Invariant expectations and vanishing of bounded cohomology for exact groups

We study exactness of groups and establish a characterization of exact groups in terms of the existence of a continuous linear operator, called an invariant expectation, whose properties make it a weak counterpart of an invariant mean on a group. We apply this operator to show that exactness of a finitely generated group $G$ implies the vanishing of the bounded cohomology of $G$ with coefficients in a new class of modules, which are defined using the Hopf algebra structure of $\ell_1(G)$.Comment: Final version, to appear in the Journal of Topology and Analysi

Modifying monolayer behaviour by incorporating subphase additives and improving Langmuir–Blodgett thin film deposition on optical fibres

Experiments showing the possibility of modifying the behaviour of calix[4]resorcinarene monolayers at the air–water interface and optimising the deposition of multilayer coatings onto optical fibres are presented. The nature of the subphase is fundamental to the behaviour of monolayers and their utility in coating and sensing applications. Here we show initial studies exploring the modification of the calix[4]resorcinarene monolayer–water interaction through the introduction of dipole altering alcohol additives to the aqueous subphase. We explored the effect of this modification for three small alcohols. The resulting isotherms of the materials showed a reduction in the surface pressure and area per molecule required in order for the monolayer to reach its point of collapse. Incorporation of alcohols shifted the point of collapse, leading to the application of ethanol being successful in improving the transfer of material via Langmuir–Blodgett coating onto optical fibres at lower pressures. This method may prove useful in allowing greater control over future sensor surface coatings

Detection of volatile organic compounds (VOCs) using an optical fibre long period grating with a calixarene anchored mesoporous thin film

A long period grating (LPG) modified with a mesoporous film infused with a functional compound, calix[4]arene, was employed for the detection of volatile organic compounds (VOCs). The mesoporous film consisted of an inorganic part, of SiO2 nanoparticles (NPs) along with an organic moiety of poly(allylamine hydrochloride) polycation PAH, which was finally infused with functional compound, p-sulphanatocalix[4]arene (CA[4]). The LPG sensor was designed to operate at the phase matching turning point to provide the highest sensitivity. The sensing mechanism is based on the measurement of the refractive index (RI) change induced by the complexion of the VOCs with calix[4]arene (CA). The LPG modified with 5 cycles of (SiO2 NPs/PAH)5PAA responded to exposure to chloroform and benzene vapours. The sensitivity to humidity as an interfering parameter was also investigated

Towards a commerical microelectrode array based sensor for improved chlorine detection

The commercial development of a disposable aqueous chlorine sensor based on a novel microelectrode array fabrication process is described. Non-conducting poly(o-phenylenediamine) films are firstly used to passivate conductive surfaces. Ultrasonic ablation of passivated electrode assemblies then results in the formation of a plurality of wells to expose the underlying conductive substrate, thereby forming a microelectrode array. Microelectrode arrays produced in this manner can be exploited within many electrochemical sensing applications; however, portable aqueous chlorine detection has been selected by Microarray Limited (the industrial sponsors of this project) as a primary vehicle for launching its generic production technology. The scale of microelectrode array production has been extended from that of individual gold sputtercoated glass slide electrodes - to the simultaneous production of hundreds of low-cost screen printed carbon-ink based sensors. A focus has been directed at all stages towards permitting the cost-effective large-scale mass production of sensors with a view to challenging existing portable aqueous chlorine measurement technologies both in terms of performance and unit cost. Based on volume batches of 250,000, it has been calculated that Microarray Limited sensors can be manufactured for a unit cost of approximately 2.5 pence, sufficiently low to provide scope for a competitive yet profitable sale price. The Microarray Limited aqueous chlorine detection system has improved the limit of detection from 0.01 ppm to 0.005 ppm total chlorine without sacrificing accuracy. Furthermore, this novel approach to aqueous chlorine detection offers numerous key benefits to the customer including reduced testing time, a more straightforward operation and the elimination of harmful reagents. Product development has been described from an initial concept through to a pre-production phase. The development of an innovative generic sensor packaging technology is also described.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

D-branes, KK-theory and duality on noncommutative spaces

We present a new categorical classification framework for D-brane charges on noncommutative manifolds using methods of bivariant K-theory. We describe several applications including an explicit formula for D-brane charge in cyclic homology, a refinement of open string T-duality, and a general criterion for cancellation of global worldsheet anomalies

Elliptic operators on manifolds with singularities and K-homology

It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula for K-groups of algebras of pseudodifferential operators.Comment: revised version; 25 pages; section with applications expande

KO-Homology and Type I String Theory

We study the classification of D-branes and Ramond-Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the Cl(n)-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond-Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond-Ramond fields in Type II string theory in terms of topological K-chains.Comment: 40 pages; v4: Clarifying comments added, more detailed proof of main isomorphism theorem given; Final version to be published in Reviews in Mathematical Physic

Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles

We apply some methods of homology and K-theory to special classes of branes wrapping homologically nontrivial cycles. We treat the classification of four-geometries in terms of compact stabilizers (by analogy with Thurston's classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. More complicated examples of T-duality and topology change from fluxes are also considered. We analyse D-branes and fluxes in type II string theory on ${\mathbb C}P^3\times \Sigma_g \times {\mathbb T}^2$ with torsion $H-$flux and demonstrate in details the conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In the simple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces to duality between ${\mathbb C}P^3\times {\mathbb T}^2 \times {\mathbb T}^2$ with $H-$flux and ${\mathbb R}P^7\times {\mathbb T}^3$ with no flux.Comment: 27 pages, tex file, no figure

Extensions and degenerations of spectral triples

For a unital C*-algebra A, which is equipped with a spectral triple and an extension T of A by the compacts, we construct a family of spectral triples associated to T and depending on the two positive parameters (s,t). Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact quantum metric spaces it is possible to define a metric on this family of spectral triples, and we show that the distance between a pair of spectral triples varies continuously with respect to the parameters. It turns out that a spectral triple associated to the unitarization of the algebra of compact operators is obtained under the limit - in this metric - for (s,1) -> (0, 1), while the basic spectral triple, associated to A, is obtained from this family under a sort of a dual limiting process for (1, t) -> (1, 0). We show that our constructions will provide families of spectral triples for the unitarized compacts and for the Podles sphere. In the case of the compacts we investigate to which extent our proposed spectral triple satisfies Connes' 7 axioms for noncommutative geometry.Comment: 40 pages. Addedd in ver. 2: Examples for the compacts and the Podle`s sphere plus comments on the relations to matricial quantum metrics. In ver.3 the word "deformations" in the original title has changed to "degenerations" and some illustrative remarks on this aspect are adde

Fluxes, Brane Charges and Chern Morphisms of Hyperbolic Geometry

The purpose of this paper is to provide the reader with a collection of results which can be found in the mathematical literature and to apply them to hyperbolic spaces that may have a role in physical theories. Specifically we apply K-theory methods for the calculation of brane charges and RR-fields on hyperbolic spaces (and orbifolds thereof). It is known that by tensoring K-groups with the rationals, K-theory can be mapped to rational cohomology by means of the Chern character isomorphisms. The Chern character allows one to relate the analytic Dirac index with a topological index, which can be expressed in terms of cohomological characteristic classes. We obtain explicit formulas for Chern character, spectral invariants, and the index of a twisted Dirac operator associated with real hyperbolic spaces. Some notes for a bivariant version of topological K-theory (KK-theory) with its connection to the index of the twisted Dirac operator and twisted cohomology of hyperbolic spaces are given. Finally we concentrate on lower K-groups useful for description of torsion charges.Comment: 26 pages, no figures, LATEX. To appear in the Classical and Quantum Gravit