On the near periodicity of eigenvalues of Toeplitz matrices

Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Recently, Levitin and Shargorodsky, on the basis of some numerical experiments, conjectured, for symbols $f$ with two discontinuities located at rational multiples of $\pi$, that the eigenvalues of $A_N$ located in the gap of $f$ asymptotically exhibit periodicity in $N$, and suggested a formula for the period as a function of the position of discontinuities. In this paper, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$.

Molecular t-matrices for low-energy electron diffraction (TMOL v1.1)*1

We describe a FORTRAN-90 program that computes scattering t-matrices for a molecule. These can be used in a Low-Energy Electron Diffraction program to solve the molecular structural problem very efficiently. The intramolecular multiple scattering is computed within a Dyson-like approach, using free space Green propagators in a basis of spherical waves. The advantage of this approach is related to exploiting the chemical identity of the molecule, and to the simplicity to translate and rotate these t-matrices without performing a new multiple-scattering calculation for each configuration. FORTRAN-90 routines for rotating the resulting t-matrices using Wigner matrices are also provided.

Synthesis of broadband anti-reflection coatings for spaceflight infrared optics

A synthesis method is outlined for the design of broadband anti-reflection coatings for use in spaceborne infrared optics. The Golden Section optimisation routine is used to make a search, using designated non-absorptive dielectric thin film combinations, for the coating design which fulfils the required spectral requirements using the least number of layers and different materials. Three examples are given of coatings designed by this method : (I) 1µm to 12µm anti-reflection coating on Zinc Sulphide using Zinc Sulphide and Yttrium Fluoride thin film materials. (ii) 2µm to 14µm anti-reflection coating on Germanium using Germanium and Ytterbium Fluoride thin film materials. (iii) 6µm to 17µm anti-reflection coating on Germanium using Lead Telluride, Zinc Selenide and Barium Fluoride. The measured spectral performance of the manufactured 6µm to 17µm coating on Germanium is given. This is the anti-reflection coating for the germanium optics in the NASA Cassini Orbiter CIRS instrument.

Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.

Precision of quasi-optical null-balanced bridge techniques for transmission and reflection coefficient measurements

The precision of quasioptical null-balanced bridge instruments for transmission and reflection coefficient measurements at millimeter and submillimeter wavelengths is analyzed. A Jones matrix analysis is used to describe the amount of power reaching the detector as a function of grid angle orientation, sample transmittance/reflectance and phase delay. An analysis is performed of the errors involved in determining the complex transmission and reflection coefficient after taking into account the quantization error in the grid angle and micrometer readings, the transmission or reflection coefficient of the sample, the noise equivalent power of the detector, the source power and the post-detection bandwidth. For a system fitted with a rotating grid with resolution of 0.017 rad and a micrometer quantization error of 1 μm, a 1 mW source, and a detector with a noise equivalent power 5×10−9 W Hz−1/2, the maximum errors at an amplitude transmission or reflection coefficient of 0.5 are below ±0.025.

Recommendations for the viability assessment of probiotics as concentrated cultures and in food matrices

Due to the fact that probiotic cells need to be alive when they are consumed, culture-based analysis (plate count) is critical in ascertaining the quality (numbers of viable cells) of probiotic products. Since probiotic cells are typically stressed, due to various factors related to their production, processing and formulation, the standard methodology for total plate counts tends to underestimate the cell numbers of these products. Furthermore, products such as microencapsulated cultures require modifications in the release and sampling procedure in order to correctly estimate viable counts. This review examines the enumeration of probiotic bacteria in the following commercial products: powders, microencapsulated cultures, frozen concentrates, capsules, foods and beverages. The parameters which are specifically examined include: sample preparation (rehydration, thawing), dilutions (homogenization, media) and plating (media, incubation) procedures. Recommendations are provided for each of these analytical steps to improve the accuracy of the analysis. Although the recommendations specifically target the analysis of probiotics, many will apply to the analysis of commercial lactic starter cultures used in food fermentations as well.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Mountain waves in two-layer sheared flows: Critical-level effects, wave reflection, and drag enhancement

Internal gravity waves generated in two-layer stratified shear flows over mountains are investigated here using linear theory and numerical simulations. The impact on the gravity wave drag of wind profiles with constant unidirectional or directional shear up to a certain height and zero shear above, with and without critical levels, is evaluated. This kind of wind profile, which is more realistic than the constant shear extending indefinitely assumed in many analytical studies, leads to important modifications in the drag behavior due to wave reflection at the shear discontinuity and wave filtering by critical levels. In inviscid, nonrotating, and hydrostatic conditions, linear theory predicts that the drag behaves asymmetrically for backward and forward shear flows. These differences primarily depend on the fraction of wavenumbers that pass through their critical level before they are reflected by the shear discontinuity. If this fraction is large, the drag variation is not too different from that predicted for an unbounded shear layer, while if it is small the differences are marked, with the drag being enhanced by a considerable factor at low Richardson numbers (Ri). The drag may be further enhanced by nonlinear processes, but its qualitative variation for relatively low Ri is essentially unchanged. However, nonlinear processes seem to interact constructively with shear, so that the drag for a noninfinite but relatively high Ri is considerably larger than the drag without any shear at all.

Layer-by-layer coating of alginate matrices with chitosan–alginate for the improved survival and targeted delivery of probiotic bacteria after oral administration

The oral administration of probiotic bacteria has shown potential in clinical trials for the alleviation of specific disorders of the gastrointestinal tract. However, cells must be alive in order to exert these benefits. The low pH of the stomach can greatly reduce the number of viable microorganisms that reach the intestine, thereby reducing the efficacy of the administration. Herein, a model probiotic, Bifidobacterium breve, has been encapsulated into an alginate matrix before coating in multilayers of alternating alginate and chitosan. The intention of this formulation was to improve the survival of B. breve during exposure to low pH and to target the delivery of the cells to the intestine. The material properties were first characterized before in vitro testing. Biacore™ experiments allowed for the polymer interactions to be confirmed; additionally, the stability of these multilayers to buffers simulating the pH of the gastrointestinal tract was demonstrated. Texture analysis was used to monitor changes in the gel strength during preparation, showing a weakening of the matrices during coating as a result of calcium ion sequestration. The build-up of multilayers was confirmed by confocal laser-scanning microscopy, which also showed the increase in the thickness of coat over time. During exposure to in vitro gastric conditions, an increase in viability from <3 log(CFU) per mL, seen in free cells, up to a maximum of 8.84 ± 0.17 log(CFU) per mL was noted in a 3-layer coated matrix. Multilayer-coated alginate matrices also showed a targeting of delivery to the intestine, with a gradual release of their loads over 240 min.

CLSM method for the dynamic observation of pH change within polymer matrices for oral delivery

If acid-sensitive drugs or cells are administered orally, there is often a reduction in efficacy associated with gastric passage. Formulation into a polymer matrix is a potential method to improve their stability. The visualization of pH within these materials may help better understand the action of these polymer systems and allow comparison of different formulations. We herein describe the development of a novel confocal laser-scanning microscopy (CLSM) method for visualizing pH changes within polymer matrices and demonstrate its applicability to an enteric formulation based on chitosan-coated alginate gels. The system in question is first shown to protect an acid-sensitive bacterial strain to low pH, before being studied by our technique. Prior to this study, it has been claimed that protection by these materials is a result of buffering, but this has not been demonstrated. The visualization of pH within these matrices during exposure to a pH 2.0 simulated gastric solution showed an encroachment of acid from the periphery of the capsule, and a persistence of pHs above 2.0 within the matrix. This implies that the protective effect of the alginate-chitosan matrices is most likely due to a combination of buffering of acid as it enters the polymer matrix and the slowing of acid penetration.

Reflection iii: Transnationalisation, Television Formats and the Universal

This reflection argues that, despite various good reasons for approaching the notion of the ‘universal’ with caution, cultural theorists should give up their resistance to the universal. The prominence of formats in today’s television suggests that the time is ripe to do. Intentionally or not, accounts of difference implicitly also often reveal sameness; the more we probe heterogeneity, the more likely we are to encounter something that remains consistent and similar. Thus, it is time to collaborate with scholars from the numerous disciplines for which the universal has long had validity and pertinence.