DSpace Community:

Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

Tran, Chuong Van — 2013-06-01T00:00:00Z

Abstract: We study both theoretically and numerically two-dimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number $Pm$ (and the limits thereof), with an emphasis on solution regularity. For $Pm=0$, both $\norm{\omega}^2$ and $\norm{j}^2$, where $\omega$ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $\norm{\nabla j}^2$ is integrable over $[0,\infty)$. The uniform boundedness of $\norm{\omega}^2$ implies that in the presence of vanishingly small viscosity $\nu$ (i.e. in the limit $Pm\to0$), the kinetic energy dissipation rate $\nu\norm{\omega}^2$ vanishes for all times $t$, including $t=\infty$. Furthermore, for sufficiently small $Pm$, this rate decreases linearly with $Pm$. This linear behaviour of $\nu\norm{\omega}^2$ is investigated and confirmed by high-resolution simulations with $Pm$ in the range $[1/64,1]$. Several criteria for solution regularity are established and numerically tested. As $Pm$ is decreased from unity, the ratio $\norm{\omega}_\infty/\norm{\omega}$ is observed to increase relatively slowly. This, together with the integrability of $\norm{\nabla j}^2$, suggests global regularity for $Pm=0$. When $Pm=\infty$, global regularity is secured when either $\norm{\nabla\u}_\infty/\norm{\omega}$, where $\u$ is the fluid velocity, or $\norm{j}_\infty/\norm{j}$ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range $Pm\in[1,64]$ show that $\norm{\nabla\u}_\infty/\norm{\omega}$ varies slightly (with similar behaviour for $\norm{j}_\infty/\norm{j}$), thereby lending strong support for the possibility $\norm{\nabla\u}_\infty/\norm{\omega}<\infty$ in the limit $Pm\to\infty$. The peak of the magnetic energy dissipation rate $\mu\norm{j}^2$ is observed to decrease rapidly as $Pm$ is increased. This result suggests the possibility $\norm{j}^2<\infty$ in the limit $Pm\to\infty$. We discuss further evidence for the boundedness of the ratios $\norm{\omega}_\infty/\norm{\omega}$, $\norm{\nabla\u}_\infty/\norm{\omega}$ and $\norm{j}_\infty/\norm{j}$ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields. Description: LAKB was supported by an EPSRC post-graduate studentship.

Note on solution regularity of the generalized magnetohydrodynamic equations with partial dissipation

Tran, Chuong Van — 2013-07-01T00:00:00Z

Abstract: In this brief note we study the $n$-dimensional magnetohydrodynamic equations with hyper-viscosity and zero resistivity. We prove global regularity of solutions when the hyper-viscosity is sufficiently strong.

Solar magnetic carpet III : coronal modelling of synthetic magnetograms

Meyer, Karen Alison — 2013-09-01T00:00:00Z

Estimating animal population density using passive acoustics

Marques, Tiago A. — 2013-05-01T00:00:00Z

Abstract: Reliable estimation of the size or density of wild animal populations is very important for effective wildlife management, conservation and ecology. Currently, the most widely used methods for obtaining such estimates involve either sighting animals from transect lines or some form of capture-recapture on marked or uniquely identifiable individuals. However, many species are difficult to sight, and cannot be easily marked or recaptured. Some of these species produce readily identifiable sounds, providing an opportunity to use passive acoustic data to estimate animal density. In addition, even for species for which other visually based methods are feasible, passive acoustic methods offer the potential for greater detection ranges in some environments (e.g. underwater or in dense forest), and hence potentially better precision. Automated data collection means that surveys can take place at times and in places where it would be too expensive or dangerous to send human observers. Here, we present an overview of animal density estimation using passive acoustic data, a relatively new and fast-developing field. We review the types of data and methodological approaches currently available to researchers and we provide a framework for acoustics-based density estimation, illustrated with examples from real-world case studies. We mention moving sensor platforms (e.g. towed acoustics), but then focus on methods involving sensors at fixed locations, particularly hydrophones to survey marine mammals, as acoustic-based density estimation research to date has been concentrated in this area. Primary among these are methods based on distance sampling and spatially explicit capture-recapture. The methods are also applicable to other aquatic and terrestrial sound-producing taxa. We conclude that, despite being in its infancy, density estimation based on passive acoustic data likely will become an important method for surveying a number of diverse taxa, such as sea mammals, fish, birds, amphibians, and insects, especially in situations where inferences are required over long periods of time. There is considerable work ahead, with several potentially fruitful research areas, including the development of (i) hardware and software for data acquisition, (ii) efficient, calibrated, automated detection and classification systems, and (iii) statistical approaches optimized for this application. Further, survey design will need to be developed, and research is needed on the acoustic behaviour of target species. Fundamental research on vocalization rates and group sizes, and the relation between these and other factors such as season or behaviour state, is critical. Evaluation of the methods under known density scenarios will be important for empirically validating the approaches presented here

Estimating prevalence of injecting drug users and associated heroin-related death rates in England by using regional data and incorporating prior information

King, Ruth — 2013-01-01T00:00:00Z

Abstract: Injecting drug users (IDUs) have a direct social and economic effect yet can typically be regarded as a hidden population within a community. We estimate the size of the IDU population across the nine different Government Office regions of England in 2005–2006 by using capture–recapture methods with age (ranging from 15 to 64 years) and gender as covariate information. We consider a Bayesian model averaging approach using log-linear models, where we can include explicit prior information within the analysis in relation to the total IDU population (elicited from the number of drug-related deaths and injectors’ drug-related death rates). Estimation at the regional level allows for regional heterogeneity with these regional estimates aggregated to obtain a posterior mean estimate for the number of England's IDUs of 195840 with 95% credible interval (181700, 210480). There is significant variation in the estimated regional prevalence of current IDUs per million of population aged 15–64 years, and in injecting drug-related death rates across the gender × age cross-classifications. The propensity of an IDU to be seen by at least one source also exhibits strong regional variability with London having the lowest propensity of being observed (posterior mean probability 0.21) and the South West the highest propensity (posterior mean 0.46).

Decomposition tables for experiments. II. Two–one randomizations

Brien, C. J. — 2010-10-01T00:00:00Z

Abstract: We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomized-inclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used.

Decomposition tables for experiments I. A chain of randomizations

Brien, C. J. — 2009-12-01T00:00:00Z

Abstract: One aspect of evaluating the design for an experiment is the discovery of the relationships between subspaces of the data space. Initially we establish the notation and methods for evaluating an experiment with a single randomization. Starting with two structures, or orthogonal decompositions of the data space, we describe how to combine them to form the overall decomposition for a single-randomization experiment that is "structure balanced." The relationships between the two structures are characterized using efficiency factors. The decomposition is encapsulated in a decomposition table. Then, for experiments that involve multiple randomizations forming a chain, we take several structures that pairwise are structure balanced and combine them to establish the form of the orthogonal decomposition for the experiment. In particular, it is proven that the properties of the design for Such an experiment are derived in a straightforward manner from those of the individual designs. We show how to formulate an extended decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated.

Estimating seasonal abundance of a central place forager using counts and telemetry data

Sharples, RJ — 2009-01-01T00:00:00Z

Abstract: Obtaining population estimates of species that are not easily observed directly can be problematic. However, central place foragers can often be observed some of the time, e.g. when seals are hauled out. In these instances, population estimates can be derived from counts, combined with information on the proportion of time that animals can be observed. We present a modelling framework to estimate seasonal absolute abundance using counts and information from satellite telemetry data. The method was tested on a harbour seal population in an area of southeast Scotland. Counts were made monthly, between November 2001 and June 2003, when seals were hauled out on land and were corrected for the proportion of time the seals were at sea using satellite telemetry. Harbour seals (n=25) were tagged with satellite relay data loggers between November 2001 and March 2003. To estimate the proportion of time spent hauled out, time at sea on foraging trips was modelled separately from haul-out behaviour close to haul-out sites because of the different factors affecting these processes. A generalised linear mixed model framework was developed to capture the longitudinal nature of the data and the repeated measures across individuals. Despite seasonal variability in the number of seals counted at haul-out sites, the model generated estimates of abundance, with an overall mean of 846 (95% CI: 767 to 979). The methodology shows the value of using count and telemetry data collected concurrently for estimating absolute abundance, information that is essential to assess interactions between predators, fish stocks and fisheries. Description: R.J.S. was supported by a Natural Environment Research Council studentship.

Quantifying biodiversity trends in time and space

Studeny, Angelika C. — 2012-11-30T00:00:00Z

Abstract: The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale.

On global regularity of 2D generalized magnetohydrodynamic equations

Tran, Chuong Van — 2013-05-15T00:00:00Z

Abstract: In this article we study the global regularity of 2D generalized magnetohydrodynamic equations (2D GMHD), in which the dissipation terms are $- \nu \left( - \triangle \right)^{\alpha} u$ and $- \kappa \left( - \triangle \right)^{\beta} b$. We show that smooth solutions are global in the following three cases: $\alpha \geqslant 1 / 2, \beta \geqslant 1$; $0 \leqslant \alpha < 1 / 2, 2 \alpha + \beta > 2$; $\alpha \geqslant 2, \beta = 0$. We also show that in the inviscid case $\nu = 0$, if $\beta > 1$, then smooth solutions are global as long as the direction of the magnetic field remains smooth enough.

Sharp global nonlinear stability for a fluid overlying a highly porous material

Hill, Antony A. — 2010-01-08T00:00:00Z

Abstract: The stability of convection in a two-layer system in which a layer of fluid with a temperature-dependent viscosity overlies and saturates a highly porous material is studied. Owing to the difficulties associated with incorporating the nonlinear advection term in the Navier-Stokes equations into a stability analysis, previous literature on fluid/porous thermal convection has modelled the fluid using the linear Stokes equations. This paper derives global stability for the full nonlinear system, by utilizing a model proposed by Ladyzhenskaya. The nonlinear stability boundaries are shown to be sharp when compared with the linear instability thresholds.

Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers

Hill, A A — 2010-09-01T00:00:00Z

Abstract: Studies of the nonlinear stability of fluid/porous systems have been developed very recently. A two-domain modelling approach has been adopted in previous works, but was restricted to specific configurations. The extension to the more general case of a Navier–Stokes modelled fluid over a porous material was not achieved for the two-domain approach owing to the difficulties associated with handling the interfacial boundary conditions. This paper addresses this issue by adopting a one-domain approach, where the governing equations for both regions are combined into a unique set of equations that are valid for the entire domain. It is shown that the nonlinear stability bound, in the one-domain approach, is very sharp and hence excludes the possibility of subcritical instabilities. Moreover, the one-domain approach is compared with an equivalent two-domain approach, and excellent agreement is found between the two.

Instability in internal solitary waves with trapped cores

Carr, Magda — 2012-01-01T00:00:00Z

Abstract: A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to investigate instability in internal solitary waves with trapped cores. A three-layer configuration for the background stratification in which the top two layers are linearly stratified and the lower layer is homogeneous is considered throughout. The strength of the stratification in the very top layer is chosen to be sufficient so that waves of depression with trapped cores can be generated. The flow is assumed to satisfy the Dubriel-Jacotin-Long equation both inside and outside of the core region. The Brunt-Vaisala frequency is modelled such that it varies from a constant value outside of the core to zero inside the core over a sharp but continuous transition length. This results in a stagnant core in which the vorticity is zero and the density is homogeneous and approximately equal to that at the core boundary. The time dependent simulations show that instability occurs on the boundary of the core. The instability takes the form of Kelvin-Helmholtz billows. If the instability in the vorticity field is energetic enough, disturbance in the buoyancy field is also seen and fluid exchange takes place across the core boundary. Occurrence of the Kelvin-Helmholtz billows is attributed to the sharp change in the vorticity field at the boundary between the core and the pycnocline. The numerical scheme is not limited by small Richardson number unlike the other alternatives currently available in the literature which appear to be.

Shear induced breaking of large amplitude internal solitary waves

Fructus, D — 2009-02-01T00:00:00Z

Convectively induced shear instability in large amplitude internal solitary waves

Carr, Magda — 2008-12-01T00:00:00Z

Abstract: Laboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focused on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification has been varied and it is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave. It is shown that there is an interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation in the upper boundary condition is also found to have an effect on stability. In particular, the implications for convective instability are shown to be profound and a dramatic increase in wave amplitude is seen for a fixed (as opposed to free) upper boundary condition.

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Mitchell, James David — 2010-09-01T00:00:00Z

Abstract: Let ΩΩ be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of ΩΩ, then we write U≈V if there exists a finite subset F of ΩΩ such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in ΩΩ is the least cardinality of a subset A of ΩΩ such that the union of U and A generates ΩΩ. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω. The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in ΩΩ where d is the minimum cardinality of a dominating family for NN. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in ΩΩ are 0, 1, 2, and d. We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2ℵ0.

Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence

Tran, Chuong V. — 2011-10-01T00:00:00Z

Abstract: We study both theoretically and numerically surface quasi-geostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system's number of degrees of freedom N, which is defined in terms of local Lyapunov exponents, scales as Re-3/2, where R e is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system's integral scale. For general power-law energy spectra k(-alpha), a comparison of N with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields alpha = 5/3. This comparison further renders the scaling Re-1/2 for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov's theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations. Description: L.A.K.B. was supported by an EPSRC post-graduate studentship.

Coronal heating by the partial relaxation of twisted loops

Bareford, Michael — 2013-02-01T00:00:00Z

Abstract: Context: Relaxation theory offers a straightforward method for estimating the energy that is released when a magnetic field becomes unstable, as a result of continual convective driving. Aims: We present new results obtained from nonlinear magnetohydrodynamic (MHD) simulations of idealised coronal loops. The purpose of this work is to determine whether or not the simulation results agree with Taylor relaxation, which will require a modified version of relaxation theory applicable to unbounded field configurations. Methods: A three-dimensional (3D) MHD Lagrangian-remap code is used to simulate the evolution of a line-tied cylindrical coronal loop model. This model comprises three concentric layers surrounded by a potential envelope; hence, being twisted locally, each loop configuration is distinguished by a piecewise-constant current profile. Initially, all configurations carry zero-net-current fields and are in ideally unstable equilibrium. The simulation results are compared with the predictions of helicity conserving relaxation theory. Results: For all simulations, the change in helicity is no more than 2% of the initial value; also, the numerical helicities match the analytically-determined values. Magnetic energy dissipation predominantly occurs via shock heating associated with magnetic reconnection in distributed current sheets. The energy release and final field profiles produced by the numerical simulations are in agreement with the predictions given by a new model of partial relaxation theory: the relaxed field is close to a linear force free state; however, the extent of the relaxation region is limited, while the loop undergoes some radial expansion. Conclusions: The results presented here support the use of partial relaxation theory, specifically, when calculating the heating-event distributions produced by ensembles of kink-unstable loops.

Fitting complex ecological point process models with integrated nested Laplace approximation

Illian, Janine Baerbel — 2013-01-01T00:00:00Z

Abstract: Summary 1. We highlight an emerging statistical method, integrated nested Laplace approximation (INLA), which is ideally suited for fitting complex models to many of the rich spatial data sets that ecologists wish to analyse. 2. INLA is an approximation method that nevertheless provides very exact estimates. In this article, we describe the INLA methodology highlighting where it offers opportunities for drawing inference from (spatial) ecological data that would previously have been too complex to make practical model fitting feasible. 3. We use INLA to fit a complex joint model to the spatial pattern formed by a plant species, Thymus carnosus, as well as to the health status of each individual. 4. The key ecological result revealed by our spatial analysis of these data, relates to the distance-to-water covariate. We find that T. carnosus plants are generally healthier when they are further away from the water. 5. We suggest that this may be the result of a combination of (1) plants having alternative rooting strategies depending on how close to water they grow and (2) the rooting strategy determining how well the plants were able to tolerate an unusually dry summer. 6. We anticipate INLA becoming widely used within spatial ecological analysis over the next decade and suggest that both ecologists and statisticians will benefit greatly from working collaboratively to further develop and apply these emerging statistical methods.

Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures

McPhee, Jillian Dawn — 2012-11-30T00:00:00Z

Abstract: Let Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when Ω = B,Q there exist regular D-classes which contain countably many group H-classes.

A family of spatial biodiversity measures based on graphs

Rajala, T — 2012-12-01T00:00:00Z

Abstract: While much research in ecology has focused on spatially explicit modelling as well as on measures of biodiversity, the concept of spatial (or local) biodiversity has been discussed very little. This paper generalises existing measures of spatial biodiversity and introduces a family of spatial biodiversity measures by flexibly defining the notion of the individuals’ neighbourhood within the framework of graphs associated to a spatial point pattern. We consider two non-independent aspects of spatial biodiversity, scattering, i.e. the spatial arrangement of the individuals in the study area and exposure, the local diversity in an individual’s neighbourhood. A simulation study reveals that measures based on the most commonly used neigh-bourhood defined by the geometric graph do not distinguish well between scattering and exposure. This problem is much less pronounced when other graphs are used. In an analysis of the spatial diversity in a rainforest, the results based on the geometric graph have been shown to spuriously indicate a decrease in spatial biodiversity when no such trend was detected by the other types of neighbourhoods. We also show that the choice neighbourhood markedly impacts on the classification of species according to how strongly and in what way different species spatially structure species diversity. Clearly, in an analysis of spatial or local diversity an appropriate choice of local neighbourhood is crucial in particular in terms of the biological interpretation of the results. Due to its general definition, the approach discussed here offers the necessary flexibility that allows suitable and varying neighbourhood structures to be chosen.

Multispecies functional response of the minke whale Balaenoptera acutorostrata based on small-scale foraging studies

Smout, Sophie Caroline — 2007-07-01T00:00:00Z

Abstract: Atlantic minke whales are important predators in the Barents Sea ecosystem; capelin Mallotus villosus, krill Thysanoessa sp. and Meganyctephanes norvegica and herring Clupea harengus are their major prey. Their consumption of commercial species may present an economic problem for the local fishery. In order to estimate this consumption and understand the potential consequences for prey dynamics, it is essential to determine the multispecies functional response of the whales. The parameterisation of a functional response requires measurements of consumption rates and prey availability. In this localised study, undigested stomach contents were used to assess the amount of each prey that had been consumed immediately prior to capture. To determine the availability of prey to the whales, standard acoustic surveys were run in the same area within 2 d of the capture of the whales. The spatial distribution of prey was modelled using generalised additive models (GAMs). In order to generate a measure of prey availability and the uncertainty in this value, a simple model was assumed for whale movement, and prey abundance was sampled over space according to a Gaussian kernel. A multispecies functional response (MSFR) model was then fitted to the consumption and prey availability data using Bayesian methods. Simple simulations, based on the fitted MSFR, indicate that minke whales may deplete local capelin aggregations at small spatial scales. This is the first time that a multispecies functional response has been fitted for a cetacean predator, and the methods outlined here may prove useful for modelling marine mammal-fish interactions in other systems.

Estimating demographic parameters for capture-recapture data in the presence of multiple mark types

Smout, Sophie Caroline — 2011-06-01T00:00:00Z

Abstract: In mark-recapture studies, various techniques can be used to uniquely identify individual animals, such as ringing, tagging or photo-identification using natural markings. In some long-term studies more than one type of marking procedure may be implemented during the study period. In these circumstances, ignoring the different mark types can produce biased survival estimates since the assumption that the different mark types are equally catchable (homogeneous capture probability across mark types) may be incorrect.We implement an integrated approach where we simultaneously analyse data obtained using three different marking techniques, assuming that animals can be cross-classified across the different mark types. We discriminate between competing models using the AIC statistic. This technique also allows us to estimate both relative mark-loss probabilities and relative recapture efficiency rates for the different marking methods.We initially perform a simulation study to explore the different biases that can be introduced if we assume a homogeneous recapture probability over mark type, before applying the method to a real dataset. We make use of data obtained from an intensive long-term observational study of UK female grey seals (Halichoerus grypus) at a single breeding colony, where three different methods are used to identify individuals within a single study: branding, tagging and photo-identification based on seal coat pattern or pelage.

Every group is a maximal subgroup of the free idempotent generated semigroup over a band

Dolinka, I — 2013-01-01T00:00:00Z

Abstract: Given an arbitrary group G we construct a semigroup of idempotents (band) BG with the property that the free idempotent generated semigroup over BG has a maximal subgroup isomorphic to G. If G is finitely presented then BG is finite. This answers several questions from recent papers in the area.

On semigroups which are unions of finitely many copies of the free monogenic semigroup

Abughazalah, Nabilah — 2013-01-01T00:00:00Z

Abstract: Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.

Damping of kink waves by mode coupling : I. Analytical treatment

Hood, Alan William — 2013-01-01T00:00:00Z

Abstract: Aims. To investigate the spatial damping of propagating kink waves in an inhomogeneous plasma. In the limit of a thin tube surrounded by a thin transition layer, an analytical formulation for kink waves driven in from the bottom boundary of the corona is presented. Methods. The spatial form for the damping of the kink mode was investigated using various analytical approximations. When the density ratio between the internal density and the external density is not too large, a simple di.erential-integral equation was used. Approximate analytical solutions to this equation are presented. Results. For the first time, the form of the spatial damping of the kink mode is shown analytically to be Gaussian in nature near the driven boundary. For several wavelengths, the amplitude of the kink mode is proportional to (1 + exp(-z2 /L2 g))/2, where L2g = 16/ǫκ2 k2 . Although the actual value of 16 in Lg depends on the particular form of the driver, this form is very general and its dependence on the other parameters does not change. For large distances, the damping profile appears to be roughly linear exponential decay. This is shown analytically by a series expansion when the inhomogeneous layer width is small enough.

The influence of a fluid-porous interface on solar pond stability

Hill, A. A — 2013-02-01T00:00:00Z

Abstract: The linear instability of the gradient zone of a solar pond containing a fluidporous interface is investigated. It is found that the gradient zone can retain the same stability for lower values of the solute Rayleigh number with the introduction of a porous material compared with a purely fluid layer, whilst maintaining the same lower convective zone temperature. Interestingly, it is also shown that for certain parameter values the penetration of a porous medium into the gradient zone can cause the temperature of the lower convective zone to rise. However, for certain parameter ranges, when the fluid-porous interface is towards the top of the gradient zone, the solar pond can become highly unstable.

Ideals and finiteness conditions for subsemigroups

Gray, Robert Duncan — 2013-01-01T00:00:00Z

Abstract: In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals.

The geometric mean of relative abundance indices : a biodiversity measure with a difference

Buckland, Stephen Terrence — 2011-09-02T00:00:00Z

Abstract: The 2010 Biodiversity Target of the Convention on Biological Diversity (CBD), set in 2002, which stated that there should be ‘a significant reduction of the current rate of biodiversity loss' by 2010, highlighted the need for informative and tractable metrics that can be used to evaluate change in biological diversity. While the subsequent Aichi 2020 targets are more wide-ranging, they also seek to reduce the rate of biodiversity loss. The geometric mean of relative abundance indices, G, is increasingly being used to examine trends in biological diversity and to assess whether biodiversity targets are being met. Here, we explore the mathematical and statistical properties of G that make it useful for judging temporal change in biological diversity, and we discuss its advantages and limitations relative to other measures. We demonstrate that the index reflects trends in both abundance and evenness, and that it is not prone to bias when detectability of individuals varies by species. We note that it allows data from different surveys to be combined to generate a composite index. However, the index exhibits high variance and unstable behaviour when rarely-recorded species are included in the analyses. Read More: http://www.esajournals.org/doi/abs/10.1890/ES11-00186.1

Using INLA to fit a complex point process model with temporally varying effects – a case study

Illian, Janine Baerbel — 2012-07-01T00:00:00Z

Abstract: Integrated nested Laplace approximation (INLA) provides a fast and yet quite exact approach to fitting complex latent Gaussian models which comprise many statistical models in a Bayesian context, including log Gaussian Cox processes. This paper discusses how a joint log Gaussian Cox process model may be fitted to independent replicated point patterns. We illustrate the approach by fitting a model to data on the locations of muskoxen (Ovibos moschatus) herds in Zackenberg valley, Northeast Greenland and by detailing how this model is specified within the R-interface R-INLA. The paper strongly focuses on practical problems involved in the modelling process, including issues of spatial scale, edge effects and prior choices, and finishes with a discussion on models with varying boundary conditions.

A Bayesian approach to fitting Gibbs processes with temporal random effects

King, Ruth — 2012-12-01T00:00:00Z

Abstract: We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a “snapshot” of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.

Quantifying temporal change in biodiversity : challenges and opportunities

Dornelas, Maria — 2013-01-01T00:00:00Z

Abstract: Growing concern about biodiversity loss underscores the need to quantify and understand temporal change. Here, we review the opportunities presented by biodiversity time series, and address three related issues: (i) recognizing the characteristics of temporal data; (ii) selecting appropriate statistical procedures for analysing temporal data; and (iii) inferring and forecasting biodiversity change. With regard to the first issue, we draw attention to defining characteristics of biodiversity time series—lack of physical boundaries, uni-dimensionality, autocorrelation and directionality—that inform the choice of analytic methods. Second, we explore methods of quantifying change in biodiversity at different timescales, noting that autocorrelation can be viewed as a feature that sheds light on the underlying structure of temporal change. Finally, we address the transition from inferring to forecasting biodiversity change, highlighting potential pitfalls associated with phase-shifts and novel conditions.

The functional response of a generalist predator

Smout, Sophie Caroline — 2010-05-27T00:00:00Z

Abstract: Background: Predators can have profound impacts on the dynamics of their prey that depend on how predator consumption is affected by prey density (the predator's functional response). Consumption by a generalist predator is expected to depend on the densities of all its major prey species (its multispecies functional response, or MSFR), but most studies of generalists have focussed on their functional response to only one prey species. Methodology and principal findings: Using Bayesian methods, we fit an MSFR to field data from an avian predator (the hen harrier Circus cyaneus) feeding on three different prey species. We use a simple graphical approach to show that ignoring the effects of alternative prey can give a misleading impression of the predator's effect on the prey of interest. For example, in our system, a “predator pit” for one prey species only occurs when the availability of other prey species is low. Conclusions and significance: The Bayesian approach is effective in fitting the MSFR model to field data. It allows flexibility in modelling over-dispersion, incorporates additional biological information into the parameter priors, and generates estimates of uncertainty in the model's predictions. These features of robustness and data efficiency make our approach ideal for the study of long-lived predators, for which data may be sparse and management/conservation priorities pressing.

A non-technical overview of spatially explicit capture-recapture models

Borchers, David — 2012-02-01T00:00:00Z

Abstract: Most capture-recapture studies are inherently spatial in nature, with capture probabilities depending on the location of traps relative to animals. The spatial component of the studies has until recently, however, not been incorporated in statistical capture-recapture models. This paper reviews capture-recapture models that do include an explicit spatial component. This is done in a non-technical way, omitting much of the algebraic detail and focussing on the model formulation rather than on the estimation methods (which include inverse prediction, maximum likelihood and Bayesian methods). One can view spatially explicit capture-recapture (SECR) models as an endpoint of a series of spatial sampling models, starting with circular plot survey models and moving through conventional distance sampling models, with and without measurement errors, through mark-recapture distance sampling (MRDS) models. This paper attempts a synthesis of these models in what I hope is a style accessible to non-specialists, placing SECR models in the context of other spatial sampling models.

Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

Boore, Graeme — 2013-01-01T00:00:00Z

Abstract: For directed graph iterated function systems (IFSs) defined on R, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.

Topics in combinatorial semigroup theory

Maltcev, Victor — 2012-11-30T00:00:00Z

Abstract: In this thesis we discuss various topics from Combinatorial Semigroup Theory: automaton semigroups; finiteness conditions and their preservation under certain semigroup theoretic notions of index; Markov semigroups; word-hyperbolic semigroups; decision problems for finitely presented and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman’s property. We prove that an automaton semigroup generated by a Cayley machine of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups. We investigate how various semigroup finiteness conditions, linked to the notion of ideal, are preserved under finite Rees and Green indices. We obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption, hopficity is preserved by finite Rees index extensions. Still, there is an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness. This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate how the property of being Markov is preserved under finite Rees and Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and one-relator monoids.

Finite and infinite ergodic theory for linear and conformal dynamical systems

Munday, Sara Ann — 2011-11-30T00:00:00Z

Abstract: The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map Lα is the jump transformation of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fα and some Diophantine-type sets of numbers expressed in terms of the α-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic. The interest in the invariant measure for Fα lies in the fact that under a particular condition on the underlying partition α, the invariant measure associated to the map Fα is infinite. Then we proceed to introduce and examine the sequence of α-sum-level sets arising from the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric description of the Lyapunov spectra associated with each of the maps Lα and Fα. The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps Lα. These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group.

Workshop on new developments in cetacean survey methods

Borchers, David Louis — 2011-01-01T00:00:00Z

Abstract: This report contains the slides from a workshop on New Developments in Cetacean Survey Methods held on 27th November 2011 at the 19th Biennial Conference on the Biology of Marine Mammals, Tampa, Florida. Review talks were given on Passive Acoustic Density Estimation (Len Thomas); Dealing with g(0)<1: Perception Bias (Stephen Buckland); Dealing with g(0)<1: Availability Bias (Hans Skaug); Dealing with Measurement Error (David Borchers); and Density Surface Modelling (Jay Barlow). The sessions were followed by a discussion, and this is summarized at the end of the report.

The propagation and damping of slow magnetoacoustic waves in the solar atmosphere

Owen, Nicholas Robert — 2012-11-30T00:00:00Z

Abstract: The propagation and damping of slow magnetoacoustic waves in the solar atmosphere is investigated, with particular emphasis placed on waves with periodicities of five minutes. The basic model of a uniform temperature loop is extended by the addition of an equilibrium temperature gradient allowing study of wave propagation from the transition region to the corona. The inclusion of thermal conduction produces a phase shift between the perturbations in velocity, density and temperature, which for a non-uniform equilibrium temperature varies along the loop and may be observable as a phase shift between intensity and Doppler shift observations. Forward modelling of the simulation results, for both constant and non-constant equilibrium temperature profiles, is undertaken in order to establish the observational consequences for TRACE, SoHO/CDS and Hinode/EIS. Slow waves propagating in a non-uniform equilibrium temperature loop are seen to damp rapidly in the corona, however, as a result of the ionisation balance, the inclusion of damping can actually increase the amplitude of some parts of the oscillation. The ability of several data analysis techniques to identify oscillation signatures are examined. In particular, empirical mode decomposition was found to be a very useful technique for extracting oscillations from a wide range of data sets and is capable of intrinsically determining background trends. Co-spatial and cotemporal TRACE 171 A, CDS and EIS data are analysed for evidence of propagating slow waves. Slow waves with periods of 210 s to 370 s are found with amplitudes of 1.2% to 3.4% in the corona and 2.3% to 6.0% in the transition region.

Wave propagation, phase mixing and dissipation in Hall MHD

Threlfall, James W. — 2012-06-22T00:00:00Z

Abstract: In this thesis the effect of the Hall term in the generalised Ohm’s law on Alfvén (shear) and fast wave propagation and dissipation in the ion cyclotron frequency range is investigated. The damping of an initially Gaussian field perturbation in a uniform Hall MHD plasma is treated analytically. Subsequently a 2D Lagrangian remap code (Lare2d) is used to study the damping and phase mixing of initially Gaussian field perturbations and a harmonic series of boundary-driven perturbations in a uniform field (in the presence of a transverse equilibrium density gradient). The same code is then used to study a range of initially shear and fast-wave perturbations in the vicinity of a magnetic X-type null point. The magnetic energy associated with an initially Gaussian field perturbation in a uniform resistive plasma is shown to decay algebraically at a rate that is unaffected by the Hall term to leading order in kδ where k is wavenumber and δ is ion skin depth. A similar decay law applies to whistler perturbations in the limit kδ>>>1. We demonstrate that in both geometries considered, the inclusion of the Hall term reduces the effectiveness of phase-mixing in plasma heating. The reduction in the damping rate in the uniform ﬁeld (non-uniform density) cases, arising from dispersive effects, tends to zero in both the weak and strong phase mixing limits. In the Hall MHD X-point case, minimal reductions are seen for initially shear wave pulses, suggesting that little or no phase-mixing takes place. Nonlinear fast wave pulses which interact with the initial X-point destabilise the local field sufficiently to generate multiple null pairs; subsequent oscillatory current sheet behaviour appears unaffected by earlier differences between the MHD and Hall MHD cases.

Collisionless distribution function for the relativistic force-free Harris sheet

Stark, C. R. — 2012-01-01T00:00:00Z

Abstract: A self-consistent collisionless distribution function for the relativistic analogue of the force-free Harris sheet is presented. This distribution function is the relativistic generalization of the distribution function for the non-relativistic collisionless force-free Harris sheet recently found by Harrison and Neukirch [Phys. Rev. Lett. 102, 135003 (2009)], as it has the same dependence on the particle energy and canonical momenta. We present a detailed calculation which shows that the proposed distribution function generates the required current density profile (and thus magnetic field profile) in a frame of reference in which the electric potential vanishes identically. The connection between the parameters of the distribution function and the macroscopic parameters such as the current sheet thickness is discussed. (C) 2012 American Institute of Physics. [doi: 10.1063/1.3677268]

A detailed investigation of the properties of a Vlasov-Maxwell equilibrium for the force-free Harris sheet

Neukirch, Thomas — 2009-12-01T00:00:00Z

Abstract: A detailed discussion is presented of the Vlasov-Maxwell equilibrium for the force-free Harris sheet recently found by Harrison and Neukirch [Phys. Rev. Lett. 102, 135003 (2009)]. The derivation of the distribution function and a discussion of its general properties and their dependence on the distribution function parameters will be given. In particular, the distribution function can be single-peaked or multipeaked in two of the velocity components, with possible implications for stability. The dependence of the shape of the distribution function on the values of its parameters will be investigated and the relation to macroscopic quantities such as the current sheet thickness will be discussed.

A non-linear force-free field model for the solar magnetic carpet

Meyer, Karen A. — 2012-06-22T00:00:00Z

Abstract: The magnetic carpet is defined to be the small-scale photospheric magnetic field of the quiet Sun. Observations of the magnetic carpet show it to be highly dynamic, where the time taken for all flux within the magnetic carpet to be replaced is on the order of just a few hours. The magnetic carpet is continually evolving due to the Sun's underlying convection and the interaction of small-scale magnetic features with one another. Due to this, the small-scale coronal field of the magnetic carpet is also expected to be highly dynamic and complex. Previous modelling has shown that much of the flux from the magnetic carpet is stored along low-lying closed connections between magnetic features. This indicates that significant coronal heating could occur low down in the small-scale corona. In this thesis, a new two-component magnetic field model is developed for the evolution of the magnetic carpet. A 2D model is constructed to realistically simulate the evolution of the photospheric field of the magnetic carpet, where many of the parameters for the model are taken from observational studies. The photospheric model contains a granular and supergranular flow profile to describe the motion of the small-scale magnetic features, and includes the processes of flux emergence, cancellation, coalescence and fragmentation. This 2D model then couples to a 3D model as the lower boundary condition, which drives the evolution of the coronal field through a series of non-linear force-free states, via a magnetofrictional relaxation technique. We first apply the magnetofrictional technique to consider the coronal evolution of three basic small-scale photospheric processes: emergence, cancellation and flyby. We consider the interaction of the magnetic features with an overlying coronal magnetic field, and quantify magnetic energy build-up, storage and dissipation. The magnetofrictional technique is then applied to synthetic magnetograms produced from the 2D model, to simulate the evolution of the coronal field in a situation involving many hundreds of magnetic features. We conduct a preliminary analysis of the resultant 3D simulations, considering the magnetic energy stored and dissipated, as well as regions of enhanced velocity and electric current density within the coronal volume. The simulations show that the so-called 'quiet Sun' is not quiet and a significant amount of complex interactions take place.

A highly adaptive three dimensional hybrid vortex method for inviscid flows and helically symmetric vortex equilibria

Lucas, Daniel — 2012-06-22T00:00:00Z

Abstract: This thesis is concerned with three-dimensional vortex dynamics, in particular the modelling of vortex structures in an inviscid context. We are motivated by the open problem of regularity of the inviscid equations, i.e. whether or not these equations possess solutions. This problem is manifest in small scales, where vortex filaments are stretched and intensify as they are drawn into increasingly thin tendrils. This creates great difficulty in the investigation of such flows. Our only means of experimentation is to perform numerical simulations, which require exceptionally high resolution to capture the small scale vortex structures. A new numerical method to solve the inviscid Euler equations for three-dimensional, incompressible fluids is presented, with special emphasis on spatial adaptivity to resolve as broad a range of scales as possible in a completely self-similar fashion. We present a hybrid vortex method whereby we discretise the vorticity in Lagrangian filaments and perform and inversion to compute velocity on an arbitrary unstructured finite-volume grid. This allows for a two-fold adaptivity strategy. First, although naturally spatially adaptive by definition, the vorticity filaments undergo ‘renoding’. We redistribute nodes along the filament to concentrate their density in regions of high curvature. Secondly the Eulerian mesh is adapted to follow high strain by increasing resolution based on local filament dimensions. These features allow vortex stretching and folding to be resolved in a completely automatic and self-similar way. The method is validated via well known vortex rings and newly discovered helical vortex equilibria are also used to test the method. We begin by presenting this new class of three-dimensional vortex equilibria which possess helical symmetry. Such vortices are observed in propeller and wind turbine wakes, and their equilibria shapes have until now been unknown. These vortices are described by contours bounding regions of uniform axial vorticity. Material conservation of axial vorticity enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterised by their mean radius and centroid position. In the case of a single vortex, the parameter space cannot be fully filled by our numerical approach. We conjecture that multiply connected contours will characterise equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. In such cases instabilities often lead to a single helical vortex.

Spatial patterns and species coexistence : using spatial statistics to identify underlying ecological processes in plant communities

Brown, Calum — 2012-11-01T00:00:00Z

Abstract: The use of spatial statistics to investigate ecological processes in plant communities is becoming increasingly widespread. In diverse communities such as tropical rainforests, analysis of spatial structure may help to unravel the various processes that act and interact to maintain high levels of diversity. In particular, a number of contrasting mechanisms have been suggested to explain species coexistence, and these differ greatly in their practical implications for the ecology and conservation of tropical forests. Traditional first-order measures of community structure have proved unable to distinguish these mechanisms in practice, but statistics that describe spatial structure may be able to do so. This is of great interest and relevance as spatially explicit data become available for a range of ecological communities and analysis methods for these data become more accessible. This thesis investigates the potential for inference about underlying ecological processes in plant communities using spatial statistics. Current methodologies for spatial analysis are reviewed and extended, and are used to characterise the spatial signals of the principal theorised mechanisms of coexistence. The sensitivity of a range of spatial statistics to these signals is assessed, and the strength of such signals in natural communities is investigated. The spatial signals of the processes considered here are found to be strong and robust to modelled stochastic variation. Several new and existing spatial statistics are found to be sensitive to these signals, and offer great promise for inference about underlying processes from empirical data. The relative strengths of particular processes are found to vary between natural communities, with any one theory being insufficient to explain observed patterns. This thesis extends both understanding of species coexistence in diverse plant communities and the methodology for assessing underlying process in particular cases. It demonstrates that the potential of spatial statistics in ecology is great and largely unexplored.

Vessel noise affects beaked whale behavior : Results of a dedicated acoustic response study

Pirotta, Enrico — 2012-08-03T00:00:00Z

Abstract: Some beaked whale species are susceptible to the detrimental effects of anthropogenic noise. Most studies have concentrated on the effects of military sonar, but other forms of acoustic disturbance (e.g. shipping noise) may disrupt behavior. An experiment involving the exposure of target whale groups to intense vessel-generated noise tested how these exposures influenced the foraging behavior of Blainville’s beaked whales (Mesoplodon densirostris) in the Tongue of the Ocean (Bahamas). A military array of bottom-mounted hydrophones was used to measure the response based upon changes in the spatial and temporal pattern of vocalizations. The archived acoustic data were used to compute metrics the echolocation-based foraging behavior for 16 targeted groups, 10 groups further away on the range, and 26 nonexposed groups. The duration of foraging bouts was not significantly affected by the exposure. Changes in the hydrophone over which the group was most frequently detected occurred as the animals moved around within a foraging bout, and their number was significantly less the closer the whales were to the sound source. Non-exposed groups also had significantly more changes in the primary hydrophone than exposed groups irrespective of distance. Our results suggested that broadband ship noise caused a significant change in beaked whale behavior up to at least 5.2 kilometers away from the vessel. The observed change could potentially correspond to a restriction in the movement of groups, a period of more directional travel, a reduction in the number of individuals clicking within the group, or a response to changes in prey movement.

Growth of generating sets for direct powers of classical algebraic structures

Quick, Martyn — 2010-08-01T00:00:00Z

Abstract: For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.

Numerical simulation of shear-induced instabilities in internal solitary waves

Carr, Magda — 2011-09-25T00:00:00Z

Abstract: A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely Ric = 0.1 ± 0.01 , is not a sufficient condition for instability. It is also shown that the critical value of Lx/λ required for instability, where Lx is the length of the region in a wave in which Ri < 1/4 and λ is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that δiTw (where δi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri < 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely Lx/λ ≥ 0.86 and δiTw > 5 , are shown to be valid only for a limited parameter range. Description: This work was supported by the UK Engineering and Physical Sciences Research Council [grant number EP/F030622/1]

Continued fractions which correspond to two series expansions and the strong Hamburger moment problem

Sri Ranga, A. — 1984-01-01T00:00:00Z

Abstract: Just as the denominator polynomials of a J-fraction are orthogonal polynomials with respect to some moment functional, the denominator polynomials of an M-fraction are shown to satisfy a skew orthogonality relation with respect to a stronger moment functional. Many of the properties of the numerators and denominators of an M- fraction are also studied using this pseudo orthogonality relation of the denominator polynomials. Properties of the zeros of the denominator polynomials when the associated moment functional is positive definite are also considered. A type of continued fraction, referred to as a J-fraction, is shown to correspond to a power series about the origin and to another power series about infinity such that the successive convergents of this fraction include two more additional terms of anyone of the power series. Given the power series expansions, a method of obtaining such a J-fraction, whenever it exists, is also looked at. The first complete proof of the so called strong Hamburger moment problem using a continued fraction is given. In this case the continued fraction is a J-fraction. Finally a special class of J-fraction, referred to as positive definite J-fractions, is studied in detail. The four chapters of this thesis are divided into sections. Each section is given a section number which is made up of the chapter number followed by the number of the section within the chapter. The equations in the thesis have an equation number consisting of the section number followed by the number of the equation within that section. In Chapter One, in addition to looking at some of the historical and recent developments of corresponding continued fractions and their applications, we also present some preliminaries. Chapter Two deals with a different approach of understanding the properties of the numerators and denominators of corresponding (two point) rational functions and, continued fractions. This approach, which is based on a pseudo orthogonality relation of the denominator polynomials of the corresponding rational functions, provides an insight into understanding the moment problems. In particular, results are established which suggest a possible type of continued fraction for solving the strong Hamburger moment problem. In the third chapter we study in detail the existence conditions and corresponding properties of this new type of continued fraction, which we call J-fractions. A method of derivation of one of these 3-fractions is also considered. In the same chapter we also look at the all important application of solving the strong Hamburger moment problem, using these 3-fractions. The fourth and final chapter is devoted entirely to the study of the convergence behaviour of a certain class of J-fractions, namely positive definite J-fractions. This study also provides some interesting convergence criteria for a real and regular 3-fraction. Finally a word concerning the literature on continued fractions and moment problems. The more recent and up-to-date exposition on the analytic theory of continued fractions and their applications is the text of Jones and Thron [1980]. The two volumes of Baker and Graves-Morris [1981] provide a very good treatment on one of the computational aspects of the continued fractions, namely Pade approximants. There are also the earlier texts of Wall [1948] and Khovanskii [1963], in which the former gives an extensive insight into the analytic theory of continued fractions while the latter, being simpler, remains the ideal book for the beginner. In his treatise on Applied and Computational Complex Analysis, Henrici [1977] has also included an excellent chapter on continued fractions. Wall [1948] also includes a few chapters on moment problems and related areas. A much wider treatment of the classical moment problems is provided in the excellent texts of Shohat and Tamarkin [1943] and Akhieser [1965].

Presentations and efficiency of semigroups

Ayik, Hayrullah — 1998-01-01T00:00:00Z

Abstract: In this thesis we consider in detail the following two problems for semigroups: (i) When are semigroups finitely generated and presented? (ii) Which families of semigroups can be efficiently presented? We also consider some other finiteness conditions for semigroups, homology of semigroups and wreath product of groups. In Chapter 2 we investigate finite presentability and some other finiteness conditions for the O-direct union of semigroups with zero. In Chapter 3 we investigate finite generation and presentability of Rees matrix semigroups over semigroups. We find necessary and sufficient conditions for finite generation and presentability. In Chapter 4 we investigate some other finiteness conditions for Rees matrix semigroups. In Chapter 5 we consider groups as semigroups and investigate their semigroup efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups that are not groups. We first give examples of efficient and inefficient "proper" semigroups by computing their homology and finding their minimal presentations. In Chapter 7 we compute the second homology of finite simple semigroups and find a "small" presentation for them. If that "small" presentation has a special relation, we prove that finite simple semigroups are efficient. Finally, in Chapter 8, we investigate the efficiency of wreath products of finite groups as groups and as semigroups. We give more examples of efficient groups and inefficient groups.

Solar flare particle acceleration in collapsing magnetic traps

Grady, Keith J. — 2012-06-22T00:00:00Z

Abstract: The topic of this thesis is a detailed investigation of different aspects of the particle acceleration mechanisms operating in Collapsing Magnetic Traps (CMTs), which have been suggested as one possible mechanism for particle acceleration during solar flares. The acceleration processes in CMTs are investigated using guiding centre test particle calculations. Results including terms of different orders in the guiding centre approximation are compared to help identify which of the terms are important for the acceleration of particles. For a basic 2D CMT model the effects of different initial conditions (position, kinetic energy and pitch angle) of particles are investigated in detail. The main result is that the particles that gain most energy are those with initial pitch angles close to 90° and start in weak field regions in the centre of the CMT. The dominant acceleration mechanism for these particles is betatron acceleration, but other particles also show signatures of Fermi acceleration. The basic CMT model is then extended by (a) including a magnetic field component in the invariant direction and (b) by making it asymmetric. It is found that the addition of a guide field does not change the characteristics of particle acceleration very much, but for the asymmetric models the associated energy gain is found to be much smaller than in symmetric models, because the particles can no longer remain very close to the trap centre throughout their orbit. The test particle method is then also applied to a CMT model from the literature which contains a magnetic X-line and open and closed field lines and the results are compared with the previous results and the findings in the literature. Finally, the theoretical framework of CMT models is extended to 2.5D models with shear flow and to fully 3D models, allowing the construction of more realistic CMT models in the future.

Semigroups of order-decreasing transformations

Umar, Abdullahi — 1992-01-01T00:00:00Z

Abstract: Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this Thesis the study of order-increasing full (partial, partial one-to-one) transformations has been reduced to that of order-decreasing full (partial, partial one-to-one) transformations and the study of order-decreasing partial transformations to that of order-decreasing full transformations for both the finite and infinite cases. For the finite order-decreasing full (partial one-to-one) transformation semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden (1990) concerning products of idempotents (quasi-idempotents), and concerning combinatorial and rank properties. By contrast with the semigroups of order-preserving transformations and the full transformation semigroup, the semigroups of orderdecreasing full (partial one-to-one) transformations and their Rees quotient semigroups are not regular. They are, however, abundant (type A) semigroups in the sense of Fountain (1982,1979). An explicit characterisation of the minimum semilattice congruence on the finite semigroups of order-decreasing transformations and their Rees quotient semigroups is obtained. If X is an infinite chain then the semigroup S of order-decreasing full transformations need not be abundant. A necessary and sufficient condition on X is obtained for S to be abundant. By contrast, for every chain X the semigroup of order-decreasing partial one-to-one transformations is type A. The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing full (partial one-to-one) transformations have been investigated.

Semigroup presentations

Ruskuc, Nikola — 1995-01-01T00:00:00Z

Abstract: In this thesis we consider in detail the following two fundamental problems for semigroup presentations: 1. Given a semigroup find a presentation defining it. 2. Given a presentation describe the semigroup defined by it. We also establish two links between these two approaches: semigroup constructions and computational methods. After an introduction to semigroup presentations in Chapter 3, in Chapters 4 and 5 we consider the first of the two approaches. The semigroups we examine in these two chapters include completely O-simple semigroups, transformation semigroups, matrix semigroups and various endomorphism semigroups. In Chapter 6 we find presentations for the following semi group constructions: wreath product, Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids, Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in more detail presentations for subsemigroups in Chapters 7 and 10, where we prove a number of Reidemeister-Schreier type results for semigroups. In Chapter 9 we examine the connection between the semi group and the group defined by the same presentation. The general results from Chapters 6, 7, 9 and 10 are applied in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci semigroups, semigroups defined by Coxeter type presentations and one relator products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter enumeration procedure and introduce three modifications of this procedure.

Green index in semigroups : generators, presentations and automatic structures

Cain, A.J. — 2012-01-01T00:00:00Z

Abstract: The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into T-relative H -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups).

The visible part of plane self-similar sets

Falconer, Kenneth John — 2013-01-01T00:00:00Z

Abstract: Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1 then dimH VθF = 1 for almost all θ , where dimH denotes Hausdorff dimension. We conrm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected. Description: JMF was supported by an EPSRC grant whilst undertaking this work.

Global analysis of cetacean line-transect surveys : detecting trends in cetacean density

Jewell, Rebecca Lucy — 2012-05-07T00:00:00Z

Abstract: Measuring the effect of anthropogenic change on cetacean populations is hampered by our lack of understanding about population status and a lack of power in the available data to detect trends in abundance. Often long-term data from repeated surveys are lacking, and alternative approaches to trend detection must be considered. We utilised an existing database of line transect survey records to determine whether temporal trends could be detected when survey effort from around the world was combined. We extracted density estimates for 25 species and fitted generalised additive models (GAMs) to investigate whether taxonomic, spatial or methodological differences among systematic line-transect surveys affect estimates of density and whether we can identify temporal trends in the data once these factors are accounted for. The selected GAM consisted of 2 parts: an intercept term that was a complex interaction of taxonomic, spatial and methodological factors and a smooth temporal term with trends varying by family and ocean basin. We discuss the trends found and assess the suitability of published density estimates for detecting temporal trends using retrospective power analysis. In conclusion, increasing sample size through combining survey effort across a global scale does not necessarily result in sufficient power to detect trends because of the extent of variability across surveys, species and oceans. Instead, results from repeated dedicated surveys designed specifically for the species and geographical region of interest should be used to inform conservation and management.

Applications of statistics in flood frequency analysis

Ahmad, Muhammad Idrees — 1989-01-01T00:00:00Z

Abstract: Estimation of the probability of occurrence of future flood events at one or more locations across a river system is frequently required for the design of bridges, culverts, spillways, dams and other engineering works. This study investigates some of the statistical aspects for estimating the flood frequency distribution at a single site and on regional basis. It is demonstrated that generalized logistic (GL) distribution has many properties well suited for the modelling of flood frequency data. The GL distribution performs better than the other commonly recommended flood frequency distributions in terms of several key properties. Specifically, it is capable of reproducing almost the same degree of skewness typically present in observed flood data. It appears to be more robust to the presence of extreme outliers in the upper tail of the distribution. It has a relatively simpler mathematical form. Thus all the well known methods of parameter estimation can be easily implemented. It is shown that the method of probability weighted moments (PWM) using the conventionally recommended plotting position substantially effects the estimation of the shape parameter of the generalized extreme value (GEV) distribution by relocating the annual maximum flood series. A location invariant plotting position is introduced to use in estimating, by the method of PWM, the parameters of the GEV and the GL distributions. Tests based on empirical distribution function (EDF) statistics are proposed to assess the goodness of fit of the flood frequency distributions. A modified EDF test is derived that gives greater emphasis to the upper tail of a distribution which is more important for flood frequency prediction. Significance points are derived for the GEV and GL distributions when the parameters are to be estimated from the sample data by the method of PWMs. The critical points are considerably smaller than for the case where the parameters of a distribution are assumed to be specified. Approximate formulae over the whole range of the distribution for these tests are also developed which can be used for regional assessment of GEV and GL models based on all the annual maximum series simultaneously in a hydrological region. In order to pool at-site flood data across a region into a single series for regional analysis, the effect of standardization by at-site mean on the estimation of the regional shape parameter of the GEV distribution is examined. Our simulation study based on various synthetic regions reveals that the standardization by the at-site mean underestimates the shape parameter of the GEV by about 30% of its true value and also contributes to the separation of skewness of observed and simulated floods. A two parameter standardization by the at-site estimates of location and scale parameters is proposed. It does not distort the shape of the flood frequency data in the pooling process. Therefore, it offers significantly improved estimate of the shape parameter, allows pooling data with heterogeneous coefficients of variation and helps to explain the separation of skewness effect. Regions on the basis of flood statistics L-CV and USKEW are derived for Scotland and North England. Only about 50% of the basins could be correctly identified as belonging to these regions by a set of seven catchment characteristics. The alternative approach of grouping basins solely on the basis of physical properties is preferable. Six physically homogeneous groups of basins are identified by WARD's multivariate clustering algorithm using the same seven characteristics. These regions have hydrological homogeneity in addition to their physical homogeneity. Dimensionless regional flood frequency curves are produced by fitting GEV and GL distributions for each region. The GEV regional growth curves imply a larger return period for a given magnitude flood. When floods are described by GL model the respective return periods are considerably smaller.

A general discrete-time modeling framework for animal movement using multi-state random walks

McClintock, Brett Thomas — 2012-01-01T00:00:00Z

Abstract: Recent developments in animal tracking technology have permitted the collection of detailed data on the movement paths of individuals from many species. However, analysis methods for these data have not developed at a similar pace, largely due to a lack of suitable candidate models, coupled with the technical difficulties of fitting such models to data. To facilitate a general modeling framework, we propose that complex movement paths can be conceived as a series of movement strategies among which animals transition as they are affected by changes in their internal and external environment. We synthesize previously existing and novel methodologies to develop a general suite of mechanistic models based on biased and correlated random walks that allow different behavioral states for directed (e.g., migration), exploratory (e.g., dispersal), area-restricted (e.g., foraging), and other types of movement. Using this “tool-box” of nested model components, multi-state movement models may be custom-built for a wide variety of species and applications. As a unified state-space modeling framework, it allows the simultaneous investigation of numerous hypotheses about animal movement from imperfectly observed data, including time allocations to different movement behavior states, transitions between states, the use of memory or navigation, and strengths of attraction (or repulsion) to specific locations. The inclusion of covariate information permits further investigation of specific hypotheses related to factors driving different types of movement behavior. Using reversible jump Markov chain Monte Carlo methods to facilitate Bayesian model selection and multi-model inference, we apply the proposed methodology to real data by adapting it to the natural history of the grey seal (Halichoerus grypus) in the North Sea. Although previous grey seal studies tended to focus on correlated movements, we found overwhelming evidence that bias towards haul-out or foraging locations better explained seal movement than simple or correlated random walks. Posterior model probabilities also provided evidence that seals transition among directed, area-restricted, and exploratory movements associated with haul-out, foraging, and other behaviors. With this intuitive framework for modeling and interpreting animal movement, we believe the development and application of bespoke movement models will become more accessible to ecologists and non-statisticians.

Coupling of the solar wind, magnetosphere and ionosphere by MHD waves

Russell, Alexander J. B. — 2010-11-30T00:00:00Z

Abstract: The solar wind, magnetosphere and ionosphere are coupled by magnetohydrodynamic waves, and this gives rise to new and often unexpected behaviours that cannot be produced by a single, isolated part of the system. This thesis examines two broad instances of coupling: field-line resonance (FLR) which couples fast and Alfvén waves, and magnetosphere-ionosphere (MI-) coupling via Alfvén waves. The first part of this thesis investigates field-line resonance for equilibria that vary in two dimensions perpendicular to the background magnetic field. This research confirms that our intuitive understanding of FLR from 1D is a good guide to events in 2D, and places 2D FLR onto a firm mathematical basis by systematic solution of the governing equations. It also reveals the new concept of ‘imprinting’ of spatial forms: spatial variations of the resonant Alfvén wave correlate strongly with the spatial form of the fast wave that drives the resonance. MI-coupling gives rise to ionosphere-magnetosphere (IM-) waves, and we have made a detailed analysis of these waves for a 1D sheet E-region. IM-waves are characterised by two quantities: a speed v_{IM} and an angular frequency ω_{IM} , for which we have obtained analytic expressions. For an ideal magnetosphere, IM-waves are advective and move in the direction of the electric field with speed v_{IM}. The advection speed is a non-linear expression that decreases with height-integrated E-region plasma-density, hence, wavepackets steepen on their trailing edge, rapidly accessing small length-scales through wavebreaking. Inclusion of electron inertial effects in the magnetosphere introduces dispersion to IM-waves. In the strongly inertial limit (wavelength λ << λ_{e} , where λ_{e} is the electron inertial length at the base of the magnetosphere), the group velocity of linear waves goes to zero, and the waves oscillate at ω_{IM} which is an upper limit on the angular frequency of IM-waves for any wavelength. Estimates of v_{IM} show that this speed can be a significant fraction (perhaps half) of the E_{⊥} × B_{0} drift in the E-region, producing speeds of up to several hundred metres per second. The upper limit on angular frequency, ωIM , is estimated to give periods from a few hundredths of a second to several minutes. IM-waves are damped by recombination and background ionisation, giving an e-folding decay time that can vary from tens of seconds to tens of minutes. We have also investigated the dynamics and steady-states that occur when the magnetosphere-ionosphere system is driven by large-scale Alfvénic field-aligned currents. Steady-states are dominated by two approximate solutions: an ‘upper’ solution that is valid in places where the E-region is a near perfect conductor, and a ‘lower’ solution that is valid where E-region depletion makes recombination negligible. These analytic solutions are extremely useful tools and the global steady-state can be constructed by matching these solutions across suitable boundary-layers. Furthermore, the upper solution reveals that E-region density cavities form and widen (with associated broadening of the magnetospheric downward current channel) if the downward current density exceeds the maximum current density that can be supplied by background E-region ionisation. We also supply expressions for the minimum E-region plasma-density and shortest length-scale in the steady-state. IM-waves and steady-states are extremely powerful tools for interpreting MI-dynamics. When an E-region density cavity widens through coupling to an ideal, single-fluid MHD magnetosphere, it does so by forming a discontinuity that steps between the upper and lower steady-states. This discontinuity acts as part of an ideal IM-wave and moves in the direction of the electric field at a speed U = \sqrt{v_{IM}^{+} v_{IM}^{-}}, which is the geometric mean of v_{IM} evaluated immediately to the left and right of the discontinuity. This widening speed is typically several hundreds of metres per second. If electron inertial effects are included in the magnetosphere, then the discontinuity is smoothed, and a series of undershoots and overshoots develops behind it. These undershoots and overshoots evolve as inertial IM-waves. Initially they are weakly inertial, with a wavelength of about λ_{e}, however, strong gradients of ω_{IM} cause IM-waves to phase-mix, making their wavelength inversely proportional to time. Therefore, the waves rapidly become strongly inertial and oscillate at ω_{IM}. The inertial IM-waves drive upgoing Alfvén waves in the magnetosphere, which populate a region over the downward current channel, close to its edge. In this manner, the E-region depletion mechanism, that we have detailed, creates small-scale Alfvén waves in large-scale current systems, with properties determined by MI-coupling.

Topics in computational group theory : primitive permutation groups and matrix group normalisers

Coutts, Hannah Jane — 2011-11-01T00:00:00Z

Abstract: Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in S[subscript(d)] of G and the rank of N. Part II presents a new algorithm NormaliserGL for computing the normaliser in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in the computational algebra system MAGMA and employs Aschbacher's theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over MAGMA's existing algorithm.

Status assessment of the Critically Endangered Azores Bullfinch Pyrrhula murina

Ceia, Ricardo S. — 2011-01-01T00:00:00Z

Abstract: The Azores Bullfinch is endemic to the island of São Miguel (Azores, Portugal). Its status was uplisted to Critically Endangered in 2005 on the basis of an extremely small and declining population that was considered to be restricted to a very small mountain range (43 km2), in a single location, within which the spread of invasive plants constituted a threat to habitat quality. Nevertheless, information was mostly inferred, or the product of, non-systematic studies. In order to carry out a complete assessment of the conservation status we analysed: (i) population trend, calculated from annual monitoring 1991–2008, (ii) population size, and (iii) range size, obtaining estimates in a single morning study in 2008 involving the simultaneous participation of 48 observers. Contrary to previous inferences, the population is no longer decreasing, although quality of laurel forest habitat continues to decline due to the persistent threat of invasive species. Population size (mean ± SE) was estimated at 1,064 ± 304 individuals using distance sampling methods, although the estimate was very sensitive to the survey method used. Range size estimates (extent of occurrence and area of occupancy) were 144 km2 and 83 km2 respectively. Given the present information, we propose the downlisting of Azores Bullfinch to Endangered on the IUCN Red List.

Atmospheric transport and critical layer mixing in the troposphere and stratosphere

Smy, Louise Ann — 2012-06-22T00:00:00Z

Abstract: This thesis aims to improve the understanding of transport and critical layer mixing in the troposphere and stratosphere. A dynamical approach is taken based on potential vorticity which has long been recognised as the essential field inducing the flow and thermodynamic structure of the atmosphere. Within the dynamical framework of critical layer mixing of potential vorticity, three main topics are addressed. First, an idealised model of critical layer mixing in the stratospheric surf zone is examined. The effect of the shear across the critical layer on the critical layer evolution itself is investigated. In particular it is found that at small shear barotropic instability occurs and the mixing efficiency of the critical layer increases due to the instability. The effect of finite deformation length is also considered which extends previous work. Secondly, the dynamical coupling between the stratosphere and troposphere is examined by considering the effect of direct perturbations to stratospheric potential vorticity on the evolution of midlatitude baroclinic instability. Both zonally symmetric and asymmetric perturbations to the stratospheric potential vorticity are considered, the former representative of a strong polar vortex, the latter representative of the stratospheric state following a major sudden warming. A comparison of these perturbations gives some insight into the possible influence of pre or post-sudden warming conditions on the tropospheric evolution. Finally, the influence of the stratospheric potential vorticity distribution on lateral mixing and transport into and out of the tropical pipe, the low latitude ascending branch of the Brewer-Dobson circulation, is investigated. The stratospheric potential vorticity distribution in the tropical stratosphere is found to have a clear pattern according to the phase of the quasi-biennial oscillation (QBO). The extent of the QBO influence is quantified, by analysing trajectories of Lagrangian particles using an online trajectory code recently implemented in the Met Office's Unified Model.

Generation problems for finite groups

McDougall-Bagnall, Jonathan M. — 2011-11-30T00:00:00Z

Abstract: It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.

Behind and beyond a theorem on groups related to trivalent graphs

Havas, George — 2008-12-01T00:00:00Z

Abstract: In 2006 we completed the proof of a five-part conjecture that was made in 1977 about a family of groups related to trivalent graphs. This family covers all 2-generator, 2-relator groups where one relator specifies that a generator is an involution and the other relator has three syllables. Our proof relies upon detailed but general computations in the groups under question. The proof is theoretical, but based upon explicit proofs produced by machine for individual cases. Here we explain how we derived the general proofs from specific cases. The conjecture essentially addressed only the finite groups in the family. Here we extend the results to infinite groups, effectively determining when members of this family of finitely presented groups are simply isomorphic to a specific quotient.

Lower-hybrid waves generated by anomalous Doppler resonance in auroral plasmas

Bingham, R. — 2010-08-01T00:00:00Z

Abstract: This paper describes sonic aspects of lower-hybrid wave activity in space plasmas. Lower-hybrid waves are particularly important since they can transfer energy efficiently between electrons and ions in a collisionless magnetized plasma. We consider the 'fan' or anomalous Doppler resonance instability driven by energetic electron tails and show that it is responsible for the generation of lower-hybrid waves. We also demonstrate that observations of their intensity are sufficient to drive the modulational instability.

Falling towards forgetfulness : synaptic decay prevents spontaneous recovery of memory

Stone, James V. — 2008-08-22T00:00:00Z

Abstract: Long after a new language has been learned and forgotten, relearning a few words seems to trigger the recall of other words. This "free-lunch learning'' (FLL) effect has been demonstrated both in humans and in neural network models. Specifically, previous work proved that linear networks that learn a set of associations, then partially forget them all, and finally relearn some of the associations, show improved performance on the remaining (i.e., nonrelearned) associations. Here, we prove that relearning forgotten associations decreases performance on nonrelearned associations; an effect we call negative free-lunch learning. The difference between free-lunch learning and the negative free-lunch learning presented here is due to the particular method used to induce forgetting. Specifically, if forgetting is induced by isotropic drifting of weight vectors (i.e., by adding isotropic noise), then free-lunch learning is observed. However, as proved here, if forgetting is induced by weight values that simply decay or fall towards zero, then negative free-lunch learning is observed. From a biological perspective, and assuming that nervous systems are analogous to the networks used here, this suggests that evolution may have selected physiological mechanisms that involve forgetting using a form of synaptic drift rather than synaptic decay, because synaptic drift, but not synaptic decay, yields free-lunch learning.

Geometric grid classes of permutations

Albert, M.H. — 2012-01-01T00:00:00Z

Unary FA-presentable semigroups

Cain, Alan James — 2012-06-08T00:00:00Z

Abstract: Automatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.

Three-dimensional solutions of the magnetohydrostatic equations : rigidly rotating magnetized coronae in spherical geometry

Al-Salti, N. — 2010-10-01T00:00:00Z

Abstract: Context. Magnetohydrostatic (MHS) equilibria are often used to model astrophysical plasmas, for example, planetary magnetospheres or coronae of magnetized stars. However, finding realistic three-dimensional solutions to the MHS equations is difficult, with only a few known analytical solutions and even finding numerical solution is far from easy. Aims. We extend the results of a previous paper on three-dimensional solutions of the MHS equations around rigidly rotating massive cylinders to the much more realistic case of rigidly rotating massive spheres. An obvious application is to model the closed field line regions of the coronae of rapidly rotating stars. Methods. We used a number of simplifying assumptions to reduce the MHS equations to a single elliptic partial differential equation for a pseudo-potential U, from which all physical quantities, such as the magnetic field, the plasma pressure, and the density, can be derived by differentiation. The most important assumptions made are stationarity in the co-rotating frame of reference, a particular form for the current density, and neglect of outflows. Results. In this paper we demonstrate that standard methods can be used to find numerical solutions to the fundamental equation of the theory. We present three simple different cases of magnetic field boundary conditions on the surface of the central sphere, corresponding to an aligned dipole field, a non-aligned dipole field, and a displaced dipole field. Our results show that it should be possible in the future to use this method without dramatically increasing the demands on computational resources to improve upon potential field models of rotating magnetospheres and coronae.

On the relationship between equilibrium bifurcations and ideal MHD instabilities for line-tied coronal loops

Neukirch, T. — 2010-01-01T00:00:00Z

Abstract: For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated, imposing line-tied boundary conditions. Using a well-studied example based on the Gold -aEuro parts per thousand Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad -aEuro parts per thousand Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point, because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials, the first bifurcation point of the Grad -aEuro parts per thousand Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.

Three-dimensional solutions of the magnetohydrostatic equations : rigidly rotating magnetized coronae in cylindrical geometry

Al-Salti, N. — 2010-05-01T00:00:00Z

Abstract: Context. Solutions of the magnetohydrostatic (MHS) equations are very important for modelling astrophysical plasmas, such as the coronae of magnetized stars. Realistic models should be three-dimensional, i.e., should not have any spatial symmetries, but finding three-dimensional solutions of the MHS equations is a formidable task. Aims. We present a general theoretical framework for calculating three-dimensional MHS solutions outside massive rigidly rotating central bodies, together with example solutions. A possible future application is to model the closed field region of the coronae of fast-rotating stars. Methods. As a first step, we present in this paper the theory and solutions for the case of a massive rigidly rotating magnetized cylinder, but the theory can easily be extended to other geometries, We assume that the solutions are stationary in the co-rotating frame of reference. To simplify the MHS equations, we use a special form for the current density, which leads to a single linear partial differential equation for a pseudo-potential U. The magnetic field can be derived from U by differentiation. The plasma density, pressure, and temperature are also part of the solution. Results. We derive the fundamental equation for the pseudo-potential both in coordinate independent form and in cylindrical coordinates. We present numerical example solutions for the case of cylindrical coordinates.

A critical review of the literature on population modelling

Cabrelli, Abigail — 2009-01-01T00:00:00Z

Abstract: The 2005 report of the National Research Council’s ‘Committee on Characterizing Biologically Significant Marine Mammal Behavior’ proposed a framework, which they called PCAD - Population Consequences of Acoustic Disturbance, that uses a series of transfer functions to link behavioural responses to sound with life functions, vital rates, and population change. The Committee suggested that the best understood transfer functions are those linking vital rates to population change. One of the main aims of this report is to document that understanding. However, we also show how the existing frameworks for modelling the dynamics of marine mammal populations can be extended to include the effects of behavioural responses on vital rates. In Chapter 1 we introduce the central concept of the rate of increase (lambda) of a population, which we believe is the most useful measure of the effects of behavioural responses on the dynamics of a population. If the value of lambda exceeds one, then thepopulation will increase over time; if it is less than one it will decrease. We show how changes in lambda provide a measure of the impact of human activities (such as exploitation, conservation, or disturbance) on a population. We also introduce structured population models, which take account of the fact that all individuals in a population are not identical, and show how the dynamics of different parts of a population can be modelled using a population projection matrix. The mathematical properties of this projection matrix can be used to determine the sensitivity of lambda to small changes in vital rates. Finally, we provide a very brief introduction to the concept of stochasticity, and the use of lambda to predict when (and if) a population might be driven to extinction. Chapter 2 describes how lambda also provides a measure of the Darwinian fitness of the individual members of a population. An individual’s fitness, the contribution it will make to future generations, depends to a large extent on its body condition and on the risks of mortality to which it is exposed. Both of these could be affected by behaviour responses to sound. We also explain current theories about the relationship between an individual’s feeding behaviour and the abundance and distribution of prey, and how this can affect body condition. Chapter 3 provides a more detailed description of how elasticity analysis can be used to investigate the impact of changes in vital rates on lambda . Elasticity analysis is a useful tool for detecting which vital rates are most important in determining the dynamics of a population. However, its value is limited because it does not take account of random variations (stochasticity) and, in theory, it can only predict the effect of small changes in vital rates. Chapter 4 describes the fundamental concept of density dependence: the way in which vital rates change with population size or the availability of resources, such as prey. Not only is density dependence an essential prerequisite for population stability and sustainable use, but the form it takes will also determine how a population responds to behavioural changes. This is because behaviour, and particularly the effect of behavioural change on body condition, plays a central role in many of the mechanistic models of density dependence. Chapters 5 and 6 explore the way in which additional complexities, such as social structure and the way in which populations are distributed in space, can affect the dynamics of populations. Models that account for these complexities behave in a much less predictable way than the relatively simple structured models that form the core of Chapters 1-4. So far, the models of population dynamics that we have reviewed have been deterministic. That is, they have assumed that the only way in which vital rates can vary is in response to a change in abundance, via density dependent mechanisms. In Chapters 7 and 8 we investigate the effect of random variation (stochasticity) on population dynamics. We distinguish the effects of demographic stochasticity, chance variations in the number of animals that die or give birth in a time interval that occur even if vital rates do not vary over time, and environmental stochasticity, which is the result of variations in vital rates across years. Variation in abundance may also occur as a result of environmental change and changes in the ecological community of which a population is a part. The effect of all these sources of variation is to reduce the realised growth rate of a population, and therefore its risk of extinction. In Chapter 9 we consider how the basic population modelling framework described in Chapters 1-8 might be extended to take account of the life functions identified by the NRC Committee. We suggest that these life functions are useful for defining the context in which behavioural responses might affect vital rates, but that they do not need to be modelled explicitly. Removing vital functions from the PCAD framework results in a much simpler structure, which is compatible with existing population modelling frameworks. However, these will have to be extended to allow population states, like body condition, that vary continuously to be modelled. Chapter 10 describes how changes in lambda can be detected. The simple analytical frameworks that are available for this are all vulnerable to the effects of variability that we introduced in Chapter 7. However, there is a framework (state-space and hidden Markov process modelling) that can account for the effects of this variability, and we recommend its use for detecting trends. The additional benefit of this approach is that its use results in a detailed model of the dynamics of the population that is under investigation. Chapter 11 reviews the different model structures that can be used to describe the dynamics of a population, and explains when different forms of population models (e.g. discrete vs. continuous time, deterministic vs. stochastic) are most appropriate. We also discuss how these different frameworks can be extended to account for continuous population states, as recommended in Chapter 8. The final focus is on how state-space models can be fitted to time series of abundance estimates and information on vital rates. Chapter 12 looks at the relevance of the different modelling approaches described in the previous chapters for analysing the potential effects of behavioural responses to sound on population dynamics, particularly the kinds of sounds that may be generated by the oil and gas industry. We conclude that lambda , the population rate of increase, and its variation provides a useful measure of these effects. We also believe that the models used for this purpose will certainly have to account for the effects of variability and density dependence. They will probably also have to account for the effects of social structure and the way in which populations use space. The state-space modelling framework outlined in Chapter 11 can, in principle, be extended to capture all of these features although work on this is still in its infancy. Description: Final Report to the Joint Industry Project of the International Association of Oil & Gas Producers on contract JIP22 07_20

Universal scaling rules predict evolutionary patterns of myogenesis in species with indeterminate growth

Johnston, Ian Alistair — 2012-06-07T00:00:00Z

Abstract: Intraspecific phenotypic variation is ubiquitous and often associated with resource exploitation in emerging habitats. For example, reduced body size has evolved repeatedly in Arctic charr (Salvelinus alpinus L.) and threespine stickleback (Gasterosteus aculeatus L.) across post-glacial habitats of the Northern Hemisphere. Exploiting these models, we examined how body size and myogenesis evolve with respect to the 'optimum fibre size hypothesis', which predicts that selection acts to minimize energetic costs associated with ionic homeostasis by optimizing muscle fibre production during development. In eight dwarf Icelandic Arctic charr populations, the ultimate production of fast-twitch muscle fibres (FN(max)) was only 39.5 and 15.5 per cent of that in large-bodied natural and aquaculture populations, respectively. Consequently, average fibre diameter (FD) scaled with a mass exponent of 0.19, paralleling the relaxation of diffusional constraints associated with mass-specific metabolic rate scaling. Similar reductions in FN(max) were observed for stickleback, including a small-bodied Alaskan population derived from a larger-bodied oceanic stock over a decadal timescale. The results suggest that in species showing indeterminate growth, body size evolution is accompanied by strong selection for fibre size optimization, theoretically allowing resources saved from ionic homeostasis to be allocated to other traits affecting fitness, including reproduction. Gene flow between small- and large-bodied populations residing in sympatry may counteract the evolution of this trait.

An update to the methods in Endangered Species Research 2011 paper "Estimating North Pacific right whale Eubalaena japonica density using passive acoustic cue counting"

Marques, Tiago A. — 2012-01-01T00:00:00Z

Mathematics for history's sake : a new approach to Ptolemy's Geography

Mintz, Daniel V. — 2011-06-22T00:00:00Z

Abstract: Almost two thousand years ago, Claudius Ptolemy created a guide to drawing maps of the world, identifying the names and coordinates of over 8,000 settlements and geographical features. Using the coordinates of those cities and landmarks which have been identified with modern locations, a series of best-fit transformations has been applied to several of Ptolemy’s regional maps, those of Britain, Spain, and Italy. The transformations relate Ptolemy’s coordinates to their modern equivalents by rotation and skewed scaling. These reflect the types of error that appear in Ptolemy’s data, namely those of distance and orientation. The mathematical techniques involved in this process are all modern. However, these techniques have been altered in order to deal with the historical difficulties of Ptolemy’s maps. To think of Ptolemy’s data as similar to that collected from a modern random sampling of a population and to apply unbiased statistical methods to it would be erroneous. Ptolemy’s data is biased, and the nature of that bias is going to be informed by the history of the data. Using such methods as cluster analysis, Procrustes analysis, and multidimensional scaling, we aimed to assess numerically the accuracy of Ptolemy’s maps. We also investigated the nature of the errors in the data and whether or not these could be linked to historical developments in the areas mapped.

Substitution-closed pattern classes

Atkinson, M.D. — 2011-02-01T00:00:00Z

Abstract: The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classied by listing them as a set of explicit families.

Automatic presentations and semigroup constructions

Cain, Alan J. — 2010-08-01T00:00:00Z

Abstract: An automatic presentation for a relational structure is, informally, an abstract representation of the elements of that structure by means of a regular language such that the relations can all be recognized by finite automata. A structure admitting an automatic presentation is said to be FA-presentable. This paper studies the interaction of automatic presentations and certain semigroup constructions, namely: direct products, free products, finite Rees index extensions and subsemigroups, strong semilattices of semigroups, Rees matrix semigroups, Bruck-Reilly extensions, zero-direct unions, semidirect products, wreath products, ideals, and quotient semigroups. For each case, the closure of the class of FA-presentable semigroups under that construction is considered, as is the question of whether the FA-presentability of the semigroup obtained from such a construction implies the FA-presentability of the original semigroup[s]. Classifications are also given of the FA-presentable finitely generated Clifford semigroups, completely simple semigroups, and completely 0-simple semigroups.

Automatic presentations for semigroups

Cain, Alan J. — 2009-11-01T00:00:00Z

Abstract: This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved. Description: Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)

On residual finiteness of direct products of algebraic systems

Gray, R. — 2009-09-01T00:00:00Z

Abstract: It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.

The Bergman property for semigroups

Maltcev, V. — 2009-08-01T00:00:00Z

Abstract: In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.

Green index and finiteness conditions for semigroups

Gray, R. — 2008-10-15T00:00:00Z

Abstract: Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left and by right multiplication. If the complement S \ T has finitely many strong orbits by both these actions we say that T has finite Green index in S. This notion of finite index encompasses subgroups of finite index in groups, and also subsemigroups of finite Rees index (complement). Therefore, the question of S and T inheriting various finiteness conditions from each other arises. In this paper we consider and resolve this question for the following finiteness conditions: finiteness, residual finiteness, local finiteness, periodicity, having finitely many right ideals, and having finitely many idempotents. (c) 2008 Elsevier Inc. All rights reserved.

Properties of the subsemigroups of the bicyclic monoid

Descalco, L. — 2008-06-01T00:00:00Z

Abstract: In this paper we study some properties of the subsemigroups of the bicyclic monoid B, by using a recent description of its subsemigroups. We start by giving necessary and sufficient conditions for a subsemigroup to be finitely generated. Then we show that all finitely generated subsemigroups are automatic and finitely presented. Finally we prove that a subsemigroup of B is residually finite if and only if it does not contain a copy of B.

Pattern classes of permutations via bijections between linearly ordered sets

Huczynska, Sophie — 2008-01-01T00:00:00Z

Abstract: A pattern class is a set of permutations closed under pattern involvement or, equivalently, defined by certain subsequence avoidance conditions. Any pattern class X which is atomic, i.e. indecomposable as a union of proper subclasses, has a representation as the set of subpermutations of a bijection between two countable (or finite) linearly ordered sets A and B. Concentrating on the situation where A is arbitrary and B = N, we demonstrate how the order-theoretic properties of A determine the structure of X and we establish results about independence, contiguousness and subrepresentations for classes admitting multiple representations of this form.

Cancellative and Malcev presentations for finite Rees index subsemigroups and extensions

Cain, Alan James — 2008-02-01T00:00:00Z

Abstract: It is known that, for semigroups, the property of admitting a finite presentation is preserved on passing to subsemigroups and extensions of finite Rees index. The present paper shows that the same holds true for Malcev, cancellative, left-cancellative and right-cancellative presentations. (A Malcev (respectively, cancellative, left-cancellative, right-cancellative) presentation is a presentation of a special type that can be used to define any group-embeddable (respectively, cancellative, left-cancellative, right-cancellative) semigroup.).

Growth rates for subclasses of Av(321)

Albert, M.H. — 2010-10-22T00:00:00Z

Abstract: Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.

On generators and presentations of semidirect products in inverse semigroups

Dombi, E. R. — 2009-06-01T00:00:00Z

Abstract: In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

Maximal subgroups of free idempotent-generated semigroups over the full transformation monoid

Gray, R — 2012-01-01T00:00:00Z

Abstract: Let Tn be the full transformation semigroup of all mappings from the set {1, . . . , n} to itself under composition. Let E = E(Tn) denote the set of idempotents of Tn and let e ∈ E be an arbitrary idempotent satisfying |im (e)| = r ≤ n − 2. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group Sr.

Generators and relations for subsemigroups via boundaries in Cayley graphs

Gray, R — 2011-11-01T00:00:00Z

Abstract: Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.

On the growth of generating sets for direct powers of semigroups

Hyde, James Thomas — 2012-01-01T00:00:00Z

Abstract: For a semigroup S its d-sequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.

On maximal subgroups of free idempotent generated semigroups

Gray, R — 2012-01-01T00:00:00Z

Abstract: We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister–Schreier type rewriting.

A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA)

Illian, Janine Baerbel — 2012-12-01T00:00:00Z

Abstract: This paper develops methodology that provides a toolbox for routinely fitting complex models to realistic spatial point pattern data. We consider models that are based on log-Gaussian Cox processes and include local interaction in these by considering constructed covariates. This enables us to use integrated nested Laplace approximation and to considerably speed up the inferential task. In addition, methods for model comparison and model assessment facilitate the modelling process. The performance of the approach is assessed in a simulation study. To demonstrate the versatility of the approach, models are tted to two rather dierent examples, a large rainforest data set with covariates and a point pattern with multiple marks.

Directed graph iterated function systems

Boore, Graeme C. — 2011-11-30T00:00:00Z

Abstract: This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μ[subscript(u)], the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μ[subscript(u)] to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μ[subscript(u)], of the set consisting of all the intersections of the components of F[subscript(u)], is strictly less than the multifractal q Hausdorff dimension with respect to μ[subscript(u)] of F[subscript(u)].

The investigation of quasi-separatrix layers in solar magnetic fields

Restante, Anna Lisa — 2011-06-20T00:00:00Z

Abstract: The structure of the magnetic field is often an important factor in many energetic processes in the solar corona. To determine the topology of the magnetic field features such as null points, separatrix surfaces, and separators must be found. It has been found that these features may be preferred sites for the formation of current sheets associated with the accumulation of free magnetic energy. Over the last decade, it also became clear that the geometrical analogs of the separatrices, the so-called quasi separatrix layers, have similar properties. This thesis has the aim of investigating these properties and to find correlations between these quantities. Our goal is to determine the relation between the geometrical features associated with the QSLs and with current structures, sites of reconnection and topological features. With these aims we conduct three different studies. First, we investigate a non linear force free magnetic field extrapolation from observed magnetogram data taken during a solar flare eruption concentrating our attention on two snapshots, one before the event and one after. We determine the QSLs and related structures and by considering carefully how these change between the two snapshots we are able to propose a possible scenario for how the flare occurred. In our second project we consider potential source distributions. We take different potential point source models: two four sources models already presented in the literature and a random distribution of fifteen sources. From these potential models we conduct a detailed analysis of the relationship between topological features and QSLs. It is found that the maxima of the Q-factor in the photosphere are located near and above the position of the subphotospheric null points (extending part way along their spines) and that their narrow QSLs are associated with the curves defined by the photospheric endpoints of all fan field lines that start from subphotospheric sources. Our last study investigates two different flux rope emergence simulations. In particular, we take one case with and one without an overlying magnetic field. Here, we can identify the QSLs, current, and sites of reconnection and determine the relation between them. From this work we found that not all high-Q regions are associated with current and/or reconnection and vice-versa. We also investigated the geometry of the field lines associated with high-Q regions to determine which geometrical behaviour of the magnetic field they are associated with. Those that are associated with reconnection also coincide with topological features such as separators.

The period ratio P₁/2P₂ in coronal waves

Macnamara, Cicely K. — 2011-11-30T00:00:00Z

Abstract: Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P₁/2P₂, between the period P₁ of the fundamental mode and the period P₂ of its first overtone is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. In this thesis we consider the period ratio P₁/2P₂ of coronal loops from a theoretical standpoint. Previous theory and observations indicate that the period ratio is likely to be less than unity for oscillations of coronal loops. We consider the role of damping and density structuring on the period ratio. In Chapter 2 we consider analytically the one-dimensional wave equation with the inclusion of a generic damping term for both uniform and non-uniform media. Results suggest that the period ratio is dominated by longitudinal structuring rather than damping. In Chapter 3 we consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. We find that damping by either thermal conduction or compressive viscosity typically has a small effect on the period ratio. For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. In Chapter 4 we extend the analysis of Chapter 3 to include radiative cooling and find that it too has a negligible effect on the period ratio for typical coronal values. As an extension to the investigation, damping rates are considered for thermal conduction, compressive viscosity and radiative cooling. The damping time is found to be optimal for each mechanism in a different temperature range, namely below 1 MK for radiative cooling, 2 − 6 MK for thermal conduction and above 6 MK for compressive viscosity. In Chapter 5 we consider analytically the period ratio for the fast kink, sausage and n = N modes of a magnetic slab, discussing both an Epstein density profile and a simple step function profile. We find that transverse density structuring in the form of an Epstein profile or a step function profile may contribute to the shift of the period ratio for long thin slab-like structures. The similarity in the behaviour of the period ratio for both profiles means either can be used as a robust model. We consider also other profiles numerically for the kink mode, which are found to be either slab-like or Epstein-like suggesting again that it is not necessary to distinguish the nature of the density profile when considering the period ratio.

The steady-state form of large-amplitude internal solitary waves

King, Stuart Edward — 2011-01-10T00:00:00Z

Abstract: A new numerical scheme for obtaining the steady-state form of an internal solitary wave of large amplitude is presented. A stratified inviscid two-dimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver, where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear time-dependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.

Small-scale magnetic feature evolution as observed by Hinode/NFI and SOHO/MDI

Thornton, L. M. — 2011-11-30T00:00:00Z

Abstract: The surface (photosphere) of the Sun is threaded throughout by magnetic fields. Groups of magnetic fields form magnetic features (of a wide range of sizes in flux and area) on the surface where the fields are directed into or out of the Sun. The aim of this thesis is to examine in detail the four key processes, emergence, cancellation, fragmentation and coalescence, which determine the behaviour of small-scale magnetic features, in the Sun’s photosphere. I identify features in both Hinode/NFI and SOHO/MDI full-disk to enable these processes to be examined at the currently smallest observable scales and over an entire solar cycle. The emerging event frequency versus flux distribution, for intranetwork emerging regions to active regions, is found to follow a power-law distribution with index -2.50, which spans nearly 7 orders of magnitude in flux (10¹⁶ - 10²³ Mx) and 18 orders of magnitude in frequency. The global rate of flux emergence is found to be 3.9 x 10²⁴ Mx day⁻¹. Since the slope of all emerged fluxes is less than -2 this implies that most of the new flux that is fed into the solar atmosphere is from small-scale emerging events. This single power-law distribution over all emerged fluxes suggest a scale-free dynamo, therefore indicating that in addition to dynamo actions in the tachocline producing sunspots, a turbulent dynamo may act throughout the convection zone. Similarly for cancellations I find a power-law relationship between the frequency of cancellation and the peak flux lost per cancelling event (for events detected in both Hinode/NFI and SOHO/MDI full-disk), with slope -2.10. Again, the process of cancellation appears to be scale free and the slope is less than -2 indicating that numerous small-scale features are cancelling the majority of flux on the Sun. I also estimate the frequency of all surface processes at solar maximum and find, 1.3 x 10⁸, 4.5 x 10⁷, 4.0 x 10⁷ and 3.6 x 10⁶ events per day over the whole surface for emergence, cancellation, fragmentation and coalescence events, respectively. All the surface processes are found to behave in a similar manner over all flux scales. The majority of events for all processes occur in features with flux below 10²º Mx, which highlights the dynamic nature of the magnetic carpet. Using SOHO/MDI full-disk data I investigate the cyclic variation of the 4 key processes throughout cycle 23. It is found that the rate of emerging events, cancellations, fragmentations and coalescences varied in anti-phase with the solar cycle by factors of 3.4, 3.1, 2.4 and 2.2, respectively over the cycle. Not surprisingly, therefore, the number of network features detected throughout the cycle also exhibits an anti-phase variation over the solar cycle by a factor of 1.9. The mean peak flux of tracked small-scale network, fragmenting, coalescing and cancelling features showed in-phase relationships with the solar cycle by factors of 1.4, 1.7, 2.4 and 1.2, respectively. The total flux which is emerged and cancelled by small-scale events, varied in anti-phase with the solar cycle, by factors of 1.9 and 3.2. This is clearly due to the variation in the number of emerging and cancelling events and the fact that the flux of individual emerging events showed no cyclic variation. The results in this thesis show that the large-scale solar cycle plays a complex role in the surface processes features undergo. The fact that the number of ephemeral regions emerging has an anti-phase variation to the solar cycle has a knock-on effect in the number of features which are available to undergo surface processes. Also decaying active regions, during more active periods, contribute more small-scale features, with high flux density, into the network which has an effect on the surface processes. This work has revealed the significant importance of small-scale features in the flux budget through continual emergence and cancellation, plus highlighted how through dynamic surface motions, small-scale features form the fundamental components with which the network is developed.

Current sheets in the solar corona : formation, fragmentation and heating

Bowness, Ruth — 2011-11-30T00:00:00Z

Abstract: In this thesis we investigate current sheets in the solar corona. The well known 1D model for the tearing mode instability is presented, before progressing to 2D where we introduce a non-uniform resistivity. The effect this has on growth rates is investigated and we find that the inclusion of the non-uniform term in η cause a decrease in the growth rate of the dominant mode. Analytical approximations and numerical simulations are then used to model current sheet formation by considering two distinct experiments. First, a magnetic field is sheared in two directions, perpendicular to each other. A twisted current layer is formed and we find that as we increase grid resolution, the maximum current increases, the width of the current layer decreases and the total current in the layer is approximately constant. This, together with the residual Lorentz force calculated, suggests that a current sheet is trying to form. The current layer then starts to fragment. By considering the parallel electric field and calculating the perpendicular vorticity, we find evidence of reconnection. The resulting temperatures easily reach the required coronal values. The second set of simulations carried out model an initially straight magnetic field which is stressed by elliptical boundary motions. A highly twisted current layer is formed and analysis of the energetics, current structures, magnetic field and the resulting temperatures is carried out. Results are similar in nature to that of the shearing experiment.

Magnetic flux transport simulations : applications to solar and stellar magnetic fields

Cook, Graeme Robert — 2011-11-01T00:00:00Z

Abstract: Magnetic fields play a key role in a wide variety of phenomena found on the Sun. One such phenomena is the Coronal Mass Ejection (CME) where a large amount of material is ejected from the Sun. CME’s may directly affect the earth, therefore understanding their origin is of key importance for space weather and the near-Earth environment. In this thesis, the nature and evolution of solar magnetic fields is considered through a combination of Magnetic Flux Transport Simulations and Potential Field Source Surface Models. The Magnetic Flux Transport Simulations produce a realistic description of the evolution and distribution of the radial magnetic field at the level of the solar photosphere. This is then applied as a lower boundary condition for the Potential Field Source Surface Models which prescribe a coronal magnetic field. Using these two techniques, the location and variation of coronal null points, a key element in the Magnetic Breakout Model of CMEs, are determined. Results show that the number of coronal null points follow a cyclic variation in phase with the solar cycle. In addition, they preferentially form at lower latitudes as a result of the complex active latitude field. Although a significant number of coronal nulls may exist at any one time (≈ 17), it is shown that only half may satisfy the necessary condition for breakout. From this it is concluded that while the Magnetic Breakout Model of CMEs is an important model in understanding the origin of the CMEs, other processes must occur in order to explain the observed number of CMEs. Finally, the Magnetic Flux Transport Simulations are applied to stellar magnetic fields and in particular to the fast rotating star HD171488. From this speculative study it is shown that the Magnetic Flux Transport Simulations constructed for the Sun may be applied in very different stellar circumstances and that for HD171488 a significantly higher rate of meridional flow (1200-1400 ms⁻¹) is required to match observed magnetic field distributions.

Estimating abundance of rare, small mammals: A case study of the Key Largo woodrat (Neotoma floridana smalli)

Potts, Joanne M. — 2011-01-01T00:00:00Z

Abstract: Estimates of animal abundance or density are fundamental quantities in ecology and conservation, but for many species such as rare, small mammals, obtaining robust estimates is problematic. In this thesis, I combine elements of two standard abundance estimation methods, capture-recapture and distance sampling, to develop a method called trapping point transects (TPT). In TPT, a "detection function", g(r) (i.e. the probability of capturing an animal, given it is r m from a trap when the trap is set) is estimated using a subset of animals whose locations are known prior to traps being set. Generalised linear models are used to estimate the detection function, and the model can be extended to include random effects to allow for heterogeneity in capture probabilities. Standard point transect methods are modified to estimate abundance. Two abundance estimators are available. The first estimator is based on the reciprocal of the expected probability of detecting an animal, ^P, where the expectation is over r; whereas the second estimator is the expectation of the reciprocal of ^P. Performance of the TPT method under various sampling efforts and underlying true detection probabilities of individuals in the population was investigated in a simulation study. When underlying probability of detection was high (g(0) = 0:88) and between-individual variation was small, survey effort could be surprisingly low (c. 510 trap nights) to yield low bias (c. 4%) in the two estimators; but under certain situations, the second estimator can be extremely biased. Uncertainty and relative bias in population estimates increased with decreasing detectability and increasing between-individual variation. Abundance of the Key Largo woodrat (Neotoma floridana smalli), an endangered rodent with a restricted geographic range, was estimated using TPT. The TPT method compared well to other viable methods (capture-recapture and spatially-explicit capture-recapture), in terms of both field practicality and cost. The TPT method may generally be useful in estimating animal abundance in trapping studies and variants of the TPT method are presented.

Complex Region Spatial Smoother (CReSS)

Scott Hayward, Lindesay Alexandra Sarah — 2011-01-01T00:00:00Z

Abstract: Conventional smoothing over complicated coastal and island regions may result in errors across boundaries, due to the use of Euclidean distances to represent inter-point similarity. The new Complex Region Spatial Smoother (CReSS) method presented here, uses estimated geodesic distances, model averaging and a local radial basis function to provide improved smoothing over complex domains. CReSS is compared, via simulation, to recent related smoothing techniques, Thin Plate Splines (TPS, Harder and Desmarais, 1972), geodesic low rank TPS [Wang and Ranalli, 2007] and the Soap film smoother [Wood et al., 2008]. The GLTPS method cannot be used in areas with islands and SOAP can be hard to parameterize. CReSS is comparable with, if not better than, all considered methods on a range of simulations. Supplementary materials for this article are available online.

The primitive permutation groups of degree less than 4096

Coutts, Hannah Jane — 2011-10-14T00:00:00Z

Abstract: In this paper we use the Classification of the Finite Simple Groups, the O’Nan– Scott Theorem and Aschbacher’s theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and Magma. Description: The first author is supported by an EPSRC doctoral training grant. The second and third authors acknowledge the support of EPSRC grant number EP/C523229/1.

Groups with the basis property

McDougall-Bagnall, Jonathan M. — 2011-11-15T00:00:00Z

Abstract: We study finite groups for which every minimal generating set has the same cardinality. A group has the basis property if it and every subgroup satisfies this condition on minimal generating sets. We classify all finite groups with the basis property.

Comparing pre- and post-construction distributions of long-tailed ducks Clangula hyemalis in and around the Nysted offshore wind farm, Denmark : a quasi-designed experiment accounting for imperfect detection, local surface features and autocorrelation

Petersen, Ib Krag — 2011-01-01T00:00:00Z

Abstract: We report a novel technique to model abundance patterns of wintering seaducks in relation to the construction of an offshore wind farm (OWF) based on seven years of aerial survey transect data. Distance sampling was used to estimate seaduck densities adjusted for covariates affecting detection probabilities. A generalized additive model (GAM) generated seaduck densities in sampling units in relation to spatially explicit covariates, using bootstrapping to account for uncertainties in both processes. Generalized estimating equations generated precision measures for the GAM robust to spatial and temporal autocorrelation. Comparison of pre- and post-construction model generated surfaces showed significant reductions in long-tailed duck numbers only within the OWF (despite the fact that the model was uninformed about the OWF location), although the absolute numbers involved were trivial in a flyway population context. This method provides quantification of distributional effects on organisms over a gradient in space and time that offers an alternative to Before-After/Control-Impact designs in environmental impact assessment.

Finite groups are big as semigroups

Dolinka, Igor — 2011-09-01T00:00:00Z

Abstract: We prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if |G| ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.

On convex permutations

Albert, M.H. — 2011-05-01T00:00:00Z

Abstract: A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.

Presentations of inverse semigroups, their kernels and extensions

Carvalho, C.A. — 2011-06-01T00:00:00Z

Abstract: Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.

Simple extensions of combinatorial structures

Brignall, R — 2011-07-01T00:00:00Z

Abstract: An interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.

A first survey of the global population size and distribution of the Scottish Crossbill Loxia scotica

Summers, Ron W — 2011-06-01T00:00:00Z

Abstract: A survey of Scottish Crossbills Loxia scotica was carried out in 3,506 km2 of conifer woodland in northern Scotland during January to April 2008 to provide the first estimate of the global population size for this endemic bird. Population estimates were also made for Common Crossbills L. curvirostra and Parrot Crossbills L. pytyopsittacus within this range. Crossbills were lured to systematically selected survey points for counting, sexing and recording their calls for later call-type (species) identification from sonograms. Crossbills were located at 451 of the 852 survey points, and adequate tape-recordings made at 387 of these. The Scottish Crossbill had a disjunct distribution, occurring largely within the eastern part of the study area, but also in the northwest. Common Crossbills had a mainly westerly distribution. The population size of postjuvenile Scottish Crossbills was estimated as 13,600 (95%C.I. 8,130–22,700), which will approximate to 6,800 (4,065–11,350) pairs. Common Crossbills were more abundant within this range (27,100, 95% C.I. 14,700–38,400) and Parrot Crossbills rare (about 100). The sex ratio was not significantly different from parity for Scottish Crossbills. The modal number at survey points was two but numbers were larger in January than later in the survey. The numbers and distribution of all crossbill species are likely to vary between years, depending upon the size of the cone crops of the different conifers: all were coning in 2008. Common Crossbill and Parrot Crossbill numbers will also be affected by irruptions from continental Europe. A monitoring scheme is required to detect any population trend, and further work on their habitat requirement (e.g. conifer selection at different seasons) is needed to inform habitat management of native and planted conifer forests to ensure a secure future for this endemic bird.

The horizon problem for prevalent surfaces

Falconer, Kenneth John — 2011-01-01T00:00:00Z

Abstract: We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2 $. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most $\alpha$, for $\alpha \in [2,3)$. In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

Estimating bird abundance : making methods work

Buckland, Stephen T. — 2008-09-01T00:00:00Z

Abstract: In many bird monitoring Surveys, no attempt is made to estimate bird densities or abundance. instead, counts of one form or another are made, and these are assumed to correlate with bird density. Unless complete Counts Oil Sample plots are feasible, this approach can easily lead to false conclusions, because detectability of birds varies by species, habitat, observer and many other factors. Trends in time of counts often reflect trends in detectability, rather than trends in abundance. Conclusions are further compromised when surveys are conducted at unrepresentative sites. We consider how to avoid these problems. We give a brief description of distance sampling methods, which allow detectability to be estimated. We consider strategies to ease their implementation, to enhance their reliability, to adapt the methods for difficult species, and to deal with circumstances in which representative sampling is problematic. We also consider some of the common problems encountered, and suggest solutions.

Double-observer line transect methods : levels of independence

Buckland, Stephen T. — 2010-03-01T00:00:00Z

Abstract: Double-observer line transect methods are becoming increasingly widespread, especially for the estimation of marine mammal abundance from aerial and shipboard surveys when detection of animals on the line is uncertain. The resulting data supplement conventional distance sampling data with two-sample mark–recapture data. Like conventional mark–recapture data, these have inherent problems for estimating abundance in the presence of heterogeneity. Unlike conventional mark–recapture methods, line transect methods use knowledge of the distribution of a covariate, which affects detection probability (namely, distance from the transect line) in inference. This knowledge can be used to diagnose unmodeled heterogeneity in the mark–recapture component of the data. By modeling the covariance in detection probabilities with distance, we show how the estimation problem can be formulated in terms of different levels of independence. At one extreme, full independence is assumed, as in the Petersen estimator (which does not use distance data); at the other extreme, independence only occurs in the limit as detection probability tends to one. Between the two extremes, there is a range of models, including those currently in common use, which have intermediate levels of independence. We show how this framework can be used to provide more reliable analysis of double-observer line transect data. We test the methods by simulation, and by analysis of a dataset for which true abundance is known. We illustrate the approach through analysis of minke whale sightings data from the North Sea and adjacent waters.

Design and analysis of line transect surveys for primates

Buckland, Stephen Terrence — 2010-10-01T00:00:00Z

Abstract: Line transect surveys are widely used for estimating abundance of primate populations. The method relies on a small number of key assumptions, and if these are not met, substantial bias may occur. For a variety of reasons, primate surveys often do not follow what is generally considered to be best practice, either in survey design or in analysis. The design often comprises too few lines (sometimes just one), subjectively placed or placed along trails, so lacks both randomization and adequate replication. Analysis often involves flawed or inefficient models, and often uses biased estimates of the locations of primate groups relative to the line. We outline the standard method, emphasizing the assumptions underlying the approach. We then consider options for when it is difficult or impossible to meet key assumptions. We explore the performance of these options by simulation, focusing particularly on the analysis of primate group sizes, where many of the variations in survey methods have been developed. We also discuss design issues, field methods, analysis, and potential alternative methodologies for when standard line transect sampling cannot deliver reliable abundance estimates. Description: An erratum to this article can be found at http://dx.doi.org/10.1007/s10764-010-9470-y

Line transect sampling of primates : can animal-to-observer distance methods work?

Buckland, Stephen Terrence — 2010-06-01T00:00:00Z

Abstract: Line transect sampling is widely used for estimating abundance of primate populations. Animal-to-observer distances (AODs) are commonly used in analysis, in preference to perpendicular distances from the line. This is in marked contrast with standard practice for other applications of line transect sampling. We formalize the mathematical shortcomings of approaches based on AODs, and show that they are likely to give strongly biased estimates of density. We review papers that claim good performance for the method, and explore this performance through simulations. These confirm strong bias in estimates of density using AODs. We conclude that AOD methods are conceptually flawed, and that they cannot in general provide valid estimates of density. Description: An erratum to this article can be found at http://dx.doi.org/10.1007/s10764-010-9469-4

An investigation into the use of balance in operational numerical weather prediction

Devlin, David J.J. — 2011-01-01T00:00:00Z

Abstract: Presented in this study is a wide-ranging investigation into the use of properties of balance in an operational numerical weather prediction context. Initially, a joint numerical and observational study is undertaken. We used the Unified Model (UM), the suite of atmospheric and oceanic prediction software used at the UK Met Office (UKMO), to locate symmetric instabilities (SIs), an indicator of imbalanced motion. These are areas of negative Ertel potential vorticity (in the Northern hemisphere) calculated on surfaces of constant potential temperature. Once located, the SIs were compared with satellite and aircraft observational data. As a full three-dimensional calculation of Ertel PV proved outwith the scope of this study we calculated the two-dimensional, vertical component of the absolute vorticity, to assess the inertial stability criterion. We found that at the synoptic scale in the atmosphere, if there existed a symmetric instability, it was dominated by an inertial instability. With the appropriate observational data, evidence of inertial instability from the vertical component of the absolute vorticity, predicted by the UM was found at 12km horizontal grid resolution. Varying the horizontal grid resolution allowed the estimation of a grid length scale, above which, the inertial instability was not captured by the observational data, of approximately 20km. Independently, aircraft data was used to estimate that horizontal grid resolutions above 20-25km should not model any features of imbalance providing a real world estimate of the lower bound of the grid resolution that should be employed by a balanced atmospheric prediction model. A further investigation of the UM concluded that the data assimilation scheme and time of initialisation had no effect on the generation of SIs. An investigation was then made into the robustness of balanced models in the shallow water context, employing the contour-advective semi-Lagrangian (CASL) algorithm, Dritschel & Ambaum (1997), a novel numerical algorithm that exploits the underlying balance observed within a geophysical flow at leading order. Initially two algorithms were considered, which differed by the prognostic variables employed. Each algorithm had their three-time-level semi-implicit time integration scheme de-centred to mirror the time integration scheme of the UM. We found that the version with potential vorticity (PV), divergence and acceleration divergence, CA[subscript(δ,γ)], as prognostic variables preserved the Bolin-Charney balance to a much greater degree than the model with PV, divergence and depth anomaly CA[subscript(tilde{h},δ)], as prognostic variables. This demonstrated that CA[subscript(δ,γ)] was better equipped to benefit from de-centring, an essential property of any operational numerical weather prediction (NWP) model. We then investigate the robustness of CA[subscript(δ,γ)] by simulating flows with Rossby and Froude number O(1), to find the operational limits of the algorithm. We also investigated increasing the efficiency of CA[subscript(δ,γ)] by increasing the time-step Δt employed while decreasing specific convergence criteria of the algorithm while preserving accuracy. We find that significant efficiency gains are possible for predominantly mid-latitude flows, a necessary step for the use of CA[subscript(δ,γ)] in an operational NWP context. The study is concluded by employing CASL in the non-hydrostatic context under the Boussinesq approximation, which allows weak stratification to be considered, a step closer to physical reality than the shallow water case. CASL is compared to the primitive equation pseudospectral (PEPS) and vorticity-based pseudospectral (VPS) algorithms, both as the names suggest, spectral-based algorithms, which again differ by the prognostic variables employed. This comparison is drawn to highlight the computational advantages that CASL has over common numerical methods used in many operational forecast centres. We find that CASL requires significantly less artificial numerical diffusion than its pseudospectral counterparts in simulations of Rossby number ~O(1). Consequently, CASL obtains a much less diffuse, more accurate solution, at a lower resolution and therefore lower computational cost. At low Rossby number, where the flow is strongly influence by the Earth's rotation, it is found that CASL is the most cost-effective method. In addition, CASL also preserves a much greater proportion of balance, diagnosed with nonlinear quasigeostrophic balance (NQG), another significant advantage over its pseudospectral counterparts.

MHD evolution of magnetic null points to static equilibria

Fuentes Fernández, Jorge — 2011-05-18T00:00:00Z

Abstract: In magnetised plasmas, magnetic reconnection is the process of magnetic field merging and recombination through which considerable amounts of magnetic energy may be converted into other forms of energy. Reconnection is a key mechanism for solar flares and coronal mass ejections in the solar atmosphere, it is believed to be an important source of heating of the solar corona, and it plays a major role in the acceleration of particles in the Earth's magnetotail. For reconnection to occur, the magnetic field must, in localised regions, be able to diffuse through the plasma. Ideal locations for diffusion to occur are electric current layers formed from rapidly changing magnetic fields in short space scales. In this thesis we consider the formation and nature of these current layers in magnetised plasmas. The study of current sheets and current layers in two, and more recently, three dimensions, has been a key field of research in the last decades. However, many of these studies do not take plasma pressure effects into consideration, and rather they consider models of current sheets where the magnetic forces sum to zero. More recently, others have started to consider models in which the plasma beta is non-zero, but they simply focus on the actual equilibrium state involving a current layer and do not consider how such an equilibrium may be achieved physically. In particular, they do not allow energy conversion between magnetic and internal energy of the plasma on their way to approaching the final equilibrium. In this thesis, we aim to describe the formation of equilibrium states involving current layers at both two and three dimensional magnetic null points, which are specific locations where the magnetic field vanishes. The different equilibria are obtained through the non-resistive dynamical evolution of perturbed hydromagnetic systems. The dynamic evolution relaxes via viscous damping, resulting in viscous heating. We have run a series of numerical experiments using LARE, a Lagrangian-remap code, that solves the full magnetohydrodynamic (MHD) equations with user controlled viscosity and resistivity. To allow strong current accumulations to be created in a static equilibrium, we set the resistivity to be zero and hence simply reach our equilibria by solving the ideal MHD equations. We first consider the relaxation of simple homogeneous straight magnetic fields embedded in a plasma, and determine the role of the coupling between magnetic and plasma forces, both analytically and numerically. Then, we study the formation of current accumulations at 2D magnetic X-points and at 3D magnetic nulls with spine-aligned and fan-aligned current. At both 2D X-points and 3D nulls with fan-aligned current, the current density becomes singular at the location of the null. It is impossible to be precisely achieve an exact singularity, and instead, we find a gradual continuous increase of the peak current over time, and small, highly localised forces acting to form the singularity. In the 2D case, we give a qualitative description of the field around the magnetic null using a singular function, which is found to vary within the different topological regions of the field. Also, the final equilibrium depends exponentially on the initial plasma pressure. In the 3D spine-aligned experiments, in contrast, the current density is mainly accumulated along and about the spine, but not at the null. In this case, we find that the plasma pressure does not play an important role in the final equilibrium. Our results show that current sheet formation (and presumably reconnection) around magnetic nulls is held back by non-zero plasma betas, although the value of the plasma pressure appears to be much less important for torsional reconnection. In future studies, we may consider a broader family of 3D nulls, comparing the results with the analytical calculations in 2D, and the relaxation of more complex scenarios such as 3D magnetic separators.

Estimating the Barents Sea polar bear subpopulation size

Aars, J — 2009-01-01T00:00:00Z

Abstract: A large scale survey was conducted in August 2004 to estimate the size of the Barents Sea polar bear subpopulation. We combined helicopter line transect distance sampling surveys in most of the survey area with total counts in small areas not suitable for distance sampling. Due to weather constraints we failed to survey some of the areas originally planned to be covered by distance sampling. For those, abundance was estimated using a ratio estimator, in which the auxiliary variable was the number of satellite telemetry fixes (in previous years). We estimated that the Barents Sea subpopulation had approximately 2650 (95% CI approx 1900 to 3600) bears. Given current intense interest in polar bear management due to the potentially disastrous effects of climate change, it is surprising that many subpopulation sizes are still unknown. We show here that line transect sampling is a promising method for addressing the need for abundance estimates.

The effect of sea-ice loss on beluga whales (Delphinapterus leucas) in West Greenland

Heide-Jørgensen, M. P. — 2010-01-01T00:00:00Z

Abstract: An aerial survey was conducted to estimate the abundance of belugas (Delphinapterus leucas) on their wintering ground in West Greenland in March–April 2006 and 2008. The survey was conducted as a double platform aerial line transect survey, and sampled approximately 17% of the total survey area of ca. 125 000 km2. The abundance of belugas was 10 595 (95% confidence interval 4904–24 650). The largest abundance was found at the northern part of Store Hellefiske Bank, at the eastern edge of the Baffin Bay pack ice, a pattern similar to that found in eight systematic surveys conducted since 1981. A clear relationship between decreasing sea-ice cover and increasing offshore distance of beluga sightings was established from all previous surveys, suggesting that belugas expand their distribution westward as new areas on the banks of West Greenland open up earlier in spring with reduced sea-ice coverage or early annual ice recession. This is in contrast to the relatively confined distribution of belugas near the coast in limited open areas in the early 1980s, when sea-ice cover was greater. However, the effects of the changes in coastal availability of belugas can also be observed with the correlation between catches from the local Inuit hunt and sea-ice cover, where the catches increased significantly with increasing sea-ice coverage during the period 1954–2006. These results, based on nearly 30 years of dedicated survey effort, are among the first available evidence showing a shift in distribution of an Arctic cetacean in response to changes in sea-ice coverage.

A commutative noncommutative fractal geometry

Samuel, Anthony — 2010-01-01T00:00:00Z

Abstract: In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

Numerical modelling of two HMX-based plastic-bonded explosives at the mesoscale

Handley, Caroline A. — 2011-06-22T00:00:00Z

Abstract: Mesoscale models are needed to predict the effect of changes to the microstructure of plastic-bonded explosives on their shock initiation and detonation behaviour. This thesis describes the considerable progress that has been made towards a mesoscale model for two HMX-based explosives PBX9501 and EDC37. In common with previous work in the literature, the model is implemented in hydrocodes that have been designed for shock physics and detonation modelling. Two relevant physics effects, heat conduction and Arrhenius chemistry, are added to a one-dimensional Lagrangian hydrocode and correction factors are identified to improve total energy conservation. Material models are constructed for the HMX crystals and polymer binders in the explosives, and are validated by comparison to Hugoniot data, Pop-plot data and detonation wave profiles. One and two-dimensional simulations of PBX9501 and EDC37 microstructures are used to investigate the response of the bulk explosive to shock loading. The sensitivity of calculated temperature distributions to uncertainties in the material properties data is determined, and a thermodynamic explanation is given for time-independent features in temperature profiles. Hotspots are widely accepted as being responsible for shock initiation in plastic-bonded explosives. It is demonstrated that, although shock heating of crystals and binder is responsible for temperature localisation, it is not a feasible hotspot mechanism in PBX9501 and EDC37 because the temperatures generated are too low to cause significant chemical reaction in the required timescales. Critical hotspot criteria derived for HMX and the binders compare favourably to earlier studies. The speed of reaction propagation from hotspots into the surrounding explosive is validated by comparison to flame propagation data, and the temperature of the gaseous reaction products is identified as being responsible for negative pressure dependence. Hotspot size, separation and temperature requirements are identified which can be used to eliminate candidate mechanisms in future.

The lesser names : the teachers of the Edinburgh Mathematical Society and other aspects of Scottish mathematics, 1867–1946

Hartveit, Marit — 2011-06-22T00:00:00Z

Abstract: The Edinburgh Mathematical Society started out in 1883 as a society with a large proportion of teachers. Today, the member base is mainly academical and there are only a few teachers left. This thesis explores how and when this change came about, and discusses what this meant for the Society. It argues that the exit of the teachers is related to the rising standard of mathematics, but even more to a change in the Society’s printing policy in the 1920s, that turned the Society’s Proceedings into a pure research publication and led to the death of the ‘teacher journal’, the Mathematical Notes. The thesis also argues that this change, drastic as it may seem, does not represent a change in the Society’s nature. For this aim, the role of the teachers within the Society has been studied and compared to that of the academics, from 1883 to 1946. The mathematical contribution of the teachers to the Proceedings is studied in some detail, in particular the papers by John Watt Butters. A paper in the Mathematical Notes by A. C. Aitken on the Bell numbers is considered in connection with a series of letters on the same topic from 1938–39. These letters, written by Aitken, Sir D’Arcy Thompson, another EMS member, and the Cambridge mathematician G. T. Bennett, explores the relation between the three and gives valuable insight into the status of the Notes. Finally, the role of the first women in the Society is studied. The first woman joined without any official university education, but had received the necessary mathematical background from her studies under the Edinburgh Association for the University Education of Women. The final chapter is largely an assessment of this Association’s mathematical classes.

Theoretical magnetic flux emergence

MacTaggart, David — 2011-01-01T00:00:00Z

Abstract: Magnetic flux emergence is the subject of how magnetic fields from the solar interior can rise and expand into the atmosphere to produce active regions. It is the link that joins dynamics in the convection zone with dynamics in the atmosphere. In this thesis, we study many aspects of magnetic flux emergence through mathematical modelling and computer simulations. Our primary aim is to understand the key physical processes that lie behind emergence. The first chapter introduces flux emergence and the theoretical framework, magnetohydrodynamics (MHD), that describes it. In the second chapter, we discuss the numerical techniques used to solve the highly non-linear problems that arise from flux emergence. The third chapter summarizes the current literature. In the fourth chapter, we consider how changing the geometry and parameter values of the initial magnetic field can affect the dynamic evolution of the emerging magnetic field. For an initial toroidal magnetic field, it is found that its axis can emerge to the corona if the tube’s initial field strength is large enough. The fifth chapter describes how flux emergence models can produce large-scale solar eruptions. A 2.5D model of the breakout model, using only dynamic flux emergence, fails to produce any large scale eruptions. A 3D model of toroidal emergence with an overlying magnetic field does, however, produce multiple large-scale eruptions and the form of these is related to the breakout model. The sixth chapter is concerned with signatures of flux emergence and how to identify emerging twisted magnetic structures correctly. Here, a flux emergence model produces signatures found in observations. The signatures from the model, however, have different underlying physical mechanisms to the original interpretations of the observations. The thesis concludes with some final thoughts on current trends in theoretical magnetic flux emergence and possible future directions.

Application of stochastic differential equations and real option theory in investment decision problems

Chavanasporn, Walailuck — 2010-11-30T00:00:00Z

Abstract: This thesis contains a discussion of four problems arising from the application of stochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly with the author’s supervisor. In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process. The model is an adaptation to a continuous-time framework of a discrete evolutionary stock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). We consider the case of fix-mix strategies and derive the stochastic differential equations which determine the evolution of the wealth processes of the various market players. The wealth dynamics for various initial set-ups of the market are simulated. In the second problem, we apply an entry-exit model in real option theory to study concessionary agreements between a private company and a state government to run a privatised business or project. The private company can choose the time to enter into the agreement and can also choose the time to exit the agreement if the project becomes unprofitable. An early termination of the agreement by the company might mean that it has to pay a penalty fee to the government. Optimal times for the company to enter and exit the agreement are calculated. The dynamics of the project are assumed to follow either a geometric mean reversion process or geometric Brownian motion. A comparative analysis is provided. Particular emphasis is given to the role of uncertainty and how uncertainty affects the average time that the concessionary agreement is active. The effect of uncertainty is studied by using Monte Carlo simulation. In the third problem, we study numerical methods for solving stochastic optimal control problems which are linear in the control. In particular, we investigate methods based on spline functions for solving the two-point boundary value problems that arise from the method of dynamic programming. In the general case, where only the value function and its first derivative are guaranteed to be continuous, piecewise quadratic polynomials are used in the solution. However, under certain conditions, the continuity of the second derivative is also guaranteed. In this case, piecewise cubic polynomials are used in the solution. We show how the computational time and memory requirements of the solution algorithm can be improved by effectively reducing the dimension of the problem. Numerical examples which demonstrate the effectiveness of our method are provided. Lastly, we study the situation where, by partial privatisation, a government gives a private company the opportunity to invest in a government-owned business. After payment of an initial instalment cost, the private company’s investments are assumed to be flexible within a range [0, k] while the investment in the business continues. We model the problem in a real option framework and use a geometric mean reversion process to describe the dynamics of the business. We use the method of dynamic programming to determine the optimal time for the private company to enter and pay the initial instalment cost as well as the optimal dynamic investment strategy that it follows afterwards. Since an analytic solution cannot be obtained for the dynamic programming equations, we use quadratic splines to obtain a numerical solution. Finally we determine the optimal degree of privatisation in our model from the perspective of the government.

Density estimation implications of increasing ambient noise on beaked whale click detection and classification

Marques, Tiago Andre Lamas Oliveira — 2010-01-01T00:00:00Z

Abstract: Acoustic based density estimates are being increasingly used. Usually density estimation methods require one to evaluate the eﬀective survey area of the acoustic sensors, or equivalently estimate the mean detection probability of detecting the animals or cues of interest. This is often done based on an estimated detection function, the probability of detecting an object of interest as a function of covariates, usually distance and additional covariates. If the actual survey data and the data used to estimate a detection function are not collected simultaneously, as in Marques et al. (2009), the estimated detection function might not correspond to the detection process that generated the survey data. This would lead to biaseddensity estimates. Here we evaluate the inﬂuence of ambient noise in the detection and classiﬁcation of beaked whale clicks at the Atlantic Undersea Test and Evaluation Center (AUTEC) hydrophones, to assess if the density estimates reported in Marques et al. (2009) might have been biased. To do so we contaminated a data set with increasing levels of ambient noise, and then estimated the detection function accounting for the noise level as an additional covariate. The results obtained suggest that for the particular results obtained at AUTEC’s deep water hydrophones the inﬂuence of ambient noise on the beaked whale’s click detection probability might have been minor, and hence unlikely to have had an impact on density estimates. However, we do not exclude the possibility that the results could be diﬀerent under other scenarios.

Bayesian modelling of integrated data and its application to seabird populations

Reynolds, Toby J. — 2010-11-30T00:00:00Z

Abstract: Integrated data analyses are becoming increasingly popular in studies of wild animal populations where two or more separate sources of data contain information about common parameters. Here we develop an integrated population model using abundance and demographic data from a study of common guillemots (Uria aalge) on the Isle of May, southeast Scotland. A state-space model for the count data is supplemented by three demographic time series (productivity and two mark-recapture-recovery (MRR)), enabling the estimation of prebreeder emigration rate - a parameter for which there is no direct observational data, and which is unidentifiable in the separate analysis of MRR data. A Bayesian approach using MCMC provides a flexible and powerful analysis framework. This model is extended to provide predictions of future population trajectories. Adopting random effects models for the survival and productivity parameters, we implement the MCMC algorithm to obtain a posterior sample of the underlying process means and variances (and population sizes) within the study period. Given this sample, we predict future demographic parameters, which in turn allows us to predict future population sizes and obtain the corresponding posterior distribution. Under the assumption that recent, unfavourable conditions persist in the future, we obtain a posterior probability of 70% that there is a population decline of >25% over a 10-year period. Lastly, using MRR data we test for spatial, temporal and age-related correlations in guillemot survival among three widely separated Scottish colonies that have varying overlap in nonbreeding distribution. We show that survival is highly correlated over time for colonies/age classes sharing wintering areas, and essentially uncorrelated for those with separate wintering areas. These results strongly suggest that one or more aspects of winter environment are responsible for spatiotemporal variation in survival of British guillemots, and provide insight into the factors driving multi-population dynamics of the species.

Generating continuous mappings with Lipschitz mappings

Cichon, J — 2007-05-01T00:00:00Z

Abstract: If X is a metric space, then C-X and L-X denote the semigroups of continuous and Lipschitz mappings, respectively, from X to itself. The relative rank of C-X modulo L-X is the least cardinality of any set U\L-X where U generates C-X. For a large class of separable metric spaces X we prove that the relative rank of C-X modulo L-X is uncountable. When X is the Baire space N-N, this rank is N-1. A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

Primitive free cubics with specified norm and trace

Huczynska, Sophie — 2003-08-01T00:00:00Z

Abstract: The existence of a primitive free (normal) cubic x(3) ax(2) + cx b over a finite field F with arbitrary specified values of a (not equal 0) and b (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

Impeded inverse energy transfer in the Charney--Hasegawa--Mima model of quasi-geostrophic flows

Tran, Chuong Van — 2006-03-25T00:00:00Z

Abstract: The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney-Hasegawa-Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber lambda (or free-surface elasticity). Herein, we derive a bound oil the inverse energy transfer, specifically on the growth rate dl/dt of the characteristic length scale e representing the energy centroid. It is found that dl/dt <= 2 parallel to q parallel to(infinity)/(l(s)lambda(2)), where parallel to q parallel to(infinity) is the supremum of the potential vorticity and l(s) represents the potential enstrophy centroid of the reservoir, both invariant. This result implies that in the potential-energy-dominated regime (l >= l(s) >> lambda(-1)) the inverse energy transfer is strongly impeded, in the sense that under the usual time scale no significant transfer of energy to larger scales occurs. The physical implication is that the elasticity of the free surface impedes turbulent energy transfer in wavenumber space, effectively rendering large-scale vortices long-lived and inactive. Results from numerical simulations of forced-dissipative turbulence confirm this prediction.

Vanishing enstrophy dissipation in two-dimensional Navier--Stokes turbulence in the inviscid limit

Tran, Chuong Van — 2006-07-25T00:00:00Z

Abstract: Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) developed a theory of two-dimensional turbulence based on the assumption that the dissipation of enstrophy (mean-square vorticity) tends to a finite non-zero constant in the limit of infinite Reynolds number Re. Here, by assuming power-law spectra, including the one predicted by Batchelor's theory, we prove that the maximum dissipation of enstrophy is in fact zero in this limit. Specifically, as Re -> infinity, the dissipation approaches zero no slower than (ln Re)(-1/2). The physical reason behind this result is that the decrease of viscosity enhances the production of both palinstrophy (mean-square vorticity gradients) and its dissipation - but in such a way that the net growth of palinstrophy is less rapid than the decrease of viscosity, resulting in vanishing enstrophy dissipation. This result generalizes to a rich class of quasi-geostrophic models as well as to the case of a passive tracer in layerwise-two-dimensional turbulent flows having bounded enstrophy.

Quasi-geostrophic vortices in compressible atmospheres

Scott, Richard Kirkness — 2005-05-10T00:00:00Z

Abstract: This paper considers the effect of an exponential variation in the background density field (as exists in compressible atmospheres) on the structure and dynamics of the quasi-geostrophic system, and compares the results with the corresponding Boussinesq limit in which background density variations are assumed small. The behaviour of the compressible system is understood via a closed-form analytic expression for the Green's function of the inversion operator relating potential vorticity and streamfunction. This expression makes explicit the anisotropy of the Green's function, inherited from the density profile, which has a slow, algebraic decay directly above the source and an exponential decay in all other directions. An immediate consequence for finite-volume vortices is a differential rotation of upper and lower levels that results in counterintuitive behaviour during the nonlinear evolution of ellipsoidal vortices, in which vortex destruction is confined to the lower vortex and wave activity is seen to propagate downwards. This is in contrast to the Boussinesq limit, which exhibits symmetric destruction of the upper and lower vortex, and in contrast to naive expectations based on a consideration of the mass distribution alone, which would lead to greater destruction of the upper vortex. Finally, the presence of a horizontal lower boundary introduces a strong barotropic component that is absent in the unbounded case (the presence of an upper boundary has almost no effect). The lower boundary also alters the differential rotation in the lower vortex with important consequences for the nonlinear evolution: for very small separation between the lower boundary and the vortex, the differential rotation is reversed leading to strong deformations of the middle vortex; for a critical separation, the vortex is stabilized by the reduction of the differential rotation, and remains coherent over remarkably long times.

Subsemigroups of virtually free groups : finite Malcev presentations and testing for freeness

Cain, AJ — 2006-07-01T00:00:00Z

Abstract: This paper shows that, given a finite subset X of a finitely generated virtually free group F, the freeness of the subsemigroup of F generated by X can be tested algorithmically. (A group is virtually free if it contains a free subgroup of finite index.) It is then shown that every finitely generated subsemigroup, of F has a finite Malcev presentation (a type of semigroup presentation which can be used to define any semigroup that embeds in a group), and that such a presentation can be effectively found from any finite generating set.

The critical merger distance between two co-rotating quasi-geostrophic vortices

Reinaud, Jean Noel — 2005-01-10T00:00:00Z

Abstract: This paper examines the critical merger or strong interaction distance between two equal-potential-vorticity quasi-geostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their height-to-width aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasi-geostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.

The shape of vortices in quasi-geostrophic turbulence

Reinaud, Jean Noel — 2003-01-10T00:00:00Z

Abstract: The present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasi-geostrophic approximation. In decaying turbulence, these vortices-coherent regions of the materially-invariant potential vorticity-dominate the flow evolution, and indeed the flow evolution is governed by their interactions. An analysis of several exceptionally high-resolution simulations of quasi-geostrophic turbulence is performed. The results indicate that the population of vortices exhibits a mean height-to-width aspect ratio less than unity, in fact close to 0.8. This finding is justified here by a simple model, in which vortices are taken to be ellipsoids of uniform potential vorticity. The model focuses on steady ellipsoids within a uniform background strain flow. This background flow approximates the effects of surrounding vortices in a turbulent flow on a given vortex. It is argued that the vortices which are able to withstand the highest levels of strain are those most likely to be found in the actual turbulent flow. Our calculations confirm that the optimal height-to-width aspect ratio is close to 0.8 for a wide range of background straining flows.

The quasi-geostrophic ellipsoidal vortex model

Dritschel, David Gerard — 2004-04-25T00:00:00Z

Abstract: We present a simple approximate model for studying general aspects of vortex interactions in a rotating stably-stratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak non-ellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.

The merger of vertically offset quasi-geostrophic vortices

Reinaud, Jean Noel — 2002-10-25T00:00:00Z

Abstract: We examine the critical merging distance between two equal-volume, equal-potential-vorticity quasi-geostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean height-to-width aspect ratio. The vertical direction is special in the quasi-geostrophic model (used to capture the leading-order dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection. In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasi-geostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.

Generating the full transformation semigroup using order preserving mappings

Higgins, PM — 2003-09-01T00:00:00Z

Abstract: For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O-X on X modulo the full transformation semigroup Ex. In other words, we ask what is the smallest cardinality of a set A of mappings such that = T-X. When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable.

The persistence of balance in geophysical flows

Dritschel, David Gerard — 2007-01-10T00:00:00Z

Abstract: Rotating stably stratified geophysical flows can exhibit a near 'balanced' evolution controlled by the conservative advection of a single scalar quantity, the potential vorticity (PV). This occurs frequently in the Earth's atmosphere and oceans where motions tend to be weak compared with the background planetary rotation and where stratification greatly inhibits vertical motion. Under these circumstances, both high-frequency acoustic waves and lower-frequency inertia-gravity waves (IGWs) contribute little to the flow evolution compared with the even-lower-frequency advection of PV. Moreover, this 'slow' PV-controlled balanced evolution appears unable to excite these higher-frequency waves in any significant way-i.e. balance persists. The present work pushes the limits of balance by systematically exploring the evolution of a range of highly nonlinear flows in which motions are comparable with the background rotation. These flows do not possess a frequency separation between PV advection and IGWs. Nonetheless, the flows exhibit a remarkable persistence of balance. Even when flows are not initialized to minimize the amount of IGWs initially present, and indeed even when flows are deliberately seeded with significant IGW amplitudes, the flow evolution-over many inertial periods (days)-remains strongly controlled by PV advection. Description: This paper introduces a novel, powerful way to understand the why geophysical flows are largely under the control of a single scalar field, the potential vorticity, a materially conserved dynamical tracer in the absence of viscous and diabatic effects.

Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid

Billant, Paul — 2006-08-25T00:00:00Z

Abstract: In this paper, we investigate the three-dimensional stability of the Moore-Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability). In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate gamma and by the rotation rate Omega of the principal axes of the elliptical vortex relative to the rotating frame of reference. For the m = 1 mode, it is shown that when Omega < -gamma, the vortex is stable to any long-wavelength disturbances, when -gamma < Omega less than or similar to 0, two-dimensional perturbations are most unstable, when 0 less than or similar to Omega < gamma, long-wavelength three-dimensional disturbances are the most unstable, and finally when gamma < Omega, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the m = 2 instability is two-dimensional or three-dimensional depending only on gamma and Omega, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids. The present results for the azimuthal mode m = 1 suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related. Description: This is a comprehensive analysis of the linear stability of columnar elliptical vortices subject to two-dimensional strain in a rotating, stratified fluid. It is the culmination of two lines of research, one started by Dritschel involving the tall-column instability, and another started by Billant and Chomaz involving the zigzag instability. Our joint work unifies these instabilities, and shows that they exist over a vast parameter space. This work represents over 7 years of collaborative effort.

Revisiting Batchelor's theory of two-dimensional turbulence

Dritschel, David Gerard — 2007-11-25T00:00:00Z

Abstract: Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a X-2/3 k(-1) enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation X in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes. We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing X in the limit Re -> infinity. Our proposal is supported by high Reynolds number simulations which confirm that X decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy (omega(2))/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing X, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum X(2/3)k(-1) with (omega(2))k(-1)(In Re)(-1).

A balanced approach to modelling rotating stably stratified geophysical flows

Dritschel, David Gerard — 2003-08-10T00:00:00Z

Abstract: We describe a new approach to modelling three-dimensional rotating stratified flows under the Boussinesq approximation. This approach is based on the explicit conservation of potential vorticity, and exploits the underlying leading-order geostrophic and hydrostratic balances inherent in these equations in the limit of small Froude and Rossby numbers. These balances are not imposed, but instead are used to motivate the use of a pair of new variables expressing the departure from geostrophic and hydrostratic balance. These new variables are the ageostrophic horizontal vorticity components, i.e. the vorticity not directly associated with the displacement of isopycnal surfaces. The use of potential vorticity and ageostrophic horizontal vorticity, rather than the usual primitive variables of velocity and density, reveals a deep mathematical structure and appears to have advantages numerically. This change of variables results in a diagnostic equation, of Monge-Amp re type, for one component of a vector potential phi, and two Poisson equations for the other two components. The curl of phi gives the velocity field while the divergence of phi is proportional to the displacement of isopycnal surfaces. This diagnostic equation makes transparent the conditions for both static and inertial stability, and may change form from (spatially) elliptic to (spatially) hyperbolic even when the flow is statically and inertially stable. A numerical method based on these new variables is developed and used to examine the instability of a horizontal elliptical shear zone (modelling a jet streak). The basic-state flow is in exact geostrophic and hydrostratic balance. Given a small perturbation however, the shear zone destabilizes by rolling up into a street of vortices and radiating inertia-gravity waves. Description: This work was the first to show how one can rewrite the equations for a rotating stratified fluid in a way which makes potential vorticity conservation explicit. Potential vorticity is linked closely to balance, a state void of high-frequency gravity waves. The mathematical transformation reveals a deep underlying mathematical structure, including explicit conditions for inertial and static stability as well as a new double Monge-Ampere equation. This work forms the cornerstone of much subsequent research into the fundamental nature of rotating stratified fluids.

Penetrative convection in a superposed porous-medium–fluid layer via internal heating

Carr, Magda — 2004-06-01T00:00:00Z

On defining groups efficiently without using inverses

Campbell, Colin Matthew — 2002-07-01T00:00:00Z

Abstract: Let G be a group, and let be a finite group presentation for G with \R\ greater than or equal to \A\. Then there exists a, finite semigroup, presentation for G such that \Q\ - \B\ = \R\ - \A\. Moreover, B is either the same generating set or else it contains one additional generator.