On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

Moisture induced solid phase degradation of l-ascorbic acid Part 1: a kinetic study using tristimulus colorimetry and a quantitative HPLC assay

l-Ascorbic acid was found to degrade in the solid phase with discoloration under the influence of moisture in proportion to the moisture content. This degradation pattern was different to that in solution and followed zero order kinetics. The exclusion of air reduced the rate of reaction suggesting the degradation may proceed via an oxidative route but no evidence was found for the presence of dehydroascorbic acid. A method was developed for the determination of dehydroascorbic acid using an automated precolumn reduction reaction with dl-dithiothreitol. The degradation was found to be zero order and activation energy was been measured at 37.57 kJ mol(-1) by high performance liquid chromotography (HPLC) assay and 33.30 kJ mol(-1) by tristimulus colorimetry, resulting in a 12.8% difference between the two methods. Tristimulus colorimetry was more sensitive to the onset of degradation than HPLC assay, but it is non-specific. The purpose of this study was to obtain kinetic data on the rate of degradation of l-ascorbic acid alone under the influence of moisture and air and to identify whether tristimulus colorimetry could be used as a rapid and non-destructive means of monitoring for the degradation of l-ascorbic acid in the solid phase. Further studies to determine the degradation pathway and to identify the degradation products are to be reported in subsequent papers.     

Evaluation of protocols for reproducible electrospray in-source collisionally induced dissociation on various liquid chromatography/mass spectrometry instruments and the development of spectral libraries

Mass spectral libraries provide a tool for identifying unknown compounds using both molecular weight and fragmentation information. Mass spectrometers with electrospray ionisation (ESI) and atmospheric chemical ionisation (ApCI) sources have the capability to produce data of this type using in-source collisionally induced dissociation (CID), and in-source CID libraries can be created. Due to the variation in electrospray source design from different instrument manufacturers, the production of reproducible in-source CID spectra that can be used in libraries for all instrument types is not a trivial task. To date, the evaluation of the production of in-source CID libraries has tended to focus on similar instruments from one manufacturer. The studies have also tended to focus on specific compound classes, with a limited molecular weight range.This report describes the findings from the investigation of protocols for the creation of mass spectral libraries using ESI in-source CID on six instruments from four different manufacturers. The overall goal was to create a spectral library for the identification of unknowns. The library could then be applied across all manufacturers' electrospray instruments.Two different experimental protocols were attempted. The first used a tuning compound to establish standard ESI source conditions, with fixed fragmentation potentials. The second involved the attenuation of the [M + H](+) ion to a known degree. A diverse range of compounds (pharmaceutical, photographic, pesticides) was tested to establish the reproducibility of the spectra on the six instruments. Both protocols produced spectra on the various instruments that in many cases were very similar. In other examples, the spectra differed not only in their relative ion abundances, but also in terms of the spectral content. Important observations regarding the effect of ion source design are also reported.The degree of spectral reproducibility was calculated off-line by comparing the five most abundant ions (20% for each ion that matches) from each spectrum on each instrument. This approach was adopted, as we do not possess a software package that met our requirements for spectral comparison. Match factors (% fit) were calculated by comparing each spectrum against the spectra recorded for the same compound and then for all other compounds, on each instrument. The % fit values derived by the off-line approach gave a clear view of the spectral reproducibility from instrument to instrument and also discriminated the spectra of the various compounds from each other. The applicability of this approach was tested using a blind trial in which several compounds were presented as unknowns, their in-source CID spectra recorded and the five-ion approach used for identification.     

Effect of hydrostatic pressure on the elastic moduli of As2S3 and As2Se3 glasses at 195 and 296 K

Velocities of 30 MHz longitudinal and shear ultrasonic waves have been measured in As₂S₃ and As₂Se₃ glasses as a function of hydrostatic pressure up to 1.5 kbar at 195 K and 3 kbar at 296 K. The elastic stiffness moduli are found to have relatively large, positive, pressure dependences which are about the same at both temperatures for both glasses. This behavior is attributed to the weakness of bonding between layers comprised of AsS₃ and AsS₃ pyramids.Inspection of data for a variety of glasses reveals a correlation between the value of C_L/3C_T and whether the elastic moduli are increased or decreased by pressure. (C_L is the longitudinal modulus and C_T the shear modulus.)Using the pressure dependences of the elastic moduli obtained in the present work, it is found that volume change is responsible for most of the temperature dependences of the moduli. In addition elastic gammas are obtained which are consistent with thermal Grüneisen gammas at 12 K. The pressure dependence of the volume of As₂S₃ glass at 296 K is calculated using the present results in the Murnagham equation. Agreement with volumetric data of Weir is obtained.     

Pressure dependence of the electrical resistivity and the ionization energy of Cr In n-type InP

The electrical resistance of chromium-doped, n-type InP has been found to increase exponentially with hydrostatic pressure (up to 4.7 kbar) at 273 K, 301 K, and 344 K. The resistivity activation energy increases by 6.5 × 10^-6 eV/bar. This rate equals the difference between the pressure dependences of the lowest conduction band minimum at $\text{k}$ = (000) and the <111> subsidiary minima determined by others. Our results indicate that pressure increases the ionization energy of a Cr donor level whose wave function has a large contribution from the <111> minima. It is suggested the Cr donor level is due to Cr⁺³ occupying an indium site.     

Self-injection length in La0.7Ca0.3MnO3-YBa2Cu3O7-δ ferromagnet-superconductor multilayer thin films

We have carried out extensive studies on the self-injection problem in barrierless heterojunctions between La_0.7Ca_0.3MnO₃ (LCMO) and YBa₂Cu₃O_7-δ (YBCO) thin films. The heterojunctions were formed in situ by sequentially growing LCMO and YBCO films on 〈100〉 LaAlO₃ (LAO) substrate using a pulsed laser deposition (PLD) system. YBCO micro-bridges with 64 μm width were patterned both on the LAO (control) and LCMO side of the substrate. Critical current, I _c, was measured at 77 K on both the control side as well as the LCMO side for different YBCO film thickness. It was observed that while the control side showed a J _c of ∼ 2 × 10⁶ A/cm², the LCMO side showed about half the value for the same thickness (1800 ?). The difference in J _c indicates that a certain thickness of YBCO has become ‘effectively’ normal due to self-injection. From the measurement of J _c at two different thicknesses (1800 ? and 1500 ?) of YBCO films both on the LAO as well as the LCMO side, the value of self-injection length (at 77 K) was estimated to be ∼ 900 ?. To the authors’ best knowledge, this is the first time that self-injection length has been quantified. A control experiment carried out with LaNiO₃ deposited by PLD on YBCO did not show any evidence of self-injection.     

Mueller matrices in fluorescence scattering

A reappraisal of the 1929 analysis of luminescence by Soleillet reveals the form of the Mueller matrix for fluorescence scattering whose parameters are directly defined in terms of the now-familiar fluorescence anisotropy parameter. If the scattering analyte is optically active, it is further shown how fluorescence detected circular dichroism and circularly polarized luminescence can be recovered, simultaneously and free of artifacts.     

L1-regularisation for ill-posed problems in variational data assimilation

We consider four-dimensional variational data assimilation (4DVar) and show that it can be interpreted as Tikhonov or L₂-regularisation, a widely used method for solving ill-posed inverse problems. It is known from image restoration and geophysical problems that an alternative regularisation, namely L₁-norm regularisation, recovers sharp edges better than L₂-norm regularisation. We apply this idea to 4DVar for problems where shocks and model error are present and give two examples which show that L₁-norm regularisation performs much better than the standard L₂-norm regularisation in 4DVar. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)