Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

On instabilities in data assimilation algorithms

Data assimilation algorithms are a crucial part of operational systems in numerical weather prediction, hydrology and climate science, but are also important for dynamical reconstruction in medical applications and quality control for manufacturing processes. Usually, a variety of diverse measurement data are employed to determine the state of the atmosphere or to a wider system including land and oceans. Modern data assimilation systems use more and more remote sensing data, in particular radiances measured by satellites, radar data and integrated water vapor measurements via GPS/GNSS signals. The inversion of some of these measurements are ill-posed in the classical sense, i.e. the inverse of the operator H which maps the state onto the data is unbounded. In this case, the use of such data can lead to significant instabilities of data assimilation algorithms. The goal of this work is to provide a rigorous mathematical analysis of the instability of well-known data assimilation methods. Here, we will restrict our attention to particular linear systems, in which the instability can be explicitly analyzed. We investigate the three-dimensional variational assimilation and four-dimensional variational assimilation. A theory for the instability is developed using the classical theory of ill-posed problems in a Banach space framework. Further, we demonstrate by numerical examples that instabilities can and will occur, including an example from dynamic magnetic tomography.

The signature of a rough path: uniqueness

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly–Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.

Estimating surface CO2 fluxes from space-borne dry air mole fraction observations using an Ensemble Kalman filter

We have developed an ensemble Kalman Filter (EnKF) to estimate 8-day regional surface fluxes of CO2 from space-borne CO2 dry-air mole fraction observations (XCO2) and evaluate the approach using a series of synthetic experiments, in preparation for data from the NASA Orbiting Carbon Observatory (OCO). The 32-day duty cycle of OCO alternates every 16 days between nadir and glint measurements of backscattered solar radiation at short-wave infrared wavelengths. The EnKF uses an ensemble of states to represent the error covariances to estimate 8-day CO2 surface fluxes over 144 geographical regions. We use a 12×8-day lag window, recognising that XCO2 measurements include surface flux information from prior time windows. The observation operator that relates surface CO2 fluxes to atmospheric distributions of XCO2 includes: a) the GEOS-Chem transport model that relates surface fluxes to global 3-D distributions of CO2 concentrations, which are sampled at the time and location of OCO measurements that are cloud-free and have aerosol optical depths <0.3; and b) scene-dependent averaging kernels that relate the CO2 profiles to XCO2, accounting for differences between nadir and glint measurements, and the associated scene-dependent observation errors. We show that OCO XCO2 measurements significantly reduce the uncertainties of surface CO2 flux estimates. Glint measurements are generally better at constraining ocean CO2 flux estimates. Nadir XCO2 measurements over the terrestrial tropics are sparse throughout the year because of either clouds or smoke. Glint measurements provide the most effective constraint for estimating tropical terrestrial CO2 fluxes by accurately sampling fresh continental outflow over neighbouring oceans. We also present results from sensitivity experiments that investigate how flux estimates change with 1) bias and unbiased errors, 2) alternative duty cycles, 3) measurement density and correlations, 4) the spatial resolution of estimated flux estimates, and 5) reducing the length of the lag window and the size of the ensemble. At the revision stage of this manuscript, the OCO instrument failed to reach its orbit after it was launched on 24 February 2009. The EnKF formulation presented here is also applicable to GOSAT measurements of CO2 and CH4.

Diversity and difference in the everyday lives of Ugandan street children: The significance of age and gender for understanding the use of space

Childhood is characterised by diversity and difference across and within societies. Street children have a unique relationship to the urban environment evident through their use of the city. The everyday geographies that street children produce are diversified through the spaces they frequent and the activities they engage in. Drawing on a range of children-centred qualitative methods, this article focuses on street children's use of urban space in Kampala, Uganda. The article demonstrates the importance of considering variables such as gender and age in the analysis of street children's socio-spatial experiences, which, to date, have rarely been considered in other accounts of street children's lives. In addition the article highlights the need for also including street children's individuality and agency into understanding their use of space. The article concludes by arguing for policies to be sensitive to the diversity that characterises street children's lives and calls for a more nuanced approach where policies are designed to accommodate street children's age and gender differences, and their individual needs, interests and abilities.

Remaking place and securitising space: Urban regeneration and the strategies, tactics and practices of policing in the UK

Urban regeneration programmes in the UK over the past 20 years have increasingly focused on attracting investors, middle-class shoppers and visitors by transforming places and creating new consumption spaces. Ensuring that places are safe and are seen to be safe has taken on greater salience as these flows of income are easily disrupted by changing perceptions of fear and the threat of crime. At the same time, new technologies and policing strategies and tactics have been adopted in a number of regeneration areas which seek to establish control over these new urban spaces. Policing space is increasingly about controlling human actions through design, surveillance technologies and codes of conduct and enforcement. Regeneration agencies and the police now work in partnerships to develop their strategies. At its most extreme, this can lead to the creation of zero-tolerance, or what Smith terms 'revanchist', measures aimed at particular social groups in an effort to sanitise space in the interests of capital accumulation. This paper, drawing on an examination of regeneration practices and processes in one of the UK's fastest-growing urban areas, Reading in Berkshire, assesses policing strategies and tactics in the wake of a major regeneration programme. It documents and discusses the discourses of regeneration that have developed in the town and the ways in which new urban spaces have been secured. It argues that, whilst security concerns have become embedded in institutional discourses and practices, the implementation of security measures has been mediated, in part, by the local socio-political relations in and through which they have been developed.

Changing times, changing places: urban development and the politics of space - time

Much of the writing on urban regeneration in the UK has been focused on the types of urban spaces that have been created in city centres. Less has been written about the issue of when the benefits of regeneration could and should be delivered to a range of different interests, and the different time frames that exist in any development area. Different perceptions of time have been reflected in dominant development philosophies in the UK and elsewhere. The trickle-down agendas of the 1980s, for example, were criticised for their focus on the short-term time frames and needs of developers, often at the expense of those of local communities. The recent emergence of sustainability discourses, however, ostensibly changes the time focus of development and promotes a broader concern with new imagined futures. This paper draws on the example of development in Salford Quays, in the North West of England, to argue that more attention needs to be given to the politics of space-time in urban development processes. It begins by discussing the importance and relevance of this approach before turning to the case study and the ways in which the local politics of space-time has influenced development agendas and outcomes. The paper argues that such an approach harbours the potential for more progressive, far-reaching, and sustainable development agendas to be developed and implemented.

Diversity and difference in the everyday lives of Ugandan street children: The significance of age and gender for understanding the use of space

Childhood is characterised by diversity and difference across and within societies. Street children have a unique relationship to the urban environment evident through their use of the city. The everyday geographies that street children produce are diversified through the spaces they frequent and the activities they engage in. Drawing on a range of children-centred qualitative methods, this article focuses on street children's use of urban space in Kampala, Uganda. The article demonstrates the importance of considering variables such as gender and age in the analysis of street children's socio-spatial experiences, which, to date, have rarely been considered in other accounts of street children's lives. In addition the article highlights the need for also including street children's individuality and agency into understanding their use of space. The article concludes by arguing for policies to be sensitive to the diversity that characterises street children's lives and calls for a more nuanced approach where policies are designed to accommodate street children's age and gender differences, and their individual needs, interests and abilities.

Space-efficient Indexing of Chess Endgame Tables

Chess endgame tables should provide efficiently the value and depth of any required position during play. The indexing of an endgame’s positions is crucial to meeting this objective. This paper updates Heinz’ previous review of approaches to indexing and describes the latest approach by the first and third authors. Heinz’ and Nalimov’s endgame tables (EGTs) encompass the en passant rule and have the most compact index schemes to date. Nalimov’s EGTs, to the Distance-to-Mate (DTM) metric, require only 30.6 × 10^9 elements in total for all the 3-to-5-man endgames and are individually more compact than previous tables. His new index scheme has proved itself while generating the tables and in the 1999 World Computer Chess Championship where many of the top programs used the new suite of EGTs.

Space-efficient indexing of endgame tables for chess

Chess endgame tables should provide efficiently the value and depth of any required position during play. The indexing of an endgame’s positions is crucial to meeting this objective. This paper updates Heinz’ previous review of approaches to indexing and describes the latest approach by the first and third authors. Heinz’ and Nalimov’s endgame tables (EGTs) encompass the en passant rule and have the most compact index schemes to date. Nalimov’s EGTs, to the Distance-to-Mate (DTM) metric, require only 30.6 × 109 elements in total for all the 3-to-5-man endgames and are individually more compact than previous tables. His new index scheme has proved itself while generating the tables and in the 1999 World Computer Chess Championship where many of the top programs used the new suite of EGTs.

Influences of space, soil, nematodes and plants on microbial community composition of chalk grassland soils

Microbial communities respond to a variety of environmental factors related to resources (e.g. plant and soil organic matter), habitat (e.g. soil characteristics) and predation (e.g. nematodes, protozoa and viruses). However, the relative contribution of these factors on microbial community composition is poorly understood. Here, we sampled soils from 30 chalk grassland fields located in three different chalk hill ridges of Southern England, using a spatially explicit sampling scheme. We assessed microbial communities via phospholipid fatty acid (PLFA) analyses and PCR-denaturing gradient gel electrophoresis (DGGE) and measured soil characteristics, as well as nematode and plant community composition. The relative influences of space, soil, vegetation and nematodes on soil microorganisms were contrasted using variation partitioning and path analysis. Results indicate that soil characteristics and plant community composition, representing habitat and resources, shape soil microbial community composition, whereas the influence of nematodes, a potential predation factor, appears to be relatively small. Spatial variation in microbial community structure was detected at broad (between fields) and fine (within fields) scales, suggesting that microbial communities exhibit biogeographic patterns at different scales. Although our analysis included several relevant explanatory data sets, a large part of the variation in microbial communities remained unexplained (up to 92% in some analyses). However, in several analyses, significant parts of the variation in microbial community structure could be explained. The results of this study contribute to our understanding of the relative importance of different environmental and spatial factors in driving the composition of soil-borne microbial communities.