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 .

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.

Geographical variation in total and inorganic arsenic content of polished (white) rice

An extensive data set of total arsenic analysis for 901 polished (white) grain samples, originating from 10 countries from 4 continents, was compiled. The samples represented the baseline (i.e., not specifically collected from arsenic contaminated areas), and all were for market sale in major conurbations. Median total arsenic contents of rice varied 7-fold, with Egypt (0.04 mg/kg) and India (0.07 mg/kg) having the lowest arsenic content while the U.S. (0.25 mg/kg) and France (0.28 mg/kg) had the highest content. Global distribution of total arsenic in rice was modeled by weighting each country’s arsenic distribution by that country’s contribution to global production. A subset of 63 samples from Bangladesh, China, India, Italy, and the U.S. was analyzed for arsenic species. The relationship between inorganic arsenic content versus total arsenic content significantly differed among countries, with Bangladesh and India having the steepest slope in linear regression, and the U.S. having the shallowest slope. Using country-specific rice consumption data, daily intake of inorganic arsenic was estimated and the associated internal cancer risk was calculated using the U.S. Environmental Protection Agency (EPA) cancer slope. Median excess internal cancer risks posed by inorganic arsenic ranged 30-fold for the 5 countries examined, being 0.7 per 10,000 for Italians to 22 per 10,000 for Bangladeshis, when a 60 kg person was considered.

Gallium-sulphide supertetrahedral clusters as building blocks of covalent organic-inorganic networks

The synthesis and characterisation of novel covalent organic-inorganic architectures containing organically-functionalised supertetrahedra is described. The structures of these unique materials consist of one-dimensional zigzag chains or of honeycomb-type layers, in which gallium-sulfide supertetrahedral clusters and dipyridyl ligands alternate.

Controls on inorganic nitrogen leaching from Finnish catchments assessed using a sensitivity and uncertainty analysis of the INCA-N model

The semi-distributed, dynamic INCA-N model was used to simulate the behaviour of dissolved inorganic nitrogen (DIN) in two Finnish research catchments. Parameter sensitivity and model structural uncertainty were analysed using generalized sensitivity analysis. The Mustajoki catchment is a forested upstream catchment, while the Savijoki catchment represents intensively cultivated lowlands. In general, there were more influential parameters in Savijoki than Mustajoki. Model results were sensitive to N-transformation rates, vegetation dynamics, and soil and river hydrology. Values of the sensitive parameters were based on long-term measurements covering both warm and cold years. The highest measured DIN concentrations fell between minimum and maximum values estimated during the uncertainty analysis. The lowest measured concentrations fell outside these bounds, suggesting that some retention processes may be missing from the current model structure. The lowest concentrations occurred mainly during low flow periods; so effects on total loads were small.

Spectra of a class of non-self-adjoint matrices

We consider a new class of non-self-adjoint matrices that arise from an indefinite self- adjoint linear pencil of matrices, and obtain the spectral asymptotics of the spectra as the size of the matrices diverges to infinity. We prove that the spectrum is qualitatively different when a certain parameter c equals 0, and when it is non-zero, and that certain features of the spectrum depend on Diophantine properties of c.

On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators

In this paper we develop and apply methods for the spectral analysis of non-selfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p-norm ε - pseudospectra (ε > 0, p ∈ [1,∞] ) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n×n matrices. We propose similar convergent approximations for the 2-norm ε -pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.

Synthesis and characterization of inorganic-organic hybrid gallium selenides

Two semiconducting hybrid gallium selenides, [Ga6Se9(C6H14N2)4][H2O] (1) and [C6H14N2][Ga4Se6(C6H14N2)2] (2), were prepared using a solvothermal method in the pres-ence of 1,2-diaminocyclohexane (1,2-DACH). Both materials consist of neutral inorganic layers, in which 1,2-DACH is co-valently bonded to gallium. In (1), the organic amine acts as a monodentate and a bidentate ligand, while in (2) bidentate and uncoordinated 1,2-DACH molecules coexist.

Sample preparation: a crucial factor for the analytical performance of rationally designed MALDI matrices

Evidence is presented that the performance of the rationally designed MALDI matrix 4-chloro-α-cyanocinnamic acid (ClCCA) in comparison to its well-established predecessor α-cyano-4-hydroxycinnamic acid (CHCA) is significantly dependent on the sample preparation, such as the choice of the target plate. In this context, it becomes clear that any rational designs of MALDI matrices and their successful employment have to consider a larger set of physicochemical parameters, including sample crystallization and morphology/topology, in addition to parameters of basic (solution and/or gas-phase) chemistry.

Commercial thickeners used by patients with dysphagia: rheological and structural behaviour in different food matrices

In order to achieve a safe swallowing in patients with dysphagia, liquids must be thickened. In this work, two commercial starch based thickeners dissolved in water, whole milk, apple juice and tomato juice were studied. The thickeners were Resource®, composed of modified maize starch and Nutilis®, composed of modified maize starch and gums. They were formulated at two different concentrations corresponding to nectar- and pudding-like consistencies. Influence of composition, concentration and food matrix on rheological properties and structure of the resulting pastes were analysed. Viscoelastic measurements and microscopic observations of the thickeners dissolved in water revealed structural differences due to the presence of gums. When the thickeners were dissolved in the other food matrices significant statistical interactions were found between the matrix and the thickener-type in both the viscoelastic and flow parameters. The most relevant differences were observed for the nectar-like consistency with Nutilis® thickener in milk and apple juice. These samples had lower zero viscosity values and higher loss tangent values, that corresponded to weaker structured systems. Light microscopy images showed that the matrix formed by swollen starch granules was interrupted by the presence of gums. The structure of the matrices in pudding-like formulations became more continuous irrespectively of the matrix employed, and also differences in viscoelasticity among samples diminished. Although differences were observed in zero shear viscosity values among samples, the viscosity of the beverages at 50 s−1 – commonly used as a reference for swallowing – was similar for all samples regardless of the matrix used.

The effect of temperature and inorganic phosphorus supply on growth and acid phosphatase production in arctic and temperate strains of ectomycorrhizal Hebeloma spp. in axenic culture

Acid phosphatase production by 12 Hebeloma strains was usually derepressed when inorganic phosphorus in the growth medium was limited, but appeared to be constitutive in some strains. At low temperatures (≤ 12°) arctic strains produced more extracellular and wall-bound acid phosphatase, yet grew more slowly than the temperate strains. We suggest that low growth rates in arctic strains may be a physiological response to cold whereby resources are diverted into carbohydrate accumulation for cryoprotection. At near freezing temperatures, increased extracellular phosphatase production may compensate for a loss of enzyme activity at low temperature and serve to hydrolyse organic phosphorus in frozen soil over winter.

Comparative growth of ectomycorrhizal fungi (Hebeloma spp.) on organic and inorganic nitrogen

In the largely organic soils in which ectomycorrhizas are commonly found, a preference for absorbing organic nitrogen over mineral forms is likely to be an advantage, especially where mineralisation rates are low. To determine rates of both independent and preferential growth of ectomycorrhizal basidiomycetes on organic and inorganic nitrogen, strains of Hebeloma were grown on nutrient agar media containing either NH4+ or glutamic acid as the sole source of nitrogen, on both single medium and split plate Petri dishes. Growth rates on the split plate Petri dishes, where the fungi had access to both nitrogen sources, were generally greater than on the single medium dishes. Growth on glutamic acid was at least equal to, and usually greater than, that on NH4+. In some cases growth on NH4+ alone appeared severely inhibited, a condition that was partially alleviated by access to glutamic acid on the split plates Petri dishes. This highlights a potential pitfall of single nitrogen source growth studies. The greater growth of most strains on glutamic acid suggests an adaptation to organic nitrogen utilisation in these strains. If this is so in soils with low mineralisation rates, direct uptake of amino acids by ectomycorrhizal plants could by-pass the bottle neck that requires mineral nitrogen to be made available for plant uptake.

Low rank representation on Riemannian manifold of symmetric positive definite matrices

Sparse coding aims to find a more compact representation based on a set of dictionary atoms. A well-known technique looking at 2D sparsity is the low rank representation (LRR). However, in many computer vision applications, data often originate from a manifold, which is equipped with some Riemannian geometry. In this case, the existing LRR becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to applications. In this paper, we generalize the LRR over the Euclidean space to the LRR model over a specific Rimannian manifold—the manifold of symmetric positive matrices (SPD). Experiments on several computer vision datasets showcase its noise robustness and superior performance on classification and segmentation compared with state-of-the-art approaches.