Rank constrained homotopies of matrices and the Blackadar-Handelman conjectures on C*-algebras

Rank constrained homotopies of matrices: For any n ≥ k ≥ l ∈ N, let S( n,k,l) be the set of all non-negative definite matrices a ∈ Mn(C) with l ≤ rank a ≤ k. We investigate homotopy equivalence of continuous maps from a compact Hausdorff space X into sets of the form S(n,k,l). From [37] it is known that for any n, if 4dim X ≤ k-l where dim X denote the covering dimension of X, then there is exactly one homotopy class of maps from X into S(n,k,l). In Section 3.1 we improve this bound by a factor of 8 by confirming C(X,S( n,k,l)) to have exactly one homotopy class of maps when [floor bracket] (dim X/2[end floor bracket] ≤ k - l.) This in particular means πr(S( n,k,l))=0 Blackadar-Handelman conjectures on C*-algebras: Let DF(A) denote the set of all dimension functions on a C*-algebra A and let LDF(A) be the set of all s ∈ DF(A) which are lower semicontinuous. It is well known that DF(A) is naturally identified with the state space of the Cuntz semigroup W(A). From [6], LDF(A) bijectively corresponds to the space of all normalized quasitraces QT(A) through a continuous affine map. [6] conjectures LDF( A) to be pointwise dense in DF(A) and DF(A) to be a Choquet simplex. In Theorem 5.1.1 we provide an equivalent condition for the first of these conjectures for unital A. Applying this condition we confirm the first conjecture for all unital A for which either the radius of comparison is finite or the semigroup W( A) is almost unperforated (Theorem 5.2.5). for every $r\leq 2(k-l)+1$. Our results are achieved through applications of the techniques developed in [8] and [33]. If LDF(A) is dense in DF(A) for an unital A that has only finitely many extreme points in QT(A), through a simple application of Krein-Milman Theorem we note that DF(A)=LDF(A) and that DF(A) is affinely homeomorphic to QT(A). Together with results on the first conjecture this confirms the second conjecture for a new class of C*-algebras. Possibility of extending these results to inductive limits remain an open question. In general the second conjecture is true for any unital A for which (ordered) Grothendieck group K0( A) of W(A) has Riesz interpolation property [15] and every known confirmation of the second conjecture is achieved by showing Riesz interpolation hold for K0( A) [1,9,29]. We consider a stably approximate version of interpolation that is weaker than the classical Riesz interpolation. In fact it is easily seen that this property is even weaker than the asymptotic interpolation property considered in [28]. In Corollary 6.4.3 we confirm DF(A) to be a Choquet simplex for any unital A for which W(A) satisfies this weaker notion of interpolation. While Corollary 6.4.3 has the scope of confirming the second conjecture for a broader class of C*-algebras, finding a `good\u27 class of C*-algebras in which W(A) exhibits stably approximate interpolation but does not satisfy Riesz interpolation remains open

Evaluate the blend quality of powders using a simplified mixing law

The research presents an easy to follow approach to evaluate blend quality of particle blends which are complex in shape and vary in size. For some materials simulations with discrete modelling approaches are challenging because of size and shape constrains but also of the access to recourses and knowhow of practitioners in the field using blending equipment for evaluating the blend quality of their products. A significant proportion of bulk handling operations happens in agriculture and related industries e.g. producing fertilisers, handling and storing food such as beans, lentils and rice. Different materials such as mung beans, black eye beans, lentils and rice were blended in a variable speed screw blender. The blender could vary the screw speed and then results taken from experiments were analysed to quantify the blend quality. It was found that a screw speed of 120 rpm gave the highest blend “quality” defined as a50 / 50 percent mix for different materials and blending time intervals. It was also found that the sampling position in the variable speed screw blender mattered because of axial and radial segregation occurring in the blender

Predicted and in situ performance of a solar air collector incorporating a translucent granular aerogel cover

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierThere is an opportunity to improve the efficiency of flat plate solar air collectors by replacing their conventional glass covers with lightweight polycarbonate panels filled with high performance aerogel insulation. The in situ performance of a 5.4m2 solar air collector containing granular aerogel is simulated and tested. The collector is incorporated into the external insulation of a mechanically ventilated end terrace house, recently refurbished in London, UK. During the 7 day test period, peak outlet temperatures up to 45 °C are observed. Resultant supply and internal air temperatures peak at 25–30 and 21–22 °C respectively. Peak efficiencies of 22–36% are calculated based on the proposed design across a range of cover types. Measured outlet temperatures are validated to within 5% of their predicted values. Estimated outputs range from 118 to 166 kWh/m2/year for collectors with different thickness granular aerogel covers, compared to 110 kWh/m2/year for a single glazed collector, 140 k h/m2/year for a double glazed collector and 202 kWh/m2/year for a collector incorporating high performance monolithic aerogel. Payback periods of 9–16 years are calculated across all cover types. An efficiency up to 60% and a payback period as low as 4.5 years is possible with an optimised collector incorporating a 10 mm thick granular aerogel cover.This work is supported by the EPSRC, Brunel University, Buro Happold Ltd. and the Technology Strategy Board

Direct patterning of gold nanoparticles using flexographic printing for biosensing applications

In this paper, we have presented the use of flexographic printing techniques in the selective patterning of gold nanoparticles (AuNPs) onto a substrate. Highly uniform coverage of AuNPs was selectively patterned on the substrate surface, which was subsequently used in the development of a glucose sensor. These AuNPs provide a biocompatible site for the attachment of enzymes and offer high sensitivity in the detection of glucose due to their large surface to volume ratio. The average size of the printed AuNPs is less than 60 nm. Glucose sensing tests were performed using printed carbon-AuNP electrodes functionalized with glucose oxidase (GOx). The results showed a high sensitivity of 5.52 μA mM−1 cm−2 with a detection limit of 26 μM. We have demonstrated the fabrication of AuNP-based biosensors using flexographic printing, which is ideal for low-cost, high-volume production of the devices

A comparison between the pressure gradients in vertical and horizontal pneumatic conveying, with an investigation into the effect of pipeline bore in vertical conveying

This study was initiated by the need to improve current techniques used in the design of pneumatic conveying systems. At present, the commonly used method to predict the pressure drop in a vertical pipeline in a pneumatic conveying system is to obtain the pressure gradient in a horizontal for the identical conveying conditions and double the pressure gradient in the horizontal pipeline to give the pressure gradient in the vertical pipeline. In addition, scaling for pipeline bore in vertical pipelines is simply undertaken by considering the change in cross-sectional area. This is another area where sufficient investigation has not been undertaken. Therefore, as part of remit of this research study into improving current design techniques, an investigation into the effect of pipeline bore in vertical conveying would also be undertaken. This thesis documents the systematic approach that was used in order to produce some usable models that may be used in improving the understanding and design of pneumatic conveying systems. The models produced were based on data obtained by testing a range of products in an industrial scale pneumatic conveying test facility. The experimental data that was obtained from the pneumatic conveying tests form the basis of the ensuing analysis. The basic experimental data, which is primarily in the form of pressure gradient data for pneumatic conveying in horizontal and vertical pipelines, is explained in detail, along with the subsequent analysis of the data