The Nonstraddling Lemma and a new look at the Heine-Borel Theorem

Any family of intervals in the real line determines a linking equivalence relation on its union. The equivalence classes are order convex so are therefore themselves intervals, partitioning the union of the original family. If one starts from a cover of a bounded closed interval by open intervals one can then apply the Nonstraddling Lemma, a result of the utmost simplicity, to clinch the proof of the Heine-Borel Theorem. The Structure Theorem for Open Sets emerges naturally from this discussion. The natural setting for this scheme of proof is in the wider context of complete linearly ordered spaces where the generalisation of the Heine-Borel Theorem is known as the Haar-König Theorem

Numerical range of a simple compression

The numerical range of a contraction acting on L(C2) is identified, so allowing one to exhibit a hermitian projection that is not ultrahermitian. An explicit formula for the norm of the associated operator κm translates into a family of inequalities in four complex variables

Ultrahermitian Projections on Banach Spaces

I introduce a new class of projections on Banach spaces and analyse some of their prop- erties. I also present improvements on known results for hermitian projections (in Theorem 2.6) and on their order properties (see Theorem 3.3). This leads to results on the compression and patching of hermitian operators: see Theorem 3.1 & Theorem 4.6 and its corollaries

The NC property for Banach algebras, and C^*-equivalence

Consider a unital Banach algebra A having the NC property: that 1 − A + 1 ⊆ A + 1 . Then | h | σ = ||h|| (h ∈ A+). Any unital hermitian Banach ∗-algebra with this property is therefore C ∗ -equivalent

Controlled synthesis and properties of Neoglycoconjugates

Glucosyl and galactosyl bearing methacrylates were synthesised using a combination of well known carbohydrate chemistry and ultrasonic chemistry. Polymerisation of these monomers by aqueous reversible addition-fragmentation chain transfer polymerisation yielded glycosylated polymers in high yield with a high degree of control over molecular weight and narrow polydispersity. Polymers of varying levels of glycosylation could be achieved through addition of a comonomer. Block copolymers were synthesised through sequential monomer addition. Binding studies of these polymers to Ricinus communis agglutinin 120 by both isothermal titration calorimetry and surface plasmon resonance found a high degree of molecular weight dependence on avidity. Studies also suggest that the increase in avidity through multivalency is due to both simultaneous binding to multiple receptors and the increased probability of binding due to the high local concentration of ligands. Polymers synthesised have been used in the synthesis of glycosylated gold nano-particles. Some degree of control over nanoparticle size was achieved through adjustment of the gold:polymer ratio but was found to be limited. Agglutination assays show the particles display specific lectin binding but in vitro cell studies were unable to give conclusive information regarding the uptake of particles into cells

How far is (1 + a/n)^n from e^a ?

We present effective upper and lower bounds for the distance from (1 +a/n)^n to e^a for an element a of a complex unital Banach algebra and positive integer n

Effects of ErbB2 overexpression on the proteome and ErbB ligand-specific phosphosignalling in mammary luminal epithelial cells

Most breast cancers arise from luminal epithelial cells and 25-30% of these tumours overexpress the ErbB2/HER2 receptor which correlates with disease progression and poor prognosis. The mechanisms of ErbB2 signalling and the effects of its overexpression are not fully understood. Herein, SILAC expression profiling and phosphopeptide enrichment of a relevant, non-transformed, immortalized human mammary luminal epithelial cell model were used to profile ErbB2-dependent differences in protein expression and phosphorylation events triggered via EGFR (EGF treatment) and ErbB3 (HRG1β treatment) in the context of ErbB2 overexpression. Bioinformatics analysis was used to infer changes in cellular processes and signalling events. We demonstrate the complexity of the responses to oncogene expression and growth factor signalling and identify protein changes relevant to ErbB2-dependent altered cellular phenotype, in particular cell cycle progression and hyper-proliferation, reduced adhesion and enhanced motility. Moreover, we define a novel mechanism by which ErbB signalling suppresses basal interferon signalling that would promote the survival and proliferation of mammary luminal epithelial cells. Numerous novel sites of growth factor-regulated phosphorylation were identified that were enhanced by ErbB2 overexpression and we putatively link these to altered cell behaviour and also highlight the importance of performing parallel protein expression profiling alongside phosphoproteomic analysis

The application of active controls technology to a generic hypersonic aircraft configuration

Analytical methods are described for the prediction of aerothermoelastic stability of hypersonic aircraft including active control systems. Thermal loads due to aerodynamic heating were applied to the finite element model of the aircraft structure and the thermal effects on flutter were determined. An iterative static aeroelastic trim analysis procedure was developed including thermal effects. And active control technology was assessed for flutter suppression, ride quality improvement, and gust load alleviation to overcome any potential adverse aeroelastic stability or response problems due to aerodynamic heating. A generic hypersonic aircraft configuration was selected which incorporates wing flaps, ailerons, and all moveable fins to be used for active control purposes. The active control system would use onboard sensors in a feedback loop through the aircraft flight control computers to move the surfaces for improved structural dynamic response as the aircraft encounters atmospheric turbulence