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 .

Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

The Fredholm properties of Toeplitz operators on the Bergman space A2 have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on Ap that arise when we no longer have 1of Hankel operators on A1.

A theory of rational spatial agglomerations

We model the behavior of rational forward-looking agents in a spatial economy. The economic geography structure is built on Fujita et al. (1999)'s racetrack economy. Workers choose optimally what to consume at each period, as well as which spatial itinerary to follow in the geographical space. The spatial extent of the resulting agglomerations increases with the taste for variety and the expenditure share on manufactured goods, and decreases with transport costs. Because forward-looking agents anticipate the future formation of agglomerations, they are more responsive to spatial utility differentials than myopic agents. As a consequence, the emerging agglomerations are larger under perfect foresight spatial adjustments than under myopic ones.

The economic theory of international supply chains: a systems view

This paper offers an integrated analysis of out-sourcing, off-shoring and foreign direct investment within a systems view of international business. This view takes the supply chain rather than the firm as the basic unit of analysis. It argues that competition in the global economy selects supply chains that maximise the joint profit of all the firms in the chain. The systems view is compared with the firm-centred view commonly used in strategy literature. The paper shows that a firm’s strategy must be embedded within an efficient supply chain strategy, and that this strategy must be negotiated with, rather than imposed upon, other firms. The paper analyses the conditions under which various supply chain strategies - and by implication various firm-level strategies - are efficient. Only by adopting a systems view of supply chains is it possible to determine which firm-level strategies will succeed in a volatile global economy.

A method for incorporating climate variability in climate change impact assessments: sensitivity of river flows in the Eden catchment to precipitation scenarios

Interest in the impacts of climate change is ever increasing. This is particularly true of the water sector where understanding potential changes in the occurrence of both floods and droughts is important for strategic planning. Climate variability has been shown to have a significant impact on UK climate and accounting for this in future climate cahgne projections is essential to fully anticipate potential future impacts. In this paper a new resampling methodology is developed which includes the variability of both baseline and future precipitation. The resampling methodology is applied to 13 CMIP3 climate models for the 2080s, resulting in an ensemble of monthly precipitation change factors. The change factors are applied to the Eden catchment in eastern Scotland with analysis undertaken for the sensitivity of future river flows to the changes in precipitation. Climate variability is shown to influence the magnitude and direction of change of both precipitation and in turn river flow, which are not apparent without the use of the resampling methodology. The transformation of precipitation changes to river flow changes display a degree of non-linearity due to the catchment's role in buffering the response. The resampling methodology developed in this paper provides a new technique for creating climate change scenarios which incorporate the important issue of climate variability.

Commodity futures prices: more evidence on forecast power, risk premia and the theory of storage

In this paper, we examine the temporal stability of the evidence for two commodity futures pricing theories. We investigate whether the forecast power of commodity futures can be attributed to the extent to which they exhibit seasonality and we also consider whether there are time varying parameters or structural breaks in these pricing relationships. Compared to previous studies, we find stronger evidence of seasonality in the basis, which supports the theory of storage. The power of the basis to forecast subsequent price changes is also strengthened, while results on the presence of a risk premium are inconclusive. In addition, we show that the forecasting power of commodity futures cannot be attributed to the extent to which they exhibit seasonality. We find that in most cases where structural breaks occur, only changes in the intercepts and not the slopes are detected, illustrating that the forecast power of the basis is stable over different economic environments.

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.

A unified theory of balance in the extratropics

Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number R or the Froude number F; either R ≪ 1 or F ≪ 1. We examined the shallow-water equations and Boussinesq equations on an f -plane and determined a dimensionless parameter _, small values of which imply a time-scale separation. In terms of R and F, ∈= RF/√(R^2+R^2 ) We then developed a unified theory of (extratropical) balance based on _ that includes all cases of small R and/or small F. The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a timescale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.

Self-organized 40Hz synchronization in a physiological theory of EEG

We present evidence that large-scale spatial coherence of 40 Hz oscillations can emerge dynamically in a cortical mean field theory. The simulated synchronization time scale is about 150 ms, which compares well with experimental data on large-scale integration during cognitive tasks. The same model has previously provided consistent descriptions of the human EEG at rest, with tranquilizers, under anesthesia, and during anesthetic-induced epileptic seizures. The emergence of coherent gamma band activity is brought about by changing just one physiological parameter until cortex becomes marginally unstable for a small range of wavelengths. This suggests for future study a model of dynamic computation at the edge of cortical stability.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Another Waltz? Methodological rhetoric and practice in theory of international politics

Although Theory of International Politics is a standard-bearer for explanatory theory in international relations (IR), Waltz’s methodology has been subject to numerous quite disparate analyses. One reason why it has proved hard to pin down is that too little attention has been paid to how, in practice, Waltz approaches real-world problems. Despite his neopositivist rhetoric, Waltz applies neorealism in a notably loose, even indeterminate, fashion. There is therefore a disjunction between what he says and what he does. This is partly explained by his unsatisfactory attempt to reconcile his avowed neopositivism with his belief that international politics is characterized by organized complexity. The inconsistencies thus created also help to make sense of why competing interpretations of his methodology have emerged. Some aspects of his work do point beyond these particular methodological travails in ways that will continue to be of interest to IR theorists, but its most enduring methodological lesson may be that rhetoric and practice do not necessarily fit harmoniously together.

Computer simulations of pyroclastic flows from dome collapse

We present a method of simulating both the avalanche and surge components of pyroclastic flows generated by lava collapsing from a growing Pelean dome. This is used to successfully model the pyroclastic flows generated on 12 May 1996 by the Soufriere Hills volcano, Montserrat. In simulating the avalanche component we use a simple 3-fold parameterisation of flow acceleration for which we choose values using an inverse method. The surge component is simulated by a 1D hydraulic balance of sedimentation of clasts and entrainment of air away from the avalanche source. We show how multiple simulations based on uncertainty of the starting conditions and parameters, specifically location and size (mass flux), could be used to map hazard zones.