DSpace Community:
http://hdl.handle.net/2381/445
2016-02-13T02:34:15ZMultiscale principal component analysis
http://hdl.handle.net/2381/36616
Title: 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 filters
http://hdl.handle.net/2381/36471
Title: 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 processes
http://hdl.handle.net/2381/36413
Title: 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 systems
http://hdl.handle.net/2381/36295
Title: 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 exponents
http://hdl.handle.net/2381/36266
Title: 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 Averaging
http://hdl.handle.net/2381/36247
Title: 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)
http://hdl.handle.net/2381/36243
Title: 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 investment
http://hdl.handle.net/2381/36189
Title: 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 spaces
http://hdl.handle.net/2381/36151
Title: 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 convolution
http://hdl.handle.net/2381/36144
Title: 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 convolution
http://hdl.handle.net/2381/36143
Title: 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 measures
http://hdl.handle.net/2381/36136
Title: 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 Fleets
http://hdl.handle.net/2381/36130
Title: 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 Degeneration
http://hdl.handle.net/2381/36118
Title: 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 stacks
http://hdl.handle.net/2381/36102
Title: 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 Change
http://hdl.handle.net/2381/36058
Title: 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 cones
http://hdl.handle.net/2381/36037
Title: 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 systems
http://hdl.handle.net/2381/35950
Title: 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.
http://hdl.handle.net/2381/34581
Title: 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.
http://hdl.handle.net/2381/34582
Title: 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.
http://hdl.handle.net/2381/34580
Title: 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.
http://hdl.handle.net/2381/34579
Title: 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.
http://hdl.handle.net/2381/34578
Title: 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.
http://hdl.handle.net/2381/34577
Title: 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.
http://hdl.handle.net/2381/34575
Title: 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.
http://hdl.handle.net/2381/34576
Title: 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.
http://hdl.handle.net/2381/34574
Title: 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.
http://hdl.handle.net/2381/34573
Title: 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.
http://hdl.handle.net/2381/34572
Title: 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.
http://hdl.handle.net/2381/34570
Title: 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.
http://hdl.handle.net/2381/34571
Title: 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.
http://hdl.handle.net/2381/34569
Title: 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.
http://hdl.handle.net/2381/34565
Title: 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.
http://hdl.handle.net/2381/34566
Title: 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.
http://hdl.handle.net/2381/34568
Title: 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.2015-11-19T08:55:41ZLogical systems with finitely many truth values.
http://hdl.handle.net/2381/34567
Title: Logical systems with finitely many truth values.
Authors: Rousseau, G.
Abstract: Abstract not provided.2015-11-19T08:55:41ZTransmission of guided sound waves through a layer of fluid or solid.
http://hdl.handle.net/2381/34564
Title: Transmission of guided sound waves through a layer of fluid or solid.
Authors: Romilly, N.
Abstract: The thesis considers the transmission of sound waves through a layer of fluid or solid contained in a wave-guide of a simple form. The main aim is to find the transmission coefficient for a lowest order incident mode and to fine the lengths of the layer for which the transmission is a maximum or minimum. The first part of the thesis gives the exact solution for transmission through a layer of inviscid fluid, and for transmission through a layer of viscous fluid when the boundaries of the guide are rigid and lubricated. It also gives approximate solutions for transmission through a layer of viscous fluid when the boundaries of the guide are pressure-free and when they are rigid but not lubricated. The second part of the thesis considers transmission through a layer of solid. It gives the exact solution, in infinite series form, to the problem of the transmission of any incident waveguide mode through a stretched membrane contained in a rigid circular guide. It is shown that above a certain frequency an incident plane wave can never be completely transmitted or completely reflected. Below this frequency complete transmission or reflection can occur, but the frequencies at which it does occur depend on the medium surrounding the membrane. The solution is discussed and results are given for a particular case and compared with approximate solutions obtained by other authors. The same analysis is applied to transmission through a thin plate. The second part of the thesis also contains work on transmission through a thick layer of elastic solid. An exact solution is found using an approximate equation of motion for the solid which should be valid at low frequencies. An attempt is made to find a solution based on the exact equations for the solid, but it is necessary to use an approximation.2015-11-19T08:55:39ZThe classification of ultrafilters on N.
http://hdl.handle.net/2381/34562
Title: The classification of ultrafilters on N.
Authors: Pitt, R. A.
Abstract: This thesis has as its aim the classification of ultrafilters on N, by use of partitions and collections of partitions of N, and the investigation of the operations under which each class is closed and the inclusion/exclusion relationships between them. Choquet (1) asked whether there was a n.p.u.f ? on N such that for no map 0 : N ? N is 0 absolute; Mathias answered that there was by constructing a n.p.u.f. ? on N with the stronger property that for no map 0 : N ? N is 0 a P point (of ?N-N). In Chapter II we construct a P point ? such that for no map 0 : N ? N is 0 rare and, using this result, we construct a n.p.u.f. ? on N such that for no map 0 : N ? N is 0 a P point or a rare ultrafilter. We also show that if ? is a n.p.u.f. on N that cannot be mapped to a rare ultrafilter then ? cannot be mapped to a countable limit of absolute ultrafilters.;NOTATION: Let ? be a n.p.u.f. on N and s be a partition of N into finite sets (p.o.N.i.f.s.); we will write ? ? s whenever F ? ? implies |F ? A| ? 1 for each A ? s. Let S1,S2 be p.o.N.i.f.s.; we will write S1?S2 whenever A E S1 and B ? S2 imply |A ? B| ? 1.;DEFINITION: A n.p.u.f. ? on N is an n(a) point (point of degree of complexity n) if for every collection S = S1,S2,...,Sn+1 of p.o.N.i.f.s. satisfying Si?sj for 1 ? i < j ? n+1 there is a t, 1 ? t ? n+1 such that ??st, and this is the least n for which it is true. Chapter III is devoted to the investigation of this and allied notions. We extend the idea to allow infinite degrees of complexity (S. and c) and show that for any n.p.u.f. ? on N there is a n ? {lcub}0,1,.., s.,c{rcub} such that ? is a n(a) point. We also show that for any n,n(a) P points exist. Many of the theorems give bounds for the degree of complexity of ultrafilters of the form ? = ? lim ?i given the degree of complexity of ?, ?1, ?2,.. and given that ? has a certain property (e.g. ? is a P point). The first section of Chapter IV gives counterexamples to the following plausible hypotheses: 1) each 1(a) point is rapid; 2) each c(a ) point is not rapid. The final section of the thesis deals with a concept appearing in a letter from Professor G. Choquet to Dr. R.O.Davies.;DEFINITION: A n.p.u.f. ? on N has property c if for any pair of maps 0,? : N ? N, 0 = ? implies 0 and ? agree on some member of ?. We show that there is a n.p.u.f. ? on N that is neither a P point nor a rare ultrafilter with property c and a n.p.u.f. on N that is a P point without property c. We investigate the relationships between the class of n.p.u.f.'s with property c and the classes defined previously. 1) G. Choquet, Deux classes remarquables d'ultrafiltres sur N, Bull.Sci.Math.(2) 92 (1968), 143-153.2015-11-19T08:55:38ZThe water wave - ice floe interaction and associated integral equation problems.
http://hdl.handle.net/2381/34563
Title: The water wave - ice floe interaction and associated integral equation problems.
Authors: Porter, D.
Abstract: The water wave - ice floe interaction is introduced by reviewing the work done to date on the problem. Several mathematical models, incorporating hitherto unexplored and possibly significant mechanisms of the interaction, are then constructed and investigated. In the first place, the effect of a plane wave incident at any angle upon a semi-infinite elastic sheet of constant thickness is considered, using linearised shallow water theory. The solution for the velocity potential under the ice is discussed for various values of the physical parameters, and in the most interesting case, numerical calculations are made to determine the relevance of such factors as ice thickness and angle of incidence. Secondly, a semi-infinite sheet of variable thickness is examined and the particular case treated when this thickness has a sinusoidal form. Ranges of incident wavelengths corresponding to a progressive wave solution under the ice are calculated. Also, an ice thickness having a rectangular wave form is considered with similar results. Attention is then turned to the problem of the existence of a progressive wave in an infinite array of rigidly held, equally spaced floes. Two different approaches are employed to reduce the resulting potential problem to weakly singular integral equations, which in turn are solved by a perturbation method, and, in the general case, by a numerical technique. It is found that complex wave groups can be constructed satisfying the problem, but that simple progressive waves do not exist. In an attempt to make analytic inroads on the above mentioned integral equations, some aspects of singular integro-differential equations are investigated, and methods developed by which these may be solved. The closely associated generalised Riemann-Hilbert problem is also discussed and two integro-differential equations arising in aerodynamic theory are solved as examples of the techniques proposed.2015-11-19T08:55:38ZThe Hamiltonian formulation in relativity.
http://hdl.handle.net/2381/34560
Title: The Hamiltonian formulation in relativity.
Authors: Palfreyman, Niall M.
Abstract: Like any major breakthrough in thinking, the theory of relativity caused a great upheaval in our attitude to science. Seventy years after the advent of relativity we are still coming to terms with the changes it has brought in our outlook. Part of this process is simply the valid translation of pre-relativistic laws and concepts into the 4-dimensional language of relativity - a problem by no means as easy as would at first seem; the aim of this thesis is to survey the ways in which the methods of analytical mechanics may be translated into a relativistic setting. Chapter 1 provides an introduction to the work in the form of a non-rigorous discussion of the historical and mathematical development of electromagnetism, analytical mechanics and relativity, and ends with a presentation of the basics of the functional calculus. This is needed in the presentation of field theory given in chapter 2. We see two possibilities for the relativistic formulation of analytical mechanics, and field theory represents the first of these possibilities. In the absence of any real grounds for continuing on this tack we then move on to the other possibility in chapter 3, where we review the attempts of a number of authors to formulate relativistic particle mechanics as a Hamiltonian system. This then leads in chapter 4 to our own such attempt, based mainly on the work of Synge, which we have named homogeneous mechanics. After the main exposition of the theory the work of the remaining chapters 5 and 6 is then to apply the above theory (not always successfully) to a number of cases where analytical mechanics has in the past proven itself an invaluable tool: namely, the areas of symmetries and quantum theory.2015-11-19T08:55:37ZCommutative multiple successor recursive arithmetics.
http://hdl.handle.net/2381/34561
Title: Commutative multiple successor recursive arithmetics.
Authors: Partis, M. T. (Michael T)
Abstract: Recursive arithmetics are usually based on three initial functions, namely the zero, successor and identity functions. In this thesis recursive arithmetics are considered which instead of having just one successor function have a number of different successor functions. These will be represented by Sv where v ranges from 1 to n. The system is made commutative by stipulating that SuSvx = SvSux for all u and v. The notion of a primitive recursive function is introduced into this arithmetic and various basic functions are defined. Another recursive arithmetic is then constructed in which the elements are ordered sets of natural numbers. It is shown that a complete isomorphism, both functional and deductive, exists between this arithmetic and the arithmetic with n successors. It is then shown by using this isomorphism that a proof can be constructed of the key equation x + (y - x) = y + (x - y) in multiple successor recursive arithmetic. A formal equation calculus is then developed for multiple successor recursive arithmetic in which the proof of the key equation given above is derived without resource to a doubly recursive uniqueness rule. The properties of the basic primitive recursive functions are also established. The problem of avoiding irregular models of this equation calculus is then examined and it is shown that this can be done by using relatively simple axioms. An inequality relationship is then defined and it is shown that with respect to this relationship the numbers of a multiple successor recursive arithmetic form a lattice. It is then shown that this lattice is modular and distributive. The problem of introducing limited universal and existential quantifiers is then considered. It is shown that this can be done in an arithmetic of ordered sets and hence, by the isomorphism established earlier, they can also be introduced into a multiple successor recursive arithmetic. Three different logical models in multiple successor recursive arithmetic are then considered. The models are of classical two-valued logic, a modified form of Heyting's intuitionist logic, and a many-valued logic. The connection between these models is examined.2015-11-19T08:55:37ZIntermediate propositional logics.
http://hdl.handle.net/2381/34558
Title: Intermediate propositional logics.
Authors: McKay, C. G.
Abstract: The main object of the thesis is to investigate a variety of questions relating to the set of intermediate prepositional logics. Let H denote the set of words which are intuitionist theses and let K denote the set of words which are classical theses. Then a set of words X is an intermediate (prepositional) logic iff 1) HcXcK and 2) X is closed wrt modus ponens and substitution. Of special interest among intermediate logics, are those which are characterised by a finite pseudocomplemented lattice. We prove the important result that every such finite logic is finitely axiomatisable. This result is one of the many consequences of the fundamental representation theorem for pseudocomplemented lattices (PLs) whereby every PL is subdirectly reducible to a set of so-called strongly compact PLs. In addition we provide a neat syntactic characterisation of finite logics, and show that H is the limit of a certain sequence of explicitly axiomatised finite logics. In addition we consider more restricted types of intermediate logics, in particular intermediate ICN logics. By generalising a result of DIEGO, to show that every ICN algebra with a finite number of generators, is finite, we manage to prove that every finitely axiomatised intermediate ICN logic is decidable with primitive recursive bound. This generalises and completes earlier work of BULL. The same methods are then applied to obtain a proof of the decidability of all those intermediate logics, obtained by adding a finite set of disjunction-free words, as additional axioms to H. Many older results in the literature are then seen to be special cases of this general result. We introduce the new concept of strong undefinability of a prepositional connective, and examine its relation to McKINSEY'S related notion. It is shown that the connectives of implication, disjunction and negation, are all strongly undefinable in H, whereas conjunction is weakly definably. Lastly we investigate the scope of the so-called Separation theorem in the field of intermediate logics. It is shown that certain intermediate logics treated in the literature do not possess any axiomatisation for which the Separation theorem can be proved.2015-11-19T08:55:36ZOperations on generalized functions.
http://hdl.handle.net/2381/34559
Title: Operations on generalized functions.
Authors: Özçag, Emin.
Abstract: In Chapter 1, we give some properties distributions and introduce the notions of neutrix and neutrix limit with examples, in order to study the problem of defining the convolution product and the product of distributions. The problem of defining the distribution such that the ordinary derivative formula is satisfied for all and s = 0,1,2,... is studied in Chapter 2. In Chapter 3, we define the Beta function Bp,q (,) using the neutrix limit and prove that this neutrix limit exists for all . In Chapter 4 we let f and g be distributions and let fn(x) = f(x)Tn(x), where Tn(x) is a certain function which converges to the identity function as n tends to infinity. We then define the neutrix convolution product fg as the neutrix limit of the sequence {lcub}fn * g{rcub}, provided the limit h exists in the sense that N - limn fn * g,? = h, for all in D. The neutrix convolution products In are evaluated, from which other neutrix convolution products are deduced. The neutrix convolution product of distributions in Chapter 4 is not commutative. Therefore, in Chapter 5, we consider the commutative neutrix convolution product of distributions, *, and also evaluate the neutrix convolution product. The problem of defining the product of ultradistributions is considered in Chapter 6, and the neutrix product (Ff) (Fg) in Z', where F denotes the Fourier transform, is defined as the neutrix limit of {lcub}F(fTn).F(gTn). Later, we prove that the exchange formula holds. We finally define the neutrix product F(f)0G(g) of F(f) and G(g), where F and G are distributions and f and g are locally summable functions. It is proved that if f is infinitely differentiable function with f'(x) 0 and if the neutrix product F o G exists and equals H, then the neutrix product F(f) o G(f) exists and equals H(f). We also give an alternative approach to the form F(f(x)) in D', where F and f are distributions.2015-11-19T08:55:36ZAn investigation of the propagation of electromagnetic waves in some circular cylindrical waveguides using a finite difference formulation.
http://hdl.handle.net/2381/34556
Title: An investigation of the propagation of electromagnetic waves in some circular cylindrical waveguides using a finite difference formulation.
Authors: Lawrence, P. J.
Abstract: This thesis is concerned with the propagation of electromagnetic waves through circular cylindrical waveguide having perfectly conducting walls. A finite difference approximation method is used to evaluate the propagation constant of the waves. The method is one of great generality. It may be used for any coaxial configuration of media inside the waveguide. In particular, the effects on propagating electromagnetic waves of a transversely magnetised ferrite tube adjacent to the waveguide wall are studied. Ferrite material is taken to have a permeability tensor of the form [image] when it is subjected to a static magnetic field along its third coordinate axis. The ferrite tube is subject to a static magnetic field formed by four magnetic poles at the corners of a square centred on the axis of the guide, like poles being at opposite corners. In the ferrite, this field leads to a permeability tensor which is dependent upon the angle in cylindrical polar coordinates when the z-axis is taken along the guide and Maxwell's equations reduce to two simultaneous second order partial differential equations with non-constant coefficients in the EZ and HZ components of the propagating electromagnetic wave. The finite difference approximation method reduces the problem to one of solving the condition for consistency of a large number of difference equations. Values of the propagation constant which satisfy this condition are found by a trial method which involves evaluating a determinant of very high order. This evaluation is carried out by computer and use is made of the banded nature of the determinant to prevent the amount of computer store required becoming prohibitive. The validity of the method is tested by applying it to several special cases with known results and its limitations and accuracy are discussed. A hypothesis is suggested to explain the numerical results.2015-11-19T08:55:36ZDecidable classes of recursive equations.
http://hdl.handle.net/2381/34557
Title: Decidable classes of recursive equations.
Authors: Lee, R. D.
Abstract: Many different formalisations of recursive arithmetic have been proposed, and this thesis is concerned mainly with the system proposed by R.L. Goodstein and known as the Axiom - Free Equation Calculus. As with all other formal systems of arithmetic with sufficient content, the system is incomplete and recursively undecidable. The interesting questions lie in the completeness and decidability, or otherwise, of fragments of the system. I attempt to answer some of these questions. It happens that some of the problems lead to well known questions in the theory of diophantine equations namely, Hilbert's 10th Problem, The Undecidability of Exponential Diophantine Equations, and the Integer Linear Programming Problem. In 1943 Kalmar proposed a set of functions called elementary functions, and Ilona Bereczki showed effectively that the class of equations F = 0, where F is any elementary function, is undecidable. The class of functions given by Kalmar was, variables, l,+,., |a - b|, [a/b], but it can easily be shown that this is the same as those formed by composition from +,.,? This latter definition is the one we use. In his paper, A Decidable Fragment of Recursive Arithmetic, Goodstein showed the class of equations F = 0 where F is any function formed by composition from x + y, x.y and 1 ? x is decidable. So I have attempted to extend Goodstein's result with the upper bound provided by the undecidability of the elementary equations. The main results I have obtained are 1. If F is any function formed by composition from x + y, x.y, 1 ? x, ? 1, E y=w, II y=w, then F = 0 is decidable, and furthermore the provability in the equation calculus of F = 0 is decidable and that this class of equations is complete. 2. If F,G are any functions formed from x + y, x.y, 1 ? x, x ? 1, by composition, then the class of equations F = G is decidable. 3. If F,G are any functions formed by composition from x + y, x ? y then the class of equations F = G is decidable. 4. If F.G are any functions formed by composition from x + y, x ? y, x.y, then the class of equations F = G is decidable if and only if Hilbert's 10th Problem is decidable. 5. If F,G are any functions formed by composition from x + y, x.y, II y=w then the class of equations F = G is undecidable. 6. Presburger's Algorithm can be used to solve the Integer Linear Programming Problem - the problem was not solved until 1958.2015-11-19T08:55:36ZLattices and topologies on Newman algebras.
http://hdl.handle.net/2381/34555
Title: Lattices and topologies on Newman algebras.
Authors: Beazer, R.
Abstract: In what was almost certainly an attempt to find a new axiom system for Boolean algebra based on distributivity and the existence of complements MHA Newman discovered a remarkable set of independent postulates defining an algebra which may be regarded as a generalization of Boolean algebra and now bears his name. Shortly after publication of his paper Newman extended his discussion to a wider class of relatively complemented algebras which we call Generalized Newman algebra. Recently K.Roy investigated the properties of an algebra closely related to Newman algebra, called Dual Newman algebra, and found that it has similar properties to its progenitor the opening chapters of the thesis are devoted to a discussion of the properties of the lattices of ideals, congruence relations and filters in Newman algebra and the relationships between them. The concepts of inverse and sub-inverse limits of Newman algebras are introduced, some general properties proved, and a sub-inverse limit representation established for a particular class of Newman algebras together with an inverse limit representation for the class of infinite, couplet, Boolean algebras. Furthermore, it is proved that a Newman algebra can be represented as a direct product of simple algebras if and only if its ideal lattice is a finite Boolean algebra. In the following chapter we investigate, within the framework of Newman algebras, the analogues of the auto and ideal topologies on Boolean algebra discovered by P.S. Rema. It is shown that the set of all ideal topologies L1 on a Newman algebra N is a complete, Brouwerian, dually atomic lattice containing the set L0 of all auto topologies as a complete sub-lattice and that L0 is completely isomorphic to the lattice of filters of N. Some important types of filters in N are characterized in terms of properties of the associated auto topologies on N and the auto topology associated with a given filter characterized within the lattice of ideal topologies on N. Amongst the more general properties proved we mention that the property of a topology, compatible with the fundamental operations on N, being an ideal (auto) topology is, in the algebraic and topological sense, hereditary, productive and divisible. The various connectedness properties of ideal topologized Newman algebra N;J are considered in some detail; the components being exhibited as certain congruence classes of N and necessary and sufficient conditions found for N;J to be connected, locally connected and totally disconnected. Some results are obtained concerning complete ideal uniformities and compact ideal uniformities. The properties of a particular class of ideal uniformities, called chain uniformities, are investigated and a clear cut family of metrizable chain uniformities are exhibited. Necessary and sufficient conditions are then established for a Newman algebra endowed with a separated ideal uniformity to the metrizable. In the closing chapters of the thesis we are concerned with the axiomatics of Dual and Generalized Newman algebras. Two new sets of axioms for Dual Newman algebra are exhibited each containing one less axiom than the system due to K. Roy. A new set of axioms for Generalized Boolean algebra is found containing one less axiom than the system discovered by lawmen together with a new set of independent postulates, characterizing the direct product of an arbitrary Generalized Boolean algebra and Boolean ring, which contains two fewer axioms than the system discovered by Newman.2015-11-19T08:55:35ZNon-recurrent stationary stochastic point processes.
http://hdl.handle.net/2381/34554
Title: Non-recurrent stationary stochastic point processes.
Authors: Lawrance, Anthony J.
Abstract: The work is mainly concerned with the general theory of stationary point processes and the theory of some particular stationary point processes. The intervals separating events are dealt with by the introduction of 'average events' and 'arbitrary events'. An average event is based on the average of the first n events as n tends to infinity; an arbitrary event is based on the notion of an instant of at least one event. The latter leads to modifications and extensions of some work contained in Khintchine (1960, Mathematical Methods in the Theory of Queuing). A general theory of stationary point processes is built up in which the assumptions for the results of McFadden (1962, J.R.Statist.Soc.B.,28) are clarified. A relation connecting the arbitrary and basic random variables is obtained which does not depend on the events occurring distinctly or the arbitrary intervals forming a stationary sequence (Wold stationarity). Four particular point processes are then discussed in detail. The pooling of point process, in the sense of Cox and Smith (1954, Biometrika, 41), is considered both by the average and arbitrary event approaches. The joint distributions of up to four average intervals are obtained for the pooling of two renewal processes, and Wold stationarity is verified. The pooling of any number of general stationary point processes is then dealt with by the arbitrary event approach. Next the 'renewal inhibited Poisson process' of Ten Hoopen and Reuver (l965, J.Appl.Prob.,2) is treated as a point process. The joint average interval distribution, indicating its Wold stationarity, is obtained and the counting processes of events, both in the stationary and synchronous cases, are derived. A joint process covering all aspects of the process is investigated. The work on special process is completed with the stationary point theories of semi-Markov processes and the 'random hazard process' of Gaver (1963, Technometrics, 5). Computer simulations and extensions of the processes are discussed.2015-11-19T08:55:33ZOptimum heating and optimum shape problems in distributed parameter control theory.
http://hdl.handle.net/2381/34553
Title: Optimum heating and optimum shape problems in distributed parameter control theory.
Authors: Kongphrom, S.
Abstract: In Part I, the problem of heating a thin plate or material travelling through a furnace, in which the system is described by first order linear partial differential equations, is introduced as an example of optimal control theory in distributed parameter systems. The variational technique in a fixed domain is used to obtain the necessary conditions for optimality. Many cases of the problem with the state equation described by first order linear partial differential equations are discussed, in which the control function enters into the state equation in different positions. The problems are analysed and solved by making use of characteristic curves. In Part II, we have studied the variation of a functional defined on a variable domain, and we apply it to the problem of finding the optimum shape of the domain in which some performance criterion has an extremum. The problem in which the state equation is Laplace's equation defined on the variable domain of an annular shape with given boundary conditions is discussed and completely solved for the case when the inner boundary of the domain is only a small departure from a circle. We also introduce the method of logarithmic potential of a single layer to solve the boundary value problem of Laplace's equation with mixed boundary conditions and two simple examples are solved by using this method which leads to coupled integral equations.2015-11-19T08:55:32ZGeneralized functions using the neutrix calculus.
http://hdl.handle.net/2381/34551
Title: Generalized functions using the neutrix calculus.
Authors: Kilicman, Adem.
Abstract: In Chapter 1, we give a brief review of the basic properties of distributions and we also present neutrices and neutrix limits which are needed to define the product and convolution product of distributions. In Chapter 2, the product of two distributions f and g is defined to be the neutrix limit of the {lcub}fgn{rcub}, provided this limit exists, where gn = 9 n and is a regular sequence converging to the Dirac delta function. The neutrix product f o g is said to exist and be equal to h if [Mathematical equation removed] for all in D. Some theorems about the existence of this product for distributions are proved. The neutrix product of distributions in this chapter is in general non-commutative. In Chapter 3, we define the commutative neutrix product of distributions. Neutrix products of the form [Mathematical equation removed] are evaluated from which further neutrix products are obtained. In Chapter 4, we let f and g be distributions in D' and let fn{lcub}x) = f{lcub}x)Tn(x), where {lcub}Tn(x){rcub} is a certain sequence of functions which converges to the identity functions as n tends to infinity. The neutrix convolution product f g is then defined as the neutrix limit of the sequence {lcub}fn g{rcub}, provided the limit h exists in the sense that [Mathematical equation removed] for all in D. The neutrix convolution product is evaluated for [Mathematical equation removed] The convolution product of distributions in this chapter, is in general non-commutative. In Chapter 5, we consider a commutative neutrix convolution product f * g of distributions and evaluate various neutrix convolution products of distributions. Finally, in Chapter 6, we define a new neutrix product f g on the space of ultradistributions Z'. A new commutative neutrix convolution product f g of two distributions f and g in D' has recently been defined. If f = f (f) and g = f (g) are the Fourier transforms of f and g respectively, then the neutrix product f g is defined by the exchange formula [Mathematical equation removed].2015-11-19T08:55:32ZBoundary value problems for a differential equation of the mixed type.
http://hdl.handle.net/2381/34552
Title: Boundary value problems for a differential equation of the mixed type.
Authors: King, S. P.
Abstract: Boundary value problems for differential equations of the mixed type were first considered by tricomi in 1923. In his paper (1) he resolves the Dirichlet type problem for the equation yzxx + zyy = 0, (E) in a mixed domain by reducing it to a singular integral equation. Later Holmgren (2) and Gellerstedt (3), (4) generalized some of Tricomi's results to the equation ym a2z/ax2 + a2z/ay2 = 0 m being an odd positive integer. Due to the breathily of Holmgrens paper we have felt it necessary to obtain his results in detail (chapters 4 and 5). In Part I we completely resolve the Dirichlet type problem for the equation (E) in a mixed domain by reducing it to a singular integral equation. We solve this integral equation in a closed form by using the elegant theory developed by Gakhov and Chibrikova (5). In part II we consider a boundary value problem for (E) of the mixed type in a mixed domain. That is to say, we suppose that the value of the unknown solution is given on part of the boundary, and the value of a directional derivative of the solution is given on another part. Once again we obtain a singular integral equation, but in this case it cannot be solved in a closed form. We show the existence of a solution to the integral equation by reducing it to an equivalent fredholm equation, using the regularization method of Carleman-Vekua (see for example Gakhov (6)). I would like to thank my supervisor or professor T.V. Davies for his help and encouragement, and for suggesting the problems considered here.2015-11-19T08:55:32Z