Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds

In the first part of this article, we prove an explicit lower bound on the distance to the cut point of an arbitrary geodesic in a simply connected two-step nilpotent Lie group G with a lieft invariant metric. As a result, we obtaine a lower bound on the injectivity radius of a simply connected two-step nilpotent Lie group with a left invariant metric. We use this lower bound to determine the form of certain length minimizing geodesics from the identity to elements in the center of G. We also give an example of a two-step nilpotent Lie group G such that along most geodesics in this group, the cut point and the first conjugate point do not coincide. In the second part of this article, we examine the relation between the Laplace spectrum and the length spectrum on nilmanifolds by showing that a method developed by Gordon and Wilson for constructing families of isospectral two-step nilmanifolds necessarily yields manifolds with the same length spectrum. As a consequence, all known methods for constructing families of isospectral two-step nilmanifolds necessarily yield manifolds with the same length spectrum. In memory of Robert Brooks     

Tangential Boundary Behaviour of Harmonic Functions on Trees

We study the behaviour of harmonic functions on a homogeneous tree from the point of view of the tangential boundary covergence.     

Deactivation of a ruthenium(II) <Emphasis Type="Italic">N</Emphasis>-heterocyclic carbene <Emphasis Type="Italic">p</Emphasis>-cymene complex during transfer hydrogenation catalysis

A ruthenium (II) N-heterocyclic carbene (NHC) complex was synthesized to investigate ligand dissociation as a possible deactivation pathway for the catalytic cycle of a transfer hydrogenation reaction. Diiodo(1,3-dimethylbenzimidazole-2-ylidene)(p-cymene)ruthenium(II) was synthesized for use as the catalytic species and characterized using physico-chemical, spectroscopic methods, and single crystal X-ray diffraction. The transfer of hydrogen from isopropanol to acetophenone was followed using ¹H NMR. We observed 94% conversion of the substrate to the alcohol product after 1 h. We also found that the p-cymene complex decomposed during the catalytic reaction to the extent of 80% deactivation after 1 h, based on ¹H NMR spectrometry. From Gaussian calculations, an ultraviolet–visible spectrum that is in excellent agreement with the actual spectrum was computed, giving insight into the nature of the absorptions observed experimentally.     

A study on the coordinative versatility of new N,S-donor macrocyclic ligands: XAFS, and Cu2+ complexation thermodynamics in solution

We investigated, both in the solid state and in aqueous solution, the coordination environment and stability behavior of four macrocyclic ligands (three N(2)S(2) and one N(3)S(2)) and of the corresponding Cu(II) complexes. The structural characterization in the solid state of the copper derivatives was performed by X-Ray Absorption Spectroscopy. Copper is found to be 4-fold coordinated with a CuN(2)S(2) environment with different Cu-S distances depending on the size of the macrocyclic ring. The EXAFS technique has indicated that nitrogen and sulfur atoms are more preferable to oxygen atoms as donor systems, without the evidence of coordination of the carboxylic moieties to copper in the first shell. The joint EXAFS and XANES study of the copper(II) complex with the N(3)S(2) ligand confirms the 4-fold coordination with an additional, long Cu-N interaction. The Cu(2+) complexation constants for one ligand were determined in aqueous solution. The results indicate that the species [CuL], although isolated in the solid state, is not the most abundant at the pH of blood serum. Instead, at pH 7.4 the protonated [Cu(HL)](+) species was found to be the most relevant. The behaviour of the copper complexes in the presence of the strong copper chelating bioagent human serum albumin was also examined in order to gain information on the stability of these compounds in biological fluids.     

Novel Copolymers of Styrene. 1. Alkyl Ring-Substituted Butyl 2-Cyano-3-Phenyl-2-Propenoates

Novel trisubstituted ethylenes, alkyl ring-substituted butyl 2-cyano-3-phenyl-2-propenoates, RPhCH=C(CN)CO₂C₄H₉ (where R is 2-methyl, 3-methyl, 4-methyl, 2-ethyl, 4-ethyl, 4-butyl, 4-t-butyl, 4-i-butyl) were prepared and copolymerized with styrene. The monomers were synthesized by the piperidine catalyzed Knoevenagel condensation of ring-substituted benzaldehydes and butyl cyanoacetate, and characterized by CHN analysis, IR, ¹H and ¹³C-NMR. All the ethylenes were copolymerized with styrene (M₁) in solution with radical initiation (ABCN) at 70°C. The compositions of the copolymers were calculated from nitrogen analysis and the structures were analyzed by IR, ¹H and ¹³C-NMR. The order of relative reactivity (1/r₁) for the monomers is 4-ethyl (4.69) > 3-methyl (4.18) > 4-t-butyl (2.98) > 2-ethyl (2.52) > 4-butyl (2.47) > 4-methyl (1.86) > 4-i-butyl (0.94) > 2-methyl (0.87). Decomposition of the copolymers in nitrogen occurred in two steps, first in the 200–500°C range with residue (3–8% wt), which then decomposed in the 500–800°C range.     

Weighted external difference families and R-optimal AMD codes

In this paper, we provide a mathematical framework for characterizing AMD codes that are R-optimal. We introduce a new combinatorial object, the reciprocally-weighted external difference family (RWEDF), which corresponds precisely to an R-optimal weak AMD code. This definition subsumes known examples of existing optimal codes, and also encompasses combinatorial objects not covered by previous definitions in the literature. By developing structural group-theoretic characterizations, we exhibit infinite families of new RWEDFs, and new construction methods for known objects such as near-complete EDFs. Examples of RWEDFs in non-abelian groups are also discussed.     

Aerobic Fujiwara-Moritani alkenylation and dienylation of ferrocene

Pd(OAc)₂-catalyzed aerobic alkenylation and dienylation of ferrocene with electron-poor olefins and dienes (ethyl acrylate, methyl (E)-cinnamate, diethyl fumarate and maleate, ethyl sorbate and sorbic acid) is reported. In the case of acrylate and cinnamate the reaction leads to mixtures of mono- and dialkenylated ferrocenes. In other cases the products of mono alkenylation and dienylation are formed in low yield. The relatively slow (∼1 week) reaction takes place in AcOH solutions at room temperature.     

A new approach to the decontamination of asbestos-polluted waters by treatment with oxalic acid under power ultrasound

A suspension of chrysotile asbestos fibres in aqueous 0.5M oxalic acid was subjected to power ultrasound with the aim to disrupt and detoxify the mineral by the leaching action of oxalic acid on its structural cations acting simultaneously with a vigorous acoustic cavitation. Sonication was performed in a "cavitating tube", a vertical hollow vibrating cylinder made of titanium, operating at 19.2 kHz and 150 W. Treatment lasted from 2.5 to 21 h. Scanning electron microscopy (SEM) showed that the joint action of the chelating agent and ultrasound (though not of either when applied independently) mostly converted asbestos fibres into micrometric aggregates and nano-sized debris, whose morphology totally differed from asbestos fibres. When treated suspensions were filtered through CA membranes (pore size 0.20 microm), more than half of the asbestos went through the filter because it had either been brought in solution or dispersed in the form of extremely small particles. Most of the structural metal ions were brought into solution (ICP-AES). After the treatment the BET surface area of the recovered solid was tenfold greater than the original. The crystalline fraction of residual solids, though resembling the original sample in XRD, was shown by micro-Raman spectra to be made of antigorite, a polymorph form of serpentine. Furthermore, as the length of these antigorite fibrils lay outside the fibre range rated as a health hazard under worldwide regulations, our procedure can be employed for the decontamination of chrysotile-polluted waters and sediments.     

Hong-Ou-Mandel characterization of multiply charged Levitons

We review and develop recent results regarding Leviton excitations generated in topological states of matter – such as integer and fractional quantum Hall edge channels – and carrying a charge multiple of the electronic one. The peculiar features associated with these clean and robust emerging excitations can be detected through current correlation measurements. In particular, relevant information can be extracted from the noise signal in generalized Hong-Ou-Mandel experiments, where Levitons with different charges collide against each other at a quantum point contact. We describe this quantity both in the framework of the photo-assisted noise formalism and in terms of a very interesting and transparent picture based on wave-packet overlap.