Toeplitz operators on Dirichlet-Besov spaces

We study Toeplitz operators on the Besov spaces in the case of the open unit disk. We prove that a symbol satisfying a weak Lipschitz type condition induces a bounded Toeplitz operator. Such symbols do not need to be bounded functions or have continuous extensions to the boundary of the open unit disk. We discuss the problem of the existence of nontrivial compact Toeplitz operators, and also consider Fredholm properties and prove an index formula.

Conceptualizing change in the Portuguese footwear industry: regional voices and cognitive spaces of agency

Expanding national services sectors and global competition aggravate current and perceived future market pressures on traditional manufacturing industries. These perceptions of change have provoked a growing intensification of geo-political discourses on technological innovation and ‘learning’, and calls for competency in design among other professional skills. However, these political discourses on innovation and learning have paralleled public concerns with the apparent ‘growth pains’ from factory closures and subsequent increases in unemployment, and its debilitating social and economic implications for local and regional development. In this respect the following investigation sets out to conceptualize change through the complementary and differing perceptions of industry and regional actors’ experiences or narratives, linking these perceptions to their structure-determined spheres of agent-environment interactivity. It aims to determine whether agents’ differing perceptions of industry transformation can have a role in the legitimization of their interests in, and in sustaining their organizational influence over the process of industry-regional transformation. It argues that industry and regional agent perceptions are among the cognitive aspects of agent-environment interactivity that permeate agency. It stresses agents’ ability to reason and manipulate their work environments to preserve their self-regulating interests in, and task representative influence over the multi-jurisdictional space of industry-regional transformation. The contributions of this investigation suggest that agents’ varied perceptions of industry and regional change inform or compete for influence over the redirection of regional, industry and business strategies. This claim offers a greater appreciation for the reflexive and complex institutional dimensions of industry planning and development, and the political responsibility to socially just forms of regional development. It positions the outcomes of this investigation at the nexus of intensifying geo-political discourses on the efficiency and equity of territorial development in Europe.

Universally reversible JC*-triples and operator spaces

Abstract. We prove that the vast majority of JC∗-triples satisfy the condition of universal reversibility. Our characterisation is that a JC∗-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW∗-triples and of TROs (regarded as JC∗-triples). We show that the distinct natural operator space structures on a universally reversible JC∗-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.

Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

Young caregiving and HIV in the UK: caring relationships and mobilities in African migrant families

Recent research in Sub-Saharan Africa has revealed the importance of children’s caring roles in families affected by HIV and AIDS. However, few studies have explored young caregiving in the context of HIV in the UK, where recently arrived African migrant and refugee families are adversely affected by the global epidemic. This paper explores young people’s socio-spatial experiences of caring for a parent with HIV, based on qualitative research with 37 respondents in London and other urban areas in England. In-depth semi-structured interviews were conducted with young people with caring responsibilities and mothers with HIV, who were predominantly African migrants, as well as with service providers. Drawing on their perspectives, the paper discusses the ways that young people and mothers negotiate the boundaries of young people’s care work within and beyond homespace, according to norms of age, gender, generational relations and cultural constructions of childhood. Despite close attachments within the family, the emotional effects of living with a highly stigmatised life-limiting illness, pressures associated with insecure immigration status, transnational migration and low income undermined African mothers’ and young people’s sense of security and belonging to homespace. These factors also restricted their mobility and social participation in school/college and neighbourhood spaces. While young people and mothers valued supportive safe spaces within the community, the stigma surrounding HIV significantly affected their ability to seek support. The article identifies security, privacy, independence and social mobility as key dimensions of African young people’s and mothers’ imagined futures of ‘home’ and ‘family’.

The price of space: the convergence of value in use and value in exchange

This paper discusses concepts of value from the point of view of the user of the space and the counter view of the provider of the same. Land and property are factors of production. The value of the land flows from the use to which it is put, and that in turn, is dependent upon the demand (and supply) for the product or service that is produced/provided from that space. If there is a high demand for the product (at a fixed level of supply), the price will increase and the economic rent for the land/property will increase accordingly. This is the underlying paradigm of Ricardian rent theory where the supply of land is fixed and a single good is produced. In such a case the rent of land is wholly an economic rent. Economic theory generally distinguishes between two kinds of price, price of production or “value in use” (as determined by the labour theory of value), and market price or “value in exchange” (as determined by supply and demand). It is based on a coherent and consistent theory of value and price. Effectively the distinction is between what space is ‘worth’ to an individual and that space’s price of exchange in the market place. In a perfect market where any individual has access to the same information as all others in the market, price and worth should coincide. However in a market where access to information is not uniform, and where different uses compete for the same space, it is more likely that the two figures will diverge. This paper argues that the traditional reliance of valuers to use methods of comparison to determine “price” has led to an artificial divergence of “value in use” and “value in exchange”, but now such comparison are becoming more difficult due to the diversity of lettings in the market place, there will be a requirement to return to fundamentals and pay heed to the thought process of the user in assessing the worth of the space to be let.

The theme that dare not speak its name: Geopolitik, geopolitics and German foreign policy since unification

The purpose of this chapter is to trace the emergence of a new security imaginary in the foreign policy discourse in Germany during the 1990s and to determine whether it constitutes a return of Geopolitik in German foreign policy making. Does the re- appearance of geopolitical terms and expressions in the official and the academic discourses in post-unification Germany indicate such a shift? The essay will argue that the claims about a return of Geopolitik cannot be sustained. To the extent that the rhetoric of German government officials changes during the 1990s, this does not produce a coherent geopolitical security imaginary that stands diametrically opposed to the definition of political and institutional spaces of the Bonner Republik.

Panaméricas Utópicas: entranced and transient nations in 'I Am Cuba' and 'Land in Trance'

This Chapter looks at two political films, Land in Trance (Glauber Rocha, 1967) and I Am Cuba (Mikhail Kalatozov, 1964), which address the subject of the nation through the enactment of trance. Rejecting all forms of naturalistic account, both films adopt a series of anti-realist devices, such as poetic language, synecdoche, personification, parable and allegory, as a means of expanding the concept of the nation beyond territorial borders and conveying the meaning of revolution through the film form rather than its content.

‘Getting known’: Beckett, Ireland and the creative industries

In the context of national and global trends of producing Beckett’s work, this essay will investigate recent productions of Beckett’s drama which originate in Ireland and tour internationally, examining how these relate to the concept of national identity and its marketability, as well the conceptual and material spaces provided by large-scale festival events. In the last few months, Pan Pan has toured its production of All that Fall from Dublin to the Beckett festival in Enniskillen to New York’s BAM. The Gate Theatre, always a powerhouse of Beckett productions, continues its revival of Barry McGovern’s adaptation of Watt; after the Edinburgh festival, the show will play London’s Barbican in March 2013. While originating in Ireland, these productions – those of the Gate in particular – have an international, as well as domestic, appeal. Examining these and forthcoming Gate productions, I query to what extent a theatre company’s cultural origins and international profile may create a perceived sense of authenticity or definitiveness among critical discourses at ‘home’ and abroad, and how such markers of identity are utilized by the marketing strategies which surround these productions. This article will interrogate the potential convergence of the globalized branding of both Beckett’s work and Irish identity, drawing on the writings of Bourdieu to elucidate how identity may be converted into economic and cultural capital, as well as examining the role that the festival event plays in this process.

Applications of micromorphology to understanding activity areas and site formation processes in experimental hut floors

Experimental buildings at Butser Ancient Farm and St. Fagans (UK) and Lejre (Denmark) were sampled to investigate micromorphology of known activity areas, to contribute to our understanding of the internal use of space in excavated buildings and formation processes of house floor deposits. The experimental buildings provided important information relating to activity residues and sediments over the 16 years that the buildings were in use. Specifically, these results contribute to our understanding of the routes and cycles for transportation of materials in occupation contexts, which can be used to inform archaeological studies. It has been possible to identify internal ‘hot spots’ within the buildings for the deposition of activity residues and for the formation of specific deposit types. Analysis also highlighted postdepositional alterations occurring in internal occupation deposits, which has provided a means of identifying roofed and unroofed spaces in the archaeological record.

Tensor LRR and sparse coding-based subspace clustering

Subspace clustering groups a set of samples from a union of several linear subspaces into clusters, so that the samples in the same cluster are drawn from the same linear subspace. In the majority of the existing work on subspace clustering, clusters are built based on feature information, while sample correlations in their original spatial structure are simply ignored. Besides, original high-dimensional feature vector contains noisy/redundant information, and the time complexity grows exponentially with the number of dimensions. To address these issues, we propose a tensor low-rank representation (TLRR) and sparse coding-based (TLRRSC) subspace clustering method by simultaneously considering feature information and spatial structures. TLRR seeks the lowest rank representation over original spatial structures along all spatial directions. Sparse coding learns a dictionary along feature spaces, so that each sample can be represented by a few atoms of the learned dictionary. The affinity matrix used for spectral clustering is built from the joint similarities in both spatial and feature spaces. TLRRSC can well capture the global structure and inherent feature information of data, and provide a robust subspace segmentation from corrupted data. Experimental results on both synthetic and real-world data sets show that TLRRSC outperforms several established state-of-the-art methods.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Hemispace differences in the visual perception of size in left hemi-Parkinson's disease

The visual perception of size in different regions of external space was studied in Parkinson's disease (PD). A group of patients with worse left-sided symptoms (LPD) was compared with a group with worse right-sided symptoms (RPD) and with a group of age-matched controls on judgements of the relative height or width of two rectangles presented in different regions of external space. The relevant dimension of one rectangle (the 'standard') was held constant, while that of the other (the 'variable') was varied in a method of constant stimuli. The point of subjective equality (PSE) of rectangle width or height was obtained by probit analysis as the mean of the resulting psychometric function. When the standard was in left space, the PSE of the LPD group occurred when the variable was smaller, and when the standard was in right space, when the variable was larger. Similarly, when the standard rectangle was presented in upper space, and the variable in lower space, the PSE occurred when the variable was smaller, an effect which was similar in both left and right spaces. In all these experiments, the PSEs for both the controls and the RPD group did not differ significantly, and were close to a physical match, and the slopes of the psychometric functions were steeper in the controls than the patients, though not significantly so. The data suggest that objects appear smaller in the left and upper visual spaces in LPD, probably because of right hemisphere impairment. (C) 2002 Elsevier Science Ltd. All rights reserved.