DSpace Community:http://hdl.handle.net/2381/4452016-05-06T03:40:56Z2016-05-06T03:40:56ZStability modes in vortex structure formation: Canonical examples for rotating componentsGostelow, J. PaulGarrett, Stephen J.Rona, AldoAdebayo, David S.http://hdl.handle.net/2381/374762016-05-05T02:09:18Z2016-05-04T14:50:44ZTitle: Stability modes in vortex structure formation: Canonical examples for rotating components
Authors: Gostelow, J. Paul; Garrett, Stephen J.; Rona, Aldo; Adebayo, David S.
Abstract: Three rather different physical cases have been studied. All represent very practical geometries for which the modal behavior of vortex structures is not completely understood. The work on these problems is ongoing with the objective of obtaining physical confirmation, enhanced understanding and predictive capability for the vortex structures encountered in rotating machines.2016-05-04T14:50:44ZCohomology of tiling spaces: beyond primitive substitutionsRust, Daniel Georgehttp://hdl.handle.net/2381/374692016-04-30T02:28:11Z2016-04-29T13:40:19ZTitle: Cohomology of tiling spaces: beyond primitive substitutions
Authors: Rust, Daniel George
Abstract: This thesis explores the combinatorial and topological properties of tiling spaces
associated to 1-dimensional symbolic systems of aperiodic type and their associated
algebraic invariants. We develop a framework for studying systems which are more
general than primitive substitutions, naturally partitioned into two instances: in the
first instance we allow systems associated to sequences of substitutions of primitive
type from a finite family of substitutions (called mixed substitutions); in the second
instance we concentrate on systems associated to a single substitution, but where
we entirely remove the condition of primitivity.
We generalise the notion of a Barge-Diamond complex, in the one-dimensional case,
to any mixed system of symbolic substitutions. This gives a way of describing
the associated tiling space as an inverse limit of complexes. We give an effective
method for calculating the Cech cohomology of the tiling space via an exact sequence
relating the associated sequence of substitution matrices and certain subcomplexes
appearing in the approximants. As an application, we show that there exists a
system of substitutions on two letters which exhibit an uncountable collection of
minimal tiling spaces with distinct isomorphism classes of Cech cohomology.
In considering non-primitive substitutions, we naturally divide this set of substitutions
into two cases: the minimal substitutions and the non-minimal substitutions.
We provide a detailed method for replacing any non-primitive but minimal substitution
with a topologically conjugate primitive substitution, and a more simple
method for replacing the substitution with a primitive substitution whose tiling
space is orbit equivalent. We show that an Anderson-Putnam complex with a collaring
of some appropriately large radius suffices to provide a model of the tiling
space as an inverse limit with a single map. We apply these methods to effectively
calculate the Cech cohomology of any substitution which does not admit a periodic
point in its subshift. Using its set of closed invariant subspaces, we provide a pair
of invariants which are each strictly finer than the usual Cech cohomology for a
substitution tiling space.2016-04-29T13:40:19ZTiling spaces, codimension one attractors and shapeClark, AlexanderHunton, J.http://hdl.handle.net/2381/373462016-04-19T02:15:49Z2016-04-18T11:58:33ZTitle: Tiling spaces, codimension one attractors and shape
Authors: Clark, Alexander; Hunton, J.
Abstract: We establish a close relationship between, on the one hand, expanding, codimension one attractors of diffeomorphisms on closed manifolds (examples of so-called strange attractors), and, on the other, spaces which arise in the study of aperiodic tilings. We show that every such orientable attractor is homeomorphic to a tiling space of either a substitution or a projection tiling, depending on its dimension. We also demonstrate that such an attractor is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the nonorientable case, to a space with a two-to-one covering by such a torus-less-points. This puts considerable constraints on the topology of codimension one attractors, and constraints on which manifolds tiling spaces may be embedded in. In the process we develop a new invariant for aperiodic tilings, which, for 1-dimensional tilings is in many cases finer than the cohomological or K-theoretic invariants studied to date.2016-04-18T11:58:33ZFrom Ambiguity Aversion to a Generalized Expected Utility. Modeling Preferences in a Quantum Probabilistic FrameworkAerts, D.Sozzo, Sandrohttp://hdl.handle.net/2381/372652016-04-13T02:22:20Z2016-04-12T11:41:18ZTitle: From Ambiguity Aversion to a Generalized Expected Utility. Modeling Preferences in a Quantum Probabilistic Framework
Authors: Aerts, D.; Sozzo, Sandro
Abstract: Ambiguity and ambiguity aversion have been widely studied in decision theory and economics both at a theoretical and an experimental level. After Ellsberg's seminal studies challenging subjective expected utility theory (SEUT), several (mainly normative) approaches have been put forward to reproduce ambiguity aversion and Ellsberg-type preferences. However, Machina and other authors have pointed out some fundamental difficulties of these generalizations of SEUT to cope with some variants of Ellsberg's thought experiments, which has recently been experimentally confirmed. Starting from our quantum modeling approach to human cognition, we develop here a general probabilistic framework to model human decisions under uncertainty. We show that our quantum theoretical model faithfully represents different sets of data collected on both the Ellsberg and the Machina paradox situations, and is flexible enough to describe different subjective attitudes with respect to ambiguity. Our approach opens the way toward a quantum-based generalization of expected utility theory (QEUT), where subjective probabilities depend on the state of the conceptual entity at play and its interaction with the decision-maker, while preferences between acts are determined by the maximization of this 'state-dependent expected utility'.
Description: The file associated with this record is under a 24-month embargo from publication in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-04-12T11:41:18ZA classification of the point spectrum of constant length substitution tiling spaces and general fixed point theorems for tilingsAbuzaid, Dina Asaadhttp://hdl.handle.net/2381/370272016-03-12T04:27:12Z2016-03-11T16:11:58ZTitle: A classification of the point spectrum of constant length substitution tiling spaces and general fixed point theorems for tilings
Authors: Abuzaid, Dina Asaad
Abstract: We examine the point spectrum of the various tiling spaces that result from
different choices of tile lengths for substitution of constant length on a two or a three letter
alphabet. In some cases we establish insensitivity to changes in length. In a wide range
of cases, we establish that the typical choice of length leads to trivial point spectrum.
We also consider a problem related to tilings of the integers and their connection to fixed
point theorems. Using this connection, we prove a generalization of the Banach Contraction
Principle.2016-03-11T16:11:58ZThe centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free streamHussain, Z.Garrett, Stephen J.Stephen, S. O.Griffiths, Paul Travishttp://hdl.handle.net/2381/369862016-03-08T03:20:27Z2016-03-07T11:36:56ZTitle: The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free stream
Authors: Hussain, Z.; Garrett, Stephen J.; Stephen, S. O.; Griffiths, Paul Travis
Abstract: In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed, leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, yielding favourable comparisons with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in a sufficiently strong axial flow, the instability mode breaks down into Görtler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise still fluid.
Description: The file associated with this record is under a 6-month embargo from publication in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-03-07T11:36:56ZThe neutral curve for stationary disturbances in rotating disk flow for power-law fluidsGriffiths, P. T.Garrett, Stephen JohnStephen, S. O.http://hdl.handle.net/2381/369852016-03-08T03:19:26Z2016-03-07T11:32:29ZTitle: The neutral curve for stationary disturbances in rotating disk flow for power-law fluids
Authors: Griffiths, P. T.; Garrett, Stephen John; Stephen, S. O.
Abstract: This paper is concerned with the convective instabilities associated with the boundary-layer flow due to a rotating disk. Shear-thinning fluids that adhere to the power-law relationship are considered. The neutral curves are computed using a sixth-order system of linear stability equations which include the effects of streamline curvature, Coriolis force and the non-Newtonian viscosity model. Akin to previous Newtonian studies it is found that the neutral curves have two critical values, these are associated with the type I upper-branch (cross-flow) and type II lower-branch (streamline curvature) modes. Our results indicate that an increase in shear-thinning has a stabilising effect on both the type I and II modes, in terms of the critical Reynolds number and growth rate. Favourable agreement is obtained between existing asymptotic predictions and the numerical results presented here.
Description: The file associated with this record is under a 24-month embargo from publication in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-03-07T11:32:29ZStability of the boundary layer on a rotating disk for power-law fluidsGriffiths, P. T.Stephen, S. O.Bassom, A. P.Garrett, Stephen Johnhttp://hdl.handle.net/2381/369632016-03-04T03:22:45Z2016-03-03T11:51:44ZTitle: Stability of the boundary layer on a rotating disk for power-law fluids
Authors: Griffiths, P. T.; Stephen, S. O.; Bassom, A. P.; Garrett, Stephen John
Abstract: The stability of the flow due to a rotating disk is considered for non-Newtonian fluids, specifically shear-thinning fluids that satisfy the power-law (Ostwald-de Waele) relationship. In this case the basic flow is not an exact solution of the Navier–Stokes equations, however, in the limit of large Reynolds number the flow inside the three-dimensional boundary layer can be determined via a similarity solution. An asymptotic analysis is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally in the Newtonian case can be described for shear-thinning fluids by a linear stability analysis. Predictions for the wavenumber and wave angle of the disturbances suggest that shear-thinning fluids may have a stabilising effect on the flow.2016-03-03T11:51:44ZGlobal linear stability of the boundary-layer flow over a rotating sphereBarrow, A.Garrett, Stephen J.Peake, N.http://hdl.handle.net/2381/368892016-02-26T03:26:22Z2016-02-25T13:44:07ZTitle: Global linear stability of the boundary-layer flow over a rotating sphere
Authors: Barrow, A.; Garrett, Stephen J.; Peake, N.
Abstract: We consider the linear global stability of the boundary-layer flow over a rotating sphere. Our results suggest that a self-excited linear global mode can exist when the sphere rotates sufficiently fast, with properties fixed by the flow at latitudes between approximately 55°–65° from the pole (depending on the rotation rate). A neutral curve for global linear instabilities is presented with critical Reynolds number consistent with existing experimentally measured values for the appearance of turbulence. The existence of an unstable linear global mode is in contrast to the literature on the rotating disk, where it is expected that nonlinearity is required to prompt the transition to turbulence. Despite both being susceptible to local absolute instabilities, we conclude that the transition mechanism for the rotating-sphere flow may be different to that for the rotating disk.2016-02-25T13:44:07ZThe centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free-streamHussain, Z.Garrett, Stephen J.Stephen, S. O.Griffiths, Paul T.http://hdl.handle.net/2381/368882016-02-26T03:26:36Z2016-02-25T13:41:12ZTitle: The centrifugal instability of the boundary-layer flow over a slender rotating cone in an enforced axial free-stream
Authors: Hussain, Z.; Garrett, Stephen J.; Stephen, S. O.; Griffiths, Paul T.
Abstract: In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed, leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, yielding favourable comparisons with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in a sufficiently strong axial flow, the instability mode breaks down into Görtler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise still fluid.
Description: The file associated with this record is under a 6-month embargo from publication in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-02-25T13:41:12ZThe effect of surface roughness on the convective instability of the BEK family of boundary-layer flowsAlveroglu, BurhanSegalini, A.Garrett, Stephen J.http://hdl.handle.net/2381/368872016-02-26T03:26:38Z2016-02-25T13:34:13ZTitle: The effect of surface roughness on the convective instability of the BEK family of boundary-layer flows
Authors: Alveroglu, Burhan; Segalini, A.; Garrett, Stephen J.
Abstract: A Chebyshev polynomial discretisation method is used to investigate the effect of both anisotropic (radially and azimuthally) and isotropic surface roughnesses on the convective instability of the BEK family of rotating boundary-layer flows. The mean-flow profiles for the velocity components are obtained by modelling surface roughness with a partial-slip approach. A linear stability analysis is then performed to investigate the effect of roughness on the convective instability characteristics of the inviscid Type I (cross-flow) instability and the viscous Type II instability. It is revealed that all roughness types lead to a stabilisation of the Type I mode in all flows within the BEK family, with the exception of azimuthally-anisotropic roughness (radial grooves) within the Bödewadt layer which causes a mildly destabilising effect. In the case of the Type II mode, the results reveal the destabilising effect of radially-anisotropic roughness (concentric grooves) on all the boundary layers, whereas both azimuthally-anisotropic and isotropic roughnesses have a stabilising effect on the mode for Ekman and von Kármán layers. Complementary results are also presented by considering the effects of roughness on the growth rates of each instability mode within the Ekman layer.
Description: The file associated with this record is under a 24-month embargo from publication in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-02-25T13:34:13ZThe effect of anisotropic and isotropic roughness on the convective stability of the rotating disk boundary layerCooper, A. J.Harris, J. H.Garrett, Stephen JohnOezkan, M.Thomas, P. J.http://hdl.handle.net/2381/368862016-02-26T03:26:38Z2016-02-25T13:28:22ZTitle: The effect of anisotropic and isotropic roughness on the convective stability of the rotating disk boundary layer
Authors: Cooper, A. J.; Harris, J. H.; Garrett, Stephen John; Oezkan, M.; Thomas, P. J.
Abstract: A theoretical study investigating the effects of both anisotropic and isotropic surface roughness on the convective stability of the boundary-layer flow over a rotating disk is described. Surface roughness is modelled using a partial-slip approach, which yields steady-flow profiles for the relevant velocity components of the boundary-layer flow which are a departure from the classic von Kármán solution for a smooth disk. These are then subjected to a linear stability analysis to reveal how roughness affects the stability characteristics of the inviscid Type I (or cross-flow) instability and the viscous Type II instability that arise in the rotating disk boundary layer. Stationary modes are studied and both anisotropic (concentric grooves and radial grooves) and isotropic (general) roughness are shown to have a stabilizing effect on the Type I instability. For the viscous Type II instability, it was found that a disk with concentric grooves has a strongly destabilizing effect, whereas a disk with radial grooves or general isotropic roughness has a stabilizing effect on this mode. In order to extract possible underlying physical mechanisms behind the effects of roughness, and in order to reconfirm the results of the linear stability analysis, an integral energy equation for three-dimensional disturbances to the undisturbed three-dimensional boundary-layer flow is used. For anisotropic roughness, the stabilizing effect on the Type I mode is brought about by reductions in energy production in the boundary layer, whilst the destabilizing effect of concentric grooves on the Type II mode results from a reduction in energy dissipation. For isotropic roughness, both modes are stabilized by combinations of reduced energy production and increased dissipation.2016-02-25T13:28:22ZOn the stability of von Kármán rotating-disk boundary layers with radial anisotropic surface roughnessGarrett, Stephen J.Cooper, A. J.Harris, J. H.Ozkan, M.Segalini, A.Thomas, P. J.http://hdl.handle.net/2381/368852016-03-02T03:26:01Z2016-02-25T13:21:15ZTitle: On the stability of von Kármán rotating-disk boundary layers with radial anisotropic surface roughness
Authors: Garrett, Stephen J.; Cooper, A. J.; Harris, J. H.; Ozkan, M.; Segalini, A.; Thomas, P. J.
Abstract: We summarise results of a theoretical study investigating the distinct convective instability properties of steady boundary-layer flow over rough rotating disks. A generic roughness pattern of concentric circles with sinusoidal surface undulations in the radial direction is considered. The goal is to compare predictions obtained by means of two alternative, and fundamentally different, modelling approaches for surface roughness for the first time. The motivating rationale is to identify commonalities and isolate results that might potentially represent artefacts associated with the particular methodologies underlying one of the two modelling approaches. The most significant result of practical relevance obtained is that both approaches predict overall stabilising effects on type I instability mode of rotating disk flow. This mode leads to transition of the rotating-disk boundary layer and, more generally, the transition of boundary-layers with a cross-flow profile. Stabilisation of the type 1 mode means that it may be possible to exploit surface roughness for laminar-flow control in boundary layers with a cross-flow component. However, we also find differences between the two sets of model predictions, some subtle and some substantial. These will represent criteria for establishing which of the two alternative approaches is more suitable to correctly describe experimental data when these become available.2016-02-25T13:21:15ZGlobal linear stability of the boundary-layer flow over a rotating sphereBarrow, A.Garrett, Stephen JohnPeake, N.http://hdl.handle.net/2381/368842016-02-26T03:28:14Z2016-02-25T13:07:28ZTitle: Global linear stability of the boundary-layer flow over a rotating sphere
Authors: Barrow, A.; Garrett, Stephen John; Peake, N.
Abstract: We consider the linear global stability of the boundary-layer flow over a rotating sphere. Our results suggest that a self-excited linear global mode can exist when the sphere rotates sufficiently fast, with properties fixed by the flow at latitudes between approximately 55°-65° from the pole (depending on the rotation rate). A neutral curve for global linear instabilities is presented with critical Reynolds number consistent with existing experimentally measured values for the appearance of turbulence. The existence of an unstable linear global mode is in contrast to the literature on the rotating disk, where it is expected that nonlinearity is required to prompt the transition to turbulence. Despite both being susceptible to local absolute instabilities, we conclude that the transition mechanism for the rotating-sphere flow may be different to that for the rotating disk.2016-02-25T13:07:28ZOn Streamwise Vortices in Large Eddy Simulations of Initially Laminar Plane Mixing LayersMcMullan, William AndrewGarrett, Stephen J.http://hdl.handle.net/2381/367902016-02-19T03:24:45Z2016-02-18T12:27:41ZTitle: On Streamwise Vortices in Large Eddy Simulations of Initially Laminar Plane Mixing Layers
Authors: McMullan, William Andrew; Garrett, Stephen J.
Abstract: x
Description: The file associated with this record is under a 24-month embargo from publication in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-02-18T12:27:41ZMultiscale principal component analysisAkinduko, Ayodeji Akinwumihttp://hdl.handle.net/2381/366162016-02-10T03:35:14Z2016-02-09T10:10:00ZTitle: Multiscale principal component analysis
Authors: Akinduko, Ayodeji Akinwumi
Abstract: The problem of approximating multidimensional data with objects of lower dimension is a classical problem in complexity reduction. It is important that data approximation capture the structure(s) and dynamics of the data, however distortion to data by many methods during approximation implies that some geometric structure(s) of the data may not be preserved during data approximation. For methods that model the manifold of the data, the quality of approximation depends crucially on the initialization of the method. The first part of this thesis investigates the effect of initialization on manifold modelling methods. Using Self Organising Maps (SOM) as a case study, we compared the quality of learning of manifold methods for two popular initialization methods; random initialization and principal component initialization. To further understand the dynamics of manifold learning, datasets were further classified into linear, quasilinear and nonlinear.
The second part of this thesis focuses on revealing geometric structure(s) in high dimension data using an extension of Principal Component Analysis (PCA). Feature extraction using (PCA) favours direction with large variance which could obfuscate other interesting geometric structure(s) that could be present in the data. To reveal these intrinsic structures, we analysed the local PCA structures of the dataset. An equivalent definition of PCA is that it seeks subspaces that maximize the sum of pairwise distances of data projection; extending this definition we define localization in term of scale as maximizing the sum of weighted squared pairwise distances between data projections for various distributions of weights (scales). Since for complex data various regions of the dataspace could have different PCA structures, we also define localization with regards to dataspace. The resulting local PCA structures were represented by the projection matrix corresponding to the subspaces and analysed to reveal some structures in the data at various localizations.2016-02-09T10:10:00ZOut-of-band and adjacent-channel interference reduction by analog nonlinear filtersNikitin, A. V.Davidchack, Ruslan L.Smith, J. E.http://hdl.handle.net/2381/364712016-02-02T03:14:45Z2016-02-01T09:58:41ZTitle: Out-of-band and adjacent-channel interference reduction by analog nonlinear filters
Authors: Nikitin, A. V.; Davidchack, Ruslan L.; Smith, J. E.
Abstract: In a perfect world, we would have ‘brick wall’ filters, no-distortion amplifiers and mixers, and well-coordinated spectrum operations. The real world, however, is prone to various types of unintentional and intentional interference of technogenic (man-made) origin that can disrupt critical communication systems. In this paper, we introduce a methodology for mitigating technogenic interference in communication channels by analog nonlinear filters, with an emphasis on the mitigation of out-of-band and adjacent-channel interference.
Interference induced in a communications receiver by external transmitters can be viewed as wide-band non-Gaussian noise affecting a narrower-band signal of interest. This noise may contain a strong component within the receiver passband, which may dominate over the thermal noise. While the total wide-band interference seen by the receiver may or may not be impulsive, we demonstrate that the interfering component due to power emitted by the transmitter into the receiver channel is likely to appear impulsive under a wide range of conditions. We give an example of mechanisms of impulsive interference in digital communication systems resulting from the nonsmooth nature of any physically realizable modulation scheme for transmission of a digital (discontinuous) message.
We show that impulsive interference can be effectively mitigated by nonlinear differential limiters (NDLs). An NDL can be configured to behave linearly when the input signal does not contain outliers. When outliers are encountered, the nonlinear response of the NDL limits the magnitude of the respective outliers in the output signal. The signal quality is improved in excess of that achievable by the respective linear filter, increasing the capacity of a communications channel. The behavior of an NDL, and its degree of nonlinearity, is controlled by a single parameter in a manner that enables significantly better overall suppression of the noise-containing impulsive components compared to the respective linear filter. Adaptive configurations of NDLs are similarly controlled by a single parameter and are suitable for improving quality of nonstationary signals under time-varying noise conditions. NDLs are designed to be fully compatible with existing linear devices and systems and to be used as an enhancement, or as a low-cost alternative, to the state-of-art interference mitigation methods.2016-02-01T09:58:41ZAsymptotic variance of stationary reversible and normal Markov processesDeligiannidis, G.Peligrad, M.Utev, Sergeyhttp://hdl.handle.net/2381/364132016-01-28T03:09:24Z2016-01-27T10:37:50ZTitle: Asymptotic variance of stationary reversible and normal Markov processes
Authors: Deligiannidis, G.; Peligrad, M.; Utev, Sergey
Abstract: We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in n.2016-01-27T10:37:50ZThe HELP inequality for Hamiltonian systemsBrown, B. M.Evans, W. D.Marletta, M.http://hdl.handle.net/2381/362952016-01-16T03:15:05Z2016-01-15T15:52:52ZTitle: The HELP inequality for Hamiltonian systems
Authors: Brown, B. M.; Evans, W. D.; Marletta, M.
Abstract: We extend the Hardy–Everitt–Littlewood–Polya inequality, hitherto established for 2nth order formally selfadjoint ordinary differential equations, to a wide class of linear Hamiltonian systems. The method follows Dias (Ph.D. thesis, Cardiff: University of Wales, 1994) but without the Hilbert space setting which he uses.2016-01-15T15:52:52ZRegularized semiclassical limits: linear flows with infinite Lyapunov exponentsAthanassoulis, AgisilaosKyza, I.Katsaounis, T.http://hdl.handle.net/2381/362662016-01-13T03:10:37Z2016-01-12T12:48:18ZTitle: Regularized semiclassical limits: linear flows with infinite Lyapunov exponents
Authors: Athanassoulis, Agisilaos; Kyza, I.; Katsaounis, T.
Abstract: Semiclassical asymptotics for Schrodinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P. L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posterior error controal. Thus rigorous uppen bounds for the asymptotic error on concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM.
Description: The file associated with this record is under a permanent embargo while publication is In Press in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.2016-01-12T12:48:18ZOn the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time AveragingAthanassoulis, AgisilaosAntonelli, P.Markowich, P. A.Hajaiej, H.http://hdl.handle.net/2381/362472016-01-12T03:09:29Z2016-01-11T11:41:27ZTitle: On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging
Authors: Athanassoulis, Agisilaos; Antonelli, P.; Markowich, P. A.; Hajaiej, H.
Abstract: We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree–Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray free electron laser. We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.2016-01-11T11:41:27ZNumerical Simulations of X-Ray Free Electron Lasers (XFEL)Athanassoulis, AgisilaosMarkowich, P. A.Antonelli, P.Huang, Z.http://hdl.handle.net/2381/362432016-01-12T03:09:53Z2016-01-11T10:48:23ZTitle: Numerical Simulations of X-Ray Free Electron Lasers (XFEL)
Authors: Athanassoulis, Agisilaos; Markowich, P. A.; Antonelli, P.; Huang, Z.
Abstract: We study a nonlinear Schrödinger equation which arises as an effective single particle model in X-ray free electron lasers (XFEL). This equation appears as a first principles model for the beam-matter interactions that would take place in an XFEL molecular imaging experiment in [A. Fratalocchi and G. Ruocco, Phys. Rev. Lett., 106 (2011), 105504]. Since XFEL are more powerful by several orders of magnitude than more conventional lasers, the systematic investigation of many of the standard assumptions and approximations has attracted increased attention. In this model the electrons move under a rapidly oscillating electromagnetic field, and the convergence of the problem to an effective time-averaged one is examined. We use an operator splitting pseudospectral method to investigate numerically the behavior of the model versus that of its time-averaged version in complex situations, namely the energy subcritical/mass supercritical case and in the presence of a periodic lattice. We find the time-averaged model to be an effective approximation, even close to blowup, for fast enough oscillations of the external field. This work extends previous analytical results for simpler cases [P. Antonelli, A. Athanassoulis, H. Hajaiej, and P. Markowich, Arch. Ration. Mech. Anal., 211 (2014), pp. 711--732].2016-01-11T10:48:23ZSynergy effect of cooperative investmentGrechuk, BogdanZabarankin, M.http://hdl.handle.net/2381/361892016-01-07T03:09:07Z2016-01-06T12:56:39ZTitle: Synergy effect of cooperative investment
Authors: Grechuk, Bogdan; Zabarankin, M.
Abstract: Cooperative investment consists of two problems: finding an optimal cooperative investment strategy and fairly dividing investment outcome among participating agents. In general, the two problems cannot be solved separately. It is known that when agents’ preferences are represented by mean-deviation functionals, sharing of optimal portfolio creates instruments that, on the one hand, satisfy individual risk preferences but, on the other hand, are not replicable on an incomplete market, so that each agent is strictly better off in participating in cooperative investment than investing alone. This synergy effect is shown to hold when agents’ acceptance sets are represented by cash-invariant utility functions in the case of multiperiod investment with an arbitrary feasible investment set. In this case, a set of all Pareto-optimal allocations is characterized, and an equilibrium-based method for selecting a “fair” Pareto-optimal allocation is suggested. It is also shown that if exists, the “fair” allocation belongs to the core of the corresponding cooperative game. The equilibrium-based method is then extended to the case of arbitrary utility functions. The obtained results are demonstrated in a multiperiod cooperative investment problem with investors imposing drawdown constraints on investment strategies.
Description: The file associated with this record is under a 12-month embargo from publication in accordance with the publisher's self-archiving policy, available at http://www.springer.com/gp/open-access/authors-rights/self-archiving-policy/2124. The full text may be available through the publisher links provided above.2016-01-06T12:56:39ZRelated fixed points for set-valued mappings on two uniform spacesTürkoğlu, D.Fisher, Brianhttp://hdl.handle.net/2381/361512016-01-06T03:09:34Z2016-01-05T10:28:25ZTitle: Related fixed points for set-valued mappings on two uniform spaces
Authors: Türkoğlu, D.; Fisher, Brian
Abstract: Some related fixed point theorems for set-valued mappings on two complete and compact uniform spaces are proved.
Description: 2000 Mathematics Subject Classification: 54H25, 47H10.2016-01-05T10:28:25ZOn the Fresnel integrals and the convolutionKiliçman, A.Fisher, Brianhttp://hdl.handle.net/2381/361442016-01-05T03:12:31Z2016-01-04T15:10:01ZTitle: On the Fresnel integrals and the convolution
Authors: Kiliçman, A.; Fisher, Brian
Abstract: The Fresnel cosine integral C(x), the Fresnel sine integral S(x), and the associated functions C+(x), C−(x), S+(x), and S−(x) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the Fresnel cosine integral and its associated functions with x+r and xr are evaluated.
Description: 2000 Mathematics Subject Classification: 33B10, 46F102016-01-04T15:10:01ZOn the sine integral and the convolutionFisher, BrianAl-Sirehy, F.http://hdl.handle.net/2381/361432016-01-05T03:12:29Z2016-01-04T14:57:48ZTitle: On the sine integral and the convolution
Authors: Fisher, Brian; Al-Sirehy, F.
Abstract: The sine integral Si(λx) and the cosine integral Ci(λx) and their associated functions Si+(λx), Si−(λx), Ci+(λx), Ci−(λx) are defined as locally summable functions on the real line. Some convolutions of these functions and sin(μx), sin+(μx), and sin−(μx) are found.
Description: 2000 Mathematics Subject Classification: 33B10, 46F10.2016-01-04T14:57:48ZInverse portfolio problem with coherent risk measuresGrechuk, BogdanZabarankin, M.http://hdl.handle.net/2381/361362015-12-24T03:08:14Z2015-12-23T13:36:29ZTitle: Inverse portfolio problem with coherent risk measures
Authors: Grechuk, Bogdan; Zabarankin, M.
Abstract: In general, a portfolio problem minimizes risk (or negative utility) of a portfolio of financial assets with respect to portfolio weights subject to a budget constraint. The inverse portfolio problem then arises when an investor assumes that his/her risk preferences have a numerical representation in the form of a certain class of functionals, e.g. in the form of expected utility, coherent risk measure or mean-deviation functional, and aims to identify such a functional, whose minimization results in a portfolio, e.g. a market index, that he/she is most satisfied with. In this work, the portfolio risk is determined by a coherent risk measure, and the rate of return of investor’s preferred portfolio is assumed to be known. The inverse portfolio problem then recovers investor’s coherent risk measure either through finding a convex set of feasible probability measures (risk envelope) or in the form of either mixed CVaR or negative Yaari’s dual utility. It is solved in single-period and multi-period formulations and is demonstrated in a case study with the FTSE 100 index.2015-12-23T13:36:29ZImplementing Automotive Telematics for Insurance Covers of FleetsAzzopardi, M.Cortis, Dominichttp://hdl.handle.net/2381/361302015-12-23T03:05:28Z2015-12-22T16:43:17ZTitle: Implementing Automotive Telematics for Insurance Covers of Fleets
Authors: Azzopardi, M.; Cortis, Dominic
Abstract: The advantages of Usage-Based Insurance for automotive covers over conventional rating methods have been discussed in literature for over four decades. Notwithstanding their adoption in insurance markets has been slow. This paper seeks to establish the viability of introducing fleet Telematics-Based Insurance by investigating the perceptions of insurance operators, tracking service providers and corporate fleet owners. At its core, the study involves a SWOT-analysis to appraise Telematics-Based Insurance against conventional premium rating systems. Twenty five key stakeholders in Malta, a country with an insurance industry that represents others in microcosm, were interviewed to develop our analysis. We assert that local insurers have interests in such insurance schemes as enhanced fleet management and monitoring translate into an improved insurance risk. The findings presented here have implications for all stakeholders as we argue that telematics enhance fleet management, TBI improves risk management for insurers and adoption of this technology is dependent on telematics providers increasing the perceived control by insurers over managing this technology.2015-12-22T16:43:17ZPhacoemulsification Surgery in Eyes with Neovascular Age-Related Macular DegenerationGrixti, A.Papavasileiou, E.Cortis, DominicKumar, B. V.Prasad, S.http://hdl.handle.net/2381/361182015-12-23T03:05:25Z2015-12-22T14:47:21ZTitle: Phacoemulsification Surgery in Eyes with Neovascular Age-Related Macular Degeneration
Authors: Grixti, A.; Papavasileiou, E.; Cortis, Dominic; Kumar, B. V.; Prasad, S.
Abstract: Purpose. To evaluate the visual outcomes and effect of phacoemulsification surgery on the progression of neovascular agerelated
macular degeneration (AMD). Methods. Retrospective, noncomparative, and interventional case series. Thirty eyes from 29
subjects with neovascular AMD treated with intravitreal antivascular endothelial growth factor (VEGF) injections who underwent
phacoemulsification and had a postsurgery follow-up of 6 months were included. LogMAR best corrected visual acuity (BCVA)
was assessed preoperatively; 1 month, 3 months, and 6 months postoperatively; and finally at the last visit. The frequency of antiVEGF
therapy, calculated as the number of intravitreal injections per month, and central macular thickness (CMT) before and
after cataract surgery were determined. Results. Median (range) logMAR BCVA was 0.69 (0.16 to 1.32) preoperatively; 0.55 (−0.04
to 1.32) at 1 month, 0.52 (−0.1 to 1.32) at 3 months, and 0.50 (0.0 to 1.32) at 6 months postoperatively; and 0.6 (0.0 to 1.4) at final
visit (𝑃 = 0.0011). There was no difference in the frequency of anti-VEGF injections between the immediate 6 months before
and after phacoemulsification, which was equal to 0.1667 injections per month (𝑃 = 0.6377). Median CMT measured 203 𝜇m
preoperatively, which temporarily increased to 238 𝜇m at 1 month after surgery (𝑃 = 0.0093) and then spontaneously returned
to baseline, measuring 212.5 𝜇m at 3 months postoperatively (𝑃 = 0.3811). Conclusion. Phacoemulsification surgery significantly
improved vision in patients with neovascular AMD, with no increased need for anti-VEGF injections to keep the macula dry
postoperatively2015-12-22T14:47:21ZAtiyah sequences, connections and characteristic forms for principal bundles over groupoids and stacksBiswas, IndranilNeumann, Frankhttp://hdl.handle.net/2381/361022015-12-19T03:18:59Z2015-12-18T12:05:34ZTitle: Atiyah sequences, connections and characteristic forms for principal bundles over groupoids and stacks
Authors: Biswas, Indranil; Neumann, Frank
Abstract: We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah exact sequences associated with transversal tangential distributions.; Nous construisons les connexions et formes caractéristiques pour les fibrés principaux sur
les groupoïdes et les champs dans la catégorie différentiable, holomorphe et algébrique à
l’aide des suites d’Atiyah associées aux distributions transversales tangentielles.2015-12-18T12:05:34ZMathematical Modelling of Plankton-Oxygen Dynamics Under the Climate ChangeSekerci, YadigarPetrovskii, Sergeihttp://hdl.handle.net/2381/360582015-12-12T03:11:55Z2015-12-11T16:00:50ZTitle: Mathematical Modelling of Plankton-Oxygen Dynamics Under the Climate Change
Authors: Sekerci, Yadigar; Petrovskii, Sergei
Abstract: Ocean dynamics is known to have a strong effect on the global climate change and on the composition of the atmosphere. In particular, it is estimated that about 70 % of the atmospheric oxygen is produced in the oceans due to the photosynthetic activity of phytoplankton. However, the rate of oxygen production depends on water temperature and hence can be affected by the global warming. In this paper, we address this issue theoretically by considering a model of a coupled plankton-oxygen dynamics where the rate of oxygen production slowly changes with time to account for the ocean warming. We show that a sustainable oxygen production is only possible in an intermediate range of the production rate. If, in the course of time, the oxygen production rate becomes too low or too high, the system's dynamics changes abruptly, resulting in the oxygen depletion and plankton extinction. Our results indicate that the depletion of atmospheric oxygen on global scale (which, if happens, obviously can kill most of life on Earth) is another possible catastrophic consequence of the global warming, a global ecological disaster that has been overlooked.2015-12-11T16:00:50ZThe centrifugal instability of the boundary-layer flow over slender rotating conesHussain, Z.Garrett, Stephen J.Stephen, S. O.http://hdl.handle.net/2381/360372015-12-11T03:05:26Z2015-12-10T09:45:42ZTitle: The centrifugal instability of the boundary-layer flow over slender rotating cones
Authors: Hussain, Z.; Garrett, Stephen J.; Stephen, S. O.
Abstract: Existing experimental and theoretical studies are discussed which lead to the clear hypothesis of a hitherto unidentified convective instability mode that dominates within the boundary-layer flow over slender rotating cones. The mode manifests as Görtler-type counter-rotating spiral vortices, indicative of a centrifugal mechanism. Although a formulation consistent with the classic rotating-disk problem has been successful in predicting the stability characteristics over broad cones, it is unable to identify such a centrifugal mode as the half-angle is reduced. An alternative formulation is developed and the governing equations solved using both short-wavelength asymptotic and numerical approaches to independently identify the centrifugal mode.2015-12-10T09:45:42ZUsing partially specified models to detect and quantify structural sensitivity in biological systemsAdamson, Matthew Williamhttp://hdl.handle.net/2381/359502015-11-26T03:03:19Z2015-11-25T15:53:12ZTitle: Using partially specified models to detect and quantify structural sensitivity in biological systems
Authors: Adamson, Matthew William
Abstract: Mathematical models in ecology and evolution are highly simplified representations of a complex underlying reality. For this reason, there is always a high degree of uncertainty with regards to the model specification—not just in terms of parameters, but also in the form taken by the model equations themselves. This uncertainty becomes critical for models in which the use of two different functions fitting the same dataset can yield substantially different model predictions—a property known as structural sensitivity. In this case, even if the model is purely deterministic, the uncertainty in the model functions carries through into uncertainty in the model predictions, and new frameworks are required to tackle this fundamental problem. Here, we construct a framework that uses partially specified models: ODE models in which unknown functions are represented not by a specific functional form, but by an entire data range and constraints of biological realism. Partially specified models can be used to rigorously detect when models are structurally sensitive in their predictions concerning the character of an equilibrium point by projecting the data range into a generalised bifurcation space formed of equilibrium values and derivatives of any unspecified functions. The key question of how to carry out this projection is a serious mathematical challenge and an obstacle to the use of partially specified models. We address this challenge by developing several powerful techniques to perform such a projection.2015-11-25T15:53:12ZOptimum shape problems in distributed parameter control theory.Girgis, Siham Boctor.http://hdl.handle.net/2381/345812015-11-19T08:55:48Z2015-11-19T08:55:48ZTitle: Optimum shape problems in distributed parameter control theory.
Authors: Girgis, Siham Boctor.
Abstract: The work is concerned with optimum shape problems in the distributed parameter area and it consists of four parts. In Part I we consider first the basic variational theory due to Gelfand and Fomin emphasising the importance of the transversality condition in optimum shape situations; also in Part I we discuss an application of the basic theory in a particular problem where the state equations (the constraints) are hyperbolic in character. In Part II we consider a heat transfer problem between two streams of different temperatures, moving parallel to one another and with constant speeds, the aim being to choose the inlet conditions of one stream in order to achieve desired outlet conditions for the other stream. Two different aspects of the heat transfer problem are considered. In Part III we consider a hydrodynamic problem using shallow water theory in which we seek the optimum shape of a harbour boundary in order to redistribute the liquid energy in some desired way. Here one-dimensional and two-dimensional aspects of the problem are discussed, in the former fairly precise results are achieved, and in the latter the solution of the problem is shown to depend on the solution of coupled integral equations. In Part IV we consider the problem of optimum shape of an axially symmetric elastic body (subject to the classical equations of elasticity) in order to minimise the axial moment of inertia or the weight of the body. An approximate method for finding the optimum shape is presented though considerable work remains to be done in this problem.2015-11-19T08:55:48ZDistributed parameter theory in optimal control.Gregson, M. J.http://hdl.handle.net/2381/345822015-11-20T03:16:41Z2015-11-19T08:55:48ZTitle: Distributed parameter theory in optimal control.
Authors: Gregson, M. J.
Abstract: The main result of this work is the solution of open loop optimal control problems for counterflow diffusion processes, which occur very widely in chemical and mechanical engineering. In these processes two fluids pass each other moving in opposite directions separated by a membrane which is permeable to heat or a chemical solute. The membrane may also take the form of a liquid-gas interface. Subject to certain simplifying assumptions, the equations describing such processes are 01 (x,t), 02 (x,t) are the temperatures, or concentrations of solute, of the two fluids and u(t), v(t) are time dependent flow rates. k is a transfer coefficient which is assumed constant, and C1, C2 are thermal or solute capacities of the fluids per unit length of tube. h is an equilibrium constant; h = 1 for heat transfer. Possible controls are the inlet temperature or concentration of one stream and the flow rates, while possible objectives are the regulation of the outlet temperature or concentration of the other stream, or the maximisation of heat or solute transfer. Subsidiary results are the optimal control of simpler but related hyperbolic systems. One of these is the restricted counterflow problem in which the controlling stream is assumed to be so massive that it is unaffected by giving up heat or solute to the controlled stream, i.e. the system is described by the equations ; Another is the furnace equation in which u and w are possible controls. Different classes of problem arise according to whether the multiplicative controls u and v are subject to rigid constraints (frequently leading to "bang-bang" controls), or whether they are constants, functions of x and t, or functions of t only. Variational methods based on the maximum principle of A.I. Egorov are employed. Analytic solutions and numerical solutions using finite differences are obtained to the various problems. The simplifying assumptions made are probably too severe for many of the results to be directly applicable to industry. However the qualitative features of the optimal control of these processes are explained, and it is not too difficult to build more complex models.2015-11-19T08:55:48ZApplications of variational theory in certain optimum shape problems in hydrodynamics.Essawy, Abdelrahman Hussein.http://hdl.handle.net/2381/345802015-11-20T03:16:56Z2015-11-19T08:55:47ZTitle: Applications of variational theory in certain optimum shape problems in hydrodynamics.
Authors: Essawy, Abdelrahman Hussein.
Abstract: PART I In a recent paper Wu, T.Y. & Whitney, A.K., the authors studied optimum shape problems in hydrodynamics. These problems are stated in the form of a singular integral equation depending on the unknown shape and an unknown singularity distribution; the shape is then to be determined so that some given performance criterion has to be {lcub}maximized/minimized{rcub} In the optimum problem to be studied in this part we continue to assume that the state equation is a linear integral equation but we generalize the Wu & Whitney theory in two different ways. This method is applied to functional of quadratic form and a necessary condition for the extremum to be a minimum is derived. PART II The purpose of this part is to evaluate the optimum shape of a two-dimensional hydrofoil of given length and prescribed mean curvature which produces {lcub}maximum lift/minimum drag{rcub} The problem is discussed in three cases when there is a {lcub}full/partial/zero{rcub} cavity flow past the hydrofoil. The liquid flow is assumed to be two-dimensional steady, irrotational and incompressible and a linearized theory is assumed. Two-dimensional vortex and source distributions are used to simulate the two dimensional {lcub}full/partial/zero{rcub} cavity flow past a thin hydrofoil. This method leads to a system of integral equations and these are solved exactly using the Carleman-Muskhelishvili technique. This method is similar to that used by Davies, T.V. We use variational calculus techniques to obtain the optimum shape of the hydrofoil in order to {lcub}maximized/minimized{rcub} the {lcub}lift/drag{rcub} coefficient subject to constraints on curvature and given length. The mathematical problem is that of extremizing a functional depending on {lcub}? vortex strength/ ? source strength{rcub} these three functions are related by singular integral equations. The analytical solution for the unknown shape z and the unknown singularity distribution y has branch-type singularities at the two ends of the hydrofoil. Analytical solution by singular integral equations is discussed and the approximate solution by the Rayleigh-Ritz method is derived. A sufficient condition for the extremum to be a minimum is derived from consideration of the second variation. PART III The purpose of this work is to evaluate the optimum shape of a two-dimensional hydrofoil of given length and prescribed mean curvature which produces minimum drag. A thin hydrofoil of arbitrary shape is in steady, rectilinear, horizontal motion at a depth h beneath the free surface of a liquid. The usual assumptions in problems of this kind are taken as a basis, namely, the liquid is non-viscous and moving two-dimensionally, steadily and without vorticity, the only force acting on it is gravity. With these assumptions together with a linearization assumption we determine the forces, due to the hydrofoil beneath a free surface of the liquid. We use variational calculus techniques similar to those used in Part II to obtain the optimum shape so that the drag is minimized. A sufficient condition for the extremum to be a minimum is derived from consideration of the second variation. In this part some general expressions are established concerning the forces acting on a submerged vortex and source element beneath a free surface using Blasius theorem.2015-11-19T08:55:47ZOptimum shape problems for distributed parameter systems.Edwards, Janet Mhttp://hdl.handle.net/2381/345792015-11-20T03:16:33Z2015-11-19T08:55:47ZTitle: Optimum shape problems for distributed parameter systems.
Authors: Edwards, Janet M
Abstract: In this thesis the variation of a functional defined on a variable domain has been studied and applied to the problem of finding the optimum shape of the domain in which some performance criterion has an extreme. The method most frequently used is one due to Gelf and Fomin. It is applied to problems governed by first and second order partial differential equations, unsteady one dimensional gas movements and the problem of minimum drag on a body with axial symmetry in Stokes flow.2015-11-19T08:55:47ZMany-valued logics. a study of the relationship of propositional calculi and algebraic systems.Cuninghame-Green, Raymond.http://hdl.handle.net/2381/345782015-11-19T08:55:46Z2015-11-19T08:55:46ZTitle: Many-valued logics. a study of the relationship of propositional calculi and algebraic systems.
Authors: Cuninghame-Green, Raymond.
Abstract: This thesis sets out to examine the possibility of devising a theory which will give a unified account of prepositional calculi and algebraic systems. Starting from a historical account of the principal ideas tributary to the main stream of theory from Boole to the present day, it presents a technical- language framework within which it is possible to develop in a uniform format substantial portions of the theories of both sorts of system. The idea of an Interpretation then leads to a discussion of Functional Completeness, and the use of Galois fields in the algebraic representation of functions. Two particular families of systems, the Protomodules and Protorings, are selected for more detailed study. Their principal decision problems are considered, their structure examined, and their relationship to familiar systems of algebra and prepositional calculus displayed. The discussion then specialises again to the use of Galois fields in the solution of computational problems arising in connection with an important class of protorings, the so- called Galois Logics. One of these problems is of sufficient complexity to warrant the use of an automatic digital computer, and details of the computer program are presented in an appendix. Three other appendices are devoted to the presentation of material which evolved as by-products during the contemplation of the main issues; they are concerned with closely related topics, and are given here in support of the thesis rather than as part of the theory.2015-11-19T08:55:46ZParameter reduction in definition by multi-successor recursion.Burville, J. C.http://hdl.handle.net/2381/345772015-11-20T03:16:57Z2015-11-19T08:55:45ZTitle: Parameter reduction in definition by multi-successor recursion.
Authors: Burville, J. C.
Abstract: It is well known that in primitive recursive arithmetic with a single successor the number of parameters in a definition by recursion may be successively reduced. In this thesis I examine the possibility of effecting a similar reduction in the number of parameters in a definition by recursion in a multi-successor arithmetic. The reduction process involves the discovery in multi-successor arithmetic of analogues of pairing functions and of functions which select the elements of an ordered pair. One of the difficulties in finding such functions is the construction within multi-successor arithmetic of suitable product and square foot functions and establishing the properties of these functions, and the pairing functions, within a formalisation of multi-successor arithmetic. The reduction process involves of course an examination of what functions, if any, need to be adjoined to the initial functions to secure the generality of the reduction.2015-11-19T08:55:45ZComposition algebras and their generators.Wheeler, Roger F.http://hdl.handle.net/2381/345752015-11-19T08:55:44Z2015-11-19T08:55:44ZTitle: Composition algebras and their generators.
Authors: Wheeler, Roger F.
Abstract: The aim of this thesis is to show how the study of composition algebras and their generators has developed from a simple observation in logic made by Henry Maurice Sheffer nearly 60 years ago. The results in the algebra on 2 marks, which corresponds to classical 2-value sentence logic, were firmly established when Emil Post wrote a monograph on the subject 30 years ago. In this dissertation, however, they are developed in a more coherent and systematic way than has been attempted before and it is hoped that some novelty can be claimed for this exposition. More recent work has concentrated on the algebra on 3 marks (to which the author has made a published contribution) and on the general algebra. The outstanding problem in the general case has, in fact, been solved quite recently by Ivo Rosenberg. This thesis does not try to cover these later developments comprehensively; it concentrates on investigating and elucidating aspects of the subject that the author has found interesting and elegant.2015-11-19T08:55:44ZA formalisation of the arithmetic of transfinite ordinals in a multisuccessor Equation calculus.Williams, H. P.http://hdl.handle.net/2381/345762015-11-20T03:16:46Z2015-11-19T08:55:44ZTitle: A formalisation of the arithmetic of transfinite ordinals in a multisuccessor Equation calculus.
Authors: Williams, H. P.
Abstract: This thesis presents a syntactic development of the arithmetic of ordinal numbers less than This is done by means of an Equation calculus v/here.all statements are given in the form of equations. There are rules of inference for deriving; one equation from another. Certain functions, including a countably infinite number of successor functions are taken as primitive. New functions are defined by substitution and primitive recursion starting with the primitive functions. Such definitions constitute some of the axioms of the system. The only other axioms are two rules concerning the combination of successor functions, Fundamental for this development is the axiom. In this system a multisuccessor arithmetic is developed in which it is possible to prove many of the familiar results concerning trans-finite ordinal numbers. In particular the associativity of addition and multiplication as well as multiplication being left distributive with respect to addition are proved. It is shown that each ordinal in the system can be represented in Cantor's Normal Form. An ordinal subtraction is defined and a number of results involving this are proved. It is shown that this subtraction is, in a number of respects, an inverse to addition. In particular the key-equation is proved. As in Professor Goodstein's formalisation of the primitive recursive arithmetic of the natural numbers this equation is important as it allows a difference function to be defined for which a zero value is equivalent to equality of the arguments. Inequality relations are defined and some results concerning them proved. In Chapter II it is shown, using a suitable coding, that this arithmetic can be reduced to the primitive recursive arithmetic of the natural numbers. Chapter III gives a meta-proof of the consistency of the system. Also submitted with this thesis is a paper The Synthesis of Logical Nets consisting of NOR units which is the result of work on a logical problem which was done at the same time as work for the thesis.2015-11-19T08:55:44ZTowards a theory of multivariate interpolation using spaces of distributions.Wayne, Henry.http://hdl.handle.net/2381/345742015-11-20T03:17:00Z2015-11-19T08:55:43ZTitle: Towards a theory of multivariate interpolation using spaces of distributions.
Authors: Wayne, Henry.
Abstract: The research contained in this thesis concerns the study of multivariate interpolation problems. Given a discrete set of possibly complex-valued data, indexed by a set of interpolation nodes in Euclidean space, it is desirable to generate a function which agrees with the data at the nodes. Within this general framework, this work pursues and generalizes one approach to the problem. Based on a variational theory, we construct a parameterised family of Hilbert spaces of tempered distributions, detail the necessary evolution of the interpolation problem, and provide a general error analysis. Some of the more popular applications from the theory of radial basis functions are shown to arise naturally, but the theory admits many more examples, which are not necessarily radial. The general error analysis is applied to each of the applications, and taken further where possible. Connections with the theory of conditionally positive definite functions are highlighted, but are not central to the theme.2015-11-19T08:55:43ZSome problems in the kinetic theory of plasmas.Tapp, M. C.http://hdl.handle.net/2381/345732015-11-20T03:16:55Z2015-11-19T08:55:43ZTitle: Some problems in the kinetic theory of plasmas.
Authors: Tapp, M. C.
Abstract: This thesis covers essentially two problems in the kinetic theory of plasmas. The first concerns the investigation of plasma oscillations in a constant electric field - a topic investigated by Akheizer and Sitenko as early as 1956 [1] More recently Stenflo [2] has considered the problem in which he replaces the collision integral of Boltzmann's equation by a Fokker-Planck term and a B.G.K. term. The dispersion relations derived by Stenflo contained a number of parameters the relative importance of which he did not clearly define. We have undertaken here a stability study of longitudinal oscillations of a weakly ionised gas permeated by a uniform electric field. A dispersion relation is formulated in terms of error-type functions and some computational studies are carried out for various plasma parameters of interest. The results are exhibited graphically in the form of Nyquist plots. The conclusions made by Stenflo and others regarding possible instabilities of the plasma needs modification, certainly in the context of a weakly ionised electron-ion gas. The second topic covered here concerns the transport theory of relativistic gases. This has received increasing attention in recent years [3,4]. Much attention has been devoted to calculating the first order relativistic effects on the transport coefficients. Up to now only the 'Maxwellian' model, investigated by Israel [3], has been considered. The method of attack is via the Chapman-Enskog approach. In this second topic we develop a more general approach to the problem by generalising the classical spherical harmonic solution of the Boltzmann equation to the relativistic case. The theory is applied to transport problems of fully ionised plasmas in the Coulomb field.2015-11-19T08:55:43ZIncomplete data in event history analysis.Sutton, Christopher Julian.http://hdl.handle.net/2381/345722015-11-20T03:16:37Z2015-11-19T08:55:42ZTitle: Incomplete data in event history analysis.
Authors: Sutton, Christopher Julian.
Abstract: Incomplete data present a serious problem in the modelling of event histories. Two particular forms of incompleteness are in evidence for data of this form. The first is due to recording of event times in interval-censored form. For single non-repeatable events this can be accommodated by using methods for modelling grouped survival times, such as those of Prentice and Gloeckler (1978) and Finkel- stein (1986). The other, more serious, problem relates to incomplete recording of follow-up measurements which would typically be included as time-dependent covariates in survival models. A number of methods exist for handling incomplete data. These include multiple imputation for variables subject to incompleteness and the application of iterative algorithms such as EM and the data augmentation algorithm. In this thesis, a method for handling both these types of incompleteness is derived based on multiple imputation combined with an adaptation of Finkelstein's method to handle time-varying covariates. This method is then investigated via Monte Carlo simulation and applied to data arising from the annual screening of those aged 75 years and over in the town of Melton Mowbray, as performed through the local general practice. Its performance is compared with that of more traditional approaches to modelling data collected in studies of this type. It is shown that parameter estimates can be considerably affected by the choice of approach to modelling. Whilst there are some problems with the implementation of this technique, particularly with reference to the model for the multiple imputation of the repeated risk factor values, it shows promise for application to studies of this form, particularly if combined with improved models for multiple imputations. The data from the annual screenings are assumed missing at random, but the techniques used could be extended to cover non-ignorable missing data mechanisms of known form.2015-11-19T08:55:42ZAlglat for modules over fsi rings and reflexivity.Snashall, Nicole Jane.http://hdl.handle.net/2381/345702015-11-20T03:16:57Z2015-11-19T08:55:42ZTitle: Alglat for modules over fsi rings and reflexivity.
Authors: Snashall, Nicole Jane.
Abstract: For a bimodule RMDelta where R and Delta are rings with unity, alglat RMDelta is the ring of all Delta-endomorphisms of M leaving invariant every R-submodule of M. The bimodule is said to be reflexive if the elements of alglat RMDelta are precisely the left scalar multiplications by elements of R. For most of the thesis Delta = R, a commutative ring with unity. However, in the early work, some results on the general structure of alglat are obtained, and in particular, Theorem 1.9 shows that it is an inverse limit. The next section of the thesis is concerned with reflexivity, and considers rings R for which all non-torsion or all finitely generated R-modules are reflexive. Theorem 3.4 gives eight equivalent conditions on an h-local domain R to the assertion that every finitely generated R-module is reflexive, that is R is scalar- reflexive. A local version of this property is introduced, and it is shown in Theorem 2.17 that a locally scalar-reflexive ring is scalar-reflexive. The remainder of this thesis considers alglat for all modules over an FSI ring. The local FSI rings are precisely the almost maximal valuation rings, and this is the first case to be settled. More details are then given of the structure of FSI rings and related rings. A completion is introduced in 6.4 to enable alglat to be determined for certain torsion modules over an indecomposable FSI ring. Theorem 7.3, in summarising the work of the last two chapters of the thesis, gives a complete characterisation of alglat for all modules over an FSI ring.2015-11-19T08:55:42ZSuccessor systems. An investigation into the primitive recursive functions of generalised multisuccessor arithmetics, with applications to constructive algebra.Stanford, Paul Hudson.http://hdl.handle.net/2381/345712015-11-20T03:16:16Z2015-11-19T08:55:42ZTitle: Successor systems. An investigation into the primitive recursive functions of generalised multisuccessor arithmetics, with applications to constructive algebra.
Authors: Stanford, Paul Hudson.
Abstract: An investigation into the primitive recursive functions of generalised multisuccessor arithmetics, with applications to constructive algebra.' Submitted for the degree of Doctor of Philosophy by Paul Hudson Stanford* at Leicester University, England, in 1975. The above named thesis is concerned with the extension of the notion of primitive recursion to structures other than the natural numbers. Successor systems are generalisations of the arithmetics of Vu?kovi? [2], and as a class are closed under operations corresponding to direct products and quotient formation. Given a system ? we can also define a system a* which has successor functions Sax for each numeral a of ?. The formalisation used is derived from the free variable calculus of Goodstein [1]. Various forms of recursion are considered, none of which employ more than a small number of known functions. For example, given a function g from ? x ? to ? we can define f from ?* to ? as follows. f(0) = 0; f(Sax) = g(a,f(x)) Algebraic applications include the construction of groups and rings: actual examples range from the integers and polynomials to permutations, finite sets and ordinal numbers. Several relations which may hold between systems are investigated, as are the notions of anchored and decidable systems.*(supported by a Science Research Council grant) One chapter deals with the case of commuting successor functions, and another considers systems with only one successor. In an appendix we briefly investigate the further generalisation obtained by using non-unary successor functions. The author expresses his thanks to all concerned, especially his supervisor. Professor R. L. Goodstein. Contents of thesis: (1) Introduction, (2) The Integers, (3) Products, (4) Recursion, (5) The Star Operation, (6) Commutative systems, (7) Homomorphisms, (8) Groups, (9) Further recursion, (10) Decidable systems, (11) Single successor systems, (12) Polynomials; (A1) Small systems, (A2) Joint successor arithmetics, (A3) Polish Circles, (A4) A Formalisation of the Integers. References to abstract: [1] Goodstein, R.L., Recursive Number Theory, Amsterdam (1957) [2] Vu?kovi?, V., Partially ordered recursive arithmetics, Math.Scand. 7 (1959), 305-320.2015-11-19T08:55:42ZFunctional-completeness criteria for finite domains.Schofield, P. (Peter)http://hdl.handle.net/2381/345692015-11-20T03:16:44Z2015-11-19T08:55:42ZTitle: Functional-completeness criteria for finite domains.
Authors: Schofield, P. (Peter)
Abstract: Necessary and sufficient conditions for the functional completeness of a set F of functions with variables and values ranging over N = {lcub}0,1,...,n{rcub}, where n ? 1, are investigated and in particular, completeness criteria for a single function are determined. Complete solutions are known in the special cases n = 1,2, and results about these special cases which are of use in formulating general theorems are discussed. Proceeding to the general case some preliminary criteria (which presuppose that certain 2-place functions are generated by F) for the functional completeness of F are derived. These results show that the set consisting of all 2-place functions is complete. In the special case n + 1 = p (a prime number) the functions of F are shown to have a special form, and this is used in some illustrations of complete subsets. The value sequence of a function satisfying the Stupecki conditions (that is, depending on at least 2 of its argument places, and taking all n + 1 values of N) is now examined, and some properties of such a function are found. These results are then used in demonstrating the completeness of a set F which generates all 1-place functions, together with a function satisfying the Stupecki conditions. Our main results give improved sufficient conditions for the completeness of F. In particular a set F is complete if it generates a triply transitive group of permutations of N and contains either (i) only a single function or (ii) at least one function satisfying the Stupecki conditions, the latter apart from certain exceptional cases. A detailed investigation shows that these occur only when n = 2 or when n + 1 is a power of 2 and all functions of F are linear in each variable, relative to some mapping of N as a vector space over Z2. Finally a different mapping of N into Z42 is considered, and it is shown that the functions of F can be given a unique representation relative to this mapping. This representation is then used to find some examples of complete subsets.2015-11-19T08:55:42ZFormalisations of recursive arithmetic.Rose, H. E. (Harvey Ernest)http://hdl.handle.net/2381/345652015-11-19T08:55:41Z2015-11-19T08:55:41ZTitle: Formalisations of recursive arithmetic.
Authors: Rose, H. E. (Harvey Ernest)
Abstract: In this thesis we shall present a new formalisation of the theory of primitive recursive functions, which is called Ternary Recursive Arithmetic. In a recent paper, Alonzo Church described a formalisation of recursive arithmetic in which single axioms of recursion and composition (i.e. definition by explicit substitution) took the place of an infinity of such axioms in earlier codifications. Church's system, however, postulates axioms of the propositional calculus and of mathematical induction, in Ternary Recursive Arithmetic these axioms have been eliminated in the manner of Goodstein. In chapter 1 a full statement of the primitive basic of the system will be given and in chapters 2, 3 and 4 we shall present a development of it and state precisely in what sense it may be considered a formalisation of the theory of primitive recursive functions. The main motivation of this work is that it enable us to give a proof of the consistency of primitive recursive arithmetic in a much simpler system than was hitherto possible; that is, in the system consisting of Ternary Recursive Arithmetic with one additional axiom. This proof and a discussion of the Godel incompleteness theorems are given in chapters 6 and 7. In presenting these results we have given the more routine work, which is necessary but does not form an essential part of the development, at the ends of the corresponding chapters, sections 3.7, 4.5 and 5.4 fall into this category. (Abstract shortened by UMI.).2015-11-19T08:55:41ZThe metatheory of the elementary equation calculus.Bundy, A.http://hdl.handle.net/2381/345662015-11-20T03:16:51Z2015-11-19T08:55:41ZTitle: The metatheory of the elementary equation calculus.
Authors: Bundy, A.
Abstract: Abstract not available.2015-11-19T08:55:41ZThe convective instability of the boundary-layer flow over families of rotating spheroids.Samad, Abdul.http://hdl.handle.net/2381/345682015-11-20T03:16:29Z2015-11-19T08:55:41ZTitle: The convective instability of the boundary-layer flow over families of rotating spheroids.
Authors: Samad, Abdul.
Abstract: The majority of this work is concerned with the local-linear convective instability analysis of the incompressible boundary-layer flows over prolate spheroids and oblate spheroids rotating in otherwise still fluid. The laminar boundary layer and the perturbation equations have been formulated by introducing two distinct orthogonal coordinate systems. A cross-sectional eccentricity parameter e is introduced to identify each spheroid within its family. Both systems of equations reduce exactly to those already established for the rotating sphere boundary layer. The effects of viscosity and streamline-curvature are included in each analysis. We predict that for prolate spheroids at low to moderate latitudes, increasing eccentricity has a strong stabilizing effect. However, at high latitudes of 0 60, increasing eccentricity is seen to have a destabilizing effect. For oblate spheroids, increasing eccentricity has a stabilizing effect at all latitudes. Near the pole of both types of spheroids, the critical Reynolds numbers approach that for the rotating disk boundary layer. However, in prolate spheroid case near the pole for very large values of e, the critical Reynolds numbers exceed that for the rotating disk. We show that high curvature near the pole of prolate spheroids is responsible for the increase in critical Reynolds number with increasing eccentricity. For both types of spheroids at moderate eccentricity, we predict that the most amplified modes travel at approximately 76% of the surface speed at all latitudes. This is consistent with the existing studies of boundary-layer flows over the related rotating-disk, -sphere and -cone geometries. However, for large values of eccentricity, the traveling speed of the most amplified modes increases up to approximately 90% of the surface speed of oblate spheroids and up to 100% in the prolate spheroid case.2015-11-19T08:55:41Z