comparison of atomic scale intermixing at the amorphous Al2O3/semiconductor interface for a-Al2O3/Ge, a-Al2O3/InGaAs, and a-Al2O3/InAlAs/InGaAs

The structural properties of a-Al₂O₃/Ge, a-Al₂O₃/In_0.5Ga_0.5As and a-Al₂O₃/In_0.5Al_0.5As/InGaAs interfaces were investigated by density-functional theory (DFT) molecular dynamics (MD) simulations. Realistic a-Al₂O₃ samples were generated using a hybrid classical-DFT MD “melt and quench” approach. The interfaces were formed by annealing at 700 K/800 K and 1100 K with subsequent cooling and relaxation. The a-Al₂O₃/Ge interface demonstrates pronounced interface intermixing and interface bonding exclusively through Al–O–Ge bonds generating high interface polarity. In contrast, the a-Al₂O₃/InGaAs interface has no intermixing, Al–As and O–In/Ga bonding, low interface polarity due to nearly compensating interface dipoles, and low substrate deformation. The a-Al₂O₃/InAlAs interface demonstrated mild intermixing with some substrate Al atoms being adsorbed into the oxide, mixed Al–As/O and O–Al/In bonding, medium interface polarity, and medium substrate deformation. The simulated results demonstrate strong correlation to experimental measurements and illustrate the role of weak bonding in generating an unpinned interface for metal oxide/semiconductor interfaces.     

Electropolymerizable 2,2'-Carboranyldithiophenes. Structure-Property Investigations of the Corresponding Conducting Polymer Films by Electrochemistry, UV-Visible Spectroscopy and Conducting Probe Atomic Force Microscopy

Carborane-functionalized conducting polymer films have been electrogenerated in dichloromethane from the anodic oxidation of ortho- (1), meta- (3) and para-carborane (4) isomers linked to two 2-thienyl units. The corresponding electrochemical response was characterized by a broad reversible redox system corresponding to the p-doping/undoping of the polythiophene backbone, the formal potential of which increased in the order poly(1) < poly(3) < poly(4), from ca. 0.50 to 1.15 V vs Ag/Ag(+) 10(-2) M. From further UV-visible spectroscopy analysis, the optical band gap was estimated at 1.8, 2.0 and 2.2 eV for poly(1), poly(3) and poly(4), respectively. The more conjugated and electroconductive character of poly(1) is ascribed to a more planar conformation of the conjugated backbone resulting from an intramolecular β-β' cyclization reaction in the monomer, consequently yielding a fused conjugated polymer. Molecular modeling calculations using the DFT method support this hypothesis. The surface topography and maps of the conductive domains of the electropolymerized films were evaluated by conducting probe AFM. The three polymers exhibit fairly similar morphological characteristics and a surface roughness of ~2 nm. Current-voltage (I-V) characteristics of conducting AFM tip-carborane polymer-ITO junctions showed that poly(1) had the highest conductivity.     

Avoiding the classical resolution during the synthesis of MeO-BIPHEP and 3,3′-disubstituted derivatives

The Ullmann coupling of 1 (R = H) gives a 2:1 mixture of diastereomers 2 (R = H) in 81% yield that are easily separated by silica gel chromatography. This procedure avoids the generally cumbersome and sometimes difficult resolution step with DBTA. Similar Ullmann couplings and separation of the corresponding diastereomers are employed with other derivatives of 1 (R = OtBu, iPr, Ph, and mesityl) ultimately affording a new series of 3,3′-disubsituted-MeO-BIPHEP derivatives. The use of these new derivatives in palladium-catalyzed asymmetric Heck reaction, Pd-catalyzed polyene cyclizations and rhodium-catalyzed hydrogenations is also reported.     

Stochastic radiative transfer in multilayer broken clouds. Part I: Markovian approach

A two-part paper describes the statistical treatment of solar radiative transfer in multilayer broken clouds. The proposed approach is a logical development of the statistical ones originally suggested for a single-layer broken clouds. This first part introduces a new statistically inhomogeneous Markovian model that allows one to properly account for different combinations of the random and maximum overlap of broken clouds at distinct vertical layers. The statistically inhomogeneous Markovian model and the stochastic radiative transfer equation have been used to derive equations for the mean radiance of solar radiation. It was demonstrated that in extreme cases the obtained equations agree with corresponding equations previously derived for (i) the statistically homogeneous broken clouds and (ii) the vertically inhomogeneous overcast clouds.