On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators

In this paper we develop and apply methods for the spectral analysis of non-selfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p-norm ε - pseudospectra (ε > 0, p ∈ [1,∞] ) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n×n matrices. We propose similar convergent approximations for the 2-norm ε -pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.

Network optimization for enhanced resilience of urban heat island measurements

The urban heat island is a well-known phenomenon that impacts a wide variety of city operations. With greater availability of cheap meteorological sensors, it is possible to measure the spatial patterns of urban atmospheric characteristics with greater resolution. To develop robust and resilient networks, recognizing sensors may malfunction, it is important to know when measurement points are providing additional information and also the minimum number of sensors needed to provide spatial information for particular applications. Here we consider the example of temperature data, and the urban heat island, through analysis of a network of sensors in the Tokyo metropolitan area (Extended METROS). The effect of reducing observation points from an existing meteorological measurement network is considered, using random sampling and sampling with clustering. The results indicated the sampling with hierarchical clustering can yield similar temperature patterns with up to a 30% reduction in measurement sites in Tokyo. The methods presented have broader utility in evaluating the robustness and resilience of existing urban temperature networks and in how networks can be enhanced by new mobile and open data sources.

A scalable expressive ensemble learning using Random Prism: a MapReduce approach

The induction of classification rules from previously unseen examples is one of the most important data mining tasks in science as well as commercial applications. In order to reduce the influence of noise in the data, ensemble learners are often applied. However, most ensemble learners are based on decision tree classifiers which are affected by noise. The Random Prism classifier has recently been proposed as an alternative to the popular Random Forests classifier, which is based on decision trees. Random Prism is based on the Prism family of algorithms, which is more robust to noise. However, like most ensemble classification approaches, Random Prism also does not scale well on large training data. This paper presents a thorough discussion of Random Prism and a recently proposed parallel version of it called Parallel Random Prism. Parallel Random Prism is based on the MapReduce programming paradigm. The paper provides, for the first time, novel theoretical analysis of the proposed technique and in-depth experimental study that show that Parallel Random Prism scales well on a large number of training examples, a large number of data features and a large number of processors. Expressiveness of decision rules that our technique produces makes it a natural choice for Big Data applications where informed decision making increases the user’s trust in the system.

Effects of water availability on free amino acids, sugars, and acrylamide-forming potential in potato

Irrigation is used frequently in potato cultivation to maximize yield, but water availability may also affect the composition of the crop, with implications for processing properties and food safety. Five varieties of potatoes, including drought-tolerant and -sensitive types, which had been grown with and without irrigation, were analyzed to show the effect of water supply on concentrations of free asparagine, other free amino acids, and sugars and on the acrylamide-forming potential of the tubers. Two varieties were also analyzed under more severe drought stress in a glasshouse. Water availability had profound effects on tuber free amino acid and sugar concentrations, and it was concluded that potato farmers should irrigate only if necessary to maintain the health and yield of the crop, because irrigation may increase the acrylamide-forming potential of potatoes. Even mild drought stress caused significant changes in composition, but these differed from those caused by more extreme drought stress. Free proline concentration, for example, increased in the field-grown potatoes of one variety from 7.02 mmol/kg with irrigation to 104.58 mmol/kg without irrigation, whereas free asparagine concentration was not affected significantly in the field but almost doubled from 132.03 to 242.26 mmol/kg in response to more severe drought stress in the glasshouse. Furthermore, the different genotypes were affected in dissimilar fashion by the same treatment, indicating that there is no single, unifying potato tuber drought stress response.

Endophytic bacterial community composition in wheat (Triticum aestivum) is determined by plant tissue type, developmental stage and soil nutrient availability

Aims: To understand effects of tissue type, growth stage and soil fertilisers on bacterial endophyte communities of winter wheat (Triticum aestivum cv. Hereward). Methods: Endophytes were isolated from wheat grown under six fertiliser conditions in the long term Broadbalk Experiment at Rothamsted Research, UK. Samples were taken in May and July from root and leaf tissues. Results: Root and leaf communities differed in abundance and composition of endophytes. Endophytes were most abundant in roots and the Proteobacteria were most prevalent. In contrast, Firmicutes and Actinobacteria, the Gram positive phyla, were most prevalent in the leaves. Both fertiliser treatment and sample time influenced abundance and relative proportions of each phylum and genus in the endosphere. A higher density of endophytes was found in the Nil input treatment plants. Conclusions: Robust isolation techniques and stringent controls are critical for accurate recovery of endophytes. The plant tissue type, plant growth stage, and soil fertiliser treatment all contribute to the composition of the endophytic bacterial community in wheat. These results should help facilitate targeted development of endophytes for beneficial applications in agriculture.

The equivalent-weights particle filter in a high-dimensional system

In general, particle filters need large numbers of model runs in order to avoid filter degeneracy in high-dimensional systems. The recently proposed, fully nonlinear equivalent-weights particle filter overcomes this requirement by replacing the standard model transition density with two different proposal transition densities. The first proposal density is used to relax all particles towards the high-probability regions of state space as defined by the observations. The crucial second proposal density is then used to ensure that the majority of particles have equivalent weights at observation time. Here, the performance of the scheme in a high, 65 500 dimensional, simplified ocean model is explored. The success of the equivalent-weights particle filter in matching the true model state is shown using the mean of just 32 particles in twin experiments. It is of particular significance that this remains true even as the number and spatial variability of the observations are changed. The results from rank histograms are less easy to interpret and can be influenced considerably by the parameter values used. This article also explores the sensitivity of the performance of the scheme to the chosen parameter values and the effect of using different model error parameters in the truth compared with the ensemble model runs.

The effect of the equivalent-weights particle filter on dynamical balance in a primitive equation model

The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed. The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed. This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η =

Twin experiments with the equivalent weights particle filter and HadCM3

This paper investigates the use of a particle filter for data assimilation with a full scale coupled ocean–atmosphere general circulation model. Synthetic twin experiments are performed to assess the performance of the equivalent weights filter in such a high-dimensional system. Artificial 2-dimensional sea surface temperature fields are used as observational data every day. Results are presented for different values of the free parameters in the method. Measures of the performance of the filter are root mean square errors, trajectories of individual variables in the model and rank histograms. Filter degeneracy is not observed and the performance of the filter is shown to depend on the ability to keep maximum spread in the ensemble.

Heathland restoration on former agricultural land: effects of artificial acidification on the availability and uptake of toxic metal cations

Lowland heath is an internationally important habitat type that has greatly declined in abundance throughout Western Europe. In recent years this has led to a growing interest in the restoration of heathland on agricultural land. This generally requires the use of chemical treatments to return soil chemical conditions to those appropriate for the support of heathland ecosystems. However, the potential for negative impacts on the environment due to the potential of these treatments to increase the availability of trace metals via raised soil acidity requires investigation. A large-scale field study investigated the effect of two chemical treatments used in heathland restoration, elemental sulphur and ferrous sulphate, on soil acidity and whether it is possible to predict the effect of the treatments on availability of two potentially toxic cations (Al and Cd) in the soil along with their subsequent accumulation in the shoots of the grass Agrostis capillaris. Results showed that both treatments decreased soil pH, but that only elemental sulphur produced a pH similar to heathland soil. The availability of Al, measured by extraction with 1 M ammonium nitrate, could not be predicted by soil pH, depth in the soil and total Al concentration in the soil. By contrast, availability of Cd could be predicted from these three variables. Concentrations of both Al and Cd in the shoots of A. capillaris showed no significant relationship with the extractable concentration in the soil. Results are discussed in light of the possible environmental impacts of the chemical restoration techniques.

Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of ℝ d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called quenched random Lorentz tube. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.

Recurrence for quenched random Lorentz tubes

We consider the billiard dynamics in a striplike set that is tessellated by countably many translated copies of the same polygon. A random configuration of semidispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global choice of scatterers, is called quenched random Lorentz tube. We prove that under general conditions, almost every system in the ensemble is recurrent.

A global empirical system for probabilistic seasonal climate prediction

Preparing for episodes with risks of anomalous weather a month to a year ahead is an important challenge for governments, non-governmental organisations, and private companies and is dependent on the availability of reliable forecasts. The majority of operational seasonal forecasts are made using process-based dynamical models, which are complex, computationally challenging and prone to biases. Empirical forecast approaches built on statistical models to represent physical processes offer an alternative to dynamical systems and can provide either a benchmark for comparison or independent supplementary forecasts. Here, we present a simple empirical system based on multiple linear regression for producing probabilistic forecasts of seasonal surface air temperature and precipitation across the globe. The global CO2-equivalent concentration is taken as the primary predictor; subsequent predictors, including large-scale modes of variability in the climate system and local-scale information, are selected on the basis of their physical relationship with the predictand. The focus given to the climate change signal as a source of skill and the probabilistic nature of the forecasts produced constitute a novel approach to global empirical prediction. Hindcasts for the period 1961–2013 are validated against observations using deterministic (correlation of seasonal means) and probabilistic (continuous rank probability skill scores) metrics. Good skill is found in many regions, particularly for surface air temperature and most notably in much of Europe during the spring and summer seasons. For precipitation, skill is generally limited to regions with known El Niño–Southern Oscillation (ENSO) teleconnections. The system is used in a quasi-operational framework to generate empirical seasonal forecasts on a monthly basis.

Systematic and random errors between collocated satellite ice water path observations

There remains large disagreement between ice-water path (IWP) in observational data sets, largely because the sensors observe different parts of the ice particle size distribution. A detailed comparison of retrieved IWP from satellite observations in the Tropics (!30 " latitude) in 2007 was made using collocated measurements. The radio detection and ranging(radar)/light detection and ranging (lidar) (DARDAR) IWP data set, based on combined radar/lidar measurements, is used as a reference because it provides arguably the best estimate of the total column IWP. For each data set, usable IWP dynamic ranges are inferred from this comparison. IWP retrievals based on solar reflectance measurements, in the moderate resolution imaging spectroradiometer (MODIS), advanced very high resolution radiometer–based Climate Monitoring Satellite Applications Facility (CMSAF), and Pathfinder Atmospheres-Extended (PATMOS-x) datasets, were found to be correlated with DARDAR over a large IWP range (~20–7000 g m -2 ). The random errors of the collocated data sets have a close to lognormal distribution, and the combined random error of MODIS and DARDAR is less than a factor of 2, which also sets the upper limit for MODIS alone. In the same way, the upper limit for the random error of all considered data sets is determined. Data sets based on passive microwave measurements, microwave surface and precipitation products system (MSPPS), microwave integrated retrieval system (MiRS), and collocated microwave only (CMO), are largely correlated with DARDAR for IWP values larger than approximately 700 g m -2 . The combined uncertainty between these data sets and DARDAR in this range is slightly less MODIS-DARDAR, but the systematic bias is nearly an order of magnitude.

High performance polyester-based materials

Biaxially oriented films produced from semi-crystalline, semi-aromatic polyesters are utilised extensively as components within various applications, including the specialist packaging, flexible electronic and photovoltaic markets. However, the thermal performance of such polyesters, specifically poly(ethylene terephthalate) (PET) and poly(ethylene-2,6-naphthalate) (PEN), is inadequate for several applications that require greater dimensional stability at higher operating temperatures. The work described in this project is therefore primarily focussed upon the copolymerisation of rigid comonomers with PET and PEN, in order to produce novel polyester-based materials that exhibit superior thermomechanical performance, with retention of crystallinity, to achieve biaxial orientation. Rigid biphenyldiimide comonomers were readily incorporated into PEN and poly(butylene-2,6-naphthalate) (PBN) via a melt-polycondensation route. For each copoly(ester-imide) series, retention of semi-crystalline behaviour is observed throughout entire copolymer composition ratios. This phenomenon may be rationalised by cocrystallisation between isomorphic biphenyldiimide and naphthalenedicarboxylate residues, which enables statistically random copolymers to melt-crystallise despite high proportions of imide sub-units being present. In terms of thermal performance, the glass transition temperature, Tg, linearly increases with imide comonomer content for both series. This facilitated the production of several high performance PEN-based biaxially oriented films, which displayed analogous drawing, barrier and optical properties to PEN. Selected PBN copoly(ester-imide)s also possess the ability to either melt-crystallise, or form a mesophase from the isotropic state depending on the applied cooling rate. An equivalent synthetic approach based upon isomorphic comonomer crystallisation was subsequently applied to PET by copolymerisation with rigid diimide and Kevlar®-type amide comonomers, to afford several novel high performance PET-based copoly(ester-imide)s and copoly(ester-amide)s that all exhibited increased Tgs. Retention of crystallinity was achieved in these copolymers by either melt-crystallisation or thermal annealing. The initial production of a semi-crystalline, PET-based biaxially oriented film with a Tg in excess of 100 °C was successful, and this material has obvious scope for further industrial scale-up and process development.