Search

We study multi-frequency quasiperiodic Schrodinger operators on $${\mathbb {Z}}$$ . We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non-perturbatively in the regime of positive Lyapunov exponent.

We study the quasi-periodic Schr\"odinger operator $$ -\psi"(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \mathbb{R} $$ in the regime of "small" $V(x) = \sum_{m\in\mathbb{Z}^\nu}c(m)\exp (2\pi i m\omega x)$, $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb{R}^\nu$, $|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$. We show that the set of reflectionless potentials isospectral with $V$ is homeomorphic to a torus. Moreover, we prove that any reflectionless potential $Q$ isospectral with $V$ has the form $Q (x) = \sum_{m \in \mathbb{Z}^\nu} d(m) \exp (2\pi i m\omega x)$, with the same $\omega$ and with $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{\kappa_0}{2} |m|)$. Our derivation relies on the study of the approximation via Hill operators with potentials $\tilde V (x) = \sum_{m \in \mathbb{Z}^\nu} c(m) \exp (2 \pi i m \tilde \omega x)$, where $\tilde \omega$ is a rational approximation of $\omega$. It turns out that the multi-scale analysis method of \cite{DG} applies to these Hill operators. Namely, in \cite{DGL} we developed the multi-scale analysis for the operators dual to the Hill operators in question. The main estimates obtained in \cite{DGL} allow us here to establish the estimates for the gap lengths and the Fourier coefficients in a form which is considerably stronger than the estimates known in the theory of Hill operators with analytic potentials in the general setting. Due to these estimates, the approximation procedure for the quasi-periodic potentials is effective, despite the fact that the rate of approximation $|\omega - \tilde \omega| \thicksim \tilde T^{-\delta}$, $0 < \delta < 1/2$ is slow, on the scale of the period $\tilde T$ of the Hill operator., Comment: 53 pages, to appear in Invent. Math

We study the quasi-periodic Schrodinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \quad x \in{ \mathbf {R}} $$ in the regime of “small” V. Let $(E_{m}',E''_{m})$ , m∈Z ν , be the standard labeled gaps in the spectrum. Our main result says that if $E''_{m} - E'_{m} \le\varepsilon\exp(-\kappa_{0} |m|)$ for all m∈Z ν , with e being small enough, depending on κ 0>0 and the frequency vector involved, then the Fourier coefficients of V obey $|c(m)| \le \varepsilon^{1/2} \exp(-\frac{\kappa_{0}}{2} |m|)$ for all m∈Z ν . On the other hand we prove that if |c(m)|≤eexp(−κ 0|m|) with e being small enough, depending on κ 0>0 and the frequency vector involved, then $E''_{m} - E'_{m} \le2 \varepsilon\exp(-\frac {\kappa_{0}}{2} |m|)$ .

We apply an established statistical methodology called history matching to constrain the parameter space of a coupled non-flux-adjusted climate model (the third Hadley Centre Climate Model; HadCM3) by using a 10,000-member perturbed physics ensemble and observational metrics. History matching uses emulators (fast statistical representations of climate models that include a measure of uncertainty in the prediction of climate model output) to rule out regions of the parameter space of the climate model that are inconsistent with physical observations given the relevant uncertainties. Our methods rule out about half of the parameter space of the climate model even though we only use a small number of historical observations. We explore 2 dimensional projections of the remaining space and observe a region whose shape mainly depends on parameters controlling cloud processes and one ocean mixing parameter. We find that global mean surface air temperature (SAT) is the dominant constraint of those used, and that the others provide little further constraint after matching to SAT. The Atlantic meridional overturning circulation (AMOC) has a non linear relationship with SAT and is not a good proxy for the meridional heat transport in the unconstrained parameter space, but these relationships are linear in our reduced space. We find that the transient response of the AMOC to idealised CO2 forcing at 1 and 2 % per year shows a greater average reduction in strength in the constrained parameter space than in the unconstrained space. We test extended ranges of a number of parameters of HadCM3 and discover that no part of the extended ranges can by ruled out using any of our constraints. Constraining parameter space using easy to emulate observational metrics prior to analysis of more complex processes is an important and powerful tool. It can remove complex and irrelevant behaviour in unrealistic parts of parameter space, allowing the processes in question to be more easily studied or emulated, perhaps as a precursor to the application of further relevant constraints.

We develop methods for analysing decision problems based on multi-attribute utility hierarchies, structured by mutual utility independence, which are not precisely specified due to unwillingness or inability of an individual or group to agree on precise values for the trade-offs between the various attributes. Instead, our analysis is based on whatever limited collection of preferences we may assert between attribute collections. These preferences identify a class of Pareto optimal decisions. We show how to reduce the class further by combining rules which are almost equivalent and introduce general principles appropriate to selecting decisions in an imprecise hierarchy. The approach is illustrated by the design of a university course module.

To fully exploit the potential of kinases as drug targets, novel strategies for the efficient discovery of inhibitors are required. In contrast to the traditional, linear process of inhibitor discovery, high-throughput kinase profiling enables a parallel approach by interrogating compounds against hundreds of targets in a single screen. Compound potency and selectivity are determined simultaneously, providing a choice of targets to pursue that is guided by the quality of lead compounds available, rather than by target biology alone.

In this paper we study one-dimensional Schrodinger operators on the lattice with a potential given by the skew shift. We show that Anderson localization takes place for most phases and frequencies and sufficiently large disorders.

The problem of representing and analysing partial aspects of uncertainty is examined using a geometric approach. A Hilbert space of random objects is constructed, where the inner product captures aspects of beliefs about the relationship between the objects. Orthogonal direct sums of the Hilbert space are used to restrict the amount of detail that is required for the prior specification. Using minimal assumptions of temporal consistency, this geometric space is adapted to derive the stochastic relationships between the formal restricted partial belief analysis and the corresponding posterior uncertainty judgements. To illustrate the methodology, a simple multivariate time series dynamic linear model is developed to represent the sales of leading brands of soft-drink from “cash-and-carry” depots. Restricted prior inferences are developed for the pair of variance matrices underlying this model, where uncertainty for a given depot is decomposed into aspects which may be explained with data from that depot, those which may be explained using data from related depots, and those aspects of our uncertainty for our posterior judgements which cannot be explained a priori.

This paper concerns the geometric treatment of graphical models using Bayes linear methods. We introduce Bayes linear separation as a second order generalised conditional independence relation, and Bayes linear graphical models are constructed using this property. A system of interpretive and diagnostic shadings are given, which summarise the analysis over the associated moral graph. Principles of local computation are outlined for the graphical models, and an algorithm for implementing such computation over the junction tree is described. The approach is illustrated with two examples. The first concerns sales forecasting using a multivariate dynamic linear model. The second concerns inference for the error variance matrices of the model for sales, and illustrates the generality of our geometric approach by treating the matrices directly as random objects. The examples are implemented using a freely available set of object-oriented programming tools for Bayes linear local computation and graphical diagnostic display.

We demonstrate how Bayes linear methods, based on partial prior specifications, bring us quickly to the heart of otherwise complex problems, giving us natural and systematic tools for evaluating our analyses which are not readily available in the usual Bayes formalism. We illustrate the approach using an example concerning problems of prediction in a large brewery. We describe the computer language [B/D] (an acronym for beliefs adjusted by data), which implements the approach. [B/D] incorporates a natural graphical representation of the analysis, providing a powerful way of thinking about the process of knowledge formulation and criticism which is also accessible to non-technical users.