DSpace Community:
http://hdl.handle.net/2381/445
2015-04-28T14:26:32Z
2015-04-28T14:26:32Z
Computational diagnosis of canine lymphoma
Mirkes, E. M.
Alexandrakis, I.
Slater, K.
Tuli, R.
Gorban, A. N.
http://hdl.handle.net/2381/32072
2015-04-28T02:00:14Z
2015-04-27T14:00:05Z
Title: Computational diagnosis of canine lymphoma
Authors: Mirkes, E. M.; Alexandrakis, I.; Slater, K.; Tuli, R.; Gorban, A. N.
Editors: Vagenas, E. C.; Vlachos, D. S.
Abstract: One out of four dogs will develop cancer in their lifetime and 20% of those will be lymphoma cases. PetScreen developed a lymphoma blood test using serum samples collected from several veterinary practices. The samples were fractionated and analysed by mass spectrometry. Two protein peaks, with the highest diagnostic power, were selected and further identified as acute phase proteins, C-Reactive Protein and Haptoglobin. Data mining methods were then applied to the collected data for the development of an online computer-assisted veterinary diagnostic tool. The generated software can be used as a diagnostic, monitoring and screening tool. Initially, the diagnosis of lymphoma was formulated as a classification problem and then later refined as a lymphoma risk estimation. Three methods, decision trees, kNN and probability density evaluation, were used for classification and risk estimation and several preprocessing approaches were implemented to create the diagnostic system. For the differential diagnosis the best solution gave a sensitivity and specificity of 83.5% and 77%, respectively (using three input features, CRP, Haptoglobin and standard clinical symptom). For the screening task, the decision tree method provided the best result, with sensitivity and specificity of 81.4% and >99%, respectively (using the same input features). Furthermore, the development and application of new techniques for the generation of risk maps allowed their user-friendly visualization.
2015-04-27T14:00:05Z
Adaptive discontinuous Galerkin methods for nonlinear parabolic problems
Metcalfe, Stephen Arthur
http://hdl.handle.net/2381/32041
2015-04-23T02:00:25Z
2015-04-22T14:56:10Z
Title: Adaptive discontinuous Galerkin methods for nonlinear parabolic problems
Authors: Metcalfe, Stephen Arthur
Abstract: This work is devoted to the study of a posteriori error estimation and adaptivity
in parabolic problems with a particular focus on spatial discontinuous Galerkin
(dG) discretisations.
We begin by deriving an a posteriori error estimator for a linear non-stationary
convection-diffusion problem that is discretised with a backward Euler dG method.
An adaptive algorithm is then proposed to utilise the error estimator. The
effectiveness of both the error estimator and the proposed algorithm is shown
through a series of numerical experiments.
Moving on to nonlinear problems, we investigate the numerical approximation
of blow-up. To begin this study, we first look at the numerical approximation
of blow-up in nonlinear ODEs through standard time stepping schemes. We
then derive an a posteriori error estimator for an implicit-explicit (IMEX) dG
discretisation of a semilinear parabolic PDE with quadratic nonlinearity. An
adaptive algorithm is proposed that uses the error estimator to approach the
blow-up time. The adaptive algorithm is then applied in a series of test cases to
gauge the effectiveness of the error estimator.
Finally, we consider the adaptive numerical approximation of a nonlinear
interface problem that is used to model the mass transfer of solutes through
semi-permiable membranes. An a posteriori error estimator is proposed for the
IMEX dG discretisation of the model and its effectiveness tested through a series
of numerical experiments.
2015-04-22T14:56:10Z
Is it possible to predict long-term success with k-NN? Case study of four market indices (FTSE100, DAX, HANGSENG, NASDAQ)
Shi, Y.
Gorban, A. N.
Yang, T. Y.
http://hdl.handle.net/2381/32035
2015-04-22T02:00:17Z
2015-04-21T14:32:46Z
Title: Is it possible to predict long-term success with k-NN? Case study of four market indices (FTSE100, DAX, HANGSENG, NASDAQ)
Authors: Shi, Y.; Gorban, A. N.; Yang, T. Y.
Editors: Vagenas, E. C.; Vlachos, D. S.
Abstract: This case study tests the possibility of prediction for 'success' (or 'winner') components of four stock & shares market indices in a time period of three years from 02-Jul-2009 to 29-Jun-2012.We compare their performance ain two time frames: initial frame three months at the beginning (02/06/2009-30/09/2009) and the final three month frame (02/04/2012-29/06/2012).To label the components, average price ratio between two time frames in descending order is computed. The average price ratio is defined as the ratio between the mean prices of the beginning and final time period. The 'winner' components are referred to the top one third of total components in the same order as average price ratio it means the mean price of final time period is relatively higher than the beginning time period. The 'loser' components are referred to the last one third of total components in the same order as they have higher mean prices of beginning time period. We analyse, is there any information about the winner-looser separation in the initial fragments of the daily closing prices log-returns time series.The Leave-One-Out Cross-Validation with k-NN algorithm is applied on the daily log-return of components using a distance and proximity in the experiment. By looking at the error analysis, it shows that for HANGSENG and DAX index, there are clear signs of possibility to evaluate the probability of long-term success. The correlation distance matrix histograms and 2-D/3-D elastic maps generated from ViDaExpert show that the 'winner' components are closer to each other and 'winner'/'loser' components are separable on elastic maps for HANGSENG and DAX index while for the negative possibility indices, there is no sign of separation.
2015-04-21T14:32:46Z
Multiscale principal component analysis
Akinduko, A. A.
Gorban, Alexander N.
http://hdl.handle.net/2381/32009
2015-04-17T02:00:19Z
2015-04-16T13:57:07Z
Title: Multiscale principal component analysis
Authors: Akinduko, A. A.; Gorban, Alexander N.
Editors: Vagenas, E. C.; Vlachos, D. S.
Abstract: Principal component analysis (PCA) is an important tool in exploring data. The conventional approach to PCA leads to a solution which favours the structures with large variances. This is sensitive to outliers and could obfuscate interesting underlying structures. One of the equivalent definitions of PCA is that it seeks the subspaces that maximize the sum of squared pairwise distances between data projections. This definition opens up more flexibility in the analysis of principal components which is useful in enhancing PCA. In this paper we introduce scales into PCA by maximizing only the sum of pairwise distances between projections for pairs of datapoints with distances within a chosen interval of values [l,u]. The resulting principal component decompositions in Multiscale PCA depend on point (l,u) on the plane and for each point we define projectors onto principal components. Cluster analysis of these projectors reveals the structures in the data at various scales. Each structure is described by the eigenvectors at the medoid point of the cluster which represent the structure. We also use the distortion of projections as a criterion for choosing an appropriate scale especially for data with outliers. This method was tested on both artificial distribution of data and real data. For data with multiscale structures, the method was able to reveal the different structures of the data and also to reduce the effect of outliers in the principal component analysis.
2015-04-16T13:57:07Z
Multiscale approach to pest insect monitoring: Random walks, pattern formation, synchronization, and networks
Petrovskii, Sergei
Petrovskaya, N.
Bearup, Daniel
http://hdl.handle.net/2381/31970
2015-04-11T02:03:17Z
2015-04-10T08:44:10Z
Title: Multiscale approach to pest insect monitoring: Random walks, pattern formation, synchronization, and networks
Authors: Petrovskii, Sergei; Petrovskaya, N.; Bearup, Daniel
Abstract: Pest insects pose a significant threat to food production worldwide resulting in annual losses worth hundreds of billions of dollars. Pest control attempts to prevent pest outbreaks that could otherwise destroy a sward. It is good practice in integrated pest management to recommend control actions (usually pesticides application) only when the pest density exceeds a certain threshold. Accurate estimation of pest population density in ecosystems, especially in agro-ecosystems, is therefore very important, and this is the overall goal of the pest insect monitoring. However, this is a complex and challenging task; providing accurate information about pest abundance is hardly possible without taking into account the complexity of ecosystems' dynamics, in particular, the existence of multiple scales. In the case of pest insects, monitoring has three different spatial scales, each of them having their own scale-specific goal and their own approaches to data collection and interpretation. In this paper, we review recent progress in mathematical models and methods applied at each of these scales and show how it helps to improve the accuracy and robustness of pest population density estimation.
2015-04-10T08:44:10Z
Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration.
Bearup, D.
Petrovskaya, N.
Petrovskii, Sergei
http://hdl.handle.net/2381/31969
2015-04-11T02:03:12Z
2015-04-10T08:35:29Z
Title: Some analytical and numerical approaches to understanding trap counts resulting from pest insect immigration.
Authors: Bearup, D.; Petrovskaya, N.; Petrovskii, Sergei
Abstract: Monitoring of pest insects is an important part of the integrated pest management. It aims to provide information about pest insect abundance at a given location. This includes data collection, usually using traps, and their subsequent analysis and/or interpretation. However, interpretation of trap count (number of insects caught over a fixed time) remains a challenging problem. First, an increase in either the population density or insects activity can result in a similar increase in the number of insects trapped (the so called "activity-density" problem). Second, a genuine increase of the local population density can be attributed to qualitatively different ecological mechanisms such as multiplication or immigration. Identification of the true factor causing an increase in trap count is important as different mechanisms require different control strategies. In this paper, we consider a mean-field mathematical model of insect trapping based on the diffusion equation. Although the diffusion equation is a well-studied model, its analytical solution in closed form is actually available only for a few special cases, whilst in a more general case the problem has to be solved numerically. We choose finite differences as the baseline numerical method and show that numerical solution of the problem, especially in the realistic 2D case, is not at all straightforward as it requires a sufficiently accurate approximation of the diffusion fluxes. Once the numerical method is justified and tested, we apply it to the corresponding boundary problem where different types of boundary forcing describe different scenarios of pest insect immigration and reveal the corresponding patterns in the trap count growth.
2015-04-10T08:35:29Z
Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect
Jankovic, Masha
Petrovskii, Sergei
http://hdl.handle.net/2381/31968
2015-04-11T02:03:07Z
2015-04-10T08:27:00Z
Title: Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect
Authors: Jankovic, Masha; Petrovskii, Sergei
Abstract: One of the main challenges in ecology is to determine the cause of population fluctuations. Both theoretical and empirical studies suggest that delayed density dependence instigates cyclic behavior in many populations; however, underlying mechanisms through which this occurs are often difficult to determine and may vary within species. In this paper, we consider single species population dynamics affected by the Allee effect coupled with discrete time delay. We use two different mathematical formulations of the Allee effect and analyze (both analytically and numerically) the role of time delay in different feedback mechanisms such as competition and cooperation. The bifurcation value of the delay (that results in the Hopf bifurcation) as a function of the strength of the Allee effect is obtained analytically. Interestingly, depending on the chosen delayed mechanism, even a large time delay may not necessarily lead to instability. We also show that, in case the time delay affects positive feedback (such as cooperation), the population dynamics can lead to self-organized formation of intermediate quasi-stationary states. Finally, we discuss ecological implications of our findings.
2015-04-10T08:27:00Z
On the composition of the distributions x-s+ lnmx+ and xμ+
Fisher, Brian
http://hdl.handle.net/2381/31949
2015-04-01T02:00:13Z
2015-03-31T10:48:47Z
Title: On the composition of the distributions x-s+ lnmx+ and xμ+
Authors: Fisher, Brian
Abstract: Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {Fn(f)}, where Fn(x) = F(x)*δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x-s + lnm x+ and xμ + is proved to exist and be equal to μmx-sμ + lnm x+ for μ > 0 and s,m = 1, 2,....
Description: 2000 Mathematics Subject Classification. 46F10
2015-03-31T10:48:47Z
New Langevin and Gradient Thermostats for Rigid Body Dynamics
Davidchack, Ruslan L.
Ouldridge, T. E.
Tretyakov, M. V.
http://hdl.handle.net/2381/31940
2015-03-31T02:00:19Z
2015-03-30T09:07:46Z
Title: New Langevin and Gradient Thermostats for Rigid Body Dynamics
Authors: Davidchack, Ruslan L.; Ouldridge, T. E.; Tretyakov, M. V.
Abstract: We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.
Description: AMS 2000 subject classification. 65C30, 60H35, 60H10.
2015-03-30T09:07:46Z
Segal-type algebraic models of n-types
Blanc, D.
Paoli, Simona
http://hdl.handle.net/2381/31938
2015-04-01T02:00:14Z
2015-03-30T08:45:24Z
Title: Segal-type algebraic models of n-types
Authors: Blanc, D.; Paoli, Simona
Abstract: For each n ≥ 1, we introduce two new Segal-type models of n-types of topological
spaces: weakly globular n-fold groupoids, and a lax version of these. We show
that any n-type can be represented up to homotopy by such models via an explicit
algebraic fundamental n-fold groupoid functor. We compare these models to
Tamsamani’s weak n-groupoids, and extract from them a model for (k − 1)-
connected n-types.
Description: Mathematical Subject Classification 2000
Primary: 55S45
Secondary: 18G50, 18B40
2015-03-30T08:45:24Z
The weakly globular double category of fractions of a category
Paoli, Simona
Pronk, D.
http://hdl.handle.net/2381/31937
2015-04-01T01:45:07Z
2015-03-30T08:34:24Z
Title: The weakly globular double category of fractions of a category
Authors: Paoli, Simona; Pronk, D.
Abstract: This paper introduces the construction of a weakly globular double category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples.
Description: 2010 Mathematics Subject Classification: 18D05, 18E35
2015-03-30T08:34:24Z
A circular order on edge-coloured trees and RNA m-diagrams
Marsh, Robert J.
Schroll, Sibylle
http://hdl.handle.net/2381/31820
2015-03-10T02:02:11Z
2015-03-09T10:16:25Z
Title: A circular order on edge-coloured trees and RNA m-diagrams
Authors: Marsh, Robert J.; Schroll, Sibylle
Abstract: We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k, combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises as an induction in the context of interval exchange transformations. © 2013 Elsevier Inc.
Description: 2010 Mathematics Subject Classification: Primary: 05C05, 05A15; Secondary: 37B10
2015-03-09T10:16:25Z
Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces
Canakci, Ilke
Schroll, Sibylle
http://hdl.handle.net/2381/31819
2015-03-10T02:02:13Z
2015-03-09T10:08:08Z
Title: Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces
Authors: Canakci, Ilke; Schroll, Sibylle
Abstract: In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and almost all aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We give bases of their extension spaces and show that the dimensions of these extension spaces are given in terms of crossing arcs in the surface. Our approach is new and consists of interpreting snake graphs as indecomposable modules. To give a complete answer, we need to work in the associated cluster category where we explicitly calculate the middle terms of extensions and give a basis of the extension space. We note that not all extensions in the cluster category give rise to extensions for the Jacobian algebra.
Description: Generalized the results to include self-extensions, Added a new section containing an example, New abstract, Added a new result on snake graphs, Minor corrections, 31 pages, 14 figures. 2000 Mathematics Subject Classification. Primary: 13F60, 16P10, 18G15, 18E30
2015-03-09T10:08:08Z
Trivial Extensions of Gentle Algebras and Brauer Graph Algebras
Schroll, Sibylle
http://hdl.handle.net/2381/31818
2015-03-10T02:02:12Z
2015-03-09T10:05:05Z
Title: Trivial Extensions of Gentle Algebras and Brauer Graph Algebras
Authors: Schroll, Sibylle
Abstract: We show that two well-studied classes of tame algebras coincide: namely, the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. We then explore the connection between gentle algebras and symmetric special biserial algebras by explicitly determining the trivial extension of a gentle algebra by its minimal injective co-generator. This is a symmetric special biserial algebra and hence a Brauer graph algebra of which we explicitly give the Brauer graph. We further show that a Brauer graph algebra gives rise, via admissible cuts, to many gentle algebras and that the trivial extension of a gentle algebra obtained via an admissible cut is the original Brauer graph algebra. As a consequence we prove that the trivial extension of a Jacobian algebra of an ideal triangulation of a Riemann surface with marked points in the boundary is isomorphic to the Brauer graph algebra with Brauer graph given by the arcs of the triangulation.
Description: Added an example. 2010 Mathematics Subject Classification. Primary 16G10, 16G20; Secondary 16S99, 13F60
2015-03-09T10:05:05Z
The geometry of Brauer graph algebras and cluster mutations
Marsh, Robert J.
Schroll, Sibylle
http://hdl.handle.net/2381/31817
2015-03-10T02:02:12Z
2015-03-09T09:51:31Z
Title: The geometry of Brauer graph algebras and cluster mutations
Authors: Marsh, Robert J.; Schroll, Sibylle
Abstract: In this paper we establish a connection between ribbon graphs and Brauer graphs. As
a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer
graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual
construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise
to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related
Brauer graph algebras reflecting this geometrical move.
Description: MSC
primary, 16G10, 16G20, 16E35; secondary, 13F60, 14J10
2015-03-09T09:51:31Z
A circular order on edge-coloured trees and RNA m-diagrams
Marsh, Robert J.
Schroll, Sibylle
http://hdl.handle.net/2381/31816
2015-03-10T02:02:14Z
2015-03-09T09:45:40Z
Title: A circular order on edge-coloured trees and RNA m-diagrams
Authors: Marsh, Robert J.; Schroll, Sibylle
Abstract: We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k , combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises as an induction in the context of interval exchange transformations.
2015-03-09T09:45:40Z
The Ext algebra of a Brauer graph algebra
Green, Edward L.
Schroll, Sibylle
Snashall, Nicole
Taillefer, Rachel
http://hdl.handle.net/2381/31815
2015-03-10T02:02:14Z
2015-03-09T09:37:28Z
Title: The Ext algebra of a Brauer graph algebra
Authors: Green, Edward L.; Schroll, Sibylle; Snashall, Nicole; Taillefer, Rachel
Abstract: In this paper we study finite generation of the Ext algebra of a Brauer graph algebra by determining the degrees of the generators. As a consequence we characterize the Brauer graph algebras that are Koszul and those that are K_2.
Description: Minor changes only. 2010 Mathematics Subject Classification. 16G20, 16S37, 16E05, 16E30
2015-03-09T09:37:28Z
Group actions and coverings of Brauer graph algebras
Green, E. L.
Schroll, Sibylle
Snashall, Nicole
http://hdl.handle.net/2381/31808
2015-03-07T02:02:39Z
2015-03-06T16:08:02Z
Title: Group actions and coverings of Brauer graph algebras
Authors: Green, E. L.; Schroll, Sibylle; Snashall, Nicole
Abstract: We develop a theory of group actions and coverings on Brauer graphs that parallels
the theory of group actions and coverings of algebras. In particular, we show that any Brauer
graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has
multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify
the coverings of Brauer graph algebras that are again Brauer graph algebras.
Description: 2010 Mathematics Subject Classification. Primary 05E18, 16G20; Secondary 14E20, 16W50, 58E40
2015-03-06T16:08:02Z
Gaussian process regression with multiple response variables
Wang, Bo
Chen, Tau
http://hdl.handle.net/2381/31763
2015-03-05T02:02:22Z
2015-03-04T15:57:18Z
Title: Gaussian process regression with multiple response variables
Authors: Wang, Bo; Chen, Tau
Abstract: Gaussian process regression (GPR) is a Bayesian non-parametric technology that has
gained extensive application in data-based modelling of various systems, including
those of interest to chemometrics. However, most GPR implementations model only a
single response variable, due to the difficulty in the formulation of covariance function
for correlated multiple response variables, which describes not only the correlation
between data points, but also the correlation between responses. In the paper we
propose a direct formulation of the covariance function for multi-response GPR, based
on the idea that its covariance function is assumed to be the “nominal” uni-output
covariance multiplied by the covariances between different outputs. The effectiveness
of the proposed multi-response GPR method is illustrated through numerical examples
and response surface modelling of a catalytic reaction process.
2015-03-04T15:57:18Z
n-Fold groupoids, n-types and n-track categories
Blanc, David
Paoli, Simona
http://hdl.handle.net/2381/31752
2015-03-05T02:02:32Z
2015-03-04T11:33:24Z
Title: n-Fold groupoids, n-types and n-track categories
Authors: Blanc, David; Paoli, Simona
Abstract: For each n ≥ 1, we introduce two new Segal-type models of ntypes
of topological spaces: weakly globular n-fold groupoids, and a lax version
of these. We show that any n-type can be represented up to homotopy by
such models via an explicit algebraic fundamental n-fold groupoid functor.
We compare these models to Tamsamani’s weak n-groupoids, and extract from
them a model for (k − 1)-connected n-types.
Description: 1991 Mathematics Subject Classification. 55S45; 18G50, 18B40
2015-03-04T11:33:24Z
Minimum Distance Estimation of Milky Way Model Parameters and Related Inference
Banerjee, S.
Bhattacharya, S.
Basu, A.
Bose, S.
Chakrabarty, Dalia
Mukherjee, S.
http://hdl.handle.net/2381/31616
2015-02-06T02:01:52Z
2015-02-05T14:31:58Z
Title: Minimum Distance Estimation of Milky Way Model Parameters and Related Inference
Authors: Banerjee, S.; Bhattacharya, S.; Basu, A.; Bose, S.; Chakrabarty, Dalia; Mukherjee, S.
Abstract: We propose a method to estimate the location of the Sun in the disk of the Milky Way using a
method based on the Hellinger distance and construct confidence sets on our estimate of the unknown
location using a bootstrap based method. Assuming the Galactic disk to be two-dimensional, the
sought solar location then reduces to the radial distance separating the Sun from the Galactic center
and the angular separation of the Galactic center to Sun line, from a pre-fixed line on the disk. On
astronomical scales, the unknown solar location is equivalent to the location of us earthlings who
observe the velocities of a sample of stars in the neighborhood of the Sun. This unknown location
is estimated by undertaking pairwise comparisons of the estimated density of the observed set of
velocities of the sampled stars, with the density estimated using synthetic stellar velocity data
sets generated at chosen locations in the Milky Way disk. The synthetic data sets are generated
at a number of locations that we choose from within a constructed grid, at four different base
astrophysical models of the Galaxy. Thus, we work with one observed stellar velocity data and
four distinct sets of simulated data comprising a number of synthetic velocity data vectors, each
generated at a chosen location. For a given base astrophysical model that gives rise to one such
simulated data set, the chosen location within our constructed grid at which the estimated density of
the generated synthetic data best matches the density of the observed data, is used as an estimate
for the location at which the observed data was realized. In other words, the chosen location
corresponding to the highest match offers an estimate of the solar coordinates in the Milky Way
disk. The “match” between the pair of estimated densities is parameterized by the affinity measure
based on the familiar Hellinger distance. We perform a novel cross-validation procedure to establish
a desirable “consistency” property of the proposed method.
2015-02-05T14:31:58Z
Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data
Chakrabarty, Dalia
Saha, P.
http://hdl.handle.net/2381/31604
2015-02-05T02:02:06Z
2015-02-04T17:07:47Z
Title: Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data
Authors: Chakrabarty, Dalia; Saha, P.
Abstract: In this paper, we focus on a type of inverse problem in which the data are expressed as an unknown function of
the sought and unknown model function (or its discretised representation as a model parameter vector). In particular,
we deal with situations in which training data are not available. Then we cannot model the unknown
functional relationship between data and the unknown model function (or parameter vector) with a Gaussian
Process of appropriate dimensionality. A Bayesian method based on state space modelling is advanced instead.
Within this framework, the likelihood is expressed in terms of the probability density function (pdf) of the state
space variable and the sought model parameter vector is embedded within the domain of this pdf. As the measurable
vector lives only inside an identified sub-volume of the system state space, the pdf of the state space variable
is projected onto the space of the measurables, and it is in terms of the projected state space density that the
likelihood is written; the final form of the likelihood is achieved after convolution with the distribution of measurement
errors. Application motivated vague priors are invoked and the posterior probability density of the
model parameter vectors, given the data are computed. Inference is performed by taking posterior samples with
adaptive MCMC. The method is illustrated on synthetic as well as real galactic data.
2015-02-04T17:07:47Z
Bayesian Density Estimation via Multiple Sequential Inversions of 2-D Images with Application in Electron Microscopy
Chakrabarty, Dalia
Rigat, F.
Gabrielyan, N.
Beanland, R.
Paul, S.
http://hdl.handle.net/2381/31575
2015-02-05T02:02:00Z
2015-02-04T10:02:57Z
Title: Bayesian Density Estimation via Multiple Sequential Inversions of 2-D Images with Application in Electron Microscopy
Authors: Chakrabarty, Dalia; Rigat, F.; Gabrielyan, N.; Beanland, R.; Paul, S.
Abstract: We present a new Bayesian methodology to learn the unknown material density of
a given sample by inverting its two-dimensional images that are taken with a Scanning Electron
Microscope. An image results from a sequence of projections of the convolution of the density
function with the unknown microscopy correction function that we also learn from the data;
thus learning of the unknowns demands multiple inversions. We invoke a novel design of experiment,
involving imaging at multiple values of the parameter that controls the sub-surface
depth from which information about the density structure is carried, to result in the image.
Real-life material density functions are characterized by high density contrasts and are highly
discontinuous, implying that they exhibit correlation structures that do not vary smoothly. In
the absence of training data, modeling such correlation structures of real material density functions
is not possible. So we discretize the material sample and treat values of the density function
at chosen locations inside it as independent and distribution-free parameters. Resolution
of the available image dictates the discretization length of the model; three models pertaining
to distinct resolution classes (at μm to nano metre scale lengths) are developed. We develop
priors on the material density, such that these priors adapt to the sparsity inherent in the density
function. The likelihood is defined in terms of the distance between the convolution of the unknown
functions and the image data. The posterior probability density of the unknowns given
the data is expressed using the developed priors on the density and priors on the microscopy
correction function as elicited from the Microscopy literature. We achieve posterior samples
using an adaptive Metropolis-within-Gibbs inference scheme. The method is applied to learn
the material density of a 3-D sample of a nano-structure, using real image data. Illustrations on
simulated image data of alloy samples are also included
2015-02-04T10:02:57Z
Bayesian Learning of Material Density Function by Multiple Sequential Inversions of 2-D Images in Electron Microscopy
Chakrabarty, Dalia
Paul, S.
http://hdl.handle.net/2381/31574
2015-02-05T02:01:46Z
2015-02-04T09:51:38Z
Title: Bayesian Learning of Material Density Function by Multiple Sequential Inversions of 2-D Images in Electron Microscopy
Authors: Chakrabarty, Dalia; Paul, S.
Editors: Polpo de Campos, A; Neto,; Ramos Rifo,; Stern,; Lauretto,
Abstract: We discuss a novel inverse problem in which the data is generated by the sequential contractive projections
of the convolution of two unknown functions, both of which we aim to learn. The method is illustrated
using an application that relates to the multiple inversions of image data recorded with a Scanning Electron
Microscope, with the aim of learning the density of a given material sample and the microscopy correction
function. Given the severe logistical difficulties in this application of taking multiple images at different
viewing angles, a novel imaging experiment is undertaken, resulting in expansion of information. In lieu of
training data, it is noted that the highly discontinuous material density function cannot be modelled using a
Gaussian Process (GP) as the parametrisation of the required non-stationary covariance function of such a
GP cannot be achieved without training data. Consequently, we resort to estimating values of the unknown
functions at chosen locations in their domain–locations at which an image data are available. Image data
across a range of resolutions leads to multiscale models which we use to estimate material densities from the
micro-metre to nano-metre length scales. We discuss applications of the method in non-destructive learning
of material density using simulated metallurgical image data, as well as perform inhomogeneity detection in
multi-component composite on nano metre scales, by inverting real image data of a brick of nano-particles.
2015-02-04T09:51:38Z
Simple Locally Finite Lie Algebras of Diagonal Type
Baranov, Alexander
http://hdl.handle.net/2381/31502
2015-01-27T14:40:49Z
2015-01-27T14:38:49Z
Title: Simple Locally Finite Lie Algebras of Diagonal Type
Authors: Baranov, Alexander
Abstract: We discuss various characterizations of simple locally finite Lie algebras of diagonal type over an algebraically closed field of characteristic zero.
2015-01-27T14:38:49Z
On time scale invariance of random walks in confined space.
Bearup, Daniel
Petrovskii, Sergei
http://hdl.handle.net/2381/31448
2015-01-21T02:01:39Z
2015-01-20T14:49:30Z
Title: On time scale invariance of random walks in confined space.
Authors: Bearup, Daniel; Petrovskii, Sergei
Abstract: Animal movement is often modelled on an individual level using simulated random walks. In such applications it is preferable that the properties of these random walks remain consistent when the choice of time is changed (time scale invariance). While this property is well understood in unbounded space, it has not been studied in detail for random walks in a confined domain. In this work we undertake an investigation of time scale invariance of the drift and diffusion rates of Brownian random walks subject to one of four simple boundary conditions. We find that time scale invariance is lost when the boundary condition is non-conservative, that is when movement (or individuals) is discarded due to boundary encounters. Where possible analytical results are used to describe the limits of the time scaling process, numerical results are then used to characterise the intermediate behaviour.
2015-01-20T14:49:30Z
Modelling biological invasions : population cycles, waves and time delays
Jankovic, Masha
http://hdl.handle.net/2381/31392
2015-01-09T02:02:08Z
2015-01-08T12:53:35Z
Title: Modelling biological invasions : population cycles, waves and time delays
Authors: Jankovic, Masha
Abstract: Biological invasions are rapidly gaining importance due to the ever-increasing number
of introduced species. Alongside the plenitude of empirical data on invasive
species there exists an equally broad range of mathematical models that might be
of use in understanding biological invasions.
This thesis aims to address several issues related to modelling invasive species
and provide insight into their dynamics. Part I (Chapter 2) documents a case
study of the gypsy moth, Lymantria dispar, invasion in the US. We propose an
alternative hypothesis to explain the patchiness of gypsy moth spread entailing
the interplay between dispersal, predation or a viral infection and the Allee effect.
Using a reaction-diffusion framework we test the two models (prey-predator and
susceptible-infected) and predict qualitatively similar patterns as are observed in
natural populations. As high density gypsy moth populations cause the most
damage, estimating the spread rate would be of help in any suppression strategy.
Correspondingly, using a diffusive SI model we are able to obtain estimates of the
rate of spread comparable to historical data.
Part II (Chapters 3, 4 and 5) is more methodological in nature, and in a single
species context we examine the effect of an ubiquitous phenomenon influencing
population dynamics time delay. In Chapter 3 we show that contrary to the
general opinion, time delays are not always destabilising, using a delay differential
equation with discrete time delay. The concept of distributed delay is introduced
in Chapter 4 and studied through an integrodifferential model. Both Chapters 3
and 4 focus on temporal dynamics of populations, so we further this consideration
to include spatial effects in Chapter 5. Using two different representations of movement,
we show that the onset of spatiotemporal chaos in the wake of population
fronts is possible in a single species model.
2015-01-08T12:53:35Z
Special functions and generalized functions
Al-Sirehy, Fatma.
http://hdl.handle.net/2381/30544
2014-12-16T02:27:59Z
2014-12-15T10:40:16Z
Title: Special functions and generalized functions
Authors: Al-Sirehy, Fatma.
Abstract: In 1950, Laurent Schwartz marked a convenient starting point for the theory of generalized functions as a subject in its own right. He developed and unified much of the earlier work by Hadamard, Bochner, Sobolev and others. Since then an enormous literature dealing with both theory and applications has grown up, and the subject has undergone extensive further development. The original Schwartz treatment defined a distribution as a linear continuous functional on a space of test functions.;This thesis can be considered a part of the development going in that direction. It is partly an extension of earlier contributions by Fisher, Kuribayashi, Itano and others.;After introducing the background and basic definitions in Chapter One, we developed some basic results concerning the cosine integral Ci(lambda x) and its associated functions Ci+(lambda x) and Ci-(lambdax) as well as the neutrix convolution products of the cosine integral.;Chapter Three is devoted to similar results concerning the sine integral Si(lambdax).;In Chapter Four, we generalize some earlier results by Fisher and Kuribayashi concerning the product of the two dimensions xl+ and x-l-r+ . Moreover, other results are obtained concerning the neutrix product of |x|lambda-r lnp |x| and sgn x| x|lambda-r. Other theorems are proved about the matrix product of some other distributions such as xl+ ln x+ and x-l-r- .;Chapter Five contains new results about the composition of distributions. It involves the applications of the neutrix limit to establish such relationships between different distributions.
2014-12-15T10:40:16Z
Ancient Egyptian astronomy : timekeeping and cosmography in the New Kingdom
Symons, Sarah.
http://hdl.handle.net/2381/30543
2014-12-16T02:27:58Z
2014-12-15T10:40:15Z
Title: Ancient Egyptian astronomy : timekeeping and cosmography in the New Kingdom
Authors: Symons, Sarah.
Abstract: The first part of this study analyses and discusses astronomical timekeeping methods used in the New Kingdom. Diagonal star clocks are examined first, looking at classification of sources, decan lists, and the updating of the tables over time. The date list in the Osireion at Abydos is discussed, and issues concerning its place in the history of astronomical timekeeping are raised. The final stellar timekeeping method, the Ramesside star clock, is then examined. The conventional interpretation of the observational method behind the tables is challenged by a new theory, and a system of analysing the tables is introduced. The conclusions of the previous sections are then gathered together in a discussion of the development of stellar timekeeping methods.;The small instruments known as shadow clocks, and their later relatives the sloping sundials, are also examined. The established hypothesis that the shadow clock was completed by the addition of a crossbar is challenged and refuted.;The second part of this study is based on New Kingdom representations of the sky. Two major texts and several celestial diagrams are discussed in detail, beginning with the Book of Nut, which describes the motions of the sun and stars. New translations of the vignette and dramatic text are presented and discussed. Portions of the Book of the Day describing the behaviour of the sun and circumpolar group of stars are analysed.;Finally, celestial diagrams dating from the New Kingdom are discussed. Their composition and significance is discussed and the conceptual framework behind the diagrams is recreated. By introducing new theories and analysis methods, and using a modern but sympathetic approach to the original sources, this study attempts to update and extend our knowledge of these areas of ancient astronomy..
2014-12-15T10:40:15Z
Data structures and implementation of an adaptive hp finite element method
Senior, Bill.
http://hdl.handle.net/2381/30541
2014-12-16T02:27:57Z
2014-12-15T10:40:15Z
Title: Data structures and implementation of an adaptive hp finite element method
Authors: Senior, Bill.
Abstract: For a fully adaptive hp finite element programme to be implemented it is necessary to implement n-irregular meshes efficiently. This requires a sufficiently flexible data structure to be implemented. Because such flexibility is required, the traditional array based approach cannot be used because of its limited applicability. In this thesis this traditional approach has been replaced by an object orientated design and implementation. This leads to an implementation that can be extended easily and safely to include other problems for which it was not originally designed.;The problems with maintaining continuity on such a diverse variety of meshes and how continuity is maintained are discussed. Then the main structure of the mesh is described in the form of domain, subdomains and elements. These are used in conjunction with constraint mappings to give a conforming approximation even with the most irregular of meshes.;There are several varieties of matrix generated by the method each with its own problems of storage. Sparse matrices, with perhaps more than 95% of zero entries, need to be used along side dense matrices. In this thesis an object oriented matrix library is implemented that enables this variety of matrices to be used.;An hp finite element algorithm is then implemented using the data structures, and is tested on a range of test problems. The method is shown to be effective on these problems.
2014-12-15T10:40:15Z
On finite groups of p-local rank one and a conjecture of Robinson
Eaton, Charles.
http://hdl.handle.net/2381/30542
2014-12-16T02:27:57Z
2014-12-15T10:40:15Z
Title: On finite groups of p-local rank one and a conjecture of Robinson
Authors: Eaton, Charles.
Abstract: We use the classification of finite simple groups to verify a conjecture of Robinson for finite groups G where G/Op(G) has trivial intersection Sylow p-subgroups. Groups of this type are said to have p-local rank one, and it is hoped that this invariant will eventually form the basis for inductive arguments, providing reductions for the conjecture, or even a proof using the results presented here as a base. A positive outcome for Robinson's conjecture would imply Alperin's weight conjecture.;It is shown that in proving Robinson's conjecture it suffices to demonstrate only that it holds for finite groups in which Op(G) is both cyclic and central.;Part of the proof of the former result is used to complete the verification of Dade's inductive conjecture for the Ree groups of type G2.;.
2014-12-15T10:40:15Z
A special numerical method for solving Hamiltonian eigenproblems
Maple, Carsten R.
http://hdl.handle.net/2381/30540
2014-12-16T02:27:56Z
2014-12-15T10:40:15Z
Title: A special numerical method for solving Hamiltonian eigenproblems
Authors: Maple, Carsten R.
Abstract: In this thesis we develop and implement a new algorithm for finding the solutions of linear Hamiltonian systems arising from ordinary differential equation (ODE) eigenproblems; a large source of these systems is Sturm-Liouville problems, and these will provide the angle of approach. The convergence properties of the algorithm will be analysed, as will the performance of the algorithm for large values of eigenparameter.;An algorithm is also proposed to find high-index eigenvalues of general Sturm-Liouville problems.
2014-12-15T10:40:15Z
Multivariate hermite interpolation in euclidean space and its unit sphere by radial basis functions
Luo, Zuhua.
http://hdl.handle.net/2381/30539
2014-12-16T02:27:56Z
2014-12-15T10:40:14Z
Title: Multivariate hermite interpolation in euclidean space and its unit sphere by radial basis functions
Authors: Luo, Zuhua.
Abstract: In this thesis, we consider radial basis function interpolations in d-dimensional Euclidean space Hd and the unit sphere 5d_1, where the data is generated not only by point-evaluations, but also by the derivatives, or differential/pseudo-differential operators. Some sufficient and necessary conditions for the well-posedness of the interpolations are given. The results on sensitivity and sta bility of the interpolation systems are obtained. The optimal properties of the interpolants are analysed through the variational framework and reproducing kernel Hilbert space property, the error bounds and convergence rates of the interpolants are derived. The admissible reproducing kernel Hilbert spaces are also characterised.
2014-12-15T10:40:14Z
On indecomposable modules over cluster-tilted algebras of type A
Parsons, Mark James
http://hdl.handle.net/2381/30538
2014-12-16T02:27:55Z
2014-12-15T10:40:14Z
Title: On indecomposable modules over cluster-tilted algebras of type A
Authors: Parsons, Mark James
Abstract: Gabriel's Theorem describes the dimension vectors of the finitely generated indecomposable modules over the path algebra of a simply-laced Dynkin quiver. It shows that they can be obtained from the expressions for the positive roots of the corresponding root system in terms of the simple roots. Here, we present a method for finding the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A.;It is known that the quiver of a cluster-tilted algebra of Dynkin type A is given by an exchange matrix of the corresponding cluster algebra. We define a companion basis for such a quiver to be a Z -basis of roots of the integral root lattice of the corresponding root system whose associated matrix of inner products is a positive quasi-Cartan companion of the corresponding exchange matrix.;Our main result establishes that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A arise from expressions for the positive roots of the corresponding root system in terms of a companion basis (for the quiver of that algebra). This can be regarded as a generalisation of part of Gabriel's Theorem in the Dynkin type A case. The proof uses the fact that the quivers of the cluster-tilted algebras of Dynkin type A have a particularly nice description in terms of triangulation of regular polygons.
2014-12-15T10:40:14Z
Anisotropic adaptive refinement for discontinuous galerkin methods
Hall, Edward John Cumes
http://hdl.handle.net/2381/30536
2014-12-16T02:27:54Z
2014-12-15T10:40:14Z
Title: Anisotropic adaptive refinement for discontinuous galerkin methods
Authors: Hall, Edward John Cumes
Abstract: We consider both the a priori and a posteriori error analysis and hp-adaptation strategies for discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched polynomial degrees. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution, exploiting duality based arguments.;The a priori error analysis is carried out in two settings. In the first, full orientation of elements is allowed but only (possibly high-order) isotropic polynomial degrees considered; our analysis, therefore, extends previous results, where only finite element spaces comprising piecewise linear polynomials were considered, by utilizing techniques from tensor analysis. In the second case, anisotropic polynomial degrees are allowed, but the elements are assumed to be axiparallel; we thus apply previously known interpolation error results to the goal-oriented setting.;Based on our a posteriori error bound we first design and implement an adaptive anisotropic h-refinement algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement, chosen on a competitive basis requiring the solution of local problems. The superiority of the proposed algorithm in comparison with a standard h-isotropic mesh refinement algorithm and a Hessian based h-anisotropic adaptive procedure is illustrated by a series of numerical experiments. We then describe a fully hp -adaptive algorithm, once again using a competitive refinement approach, which, numerical experiments reveal, offers considerable improvements over both a standard hp-isotropic refinement algorithm and an h-anisotropic/p-isotropic adaptive procedure.
2014-12-15T10:40:14Z
On the quiver and relations of the Borel Schur algebras
Liang, Degang
http://hdl.handle.net/2381/30537
2014-12-16T02:27:55Z
2014-12-15T10:40:14Z
Title: On the quiver and relations of the Borel Schur algebras
Authors: Liang, Degang
Abstract: Let K be an infinite field of characteristic p â‰¥ 0 and let n, r be positive integers. Let S+(n, r) be the Borel Schur algebra over K, which is a sub-algebra of the Schur algebra S(n, r). We aim to give a description of the Borel Schur algebra S+ (n, r) by finding its quiver and relations. We give a complete description of the quiver and relations for S+(2, r). We also construct a family of embedding from S+(2, r) to S+(n, r + s) which induce embeddings of the corresponding quivers. This gives us some relations for S+(n, r) for n > 2.;We describe the quiver of S+( n, r) for both p = 0 and p > 0. We also describe some relations of special type for p > 0 and find all relations for p = 0.
2014-12-15T10:40:14Z
Uncertainty propagation and reduction in reservoir forecasting
Busby, Daniel
http://hdl.handle.net/2381/30534
2014-12-16T02:27:52Z
2014-12-15T10:40:13Z
Title: Uncertainty propagation and reduction in reservoir forecasting
Authors: Busby, Daniel
Abstract: In this work we focus on nonparametric regression techniques based on Gaussian process, considering both the frequentist and the Bayesian approach. A new sequential experimental design strategy referred to as hierarchical adaptive experimental design is proposed and tested on synthetic functions and on realistic reservoir models using a commercial oil reservoir multiphase flow simulator. Our numerical results show that the method effectively approximate the simulators output with the required approximation accuracy using an affordable number of simulator runs. Moreover, the number of simulations necessary to reach a given approximation accuracy is sensibly reduced respect to other existing experimental designs such as maximin latin hypercubes, or other classical designs used in commercial softwares.;Once an accurate emulator of the simulator output is obtained, it can be also used to calibrate the simulator model using data observed on the real physical system. This process, referred to as history matching in reservoir forecasting, is fundamental to tune input parameters and to consequently reduce output uncertainty. An approach to model calibration using Bayesian inversion is proposed in the last part of this work. Here again a hierarchical emulator is adopted. An innovative sequential design is proposed with the objective of increasing the emulator accuracy around possible history matching solutions. The excellent performances obtained on a very complicated reservoir test case, suggest the high potential of the method to solve complicated inverse problems. The proposed methodology is about to be commercialized in an industrial environment to assist reservoir engineers in uncertainty analysis and history matching.
2014-12-15T10:40:13Z
Euler characteristics and cohomology for quasiperiodic projection patterns
Irving, Claire Louise
http://hdl.handle.net/2381/30533
2014-12-16T02:27:52Z
2014-12-15T10:40:13Z
Title: Euler characteristics and cohomology for quasiperiodic projection patterns
Authors: Irving, Claire Louise
Abstract: This thesis investigates quasiperiodic patterns and, in particular, polytopal projection patterns, which are produced using the projection method by choosing the acceptance domain to be a polytope. Cohomology theories applicable in this setting are defined, together with the Euler characteristic.;Formulae for the Cech cohomology Hˇ* ( M P ) and Euler characteristic eP are determined for polytopal projection patterns of codimension 2 and calculations are carried out for several examples. The Euler characteristic is shown to be undefined for certain codimension 3 polytopal projection patterns. The Euler characteristic eP is proved to be always defined for a particular class of codimension n polytopal projection patterns P and a formula for eP for such patterns is given. The finiteness or otherwise of the rank of Hˇm(M P ) âŠ— Q for m â‰¥ 0 is also discussed for various classes of polytopal projection patterns. Lastly, a model for M P is considered which leads to an alternative method for computing the rank of Hˇm(M P ) âŠ— Q for P a d-dimensional codimension n polytopal projection pattern with d > n..
2014-12-15T10:40:13Z
Efficient method for detection of periodic orbits in chaotic maps and flows
Crofts, Jonathan J.
http://hdl.handle.net/2381/30535
2014-12-16T02:27:53Z
2014-12-15T10:40:13Z
Title: Efficient method for detection of periodic orbits in chaotic maps and flows
Authors: Crofts, Jonathan J.
Abstract: An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional system. The difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of stabilising transformations grows extremely fast with increasing system dimension. in this thesis, we propose to construct stabilising transformations based on the knowledge of the stability matrices of already detected periodic orbits (used as seeds). The advantage of our approach is in a substantial reduction of the number of transformations, which increases the efficiency of the detection algorithm, especially in the case of high-dimensional systems. The dependence of the number of transformations on the dimensionality of the unstable manifold rather than on system size enables us to apply, for the first time, the method of stabilising transformations to high-dimensional systems. Another important aspect of our treatment of high-dimensional flows is that we do not restrict to a Poincare surface of section. This is a particularly nice feature, since the correct placement of such a section in a high-dimensional phase space is a challenging problem in itself. The performance of the new approach is illustrated by its application to the four-dimensional kicked double rotor map, a six-dimensional system of three coupled Henon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.
2014-12-15T10:40:13Z
Logic, computation and constraint satisfaction
Martin, Barnaby D.
http://hdl.handle.net/2381/30530
2014-12-16T02:27:50Z
2014-12-15T10:40:12Z
Title: Logic, computation and constraint satisfaction
Authors: Martin, Barnaby D.
Abstract: We study a class of non-deterministic program schemes with while loops: firstly, augmented with a priority queue for memory; secondly, augmented with universal quantification; and, thirdly, augmented with universal quantification and a stack for memory. We try to relate these respective classes of program schemes to well-known complexity classes and logics.;We study classes of structure on which path system logic coincides with polynomial time P.;We examine the complexity of generalisations of non-uniform boolean constraint satisfaction problems, where the inputs may have a bounded number of quantifier alternations (as opposed to the purely existential quantification of the CSP). We prove, for all bounded-alternation prefixes that have some universal quantifiers to the outside of some existential quantifiers (i.e. 2 and above), that this generalisation of boolean CSP respects the same dichotomy as that for the non-uniform boolean quantified constraint satisfaction problem.;We study the non-uniform QCSP, especially on digraghs, through a combinatorial analog - the alternating-homomorphism problem - that sits in relation to the QCSP exactly as the homomorphism problem sits with the CSP. We establish a trichotomy theorem for the non-uniform QCSP when the template is restricted to antireflexive, undirected graphs with at most one cycle. Specifically, such templates give rise to QCSPs that are either tractable, NP-complete or Pspace-complete.;We study closure properties on templates that respect QCSP hardness or QCSP equality. Our investigation leads us to examine the properties of first-order logic when deprived of the equality relation.;We study the non-uniform QCSP on tournament templates, deriving sufficient conditions for tractablity, NP-completeness and Pspace-completeness. In particular, we prove that those tournament templates that give rise to tractable CSP also give rise to tractable QCSP.
2014-12-15T10:40:12Z
Embeddings, fault tolerance and communication strategies in k-ary n-cube interconnection networks
Ashir, Yaagoub A.
http://hdl.handle.net/2381/30532
2014-12-16T02:27:51Z
2014-12-15T10:40:12Z
Title: Embeddings, fault tolerance and communication strategies in k-ary n-cube interconnection networks
Authors: Ashir, Yaagoub A.
Abstract: The k-ary n-cube interconnection network Qkn, for k 3 and n 2, is n-dimensional network with k processors in each dimension. A k-ary n-cube parallel computer consists of kn identical processors, each provided with its own sizeable memory and interconnected with 2n other processors. The k-ary n-cube has some attractive features like symmetry, high level of concurrency and efficiency, regularity and high potential for the parallel execution of various algorithms. It can efficiently simulate other network topologies. The k-ary n-cube has a smaller degree than that of its equivalent hypercube (the one with at least as many nodes) and it has a smaller diameter than its equivalent mesh of processors.;In this thesis, we review some topological properties of the k-ary n-cube Qkn and show how a Hamiltonian cycle can be embedded in Qkn using the Gray codes strategy. We also completely classify when a Qkn contains a cycle of some given length.;The problem of embedding a large cycle in a Qkn with both faulty nodes and faulty links is considered. We describe a technique for embedding a large cycle in a k-ary n-cube Qkn with at most n faults and show how this result can be extended to obtain embeddings of meshes and tori in such a faulty k-ary n-cube.;Embeddings of Hamiltonian cycles in faulty k-ary n-cubes is also studied. We develop a technique for embedding a Hamiltonian cycle in a k-ary n-cube with at most 4n-5 faulty links where every node is incident with at least two healthy links. Our result is optimal as there exist k-ary n-cubes with 4n - 4 faults (and where every node is incident with at least two healthy links) not containing a Hamiltonian cycle. We show that the same technique can be easily applied to the hypercube. We also show that the general problem of deciding whether a faulty k-ary n-cube contains a Hamiltonian cycle is NP-complete, for all (fixed) k 3.
2014-12-15T10:40:12Z
On a construction of young modules
Vernon, Marie
http://hdl.handle.net/2381/30531
2014-12-16T02:27:50Z
2014-12-15T10:40:12Z
Title: On a construction of young modules
Authors: Vernon, Marie
Abstract: Let n be a natural number and E an n-dimensional vector space over a field K. The symmetric group acts by place permutation on the tensor space E âŠ—r. The Sigmar-module EâŠ—r can be decomposed into a direct sum of permutation modules Mlambda where lambda is a composition of r into at most n parts.;Each permutation module labelled by such a composition is isomorphic to one labelled be a partition of r into at most n parts, and therefore we assume that lambda is such a partition. The indecomposable direct summands of the permutation module M lambda are called Young modules, and they are labelled by partitions of r into at most n parts.;Throughout this thesis we consider the case where E has dimension two. For lambda a two-part partition of r, we explicitly decompose the module M lambda into a direct sum of Young modules by providing spanning sets for the Young modules.;Moreover, we consider the problem of finding a basis or an algorithm for a basis for the Young modules in this case and, although we have not been able to solve this in general, we give some conjectures and examples showing in which cases we can find a basis.
2014-12-15T10:40:12Z
Mesh-free radical basis function methods for advection-dominated diffusion problems
Hunt, David Patrick
http://hdl.handle.net/2381/30529
2014-12-16T02:27:49Z
2014-12-15T10:40:11Z
Title: Mesh-free radical basis function methods for advection-dominated diffusion problems
Authors: Hunt, David Patrick
Abstract: This thesis is concerned with the numerical solution of advection-dominated diffusion problems. There are essentially two key aspects to this work: the derivatives of an a priori error estimate for a semi-Lagrangian mesh-free method using radial basis function interpolation to numerically approximate the first-order linear transport problem; and the design and testing of a semi-Lagrangian mesh-less method to numerically solve the full parabolic advection-diffusion problem, using radial basis function Hermite interpolation. We begin by establishing the theory of radical basis function interpolation, including new results for the stability of interpolation via the class of radial basis functions known as polyharmonic splines, as well as error estimates for interpolation by the same class of function. These results provide us with the necessary tools to prove the a priori error estimate for the semi-Lagrangian advection scheme, given certain assumptions on the smoothness of the solution. We then validate both the scheme and the analysis with a series of numerical experiments. By introducing the concept of Hermite interpolation, we develop and implement a new semi-Lagrangian method for the numerical approximation of advection-dominated diffusion problems, which is validated through two numerical experiments..
2014-12-15T10:40:11Z
Self-injective algebras and the second Hochschild cohomology group
Al-Kadi, Deena
http://hdl.handle.net/2381/30528
2014-12-16T02:27:49Z
2014-12-15T10:40:11Z
Title: Self-injective algebras and the second Hochschild cohomology group
Authors: Al-Kadi, Deena
Abstract: In this thesis we study the second Hochschild cohomology group HH 2(Lambda) of a finite dimensional algebra Lambda. In particular, we determine HH2(Lambda) where Lambda is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field K and show that this group is zero for most such Lambda; we give a basis for HH2(Lambda) in the few cases where it is not zero.;Then we consider algebras of tame representation type; more specifically, we study finite dimensional self-injective one parametric tame algebras which are not weakly symmetric. Here we show that HH2(Lambda) is non-zero and find a non-zero element eta in HH2(Lambda) and an associative deformation Lambdaeta of Lambda.
2014-12-15T10:40:11Z
Finiteness conditions on the Ext-algebra
Davis, Gabriel.
http://hdl.handle.net/2381/30527
2014-12-16T02:27:48Z
2014-12-15T10:40:11Z
Title: Finiteness conditions on the Ext-algebra
Authors: Davis, Gabriel.
Abstract: Let A be a finite-dimensional algebra given by quiver and monomial relations. In [18] we see that the Ext-algebra of A is finitely generated only if all the Ext-algebras of certain cycle algebras overlying A are finitely generated. Here a cycle algebra Lambda is a finite-dimensional algebra given by quiver and monomial relations where the quiver is an oriented cycle. The main result of this thesis gives necessary and sufficient conditions for the Ext-algebra of such a Lambda to be finitely generated; this is achieved by defining a computable invariant of Lambda, the smo-tube. We also give necessary and sufficient conditions for the Ext-algebra of Lambda to be Noetherian.;Let Lambda be a triangular matrix algebra, defined by algebras T and U and a T-U-bimodule M. Under certain conditions we show that if the Ext-algebras of T and U are right (respectively left) Noetherian rings, then the Ext-algebra of Lambda is a right (respectively left) Noetherian ring. An example shows the hypotheses used cannot be improved. We also specialise to the case where Lambda is a one-point extension: we give a specific presentation of a result that parallels a similar theorem for the more general case above.
2014-12-15T10:40:11Z
Hamilton thermostatting techniques for molecular dynamics simulation
Sweet, Christopher Richard
http://hdl.handle.net/2381/30526
2014-12-16T02:27:48Z
2014-12-15T10:40:11Z
Title: Hamilton thermostatting techniques for molecular dynamics simulation
Authors: Sweet, Christopher Richard
Abstract: Molecular dynamics trajectories that sample from a Gibbs, or canonical, distribution can be generated by introducing a modified Hamiltonian with additional degrees of freedom as described by Nose [46]. Although this method has found widespread use in its time re-parameterized Nose-Hoover form, the lack of a Hamiltonian, and the need to 'tune' thermostatting parameters has limited, its use compared to stochastic methods. In addition, since the proof of the correct sampling is based on an ergodic assumption, thermostatting small of stiff systems often does not given the correct distributions unless the Nose-Hoover chains [43] method is used, which inherits the Nose-Hoover deficiencies noted above. More recently the introduction of the Hamiltonian Nose-Poincare method [11], where symplectic integrators can be used for improved long term stability, has renewed interest in the possibility of Hamiltonian methods which can improve dynamical sampling. This class of methods, although applicable to small systems, has applications in large scale systems with complex chemical structure, such as protein-bath and quantum-classical models.;For Nose dynamics, it is often stated that the system is driven to equilibrium through a resonant interaction between the self-oscillation frequency of the thermostat variable and a natural frequency of the underlying system. By the introduction of multiple thermostat Hamiltonian formulations, which are not restricted to chains, it has been possible to clarify this perspective, using harmonic models, and exhibit practical deficiencies of the standard Nose-chain approach. This has led to the introduction of two Hamiltonian schemes, the Nose-Poincare chains method and the Recursive Multiple Thermostat (RMT) method. The RMT method obtains canonical sampling without the stability problems encountered with chains with the advantage that the choice of Nose mass is independent of the underlying system.
2014-12-15T10:40:11Z
Blocks of fat category O
Fonseca, Andre
http://hdl.handle.net/2381/30525
2014-12-16T02:27:47Z
2014-12-15T10:40:11Z
Title: Blocks of fat category O
Authors: Fonseca, Andre
Abstract: We generalize the category O of Bernstein, Gelfand and Gelfand to the so called fat category O, On and derive some of its properties. From a Lie theoretic point of view, contains a significant amount of indecomposable representations that do not belong to O (although it fails to add new simple ones) such as the fat Verma modules. These modules have simple top and socle and may be viewed as standard objects once a block decomposition of is obtained and each block is seen to be equivalent to a category of finite dimensional modules over a finite dimensional standardly stratified algebra. We describe the Ringel dual of these algebras (concluding that principal blocks are self dual) and we obtain the character formulae for their tilting modules. Furthermore, a double centralizer property is proved, relating each block with the corresponding fat algebra of coinvariants. As a byproduct we obtain a classification of all blocks of in terms of their representation type. In the process of determining the quiver and relations which characterize the basic algebras associated to each block of On we prove (for root systems of small rank) a formula establishing the dimension of the Ext1 spaces between simple modules. By borrowing from Soergel some results describing the behaviour of the combinatorial functor V, we are able to compute examples.
2014-12-15T10:40:11Z
Constraint satisfaction problems and related logic
Madelaine, Florent
http://hdl.handle.net/2381/30524
2014-12-16T02:27:47Z
2014-12-15T10:40:10Z
Title: Constraint satisfaction problems and related logic
Authors: Madelaine, Florent
Abstract: Feder and Vardi have proved that the class captured by a monadic fragment of existential second-order logic, MMSNP, is computationally equivalent (via randomised reductions) to the class of constraint satisfaction problems (CSP) while the latter is strictly included in the former. I introduce a new class of combinatorial problems, the so-called forbidden patterns problems (FP), that correspond exactly to the logic MMSNP and introduce some novel algebraic tools like the re-colouring that allow me to construct a normal form. This leads to a constructive characterisation of the borderline of CSP within FP: a given problem in FP is either given as a problem in CSP or we build counter-examples. I relate this result to a recent and independent work by Tardif and Nesetril which relies heavily on a correspondence between duality and density. I generalise this approach to FP. Finally, I investigate homomorphism problems for unary algebras.
2014-12-15T10:40:10Z
Monads in coalgebra
Di Marchi, Federico
http://hdl.handle.net/2381/30522
2014-12-16T02:27:45Z
2014-12-15T10:40:10Z
Title: Monads in coalgebra
Authors: Di Marchi, Federico
Abstract: Universal algebra has long been regarded as a fundamental tool in studying semantics of programming languages. Within this paradigm, one can formulate statements regarding the correctness of a program by looking at the interpretations of the code in any model for the language.;While this provides a description of finite computations, other models have to be introduced in order to provide a semantics for recursion and infinite computations in general. This leads to a study of rational and infinite terms. Such terms arise by a dual construction to that of the finite ones. namely, while the latter form an initial algebra, the former are a final coalgebra.;For this reason, it is natural to approach the study of infinite terms by dualising the categorical model of universal algebra. This leads to various different constructions, which are worth of investigation. In this thesis we approach two of them. In one case, we introduce the notions of cosignature, coequation and comodel, in the spirit of the theory of coalgebraic specification. In the second we focus on the properties of monads which can model infinitary computations. Such monads we call guarded, and include, amongst others, the monads of finite terms, infinite terms, rational terms and term graphs. As a byproduct of identifying this notion, we can solve algebraic systems of equation, which are an abstract counterpart to the notion of a recursive program scheme.;Many guarded monads we encounter are obtained by collecting, in an appropriate sense, a suitable family of coagebras. These examples are all instances of a general theorem we present, which tells under which conditions we can define a monad by a colimit operation, and when such comonads are guarded.;The level of abstraction allowed by the use of the categorical formalism allows us to instantiate some of the results in different categories, obtaining a monadic semantics for rational and infinite parallel term rewriting.
2014-12-15T10:40:10Z
Hierarchic modelling of separable elliptic boundary value problems on thin domains
Arnold, Mark Edward.
http://hdl.handle.net/2381/30521
2014-12-16T02:27:44Z
2014-12-15T10:40:10Z
Title: Hierarchic modelling of separable elliptic boundary value problems on thin domains
Authors: Arnold, Mark Edward.
Abstract: The dimensional reduction method for solving Laplace's equation on a flat plate, an arch and a spherical shell is investigated, extending previous work on laminated plates by Vogelius and Babuska (1981). Convergence rates for the error in energy are obtained, extending previous results by deriving explicit values for the constant of approximation and its dependence on the thickness of the domain and the model order. The framework for laminated plates is shown to easily extend to other geometries. Numerical results are given which verify the convergence rates in terms of the thickness. Details are given as to how to implement the dimensional reduction technique and in particular, for a spherical shell, a method is given which reduces the problem to that of inverting relatively small matrices.;A posteriori error estimators are given for each of the geometries under consideration. Error estimators are already known for flat plates. It is shown how the estimators for flat plates can be modified for use in the arch case, and for shells, techniques for estimating the discretization error and modelling error are presented. The a posteriori estimators are then used to derive a refinement algorithm for adaptively constructing hierarchic models for representative problems on each of the geometries under consideration.
2014-12-15T10:40:10Z