Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Hankel operators on Fock spaces

We study Hankel operators on the weighted Fock spaces Fp. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu's characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

Acetone enhances the direct analysis of procyanidin- and prodelphinidin-based condensed tannins in lotus species by the butanol-HCl-iron assay

The butanol-HCl spectrophotometric assay is widely used for quantifying extractable and insoluble condensed tannins (CT, syn. proanthocyanidins) in foods, feeds, and foliage of herbaceous and woody plants, but the method underestimates total CT content when applied directly to plant material. To improve CT quantitation, we tested various cosolvents with butanol-HCl and found that acetone increased anthocyanidin yields from two forage Lotus species having contrasting procyanidin and prodelphinidin compositions. A butanol-HCl-iron assay run with 50% (v/v) acetone gave linear responses with Lotus CT standards and increased estimates of total CT in Lotus herbage and leaves by up to 3.2-fold over the conventional method run without acetone. The use of thiolysis to determine the purity of CT standards further improved quantitation. Gel-state 13C and 1H–13C HSQC NMR spectra of insoluble residues collected after butanol-HCl assays revealed that acetone increased anthocyanidin yields by facilitating complete solubilization of CT from tissue.

Spectral theory of some non-selfadjoint linear differential operators

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 0operators, , which ensure that, if λ≠0 and λφ=Kkφ has only the trivial solution in X, for all k∈W, then, for 0⩽a⩽b, (λ−K)φ=ψ has exactly one solution φ∈Xa for every k∈W and ψ∈Xa. These conditions ensure further that is bounded uniformly in k∈W, for 0⩽a⩽b. As a particular application we consider the case when the kernel takes the form k(s,t)=κ(s−t)z(t), with , , and κ(s)=O(|s|−b) as |s|→∞, for some b>1. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

Improving EU biofuels policy? Greenhouse gas emissions, policy efficiency, and WTO compatibility

Both the EU’s Renewable Energy Directive (RED) and Article 7a of its Fuel Quality Directive (FQD) seek to reduce greenhouse gas (GHG) emissions from transport fuels. The RED mandates a 10% share of renewable energy in transport fuels by 2020, whilst the FQD requires a 6% reduction in GHG emissions (from a 2010 base) by the same date. In practice, it will mainly be biofuels that economic operators will use to meet these requirements, but the different approaches can lead to either the RED, or the FQD, acting as the binding constraint. A common set of environmental sustainability criteria apply to biofuels under both the RED and the FQD. In particular, biofuels have to demonstrate a 35% (later increasing to 50/60%) saving in life-cycle GHG emissions. This could be problematic in the World Trade Organization (WTO), as a non-compliant biofuel with a 34% emissions saving would probably be judged to be ‘like’ a compliant biofuel. A more economically rational way to reduce GHG emissions, and one that might attract greater public support, would be for the RED to reward emission reductions along the lines of the FQD. Moreover, this modification would probably make the provisions more acceptable in the WTO, as there would be a clearer link between policy measures and the objective of reductions in GHG emissions; and the combination of the revised RED and the FQD would lessen the commercial incentive to import biofuels with modest GHG emission savings, and thus reduce the risk of trade tension.

Error, reproducibility and sensitivity: a pipeline for data processing of Agilent oligonucleotide expression arrays

Background: Expression microarrays are increasingly used to obtain large scale transcriptomic information on a wide range of biological samples. Nevertheless, there is still much debate on the best ways to process data, to design experiments and analyse the output. Furthermore, many of the more sophisticated mathematical approaches to data analysis in the literature remain inaccessible to much of the biological research community. In this study we examine ways of extracting and analysing a large data set obtained using the Agilent long oligonucleotide transcriptomics platform, applied to a set of human macrophage and dendritic cell samples. Results: We describe and validate a series of data extraction, transformation and normalisation steps which are implemented via a new R function. Analysis of replicate normalised reference data demonstrate that intrarray variability is small (only around 2 of the mean log signal), while interarray variability from replicate array measurements has a standard deviation (SD) of around 0.5 log(2) units (6 of mean). The common practise of working with ratios of Cy5/Cy3 signal offers little further improvement in terms of reducing error. Comparison to expression data obtained using Arabidopsis samples demonstrates that the large number of genes in each sample showing a low level of transcription reflect the real complexity of the cellular transcriptome. Multidimensional scaling is used to show that the processed data identifies an underlying structure which reflect some of the key biological variables which define the data set. This structure is robust, allowing reliable comparison of samples collected over a number of years and collected by a variety of operators. Conclusions: This study outlines a robust and easily implemented pipeline for extracting, transforming normalising and visualising transcriptomic array data from Agilent expression platform. The analysis is used to obtain quantitative estimates of the SD arising from experimental (non biological) intra- and interarray variability, and for a lower threshold for determining whether an individual gene is expressed. The study provides a reliable basis for further more extensive studies of the systems biology of eukaryotic cells.

Evolutionary and revolutionary effects of transcomputation

Mathematics in Defence 2011 Abstract. We review transreal arithmetic and present transcomplex arithmetic. These arithmetics have no exceptions. This leads to incremental improvements in computer hardware and software. For example, the range of real numbers, encoded by floating-point bits, is doubled when all of the Not-a-Number(NaN) states, in IEEE 754 arithmetic, are replaced with real numbers. The task of programming such systems is simplified and made safer by discarding the unordered relational operator,leaving only the operators less-than, equal-to, and greater than. The advantages of using a transarithmetic in a computation, or transcomputation as we prefer to call it, may be had by making small changes to compilers and processor designs. However, radical change is possible by exploiting the reliability of transcomputations to make pipelined dataflow machines with a large number of cores. Our initial designs are for a machine with order one million cores. Such a machine can complete the execution of multiple in-line programs each clock tick

Evolution and change in the serviced office sector: a decade later

With particular reference to its role in the corporate real estate supply chain, this paper focuses on how the serviced office sector in the UK has evolved and changed over the last decade. A qualitative research approach involving 21 semi-structured interviews with corporate clients of serviced office operators was used to address a number of issues regarding the perceptions of users of serviced offices. It is concluded that the serviced office sector has become an established sector of the UK’s commercial real estate market providing an essential product for many corporate organisations. The serviced office sector has been relatively nimble and a range of operational models have emerged. It was found that corporate organisations use serviced office space and services in order to align workforce change with portfolio change, to transfer risk, for short-term project space, as temporary overflow space, to pilot a new location, to become familiar with a specific geographical marketplace or simply to gain an initial presence in an area.

Performance analysis and deployment of VoLTE mechanisms over 3GPP LTE-based networks

Long Term Evolution based networks lack native support for Circuit Switched (CS) services. The Evolved Packet System (EPS) which includes the Evolved UMTS Terrestrial Radio Access Network (E-UTRAN) and Evolved Packet Core (EPC) is a purely all-IP packet system. This introduces the problem of how to provide voice call support when a user is within an LTE network and how to ensure voice service continuity when the user moves out of LTE coverage area. Different technologies have been proposed for the purpose of providing a voice to LTE users and to ensure the service continues outside LTE networks. The aim of this paper is to analyze and evaluate the overall performance of these technologies along with Single Radio Voice Call Continuity (SRVCC) Inter-RAT handover to Universal Terrestrial Radio Access Networks/ GSM-EDGE radio access Networks (UTRAN/GERAN). The possible solutions for providing voice call and service continuity over LTE-based networks are Circuit Switched Fall Back (CSFB), Voice over LTE via Generic Access (VoLGA), Voice over LTE (VoLTE) based on IMS/MMTel with SRVCC and Over The Top (OTT) services like Skype. This paper focuses mainly on the 3GPP standard solutions to implement voice over LTE. The paper compares various aspects of these solutions and suggests a possible roadmap that mobile operators can adopt to provide seamless voice over LTE.

LTE and LTE-A interworking and interoperability with 3GPP and non-3GPP wireless networks

This paper provides a high-level overview of E-UTRAN interworking and interoperability with existing Third Generation Partnership Project (3GPP) and non-3GPP wireless networks. E-UTRAN access networks (LTE and LTE-A) are currently the latest technologies for 3GPP evolution specified in Release 8, 9 and beyond. These technologies promise higher throughputs and lower latency while also reducing the cost of delivering the services to fit with subscriber demands. 3GPP offers a direct transition path from the current 3GPP UTRAN/GERAN networks to LTE including seamless handover. E-UTRAN and other wireless networks interworking is an option that allows operators to maximize the life of their existing network components before a complete transition to truly 4G networks. Network convergence, backward compatibility and interpretability are regarded as the next major challenge in the evolution and the integration of mobile wireless communications. In this paper, interworking and interoperability between the E-UTRAN Evolved Packet Core (EPC) architecture and 3GPP, 3GPP2 and IEEE based networks are clearly explained. How the EPC is designed to deliver multimedia and facilitate interworking is also explained. Moreover, the seamless handover needed to perform this interworking efficiently is described briefly. This study showed that interoperability and interworking between existing networks and E-UTRAN are highly recommended as an interim solution before the transition to full 4G. Furthermore, wireless operators have to consider a clear interoperability and interworking plan for their existing networks before making a decision to migrate completely to LTE. Interworking provides not only communication between different wireless networks; in many scenarios it contributes to add technical enhancements to one or both environments.

Protein precipitation behaviour of condensed tannins from Lotus pedunculatus and Trifolium repens with different mean degrees of polymerization

The precipitation of bovine serum albumin (BSA), lysozyme (LYS) and alfalfa leaf protein (ALF) by two large- and two medium-sized condensed tannin (CT) fractions of similar flavan-3-ol subunit composition is described. CT fractions isolated from white clover flowers and big trefoil leaves exhibited high purity profiles by 1D/2D NMR and purities >90% (determined by thiolysis). At pH 6.5, large CTs with a mean degree of polymerization (mDP) of ~18 exhibited similar protein precipitation behaviors and were significantly more effective than medium CTs (mDP ~9). Medium CTs exhibited similar capacities to precipitate ALF or BSA, but showed small but significant differences in their capacity to precipitate LYS. All CTs precipitated ALF more effectively than BSA or LYS. Aggregation of CT-protein complexes likely aided precipitation of ALF and BSA, but not LYS. This study, one of the first to use CTs of confirmed high purity, demonstrates that mDP of CTs influences protein precipitation efficacy.

1H-13C HSQC NMR spectroscopy for estimating procyanidin/prodelphinidin and cis/trans-flavan-3-ol ratios of condensed tannin samples: correlation with thiolysis

Studies with a diverse array of 22 purified condensed tannin (CT) samples from nine plant species demonstrated that procyanidin/prodelphinidin (PC/PD) and cis/trans-flavan-3-ol ratios can be appraised by 1H-13C HSQC NMR spectroscopy. The method was developed from samples containing 44 to ~100% CT, PC/PD ratios ranging from 0/100 to 99/1, and cis/trans ratios from 58/42 to 95/5 as determined by thiolysis with benzyl mercaptan. Integration of cross-peak contours of H/C-6' signals from PC and of H/C-2',6' signals from PD yielded nuclei adjusted estimates that were highly correlated with PC/PD ratios obtained by thiolysis (R2 = 0.99). Cis/trans-flavan-3-ol ratios, obtained by integration of the respective H/C-4 cross-peak contours, were also related to determinations made by thiolysis (R2 = 0.89). Overall, 1H-13C HSQC NMR spectroscopy appears to be a viable alternative to thiolysis for estimating PC/PD and cis/trans ratios of CT, if precautions are taken to avoid integration of cross-peak contours of contaminants.