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.     

Bimodal Nanoparticle Size Distributions Produced by Laser Ablation of Microparticles in Aerosols

Silver nanoparticles were produced by laser ablation of a continuously flowing aerosol of microparticles in nitrogen at varying laser fluences. Transmission electron micrographs were analyzed to determine the effect of laser fluence on the nanoparticle size distribution. These distributions exhibited bimodality with a large number of particles in a mode at small sizes (3–6-nm) and a second, less populated mode at larger sizes (11–16-nm). Both modes shifted to larger sizes with increasing laser fluence, with the small size mode shifting by 35% and the larger size mode by 25% over a fluence range of 0.3–4.2-J/cm². Size histograms for each mode were found to be well represented by log-normal distributions. The distribution of mass displayed a striking shift from the large to the small size mode with increasing laser fluence. These results are discussed in terms of a model of nanoparticle formation from two distinct laser–solid interactions. Initially, laser vaporization of material from the surface leads to condensation of nanoparticles in the ambient gas. Material evaporation occurs until the plasma breakdown threshold of the microparticles is reached, generating a shock wave that propagates through the remaining material. Rapid condensation of the vapor in the low-pressure region occurs behind the traveling shock wave. Measurement of particle size distributions versus gas pressure in the ablation region, as well as, versus microparticle feedstock size confirmed the assignment of the larger size mode to surface-vaporization and the smaller size mode to shock-formed nanoparticles.