Three paths toward the quantum angle operator

We examine mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements. We implement three methods of construction of quantum angle. The first one is based on operator theory and parallels the definition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are integral quantization generalizing in a certain sense the Berezin–Klauder approaches. One method pertains to Weyl–Heisenberg integral quantization of the plane viewed as the phase space of the motion on the line. It depends on a family of “weight” functions on the plane. The third method rests upon coherent state quantization of the cylinder viewed as the phase space of the motion on the circle. The construction of these coherent states depends on a family of probability distributions on the line.     

Solution Layer Deposition: A Technique for the Growth of Ultra‐Pure Manganese Oxides on Silica at Room Temperature

With the ever increasing miniaturization in microelectronic devices, new deposition techniques are required to form high‐purity metal oxide layers. Herein, we report a liquid route to specifically produce thin and conformal amorphous manganese oxide layers on silicon substrate, which can be transformed into a manganese silicate layer. The undesired insertion of carbon into the functional layers is avoided through a solution metal–organic chemistry approach named Solution Layer Deposition (SLD). The growth of a pure manganese oxide film by SLD takes place through the decoordination of ligands from a metal–organic complex in mild conditions, and coordination of the resulting metal atoms on a silica surface. The mechanism of this chemical liquid route has been elucidated by solid‐state ²⁹Si MAS NMR, XPS, SIMS, and HRTEM.     

A Bis(Diphosphanyl N‐Heterocyclic Carbene) Gold Complex: A Synthon for Luminescent Rigid AuAg2 Arrays and Au5 and Cu6 Double Arrays

A mononuclear bis(NHC)/Au^I (NHC=N‐heterocyclic carbene) cationic complex with a rigid bis(phosphane)‐functionalized NHC ligand (PC_NHCP) was used to construct linear Au₃ and Ag₂Au arrays, a Au₅ cluster with two intersecting crosslike Au₃ arrays, and an unprecedented Cu₆ complex with two parallel Cu₃ arrays. The impact of metallophilic interactions on photoluminescence was studied experimentally.     

Recovering Invisible Signals by Two‐Field NMR Spectroscopy

Nuclear magnetic resonance (NMR) studies have benefited tremendously from the steady increase in the strength of magnetic fields. Spectacular improvements in both sensitivity and resolution have enabled the investigation of molecular systems of rising complexity. At very high fields, this progress may be jeopardized by line broadening, which is due to chemical exchange or relaxation by chemical shift anisotropy. In this work, we introduce a two‐field NMR spectrometer designed for both excitation and observation of nuclear spins in two distinct magnetic fields in a single experiment. NMR spectra of several small molecules as well as a protein were obtained, with two dimensions acquired at vastly different magnetic fields. Resonances of exchanging groups that are broadened beyond recognition at high field can be sharpened to narrow peaks in the low‐field dimension. Two‐field NMR spectroscopy enables the measurement of chemical shifts at optimal fields and the study of molecular systems that suffer from internal dynamics, and opens new avenues for NMR spectroscopy at very high magnetic fields.     

The detection of nicotine in e-liquids using ion mobility spectrometry

Ion mobility spectrometry (IMS) is a well known field technique for the detection of various materials such as explosives and narcotics. IMS has been used for the detection and identification of nicotine, and this paper describes a simple preparation method and analysis using an Ionscan 500DT which could be used in a field environment for the detection of nicotine in e-liquids. E-liquids containing nicotine are presently a topic of much debate in many countries and their shipment across the Canadian border is prohibited. The method described here would allow border officers or any other operators of IMS instruments to use this technique to correctly determine the presence or absence of nicotine in the e-liquid; this would allow the timely importation of the e-liquids with no nicotine and restrict the laboratory analysis only to those liquids containing nicotine. The IMS method has been used on a number of samples received from manufacturers of e-liquids as well as samples seized at the border. The results of the IMS analysis correspond well with those obtained using a Gas Chromatograph – Mass Spectrometer (GC-MS) method of analysis for nicotine.     

On a Constant Related to the Prime Counting Function

Let \({\pi(x)}\) be the number of primes not exceeding x. We produce new explicit bounds for \({\pi(x)}\) and we use them to obtain a fine frame for the remainder term in the asymptotic formula of the sum \({\sum_{2\leq n\leq x}1/\pi(n)}\).     

Courant-Sharp Eigenvalues for the Equilateral Torus,and for the Equilateral Triangle

We address the question of determining the eigenvalues \({\lambda_{n}}\) (listed in nondecreasing order, with multiplicities) for which Courant’s nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with \({n}\) nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.     

Pulsed lasers versus continuous light sources in capillary electrophoresis and fluorescence detection studies: Photodegradation pathways and models

Pulsed lasers are widely used in capillary electrophoresis (CE) studies to provide laser induced fluorescence (LIF) detection. Unfortunately pulsed lasers do not give linear calibration curves over a wide range of concentrations. While this does not prevent their use in CE/LIF studies, the non-linear behavior must be understood. Using 7-hydroxycoumarin (7-HC) (10–5000 nM), Tamra (10–5000 nM) and tryptophan (1–200 μM) as dyes, we observe that continuous lasers and LEDs result in linear calibration curves, while pulsed lasers give polynomial ones. The effect is seen with both visible light (530 nm) and with UV light (355 nm, 266 nm). In this work we point out the formation of byproducts induced by pulsed laser upon irradiation of 7-HC. Their separation by CE using two Zeta LIF detectors clearly shows that this process is related to the first laser detection. All of these photodegradation products can be identified by an ESI-/MS investigation and correspond to at least two 7HC dimers. By using the photodegradation model proposed by Heywood and Farnsworth (2010) and by taking into account the 7-HC results and the fact that in our system we do not have a constant concentration of fluorophore, it is possible to propose a new photochemical model of fluorescence in LIF detection. The model, like the experiment, shows that it is difficult to obtain linear quantitation curves with pulsed lasers while UV-LEDs used in continuous mode have this advantage. They are a good alternative to UV pulsed lasers. An application involving the separation and linear quantification of oligosaccharides labeled with 2-aminobezoic acid is presented using HILIC and LED (365 nm) induced fluorescence.