The Dirichlet ring and unconditional bases in L 2[0, 2π]

It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms.     

Bis(μ‐6‐hydroxy­picolinato)‐μ‐oxo‐bis­[di­pyridine­manganese(III)] monohydrate

The title compound, [Mn₂(μ‐O)(C₆H₃NO₃)₂(C₅H₅N)₄]·H₂O, was isolated from the reaction of 2,6‐pyridine­di­carboxylic acid with [Mn₁₂O₁₂(CH₃COO)₁₆(H₂O)₄] in pyridine. The dimanganese complex has twofold symmetry; the Mn^III atoms are bridged by one oxo and two amidate ligands and show compressed octahedral Jahn–Teller distortion. The molecular packing comprises a three‐dimensional structure constructed by means of extensive intermolecular interactions, including three kinds of hydrogen bonds and π–π interactions.     

A 1:1 cocrystal of baicalein with nicotinamide

Cocrystallization of baicalein with nicotinamide yields a 1:1 cocrystal [systematic name: pyridine‐3‐carboxamide–5,6,7‐trihydroxy‐2‐phenyl‐4H‐chromen‐4‐one (1/1)], C₆H₆N₂O·C₁₅H₁₀O₅. The asymmetric unit contains one baicalein and one nicotinamide molecule, both in neutral forms. Molecules in the cocrystal form column motifs stabilized by an array of intermolecular hydrogen bonds.     

Centerband-Only Detection of Exchange NMR with Natural-Abundance Correction Reveals an Expanded Unit Cell in Phenylalanine Crystals

The NMR pulse sequence CODEX (centerband-only detection of exchange) is a widely used method to report on the number of magnetically inequivalent spins that exchange magnetization via spin diffusion. For crystals, this rules out certain symmetries, and the rate of equilibration is sensitive to distances. Here we show that for ¹³C CODEX, consideration of natural abundance spins is necessary for crystals of high complexity, demonstrated here with the amino acid phenylalanine. The NMR data rule out the C₂ space group that was originally reported for phenylalanine, and are only consistent with a larger unit cell containing eight magnetically inequivalent molecules. Such an expanded cell was recently described based on single crystal data. The large unit cell dictates the use of long spin diffusion times of more than 200 seconds, in order to equilibrate over the entire unit cell volume of 1622 ų.     

Precision of Raman depolarization and optical attenuation measurements of sound tooth enamel

The demineralization of enamel that is associated with early caries formation affects the optical properties of the enamel. Polarized Raman spectroscopy and optical coherence tomography have been used to detect these changes and potentially offer a means to detect and monitor early caries development. The total optical attenuation coefficient as measured by optical coherence tomography and the polarization anisotropy of the Raman peak arising from the symmetric ν₁ vibration of at approximately 959 cm⁻¹ have been demonstrated as being sensitive markers of early caries. This ex vivo study on extracted human teeth demonstrates that these measurements can be made with reasonable precision with concomitantly good repeatability and reproducibility in sound enamel. Such reliability is crucial for these techniques to have a practical clinical value.     

Determination of vanillin and related flavor compounds in cocoa drink by capillary electrophoresis

In the first part of this study, the stability of five terpenes (alpha-pinene, limonene, camphor, citronellol, and carvacrol) under subcritical water conditions was investigated. The stability studies were carried out at four different temperatures (100, 150, 200, and 250 degrees C) with two different heating times (30 and 300 min). When water temperature was increased, the degradation of terpenes became more serious. Prolonged exposure time to each heating temperature also caused decreased terpene stability. The terpene recoveries were determined by conducting subcritical water extraction of sand spiked with terpenes. The recoveries are typically around 70 to 80% for extractions at 100 degrees C. Terpene recoveries were decreased with increasing water temperature due to poorer stability of terpenes. After the degradation and recovery studies, basil and oregano leaves were extracted using water at both 100 and 150 degrees C. The concentrations of each individual terpene in the water extract generally ranged from trace quantity to 65 microg terpene/g herb. However, the concentration of carvacrol in the oregano-water extract at 150 degrees C was found to be as high as 4270 microg carvacrol/g oregano.     

Spectra of nonlocally bound quantum systems

We discuss a class of nonlinear and nonlocal models for the dynamics of a composite quantum system. The models in question depend on the following constituents: on two subsystem Hamiltonians (denoted by H and ?), an analytic function (f), and a real parameter (s). As demonstrated elsewhere before, the stationary states can be described in these models fairly explicitly. In this article, we build upon that result, and discuss the topological as well as statistical characteristics of the spectra. Here, we concentrate on the special case f = log. It turns out that an energy spectrum of the nonlocally bound system substantially differs from that of its components. Indeed, we show rigorously that, if H is the harmonic oscillator and ? is completely degenerate with one energy level, then the energy spectrum of the composite system has the topology of the Cantor set (for s > 2). In addition, we show that, if H is replaced by the logarithm of the harmonic oscillator, then the spectrum consists of finitely many intervals separated by gaps (for s sufficiently large). In the last case, the key analytic object is the series Σn ^?s . In particular, as an interesting offshoot, this structure furnishes a nontautological immersion of fundamental number-theoretic functions into the quantum formalism.     

Single crystal diffraction under high pressure with synchrotron radiation and a diamond anvil cell

Abstract

A single crystal study on AlPO₄ was performed at 2.90 GPa with synchrotron radiation with a wavelength of only 0.54 ?. The diffracted intensity was high enough to measure even weak reflections with sufficient counting statistics. However, the search for the reflections needed to setup the orientation matrix required a lot of beamtime. A feasibility study was carried out using a proportional area counter to reduce this search time. The results demonstrate that such counters can considerably reduce the time needed for the orientation of the crystal and the data collection.

Presented at the IUCr Workshop on ‘Synchrotron Radiation Instrumentation for High Pressure Crystallography’. Daresbury Laboratory 20-21 July 1991