Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.     

Anion–π Interactions in Salts with Polyhalide Anions: Trapping of I42−

The directionality of interaction of electron‐deficient π systems with spherical anions (e.g,. halides) can be controlled by secondary effects like NH or CH hydrogen bonding. In this study a series of pentafluorophenyl‐substituted salts with polyhalide anions is investigated. The compounds are obtained by aerobic oxidation of the corresponding halide upon crystallization. Solid‐state structures reveal that in bromide 2 , directing NH–anion interactions position the bromide ion in an η¹‐type fashion over but not in the center of the aromatic ring. The same directing forces are effective in corresponding tribromide salt 3 . In the crystal, the bromide ion is paneled by four electron‐deficient aromatic ring systems. In addition, compounds 4 and 6 , which have triiodide and the rare tetraiodide dianion as anions, are described. Computational studies reveal that the latter is highly unstable. In the present case it is stabilized by the crystal lattice, for example, by interaction with electron‐deficient π systems.     

An organocatalytic approach to 2-hydroxyalkyl- and 2-aminoalkyl furanes

The first enantioselective methodology for the synthesis of electron-poor 2-hydroxyalkyl- and 2-aminoalkyl furanes is demonstrated in this study. It utilizes a highly stereoselective organocatalytic one-pot reaction cascade: epoxidation or aziridination of α,β-unsaturated aldehydes followed by Feist-Be?nary reaction of various 1,3-dicarbonyl compounds to give the target furanes. This efficient multibond forming reaction cascade benefits from low catalyst loadings and readily available starting materials. Furthermore, the possibility to interrupt the reaction sequence at the stage of the corresponding optically active 2-hydroxyalkyl- and 2-aminoalkyl 2,3-dihydrofuranes with three stereogenic centers is also presented. Finally, models which account for the formation of the optically active 2,3-dihydrofuranes are proposed.     

Magnetic Coupling in Enantiomerically Pure Trinuclear Helicate‐Type Complexes Formed by Hierarchical Self‐Assembly

Based on chiral, enantiomerically pure 7‐[(S)‐phenylethylurea]‐8‐hydroxyquinoline ( 1 ‐H), trinuclear helicate‐type complexes 2 – 5 are formed with divalent transition‐metal cations. X‐ray structural analyses reveal the connection of two monomeric complex units [M( 1 )₃]^? (M=Zn, Mn, Co, Ni) by a central metal ion to form a “dimer”. Due to the enantiopurity of the ligand, the complexes are obtained as pure enantiomers, resulting in pronounced circular dichroism (CD) spectra. Single‐ion effects and intra‐ and intermolecular coupling are observed with dominating ferromagnetic coupling in the case of the cobalt(II) and nickel(II) and dominating antiferromagnetic coupling in the case of the manganese(II) complex.     

Cosmic curvature from de Sitter equilibrium cosmology

I show that the de Sitter equilibrium cosmology generically predicts observable levels of curvature in the Universe today. The predicted value of the curvature, Ω(k), depends only on the ratio of the density of nonrelativistic matter to cosmological constant density ρ(m)(0)/ρ(Λ) and the value of the curvature from the initial bubble that starts the inflation, Ω(k)(B). The result is independent of the scale of inflation, the shape of the potential during inflation, and many other details of the cosmology. Future cosmological measurements of ρ(m)(0)/ρ(Λ) and Ω(k) will open up a window on the very beginning of our Universe and offer an opportunity to support or falsify the de Sitter equilibrium cosmology.     

Arrays of Ru nanoclusters with narrow size distribution templated by monolayer graphene on Ru

Ru nanoclusters self-assemble over macroscopic sample areas during vapor deposition of Ru on monolayer graphene (MLG) on Ru(0001). The Ru nanoclusters form arrays with a mean lateral cluster diameter of ~ 20 Å, cluster heights of 1 or 2 ML, and a size distribution that remains nearly constant with increasing coverage. Combined scanning tunneling microscopy and density functional theory (DFT) show that the clusters are templated by the MLG/Ru(0001) moiré unit cell and identify the preferred binding site of the clusters as the low fcc region of the moiré. Cross-sectional transmission electron microscopy (TEM) and high-resolution TEM contrast simulations experimentally demonstrate that the interaction of the Ru clusters with the underlying MLG/Ru(0001) leads to a local lifting of the graphene layer of the template. DFT calculations confirm this mechanism of interaction of the Ru clusters with the strongly coupled MLG/Ru(0001). Weakening of the graphene-support coupling via oxygen intercalation is shown to have a major effect on the assembly of Ru nanocluster arrays. With a preferred binding site lacking on decoupled graphene, the Ru nanoclusters grow significantly larger, and clusters with 1 to 4 ML height can coexist.     

A class of distributed optimization methods with event-triggered communication

We present a class of methods for distributed optimization with event-triggered communication. To this end, we extend Nesterov’s first order scheme to use event-triggered communication in a networked environment. We then apply this approach to generalize the proximal center algorithm (PCA) for separable convex programs by Necoara and Suykens. Our method uses dual decomposition and applies the developed event-triggered version of Nesterov’s scheme to update the dual multipliers. The approach is shown to be well suited for solving the active optimal power flow (DC-OPF) problem in parallel with event-triggered and local communication. Numerical results for the IEEE 57 bus and IEEE 118 bus test cases confirm that approximate solutions can be obtained with significantly less communication while satisfying the same accuracy estimates as solutions computed without event-triggered communication.     

Separation and Thermal Racemization of the Enantiomers of Spiro[cyclohexadiene-dihydroacridines]

 The enantiomers of substituted spiro[cyclohexadiene-dihydroacridines] were separated by enantioselective liquid chromatography with the sorbent/solvent systems triacetylcellulose/methanol, tris-(3,5-dimethylphenylcarbamoyl)-cellulose/silica gel (Chiralcel OD^TM)/n-heptane/2-propanol, and (+)-poly-(trityl methacrylate)/silica gel/n-heptane/2-propanol. Interconversion barriers of the enantiomers were determined for a series of derivatives by thermal racemization. Electrocyclic ring opening/ring closure in terms of the Woodward-Hoffmann rules is discussed for the enantiomerization mechanism; the interconversion of the enantiomers by enolization is ruled out by deuterium exchange experiments.